high resolution spectroscopy with the neutron resonance spin … · 2017. 10. 26. · trometers....
TRANSCRIPT
High Resolution Spectroscopy with theNeutron Resonance Spin Echo Method
vorgelegt von
Diplom-Physiker
Felix Groitl
aus Erlangen
von der Fakultät II - Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. M. Kneissl
Gutachter: Prof. Dr. D. A. Tennant
Gutachter: Prof. Dr. P. Böni
Gutachter: Dr. K. Habicht
Tag der wissenschaftlichen Aussprache: 18.12.2012
Berlin 2013
D 83
Abstract
The first part of this thesis is dedicated to explore new territory for high resolution Neu-
tron Resonance Spin Echo (NRSE) spectroscopy beyond measuring lifetimes of elementary
excitations. The data analysis of such experiments requires a detailed model for the echo
amplitude as a function of correlation time. The model also offers guidance for planning
NRSE experiments in terms of a sensible choice of parameters and allows predicting quan-
titatively the information content of NRSE spectroscopy for line shape analysis or energy
level separation. Major generalizations of the existing formalism, developed in this thesis,
allow for violated spin echo conditions, arbitrary local gradient components of the dispersion
surface and detuned parameters of the background triple axis spectrometer (TAS) giving
rise to important additional depolarizing effects, which have been neglected before. Fur-
thermore, the formalism can now be applied to any crystal symmetry class. The model was
successfully tested by experiments on phonons in a high quality single crystal of Pb and the
results demonstrate the stringent necessity to consider second order depolarization effects.
The formalism was subsequently extended to analyze mode doublets. As a major step for-
ward, detuning effects for both modes are taken into account here. The model was verified
by NRSE measurements on a unique tunable double dispersion setup. The results prove the
potential of NRSE spectroscopy to resolve mode doublets with an energy separation smaller
than the typical energy resolution of a standard TAS.
The second class of NRSE experiments was dedicated to line shape analysis of temperature
dependent asymmetric line broadening. Inelastic NRSE spectroscopy was performed on two
systems, Cu(NO3)2·2.5D2O and Sr3Cr2O8. For this purpose high quality single crystals
of Cu(NO3)2·2.5D2O were grown in the course of this thesis. As a proof of principle the
results clearly show that the NRSE method can be used to detect temperature dependent
asymmetric line broadening. For the first time this effect was measured with NRSE.
The second major part of this thesis was the upgrade of the NRSE option of FLEXX at the
BER II neutron source at HZB, Berlin. Redesigned NRSE bootstrap coils allow for a more
efficient exploitation of the larger beam cross section, given due to the overall upgrade of
FLEXX. Higher accessible coil tilt angles enable measurements on steeper dispersions. The
newly designed spectrometer arms result in a more compact instrument, enabling direct
beam calibration measurements for the entire accessible wavevector range. In combination
with higher coil tilt angles the accessible Q-range in Larmor diffraction geometry is en-
hanced. Extensive calibration measurements were performed and the results clearly demon-
strate the reliable performance of the new NRSE option, now available for the broader user
community at FLEXX.
Zusammenfassung
Der erste Teil dieser Arbeit erkundet Neuland für die hochauflösende Neutronen Resonanz
Spin-Echo (NRSE) Spektroskopie über die Messung von Lebensdauern elementarer Anre-
gungen hinaus. Die Datenanalyse solcher Experimente erfordert ein detailliertes Modell der
Echoamplitude als Funktion der Korrelationszeit. Das Modell bietet zudem eine Hilfestel-
lung für die Experimentplanung in Bezug auf die Wahl der Parameter. Des Weiteren erlaubt
es eine quantitative Vorhersage des Informationsgehaltes von NRSE Messungen, z.B. im Be-
reich der Linienformanalyse oder der Aufspaltung von Anregungsenergien. Wichtige, in die-
ser Arbeit entwickelte Verallgemeinerungen des existierenden Formalismus berücksichtigen
Depolarisationseffekte durch Spin-Echo-Bedingungen, die nicht exakt erfüllt sind. Lokale
Gradienten der Dispersion mit einer Orientierung, die nicht parallel zum Wellenvektor q
sein muss, und geringfügige Abweichungen der Parameter des Dreiachsen-Spektrometers
(DAS), welche zu zusätzlichen, zuvor vernachlässigten Depolarisationseffekten führen, wer-
den jetzt berücksichtigt. Ferner kann der Formalismus nun auf beliebige Symmetrieklassen
angewendet werden. Das Modell wurde erfolgreich mit Experimenten an Phononen in einem
Pb-Einkristall mit exzellenter Mosaizität überprüft. Die Ergebnisse demonstrieren die Not-
wendigkeit, Depolarisationseffekte zweiter Ordnung zu berücksichtigen.
Der Formalismus wurde dahingehend erweitert, die Analyse von Anregungsdubletts zu er-
möglichen. Dadurch werden nun Dejustageeffekte für beide Anregungen berücksichtigt. Das
Modell wurde durch elastische und inelastische NRSE Messungen an einem eigens dafür
entwickelten Aufbau, welcher künstlich aufgespaltene Moden realisiert, überprüft. Die Er-
gebnisse zeigen das Potenzial der NRSE Spektroskopie, Anregungsdubletts aufzulösen, deren
Energieaufspaltung unter der Energieauflösung eines Standard-DAS liegt.
Weitere hier durchgeführte NRSE Experimente widmeten sich der Linienformanalyse tem-
peraturabhängiger asymmetrischer Linienverbreiterungen. Dafür wurden inelastische NRSE
Messungen an Cu(NO3)2·2.5D2O sowie an Sr3Cr2O8 durchgeführt. Hierfür wurden eigens
hochwertige Cu(NO3)2·2.5D2O-Einkristalle gezüchtet. Die Ergebnisse zeigen deutlich, dass
die NRSE Methode in der Lage ist, eine temperaturabhängige asymmetrische Linienverbrei-
terung zu bestimmen. Erstmalig wurde dieser Effekt mit NRSE gemessen.
Im Zuge dieser Arbeit wurde außerdem die NRSE-Option des kalten Dreiachsen-Spektrome-
ters FLEXX an der Neutronenquelle BER II am HZB, Berlin, aufgerüstet. Die dafür neu ge-
fertigten NRSE Bootstrap-Spulen erlauben eine effektivere Ausnutzung des größeren Strahl-
querschnitts, der durch das FLEXX Upgrade zur Verfügung steht. Höhere erreichbare Spu-
lenkippwinkel bieten zusätzlich Zugang zu steileren Dispersionen. Das durch die neu ent-
wickelten Spektrometerarme kompaktere Instrument ermöglicht Kalibrationsmessungen im
direkten Strahl für den gesamten zugänglichen Wellenvektor-Bereich. In Kombination mit
höheren Spulenkippwinkeln wird der zugängliche Q-Bereich in der Larmor Diffraktionsgeo-
metrie vergrößert. Umfangreiche Kalibrationsmessungen zeigen deutlich die Zuverlässigkeit
und Leistungsfähigkeit der neuen NRSE-Option, die nun einer breiten Nutzerschaft zu Ver-
fügung steht.
Table of contents
1 Introduction 1
2 NRSE resolution theory 5
2.1 Principle of inelastic neutron spin echo spectroscopy . . . . . . . . . . . . . 5
2.2 Principle of neutron resonance spin echo . . . . . . . . . . . . . . . . . . . . 8
2.2.1 The π-coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 NRSE instrument with 4 π-coils . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Spin echo phonon focusing . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Extended NRSE resolution function . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Generalized spin echo phase - violated spin echo conditions . . . . . 17
2.3.2 The τ dependence of the polarization . . . . . . . . . . . . . . . . . . 24
2.3.3 Quantitative description of depolarization due to sample imperfections 28
2.3.4 Quantitative description of depolarization due to curvature of the dis-
persion surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.5 Quantitative description of depolarization due to sample imperfections
and curvature of the dispersion surface . . . . . . . . . . . . . . . . . 35
2.3.6 Dispersion surface not coinciding with the center of the TAS resolution
ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Experimental test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 NRSE investigations on split modes 47
3.1 Two modes within the TAS resolution ellipsoid - Simplified NRSE model . . 47
3.2 Second dispersion surface within the TAS resolution ellipsoid - General model 51
3.3 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Niobium dispersion models . . . . . . . . . . . . . . . . . . . . . . . 55
i
ii Table of contents
3.3.3 Elastic measurements on split modes . . . . . . . . . . . . . . . . . . 56
3.3.4 Inelastic measurements on split modes . . . . . . . . . . . . . . . . . 61
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 NRSE line shape analysis 69
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O . . . . . . . . . . 70
4.1.1 Properties of Cu(NO3)2·2.5D2O . . . . . . . . . . . . . . . . . . . . . 70
4.1.2 Sample deuteration and growth of single crystals . . . . . . . . . . . 71
4.1.3 Inelastic NRSE measurements . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Asymmetric line shape of excitations in Sr3Cr2O8 . . . . . . . . . . . . . . . 82
4.2.1 Properties of Sr3Cr2O8 . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.2 Inelastic NRSE measurements . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Upgrade of the NRSE option at FLEXX 89
5.1 Bootstrap coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.1 B0 coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 Cooling circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.3 RF coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Spin echo instrument arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.1 Magnetic shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.2 Coupling coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.3 Motors and encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Calibration of the new NRSE option at FLEXX . . . . . . . . . . . . . . . . 102
5.3.1 Calibration of currents and HF voltage . . . . . . . . . . . . . . . . . 103
5.3.2 Spin echo curve and echo point . . . . . . . . . . . . . . . . . . . . . 106
5.3.3 Phase stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.4 Calibration of coil tilt angles . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Conclusion and perspectives 119
Appendix 123
References 139
Chapter 1
Introduction
To investigate the structure and dynamics of matter on the atomic scale, neutron physics
and the field of neutron scattering have produced many distinguished methods and instru-
ments. The unique properties of thermal neutrons - the electrical charge neutrality, the
magnetic moment, the wavelength being of the order of inter-atomic distances, a high pene-
tration power for many materials and an energy of the order of excitations in condensed
matter - emphasizes it as an excellent probe for structure and dynamics in matter.
Energy resolving neutron scattering techniques have made essential contributions to the un-
derstanding of matter, e.g. the development of the triple axis spectrometer (TAS) enabled
measuring dispersive excitations. Allowing a reasonable intensity for inelastic experiments,
the energy resolution of a conventional TAS is limited to 0.1meV. However, the investigations
of linewidths of dispersive excitations and slow dynamics require much higher resolution in
the range of µeV. This is only accessible using specially designed, highest resolution spec-
trometers. Neutron spin echo (NSE) introduced by Mezei [1, 2] is one of the outstanding
methods providing the necessary energy resolution in the µeV-range. The method of neutron
resonance spin echo (NRSE), developed by Gähler and Golub [3], in combination with TAS,
as suggested by Mezei [5], allows to measure linewidths of dispersive excitations with highest
resolution. This is still a young technique getting more and more established. Furthermore,
Rekveldt [44] pointed out that the technique can be used to perform high precision mea-
surements in Larmor diffraction geometry.
A major part of this thesis focuses on methodological development and experiments, explor-
ing new territory of NRSE and pushing the method beyond “standard” linewidth measure-
ments. One class of experiments, which demands highest resolution, are those seeking to
resolve mode doublets with an energy separation smaller than the typical energy resolution
of standard TAS. The data analysis of such experiments requires a model for the echo ampli-
1
2 Introduction
tude as a function of the correlation time. A model first developed in [6, 7] expands the spin
echo phase to second order and includes second order effects arising from the TAS resolution
function, sample imperfections and the curvature of the dispersion surface. However, the
model is restricted to cubic systems. In this thesis a more generalized formalism is devel-
oped allowing for all crystal symmetry classes. Furthermore, violated spin-echo conditions,
arbitrary oriented local gradients of the dispersion surface and a detuning of the instrument
parameters are included in the generalized description. The derivation of the model, the
resolution function and experimental tests on phonons in Pb are discussed in chapter 2.
Subsequently the model is extended to include the interesting case of mode doublets. Exper-
imental tests were performed on Nb as a well understood model system. A unique tunable
double dispersion setup, allowing to generate artificially split modes, was realized. The
model and the experimental verifications are presented in chapter 3.
Another class of new NRSE-TAS experiments investigates temperature dependent asymme-
tric line broadening. This effect has recently been observed in Cu(NO3)2·2.5D2O, a model
system for a 1-D bond alternating Heisenberg chain [53], and Sr3Cr2O8, a 3-D gapped quan-
tum spin dimer [65], using time-of-flight (ToF) and triple axis spectrometer (TAS) neutron
scattering instruments. In the course of this thesis inelastic NRSE spectroscopy was per-
formed on both material systems. For the first time the effect of temperature dependent
asymmetric line broadening was measured with NRSE. The particular advantage of the
NRSE method is the direct access to the line shape, since there is no convolution of the sig-
nal with the resolution function of the spectrometer. The results are discussed in chapter 4.
The instrumental part of this thesis was the upgrade of the NRSE option available at the
cold TAS FLEXX at BER II at HZB, Berlin. The upgrade of the NRSE option benefits from
the new features available at FLEXX after its major upgrade. In the course of this thesis
the NRSE spectrometer arms were redesigned to enable a more compact construction of the
overall NRSE option. This gives access to higher scattering angles, allowing a larger Q-range
for Larmor diffraction experiments. In combination with the increased instrument area, the
direct beam geometry is now accessible for the entire wavelength-range of the NRSE-TAS
instrument. Furthermore, new NRSE bootstrap coils were manufactured in collaboration
with the Max Planck Institute For Solid State Research, Stuttgart. The new design of the
NRSE coils allows for larger beam cross sections exploiting the increased polarized neutron
flux. In addition, larger accessible coil tilt angles enable measurements of steeper dispersions.
All necessary calibration measurements for the NRSE option were performed with the new
instrument components. The individual experimental tests demonstrate the high quality,
the high manufacturing accuracy and the excellent performance of the new coils. Calibration
Introduction 3
tables for currents and impedance matching, essential for a successful user operation, were
measured and implemented successfully. Finally, spin echo measurements were performed,
demonstrating high polarization for a wide range of spin echo times virtually independent
of coil tilt angle and thus excellent performance of the NRSE option. All measurements are
presented in chapter 5.
4 Introduction
Chapter 2
NRSE resolution theory
In this chapter the principles of classical neutron spin echo [5], the neutron resonance spin
echo technique [3, 4] and the so called “phonon focusing” method [5], used to investigate
dispersive excitations, will be introduced. The theory of inelastic neutron spin echo spec-
troscopy will be extended to second order including depolarizing effects [6]. In order to
develop a more general resolution function, the existent formalism will be extended here
by dropping limiting restrictions. The formalism will be subsequently extended to include
a detuning of the background TAS spectrometer. At the end of this chapter experimental
tests of the model performed at the cold triple axis spectrometer V2/FLEX at BER II at
HZB, Berlin, will be discussed.
A classical point of view is sufficient to develop the resolution function for inelastic neutron
spin echo. Therefore, all principles introduced in this chapter are discussed in a classical
frame.
2.1 Principle of inelastic neutron spin echo spectroscopy
In neutron spin echo (NSE), first introduced by Mezei [5], the velocity change of neutrons
due to scattering is measured by comparing their Larmor precession angle in magnetic fields
before and after the sample. The experimental setup for a classical NSE spectrometer is
shown in Fig. 2.1.
A polarizer P and an analyzer A are placed before and after the spectrometer, respectively.
Propagating in the y-direction the monochromatic neutron beam (∆λ/λ = 10% − 20%) is
polarized in the x-direction. Neutrons enter the first precession field region of length L1,
with a well defined homogenous magnetic field B1 oriented in z-direction perpendicular to
the polarization vector of the beam. In the magnetic field the magnetic moments of the
neutrons start to precess with the Larmor frequency ω1 = γB1, where γ = 2π · 2.913 · 103HzG
5
6 2 NRSE resolution theory
L1 L2
Ppolarized
beam
B1
A
Det
ecto
r
y
z
x
0
-1
1
Pola
riza
tion
B2Sample
Fig. 2.1: Top: Schematic NSE setup. The beam is polarized by a polarizerP. The average polarization decays in the first spectrometer arm. For elasticscattering and equal field integrals the polarization is restored in a spin echoafter the second spectrometer arm (black) and analyzed by an analyzer A.For the inelastic case (red) the polarization at the echo point decreases asa function of the spin echo time τ and the phase is shifted. Bottom: Theaverage beam polarization as a function of the instrument length for theelastic case (black) and the inelastic case (red).
is the gyromagnetic ratio of the neutron. A neutron with a velocity v1 needs the time
t1 = L1/v1 to pass the first region, where its magnetic moment Larmor precesses by the
angle Φ1 = ω1t1. Due to the velocity spread of the incoming beam the distribution of the
Larmor precession angles becomes more and more spread out. For a typical configuration of
B = 300G, L = 1m, λ = 6Å and a wavelength distribution of ∆λλ = 10% the spread dΦ is of
the order of 102 · 2π. Since the polarization of the beam is the average of the analyzed spin
component and given by P = 〈cosΦ〉, it is zero after the first field region. By scattering at
the sample the neutron velocity changes to v2. The orientation of the second magnetic field
B2 is opposite to B1. Here B2 and B1 denote the magnitudes of the magnetic fields. By
passing through the second field region, the magnetic moment of the neutron precesses by
an angle of φ2 = −ω2t1, resulting in a total angle of
ΦNSE =γB1L1
v1− γB2L2
v2(2.1)
after the second spectrometer arm.
For elastic scattering (v1 = v2), a symmetric instrument configuration (|B1L1| = |B2L2|)and ~B1 = − ~B2, the phase φNSE will be zero for all velocities after the second field region.
Since all magnetic moments are then oriented in the original direction, this effect is called
“spin echo”. As pointed out by Mezei [1, 2] for inelastic scattering processes, the idea of
2.1 Principle of inelastic neutron spin echo spectroscopy 7
neutron spin echo is the use of ΦNSE to determine the energy transfer
~ω =m
2
(v21 − v22
). (2.2)
To get a quantitative relation between ΦNSE and ω, the scattering function S (ω) is consid-
ered to be independent of the momentum transfer Q and distributed around a mean energy
transfer of ~ω0. The energy and the velocity of the incoming neutron beam then read:
ω = ω0 + dω and v1 = v1 + dv1 (2.3)
Using equation (2.2) v2 can be replaced in equation (2.1):
ΦNSE =γB1L1
v1 + dv1− γB2L2√
(v1 + dv1)2 − 2~
m (ω0 + dω)(2.4)
Expanding equation (2.4) to first order in dv1 and dω yields:
ΦNSE = ΦNSE,0 −(γB1L1
(v1)2 − γB2L2v1
(v2)3
)dv1 −
~
m
γB2L2
(v2)2 dω (2.5)
where ΦNSE,0 corresponds to the mean velocities v1 and v2
ΦNSE,0 =γB1L1
v1− γB2L2
v2(2.6)
and v2 is the mean neutron velocity after the sample
v2 =
√(v1)
2 − ~
2mω0. (2.7)
If the so called spin echo condition
B1L1
B2L2=
(v1)3
(v2)3 (2.8)
holds, the phase ΦNSE becomes independent of the velocity spread dv1 and dΦNSE is
proportional to dω:
dΦNSE = ΦNSE − ΦNSE,0 = −τNSEdω (2.9)
The instrumental constant
τNSE =~γB1L1
m (v1)3 =
~γB2L2
m (v2)3 (2.10)
8 2 NRSE resolution theory
is called “spin echo time”.
After the second field region the neutron beam passes an analyzer transmitting neutrons
polarized in the x-direction. The intensity at the detector is given by:
I =I02(1 + P ) (2.11)
where I0 is the intensity of the incident neutron beam. The mean beam polarization P at
the analyzer, i.e. the echo amplitude AE , is the cosine Fourier transform of the scattering
function S (ω)
AE = 〈cosΦNSE〉 =∫
S (dω) cos (ΦNSE,0 − dω · τNSE) d (dω) . (2.12)
As a classical example from quasi elastic scattering let the scattering function S (ω) be a
Lorentzian line with a linewidth of Γ:
S (ω) ∝ Γ
Γ2 + ~2ω2(2.13)
Equation (2.12) then yields:
AE (τNSE) =
∫S (ω) cos (ωτNSE) dω∫
S (ω) dω= e−
ΓτNSE~ (2.14)
2.2 Principle of neutron resonance spin echo
The method of neutron resonance spin echo (NRSE) was suggested by Gähler and Golub
[3, 4] in 1987. NRSE replaces each of the long static fields of the NSE method by two
resonance spin flippers with sharp field boundaries placed in a zero field region.
2.2.1 The π-coil
The basic component of NRSE is a resonance flipper called π-coil. Within a resonance spin
flipper two magnetic fields are superimposed: a static field ~B0 pointing in the z-direction
and a rotating field ~BRF in the xy-plane with the frequency ωRF . The superposed magnetic
field is described by the vector
~B1 (t) = (BRF cos (ωRF t) , BRF sin (ωRF t) , B0) . (2.15)
To understand the movement of the magnetic moment of a neutron entering the coil with an
initial orientation of the magnetic moment in the xy-plane, it is useful to change the reference
2.2 Principle of neutron resonance spin echo 9
B1
x
z
y
Fig. 2.2: Principle of the π-coil. The incoming neutron polarization vector(green) precesses around the effective magnetic field B1 and leaves the coilwith a new orientation (blue). Setting B1 = πv0
γdthe polarization vector
performs a π-flip and stays in the scattering plane
frame to a system rotating with the frequency ωRF around the z-axis. The effective field
will be [10]:
~B1 =
(BRF , 0, B0 −
ωRF
γ
). (2.16)
If the frequency of the rotating field is tuned to the Larmor frequency of the static field
(ωRF = γB0), the z-component vanishes and the static field BRF remains. Thus, the
magnetic moment of the neutron will precess around BRF with the frequency ωRF = γBRF .
The magnitude of BRF is chosen such that the magnetic moment precesses by an angle π
while the neutron passes the coil:
BRF =πv0γd
(2.17)
where v0 is the velocity of the neutron and d is the thickness of the coil. Therefore, after
leaving the coil the magnetic moment is again in the xy-plane (see Fig 2.2).
In the lab-system the neutron enters the coil at a time t1. At this time the rotating field
has a phase of ΦRF = ωRF t1 and while passing the coil, the magnetic moment precesses an
angle π around ~B1. Simultaneously ~B1 precesses around the z-axis by an angle of ωRFdv0
.
Thus, after leaving the coil the phase of the magnetic moment is:
Φ1 = 2ωRF t1 + ωRFd
v0− Φ0 (2.18)
where Φ0 is the angle between the magnetic moment and the x-axis.
Since the realization of a rotating field setup is challenging, it is favorable to use a linearly
oscillating field instead. The oscillating field B1 can be divided into two counter rotating
parts B+1 and B−
1 with an amplitude of B12 . Only the part rotating in the same direction as
the magnetic moment of the neutron contributes to the π-flip. The counter rotating part
10 2 NRSE resolution theory
can be neglected for B0 ≫ B1. This causes a small shift of the resonance frequency, the so
called “Bloch-Siegert-Shift” [11]∆ω
ω=
B2RF
4B20
, (2.19)
which can be neglected for frequencies higher than 50kHz.
The resonance condition given by equation (2.17) is only valid for one v0. The magnetic
moments of faster (slower) neutrons will precess less (more) than an angle of π. Therefore,
the magnetic moment is oriented slightly out of the xy-plane after the coil and the average
polarization decreases. The estimates in [12] show that for small velocity spreads the echo
amplitude is:
AE = 1−Mπ2
4
(∆v
v0
)2
(2.20)
where M is the number of π-flippers used. For a velocity spread of 1.5%, which is typical
for a TAS using a crystal monochromator, and 8 flippers, the loss of average polarization
will be < 1%.
2.2.2 NRSE instrument with 4 π-coils
Using a NRSE instrument with a setup of 4 π-coils (see Fig. 2.3) provides the same function
as a conventional NSE spectrometer with the same length and the static magnetic fields twice
as strong.
The two coils before the sample region (A and B) have a static field ~B1 and are driven with
a frequency of ω1 = γB1 while the coils C and D after the sample region have the static
field ~B2 with a frequency of ω2 = γB2. The two fields ~B1 and ~B2 are oriented in opposite
directions. Therefore, the precession direction of the magnetic moment of the neutron and
the rotating fields is opposite to each other. The regions between the coils are zero field
regions and the stray fields of the coils have to be minimized.
A neutron polarized in x-direction (Φ0 = 0 in equation (2.18)) with a velocity v1 enters the
first coil at the time tA. After passing the coil, the phase angle of the magnetic moment is
ΦA = 2ω1tA + ω1d
v1. (2.21)
In the zero field region, between the coils A and B, the magnetic moment of the neutron does
not change its phase angle and enters the second coil at the time tB = tA + L1v1
. Therefore,
the phase angle of the magnetic moment after the second coil is:
ΦB = 2ω1tB + ω1d
v1− ΦA = 2ω1
L1
v1. (2.22)
2.2 Principle of neutron resonance spin echo 11
Spin
x
y
z
L1 L2
A B C D
zero-field regions
sample regiond
Fig. 2.3: Schematic drawing of a NRSE setup with 4 π-coils: The orienta-tion of the magnetic moment of the neutron entering each coil is illustratedin green. Within the coil the magnetic moment precesses around the ef-fective magnetic field Beff (red) and has a new orientation (blue) leavingeach coil. The flight path between the coils are regions of zero magneticfield.
ΦB becomes independent from the time tA and only depends on the time of flight L1v1
between
the two coils. Compared to a conventional NSE instrument with the same magnitude of the
magnetic field and the same length, one obtains twice the phase angle. In other words, the
two π-coils give the same Larmor precession angle as a NSE spectrometer arm with length
L1 and magnetic field 2B1.
During the scattering process the neutron velocity changes to v2. Since the sample region
is a field free region, the magnetic moment enters coil C with the phase ΦB at time tC and
leaves coil C with an angle of
ΦC = −2ω2tC − ω2d
v2− ΦB. (2.23)
Hence, after the last coil D the phase is
ΦD = −2ω2tD − ω2d
v2− ΦC
= 2ω1L1
v1− 2ω2
L2
v2
= ΦNRSE = 2ΦNSE .
(2.24)
12 2 NRSE resolution theory
Therefore, the previous discussion on spin echo applies directly to NRSE if B1 and B2
are substituted by 2ω1γ and 2ω2
γ , respectively. However, in a typical NRSE experiment the
minimum accessible spin echo time is determined by a frequency of 50 kHz in the π-coils,
since for lower frequencies the effect of the above mentioned Bloch-Siegert-Shift (see equation
(2.19)) increases leading to a strong depolarization.
Enhancement of NRSE by the bootstrap technique
The so called bootstrap technique was first introduced in [4]. By replacing each π-coil by a
pair of resonance coils with the static fields of each pair oriented in opposite directions, the
effective precession angle and hence the resolution is doubled. Within this configuration the
first coil within a bootstrap coil provides a return path for the magnetic flux of the second
coil and vice versa. This reduces the stray field outside the coils significantly. The phase
angle after four bootstrap coils, each containing a series of N resonance coils, is
ΦN = 2N
(ω1
L1
v1− ω2
L2
v2
)= NΦNRSE (2.25)
All existing NRSE spectrometers operate with double bootstrap coils (N = 2). A discussion
of the limits of the bootstrap technique can be found in [4].
2.2.3 Spin echo phonon focusing
A triple axis spectrometer (TAS) is a classic instrument to determine dispersion curves
in the entire Brillouin zone. However, the instrumental resolution of a TAS is in rare
cases sufficient to measure the natural linewidth of excitations and in addition needs to
be known very accurately at the inelastic signal. NSE and NRSE spectrometers provide
the required high resolution in the µeV range. Mezei [5] suggested to combine TAS and
spin echo spectrometers and to use the tilted field technique, which is necessary to measure
linewidths of dispersive excitations.
For this technique to work, the field boundaries need to be perpendicular to the flight path of
the neutrons. Then the phase Φ (ω) is proportional to the energy transfer ω and independent
of the momentum transfer q. In the quasielastic case with a mean excitation energy of zero
and a broad wavelength distribution ∆λλ = 10%, a relaxed q resolution does not affect the
energy resolution. However, for dispersive excitations where the excitation energy ω is a
function of q, ω = ω (q), a finite momentum resolution results in a spread in ω and hence
in Φ (ω) even for sharp excitations with zero linewidth. To allow measurements of the
intrinsic linewidth of excitations, the spin echo phase needs to be tuned to the slope ~∇qω
of the dispersion (see Fig. 2.4). Hence, all scattering events lying within the resolution
2.2 Principle of neutron resonance spin echo 13
ω
q
ω0
q0
Fig. 2.4: The resolution function of the TAS selects a certain region inthe (q, ω)-space. The lines of constant Larmor phase (Φ = const, paleblue) have to be parallel to the dispersion curve. This can be reached byinclining the precession field boundaries.
ellipsoid of the background TAS and on the dispersion surface have the same spin echo
phase Φ (ω,q). As pointed out by Mezei [5] the spin echo phase can be tuned to the slope
of the dispersion surface by tilting the boundaries of the precession fields relative to the
neutron flight path. In such a configuration, for a linear dispersion with zero linewidth, all
scattered neutrons have the same total Larmor precession phase. A finite linewidth would
cause a spread in Φ. If the lineshape F (ω) within the resolution ellipsoid is independent
of q (S (q, ω) = S (q0)F (ω)) and significantly smaller than the width of the resolution
ellipsoid, Φ is proportional to the Fourier transform of the line shape as explained above.
Spin echo conditions
In this section the spin echo conditions for the phonon focusing technique are introduced
briefly in a classical picture. The discussion of an alternative quantum mechanical framework
where Φ is a phase shift between two spin states can be found in [8, 13]. The quantum
mechanical approach shows that the spin echo time τNSE is equal to the correlation time in
the time dependent van Hove density-density correlation function [9].
Assuming that the variation of the scattering law on the dispersion surface depends on
ω (q) only, S (q, ω) can be considered as a function of energy deviation from the dispersion
surface:
S (q, ω) = S (ω (q)− ω0 (q)) . (2.26)
In order to measure equation (2.26) the Larmor precession angle needs to be proportional
to (ω (q)− ω0 (q)):
Φ− Φ0 = −τ (ω (q)− ω0 (q)) = −τ∆ω. (2.27)
14 2 NRSE resolution theory
kF
B2
B1
kI
nF
nI
i2
j2
l2
j1
i1
l1
L2
precession fieldregions
sample
L1
kI
ni
ti
Q
sf
si
kF
l0
j0
i0
nI
nF
nf
tf
Θ2
Θ1
Fig. 2.5: Schematic drawing of the tilted precession field regions in aninelastic NSE setup. The defined Cartesian coordinate systems are usedthroughout.
Assuming a planar dispersion and expanding the dispersion to first order in (q− q0) yields:
∆ω = ω (q)− ω0 (q) = ω (q)− [ω0 (q0) + (q− q0) · ∇qω (q0)] . (2.28)
Using momentum and energy conservation with mean wavevectors kI and kF before and
after the scattering:
ki = kI +∆ki and kf = kF +∆kf
ω(q) =~2
2m
(k2i − k2
f
)and ω0(q) =
~2
2m
(k2I − k2
F
),
(2.29)
∆ω is expressed as a function of the deviations from the mean wavevectors ∆ki and ∆kf :
∆ω = ω (q)− ω0 (q) =~
m[(kI ·∆ki)− (kF ·∆kf )] . (2.30)
Using the relation for the momentum in a perfect crystal
q = ki − kf −G0 and q0 = kI − kF −G0 (2.31)
where G0 is the reciprocal lattice vector and combining equations (2.28) and (2.30) yields:
∆ω = ∆ki
[~
mkI −∇qω0(q0)
]−∆kf
[~
mkF −∇qω0(q0)
](2.32)
2.2 Principle of neutron resonance spin echo 15
The Larmor precession angle after two tilted field regions is
Φ = Φ0 +∆Φ =m
~
(ωL1L1 cosΘ1
ki cosΘ1− ωL2L2 cosΘ2
kf cosΘ2
)
=m
~
(ωL1L1 cosΘ1
ki · ni− ωL2L2 cosΘ2
kf · nf
)=
A1
ki · ni− A2
kf · nf
(2.33)
where ni,f are the unit vectors normal to the precession field boundaries before and after
the sample tilted by an angle Θ1,2 w.r.t. the neutron flight path (see Fig. 2.5). The Larmor
frequency is ωL1,2 = NγB1,2 with N = 1 for conventional NSE, N = 2 for NRSE with 4
flipper coils and N = 4 for NRSE in bootstrap mode (8 flipper coils). Using
Φ0 =A1
kI · ni− A2
kF · nf
(2.34)
and expanding the total Larmor precession angle to first order yields
Φ− Φ0 = − A1
(kI · ni)2 (∆ki · ni) +
A2
(kF · nf )2 (∆kf · nf ) . (2.35)
For equation (2.27) to hold, the coefficients in equations (2.32) and (2.35) have to be com-
pared. The result for the normal vectors of the precession field boundaries is:
ni,f =~
mkI,F −∇qω0(q0)∣∣ ~mkI,F −∇qω0(q0)
∣∣ =~
mkI,F −∇qω0(q0)
NI,F(2.36)
and for the spin echo time
τ =A1,2
(kI,F · ni,f )∣∣ ~mkI,F −∇qω0(q0)
∣∣ =A1,2
(kI,F · ni,f )NI,F. (2.37)
The condition for the adjustment of the field integrals is then:
ωL1L1
ωL2L2=
cosΘ2 (kI · ni)2∣∣ ~mkI −∇qω0(q0)
∣∣
cosΘ1 (kF · nf )2∣∣ ~mkF −∇qω0(q0)
∣∣ =cosΘ2 (kI · ni)
2NI
cosΘ1 (kF · nf )2NF
. (2.38)
If the spin echo conditions given by equations (2.36) and (2.38) are satisfied, τ is the same
for both spectrometer arms and the total Larmor phase is independent of the momentum
transfer in the presence of a finite dispersion. Note that this is only true for a first order
approximation of the total Larmor phase.
In the general case the echo amplitude, measured by a spin echo spectrometer combined
16 2 NRSE resolution theory
with a background TAS, is given by [6]
AE =1
N
∫S (Q, ω)RTAS (ki,kf ) e
iΦ(ki,kf)d3kid3kf + c.c. (2.39)
where RTAS (ki,kf ) is the transmission function of the background TAS and Φ (ki,kf ) is the
sum of Larmor precession angles before and after the sample. Assuming that the linewidth
and S (Q) are independent of Q within the TAS resolution ellipsoid, the integral over the
momentum components yields
AE ∝∫
S (∆ω)RTAS (∆ω) eiΦd∆ω + c.c. (2.40)
with ∆ω given by equation (2.28). The application of spin echo spectroscopy is only rea-
sonable if the linewidth S (∆ω) of the excitation is much smaller than the energy resolution
T (∆ω) of the background TAS. Therefore, RTAS (∆ω) is assumed to be constant. Thus,
the result for the echo amplitude is again the cosine Fourier transform of the line shape:
AE ∝∫
S (∆ω) cos (τ∆ω) d∆ω (2.41)
2.3 Extended NRSE resolution function
In this section the resolution function for a NRSE spectrometer with a background TAS is
extended by expanding the spin echo phase to second order following the approach of [6]
and expressing the resolution function in a covariance matrix formalism. This will include
second order effects arising from the TAS resolution function, sample imperfections, the
curvature of the dispersion surface and a detuning of the instrument parameters. Major
differences between the treatment given in [6] and the treatment given in the present thesis
are the following generalizations [14, 15]:
1. Allow for violated spin echo conditions
2. The formalism is extended to treat systems with lower crystallographic symmetry than
cubic
3. The local gradient of the dispersion surface ∇ω(qc0) may have components out of the
scattering plane, i.e. the center of the TAS resolution ellipsoid must not necessarily be
located at a point with high crystallographic symmetry. Here qc0 is the momentum
transfer expressed in a Cartesian coordinate system attached to the reciprocal lattice
system. q0 is transformed into qc0 by the so called B-matrix [16, 17]. The UB matrix
formalism is explained in Appendix A.
2.3 Extended NRSE resolution function 17
One advantage of the generalizations made here is the applicability of the formalism to a
broader range of systems. The allowance of violated spin echo conditions enables the treat-
ment of cases where a second excitation is present within the TAS resolution ellipsoid. In
such cases the spin echo conditions are at least violated for one excitation. Investigations on
split modes using NRSE are treated in chapter 3. Without these generalizations, depolar-
ization effects would be neglected. This would result in an overestimation of the linewidth Γ
of the excitation.
In addition minor errors in [6] in the part discussing lattice imperfections and the curvature
of the dispersion surface are corrected.
2.3.1 Generalized spin echo phase - violated spin echo conditions
A simplified situation is discussed by making the following assumptions:
1. A single dispersion surface is located within the TAS resolution ellipsoid.
2. The center of the TAS resolution ellipsoid coincides with the dispersion surface.
3. The precession field boundaries are exactly perpendicular to the scattering plane.
4. The precession field boundaries can only be tilted around axes that are perpendicular
to the scattering plane.
5. All instrument components are ideal, i.e. field boundaries are ideal planes, no stray
fields are present and RF flippers provide exact π-flips.
Following [6] and expanding equation (2.35) to second order yields:
Φ (ki,kf )− Φ0 =− A1
(kI · ni)2 (∆ki · ni) +
A2
(kF · nf )2 (∆kf · nf )
+A1
(kI · ni)3 (∆ki · ni)
2 − A2
(kF · nf )3 (∆kf · nf )
2 .
(2.42)
The aim is to express the total Larmor precession angle in terms of the variable vector
J = (∆ω,∆kin, y1, y2, z1, z2) with
∆ki = x1i1 + y1j1 + z1l1 (2.43)
∆kf = x2i2 + y2j2 + z2l2 (2.44)
∆kin = ∆ki · ni (2.45)
18 2 NRSE resolution theory
where the variables of J are defined as in [6]. According to Fig. 2.5 i, j and l are the basic
vectors of a right handed coordinate system with i1 and i2 pointing along the direction of
the mean wavevectors kI and kF , respectively. First, the dispersion relation is expanded to
second order
ω (qc) = ∆ω + ω (qc0) + ∆qc0 · ∇ω (qc0) +1
2∆qT
c Hc|qc0∆qc (2.46)
assuming that the line broadening is the same for every qc within the TAS resolution
ellipsoid, i.e. ∆ω is independent from qc. Here Hc is the curvature matrix of the dispersion
surface at qc0, which is defined in section 2.3.4. In a real crystal lattice imperfections are
present and therefore variations of the lattice vector Gc need to be included. The following
relation
∆q′c = ∆qc −∆Gc = ∆ki −∆kf −∆Gc (2.47)
with the total wavevector transfer
Qc = Gc + qc (2.48)
and
Gc = Gc0 +∆Gc (2.49)
leads to
ω (qc) =∆ω + ω (qc0) + ∆ki · ∇ω (qc0)−∆kf · ∇ω (qc0)
−∆Gc · ∇ω (qc0) +1
2∆q′T
c Hc|qc0∆q′c
. (2.50)
Note again that all variables with the index c are expressed in a Cartesian coordinate system
attached to the reciprocal lattice. The relation between these quantities and their expression
in the basis of the reciprocal lattice are explained in Appendix A. For now the last two
terms will be left unchanged, since effects from sample imperfections and curvature of the
dispersion surface will be treated separately in sections 2.3.3 and 2.3.4, respectively.
The energy conservation in second order reads
ω (qc) =~
2m
(k2i − k2
f
)(2.51)
= ω (qc0) +~
mkI ·∆ki −
~
mkF ·∆kf
+~
2m∆k2
i −~
2m∆k2
f (2.52)
2.3 Extended NRSE resolution function 19
using
qc0 = kI − kF (2.53)
and
ω (qc0) =~
2m
(k2I − k2
F
). (2.54)
Combining equations (2.50) and (2.51) yields
∆ω =
(~
mkI −∇ω (qc0)
)·∆ki −
(~
mkF −∇ω (qc0)
)·∆kf
+~
2m∆k2
i −~
2m∆k2
f +∆Gc · ∇ω (qc0)−1
2∆q′T
c Hc|qc0∆q′
c. (2.55)
Now unit vectors are defined as
ǫi =~
mkI −∇ω (qc0)∣∣ ~mkI −∇ω (qc0)
∣∣ =~
mkI −∇ω (qc0)
NI(2.56)
ǫf =~
mkF −∇ω (qc0)∣∣ ~mkF −∇ω (qc0)
∣∣ =~
mkF −∇ω (qc0)
NF. (2.57)
Since these unit vectors, defined by the crystal properties, in general may have components
out of the scattering plane, they cannot be assumed to be equal with the normal vectors
of the precession field boundaries ni,f , which are defined by the instrument settings. As
mentioned above in this subsection kI,F correspond to a point on the dispersion surface
coinciding with the center of the TAS resolution ellipsoid. This assumption will be dropped
later in section 2.3.6. In the next step the spin echo times in both spectrometer arms τ1,2
given by the instrument parameters are defined:
τ1,2 =A1,2
(kI,F ·ni,f )2∣∣ ~mkI,F −∇ω (qc0)
∣∣ =A1,2
(∆kI,F ·ni,f )2NI,F
. (2.58)
Here τ1,2 are not longer identical to the correlation time τ in the time-dependent van Hove
density-density correlation function. The inclusion of instrument alignment errors in the
tilt angle of the precession field boundaries and the ratio of the field integrals means that in
general τ1,2 are not equal in the two spectrometer arms. Multiplying equation (2.55) with
τ2 yields:
20 2 NRSE resolution theory
τ2∆ω =A2
(kF ·nf )2NF
(~
mkI −∇ω (qc0)
)·∆ki
− A2
(kF ·nf )2
~
mkF −∇ω (qc0)
NF·∆kf
+~
2mτ2∆k2
i −~
2mτ2∆k2
f
+τ2∆Gc · ∇ω (qc0)−1
2τ2∆q′T
c Hc|qc0∆q′
c (2.59)
=A2
(kF ·nf )2
NI
NFǫi ·∆ki −
A2
(kF ·nf )2 ǫf ·∆kf
+~
2mτ2∆k2
i −~
2mτ2∆k2
f
+τ2∆Gc · ∇ω (qc0)−1
2τ2∆q′T
c Hc|qc0∆q′
c. (2.60)
In the general case the unit vectors ǫi,f are expressed in a right handed Cartesian coordinate
system with the basis vectors ni,f , ti,f and si,f according to Fig. 2.5. The unit vectors ni,f ,
ti,f and si,f are the same as u1,2, v1,2 and w1,2 in [6].
ǫi,f = ei,f1ni,f + ei,f2ti,f + ei,f3si,f (2.61)
Here ni,f and ti,f are the normal vector and the tangent of the precession field boundaries
lying in the scattering plane, respectively. si,f is the unit vector perpendicular to the
scattering plane defined by si,f = ni,f × ti,f . Note that the unit vectors si,f are identical
since they are both perpendicular to the scattering plane. The components ei,f1, ei,f2 and
ei,f3 are defined by
ei1 = ǫi · ni ei2 = ǫi · ti ei3 = ǫi · si (2.62)
ef1 = ǫf · nf ef2 = ǫf · tf ef3 = ǫf · sf . (2.63)
As a generalization compared to [6], the case where the gradient vector of the dispersion
∇ω (qc0) has a component out of the scattering plane is included. Inserting equation (2.61)
in equation (2.59) yields
2.3 Extended NRSE resolution function 21
A2
(kF ·nf )2∆kf ·nf = −τ ′′2∆ω + τ ′′2NIei1∆kin
+τ ′′2NIei2
(−∆kin tan θ1 + y1
1
cos θ1
)
+τ ′′2NIei3z1 − τ ′′2NF ef3z2
−τ ′′2NF ef21
cos θ2y2
+~
2mτ ′′2∆k2
i −~
2mτ ′′2∆k2
f
+τ ′′2∆Gc · ∇ω (qc0)−1
2τ ′′2∆q′T
c Hc|qc0∆q′
c (2.64)
with
τ ′′2 =τ2
(ef1 − ef2 tan θ2). (2.65)
For the calculation of equation (2.64) the following relations for ∆ki,f expressed in the basis
of ni,f , ti,f and si,f were used:
si ·∆ki = z1 (2.66)
sf ·∆kf = z2 (2.67)
ni ·∆ki = ∆kin. (2.68)
∆ki = (x1 cos θ1 + y1 sin θ1)ni + (−x1 sin θ1 + y1 cos θ1) ti + z1si (2.69)
= ∆kinni +
(−∆kin tan θ1 + y1
1
cos θ1
)ti + z1si (2.70)
∆kf = (∆kf ·nf )nf +
(− (∆kf ·nf ) tan θ2 + y2
1
cos θ2
)tf + z2sf . (2.71)
ti ·∆ki = −∆kin tan θ1 + y11
cos θ1(2.72)
and
tf ·∆kf = − (∆kf ·nf ) tan θ2 + y21
cos θ2. (2.73)
Substituting equation (2.64) into equation (2.42) introduces the energy deviation from the
dispersion surface due to the linewidth broadening ∆ω into the Larmor phase. This is re-
quired to provide the Fourier transform of the scattering function S (Q, ω):
22 2 NRSE resolution theory
φ(ki,kf )− φ0 = −τ1NI∆kin − τ ′′2∆ω + τ ′′2NIei1∆kin
+τ ′′2NIei2
(−∆kin tan θ1 + y1
1
cos θ1
)+ τ ′′2NIei3z1
−τ ′′2NF ef21
cos θ2y2 − τ ′′2NF ef3z2
+τ ′′2∆Gc · ∇ω (qc0)−1
2τ ′′2∆q′T
c Hc|qc0∆q′
c
+~
2mτ ′′2∆k2
i −~
2mτ ′′2∆k2
f
+τ1NI
kI · ni∆k2in − τ2
NF
kF ·nf(∆kf ·nf )
2 . (2.74)
The Larmor phase now takes into account second order effects and the above mentioned
generalizations. Contributions from components of the gradient vector of the dispersion
surface, which are out of the scattering plane are considered by the terms proportional to
ei2,3 and ef2,3. If the gradient vector of the dispersion surface lies within the scattering
plane, these terms will vanish since then ei2,3 = 0 and ef2,3 = 0. The terms proportional to
∆Gc and Hc account for contributions arising from sample imperfections and the curvature
of the dispersion surface, respectively.
The term (∆kf · nf )2 is now substituted in equation (2.74) by using equation (2.64). Since
only second order effects are considered and higher order terms are neglected, it is sufficient
to consider equation (2.64) to first order only:
∆kf ·nf = − 1
CfNF∆ω +
NICi
NFCf∆kin
+NI
CfNFei2
1
cos θ1y1 −
1
Cfef2
1
cos θ2y2
+NI
CfNFei3z1 −
1
Cfef3z2 +
1
CfNF∆G′
c · ∇ω (qc0) (2.75)
using
τ2 =A2
(kf · nf )2NF
(2.76)
Ci = ei1 − ei2 tan θ1 (2.77)
Cf = ef1 − ef2 tan θ2. (2.78)
2.3 Extended NRSE resolution function 23
The term + 1CfNF
∆G′c ·∇ω (qc0) considers only the first order terms arising from the lattice
imperfections. Since lattice imperfections have been introduced from the beginning, it is
considered here. This term introduces cross terms between the lattice imperfection variables
∆η, ∆ν and ∆Gc and the variables of the 6 component vector J = (∆ω,∆kin, y1, y2, z1, z2).
∆G′c is defined in equation (2.99). ∆k2
i and ∆k2f are substituted in equation (2.74) using
equations (2.69) and (2.71):
∆k2i = ∆k2in +
(−∆kin tan θ1 + y1
1
cos θ1
)2
+ z21 (2.79)
∆k2f = (∆kf ·nf )
2 +
(− (∆kf ·nf ) tan θ2 + y2
1
cos θ2
)2
+ z22 . (2.80)
Inserting equations (2.79) and (2.80) into equation (2.74) and using again equation (2.75)
to substitute all ∆kf · nf terms allows to express the total Larmor precession angle as a
function of squared and cross terms of the six variables (∆ω,∆kin, y1, y2, z1, z2). The total
Larmor phase can conveniently be expressed in matrix notation [13]:
φ(ki,kf )− φ0 = τ ′′2TTJ− 1
2τ ′′2 J
TΨJ
+τ ′′2∆Gc · ∇ω (qc0)
−1
2τ ′′2∆q′T
c Hc|qc0∆q′
c +X(∆Gc) (2.81)
Here X(∆Gc) denotes all cross terms introduced by ∆η, ∆ν and ∆Gc. Effects from sample
imperfections are treated in section 2.3.3. The components of the 6-dimensional column
vector T are:
T1 = −1 T2 = NI
(Ci −
τ1τ2Cf
)T3 =
NIei2cos θ1
(2.82)
T4 = −NF ef2cos θ2
T5 = NIei3 T6 = −NF ef3. (2.83)
The elements of the symmetric (6× 6) matrix Ψ are given in Appendix B.
For the special case, where the spin echo conditions are satisfied, and the gradient of the
dispersion surface lies in the scattering plane:
ei1 = 1 ei2 = ei3 = 0 (2.84)
ef1 = 1 ef2 = ef3 = 0 (2.85)
Ci = Cf = 1 (2.86)
24 2 NRSE resolution theory
and thus
τ ′′2 = τ2 (2.87)
τ1 = τ2. (2.88)
The matrix Ψ then reduces to the matrix given in [6]. Since only T1 = −1 remains non-zero,
the terms, which are linear in J, vanish, leaving only the desired term −τ∆ω in the spin
echo phase.
2.3.2 The τ dependence of the polarization
In this section the τ dependence of the polarization for violated spin echo conditions for a
sample without lattice imperfections (∆Gc = 0) and a planar dispersion (Hc|qc0 = 0) is
derived. With these assumptions equation (2.81) reduces to
φ(ki,kf )− φ0 = τ ′′2TTJ− 1
2τ ′′2 J
TΨJ. (2.89)
The TAS transmission function is derived, following the approach of Popovici [51]. In order
to have a consistent nomenclature, this approach is summarized in Appendix C. It reads:
RTAS (ki,kf ) = exp
(−1
2JTLTASJ
). (2.90)
Substituting the equations (2.89) and (2.90) into the fundamental equation (2.39) yields:
AE =1
N
∫S (Q, ω) exp
(iτ ′′2T
TJ)exp
(−1
2JTLIJ
)d6Jn + c.c. (2.91)
with
LI = LTAS + iτ ′′2Ψ. (2.92)
Terms in ∆ω higher than linear can be neglected, since the integral over the energy coordi-
nate will be dominated by the term exp (−iτ∆ω). The only reasonable application of spin
echo is the situation where ∆ω is very small and hence S (∆ω) is very narrow compared to
the TAS energy resolution. Therefore, rewriting equation (2.91) yields:
AE =1
N
∫S (Q, ω) exp
(−iτ ′′2∆ω
)d∆ω
×∫
exp(iτ ′′2 T
T J)exp
(−1
2JT LI J
)d5Jn (2.93)
2.3 Extended NRSE resolution function 25
where J = (∆kin, y1,y2, z1, z2) and T are the 5-D sub vectors of J and T without the energy
variable ∆ω, respectively, and LI is the corresponding symmetric (5× 5) sub matrix of LI .
Following [6] and using the general matrix theorem [18]
∞∫
−∞
exp(KTJ
)exp
(−1
2JTMJ
)dnJn
=(2π)n/2
(detM)1/2exp
(−1
2KTM−1K
)(2.94)
the resolution function is obtained
FI
(τ ′′2)=
∣∣∣∣∣
√det LI (τ ′′2 = 0)
det LI (τ ′′2 )exp
(−1
2τ ′′22 TT L−1
I
(τ ′′2)T
)∣∣∣∣∣ . (2.95)
For strongly violated spin echo conditions the exponential term will dominate the decay of
the polarization. For perfectly satisfied spin echo conditions this term will be unity since T
becomes zero.
Example 1: Numerical calculations for RbMnF3
Fig. 2.6 and 2.7 show numerical examples for depolarization effects arising from a detuning
of the tilt angles Θ1,2 and a detuning of the frequency f1,2. Here the echo amplitude is
normalized to 1 for the optimum value for each parameter. The numerical calculations were
done for RbMnF3 at the zone boundary excitation Q = [0.5 0.5 −1], E = 8.287meV, for
two different spin echo times τ = 12.4ps (Fig. 2.6) and τ = 80ps (Fig. 2.7). Details about
the dispersion relation properties of RbMnF3 are listed in Appendix D. At the maximum of
the dispersion the slope of the dispersion is equal to zero and hence Θ1,2 = 0. Due to the
symmetry of the dispersion at the maximum and the first order dependence on the cosine
of the coil tilt angles, the depolarizing effects arising from a detuning of Θ1,2 are symmetric
with respect to the optimum value. Note that this is different for other values of Q, where
the slope of the dispersion is different from zero. The depolarizing effect is slightly stronger
for a detuning of Θ1. For the case of Θ1,2 = 0 any detuning of the coil tilt angles increases
the path length of the neutron within the π-coil. Therefore, a higher Larmor phase is ob-
tained and the depolarizing effect due to detuned coil tilt angles increases with increasing
frequencies. Since f1 > f2 for the inelastic case, the depolarizing effect is stronger for the
first spectrometer arm. In the case of the minimum experimentally accessible τ = 12.4ps a
detuning of dΘ1,2 = ±9 would decrease the echo amplitude by about 3%.
26 2 NRSE resolution theory
−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
dΘ1 [deg]
echo
am
plitu
de
a−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
dΘ2 [deg]
echo
am
plitu
de
b
−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
df1 [kHz]
echo
am
plitu
de
c−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
df2 [kHz]
echo
am
plitu
de
d
Fig. 2.6: Depolarization effects due to a detuning of spin echo parametersat τ = 12.4ps corresponding to f2 = 50kHz. The detuned Parameters are:Θ1 (a), Θ2 (b), f1 (c) and f2 (d).
In contrast, the depolarizing effect arising from a frequency detuning is more sensitive to a
detuning of the second spectrometer arm. Since f1 > f2 the relative detuning is smaller for
higher frequencies and therefore the depolarizing effect decreases with increasing frequency.
The calculated frequency ratio of this numerical example is f1f2
≈ 3. For τ = 12.4ps a de-
tuning of df1 = ±25kHz and df2 = ±8kHz would decrease the echo amplitude by about 3%.
The detuning of f1 needs to be a factor of 3 higher compared to a detuning of f2 in order to
obtain the same effect. This is in very good agreement with the calculated frequency ratio.
The depolarizing effect arising from detuned coil tilt angles increases with increasing fre-
quency. Hence, the echo amplitude is more sensitive to a detuning of Θ1,2 for higher spin
echo times (see Fig. 2.7 a and b). For τ = 80ps a detuning of dΘ1,2 = ±1.5 would de-
crease the echo amplitude by about 3%. In contrast the echo amplitude gets less sensitive
for a detuning of the frequencies (see Fig. 2.7 c and d), since the frequency increases with
increasing spin echo time τ . For τ = 80ps a detuning of ±30kHz for f1 and ±10kHz for f2
would decrease the echo amplitude by about 3%.
A special case of detuning occurs if no spin-flip scattering occurs in the sample while the
second precession field region is tuned to spin-flip-scattering. This case is easily treated if
the second frequency f2 is assumed to be detuned to −f2. A numerical example for the zone
boundary excitation Q = [0.5 0.5 −1], E = 8.287meV, in RbMnF3 is shown in Fig. 2.8.
2.3 Extended NRSE resolution function 27
−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
dΘ1 [deg]
echo
am
plitu
de
a−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
dΘ2 [deg]
echo
am
plitu
de
b
−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
df1 [kHz]
echo
am
plitu
de
c−40 −20 0 20 400
0.2
0.4
0.6
0.8
1
df2 [kHz]
echo
am
plitu
de
d
Fig. 2.7: Depolarization effects due to a detuning of spin echo parametersat τ = 80ps. The detuned Parameters are: Θ1 (a), Θ2 (b), f1 (c) and f2(d).
It is obvious that the non-spin-flip signal is already completely depolarized at the lowest
accessible spin echo time τ = 12.4ps. Note that this is not the general case and the signal
arising from the occurrence of non-spin-flip scattering might be not completely depolarized
at the lowest accessible spin echo time for other materials.
0 5 10 15 200
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
Fig. 2.8: Depolarization effects for spin-flip scattering without reversingthe second field region. The example shown applies to the zone boundaryexcitation Q = [0.5 0.5 −1],E = 8.287meV, in RbMnF3.
28 2 NRSE resolution theory
Example 2: Transverse acoustic phonon in Pb
Using equation (2.95) and assuming an excitation with zero linewidth and perfectly satisfied
spin echo conditions the instrumental resolution function for the transverse acoustic phonon
[2 ±0.1 0] in Pb can be calculated numerically (see Fig. 2.9). For the calculations an energy
of ~ω0 = 0.88meV, a slope of the dispersion of |~∇qω0 (q0)| = 6.9meVÅ at T = 290K and
an incident wavevector of ki = 1.7Å−1 were assumed. The resolution function changes only
slightly with different TAS configurations. This is due to the small dependence of the beam
divergence on the scattering senses. For this specific phonon the spin echo time for the
NRSE option of the cold TAS V2/FLEX at BER II is limited to τ = 230ps. Therefore, the
depolarizing effects arising from the scattering senses of the background TAS do not play a
role in the accessible τ -range of the NRSE spectrometer.
0 1000 2000 3000 40000
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
SM=−1 SS=−1 SA=−1SM=−1 SS=−1 SA=+1SM=−1 SS=+1 SA=−1SM=−1 SS=+1 SA=+1
Fig. 2.9: Calculated instrumental resolution for the [2 ±0.1 0] TA phononin Pb assuming zero linewidth and perfectly satisfied spin echo conditions.The depolarization is calculated for different scattering senses. The smalldifference for different instrument settings arise from small changes of thebeam divergence due to the scattering senses. Note that for this specificphonon the spin echo time for the NRSE option of the TAS V2/FLEX atBER II is limited to τ = 230ps.
2.3.3 Quantitative description of depolarization due to sample imperfec-
tions
In this section the additional term +τ ′′2∆Gc · ∇ω (qc0) in the total Larmor phase given by
equation (2.81) arising from sample imperfections is considered in more detail. As pointed
out by Pynn [19] the mosaicity of the sample introduces a further limit to the resolution.
Fig. 2.10 shows a schematic drawing of this effect for a transverse phonon. For each lattice
vector G within the mosaic spread the dispersion will have a different orientation in q-space.
2.3 Extended NRSE resolution function 29
This smearing of the dispersion leads to a broadening of the linewidth, which is different
from the intrinsic linewidth of the excitation.
ω
q
Δω
Δq
lattice vector G
Fig. 2.10: Effect of the mosaic spread of the sample on the width of thedispersion. An angular variation in the lattice vector G (mosaic) leadsto a variation of the linewidth of the dispersion and thus to an artificialbroadening of the linewidth.
Correction for sign errors and using the correct second order term rather than that stated
in [6] yield:
∆Gc = Gc −Gc0 (2.96)
=
(Gc0 +∆Gc) cos∆ν cos∆η
(Gc0 +∆Gc) cos∆ν sin∆η
(Gc0 +∆Gc) sin∆ν
. (2.97)
Expanding small variations in the lattice vector Gc and neglecting terms higher than second
order leads to
∆Gc =
∆Gc − 12Gc0
(∆ν2 +∆η2
)
Gc0∆η +∆Gc∆η
Gc0∆ν +∆Gc∆ν
. (2.98)
Since the scattering function S (Q, ω) is proportional to (Q · ξ)2 where ξ is the phonon
polarization vector, the optimum choice for q0 for transverse phonons is an orientation
perpendicular to G0, whereas for a longitudinal phonon the optimum choice for q0 is parallel
30 2 NRSE resolution theory
to G0. Equation (2.98) shows that for longitudinal phonons the mosaic spread contributes
only in second order. Therefore, the sample can have a rather large mosaic spread before
these effects limit the resolution. For a more detailed discussion see [6]. Since this vector
has linear and quadratic terms in the variables ∆η, ∆ν and ∆Gc the linear and quadratic
parts are defined separately:
∆G′c =
∆Gc
G0c∆η
G0c∆ν
(2.99)
∆G′′c =
−12G0c
(∆ν2 +∆η2
)
∆Gc∆η
∆Gc∆ν
. (2.100)
The complex resolution matrix can now be written as
LM = I−1LII+N+W, (2.101)
with the non-zero elements of the (6× 9) matrix I:
I11 = I22 = I33 = I44 = I55 = I66 = 1, (2.102)
I17 = Cx, I18 = CyG0c, I19 = CzG0c (2.103)
with the definition C = ∇qω (qc0). The non-zero elements of the (9× 9) matrix N are:
N77 =1
Υ2S
, N88 =1
η2S, N99 =
1
ν2S. (2.104)
Here ΥS is the 1σ standard deviation for the Gaussian distribution of lattice vectors. The
quadratic terms in ∆η, ∆ν and ∆Gc are taken into account by the symmetric (9× 9) matrix
W. Its non-zero elements are:
W78 = W87 = −iτ ′′2Cy, W79 = W97 = −iτ ′′2Cz (2.105)
W88 = +iτ ′′2CxG0c, W99 = +iτ ′′2CxG0c. (2.106)
Note that equation (2.106) is the corrected version of equation (83) in [6]. The linear terms
in ∆η, ∆ν and ∆Gc arising from ∆G′c can be written as +iτ ′′2T
Tg JM , with the column
2.3 Extended NRSE resolution function 31
vectors
Tg = (0, 0, 0, 0, 0, 0, Cx, CyG0c, CzG0c) (2.107)
JM = (∆ω,∆kin, y1, y2, z1, z2,∆Gc,∆η,∆ν) . (2.108)
All linear terms can be taken into account by introducing
TM =
−1, NI
(Ci − τ1
τ2Cf
), NIei2cos θ1
,−NF ef2cos θ2
, NIei3,−NF ef3,
Cx,−CyG0c,−CzG0c
. (2.109)
Analog to the treatment in the previous subsection, terms in ∆ω higher than linear are
neglected, leading to
AE =1
N
∫S (Q, ω) exp
(−iτ ′′2∆ω
)d∆ω (2.110)
×∫
exp(iτ ′′2 T
TM JM
)exp
(−1
2JTM LM JM
)d8Jn, (2.111)
where JM = (∆kin, y1,y2, z1, z2,∆Gc,∆η,∆ν) and TM are the 8-dimensional sub vectors of
JM and TM , respectively. LM is the corresponding symmetric (8× 8) sub matrix of LM .
Applying the general matrix theorem (2.94), leads to the resolution function in the form
FM
(τ ′′2)=
∣∣∣∣∣
√det LM (τ ′′2 = 0)
det LM (τ ′′2 )exp
(−1
2τ ′′22 TT
M L−1M
(τ ′′2)TM
)∣∣∣∣∣ , (2.112)
which includes general lattice imperfections.
Figure 2.11 shows numerical calculations for the depolarizing effects of different mosaic
spreads using (2.112). The calculations were done for the [2 0.1 0] TA phonon in Pb at
the energy E = 0.879meV, assuming a zero linewidth. Note that the mosaic spread ηs is
the FWHM. As discussed above, the higher the mosaic spread of the sample, the faster the
polarization decreases.
2.3.4 Quantitative description of depolarization due to curvature of the
dispersion surface
In this subsection the additional term −12τ
′′2∆q′T
c Hc|qc0∆q′
c in the total Larmor phase given
by equation (2.81), arising from the curvature matrix of the dispersion surface, is discussed.
A curved dispersion surface within the TAS resolution ellipsoid leads to additional depo-
larization effects, since neutrons corresponding to different points on the dispersion surface
32 2 NRSE resolution theory
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
η
S=1’
ηS=2.5’
ηS=5’
Fig. 2.11: Depolarization effects due to mosaic spread calculated for the[2 0.1 0] TA phonon in Pb. The mosaic spread ηs is the FWHM.
will have a different spin echo phase Φ. The formalism discussed here is also applicable for
crystallographic systems with a symmetry lower than cubic. The Hessian matrix Hc is ex-
pressed in the Cartesian coordinate system related to the reciprocal lattice by the B-matrix
[16]. The elements of the Hessian H are defined by
Hijc =∂2
∂qic∂qjcω (qc) i, j = 1, 2, 3. (2.113)
If the dispersion ω (q) is differentiated in the frame of the reciprocal lattice with basis vectors
b1, b2 and b3, i.e. the elements of H are defined as
Hij =∂2
∂qi∂qjω (q) , (2.114)
and the Hessian H needs to be transformed into Cartesian coordinates using the B matrix:
Hc = BHB−1. (2.115)
Here only second order curvature terms and no sample imperfections are considered. Thus
∆Gc = 0, ∆q′c reduces to ∆qc and LI and J are used. Terms arising from sample imper-
fections will be included in the next subsection. Following [6], the modified matrix reads:
LC = LI + iτ ′′2 I−1C Θ−1
C Hc|qc0ΘCIC (2.116)
2.3 Extended NRSE resolution function 33
where the matrix ΘCIC transforms the Hessian Hc into the coordinate space of LI . The
matrix ΘC describes the transformation
∆qθ = ΘCY (2.117)
with the column vector Y = (x1, y1, z1, x2, y2, z2). The variation of the wavevector ∆qθ =
∆ki−∆kf is expressed in the basis of the Cartesian system (i0, j0 and l0). In this coordinate
system the total wavevector transfer Qθ is parallel to i0 (see Fig. 2.5):
Qθ =
QM
2π
0
0
. (2.118)
Note that (i0, j0 and l0) are identical to the coordinate system as defined as θ-coordinate
system in [17]. Here the same notation is adopted. In order to express the wavevector
variation ∆qθ in the θ-coordinate system, ∆ki and ∆kf have to be rotated into the Qθ-
frame. Using equations (2.43), (2.44) and the definitions made in Fig. 2.5 yields
∆ki = (x1 cosφ− y1 sinφ) i0 + (x1 sinφ+ y1 cosφ) j0 + z1l0 (2.119)
∆kf = (x2 cos Ξ− y2 sin Ξ) i0 + (x2 sin Ξ + y1 cos Ξ) j0 + z2l0 (2.120)
where φ is defined as the angle between Qθ and ki and Ξ is defined as the angle between
Qθ and kf . For equation (2.117) to hold
ΘC =
cosφ − sinφ 0 − cos Ξ sinΞ 0
sinφ cosφ 0 − sin Ξ − cos Ξ 0
0 0 1 0 0 −1
. (2.121)
Note that here the definitions of φ and Ξ are different to [6] and the sign errors in the
definition of ΘC are corrected. In order to evaluate the expression ∆qTc Hc|qc0
∆qc, the
Hessian Hc needs to be rotated into Hθ. According to [17] the transform of the vector Q is
defined as
Qθ = ΩMNUBQ (2.122)
34 2 NRSE resolution theory
and that of the matrix H is
Hθ = ΩMNUBHB−1U−1N−1M−1Ω−1, (2.123)
since QTθ HθQθ = QTHQ must be invariant. Equation (2.123) is the generalization of the
matrix transform given by equation (91) in [6].
In order to evaluate the resolution matrix, the Hessian needs a further transformation
to the variable space of the six dimensional vector J. The matrix IC relates Y and
J = (∆ω,∆kin, y1, y2, z1, z2) in the linear transformation [6]
Y = ICJ. (2.124)
Since the aim is an expression for the linear relation between the variable vectors J and
Y, the linear term in the expansion of the dispersion relation in combination with energy
conservation relation (see equation 2.55) is used:
∆ω =
(~
mkI −∇ω (qc0)
)·∆ki −
(~
mkF −∇ω (qc0)
)·∆kf (2.125)
= NIǫi ·∆ki −NF ǫf ·∆kf . (2.126)
The substitution of ki,f and ǫi,f by using equations (2.69), (2.71) and (2.61) yields
∆ω = x1 (NIei1 cos θ1 −NIei2 sin θ1)
+y1 (NIei1 sin θ1 +NIei2 cos θ1) + ei3NIz1
−x2 (NF ef1 cos θ2 −NF ef2 sin θ2)
−y2 (NF ef1 sin θ2 −NF ef2 cos θ2)− ef3NF z2 (2.127)
and
∆kin = x1 cos θ1 + y1 sin θ1. (2.128)
According to equation (2.124) the transformation reads
J = I−1c Y. (2.129)
The elements of the matrices I−1C and IC are defined in Appendix B. For perfectly satisfied
spin echo conditions equations (2.84) and (2.85) hold and IC reduces to the matrix given
in [6].
With the equations (2.124) and (2.117) the additional term in the Larmor phase due to a
2.3 Extended NRSE resolution function 35
curved dispersion surface is given by
∆qTc Hc|qc0
∆qc = YTΘ−1C Hc|qc0
ΘCY = JT I−1C Θ−1
C Hc|qc0ΘCICJ. (2.130)
The resolution matrix as given in equation (2.116) can now be evaluated. Following the
procedure described in section 2.3.2 the polarization can be written as
AE =1
N
∫S (Q, ω) exp
(−iτ ′′2∆ω
)d∆ω
×∫
exp(iτ ′′2 T
T J)exp
(−1
2JT LC J
)d5Jn. (2.131)
Using equation (2.94) the resolution function taking into account curvature effects and
neglecting contributions from sample imperfections is then given by
FC
(τ ′′2)=
∣∣∣∣∣
√det LC (τ ′′2 = 0)
det LC (τ ′′2 )exp
(−1
2τ ′′22 TT L−1
C
(τ ′′2)T
)∣∣∣∣∣ . (2.132)
2.3.5 Quantitative description of depolarization due to sample imperfec-
tions and curvature of the dispersion surface
The results of the previous two subsections can now be combined to give the resolution
matrix
LMC = ITLCI+N+W (2.133)
The resolution function can be expressed as
FMC
(τ ′′2)=
∣∣∣∣∣
√det LMC (τ ′′2 = 0)
det LMC (τ ′′2 )exp
(−1
2τ ′′22 TT
M L−1MC
(τ ′′2)TM
)∣∣∣∣∣ (2.134)
This resolution function FMC includes general lattice imperfections and the curvature of
the dispersion surface. Here, the advantages of the matrix formalism are obvious, since
cross terms arising from sample imperfections and the curvature are automatically taken
into account by using (2.133)
∆q′Tc Hc|qc0
∆q′c =
(∆qc −∆G′
c
)THc|qc0
(∆qc −∆G′
c
)(2.135)
= ∆qTc Hc|qc0
∆qc −∆G′Tc Hc|qc0
∆G′c
−2∆qTc Hc|qc0
∆G′c (2.136)
36 2 NRSE resolution theory
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
only instrinstr + sampleinstr + curvinstr + sample + curv
Fig. 2.12: Numerical results for different depolarizing effects for the [20.1 0] TA phonon in Pb. Considering only instrumental resolution (black),instrumental resolution and sample imperfections (red), instrumental res-olution and curvature of the dispersion surface (blue). The depolarizationdue to all effects is shown by the green curve.
A comparison of the depolarization arising from the different effects discussed in the previous
subsections is shown in Fig. 2.12. Since experimental tests on the numerical model discussed
here were performed with Pb (see section 2.4), the numerical calculations were done for the
[2 0.1 0] TA phonon in Pb, making the same assumptions as in section 2.3.2. Within the
considered τ -range the depolarizing effects arising from the instrumental configuration of
the background TAS (black) are negligible. The depolarizing effects due to sample mosaicity
(red) are stronger compared to effects arising from the curvature of the dispersion surface
(blue). However, the green curve shows, that it is important to consider all effects.
2.3.6 Dispersion surface not coinciding with the center of the TAS reso-
lution ellipsoid
In this subsection the simplifying assumption that the center of the TAS resolution ellipsoid
coincides with the dispersion surface of the excitation is dropped. The TAS is set such that
the wavevector Q0 satisfies the scattering condition for a given set of wavevectors kI0 and
kF0 and an energy of ~ω0TAS . The wavevector transfer is
Q0 = kI0 − kF0 (2.137)
2.3 Extended NRSE resolution function 37
and the TAS resolution ellipsoid is centered at the energy of (Q0, ω0TAS), where
ω0TAS (kI0,kF0) =~
2m
(k2I0 − k2
F0
). (2.138)
This center does not need necessarily to coincide with the energy of the excitation ~ω0S (Q0),
which is given by the dispersion relation
ω0S (Q0) = ω0 (qc0) . (2.139)
As a consequence all “mean” quantities, used in the derivation of the spin echo phase so far,
have to be taken with respect to the sample dispersion. Since the TAS resolution ellipsoid is
offset, the TAS resolution function needs to take into account the finite energy shift ∆ΩTAS :
∆ω → ∆ω −∆ΩTAS (2.140)
where
∆ΩTAS = ω0TAS − ω0S (Q0) . (2.141)
The resolution matrix reads
LMC = ITLCI+N+W (2.142)
= IT(LTAS + iτ ′′2Ψ+ iτ ′′2 I
−1C Θ−1
C Hc|qc0ΘCIC
)I+N+W. (2.143)
Since only the TAS transmission function changes, the following substitution has to be made
JT ITLTASIJ → JTTASI
TLTASIJTAS (2.144)
with the substitution
JTAS = J− J′=(∆ω −∆ΩTAS ,∆kin, y1, y2, z1, z2), (2.145)
where J′ = (∆Ω, 0, 0, 0, 0, 0). Therefore, the new TAS resolution matrix is defined as
LS = ITLTASI. (2.146)
The Matrix LS is symmetric and therefore J′TLSJ = JTLSJ′. Using this symmetry yields
JTTASLSJTAS = JTLSJ− 2JTLSJ
′ + J′TLSJ′ (2.147)
= JTLSJ− 2∆ΩTAS (LS)1n J+ (LS)11∆Ω2TAS . (2.148)
38 2 NRSE resolution theory
where (LS)1n defines the 6 dimensional row vector of the matrix LS and (LS)11 is the
(1,1) element of the matrix LS . The term proportional to (LS)11∆Ω2TAS only produces a
constant, which can be absorbed in the normalization factor N. With
−1
2JTTASLMC,TASJTAS = −1
2JTLMCJ+∆ΩTAS (LS)1n J
−1
2(LS)11∆Ω2
TAS (2.149)
the modified expression for the polarization reads
AE =1
N
∫S (Q, ω) exp
(−iτ ′′2∆ω +∆ΩTAS (LS)11∆ω
)d∆ω
×∫
exp(TT
TASJ)exp
(−1
2JT LMC J
)d5Jn. (2.150)
Here the definition
TTAS = iτ ′′2 TM +∆ΩTAS
(LS
)
1n. (2.151)
was used. Therefore, the final result for the resolution function, including its normalization
to 1 at τ = 0, is
FMC,TAS
(τ ′′2)
=
∣∣∣∣∣∣∣∣∣∣∣
√det LMC(τ ′′2 =0)det LMC(τ ′′2 )
× exp(12T
TTASL
−1MC (τ ′′2 ) TTAS
)
× exp(−1
2TTTAS (0) L−1
MC (τ ′′2 = 0) TTAS (0))
∣∣∣∣∣∣∣∣∣∣∣
. (2.152)
Since
exp(TT
TASL−1MCTTAS
)= exp
(−τ ′′22 TT
M L−1MCTM
)
× exp
(+∆Ω2
TAS
(LS
)T1n
L−1MC
(LS
)
1n
)
× exp(+i2τ ′′2∆ΩTASTM L−1
MC
(LS
)
1n
), (2.153)
2.3 Extended NRSE resolution function 39
equation (2.152) yields
FMC,TAS
(τ ′′2)
=
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
√det LMC(τ ′′2 =0)det LMC(τ ′′2 )
× exp
(12∆Ω2
TAS
(LS
)T1n
L−1MC (τ ′′2 )
(LS
)
1n
)
× exp
(−1
2∆Ω2TAS
(LS
)T1n
L−1MC (τ ′′2 = 0)
(LS
)
1n
)
× exp(−1
2τ′′22 TT
M L−1MC (τ ′′2 ) TM
)
× exp(iτ ′′22 ∆ΩTAST
TM L−1
MC (τ ′′2 )(LS
)
1n
)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
. (2.154)
Note that the phase term arising from
exp
(−1
2TT
TAS (0) L−1MC
(τ ′′2 = 0
)TTAS (0)
)(2.155)
= exp
(−1
2∆Ω2
TAS
(LS
)T1n
L−1MC
(τ ′′2 = 0
) (LS
)
1n
)(2.156)
is of no interest in this context, since the term is independent from τ ′′2 and will be absorbed
in the normalization factor.
The Fourier transform is no longer as simple as in the previous subsections, since there is
an additional term linear in ∆ω
∫exp (∆ΩTAS (LS)11∆ω)S (∆ω) exp
(−iτ ′′2∆ω
)d∆ω. (2.157)
This term
exp (∆ΩTAS (LS)11∆ω) (2.158)
multiplied with the scattering function introduces a small asymmetry. However, in practical
cases
∆ΩTAS (LS)11∆ω ≪ 1 (2.159)
Thus, the exponential
exp (∆ΩTAS (LS)11∆ω) (2.160)
is only slowly varying with ∆ω and close to 1 over the ∆ω-range of the scattering function
S (∆ω). Therefore, this factor can be neglected.
Note that it is very important to determine the correct ∆ΩTAS with respect to the exci-
tation energy. Since any detuning of the background triple axis spectrometer changes the
effective Q0 vector for the sample, all the excitation parameters might change, which leads
40 2 NRSE resolution theory
automatically to a detuning of the spin echo parameters. A detuning of the TAS effectively
corresponds to a new Q′0, which can be derived using the UB matrix formalism described
in Appendix A. Q′0 then gives rise to a new excitation energy, dispersion gradient and
curvature matrix. Considering the change in the excitation energy one can easily derive the
correct ∆ΩTAS . Numerical examples for detuning effects arising from detuned instrument
parameters are discussed in the next section.
2.3.7 Numerical examples
The numerical examples presented here show the depolarizing effects arising from a detuning
of the background TAS. Using the results of the previous sections, all numerical calcula-
tions were performed for the zone boundary excitation Q = [0.5 0.5 −1], E = 8.287meV,
in RbMnF3. This excitation has been measured in previous NRSE experiments (see for
example [23]). The investigated TAS parameters were:
• A3: A detuning in A3 corresponds to a rotation of the sample around the z-axis.
• A4: The scattering angle A4 is the angle between the wavevectors kI and kF .
• ν and µ: These parameters are the angular variables of the sample goniometers.
• ∆ΩTAS : A detuning in ∆ΩTAS corresponds to a shift of the center of the TAS reso-
lution ellipsoid in energy w.r.t. the nominal energy of the excitation, while the total
wavevector transfer Q is kept fixed.
The zone boundary excitation corresponds to the maximum of the dispersion. Here the
slope of the dispersion is zero and thus, no depolarizing effects due to sample mosaicity
are expected. However, by considering second order effects, the curvature of the dispersion
surface leads to a depolarization even for the case of a tuned instrument. If the instrument
is detuned, the total wavevector transfer Q changes in general. Thus, the local dispersion
parameters, i.e. energy, slope and curvature matrix, need to be recalculated. As a result
the slope of the dispersion becomes non-zero. Hence, depolarizing effects due to sample
mosaicity contribute to the total effect. Since in the case of RbMnF3 the dispersion around
the maximum is flat, small changes in Q lead only to small changes in the excitation energy
and the slope of the dispersion. Hence, depolarization effects arising from a detuning of
the spin echo parameters are negligible (see section 2.3.2). Note that a detuning at steeper
parts of the dispersion will increase the depolarization effects. A decrease of the intensity
at the detector due to a detuning of the TAS parameters is not taken into account here.
2.3 Extended NRSE resolution function 41
Detuning the sample angle A3 and the scattering angle A4
0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
dA3 = 0°dA3 = 0.1°dA3 = 0.5°dA3 = 1°
0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
dA4 = 0°dA4 = 0.1°dA4 = 0.5°dA4 = 1°
Fig. 2.13: Calculated depolarization effects for RbMnF3 at the zoneboundary excitation Q = [0.5 0.5 −1], E = 8.457meV, arising from adetuned A3 angle (left) and a detuned scattering angle A4 (right).
Fig. 2.13 (left) shows the depolarizing effects arising from a detuning of A3. The depolariz-
ing effects arising from a detuning of the scattering angle A4 are shown in Fig. 2.13 (right).
The depolarizing effect increases with an increasing detuning of the angles. A comparison
between the two plots shows that a detuning of A3 and A4 give almost the same depolar-
izing effects (see Fig. 2.16). Due to the small changes in Q the depolarizing effects due to
a detuning of the spin echo parameters are negligible.
Detuning the sample goniometers
0 20 40 60 800
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
dν=0°dν=0.5°dν=1°dν=2°
0 20 40 60 800
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
dµ=0°dµ=0.5°dµ=1°dµ=2°
Fig. 2.14: Calculated depolarization effects for RbMnF3 at the zoneboundary excitation Q = [0.5 0.5 −1], E = 8.457meV, arising from adetuning of the sample goniometers.
42 2 NRSE resolution theory
Fig. 2.14 shows the depolarizing effects arising from a detuning of the sample goniometers.
Here the rotation axis µ was perpendicular to the [1 1 0] direction (left) while the rotation
axis of ν was perpendicular to the [0 0 1] direction (right). The depolarizing effects due
to a misalignment in µ (ν) increase with increasing detuning. However, the depolarizing
effects are small. Again the changes in Q are small, allowing to neglect effects arising from
detuned spin echo parameters.
Detuning the center of the TAS resolution ellipsoid in energy
0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
∆ΩTAS
= 0 meV
∆ΩTAS
= 0.1 meV
∆ΩTAS
= 0.5 meV
∆ΩTAS
= 1 meV
Fig. 2.15: Calculated depolarization effects for RbMnF3 at the zoneboundary excitation Q = [0.5 0.5 −1], E = 8.287meV, arising from a shiftof the center of the TAS resolution ellipsoid in energy.
Fig. 2.15 shows the depolarizing effects arising from a detuning of the TAS resolution
ellipsoid in energy. The depolarizing effects increase with increasing detuning. However,
a detuning of ∆ΩTAS = 0.1meV has almost no effect on the echo amplitude. A resulting
systematic error arising from a slight detuning of the TAS resolution ellipsoid can therefore
be neglected. This is simultaneously an indicator for the required accuracy of the dispersion
data. The required accuracy of the center of the TAS resolution ellipsoid in energy can easily
be achieved for a triple axis spectrometer. By increasing the detuning to ∆ΩTAS = 0.5meV
the depolarizing effects become significant. For ∆ΩTAS = 1meV the echo amplitude drops
drastically. However, the signal is not completely depolarized for accessible spin echo times
τ , which are above τ ≈ 12ps for fRF = 50kHz frequency of the RF coils. If the instrument
is tuned to a certain mode, a strong second mode can still contribute to the echo amplitude
for small spin echo times τ , even if the instrument is strongly detuned for the second mode
(see chapter 3 and section 4.1.3 where it is exactly this effect, which plays a dominant role
in phase sensitive NRSE measurements).
2.4 Experimental test 43
Comparison of different detuned TAS parameters
0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
no detuningdΩ=0.5meVdA3=0.5°dA4=0.5°
Fig. 2.16: Depolarization effects for RbMnF3 at the zone boundary exci-tation Q = [0.5 0.5 −1], E = 8.287meV, for different detuned TAS param-eters.
Fig. 2.16 shows the numerically calculated depolarization effects arising from a detuning
of different TAS parameters. It is obvious that a shift of the TAS resolution ellipsoid in
energy (∆ΩTAS) has the largest effect. A rotation of the sample around the z-axis (A3) and
a detuning of the scattering angle A4 have smaller but still significant effects. A detuning
of the center of the TAS resolution ellipsoid in energy causes a decay of the echo amplitude
different from the expected behavior as seen for a detuning of A3 and A4. The deviations
of the echo amplitude are similar to the spin echo signal corresponding to an asymmetric
line shape of the scattering function (see section 4.1.3). Since the energy resolution of the
background TAS is multiplied with the signal a detuning of the TAS resolution ellipsoid
introduces an artificial asymmetry and thus modifies the decay of the echo amplitude.
2.4 Experimental test
In this section the results of experimental tests [20] performed on the extended resolution
model described in the previous section are discussed. The results presented here have been
published in [21]. If the spin echo parameters of an inelastic NRSE experiment are kept fix
while the sample is rotated, the instrumental settings are effectively detuned, which can be
described by the treatment given in the previous subsection. By rotating the sample, the
dispersion surface is moved through the resolution ellipsoid of the background TAS giving
rise to an energy offset ∆ΩTAS . Therefore, the instrument probes different portions of the
dispersion, leading to a decay of the polarization according to the resolution function given
by equation (2.152) and a phase shift due to additional second order terms in the spin echo
44 2 NRSE resolution theory
phase. These additional terms predict a non-linear behavior of the phase, compared to a
linear relation in the first order expansion of the Larmor phase.
Fig. 2.17: The cold triple axis spectrometer V2/FLEX at the BER IIof HZB, Berlin, with its spin echo option mounted (before the upgrade in2010-2011).
An experiment was performed at a spin echo time τ = 20ps focusing on the phase shift arising
from the detuning of the apparatus. For the present purpose it was beneficial to perform an
experiment on a simple, well understood model system. Therefore, the [2 0.1 0] TA phonon
in a large single crystal of Pb was chosen. This phonon has an energy of E = 0.97meV, well
suited to a cold TAS. The measurements were performed with the NRSE option of the cold
neutron TAS V2/FLEX at the BER II reactor of HZB, Berlin [76] (see Fig. 2.17) at a fixed
incident wavevector kI = 1.7Å−1. The sample was kept at a temperature of T = 100K to
enhance the echo amplitude and the polarization, respectively. At T > 100K the lifetime is
reduced due to phonon-phonon interactions, while for T < 100K the intensity is reduced by
the Bose factor. Since the intensity changes with moving the dispersion surface through the
TAS resolution ellipsoid, the only way to experimentally access the phase, is to record a full
spin echo curve for each rotation angle of the sample. For a more accurate determination
of the phase angle about 2.5 periods of the spin echo signal were recorded for each rotation
angle. As an example, Fig. 2.18 shows the measured spin echo signal for τ = 20ps at
∆A3 = 0. Here ∆A3 is the deviation of the sample angle A3 (corresponding to a rotation
around the z-axis) from the tuned position of A3 corresponding to the excitation at [2 0.1 0].
2.4 Experimental test 45
10 15 20 25100
120
140
160
180
200
220
240
dl2 [mm]
Cou
nts
per
mon
itor
Fig. 2.18: NRSE scan at τ = 20ps for ∆A3 = 0.
Fig. 2.19 shows the total accumulated Larmor phase as a function of the rocking angle ∆A3
using the extended resolution model. The experimental data is in full agreement with the
“parameter free” calculations from our model (black) and shows the non-linear behavior of
the Larmor phase as predicted.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2−600
−500
−400
−300
−200
−100
0
100
200
∆ A3 [°]
phas
e sh
ift [°
]
Fig. 2.19: Phase shift of the [2 0.1 0] TA phonon while detuning the setupwith rotating the sample around ∆A3. Each phase angle is obtained froma full NRSE scan at each rocking angle. The blue line shows a fit to alinear function. The black line is obtained from the extended model and isin full agreement with the experimental data.
Using a simplified approach of a first-order expansion of the Larmor phase, allows to extract
the slope of the dispersion dωdq from the linear coefficient of the phase dependence. A linear
46 2 NRSE resolution theory
fit [6] gives dωdq = 6.42 (14)meVÅ. Though using a full force-constant parameterization of
the dispersion including three-body interactions [22] gives a significantly higher value ofdωdq = 7.43meVÅ. This result demonstrates that higher order terms in the total Larmor
phase are generally significant and important to consider.
2.5 Summary
The theory of inelastic neutron spin echo spectroscopy has been introduced and extended to
second order. As a result depolarizing effects arising from the TAS instrumental resolution,
sample imperfections and curvature of the dispersion surface are included. By allowing
for violated spin echo conditions and crystal symmetries lower than cubic, the existing
formalism has been generalized. A further important extension of the formalism includes
a detuning of the background TAS spectrometer. In order to test the extended resolution
theory, phase sensitive measurements were performed at the cold triple axis spectrometer
V2/FLEX at BER II at HZB, Berlin. By rotating the sample, the background TAS is
automatically detuned and the single dispersion surface is effectively moved through the
resolution ellipsoid of the background TAS. The results are in good agreement with the
extended resolution model and show that second order effects need to be considered. A
resolution model, which accounts for a violation of the spin echo conditions for inelastic
scattering, is appropriate for high resolution spin echo measurements on mode doublets.
Such experiments and the required theoretical framework will be discussed in the next
chapter.
Chapter 3
NRSE investigations on split modes
The high resolution of NRSE spectroscopy potentially allows to resolve excitations separated
in energy, which are unresolved by standard neutron scattering techniques [21, 23]. Pos-
sible applications are for example splittings of magnon excitations as observed by Náfrádi
et al. [24], hybridized magnon-phonon modes, which are existent in multiferroics [25, 26], or
excitations with small energy separations, which are found in orbital Peierls systems [27].
However, prior to application of the method to complex systems with interesting physical
properties, a basic understanding of the potential of the method is required. Since it is not
possible for two excitations separated in energy to satisfy the spin echo conditions simulta-
neously, unavoidable depolarization effects are introduced.
First, a simplified model describing the signature of split modes, a modulation of the echo
amplitude, is introduced. In the next section the results obtained in the previous chapter
will be used to develop a numerical model for the most general case. To test this model,
experimental tests were performed with a so called tunable double dispersion setup. The
results of inelastic and elastic measurements are presented here and have been published
in [21].
3.1 Two modes within the TAS resolution ellipsoid - Simplified
NRSE model
A first approach considers a simplified NRSE model with two modes within the TAS reso-
lution ellipsoid [23]. In order to describe the echo amplitude as a function of the spin echo
time τ , the spin echo conditions are assumed to be perfectly satisfied for both excitations.
Although this is strictly not possible, this assumption can be useful in practical cases, where
the violation of the spin echo conditions is very small. In this minimal model the echo am-
47
48 3 NRSE investigations on split modes
plitude AE is described by the Fourier transform of two Lorentzians. Including the relevant
energy offsets yields:
|AE | =∣∣∣∣1
N
∫S (∆ω − ω0S1)RTAS (∆ω − ω0TAS) exp (−i (∆ω − ω0NRSE) τ) d∆ω
∣∣∣∣ . (3.1)
Here S (∆ω − ω0S1) is the scattering function describing two Lorentzian profils, given by
equation (3.3). The NRSE instrument is tuned to ω0NRSE and N is a normalization factor
ensuring that AE = 1 for τ = 0. As in chapter 2.2 it is assumed that the scattering function
S (Q, ω) does not vary in Q within the TAS resolution ellipsoid. Therefore, the scattering
function can be factorized. S (Q) then only contributes to the absolute intensity and can
be neglected. As an additional weighting function the energy resolution of the background
TAS is
RTAS (∆ω − ω0TAS) = exp
(−4 ln 2
~2 (∆ω − ω0TAS)
2
E2TAS
), (3.2)
where the TAS resolution ellipsoid is centered at ω0TAS with a FWHM of ETAS . The
simplified model function considers no dispersion surface at all and effects arising from TAS
resolution in Q-space and NRSE resolution are neglected. Two excitations are allowed and
parameterized as
S (∆ω − ω0S1) = A1Γ1
Γ21 + (∆ω − ω0S1)
2 +A2Γ2
Γ22 + (∆ω − ω0S1 −∆Ω)2
, (3.3)
where A1,2 are amplitude factors representing the relative intensity weight of the Lorentzians.
The linewidths are given as HWHM Γ1,2. The two excitations are separated in energy
by ∆Ω. Note that the scattering function is centered at the first excitation with an energy
of ~ω0S1.
In an inelastic scattering process the total Larmor phase will have a constant term propor-
tional to ω0NRSE . This energy will be equal to the mean energy of the first excitation ω0S1,
if the apparatus is tuned to this mode. The Fourier transform of the echo amplitude then
reduces to
AE =1
N
∫S (∆ω)RTAS (∆ω − (ω0TAS − ω0S1)) exp (−i∆ωτ) d∆ω, (3.4)
representing the fact that the scattering function is centered at the first mode and the center
of the TAS resolution is at ω0TAS − ω0S1. The normalization factor is then given by
N =
∫S (∆ω)RTAS (∆ω − (ω0TAS − ω0S1)) d∆ω. (3.5)
3.1 Two modes within the TAS resolution ellipsoid - Simplified NRSE model 49
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
2
4
6
8
10
∆ ω [meV]
S(∆
ω)
[a.u
.]
0 5 10 15 20 25 30
10−1
100
τ [ps]
echo
am
plitu
de
Fig. 3.1: Top: Scattering function S (∆ω) (blue solid), assuming Γ = Γ1 =Γ2 = 20µeV, A1=1, A2=2 and ∆Ω = 0.5meV, scattering function multi-plied with the TAS energy resolution function assuming ETAS = 1.25meV(red solid) and scattering function convoluted with the TAS resolution func-tion (grey dashed). Bottom: Single exponential decay corresponding to Γ(black solid), Fourier transform of S (∆ω) (blue solid), Fourier transformof R ·S (∆ω) (red) and the approximation of equation (3.6) (green dashed)
Fig. 3.1 (top) shows the scattering function described by equation (3.3), the scattering
function multiplied and convoluted with the resolution function of the background TAS (see
equation (3.2)) and the corresponding Fourier transform (bottom). Note that the maxima
of the modulation follow the single exponential decay. For the case of identical linewidths
the echo amplitude can be described by the analytical expression of an exponential decay
modulated by a cosine term [23]
AE =|AM |+ (1− |AM |)
∣∣∣cos(2π
τ
T
)∣∣∣exp
(−τΓ
~
), (3.6)
where the period of the modulation is inversely proportional to the separation in energy
T =4π~
∆Ω(3.7)
50 3 NRSE investigations on split modes
and the amplitude of the modulation is determined by the intensity ratio of the modes A1A2
AM =A′
1 −A′2
A′1 +A′
2
(3.8)
with
A′1 = A1RTAS (−ω0TAS) (3.9)
A′2 = A2RTAS (∆Ω− ω0TAS) . (3.10)
Another approximation is
AE =|AM |+ (1− |AM |) cos2
(2π
τ
T
)exp
(−τΓ
~
), (3.11)
which describes the Fourier transform of the scattering function almost as good as equation
(3.6).
To demonstrate the effects of a change in the energy separation and the amplitude ratio
of the two modes on the echo amplitude, Fig. 3.2 shows numerical examples for different
scattering functions and the corresponding Fourier transforms. As seen from equation (3.7),
the period T of the modulation of the echo amplitude decreases with increasing energy sepa-
ration of the excitations (comparing top to middle in Fig. 3.2). Whereas the amplitude AM
of the modulation given by (3.8) reaches a minimum if the amplitude ratio of both modes
equals 1 (comparing middle to bottom in Fig. 3.2).
Beside the scattering probability in this simple model the amplitude parameters A1 and A2
should also contain the information about the fraction of the signal giving rise to a polarized
signal in the detector. As mentioned in [23] two limiting cases can be seen directly:
First, if the spin echo conditions are strongly violated for the second excitation, the depo-
larized background, independent from the spin echo time τ , will increase and A2 can be set
to zero. For the case that the signal from the second excitation is completely depolarized
for the smallest accessible spin echo time, the initial polarization for τ = 0, P0, will be less
than 1. In the second case the spin echo conditions are perfectly satisfied for both modes.
The amplitudes A1,2 are then determined by the structure factor. In this case the contrast of
the modulation is independent from the spin echo time τ . Note that in general the contrast
of the modulation is a function of the spin echo time τ . If the depolarizing effects arising
from violated spin echo conditions are different for both modes, the amplitude ratio A1A2
, i.e.
the contrast, becomes τ -dependent.
3.2 Second dispersion surface within the TAS resolution ellipsoid - General model 51
Any intermediate case is beyond this model. A general model taking into account the decay-
ing contributions of the excitations arising from violated spin echo conditions is discussed
in the next subsection.
−0.2 −0.1 0 0.1 0.20
5
10
15
∆ ω [meV]
S(∆
ω)
[a.u
.]
0 20 40 60
10−1
100
τ [ps]
echo
am
plitu
de
−0.2 −0.1 0 0.1 0.20
5
10
15
∆ ω [meV]
S(∆
ω)
[a.u
.]
0 20 40 60
10−1
100
τ [ps]
echo
am
plitu
de
−0.2 −0.1 0 0.1 0.20
10
20
∆ ω [meV]
S(∆
ω)
[a.u
.]
0 20 40 60
10−1
100
τ [ps]
echo
am
plitu
de
Fig. 3.2: Calculated examples for different model scattering functionsS (∆ω) (left) and the corresponding Fourier transform (right), assumingΓ = Γ1 = Γ2 = 20µeV. Top: A1 = A2, ∆Ω = 0.1meV. Middle: A1 = A2,∆Ω = 0.2meV. Bottom: A1 = 2 ·A2, ∆Ω = 0.2meV.
3.2 Second dispersion surface within the TAS resolution ellip-
soid - General model
In this section the general case of a second dispersion surface within the TAS resolution
ellipsoid is discussed. The model introduced in [23] will be generalized such that the center
of the TAS resolution ellipsoid ωTAS is not required to coincide with neither the energy
of the first dispersion surface ω0S1 (qc0) nor with the second ω0S2 (qc0). In addition, the
spin echo conditions may be violated for both modes. The general expression for the echo
amplitude is given by
AE =1
N
∫S1 (Q, ω)R1(ki,kf )e
iφ1(ki,kf )d3kid3kf
+
∫S2 (Q, ω)R2(ki,kf )e
iφ2(ki,kf )d3kid3kf
. (3.12)
52 3 NRSE investigations on split modes
The echo amplitude at the spin echo point will be given by |AE |. Using the results from
section 2.3.6 AE is
AE =1
N
∫S1 (∆ω)RTAS(∆ω −∆Ω1, Jn)e
iφ1(∆ω,Jn)d∆ωd5Jn
+
∫S2 (∆ω)RTAS(∆ω −∆Ω2, Jn)e
iφ2(∆ω,Jn)d∆ωd5Jn
, (3.13)
where
∆Ω1 = ω0TAS − ω0S1 (qc0) (3.14)
∆Ω2 = ω0TAS − ω0S2 (qc0) . (3.15)
The scattering function does not contain the energy offsets:
S1 (∆ω) = A1Γ1
Γ21 +∆ω2
(3.16)
S2 (∆ω) = A2Γ2
Γ22 +∆ω2
. (3.17)
Substituting the TAS resolution function RTAS in equation (3.13), the most general case
reads:
AE =1
N
∫S1 (∆ω) exp
(−iτ ′′2,1∆ω +∆Ω1 (LS)11∆ω
)d∆ω
×∫
exp(TT
TAS1J)exp
(−1
2JT LMC1J
)d5Jn
+
∫S2 (∆ω) exp
(−iτ ′′2,2∆ω +∆Ω2 (LS)11∆ω
)d∆ω
×∫
exp(TT
TAS2J)exp
(−1
2JT LMC2J
)d5Jn
(3.18)
where
TTAS1,2 = iτ ′′2,1,2TM1,2 +∆Ω1,2
(LS
)
1n. (3.19)
In principle this formalism can be extended to a multiple dispersion case, which can be
useful for mode multiplets with more than 2 excitations. However, this is beyond the scope
of this thesis.
Any mixture of spin-flip and non-spin-flip scattering is included in the general expression
given by equation (3.18). In that case there would be no energy offset for any of the two
excitations. The TAS resolution ellipsoid would be centered and the spin echo conditions
3.2 Second dispersion surface within the TAS resolution ellipsoid - General model 53
for one of the two modes would be violated by a sign reversal in τ2, resulting in a rapid de-
polarization of the “wrong” spin state. An example for this was already given in section 2.3.2.
In order to test the developed general model, existing experimental data of RbMnF3 [23, 15]
were analyzed. The investigated crystal consisted of two grains of comparable size. The
relative orientation of the two crystallites has been obtained by standard procedures [15].
The measurement was performed at a fixed kf = 2.51Å−1 and a temperature T = 3.18K
for the zone boundary magnon at Q = [0.5 0.5 −1], E = 8.46meV. The data points and the
calculated time dependence of the echo amplitude given in [23] is shown in Fig. 3.3 (dashed
blue). Note that in [23] it has been assumed that one mode is perfectly tuned while the
instrument is detuned for the second mode. The black line displays the calculations for the
time dependence of the echo amplitude according to the general model developed in this
thesis. Here, depolarization effects for both excitations due to a detuning of the instrument
are considered and minor sign errors in [6] were corrected for. The calculations give a much
better description of the modulated signal compared to the results in [23]. This proves
the importance of taking depolarization effects for all excitations into account. The energy
separation between the two modes was calculated to be 0.386meV using the UB matrix
formalism. However, as stated in [23], for a fit of the energy split of the two modes more
data points are required.
0 5 10 15 20 25 3010
−1
100
τ [ps]
echo
am
plitu
de
Fig. 3.3: Experimental data measured on the zone boundary magnonQ = [0.5 0.5 −1], E = 8.46meV, T = 3.18K, in RbMnF3 at TRISP, FRM-II [23]. The black line shows the calculations according to the generalmodel developed in this thesis. Calculations assuming several simplifica-tions [23] are given by the dashed blue line. The general model gives abetter description of the data and demonstrates the importance of takingdepolarization effects for both excitations into account.
54 3 NRSE investigations on split modes
3.3 Experimental verification
Before applying the general model for NRSE spectroscopy on multiple modes to complex
systems it is crucial to gain a basic understanding of the potential of the method on a well
understood model system. Therefore, phonon excitations in Nb were chosen as a simple
system to verify the numerical model described in the previous section experimentally.
3.3.1 Experimental setup
The experiments were performed using the NRSE option of the cold triple axis spectrometer
V2/FLEX at the BER II of HZB, Berlin [76] (see Fig. 2.17). In order to study a mode dou-
blet with NRSE, a unique tunable double dispersion setup was realized using 2 Nb crystals
(see Fig. 3.4). In this setup the lower crystal was rigidly fixed, while the upper crystal was
mounted on a stack of piezoelectric devices (purchased from Attocube Systems AG [28]),
consisting of two goniometers and a rotational stage. Thus, the setup allowed for rotating
the upper crystal with respect to the lower crystal and hence provided the opportunity to
generate artificially split modes with a tunable separation in energy. The whole setup was
mounted on a sample stick and both crystals were simultaneously illuminated by the neu-
tron beam. Using the two goniometers of the Attocube setup and the sample goniometers
of the background TAS, both crystals were oriented in the (h h l) scattering plane.
Rotation stage: ( A3)ΔΩ Δ
Goniometer 1: tiltμ
Goniometer 2: tiltν
Nb crystal 1: movablecylinder axis (100)
Nb crystal 2: fixedcylinder axis (110)
0 0 11 1 0
μ
ν
Ω
Fig. 3.4: Left: The double crystal setup mounted on a 3-stage Attocube[28] module (1 rotational stage and 2 goniometers) and attached to thesample stick. The Nb crystals are oriented in the (h h l) scattering plane.Right: Sketch of the sample rotations provided by the module.
3.3 Experimental verification 55
3.3.2 Niobium dispersion models
For the extended resolution model the dispersion parameters enter into the calculation of
the depolarization in the case of detuned instrument parameters. Hence, a numerical model
for the dispersion of Nb is required. Several different models for the dispersion of transition
metals such as Nb are described in the literature (see [29, 30, 32, 33, 34, 35] and a brief
overview is given in [36]). The approaches differ in their ability to reproduce features of the
Nb dispersion such as Kohn anomalies [38]. However, for small values of q all models are
well suited to describe the dispersion relation. Since the inelastic experiments were carried
out for a small q (q = 0.05r.l.u.) the following models were implemented:
• CGW model: This model, proposed by Clark, Gazis and Wallis [39], is based on
angular forces. Modifications suggested by Behari et al. [41, 42] to consider electron-
ion interactions were included by Bose et al. [29].
• DAF model: This model, proposed by deLaunay [40], is based on angular forces.
Modifications suggested by Behari et al [41, 42] to include electron-ion interactions
are realized in [29]. It is shown that the modified CGW and the modified DAF model
give the same results for bcc metals such as Nb [29].
• Modified axially-symmetric model: This model discussed by Bajpai et al. [30]
considers ion-ion and electron-ion interactions. In order to correct smaller typos in
the formulas [37] was used.
0 0.05 0.1 0.15 0.20
1
2
3
4
5
6
q=[0 0 l] [r.l.u]
E [m
eV]
DAFCGWBaypai
Fig. 3.5: Comparison of the dispersion data obtained from the differentimplemented models for Nb. The wavevector of the excitation q is directedalong the [0 0 l] direction. For small values of q the models give almost thesame result.
56 3 NRSE investigations on split modes
Fig. 3.5 shows the dispersion values obtained from the different models for the wavevector of
the excitation q directing along the [0 0 1] direction. For small values of q, here q = 0.05r.l.u.,
all implemented models give the same result within 0.3% except for the (3,3)-component of
the curvature matrix where the value given by the Bajpai is smaller (see Tab. 3.1). There-
fore, for the following analysis of the experimental results only one model, namely the DAF
model, was used.
Model E[meV]
Slope[meV·Å]
H(1,1)[meV·Å2]
H(2,2)[meV·Å2]
H(3,3)[meV·Å2]
CGW 1.1670 12.4389 541.91 541.97 5.61
DAF 1.1656 12.4244 542.60 542.60 5.61
Bajpai 1.1661 12.4017 542.09 542.09 4.74
Tab. 3.1: Dispersion parameters for Nb extracted from the different mod-els for q = 0.05r.l.u. . Elements of the curvature matrix H not listedare zero. The elements of H are expressed in the Cartesian system of thereciprocal lattice.
3.3.3 Elastic measurements on split modes
Since separated Bragg peaks also cause a modulation of the echo amplitude in the spin
echo length domain, accompanying elastic measurements with a mosaic-sensitive Larmor
diffraction setup [44] were performed. This allows for an independent accurate calibration
of the attocube parameters, which determine the dispersion splitting in the inelastic case.
Larmor diffraction
For completeness a short summary of the principles of Larmor diffraction is given here.
A more detailed discussion can be found in [8, 44, 45]. In contrast to the main idea of
neutron spin echo spectroscopy that the Larmor precessions can be used to cancel for each
velocity separately to preserve the polarization, Rekveldt [44, 45] pointed out that useful
measurements can be done by adding the Larmor precessions instead of reversing them.
The key point of this so called Larmor diffraction is that all neutrons fulfilling the Bragg
condition will have the same Larmor precession angle. If the precession field boundaries
are oriented parallel to the diffraction planes (see Fig. 3.6 left), every neutron satisfying
Bragg’s law will have the same perpendicular component ~k⊥ and therefore accumulate the
same Larmor precession angle, while passing through the fields before and after the sample.
3.3 Experimental verification 57
kI
L
θ
kF
k
kI
θ
kF
Fig. 3.6: Left: Larmor diffraction setup. The precession field boundariesare parallel to the lattice planes and the magnetic fields are oriented inthe same direction. All neutrons fulfilling the Bragg condition have thesame k⊥ and therefore undergo the same Larmor precession. This setupis sensitive to a spread of the lattice spacing. Right: Mosaicity sensitiveLarmor diffraction geometry. The lattice planes of the crystal are slightlytilted with respect to each other. Hence, neutrons acquire a different totalLarmor precession angle, as their paths depend on which crystallite theyare scattering off. The magnetic fields of the precession regions are orientedantiparallel.
It can be shown that for parallel oriented magnetic fields the total Larmor phase reads:
Φtot =2mωLL
π~d (3.20)
where d is the lattice spacing of the crystal, ωL is the Larmor frequency and L is the distance
between the precession field boundaries (see Fig. 3.6 left). Therefore, a change in the lattice
spacing results in a phase shift
∆Φ = Φtot∆d
d. (3.21)
Since Φtot can be up to 105rad, a relative resolution of ∆dd
∼= 10−6 can be realized. Analog
to neutron spin echo spectroscopy in Larmor diffraction, the echo amplitude at the analyzer
is the cosine Fourier transform of the scattering function S(Q, ω) w.r.t. Q. This directly
gives the normalized distribution of the lattice spacing variations f (∆d). Note that the
total Larmor precession angle is to first order independent of a tilt of the diffraction planes
(due to mosaicity) with respect to the field boundaries.
If the magnetic fields are oriented in opposite directions as in conventional spin echo mea-
surements, the setup becomes sensitive to mosaic spread, rather than to the distribution of
lattice constants [45] (see Fig. 3.6 right).
58 3 NRSE investigations on split modes
Experimental results
The elastic part of the measurements using the tunable double dispersion setup was per-
formed in a mosaicity sensitive Larmor diffraction setup, i.e. all precession field boundaries
were kept perpendicular to the reciprocal lattice vector QB and were oriented antiparallel.
The spin echo signal was measured using the [1 1 0] Bragg peaks of the two crystals with
an incident wavevector of kI = 1.9Å−1.
First, the [1 1 0] Bragg peak of each crystal was measured individually. This was realized
by rotating the other crystal by more than 4 in A3. In Fig. 3.7 the echo amplitude of
the individual Bragg peaks is shown as a function of the spin echo length δ [46], which is
identical to the van-Hove correlation length [9]. The data were corrected for instrumental
depolarization using direct beam calibration measurements. A Gaussian distribution was
fitted to each data set (see Fig. 3.7) resulting in a FWHM of 564Å (fixed crystal, black)
and 1026Å (movable crystal, blue). This corresponds to a mosaic spread of 5.12arcmin and
7.82arcmin, respectively. The results are in good agreement with previous gamma diffrac-
tion measurements performed at HZB, Berlin.
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
δ [Å]
echo
am
plitu
de
Fig. 3.7: Gaussian distribution fitted to the elastic NRSE data measuredat the [1 1 0] Bragg peak of the individual crystals. The black and blueline correspond to the fixed crystal and the movable crystal, respectively.
In order to investigate the spin echo signal of two Gaussian mosaic distributions the Bragg
peaks of the two crystals were then separated by a small angle ∆A3 in the scattering plane.
The echo amplitude of the exact Fourier transform of the two Bragg peaks can be approxi-
mated with a Gaussian, modulated by a cosine term [21]:
AE =
∣∣∣∣|AM |+ (1− |AM |) cos(2π
δ
∆s
)∣∣∣∣ exp(−4ln2
δ2
w2S
), (3.22)
3.3 Experimental verification 59
0 200 400 600 8000
0.2
0.4
0.6
0.8
1
δ [Å]
echo
am
plitu
de
Fig. 3.8: Elastic NRSE data measured at the [1 1 0] Bragg peaks of two Nbcrystals. The red (green) data points were measured at a nominal relativeangle of 0.3 (0.39) between the [1 1 0] Bragg peaks. The model describedby equation (3.22) is fitted to the data.
where δ is the spin echo length. The modulation of the echo amplitude is inversely propor-
tional to the difference in sample rotation angle ∆A3 and the magnitude of the reciprocal
lattice vector QB
∆s =4π
QB∆A3. (3.23)
The FWHM wS of the distribution function in correlation length may be obtained from the
FWHM of the sample mosaicity ηS .
wS =2π
QBηS. (3.24)
In a mosaicity sensitive Larmor diffraction setup using NRSE, the correlation length δ reads:
δ = 4πm
~
LtanΘ
kIfeff
1
QB, (3.25)
where L is the distance between the NRSE coils operated with effective frequencies feff
and Θ is the tilt angle of the NRSE coils with respect to the incident wavevector with
magnitude kI .
The A3-positions of the Bragg peaks were determined by TAS rocking scans. In order to
obtain a good contrast of the modulation of the echo amplitude the measurements were
performed at a sample angle A3 close to the center between the two Bragg peaks.
Fig. 3.8 shows the results of Larmor diffraction measurements for nominal angular sepa-
rations of 0.30 (red) and 0.39 (green). As expected, the modulation period of the signal
60 3 NRSE investigations on split modes
decreases with increasing angular separation. The approximation described by equation
(3.22) is fitted to the data. Modulation periods of ∆s = 611(1)Å (red) and ∆s = 486(1)Å
(green) are obtained from the fits. According to equation (3.23) a corresponding angular
peak separation of ∆ω = 0.357(1) (red) and ∆ω = 0.449(1) (green) can be extracted. This
is in agreement with an independent calibration of the angular encoders of the Attocube
device using standard TAS scans at larger peak separations. The results demonstrate that
the echo amplitude in the spin echo length space is in good agreement with the prediction.
The development of the contrast of the modulation as a function of the amplitude ratio
of both [1 1 0] Bragg peaks was investigated. Elastic measurements were performed for
different TAS sample angles A3 in Larmor diffraction geometry with an angular separation
between the two Bragg peaks of 0.39. By varying the TAS sample angle A3 the respective
contributions of the Bragg peaks, i.e. the amplitude ratio, changes. Hence, the contrast of
the modulation changes. With a decreasing amplitude ratio the contrast of the modulation
of the echo amplitude should also decrease. Fig. 3.9 shows the accompanying TAS rocking
scan for the experiment. Two Gaussian functions (dashed blue lines) are fitted to the data.
The sum of both functions is shown by the black curve. Larmor diffraction measurements
were performed for A3 values of 45.05, 45.10 and 45.15. Corresponding amplitude ratios
of A1A2 = 0.63, A1
A2 = 0.16 and A1A2 = 0.04 are obtained from the Gaussian fits. Thus, the
contrast of the modulation should decrease with increasing A3.
44 44.5 45 45.5 460
200
400
600
800
1000
1200
rocking angle A3 [°]
coun
ts /
1 s
Fig. 3.9: TAS rocking scan at an angular separation of 0.39 betweenthe two Bragg peaks. Two Gaussians (dashed blue lines) are fitted too thedata. The black line shows the superimposed fit.
Fig. 3.10 shows the result of the Larmor diffraction measurements for A3 values of 45.05
(green), 45.10 (red) and 45.15 (black). As expected, the modulation decreases for larger
values of A3. The approximation given by equation (3.22) is fitted to the data. Correspond-
3.3 Experimental verification 61
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
δ [Å]
echo
am
plitu
de
Fig. 3.10: Elastic NRSE data measured at the [1 1 0] Bragg peaks of twoNb crystals with an angular separation of 0.39. The measurements wereperformed for A3 values of 45.05 (green), 45.10 (red) and 45.15 (black).The approximation (3.22) is fitted to the data. With increasing A3 theamplitude ratio, i.e. the contrast of the modulation, decreases.
ing amplitude ratios A1A2 of 0.553(3), 0.109(4) and 0.066(3) are obtained from the fits. The
results are in good agreement with the amplitude ratios extracted from the TAS rocking
scan. The first minimum of the modulation is reasonably well fitted by the simplified ap-
proximation. However, for lower amplitude ratios the approximation fails to describe the
rise of the echo amplitude beyond δ = 700Å. Thus, the approximation described by equa-
tion (3.22) is suitable to determine the splitting of the Bragg peaks only if A1A2 > 0.5. For
smaller amplitude ratios the approximation fails. Hence, a model including a more detailed
description of the echo amplitude for Larmor diffraction geometry would be needed. How-
ever, a discussion of such a model is beyond the scope of this thesis.
3.3.4 Inelastic measurements on split modes
In this subsection the results from the inelastic measurements performed on the tunable
double dispersion setup during two beamtimes on Nb single crystals are presented. The
simplified model as discussed in section 3.1 and the general model of section 3.2 have been
fitted to the data. A discussion of the results obtained includes a comparison between the
two approaches.
The cold triple axis spectrometer V2/FLEX was operated in a configuration with scat-
tering senses SM = −1 (monochromator), SS = −1 (sample) and SA = +1 (analyzer)
with an experimental transverse Q-resolution of about 0.006Å−1 FWHM at fixed incident
ki = 1.9Å−1 [43]. The two Nb crystals were aligned with [h h l] plane as the scattering
62 3 NRSE investigations on split modes
plane. The energy of the investigated excitation [1 1 0.05] phonon was experimentally found
at E = 1.144(8)meV at the temperature T = 65K (see Fig. 3.11). This is in good agreement
with the excitation energy obtained from the DAF model. The energy scan was performed
at [1 1 -0.05] since this is the focused TAS configuration leading to higher intensities. Due to
resolution effects of the background TAS the signal due to incoherent scattering is not cen-
tered around zero. The required tilt angles of the precession field boundaries, Θ1 = −31.60
and Θ2 = +27.27, were calculated using SERESCAL [15].
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20
200
400
600
800
1000
1200
1400
1600
E [meV]
Inte
nsity
[Cou
nts
/ 15m
in]
Fig. 3.11: TAS energy scan to determine the energy of the [1 1 -0.05]excitation for T = 68K. The scan was done for [1 1 -0.05] since this is thefocused TAS configuration leading to higher intensities. The energy of theexcitation is E = 1.144(8)meV. Due to resolution effects of the backgroundTAS the incoherent scattering signal is not centered around zero.
In order to calibrate the Attocube modules, several TAS energy scans for different angular
separations of the crystals were performed for each module. An examples for the separation
44 44.5 45 45.5 46 46.5 470
500
1000
1500
2000
A3 [°]
Inte
nsity
[Cou
nts
/ 1s]
−0.5 0 0.5 1 1.5 20
500
1000
1500
E [meV]
Inte
nsity
[Cou
nts
/ 12.
3min
]
Fig. 3.12: Left: A3 rocking scan giving a split in A3 of −0.84. The mov-able peak (A3 = 45.72) is defined as the [1 1 0] Bragg peak. Right: Thecorresponding TAS energy scan at [1 1 -0.05] gives an energy of 1.607meVfor the shifted excitation resulting in an energy split of 0.463meV.
3.3 Experimental verification 63
of the two crystals in the rocking angle Ω (A3) is shown in Fig. 3.12. The movable peak
(A3 = 45.73) was defined as the [1 1 0] Bragg peak leading to a split of ∆A3 = −0.84
for the fixed peak. The left plot displays the A3 rocking scan to determine the separation
in A3, while the right plot shows the corresponding TAS energy scan. The energy of the
shifted excitation is fitted to 1.60(2)meV. Here a negative tilt in A3 results in a shift towards
higher energies. Note that the relation between the sign of the split in A3 and the sign of
the energy shift is reversed for the unfocused TAS configuration, which is used for the NRSE
scans at [1 1 0.05].
0 5 10 15 20 25 300.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
de
exponential decayextended model
Fig. 3.13: Inelastic NRSE data of the fixed peak at [1 1 0.05], E =1.144meV. A single exponential decay of the spin echo time τ (red dashed)and the extended resolution model (blue solid) are fitted to the data. Theextended resolution model gives a linewidth, which is zero within error. Incontrast, the linewidth obtained from the exponential decay would suggestan unphysical linewidth of Γ = 40(3)µeV.
In order to determine the linewidth of the excitation at T = 65K, inelastic NRSE measure-
ments were performed on the fixed crystal, while the movable crystal was rotated in A3 by
more than 3 ensuring that only the inelastic signal from the fixed crystal contributes to
the NRSE signal. The data were corrected using tilt angle calibration scans of the direct
beam (see section 5.3.4) allowing to fix P0 = P (τ = 0) = 1 in the fits. Since the disper-
sion at [1 1 0.05], E = 1.144meV, is very steep and close to the Bragg peak the curvature
parameters of the dispersion surface at this point in the (Q, ω)-space have large values.
Therefore, as it can be calculated by the formalism outlined in the previous chapter strong
depolarization effects arising from curvature effects are expected. In a first step to verify
these assumption, a simple exponential decay was fitted to the data resulting in a very broad
linewidth of Γ = 40(3)µeV. A second fit using the extended resolution model, i.e. equation
(2.134), and the DAF model, to calculate the dispersion parameters, gives Γext = 2(3)µeV.
64 3 NRSE investigations on split modes
This is within error in agreement with a vanishing intrinsic linewidth or decay mechanism
(anharmonic interactions, i.e. phonon-defect interactions) giving rise to a linewidth of less
than 3µeV. Both fits are plotted in Fig. 3.13. This demonstrates again that higher order
terms in the Larmor phase are generally significant and important to consider. As the ex-
ample of Nb shows, the depolarization effects increase for steeper and more strongly curved
dispersion surfaces.
For the sake of completeness both models were fitted to the data, leaving P0 as an additional
free parameter. The exponential decay then gives a linewidth of Γ = (26.46± 6.67)µeV and
P0 = (0.71±0.11). Since the data is corrected using the calibration scans, a P0 significantly
below 1 does not make any physical sense. The fit of the general model however, gives a
linewidth of Γ = (4.24 · 10−7 ± 0.008)µeV and P0 = (0.94± 0.17). The results for Γ of both
fits using the general model agree within the error. Furthermore, leaving P0 as a free fit
parameter gives a result close to 1 confirming the validity of the model assumed.
44 44.5 45 45.5 46 46.50
100
200
300
400
500
600
700
800
A3 [°]
Inte
nsity
[Cou
nts
/ 1s]
Fig. 3.14: TAS rocking scan to determine the splitting in A3 of the twomodes.
For a second set of inelastic measurements, both crystals were separated in A3 providing
an artificially split dispersion. The period of the modulation of the echo amplitude should
decrease with increasing separation of the two modes (see equation (3.7)). Since the inelastic
signal decays very fast due to the depolarizing effects arising from the dispersion parame-
ters, a rather large separation in A3 was chosen to ensure that at least one minimum of the
modulated echo amplitude is within the accessible range of spin echo times τ . Therefore,
the crystals have been aligned to have a separation of ∆A3 = 0.5, as determined by a
TAS rocking scan (see Fig. 3.14). Since the contrast of the modulation should be largest
if both modes contribute with the same intensity, the NRSE measurements were done with
an nominal TAS energy of E = 1.28meV. This energy value was obtained as the center
3.3 Experimental verification 65
between the two excitations using the chosen Attocube parameters and the independent
TAS calibrations scans of the Attocube modules.
Simplified model
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
Fig. 3.15: The inelastic data at [1 1 0.05], E = 1.28meV, at T = 68K showsa clear modulation of the echo amplitude. Left: Fit of the simple model(equation (3.6)). Right: Fit of the two other approximations (equation(3.11), red dotted, and equation (3.26), green).
The results of the inelastic measurements with the split dispersion are shown in Fig. 3.15.
The data were corrected using direct beam calibration measurements and show a clear
modulation of the echo amplitude. As a first step, the simple model discussed in section 3.1
(see equation (3.6)) was fitted to the data (left). In addition the second approximation (see
equation (3.11)) and the similar approximation
AE =∣∣∣|AM |+ (1− |AM |) cos
(2π
τ
T
)∣∣∣ exp(−τΓ
~
)(3.26)
were fitted to the data (right). The results of the different fits are listed in Tab. 3.2. All
simplified models give almost the same result for the energy separation ∆Ω of the two exci-
tation of about 260µeV, which is in good agreement with the value of 280µeV, as obtained
from ∆A3 and the calculated dispersion values. The results for the linewidth Γ are far too
high compared to the fit of of the extended model to the single dispersion data. This is
similar to the result from the exponential decay fitted to the data of the single excitation.
The first two simplified models in Tab. 3.2 give an amplitude ratio A1A2
below 0.5, which
is incompatible with the accompanying TAS scans giving an amplitude ratio close to 1.
Whereas the result of the last simplified model is at least closer to 1 compared to the other
models. However, due to the low count rate, the statistics of the data is low, which obscures
66 3 NRSE investigations on split modes
the contrast of the modulation. Since the amplitude ratio is derived from the contrast, this
contributes to a large error in A1A2
.
Model (equation) Amplitude ratio A1A2 Γ [µeV] ∆Ω [meV] χ2
(3.6), blue 0.49± 0.09 47.0± 3.7 0.26± 0.01 0.67
(3.11), red dotted 0.41± 0.08 44.4± 4.0 0.26± 0.01 0.74
(3.26), green 0.74± 0.08 25.5± 8.2 0.27± 0.01 1.54
Tab. 3.2: Fit results of the different approximations of the simple modelcorresponding to Fig. 3.15.
In conclusion, the simple models can be used to determine the split in energy of the two
dispersions. However, the deduced amplitude ratio is very inaccurate. Since the simplified
models do not take into account any depolarizing effects arising from sample imperfections
and curvature of the dispersion surface, the obtained results for the linewidth Γ appear
unphysically large. Note that here it is assumed that the two excitations have the same
linewidth.
General model
In order to test the extended resolution model, the results from section 3.2 were fitted to
the data. Since there is only one maximum of the modulation of the echo amplitude within
the accessible τ -range, it would be very difficult to extract the linewidth Γ out of the data.
Therefore, Γ was fixed to zero, which is in good agreement with the result obtained from the
single dispersion fit with the extended resolution model. Note that it is again assumed that
both excitations have the same linewidth, since in order to extract linewidths out of inelastic
measurements on split modes, many more maxima of the echo amplitude would be needed
within the accessible τ -range to perform proper fits. From the TAS calibration scans using
the UB matrix formalism and the DAF model for the Nb dispersion parameters the energy
separation was calculated to ∆ΩUB = 0.294meV. The result of the fit of the extended model
is shown in Fig. 3.16 (left). The split in energy gives ∆Ω = 0.274± 0.026meV, which is in
very good agreement with the calculated value. The amplitude ratio gives A1A2
= 0.35±0.86.
The large error in the amplitude ratio is a consequence of the rather large error of the echo
amplitude in the minima. The amplitude ratio is extracted from the contrast of the mod-
ulation of the echo amplitude. However, the value obtained for the amplitude ratio is in
agreement with the TAS calibration scans within the error.
3.4 Summary 67
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
τ [ps]
echo
am
plitu
de
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
τ [ps]ec
ho a
mpl
itude
single dispersiondouble dispersion
Fig. 3.16: Left: The general model discussed in section 3.2 fitted to theinelastic data. The fit is in very good agreement with the data and gives aseparation in energy of the modes of ∆Ω = 0.274 ± 0.026meV. Right: Fitresults of the single and the double dispersion using the extended model.The maxima of the modulation follow the decay of the single dispersionsignal.
For comparison the fit curves for the single dispersion and the split dispersion using the
extended resolution model are shown in Fig. 3.16 (right). The plot shows that the maxima
of the modulation of the echo amplitude follow the decay of the single mode as mentioned
in section 3.1.
The results show that the NRSE method can be used to resolve modes split in energy. How-
ever, the achievable resolution depends strongly on the dispersion of the excitation. In the
case of Nb and small q only larger splittings can be resolved, since the depolarization arising
from the dispersion parameters masks the larger modulation periods for smaller separations.
Both models presented here, the simplified and the general model, give comparable results
for the energy splitting of the two excitations. The general model considers depolarizing
resolution effects and describes the linewidth of the excitation properly in contrast to the
phenomenological function of the simplified model. However, since the computation time
for a fit of the general model is much longer, the simplified model is sufficient and preferable
for a fast estimation of the energy splitting. For a proper physical interpretation of the
splitting the generalized model should be chosen.
3.4 Summary
The resolution model discussed in chapter 2 accounts for a violation of the spin echo condi-
tions for inelastic scattering and is appropriate for high resolution spin echo measurements
68 3 NRSE investigations on split modes
on mode doublets. A simplified model describing the signature of split modes, a modula-
tion of the echo amplitude, was discussed and the results obtained in the previous chapter
were used to develop a model for the most general case. In order to test the models a
unique setup was realized, allowing to tune a double crystal arrangement to any desired
splitting of elastic or inelastic TAS and NRSE signals. Echo amplitudes for this tunable
double dispersion setup were measured in Larmor diffraction geometry for two neighboring
Bragg peaks. The results demonstrate the echo modulation in spin echo length space to
be in good agreement with the predictions. Inelastic NRSE spectroscopy on an effectively
split dispersion clearly shows the modulation and agrees with the simplified and the general
model, indicating persistence of the modulation over the entire spin echo time range, probed
by the experiment.
Chapter 4
NRSE line shape analysis
Another class of experiments, where the NRSE method with its high resolution opens up
entirely new perspectives, is dedicated to line shape analysis. In particular, this is relevant
for phenomena related to temperature dependent asymmetric line broadening, as has been
studied in this thesis. Here, the high resolution of NRSE allows to resolve deviations of the
scattering function from symmetric Lorentzian line shape by directly accessing correlations
in the time domain.
The phenomenological function [53]
S (ω) =1
π
1
1 +(ωΓ − α
(ωΓ
)2+ γ
(ωΓ
)3)2 (4.1)
takes the form of a modified Lorentzian and describes an asymmetric line shape. Here, the
usual argument is replaced by a polynomial, that includes an asymmetry term α(ωΓ
)2 and
a damping term γ(ωΓ
)3. A numerical example for the NRSE fingerprint of an asymmetric
line shape using equation (4.1) is shown in Fig. 4.1. The particular advantage of the NRSE
method is its direct access to the line shape in the time domain, since there is no convolu-
tion of the signal with the resolution function of the spectrometer. From the methodological
point of view the challenge of the experiments presented in this chapter is to explore new
territory for NRSE beyond standard linewidth measurements.
A quasi-particle, which is interacting with other thermally excited quasi-particles, will have
a limited lifetime. Conventional theory describes the loss of correlation in the time-domain
as an exponential decay, i.e. the scattering function is a Lorentzian in the energy domain.
Recent time-of-flight (ToF) experiments performed on OSIRIS, ISIS Facility, Rutherford
Appleton Laboratory, UK, on Cu(NO3)2·2.5D2O (copper nitrate), a model system for a 1-D
bond alternating Heisenberg chain, have established thermal development of an asymmetric
69
70 4 NRSE line shape analysis
−0.2 −0.1 0 0.1 0.2 0.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
∆ω [meV]
Sca
tterin
g fu
nctio
n
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ [ps]
echo
am
plitu
deFig. 4.1: Left: Model scattering function with an asymmetric line shape(black) using the phenomenological function (see equation (4.1)) and anormal Lorentzian scattering function (dashed blue). Right: Calculatedecho amplitude for the asymmetric line shape (black) and the Lorentzian(dashed blue).
continuum of scattering, which differs strongly from the proposed universality of Lorentzian-
type linewidths for one-dimensional quantum systems ([53] and references therein). Recent
theoretical work [54] relates this asymmetric line broadening to hard-core constraints and
quasi-particle interactions. This is proposed to apply to a broad range of quantum systems
[53].
Investigations on Sr3Cr2O8, a gapped 3-dimensional antiferromagnet, have shown that
magnons in this compound likewise develop a temperature dependent asymmetric line broad-
ening [65]. The results of [65] prove that the effect of asymmetric thermal line broadening is
not only confined to the special case of a highly gapped alternating chain like copper nitrate,
but is realized for the whole system of dimerized magnets, independent of dimensionality.
A major part of this thesis was to explore the potential of NRSE as a method to resolve such
effects. High resolution inelastic NRSE measurements were performed for both compounds,
Cu(NO3)2·2.5D2O and Sr3Cr2O8, at TRISP at the FRM-II, Garching. For the first time,
this effect was measured with NRSE.
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O
4.1.1 Properties of Cu(NO3)2·2.5D2O
High field magnetization and inelastic neutron scattering measurements have been per-
formed in the past [56, 57, 58, 60, 55] and provide a consistent picture of Cu(NO3)2·2.5D2O
as a 1-D dimerized spin-12 antiferromagnet. The compound was shown to have a monoclinic
structure with space group I12/c1 [56, 57] and the low temperature (T = 3K) lattice pa-
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 71
rameters a = 16.1Å, b = 4.9Å, c = 18.8Å and β = 92.9 [58]. Copper nitrate closely realizes
the alternating Heisenberg chain with the spin-1/2 moments of its Cu2+ ions (see Fig. 4.2).
Equivalent chains lie along the [12 ,12 ,
12 ] and [12 ,−1
2 ,12 ] directions and project onto the same
direction on the (h 0 l)-plane. The dimerization gives rise to a singlet ground state and the
elementary excitation is a triplet of spin-1 states. The low alternation parameter ensures an
energy separation between the spin-1 bound state and the 2-magnon continuum [59]. The
inter-dimer coupling allows the excitation to hop from site-to-site along the chain. For the
dominant exchange couplings (J = 0.443meV, J ′ = 0.101meV) the magnon bandwidth is
small compared to the gap and due to the smallness of the alternation ratio α = J ′
J ≈ 0.227,
there is a clear energy separation of about 0.5meV between single magnon excitations and
two magnon continuum even at higher temperatures [53].
chain direction
JJ´
d
ρ
Fig. 4.2: Alternating chain layout with the dimer coupling strength pro-vided by J and the intradimer coupling provided by J ′. ρ donates theseparation between dimer spins and the chain repeat vector is given by d
[62].
4.1.2 Sample deuteration and growth of single crystals
In order to increase the intensity of the scattered signal large single crystals are needed.
Since impurities can mask the effect of asymmetric line broadening in the spin echo signal,
the investigated single crystals must be of high purity. High quality single crystals of copper
nitrate were grown from powder samples of Cu(NO3)2·2.5H2O. In its original composition
Cu(NO3)2·2.5H2O copper nitrate produces a high incoherent cross section due to the large
amount of associated crystal water. To avoid a large incoherent contribution to the total
cross section hydrogen needs to be replaced with deuterium, as the incoherent cross section
of deuterium is comparably low: σincH = 80.26b and σinc
D = 2.05b [61]. Quantitative analysis
of the incoherent cross sections of Cu(NO3)2·2.5H2O, Cu(NO3)2·2.5D2O and a mixture of
both show that a deuteration level of at least 98.5% results in an incoherent cross section
comparable within a factor of 1.5 to the incoherent cross section of 100% deuterated copper
nitrate (see Appendix E).
72 4 NRSE line shape analysis
The substitution of the two isotopes was performed in a distillation process similar to the
processes outlined by Xu [60] and Notbohm [62]. To achieve the desired deuteration ratio,
several distillation runs were necessary. In order to reduce the amount of heavy water used,
more distillation steps using less D2O compared to [62] were performed. In order to lower
the boiling point of the solution the distillation was performed under lower pressure, which
could be adjusted using a vacuum pump. The flask containing the solution was kept at
T=65C in a temperature controlled oil bath using a LakeShore temperature controller (see
Fig. 4.3).
vacuum
on on
argon
cooling water
vacuum
argon
vacuum
cooling water
Liebig condenser
oil bath
solution
Fig. 4.3: Sketch of the distillation setup. The solution (dark blue) iskept in a temperature controlled oil bath. During the vacuum pumpingprocess a mixture of normal and heavy water is removed from the solution,condensed in the Liebig cooler and collected in a second flask. To avoidair contact during the refill of heavy water the whole setup is flooded withargon.
140.99g of Cu(NO3)2·2.5H2O powder were dissolved in D2O of high purity (>99.9%). Ac-
cording to the literature [64], 140.99g of Cu(NO3)2·2.5H2O can be dissolved in 29.52g of
H2O and 32.68g D2O, respectively. Using 47.56g of D2O for the starting solution, ensured
the compound to be in solution throughout the whole distillation process. During the vac-
uum pumping process, a mixture of normal and heavy water was removed from the solution
and the condensate was weighed. The deuteration ratio of the solution should not change
significantly during this process, if at all, a small shift to a higher ratio is possible, since the
boiling point of H2O is 1.4C lower compared to D2O. After the pumping additional D2O
was added to the solution a to increase the D2O:H2O-ratio. During the refill, the setup was
flooded with argon to avoid air contact of the solution and thus, an exchange of deuterium
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 73
Fig. 4.4: Highly deuterated copper nitrate single crystals grown usingenrichment and solution growth method. Large high quality crystals can begrown using this method. Left: 2×2×1cm3, m ≈ 4.1g. Right: 2×1×1cm3,m ≈ 2g.
and hydrogen. These steps were repeated several times (see Appendix E), resulting in a
calculated deuteration ratio of >99.38%.
Single crystals were then grown by cooling a saturated solution of Cu(NO3)2·2.5D2O. The
highly hygroscopic nature of copper nitrate stabilizes a second phase (Cu(NO3)2·6H2O)
below T = 26C makes the growth process rather complicated. To limit any substitution
with hydrogen the whole growth process was performed under argon atmosphere. The final
growth products were kept sealed under argon atmosphere. A slow cooling rate is essential
for the homogeneous growth of single crystals. The best parameters found for the growth
process here, were reducing the temperature from T = 85C to T = 40C in discrete steps
with a cooling rate dTdt = 0.05C
12min. During the cooling process needle shaped seed crystals
crystallize in the solution, providing the starting point of the single crystal growth. Apply-
ing this sequence, seven large single crystals with a mass of up to 4.1g each were grown (see
Fig. 4.4). The crystals tend to grow along their b-axis as has been previously found [60].
Subsequently all single crystals were orientated with X-ray Laue in the (h 0 l)-plane within
less than 1 deviation.
4.1.3 Inelastic NRSE measurements
Experiments were performed on Cu(NO3)2·2.5D2O using the thermal NRSE-TAS spectrom-
eter TRISP [66] at the FRM-II, Garching. TRISP was operated in a configuration with
scattering senses SM = −1, SS = −1 and SA = +1 and a fixed incident ki = 1.7Å−1. The
decay of the echo amplitude as a function of the spin echo time τ was investigated at the
minimum of the dispersion of the one-magnon mode at Q=(1 0 1) r.l.u., E = 0.385meV.
Here the intensity is highest and the slope of the dispersion is zero. Since the energy of
the excitation is small compared to the background TAS resolution, TAS energy scans were
performed at the base temperature T = 0.5K to determine the energy of the excitation. Due
74 4 NRSE line shape analysis
to the fact that the polarization in spin-echo mode (precession fields of the spectrometer
arms are oriented antiparallel) was experimentally found to be higher compared to Larmor
mode (parallel orientation), TRISP was operated in spin-echo mode.
−0.2 −0.1 0 0.1 0.20
0.1
0.2
0.3
E [meV]
Inte
nsity
Scattering function T=0.5K
0 20 40 60 80 100 120
0.2
0.4
τ [ps]
echo
am
plitu
de
Spin echo signal T=0.5K
−0.2 −0.1 0 0.1 0.20
0.1
0.2
0.3
E [meV]
Inte
nsity
Scattering function T=2K
0 20 40 60 80 100 120
0.2
0.4
0.6
τ [ps]
echo
am
plitu
de
Spin echo signal T=2K
−0.2 −0.1 0 0.1 0.20
0.1
0.2
0.3
E [meV]
Inte
nsity
Scattering function T=2.5K
0 20 40 60 80 100 120
0.2
0.4
0.6
τ [ps]
echo
am
plitu
de
Spin echo signal T=2.5K
−0.2 −0.1 0 0.1 0.20
0.1
0.2
0.3
E [meV]
Inte
nsity
Scattering function T=3K
0 20 40 60 80
0.2
0.4
0.6
τ [ps]
echo
am
plitu
de
Spin echo signal T=3K
Fig. 4.5: Left: Echo amplitude as a function of spin echo time τ fordifferent temperatures (from the top: T = 0.5K, T = 2K, T = 2.5K andT = 3K) fitted with the phenomenological model (see equation (4.1)).Right: Line shape calculated from the fit parameters obtained from thespin echo data.
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 75
The decay of the echo amplitude as a function of correlation time τ was measured for 4
temperatures (T = 0.5K, 2K, 2.5K and 3K) in the τ -range of 14.5ps to 112.6ps. The data
were corrected using a calibration obtained in the direct beam. Resolution effects arising
from the sample were corrected for. The results are shown in Fig. 4.5.
In order to fit the data, the phenomenological function described by equation (4.1) was
used. Note that here it is assumed that the scattering function does not vary with Q within
the TAS resolution function. While ToF methods need to take the convolution with the
resolution function explicitly into account, NRSE gives direct access to the line shape in the
time domain. Since the asymmetry vanishes for T → 0K [53], i.e. α, γ → 0, the line shape
becomes Lorentzian again. In this case the echo amplitude decays exponentially with τ .
Within the measured τ -range at the base temperature T = 0.5K no noticeable decay of
the echo amplitude was measured. This suggests, that the lifetime of the mode is at least
6.5± 52.6ns corresponding to a linewidth of Γ0.5K = 0.1± 0.8µeV. The sharp excitation at
base temperature indicates that no asymmetry is introduced by the shape of the dispersion
and resolution effects.
P0 0.404± 0.033
Γ0.5K 0.1± 0.8µeV
α0.5K 0
γ0.5K 0
Γ2K 0.009± 0.003meV
α2K 0.119± 0.087
γ2K 0.007± 0.008
Γ2.5K 0.006± 0.002meV
α2.5K 0.121± 0.026
γ2.5K 0.005± 0.002
Γ3K 0.022± 0.009meV
α3K 0.420± 0.166
γ3K 0.071± 0.064
Tab. 4.1: Fit results of the phenomenological model (see equation (4.1))applied to the spin echo data for different temperatures.
The numerical Fourier transform of the applied phenomenological model was fitted for
T = 2K, T = 2.5K and T = 3K and the results are shown in Fig. 4.5. For all 4 tem-
peratures, the parameter P (τ = 0) = P0 was used as a shared fit parameter, since P0
76 4 NRSE line shape analysis
should not change with temperature for τ = 0. The fits are in good agreement with the
data and a clear deviation from an exponential decay can be seen. The fit parameters (see
Tab. 4.1) obtained from the spin echo data and equation (4.1) were used to determine
the temperature dependent asymmetric broadening of the excitation in energy space (see
Fig. 4.5). The value obtained for P0 of approximately 0.4 (see Tab. 4.1) is significantly
smaller than 1. For the chosen scattering plane an equal contribution of spin-flip (SF) and
non-spin-flip (NSF) scattering is expected. This can be seen by determining the scattering
amplitudes for SF and NSF scattering using the treatment of Moon, Riste and Koehler
[63], taking into account that the neutron polarization at the sample adopts all orientations
in the scattering plane for NRSE. Since the instrument was operated in the NSF sensitive
spin echo mode, the whole apparatus was detuned for the SF scattering case. This results
in a depolarized background, yielding a P0 below 1. Since SF and NSF are expected to
contribute by the same amount, an ideal P0 = 0.5 is expected. The value of P0 = 0.40(3)
obtained from the fit is lower. This is in agreement with an unpolarized background rate
of 1 countmin
.
The result of this experiment unambiguously proves an increasing asymmetry with increas-
ing temperature. A comparison of the fit parameters obtained from the spin echo data and
data from time-of-flight measurements at OSIRIS [53] is shown in Fig. 4.6. The results are
in good agreement, except the result for T = 2.5K.
0 1 2 3 4 5−5
0
5
10
15
20
25
30
35
T [K]
Γ [µ
eV]
0 1 2 3 4 5−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
T [K]
α
0 1 2 3 4 5−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
T [K]
γ
Fig. 4.6: Comparison of the fit parameters of the phenomenological modelobtained from the spin echo data (black) and the fit parameters from [53](red).
The NRSE data suggests a double peaked line shape rather than a continuous asymme-
try. The statistical error of the data is a limiting effect of the NRSE method. However,
the method does not depend on a systematic error arising from the convolution with the
resolution function as is present in time-of-flight measurements. The errors of the NRSE
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 77
results increase drastically with increasing temperature since the statistical error of the spin
echo data worsens with increasing temperature and τ . This is due to the lower intensity of
the excitation at higher temperatures and a faster decay of the spin echo signal due to a
broadening of the linewidth. In addition, less data points are available for the fit compared
to the ToF data. In order to reduce the errors and to get a better fit of the profile of the
line shape, better statistics of the spin echo data are needed.
As a proof of principle it could be shown that temperature dependent asymmetric line
broadening can be determined using high resolution NRSE. This is the first time this effect
was measured with NRSE. The performed measurements explored new territory for the
NRSE method and enhanced the potential of the NRSE method beyond standard linewidth
measurements. The particular advantage of the NRSE method is its direct access to the
line shape, since there is no convolution of the signal with the resolution function of the
background spectrometer.
The results presented above, show that NRSE can be used to determine the asymmetric
broadening of an excitation with increasing temperature by measuring the decay of the echo
amplitude as a function of the spin echo time τ . Using such measurements, an asymmetry
can be determined. However, it cannot be extracted if the asymmetry develops towards
lower or higher energies relative to the excitation energy. If the mean energy
Em =
∫ωS(ω)dω∫S(ω)dω
(4.2)
of an excitation is shifted due to an asymmetric broadening of the line shape, the phase of the
spin echo signal for a fixed τ is shifted accordingly. In order to determine the orientation
of the asymmetry, phase sensitive measurements were performed at the minimum of the
dispersion Q =(1 0 1) r.l.u., E = 0.385meV for a temperature range of T = 0.5 − 3K.
The accuracy of the energy shift extracted from the phase sensitive measurements increases
when they are performed at a larger τ . Simultaneously the echo amplitude decreases with
increasing τ increasing the error of the phase. Therefore, τ = 47.23ps was chosen as a
good compromise between the accuracy of the energy shift and the magnitude of the echo
amplitude. The phase shifts relative to the phase at base temperature were converted to
energy shifts using
∆E(T ) = ~φ(T )− φ(T = 0.5K)
τ. (4.3)
Here the phase φ is expressed in radians. The results are shown in Fig. 4.7. Instead of the
expected shift to higher energies according to the results from [53] the data clearly show
78 4 NRSE line shape analysis
a shift to smaller energies. The separation in energy between the magnetic (1 0 1) Bragg
peak and the excitation is large and a contribution of the nuclear (2 0 2) Bragg peak is not
expected due to a velocity selector installed suppressing second order scattering efficiently
at TRISP. Hence, a priori no contamination arising from the Bragg peak is expected since
the intensity of the Bragg peak is reduced by the instrument resolution. Furthermore, the
instrument is tuned to the excitation and therefore detuned for the Bragg peak. Thus,
the remaining contribution from the Bragg peak is expected to be completely depolarized.
However, the results of a more detailed quantitative analysis show, that it is necessary to
consider a contamination from the polarized fraction of the magnetic (1 0 1) Bragg peak in
the present case. As the contribution of this Bragg peak is temperature independent in the
investigated range and the amplitude of the one-magnon excitation decreases, the center of
mass is shifted to smaller energies. This is in very good agreement with the experimental
results.
0.5 1 1.5 2 2.5 3−14
−12
−10
−8
−6
−4
−2
0
2
T[K]
∆ E
[µ e
V]
Fig. 4.7: Temperature dependent data from phase sensitive measure-ments. Equation (4.5) considering a contribution of the Bragg peak isfitted to the data. A polarized fraction of 1.9(2)% of the Bragg peak con-taminating the signal is obtained.
The contribution of the Bragg peak can be estimated by the resolution function of the TAS
at the excitation energy and the intensity of the Bragg peak IBragg,calib from calibration
measurements:
IBragg = e−(
EBragg−Eexc)2
2σ2TAS · IBragg,calib = e
− (0−0.385meV)2
2(0.126meV)2 · IBragg,calib = 538.68counts
sec(4.4)
with σTAS = 0.126meV obtained from TAS calibration measurements. The function Iexc (T )
giving the intensity of the one-magnon excitation as a function of temperature T was ob-
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 79
tained for τ = 47.23ps from the temperature dependent measurements described above.
Using (4.1) and the results in Tab. 4.1 gives the function C (T ), which describes the mean
energy of the asymmetric excitation.
Using these functions and considering the contribution of the Bragg peak, the polarized
fraction f of the Bragg peak, contaminating the phase sensitive measurements, the energy
shift, with respect to ∆E (T = 0.5K) at base temperature, can be fitted by
∆E (T ) =
(1− IBragg · f
IBragg · f + Iexc (T )
)· (C (T ) + 0.385meV)−∆E (T = 0.5K) . (4.5)
The fit of equation (4.5) to the data is shown in Fig. 4.7 and gives a polarized fraction of
f = 1.9± 0.2% of the Bragg peak.
An independent approach to estimate the polarized fraction uses the second order expansion
of the spin echo phase (see equation (2.42). ∆kf is substituted by ∆kf + δkf where ∆kf
is still the distribution of the kf and δkf is the change of kf due to the detuning. An
excitation energy of E = 0.385meV and an incident ki = 1.7Å−1 results in kf = 1.644Å.
Therefore, to change to kf,Bragg = 1.7Å−1 of the elastic case a δkf = 0.056Å−1 is needed.
Substituting ∆kf → ∆kf + δkf into equation (2.42) the last term, A2
(kF ·nf)3 (∆kf · nf )
2, is
replaced by 3 new terms. The term proportional to δk2f results in a constant phase, which
can be neglected. The term proportional to ∆k2f stays the same as before and the only
relevant new term is A2
|kF |3 (2∆kf · δkf )2. Here it is taken into account that the slope of the
dispersion at the minimum is zero and therefore kF ·nf = |kF |. Assuming ∆kf
kf= 1% leads
to an
FWHM∆kf ≈ 0.01 · kF ≈ 0.01644Å−1 and σ∆kf ≈ 0.00698Å−1. (4.6)
The polarization at the Bragg peak is then estimated by
P =
∣∣∣∣∣∣∣
∫e−
∆k2f
2σ2∆kf e
iA22k3
F
2δkf∆kfd∆kf
∣∣∣∣∣∣∣, (4.7)
resulting in a remaining polarization of 1.75%, which is in good agreement with the result
from the fit shown in Fig. 4.7.
Thus, it can be convincingly concluded that for Q=(1 0 1) r.l.u. a relative shift of the mean
energy of the excitation due to an asymmetric broadening cannot be measured with NRSE,
given the energy resolution of TRISP at ki = 1.7Å−1. The shift is masked due to a polarized
fraction of the (1 0 1) Bragg peak contaminating the signal.
80 4 NRSE line shape analysis
In order to investigate the shift of the mean energy of the excitation as a function of tem-
perature, supplementary phase sensitive measurements were performed at the minimum of
the dispersion Q =(1.11 0 0.855) r.l.u., E = 0.385meV, avoiding a contamination by the
(1 0 1) Bragg peak. This was experimentally confirmed by TAS energy scans. The phase
sensitive measurements were performed in a temperature range from T = 0.5k to T = 3K
at two different spin echo times τ = 24.01ps and τ = 47.57ps each. The phase shift ∆φ
relative to the phase at base temperature T = 0.5K extracted from the data was converted
using equation (4.3).
0 1 2 3 4−10
0
10
20
30
40
50
60
70
T [K]
Ene
rgy
shift
∆ E
[µeV
]
0 1 2 3 4−30
−20
−10
0
10
20
30
T [K]
Ene
rgy
shift
∆ E
[µeV
]
Fig. 4.8: Temperature dependent data (black) from phase sensitive mea-surements for spin echo time τ = 24.01ps (left) and τ = 47.57ps (right).The data is compared to calculated mean energies using equation (4.1)and (4.2) and the fit parameters displayed in Tab. 4.1 (blue). Energyshifts calculated from the cosine Fourier transforms of the correspondingasymmetric scattering functions (see Fig. 4.10) are displayed in red.
The results for both data sets are shown in Fig. 4.8. The experimental energy shift (black
data points) is compared to calculated mean energies using equations (4.1) and (4.2) and
the fit parameters displayed in Tab. 4.1 (blue data points). Both data sets display a clear
deviation from the calculated values. Due to the absence of a second strong mode, e.g. a
Bragg peak, the deviations cannot be described by a contamination. Therefore, the data
suggest that the phase shift ∆φ is not directly proportional to the energy shift ∆E in the
case of an asymmetric line shape. Due to the difference between both data sets, the mea-
surements indicate that the measured phase shift is a non-linear function of the spin echo
time τ .
As a numerical example Fig. 4.9 displays the cosine Fourier transform (right) of four scat-
tering functions with a Lorentzian shape (left). The functions are successively shifted by
4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 81
0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
E [meV]
Inte
nsity
0 5 10 15 20 25 30 35 40 45 50−1
−0.5
0
0.5
1
τ [ps]
echo
am
plitu
de
Fig. 4.9: Left: Lorentzian shaped scattering function successively shiftedby 10µeV. The functions are centered at: Ec = 0.385meV (black), Ec =0.395meV (blue), Ec = 0.405meV (green) and Ec = 0.415meV (red.).Right: Corresponding numerical calculation of the cosine Fourier trans-form.
10µeV. A comparison of the phases of the different cosine Fourier transforms shows that the
phase shift for a fixed τ is a linear function of the energy shift, i.e. the phase is a linear
function of the spin echo time τ .
Results for the fit parameters of the temperature dependent asymmetric scattering func-
tions are shown in Tab. 4.1 and the corresponding numerically calculated cosine Fourier
transforms are shown in Fig. 4.10. For the minimum close to τ = 24.01ps, a phase shift
between the signals is clearly visible. The phase shift increases with increasing temperature
as expected from the calculations of the mean energy of the asymmetric line shapes. The
energy shift calculated from the cosine Fourier transform follows the trend of the shift of
the mean energy as displayed in Fig. 4.8 (left). The experimental energy shifts disagree
with the theoretical predictions from both the calculated mean energy and the energy shift
extracted from the cosine Fourier transforms.
0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
E [meV]
Inte
nsity
T=0.5KT=2KT=2.5KT=3K
0 5 10 15 20 25 30 35 40 45 50−1
−0.5
0
0.5
1
τ [ps]
echo
am
plitu
de
T=0.5KT=2KT=2.5KT=3K
Fig. 4.10: Left: asymmetric scattering function calculated from the fitparameters given in Tab. 4.1. Right: Corresponding numerical calculationof the cosine Fourier transform.
82 4 NRSE line shape analysis
In contrast for the minimum close to τ = 47.57ps almost no phase shift between the signals
is visible. The energy shift calculated from the signal according to equation (4.3) is shown
in Fig. 4.8 (right). This is in agreement with the experimental results for all temperatures
except at T = 3K, where the error is already quite large. However, it strongly disagrees with
the predictions of the calculations of the mean energy (see equation (4.2)). The difference
between the two data sets clearly demonstrates that the phase is a non-linear function of the
spin echo time τ for a line shape, differing from a Lorentzian shape. In such cases the cosine
Fourier transform no longer follows a simple exponential decay. Thus, additional terms,
which depend strongly on the line shape of the excitation, need to be taken into account
carefully for the data analysis.
As a result it could be shown that the shift of the mean energy of an excitation due to an
asymmetric line broadening cannot be measured in a straight forward manner with phase
sensitive NRSE measurements, since the phase seems to become a non-linear function of
the spin echo time τ . These results are a counter example to the assertions of a linear
dependence between phase shift and spin echo time τ . Thus, a careful treatment of results
is necessary, since these effects could easily lead to wrong data interpretation. This result
can be used to determine a deviation of the scattering function from Lorentzian shape.
4.2 Asymmetric line shape of excitations in Sr3Cr2O8
The 3-D gapped quantum spin dimer Sr3Cr2O8 has been extensively characterized recently
[65, 67, 68]. Standard TAS experiments performed at V2/FLEX, HZB, Berlin, and at the
cold triple axis spectrometer TASP, Paul-Scheerer Institut (PSI), Switzerland, showed that
the effect of temperature dependent asymmetric line broadening is also present in this system
[65, 67]. Using the high resolution NRSE technique, the TAS results were confirmed and it
was shown that the temperature development of the line shape in Sr3Cr2O8 does not have
a Lorentzian line shape.
4.2.1 Properties of Sr3Cr2O8
DC susceptibility, high field magnetization and inelastic neutron scattering measurements
have been performed by Quintero-Castro et al. [65, 67, 68] and provide a consistent picture
of Sr3Cr2O8 as a dimerized spin-12 antiferromagnet. Sr3Cr2O8 consists of a lattice of spin-12Cr5+ ions, which form hexagonal bilayers at room temperature and are paired into dimers
by the dominant antiferromagnetic intrabilayer coupling J0 [70] (see Fig.4.11). The dimers
4.2 Asymmetric line shape of excitations in Sr3Cr2O8 83
J0
J2'
J2''
J2'''
J4'
m
m
m
J1'
J1''
J1''
J3' J
3''
J3'''
J0 5.55(1) meV
J1' -0.04(1) meV
J1'' 0.25(1) meV
J2'- J3' 0.75(1) meV
J2''- J3'' -0.54(1) meV
J2'''- J3''' -0.12(1) meV
J4' 0.06(2) meV
J4'' -0.05(1) meV
Fig. 4.11: Lowtemperature (monoclinic) crystal structure of Sr3Cr2O8
showing the magnetic Cr5+ ions only. The exchange interactions are labeledon the diagram and listed in the table [67].
are coupled three-dimensionally by frustrated interdimer interactions. Sr3Cr2O8 undergoes
a structural Jahn-Teller distortion below TJT = 285K [71]. This lowers the symmetry from
hexagonal (R3m) to monoclinic (C2/c), which is stable for T < 120K [72, 73]. The orbital or-
der and lifting of the frustration gives rise to spatially anisotropic exchange interactions. The
hexagonal lattice parameters at room temperature are a = b = 5.57Å and c = 20.17Å, while
the monoclinic lattice parameters at T = 1.6K are a = 9.66Å, b = 5.5437Å, c = 13.7882Å
and β = 103.66.
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0 M
(-1/2,1/2,-1)h
(1/2,1/2,0)h
(1,1,-3/2)h (-1/2,1/2,0)
h(1,1,0)
h (0,0,0)h
Ener
gy (m
eV)
Twin 1 Twin 2 Twin 3Flex Data
M'
Fig. 4.12: Dispersion relation of Sr3Cr2O8 (from [67]). The blue, ma-genta and black lines correspond to the fitted dispersion relations of threemonoclinic twins. Green points are fitted centers of peaks measured atV2/FLEX at HZB.
84 4 NRSE line shape analysis
The neutron scattering experiments [67] reveal three gapped and dispersive singlet to triplet
modes arising from the three twinned domains formed below the transition (see Fig. 4.12).
The exchange constants were extracted using Random-Phase-Approximation [67]. The in-
tradimer exchange constant is identified as the intrabilayer interaction J0, which has a value
of 5.55meV as found from both, susceptibility and inelastic neutron scattering measurements
[67]. This dimerization gives rise to a singlet ground state and gapped one-magnon exci-
tations. Additionally there are significant interdimer interactions, which allow the dimer
excitations to hop and develop dispersion. The dispersion of the one-magnon mode pro-
duces a bandwidth, extending between the gap energy of 3.5meV and the maximum value
of 7.0meV, as found by magnetization and inelastic neutron scattering measurements [67].
4.2.2 Inelastic NRSE measurements
The experiments on Sr3Cr2O8 were performed at the thermal NRSE-TAS spectrometer
TRISP [66] at the FRM-II, Garching, using a single crystal grown at the HZB, Berlin [69].
TRISP was operated in a configuration with a fixed kf = 2.51Å−1 and scattering senses
SM = −1, SS = −1 and SA = +1. The investigations were done on the lowest energy
mode at the center of the Brillouin zone at Q=(0.5 0.5 3) r.l.u. where the most favorable
conditions are met. The intensity is highest, the slope of the dispersion is zero and the sepa-
ration between the modes of the three existing domains is largest (see Fig. 4.12). Since the
energy of the excitation shifts to higher values with increasing temperature [65], the back-
ground TAS parameters and the NRSE parameters were adjusted to their nominal values for
the excitation energy, appropriate for each temperature. Therefore, TAS energy scans were
performed at T = 0.5K and 15K. For intermediate temperatures TAS-data available from
measurements at V2/FLEX, HZB, Berlin, [65] was interpolated. Since the experimentally
determined polarization in Larmor mode was higher than in spin-echo mode TRISP was
operated in Larmor mode.
The decay of the echo amplitude as a function of correlation time τ was measured for 4
temperatures (T = 0.5K, 10K, 15K and 20K). The data were corrected, using a direct beam
calibration and resolution effects arising from the sample were corrected for. However, the
corrections, which are mainly due to curvature effects, are small (≈ 3%). The results are
shown in Fig. 4.13. With increasing temperature the depolarization increases for higher
values of τ . As in the previous section the phenomenological model given in equation 4.1
was fitted to the data. Again it is assumed that the scattering function does not vary in Q
within the TAS resolution ellipsoid. The asymmetry vanishes for T → 0K [53], i.e. α, γ → 0,
and the line shape becomes Lorentzian again. For this case the spin echo signal is an expo-
nential decay of the echo amplitude. Due to the statistical error of the data at T = 0.5K a
4.2 Asymmetric line shape of excitations in Sr3Cr2O8 85
clear discrimination between an exponential decay and a signal corresponding to an already
developed asymmetry of the line shape is not possible. However, the TAS measurements
performed by Quintero-Castro et al. [65] suggest a symmetric Lorentzian shape of the scat-
tering function for T ≤ 1.6K. Thus, the fit parameters α and γ were fixed to zero for the
base temperature T = 0.5K. A clear determination of the line shape at T = 0.5K would
require further measurements and better statistics.
−0.5 0 0.50
0.2
0.4
E [meV]
Inte
nsity
Scattering function T=0.5K
0 10 20 30 40 500
0.2
0.4
τ [ps]
echo
am
plitu
de
Spin echo signal T=0.5K
−0.5 0 0.50
0.2
0.4
E [meV]
Inte
nsity
Scattering function T=10K
0 10 20 30 40 500
0.2
0.4
0.6
τ [ps]
echo
am
plitu
de
Spin echo signal T=10K
−0.5 0 0.50
0.2
0.4
E [meV]
Inte
nsity
Scattering function T=15K
0 10 20 30 40 500
0.2
0.4
0.6
τ [ps]
echo
am
plitu
de
Spin echo signal T=15K
−0.5 0 0.50
0.2
0.4
E [meV]
Inte
nsity
Scattering function T=20K
0 10 20 30 40 500
0.2
0.4
0.6
τ [ps]
echo
am
plitu
de
Spin echo signal T=20K
Fig. 4.13: Left: Echo amplitude as a function of spin echo time τ fordifferent temperatures (from the top: T = 0.5K, T = 10K, T = 15K andT = 20K) fitted with the phenomenological model (see equation (4.1)).Right: Line shape of the excitation in the energy domain calculated fromthe fit parameters obtained from the spin echo data.
86 4 NRSE line shape analysis
The results fitting the Fourier transform of the model function given by equation 4.1 for the
temperatures T = 10K, T = 15K and T = 20K are shown in Fig. 4.13. The fits are in good
agreement with the data. The parameter P (τ = 0) = P0 was used as a shared fit parameter,
assuming that the polarization at τ = 0ps does not depend on sample temperature. The
obtained value for P0 ≈ 0.4 (see Tab. 4.2) is significantly smaller than 1. For the chosen
experiment parameters equal amounts of spin-flip (SF) and non-spin-flip (NSF) scattering
are expected. Since the instrument has been operated in the SF sensitive Larmor mode,
the whole apparatus has been detuned for any signal arising from NSF scattering. This
results in a depolarized background yielding a P0 below 1. Since SF and NSF scattering
are expected to contribute by the same amount, an ideal P0 = 0.5 is expected. The value
of P0 = 0.426(45) obtained from the fit is lower and is in agreement with an unpolarized
background rate of 1 countmin
.
The fit parameters are listed in Tab. 4.2. Using the fit results and the phenomenological
model, the scattering function and thus, the temperature dependent asymmetric line broad-
ening of the excitation in the energy domain can be calculated (see Fig. 4.13). The results
of this experiment suggest an increasing asymmetry of the line shape of the excitation with
increasing temperature.
P0 0.426± 0.045
Γ0.5K 0.027± 0.006meV
α0.5K 0
γ0.5K 0
Γ10K 0.028± 0.011meV
α10K 0.266± 0.089
γ10K 0.028± 0.020
Γ15K 0.038± 0.014meV
α15K 0.265± 0.116
γ15K 0.030± 0.025
Γ20K 0.058± 0.023meV
α20K 0.290± 0.14
γ20K 0.032± 0.032
Tab. 4.2: Fit results of the phenomenological model (4.1) applied to thespin echo data for different temperatures.
4.2 Asymmetric line shape of excitations in Sr3Cr2O8 87
A comparison of the fit parameters from the spin echo data and TAS data [65] is shown
in Fig. 4.14. The increase of the linewidth Γ of the excitation for increasing temperature
obtained from the spin echo data is less steep compared to the TAS data. The errors of the
linewidths for the different temperatures are of the same order of magnitude. The difference
in the results for the asymmetry parameters α and γ is larger. The larger errors for the
NRSE asymmetry parameters arise from the fact that there are less data points available
for the fit compared to the TAS data. To reduce the errors and to get a better fit of the
profile of the line shape, better statistics of the spin echo data would be crucial.
0 10 20 300
50
100
150
T [K]
Γ [µ
eV]
0 10 20 300
0.1
0.2
0.3
0.4
0.5
T [K]
α
0 10 20 30
0
0.02
0.04
0.06
0.08
T [K]
γ
Fig. 4.14: Comparison of the fit parameters of the phenomenologicalmodel obtained from the spin echo data (black) and the fit parametersfrom [65] (red).
In contrast to copper nitrate the energy of the low energy mode of Sr3Cr2O8 shifts towards
higher energies with increasing temperature [68]. Since this effect masks the small shift
of the center of mass of the scattering function due to an asymmetric line broadening,
phase sensitive measurements can be used to determine the shift of the mode. In order to
check and extend previous measurements on the temperature dependent energy shift of the
mode, additional phase sensitive measurements were performed with NRSE spectroscopy
at Q=(0.5 0.5 3) r.l.u. at an energy of E = 3.6meV and τ = 9.5ps for a temperature
range of T = 0.5 − 20K. The results are shown in Fig. 4.15 and compared with TAS data
taken from [65]. Both data sets are in good agreement. With increasing temperature the
excitation energy is shifted towards higher energies. The dimers become thermally excited
as the temperature is increased. This interferes with the intersite interactions resulting in
a shift of the dispersion towards the intradimer exchange energy J0 = 5.55meV [65]. The
results of the phase sensitive NRSE measurements confirm the TAS measurements and both
data sets are in good agreement with the predictions of the Random Phase Approximation
calculations carried out in [68].
88 4 NRSE line shape analysis
0 10 20 30 40 50 60−0.2
0
0.2
0.4
0.6
0.8
1
1.2
T [K]
∆ E
[meV
]
NRSETAS
Fig. 4.15: Energy shift ∆E of the low energy excitation at (0.5 0.5 3) fordifferent temperatures. The NRSE data and the TAS data [68] are in goodagreement.
4.3 Summary
Two systems, the 1-D Cu(NO3)2·2.5D2O and the 3-D Sr3Cr2O8, in which the effect of tem-
perature dependent asymmetric line broadening has been observed were investigated using
the NRSE technique on the thermal triple axis spectrometer TRISP. For the measurements
on Cu(NO3)2·2.5D2O large single crystals were grown from D2O-enriched solution. For the
experiment with Sr3Cr2O8 a single crystal grown at the HZB was used.
As a proof of principle it could be shown that the high resolution method of NRSE allows to
resolve line shapes, which are not Lorentzian type. Such line shape analysis is relevant for
the phenomenon of temperature dependent asymmetric line broadening as has been first ob-
served in gapped quantum magnets. In this thesis, this effect was measured for the first time
with NRSE in the time-domain. The method of NRSE gives direct access to the scattering
function without a convolution with the resolution function of the background spectrometer.
The statistical error of the NRSE data is a limiting effect, however the method does not
depend on systematic errors arising from the convolution with the resolution function as
it is the case for time-of-flight measurements. As an important result it could be shown,
that the phase shift due to an asymmetric line broadening is a non-linear function of the
spin echo time τ . This effect can also be used to determine a line shape deviation from
Lorentzian shape.
From the methodological point of view the challenge of these experiments was to explore
new territory for NRSE beyond standard linewidth measurements. It could be shown suc-
cessfully that NRSE spectroscopy can be used to reveal the effect of temperature dependent
asymmetric line broadening.
Chapter 5
Upgrade of the NRSE option at
FLEXX
The cold triple axis spectrometer at the BER II reactor at HZB, Berlin, [74] has recently
been upgraded and rechristened FLEXX [75]. In order to benefit from the enhanced TAS
parameters the NRSE option of the spectrometer [76, 77, 78] was upgraded, which was a
major part of this thesis.
The TAS spectrometer was moved to the end of the rebuilt NL1B guide, which extends the
accessible range of wavevector transfer. The guide section has been upgraded with m = 3
neutron supermirrors increasing the neutron flux at the instrument. An elliptical guide
section focuses neutrons onto a virtual source, which is imaged on the monochromator. A
wavelength band is selected by a velocity selector obviating the need for a second order
filter and reducing the background signal. Due to the change of the beam geometry, the
former vertically focusing monochromator has been replaced by a larger double focusing
PG002 monochromator, increasing the neutron flux at the sample position. A major goal
of the upgrade was to improve the polarized neutron capabilities [75] of the spectrometer.
A vertical guide changer has been incorporated and allows to replace a part of the regular
supermirror guide with an S-bender polarizer. Subsequently a guide field is provided for the
remaining neutron flight path.
This chapter deals with the upgrade of the NRSE option of FLEXX. The major goal was
to increase the performance and the accessible parameter range of the combined NRSE-
TAS instrument. In order to feed the larger beam cross section without losses from the
monochromator to the sample and from there to the analyzer, basic components of the
NRSE option were rebuilt. New NRSE bootstrap coils, allowing for larger beam cross
sections and larger coil tilt angles, were manufactured and tested in collaboration with the
89
90 5 Upgrade of the NRSE option at FLEXX
Max Planck Institute for Solid State Research, Stuttgart. The spectrometer arms were
redesigned making the instrument more compact and extending the accessible range in
scattering angle. New, more compact mu-metal shielding provides a better shielding of the
field free region between the NRSE coils. In contrast to the previously used spectrometer
arms, all mechanical and electrical components (e.g. goniometers) are relocated beneath
the shielding boxes to reduce scattering fields inside the field free zone. The coupling coils,
providing the non-adiabatic transition in and out of the field free regions, were redesigned
in order to increase the beam cross section. Finally, the results of successful first calibration
measurements of the NRSE option available at FLEXX are presented at the end of this
chapter.
5.1 Bootstrap coils
The centerpiece of a NRSE spectrometer are the π-coils described in section 2.2.1. Each
π-coils consist of two coils, a B0 coil providing a static magnetic field perpendicular to the
scattering plane and an RF coil creating an oscillating field within the scattering plane. For
the spin echo option of FLEXX new π-coils were manufactured at the FRM-II reactor in
collaboration with the Max Planck Institute For Solid State Research, Stuttgart. The π-coils
were assembled in pairs to bootstrap coils (see Fig. 5.1), as introduced in section 2.2.2. In a
bootstrap coil the static fields of the two B0 coils are oriented in opposite directions, which
allows to minimize stray fields outside of the coils in the field free regions. The manufacturing
of the components of the bootstrap coils is described in the following subsections.
Fig. 5.1: Design of one bootstrap coil. The two B0 coils (light blue) aremounted between cooling plates (light green) [85].
5.1 Bootstrap coils 91
5.1.1 B0 coils
The static B0 fields are created by using a vertical air coil wound on an aluminum body (see
Fig. 5.2). The required precision of the coils is very high. Following the estimation in [12]
the required accuracy of the coils can be calculated. The Larmor precession angle after one
coil is given by
Φ = γB0lmλ
2π~(5.1)
where B0 is the applied magnetic field, l is the mean coil thickness and λ is the neutron
wavelength. If ∆Φ1 is the mean deviation from the Larmor precession angle after one coil,
then the mean deviation after M coils is
∆ΦM =√M ·∆Φ1. (5.2)
Therefore, for a polarization of P = cos(∆Φ8) > 95% after 8 π-coils, it follows that
∆Φ8 < 18 and ∆Φ1 < 6.5. Assuming a maximum magnetic field of B0,max = 165G,
a wavelength of λ = 4Å and 8 π-coils with a thickness of l = 42mm each, the relative
deviation of the phase angle should not exceed
∆Φ1
Φ1= 1 · 10−3 =
√(∆B0
B0
)2
+
(∆l
l
)2
. (5.3)
The relative deviations of the magnetic field (∆B0B0
) and of the coil thickness (∆ll ) should
contribute equally with 1·10−3√2
= 7.1 · 10−4. Assuming B0 = 165G and l = 42mm yields for
the tolerable deviations of the magnetic field and the coil thickness:
∆B0 < 115mG ∆l < 30µm (5.4)
A coil wound with wire would not satisfy the required surface flatness. Therefore, aluminum
band is used as lead. The band has a purity of about 99.9% and for the purpose of electrical
insulation it has an anodized layer of about 5µm. Aluminum has a high neutron transmis-
sion and a high conductivity. During the anodization of the aluminum band crystal water is
embedded in the surface. Since this effect causes small angle scattering [12, 79] the crystal
water is replaced with D2O. This is realized by heating the band in a D2O atmosphere in a
pressure oven at a pressure of p = 5bar. Such treatment reduces the small angle scattering
and improves the transmission of the coils [12, 79].
The aluminum body(253× 204× 42mm3
)(see Fig. 5.2) of the coil consists of two anodized
parts with windows (inner part: 64× 52mm2, outer part: 140× 52mm2) at the beam area.
92 5 Upgrade of the NRSE option at FLEXX
Fig. 5.2: Left: Sketch of the aluminum coil body of the B0 coil [85]. Right:B0 coil wound with two layers of anodized aluminum band.
The accessible beam cross section of 56 × 52mm2 is defined by the inner cooling plate
between two B0-coils (see section 5.1.2). Note that the accessible beam cross section of
the previously used coils was 50 × 32mm2 and thus the maximum beam cross section is
enhanced by a factor of ≈ 1.6. Using a beam cross section of 56 × 52mm2 the openings
of the new B0-coils allow for a maximum coil tilt angle for the whole bootstrap coil of 45
before reducing the beam cross section by more than a factor of 0.5 (see section 5.3.4).
To improve the insulation between the coil body and the aluminum band, the whole coil
body is covered with Kapton foil. The winding process has to be done very careful. Since
the aluminum band is not very flexible, it has to be wound under very high tension. Forces
parallel to the surface could bend the band and lead to gaps between the windings resulting
in field inhomogeneities.
For the winding the body was mounted on a lathe using dedicated adapter parts. The
tension of the aluminum band was realized by a guide consisting of two Teflon blocks pressed
together with several springs. The whole guidance system was mounted on a rotary table
to avoid forces, which could twist the band. During the winding one needs to take care that
the edges of the band do not yaw against each other to avoid short circuits between the
windings. At the same time, neighboring windings should lie closely to each other to get a
homogeneous magnetic field. Due to its width, the band has a slope of about 1 with respect
to the coil axis. To avoid cross components of the magnetic field, each coil has two layers
of windings with opposite slopes. Therefore, the cross components of the different layers
compensate. Each layer is fixed with a thin layer of two-component adhesive (Araldit 2015).
The adhesive was hardened for one hour in an oven at T = 120C, while the coil was pressed
between two steel plates with a very level surface. In the beam area the flatness of the coil
surface is of highest importance. Since the aluminum band does not lie on the coil body
in this region, the band was supported from the inside during the winding process with
5.1 Bootstrap coils 93
even plates, which fit into the coil windows. The support parts were removed afterwards.
Following this process the required flatness of the surface was reached as measured using a
straight-edge.
Static fields
Measurements of the static magnetic fields B0 yielded:
B0[G] = 3.08 · I0[A]. (5.5)
The vertical component of the B0 field along the coil axis is shown in Fig. 5.3. If both
B0 coils of a bootstrap pair are powered (blue line) with the fields oriented in opposite
directions, the return field of one coil is fed back through the other coil. Therefore, the field
profile approaches the profile of a long coil. The total field becomes more homogeneous and
slightly stronger compared to a single B0 coil (black line). The magnitude of the magnetic
field at the edges of the beam area (dashed red line) is slightly smaller (∆B0 = −30mG)
compared to the center of the beam area (red line).
−20 −18 −16 −14 −12 −10 −8 −6 −40
2
4
6
8
10
12
14
16
distance [cm]
B0 [G
]
Fig. 5.3: Vertical component of the B0 field along the coil axis in a boot-strap pair. If two coils are powered (blue line) the field is more homoge-neous and slightly enhanced as compared to a single powered coil (blackline). The center of the beam area is marked red while the edges of thebeam area are marked dashed red.
In order to reduce the stray fields of the bootstrap coil mu-metal plates with a thickness
of 3mm are screwed to the bottom and to the top of the bootstrap coil to improve the closure
of the magnetic circuit. At the maximum field strength of B0,max = 165G the magnetic flux
density within the mu-metal plates is approximately 42mm3mm
·165G = 2.3kG, which is far from
saturation. To decrease the stray fields further a second layer of mu-metal is mounted on
the bootstrap coil. This layer consists of 0.5mm thick mu-metal foil attached to the front
94 5 Upgrade of the NRSE option at FLEXX
faces and u-shaped, 1mm thick covers on top and bottom of the coil. The second layer is
about 1cm separated from the strong magnetic flux of the coils within the inner mu-metal
plates. It has been shown that a second layer reduces the stray fields significantly by a
factor of ≈ 14 [12].
5.1.2 Cooling circuit
The heat generated by the coils at higher fields needs to be dissipated. Therefore, cooling
plates with flowing water and air are installed in front, in between and behind the coils.
The cooling plates consist of two aluminum parts (outer plates: 10mm and 2mm thick,
inner plates: 4mm and 2mm thick), in which channels for the cooling water and the air
flow are milled into. The two parts are anodized and bonded together with heat sealable
film (Collano 22.100 [80]). The cooling plates are glued to the coils with a heat conducting
adhesive (Gap Filler 1000 [81]) with a heat conductivity of λ = 1 WKm
.
The resistance of one B0 coil is R = 0.12Ω. Using P = I2R and equation (5.5) the thermal
dissipation is:
P [W] = 1.26 · 10−2B20 [G], (5.6)
which is P = 343W for B0,max = 165G. The thermal dissipation losses are transferred by
the cooling water. The cooling plates cover the coil with an area of 800cm2 where the heat
flow of about q = 0.43 Wcm2 at B0,max needs to be transferred. With an adhesive thickness of
d = 0.2mm this results in a temperature difference between coil and plates of
∆T =qd
λ≈ 1K. (5.7)
The cooling water passes 3 cooling plates one after another. A flow rate of 2− 3 lmin
is more
than sufficient and even higher dissipation losses could be easily transferred.
The beam area of the coils is not covered by the cooling plates. Without additional cooling
this area would become excessively hot. This could result in a liquefaction of the glue and a
deformation of the band. Therefore, the beam area is additionally cooled with compressed
air using a similar design as in [12]. A sketch of the cooling system of the bootstrap coils is
shown in Fig. 5.4.
The brass components of the previously used cooling circuit reacted electrochemically with
the aluminum parts of the coils. Due to the formation of aluminum oxide, the cooling plates
of the coils became blocked resulting in a cooling failure. In order to avoid electrochemical
reactions to a large extend, all brass parts were replaced by stainless steel and plastic parts
throughout.
5.1 Bootstrap coils 95
Air
AirH O2
Fig. 5.4: Cooling system of the bootstrap coils. Cooling plates flowedthrough by water are glued to the B0 coils and the whole stack is pressedtogether. The beam areas not covered by the cooling plates are cooled withpressurized air.
5.1.3 RF coils
The tolerance of the mechanical accuracy of the RF coils is lower since deviations from
the π-flip of the magnetic moment of the neutron within the coil only contribute in second
order [12]. Following the estimate in [12], the polarization after N RF-coils with an applied
field BRF and a thickness of l can be approximated by
Pn = 1−Nπ2
4
[(∆BRF
BRF
)2
+
(∆l
l
)2]. (5.8)
Assuming a tolerable reduction of the polarization of 5% after N = 8 coils and equal relative
deviations of the magnetic field and the coil thickness yields
∆BRF
BRF=
∆l
l< 4%. (5.9)
According to equation (2.17) the amplitude of the HF field is:
BRF = 4~
m
π2
γ
1
lλ. (5.10)
The linear oscillating field BRF is divided into two counter rotating parts B+1 and B−
1 with
an amplitude of BRF
2 (see section 2.2.1). Only the part rotating in the same sense as the
precession of magnetic moment of the neutron contributes to the π-flip. For a wavelength
of λ = 2.4Å and l = 35mm the amplitude of the RF field is BRF = 15.7G. Therefore, the
accuracy needed is:
∆BRF = 0.63G ∆l = 1.4mm. (5.11)
96 5 Upgrade of the NRSE option at FLEXX
Manufacturing
The design of the RF coils follows closely the designs discussed in [12, 79, 82]. The RF coils
consist of a rectangular coil L1 wound with one layer of round aluminum wire, which covers
the beam area and two u-shaped rectangular coils L2 wound with one layer of copper wire
and litz wire (see Fig. 5.5). The coils L2 are not in the neutron beam and feed back the
magnetic field of coil L1. This design simultaneously reduces the stray fields in the field
free region and the eddy currents induced in the cooling plates and thus the power losses
by dissipation. Since the new RF coils are thicker than the previous coils used at FLEXX,
smaller magnetic fields and therefore less current is needed for a π-flip of the magnetic mo-
ment of the neutron.
L1
L2
L2
B1
B1
B1
Fig. 5.5: Left: Sketch of the RF coil as an assembly of three coils. Themagnetic field of the inner coil L1 is actively fed back by the outer coils L2.Right: Side view of the complete RF coil.
The whole coil body consists of a glass-reinforced plastic frame for L1 and two massive PVC
bodies for L2. To feed back the field BRF of L1, the coils L2 need to produce the same field
BRF . The wiring diagram is shown in Fig. 5.6. In order to reduce the total inductance, the
two coils L2 are connected parallel to each other and in series with L1. Since the current
passing through each L2 coil is half of the current passing through L1, the winding density
of L2 needs to be twice as high as for L1 to produce the same magnetic field BRF .
The inner coil L1 is wound with four aluminum wires in parallel with a cross section of 1mm2
each. The straight part of the outer coils L2 is wound with three copper wires in parallel
with a cross section of 0.75mm2 each. The curved parts of L2 are wound with litz wire such
that the winding density is the same as for the straight section. The total inductance of the
RF coils was measured to be L = 36.6µH and the DC resistance is R = 0.35Ω.
5.2 Spin echo instrument arms 97
L1
L2
L2
I
I/2
I/2
Fig. 5.6: RF coil wiring diagram.
Since the inductance of the new RF coils is different compared to the previously used
coils it was necessary to adapt the parameters for the impedance matching. Details of the
impedance matching procedure are described in Appendix F.
5.2 Spin echo instrument arms
The base frames of the new spin echo spectrometer arms were completely rebuilt in a more
compact design. They are assembled using an aluminum ITEM profile [83]. The NRSE
spectrometer arms are attached to the TAS components by coupling devices and are mov-
able by four air pads. The frames are designed such that every part, which potentially needs
maintenance, is easily accessible. Beside the valves for the air pads and the air cooling of
the bootstrap coils, the bottom part of the frames hosts the valves and the flow meter of the
cooling water circuit. If the flow of the cooling water is reduced below a certain level, the
flow meters activate the interlock signal of the power supplies to ensure that overheating of
the bootstrap coils is avoided. At the top part of the base frames, all rotary and translation
stages (see section 5.2.3) are mounted on an aluminum baseplate. On top of the base frame
the magnetic shielding boxes for the bootstrap coils are installed. The shielding performance
is described in section 5.2.1. The separation of the coils and the dimensions of the magnetic
shielding are chosen such that the coils can be rotated by 360. Before the upgrade, the hard
limit of the coil tilt angles was ≈ 45 while intensity losses due to geometrical restrictions
of the available beam cross section were encountered from Θ1 = Θ2 ≈ 35. Combined with
the new design of the coils the instrument now allows for much larger coil tilt angles (see
section 5.3.4). The new coils enhance this value to Θ ≈ 45. The housing of the impedance
matching capacitors (see Appendix F) is mounted on the back side of the base frame.
The new base frames are slimmer compared to the previously used frames and allow for
larger scattering angles. Here the maximum value for the scattering angle A4 is enhanced
from 110 to 120. The previous spin echo spectrometer could not be positioned in di-
rect beam geometry, i.e. A4 = 0 and A6 = 0 (detector angle), in the ki-range from
98 5 Upgrade of the NRSE option at FLEXX
ki = 1.3Å−1 to ki = 1.7Å−1 since the maximum radius of the instrument was too large
for the old instrument floor. Therefore, direct beam calibration measurements (see section
5.3.4) were not possible in this ki-range. The new, more compact design shortens the spin
echo spectrometer arms, decreasing the maximum length of the instrument. As FLEXX is
installed at a new instrument position in the neutron guide hall of BER II, which provides a
larger instrument floor area, direct beam calibration measurements are now possible for the
whole ki-range of the instrument. Hence, the base frames hosting the NRSE components
increase the measurement range for the spin echo option of FLEXX. The accessible ki-range
is between 1.37Å−1 and 2.37Å−1 corresponding to a maximum λ = 4.57Å and a minimum
λ = 2.77Å, respectively. This range is determined by the monochromator shielding and was
limited by mechanical constraints at the time of the experiment.
The combination of more compact spectrometer arms and larger accessible coil tilt angles
(see section 5.3.4) allows for a larger range of scattering angles in Larmor diffraction geome-
try (see section 3.3.3). The previous NRSE option allowed for a |Q|-range in Larmor diffrac-
tion geometry from 2.69Å−1 to 3.11Å−1. These values are calculated for ki = kf = 1.9Å−1.
Given the new geometry of the spectrometer arms, the enhanced |Q|-range is now from
2.44Å−1 to 3.29Å−1 for ki = kf = 1.9Å−1.
5.2.1 Magnetic shielding
Between the monochromator and the analyzer no depolarization of the neutron beam due
to uncontrolled magnetic fields should occur. The earth’s magnetic field (Be ≈ 0.3G) gives
rise to a field integral of about 1Gm over the whole instrument and would lead to four
360-precessions of the polarization vector for a wavelength of λ = 6Å and about three
360-precessions for a wavelength of λ = 4Å. A tolerable variation in precession angle of
Φ = 25 reduces the polarization by 10% and corresponds to a residual mean magnetic field
of Br ≈ 5mG for λ = 6Å and Br ≈ 8mG for λ = 4Å. The shielding factor is defined by:
S =Ba
Br(5.12)
where Ba is the mean magnetic field without any shielding. Therefore, a shielding factor of
at least 60 is required.
The design of the NRSE option previously used at FLEXX provided the “field free” region
by shielding the instrument with a single layer of standard mu-metal with a high relative
permeability of 50000 [84]. The new shielding design, which has been basically adopted
from the previously used NRSE components, consists of three parts: Each spectrometer
arm has a square tube with a rectangular cross section (780 × 380 × 380mm3, thickness
5.2 Spin echo instrument arms 99
d = 2mm) to shield a pair of bootstrap coils. The covers are removable and the end caps are
properly connected by screws ensuring a good magnetic contact. A split vertical cylinder
(∅500mm × 500mm each, thickness d = 2mm) shields the sample position (see Fig. 5.7).
The rectangular tubes of the spectrometer arms are closed with openings for the neutron
beam and have a removable cover. Compensation coils at the outer faces allow for an addi-
tional active shielding.
Fig. 5.7: Sketch of the magnetic shielding design of the NRSE option ofFLEXX.
Smaller mu-metal cylinders (∅100mm × 300mm each) shield the region between the spec-
trometer arms and the sample position. The mu-metal cylinder towards the sample region
is flange-mounted to the caps. At the entrance face of the first and the exit of the second
spectrometer arm a mu-metal square tube (80 × 80 × 100mm3) is flange-mounted to the
caps as a shielding housing for the coupling coils (see section 5.2.2) of the spectrometer.
Since the thickness of the shielding of the bootstrap coils is increased from d = 1.5mm to
d = 2mm, the shielding factor is enhanced. The shielding factor of a box with a quadratic
cross section can be calculated to good approximation by the ballistic demagnetization fac-
tor N and the relative flux concentration in the walls of the shielding [86]. The ballistic
demagnetization factor for this geometry is given by:
N =1
1 + p
(1−
43
q + 203
)1− 3
40
(q − 9
5
)(q − 20)
(q + 13
2
) 135
, (5.13)
100 5 Upgrade of the NRSE option at FLEXX
where p = ab is the ratio of the edge lengths a and b of the box and q = 1
p+p. Approximations
for the perpendicular and longitudinal shielding factor are [86]:
Sp = 0.7µd
a(5.14)
Sl =2.52N
1 + 0.56 ba
µd
b. (5.15)
For the new dimensions of the shielding the approximations yield Nl ≈ 0.58 and Sl ≈ 87 for
the longitudinal case and Np ≈ 0.42 and Sp ≈ 184 for the perpendicular case. The calculated
shielding factors for the old dimensions (1000 × 300 × 300mm3, thickness d = 1.5mm) are
Sl ≈ 40 and Sp ≈ 175. The longitudinal shielding factor for the spectrometer arms is
therefore substantially increased by a factor of 2 while the perpendicular shielding factor
is slightly enhanced. Measurements showed that the residual fields in the horizontal and
vertical directions within the new shielding do not exceed the critical value of Br = 5mG in
both arms in the magnetic field of the earth.
The vertical field component at the sample position needs to be actively shielded since the
two cylinders are separated. Therefore, compensation coils at the outer ends are installed
to reduce this field component. The vertical field at the gap between the cylinders is
higher since the field components are bundled within the shielding walls [12]. This region
is bypassed using small, flange-mounted mu-metal cylinders extending into the vertical
cylinder. The horizontal field component at the sample region is compensated using the
coils mounted to the caps of the shielding of the spectrometer arms.
Important for the operation of the NRSE option are stray fields arising from a possible
operation of magnets, vertical or horizontal, at the neighboring V4/SANS instrument, which
may give rise to unwanted spin precessions. The sample position of the V4/SANS instrument
is about 11m away from the beginning of the field free region of the first spectrometer arm.
Stray field measurements for the strongest vertical magnet available at the HZB, the VM-1B
with a maximum field of 17.5T, show that the remaining vertical field at the position of the
NRSE option is about 200mG. Since Sp ≈ 184, such a field is easily shielded by the actual
mu-metal shielding. However, stray field measurements for the strongest horizontal magnet
available at the HZB, the HM-1 with a maximum field of 6T, show that the stray field
components parallel to the neutron flight path are an issue. Depending on the orientation
and on the chosen geometry of the NRSE instrument the stray field components parallel
and perpendicular to the neutron flight path are in the range of 0.5−1G. The perpendicular
component is sufficiently shielded. Since Sl ≈ 87 the remaining field within the field free
zone exceeds 5mG and will cause significant depolarization. Therefore, a parallel operation
5.2 Spin echo instrument arms 101
of the horizontal magnet at V4/SANS and the NRSE option at FLEXX should be avoided
and be considered for the scheduling of the experiments of both instruments.
5.2.2 Coupling coils
Fig. 5.8: Side view of the new coupling coils.
The neutron beam is polarized and analyzed in the vertical z-direction perpendicular to
the scattering plane. Thus, the polarization vector needs to be turned by 90 into the
scattering plane before entering the first field free region. After the second field free region
the process has to be reversed by turning the polarization vector into the vertical direction.
This is realized by coupling coils. The coupling coils are positioned at the entrance of the
first shielding and the exit of the second shielding box. In standard operation the first
coupling coil turns the incoming neutron spin (polarized in the vertical z-direction) into the
horizontal plane by an adiabatic transition and transports the neutron spin into the screened
area by a non-adiabatic transition (see Appendix G). The second coupling coil at the exit
of the second shielding provides the transition from a field free region to a region with guide
field and rotates the neutron polarization vector again by 90 to be perpendicular to the
scattering plane. In order to tune the single RF-coils of the NRSE option the coupling coils
are turned by 90 around the beam axis. The neutron polarization vector is then left aligned
in the vertical direction providing only the non-adiabatic transition into the screened area.
The coupling coils manufactured at HZB have a new design. Each coil (see Fig. 5.8) is
wound on an anodized aluminum body with an Al-wire (∅0.5mm). In order to have an
adiabatic transition at the side outside the magnetic shielding the wire is bent out of the
neutron beam. For the non-adiabatic transition the neutrons pass through the wire within
the magnetic shielding. To avoid a leakage of the return field into the screened area, the coil
is covered with mu-metal to guide the return field. Compared to the former design [12] the
102 5 Upgrade of the NRSE option at FLEXX
adiabatic transition is realized by a diagonal wound opening of the coil. Simultaneously the
wire is everywhere properly attached to the coil body. This makes the design more compact
and robust at the same time.
5.2.3 Motors and encoders
Each bootstrap coil is mounted on its own rotary stage allowing to rotate the coil around its
vertical axis for measuring dispersive excitations or for measurements in Larmor diffraction
geometry. The rotary stages of the first and fourth coil are mounted on a linear stage. This
allows for translating the coils along the beam direction to fine tune or to scan the field
integral L ·B. Due to the upgrade it is now possible to translate both the first and the last
NRSE coils. With the new, more compact magnetic shielding the rotary and translation
stages were positioned outside of the shielding and placed below. Hence, the motors and
encoders do not produce any stray fields within the magnetic shielding. The new position
of the mechanical devices also facilitates maintenance work.
5.3 Calibration of the new NRSE option at FLEXX
Prior to performing any measurements on samples with a completely new NRSE spectrom-
eter, several calibrations measurements have to be done. Since the parameters of all coils of
the instrument changed, the optimum current parameters will also change. Therefore, these
parameters have to be determined first. For the calibration measurements the coils in each
of the spectrometer arms were aligned parallel using a theodolite. The S-bender was moved
into the FLEXX neutron guide system. The S-bender has a magnetization field perpendic-
ular to the scattering plane and the vertical directions of the polarization is maintained up
to the coupling coils. A bender was mounted in front of the detector. In future for polarized
neutron work a Heussler analyzer will be installed replacing the PG analyzer. However, the
device was not available at the time of the calibration measurements described here.
Conventional NRSE spectrometers are run in two modes. In the so called 4π-mode only the
inner π-coils of each spectrometer arm are used for the measurement (4 π-coils in total are
in use). This mode gives access to smaller spin echo times. The so called 8π-mode uses all
π-coils to benefit from the bootstrap enhancement factor and to give access to higher spin
echo times.
5.3 Calibration of the new NRSE option at FLEXX 103
5.3.1 Calibration of currents and HF voltage
New NRSE look-up table
The optimum parameters to perform a π-flip within a π-coil need to be determined experi-
mentally. The look-up table for experimentally optimized values of the currents I0 applied
to the B0-coils and the pick-up Voltage Upu of the RF coils was measured for a wavelength
of λi = 3.3Å. Since the neutron velocity and therefore the time of the neutron spent in
the magnetic field is linear in λ, the optimum values can easily be extrapolated for other
wavelengths. The same argument holds for a change in the coil tilt angles.
8 10 120
2
4
6
8
10x 10
4
I0 [A]
Inte
nsity
[Cou
nts
/ 5s]
Coil #1Coil #2Coil #3Coil #4
8 10 120
2
4
6
8
10x 10
4
I0 [A]
Inte
nsity
[Cou
nts
/ 5s]
Coil #5Coil #6Coil #7Coil #8
Fig. 5.9: Detector counts as a function of the current of the B0-coils spec-trometer arm 1 (left) and 2 (right). If the Larmor frequency of the staticfield is equal to the frequency f of the RF coils the detector counts reacha minimum. The results show very little variation between the individualcoils manufactured.
First, the optimum currents for the B0-coils as a function of the frequency applied to the
RF-coils were determined. For these measurements the incoming polarization vector is ori-
ented parallel to the axis of the B0-coil. This is realized by operating the coupling coils in
a mode to provide a field perpendicular to the scattering plane. Hence, the initially verti-
cally oriented polarization vector is not turned into the horizontal plane and the coupling
coil only provides the non-adiabatic transition into the screened region. The current of a
single static field coil is then scanned for a fixed frequency f of the RF coil. If the Larmor
frequency corresponding to B0 is the same as f , the magnetic moment of the neutron will
perform the largest possible rotation around the RF field and therefore causes a minimum
in the detector counts. The RF amplitude may differ from its optimum value and the mini-
mum of the polarization becomes more pronounced with the RF amplitude approaching the
optimum value. Fig. 5.9 shows the intensity at the detector as a function of the B0-coil
104 5 Upgrade of the NRSE option at FLEXX
current I0 for all 4 coils of each spectrometer arm for f = 100kHz. The plot shows very
little variation between the individual coils manufactured. This is very important since all
B0-coils of a spectrometer arm are operated in series using a single power supply. Due to
the negligible variation in the optimum I0 currents no additional resistors were needed to
adjust the static magnetic field amplitudes.
0 50 100 150 200 250 300 350 4000
10
20
30
40
50
f [kHz]
B0 c
urre
nt [A
]
data 1st armdata 2nd armlinear fit
0 50 100 150 200 250 300 350 400−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
f [kHz]
Diff
eren
ce D
ata
− F
it [A
]
data 1st armdata 2nd arm
Fig. 5.10: Left: Optimum currents for the first B0-coils of each spectrom-eter arm as function of RF frequency. A linear function is fitted to thedata. Right: Difference between data and linear fit.
The calibration measurements for I0 as a function of frequency were performed using the
first B0-coil of each spectrometer arm. The results for the optimum current I0 over the
frequency range probed are shown in Fig. 5.10, displaying the expected linear behavior.
They are almost identical for both arms. A linear function is fitted to the data and the
deviations from the fit are shown. The deviations are larger for small frequencies but since
the minimum in intensity is quite broad, depolarization effects arising from variations in B0
can be neglected.
Similar measurements were performed to obtain the optimum values for the amplitude of
the RF signal. The signal from a small pick-up coil located at one of the U-shaped parts of
the RF coil, directly probing th RF field by the induced voltage Upu, is used as a control
variable (see Appendix F). The RF amplitude is scanned for a fixed frequency f at the op-
timum value of B0(f). At the optimum the magnetic moment of the neutron will perform a
π-flip resulting in a minimum of the detector counts. The measurements for each frequency
were performed using the first π-coil of each spectrometer arm. The results are shown in
Fig. 5.11 and are almost identical for both arms. A linear function is fitted to the data
and the deviations from the fit are shown. The deviations are larger for small frequencies
but since the minimum of the detector counts is quite broad, depolarization effects can be
neglected.
5.3 Calibration of the new NRSE option at FLEXX 105
0 50 100 150 200 250 300 350 4000
5
10
15
20
f [kHz]
Pic
k−U
p V
olta
ge [V
]
data 1st armdata 2nd armlinear fit
0 50 100 150 200 250 300 350 400−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
f [kHz]D
iffer
ence
Dat
a −
Fit
[V]
data 1st armdata 2nd arm
Fig. 5.11: Left: Optimum pick-up voltages Upu for the first RF coils ofeach spectrometer arm as function of RF frequency. A linear function isfitted to the data. Right: Difference between data and linear fit.
Coupling coils
The coupling coils provide the transition of the polarization vector into and out of the
field free region. In the limit of large coupling coil currents the adiabatic transition is best
realized. However, the stray fields increase with increasing coupling coil current. An exper-
imental calibration of the currents is therefore needed. For the calibration measurements
the coupling coils were operated in their standard orientation, i.e. the incoming vertical po-
larization vector is guided into the scattering plane by the first coupling coil and is guided
back to the vertical direction by the second coupling coil at the end of the field free region.
Depending on the relative orientation of the magnetic fields, i.e. the relative sign of the
currents, of the coupling coils the detector counts will either reach a maximum plateau (the
polarization vector is restored in vertical direction) or a minimum plateau (the incoming
polarization vector is rotated by 180). The current of one coil was scanned while the cur-
rent of the other coil was fixed at +0.9A.
The results for both coupling coils are shown in Fig. 5.12. The current scans for both cou-
pling coils clearly show the two plateaus. With increasing coil current the detector counts
decrease slightly because of depolarization effects of the increasing stray field of the coupling
coils. Components of the polarization vector perpendicular to the nominal direction cause a
slight oscillation of the signal. This effect is due to a rotation of perpendicular components
around the effective magnetic field. Stray field effects decrease the intensity with increasing
coil current. Apparently this effect is stronger for the second coupling coil (right) due to the
different transition geometry between coupling coil and guide field. For further calibration
measurements Ic1 and Ic2 were set to +1.5A.
106 5 Upgrade of the NRSE option at FLEXX
−5 −4 −3 −2 −1 0 1 2 3 4 50
2
4
6
8
10x 10
4
Ic1
[A]
Inte
nsity
[Cou
nts
/ 10s
]
−5 −4 −3 −2 −1 0 1 2 3 4 50
2
4
6
8
10x 10
4
Ic2
[A]
Inte
nsity
[Cou
nts
/ 10s
]Fig. 5.12: Intensity at the detector as a function of the current of thecoupling coils of spectrometer arm 1 (left) and 2 (right). While scanning thecurrent Ic1 (Ic2) the current Ic2 (Ic1) was set to Ic2 = +0.9A (Ic1 = +0.9A).
5.3.2 Spin echo curve and echo point
According to [2] the envelope of the spin echo curve is the Fourier transform of the incoming
wavelength distribution
AE = 〈cosΦNSE〉 =∫
f (λ) cos(ωeffm
2π~∆Lλ
)dλ, (5.16)
where ωeff = γBeff = 2NγB0 is the effective Larmor frequency, N the number of active
π-coils within one bootstrap coil (see section 2.2.2) and ∆L = L1 −L2. Approximating the
wavelength distribution with a Gaussian shape
f(λ) =1
σλ√2π
e− (λ−λ0)
2
2σ2λ (5.17)
with a mean wavelength λ0, the Fourier transform yields
AE = e− 1
8
(ωeffm
2π~∆Lσλ
)2
cos(ωeffm
2π~∆Lλ0
)(5.18)
According to equation (2.11) the beam intensity after the analyzer is
I =I02(1 +AE) (5.19)
Fig. 5.13 shows a typical spin echo curve obtained by scanning the position of the fourth
bootstrap coil using all π-coils (N = 2) with a frequency of 400kHz. The accessible range
of the spin echo curve is determined by the range of the translation table.
5.3 Calibration of the new NRSE option at FLEXX 107
−25 −20 −15 −10 −5 0 5 10 15 20 250
1
2
3
4
5
6
7
8
x 104
Position of fourth bootstrap coil [mm]
Inte
nsity
[Cou
nts
/ 10s
]
Fig. 5.13: Spin echo curve obtained by direct beam measurements. Thefit yields a wavelength of λ = 3.3498(1)Å and ∆λ
λ≈ 1.3%.
The cosine term in equation (5.18) causes the rapid oscillations. By fitting the period length
of one oscillation
∆L =2π~
ωeffmλ0(5.20)
the mean wavelength λ0 can be determined. In this case for a frequency of 400kHz, N = 2
and a period length ∆L = 0.73812(3)mm the obtained wavelength is λ = 3.3498(1)Å. The
spectral width is fitted to ∆λλ ≈ 1.3%. This is in good agreement with the properties of the
focusing monochromator used during the experiment.
Supplementary spin echo curves were measured in the direct beam for nominal wavevectors
ki = 1.37Å−1 and ki = 2.37Å−1. Fitting the data results in ki,fit = 1.3762(1)Å−1 (λ =
4.8686(2)Å) with a corresponding ∆λλ ≈ 0.8% and ki,fit = 2.3666(3)Å−1 (λ = 2.2666(3)Å)
with a corresponding ∆λλ ≈ 1.4%. These two wavevectors are the limits of the measur-
able ki-range due to mechanical constraints of the monochromator shielding at the time of
the experiment. The value of ∆λλ decreases from ∆λ
λ ≈ 1.4% (λ = 2.27Å) to ∆λλ ≈ 0.8%
(λ = 4.87Å) due to the fact, that the same angular variation of neutron trajectories emanat-
ing from the monochromator corresponds to a larger wavelength spread at lower wavelengths.
The condition of the echo point is given by equation (2.8). In the case of an unscattered
neutron beam, there is no energy transfer and therefore no change in the neutron velocity.
Hence, the field integrals B1L1 and B2L2 of the two spectrometer arms need to be the same.
In the present experimental setup the effective precession fields B1 and B2 are the same in
both spectrometer arms and therefore the lengths L1 and L2 need to be equal too. It is
108 5 Upgrade of the NRSE option at FLEXX
best to determine the distances between the coils using neutron experiments. Hence, the
echo amplitude for a fixed frequency f was measured as a function of the position of the
last bootstrap coil by varying the length L2 in direct beam geometry. The maximum of this
spin echo curve is the echo point where L1 = L2. For conventional spin echo spectrometers
with a broad wavelength distribution (∆λλ ≈ 10%), where ∆λ is a FWHM, the spin echo
point can easily be identified. However, if f(λ) becomes small the spin echo curve becomes
very broad and the echo point cannot be determined with high precision.
In the case of a broad spin echo curve, the spin echo point can be determined by measuring
the spin echo curve for several frequencies, which should be coprime. In the case of ∆L =
0 all curves should have a common maximum or a minimum depending on the relative
orientation of the fields provided by the coupling coils (see equation (5.18)). During the
measurements the coupling coils were operated in an antiparallel configuration. This leads to
a phase shift of π since the incoming polarization vector oriented in the positive z-direction
is turned by 180 so that the outgoing polarization vector points in the negative z-direction.
Thus, in this case all curves should have a minimum in the spin echo point. The fitted
curves of such a measurement are shown in Fig. 5.14. The data were collected in 8π-mode
with a wave length of λ = 3.35Å for the frequencies 271kHz, 399kHz and 437.5kHz. The
minima of the curves are at x = −3.26± 0.01mm. The small deviations of the curves from
the common minimum arise from the slightly different static fields of the coils leading to a
small phase shift of the spin echo curve.
−10 −8 −6 −4 −2 0 2 4 60
1
2
3
4
5
6
7
8
x 104
Position of fourth coil [mm]
Inte
nsity
[Cou
nts
/ 10s
]
437.5 kHz399 kHz271 kHz
Fig. 5.14: At the spin echo point x = −3.26 ± 0.01mm (∆L = 0) spinecho curves for different frequencies have a common minimum. The mea-surements were performed in 8π-mode using a wavelength of λ = 3.35Å forthe frequencies 271kHz, 399kHz and 437.5kHz.
5.3 Calibration of the new NRSE option at FLEXX 109
The position of the spin echo point can be used to determine the absolute distance L1 = L2
between the coils, i.e. the length of the effective magnetic field, at the spin echo point. In
the first measurement the same frequency was applied to the coils in both spectrometer
arms. By decreasing the frequency of the second arm by a factor of 0.97 the length L′2
should increase to
L′2 =
L
0.97= L+∆x (5.21)
to satisfy the spin echo condition. Here ∆x is the shift of the spin echo point. The distance
between the coils can then be calculated from
L =0.97
0.03∆x. (5.22)
2 4 6 8 10 12 14 160
1
2
3
4
5
6
7
8
x 104
Position of fourth coil [mm]
Inte
nsity
[Cou
nts
/ 10s
]
f1=437.5 kHz, f
2=424.375 kHz
f1=399 kHz, f
2=387.03 kHz
f1=271 kHz, f
2=262.87 kHz
Fig. 5.15: At the spin echo point x = 8.86 ± 0.01mm (∆L = 0) all spinecho curves have a minimum. The measurements were performed in 8π-mode with a wavelength of λ = 3.35Å. The frequencies applied to thesecond spectrometer arm were lowered by a factor of 0.97 compared to thefirst arm.
The results for the measurements of the shifted spin echo point are shown in Fig. 5.15. The
frequency of the second arm was lowered by a factor of 0.97. The spin echo point is now at
x = 8.86 ± 0.01mm leading to ∆x = 12.12 ± 0.01mm, which yields L8π = 391.9 ± 0.3mm,
which is in very good agreement with the distance between coils measured mechanically.
The same measurement was performed in 4π-mode to determine the distance between the
inner π-coils at the spin echo point. The data measured at λ = 3.35Å with the frequencies
122kHz, 225kHz and 297kHz applied to the RF-coils in both spectrometer arms is shown in
110 5 Upgrade of the NRSE option at FLEXX
Fig. 5.16 (top). The spin echo point is at x = −2.78 ± 0.01mm. The signal corresponding
to f = 122kHz is slightly shifted. This can caused by residual magnetic fields. However, the
shifted minimum is still consistent with the correct common minimum of the other signals.
−20 −15 −10 −5 0 5 10 150
1
2
3
4
5
6
7
8
9x 10
4
Position of fourth coil [mm]
Inte
nsity
[Cou
nts
/ 10s
]
122 kHz225 kHz297 kHz
−10 −5 0 5 10 15 20 250
1
2
3
4
5
6
7
8
9x 10
4
Position of fourth coil [mm]
Inte
nsity
[Cou
nts
/ 10s
]
f1=122 kHz, f
2=118.34
f1=225 kHz, f
2=218
f1=297 kHz, f
2=288.09
Fig. 5.16: Top: At the spin echo point x = −2.78± 0.01mm all spin echocurves have a common minimum. The measurements were performed in 4π-mode using a wavelength of λ = 3.35Å for the frequencies 122kHz, 255kHzand 297kHz. Bottom: The frequencies applied to the second spectrometerarm were lowered by a factor of 0.97 compared to the first arm and thespin echo point is shifted to x = 7.63± 0.01mm.
To determine the shift of the spin echo point the frequencies applied to the second arm
were again lowered by a factor of 0.97 as for the measurement in 8π-mode. In 4π-mode the
RF-coils of the outer π-coils of each spectrometer arm are not connected though the B0-
coils are powered and the magnetic moments precess in these static fields with the Larmor
frequency. If the same frequencies are applied in all coils, i.e. the same current is applied to
5.3 Calibration of the new NRSE option at FLEXX 111
the B0-coils, the additional Larmor phase collected in the static fields cancels out after the
second arms. However, if the frequency in the second arm and thus the B0-coil current is
lower a difference in the total Larmor phase ∆Φ collected in the static fields remains. Due
to the additional phase the resulting spin echo curve is shifted and the measured curves
need to be corrected for each frequency separately. Since the phases corresponding to the
inner π-coils cancel at the spin echo point, ∆Φ at the spin echo point can be calculated as:
∆Φ = 2πf12Lcoil
v− 2πf2
2Lcoil
v= 4π
Lcoil
v(f1 − f2) (5.23)
=2Lcoilmλ
~(f1 − 0.97f1) = 0.03
2Lcoilmλ
~f1. (5.24)
(5.25)
Here f1 and f2 are the frequencies applied to the spectrometer arms and Lcoil is the thickness
of one π-coil. Using the period of one oscillation
L0(f) =2π~
2Nfmλ(5.26)
the correction in distance can be calculated for each frequency by
∆Lcorr(f) = L0∆Φ
2π. (5.27)
Fig. 5.16 (bottom) shows the corrected data using the coil thickness of 52mm. The spin
echo point is at x = 7.63 ± 0.01mm leading to ∆x = 10.41 ± 0.01mm, which yields L4π =
336.6± 0.3mm. This is in good agreement with direct distance measurements.
5.3.3 Phase stability
The phase stability of the NRSE spectrometer was tested by repeating a spin echo measure-
ment several times and comparing the phases of the individual scans. The measurements
were done in 8π-mode using λ = 3.35Å and a frequency of 300kHz. Each spin echo run
took about 10 minutes. Fig. 5.17 (left) shows the fits of the single scans. The results for
the phases are plotted in Fig. 5.17 (right). During the measuring time of one hour the
phase shifted by ∆Φ = 0.058rad. If the phase shifts while recording a spin echo signal, the
contrast of the resulting spin echo signal is decreased, i.e. the echo amplitude decreases.
For a long spin echo scan with a typical total measurement time (here assumed to be 3
hours), assuming a perfectly polarized beam and neglecting all other depolarizing effects,
the remaining polarization would be 1 − cos(0.058 radh
· 3h) = 98.5%. Thus, this effect is
negligible compared to the statistical error in most cases.
112 5 Upgrade of the NRSE option at FLEXX
The phase shift of the signal is due to thermal drifts. During a measurement the coils heat
up, which changes the resistance slightly, i.e. the currents, the magnetic fields and the
Larmor frequencies change slightly, too. The estimations for the depolarization made here
present a conservative limit since for longer measurements the phase reaches a plateau when
the coils and the cooling plates reach thermal equilibrium. For more precise estimations of
systematic errors longer test measurements would be needed, which could not be performed
due to the limited beam time at FLEXX.
−0.5 0 0.50.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8x 10
5
Position of fourth coil [mm]
Inte
nsity
[cou
nts
/ 20s
]
Run 1Run 2Run 3Run 4Run 5Run 6
0 2 4 6 81.74
1.75
1.76
1.77
1.78
1.79
1.8
1.81
1.82
Run number
Pha
se [r
ad]
Fig. 5.17: Left: Fits of the spin echo scans performed at 300kHz in 8π-mode with λ = 3.35Å. Right: Phase as a function of run number. Within1 hour the phase shifted by ∆Φ = 0.058rad.
5.3.4 Calibration of coil tilt angles
Mezei pointed out [2] that the correction for instrumental depolarization effects can be
performed by dividing the measured echo amplitude from inelastic scattering
AE,meas(τ) = AE,meas(f1, f2,Θ1,Θ2, ki, kf ) (5.28)
by the echo amplitude AE0 from direct beam measurements corresponding to the same spin
echo parameters:
AE(τ) = AE(f1, f2,Θ1,Θ2, ki, kf ) =AE,meas(f1, f2,Θ1,Θ2, ki, kf )
AE0(f1, f2,Θ1,Θ2, ki, kf ). (5.29)
Since a convolution of the signal and the instrumental resolution in real space correspond
to a multiplication in Fourier space and vice versa, this simplifies the correction compared
to classical neutron spectroscopy where a deconvolution is performed.
5.3 Calibration of the new NRSE option at FLEXX 113
For quasielastic scattering experiments a purely elastic scattering sample can be used to
obtain AE0. However, for inelastic scattering no standard samples exist since there is in
principle no dispersive excitation with zero linewidth. Therefore, AE0 is obtained by direct
beam measurements with f1 = f2, Θ1 = Θ2 and ki = kf . For direct beam measurements
the echo amplitude AE,calib can be written as a product of the echo amplitudes AE1 and
AE2 after each spectrometer arm:
AE,calib(f1,Θ1, ki) = AE1(f1,Θ1, ki) ·AE2(f1,Θ1, ki). (5.30)
Assuming that the depolarizing effects are the same in both arms yields
AE1 = AE2 =√AE,calib(f1,Θ1, ki). (5.31)
In inelastic spin echo measurements the spin echo parameters of both spectrometer arms
are different, i.e. AE1(f1,Θ1, ki) and AE2(f2,Θ2, kf ). The corresponding echo amplitude
AE0 is then obtained by:
AE0(f1, f2,Θ1,Θ2, ki, kf ) = AE1(f1,Θ1, ki) ·AE2(f2,Θ2, kf ) (5.32)
=√
AE,calib(f1,Θ1, ki) ·√AE,calib(f2,Θ2, kf ). (5.33)
AE,calib is measured for a discrete set of frequencies and coil tilt angles according to the
chosen ki and kf of the experiment. If the actual spin echo parameters of the experiment
differ from the parameters of the calibration measurements, the corresponding AE,calib is
obtained by interpolation between the two closest calibration sets.
Fig. 5.18 shows calibration data for 4π-mode (left) and 8π-mode (right). The set was
recorded for coil tilt angles of 0, 10, 20, 30, 40 and 50 for ki = 1.8757Å−1. The effec-
tive frequency feff is the actual frequency f applied to the π-coils times the enhancement
factor 2N (N = 1 for 4π-mode and N = 2 for 8π-mode). As the frequency increases, the
magnetic moment of the neutron performs more precessions within the magnetic field and
the stray fields increase. The depolarization increases with increasing frequency. This can
be seen for both modes in Fig. 5.18 where the echo amplitude decreases with increasing feff
for each coil tilt angle. Since the flight path of the neutrons within the coil and therefore
the time spent in the magnetic field increases with increasing coil tilt angle, the influence
of depolarizing effects, e.g. due to field inhomogeneities, increases. As seen in Fig. 5.18
especially for high frequencies in 8π-mode the echo amplitude decreases with increasing coil
tilt angle. However, the similar values for the echo amplitude for all coil tilt angles in the low
and middle regime of feff display the accurate alignment of the coils. If the field boundaries
114 5 Upgrade of the NRSE option at FLEXX
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
feff
[kHz]
echo
am
plitu
de
θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°
0 500 1000 15000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
feff
[kHz]ec
ho a
mpl
itude
θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°
Fig. 5.18: NRSE calibration data for ki = 1.8757Å−1
. The data set wasmeasured for coil tilt angles of 0, 10, 20, 30, 40 and 50 for both NRSEmodes, 4π (left) and 8π (right).
of the coils are not parallel to each other, the flight paths in the coils would differ and the
depolarization would increase more strongly with increasing frequency and increasing coil
tilt angle. The comparably low echo amplitude at feff = 100kHz in 4π-mode corresponds
to a frequency of f = 50kHz and is a result of the Bloch-Siegert-Shift (see section 2.2.1).
Similarly the effect is noticeable in the 8π data at feff = 400kHz. Supplementary calibra-
tion sets were measured for ki = 2.3666Å−1 for both NRSE modes, 4π and 8π. The results
are shown in Fig. 5.19.
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
feff
[kHz]
echo
am
plitu
de
θ = 0°θ = 20°θ = 30°θ =40°θ = 50°
0 500 1000 15000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
feff
[kHz]
echo
am
plitu
de
θ = 0°θ = 20°θ = 30°θ =40°θ = 50°
Fig. 5.19: NRSE calibration set for ki = 2.3666AA−1. The set wasmeasured for coil tilt angles of 0, 20, 30, 40 and 50 for both NRSEmodes, 4π (left) and 8π (right).
5.3 Calibration of the new NRSE option at FLEXX 115
Fig. 5.20 shows the fitted intensities I0 for ki = 1.8757Å−1 according to equation (5.19)
for all measured spin echo signals of the calibration set. The variation in intensity within
one coil tilt angle set is due to variation in reactor power since the data was taken for fixed
count time and is shown not normalized to monitor counts.
0 200 400 600 8000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
feff
[kHz]
Inte
nsity
[Cou
nts
/ s]
θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
feff
[kHz]
Inte
nsity
[Cou
nts
/ s]
θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°
Fig. 5.20: Fitted intensities I0 according to equation (5.19) for all mea-sured spin echo signals of the calibration set shown in Fig. 5.18 separatedaccording to the mode used (left: 4π, right: 8π).
With increasing coil tilt angle the intensity decreases since the neutron beam cross section is
gradually reduced by the finite window width of the NRSE coils. Fitting an average intensity
for every coil tilt angle set and normalizing the average intensity to the value for Θ = 0
the effect of reduced effective beam cross section on intensity at the detector is shown in
Fig. 5.21. Here, the mean intensities for both modes obtained from the calibration set as
a function of the coil tilt angle are displayed. The black curve is the calculated normalized
beam cross section as calculated from the geometric dimensions of the bootstrap coils. The
data is in very good agreement with the calculations, however, the steep reduction of the
neutron beam seems to set in at smaller tilt angles than predicted. This effect can arise
from a dislocation of the coils to each other perpendicular to the neutron beam. Such
a dislocation would reduce the accessible beam cross section, but would not change the
length of the flight path within the coil. The green curve displays the beam cross section
as calculated from the geometric dimensions of the previously used bootstrap coils. Note
that the enhancement of the coil window height from 32mm to 52mm is taken into account
for the calculations. The previously used bootstrap coils had a hard limit for the coil tilt
angle of Θ ≈ 45 and at Θ0.5 ≈ 35 the beam cross section was reduced by a factor of 0.5.
116 5 Upgrade of the NRSE option at FLEXX
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
coil tilt angle [°]
norm
aliz
ed b
eam
cro
ss s
ectio
n
I0mean
4π
I0mean
8π
Fig. 5.21: Normalized mean intensities fitted to the calibration set forboth modes as a function of coil tilt angle. This displays the cropping ofthe neutron beam. The black (green) line is a rough estimation accordingto the geometric dimensions of the current (previously used) bootstrap coiland is in good agreement with the fit results.
The new coils have no hard limit for the coil tilt angle and the value for Θ0.5 was enhanced
to ≈ 45. The measurements show that the new bootstrap coils allow for much higher tilt
angles compared to the previously used bootstrap coils.
A comparison between calibration data from experiments using the previous NRSE option
and the calibration measurements presented here show, that the polarized neutron flux at
the sample position is increased by a factor of 5. This was measured for ki = 1.9Å−1 in
direct beam geometry. The enhancement is due to the larger beam cross section available
at FLEXX now using a double focusing monochromator, the more compact instrument and
the larger accessible beam cross section of the coupling and the bootstrap coils.
5.4 Summary
The upgrade of the NRSE option of the cold TAS FLEXX was a major part of this thesis.
Redesigned NRSE bootstrap coils were manufactured in collaboration with the Max Planck
Institute For Solid State Research, Stuttgart. The new coils allow for larger beam cross sec-
tions and larger coil tilt angles. Hence, steeper dispersions are accessible. The spectrometer
arms were redesigned, making the instrument more compact, leading to an extended range
in scattering angle. In combination with larger coil tilt angles, the range of scattering an-
5.4 Summary 117
gles in Larmor diffraction geometry is increased. A new, more compact mu-metal shielding
guarantees a better shielding of the field free area between the NRSE coils. In contrast to
the previously used spectrometer arms, all mechanical and electrical components (e.g. go-
niometers) were relocated beneath the shielding boxes to reduce scattering fields inside the
field free zone. In order to benefit from the larger beam cross section available at FLEXX,
now using a double focusing monochromator, the coupling coils were redesigned with a
larger cross section and a more compact design. The larger beam cross section available at
FLEXX in combination with the new geometries of the new NRSE result in an increase of
the polarized neutron flux at the sample position by a factor of 5.
Extensive calibration measurements were performed in order to demonstrate the functional-
ity of the instrument and to obtain the parameters necessary to make the TAS-NRSE spec-
trometer available for future experiments. In particular, look-up tables for the impedance
matching and coil currents were measured. The echo point and the distance between the
coils were determined. The measurements prove the phase stability and calibration sets for
different coil tilt angles show that the properties of the individually produced bootstrap
coils are practically identical.
The calibration measurements clearly demonstrate that the NRSE option works without
any major technical problems and demonstrated its reliable performance.
118 5 Upgrade of the NRSE option at FLEXX
Chapter 6
Conclusion and perspectives
Two new kinds of NRSE experiments were investigated in this thesis: Resolving mode dou-
blets with an energy separation smaller than the typical energy resolution of a standard
triple axis spectrometer and line shape analysis of temperature dependent asymmetric line
broadening. The data analysis of these experiments requires a detailed model for the echo
amplitude as a function of correlation time as introduced by Habicht et al. [6]. This model
includes depolarization effects due to sample imperfections, the curvature of the dispersion
surface and the TAS resolution function. This is essential for a thorough understanding of
the results. Up to now the existent model was limited to cubic systems.
In this thesis, major generalizations of the existing formalism were developed. As a result,
the model can now be applied to any crystal symmetry class. Furthermore, the extended
model allows for violated spin echo conditions, arbitrary local gradient components of the
dispersion surface and detuned TAS parameters, giving rise to important additional depo-
larizing effects, which have been neglected before. The model was successfully tested by
experiments on phonons in a high quality single crystal of Pb. For a detuned background
TAS, realized by rotating the sample, the model predicts additional phase terms and a
non-linear behavior of the spin echo phase. The results of the experiment are in very good
agreement with the extended resolution model and demonstrate the stringent necessity to
consider second order effects.
Subsequently, the formalism was extended to analyze mode doublets. As a major general-
ization, detuning effects for both modes are taken into account in contrast to the previous
treatment [23]. Nb was chosen for experimental verification as it is a well understood model
system. The realization of a unique tunable double dispersion setup allowed to generate
artificially split modes. Echo amplitudes were measured in elastic Larmor diffraction ge-
ometry for two neighboring Bragg peaks. The results demonstrate the echo modulation in
119
120 Conclusion and perspectives
spin echo length space to be in good agreement with a simplified phenomenological model
describing the signature of split modes [21]. Inelastic NRSE spectroscopy on an effectively
split dispersion clearly shows the modulation of the echo amplitude. The results agree with
the developed extended model predictions, indicating persistence of the modulation over
the entire spin echo time range, probed by the experiment. It was proven once more that
inelastic NRSE measurements on a single crystal require the consideration of second order
effects. Since the dispersion of Nb is steep for the chosen experimental parameters, strong
depolarizing effects occur. Hence, neglecting these effects would result in a significant over-
estimation of the linewidths. It was shown that the phenomenological model is sufficient for
the determination of the splitting. However, for a detailed analysis the use of the generalized
resolution function, developed in this thesis, is inevitable.
The results prove the potential of NRSE spectroscopy to resolve mode doublets with an
energy separation smaller than the typical energy resolution of a standard TAS. Hence,
the NRSE method can give valuable input to investigations on effects like the splitting of
magnon excitations as observed by Náfrádi et al. [24], hybridized magnon-phonon modes
in multiferroics [25, 26], and excitations with small energy separations, which are found in
orbital Peierls systems [27].
The second class of new NRSE experiments treated in this thesis were dedicated to line
shape analysis of temperature dependent asymmetric line broadening. This effect has been
observed in two systems: Cu(NO3)2·2.5D2O, a model system for a 1-D bond alternating
Heisenberg chain [53], and Sr3Cr2O8, a 3-dimensional gapped quantum spin dimer [65],
with standard ToF and TAS techniques. In order to explore the potential of NRSE to
investigate the effect, inelastic spin echo measurements were performed. For this purpose
high quality single crystals of Cu(NO3)2·2.5D2O were grown in the course of this thesis.
Temperature dependent inelastic NRSE measurements were performed and analyzed using
a phenomenological model for asymmetric line shapes for both systems. For the first time
this effect was measured with NRSE. As a proof of principle the results clearly show that the
NRSE method can be used to detect temperature dependent asymmetric line broadening.
Since there is no convolution of the signal with the resolution function of the spectrometer,
the NRSE method gives direct access to the line shape in the time domain. This is an
important advantage compared to other high resolution methods, such as ToF. It was shown,
that for a line shape differing from Lorentzian shape, the phase of the spin echo signal
becomes a non-linear function of the spin echo time τ . The results are a counter example
to the assertion of a general linear dependence between phase shift and spin echo time τ .
Thus, phase sensitive measurements can be used to determine a deviation of the scattering
Conclusion and perspectives 121
function from a Lorentzian shape. Exceptions are temperature dependent excitation energies
as present in Sr3Cr2O8.
Applications of NRSE are very interesting and might stimulate further experiments on
different systems, since the results for Cu(NO3)2·2.5D2O and Sr3Cr2O8 are suggested to be
applicable to a broad range of quantum systems [53, 65].
The second major part of this thesis was the upgrade of the NRSE option of FLEXX at
the BER II neutron source at HZB, Berlin. In the course of this project, redesigned NRSE
bootstrap coils were manufactured in collaboration with the Max Planck Institute For Solid
State Research, Stuttgart. The new coils allow for larger beam cross sections and higher coil
tilt angles. Hence, steeper dispersions are accessible. A major advantage is the exploitation
of the larger beam cross section, now available at FLEXX. The newly designed spectrometer
arms resulted in a more compact instrument. Thus, direct beam calibration measurements
are now feasible for the entire accessible wavevector range. In combination with higher coil
tilt angles the accessible Q-range in Larmor diffraction geometry is enlarged. A new, more
compact mu-metal shielding guarantees a better shielding of the field free area between the
NRSE coils. All mechanical and electrical components (e.g. rotary tables) were relocated
beneath the shielding boxes to reduce stray fields inside the field free zone. In order to
benefit from the larger beam cross section, the coupling coils were redesigned.
Extensive calibration measurements were performed. The parameters, necessary for user op-
eration of the instrument, were obtained and look-up tables, important for the impedance
matching and coil currents, are available now. The echo point and the distance between the
coils were determined. Measurements were successfully done to prove the phase stability.
Calibration sets for different coil tilt angles show that the properties of the individually
manufactured bootstrap coils are practically identical. The measurements clearly demon-
strate that the NRSE option works without any major technical problems and prove its
reliable performance.
The upgrade of the NRSE option of FLEXX was performed during the scheduled shut down
and upgrade of the BER II neutron source at HZB, Berlin. Since originally only a six months
shut down period until April 2011 was planned, NRSE measurements at the cold triple axis
spectrometer FLEXX on samples were foreseen in the course of this thesis. Due to the
unexpected prolongation of the shut down of BER II until April 2012, the available beam
time was severely limited. The shortage of beam time restricted the NRSE experiments
to commissioning of the NRSE option and calibration measurements. Hence, the thermal
NRSE-TAS TRISP at FRM-II, Garching, was chosen to perform NRSE line shape analysis
on Cu(NO3)2·2.5D2O.
122 Conclusion and perspectives
A feasible future advancement could be the upgrade of the NRSE-TAS instrument to be
able to perform MIEZE I&II measurements (Modulated IntEnsity by Zero Effort) [8].
MIEZE is a quasi-elastic variant of the NRSE technique generating a high frequent, time-
dependent sinusoidal signal at the detector and can be regarded as a high resolution ToF
spectrometer [8]. MIEZE I provides the advantage that the signal modulation is achieved
before the sample. Hence, the measurement becomes insensitive to depolarizing effects at
the sample region. Thus, the sample region does not need to be shielded magnetically
and measurements including magnetic fields and ferromagnetic samples are feasible. Such
an upgrade would require a detector with a high time resolution suitable for resolving the
time-dependent sinusoidal signal and minor changes of the instrument control software.
A MIEZE spectrometer is a promising instrument candidate [91, 92] for the future European
Spallation Source (ESS), Lund, Sweden. Applying the method to pulsed sources is techni-
cally challenging and requires major efforts and experimental verification. Such tests could
be performed at the ESS test beam line currently being built at the BER II, Berlin. Here,
the existing upgraded NRSE option of FLEXX can be used. The highly flexible spectrom-
eter arms could be adapted to other instrument components without major modifications
and provide ready to use NRSE components. A cost intensive and time consuming design
and manufacturing process would thus be redundant.
Appendix
A UB matrix formalism
The UB matrix formalism as discussed in [17] extends the original formalism introduced for
four-circle diffractometers [16] to TAS and ToF spectrometers. The UB matrix formalism
allows to calculate the instrument parameters (ν, µ, ω, φ) for a given (Q, E) and vice versa.
Here µ and ν are the sample goniometer angles, φ is the angle between kI and kF and ω is
related to the sample angle A3 by
ω = A3− θ (A.1)
with
θ =kI − kF cosφ
kF sinφ. (A.2)
A.1 The B matrix
The B matrix transforms any wavevector Q expressed in the basis of the reciprocal lattice
vectors b1,2,3 into a Cartesian system attached in a defined way to the reciprocal lattice.
The definition of Q reads:
Q =
h
k
l
= hb1 + kb2 + lb3. (A.3)
Follow Busing and Levy [16] here the x-axis is chosen to be parallel to b1, the y-axis to be in
the plane of b1 and b2 and the z-axis to be perpendicular to this plane. The transformation
of Q is then described by
Qc = BQ. (A.4)
123
124 Appendix
This equation follows the crystallographic convention, i.e. the factor 2π is omitted. The
magnitude of the wavevector is given by QM = 2π |Qc| and B is given by
B =
(B1 B2 B3
)=
a∗ b∗ cos γ∗ B13
0 b∗ sin γ∗ B23
0 0 B33
(A.5)
with
B13 = c∗ cosβ∗ (A.6)
B23 = c∗(cosα∗ − cos γ∗ cosβ∗)
sin γ∗= −c∗ sinβ∗ cosα∗ (A.7)
B33 =√c∗2 −B2
3x −B23y = c∗ sinβ∗ sinα∗. (A.8)
Note that Busing and Levy give B33 = 1/c. The usual crystallographic relations apply [47]
for the reciprocal lattice
a∗ =bc
Vsinα, b∗ =
ca
Vsinβ, c∗ =
ab
Vsin γ (A.9)
using
V = abc√1− cos2 α− cos2 β − cos2 γ + 2 cosα cosβ cos γ (A.10)
and
cosα∗ =cosβ cos γ − cosα
sinβ sin γ(A.11)
cosβ∗ =cos γ cosα− cosβ
sin γ sinα(A.12)
cos γ∗ =cosα cosβ − cos γ
sinα sinβ. (A.13)
A.2 The U matrix
The procedure introduced by Busing and Levy uses two non-collinear reflections Q1 and Q2
to calculate the orientation matrix U.
U = TνT−1c , (A.14)
A UB matrix formalism 125
where Tc is defined by a triple of right-handed orthogonal unit vectors t1c, t2c, t3c and Tν
is defined by a triple of right-handed orthogonal unit vectors t1ν , t2ν , t3ν . The matrix Tc
then reads
Tc =
(BQ1
|BQ1|,(BQ1 ×BQ2)×BQ1
|(BQ1 ×BQ2)×BQ1|,BQ1 ×BQ2
|BQ1 ×BQ2|
)(A.15)
where t1c is parallel to BQ1, t2c is in the plane of BQ1 and BQ2 and t3c is perpendicular to
this plane. Note that the matrix T−1c transforms from the crystal lattice Cartesian system
into the Cartesian system defined by the scattering system where Q1 points along the x-axis.
The matrix Tν is calculated in a similar way using the vectors u1ν and u2ν , given by
uiν =
cosωi cosµi
− sinωi cos νi + cosωi sinµi sin νi
sinωi sin νi + cosωi sinµi cos νi
. (A.16)
The vectors u1ν and u2ν represent the two non-collinear reflections expressed in the labora-
tory system by the experimental values (νi, µi, ωi) (i = 1, 2) of each reflection. The matrix
Tν therefore transforms from the Cartesian system defined by the scattering system into
the laboratory system and reads
Tν =
(u1ν
|u1ν |,(u1ν × u2ν)× u1ν
|(u1ν × u2ν)× u1ν |,u1ν × u2ν
|u1ν × u2ν |
). (A.17)
With the matrices U and B determined, the vector Qn for any given set of instrumental
positioning parameters (ν, µ ,ω, ϕ) can be calculated using
Qn =QM
2π(UB)−1
unν . (A.18)
A.3 The R matrix
As pointed out in [17] there are ambiguities for calculating the instrument parameters for
a given parameter set (Qn, E), since the Q-vector can be rotated around its own axis via
the sample goniometers without changing the scattering condition. Here, the condition
“keep a reference plane as horizontal as possible” is adopted from [17]. The matrix R then
transforms from the ν-coordinate system, where the rotation axis of the upper goniometer
points along the x-axis, into the θ-coordinate system, with Q pointing along the x-axis. The
126 Appendix
transformation reads
RQν = Qθ =1
2π
QM
0
0
. (A.19)
The matrix R is related to the elementary rotation matrices
R = ΩMN, (A.20)
where
Ω =
cosω − sinω 0
sinω cosω 0
0 0 1
(A.21)
M =
cosµ 0 sinµ
0 1 0
− sinµ 0 cosµ
(A.22)
N =
1 0 0
0 cos ν − sin ν
0 sin ν cos ν
(A.23)
represent the rotations of the sample table and the sample goniometers, respectively. The
matrix elements of R are explicitly
R =
R11 R12 R13
R21 R22 R23
R31 R32 R33
=
ωcµc ωcµsνs − ωsνc ωcµsνc + ωsνs
ωsµc ωsµsνs + ωcνc ωsµsνc − ωcνs
−µs µcνs µcνc
, (A.24)
A UB matrix formalism 127
where the instrumental positioning parameters can be obtained from
µs = sinµ = −R31 (A.25)
µc = cosµ =(R2
11 +R221
)1/2(A.26)
νs = sin ν = R32/(R2
11 +R221
)1/2(A.27)
νc = cos ν = R33/(R2
11 +R221
)1/2(A.28)
ωs = sinω = R21/(R2
11 +R221
)1/2(A.29)
ωc = cosω = R11/(R2
11 +R221
)1/2. (A.30)
Note that the proper quadrants are obtained from
µ = arcsin (−R31) (A.31)
ν = arcsin(R32/
(R2
11 +R221
)1/2)(A.32)
and
ω =
arcsin(R21/
(R2
11 +R221
)1/2) if R11 > 0
π − arcsin(R21/
(R2
11 +R221
)1/2) if R11 < 0.(A.33)
In order to calculate the matrix elements of R and following Lumsden et al. [17], the unit
vector perpendicular to the reference plane is defined by
uν⊥ =
− sinµplane
cosµplane sin νplane
cosµplane cos νplane
. (A.34)
Using the unit vectors Qν
|Qν | and uν⊥, a right-handed orthogonal set of vectors is constructed
by
t1ν = u1ν =Qν
|Qν |=
UBQ
|UBQ| (A.35)
t2ν = uν⊥ × u1ν (A.36)
t3ν = t1ν × t2ν . (A.37)
With
T1ν =
(t1ν t2ν t3ν
)(A.38)
128 Appendix
the result for R is:
R = T−11ν . (A.39)
Now the matrix elements of R are known and equations (A.31 – A.33) together with equa-
tions (A.1) and (A.2) allow to calculate the instrument positioning parameters for a given
Q.
B Matrix elements
B.1 The Ψ-matrix
The non-zero elements of the symmetric (6× 6)-matrix Ψ in the resolution function given
by equation (2.81) in section 2.3 are:
Ψ11 = − 2D2
C2fN
2F τ
′′2
(B.1)
Ψ22 = − ~
m− ~
mtan2 θ1 − 2
τ1τ ′′2
NI
kI ·ni− 2
D2
τ ′′2
C2i N
2I
C2fN
2F
(B.2)
Ψ33 = − ~
m
1
cos2 θ1− 2
D2
τ ′′2
e2i2N2I
C2fN
2F
1
cos2 θ1(B.3)
Ψ44 =~
m
1
cos2 θ2+ 2
~
m
tan θ2cos2 θ2
1
Cfef2 − 2
D2
τ ′′2
e2f2C2f
1
cos2 θ2(B.4)
Ψ55 = − ~
m− 2
D2
τ ′′2
e2i3N2I
C2fN
2F
(B.5)
Ψ66 =~
m− 2
D2
τ ′′2
e2f3C2f
(B.6)
Ψ12 = 2D2
τ ′′2
CiNI
C2fN
2F
(B.7)
Ψ13 = 2D2
τ ′′2
ei2NI
C2fN
2F
1
cos θ1(B.8)
Ψ14 = −2D2
τ ′′2
ef2C2fNF
1
cos θ2+
~
m
tan θ2cos θ2
1
CfNF(B.9)
Ψ15 = 2D2
τ ′′2
ei3NI
C2fN
2F
(B.10)
Ψ16 = −2D2
τ ′′2
ef3NFC2
f
(B.11)
Ψ23 =~
m
tan θ1cos θ1
− 2D2
τ ′′2
CiN2I
C2fN
2F
ei21
cos θ1(B.12)
B Matrix elements 129
Ψ24 = − ~
m
tan θ2cos θ2
CiNI
CfNF+ 2
D2
τ ′′2
CiNI
C2fNF
ef21
cos θ2(B.13)
Ψ25 = −2D2
τ ′′2
CiN2I
C2fN
2F
ei3 (B.14)
Ψ26 = 2D2
τ ′′2
CiNI
C2fNF
ef3 (B.15)
Ψ34 = 2D2
τ ′′2
ei2ef2NI
C2fNF
1
cos θ1 cos θ2− ~
m
tan θ2cos θ2 cos θ1
NI
CfNFei2 (B.16)
Ψ35 = 0 (B.17)
Ψ36 = −2D2
τ ′′2
ei2ei3N2I
C2fN
2F
1
cos θ1+ 2
D2
τ ′′2
ei2ef3NI
C2fNF
1
cos θ1(B.18)
Ψ45 = 2D2
τ ′′2
ef2ei3NI
C2fNF
1
cos θ2− ~
m
tan θ2cos θ2
1
Cf
NI
NFei3 (B.19)
Ψ46 = −2D2
τ ′′2
ef2ef3C2f
1
cos θ2+
~
m
tan θ2cos θ2
1
Cfef3 (B.20)
Ψ56 = 2D2
τ ′′2
ei3ef3NI
C2fNF
(B.21)
with
D2 =
(−τ2
NF
kF ·nf− ~
2mτ ′′2 − ~
2mτ ′′2 tan2 θ2
). (B.22)
B.2 The I−1C -matrix
The non-zero elements of the (6× 6)-matrix I−1C in the resolution function given by equation
(2.116) in section 2.3.4 are:
I−1C,11 = NI (ei1 cos θ1 − ei2 sin θ1) (B.23)
I−1C,12 = NI (ei1 sin θ1 + ei2 cos θ1) (B.24)
I−1C,13 = ei3NI (B.25)
I−1C,14 = −NF (ef1 cos θ2 + ef2 sin θ2) (B.26)
I−1C,15 = −NF (ef1 sin θ2 − ef2 cos θ2) (B.27)
I−1C,16 = −ef3NF (B.28)
I−1C,21 = cos θ1 (B.29)
130 Appendix
I−1C,22 = sin θ1 (B.30)
I−1C,32 = I−1
C,45 = I−1C,53 = I−1
C,66 = 1. (B.31)
B.3 The IC-matrix
The non-zero elements of the (6× 6)-matrix IC in the resolution function given by equation
(2.116) in section 2.3.4 are:
IC,12 =1
cos θ1(B.32)
IC,13 = − tan θ1 (B.33)
IC,41 = − 1
NF ef2 sin θ2 +NF ef1 cos θ2(B.34)
IC,42 = − ei2NI sin θ1 − ei1NI cos θ1ef1NF cos θ1 cos θ2 + ef2NF cos θ1 sin θ2
(B.35)
IC,43 =ei2NI cos
2 θ1 + ei2NI sin2 θ1
ef1NF cos θ1 cos θ2 + ef2NF cos θ1 sin θ2(B.36)
IC,44 = −ef1 sin θ2 − ef2 cos θ2ef1 cos θ2 + ef2 sin θ2
(B.37)
IC,45 =NIei3
ef2NF sin θ2 + ef1NF cos θ2(B.38)
IC,46 =ef3
ef1 cos θ2 + ef2 sin θ2(B.39)
IC,23 = IC,35 = IC,54 = IC,66 = 1. (B.40)
C Triple axis transmission function in matrix notation
The standard approximation of the resolution function for a conventional TAS has been
derived by Cooper and Nathans [48] (corrections by Dorner [49]) and reformulated in a
covariant matrix formalism by Stoica [50]. The formalism has been extended by Popovici
[51] to include spatial effects, such as monochromator and analyzer focusing and finite spatial
dimensions of the beam optical elements and the sample. Explicitly the TAS resolution
function reads [51]:
RTAS (X) =R0
(2π)2
√detMTASe
− 12XTMTASX (C.1)
where X is the four component column vector (X1, X2, X3, X4) = (Q−Q0, ω(q)− ω0(q0)),
i.e. the resolution function is defined in (Q, ω)-space. Note that X4 is the fourth com-
D Dispersion relation properties of RbMnF3 131
ponent of the vector X and the quantity ∆ω = ω(q) − ω0(q) used in the calculations is
different and refers to energy deviations from the dispersion surface for a given wavevec-
tor q. R0
(2π)2
√detMTAS is a normalization factor and R0 is given in equation (16) in [51].
Following Popovici the resolution matrix is given by:
M−1TAS = BA
[D(S+TTFT
)−1DT]−1
+G
−1
ATBT (C.2)
where G−1 is the covariance matrix of the distribution of the angular variables and F−1
is the covariance matrix of the reflectivity function. The matrices A, D, S, T, and G are
defined as in Appendix I and II in [51]. The linearized relation
X = BY (C.3)
holds, where the six component column vector Y = (∆ki,∆kf ) (see section (2.3.4). The
matrix B transforms from the coordinate space of Y into the coordinate space of X. In
order to express the exponent of the resolution function as a function of the six component
column vector J = (∆ω,∆kin, y1, y2, z1, z2), equation (2.124) is used
Y = ICJ. (C.4)
The (6× 6)-matrix IC is defined in Appendix B.3. Therefore, equation (C.2) can be refor-
mulated yielding for the TAS resolution matrix in the frame of J:
LTAS =
I−1C A
[D(S+TTFT
)−1DT−1
+G
]−1
AT(I−1C
)T−1
− (C.5)
D Dispersion relation properties of RbMnF3
The dispersion relation of RbMnF3 can be obtained from linear spin wave theory [52]:
E(q) =[gβHA + 2S (6J1 − 12J2 + 8J3) + 4SJ2B2 − 4SJ1A+ 4SJ3C2
] 12, (D.1)
where
A = cos qha0 + cos qka0 + cos qla0 (D.2)
B = cos (qh + qk) a0 + cos (qh − qk) a0 + cos (qk + ql) a0
+cos (qk − ql) a0 + cos (ql + qh) a0 + cos (ql − qh) a0 (D.3)
132 Appendix
C = cos (qh + qk + ql) a0 + cos (qh − qk − ql) a0
+cos (qh + qk − ql) a0 + cos (qh − qk − ql) a0. (D.4)
J1, J2 and J3 are the exchange interactions between first, second and third neighbors and
HA is the anisotropy field. Here qh,k,l = (2π/a0 ·Qh,k,l) and Qh,k,l are in r.l.u.. Magnetic
zone centers are located at (±0.5,±0.5,±0.5). According to [52] it is assumed that HA = 0,
J2 = 0, J3 = 0, S = 5/2, a0 = 4.204Å and J1 = 0.293meV. Hence, the dispersion relation
reduces to
E(q) = ~ω(q) = 4SJ1
[9− A(q)2
] 12. (D.5)
Since RbMnF3 has cubic symmetry, there is no need for additional transformations into a
Cartesian coordinate system and the components of the gradient of the dispersion read
∂E
∂qh,k,l= 4SJ1a0 sin a0qh,k,l
A√9−A2
. (D.6)
The elements of the curvature matrix can be calculated from the analytical expressions
∂2E
∂qh∂qh= − 4Sa20J1
(9−A2)32
[9 sin2 qha0 − 9A cos qha0 +A3 cos qha0
](D.7)
and∂2E
∂qh∂qk= −36SJ1a
20 sin qha0 sin qka0
(9−A2)32
. (D.8)
Other second derivatives are obtained from cyclic permutation of h, k and l.
E Growth of copper nitrate single crystals
Required deuteration ratio
Using the incoherent cross sections listed in Tab. 6.1, the incoherent cross section σxinc of
Cu(NO3)2·2.5H2O, Cu(NO3)2·2.5D2O and a mixture of both can be calculated. Here x
is defined as the percentage fraction of Cu(NO3)2·2.5D2O in the mixture. Therefore, the
incoherent cross section σxinc of pure Cu(NO3)2·2.5D2O and pure Cu(NO3)2·2.5H2O yields:
σ100inc =
1 · 0.55 + 2 · 0.5 + 8.5 · 0 + 5 · 2.0516.5
b = 0.72 b (E.1)
σ0inc =
1 · 0.55 + 2 · 0.5 + 8.5 · 0 + 5 · 80.2616.5
b = 24.42 b. (E.2)
E Growth of copper nitrate single crystals 133
Element σinc [b]
H 80.26
D 2.05
O 0
N 0.5
Cu 0.55
Tab. 6.1: Incoherent cross sections of the elements present in the com-pounds Cu(NO3)2·2.5H2O and Cu(NO3)2·2.5D2O [61].
A satisfactory deuteration ration is achieved, if the incoherent cross section of the mixture
σxinc is comparable to the incoherent cross section σ100
inc of pure Cu(NO3)2·2.5D2O. Thus, a
deuteration ratio of 98.5% resulting in
σ98.5inc =
1 · 0.55 + 2 · 0.5 + 8.5 · 0 + 5 · (80.26 · 0.015 + 2.05 · 0.985)16.5
b
= 1.0707 b = 1.487 σ100inc , (E.3)
would fulfill the requirement.
Distillation process
In order to calculate the remaining amount of H2O after the distillation process the molar
masses of the different compounds are needed. The molar masses of Cu(NO3)2·2.5H2O,
D2O and H2O are 232.59 gmol
, 20 gmol
and 18 gmol
, respectively.
For the distillation process 140.99g Cu(NO3)2·2.5H2O powder material, containing 27.04g
H2O, was dissolved in 47.56g high purity D2O (>99.9%). Therefore, using the molar masses,
the resulting hydrogen mass ratio and the atomic ratio are 36.25% and 38.71%, respectively.
In the first destillation run 12.80g liquid, containing 4.64g H2O and 8.16g D2O, were distilled
from the mixture. 22.29g D2O were added in the next step and decreased the hydrogen molar
ratio and the atomic ratio to 26.64% and 28.75%, respectively. By repeating these two steps
17 times, the deuteration level of the solution was successively increased, resulting in a
calculated deuteration ratio of 99.38%. The results of the distillation runs are listed in Tab.
6.2. Note that the actual deuteration ratio may be slightly higher than the calculated one
due to a small amount of liquid, remaining in the Liebig condenser of the setup, which was
not weighed after each distillation run. In the last run liquid was distilled off to ensure a
saturated solution for the crystal growth.
134 Appendix
addedD2O[g]
distilledliquid
[g]
thereofH2O[g]
thereofD2O[g]
H2O insolution
[g]
D2O insolution
[g]
D2Ofraction[g-%]
D2Ofraction[mol-%]
47.56 0.00 0.00 0.00 27.04 47.56 63.75 61.29
22.29 12.80 4.64 8.16 22.40 61.69 73.36 71.25
22.71 30.50 8.12 22.38 14.28 62.02 81.29 79.63
16.62 23.20 4.34 18.86 9.93 59.79 85.75 84.41
18.04 21.20 3.02 18.18 6.91 59.65 89.61 88.59
30.29 30.90 3.21 27.69 3.70 62.25 94.38 93.80
12.05 8.90 0.50 8.40 3.20 65.90 95.36 94.87
11.50 13.70 0.64 13.06 2.57 64.33 96.16 95.75
16.45 15.00 0.58 14.42 1.99 66.36 97.08 96.77
10.92 9.30 0.27 9.03 1.72 68.25 97.54 97.27
13.11 13.10 0.32 12.78 1.40 68.58 98.00 97.78
8.47 3.70 0.07 3.63 1.33 73.42 98.23 98.03
11.32 10.90 0.19 10.71 1.13 74.04 98.49 98.33
11.48 14.40 0.22 14.18 0.92 71.33 98.73 98.59
17.31 19.80 0.25 19.55 0.66 69.10 99.05 98.94
18.12 13.80 0.13 13.67 0.53 73.55 99.28 99.20
17.70 13.90 0.10 13.80 0.43 77.45 99.44 99.38
0.00 15.80 0.09 15.71 0.35 61.73 99.44 99.38
Tab. 6.2: List of distillation steps performed to obtain a molar deuterationlevel of 99.38% of the solution.
F Impedance matching of the RF coils
In order to measure at different spin echo times τ during an NRSE experiment, different
frequencies need to be applied to the RF coils. Thus, the optimum parameters of the res-
onant circuit of the RF coils need to be adapted, i.e. the impedance needs to be matched
to the required frequency. The spin echo option of FLEXX uses an automatic impedance
matching device implemented in the circuit shown in Fig F.1. All four RF coils of each spec-
trometer arm are connected in parallel and the RF coils of each NRSE arm are connected
to a waveform generator via a power amplifier. The phases of both waveform generators
F Impedance matching of the RF coils 135
are locked. In order to have a direct control over the magnetic field amplitude in the RF
coils, a pick-up coil is attached to one of the U-shaped parts of the RF coil. The oscillating
magnetic field induces a voltage Upu, which is read by an RF voltmeter and transferred to
the control program.
C2
C1
L
R
Z
Fig. F.1: Impedance matching circuit using two capacities [90].
The impedance matching proceeds as follows: The required frequency is applied to the RF
coils using a small signal amplitude. Optimum values for the capacitances C1 and C2 are
obtained from a look-up table for the specific frequency. The capacitance C2 is kept fix
while C1 is adjusted until the pick-up voltage Upu reaches a maximum. The amplitude of
the waveform generator is then increased until Upu has reached its target value obtained
from the look-up table (see section 5.3.1).
The impedance of one RF coil is:
ZRF = iωL+R, (F.1)
where R is the DC resistance of the RF coil (≈ 0.35Ω) and L is the inductance (≈ 36.6µH).
The output resistance of the power amplifiers is 50Ω and frequency independent. In order
to maximize the forward power of the amplifier, the real part of the load impedance needs to
match the 50Ω. Otherwise the reflected power could overheat the amplifier. The impedance
matching device uses a simple matching circuit with a discrete set of capacitances for each
of the two capacitances C1 and C2 [90] (see Fig. F.1). The impedance of the circuit is:
1
Z= iωC2 +
11
iωC1+ iωL+R
. (F.2)
136 Appendix
The real part of equation (F.2) should be 50Ω while the imaginary part should be zero:
Re
(1
Z
)=
R
R2 +(ωL− 1
ωC1
)2 = 50Ω (F.3)
Im
(1
Z
)= ωC2 −
ωL− 1ωC1
R2 +(ωL− 1
ωC1
)2 = 0 (F.4)
Therefore, the resulting capacitances are:
C1(ω) =1
ω2L− ω√ZR−R2
(F.5)
C2(ω) =L− 1
ω2C1
R2 +(ωL− 1
ωC1
)2 . (F.6)
The existing range of discrete capacitors (C1: 25pF to 3.19µF, C2: 800pF to 1.63µF) is
sufficient for a frequency range from 50kHz to 500kHz in 4π-mode (2 parallel RF coils)
and 8π-mode (4 parallel RF coils). For calibration purposes, to provide a parameter set
C1(ω) and C2(ω) for NRSE measurements, the capacitance C2 was calculated according to
equation (F.6) and C1 was then optimized. By varying C1 stepwise the read out induced
voltage was maximized for a constant output amplitude of the waveform generator.
C1 [nF]
C2 [n
F]
f = 100kHz
700 750 800
300
310
320
330
340
350
360
0.9
1
1.1
1.2
1.3
C1 [nF]
C2 [n
F]
f = 200kHz
80 85 90 95145
150
155
160
165
170
175
1.2
1.4
1.6
1.8
2
2.2
2.4
C1 [nF]
C2 [n
F]
f = 300kHz
28 30 32 34
115
120
125
130
135
140
1
1.5
2
2.5
3
Fig. F.2: Resonance of the induced voltage as a function of the capacitiesC1 and C2 for frequencies of 100kHz, 200kHz and 300kHz. The measure-ments were done in 8π-mode (4 parallel RF coils). The black line showsthe optimum values according to equation (F.6).
Fig. F.2 shows the induced voltage for different sets of capacitances for frequencies of
100kHz, 200kHz and 300kHz. For the measurement all four parallel RF coils were con-
nected, resulting in Rtot ≈ 0.25Ω and Ltot ≈ 11µH. The maximum value of the resonance
shows little sensitivity to C2. For f = 100kHz the resonance is quite broad, while it gets
G Adiabatic and non-adiabatic transitions of the magnetic moment of the neutron 137
smaller with increasing frequency. The discrete steps in the voltage values as seen in Fig.
F.2 arise from the fact that the effective capacitances C1 and C2 are each a combination
of parallel capacitances. If the effective capacitance is increased stepwise, the number of
parallel capacitances may change by a factor of two or more, resulting in a step change
of the impedance of the effective capacitance. This effect does not affect the automatic
impedance matching since for every fixed C2 there is still a pronounced maximum of the
resonance, which is found by varying C1. The black line in Fig. F.2 displays the theoretical
optimum capacitance sets (C1, C2) according to equation (F.6). The experimental results
agree completely with the calculations from the model.
In order to generate a new look-up table containing the frequency dependent optimum start-
ing values of the capacitances, the resonances were scanned for different frequencies for both
spectrometer arms. The measurements were performed in 4π-mode and 8π-mode.
G Adiabatic and non-adiabatic transitions of the magnetic
moment of the neutron
If the magnetic moment of the neutron oriented parallel to the guide field enters a temporally
and/or spatially varying magnetic field, two different limiting cases are possible [2, 89]:
• Adiabatic transition: The magnetic moment follows the change of a smoothly varying
magnetic field direction
• Non-adiabatic transition: The orientation of the magnetic moment does not change if
the magnetic field direction changes abruptly
The condition for the adiabatic case can be formulated as
ωL ≫ ωB, (G.1)
where ωL = |γ| ·B is the Larmor frequency and ωB =d( ~B/| ~B|)
dt is the frequency of the change
of the magnetic field direction. If ωL is much larger than ωB, the magnetic moment of the
neutron keeps its orientation along B and follows the change of the magnetic field direction.
If the magnetic field changes by | ~B| within the distance L and the velocity of the neutron
is v, equation (G.1) can be rewritten as
ωLL
v≫ 1. (G.2)
138 Appendix
For the non-adiabatic transition the analogous condition reads:
ωL ≪ ωB (G.3)
and can be rewritten as
ωLL
v≪ 1. (G.4)
Adiabatic transition: The guide field from the monochromator to the coupling coil is ori-
ented in the vertical direction and is of the order of 30G. The change of the magnetic field
into the horizontal plane occurs along a length of 18cm. For a wavelength of λ = 2.3Å and
thus, a neutron velocity of v = 1675ms
equation (G.2) yields:
ωLL
v= 2916
HzG
· 30G 0.18m1675m
s
≈ 9.4 ≫ 1, (G.5)
which satisfies, to a good approximation, the adiabaticity condition. For larger wavelengths
the adiabaticity parameter is > 10.
Non-adiabatic transition: The maximum field in horizontal direction in the coupling coils is
30G. The maximum change of field from the inner part of the coupling coil to the shielded
region with a remaining field of Br = 5mG is therefore approximately 30G. The transmission
occurs within a distance equal to the wire thickness of 0.5mm. For a wavelength of λ = 6Å
and therefore a neutron velocity of v = 659ms
equation (G.4) yields:
ωLL
v= 2916
HzG
· 30G0.0005m659m
s
≈ 0.066 ≪ 1 (G.6)
Thus, the transition at the current sheet is non-adiabatic.
Bibliography
[1] F. Mezei, “Neutron Spin Echo: A New Concept in Polarised Thermal Neutron Tech-
niques”, Z. Physik 255, 146-160, 1972
[2] F. Mezei ed., “Neutron Spin Echo”, Lecture Notes in Physics, Springer (1980)
[3] R. Gähler, R. Golub, “A High Resolution Neutron Spectrometer for Quasielastic Scat-
tering on the Basis of Spin Echo and Magnetic Resonance”, Z. Phys. B. Condensed
Matter 65, 269, 1987
[4] R. Gähler, R. Golub, “Neutron Resonance-Spin-Echo, Bootstrap-Method for Increas-
ing the Effective Magnetic Field”, Journal de Physique 49, 1195-1205, 1988
[5] F. Mezei, in “Neutron Inelastic Scattering” 1977 (IAEA, Vienna, 1978) p. 125
[6] K. Habicht, T. Keller, R. Golub, Journal of Applied Crystallography 36 1307-1318
(2003)
[7] K. Habicht, T. Keller, R. Golub, Physica B 350 E803-E806 (2004)
[8] T. Keller, R. Golub, R. Gähler, in “Scattering” ed. by R. Pike and P. Sabatier, San
Diego: Academic Press, 2002, pp. 1264-1286
[9] L. Van Hove, Phys. Rev. 95 249 (1954)
[10] I. I. Rabi, N. F. Ramsey, J. Schwinger, Rev. Mod. Phys. 26 167 (1954)
[11] F. Bloch, S. Siegert, Phys. Rev. 57 522 (1940)
[12] T. Keller, “Höchstauflösende Neutronenspektrometer auf der Basis von Spinflippern -
neue Varianten des Spinecho-Prinzips”, Doktorarbeit, Fakultät für Physik, Technische
Universität München (1993)
[13] K. Habicht, R. Golub, R. Gähler, T. Keller, Lecture Notes in Physics 601 116-132
(2003)
139
140 References
[14] K. Habicht, unpublished manuscript
[15] Klaus Habicht, personal communication
[16] W. Busing, H. A. Levy, Acta Cryst. A 22 457 (1967)
[17] M. D. Lumsden, J. L. Robertson, M. Yethiraj, J. Appl. Cryst 38 405 (2005)
[18] K. S. Miller, “Multidimensional Gaussian distributions”, New York: John Wiley (1964)
[19] R. Pynn, J. Phys. E 11 1133-1140 (1978)
[20] K. Habicht, M. Skoulatos, F. Groitl, BENSC Experimental Report PHY-02-735-IT
(2009)
[21] F. Groitl, K. Kiefer, K. Habicht, Physica B 406 12 2342-2345 (2011)
[22] M. Zoli, J. Phys.: Condens. Matter 3 33 6249-6256 (1991)
[23] K. Habicht, M. Enderle, B. Fåk, K. Hradil, P. Böni, T. Keller, J. Phys.: Conf. Ser.
211 012028 (2010)
[24] B. Nafradi, T. Keller, H. Manaka, A. Zheludev, B. Keimer, Phys. Rev. Lett. 106 17
177202(2011)
[25] D. Senff et al., Phys. Rev. Lett. 98 13 (2007)
[26] S. Petit et al., Pramana J. Phys. 71 869 (2008)
[27] C. Ulrich et al., Phys. Rev. Lett. 91 257202 (2003)
[28] http://www.attocube.com/
[29] G. Bose, H. C. Gupta, B. B. Tripathi, J. Phys. F: Metal Phys. 2 3 426-432 (1972)
[30] R. P. Bajpai, K. Neelakandan, Il Nuovo Cimento 7B 2 177-184 (1972)
[31] B. L. Fielek, J. Phys. F: Metal Phys. 10 8 1665-1671 (1980)
[32] Y. Nakagawa, A. D. B. Woods, Phys. Rev. Lett. 11 6 271-274 (1963)
[33] P. B. Allen, Phys. Rev. B 16 12 5139-5146 (1977)
[34] A. R. Jani, V. B. Gohel, Cz. J. Phys. 34 12 1334-1338 (1984)
[35] V. P. Singh, H. L. Kharoo, J. Prakash, L. P. Pathak, Acta Physica Academiae Scien-
tiarum Hungaricae 39 1 37-42 (1975)
References 141
[36] F. Güthoff, “Neutronenstreuexperimente an γ-Lanthan, Niob und Tl83In17 zur
Untersuchung der Phononendispersion im Zusammenhang mit den Diffusions-
eigenschaften kubisch raumzentrierter Metalle”, Doktorarbeit, Mathematisch-
Naturwissenschaftliche Fakultät, Westfälische Wilhems-Universität Münster (1994)
[37] R. P. Bajpai, K. Neelakandan, Physica 62 4 587-594 (1972)
[38] W. Kohn, Phys. Rev. Lett. 2 9 393-394 (1959)
[39] B. C. Clark, D. C. Gazis, R. F. Wallis, Phys. Rev. 134 A1486-1491 (1964)
[40] J, deLaunay, in “Solid State Physics - Advances in Research Applications Vol. 2” ed.
by F. Seitz and D. Turnbull, New York: Academic Press, 1956, pp 219-303
[41] J. Behari, B. B. Tripathi, Aus. J. Phys. 23 311-318 (1970)
[42] J. Behari, B. B. Tripathi, J. Phys. C: Solid St. Phys. 3 659-665 (1970)
[43] F. Groitl, K. Habicht, K. Kiefer, BENSC Experimental Report PHY-02-765-EF (2010)
[44] M. T. Rekveldt, Material Science Forum, 321-3 258-263 (2000)
[45] M. T. Rekveldt, Europhys. Lett. 54 3 342-346 (2001)
[46] R. Pynn in [2]
[47] U. Shmueli in “International Tables for Crystallography”, Vol B, Section 1.1.3 (2006)
[48] M. J. Cooper, R. Nathans, Acta Cryst. 23 357 (1967)
[49] B. Dorner, Acta Cryst. A28 319 (1972)
[50] A. D. Stoica, Acta Cryst. 31 189 (1975)
[51] M. Popovici, Acta Cryst. A31 507 (1975)
[52] C. G. Windsor, R. W. Stevenson, Proc. Roy. Soc. 87 501 (1966)
[53] D. A. Tennant, B. Lake, A. J. A. James, F. H. L. Essler, S. Notbohm, H.-J. Mikeska,
J. Fielden, P. Kögerler, P. C. Canfield, M. T. F. Telling, Phys. Rev. B 85 014402
(2012)
[54] A. J. A. James, F. H. L. Essler, R. M. Konik, Phys. Rev. B 78 094411 (2008)
[55] D. A. Tennant, C. Broholm, D. H. Reich, S. E. Nagler, G. E. Granroth, T. Barnes,
K. Damle, G. Y. Xu, Y. Chen, B. C. Sales, Phys. Rev. B 57 054414 (2003)
142 References
[56] J. Garaj, Acta. Chem. Scand. 22 1710 (1968)
[57] B. Morosin, Acta. Cryst. B: Struct. Crystallogr. Cryst. Chem. 26 1203 (1970)
[58] G. Y. Xu, C. Broholm, D. H. Reich, M. A. Adams, Phys. Rev. Lett. 84 4465 (2000)
[59] T. Barnes, J. Riera, D. A. Tennant, Phys. Rev. B 59 11384 (1999)
[60] G. Y. Xu, “Spin Correlations and Impurities in 1-D Gapped Spin Systems”, PhD
Thesis, Johns Hopkins University, USA (2001)
[61] “Neutron Data Booklet - Second Edition” ed. by A.-J. Dianoux, G. Lander, Institut
Laue-Langevin, Grenoble, France, OCP Science imprint, Old City Publishing Group,
Philadelphia, USA (2003).
[62] S. Notbohm, “Spin dynamics of quantum spin-ladders and chains”, PhD Thesis, Uni-
versity of St. Andrews, UK (2007)
[63] R. M. Moon, T. Riste, W. C. Koehler, Phys. Rev. 181 2 920 (1969)
[64] L. Gmelin, “Handbuch der Organischen Chemie”, Verlag Chemie GmbH, Wein-
heim/Bergstrasse (1972)
[65] D. L. Quintero-Castro, B. Lake, A. T. M. N. Islam, E. M. Wheeler, C. Balz, M. Mans-
son, K. C. Rule, S. Gvasaliya, A. Zheludev, PRL 109 127206 (2012)
[66] T. Keller, K. Habicht, H. Klann, M. Ohl, H. Schneider, B. Keimer, Appl. Phys. A 74
[Suppl.] S332-S335 (2002)
[67] D. L. Quintero-Castro, B. Lake, E. M. Wheeler, A. T. M. N. Islam, T. Guidi,
K. C. Rule, Z. Izaola, M. Russina, K. Kiefer, Y. Skoursky, Phys. Rev. B 81 014415
(2010)
[68] D. L. Quintero-Castro, “Neutron Scattering Investigations on 3d and 4f Frustrated
Magnetic Insulators”, PhD Thesis, TU Berlin, Germany (2011)
[69] A. T. M. N. Islam, D. L. Quintero-Castro, B. Lake, K. Siemensmeyer, K. Kiefer,
Y. Skourski, T. Herrmannsdorfer, Cryst. Growth Des. 10 465 (2010)
[70] Y. Singh, D. C. Johnston, Phys. Rev. B 76 012407 (2007)
[71] L. C. Chapon, C. Stock, P. G. Radelli, C. Martin, arXiv:0807.0877v2 [cond-mt.mtrl-
sci] (2008)
References 143
[72] Z. Wang, M. Schmidt, A. Guenther, S. Schaile, N. Pascher, F. Mayr, Y. Goncharow,
D. L. Quintero-Castro, A. T. M. N. Islam, B. Lake, H.-A. Krug von Nidda, A. Loidl,
J. Deisenhofer, Phys. Rev. B 83 201102(R) (2011)
[73] D. Wulferding, P. Lemmens, K. Y. Choi, V. Gnezdilov, Y. G. Pashkevich, J. Deisen-
hofer, D. L. Quintero-Castro, A. T. M. N. Islam, B. Lake, Phys. Rev. B 84 064419
(2011)
[74] P. Vorderwisch, S. Hautecler, F. Mezei, U. Stuhr, Physica B 213 866-868 (1995)
[75] M. Skoulatos, K. Habicht, Nuclear Instruments and Methods in Physics Research
Section A 647 100-106 (2011)
[76] T. Keller, R. Gähler, H. Kunze, R. Golub, Neutron News 6 3 (1995)
[77] T. Keller, R. Golub, F. Mezei, R. Gähler, Physica B 234 1126 (1997)
[78] T. Keller, R. Golub, F. Mezei, R. Gähler, Physica B 241 101 (1997)
[79] P. Hank, “Aufbau des Neutronenresonanz-Spinechospektrometers in Saclay und Er-
weiterung der Meßmöglichkeit durch die MIEZE-Option”, Doktorarbeit, Fakultät für
Physik, Technische Universität München (1999)
[80] Homepage of distributing company: http://www.collano.com/
[81] Homepage of distributing company: http://www.bergquistcompany.com/
[82] S. Klimko, “ZETA, A zero field spin echo method for very high resolution study of
elementary excitations and first application”, Doktorarbeit, Fakultät für Physik, Tech-
nische Universität Berlin (2003)
[83] Homepage of distributing company: http://www.item24.de/
[84] Homepage of distributing company: http://www.vacuumschmelze.de/
[85] Design courtesy of Max Planck Institute For Solid State Research,
http://www.fkf.mpg.de/
[86] A. J. Mager, Physica B&C 80 1-4 (1975)
[87] A. J. Mager, IEEE Trans. on Magn. MAG-6 67 (1970)
[88] A. J. Mager, J. Appl. Phys. 39 1914 (1986)
144 References
[89] S. V. Maleyev, “Polarized Neutron Studies of Condensed Matter, Lecture I: Neutron
Polarization”, PNPI preprint-2083 (1995)
[90] “The ARRL Handbook”, Published by The American Radio Relay League, Newington
CT, 86. edition (1991)
[91] G. Brandl, R. Georgii, W. Häußler, S. Mühlbauer, P. Böni, Nuclear Instruments and
Methods in Physics Research Section A 654 1 394-398 (2011)
[92] G. Brandl, J. Lal, L. Crow, L. Robertson, R. Georgii, P. Böni, M. Bleuel, Nuclear
Instruments and Methods in Physics Research Section A 667 1-4 (2012)
Acknowledgments
Since a PhD thesis is no single combat, I want to thank all my friends and colleagues who
gave advice and support and made this time so special:
First of all, I want to thank my direct supervisor Dr. Klaus Habicht who welcomed me to
the HZB and gave me the opportunity to work on the FLEXX instrument. He shared all
his experience and took his time for all the invaluable discussions. I am especially grateful
for the opportunity to upgrade the NRSE option. Thanks for all the ideas, the support and
that you stayed even when the polarization already went home.
Special thanks to my supervisor Prof. Dr. Alan Tennant for giving me the opportunity to
perform this work. Thanks for the discussions and all the ideas.
Thanks to Prof. Dr. Peter Böni, who kindly accepted to be the second reviewer of my
thesis.
Many thanks to Dr. Katharina Rolfs for all the discussions, a lot of encouraged help during
the crystal growth, the motivation and the fun. I am really going to miss the coffee rein-
forced discussions.
Thanks to the whole FLEXX-Team, former and present members, for the nice and enjoyable
time I had as a part of this group. Especially I would like to thank Dr. Manh Duc Le for
sharing his Matlab knowledge, Dr. Kirrily Rule for all the explanations regarding triple
axis spectrometer and Dr. Markos Skoulatos for a really warm welcome to the group, nice
discussions and of course for the oregano.
Thanks to the coffee round, Dr. Katharina Rolfs, Dr. Daniil Nekrassov, Morten Sales, Dr.
Rasmus Toft-Peterson, Dr. Manh Duc Le, Dr. Mirko Boin, Dr. Robert Wimpory and Jen-
nifer Schulz, for all the delicate, sometimes unsettling, but always funny discussions during
the coffee breaks.
Thanks to Kathrin Buchner for all the patience and enthusiasm while manufacturing the
new, really nice bootstrap coils. Standing beside a lathe for 2 months, wearing a gas mask
and watching a coil slowly rotating can actually be very funny. It is not possible to com-
pensate your help with Haloren-Kugeln and Knusperflocken, although I tried.
Thanks to Dr. Thomas Keller for all the support during the beam time at TRISP. Thanks
145
146 Acknowledgments
for all the helpful discussions and patiently sharing your expertise and knowledge about
NRSE.
Thanks to Bernd Urban for his tremendous support designing and manufacturing compo-
nents as well as setting up the NRSE instrument.
Thanks to Norbert Beul for designing and manufacturing the coupling coils and the com-
pensation coils.
Thanks to Dr. Diana Quintero-Castro for the Sr3Cr2O8 crystal and a lot of fruitful discus-
sions.
Thanks to the team of sample environment, especially Dr. Klaus Kiefer and Martin Petsche
for setting up the Attocube setup.
Thanks to Dr. Wolfgang Jauch for the γ-ray diffraction experiments on the Nb crystals.
Thanks to the staff of the HZB workshops, solving all the urgent problems and orders.
Thanks to Dr. Klaus Habicht, Dr. Katharina Rolfs, Dr. Manh Duc Le and Dr. Diana
Quintero-Castro for proof reading this thesis.
Many thanks to my good friend Marijke Haffke for supporting me with your funny and
calming calls throughout all the busy times.
Thanks to my friends Christine Walch, Steffen Walch and Hannelore Fiebig, who made
Berlin a home to me.
Thanks to my parents and my family for their support throughout my studies and for the
interest in my work.
Many thanks to my girlfriend Jule, who had to go without me during the beam time, the
coil winding and all the other busy times. Thanks for the love, the care and being the door
to the world outside the institute.
Last but not least, thanks to my coffee machine for three years of continuous and reliable
work keeping the level up.