magnetic neutron scattering - ornl · sl e xp i q × r (l) ... pp 2 1, stot 0 stot 1,,, nn np pn pp...
TRANSCRIPT
![Page 1: Magnetic Neutron Scattering - ORNL · sl e xp i q × r (l) ... pp 2 1, Stot 0 Stot 1,,, nn np pn pp ... 0.4 0.5 0 2 4 0 2 4 6 Exp. data Single mode model e(Q ) Q (p) Spin waves in](https://reader033.vdocuments.us/reader033/viewer/2022050215/5f613cdd0223a162dc09ca30/html5/thumbnails/1.jpg)
Magnetic Neutron Scattering
Collin Broholm
*Supported by U.S. DoE, Basic Energy Sciences, Materials Sciences & Engineering DE-FG02-08ER46544
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Neutron spin meets electron spin
Structure : Magnetic diffraction
Excitations: Inelastic magnetic scattering
Tomorrow : Entangled Magnetism
Overview
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Magnetic properties of the neutron
m
meBn
The neutron has a dipole moment
n is 960 times smaller than the electron moment
960913.1
1836
en
e
m
m
A dipole in a magnetic field has potential energy
rBr
VCorrespondingly the field exerts a torque and a force
B
BF
driving the neutron parallel to high field regions
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Magnetic Neutron Optics The combined nuclear and magnetic potential for neutron
V =
2p 2
mrb ± r0M ×s( )
The index of refraction is
n = 1-
V
E= 1-
l 2
prb ± r0M ×s( )
Critical angle for total external reflection:
qc = l
rb ± r0M ×s
p
M
Polarized beam!
Q
k ¢k
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Structure of vortex matter: CeCoIn5
Bianchi et al., Science (2008)
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The transition matrix element The dipole moment of unfilled shells yield inhomog. B-field
2
ˆ
4 R
g B RSB 0
The magnetic neutron senses the field
2
20ˆ
4 Rm
mgV B
em
RSrBr
The transition matrix element in Fermi’s golden rule
m
2p 2¢k ¢s ¢l V
mksl = -r
0
g
2F q( ) ¢s ¢l s ×S
^lsl exp iq ×r
l( )Magnetic scattering length ~ nuclear scattering lengths
r0 = gm0
4p
e2
me
= 0.54 ´10-12 cm
It is sensitive to atomic dipole moment perp. to q
S
^ l= S
l- S
l×q̂( )q̂
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The magnetic scattering cross section
exp dF s i q r q r r
Spin density spread out scattering decreases at high q
The magnetic neutron scattering cross section
d2s
dWd ¢Es® ¢s
=¢k
k
m
2p 2
æ
èçö
ø÷
2
pl
l ¢l
å ¢k ¢s ¢l Vm
ksl2
d El
- E¢l- w( )
=¢k
kr
0
2 g
2F q( )
2
e-2W k( ) 1
2pdt e- iw t
ò eiq× R
l-R
¢l( )
l ¢l
å
´ s s ×S^ l
0( ) ¢s ¢s s ×S^ ¢l
t( ) s
For unspecified incident & final neutron spin states
Edd
d
Edd
d 2
21
2
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Un-polarized magnetic scattering
Spin correlation function
Squared form factor Polarization factor
Fourier transform
DW factor
d2s
dWd ¢E=
¢k
kr
0
2 g
2F q( )
2
e-2W q( )
dab
- q̂aq̂
b( )ab
å
´1
2pdt e- iwt e
iq× rl-r
¢l( )S
l
a 0( )S¢l
b t( )l ¢l
åò
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Online lectures accessible at:
http://www.sns.gov/education/qcmp/
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Magnetic neutron diffraction Time independent spin correlations elastic scattering
ds
dW= r
0
2 g
2F q( )
2
e-2W q( ) d
ab- q̂
aq̂
b( )ab
å eiq× r
l-r
¢l( )S
l
a S¢l
b
l ¢l
å
Periodic magnetic structures Magnetic Bragg peaks
The magnetic vector structure factor is
Magnetic primitive unit cell greater than chemical P.U.C. Magnetic Brillouin zone smaller than chemical B.Z.
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Diffraction from Ferromagnet
Nuclear & magnetic scattering interferes. Effective scatt. length:
b +s r0S^
The structure factor for reciprocal lattice vector, t
Fs
2= bd +s r0S^d( )eit ×d
då2
= bdeit ×d
då2
+ s r0S^deit ×d
då2
+ 2s bdr0S^deit × d- ¢d( )
d ¢dåNuclear Magnetic Nuclear-magnetic
The last term vanishes for un-polarized neutrons ( average). A welcome consequence is that the diffracted beam is polarized!
s
bd » r0S^d Þ F±
2»
4 bd
2 for s = 1
0 for s = -1
ì
íï
îï
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Simple cubic antiferromagnet
zS ˆB
smg
ab
*a
*b
ma
mb
*
ma
*
mb
ds
dW= N r
0
ms
2mB
æ
èç
ö
ø÷
2
e-2W q( )
F q( )2
1- q̂z
2( )2p( )
3
vd q -t
m( )t
m
å ‘
No magnetic diffraction for q S
qS
S
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Not so simple Heli-magnet : MnO2
ˆ ˆexp cos sinl l m l m l
S i S w R x Q R y Q R
Insert into diffraction cross section to obtain
ds
dW= N r
0S( )
2
e-2W q( ) g
2F q( )
2
1+ q̂z
2( )2p( )
3
v
´ d q + w - Qm
-t( ) +d q + w + Qm
-t( ){ }t
å
111w and 27
00m
Q characterize structure
*a
*c
a b c
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Diffuse Magnetic Scattering in FeTe0.6Se0.4 K
(r
.l.u
.)
L (
r.l.
u.)
Thampy et al PRL(2012)
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Hypothesis: Ising spins?
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Polarized neutrons:
isotropic static spin correlations
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Diffuse Magnetic Scattering in FeTe0.6Se0.4 K
(r
.l.u
.)
L (
r.l.
u.)
Thampy et al PRL(2012)
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Interstitial Iron in Fe1+yTe1-xSex
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Extract Local Spin Structure
Model by coupling itinerant spin to 5-band model through
Kang & Tesanovic
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Nakatsuji et al. Science (2005)
Unusually weak interlayer coupling
Dominant 3rd neighbor interactions
Exact triangular lattice
Weak easy plane anisotropy
J3 : AFM
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Elastic neutron scattering from NiGa2S4 powder
2 * 22
2
0 22 2
/ˆ 1 2 cos2
d g Ar F Q N
d
q q
τ
m Q m Q c
Q τ q
2 Surprises:
12 120oQ Q
1 25(3) Å
Nakatsuji et al. (2005) ds
dW powder
=dWQ
4pòds
dW xtal
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Multi crystal assembly of high quality NiGa2S4
19 NiGa2S4 crystals Grown by Yusuke Nambu co-aligned by Chris Stock
Y. Nambu et al., PRB (2009).
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NiGa2S4: Elastic Magnetic Scattering
1.5 meV
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Short range magnetic ordering
C. Stock et al. PRL (2010)
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Summary: Diffraction
• The neutron has a small dipole moment causing scattering from electrons
• Magnetic scattering is similar in magnitude to nuclear scattering
• Elastic magnetic scattering probes static magnetic structure
• Spin direction gleaned from polarization factor (see spin perpendicular to Q)
• Periodic structure determined through Bragg intensities
• Generally probe magnetic structure on 1-104 Å scale:
– flux line lattice
– short range order
− powder, single crystals, thin films
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Features of the cross section
The dynamic correlation function
Fluctuation Dissipation theorem
Sum-rules
Examples
Summary
Inelastic Magnetic Neutron Scattering
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Understanding Inelastic Magnetic Scattering:
Isolate the “interesting part” of the cross section
The dynamic correlation function (“scattering law”) :
In thermodynamic equilibrium (detailed balance):
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Useful exact sum-rules:
Total moment sum-rule (general result from Fourier analysis)
Correlations in space & time rearrange intensity in space leaving the total scattering scattering unaffected
Q-w
Definition of the equal time correlation function:
The wave vector dependence of the energy-integrated intensity probe a snap-shot of the fluctuating spin configuration
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Fluctuation Dissipation Theorem
FQ t( ) =
iSQ t( ),S-Qéë ùû
Define the following t-dependent “response” function:
c Qw( ) = - gmB( )2
lime®0+
dte-i w-ie( )t
0
¥
ò FQ t( )
The generalized susceptibility can be expressed as:
The “scattering law” can be expressed as:
From which, the “fluctuation-dissipation” theorem:
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First Moment Sum-rule Fourier transform of expression for scattering law:
Equating these leads to the first moment sum-rule:
Time derivative of left side evaluated for t=0:
Time derivative of right side evaluated at t=0:
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2
1
,2
1,
0totS
1totS
, ,
,
Weakly Interacting spin-1/2 pairs in Cu-nitrate
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Sum rules and the single mode approximation When a coherent mode dominates the spectrum:
Sum-rules link and :
0.4
0.5
0 2 4 0 2 4 6
Exp. data Single mode model
e(Q)
Q p( )
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Spin waves in a ferromagnet
ACNS 6/29/10
Magnon creation Magnon destruction
n w( ) =
1
eb w -1
Dispersion relation
Occupation number
Gadolinium
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Spin waves in an antiferromagnet
ACNS 6/29/10
S^ Q,w( ) =S
2
J 1- 1z eiQ×d
då( )e Q( )
´ d e Q( ) - w( ) n w( ) +1( ) +d e Q( ) + w( )n w( ){ }
Dispersion relation
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J = 4.38 meV ¢J = 0.15 meV D = 0.107 meV
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Probing matter through scattering
fp
ip
Single Particle Process
Multi Particle Process
fi
fi
EE
21
21
ppQQ
1Q
2Q fi
fi
EE
Q
ppQ
Q
Q
ћ
Intensity
Intensity
Q
ћ
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Disintegration of a spin flip in 1D
Spinon Spinon
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From spinon band-structure
to bounded continuum
2
2, QSGQS
Q (p )q (p )
e/J
w
/J
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Quantum critical spin-1/2 chain
Hammar et al. (1999)
Cu(C4H4N2)(NO3)2
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Copper Pyrazine dinitrate
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Summary
• Inelastic Magnetic scattering probes dynamic spin
correlation function:
• Exact results are critical to interpret data
• The full dynamic correlation function as opposed
to just the dispersion relation tells the story
– Intensity gives equal time correlation function
– Energy width in resonant excitations probes life time
– Broad continuum reveals multiparticle excitations
• Know your sum-rules and happy scattering!