RYKOV Alexandre / Hyperfine Interactions / February 12 (2004)

Frequency spectra of quantum beats in nuclear forward scattering

of 57Fe: The Mössbauer spectroscopy with superior energy

resolution

A.I. Rykov∗

School of Engineering, The University of Tokyo,

Hongo 7-3-1, Bunkyo-ku, Hongo 113-8656, Japan∗

I. A. Rykov

The Institute of Mathematics, Siberian Branch of the Russian Academy of Science,

Universitetskii prosp. 4, 630090, Novosibirsk, Russia and

The Faculty of Mathematics, Novosibirsk State University,

Pirogova 1, 630090, Novosibirsk, Russia

K. Nomura

The School of Engineering, The University of Tokyo,

Hongo 7-3-1, Bunkyo-ku, Hongo 113-8656, Japan

X. Zhang

High Energy Accelerator Research Organization,

Institute of Materials Structure Science, Photon Factory,

Oho,Tsukuba-shi, Ibaraki 305-0801, Japan

(Dated: February 13, 2004)

Frequency spectra of quantum beats (QB) in nuclear forward scattering (NFS) are

analysed and compared to Mössbauer spectra. Lineshape, number of lines, sensitivity

to minor sites, and other specific properties of the frequency spectra are discussed.

The most characteristic case of combined magnetic and quadrupole interactions is

considered in detail for 57Fe. Pure magnetic Zeeman splitting corresponds to a

eight-line spectrum of QB, six of which show the same energy separation as the

six lines in Mossbauer spectra. Two other lines (called 2´ and 3´) are the lower-

energy sattelites of the lines 2 and 3. As the quadrupole interaction EQ appears,

the sattelites remain unsplit in the quantum beat frequency spectra, as well as the

2

first (zero-frequency) and the 6th (largest frequency) lines. Each of the lines 3 and

5 generates the doublet split by 2EQ, and the lines 2 and 4 generate the triplets.

In QB frequency spectra (QBFS) of thin absorbers of GdFeO3 we demonstrate the

enhanced spectral resolution compared to Mössbauer spectra. Small particle size

in a antiferromagnet (Fe2O3) was found to affect the QBFS via enhancement of

intensity around zero-frequencies. Asymmetric hyperfine field distribution mixes

up into hybridization with dynamical beats that hardens the low-lying QBFS lines

relative to the highest-frequency line.

PACS numbers: 75.47.Gk , 76.80.+y, 71.38.-k, 63.50.+x

I. INTRODUCTION

The synchrotron radiation (SR) scattered elastically by an ensemble of bound Mössbauer

nuclei exhibits temporal modulations of intensity resulting from interference between differ-

ent hyperfine transitions as well as within a single transition. In the time spectra, collected

typically with a large number of SR pulses, two kinds of intensity beating occur depending

on type of the resonant scatterers. Both of them, the so-called quantum beats (QB) and

dynamic beats (DB), specifically to directional SR scattering and in strong contrast with

isotropic single-nucleus decay, were predicted theoretically 25 years ago [1—3].The QB origi-

nating from quantum splitting of nuclear levels by hyperfine interaction were soon observed

with SR source at HASYLAB [4], shortly after the first observation of the non-exponential

"speeded up" decay in Novosibirsk[5]. The first experiments were performed with using pure

nuclear Bragg diffraction from the 57Fe-enriched single crystals YIG[4], and α−Fe2O3[5, 6],having the electronic structure symmetric enough to make some of electronic reflections

forbidden. Subsequent progress in the monochromatization technique has allowed detecting

the delayed nuclear scattering in forward direction (NFS) that exhibits a combined QB-DB

pattern even in absence of any crystalline structure[7].

Multiple scattering, taken into account in conventional Mössbauer spectra via transmis-

∗Corresponding author, a-rykov@t-adm.t.u-tokyo.ac.jp

3

sion integral[8], underlies the DB, i.e. the beating pattern resulting from single-resonance

(intraresonance) interference. In pure form, the DB are evidenced, e.g., in thick stainless steel

foils[9]. The DB must be explicitly taken into account in thick samples in one of two ways,

either directly in time and space[10], or via Fourrier transformation technique[2, 3, 9, 11].

The period of DB is typically comparable to lifetime ' ~/Γ0 or ' ~/10Γ0, that is quite

similar to the typical period of quadrupole QB, however, unlike the latter the DB period

increases with time. The period of DB shortens with increasing sample thickness. Addi-

tional complexity of time spectra in thick samples, related to DB, arises not only from the

intra-resonance interference, but also from the interference between resonances with small

energy separation. In this case, an asymmetric hyperfine field distribution exemplifies the

situation whenQB and DB blend into a hybrid beat (HB), which is as fast as magnetic QB,

but shows a thickness-dependent period[10]. With reducing the sample thickness the DB is

postponed and even entirely replaced by a quasiexponential "speed-up of decay". In a thin

absorber approximation, the QB alone form the NFS time spectra thus providing in thin

samples a full "time analog" of the conventional Mössbauer spectroscopy[7].

Although both NFS and Mössbauer spectra are described by the same set of parameters

of hyperfine interactions, the understanding of the QB patterns is not straightforward even

in the thin sample approximation. In a general treatment of NFS for arbitrary thickness,

it is most convenient to start building up the theoretical spectra from the energy domain,

because in E-domain the scattering amplitudes An(E) of nuclei in different quantum states

contribute additively into refraction index n(E), while the amplitudes of the incoming and

transmitted waves are related via simple waveform of n(E)[2, 3]. The Fourier transformation

of An(E) from energy into time domain (F AE−→t) allows one obtaining the time spectra of

the radiation field A(t). The latter must be squared to compare with intensity observed in

experiment. Clearly, for the energy of Mössbauer transition E0,e.g. 14.4125 keV in 57Fe, the

phase is predicted to vary with huge frequency ω0 = E0/~. This excessive phase information

is lost exactly with the squaring operation. What remains in the intensity of QB observable

in time spectra is the phase difference. One may see how this experimental information

looks in frequency domain by applying a useful procedure of reverse Fourrier transformation

of intensities (eI) from time to energy domain (F It−→ω). Obtained in this way QB frequency

spectra (QBFS) are easily comparable with transmission Mössbauer spectra (TMS) as well

as with the theoretical QBFS. The latter can be built either in the simplified approximation

4

of thin scatterer or with taking complete account for the multiple scattering events, e.g., via

FAE−→t transformation. The analysis of the comprehensive QBFS instead of raw QB time

spectra is advantageous owing to countable number of spectral features. Limitations of this

approach are related to the finite time window effects. Another drawback of the QBFS,

different from Mössbauer spectra, is that they are not well described in the literature. In

this paper, we apply the transformation F It−→ω for several most typical cases, and describe

the correspondence between QBFS and TMS.

Several universal programs were suggested previously[12, 13] in which the fitting of NFS

spectra was implemented via different algorithms. The claimed advantages of these programs

is in their versatility, allowing to be applied to the arbitrary sample in single-crystalline or

powder form. On the other hand, the methods of external testing of the data quality and

correctness of spectra treatment remains underdeveloped. Generally speaking, the number

of features in NFS spectra is larger than that in corresponding TMS. Here we address the

question of how many frequencies are contained in typical QBFS. One previous work[14]

displays the NFS time spectra of hematite α−Fe2O3, which are entirely different from the

high quality NFS spectra of hematite, as shown below. The authors of Ref. 14 claimed to

succeed in fitting the NFS spectra with the programs created by Sturhahn and Gerdau[12],

and Shvyd’ko[13]. However, although the standard Mössbauer spectrum of hematite was

also shown in Ref.14, the comparison of the NFS spectra with conventionally measured

Mössbauer spectra is unfeasible unless one applies the transformation F It−→ω, that we are

suggesting herewith.

In this work, we obtained the simple theoretical expressions for the most characteristic

hyperfine interactions known for 57Fe Mössbauer spectra. They can be applied for analysis

of the typical NFS spectra with any kind of least-square fitting program. Applying the

transformation F It−→ω to both theoretical result and experimental data allowed us to compare

directly the NFS data with conventional Mössbauer spectra.

We demonstrate the usefulness of analyzing the QBFS, in particular, for the selection of a

theoretical model, that should be choosen prior fitting the NFS time spectra. A few examples

of spectra treatment are organized in ascending complexity: a superposition of quadrupole

and a singlet, two doublets, Zeeman sextets for purely magnetic and combined magnetic-

quadrupole interactions. It will be shown that the NFS spectroscopy at the Photon Factory

(PF) has a potential of having better spectral resolution, compared with TMS or other

5

"laboratory" methods. In the laboratory, the usage of radioactive source approximately

doubles the linewidth, except in measurements with resonant detector[15]. The advantage

of having no Mössbauer source at PF can be used for the most accurate determination of

hyperfine parameters, exactly in cases, when the enhanced resolution is necessary. Attempts

undertaken in two recent works have led the authors [16, 17] to deal with the QBFS spectra.

However, the routine adopted in Ref. 16 is at odds with our analysis, while the other

work[17] has at present considered only the basic two-transition and four-transition spectra

from magnetized samples.

II. EXPERIMENTAL DETAILS

Measurements of NFS spectra were performed in two experimental runs. The 6.5 GeV

storage ring PF-AR was always operating in single bunch timing mode with the period of

1.2 µ s. Such a long time window is suitable to suppress the effects of finite time window in

the QBFS spectra. The current between 60 and 30 mA and the beam lifetime of 1200 min

have allowed to accumulate the high quality spectra within ' 3 hours per sample having

natural abundance of 57Fe (hematite). Similar or longer registration time was needed for the

studies of enriched samples with accelerated decay and with enhanced electronic absorption.

The system of heat-load diamond and high-resolution Si (12 2 2) monochromators provided

the bandwidth of 6 meV. The pulse length was ' 0.2 ns. The NFS spectra in the range

of 500 ns were recorded into 4096 channels. All the measurements were performed at room

temperature.

The samples investigated included the hematite in two different forms, a large-particle

sample and a small-particle sample. The perovskite-related oxides of Sr2FeCoO6−δ, GdFeO3,

Y0.76Ca0.24Ba2Cu2.91Fe0 .09O7, YbBaCuFeO5 and YBaCuFeO5 were synthesized using stan-

dard ceramic technology. The normal spinel Zn57Fe2O4 was prepared from finely dispersed

commercial reagent 57Fe2O3 and Zn(II) acetyl acetonate at 800oC.

Mössbauer spectra were registered with the constant-acceleration spectrometers. The

chemical shifts are given relatively α−Fe.

6

III. RESULTS AND DISCUSSION

In the frequency domain, the amplitudes of the incident and transmitted waves are related

via the response function R(ω) = exp[−in(ω)kl] given by the simple waveform of refraction

index n(ω), initial wavenumber k and sample thickness l (a derivation of this expression was

presented in Ref. [18]):

Et(ω) = Ei(ω)e−in(ω)kl (1)

The scattering amplitudes of nuclei inN different quantum states contribute additively into

refraction index, which is a complex scalar in our case of fully σ−polarized synchrotronradiation[19] :

n(ω) = 1 +ξ

lk

NXn=1

AnΓ

(ωn − ω) + iΓ/2(2)

Here An are the relative scattering amplitudes of different transitions and ξ is the dimen-

sionless sample thickness ( Mössbauer thickness), ξ = 14σ0ρ fLM, expressed via resonance

cross section σ0, density of the resonant nuclei ρ and Lamb-Mössbauer factor fLM. The

temporal dependence of the amplitude of radiation field transmitted in forward direction is

obtained by Fourier transform of the Eq.(1). The solution of this transform was found [2]

to be given by the Bessel function of first kind and order one J1¡2√ξτ

¢with time τ = t/t0

in the natural units of t0 = /Γ0, t0 being the lifetime associated with natural linewidth Γ0.

For an ideal 57Fe absorber the natural linewidth Γ0 of 0.097 mm/s correspond to t0 = 141.1

ns. The simplest derivation of the thickness effects can be done for a single resonance ab-

sorber. If the hyperfine structure is represented by a series of well resolved lines, the same

temporal dependence is kept for the intensity of each hyperfine component[2, 20]. In this

case, the frequency spectra obtained by the Fourier transformation F It−→ω are given by the

sum of the individual components. The shape of each component is Lorentzian in the limit

of thin samples (ξ < 1), when Bessel function is well approximated by a simply exponential

speed-up [18].

7

A. Single and two-resonance spectra. The QBFS lineshape

The response function a thick sample made of a material having a single unsplit nuclear

transition can be written in the simple form[3]:

R(ω) ∝ exp·−iξ Γ0

(ω0− ω) + iΓ0/2

¸(3)

The Fourier transform of R(ω) gives the temporal dependence of the transmitted radiation

field amplitude:

E(t) =

∞Z−∞

dω

2πR(ω)e−iωtdω = e−iω0t−Γ0t/2

∞Z−∞

dz

2πe−iz t exp(−iξΓ0z ) (4)

where the z is replaced for ω−ω0+iΓ0/2. The integration is done by closing the integrationcontour in the complex ω−plane by a semicircle of infinite radius and obtaining the closedcontour C :

E(t) = e−iω0t−Γ0t/2ZC

dz

2πe−izt

·exp(−iξΓ0z )− 1

¸(5)

The power series equivalent to Bessel function J1 can be obtained with using the expansion

of the exponent in square brackets in powers of 1z and integration of each term of the series.

This results in

E(t)∝ ξΓ0 e−iω0t−Γ0t/2

J1¡2√ξτ

¢√ξτ

(6)

Thus, in absence of hyperfine interactions, when the Mössbauer spectra show a single-line

spectra, the NFS time spectra exhibits only the DB, described by the Bessel function of the

first kind and order one[2, 3]:

eI(t) ∝ ξ2 exp(−τ )

ÃJ1

¡2√ξτ

¢√ξτ

!2

(7)

With increasing ξ the shape of the Fourrier component evolves towards the profile having a

sharp apex and shoulders, as shown in Fig.1. In literature, such a profile was not described

yet, so that from now on we specify its shape asΛ−shape. It is well known that the frequencyof DB must increase with thickness[2, 3, 20]. This correspond to a monotonous increase of

the width of the Λ−peak. We show in Fig.1, however, that for large ξ the Λ−profile isinvariant of thickness ξ, so that all the curves

I(ω) = F It−→ω

heI(t)i (8)

8

FIG. 1: Evolution of the shape of the lines in QBFS spectra in function of thickness parame-

ter ξ. Profiles for the same values of ξ are shown with using both the linear (main panel) and

semilogarithmic (inset) scales.

are collapsed into a single master Λ−curve in the coordinates I/ξ vs. ω/2πξ. Thus, the

evolution from Lorentzian to the Λ−shaped profile takes pace in the range of ξ between 0.1and 30. The Lorentzian half-width in units of linear frequency is ≈ 7

2π MHz (= (2πt0)−1)

and the Λ−peak half-width is approximately 1.1ξ/2πt0. Clearly, the thick-sample scalingcan be observed up to infinite thickness[21] only for single unsplit resonance. In case of

doublet, for example, the two resonances start to interact at large enough thickness, so that

for ξ > ξc the frequency doublet becomes more complex function of (ω) than the simple sum

of two profiles of the Eq.(7) type.

The time spectra of the two-resonance hyperfine structure were extensively discussed

previously[14, 20, 22]. Two transition lines become equivalent in a randomly oriented poly-

crystalline material, in which the nuclei experience a pure quadrupole interaction [23], or in

a soft magnetic material fully magnetized in a magnetic field perpendicular to the plane of

storage ring[20]. The NFS intensity is typically written in the form proportional to cos2(

∆ωt/2) with the hyperfine splitting ∆ω(for the pure quadrupole interaction ∆ω = ∆EQ/).

9

For the purpose of present discussion it is relevant to convert cos2( ∆ωt/2) to the double

angle cosine and to use the notation |A(t)|2 for the pure QB factor cos2( ∆ωt/2) separatedfrom the exponential damping and thickness Bessel factors,

eI(t) = Cξ2 exp(−τ )ÃJ1

¡2√ξτ

¢√ξτ

!2

|A(t)|2 (9)

with constant C and QB modulation factor

2 |A(t)|2 = 1 + cos(∆ωt) (10)

The QB frequency spectrum thus consist of two peaks whose thickness-dependent shape

is shown in Fig.1. For the sake of clarity here and below we omit the time-independent

phase shift due to refraction[19, 24]. When the Eqs.(9,10) are used for fitting the time

spectra, this parameter is not independent of an additional (instrumental) variable, locating

the origin of the NFS time pattern. In all cases, this refraction shift was small, in fact,

comparable to the error of determination of the origin of NFS pattern. From the Eqs.(9,10)

it is seen in the limit of noninteracting resonances (∆ω À ξ/t0) that the zero-frequency

peak must have equal amplitude as the peak located at the frequency ∆ω. In practice, by

applying the transformation F It−→ω the two peaks are easily separated in experimental data

(Fig.2), however, we get zero-frequency peak twice higher than the peak at the quadrupole

frequencies of eQVzz(1 +p1 + η2/3)/2h. Clearly, this doubling of the height of the lowest

lying peak takes place at the expense of its halved width. This peculiarity of the QBFS

spectra, becoming apparent in the intensity and width of the lowest-lying line, makes theonly

distinction between the two-resonance QBFS and TMS. When more resonances are excited

in a sample, some extra-frequencies may appear in QBFS compared to TMS, however, the

salient feature of the zero-frequiency peak remains the most universal dissimilarity between

QBFS and Mössbauer spectra.

Shown in Fig.2 are the NFS data obtained from Y0 .76Ca0 .24Ba2Cu2.91Fe0.09O7−δ and

ZnFe2O4 studied previously with conventional Mössbauer spectroscopy [26—28]. The first

compound is the high-temperature superconductor with Tc of 49 K, in which 79 % of Fe

atoms were driven into Cu(2)-planar sites with eQVzz/2 = 0.62 mm/s. Iron was withdrawn

from Cu(1) site using the high-temperature annealing under Ar atmosphere followed by low-

temperature oxygenation. One makes use of the iron migration between Cu(1) and Cu(2)

10

sites to study the Fe dopant pairbreaking effect on Tc[26]. Here we reinvestigated the sam-

ple with greatest pairbreaking population in the Cu(2) site. The minor population of the

dopants (21%) residing in Cu(1)-site was found previously to be composed of two distinct

species having tetrahedral (4%) and pyramidal (17%) coordinations[26, 27]. In Mössbauer

spectra, these two environments were characterized by quadrupole splittings of 1.98 mm/s

and 1.2 mm/s, with isomer shifts of 0.07 mm/s and -0.06 mm/s, respectively[26, 27]. In the

QBFS, the signature of minor sites appears as the small peak around 17 MHz (1.5 mm/s),

however this feature do not correspond to a site quadrupolar spltting but arises from inter-

site interference. Compared to Mössbauer spectra, the contributions from minor doublets

are strongly reduced in QBFS, however, the interference between left-hand lines of minor

doublets and right-hind lines of major doublet is significant, as it will be discussed below

(see Sec.3 C and Fig.4).

In the same way, among the features of QBFS in ZnFe2O4 only the major peak near 4 MHz

correspond to the single-site two-resonance picture with characteristic Mössbauer doublet

splitting eQVzz/2 = 0.34 mm/s. One sees in Fig.2 that the major peak in ZnFe2O4 having

rather small quadrupolar frequency is better resolved than typical resolution in Mössbauer

spectra[28]. Another advantage of QBFS may appear only for specific values of splittings

eQVzz/2, like the ones in ZnFe2O4. This is the enhanced sensitivity of QBFS to minor

spectral components due to intersite interference. In the case of nanostructured powder of

normal spinel ZnFe2O4, some degree of inversion of Zn and Fe population among tetrahedral

and octahedral sites takes place[29]. Concurrently, the nanostructuring leads to increase of

quadrupolar splitting beyond 1 mm/s[28]. The sample studied in present work was not

mechanically nanostructured, however, it was synthesized at low temperature (800oC) using

highly reactive finely disperse powder of 57Fe2O3 and a sol-gel reagent Zn acetylacetonate.

As it will be shown below (Sec.3C and Fig.4), the appearance of QBFS features above the

main peak at 4 MHz can be attributed to the interference between the main well-crystallized

part of sample and the nanostructured component with quadrupole splitting of 1-1.5 mm/s.

B. Three-resonance spectra

In what follows, we discuss only the thin-limit approximation, noting that the thickness

effects in a powder sample change only the shape of the lines in the QB frequency spectra,

11

FIG. 2: The NFS time spectra and their Fourrier transforms in the superconductor

Y0.76Ca0.24Ba2Cu2 .91Fe0.09O7 and in ferrite ZnFe2O4 showing nearly single-site quadrupolar dou-

blets in Mossbauer spectra with the quadrupolar splittings ∆EQ of 0.62 mm/s and 0.34 mm/s,

respectively.

unless there exist the interference between the resonances. In thick samples, we assume that

the hyperfine structure is well resolved, however, in thin-sample limit, this assumption is not

necessary. Let us consider first the three resonances of equal intensity inMössbauer spectra,

that correspond to a doublet and a single line with the site abundance ratio 2:1.

1. Three equipopulated resonances

12

FIG. 3: NFS and Mossbauer spectra of Sr2FeCoO5.5 fitted with the Model 0 . In both spectra,

zero difference between isomer shifts of the doublet and singlet is assumed (∆δ = 0).

A typical spectrum of this kind is known for iron dodecarbonil[25] This case is simple and

instructive as it encompasses all the richness of effects in QB frequency spectra depending on

the relative chemical shifts δ of the components. Three characteristic types of QB frequency

spectra can be obtained.

a. Doublet and singlet with equal chemical shifts The modulation factor of intensity

is generally obtained by squaring the superposition of the amplitude of the radiation fields

summed up over all hyperfine transition:

9 |ASqD(t)|2 =¯ei(ω0+

∆EQ2 +δ)t + ei(ω0−

∆EQ2 +δ)t + ei(ω0+δ)t

¯2= 3 + 4cos

∆EQ

2 t + 2 cos∆EQ

t

(11)

One sees, that the QB spectrum consist of three equidistant lines with the intensity ratio

3:4:2. Their frequency spacing is exactly the same as the spacing between lines in Mössbauer

spectra, having the intensity ratio 1:1:1.

b. Doublet and singlet with different chemical shifts Using the notation ∆δ for the

difference of chemical shifts we obtain immediately

9 |AS/D(t)|2 = 3 +2 cos ∆EQ − 2∆δ

2 t +2 cos∆EQ + 2∆δ

2 t + 2 cos∆EQ

t (12)

The most intense line splits into two lines as ∆δ increases.

13

c. The case ∆δ = ∆EQ/2 One encounters the asymmetric doublet with intensity ratio

2:1 in antiferromagnetic half-metals above Tc, e.g., in Sr2FeMoO6 [30]. Thickness effects are

different for two lines of the resulting asymmetric doublet. In thin sample limit, we find

that Eq.(12) evolves continuously to

9¯ASq12D

(t)¯2= 5+ 4 cos

∆EQ

t (13)

by equating ∆δ =∆EQ/2. While the intensities of two lines in TMS differ twice, the QBFS

show the zero-frequency line exceeding the line at ∆EQ/ by 20% only.

2. Doublet and singlet two-site spectra

When the coefficients α/2 and (1 − α) are introduced ahead of each term in the su-

perposition of radiation fields (see the left part of Eq.(11)), the general expression for the

singlet-doublet two-site spectra takes the form:

|A(t)D /Sα|2 = 1+α(32α−2)+α(1−α)cos ∆EQ − 2∆δ

2 t+α(1−α)cos ∆EQ+ 2∆δ

2 t+1

2α2 cos

∆EQ

t

(14)

In this expression, the total spectral weight of all the QB components is normalized to unity

(cf. normalization factor 9 in Eqs.(11-13)). Using the three-resonance model with ∆δ = 0

(Model 0), the NFS spectra of Sr2FeCoO5 .5 can be fitted with Eq.(14) as shown in Fig.3.

This half-metallic perovskite exhibits interesting magnetoresistive properties and phonon

DOS susceptible to them[31] . Mössbauer spectra were described previously in Refs.32—35.

In these works, the spectra were evaluated with using two-site [32—34] and three-site[35]

models. In the present work, we obtained quite similar spectra, which can be represented,

in a simplest approximation, as a doublet and a single-line with equal chemical shifts of

0.11 mm/s. This value of δ corresponds to the average valence state of Fe intermediate

between Fe3+ and Fe4+. Averaging of the valence states is typically believed to result from

fast electron hopping. The fact that there exist at least two sites, a symmetric one and

a distorted one, is indicative of the polaron formation accompanying the electron hopping.

The doublet with δ0 = 0.11 mm/s, associated with polaron localization in a low-symmetric

site, can be further decomposed into two doublets with slightly different δD1,D2 = δ0± 0.1mm/s, identified with charge-disproportionated states.

14

3. Table 1.

Table 1. Comparison of the parameters of hyperfine interactions derived from fitting

the NFS spectra and conventional transmission Mössbauer spectra (TMS) with different

models. All the values are brought into mm/s scale.

Model 0*

NFS TMSAreaIAreaII

62%38%

57%43%

∆E IQ 0.72(2) 0.702(3)

δ(I) – 0.116(1)

Γ(I) 0.31(5) 0.454(4)

∆E IIQ 0 0.005(4)

δ(II) – 0.112(1)

Γ(II) 0.31(5) 0.514(8)

∆δ 0 0.004(3)

Model 1

NFS TMS56%44%

58%42%

0.77(2) 0.728(5)

– 0.116(1)

0.25(5) 0.45(5)

0.13(2) 0.181(4)

– 0.113(1)

0.25(5) 0.42(1)

0.008(2) 0.003(2)

Model 2

NFS TMS37%63%

60%40%

0.45(2) 0.498(1)

– 0.230(3)

0.21(7) 0.463(5)

0.62(3) 0.486(2)

– -0.041(3)

0.21(7) 0.395(6)

0.25(2) 0.27(1)

Model 3

NFS TMS60%40%

59%41%

0.37(2) 0.307(4)

– 0.318(4)

0.28(5) 0.451(3)

0.26(3) 0.229(5)

– -0.178(7)

0.28(5) 0.412(4)

0.50(2) 0.50(1)*NFS spectra are fitted with using Eq.(7) and TMS are fitted with doublet and singlet.

C. Four-line hyperfine spectra

The case of two doublets in TMS is very frequent, therefore, it is useful to obtain a short

expression for the QBFS. In the oxygen-deficient perovskites, for example, two doublets can

originate from sites of Fe having different coordination numbers or different oxidation states.

With two quadrupole splittings ∆E(I)Q and ∆E

(II)Q and the site abundances α and β = 1−α

we obtain:

|A2D(t)|2 = 12−αβ+α2

2cos

∆E(I )Q

t+β 2

2cos

∆E(II)Q

t+2αβ cos(∆E (I)

Q

2 t)cos(∆E (II )

Q

2 t)cos(∆δ

t)

(15)

The first four terms in Eq.(15) describe the interference between the resonances of a doublet,

and the last term corresponds to the interdoublet interference. In order to obtain the

number of QBFS lines from Eq.(15), the last term must be expanded into four terms with

equal abundances of αβ/2. This spectrum has 7 lines. Four of them, given by the last

15

FIG. 4: Relative intensity of the QBFS lines versus the relative abundance of sites for a two-site

spectrum with a doublet and a singlet (a) and two doublets (b).

term of Eq.(15), describe the intersite interference. Fig. 4 shows the dependence of the

intersite and intrasite terms in function of the relative abundance α for both three- and

four-line hyperfine spectra. Adding to main doublet a second site with small abundance can

manifest itself more dramatically in NFS than in conventional Mössbauer spectra because the

intrasite interference is strongly enhanced in QB patterns. Already at α =15% it reaches

a half of the maximum value (Fig.4). The intrersite interference is strongest in ZnFe2O4

because the quadrupole splitting for minor doublet is the multiple (×3) of that for themajor one and ∆δ = 0. On the other hand, in Y0.76Ca0.24Ba2Cu2 .91Fe0.09O7−δ only half of

the intersite interference constitutes the highest-lying QBFS peak. The other half merges

with zero-frequency peak due to the special relationship between the hyperfine parameters,

as discussed above.

In Fig.5, the NFS spectrum of the perovskite Sr2FeCoO5 .3 is fitted using the Eq.(15).

In the TMS, two doublets are merged into a broad single-band pattern, the multiresonance

character of which becomes clear from the appearance of the QB pattern. One may expect

16

that the ambiguity in choice of a model for fitting complex Mössbauer spectra can be un-

raveled with using the NFS spectra. However, various models (Figs.3,5) are fitted to NFS

spectra with nearly same quality. Choice between the models remains to be intricate matter.

Coincidence between hyperfine parameters derived from QBFS and from TMS could be a

criterion for the choice of the model. There exist three different ways of interconnecting

four Lorentzian lines into a two-doublet pattern. In Table 1, they are enumerated in or-

der of increasing ∆δ (Models 1, 2 and 3). Comparison between parameters resulting from

NFS and TMS shows that the Model 2, widely accepted previously[32—34], seems to be the

worst, as it results in the inverse population, compared to Mössbauer spectra, and gives the

large difference between ∆E(I)Q and ∆E(II)Q , which are nearly equal in Mössbauer spectra.

Even three line-model (Fig.3) gives the closer correspondence between parameters resulting

from NFS and Mössbauer spectra. Clearly, this result can be improved within the five-line

model[35]. It could be suitable to test this multiparametric model when the NFS pattern of

a better statistical quality would be available.

A particular case of four-line spectra is the spectrum of a soft magnetic material fully

magnetized in field parallel to the plane of storage ring[20]. In a internal magnetic hyperfine

field Hhf, the nuclear magnetic coupling constant µ is quantized by the nuclear magneton

times gyromagnetic ratio gµN . Considering four resonances as a pair of doublets, from zero

quadrupole lineshift we get "∆δ = 0" and from 57Fe ground (spin I = 1/2) and excited

(I = 3/2) states splitting scheme[25] we get the magnetic energies ∆EM and frequencies

∆EM /. Two values of ∆EQ in Eq.(15) must be replaced by ∆E1−6M = (g1/2− 3g3/2)µNHhfand ∆E3−4M = (g1/2 + g3/2)µNHhf for the magnetic splittings between the lines (1,6) and

(3,4), respectively. Using the ratio of the intensities of the sextet lines (1,6) to (3,4) 3:1, we

obtain after substituting α = 3/4 and β = 1/4 into Eq.(15):

|A2D q(t)|2 = 5

16+9

32cos

∆E 16M

t+1

32cos

∆E34M t+

3

16cos

∆E 16M +∆E34M2 t+

3

16cos

∆E16M −∆E 34M

2 t

(16)

The last term of Eq.(15) is expanded in Eq.(16) into two "interdoublet" terms which

give the major contribution (∼37%) to the QB pattern. Accidentally, there exist a helpfulrelationship between the g factors of the ground and excited state for 57Fe: (g3/2+g1/2)/g3/2 ≈3/4. Using the values g3/2 = −0.103542 and g1/2 = 0.181208 from Ref. 36, we find that

1 + g1/2/g3/2 deviates from 34 less than 0.012%. Therefore, it is convenient to use here the

17

FIG. 5: The correspondence between three models of fitting NFS and Mössbauer spectra. Both

NFS and TMS least-squares functionals have three minima with parameters (see Table 1) closely

corresponding to each other. A better correspondence is found for the Model 1 and Model 3 than

for Model 2.

following notations 3Ω ≈ (g1/2 + g3/2)µNHhf/ and 4Ω ≈ g3/2µNHhf/ . This defines Ω as

a single parameter (depending on Hhf), describing the QB pattern, e.g., in hematite, the

characteristic frequency Ω≈ 61 [2πMHz] for the hyperfine field Hhf = 515 kOe. With these

notations, Eq.(16) becomes:

|A2Dq(t)|2 = 5

16+9

32cos19Ωt +

1

32cos 3Ωt +

3

16cos 8Ωt +

3

16cos11Ωt (17)

From the Eqs. (15-17) one sees that the QB frequency spectrum of the four resonance

spectrum consist of 7 terms, but only 5 terms remain when ∆δ = 0. It can be shown, in

general, for N resonances, whose positions in Mössbauer spectrum are arbitrary, as in the

18

case described by Eq.(7), that the number of QB frequencies, including the zeroth frequency,

is the "triangular number" 1 + N(N − 1)/2. However, the arbitrary positions are not thecase of combined magnetic and quadrupole interaction, even when the lineshifts are taken

into account to the second order of perturbation theory.

D. Six-line hyperfine spectra

The number of frequencies is essentially reduced for a regular hyperfine structure. In

addition, the computation results can be shortened by employing the relationship between

the g factors of the ground and excited states for 57Fe: (g3/2 + g1/2)/g3/2 ≈ 3/4.

1. Pure magnetic Zeeman splitting

Using the parameter Ω introduced above, one can write the QBFS modulation factor for

the Mössbauer sextet with the thickness ratio 3:2:1:1:2:3 in the form:

72 |A6M (t)|2 = 14+cos3Ωt+16cos4Ωt+4 cos7Ωt+6 cos8Ωt+10 cos11Ωt+12 cos15Ωt+9 cos19Ωt(18)

Thus the QB spectrum for pure magnetic interaction consist of only 8 lines. Six lines in

Eq.(11) at the frequencies 0,4,8,11,15 and 19Ω show exactly the same energy separation as

the six lines in Mössbauer spectra. Two other lines are the lower-energy satellites of the

second and third lines. In what follows we enumerated them as 2´ and 3´. In the upper

panel of Fig.6, the theoretical QBFS pattern is compared with the standard Mössbauer

sextet spectrum. The effects of sample thickness can be easily taken into to the first order

of smallness of the generalized time-thickness variable ξτ. For example, the first-order

thickness-dependent correction ∆1 can be written with the same normalization factor as in

the Eq.(18) :

∆1¡72 |A6M(ξ, t)|2

¢= −4ξ2τ 2e−ξτ

× [11 +10 cos 4Ωt + cos7Ωt +3 cos 8Ωt + 5 cos11Ωt+ 9 cos 15Ωt +9 cos 19Ωt] (19)

Here we take into account the following expansions of the Bessel factors in power of x = ξτ:

J1 (2√x)√

x= exp(−x

2)

·1 − 1

24x2 + O

¡x3

¢¸(20)

19

J1¡2√3x

¢√3x

√x

J1 (2√x)= 1 − x+

1

6x2 + O

¡x3

¢(21)

J1¡2√2x

¢√2x

√x

J1 (2√x)= 1 − 1

2x− 1

144x3 + O

¡x4

¢(22)

In the upper panel of Fig. 6, the pure magnetic Zeeman-split Mössbauer spectra shaped

as Lorentzians of natural linewidth are compared with QBFS resulting from Eq.(18). The

upper row of numbers in the QBFS panel shows the intensities of the QBFS lines, and the

lower row indicates the initial rates ( in units of 4ξ2τ 2e−ξτ) at which the QB decay is speeded

up.

2. Combined magnetic and quadrupole interaction

Shifts in the hyperfine levels by the quadrupole energy[25] EQ = e2qQ/4 split the QBFS

lines into doublets and triplets, except the satellites 2´ and 3 , which remain unsplit, as

well as the first (zero-frequency) and the 6th (largest frequency) lines. The quadrupole

interaction creates doublets from each of the lines 3 and 5, but triplets from the lines 2 and

4. Each triplet is composed of symmetric doublet and unshifted central singlet.

To the first order of smallness of quadrupole interaction this is expressed via the cosine

sum as follows:

72 |A6MQ1(t)|2 = 14 + cos3Ωt + 4 cos 4Ωt + 4cos7Ωt + 4 cos 11Ωt +9 cos 19Ωt+

+6 cos(2ε

t)(2 cos 4Ωt + cos8Ωt +cos 11Ωt + 2 cos15Ωt) (23)

Here ε is the first-order quadrupole lineshift determined by the polar and axial coordinates

(Θ, φ) of the direction of hyperfine field in the reference of the principal axis (X,Y, Z) of

the electric field gradient (EFG), having in its principal axis the main component VZZ and

asymmetry parameter η = (VXX − VY Y )/VZZ :

ε= (EQ/2)(3 cos2 Θ − 1+ η sin2Θ cos(2φ)) (24)

The two doublets and two triplets are described by the last term of the Eq. (23). Unshifted

components of the triplet lines 2 and 4 are given by the third and fifth terms, respectively.

20

FIG. 6: Comparison of the conventionalMossbauer spectra withQBFS spectra, obtained by Fourier

transform of the NFS patterns in case of pure Zeeman magnetic interaction (upper panel) and in

case of combined magnetic and quadrupole hyperfine interactions. According to Eqs. (18) and

(23), the relative intensities in Mössbauer spectra and in QBFS are compared with using the

×6 scale (cf. 6:12:18 instead of usual 1:2:3). This factor (×6) brings into the integer scale theQBFS line intensities in both cases of the purely magnetic 8-line spectra (14:1:16:4:6:10:12:9) and

combined magnetic-quadrupole spectra (14:7:4:6:7:3:3:4:3:6:6:9). For the sake of clarity a constraint

∆EQ = 2Ω was employed in the lower panel. The upper row of numbers in each of QBFS panels

corresponds to the intensities of the QBFS lines, and the lower row indicate the initial rates (in

units of ξ2τ2e−ξτ) at which the QB decay is speeded up(see Eq.(19).

21

The effects of sample thickness can be most easily taken into account with using the

correction similar to Eq. (19). In place Eqs.(20-22), one may employ the following power

series expansions:

Ψ = exp(−ξτ/2) (25)

J1¡2√2x

¢√2x

= Ψ2·1 − 1

6x2 + O

¡x3

¢¸(26)

J1¡2√3x

¢√3x

= Ψ3·1 − 3

8x2 + O

¡x3

¢¸(27)

Then the result analogous to that given by Eqs.(18) and (19) is

I(t) = e−τΨ2[1 + 4Ψ2 + 9Ψ4 + cos 3Ωt +4Ψ cos4Ωt + 4Ψ cos 7Ωt+

+4Ψ2 cos 11Ωt +9Ψ4 cos 19Ωt+ (28)

+6 cos(2ε

t)(2Ψ3 cos 4Ωt + Ψ2 cos8Ωt + Ψ2 cos 11Ωt +2Ψ3 cos 15Ωt)].

The Eq.(28) was applied for fitting the NFS spectra of hematite α−Fe2O3 (Fig.7). Therefined values of Hhf and ε are shown in Fig.10.

A surprising result appears from comparison between the NFS patterns and Mössbauer

spectra of rather thin hematite sample (ξ ≈ 3) with natural abundance of 57Fe and a thickersample of the fully enriched (∼90%) hematite sample (Fig.7 and 8). It appears that thedecay in thin sample is faster than in the thick one. Another difference, however, existed

between these samples that was related to the particle size. The enriched α−57Fe2O3 wasthe commercial reagent that contained small particles to ensure their reactivity at synthesis.

For large ξ , the lineshape in TMS must follow the transmission integral form[8], unless some

additional factors give rise to evenmore complicated profile. Thickness inhomogeneity is the

most evident among such factors. However, in our enriched α−57Fe2O3 the unusual resultis also related to small particle size. This is evident from the shape of Mössbauer spectra

in Fig.8(c), which is very specific of the nanostructured hematite. While the spectra from

thin absorbers (ξ ∼ 1) coincide with standard sextets, known for hematite, the thick samplesalways produce the complex 6-line spectra, containing a broad "valley" (V-component), from

22

FIG. 7: The NFS spectrum of naturally abundant 57Fe in powder hematite in linear (a) and

logarithmic (b) scales and its Fourier tranform resulting in quantum beat frequency spectra (d)

compared to the conventional Mossbauer spectrum (c).

-20 to 20 mm/s, as shown in Fig. 8 (c). In contrast, the thick absorbers of large-particle

hematite do not exibit the V-component in TMS. This feature is likely to be explained

by wings of the hyperfine field distribution, having smaller effective ξ compared to the

distribution center. Attempts to fit such spectrawith single-component transmission integral

implemented in MOSSWINN program[37] have failed. Clearly, when two transmission-

integral components are introduced, the spectra can be fitted with approximately equal

intensities of the components(Fig.8 (c)), but the component abundance do not match the

distribution of hyperfine parameters. Samples of enriched α−57Fe2O3 annealed to grow up

23

FIG. 8: The NFS spectrum in thick sample of fully enriched in 57Fe hematite with finely dispersed

particles in linear (a) and logarithmic (b) scales and its Fourier tranform resulting in QBFS spec-

trum (d), as compared to the conventional Mössbauer spectrum (c). The arrow belongs to the

lower panel (d) and indicates the top of the zero-frequency QBFS peak.

the particles were also measured, and the V-shaped contributions were found to disappear

from the spectra of thick samples. TMS obtained from these annealed samples were fitted

much better with using the single-component transmission integral option[37].

Comparing the time spectra in Figs. 6 (a,b) and 7 (a,b) we observe a similarity between

these NFS patterns, however, in the enriched α−57Fe2O3 (Fig.8), the QB pattern is smearedcompared to the theoretical curve. Fourrier transformation reveals the origin of this smear-

ing. Comparison of the Fourrier spectra in Figs. 6 (d) and 7(d) shows that all the features,

except the zero-frequency line, coincide. Note, in the experimental QBFS, obtained via fast

24

Fourrier transform, the zero-frequency peak is twice more intense compared to the relative

spectral weight given by the coefficients in Eqs.(18) , (23) or (28). This enhancement results

from absence of negative frequencies. Therefore, in the datapoints of the first peak at ω > 0

(ω ≈ 0) the intensity is doubled. However, additionally, in the fully enriched α−57Fe2O3,the zero-field peak is enhanced by 66%, equivalently to the 33%-enhancement of the cor-

responding term in Eqs. (18) , (23) or (28). In these equations, the relative weight of the

zero-frequency peak is 14/72. Therefore, the contribution of this extra-enhancement into the

total spectrum amounts to 6.4%. Because the other spectral features in Fig. 8(d) are not

affected, the probability of incoherent scattering into the total NFS countrate is estimated

to be ∼6% . It is quite likely that this smoothly and slowly decaying scattering is related

to small-angle nuclear scattering [38], because the latter depends on the particle size. On

the other hand, since the broad V-component feature exists in the transmission spectra of

thick samples, one may also ascribe such incoherent (or partially coherent) scattering to the

quasielastic scattering. The slowly decaying component in the time spectrum (Fig. 8) must

correspond to a narrow peak in the QBFS. Indeed, the zero-frequency peak half-width is 3

MHz in thin natural sample and 1.7 MHz in thick enriched sample. However, noting that

zero-frequency peak has two different contributions, it is possible to represent this peak as

a superposition of a 66% broader (>3 MHz) and a 33% narrower peaks. More experiments

are needed to clarify the origin of the narrower peak. It appears that the width of this peak

approaches to the natural linewidth (cf. (2πt0)−1 ≈ 72πMHz pertinent to free nuclei decay.

The fact that the free-decay QBFS linewidth (1.128MHz corresponding to 0.097 mm/s for57Fe) is only half of the minimum linewidth in conventional Mössbauer spectroscopy (0.194

mm/s) makes it possible to obtain, in the limit of thin samples, the improved spectral

resolution. From the thin-limit linewidths (see Fig.1, and Eq.(28))

Γξ =1+ ξ

2πt0(29)

it follows that the QBFS spectral resolution is higher than the best possible resolution

in conventional Mössbauer spectroscopy when ξ < 1. In reality, because of instrumental

factors, the linewidth of 0.194 mm/s is hardly achievable even in a perfect sample. We show

in Fig.9(c,d) the narrowest QBFS lines achieved in a thin sample of GdFeO3, the prototype

of orthoperovskite series. The GdFeO3 sample showed a better resolution in QBFS than in

the Mössbauer spectra. To our knowledge, such a possibility is demonstrated here for the

25

FIG. 9: The NFS spectrum in thin sample of GdFeO3 in linear (a) and logarithmic (b) scales and

its Fourier tranform resulting in QBFS spectrum (d), as compared to the conventional Mossbauer

spectrum (c). The arrow belongs to the lower panel (d) and indicates the top of the zero-frequency

QBFS peak.

first time.

The orthoperovskites of rare-earth elements all are known to exhibit a particularly small

quadrupole splitting[39], such that the typical measurement error is similar to value of

ε(Fig.10). The smallness of ε reflects the fact that the Fe-O and O-O distances are the same

in all orthoferrites, and only Fe-O-Fe angle ϑ is changed in the series[40]. In Fig. 10, we

used this variable to plot the trends in ε and Hhf along the orthoferrite series. Our results

are compared with the data of Refs.39, 41. Since the antiferromagnetic Mott-Hubbard state

26

FIG. 10: Dependence of the hyperfine parameters ε and Hhf on the Fe-O-Fe angle ϑ in the series of

rare-earth ortoferrites and α−Fe2O3. The accuracy in determination of fitting parameters is betterby an order of magnitude in QBFS (large symbols) than in the data taken from the Refs.39, 41

(small symbols). Line is drawn to guide the eye.

breaks down under pressure in both RFeO3[42] and in Fe2O3[43], the latter was considered

as an extreme member of the perovskite series[44]. The correlation between ε and ϑ plotted

in Fig.10 may support this viewpoint, although the high-pressure phase of Fe2O3 differs from

perovskite[43]. The higher accuracy in ε and Hhf derived from NFS spectra is particularly

suitable for measuring small values and small variation of parameters,e.g, in the rare-earth

series.

The experimental data points shown in Figs.7-9 follow closely the theoretical lines drawn

according to obtained analytical expression (Eq.28) both in time and frequency domains.

The comparison between GdFeO3 and α-Fe2O3 clarifies the role of quadrupolar lineshift in

the experimental QBFS. Very small ε in GdFeO3 leads to the eight-line spectrumof quantum

beats, six of which show the same energy separation as the six lines in TMS (cf. theoretically

obtained Fig.6 and Fig.9). As the quadrupole interaction appears α-Fe2O3, the satellites,

27

called 2´ and 3´, remain unsplit in the quantum beat frequency spectra, as well as the first

(zero-frequency) and the 6th (largest frequency) lines. The quadrupole interaction create

doublets split by ∆EQ from each of the lines 3 and 5, but triplets from the lines 2 and 4,

both of which retain a unshifted central component. In Figs. 7 and 8, the doublet lines are

clearly split, while the triplet lines are discernible by the increased linewidths. These spectra

can be easily sharpened further artificially if an exponential factor for slowing the decay is

applied to raw time spectra, before Fourrier transformation, however, this procedure may

bring into the time spectra a false cutoff and into related QBFS some extra oscillations from

thereby "virtually shrank" time window.

3. Thickness effect in QBFS spectra of polycrystalline antiferromagnets

The dependence of the NFS time spectra on sample thickness, predicted 25 years ago

[2, 3] was confirmed with extraordinary accuracy on the samples of ferromagnetic Fe and

its alloys[20]. In the present work, our purpose was to search for the similar thickness

dependence in powder antiferromagnets and eventually to tackle the question: why the DB

were never observed in polycrystalline antiferromagnets till now? Most of oxides belong to

the class of materials which show only minor changes in spectra under moderate external

fields. Very complicated QBFS spectra results from samples withmultiple magnetic sites[45].

The single-Fe-site perovskite GdFeO3 is one of the antiferromagnetic material (although

showing a weak ferromagnetism originating from antisymmetric exchange[39]), in which the

effect of thickness in QBFS can be isolated in pure form due to the very small quadrupolar

splitting.

The NFS time spectra and QBFS from three samples of GdFeO3 are shown in Fig.11.

The thickest sample contained 11 mg/cm3 of 57Fe and the thinner samples were obtained by

sequentially dividing the thicker sample on 2 and 4 parts. All the samples were thoroughly

mixed with the stuff material (MgO), to ensure that the absorbers are geometrically thick.

The minima of Bessel function (dynamic beats) were expected to appear in the measured

time range, however, were not observed (Fig.11). It is likely that the thickness homogeneity

could not be achieved in these samples because of unsatisfactory ratio between the orthofer-

rite particle size and required geometrical thickness of the powder absorber. One plausible

way to observe the dynamic effects in the powdered sample may involve the preparation of

28

FIG. 11: The NFS spectra (left ) and their Fourrier transforms (right) for three samples of GdFeO3

with different sample thickness increasing from top to bottom. Since the dynamical beats are not

observed in the left panel, the first order (in ξ) approximation (Eq. 28)was employed to fit the data

in all samples. The factor Ψ = exp(−ξτ/2) describes the thickness dependence of the linewidthin the 8-line spectrum (right panel). Neglecting the very small quadrupolar lineshift, six of the

lines shows the same energy separation as the six lines in Mossbauer spectra. We enumerate them

correspondingly. The lower-energy sattelites of the lines 2 and 3 show the slowest broadening

with increasing thickness, described by the factors Ψ2 and Ψ3, respectively. The lines 2 and 4

are quadrupole-split into three sublines each, and the lines 3 and 5 are quadrupole-split into two

sublines each. Inhomogeneous broadening takes place for the first (zero-frequency) and second

lines. Note, the shape of the (3,3´)-line changes with ξ in agreement with faster broadening of the

line 3 compared to the line 3 .

29

nanoparticles. In the nanostructured materials, however, not only the hyperfine structure

is modified, but also some new effects could be expected related to the particle size, as we

observed in the fully enriched sample of α-Fe2O3 (Fig.9).

4. Distribution of hyperfine parameters

The effects of asymmetric distribution of magnetic hyperfine field were previously in-

vestigated in the foils of soft magnetic material (Invar) that can be easily magnetized by

applying a moderate external field (~0.1 T). In this case, it was shown that the thickness

effects interfere with hyperfine splittings to produce the so-called hybrid beats in the NFS

time pattern. The HB are as fast as magnetic QB, but shows the thickness-dependent

period[10]. In this work, we report on the similar effects in the materials with antiferromag-

netic exchange RBaFeCuO5 (R=Y,Yb), which were known to exhibit the asymmetric field

distribution[46—48]. In contrast to the previous work [10], our samples are not ferromagnetic

and not easily magnetizable. Therefore, when one consider six transitions that appear in

TMS (Fig. 12 a,b), the problem becomes intractable in the way as it was considered in time

domain by Shvyd’ko et al.[10]. However, in our QBFS spectra, the effects of hybridization

between QB and DB are immediately seen in the shifts of the QBFS lines from their ideal

thin-limit positions (Fig.12 e,f). The NFS time spectra (Fig.12 (d)) were fitted with four

field components, assuming ε = 0, for each. This fitting procedure has to be performed nec-

essarily before the Fourrier transformation in order to reconstruct the missing experimental

data at short times (<20 ns). The data below 25 ns affected by the intense radiation pulse

and prompt scattering were replaced with the theoretical data, as obtained from fitting. It

can be seen in Fig. 12 (d) that these short-time data deviate from the theoretical curve.

Divergences consist merely of suppression of the intensity, but the shapes of theoretical and

experimental curves are the same. Therefore, the errors in QBFS, related to the finite time-

window effects, must vary only slowly with frequency. They cannot affect the QBFS peak

positions.

Since the magnetic field has a broad distribution, the meaning of the magnetic hyperfine

parameter Ω becomes rather conventional. It is convenient to define Ω from the position

of the highest QBFS peak located at 19Ω (cf. Fig.6). With this definition, the arrows

in Fig. 12 (e,f ) show that the other peak positions in both samples shift towards higher

30

FIG. 12: Mössbauer spectra (a,b), hyperfine field distribution (c), NFS time spectra (d) and QB

frequency spectra (e,f) in RBaCuFeO5 (R=Y,Yb). The arrows in (d) and (e) indicate the thin-

sample positions of the QBFS lines.

frequencies. Indeed, according to the definition of Ω and Fig.6., a thin sample must show

the QBFS peaks centered at frequencies multiple of Ω. However, we observe that the smaller

is the QB frequency, the stronger is its shift compared to a thin sample or to a thick sample

with symmetric field distribution. Using the developed approach in the frequency space we

thereby confirm the effect of hybridizing between QB and DB [10] for a more general case

of Zeeman sextet in a unmagnetized sample.

31

IV. CONCLUDING REMARKS

In this paper, we have shown that the quantum beat frequency spectra are worthy of

calculation and careful analysis. Since the very broad time window is accessible in ex-

periments, the QBFS contain practically the same information as is covered in the initial

time spectra, but uncover a countable number of features (lines). It is fair to compare the

frequency splittings and line intensities with the conventional Mössbauer spectra. Such a

comparison is crucial at the step of evaluation of the quality of NFS data. Prior a final

theoretical model for fitting the NFS time dependence can be chosen, the QBFS are useful

for the model selection. Rough QBFS spectra can be obtained with minimum knowledge

about intensity behavior at short times. The possible minor errors in the rough QBFS

(caused by prompt intensity suppression) are easily removable after the model is refined.

Above we considered the most popular hyperfine structures: a superposition of quadrupole

doublets, a doublet and a singlet, and Zeeman sextets for purely magnetic and combined

magnetic-quadrupole interactions. Compared to conventional methods of transmission or

emission Mössbauer spectroscopy the NFS specroscopy exhibits a potential of having better

spectral resolution. This is because the spectra are obtained without a standard Mössbauer

source, that typically doubles the linewidth, except in measurements with resonant detector.

This provides us with an obvious advantage for accurate determination of hyperfine para-

meters, especially when the parameter variation is small, e.g., in the considered rare-earth

orthoferrite series. The realization of the enhanced resolution method was achieved due to

two necessary conditions: (i) long revolution period (1.2µs) in the single-bunch mode; (ii)

preparation of a thin enough sample. From the analytic aspect, we obtained the simple

algebraic expressions for the most characteristic hyperfine interactions. They can be applied

for analysis of the typical NFS spectra with any kind of least-square fitting program. The

most customary combination magnetic and quadrupole interactions is considered in detail

for 57Fe. Pure magnetic Zeeman splitting correspond to the eight-line spectrum of quantum

beats, six of which shows the same energy separation as the six lines in Mössbauer spectra.

Two other lines (called 2´ and 3´) are the lower-energy satellites of the lines 2 and 3. As

the quadrupole interaction EQ appears, the satellites remain unsplit in the quantum beat

frequency spectra, as well as the first (zero-frequency) and the 6th (largest frequency) lines.

Each of the lines 3 and 5 generates the doublet split by 2EQ, and the lines 2 and 4 generate

32

the triplets. In thin absorber of GdFeO3 we demonstrated the enhanced spectral resolu-

tion compared to Mössbauer spectra. Small particle size in an antiferromagnet (Fe2O3) was

found to affect the QBFS via enhancement of intensity around zero-frequencies. Asymmetric

hyperfine field distribution hybridizes with dynamic beats; this effect hardens the low-lying

QBFS lines relative to the highest-frequency line. This observation confirmed similar effects

previously observed immediately in time domain for a less regular (magnetized) sample. In

addition, as the general features of this exotic kind of Fourrier spectroscopy, we introduced

the Λ−form for the QBFS lineshape, and discussed the sensitivity of QBFS to minor sites.

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(1978).

[2] Yu. Kagan, A.M. Afanas’ev, and V.G. Kohn, Physics Letters 68A, 339-341 (1978).

[3] Yu. Kagan, A.M. Afanas’ev, and V.G. Kohn, J. Phys. C12, 615-631 (1979).

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