57fe: the mössbauer spectroscopy with superior energy...
TRANSCRIPT
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, [email protected]
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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