Vij D.R. Handbook of Applied Solid State Spectroscopy

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(5.8)

by the fact that the quantization z-axes of the magnetic dipole and electric

quadrupole Hamiltonians often do not coincide. Thus, one must transform one

of the axial frames onto the other in order to obtain the eigenvalues and

eigenvectors of the combined nuclear Hamiltonian, and therefore the

transition energies and relative line intensities. Obviously, the eigenvalues

cannot depend on the choice of reference frame. This problem was treated in

detail in a classic paper by Kündig [5]. The reader is also referred to papers by

Häggström [6] and Szymanski [7] for detailed treatments of the

Fe hyperfine

Hamiltonian problem.

In a coordinate system coincident with the principal axis frame of the EFG,

the hyperfine Hamiltonian may be written



222

ˆˆˆ

cos cos sin sin

ˆˆ ˆ ˆ

4I 2I 1

QnNhfZ X Y

ZXY

HH H g B III

II I I

ªº

 P T I IT

¬¼

ªº

K

¬¼



where TI are the polar angles of the hyperfine magnetic field in the EFG

principal axis frame. Alternatively, one could rotate the EFG frame onto the

magnetic frame. In the simple case where the nuclear hyperfine Hamiltonian

is diagonal and

Șș0 the nuclear energy levels are given by



m I

 31

4I 2I 1

nNh

EmgB mII

ªº

   

¬¼



In the simple situation where this EFG can be treated as a perturbation on

the magnetic effect, the eigenvalues are given by







QV 1

µ3II13cos1.

4I 2I 1 2

EmgB m

ªº

    T

¬¼



(5.9)

The usefulness of this approach is that the angular term in the quadrupole

contribution allows one to investigate spin-reorientations, i.e., temperature-

driven changes in the atomic magnetization direction. As an example, one can

easily see that if the principal axis of the EFG is along the crystal c-axis, a

simple change from an easy magnetization axis along the crystal c-axis to one

in the basal ab-plane leads to a relative change in effective quadrupole

splitting of –1/2, which is fairly straightforward to observe in a Mössbauer

experiment. In the presence of an asymmetry parameter, the preceding

eigenvalue expression becomes







3II1

4I 2I 1

3cos 1 sin cos2 .

EmgB m

ªº

 P   

¬¼



uTKTI

The hyperfine magnetic field contains numerous contributions, including

the Fermi contact term due to the s-electron overlap with the nucleus,

conduction electron polarization (in metals), dipole and transferred fields

5.1 Introduction 207

(5.10)

(5.11)

5. Mössbauer Spectroscopy

from the surrounding lattice, and externally applied fields. In

Fe the

hyperfine field in D-Fe is 33.0 T at room temperature and D-Fe foils are the

standard method for calibrating the Doppler energy-velocity scale (see

Sections 5.2 and 5.3). Mössbauer experiments are commonly used to

“determine” the atomic magnetic moment from the nuclear hyperfine

magnetic field, which raises the problem of the conversion factor between the

two. In

Fe, a commonly accepted conversion factor is 15 T/P

where P

the Bohr magneton. However, large variations in the conversion factor can

occur between different compounds and even between different crystal sites

in the same compound.

In Figure 5.3 we show the 3/2 ļ 1/2 energy level splitting in the presence

of a magnetic hyperfine field along with the effect of an additional quadrupole

splitting.

Figure 5.3 Nuclear energy level splittings for a 3/2 ļ 1/2 Mössbauer transition (appropriate to

Fe,

119

Sn, or

169

Tm) in the presence of magnetic and electric quadrupole interactions.

To illustrate the effect of a magnetic hyperfine field, in Figure 5.4 we show

the

Fe RT spectrum of an D-Fe foil, used to calibrate the Doppler velocity

scale in a Mössbauer experiment. The high crystal symmetry of the single Fe

crystallographic site in body-centered cubic D-Fe results in a zero EFG and

the spectrum is a symmetric magnetic sextet. The velocity splitting between

the outside peaks is 10.6246 mm/s, which corresponds to a magnetic

hyperfine field of 33.0 T.

In Figure 5.5 we show the corresponding energy level splittings for 2 ļ 0

transitions such as

166

Er and

170

Yb, and in Figure 5.6 we illustrate this with the

166

Er spectrum (4.5 K) of Er

, which shows a magnetic splitting

corresponding to a hyperfine field of 663.7(5) T [8]. For such a transition, the

relative intensities of the five lines are 1:1:1:1:1.

208

Figure 5.4

Fe RT spectrum of a 25 Pm D-Fe foil [Cadogan, unpublished].

Figure 5.5 Nuclear energy level splittings for a 2 ļ 0 Mössbauer transition (appropriate to

166

Er and

170

Yb) in the presence of magnetic and electric quadrupole interactions.

Figure 5.6

166

Er spectrum of Er

obtained at 4.5 K [8].

5.1 Introduction 209

5. Mössbauer Spectroscopy

5.1.4.4 Electronic Relaxation

The hyperfine magnetic field seen at a nucleus is predominantly determined

by the electronic configuration of the parent atom, i.e., the unfilled 3d or 4f

shells for the transition metals or “rare earths,” respectively. For example, to a

first approximation, the hyperfine field acting on a rare earth nucleus is

proportional to the 4f electronic ensemble thermal average

J whereas the

nuclear quadrupole splitting is proportional to





31JJJ. It is implicit

here that the 4f electronic configuration is static within the Mössbauer

measuring time frame. However, one often observes the effects of “electronic

relaxation” in which the nucleus senses the time-dependent fluctuations of the

surrounding electronic configuration and the resulting Mössbauer spectrum

reflects this behavior. There are two important timescales involved here: the

lifetime of the nuclear excited state and the Larmor precession time.

Basically, if the nucleus is to “see” a well-resolved magnetic splitting,

unaffected by relaxation broadening, there must be a sufficient sensing time,

i.e., at least one complete Larmor precession must occur before the nuclear

state will decay. In the case of very fast electronic relaxation, the electronic

configuration changes at a rate significantly faster than that corresponding to

the nuclear Larmor precession rate. Hence, the nucleus senses a time-

averaged hyperfine environment, manifest as the collapse of the well-defined

magnetic sextet. Finally, we have a restriction on the usefulness of Mössbauer

spectroscopy to study time-dependent effects in that if the characteristic time

associated with the phenomenon under study is extremely slow and is greater

than the nuclear excited state lifetime (and hence the Larmor precession time)

then the process cannot be studied effectively by Mössbauer spectroscopy.

Time-dependent effects are observed in (i) isomer shift, (ii) magnetic field,

and (iii) EFG effects, but are most often observed via the magnetic field. A

number of approaches have been used to give a good account of the resulting

broadening and/or collapse of the magnetic spectrum. In particular we

mention the classic works of Blume and Tjon [9] and Wickman and Wertheim

[10], both of which basically treat the relaxation within a two-state electronic

model and then investigate the ensuing nuclear consequences on the

Mössbauer spectrum. In Figure 5.7 we show simulated

Fe spectra illus-

trating the collapse of the magnetic sextet with increasing electronic

relaxation rate.

There are, of course, other time-dependent effects which are amenable to

study by Mössbauer spectroscopy. For example, lattice dynamics in which the

Mössbauer “atom” diffuses, i.e., moves, during the Mössbauer process have

been studied, once again given the aforementioned time considerations.

210

5.1.4.5 Crystal-Field Effects

As mentioned above, it is a good approximation in rare earth magnetic

materials that proportionality exists between the magnetic hyperfine field at

the rare earth nucleus and

J , and the 4f contribution to the EFG is

proportional to





31JJJ. The temperature dependences of these 4f

electronic averages are determined by the interplay of magnetic exchange and

electrostatic crystal field effects, and Mössbauer spectroscopy has played an

important role over the years in the study of crystal fields acting of the 4f shell

[11]. The 4f elements are quite similar chemically but exhibit a wealth of

varied magnetic behaviour due to the progressive filling of the 4f shell. As we

shall demonstrate later in this article, the combination of rare earth isotope

Mössbauer spectroscopy and neutron diffraction has proved itself to be a

formidable means of determining the magnetic behavior of intermetallic

compounds.

5.1.4.6 Amorphous Materials

The case of disordered or amorphous materials is complicated by the fact that

the probe atoms are subject to a distribution of environments, which leads to

distributions of hyperfine parameters such as hyperfine field and electric field

gradient. In such cases, Mössbauer spectroscopy is predominantly used to

determine parameters such as the average hyperfine field (and hence the

average atomic magnetic moment) or average quadrupole splitting in the case

of a paramagnetic material, both of which can be reliably determined despite

the presence of a hyperfine parameter distribution. It is known that the form

of the hyperfine parameter distribution is strongly dependent on the method

used to calculate it. However, the average parameter values deduced are quite

5.1 Introduction

Figure 5.7 Simulated

Fe spectra as a function of increasing electronic relaxation rate (Ryan

and van Lierop, unpublished).

211

5. Mössbauer Spectroscopy

robust. The calculation proceeds as follows: One selects a histogram [12] or

Fourier series [13] approach to the form of the hyperfine parameter

distribution, field, or EFG; simulates appropriate doublets or sextets with a

common linewidth; and finally builds up the form of the distribution by

varying the relative intensities of the individual components to obtain the best

fit to the experimental spectrum. Asymmetries in the spectra are accounted for

by introducing correlations between parameters such as isomer shift and

hyperfine field in a somewhat ad hoc manner. Deviations from simple

Lorentzian lineshapes can also arise, particularly for low hyperfine field

values where the commonly employed field-EFG perturbation approach is

questionable. One must also be aware of the significant effects of histogram

size, i.e., how many steps there are in the histogram. This can lead to severe

overlap effects between “adjacent” hyperfine field sextets.

At low field values, the electric quadrupole interaction can no longer be

treated as a perturbation on the magnetic hyperfine field. This can give rise to

a peak in the lower part of the hyperfine field distribution and some authors

have considered the resulting bimodality of the distribution as a signal of two

distinct environments in amorphous materials. While this might be true, one

must always bear in mind the somewhat limited validity of the approach

employed. As shown by Keller [14], for example, one can simulate a

spectrum with a single Gaussian distribution of hyperfine field and

subsequently obtain a bimodal distribution using the commonly employed

fitting methods. Le Caër and coworkers [15] published a validity diagram that

discusses this important aspect of Mössbauer data analysis in amorphous

materials.

Mössbauer spectroscopy has also been used to great advantage to

determine the relative volume fractions of a number of commercially

important soft-magnetic materials such as Vitroperm and Finemet, where one

embedded in an Fe-rich amorphous matrix [16]. Such information is vital to

the development of a model for the random anisotropy observed in these

commercially viable soft-magnetic materials. We shall illustrate this work in

Section 5.3.

In the following Table 5.1 we present a compendium of information

relevant to the more commonly used Mössbauer transitions. This information

is taken from publications by the Mössbauer Effect Data Centre [17]. At the

end of this article we list a number of books and reviews that deal with

Mössbauer spectroscopy and hyperfine interactions in more detail than is

possible here [18–26].

5.2 METHODOLOGY

To obtain a Mössbauer spectrum one needs a suitable source of J-photons, a

detector capable of isolating the desired Js from other radiations emitted by

212

has an intimate mixture of nanocrystalline materials (usually bcc-D-Fe)

5.2 Methodology

the source and, most critically, a means to modulate the energy of the

Mössbauer J-photons. In Figure 5.8 we show a schematic of a standard

Mössbauer spectrometer and in the following sections we describe the various

aspects of spectral acquisition.

Table 5.1. Mössbauer parameters for a number of important Mössbauer transitions [17].

119

151

155

166

169

170

197

Isotopic

Abundance (%)

2.14 8.58 47.82 14.73 33.41 100 3.03 100

(keV)

14.41 23.87 21.53 86.55 80.56 8.40 84.25 77.345

3/2 3/2 7/2 5/2 2 3/2 2 1/2

1/2 1/2 5/2 3/2 0 1/2 0 3/2

Excited state half-

life (ns)

97.8 17.75 9.7 6.33 1.87 4.00 1.608 1.88

* (mm/s) FWHM

0.194 0.647 1.31 0.499 1.82 8.1 2.02 1.882

Excited state

magnetic moment

)

– 0.1553 + 0.633 +2.587 –0.529 +0.629 +0.534 +0.669 +0.416

Excited state

gyromagnetic ratio

(mm/s /T)

–0.067897 0.167 0.3244 –0.0231 0.0369 0.400 0.0375 +0.1017

Ground state

magnetic moment

)

+0.090604 –1.0461 +3.465 –0.2584 0 –0.231 0 +0.1448

Ground state

gyromagnetic ratio

(mm/s /T)

–0.118821 –0.8283 +0.6083 –0.02713 0 –0.520 0 +0.0118

Excited state

electric quadrupole

moment (barns)

0.21 – 0.06 1.50 0.32 –1.59 –1.20 –2.11 0

Ground state

electric quadrupole

moment (barns)

0 0 1.14 1.59 0 0 0 0.594

Energy equivalent

of 1 mm/s in joules

(u 10

–27

)

7.70278 12.757 11.507 46.253 43.052 4.490 45.0275 41.336

Energy equivalent

of 1 mm/s in MHz

11.62477 19.253 17.367 69.803 64.973 6.776 67.954 62.382

Source half-life

271 d 293 d 87 y 4.8 y 26 h 9.4 d 130 d 19 h

Figure 5.8 Typical layout of a transmission Mössbauer spectrometer (Ryan, unpublished).

213

5. Mössbauer Spectroscopy

5.2.1 Drives

The velocity range demanded by most commonly used Mössbauer transitions

lies below r50 mm/s and is easily achieved mechanically. Therefore, most

modern Mössbauer spectrometers employ a commercial electromechanical

drive. Such a drive is basically a pair of rigidly coupled loudspeakers

designed to provide long-term stability and extremely linear motion. The

power section of the pair is used to drive the motion, while the pick-up section

is used to measure the instantaneous velocity. A digital function generator

provides the desired velocity waveform, and feedback from the pick-up coils

is used to slave the drive motion to the velocity waveform and so control the

source velocity. Accuracies and long-term stabilities of better than 0.1% are

routinely exceeded.

Drives are normally run at or close to their natural frequency (typically 30

to 40 Hz) in order to minimize the errors that have to be corrected by the

velocity feedback loop. Two velocity waveforms are commonly used:

triangular or “constant acceleration,” and sinusoidal. The linear mapping

between time and velocity provided by constant acceleration mode simplifies

data visualization and analysis, and also yields spectra with a constant

velocity resolution and time weighting. The instantaneous change in

acceleration demanded of the drive when the “constant” acceleration changes

sign at maximum velocity puts significant demands on the mechanical and

electronic designs and some ringing in the velocity error is inevitable.

However, the error occurs at the extremal velocities where only background

data should be collected; its effects can be minimized in a properly designed

system. Constant acceleration offers many advantages and is widely used for

the modest velocity ranges needed for

Fe,

119

Sn, and

151

Eu Mössbauer

spectroscopy.

Any drive that operates well in constant acceleration mode will give better

performance when operated with a sinusoidal velocity waveform. However,

the improvements are rarely significant and generally lie well below other

mechanical or resolution limitations. Disadvantages of sine-mode operation

include a nonlinear mapping between time and velocity that adds some minor

complexity to the data analysis, and a severely nonuniform velocity coverage.

A sine wave spends most of its time out near its extrema, where we would

expect only background contributions, and much less time traversing the

region around zero velocity where most of the Mössbauer lines are expected

to occur. This decreased counting time and lower central resolution need to be

carefully compensated for during data acquisition.

There are, however, two situations in which sinusoidal velocity operation

is unavoidable: (i) high velocities, (ii) long drive couplings. When velocities

of hundreds of mm/s are needed, e.g., for

161

Dy or

169

Tm, the drive must be

operated sinusoidally as the demands placed on the mechanical system

preclude operation in constant acceleration mode: the ringing at the

acceleration reversals becomes more severe and persistent. Furthermore, the

214

harmonic content of the triangular waveform becomes increasingly

problematic at higher velocities; achieving adequate vibration isolation

between the drive/source assembly and the sample is more difficult.

Dedicated drives with much stiffer internal springs are generally used for high

velocity applications, as this moves the natural frequency up above about

100 Hz and reduces the vibration amplitude. At 30 Hz, a maximum velocity

of 1000 mm/s yields an amplitude of about 5 mm and requires an extremely

linear velocity pick-up over a very wide position range to ensure accurate

velocity control. Raising the operating frequency to 100 Hz reduces the

amplitude to around 1.6 mm.

When a source is operated within a cryostat, either to apply a large

magnetic field or to keep the source at a low temperature, the source is often

placed on the end of a long (~1 m) extension rod, well away from the actual

drive. This situation puts severe demands on the drive performance, adding

the natural vibration modes of the long rod to the mix of problems. Again,

sinusoidal drive operation is generally the only viable option, and careful

choice of operating frequency is essential so that driving natural modes in the

extension is avoided. Proper design of the drive-extension-source assembly is

important to ensure that transverse motions are prevented and the rod is stiff

enough to provide proper coupling of the drive motion to the source.

5.2.2 Detectors

The Mössbauer J originates from a nucleus in an excited state. This state is

generally populated by the prior decay of a radioactive parent nucleus. The

path to the Mössbauer level often involves one or more intermediate steps,

each leading to the emission of a

J. As most Mössbauer transitions involve

quite low energy

J-photons, the process is often quite heavily internally

converted (an s-electron is emitted instead of the

J), which creates a hole in an

inner electron shell, which if filled radiatively rather than by an Auger

process, leads to the further emission of characteristic K or L X-rays. Finally,

with many energetic

J-photons and X-rays being emitted directly by the

source, other materials in the source and shielding environment may

fluoresce, adding their own characteristic K or L X-rays to the emission

spectrum.

For example, the standard Mössbauer source for

Fe is

Co, diffused into

a metallic matrix, commonly rhodium. As shown in Figure 5.9,

Co decays

by electron capture, so that a significant emission of Co-K

X-rays at

~6.9 keV is inevitable. The

Co decay populates the 136 keV level of

which decays directly to the ground state in ~10% of cases yielding a

136 keV

J. The remaining 90% of the nuclei emit a 122-keV J and populate

the 14.4-keV Mössbauer level. Ninety percent of the time, the 14.4 keV level

decays by internal conversion, emitting a low energy (~7 keV) electron and

5.2 Methodology

215

5. Mössbauer Spectroscopy

leading subsequently to the emission of a Fe-K

X-ray at ~6.4 keV. As a

result, only about 9% of the original

Coo

Fe decays lead to the emission

of the 14.4 keV Mössbauer

J, the rest yield other J-photons and X-rays. There

is also a significant emission of Rh-K

X-rays at ~ 20 keV, and likely also at

~75 keV if lead is used in the shielding. The job of the detector and associated

nuclear electronics is to efficiently detect and isolate the 14.4-keV Mössbauer

J from the six or more other photon energies emitted with similar or

substantially higher intensity.

Figure 5.9 Decay scheme associated with

Fe Mössbauer spectroscopy (the thicker arrow

represents the Mössbauer J-ray).

The choice of detector will be determined in part by budget, but primarily

by the characteristics of the source being used. None of the detectors

discussed here, or in fact no available detector, comes close to matching the

1:10

energy resolution of a typical Mössbauer transition. The detectors used

in a Mössbauer spectrometer serve only to isolate the Mössbauer

J from other

radiations emitted by the source; the fundamental resolution of the Mössbauer

transition is only observed in the spectrum obtained by velocity modulation of

the source energy. One therefore needs only modest energy resolution, and the

simplest and most rugged detector that will isolate the

J of interest is usually

the best choice. Indeed, a “better”, i.e., more expensive, detector may actually

yield much poorer performance.

5.2.2.1 Proportional Counters

Noble gas-filled proportional counters achieve relatively poor energy

resolution (~12% at 14 keV), yet this resolution is more than sufficient to

separate the 14.4-keV Mössbauer

J from the neighboring X-rays at 7 keV and

20 keV. They are also comparably cheap, robust, and have an excellent

operational lifetime. They are the detector of choice for

Fe Mössbauer

spectroscopy.

271d

136.48 keV

14.41 keV

0 keV 1/2

3/2

5/2

216