Vij D.R. Handbook of Applied Solid State Spectroscopy

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CHAPTER 5

MÖSSBAUER SPECTROSCOPY

J.M. Cadogan

and D.H. Ryan

School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia

Centre for the Physics of Materials, Department of Physics, Rutherford Building, McGill

University, 3600 University Street, Montreal, Quebec, Canada H3A 2T8

5.1 INTRODUCTION

The Mössbauer effect was discovered in 1957 by R.L. Mössbauer [1] and

involves the recoilless emission of a gamma ray (J-ray) by an excited nucleus

with a subsequent recoilless absorption by another nucleus. Mössbauer

spectroscopy has found numerous applications in physics, chemistry and

biology and one notable early success was the measurement of the

gravitational redshift [2]. This spectroscopic technique continues to make

significant contributions, such as 2004/5’s analysis of soil on the surface of

Mars [3], with spectra collected in situ. Rudolf Mössbauer’s work was

awarded the 1961 Nobel Prize for Physics.

In this chapter we present an introduction to Mössbauer spectroscopy. The

article is divided into three main sections. We begin with an introduction to

the Mössbauer effect and the nuclear hyperfine interactions underlying the use

of this spectroscopy. In section two, we give a detailed discussion of the

various methodologies of Mössbauer spectroscopy, encompassing all aspects

of this technique from the standard transmission experiment to more exotic

techniques. Finally, in section three we illustrate the use of Mössbauer

spectroscopy with a variety of spectra taken from our work, including studies

of crystalline, nanocrystalline, and amorphous materials.

5.1.1 Recoilless Processes

To begin, we consider the recoil-free nature of Mössbauer transitions, which

is a consequence of the nuclei being embedded in a solid matrix, which takes

up the momentum associated with the J-ray emission and absorption

processes. The lattice’s mass is clearly much greater than that of a single

nucleus and the energy lost to nuclear recoil is therefore negligible. As such,

the transition may be thought of as a zero-phonon process. Naturally, not all

such J-ray transitions in a material are recoil-free and the recoil-free fraction f

varies markedly with isotope, material matrix, and temperature. The most

commonly used Mössbauer isotope,

Fe, has an f-factor ~0.7 at room

temperature. The dependence of the f-factor on the host material through the

lattice’s Debye temperature has implications for quantitative phase analysis

since the common assumption that all phases in a mixed-phase sample have

The recoil-free fraction f is related to the average squared displacement

of the probe nucleus from its equilibrium position, and for the one-

dimensional case,





expfkx  where k

is the wavevector of the

incident J-ray. At temperatures much higher than the Debye temperature (4

)

of the host lattice,

is given by

§·

¨¸

©¹

(5.1)

i.e., it is linear in T, whereas at low temperatures one finds

31ʌ

ªº

§·

«»



¨¸

«»

©¹

«»

¬¼

(5.2)

i.e., it is quadratic in temperature. Within the Debye model the latter expre-

ssion can be rewritten in terms of 4

to give the f-factor as

32ʌ

ln 1 ,

22 3

ªº

½

§·



°°

«»



®¾

¨¸

«»

©¹

°°

«»

¯¿

¬¼

(5.3)

where

M is the mass of the probe atom.

This expression for the Debye-Waller factor contains important

consequences for the observation of the Mössbauer effect. First, f decreases

with increasing recoil energy so the observation of a Mössbauer transition is

limited to J-ray energies below about 100 keV. Second, lower temperatures

are favored for the experiments (notice the T

dependence in ln f ). There is

squared velocity of the probe atom

SOD E

. (5.4)

5. Mössbauer Spectroscopy202

the same f-factor is almost certainly merely a useful simplification.

also a temperature-dependent second-order Doppler shift due to the average

5.1 Introduction

5.1.2 Doppler Velocity

The emission and absorption J-ray distributions most likely will not overlap

due to such effects as hyperfine interactions. However, they may be brought

into coincidence by Doppler shifting of the J-ray source. The resulting change

in energy of the emitted J-rays, to first-order, is

, where v is the

velocity of the J-ray source relative to the absorber, c is the speed of light, and

is the unperturbed J-ray transition energy. To give some idea of the

remarkable energy resolution provided by the Mössbauer effect, a typical

Doppler shift of 1 mm/s in the most commonly used isotope

Fe, for which

= 14.4 keV, yields 'E = 48 neV and thus'E/E

= 3.3 × 10

–12

5.1.3 Lineshape

The energy uncertainty associated with a nuclear excited state of lifetime W is

given by Heisenberg’s relationship

, where * is the natural linewidth.

The energy distribution of the emitted J-rays has a Lorentzian lineshape in

which the J-ray intensity is given by





ªº



§·



«»

¨¸

©¹

«»

¬¼

I (5.5)

where the line is centered at a relative source-absorber velocity v

and the

half-width at half-maximum intensity (HWHM) is * The lower limit on * is

due to the Heisenberg uncertainty principle and experimental considerations

such as sample thickness increase *

5.1.4 Hyperfine Interactions

The hyperfine coupling of the atomic electron charge density with the nuclear

charge density shifts the nuclear energy levels and can remove their

degeneracy either fully, partially, or not at all. The removal of nuclear level

degeneracy results in multiple J-ray transitions. There are three main types of

hyperfine interaction, namely (i) the electric monopole term, leading to

spectral displacements from zero Doppler velocity commonly referred to as

the isomer or chemical shift; (ii) the electric quadrupole term, and (iii) the

magnetic dipole term.

203

5. Mössbauer Spectroscopy

5.1.4.1 Electric Monopole Term

The monopole term is electrostatic and arises from the overlap of the atomic

s-electron density with the nuclear charge distribution. This shifts the various

nuclear energy levels by different amounts, due to their different spatial

extents (hence different degrees of overlap), without affecting their

degeneracies, and the resulting single-transition Mössbauer spectrum is

¢r

² is shifted by



6İ

r \

relative to a point nucleus, where Z is

the atomic number and



ȥ 0 is the atomic electron density at the nucleus.

The isomer shift is normally quoted relative to a standard, such as the source

or an D-Fe calibration foil. It is easy to show that resonant absorption between

an absorber (A) and a source (S) occurs at a relative Doppler velocity of:

 

res A S e g

o Ȗ

00 .

Ze c

vrr

İ E

ªº

\\ 

¬¼

(5.6)

This relative Doppler velocity is called the isomer shift (or sometimes the

chemical shift). Due to the presence of the overlap term



0\ in the

expression for the isomer shift, this measurement can often unambiguously

determine the electronic valence of the parent ion. In Figure 5.1 we show,

schematically, the effect of the electric monopole interaction on the nuclear

energy levels of

Fe and we illustrate the ensuing isomer shift with the

room temperature (RT) Mössbauer spectrum of a stainless-steel foil that

shows an isomer shift of ~ – 0.4 mm/s relative to the

CoRh source.

Figure 5.1 Left –– Effect of electric monopole interaction on the nuclear energy levels in the

source and absorber, giving rise to the isomer shift. Right––

Fe Mössbauer spectrum at RT of

a stainless-steel foil [Cadogan, unpublished].

5.1.4.2 Electric Quadrupole Interaction

A nuclear state with a spin I greater than 1/2 has an aspherical charge density

characterized by its electric quadrupole moment Q, which is a measure of the

referred to as a “singlet.” The energy of a nuclear state of mean square radius

204

5.1 Introduction

degree of asphericity; (most readers will be familiar with the terms “oblate”

and “prolate” when referring to the sign of Q). This energy state splits in the

presence of an electric field gradient (EFG). The EFG is a tensor that has

contributions from the unfilled atomic electron shells, valence electrons,

surrounding lattice charges, amongst other sources. The components V

of the

EFG tensor are defined as the second derivatives of the electrostatic potential

acting at the nucleus. The principal axis frame of the EFG tensor is

such that the off-diagonal V

terms disappear and the tensor is traceless, as a

consequence of Laplace’s equation. Thus

VVV0

xx yy

 and it is

conventional to define the axes such that

VVV

zz yy xx

tt. The electric

quadrupole Hamiltonian may be written in the form







222

Qzxy

H3Ș ,

4I 2I 1

II II

ªº



¬¼



(5.7)

where the asymmetry parameter is defined as



xx yy

and is a measure of

the deviation of the EFG from uniaxial symmetry (in the presence of uniaxial

point symmetry at the site in question, K = 0). The dependence of the

quadrupole Hamiltonian on the squares of the nuclear spin operators means

that nuclear substates such as

 m and

 m remain

degenerate. In

Fe, the ground state is unaffected by EFG effects since I

1/2 and thus

= 0. The excited state I

= 3/2 splits into two

levels

r m and

r m with the corresponding energy

shifts



eQV

E r 

§·

¨¸

©¹

and



eQV

E r  

§·

¨¸

©¹

. So, for a 3/2 ļ 1/2

system such as in

Fe,

119

Sn, or

169

Tm one observes two Mössbauer

transitions (a “doublet”) with a quadrupole energy splitting (QS) of

eQV

' 

§·

¨¸

©¹

. We note that the sign of the EFG component V

indeterminate in this case. If the excited state spin is greater than 3/2, such as

is the case of

166

Er and

170

Yb (I

= 2 and I

= 0), one can determine the sign of

[4].

In Figure 5.2 we show the splitting of the nuclear energy levels of

Fe due

to EFG effects and we illustrate this with the

Fe RT Mössbauer spectrum of

FeSO

O, which is a sharp quadrupole doublet with a large quadrupole

splitting ' ~ 3.4 mm/s, typical of Fe

ˆˆ

205

5. Mössbauer Spectroscopy

5.1.4.3 Magnetic Dipole Interaction

affected by magnetic fields acting at the nucleus via the magnetic dipole

interaction

hfNnhfM

BµgBµH  

is the hyperfine magnetic field, and g

is the g-factor of the nuclear

state in question (g

can take either a + or – sign, depending on the state). The

eigenvalues of the magnetic dipole Hamiltonian are

mInNhf

B Emg and

thus the degeneracy of this nuclear state is completely removed. The selection

rule for a magnetic dipole M1 transition (excited ļ ground) is

1,0

r 'm ,

which yields six possible transitions (a “sextet”) for a 3/2 ļ 1/2 transition, as

Fe,

119

Sn, or

169

Tm. The relative intensities of the six Mössbauer

transitions are given by the well-known Clebsch-Gordan coefficients and

depend on the angle - between the incident J-ray direction and the nuclear

magnetic moment. From lowest energy to highest one observes relative line

intensities in the ratio 3:R:1:1:R:3 where

4sin

1cos

-

and, for a “perfect”

powder averaged sample, R = 2. This angular dependence allows one to study

magnetic texturing of samples; e.g., if the sample magnetization is completely

parallel to the incident J-ray direction, R = 0 and lines 2 and 5 of the sextet

disappear. One can also use this approach to study spin-reorientations of the

atomic magnetization by using magnetically aligned powder samples.

Figure 5.2 Left––Effect of the electric quadrupole interaction on the nuclear energy levels in

Fe. Right––

Fe Mössbauer spectrum at RT of FeSO

O [Cadogan, unpublished].

the direction of the atomic magnetic moment giving rise to the field, is

indeterminate in a simple experiment. However, the application of a relatively

small external field (a few tesla) is enough to determine the sign of the

hyperfine field via its effect on the atomic magnetization. (We also note at

this point that the sign of the EFG term V

in the magnetic state can only be

determined relative to the sign of the magnetic hyperfine field.)

In general, both magnetic dipole and electric quadrupole effects are present

and this results in an asymmetric sextet. The situation is further complicated

206

It is clear that the sign of the hyperfine magnetic field, usually referred to

A nuclear state with a non-zero spin has a magnetic moment and is therefore

where P is the nuclear magneton,