BASIC THEORY OF THE MSSBAUER EFFECT

Takele Seda Western Washington University

Department of Physics and Astronomy sedat@physics.wwu.edu

July 2003

1

The Mssbauer effect is the recoilless resonant emission and absorption of -rays (quanta of photon radiation) by nuclei of atoms. The energy levels of the nuclei in the Mssbauer active atoms are sensitive to the electronic and magnetic fields present at the nuclei which permit to explore various hyperfine (magnetic dipole, electric quadrupole and electric monopole) interactions. The unique characteristic feature of the technique is its high relative energy resolution to probe fine changes in the hyperfine parameters of the nucleus and its environment; one part in 1013 (e.g. for the 14.41 keV of 57Fe it is 510-13) which was difficult before the discovery of the Mssbauer effect to measure with other techniques. Detailed elaboration of the basic foundation and applications of the technique can be found in the books by Greenwood and Gibbs (1971), Wertheim (1964), Gonser (1976), Trautwein et al. (1978), Thosar et al. (1983), Nasu and Brand (unpublished). In the following we base our discussions on these books.

1. Resonant emission and absorption of -rays The Mssbauer system can be characterised as consisting of a -ray photon and a nucleus. The properties of the nuclear transition and of the -ray radiation are coupled through the conservation of energy and momentum. We consider as a before process a nucleus of -ray source in an excited state with energy Ee and atomic number A (proton number Z and neutron number N). This nucleus in its excited state emits a -ray by transition to the ground state (resonant emission). The emitted -ray can be absorbed by another nucleus originally in the ground state (Fig. 1) with energy Eg if the nucleus is of the same kind (same atomic number) as in the emitter by the reverse process called resonant absorption (the after process).

2

Figure 1: Schematic of nuclear emission and absorption processes between the excited and ground nuclear states.

The energy difference E0 between the excited Ee and the ground Eg states is the energy of the -ray (photon) emitted or absorbed during the processes. Practically, however, this can only happen in special cases. Resonant absorption occurs only when the energy distributions (spectral lines) for the emission and absorption processes overlap. The overlap of spectral lines is determined by the magnitude of the natural line width of the nuclear transition energy and the recoil energy imparted to the nucleus by the -ray during emission or absorption.

By conservation of energy, a free nucleus of a solid (originally at rest) with mass Mn emitting a -ray receives a recoil energy ER given by

2

21

vME nR = (1)

Where v is the velocity of the nucleus with which it moves in opposite direction to the emitted photon. By conservation of momentum P P M v

Ecn n

= =

(2)

before nucleus 1 after Ee

E0 = Ee Eg -ray

Eg emission absorption nucleus 2

3

where c is the speed of light and E is the energy of the emitted -ray. From

the above two equations, the recoil energy becomes

2

2

21

cME

En

R

= (3)

A nucleus in the ground state absorbing the emitted -radiation will also receive the same amount of recoil energy as the emitting nucleus and hence will move with the same velocity v in opposite direction to the emitter as shown in Fig. 2

Figure 2: Recoil energy ER received by an isolated nucleus up on -ray emission or absorption.

Therefore, the energy spectral lines for the emission and absorption will be separated by 2ER and the energy of the -ray is not exactly E0. It is rather given by E E ER = 0 for the before (emission) process and E E E R = +0 for the after (absorption) process. The energy spectral lines for the emission and absorption do overlap only if the natural line width (determined by the mean lifetime of the nuclear energy involved through Heisenbergs uncertainty relation by = ; where is Plancks constant divided by 2pi) is greater than the recoil energy. However, the recoil energy is always greater than by considerable orders of magnitude (e.g. it is six orders of magnitude greater than for the 14.4 keV of 57Fe). Therefore, resonance condition can not be fulfilled and the spectral lines of the source and absorber can not overlap. However, if the nucleus (atom) under consideration is not free, but embedded in a solid of mass Ma, either of the following two may take places (e.g. Wertheim 1964). The nucleus will dissipate its recoil energy by phonon

ER ER

-ray emission absorption

4

creation (heating the lattice) if the recoil energy is greater than the lattice vibration, or the whole crystal (solid) will receive the recoil energy without creation of phonons if the recoil energy is less than the phonon energy. For such cases of systems where the nucleus is bound in a solid the mass of the nucleus in (3) can be replaced by the mass of the whole solid Ma. Since this mass is much greater than the corresponding mass of free nucleus (atom) the recoil energy will be very small as compared to that for a free nucleus. Hence, a certain fraction of the emitted -ray can be resonantly absorbed without phonon creation. This zero phonon process of photon emission and absorption is the core of the Mssbauer effect and makes it such a powerful practical tool.

2. Probability of resonant transitions Recoilless emission and absorption of -ray is only observed for elastic transitions in which the lattice does not change its vibrational state. The probability of this recoil free emission or absorption is determined by the Lamb-Mssbauer factor f, and is related to the general features of the lattice dynamics of the solid under consideration. In the harmonic limit, this recoil free fraction is given by (Mssbauer and Clauser 1967, Maradudin 1964)

( )f k x= exp 2 2 (4) where

k is the wave vector of the -ray and x 2 is the mean square

displacement of the resonant nucleus in the direction of the -ray propagation. For a Debye model (e.g. Freeman and Watson 1965), f will take the following form:

2

20

6 1exp

4 1

DQT

Rx

B D D

E T xf dxk e

= + (5)

5

where kB is Boltzmann constant, T is the temperature in Kelvin and D is the Debye temperature. Practically the high or low temperature limit (relative to D) is important. In such cases

+

= 2

22

23

expDDB

R Tk

Ef pi for T

6

Figure 3 (a) Production of the 57Co radioactive source for 57Fe Mssbauer spectroscopy and (b) its decay scheme.

From (6) we can see that ER should be small to get high recoil free fraction and ER is small if the resonant -ray energy is small. This implies that resonant effect is hardly observed for high-energy -rays. It is also difficult to obtain resonance effect for -rays with very small energies as atomic absorption increases with decreasing -ray energies (e.g. Long et al. 1983). The internal conversion in (9) should not be high to get maximum resonance cross-section. All these conditions are relatively best fulfilled for the 14.4 keV of 57Fe Mssbauer transition (Fig. 3) than all other Mssbauer active nuclei studied so far. The radioactive source for this transition is 57Co isotope which is produced by (d,n) reaction (Figure 3 (a)) on 56Fe (Greenwood and Gibbs 1971). Some important parameters for this transition are also given in table 1.

57Co 270 days

E.C 99%

56Fe (d,n) 57Co I 5/2 136.5 keV (89 ns)

57Fe + 0.6 MeV 9% 91%

3/2 14.4 kev (98 ns)

1/2 0

57Fe (stable)

(a) (b)

7

Table 1: Nuclear parameters of the 57Fe Mssbauer isotope for the 14.4 keV, 3/2 1/2 -ray transitions (Stevens and Stevens 1976). Parameter 2 ER t1/2 e g a

Unit 10-20 cm2 __ mm/s 10-3 eV Ns N N %

Value 257 8.2 0.192 1.957 97.8 -0.155 0.0902 2.19

3. Modulation of -ray energy In the foregoing sections we have considered the absorption line to be Lorentzian centred at Eo. In most cases, however, this does not happen. Due to hyperfine interactions (see next section) the nuclear energy levels can shift their centre and/or split into sub-levels. In order to have resonance conditions fulfilled between the source and absorber, a -ray with variable energy (within the limit of the shift and splitting of the levels) is required. This can be achieved by changing the energy of the emitted -ray along the propagation direction of the -ray by the Doppler effect.

If the source (-ray) is moving with a velocity v relative to the absorber, then the mean energy of the -ray emitted towards the absorber is

c

v

EE

c

vEvE

=

+=

00 )1()( (10)

The -ray thus emitted by the source and passing through the absorber is detected behind the absorber as a function of the velocity of the source. The principle arrangement of a Mssbauer spectrum measurement is shown in Figure 4 for horizontal transmission geometry. The number of counts N( v ) detected during a certain time interval at a Doppler velocity v is the Mssbauer spectrum. A normalized experimental spectrum

8

Figure 4 (a) Schematic arrangement for a Mssbauer spectrum measurement and (b) an example of the resulting spectrum.

)()()()(

=

NvNN

vS

(11)

is usually used to calculate the magnitude of the Mssbauer resonance effect (e.g. Spijkirman 1971), where N( ) is the count rate far off resonance.

source detector drive source absorber

(a)

N(v)

Control electronics

Velocity (mm/s)

-v +v 0

Re

l. tra

nsm

issi

on

(a)

(b)

9

4. Optimum transmission and absorber thickness The count rate and resonance effect in a Mssbauer spectrum are functions of the absorber thickness. Therefore, for reliable analysis of Mssbauer spectra, determination of the thickness of the absorber is a very important factor. The usual way of analysing Mssbauer spectra is the approximation of a thin absorber. This results in an effective line width given by the sum of the source and absorber contributions (eff = S + A ). For high pressure Mssbauer measurements in a DAC (see chap. 5), due to the minute sample area and the attenuation of the resonant -ray flux by the diamonds, the absorbers are made thick or enriched with 57Fe resonant atoms. Therefore, appropriate absorber thickness need to be calculated to avoid line distortions as a result of saturation effects (e.g. Viergers, 1976) and to obtain reasonable resonance effect. This has been briefly discussed by Hearne et al. (1994). The resonance effect of the Mssbauer spectrum depends on the amount of -rays attenuated by (or transmitted through) the absorber. The function that determines the attenuation of the -ray is given by the ratio of the intensities of transmitted radiation I and the incident radiation 0I as

)exp(0

xII

e = (12)

where e is the mass absorption coefficient for the resonant -ray (in units of cm2/g); is density of the absorber and x is the physical thickness of the absorber in cm. A transmission value of 30% is normally used as a starting guideline with background contribution suppressed (see chapter 5) so that the pulse height spectrum of the resonant -ray is at least 20% above the background. This depends on the sample composition and isotopic enrichment of the absorber. The concentration of the natural abundance of iron in the absorber is determined by de = (mFe/Fe), where mFe is the mass fraction of Fe in the absorber.

10

The principal factor that determines the absorber thickness is ta (the effective absorber thickness). It is given by (e.g. Hearne et al. 1994)

afA

dNt a

a

a 030

10

= (13)

where 30 10AdN a is the number of Mssbauer atoms per unit area; 0N is

Avogadros number; A is atomic mass; ad is absorber thickness in mg/cm2

of the Mssbauer element; a is isotopic abundance of the resonant isotope 0 is the cross-section and af is recoil-free fraction in the absorber. With a desired value of at and estimated value of af from appropriate lattice dynamics, the amount of absorber required, ad in a given area can be

calculated. Thus, ed and ad are important factors that could be used as

guideline parameters during enrichment to increase the isotopic abundance of the absorber up to the desired value of transmission.

For a thin absorber approximation at is taken to be less than unity. In this

case the resonance intensities are rather small, but the lines can be fitted to Lorentzian curves. At increasing at , line distortions occur both in relative line

area and line form. Thus fitting spectral lines to Lorentzian curves may not be appropriate. Experimental spectra are usually analysed by way of corrections made to the line width with S = A such that the effective line width is (Gutlich et al. 1978) given as

+

+=

20 0025.0145.001.1

135.01

aa

aeff

tt

t (14)

for 40 < at for 104 < at

for 4at

11

where 0 is the natural line width.

In cases where it is important to accurately determine site populations or phase fractions this correction is still unsatisfactory. Therefore, for quantitative spectral analysis where split spectral overlap is pronounced (which is the case for most mineral spectra), the so-called transmission integral (Margulies and Ehrman 1961, Viegers 1976) is used in the analysing program. This takes into account thickness effects and can be used to determine the hyperfine parameters and relative abundance of different sites. In this program it is important to be able to know (determine) a priori both the recoil free fraction fs and the line width s of the source independent of the experimental spectrum with unknown parameters. This is done usually using a standard calibration spectrum or supplied with the source from the manufacturer.

5. Hyperfine Interactions With no hyperfine interactions at all, the spectral line would always be centred at zero energy (velocity) without shift or split as in Figure 4 (b) in the last section. The presence of nuclear charge and its moments (nuclear monopole interaction, quadrupole moment and magnetic dipole moment) bring the nucleus into interactions with its environment (electrons and other ions). These interactions result in the hyperfine shift and splitting of nuclear transitions. The energy of the hyperfine interactions generally does not exceed 10-6 eV so that before Mssbauers discovery it was difficult to observe directly the hyperfine splitting of nuclear transitions their energy being ten orders of magnitude higher. The presence of nuclear hyperfine interactions make the resonance emission and absorption of nuclear radiation interesting for applications in physics and chemistry since they yield

12

many information regarding the magnetic state and electronic configuration of the environment of the nucleus of the atom under consideration.

The total hyperfine interaction energy can be given as

me HHH += (15) where the subscripts denote the electrostatic (due to the surrounding electrical charges) and the magnetic (due to magnetic moments) interactions respectively. These interactions can be described by classical electrodynamics (e.g. Jackson 1965). The nucleus has an electric charge distribution )(r , the external charges (electrons, other ions etc.) surrounding the nucleus in the solid are described by the potential )(rV . The electrostatic interaction energy between the nucleus and the surrounding charges is then given by (the origin of the co-ordinate system is the centre of mass of the nucleus):

= rdrVrH e 3)()( (16)

)(rV can be expanded in a Taylor series at 0=r :

+

+

+===

i kiki

rkii

ri

XXXX

VXXVVrV

, 0

2

0 21)0()(

(17)

Here the deviations from the position r are denoted by iX , 3,2,1=i for the

three spatial directions. Using a simple notation

ikrki

iri

VXX

V

VXV

VV

=

=

=

=

=

0

2

0

0)0(

(18)

and the integral over the nuclear charge distribution

13

= Zerdr 3)( (19)

we will arrive at

+++=i ki

kiikiie rdXXrVrdXrVZeVH,

330 )(2

1)( (20)

In this equation the first term can be neglected as it represents the Coulomb interaction energy due to point charge at the centre of the nucleus which is the same in all nuclear states. The second term consists two quantities; the charge density )(r and the spatial co-ordinate iX . The parity of the nuclei is a constant of motion (the nuclear wave functions are either even or odd) which means that the charge density (being the magnitude of the square of the wave function) is always even. Since the integrand in the second term of the above equation is the product of even (charge density) and odd ( )iX it vanishes when the integral is taken over the whole space. The only non-vanishing and very important term is the third term (all the other orders higher than the third term are very small and can be ignored). Therefore, the electrostatic interaction energy can be given by

ki

kiik rdXXrV,

3)(21 (21)

In (21) we add to and subtract from the quantity under the integral the homogenous contribution given by the trace ( ))(

31 2

rrik . This gives the

expression

( ){ } +=ki

ikkiike rdrrXXrdrrVH,

3232 )(3)(61 (22)

This has been separated into a part depending on the nuclear radius 2r and charge density )(r , and another part depending on the nuclear shape (form) respectively as

=ki

ikI rdrrVH,

32 )(61 (23a)

14

and

( ) =ki

ikkiikQ rdrrXXVH,

32 )(361 (23b)

IH yields the expression for the isomer shift and is given by the trace of eH ,

while QH yields the expression for the zero-trace electric field gradient (EFG) tensor which is related to the electric quadrupole moment of the nucleus.

(a) Electric monopole interaction (Isomer shift) To find exact expression for the interaction energy IH in (23a), we use Poisson equation )0(42 pi= V at the nuclear centre. For

kiikV

,

this will be

==ki

ik eV,

2)0(4)0(4 pipi (24)

Here 2)0()0( = e is the density of negative charges (electrons)

responsible for iiV at the centre of the nucleus and 2)0( is their probability

density.

The average square nuclear radius 2R can be written in terms of )(r as

== rdrrZerdr

rdrrR

323

322 )(1

)()(

(25)

Combining (1.24) and (1.25), the electrostatic nuclear monopole interaction energy due to finite nuclear size is

222 )0(3

2 RZeH I =pi

(26)

This interaction energy describes the electronic monopole (Coulomb) interaction between the spherically expanded (distributed) nuclear charges and the electrons surrounding it, which deviates (shifts) the nuclear energy from the level of a point nucleus.

15

Generally the excited nuclear state has different 2R from the ground state

and 2)0( for the source and absorber is also different. This difference (shift) in energy is the isomer (chemical) shift and is given by (Vrtes etal.1979)

( )2222 )0()0(5

4saeff

effR

RRZe

=

pi (27)

with ge RRR = and

=

RRR

R geeff 2

22

. Here Re and Rg are nuclear radii for

the excited and ground states and 2)0(a

and 2)0(s

are the probability

densities of electrons at the absorber and source respectively.

Figure 5 (a) Schematic of nuclear level shift due to electric monople interactions in the source and absorber (b) example of the resulting Mssbauer spectrum.

Classically only s electrons have non-vanishing density at the nuclear site. However, in some cases (particularly for the 3-d transition metals) an

(a) (b)

source

ES E0

absorber

ES +

E0

I = 3/2

I = 1/2

I = 1/2

I = 3/2 Ee

Eg

Ee

Eg Velocity (mm/s)

Re

l. tra

nsm

issi

on

-v 0 +v

16

increasing number of 3-d electrons can lead to a shielding of the s electrons and results in a decrease of the electron density at the nucleus. This in turn results in an increase or decrease of isomer shift depending on the sign of the nuclear factor (R/R). When (R/R) is positive, the electron density

increases in going from the source to the absorber if the isomer shift is positive and when (R/R) is negative, the electron density at the absorber is lower than at the source if the isomer shift is positive. For example, (R/R) is negative for the 14.4 keV transition in 57Fe and an increase in the electron density is expected if the isomer shift is decreasing at the absorber. A schematic of the energy level shift in the absorber as well as in the source is shown in Figure 5(a) and an example of the resulting Mssbauer spectrum in Figure 5(b)

Since Mssbauer spectroscopy is a relative method (source and absorber are related by Doppler motion), the isomer shift is usually reported with respect to some standard absorbers, e.g. metallic iron. From the measured value of the isomer shift one can obtain information about the chemical environment of the nucleus under investigation. Most important chemical information tractable from the isomer shift are oxidation state and electronic spin-state (high or low) of the Mssbauer active atom in the absorber.

In additional to the isomer shift, there is a contribution to the total centre shift of a Mssbauer spectrum due to relativistic time dilatation, known as second-order Doppler shift. The nuclear movement can be split up into the mechanical motion of the source and thermal vibration of the atoms in the lattice. The second-order Doppler shift or thermal shift is proportional to the

mean square velocity 2v of the vibrating atoms and is given by (e.g. Veigers 1976)

17

Ec

vE 2

2

2= (28)

The second-order Doppler shift is dependent on temperature and pressure, but its contribution to the total centre shift is small as compared to the isomer shift and is usually neglected.

(b) Electric quadrupole interaction (Quadrupole splitting) In the foregoing section the electrostatic interaction between nuclear and electric charges at the nuclear site was discussed assuming the nucleus is spherical and the charge density is uniform where the interaction energy only shifts the nuclear levels. However, if the nucleus has non-spherical symmetry, higher order electrostatic interaction energy arises. This interaction was given in (23b) and it splits the nuclear levels rather than shifting them.

In (23b), the quantity ikV represents the electric field gradient (EFG) tensor

E

because 2

2

2

XV

XE

ii

i

=

and the remaining term under the integral will be

( ) = iii eQrdrrX 322 )(3 (29) Here iE

is the electric field at the nucleus due to other charges at a distance

)3,2,1( =iX i from the nuclear centre and iiQ is nuclear quadrupole moment.

Therefore, the electrostatic interaction energy ( ) QeEQVeHi

iiiiQ

.

61

6==

arises from the interaction of the nuclear quadrupole moment with the electric field gradient at the nucleus. For a spherical charge distribution the EFG tensor is zero. However, if the nuclear charge distribution deviates from spherical symmetry the EFG tensor is a measure of this deviation. The value of the EFG tensor depends upon the choice of the co-ordinate axes. For this

18

reason a standard form has been assigned such that the off diagonal elements are zero and the diagonal elements are ordered in such away that

ZZYYXX VVV and 0=++ ZZYYXX VVV . Another parameter (the asymmetry

parameter) is conventionally defined as ZZ

YYXX

VVV

= so that the only

independent parameters used to describe the diagonal elements of the EFG tensor are ZZV and ( )10 .

The origin of the EFG at the nucleus may be the asymmetric distribution of its own valence electrons outside the inner isotropic closed shells and/or external charges of the crystal lattice (ligands). All the other electrons contribute to the EFG indirectly by polarisation of the spherically symmetric inner electrons. This effect is expressed by means of the Sternheimer shielding and anti-shielding factors R and respectively (Ingals 1962,1964,Travis 1971), thus the EFG and the asymmetry parameter may be modified as ( ) ( ) latvalZZ VVRV += 11 and ( ) ( ) latvalR += 11 . Where the subscripts val and lat denote the contributions from the nucleus own valence electrons and other charges from the crystal lattice (ligands) respectively.

The interaction of the nuclear electric quadrupole moment with the principal component of the diagonalized EFG tensor ZZV at the nuclear site splits the

( )12 +I fold degenerate nuclear state with nuclear quantum number I > into sub-levels. Quantum mechanically the eigen values of these sub-levels are given by (e.g. Gutlich et al., Gonser 1975).

[ ] 21

2

31)1(3)12(4

++

=

IImII

eQVE I

ZZQ (30)

19

where 1,,1, += IIImI , and I is the nuclear magnetic spin quantum

number. A pure electric quadrupole interaction splits the excited state of the 14.4 keV 57Fe with 2/3=I into sub-levels of eigen values

21

2

31

41

+=

ZZQ eQVE .

The separation (splitting) between these two levels is the electric quadrupole splitting and is given by

21

2

31

21

+= ZZQ eQVE (31)

This splitting is shown in Fig. 6 along with an example of the resulting Mssbauer spectrum. Note that (30) contains only the second power of the magnetic quantum number mI, so states whose mI differs only in sign remain degenerate (see the values of mI in Fig. 6(a))

a11

Figure 6 (a) Schematic of nuclear level splitting in 57Fe due to quadrupole interaction (b) example of the resulting Mssbauer spectrum.

mI

I = 3/2

1 2

I = 1/2

(a)

EQ(3/2)

EQ(1/2) EQ

mI

3/2

1/2

1/2

Velocity (mm/s)

(a) (b)

21

EQ

Re

l. tra

nsm

issi

on

-v +v 0

20

For randomly oriented polycrystalline materials, the probabilities for the individual transitions (e.g. Wertheim 1964) described by the integrals

( )( ) +pi

0

2 sincos123 d and ( )

pi

0

2 sincos23

25 d have the same value,

thus the lines do not exhibit asymmetry. However, vibrational anisotropy of the Mssbauer nucleus may give rise to directional anisotropy in the recoil free fraction (Goldanskii et al. 1963). This in turn may result in relative intensities of absorption lines that are different from those expected for a randomly oriented polycrystalline sample. This is because the various nuclear transition probabilities have an angular dependence on the relationship between the propagation direction of the -radiation and the symmetry axis of the electric field gradient.

From the quadrupole splitting one can obtain information about charge distribution around the nucleus, the local symmetry of the nucleus itself and the bond property of the solid.

(c) Magnetic dipole interaction (Magnetic hyperfine field) An atomic nucleus with nuclear spin quantum number 0>I has a magnetic

dipole moment and may interact with the effective magnetic field B

at the

nucleus. This interaction is described by

BIgBH mNm

.. == (32) where Ng is the nuclear splitting factor (Land factor), m is the nuclear magneton. The magnetic dipole interaction (nuclear Zeeman effect) splits the formerly degenerate nuclear state with spin I into (2I+1) non-degenerate sub- states with eigen values

ImNI

m BmgIBm

E == (33)

21

Figure 7(a) shows the splitting of and transitions between the excited 2/3=I and ground 2/1=I states of the 14.4 keV 57Fe and Figure 7(b) is an example of the resulting Mssbauer spectrum for purely magnetic Zeeman interaction. The splitting of the sub-states are determined by the magnitude and sign of

Im whereas the transitions between the levels and hence the number of lines

in the Mssbauer spectrum is determined by the selection rule 1,0 =m .

mI

0-v +v

(b)(a)

65

4

3

21

+1/2

-1/2I = 1/2

I = 3/2

-3/2-1/2

+3/2+1/2

65

43

21Rel

. tr

ansm

issi

on

Velocity (mm/s)

Figure 7 (a) Schematic of the Zeeman splitting for the excited (I = 3/2) and ground (I = ) nuclear levels and transitions between theses levels in 57Fe (b) example of the resulting Mssbauer spectrum.

6. Origin of the magnetic hyperfine field The effective hyperfine field B

at the nucleus arises from various

contributions (Thosar et al. 1983, Pelloth et al. 1995). With no external field applied, B

can be related to the major contributors as

cod BBBB

++= (34)

Velocity (mm/s)

22

dB

is the intra-ionic dipolar field, oB

is the orbital filed and cB

is the Fermi

contact field.

The dipolar field is due to the interaction between the spin magnetic moment of the valence shell of the parent atom (spin part of the atomic magnetic moment) and the nuclear magnetic moment. It is given by

( )5

2 .

32r

SrrSrB Bd

=

(35)

where r is the electron position from the centre of the nucleus, mc

ehB pi

4

=

( m electron mass) is Bohr magneton and S is the spin operator. The angular average (expectation value) is over the occupied atomic orbital. Here the spin-orbit coupling is considered to be weak which is true for transition metal

ions. If the spin and orbital moments of the valence shell are not separable S

in (35) is replaced by LSJ += , where J is the total angular momentum and L is the orbital angular momentum respectively. The magnitude of the dipolar field depends on the distribution of electrons of the atom concerned. A spherically symmetric charge distribution (whose angular average is zero) will not cause dipolar field at the nucleus. Therefore, for an atom whose valence shell is either completely filled or half-filled the dipolar field is zero (e.g. Fe3+:3d5). Even for non-spherical charge distribution the dipolar contribution to the hyperfine field is small.

The orbital motion of electrons in the valence shell produces orbital angular

momentum L that induces the orbital magnetic field oB

. It is given by (Pelloth et al. 1995)

23

3

2r

LB Bo = (36)

oB

is zero for closed shells or half-field shells. For most transition series (Cox 1987) the orbital motion of d-electrons is also quenched by crystal field effect and oB

is taken to be zero. Both dB

and oB

if different from zero are parallel

to the d-shell magnetic moment and hence contribute positive values to the effective hyperfine field.

The most significant contribution to the effective hyperfine field comes from the Fermi contact term, which is given by (Vrtes et al. 1983, Watson and Freeman 1961a)

( )22 )0()0(3

8 = gB Bc

pi (37)

2)0( and 2)0( are the spin-up (parallel to the d-shell moment) and

spin-down (anti-parallel to the 3d-shell moment) s-electron densities at the nucleus respectively. The Fermi contact field results from the polarisation of s-electrons by the magnetic 3d-ion itself and also by the neighbouring atoms (Watson and Freeman 1961a). The polarisation effect includes both the core electrons and conduction electrons. These have been separated as (Pelloth et al. 1995) core contribution due to local moment of the 3d-shell and transferred hyperfine field due to the neighbouring ions.

According to Paulis principle (anti-symmetrization of wave functions), the s-electrons with their spins parallel to the net unpaired 3d-spin push way from the 3d-electrons than those with anti-parallel spins due to the (negative) exchange interaction. This causes the Coulomb repulsive interaction to be stronger for electrons of anti-parallel spins (since they appear to be closer together) than those with parallel spins. Therefore, the s-electrons with anti-parallel spin are pushed more towards the nucleus. This means that when

24

the s-electrons are distributed inside the 3d-shell, the electron density 2)0( of the anti-parallel s-electrons is larger than

2)0( for the parallel s-electrons. Thus, the inner shell electrons produce a negative (relative to the 3d-moment) hyperfine field contribution. On the other hand s-electrons which are outside the 3d-shell produce positive hyperfine field. The sign and magnitude of the core electron contribution to the contact field depends on whether the inner or the outer electrons produce greater contribution.

Table 2 shows the calculated hyperfine fields (Watson and Freeman 1961a) produced by the s-electrons for Fe2+ (3d6) ion. It is clear from the table that both 1s and 2s-electrons contribute negative hyperfine fields since they totally lie inside the 3d-shell. The 2s-electrons contribute more hyperfine field than 1s-electrons because the 2s-electrons have greater overlap with the 3d-electrons than 1s-electrons and therefore, are more polarised than 1s-electrons. The 3s-electrons are even expected to be more polarised. But due to the fact that they lie partly inside and partly outside the 3d-shell, the resultant contribution for these electrons is smaller than the contribution by the 2s-electrons. The total contribution to the hyperfine field by the core electrons for this particular ion is negative implying that the electrons inside the 3d-shell predominate.

Table 2: Core electron contribution to the hyperfine field of s-electrons for Fe2+ (3d6) ion (Watson and Freeman 1961a).

Electron 1s 2s 3s Total B(T) -3 -131 +79 -55

The transferred hyperfine field contribution comes from the spin polarisation of the conduction electrons by the magnetic moment of the neighbouring atoms. The polarisation can be through the Ruderman-Kittel-Kasuya-Yosida

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(RKKY) or through the ordinary exchange interaction (Watson and Freeman 1961b). The RKKY interaction oscillates and decreases with nearest neighbour atom distance. It produces positive hyperfine field while the exchange interaction produces negative field. Hence, the transferred hyperfine field can be positive or negative depending on the type of interactions.

7. References Cox P. A., The Electronic Structure and Chemistry of Solids (Oxford University press, Oxford, 1987) p. 155. Freeman A. J. and Watson R. E.: in Magnetism II A ed.: Rado G.T. and Suhl H. (Academic Press, New York, 1965) p. 167 Goldanskii V. I., Makarov E.F. and Khrapov V.V, Phys. Lett. 3 (1963) 344. Gonser U., Mssbauer Spectroscopy (Springer-Verlag, Berlin, 1975). Greenwood N.N and Gibb T.C., Mssbauer Spectroscopy (Chapman and Hall Ltd., London, 1971). Gtlich P., Link R. and Trautwein A., Mssbauer Spectroscopy and Transition Metal Chemistry (Springer-verlag, Berlin, 1978). Hafmeister D.W.: in An Introduction to Mssbauer Spectroscopy ed.: May L. (Adam Hilger, London, 1971) P. 60. Ingals R., Phys. Rev. 128 (1962) 1155; Phys. Rev. 133A (1964) 787. Jackson J. D., Classical Electrodynamics (Academic Press, New York, 1965) P. 33. Maradudin A.A., Rev. Mod. Phys. 36 (1964) 417. Margulies S. and Ehrman J. R., Nucl. Instrum. Methods 12 (1961) 131. Mssbauer R. L. and Clauser M. J.: in Hyperfine Interactions ed.: Freeman A. J. and Frankel R.B. (Academic Press, New York, 1967) P. 497. Pelloth J., Brand R. A., Takele S., Periera de Azevedo M. M., Kleeman W., Binek Ch., Kushauer J. and Bertrand D., Phys. Rev. B52 (1995) 15372.

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Stevens J., G. and Stevens V., E., Mssbauer Effect Data Index (Plenum, New York 1976) p. 51. Thosar B. V., Srivastava J. K., Iyenger P. K. and Bhargava S. C., Advances in Mssbauer Spectroscopy: Application to Physics, Chemistry and Biology (Elsevier Scientific, Amsterdam, 1983). Travis J. C.: in An Introduction to Mssbauer Spectroscopy ed.: May L. (Adam Hilger, London, 1971) P. 75. Vrtes A. and Nagy D. L., Mssbauer Spectroscopy of Frozen Solutions (Akadmia Kiado, Budapest, 1990). Vrtes A., Korecz L. and Burger K., Mssbauer Spectroscopy (Elsevier Scientific, Amsterdam, 1979). Viegres T., PhD thesis, (Nijmegen 1976), the Netherlands. Watson R. E and Freeman A. J., Phys. Rev. 123 (1961a) 2027. Watson R. E and Freeman A. J., Phys. Rev. Lett. 6 (1961b) 277. Wertheim G. K., Mssbauer Effect: Principles and Applications (Academic Press, New York, 1964).