COVALENCY EFFECTS IN HYPERFINE INTERACTIONS

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COVALENCY EFFECTS IN HYPERFINE INTERACTIONS

G. Sawatzky, F. van der Woude

To cite this version:

G. Sawatzky, F. van der Woude. COVALENCY EFFECTS IN HYPERFINE INTERACTIONS.

Journal de Physique Colloques, 1974, 35 (C6), pp.C6-47-C6-60. �10.1051/jphyscol:1974605�. �jpa-

00215705�

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COVALENCY EFFECTS IN HYPERFINE INTERACTIONS

G. A. SAWATZKY Physical Chemistry Dept.

and F. VAN DER WOUDE

Solid State Physics Dept., Materials Research Center, University of Groningen, the Netherlands

Rksum6.

^-

Nous avons discute kes effets de covalence et de distorsion par recouvrement sur les densites de spin et de charge dans des matkriaux essentiellement ioniques, et avons relie ces effets aux interactions hyperfines. Nous avons utilise une combinaison lineaire des orbitales atomiques bien que nous ayons aussi considere les expansions et contractions des fonctions d'onde par rapport A celles d'ions libres. Nous montrons que les parametres de transfert de charge peuvent Ctre obtenus par une mesure skparee du champ magnetique supertransfer6 et une comparaison entre le champ magnetique hyperfin total et le deplacement isomerique. Nous avons aussi discute et demontrk une relation entre le champ hyperfin supertransfkre et I'interaction de superechange.

Abstract.

-

The effects of covalency and overlap distortion on the spin and charge densities in mainly ionic materials are discussed and related to the hyperfine interactions.

A

linear combination of atomic orbitals is used although expansions and contractions of the wave functions relative to the free ion wave functions are also considered. We show that the relevant charge transfer parameters can be obtained from a separate measurement of the supertransferred hyperfine magnetic field and a comparison of the total hyperfine magnetic field and the isomer shift. Also a relation between the supertransferred hyperfine field and the superexchange interaction is discussed and demonstrated.

Fine and hyperfine interactions as studied by techniques like nuclear magnetic resonance (NMR), electron spin resonance (ESR), electron-nuclear double resonance (ENDOR) and the Mossbauer effect (ME) contain a vast amount of information concerning the electronic structure of atoms, ions, and their surround- ings in solids. Magnetic dipole and electric quadrupole and monopole interactions depend strongly on the spin and charge density distributions in solids which are determined by inter- and intra-atomic coulomb and exchange interactions. The problem is how to extract information about the spin and charge density distri- bution from the measured hyperfine parameters and to interpret this in terms of the electronic structure. In this paper we will attempt to review some of the work which has been done in this direction. Special attention will be payed to the more ionic materials where a linear combination of atomic orbitals is expected to be a good approximation. Other more covalent materials should be treated by first principles calculations in which case it is difficult to parameterize the hyperfine interactions and each material has to be considered as a special case.

In addition to the LCAO approximation we will also in most instances assume that the material can be considered as clusters of a central ion plus its nearest neighbors. It will be shown however that this picture probably breaks down for the sulfides, selenides and tellurides of 3d transition metals. In these materials the

anion-anion interactions become so large that a band approach becomes more appropriate resulting in long range superexchange interactions and transferred hyperfine fields.

Using the LCAO description spin and charge density distributions will be determined in terms of overlap distortions and charge transfer parameters including the effects of the core orbitals. Overlap distortion effects on the electric field gradient and the isomer shift will be discussed in some detail. It will be shown that rearrangement effects will greatly reduce the original estimates of overlap distortion effects especially as far as the isomer shift is concerned. Going beyond the LCAO approach we will also consider the effects of radial expansions of the outermost orbitals on the spin orbit coupling parameter, the electric field gradient and the isomer shift. Estimates of these radial expansions can be obtained from the by neutron diffraction measured spin density distributions.

The possibility of correlations between the various hyperfine interactions will be discussed. We will also show that under certain conditions a correlation between the hyperfine field and superexchange interac- tion can be expected.

There are basically three types of hyperfine interac- tion which will be of interest to us, namely the electric monopole, electric quadrupole and the magnetic dipole interactions. Higher order multipole interactions can

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974605

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C6-48 G. A. SAWATZKY AND F. VAN DER WOUDE

usually be neglected. Assuming the nuclear para'meters

i. e. nuclear charge and spin distributions and the gyro- magnetic ratios for both the ground and excited states to be known we will be concerned with the determina- tion of the electronic charge density at the nucleus (p,(O)), the electric field gradient tensor (Vij), and the magnetic field (HhpE) acting on the nucleus. The relationship between these quantities and the measure- ments is described in most standard text books and will be only briefly mentioned here for notation purposes.

The Isomer Shift (IS) as measured in a ME experi- ment is described by IS

=

ap,(O) where

^u

is a constant depending only on nuclear parameters. The result of a quadrupole interaction is described by the hamiltonian

where I is the nuclear spin operator, I+

=

I,

f

iIy and Q is the nuclear quadrupole moment. The magnetic hyperfine interaction is described by

X,

= gun

Hh,,

^'I, ( 2 )

where

gu,,

Zz is the

z

component of the nuclear magnetic moment.

The isomer shift therefore involves only one quantity namely the electronic charge density at the nucleus.

The quadrupole interaction involves 2 quantities VZz and

^y⁼^Vxx^-^vyy

(the asymmetry parameter).

v z z

The magnetic hyperfine field can be split up into basically

3

contributions

where H, is the dipole field which for one electron is given by

where

^{r i}

is the position of the electron and Si is its spin.

H, is the orbital contribution and for one electron it is given by

where Li is the orbital angular momentum operator.

HcoN is the Fermi contact interaction which again for one electron is given by

where pr>J(O) is the up spin respectively down spin density at the nucleus.

The total contribution of any one of these quantities will be the expectation value of the sum of the one electron contributions.

For a single isolated free ion the electronic ground state is usually written as a Slater determinant of the occupied one electron wave functions. Such a determi- nental wave function assures the satisfaction of the Pauli exclusion principle. If the one electron wave functions are orthogonal, the expectation value of the sum of one electron operators can be written as the sum of the one electron expectation values i. e.

if 0 is a one electron operator and the q i are occupied one electron wave functions orthogonal to each other (< pi I p, >

⁼h,,).

This orshogonality condition is indeed very important since it is the basis for the so called overlap distortion contribution to the hyperfine interactions [I-71. We now bring these free isolated ions together to form a solid.

If

the interaction between the ions is small we can still use the free ion wave functions.

But in order to use the sum of one electron expectation values we must assure the orthogonality of these wave functions. Although the wave functions centered on a particular ion are orthogonal to each other they will not in general be orthogonal to those centered on neighboring ions. These wave functions must therefore first be orthogonalized. To illustrate this let us consider a diatomic molecule as shown in figure I . As usual we

FIG. I . - Diagram of a diatomic molecule showing the core and valence orbitals.

separated the free ion wave functions into a

core

region and a a

valence

region and we will be interested in determining the hyperfine fields at the nucleus of ion A.

Let I+!Ii and p i be the

occupied

orbitals on ion A and B

respectively. If the overlap is small we can Schmidt orthogonalize the orbitals to obtain new wave func- tions which are orthogonal to 2nd order in the overlap integrals. The new wave functions are then

where

Sij =

< I+!Ii I

^cpj

> is the overlap integral. We

choose to lump all the changes into wave functions

(4)

centered on B so that the wave functions centered on A remaill exactly orthogonal to each other as well as to the new wave functions centered on B. When calculat- ing the hyperfine interactions on A this is a better approximation than to lump everything into

q f

[8].

For 0 representing either

p,(O), V i j

or

pi(0)

- p j ( ~ )

we then obtain an expectation value given by

with the overlap distortion contribution

effective charge of the 'nucleus to 2:.

= Z j

+

^AZj

where

AZj

is approximated by

where

R j

is the radius of the j'th Slater basis orbital and

Ap(r)

is the total change in the charge density resulting from overlap distortion plus the consequent rearrange- ment. For

j = 1, Ap(r)

will be very nearly constant up to R, so that

where <

^{A 0}

>,, represents the first term involving only local wave functions centered on A,

AOLSNL

is the 2nd term involving both local and non local wave functions and <

0

>,,,,, involves only non local wave functions. A more detailed discussion of these contributions to IS, H,,, and

V i j

will be given below.

It should be noted however that such a calculation of overlap distortion effects will always lead to an overes- timation because of the neglect of subsequent rearran-

gement effects.

An estimate of possible rearrangement effects can be made as follows. The increase in the electronic charge at the nucleus due to overlap distortion will tend to decrease the effective nuclear charge so that the elec- trons centered on ion A will tend to move outward a little thereby decreasing the charge density at the nucleus. This will have an especially large influence on the IS. The quadrupole interaction and also the hyper- fine magnetic field will be affected to a much lesser extent.

The problem is to estimate how large this rearrange- ment effect will be. We will only concern ourselves with the charge density at the nucleus and so only the rear- rangement on the s electrons will be considered. We first of all note that the wave functions are a linear combination of Slater type orbitals

Of the Slater orbitals only

j =

1 gives a contribution to the charge density at the nucleus so for the present we need only consider the possible changes in this compo- nent of the wave function. We can estimate the effects of the relaxation by simply changing

Z j

i. e., the

The change in the charge density at the nucleus result- ing from this change in the effective charge can be obtained as follows. The new core wave functions will be given by

and the change in the charge density at the nucleus to first order in

6 Z j

will be

Using

(12)

for

AZ,

we obtain

or since

For an Fe3+ ion for example we get

AP(O)

-

O.~(APO,(O))

This clearly shows how important the rearrangement effects are and how difficult it is to change the wave function close to the nucleus.

Up to now we have only discussed the occupied orbitals and have neglected any charge transfer between the free ions. Usually we will start with cations and anions with ionic charges such that the anion is in a closed shell configuration. In this case the most impor- tant charge transfer will be that from the outermost anion orbitals to the partially empty cation orbitals.

An exception to this is the

CN-

ion which has a low lying empty n* orbital so that electron transfer can take place back from the cation to the anion. This back transfer as first suggested by Danon [9] is probably responsible for the low isomer shift in low spin iron cyanides

[ l o ] .

We will return to this problem below.

Let us denote the empty cation orbitals by

^qf.

The

charge transfer can now be taken into account by

mixing some

cp:

character into the occupied ligand wave

function $i, the final wave function $f must again be

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C6-50 G . A. SAWATZKY AND F. VAN DER WOUDE

orthogonal to all the occupied orbitals qi. Our new wave functions are

= (1

- z

^S:

+ x ^b; + 2 x ^bij

^Sij^{- ' I 2}

^x

j i j

)

X

(h

^-

C

^sij

Pj + C b i j

q;)

J j

q'. = (p.

(17)

J J '

The charge transfer parameter bij can best be taken as a parameter which is to be determined experimentally.

From lowest order perturbation theory we would obtain

where E: and E: are the anion and cation respectively ground state energies with

N and

M electrons and

E K 1 + ET is the total energy of a state where one electron has been transferred from the ith orbital of the anion to the jth orbital of the cation. This denominator is therefore directly related to the electron negativity difference between the two ions. We will also find later that in some simple cases the hyperfine interactions are linearly related to the electronegativity difference. This charge transfer is usually referred to as covalency. The expectation value of a one electron operator will now be

overlap distortion) in the molecule as compared to the free ion. This is essentially the linear combination of atomic orbitals method which is a good approximation as long as the overlap integrals are small and as long as the potential produced by the surrounding ions is quite flat so that the wave function remains essentially unaltered from that of the free ion. The importance of such a potential distortion has recently been discussed by Walch et al. [17]. For the moment we will neglect these effects and carry on with the LCAO description of covalency effects on each of the hyperfine inter- actions.

We can now extend these ideas to an actual solid.

Consider for example a central ion in an octahedral configuration (Fig. 2). We now consider the central ion plus its nearest neighbors as a molecule. The orbital \Iri

on the ion B must now be replaced by molecular orbi-

I

+ k

< " I I

^{* i}

>

⁺

<

^*i

' ^'* ^>] ⁽¹⁹⁾

^no.^2.^-Diagram of an ion in an octahedral site showing the coordinate system used for constructing the molecular orbitals.

where

The additional terms have appeared because of cova- lency effects although it should be noted that the b , also appears in N i making it impossible to completely separate covalency and overlap effects. By letting S i j

+

0 and neglecting the cross terms we can write

< 0 > in terms of the occupation numbers of the orbitals i. e.

Iron atomic orbitals and the corresponding combi- nation of ligand orbitals belonging to the various symmetry representations for iron in an octahedral coordination

(").

Represen- Metal

tations orbitals Combination of ligand orbitals (*)

- -

alg ^s NaP(01

+

⁶²

+

^a3

+

^a4

+

^{0 5}

+

⁰⁶⁾

7 % dzz N T ~ ~ ( X U I

+

^nz5

+

^{8 x 3}

+

⁸²¹⁶⁾

Where we now sum over all orbitals centered on A, n j

_{~ Y Z} N ~ ~ , ( n z z

+

^nus

+

^{8 ~ 4}

+

^{~ 2 6 )}

being the occupation number of the jth orbital. This is

dzy N r Z g ( r z l

+

^n2/2

+

^zy3

+

^{r s 4 )}

a frequently used approximation in the interpretation

of hyperfine interactions.

^(*)The normalization constants take into account ligand- ligand overlap and can be written as (1

+

^R)-1/2as used in the

The basic approximation made up is that the

_text;a refers to the a bonding orbitals likep, and (ans ; n refers

ion wave functions are essentially unaltered (except for

to the x bonding orbitals likeg, andp,.

(6)

tals formed by a linear combination of atomic orbitals of the 6 surrounding anions. In table I we give the linear combinations of these orbitals which correspond to the symmetry of the various central ion orbitals.

This is simply a convenience, because only orbitals of corresponding symmetry will have non zero overlap or will mix.

For the IS we will be concerned with the calculation of p l ( ~ ) + pf(0) i. e. the total change density at the nucleus so that only 1 type orbitals centered on ion A will contribute directly.

Using eq. (19) we can write

The first term in (21) is the free ion contribution for which we can use an interpolated value so that the effects of covalency involving other orbitals such as d orbitals on the central ion can be in'cluded. In this way we in fact

go

beyond LCAO. Let us for example consider an Fe3+ ion in an octahedron of 02- ions.

We consider charge transfer to the vacant

d

orbitals of t,, and e, symmetry as well as to the vacant 4s orbitals (a,,) from the 02- 2p orbitals. We then get

For a direct contribution to IS we only have to consider the orbital with a,, symmetry since of those listed in table I this is the only one with a non zero density at the central ion nucleus Sn, is a group overlap integral

=

& < 2 p, I qn, > and D is determined by the ligand-ligand overlap integrals and appears in the normalization constant in the tji. For iron compounds the terms involving the ligand orbitals are usually considerably less than 10 % of the first correction term and can therefore be neglected. The dominant

3

term 2 I v,,(O) l2 will depend on the occupation of

,1= 1

the 3d orbitals. From free ion Hartree Fock cal-

3

culations [12, 131 it is found that 2 Z ₁ pn,(0) 1'

n = l

decreases by about 1.8 a i 3 on going from Fe3+ to Fe2+ mainly due to the screening by the extra d electron of the 3s electrons from the nucleus. We therefore obtain an expression for

where 6n-3, is the change in the occupation of the 3d orbitals and C is the free ions electron density at the nucleus.

6n3, can be obtained from the orbitals of t,, and e, symmetry. Since all overlap and covalency effects are in our notation included in the ligand wave function, 6n3, is simply the pure d character part of I ^{$ i} which is given by

:

12

²

= 2 S;

+

2 N : ~ b;

+ 3(A'L2

S:

+

N , b,)

(23) for high spin Fe3+ with spin up. Here b, and

b,

are the group transfer parameters from the 02- 2p orbitals to the empty e, or t2, Fe 3d orbitals respectively. In this case the group transfer parameters or overlap integrals are related to the single orbitals ones by

b,

⁼

J3

B,

and b,

⁼

J4

B,

.

Let us now look at a few examples. We start with the effects of overlap distortion on iron compounds. It was first demonstrated by Simanek

et

al. [I] following a suggestion by Marshall [I41 that overlap distortion of the iron core orbitals in ionic compounds can strongly influence the s electron density at the nucleus. They showed that the density at the nucleus in the fluorides could be larger by as much as 3 a i 3 in comparison with that for the free ion. By first of all neglecting the charge transfer effects we have determined the overlap distortion contribution Apoy(0) for various trivalent iron compounds. In figure 3 we show a plot of Apov(0)

I

1.9 2.0 2.1 2.2 2.3 2.1 2.5

Fe-Anlon distance

-

FIG. 3. - Charge density at the nucleus for iron in an octahedral coordination for various ligands and as a function

of

the iron- ligand distance. The heavy solid line indicates the range of Fe3+-ligand distances found in nature and the heavy dashed line indicates the range of Fez+-ligand distances found in nature.

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C6-52 G. A. SAWATZKY AND F. VAN DER WOUDE

vs the interatomic distance for the octahedrally coordi-

nated ferric flourides, oxides and chlorides. These results were obtained using Fe3' free ion spin polarized wave functions [15], 0'- wave functions from Watson [16], F - and C1- wave functions were obtained from Clementi [12]. Some of these curves have been published by Simanek, others were obtained from Hupkes [17]. We have taken R

⁼

0.4 for the oxides 0.2 for the flourides and 0.3 for the chlorides.

Similar plots can be obtained for the divalent iron compounds by adding about 7 % to the ferric values.

Also shown in figure 3 are the ranges of Fe- ligand distances encountered in nature. It is to be noted that Apov(0) is then almost independent of the ligand taking into account the appropriate Fe-ligand distances. This is certainly the case of the monovalent ligands. That Apov(0) for the oxides is higher than that for the flourides and chlorides is a result of the larger Coulomb attraction of Fe3' and 02- ions resulting in a relati- vely shorter iron ligand distance.

Since it is expected that the divalent iron flourides are highly ionic we might in this case be justified in neglect- ing covalency effects and comparing directly Apov(0) as shown in figure 3 with the isomer shifts shown in figure 4. The solid line in figure 4 is obtained using the

FIG. 4.

-

Plot of isomer shift versus the iron-ligand distance for divalent iron flourides. The solid curve is obtained

as

explained

in the text.

relation IS

⁼

- a Apov(0) + IS, with

a =

0.10. This value of a is close to the value a

⁼

0.14 obtained by Simanek et al. [2] based on the pressure measurements on KFeF, [18]. It should be noted that the above

Compound

-

Ferrites Fe(B) Ferrites Fe(A) Orthoferrites (Rh203)x(Fe203)1

^{- x}

mentioned rearrangement effects will increase this estimate of a by a factor of 2.

The neglect of charge transfer for the oxides and other halides is certainly not justified. For these sys- tems detailed covalency effect calculations have only been done for the oxides [19-211. From the oxide study a fairly consistent picture has developed with regard to the importance of electron transfer to the vacant 3d and 4s orbitals. In table I1 are listed some of the para- meters found for the ferrites, orthoferrites and the system (Rh,03),(Fe203), -, [22]. These parameters are obtained by comparing isomer shift, hyperfine field and neutron diffraction results. From isomer shift studies alone one could not separate the various contributions.

We find iron effective charges of about + 2.3 for iron in an octahedral environment and 2.0 for a tetrahedral surrounding. We also notice a strong reduction of the spin from the free ion value which will have a strong influence on the magnetic hyperfine field. It should be noted that in doing such calculations it is important to consider the whole cluster of neighboring ions and to determine the normalization constants properly. These normalization constants are far from unity when so much charge transfer is taking place. Using the in table I1 listed values for the transfer parameters and assuming these to be proportional to the respective overlap integrals we can get an idea of how IS is expected to depend on the Fe-0 distance. This is shown in figure 5 using a

=

0.14. The values for q,,(O) were obtained from Clementi's [12] wave functions for Fe in the configuration 3d5 4s'.

Perhaps some of the best examples of the dependence of the IS of Fe on the electronegativity of the anions can be found in the halides. In figure 6 is a plot of the isomer shift vs electronegativity of the anions display- ing a linear relationship. As already mentioned this effect cannot be due to overlap distortion since this is similar for the various halides. In order to obtain information about the relevant importance of 4s and 3d charge transfer a detailed calculation like that described for the oxides would have to be done.

Another very nice correlation of isomer shift and electron negativity is to be found in Sn compounds (Fig. 7) [23]. Because of the strong 5s 5p3 hybridization expected in these compounds the atomic orbitals q i to be used in an LCAO calculation should be of such a hybrid nature. The more ionic the bond the lower will

Some results obtained using the formulas in the text

Average Total

Fe-Odist

A: A:

3d spin 4s occ. charge

- - - - -

2 . 6 ~ 0.125

^-

4.62

pB

0.13 + ^2.44

1.89 A ^0.289 ^- ^4.31 ^0.26 + 2.02

2.01 a ^0.147 ^0.027 ^4.45

^pB

^0.12 + 2.23

2.03 A 0.138

-

4.58

pB

0.13 + 2.40

(8)

1 9 2 0 2 1 2 2 2 3 2 4 2 5 Fe-0 dlstance A

-

FIG. 5. - The isomer shift and hyperfine field at a central iron nucleus in octahedral coordination calculated as a function of the Fe-0 distance. The curve marked Hhpn is for Fe3+ with six antiferromagnetically coupled Fe3+ neighbors, while that marked Hhpf - HSTHW corresponds to Fe3+ in a paramagnetic material.

Paul~ng electronegativity

-

FIG. 6. - Plot of isomer shift versus the electronegativity of the anions for trivalent iron halides.

be the occupation of these hybridized orbitals. Because of the opposing effects of s and p electrons as far as IS is concerned it is

a priori

difficult to say whether IS is expected to increase or to decrease.

Goldanski

et al. [24]

have developed an expression relating the isomer shift in Sn compounds to the popu- lation of the 5s and 5p, and 5p, orbitals. In this case NQR measurements can be of great help since from these the occupation of the p orbitals can be determined.

A rather diffe~ent and new effect seems to play an important role in the low spin iron cyanides (table 111).

Electronegativity difference

FIG. 7. - Isomer shifts (relative to grey tin) for the Sn(1V) compounds as a function of electronegativity difference.

IS

of some low spin iron complexes Shifts at room temperature relative to iron metal

Compound Fe state IS (mm/s) Reference

- - -

-

[a] KERLER, W., NEUWIRTH, W. and FLUCK, E., 2. Phys. 275 (1963) 200, melting point 254 K, corrected to R. T.

[b]

COSTA,

N. L., DANON, J. and

XAVIER,

R. M., J. Phys. Chem.

Sol. 23 (1962) 1783.

The isomer shifts for these compounds is very low and in fact negative in some cases relative to iron metal. The reason for this is probably that a CN- molecule has a low lying empty n* orbital so that some cr back trans-

fer

^>)

from iron to CN- may occur [9]. This back

transfer will cause a decrease in the 3d occupation and

therefore an increase in the charge density at the

nucleus. Very recent

ab initio

calculations done in our

laboratories [25] on low and high spin Fe(CN)4-,

Fe(CN)3- as well as FeF4- and FeF3- in fact support

this arguement to a large extend. It seems, however,

that the back transfer which can occur in both high

spin and low spin cases is larger for the low spin case

which is expected also since here more of the 3d t,,

orbitals, which can mix with the empty CN-

n*

orbi-

(9)

tals, are occupied. The charge densities at the nucleus obtained from the

ab initio

calculations are ploted vs.

the measured isomer shift in figure 8. It is interesting t o note that all the points fall close to the straight line drawn yielding a calibration constant

a =

0.31. Because these

ab initio

calculations include all effects discussed and also because the results obtained for the low spin iron cyanides fall on the same line as those of the high spin florides we feel that this is a very reliable value for

a.

electrons of the nearest neighbors and Pi is the appro- priate antishielding factor. The second term is a sum of the contributions from the nearest neighbor electrons treated as if they are point charges since this has been included in v:. We will assume that the lattice contri- bution is known and will therefore concentrate only on the first term v:. Also once we know how to treat the

zz

component of the e. f. g. we can easily determine the other components. In our method described above for determining the expectation values all the overlap and covalency effects have been lumped into the ligand wave functions leaving the central ion wave functions unaltered. ~2~ can then be further split up into a contribution from the ions own electrons as if it were a free ion VP plus a contribution from the valence ligand electrons which are of course partially delo- calized into the central ion due to charge transfer and overlap distortion effects.

V?

is simply given by

V? =

viZ where the sum is over all occupied

1

orbitals and v:, is

Values for viZ for various atomic orbitals are given in

FIG. 8. - Plot of the calculated charge density at the iron

table IV. As we did in IS we can here again take an

nucleus vs. the measured isomer shift for several high spin and

for <

llr3

> ^take

low spin iron compounds. The cyanides are low spin and the flourides are high spin.

TABLE IV

1. The quadrupole interaction. -

As mentioned in

e. f. g.

for

various atomic orbitals

the introduction the electric quadrupole interaction

involves the electric field gradient (e. f. g.) at the Orbital

- ^Vzzle

^-

nucleus produced by the surrounding electrons and

ions. Such an interaction can be measured with either

^S ⁰

the Mijssbauer effect or if the nucleus has a spin in the

₄

1 ground state larger than + with nuclear quadrupole Pz 31) ₅

_r

resonance (NQR). The total electric field gradient at a

central ion nucleus can in our cluster approximation be 2 1

built up of two distinct parts for the zz component

PX

--(T)

⁵ ^r

namely

:

arising from the surrounding ions of the lattice.

Here Zi e is the charge of the ith ion and

y,

is the Sternheimer !26] antishielding factor for an electric field gradient produced by charges far from the origin, and

where < > means to take the expectation value, the is a sum over all electrons of the central ion plus the

I

(10)

screening effects due to charge transfer into the unoccupied outer orbitals. The occupation numbers of the atomic orbitals will depend on the ligand field splittings of the outermost orbitals and will therefore depend on the crystal structure.

For VZ the approximate formula (20) where all cross terms have been neglected would be essentially the approximation used by Townes and Dailey [27]

namely that v $ ~

⁼

v:, ni whereni is now the occupa-

i

tion number of the ith orbital centered on the central ion. This occupation number will in general be different from one or zero because of the charge transfer from the anion. All matrix elements like <

^*i

I

V,,

I

$i

>

where

$i

is centered on a neighboring ion are then approximated by the point charge value. In this appro- ximation the core orbital contribution to V,, is zero and only the outer orbitals are considered.

The importance of core orbital overlap distortion effects has recently been pointed out by several authors [3-61. In fact for trivalent iron oxides it has been suggested that this overlap distortion effect is as big and in some cases larger than the effect of the rest of the crystal. In all of these calculations certain terms have been neglected. From eq. (19) and (25) the efg due to the ligand orbitals neglecting any charge transfer is given by

where P. C. is the point charge value and R is the Sternheimer antishielding factor for an electric field gradient produced by the outer electrons of an ion. For Fe203 the 3 contributions are

-

2.567,

-

.581 and 3.462 x 10-l4 esu showing a net overlap distortion contribution of .314 x 10-14 esu which is less than 10 % of the point charge contribution from the lattice.

Similar calculations have been done for the iron fluo- rides and chlorides, all of which show this cancellation effect of the various parts of an overlap distortion contributions. Although this calculation was done for iron compounds such a cancellation is probably also going to be strong for other central ions. We are now left with the real covalency effect contributions. We have just shown that at least for iron compounds we can neglect the core orbitals as well as the spatial extent of the orbitals centered on the anions since these corrections all but cancel each other. Considering then only the outermost occupied ligand and vacant cation orbitals, the total covalency contribution to the electric field gradient will be

where

AV$ is the correction for the point charge

contribution from the rest of the lattice resulting from a change in the anion as well as the cation charge. For Fe203 for example the charge of the Fe ion is about

+ 2.4 as obtained from an analysis of the hyperfine interaction data of (Rh203),(Fe203)l-, 1221. A calculation by Artman [28] yields

~,d,(l

-

y,)

= -

0.83 x lO-I4 esu .

Because of the non cubic symmetry in Fe203 the first terms in vEoV will also be non zero. Consider for example only the e, orbitals and let the z axis be thec axis in Fe203. Then because the overlap of the 3d3,2-+

orbital with the ligand 2p orbitals is different from that of the 3dX2-,2 orbital the transfer parameters will also be different and a net contribution to V,, will result. This contribution can be estimated by taking the transfer parameters to be proportional to the overlap integrals. Taking the 2p,

⁴

3d3,2-,2 (z along Fe-0 bond) transfer to be 0.30 as found for the orthoferrites we obtain for the first term in

^O^'^:^v

a value of 0.3

x

10-l4 esu. The netto effect of vEoV is then to increase the efg by about 10 %. The conclusion is that at least for trivalent iron oxides covalency as well as overlap distortion do not seem to be very important as far as the efg is concerned.

We now turn to divalent iron compounds which present a different problem. The free

ion contribution

here will depend on the crystal field splitting and the spin orbit coupling constant. Ingalls [29] has shown that the efg can be written as

where A,, A, and I are crystal field parameters and the spin orbit coupling constant. One way of bringing in covalency effects is to replace < l/r3 >,, ^and ^{1 by}

effective values. Another way is to use the LCAO approach just discussed and see how V,, will be effected by overlap and charge transfer effects including of course a possible change in < l/r3 >,,. If for example the Fez+ ion finds itself in an environment such that only one of the d orbitals is doubly occupied say d,, then charge transfer can only take place to the other empty orbitals which will always tend to decrease the efg since charge transfer effects will dominate overlap effects [30]. For example neglecting core orbitals and terms involving the ligand orbitals, V,, for Fe2+ in a tetragonally distorted octahedron will be

where S,, is the overlap of the dxy wave function

(assumed to be doubly occupied) with the outermost

ligand wave function and B, is the transfer integral to

either the empty

d,,

or

d,,

orbitals. It is rather diffi-

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cult to estimate the size of these effects. We expect

them to be small. In addition to this effect we can also have charge transfer into the vacant e, and 4s orbitals.

This kind of transfer can cause a real expansion in the 3d wave function resulting in a reduction in < l/r3 >

which can be estimated from the change in < ^l/r3 >

for different oxidation states in the free ion. In addition to these effects involving charge transfer in an LCAO description (symmetry restricted covalency) there may also be real changes in the atomic wave function due to the potential produced by the other ions of the lattice.

This kind of covalency central field covalency [31] can cause either a radial expansion or contraction of the wave functions which is in general different for wave functions with different symmetries. The importance of these can be determined from neutron diffraction measurements of the magnetic form factor. For 3d transition metal compounds both contractions of about 17 % in KNiF, [32] as well as expansions (10 %) in the highly covalent sulfides [33] have been observed. These results can be interpreted as indicating an expansion of the 3d orbitals of t2, symmetry and a contraction of those with e, symmetry [34].

Before drawing any conclusions about covalency effects from the quadrupole interactions in Fe2+

compounds one should be very careful to first correct for spin orbit coupling effects [30]. After such correc- tions are made apparent covalency effects are often much reduced.

2. Spin density and magnetic hyperfine interactions.

- Perhaps the most interesting consequences of covalency effects occur in the magnetic hyperfine interactions. Not only does covalency influence the size of the interaction as in IS or the efg but it causes distinctly new contributions which can, by doing the proper experiments, be isolated from the normal interactions. We start by considering only the contact part of the hyperfine interaction which in many cases is the dominant contribution. From eq. (6) we will therefore be interested in the spin density at the nucleus i. e. p,f(0)

-

p:(0). In NMR, ESR and ENDOR experi- ments one usually works in terms of the hyperfine coupling constant which is also proportional to p I ( ~ )

-

pt(0) for the contact part. Experimental and theoretical studies have shown that the contact part of hyperfine coupling constant for a particular atom is independent of the spin on the atom or that pf(0) - p"0) is proportional to the spin. This means that neutron diffraction determinations of the spin can be used to interpret hyperfine field data. Using the same relations as for the isomer shift except that now we subtract the spin contributions we get

neglecting again terms involving the ligand wave function. These were found to contribute little to IS and will therefore contribute even less to the spin density.

As for the IS the first term represents the free ion value which is proportional to the spin of the ion which in turn depends for an iron ion on the occupation of the 3d orbitals. For example for a high spin Fe3+ ion the change in the spin of 3d character due to charge transfer from the outermost ligand orbitals is given by

where again So,, and b,,, are the group overlap and transfer integrals and

N;,,

=

(1

-

s:,,)

-'I2

,

N:,,

=

(1 + ^b:,, + ² b,,, S,,,)-112

For Fe3+ this will result in a correction to the hyperfine field

where

So

is the free ion spin and HFree for Fe3+ is about - 630 kOe. Aside from this it should be noted that the new ligand wave functions also contain some spin density because in our example only spin down density would be transferred from the ligand. This effect gives rise to the transferred hyperfine field on the anion nucleus and contributes strongly to the hyperfine coupling constant, coupling the anion nuclear spin to the central cation electron spin. These effects have been measured by NMR, ESR and ENDOR techniques in numerous systems. From these measurements we can obtain the charge transfer parameters involving the central ion 3d orbitals.

The remaining terms in eq. (30) are contributions from overlap distortion of the spin polarized core orbitals and charge transfer to the spin polarized for Fe 4s orbitals which give a direct contribution to the contact field. The spin polarization of the 4s orbital is expected to give a field of about + 420 kOe (35) for a full 4s shell. The overlap distortion contribution itself is small.

Before we discuss these contributions in more detail

there is another new contribution which arises when we

take into account more distant cations if these also have

a spin. This is the so called super transferred hyperfine

field (HsTHF). It is called super transferred because the

spin density here is transferred from one ion to another

via an intermediate ion in a similar process as that

proposed for super-exchange and can in fact be used

to obtain information about the super-exchange

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interactions. Owen and Taylor [36] and Huang and Orbach and Simanek 1371 were the first to show that the spin transfer from one metal ion via the ligand to another metal ion can produce large contributions to H,,,. The direction of HsTHF is parallel to the magnetic moment of the neighboring cation and therefore enhances the contact field if the magnetic moments are coupled antiferromagnetically.

For Fe ions in LaFeO,, Simanek et al. [37] calculated a large value of HsTHF

=

- 55 kOe. A direct observa- tion of HsTHF

=

210 kOe at Sn119 nuclei in the material CaxY3 -,(Fe3)Fe2 -,Sn,OI2 has been reported by Belov et al. [38] and Goldanskii et al. 1391. More recently, Evans et al. [40] have found HsTHF

=

300 kOe at Sb121 nuclei in Sb-substituted ferrites. Streever and Uriano have found HsTHF

^N

20 kOe at Ga nuclei in gallium-substituted garnets [41] Goldanskii et al. have shown that the direction of the hyperfine field at Sn119 nuclei in B sites is parallel to the magnetic moment of the A-site ions in garnets 1421. Since HsThF is also dependent on the kind of nearest-neighbor cations, it also explains the experimental results in CoFe204 and MnFe,04 at 0 K [43-451.

The super transferred hyperfine field is caused by the overlap distortions of the central cation s orbitals by the ligand orbitals which have been unpaired by trans- fer into unoccupied 3d orbitals of the neighboring cations. The expressions to be derived for HsTHF here are different from those obtained by Simanek et al. 1461 in that the transfer into the iron 4s orbitals has been taken into account. The inclusion of the 4s orbitals results in normalization constants for the molecular orbitals which are much smaller than 1, so that the approximations made in reference [46] are no longer valid.

To obtain an expression for H,,,, consider a linear molecule as pictured in figure 9 where

A is now an

Fe3+ ion spin down and B is the cation for which we

FIG. 9. - Diagram of a triatomic molecule A-0-B showing the orbitals and spin transfer to be considered for calculating HSTHF.

want to determine the hyperfine field. In forming the molecular orbitals we will now have to take into account that the spin up ligand electrons can transfer into the empty Fe3+ orbitals whereas the spin down ones cannot. This means that the overlap distortion of the B cation spin up s core orbitals will be less than from that of the spin down s orbitals causing a net increase in the spin down density at the nucleus.

Consider now a cluster composed of a central Fe3+

ion in an octahedron of 6

02-

ions which in turn are

surrounded by 6 Fe3+ ions with an Fe3+ -0-Fe3+ bond angle 6 antiferromagnetically coupled to the central Fe3' ion as shown in figure 10. It has been shown

FIG. 10. - The arrangement of cations and anions as used in the calculations of the hyperfine field and the isomer shift.

using the LCAO description that the spin density at the central iron nucleus is to good approximation given by 1201

p z ( ~ ) - pf(0)

x

6 N ~ ( P '

-

P1)

^-

-

6N4 PT ⁽

(A: ^-

A:) cos2

0

+ A: ) ₍₃₂₎

where

B and Snow being the single bond transfer and overlap parameters and

+

^(B:

+ ^{2 3,}

^{S,) sin2}^B

The second term in ,o,f(0)

-

pi(0) is the so called super- transferred hyperfine field. For

B = 1800

we get a result very siniilar to that obtained from the single bond treatment. In our approach however we simul- taneously get the contribution due to the occupation of a spin polarized 4s orbital and the ordinary covalency effects on the hyperfine field.

We can now write

(13)

where

of which the dominant contribution comes from the 4s term since a,, is usually much larger than

S,,

and

is a function of the covalency parameters A, and A, of the neighboring cations and on the Fe-0-Fe bond angle. As we shall see below the bond angle dependence is really quite important. It has been shown that HsTHF is really quite insensitive to the occupation of the 4s orbitals because of the strong dependence of N on

a,,

(8). In this case we can deter- mine (A: - A:) cos2

8

+ A: directly if we can independently measure HsTHF. HsTHF can be determined independently by substituting some non magnetic ions for the Fe ions. The relation for HsTHF can in this case be written as (20)

where I is the number of iron nearest neighbors. These measurements have been done for iron in the A and B sites of the ferrimagnetic spinels [43-451. For these cases we found A:(B)

=

0.125 and using similar rela- tions for iron in the tetrahedral sites we find

A:(A) =

0.289 taking A,

=

0. Also measurements done on (Rh,O,),(Fe,O,),-, [22] yield

A: =

0.138 again taking A,

⁼

0. Once these values are known we can use them to determine the change in the 3d spin (eq. (3 1)) and using then relation (34) and the measured Hhpf we can determine

a,,

taking H,,,,

= -

630 kOe.

This value of

a,,

can then be used to determine the isomer shift. This is in fact how the covalency para- meters of table I1 were determined. From eq. (36) we also see that HsTHF and therefore H,,, should depend on the Fe-0-Fe bond angle. The orthoferrites MFeO, provide an excellent test for this theory because in this series M

=

La to M = L u the Fe-0 distance remains constant according to recent X ray measure- ments [47] while the Fe-0-Fe bond angles change considerably. Also the hyperfine field decreases from

-

564 kOe in LaFeO, to

-

548 kOe in LuFeO, [48].

Recent 'measurements of the isomer shift [20] show that it is constant to within

-t

.006 mm/s in excellent agreement with eq. (21) and the fact that the Fe-0 distance remains constant. In figure (11) we have plotted H,,, as measured vs cos2

8

obtained from X ray measurements. The agreement with eq. (36) is excellent. We can derive the same sort of expres-

cos 2 0

-

FIG. 11. -

Measured hyperfine fields

vs cos2 0

for the series of orthoferrites.

sions (20) relating the NCel temperature to cos2

8 ;

a plot of TN vs cos2 8 also results in a straight line (Fig. 12) from which we can obtain < A

⁼

0.2. Using this result and the slope of the H,,, vs cos2 A,

8

line we find A:

⁼

0.147 and from H,,, (eq. (34)) we find a,,

=

0.14. These values for A: and A: (table 11) can be used to determine the spin of the ion eq. (31) resulting in

S =

0.89 So, in agreement with the by neutron diffrac- tion measured value [49]. It should be noted that expe- riments and calculations like these yield the informa- tion about the charge of an Fe ion which we have used in the electric field gradient calculations.

In order to be a little more general we have calculated the dependence of the hyperfine field and isomer shift for iron in octahedral coordination for various Fe-0 distances. We take the Fe-0-Fe bond to be 1800 and assume that the central iron ion is antiferromagneti- cally coupled to six neighboring iron ions. The transfer parameters B,, B, and

C4,

are taken to be proportional to the respective overlap integrals using proportionality constants obtained from the orthoferrite results. In figure 5 we have plotted H,,, and IS vs Fe-0 distance using Simaneks [2] value of R

=

0.14 relating the isomer shift to the charge density at the nucleus.

Since the hyperfine field is quite sensitive to HsmF

which is strongly dependent on the kind of neighboring

cations and on the bond angle we have also shown

(14)

I

.5 .6 .7 .8 .9

2 -

cos (3

Hazony

et al. [50-541 ,have proposed a completely

different approach for explaining the hyperfine interac- tions in ferrous iron compounds. They suggest that a radial expansion of especially the t2, iron orbitals are responsible for the changes in the hyperfine interactions.

Although part of the trends observed may indeed be due to small radial expansions of the 3d orbitals we feel that most of the effects observed can be explained in terms of an LCAO picture as just discussed. The changes in the hyperfine interactions will in our picture be produced by charge transfer from the ligand to the central iron ion 3d t,, and e, as well as the 4s orbitals. These effects could of course all be lumped into an effective < l/r3 > in the expressions for the electric field gradient or in the dipole or orbital contri- butions to the magnetic hyperfine field. In this way however the real physical effects which we know do occur remain hidden.

Another important approximation is that we have assumed that the solid can be built up of clusters of the central ion plus its nearest neighbors. This approxima- tion is only valid if the interaction between the clusters is not too large as compared to the interaction with the central ion. In the more covalent compounds like sulfides, selenides and tellurides the interaction between the anions themselves is very large as can be seen from the large one electron band widths

(>

5 eV) found for

0

the sulfur 3p bands in transition metal sulfides in photoemission experiments. These band widths are probably much larger than the anion cation interaction in which case such a local cluster approximation breaks

FIG. 12. - Measured NBel temperatures vs cosz 6'for the series

of orthoferrites.

down. For these it is probably better to start with a close packed S2- lattice resulting in a fairly wide S3, H,,,-HSTHF as a function of the Fe-0 distance. I t is

obvious that the covalency effects affect Hbpf-HSTHF and the isomer shift in much the same way. Before one can compare these results with experiment however one must correct the measured hyperfine fields for the dipolar and orbital contributions and to obtain the true isomer shift corrections for the zero point motion of the nucleus have to be made.

In this review we have tried to discuss most of the important mechanisms which can influence hyperfine interactions when free ions are bound to form a compound. We have restricted ourselves to a LCAO description although we have included effects resulting from expansions or contractions of atomic orbitals as a result of the change in the total number of electrons on the ion due to covalency effects.

band and consider the small transition metal ion to be situated in the holes of this lattice with localized 3d orbitals. We now can switch on the interaction or covalent mixing between the S2- 3p band and the metal 3d orbitals. The charge transfer can now no longer be considered as a transfer of electrons from the nearest neighbor S2- ions but from delocalized elec- trons in the S2-3p band. We could ti-eat this problem in a manner similar to the Anderson [55] treatment of impurities in metals. We would then expect the spin and charge density disturbances in the S2- 3p band produced by a central metal ion to be very extended in range and probably oscillatory as found for magnetic impurities in metals. Such longe range spin density disturbances will also result in long range super- exchange interactions and supertransferred hyperfine interactions.

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