OVERLAP DISTORTION CONTRIBUTION TO THE ELECTRIC FIELD GRADIENT

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OVERLAP DISTORTION CONTRIBUTION TO THE ELECTRIC FIELD GRADIENT

G. Sawatzky, F. van der Woude, J. Hupkes

To cite this version:

G. Sawatzky, F. van der Woude, J. Hupkes. OVERLAP DISTORTION CONTRIBUTION TO THE ELECTRIC FIELD GRADIENT. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-276-C1-277.

�10.1051/jphyscol:1971192�. �jpa-00214519�

JOURNAL DE PHYSIQUE

Colloque C 1, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 276

OVERLAP DISTORTION CONTRIBUTION TO THE ELECTRIC FIELD GRADIENT (*) G. A. SAWATZKY (**), F. van der WOUDE and J. HUPKES Solid State Physics Laboratory, University of Groningen, The Netherlands

Rhurn6.

- Les contributions au gradient du champ Clectronique aperqu par les noyaux de fer cause par la distor- tion de recouvrement des couches compl&tes des atomes de fer sont discutees. En tenant compte de ces effets on obtient Q(Fe57)

= 0,174

barns.

Abstract. - The contributions to the electric field gradient at iron nuclei due to overlap distortion of the iron closed shell orbitals are discussed. Including these effects we obtain Q(Fe57)

= 0.174

barns.

It is probably well known that the quadrupole moment Q(Fe57) of the Fe57 nucleus as determined from iron compounds seems to depend on the valency state of the particular iron ions. This, of course, is only due to the uncertainty in the determination of the electric field gradient at the iron nucleus. The most recent determination of Q(Fe57) from measu- rements in Fez' compounds has led to a value of Q(Fe57)

=

0.21 + 0.03 barns [I]. On the other hand from measurements in cr-Fe,03 and electric field gradient calculations assuming point ions and induced electric dipoles

(- q a

a value of

Q(Fe57)

=

0.28 k 0.03 barns [2]

has been determined. Recent calculations by Raymond and Hafner [3] have shown that the inclusion of induced quadrupole moments on the 02- ions

may increase the electric field gradient in a-Fe203 considerably. The large uncertainty in the quadru- pole polarizability of 02- makes it imp'ossible to give a reasonable estimate for this effect. However, in the determination of the quadrupole moment of FeS7 from hyperfine interactions in compounds the contributions to the electric field gradient arising from covalency and overlap of the atomic orbitals have usually been neglected. It was first demonstrated by Taylor [4] and independently by the authors [5]

that in Al,03 overlap distortions of the closed shell aluminium orbitals result in large contributions to the electric field gradient

(- q:,)

which, when added to

ql

explains the quadrupole coupling constant measured for AlZ7 in A1203.

In ref. [5] we mentioned that q:, can also be large in Fe3+ compounds. The expression derived in ref. [5]

is only valid for a single closed p shell surrounded by N 0'- ions. The situation in iron compounds is however more complicated because all the electrons in the 2p, 3p and 3d orbitals can contribute to

qAV.

In this case the contributions to q:, from the various shells are not additive, because cross terms involving the 2p and 3p 'orbitals will appear. The orthogo- nalized one electron wave functions can be written as follows

*5

^{= N j , k}

[

^(Pj

zstj~i] ⁽⁰

1

(*)

Supported by the Foundation for Fundamental Research

of

Matter

(FOM)

of the Netherlands.

(**)

Nahonal Research Council of Canada

(NATO),

Post- doctoral Fellow.

where

(P;

is a free ion wave function centered on the kth 02- ion, xi are the wave functions centered on the central Fe ion and

sffj

^{= < X i}

I

^(Pi k

> .

Only the occupied oxygen and iron orbitals are consi- dered. The

t j f

are now orthogonal to the iron one electron wave functions. If anion-anion overlap and terms which involve the overlap integrals to second order like stj stj are neglected, the wave functions

$jk can be treated as being orthogonal.

q:,

can now be determined from the sum of the one electron contributions.

4:"

=

- eC[<(Pjkl~zOzI(Pf >I ^{- 2 C}

^x

j , k i

x

stj <

^(P51

v,",

^{I X i}

> +

+ C stj Sf,, <

^Xi

I v: I

^Xi,

> ⁽²⁾

i,i'

where e is the charge of a positron,

v,9,

⁼

^{(3 c0s2 8} - l)/r3 ,

and

B

is the angle which r makes with the main prin- cipal axis (z-axis) of the electric field gradient tensor.

In eq. (2) we have taken Nj,,

=

1. The first term on the right hand side of eq. (2) involves only oxygen orbitals and has been included in q,. The second term has been determined by numerical integration [6]

and amounts to only 1 % of the third term, so it will be neglected. We are then left with

which is zero if xi and/or xi, are s orbitals or if xiand

xi. are orbitals having different quantum numbers I or 1,. Since < ^l/r3

^{> 3 d}

is one tenth of < ^l/r3

^>3p

and the overlap integrals involving the 3d orbitals are smaller than those involving the 3p orbitals we can neglect the terms involving 3d orbitab.

Following the same procedure as described in ref. [S] we get

where the factor of 2 results from the orbitals being doubly occupied ;

8, is the angle between the z-axis

and the Fe-kth 02- direction and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971192

OVERLAP DISTORTION CONTRIBUTION TO THE ELECTRIC FIELD GRADIENT C 1

-

277

for the corresponding F e - 0 distances and the values for (3 cos2 Ok -

1)

using the same lattice parameters and 02- and Fe3+ special position parameters as Artman et a1 121. From these values we find from eq. [4] for cr-Fe203 that

q:, =

4.584 x lot4 esu.

The strong dependence of

q;,

on the oxygen and iron special position parameters, x and z respectively can be seen in figure 1 . This plot of qA, and the corres- ponding plot of the lattice contribution to the electric field gradient

-

q; [2] are very similar.

In order to find the total electric field gradient a t the ~ e nucleus the shielding of the field gradient ~ "

by the electrons of the ion has to be taken into account.

The total electric field gradient at the nucleus can be written as

($2,)

=

(s;;)~ - + (s$)~ z

iron Parameter

10 ^.350 ,352 .354 ,356 358 .360

(s;,)~

=

(s;;)~ - (s;;)* + ( s z ) ~

(sip s:,)

^{= (S;;}

^S;;) - (S;; S;;) + (SG S Z )

cx

1d4)

9

- Here,

k 8 -

SEi

=

< ^np,, I 2pE, >,, ; S :

⁼

< ^np,, I 2p,, >,

and

7 -

Szk,

=

< n p , ( 2 sk > .

6

-

The z' direction has been taken to be along the

Fe-kth 0'- direction. The orbitals with superscripts

_{5 -}

are centered on the 0'- ion.

The overlap integrals were calculated analytically

_{4 -}

using Fe3+ [7] and 02- [S] free ion wave functions.

In table I are listed the various overlap integrals

TABLE I

.-

Since the shielding factor R of q;, is not known, we

b

^{2 -}

Overlap integrals for the two Fe-0 distances in

a-Fe20,. The symbols used are defined in the text.

₁

0 .295 .300 ,305 ,310

X Oxygen Parameter

-

FIG. 1. - Dependence of q;, on the iron and oxygen special position parameters. The special position parameters for a-FetOs were taken to be x = 0.307 2 and z = 0.355 28.

have chosen two values of R for the calculation of

q

namely R

=

0 and R

=

0.32. This last value is the shielding factor of a 3d electron in Fe2+ 19, 101. Further we have taken

^y, =

- 9.14 [ l l ] and

q; =

0.491 x 1014 esu [2]. The total electric field gradient at a Fe57 nucleus and the ~ e ' ' quadrupole coupling constant using R

=

0 are q

=

9.536 x 1014 esu and Q(F~")

=

0.147 barns, and using

R =

0.32 are

q =

8.096 x lo1" esu and Q(Fe57)

=

0.174 barns. The latter value is in reaso- nable agreement with the indirectly determined value for Q(Fe57) from measurements on spinels [12].

These results show that the overlap distortion of the closed shell 2p and 3p orbitals of a Fe3+-ion strongly influences the electric field gradient a t the nucleus and that the inclusion of these effects result in a rea- sonable agreement between Q(FeS7) values determined from trivalent and divalent iron compounds.

References

INGALLS

(R.), P h y ~ . Rev., 1969, 188, 1045. [7]

CLEMENTI

(E.), I. B. M. J. Res. and Dev., 1965, 9 , 2.

ARTMAN

(J. O.),

MUIR

(A. H.)

and WIEDERSICH

(H.), [8]

WATSON

(R. E.), Phys. Rev., 1958, 111, 1108.

Phys. Rev., 1968,

173,

337. [9]

FREEMAN

(A. J.),

WATSON (R.

E.), Phys. Rev., 1963, r31

RAYMOND (M.)

and

HAFNER (S. S.).

P ~ v s . Rev.. 131. 2566.

- -

^{\ - ,> -} , - - - ., - - - - - - - .

1970, ~ ' 1 , '979. [lo] INGALL; (R.), Phys. Rev., 1962,

128,

1155.

141

TAYLOR

(D. R.), J. Chem. Phys., 1968,

48,

^536. [ l l ]

STERNHEIMER

(R. M.), Phys. Rev., 1963,

130,

1423.

151

SAWATZKY (G.

A.)

and HUPKES

(J.), Phys. Rev. 1121

ROSENBERG (M.), MANDACHE

(S.),

NICULESCU-MEJEW-

Lett., 1970, 25, 100. SKA (H.),

FILOTTI

(G.)

and GOMOLEA (V.),

J.

[6]

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(A. 3. H.), Private

communication.

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1114.