On the mass operator in a diagonally disordered binary alloy

Volume 76A, number 1

PHYSICS LETTERS

3 March 1980

ON THE MASS OPERATOR IN A DIAGONALLY DISORDERED BINARY ALLOY * J. KROPIWNICKI and L. KOWALEWSKI Solid State Division, Institute of Physics, A. Mickiewicz University, 60- 769 Poznañ, Poland Received 13 November 1979

It is shown that the multiple-occupancy corrections imply identities between skeleton diagrams generating a perturbation calculus for the mass operator M. Self-consistent equations forM 6 where z is the coordination number.11,Mi,i+i are obtained. The corrections to the obtained M beginwith terms of order z

1. In the present paper we consider a model for a binary alloy AB with diagonal disorder. The one-electron hamiltonian is of the form ~ H— ~ + + _r, IT (I\ 1a~ a1 efl1a~a1 ~o V ~ —

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corrections” into account. As a result of their analysis of the diagrams obtained forM with respect to z~ (if hopping occurs only between nearest neighbours, then G.. = O(zIi/I)), Schwartz and Siggia pointed U out the importance of these corrections, and moreover, they showed that2(Ithe corrections to site CPAoccupations start with c)2 z3 if the terms of orderindependent c are mutually (c is the concentration of B atoms). Next the authors attempted to sum the dia~ grams to obtain a result with accuracy c2(l c)2z3, in which they succeeded using additional approximations for the appearing cumulants. In the present paper we show that: (i) a Schwinger equation can be obtained in a simple manner, (ii) the identity n~ = forces identities between certain sets of diagrams, (iii) the existence of the former generates a perturbation calculus forM, (iv) the perturbation calculus is correlated with the parameter c(l c) and, if the site occupations are mutually independent, with the parameter z1, (v) in the first order of the perturbation calculus we obtain equations for M 11,M~~ The corrections to the solution of these equations 2(l c)2z6. begin with terms of order c 2. In order to simplify the formulas we consider cubic lattices and assume that the concentration c of B atoms is site independent, i.e. ~ c, and, analogously, = d 1,~= d1_1. For calculational purposes it is useful to introduce averaged Green functions G defined as follows: —

where i~1•of takes the values or 1 if site i isThe occupied by an atom species A or B,0 respectively. problem consists in finding the Green function (resolvent) averaged over all atomic configurations. Ifwe-limit ourselves to single-particle states, then the following equality is valid: ‘E H = ~E H A~ —1 = G ‘2~ 0 11/ ‘~ 0 fl/ ii ‘- ‘ where the introduced mass operator ~ (proper selfenergy) does not depend on the random variables but depends upon a complex energy and is a quantity which should be calculated, After the CPA equation for the mass operator was obtained, many efforts have been put into the derivation of a better approximation, see ref. [1] and refer~

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‘

ences therein. In the summation of the diagrams for Mthe essential difficulty site is formed the “multipleoccupancy corrections”: i can bebyoccupied by an A or a B atom but not by two atoms simultaneously, i.e., = In particular, in ref. [2] a Schwinger equation was obtained forM, the iterative solution of which automatically takes the “multiple-occupancy ~

‘~

~.

This work has been supported by the Institute of Physics of the Polish Academy of Science.

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m

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61

Volume 76A, number 1

G~j~

x

=

PHYSICS LETTERS

ni~.nj(E— H

0

exp

—

3. We will show now that the identities ~

(~

avn~)

aMfl~

=

generate relations between the derivatives of M which

)]

[exp(~

3 March 1980

—i

(3)

The parameters a have a formal character and after performing the calculations we put a = 0. If a ~ 0 then the concentration of B atoms

can be naturally exploited the(6), construction a perturbation calculus. Fromineq. due to theofidentity ~ = ~ we obtain the following relation between the diagonal and off-diagonal matrix elements of M: M~1= cc1

+e

—

~MivGvMM,ji.

(8)

c1 = aailn ~

Differentiating eq. (5a) [(Sb)] with respect to the

depends upon a and the site number. The function G11 given by eq. (3) satisfies the following equation of motion:

parameter ~ [at], after using eq. (7a) [(7b)], we derive a set of equations. With the help of these equations one can express the derivatives of the form acejMji, acs1Mji in terms of those of the form:

EG11 = ~ij~+

E

+ ~

~

=

+ +

eG~ (4a) (4b)

~

Introducing the mass operator M with the aid of the following expression: M1,

=

e ~ G~~G~’ =

0~

~ aa

(9b)

~

1

—

~

‘YkJ

k*j~jl7jf’ e~c 1(l c1) +

E

—

1 = E~ we find a solution of eq. (4) in the form G17 11 M1~,and obtain the following formula for M: —

(9a)

(5a,b)

E G~G IV

where k ~ I ~ 1,7~k arriving aMk/finally at: a~I~1 ~ 8a~ k*i1—7~i~j’

~o~i

—

~—

k,l

‘f~

(~i) 1

—

aMkl

~

~3a~1

—

.

(6)

On the other hand, taking the derivative with respect to the parameter a we have: = CkGi/ + Gck) and, therefore, the equalities (5) can be rewritten in the following form:

~G1~(aM~,Iaa1)

(7a)

ec1~1+ ~(aM~~/aaj) G~1.

(7b)

M~1= ec1t511 + =

v

The thus obtained Schwinger equations (7) may. serve to generate the skeleton diagrams. With this aim one should solve eq. (7) by successive iteration starting withM~= c1611. The equation for aajMv, can be obtamed by differentiating eqs. (7a,b); after solving them by iteration we arrive a new classofof&,~M; diagrams etc. Finding M requires theatknowledge finding ~M requires the knowledge of &~ aaM, etc.

7i~ —

(9c)

—

M~1= ec~~11 + e2G~Y/) ~MivGp~M~q

=

~‘

e(1

c) G11

—

(10)

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—

Analogously, differentiating eq. (8) with respect to ak (k * i) (11) and invoking we obtain the set of equations enabling eq. us to(9)express the derivatives of the form aakMjj in terms of those of the form ~c~~kM1j (1* k ~f ~

E IA

~~15

l,s L 11(k) l*k*s =

dik

+

A~(k) B11

-~-

(k)

—

Xjk

8M15 B~(k)] aak —

(11) _______

A11(k)

=

1(k)

=

i~j~ + r~1 +

‘e

B

—

1 ‘fsk —‘f

kk

A 11 (k) =

+

GIk

‘fkk

~ 1

(12a)

,

‘fkl7kk)\

—

2

1 62

‘fik

“fkk

GkiJ, /

(12b)

Volume 76A, number 1

2c(1 ~jk = e

r~=

—

PHYSICS LETTERS

c)(~e2G,kGkj 1 ~ kk

—

rlkrkl) ,

(l2c)

—

M1~G~1.

+

(12d~

V

After solving ea.derivatives (11) we can function of the of expressM~ the form ~(I ‘~j) (Iaszrak ‘rj z I). Repeating the process with obvious changes we can successively express the latter as functions of the derivatives aajaajMkl (with all indices different) etc. For a finite crystal the process enables us to solve the problem exactly. The treatment described above can be exploited to construct the perturbation calculus with the assumption that in the nth order of the calculus the derivatives of the nth order are equal to zero if all their indices are different from each other. In the zeroth order of the perturbation treatment we put = I ~r/. Due to eq. (8) we immediately obtain the CPA equation forM11 [3].I In kthe assume that = 0 with ‘?=/next ~ I. step This we approx0 if

imation leads toM with indices coupling all the crystal sites. the mass operator isobtains to couple nearest neighboursIf only, the procedure an additional approximation, namely ~~M,11= 0 if Il —ii ~ 2. Assuming that the derivative (a = 0) is not direction dependent and using eqs. (9a) and (7a), we obtain the following equations: M 11 = cc + -~

4. Let us now estimate the order of the corrections to the solutions of the set of equations (13) with respect to c(1 c) and z~. Expanding in diagrams we can

— ~G 11

M-

3 March 1980

~Mo~G~o

—

EMOVGVMM~o; (13a)

701

/ =euG

01 +G00

1

\

-

-

M~= [c~(1 c)2z31’l I] aa.M-- = O[c2(1 0

—

—

‘

~

acskMij = O[c3(l (I ~ ~ k r i)

—

c)3z_2_k~_~1

Taking M 2 11 = 0Calculating when I I /8~ I > 2 we neglect 0 [c X (1 c)2z6]. 1M1+1,1+1we neglect aajMj+2,j+2 and aalMk,i+1 which an maccu2(1 c)2z5] (eq. (11))leads and to ofO[c2(1 racy ofO[c c)2z6] in the calculation ofM 1 ~ (eq. (7)). The neglect of ôajMk 1+1 3(1 in the calculation ofM,,,÷1 c)3z6] (eq. (7)). Thegives rise to an error of O[c inaccuracy ofM~,is proportional to that ofM 11~1 (eq. (8)). Consequently, we conclude that the corrections to 2 the of eq. (13) begin with terms of order C X (Isolution c)2z6. We can suppose, by analogy, that in the case of spin =1 (Sj~’~ systems (S~)~’~ 0, canthe alsofollowing generate conditions: a perturbation calculus. A presentation of the analytical aspects and a more —

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detailed discussion of the perturbation theory for alloys is in preparation.

d01 +X01 X

—

.

7o~/

—

easily show that the power of c(1 c) depends upon the number of different indices, whereas the power of 1 depends upon the distance between the indices z(if the site occupations are mutually independent); in particular

(13b) [A i~(0)A~1(0) B1~(0)B~1(0)] —

M~,0,

Ii—/I~2

(13c)

.

From eqs. (13) and (12) we can derive M11 and M11÷1. The CPA equation is exact for a “one-site” crystal. Eqs. (13) are exact for a “two-site” crystal and thus we can expect them to include the bonding and antibonding states of a two-atom molecule embedded in an effective medium. -

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.

The authors would like to thank Dr. Dr. A. LehmannSzweykowska and R. Micnas for interest and discussions on this work. References

[1] R.J. Elliot, J.A. Krumhansl and P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [2] L. Schwartz and E. Siggia, Phys. Rev. B5 (1972) 383. [3] R.N. Aiyer, R.J. Elliot, J.A. Krumhansl and P.L. Leath, Phys. Rev. 181 (1969) 1006.

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