U - U _ w(p) J J V

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(5.3.61a)

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be the coherent Green's operator and let

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(5.3.61b)

be the modified potential operator for each scatterer. Then (5.3.60) becomes

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(5.3.62)

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G = Gc + Gc

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"' U G

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(5.3.63)

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is the N-particle scattering equation. Equation (5.3.63) is analogous to the -original N-particle scattering equation, with Go replaced by G c and U j replaced by U j' The new transition operator is

(5.3.64)

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The process of taking configurational averages and truncating the hierarchy of equations can be repeated, giving new dispersion relations. However, theHe new dispersion relations depend on the choice of the coherent potential operator w(p), which has not yet been determined. The consistent choice for w(P) or the coherent potential choice is choosing w(p) such that

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(5.3.65)

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Hence, the final result of the averaged Green'H operator is equal to the original coherent Green's operator. The coherent potential shall be introduced into the effective field approximation and the quasi-crystalline approximation.

3.4 Coherent Potential (CP)

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Effective Field Approximation with Coherent Potential (EFA-CP)

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When the effective field approximation is applied to the N-particle scattering equation (5.3.63) in a manner completely analogous to Section 3.2, with Ge playing the role of Go and fJ playing the role of U, the result for the averaged Green's operator in momentum representation is

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G{p)

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[G~1 (p) _ no1'p{p,P)] -1

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(5.3.66)

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The coherent potential choice of w{p) is such that (5.3.65) is satisfied, which, in view of (5.3.66), imposes the condition (5.3.67) That is, the momentum representation of T with both arguments equal to p is zero. By using (5.3.61b) and l' = fJ + l' G e fJ it follows that l' = 'fij(P) +

l' G e (U - 'fij;))

so that

~ = ~ -1 w p TGeG +v =

where

(=I+TGe) = : -= U

(5.3.68)

c = (G~1 + ~ )

Define

(5.3.69)

t that satisfies the Lippmann-Schwinger equation

i=u+uci

(1' GeC -1

Simplifying we get

(5.3.70)

Eliminating U from (5.3.68) and (5.3.70) and using (5.3.69), we have

+ ~)

(1 + c i)

i + l' GJ

and then

(1 - wp{p)G\i c Ge V e w PC GV 'LX)

-1 _

(5.3.71a)

5 MULTIPLE SCATTERING THEORY FOR DISCRETE SCATTERERS

We take in momentum representation and define (pIC elp1) -

= Ce(P)(plp1)

and (pIGlp1) = G(P)(plp1)' To normalize the resulting equation, we multiply by (l/V) e-i(P-Pl) rj and integrate over Mj, noting that J Mj(l/V) e-i(P-Pl) rJ = (P!P1)/V and J Mj = V. Thus

Tp(p,p) =

[G~1 (p) G(p) lp(p,p) - W(p)]

G(p) G~1 (p)

(5.3.71b)

Hence, the coherent potential condition of Tp(p, p) = 0 implies that w(P) =

Ge (p)G(p)tp(p,p). Furthermore, by taking V ---t 00 in (5.3.69), and noting that wpCe(p) remains finite, we have G(p) ---t Ce(p). Thus the choice of

coherent potential is

W(p) = tp(p,p)

(5.3.72)

Hence, the final results for the averaged Green's operator are (5.3.61a) with

w(p) given by (5.3.72). The quantity t satisfies (5.3.70), the momentum representation of which becomes (on letting V ---t (0) lp(P1,P2) = Up(P1,P2) +

J(~):3

Up(PllP3) Ge(P3)lp(P3,P2)

(5.3.73)

The dispersion relation for EFA-CP is det Since G e depends on (5.3.73) is nonlinear.

[G~l (p) -

nJp(p,p)] = 0

(5.3.74)

via (5.3.61a) and (5.3.72), the integral equation

3.5 Quasi-crystalline Approximation with Coherent Potential (QCA-CP) The quasi-crystalline approximation can be applied to the N-particle scattering equation (5.3.63). Manipulations are performed as in Section 3.3 with

Ge playing the role of Go and

averaged Green's operator is

playing the role of U. The result for the

G(P) =

where

[G~1 (p) _ nocp(p,P)] -1

(5.3.75)

(5.3.76)

3.5 QCA-CP

Putting

in terms of U j , we have

(5; =

Ce if

j.7i'j

Uj -

df/q(r/-rj)Uj Gi5/-

df/q(r/-rj) ~Gi5/

(5.3.77)

e-i'j).'i'j

Taking (5.3.77) in momentum representation and noting that gives

C p(pI,p2) = UP(pI' 152) +

where

J(~)3

UP(pI' p3)Gc(p3)qp(p3 - p2)Cp (p3, 152)

(5.3.78)

~ P, PI Cp(- -) =

[=1 + 'Wp(p) Gc (-) qp (- - PI . C-p(- -) + 'Wp(p) (-1-) (5 .. 79) 3 P P -)] P, PI P PI

---v-

---v-

Next, use (5.3.79) to express C p in terms of Cpo We obtain

Cp(p, PI) =

[I + 'W~p) Gc(p)qp(p - PI)] Cp(p, pd _[I + 'W~p) Gc(p)qp(O)] 'W~p) (15 I PI)

(5.3.80)

Putting (5.3.80) in (5.3.78), we then have

Cp(pI' 152) = U P(pI ,p2)G c(p2)

J(~)3

[I + Wp~2) Gc(p2)qp(0)

r G~\p2)

U P(pI' p3)Gc(p3)qp(p3 - 152)

. [1 +

= 'WP(p3)=

Gc(p3)qp(p3 - 152)

=-1 _

] -1::::Cp(p3,p2)

(5.3.81)

The QCA approximation gives (5.3.82) G (15) = G c (15) - noCp(p,p) For coherent potential condition to apply, we choose 'W p such that

Cp(p,P) = 0

(5.3.83)

which we will impose on (5.3.79). To normalize properly (5.3.79), we use the technique as in EFA-CP by multiplying both sides of the equation with (ljV)e- i (p-p\).r j and integrating over dfj . Hence, (5.3.79) becomes

Cp(p, 15) =

[I + 'W~p)

Gc(P) qp(O)] Cp(p, 15) + 'Wp(p)

(5.3.84)