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Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
ELECTRON SELF-ENERGY, FERMI SURFACE CHANGES AND GENERALIZED GOLDEN RULE IN DILUTE ALLOYS A. LODDER Natuurkundig Laboratorium, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands Received 28 February 1979
The decay of a travelling wave in a dilute alloy has been studied by evaluating the matrix element of the system's Green function diagonal in the state which is prepared to be the travelling wave at an initial time. This quantity has a pole at a complex energy equal to the sum of the unperturbed energy of the travelling wave and the electron self-energy. Various model systems are considered including the case that lattice distortion and charge transfer effects have to be accounted for. The imaginary part of the self-energy measures the decay, which cannot simply be described in terms of a generalized golden rule containing the system's total t-matrix. The real part is interpreted as a measure of the change of the Fermi surface on alloying while it is not equal to the level shift. The formalism allows for the conclusion that the Anderson and McMillan condition [2] for an effective medium of a scatterer in condensed matter contains an unphysical feature. An alternative prescription is proposed.
can be parametrized by its t-matrix elements in angular momentum representation. Once one-electron potentials are given, which will be supposed in the present treatment as well, problems ii-iv are intimately connected. This, however, does not show up in the literature dealing with dilute alloys, in which scarcely attention is paid to the averaging aspect iv. It is interesting to note that the theory of alloys has two clearly distinguishable branches due to the different role o f averaging in disordered alloys compared with dilute alloys. This in fact makes the theory for disordered alloys to have more similarities with the theory of liquid metals than with that for dilute alloys. The purpose of the present paper is to give a unified presentation of the theory for a dilute alloy accounting for the problems ii-iv. The formulation is applicable for a system with change transfer and lattice distortion in the environment of an impurity. Further the formulation is used to discuss the following. First, there is a question raised by Fenton [1 ], who states that Re tkk is not the level shift, although in practice this is used in the interpretation of Fermi surface changes measured in dilute alloys as a consequence of the presence o f impurities. Connected with
1. Introduction The development of high precision techniques for the measurement of the de Haas van Alphen effect during the last ten years has enabled to obtain detailed data on electron states and scattering in alloys. These data have demanded a description as exact as possible. Due to the very nature o f condensed matter such description h a s t o make head against several problems. We mention the requirement for i) an approximate description of many-electron effects leading to a reli. able prescription for the construction of one-electron potentials, ii) the calculation o f the scattering by such potentials, iii) a multiple scattering theory because of the treatment o f the system as a whole, and iv) some average over possible configurations of scatterers. Problem i) can be considered as a separate subject. For perfect crystals the Mattheiss prescription has proven to be a most reliable one and is commonly followed. But for alloys several degrees o f arbitrariness can be observed in practice. It is in fact one o f the aims o f the research nowadays to t~md one-electron potentials from the comparison o f the best theory available with experiment. In such theory it is supposed that a one-electron potential exists so that it 156
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
this is the problem of the reliability of the prescription of Anderson and McMillan [2] for the construction of an effective environment of a scatterer in condensed matter. Second, recently [3, 4] an expression for the electrical resistivity in liquid metals has been derived which implies that a generalized golden rule holds which contains the total t-matrix of the system. Divergencies observed in the evaluation of this expression plead in favour of an alternative. These problems can be discussed nicely in terms of the Green function of the system evaluated in the representation of eigenstates of the perfect host. The latter is an additional reason that has been chosen for treatment of a dilute alloy, while several conclusions hold for a disordered alloy and a liquid metal as well. The paper is organized as follows. To illustrate principles first the simple electron-impurity system with the impurities at random positions is considered in section 2. In section 3 the impurities are distributed randomly on a lattice. This is done because the corresponding averaging technique is useful for the dilute alloy problem. Further it provides an explicit proof that the usual averaging technique for the electronimpurity system requires point scatterers. Section 4 is devoted to the dilute alloy problem assuming that the influence of an impurity is restricted to its own Wigner-Seitz cell. In section 5 the problems of generalized golden rule, level shift and effective environment of a scatterer are discussed. In section 6 lattice distortion and charge transfer effects are accounted for. In the units to be used h = 2m = 1, where m is the electron mass.
2. The electron-impurity system The Hamiltonian H for a dilute substitutional alloy is
+ v--p 2 + w + Y ot
0, ot
(2.1) where/_/c is the Hamiltonian of the perfect host metal, Vc = ~/o~ with the summation over all lattice sites, V is the potential field of the impurity scatterers, u/a stands for the impurity at lattice position R a and the
157
summation over a runs over the random positions of the N s impurities in the system's volume ~2. The form of V if lattice distortion and charge transfer effects are to be accounted for will be given in section 6. The peculiar property of the (dilute) alloy system is that, though clearly/_/c can be considered as an unperturbed Hamiltonian, for the perturbation V no region in space exists outside a certain sphere where V equals zero, no matter how dilute the alloy is. Since conventional potential scattering theory makes explicit use of such an asymptotical property, this theory is not applicable. For example a generalized golden rule in which the total perturbation is replaced by the corresponding total t-matrix T
r-l(nk) = 21r ~
IZn,k,nkl26(En,k, - gnk )
(2.2)
n'k'
cannot be derived. Here Bloch states and their energies
Enk are labeled by a band index n and a reduced wave vector k. The total t-matrix is via the perfect crystal Green function G c = (E + _/./c)- 1,
(2.3)
related with V b y
T= V+ VGeT
(2.4)
as usually, while E + = E + ie and e is infinitesimally positive. An approach which is applicable to find the decay of a Bloch state Ink) requires the evaluation of the Laplace transform
In)
(2.5)
of the time dependent amplitude (nklexp (-iHt)lnk) of the state Ink), if one follows a state which evolves according to the system Hamiltonian from a state which at t = 0 has been prepared to be the state Ink). This general approach has been formulated by Goldberger and Watson [5] and will be applied to the dilute alloy system in the present paper. The principles can be illustrated most clearly for the electronimpurity system with the impurities at random posi-
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
158
tions. Elements of this treatment are given by Doniach and Sondheimer [6] to which we refer for certain introductory details, In this elementary model system there is no lattice and the crystal potential Vc is equal to zero. Lattice effects are studied separately in the following section. So we are going to evaluate Gkk, where the bar means an average over the configurations of the impurities. For the Green function holds
G O - (E + - k2) -1,
(2.12)
n s - N s / ~ is the density of the impurities, Okk, =- Id3r exp [-i(k - k')" r] u(r)
= G O + GO(x~ o6 + ~. vaGOv~ c~
S2(k, k') - ~_, exp [-i(k - k ' ) . R e -- i(k' - k)" RE]
~
v~GOo~GOo, r + . . . ] G 0,
(2.6)
(2.14)
/
=G
O+GO(~,ta + ~ a
(2.13)
and structure factors like
G = G O + G OVG
+
Here
and
taGOt~
a~
S3(k, k', k") - Z exp [-i(k - k')" Ra aOr
+ ~, t~GOtoGOtr + . . . ) G 0.
(2.7) -
i(k'
-
k")" Rt3 - i(k"
-
k)" R.~]
(2.15)
13:g -r Now the unperturbed states are plane waves normalized to unity in the system volume
(2.8)
contain the information about the configuration of the impurities, The average over all configurations is usually [6] calculated by applying the operation
~-11d3Ra
(2.16)
The Green function for the unperturbed system is G O = (E+ _ p2)-I
(2.9)
and the t-matrix of a single impurity at position R e is given by
ta = oa + oaGOta.
(2.10)
We first give Gkg according to (2.6) 0
Gkk = G0 + Gk {nsOkk+
~-2
X exp [-i(k - k')" (Ra - R~)I = N s + N28kk ,
'
~ okk,Gk,Ok,kS2(k,k )
k'
k'k"
O.
S 2 ( k , k ' ) = N s + [2 -2 ~. Id3R~Id3R¢
(2.17) 0
+ ~ - 3 ~, Okk, Gk,0 Ok,k,,Gk,,Ok,,kS3(k,k 0 ' ,k " ) +...}G
repeatedly to all terms in the series expansion (2.11), so to the structure factors. We only give
(2.11)
as an example, where the third member of (2.17) is equal to the second member if a factor N s - 1 is replaced by N s, which is correct in the thermodynamic limit. The procedure is straight forward. The resulting series can be represented simply in terms of diagrams of the type given in the figure. A horizontal line seg-
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
I I
I
is written as a series in the potential using (2.10). Instead of resumming expression (2.19) it is much simpler to analyze (2.7) directly. The result is (2.18) and the terms of Zk(E) to second order in n s and to third order in tkk, are given by
\
I
k
Ii
\
_k ./
k'
k \kk
b
I t I
I I I
/
k ', k I k
k ,/
/
/ ~\ / \ / \
Zk(E) = nstkk + n~ ~, tkk, Gk, 0 tk,k, Gk, 0 tk,k + . . . .
k' ~ k"". k
c
159
k'
d
(2.20) /¢
k /
I \
I I
\\
//
k' \, k } k
k //
e
/
\l
~x I ,
k' Ik'\ k f
Fig. 1. Diagrams representing different terms in the series for
~kk"
The interpretation of (2.18) is as usual [12]. In measurements in which one studies states characterized by a k, like in the de Haas van Alphen measurements, the inverse life time r - l ( k ) of such a state is connected with the imaginary part of the self-energy r - l ( k ) = - 2 Im Y,k(E) = 2n s Im tkk
ment, a cross × and a dotted line stand for a factor G 0, n s and Okk, respectively. A summation has to be carried out over intermediate k's. Every summation implies a factor [2 -1. Certain diagrams, like a, b, d and f are called irreducible or connected, while diagrams which fall apart if one horizontal line segment is cut are reducible. The sum of all connected diagrams forms the self-energy ~k(E) and one finds the closed form
a,k --
- k 2 - z,(tr)) -1,
ns
ns ~ ~ 0 , , ,,~,0,, . + - ' ~ ~ Okk'trk Ok lg Lrk Ok k k'k"
o
t7_0,, n 0 , ,
k' ~kk"-'k ~k k " - ' k " k k
+ ....
(2.21)
A shifted energy, which in practice shows up as a distortion of the Fermi surface, follows from the real part of the self energy. However, remarks of Fenton [1 ] and of Anderson and McMfllan [2] require a separate discussion to be given in section 5.
3. Impurities on a lattice
~k(E) = nsOkk +-~ ~_, Vkk'GO'vk'k k'
o
~'2
(2.18)
where the first few terms of Zk(E ) are given by
I2 k' vkk'Gk vk'k'Gk'Vk'k + " "'
2n2Im~"
(2.19)
corresponding to the diagrams a, b, d and f. It is easily seen that the first three terms are reproduced if n s tkk
In order to see the influence of a lattice or equivalently, of potentials with non-zero range, the impurities will now be distributed randomly on a lattice. This lattice can be constructed by a division of the volume of the system into N equal cubes or parallelepipeds with the effective volume of one impurity. The limit N - * ,~ corresponds to the system described in section 2 which will be shown to hold explicitly. Of course, this limit is only allowed if simultaneously the impurities shrink to point scatterers. The expression for Gkk is still given by (2.11). The difference arises in the averaging procedure. The ensemble for a random distribution is well defined
lbO
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys --
having(Ns)= N ! / ( N s ! ( N - Ns)!) members. Again we only give the average of S2(k , k') as an illustrative example,
C
C
Gnknk - Gnk + Gnk { CsUnknk + N "-2 ~
Onkn,k, GC,k, On,k,nkS2(k, k')
n'k'
S2(k, k') = N s + ~ ¢ # exp [ - i ( k - k')" (R~ - R~)]
~
n'k' n"k"
__Ns Ns Us -1 N N-
+N-3
1 ~
j-~]'
exp [ - i ( k - k ' ) ' ( R / - R / , ) ]
= Ns(1 _ Ns/N ) + Ns2 5k', k +g"
(3.1)
We remark that each term of the sum in the second member of (3.1) is averaged separately, leading to a summation over all lattice sites with the appropriate factors in the third member. Like in (2.17) 1 is neglected in comparison with N s and N, which is correct in the thermodynamic limit. One might call the factor 1 - Ns/N a finite size effect and the occurrence of the reciprocal lattice vector g a lattice effect. The latter effect implies that the self energy will be no longer diagonal in k, which in a dilute alloy means band mixing. A more detailed treatment for that case will be given in the following section. As for the present case we conclude with the observation that in the limit N-+ = the result of section 2 is obtained. In that limit Ns/N approaches 0, and the first reciprocal lattice vector g ¢ 0 becomes infinitely large, so that diagonality in k is guaranteed.
c °nkn'k' Gn'k' On'k'n"k"
X Gn,,k,,Vn,,k,,nkS3(k, c c k', k") + . .. }Gnk
= GCk +
(4.1)
k Cstnknk
+ N - 2 X tnkn'k' GC'k' tn'k'nkS'2( k, k') n'k'
+ N-3
~,
tnkn,k ' GC,k, tn,k,n,,k, '
n'k'n"k" c ' , k ,' k " ) + X Gn,,k,, tn,,k,,nk S3(k
"'"
k"
(4.2)
Here
GnCk----(E + - Enk) -1,
(4.3)
c s = Ns/N is the concentration of the impurities and N is the number of unit ceils in the crystal. The Bloch states are normalized to unity in a unit cell, so that
Onkn,k, = (nklvln'k') = Id3r ~b*k(r) o(r) ~n,k,(r)
(4.4)
and similarly 4. A dilute alloy without lattice distortion
tnkn, k, =- (nkltln'k '). In this section Gnknk , defined by eq. (2.5), will be evaluated and averaged over the impurity configurations. The series expansion of G in terms of the single scatterer potentials on and their t-matrices t o are completely similar to eqs. (2.6) and (2.7) respectively and will not be written down. The only difference is, that in eqs. (2.6), (2.7) and (2.10) the free space Green function G O has to be replaced by the perfect host Green function G c defined by eq. (2.3). We write down the two series expansions for Gnknk evaluated using the Bloch states as a complete set.
(4.5)
The structure factors S 2 and S 3 are defined by eqs. (2.14) and (2.15) re spectively, while in those with primes in (4.2) certain terms in the summations are excluded corresponding with the exclusions in the tmatrix series in (2.7) which follow in its derivation from the potential series in (2.6). In order to fred an expression for Gnknk we again need the average of the structure factors. As for S'2 the steps are given in eq. (3.1): The result is slightly different because in the description by Bloch states k
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys and k' always lie within the first BriUouin zone and never differ a reciprocal lattice vector. So we find
S2(k , k') = Ns(I - Cs) -t- N2 ~ kk , .
(4.6)
Similarly it is possible to calculate the average of other structure__ factors. We merely give the results for S~, S 3 and S~.
S'2(k, k') = - N s c s + N28kk , ,
(4.7)
while
cs)(~kk' + 8kk" + ~k'k")
+ N3s Skk,6k,k,,
and (4.5). Due to the unit cell normalization of the Bloch states each intermediate summation over k requires a factor N -1. If a v or t carries a bar its matrix element is diagonal in k. Using (4.10) and (4.1 I) in the average of (4.1) and (4.2) it is seen by mental inspection that the series for Gnknk can be summed to the form
Cnknk=
F-E(k)-~(k,E)
'
(4.12)
where the matrix E(k) is diagonal in n with elements Enk and the matrix Z(k, E) is given by
S3(k, k', k") = Ns(1 - 3c s + 2c 2) + N2(1 -
161
Znn,(k, E) = (nklcsv + Cs(1 - Cs) oGco (4.8) + Cs(1 - 3c s + 2c 2) oGCoGCv
and + %2(I - c s) oGC0GCv + . . . In'k)
(4.13)
S'3(k, k', k") = Ns(-C s + 2%2) -N2scs(Skk , + 6kk,, ) =
(4.9) + c2(1 - Cs)tGCfGCt +...In'k).
Substitution of these expressions in the average of Gnknk leads to the following form of the series between curly brackets in (4.1) and (4.2)
CsVnknk +
+ C3sOGCoGCO+ ...Ink)
(4.10)
and
(4.14)
One usually refers to a very dilute alloy so that factors like 1 - c s are approximately equal to 1. As for band mixing, which clearly shows up in the nondiagonality of ~(k, E) in the band index, alkali metals and noble metals are attractive. At the Fermi level only one band contributes significantly because other bands lie way below as well as above the Fermi energy. In general, however, particularly in transition metals, one has to be aware of possibilities of band mixing and the self energy matrix has to be diagonalized for certain points on the Fermi surface.
Cstnknk + (nkl - C2stGCt + c2fGCf 5. A generalized golden rule and the level shift + % ( - % + 2c 2) tGCtGCt _ c3(?GCtGCt + tGCtGCt-) + c2(I - cs) tGCfGCt + c3tGCfGC? +...Ink>, (4.11) respectively. A bra-ket matrix notation is used for conciseness, corresponding to the definitions (4.4)
The points to be discussed in this section are connected with the fact mentioned in section 2, that the perturbation field of scatterers is everywhere in space so that no asymptotic region exists. This is not just a peculiarity of a dilute alloy but it holds for the electron-impurity system as well. Therefore it is sufficient to give the discussion for the simpler
162
A. Lodder/Electron self energy, Fermi surface changes and generalized golden rule in dilute alloys
system. Our attention will be concentrated on the expressions for Gick, ~"k(E) and r - l ( k ) given in eqs. (2.18), (2.20) and (2.21). We first mention a well-known property. The term in (2.21) which has lowest order in the density can be rewritten using the optical theorem.
- 2 n s Im tick = 27rNs~ Iqc'ltlk)12a(k '2 k'
-
k2).
(5.1)
This equality is used [7] legally in the description of Dingle temperatures measured with the de Haas van Alphen effect. However, it is clear from the derivation in section 2 that the self-energy is not equal to the average of (klTIk), where T is the total t-matrix of the system
T= V + VG°T,
(5.2)
so that an expression 27r ~" Il2 5(k '2 - k2), k'
(5.3)
being related to - 2 Im (klTIk)
(5.4)
via the optical theorem has no physical meaning. In fact, if one tries to evaluate (5.3) and (5.4) one meets with divergencies which correspond to reducible diagrams. The self-energy has no such divergencies since in this quantity only the connected diagrams are retained. Recently Dunleavy and Jones [3] have used T in an expression for the resistivity and they indeed make mention of divergencies. The expression has been given by Dreirach et al. [4] who argue that the golden rule which has the form (5.3) with V instead of T should be generalized to the form (5.3). Divergencies never showed up as long as (5.3) was approximated to lowest order in the single scattering t-matrix, in which case in practice one used the right hand side of (5.1). Since Dunleavy and Jones wanted to include multiple scattering effects they met with the divergencies, which they have omitted on what they call "physical grounds". We conclude that if one wants to calculate the total scattering rate it is natural to use
- 2 Im Zk(E ) instead of a generalized golden rule. For the electrical resistivity one needs a partial scattering rate, from a state Ik) to a state [k'). In that case there are two possibilities. One can try to trace the consequences of the reducible diagrams for (5.3) using its equality with (5.4) and comparing (5.4) with (2.21). A more direct way is the evaluation of Kubo's formula for the electrical conductivity, which approach [8] has the same spirit as the one used in the present paper. Now we turn to a discussion of the real part of the self-energy. To that end the meaning of Gkk in view of its time-dependent origin is important. An unperturbed state in k-representation, being a travelling plane wave, is followed time-dependently under the influence of the Hamiltonian of the system. To facilitate the analysis the Laplace transform of the amplitude was evaluated leading to (2.18). Since a plane wave is not an eigenstate of the system Gkk has no pole at a real energy, but at a complex energy given by the equation
E : k 2 + ~,k(E).
(5.5)
This means that the travelling wave is seen apparently at an energy k 2 + Re Zk, if one speaks in terms of a sharp k. In practice one measures states at the Fermi surface so that one does not probe states at different or shifted energies, but effectively changes of the Fermi surface on alloying. A technical detail is that according to (5.5) the self-energy has to be evaluated at an energy E consistent with this equation, which seems to form a problem. A thorough analysis, however, of the series (2.19) shows that it is necessary to sum a part of the connected diagrams. The result is that only properly connected diagrams are left in which the internal line segments stand for the full Gkk" A properly connected diagram does not fall apart if any two internal horizontal line segments are cut. In view of this diagram f can be reduced further and has to be combined with diagram b. To lowest order in the density this leads to the remarkable result, that the correct value of the right hand side of (5.5) follows from an evaluation of nstkk on the energy shell, so at the energy E = k 2. The shifted energy which follows from (5.5) is not necessarily equal to the level shift. This latter quart-
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
tity is defined as the energy difference between an eigenstate of the system and the corresponding state of the unperturbed system. The theory of this paper is not devised to the study of eigenstates of the system, but of decaying states in the system. The difference between these states can be indicated as follows. A decaying state is prepared to be an unperturbed state at t = 0 according to the boundary conditions of the unperturbed system and develops under the influence of the scatterers present in the system of interest. An eigenstate of the system has to be constructed using the boundary conditions of the system and the properties of the potential, which is called the scattering potential from the point of view of the unperturbed system. In terms of scattering theory eigenstates contain the influence of (back)scattering by the boundaries in addition to the information of the scattering by the potential. This discussion allows for a comment on Fenton's statement [1 ], that Re tkk is not the level shift. In itself this statement is correct since the level shift is defined for eigenstates, being equilibrium states. But the statement was directed towards those [9] who study decaying states, being non-equilibrium states, so that in fact one has got involved in a confusion of tongues. Further we want to comment on the objection of Anderson and McMillan [2] against multiple scattering theory, which they typify as a "theory which deals only with how waves propagate as they scatter against many centers, not with the internal structure of the centers". We emphasize, that the internal structure is correctly accounted for, since the waves contain locally the exact partial waves belonging to each individual center. It is due to the inappropriate boundary conditions that one is not led to eigenstates. That is why it seems unphysical to require that Atkk = 0 as a prescription for the construction of an effective medium, where At stands symbolically for the t-matrix of "scatterer less medium". An alternative prescription which seems more realistic is that, in addition to a scattering condition Im Atkk = 0, an equilibrium condition is made so that the relative level shift or equivalently the relative density of states vanishes. Numerical calculations according to these lines are in progress.
163
6. Lattice distortion and change transfer effects The Hamiltonian (2.1) of a dilute alloy is useful if the influence of the impurity does not reach farther than the boundaries of its own Wigner-Seitz cell. This makes it inapplicable in cases in which an impurity induces a distortion of the lattice. Further change transfer effects are almost always present, so that it is important to generalize the treatment. To that end the potential for each scatterer o(r) = oi(r) - pC(r) in the total perturbing potential V(r) = ~ a v ( r - R ~ ) has to be replaced by
o(r) = ~ (vi(r - R~I) - oC(r - R/)). ]
(6.1)
The summation runs over all scatterers which differ from the host. The impurity atom is located at R} = 0, while u/(r - R~]) for R~. :/= 0 stands for a host atom in the environment of the impurity whose potential is changed due to change transfer and is located at a shifted position R} -- R! + Aj, A/being the vector measuring the shift. The analysis of Gnknk given in section 4 leading to (4.12) holds for the present system as well, although higher order terms in the selfenergy contain finite range factors which are smaller than those given in (4.13) and (4.14). The reason is that the scatterers are larger since they are formed by the impurity together with its distorted environment. To lowest order in the concentration it is still found that ~nn,(k, E ) = Cs(nkltln'k).
(6.2)
The right hand side of (6.2) has already been evaluated in a previous paper [10]. We merely quote the result, which holds for non overlapping muffin-tin scatterers. l ~ , f f ] /'h* /': = - Cnk, L Cn k ' , L ' J L ' L ( A ] ) j LL' X sin (r/{, - r~h) exp [i(r/{, - nh)].
(6.3)
Here K = ~/'ff, L - (l, m), c jh and c/* are host and alloy wave function coefficients at the position R/, the asterisk denotes the complex conjugate,
164
A. Lodder/Electron self.energy, Fermi surface changes and generalized golden rule in dilute alloys
JLL'(~') = 4rr 7)~ CLL'L ''il-l' + l"h"(~:A/') YL"(Aj) L"
(6.4) and ~J and ~h are the phase shifts o f the alloy and host muffin-tin scatterers, respectively. Details o f notation and the derivation can be found in ref. 10. Preparations are made for a numerical calculation o f (6.2) for dilute potassium alloys, which have recently been studied experimentally [11 ].
Acknowledgements I want to thank Mr. C. E. van Dijkum for constructive comments on the manuscript. The thesis work o f Mr. J. C. C. de Ruiter induced particular emphasis on golden rule aspects and the use o f structure factors in the formulation. I want to mention a clarifying discussion with Dr. B. L. G. Bakker at an early stage o f this work.
References [1] E. W. Fenton, Solid State Comm. 22 (1977) 63. [2] P. W. Anderson and W. L. McMillan, Proc. Int. School of Physics "Enrico Fermi" 37 (1967) 50 and pp. 71-73. [3] H. N. Dunleavy and W. Jones, J. Phys. F.' Metal Phys. 8 (1978) 1477. [4] O. Dreirach, R. Evans, H.-J. Giintherodt and H.-U. Kiinzi, J. Phys. F: Metal Phys. 2 (1972) 709. [5] M. 1. Goldberger and K. M. Watson, Collision Theory (John Wiley, New York, 1964), Chap. 8. [6] S, Doniach and E. H. Sondheimer, Green's Functions for Solid State Physicists (Benjamin, Reading, Mass., 1974), Chap. 5. [7] P. T. Coleridge, N. A. W. Holzwarth and M. J. G. Lee, Phys. Rev. BI0 (1974) 1213. [8] H. J. van Zuylen and A. Lodder, Physica 68 (1973) 1, and refs. given therein. [9] I. M. Templeton and P. T. Coleridge, J. Phys. F: Metal Phys. 5 (1975) 1307, see p. 1317 as well. [10] A. Lodder, J. Phys. F: Metal Phys. 6 (1976) 1885. [ 11 ] B. Llewellyn, D. Mc. K. Paul, D. L. Randles and M. Springford, J. Phys. F: Metal Phys. 7 (1977) 2545. [12] E. A. Stern, Phys. Rev. B7 (1973) 1303.
The decay of a travelling wave in a dilute alloy has been studied by evaluating the matrix element of the system's Green function diagonal in the state which is prepared to be the travelling wave at an initial time. This quantity has a pole at a complex energy equal to the sum of the unperturbed energy of the travelling wave and the electron self-energy. Various model systems are considered including the case that lattice distortion and charge transfer effects have to be accounted for. The imaginary part of the self-energy measures the decay, which cannot simply be described in terms of a generalized golden rule containing the system's total t-matrix. The real part is interpreted as a measure of the change of the Fermi surface on alloying while it is not equal to the level shift. The formalism allows for the conclusion that the Anderson and McMillan condition [2] for an effective medium of a scatterer in condensed matter contains an unphysical feature. An alternative prescription is proposed.
can be parametrized by its t-matrix elements in angular momentum representation. Once one-electron potentials are given, which will be supposed in the present treatment as well, problems ii-iv are intimately connected. This, however, does not show up in the literature dealing with dilute alloys, in which scarcely attention is paid to the averaging aspect iv. It is interesting to note that the theory of alloys has two clearly distinguishable branches due to the different role o f averaging in disordered alloys compared with dilute alloys. This in fact makes the theory for disordered alloys to have more similarities with the theory of liquid metals than with that for dilute alloys. The purpose of the present paper is to give a unified presentation of the theory for a dilute alloy accounting for the problems ii-iv. The formulation is applicable for a system with change transfer and lattice distortion in the environment of an impurity. Further the formulation is used to discuss the following. First, there is a question raised by Fenton [1 ], who states that Re tkk is not the level shift, although in practice this is used in the interpretation of Fermi surface changes measured in dilute alloys as a consequence of the presence o f impurities. Connected with
1. Introduction The development of high precision techniques for the measurement of the de Haas van Alphen effect during the last ten years has enabled to obtain detailed data on electron states and scattering in alloys. These data have demanded a description as exact as possible. Due to the very nature o f condensed matter such description h a s t o make head against several problems. We mention the requirement for i) an approximate description of many-electron effects leading to a reli. able prescription for the construction of one-electron potentials, ii) the calculation o f the scattering by such potentials, iii) a multiple scattering theory because of the treatment o f the system as a whole, and iv) some average over possible configurations of scatterers. Problem i) can be considered as a separate subject. For perfect crystals the Mattheiss prescription has proven to be a most reliable one and is commonly followed. But for alloys several degrees o f arbitrariness can be observed in practice. It is in fact one o f the aims o f the research nowadays to t~md one-electron potentials from the comparison o f the best theory available with experiment. In such theory it is supposed that a one-electron potential exists so that it 156
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
this is the problem of the reliability of the prescription of Anderson and McMillan [2] for the construction of an effective environment of a scatterer in condensed matter. Second, recently [3, 4] an expression for the electrical resistivity in liquid metals has been derived which implies that a generalized golden rule holds which contains the total t-matrix of the system. Divergencies observed in the evaluation of this expression plead in favour of an alternative. These problems can be discussed nicely in terms of the Green function of the system evaluated in the representation of eigenstates of the perfect host. The latter is an additional reason that has been chosen for treatment of a dilute alloy, while several conclusions hold for a disordered alloy and a liquid metal as well. The paper is organized as follows. To illustrate principles first the simple electron-impurity system with the impurities at random positions is considered in section 2. In section 3 the impurities are distributed randomly on a lattice. This is done because the corresponding averaging technique is useful for the dilute alloy problem. Further it provides an explicit proof that the usual averaging technique for the electronimpurity system requires point scatterers. Section 4 is devoted to the dilute alloy problem assuming that the influence of an impurity is restricted to its own Wigner-Seitz cell. In section 5 the problems of generalized golden rule, level shift and effective environment of a scatterer are discussed. In section 6 lattice distortion and charge transfer effects are accounted for. In the units to be used h = 2m = 1, where m is the electron mass.
2. The electron-impurity system The Hamiltonian H for a dilute substitutional alloy is
+ v--p 2 + w + Y ot
0, ot
(2.1) where/_/c is the Hamiltonian of the perfect host metal, Vc = ~/o~ with the summation over all lattice sites, V is the potential field of the impurity scatterers, u/a stands for the impurity at lattice position R a and the
157
summation over a runs over the random positions of the N s impurities in the system's volume ~2. The form of V if lattice distortion and charge transfer effects are to be accounted for will be given in section 6. The peculiar property of the (dilute) alloy system is that, though clearly/_/c can be considered as an unperturbed Hamiltonian, for the perturbation V no region in space exists outside a certain sphere where V equals zero, no matter how dilute the alloy is. Since conventional potential scattering theory makes explicit use of such an asymptotical property, this theory is not applicable. For example a generalized golden rule in which the total perturbation is replaced by the corresponding total t-matrix T
r-l(nk) = 21r ~
IZn,k,nkl26(En,k, - gnk )
(2.2)
n'k'
cannot be derived. Here Bloch states and their energies
Enk are labeled by a band index n and a reduced wave vector k. The total t-matrix is via the perfect crystal Green function G c = (E + _/./c)- 1,
(2.3)
related with V b y
T= V+ VGeT
(2.4)
as usually, while E + = E + ie and e is infinitesimally positive. An approach which is applicable to find the decay of a Bloch state Ink) requires the evaluation of the Laplace transform
In)
(2.5)
of the time dependent amplitude (nklexp (-iHt)lnk) of the state Ink), if one follows a state which evolves according to the system Hamiltonian from a state which at t = 0 has been prepared to be the state Ink). This general approach has been formulated by Goldberger and Watson [5] and will be applied to the dilute alloy system in the present paper. The principles can be illustrated most clearly for the electronimpurity system with the impurities at random posi-
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
158
tions. Elements of this treatment are given by Doniach and Sondheimer [6] to which we refer for certain introductory details, In this elementary model system there is no lattice and the crystal potential Vc is equal to zero. Lattice effects are studied separately in the following section. So we are going to evaluate Gkk, where the bar means an average over the configurations of the impurities. For the Green function holds
G O - (E + - k2) -1,
(2.12)
n s - N s / ~ is the density of the impurities, Okk, =- Id3r exp [-i(k - k')" r] u(r)
= G O + GO(x~ o6 + ~. vaGOv~ c~
S2(k, k') - ~_, exp [-i(k - k ' ) . R e -- i(k' - k)" RE]
~
v~GOo~GOo, r + . . . ] G 0,
(2.6)
(2.14)
/
=G
O+GO(~,ta + ~ a
(2.13)
and structure factors like
G = G O + G OVG
+
Here
and
taGOt~
a~
S3(k, k', k") - Z exp [-i(k - k')" Ra aOr
+ ~, t~GOtoGOtr + . . . ) G 0.
(2.7) -
i(k'
-
k")" Rt3 - i(k"
-
k)" R.~]
(2.15)
13:g -r Now the unperturbed states are plane waves normalized to unity in the system volume
(2.8)
contain the information about the configuration of the impurities, The average over all configurations is usually [6] calculated by applying the operation
~-11d3Ra
(2.16)
The Green function for the unperturbed system is G O = (E+ _ p2)-I
(2.9)
and the t-matrix of a single impurity at position R e is given by
ta = oa + oaGOta.
(2.10)
We first give Gkg according to (2.6) 0
Gkk = G0 + Gk {nsOkk+
~-2
X exp [-i(k - k')" (Ra - R~)I = N s + N28kk ,
'
~ okk,Gk,Ok,kS2(k,k )
k'
k'k"
O.
S 2 ( k , k ' ) = N s + [2 -2 ~. Id3R~Id3R¢
(2.17) 0
+ ~ - 3 ~, Okk, Gk,0 Ok,k,,Gk,,Ok,,kS3(k,k 0 ' ,k " ) +...}G
repeatedly to all terms in the series expansion (2.11), so to the structure factors. We only give
(2.11)
as an example, where the third member of (2.17) is equal to the second member if a factor N s - 1 is replaced by N s, which is correct in the thermodynamic limit. The procedure is straight forward. The resulting series can be represented simply in terms of diagrams of the type given in the figure. A horizontal line seg-
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
I I
I
is written as a series in the potential using (2.10). Instead of resumming expression (2.19) it is much simpler to analyze (2.7) directly. The result is (2.18) and the terms of Zk(E) to second order in n s and to third order in tkk, are given by
\
I
k
Ii
\
_k ./
k'
k \kk
b
I t I
I I I
/
k ', k I k
k ,/
/
/ ~\ / \ / \
Zk(E) = nstkk + n~ ~, tkk, Gk, 0 tk,k, Gk, 0 tk,k + . . . .
k' ~ k"". k
c
159
k'
d
(2.20) /¢
k /
I \
I I
\\
//
k' \, k } k
k //
e
/
\l
~x I ,
k' Ik'\ k f
Fig. 1. Diagrams representing different terms in the series for
~kk"
The interpretation of (2.18) is as usual [12]. In measurements in which one studies states characterized by a k, like in the de Haas van Alphen measurements, the inverse life time r - l ( k ) of such a state is connected with the imaginary part of the self-energy r - l ( k ) = - 2 Im Y,k(E) = 2n s Im tkk
ment, a cross × and a dotted line stand for a factor G 0, n s and Okk, respectively. A summation has to be carried out over intermediate k's. Every summation implies a factor [2 -1. Certain diagrams, like a, b, d and f are called irreducible or connected, while diagrams which fall apart if one horizontal line segment is cut are reducible. The sum of all connected diagrams forms the self-energy ~k(E) and one finds the closed form
a,k --
- k 2 - z,(tr)) -1,
ns
ns ~ ~ 0 , , ,,~,0,, . + - ' ~ ~ Okk'trk Ok lg Lrk Ok k k'k"
o
t7_0,, n 0 , ,
k' ~kk"-'k ~k k " - ' k " k k
+ ....
(2.21)
A shifted energy, which in practice shows up as a distortion of the Fermi surface, follows from the real part of the self energy. However, remarks of Fenton [1 ] and of Anderson and McMfllan [2] require a separate discussion to be given in section 5.
3. Impurities on a lattice
~k(E) = nsOkk +-~ ~_, Vkk'GO'vk'k k'
o
~'2
(2.18)
where the first few terms of Zk(E ) are given by
I2 k' vkk'Gk vk'k'Gk'Vk'k + " "'
2n2Im~"
(2.19)
corresponding to the diagrams a, b, d and f. It is easily seen that the first three terms are reproduced if n s tkk
In order to see the influence of a lattice or equivalently, of potentials with non-zero range, the impurities will now be distributed randomly on a lattice. This lattice can be constructed by a division of the volume of the system into N equal cubes or parallelepipeds with the effective volume of one impurity. The limit N - * ,~ corresponds to the system described in section 2 which will be shown to hold explicitly. Of course, this limit is only allowed if simultaneously the impurities shrink to point scatterers. The expression for Gkk is still given by (2.11). The difference arises in the averaging procedure. The ensemble for a random distribution is well defined
lbO
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys --
having(Ns)= N ! / ( N s ! ( N - Ns)!) members. Again we only give the average of S2(k , k') as an illustrative example,
C
C
Gnknk - Gnk + Gnk { CsUnknk + N "-2 ~
Onkn,k, GC,k, On,k,nkS2(k, k')
n'k'
S2(k, k') = N s + ~ ¢ # exp [ - i ( k - k')" (R~ - R~)]
~
n'k' n"k"
__Ns Ns Us -1 N N-
+N-3
1 ~
j-~]'
exp [ - i ( k - k ' ) ' ( R / - R / , ) ]
= Ns(1 _ Ns/N ) + Ns2 5k', k +g"
(3.1)
We remark that each term of the sum in the second member of (3.1) is averaged separately, leading to a summation over all lattice sites with the appropriate factors in the third member. Like in (2.17) 1 is neglected in comparison with N s and N, which is correct in the thermodynamic limit. One might call the factor 1 - Ns/N a finite size effect and the occurrence of the reciprocal lattice vector g a lattice effect. The latter effect implies that the self energy will be no longer diagonal in k, which in a dilute alloy means band mixing. A more detailed treatment for that case will be given in the following section. As for the present case we conclude with the observation that in the limit N-+ = the result of section 2 is obtained. In that limit Ns/N approaches 0, and the first reciprocal lattice vector g ¢ 0 becomes infinitely large, so that diagonality in k is guaranteed.
c °nkn'k' Gn'k' On'k'n"k"
X Gn,,k,,Vn,,k,,nkS3(k, c c k', k") + . .. }Gnk
= GCk +
(4.1)
k Cstnknk
+ N - 2 X tnkn'k' GC'k' tn'k'nkS'2( k, k') n'k'
+ N-3
~,
tnkn,k ' GC,k, tn,k,n,,k, '
n'k'n"k" c ' , k ,' k " ) + X Gn,,k,, tn,,k,,nk S3(k
"'"
k"
(4.2)
Here
GnCk----(E + - Enk) -1,
(4.3)
c s = Ns/N is the concentration of the impurities and N is the number of unit ceils in the crystal. The Bloch states are normalized to unity in a unit cell, so that
Onkn,k, = (nklvln'k') = Id3r ~b*k(r) o(r) ~n,k,(r)
(4.4)
and similarly 4. A dilute alloy without lattice distortion
tnkn, k, =- (nkltln'k '). In this section Gnknk , defined by eq. (2.5), will be evaluated and averaged over the impurity configurations. The series expansion of G in terms of the single scatterer potentials on and their t-matrices t o are completely similar to eqs. (2.6) and (2.7) respectively and will not be written down. The only difference is, that in eqs. (2.6), (2.7) and (2.10) the free space Green function G O has to be replaced by the perfect host Green function G c defined by eq. (2.3). We write down the two series expansions for Gnknk evaluated using the Bloch states as a complete set.
(4.5)
The structure factors S 2 and S 3 are defined by eqs. (2.14) and (2.15) re spectively, while in those with primes in (4.2) certain terms in the summations are excluded corresponding with the exclusions in the tmatrix series in (2.7) which follow in its derivation from the potential series in (2.6). In order to fred an expression for Gnknk we again need the average of the structure factors. As for S'2 the steps are given in eq. (3.1): The result is slightly different because in the description by Bloch states k
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys and k' always lie within the first BriUouin zone and never differ a reciprocal lattice vector. So we find
S2(k , k') = Ns(I - Cs) -t- N2 ~ kk , .
(4.6)
Similarly it is possible to calculate the average of other structure__ factors. We merely give the results for S~, S 3 and S~.
S'2(k, k') = - N s c s + N28kk , ,
(4.7)
while
cs)(~kk' + 8kk" + ~k'k")
+ N3s Skk,6k,k,,
and (4.5). Due to the unit cell normalization of the Bloch states each intermediate summation over k requires a factor N -1. If a v or t carries a bar its matrix element is diagonal in k. Using (4.10) and (4.1 I) in the average of (4.1) and (4.2) it is seen by mental inspection that the series for Gnknk can be summed to the form
Cnknk=
F-E(k)-~(k,E)
'
(4.12)
where the matrix E(k) is diagonal in n with elements Enk and the matrix Z(k, E) is given by
S3(k, k', k") = Ns(1 - 3c s + 2c 2) + N2(1 -
161
Znn,(k, E) = (nklcsv + Cs(1 - Cs) oGco (4.8) + Cs(1 - 3c s + 2c 2) oGCoGCv
and + %2(I - c s) oGC0GCv + . . . In'k)
(4.13)
S'3(k, k', k") = Ns(-C s + 2%2) -N2scs(Skk , + 6kk,, ) =
(4.9) + c2(1 - Cs)tGCfGCt +...In'k).
Substitution of these expressions in the average of Gnknk leads to the following form of the series between curly brackets in (4.1) and (4.2)
CsVnknk +
+ C3sOGCoGCO+ ...Ink)
(4.10)
and
(4.14)
One usually refers to a very dilute alloy so that factors like 1 - c s are approximately equal to 1. As for band mixing, which clearly shows up in the nondiagonality of ~(k, E) in the band index, alkali metals and noble metals are attractive. At the Fermi level only one band contributes significantly because other bands lie way below as well as above the Fermi energy. In general, however, particularly in transition metals, one has to be aware of possibilities of band mixing and the self energy matrix has to be diagonalized for certain points on the Fermi surface.
Cstnknk + (nkl - C2stGCt + c2fGCf 5. A generalized golden rule and the level shift + % ( - % + 2c 2) tGCtGCt _ c3(?GCtGCt + tGCtGCt-) + c2(I - cs) tGCfGCt + c3tGCfGC? +...Ink>, (4.11) respectively. A bra-ket matrix notation is used for conciseness, corresponding to the definitions (4.4)
The points to be discussed in this section are connected with the fact mentioned in section 2, that the perturbation field of scatterers is everywhere in space so that no asymptotic region exists. This is not just a peculiarity of a dilute alloy but it holds for the electron-impurity system as well. Therefore it is sufficient to give the discussion for the simpler
162
A. Lodder/Electron self energy, Fermi surface changes and generalized golden rule in dilute alloys
system. Our attention will be concentrated on the expressions for Gick, ~"k(E) and r - l ( k ) given in eqs. (2.18), (2.20) and (2.21). We first mention a well-known property. The term in (2.21) which has lowest order in the density can be rewritten using the optical theorem.
- 2 n s Im tick = 27rNs~ Iqc'ltlk)12a(k '2 k'
-
k2).
(5.1)
This equality is used [7] legally in the description of Dingle temperatures measured with the de Haas van Alphen effect. However, it is clear from the derivation in section 2 that the self-energy is not equal to the average of (klTIk), where T is the total t-matrix of the system
T= V + VG°T,
(5.2)
so that an expression 27r ~" I
(5.3)
being related to - 2 Im (klTIk)
(5.4)
via the optical theorem has no physical meaning. In fact, if one tries to evaluate (5.3) and (5.4) one meets with divergencies which correspond to reducible diagrams. The self-energy has no such divergencies since in this quantity only the connected diagrams are retained. Recently Dunleavy and Jones [3] have used T in an expression for the resistivity and they indeed make mention of divergencies. The expression has been given by Dreirach et al. [4] who argue that the golden rule which has the form (5.3) with V instead of T should be generalized to the form (5.3). Divergencies never showed up as long as (5.3) was approximated to lowest order in the single scattering t-matrix, in which case in practice one used the right hand side of (5.1). Since Dunleavy and Jones wanted to include multiple scattering effects they met with the divergencies, which they have omitted on what they call "physical grounds". We conclude that if one wants to calculate the total scattering rate it is natural to use
- 2 Im Zk(E ) instead of a generalized golden rule. For the electrical resistivity one needs a partial scattering rate, from a state Ik) to a state [k'). In that case there are two possibilities. One can try to trace the consequences of the reducible diagrams for (5.3) using its equality with (5.4) and comparing (5.4) with (2.21). A more direct way is the evaluation of Kubo's formula for the electrical conductivity, which approach [8] has the same spirit as the one used in the present paper. Now we turn to a discussion of the real part of the self-energy. To that end the meaning of Gkk in view of its time-dependent origin is important. An unperturbed state in k-representation, being a travelling plane wave, is followed time-dependently under the influence of the Hamiltonian of the system. To facilitate the analysis the Laplace transform of the amplitude was evaluated leading to (2.18). Since a plane wave is not an eigenstate of the system Gkk has no pole at a real energy, but at a complex energy given by the equation
E : k 2 + ~,k(E).
(5.5)
This means that the travelling wave is seen apparently at an energy k 2 + Re Zk, if one speaks in terms of a sharp k. In practice one measures states at the Fermi surface so that one does not probe states at different or shifted energies, but effectively changes of the Fermi surface on alloying. A technical detail is that according to (5.5) the self-energy has to be evaluated at an energy E consistent with this equation, which seems to form a problem. A thorough analysis, however, of the series (2.19) shows that it is necessary to sum a part of the connected diagrams. The result is that only properly connected diagrams are left in which the internal line segments stand for the full Gkk" A properly connected diagram does not fall apart if any two internal horizontal line segments are cut. In view of this diagram f can be reduced further and has to be combined with diagram b. To lowest order in the density this leads to the remarkable result, that the correct value of the right hand side of (5.5) follows from an evaluation of nstkk on the energy shell, so at the energy E = k 2. The shifted energy which follows from (5.5) is not necessarily equal to the level shift. This latter quart-
A. Lodder/Electron self-energy, Fermi surface changes and generalized golden rule in dilute alloys
tity is defined as the energy difference between an eigenstate of the system and the corresponding state of the unperturbed system. The theory of this paper is not devised to the study of eigenstates of the system, but of decaying states in the system. The difference between these states can be indicated as follows. A decaying state is prepared to be an unperturbed state at t = 0 according to the boundary conditions of the unperturbed system and develops under the influence of the scatterers present in the system of interest. An eigenstate of the system has to be constructed using the boundary conditions of the system and the properties of the potential, which is called the scattering potential from the point of view of the unperturbed system. In terms of scattering theory eigenstates contain the influence of (back)scattering by the boundaries in addition to the information of the scattering by the potential. This discussion allows for a comment on Fenton's statement [1 ], that Re tkk is not the level shift. In itself this statement is correct since the level shift is defined for eigenstates, being equilibrium states. But the statement was directed towards those [9] who study decaying states, being non-equilibrium states, so that in fact one has got involved in a confusion of tongues. Further we want to comment on the objection of Anderson and McMillan [2] against multiple scattering theory, which they typify as a "theory which deals only with how waves propagate as they scatter against many centers, not with the internal structure of the centers". We emphasize, that the internal structure is correctly accounted for, since the waves contain locally the exact partial waves belonging to each individual center. It is due to the inappropriate boundary conditions that one is not led to eigenstates. That is why it seems unphysical to require that Atkk = 0 as a prescription for the construction of an effective medium, where At stands symbolically for the t-matrix of "scatterer less medium". An alternative prescription which seems more realistic is that, in addition to a scattering condition Im Atkk = 0, an equilibrium condition is made so that the relative level shift or equivalently the relative density of states vanishes. Numerical calculations according to these lines are in progress.
163
6. Lattice distortion and change transfer effects The Hamiltonian (2.1) of a dilute alloy is useful if the influence of the impurity does not reach farther than the boundaries of its own Wigner-Seitz cell. This makes it inapplicable in cases in which an impurity induces a distortion of the lattice. Further change transfer effects are almost always present, so that it is important to generalize the treatment. To that end the potential for each scatterer o(r) = oi(r) - pC(r) in the total perturbing potential V(r) = ~ a v ( r - R ~ ) has to be replaced by
o(r) = ~ (vi(r - R~I) - oC(r - R/)). ]
(6.1)
The summation runs over all scatterers which differ from the host. The impurity atom is located at R} = 0, while u/(r - R~]) for R~. :/= 0 stands for a host atom in the environment of the impurity whose potential is changed due to change transfer and is located at a shifted position R} -- R! + Aj, A/being the vector measuring the shift. The analysis of Gnknk given in section 4 leading to (4.12) holds for the present system as well, although higher order terms in the selfenergy contain finite range factors which are smaller than those given in (4.13) and (4.14). The reason is that the scatterers are larger since they are formed by the impurity together with its distorted environment. To lowest order in the concentration it is still found that ~nn,(k, E ) = Cs(nkltln'k).
(6.2)
The right hand side of (6.2) has already been evaluated in a previous paper [10]. We merely quote the result, which holds for non overlapping muffin-tin scatterers. l ~ , f f ] /'h* /':
(6.3)
Here K = ~/'ff, L - (l, m), c jh and c/* are host and alloy wave function coefficients at the position R/, the asterisk denotes the complex conjugate,
164
A. Lodder/Electron self.energy, Fermi surface changes and generalized golden rule in dilute alloys
JLL'(~') = 4rr 7)~ CLL'L ''il-l' + l"h"(~:A/') YL"(Aj) L"
(6.4) and ~J and ~h are the phase shifts o f the alloy and host muffin-tin scatterers, respectively. Details o f notation and the derivation can be found in ref. 10. Preparations are made for a numerical calculation o f (6.2) for dilute potassium alloys, which have recently been studied experimentally [11 ].
Acknowledgements I want to thank Mr. C. E. van Dijkum for constructive comments on the manuscript. The thesis work o f Mr. J. C. C. de Ruiter induced particular emphasis on golden rule aspects and the use o f structure factors in the formulation. I want to mention a clarifying discussion with Dr. B. L. G. Bakker at an early stage o f this work.
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