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COPPER-ZINC ALLOYS AS TREATED BY VIRTUAL CRYSTAL AND RELATED APPROXIMATIONS
N. March, P. Gibbs, G. Stocks, J. Faulkner
To cite this version:
N. March, P. Gibbs, G. Stocks, J. Faulkner. COPPER-ZINC ALLOYS AS TREATED BY VIRTUAL
CRYSTAL AND RELATED APPROXIMATIONS. Journal de Physique Colloques, 1972, 33 (C3),
pp.C3-259-C3-267. �10.1051/jphyscol:1972340�. �jpa-00215074�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-259
COPPER-ZINC ALLOYS AS TREATED BY VIRTUAL CRYSTAL AND RELATED APPROXIMATIONS
N. H. MARCH, P. GIBBS
Department of Physics, The University, Sheffield, England G. M. STOCKS and J. S. FAULKNER
Oak Ridge National Laboratory, Oak Ridge, Tennessee, U. S. A.
RBsumB. - On presente des rksultats obtenus dans le modkle du cristal virtue1 pour un alliage Cu-Zn dont la concentration varie de 0 a 20 % en atomes de Zn. La methode de Koninga-Kohn- Rostoker a kt6 employee et le parametre du r6seau mesure aux rayons X pour chaque concentra- tion de l'alliage a 6tC utilise. Les potentiels VcU et Vzn ont Bte construits comme le propose Mat- theiss (1964), le potentiel du cuivre seul ktant celui employ6 par O'Sdlivan, Switendick et Schir- ber (1970). Les valeurs propres ont kt6 calculees aux points de symetrie r, X, L, K et W, vingt valeurs environ Btant presentees pour le cuivre pur et pour les alliages Cu-Zn a 1,10 et 20 %.
La densite d'Ctats du cuivre pur est aussi presentee pour le potentiel utilise dans notre travail.
Pour tester 17utilit6 d'un tel modele, nous avons compare nos rtsultats avec les mesures d'effet de Haas-van Alphen par Chollet et Templeton, pour des alliages Cu-Zn dont la concentration va jusqu'a 0,l % de Zn, avec des experiences &absorption optique et avec des mesures de chaleur specifique Bectronique.
On discute brikvement la validit6 du potentiel periodique dependant de l'knergie pour le calcul de certaines propriktes des alliages alkatoires et la relation entre le cristal virtuel, la matrice t moyenne et les mkthodes du potentiel coherent dans les cas extremes de faible concentration et differences entre les potentiels atomiques dans l'alliage.
Abstract. - Results are presented for the virtual crystal model of a Cu-Zn alloy in the range of Zn concentrations from 0-20 atomic %. These are obtained by the Korringa-Kohn-Rostoker method and at each concentration the measured X-ray lattice parameter in the alloy has been used.
The potentials VC,, and VZ, were generated following the proposals of Mattheiss (1964), the pure Cu potential, in essence, being that employed by O'Sullivan, Switendick and Schirber (1970).
Crystal eigenvalues have been calculated at symmetry points r, X, L, K and W, twenty or so eigen- values being presented for pure Cu, and for 1, 10 and 20 % Cu-Zn alloys. The density of states for pure Cu is also presented, for the potential used in our work.
To test the usefulness of such a model, we have compared our results with the de Haas-van Alphen measurements of Chollet and Templeton on Cu-Zn alloys for concentrations up to 0.1 % Zn, with optical absorption experiments and with measurements of the electronic specific heat.
A brief discussion is also included of the validity of a periodic energy-dependent potential in the calculation of certain properties of random alloys, and of the relation between virtual crystal, average t-matrix and coherent potential methods in the limiting cases of small concentration and of small differences between the atomic potentials in the alloy.
1. Introduction. - Progress in calculating the elec- tronic band structure of perfect crystals has raised hopes that it might be possible t o make correspon- dingly detailed quantitative studies on alloys. These hopes have been partially frustrated by a number of factors to date, chief among which is the lack of a well-defined k vector in an alloy. A second factor is that we need knowledge of the potentials to associate with the A and B sites in a binary A-B alloy ; this raises questions of the detailed screening charge clouds around A and B ions in metallic alloys such as we shall consider here.
Specifically, we report some calculations of the band structure of random Cu-Zn alloys, which have been carried out by the so-called virtual crystal appro- ximation. Essentially, as can be seen from eq. (4.1) below, we assign a potential V , to each atom A, V, to B and we construct an average, concentration dependent, potential from these localized potentials.
We then build up a periodic potential by placing this
concentration-dependent localized potential on each lattice site. The resulting periodic potential is often said to describe a
((virtual crystal D, whose properties it is anticipated might simulate a t least some aspects of those of the random A-B alloy.
Of course, in the ct virtual crystal D, k is re-esta- blished as a good quantum number and the methods developed for band structure calculations on perfect crystals can be applied immediately.
2. Ensemble average of electron density. - Before going on to present the results of our virtual crystal computations, we wish to comment further on the use of a periodic potential in calculations on random alloys.
It has often been implied that, in a scheme like the virtual crystal method, averaging is being carried out in the wrong order ; that is on the potential rather than on the solutions of the Schrodinger equation.
We shall therefore demonstrate that some properties
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972340
C3-260 N. H. MARCH, P. GIBBS, G. M. STOCKS AND J. S. FAULKNERj of a random alloy can indeed be properly calculated
from a periodic potential.
The argument is simply as follows. In a Cu-Zn alloy, for example, where the basic lattice is fcc, and the Zn atoms are truly random, let us consider the ensemble average < p(rE) > where p(rE) gives the number of electrons per unit volume at r with energy less than E in a given configuration of A and B atoms. The electron density < p(rE) >,=,, in the alloy, where Ef is the maximum or Fermi energy, is clearly periodic, with the period of the background lattice, in our case, face-centred cubic.
But we know from well established results in quantum-mechanical density functional theory that we can generate such a periodic density < p(rE) >,=,,
from a periodic potential V(rEf) by summing the squares of N one-body wave functions generated by V(rE,), N being the number of electrons in the system.
However, such a periodic potential V(rE,) will also generate a density of states Np(E) and we must be careful to distinguish this from the alloy density of states N(E). They cannot be the same, for the density of states Np(E) has the van Hove singularities charac- teristic of a face-centred cubic lattice, whereas N(E), being a property of the random alloy, has certainly a different singularity structure ; most probably a much more complicated form (see Kohn ; this Conference).
The conclusion is now clear. The exact ensemble average density < p(rE) >,=,f can be generated by a periodic potential V(rE,), but this potential does not generate the alloy density of states. To obtain this we must construct < p(rE) > for a general energy E.
This can again be done from a periodic potential V(rE). But the energy dependence of the periodic potential must be included if we wish to generate
correctly. Thus, we have a set of densities of states generated from a set of periodic potentials, and the alloy density of states is formed by moving from one of these curves to another through the whole set.
To get the whole singularity structure, the energy dependence of the periodic potential would presumably be complicated ; without great care we expect there- fore to get only an average account of N(E).
In this paper, we enquire, nevertheless, as a start on the problem, whether we can obtain significant results by first neglecting the energy dependence and setting up V(rEf) from a virtual crystal prescription.
We are currently investigating for some simple models how V(rE) should be constructed : but here we shall use Nordheim's physical arguments. Though we have reduced the random alloy problem basically to quan- tum mechanics based on Bloch's theorem and periodic potentials, it is hardly necessary to emphasize that to get V(rE) exactly from first principles we must face the ensemble averaging problem.
Nevertheless, it is of interest that a random alloy such as Cu-Zn, based in the dilute Zn regime on a face-centred cubic lattice, can be characterized by a periodic potential, provided this has appropriate concentration and energy dependence. This will yield the local density < p(rE) > and the density of states, but we cannot expect to obtain lifetime effects from such a method (unless we allow V(rEf) to be complex).
The comparative success of the virtual crystal calculations we report here encourages us in the view that, provided we are not too far from the Fermi energy, we may be able to obtain a rather accurate average description of < p(rE) > and hence of N(E) by writing
where the Kn are reciprocal lattice vectors, and then relating VKn(E) to VK,(Ef) by a Taylor expansion around Ef. For concentrated alloys however, it is perfectly clear that the energy dependence of V(rE) must be sufficiently great to position correctly the Cu and the Zn d bands : an energy independent poten- tial failing completely in this respect.
3. Alloy systems studied. - In the virtual crystal calculation, the systems studied were face-centred- cubic random alloys (a-brass). The alloy systems were :
1. Pure Cu with lattice parameter a = 6.830 9 a. u.
2. Cu (99 atomic %) : Zn (1 %) a = 6.835 0 a. u.
3. Cu (90 atomic %) : Zn (10 %) a = 6.874 0 a. u.
4. Cu (80 %) : Zn (20 %) a = 6.917 4 a. u.
the lattice parameters above being obtained by graphi- cal interpolation from the results of X-ray measure- ments recorded in Table 3.1.
Alloy
Atomic % Zn Lattice parameter in a. u.
4. Construction of periodic potential V(r E,). - As
discussed in the Introduction, we build up a periodic
potential by assigning potentials to each type of atom
and, in an alloy in which there is atomic concentration
c of Cu, the average crystal potential is chosen as
COPPER-ZINC ALLOYS AS TREATED BY VIRTUAL CRISTAL C3-261 V Ca is the crystal potential for a pure Cu crystal having
the lattice parameter and crystal structure appropriate to the alloy and V Za similarly is the crystal potential for a pure Zn crystal having the alloy lattice parameter and structure.
In an alloy in which metals A and B in the pure state had the same crystal structure, the choice (4.1) would ensure at least some interpolation between the band structures of the pure metals. Actually in Cu-Zn, we do not have such a common crystal structure and we shall therefore restrict ourselves to Zn concentra- tions of less than 30 %. In addition we shall incorpo- rate size effects to the extent that, at each concentration, we shall use the lattice parameters recorded in Table 3.1 in calculating the energy bands.
To return to the details of the potential, the crystal potentials V Ca and V Za were generated following the proposals of Mattheiss [1]. The calculation was performed using charge densities obtained from a Herman-Skillman programme (without however using the Latter cut-off potential; see the book by Herman and Skillman [2]).
The muffin tin radius was taken as half the near- neighbour distance while the intersphere constant was calculated by averaging the muffin tin potential between the muffin tin sphere and a sphere having the volume of the Wigner-Seitz cell. For further details of
the above procedure, we refer to the book by Loucks [3].
The pure Cu potential generated in our work is, in essence, the same as that employed by O'Sullivan, Switendick and Schirber [4]. It seems unnecessary to give tables of the potentials used for the three lattice parameters considered ; they will be made available on request.
After reporting the calculations of the energy bands in the following section, we shall consider the useful- ness of this approach for dealing with various physical properties, eg. de Haas-van Alphen oscillations in very dilute alloys, optical absorption and electronic specific heat.
5. Calculational procedure and results. — A cons- tant k search was carried out using the Korringa- Kohn-Rostoker method and a maximum orbital angular momentum quantum number 1 = 4, and the eigenvalues recorded in Table 5.1 were thus obtained.
A constant energy search with / max = 2 (for a descrip- tion of this, see Faulkner, Davis and Joy [5]) was also performed to give Fermi surface data.
Certain of the eigenvalues in Table 5.1 were used in an interpolation scheme proposed by Hodges and Ehrenreich, to give density of states histograms.
However, it seemed worthwhile only to record the result of the O'Sullivan et al. potential for pure Cu in figure 1.
TABLE 5.1
Table of energies from virtual crystal calculations.
Energies in Rydbergs
E (
A r 2S r 12
X j
x 3 x 2 x 5
x; L„
L 3 1
L 3 2
Li L 1 2
K, K t K 3
K 4
K 2
w 2
W 3
W i
w;
Cu 0.620 - 0.033
0.376 0.428 0.244 0.282 0.466 0.479 0.763 0.249 0.372 0.468 0.560 0.910 0.285 0.308 0.399 0.438 0.466 0.294 0.346 0.426 0.479
CuZn (1 %) 0.618 - 0.034
0.370 0.423 0.241 0.278 0.461 0.473 0.760 0.245 0.366 0.463 0.558 0.905 0.281 0.304 0.395 0.433 0.460 0.291 0.341 0.421 0.473
CuZn (10 %) 0.610 - 0.047
0.325 0.375 0.208 0.240 0.409 0.422 0.739 0.212 0.322 0.412 0.541 0.862 0.245 0.266 0.351 0.383 0.409 0.254
CuZn (20 %) 0.604 - 0.061
0.275
0.321
0.170
0.196
0.353
0.362
0.716
0.174
0.272
0.355
0.522
0.816
0.203
0.223
0.302
0.329
0.353
0.212
0.253
0.320
0.364
C3-262 N. H. MARCH, P. GIBBS, G. M. STOCKS AND J. S. FAULKNER
Cu to the 0.106 % Cu-Zn alloy.
Density of states histogram for pure Cu metal as given by poten-
tial of O'Sullivan et al. Dashed line centred around arrow is a The mass changes got Our
schematic representation of how the d-resonance for Zn would crystal calculations are recorded for completeness in 0
W -
t- a
0 +
LL 13
appear in a 20 % Zn alloy. Table 6.2.
We must stress again that the results for 10 and 20 % Zn represent the density of states N,(E) corresponding to the periodic potential V(rE,). Even if, as seems the case from comparison with experiment below, there is some agreement with the alloy density of states N(E) near the Fermi level, the d-bands as given by N,(E) are not at all a reasonable description for the random alloy : the energy dependence of V(rE) will have to be included to get these at all realistically.
On a technical detail, the interpolation scheme used above is not sufficiently good to obtain Fermi surface data. However, such data on Fermi surface parameters was got by the method desordered above and will be used below in the comparison we make with experiment.
-
- - -
6. Comparison of virtual crystal calculations with experiment. - a) DE HAAS-VAN ALPHEN RESULTS. - De Haas-van Alphen measurements on dilute Cu-Zn alloys have been made by Chollet and Templeton [6]
in the Zn concentration range 0-0.1 %.
If we consider first the de Haas-van Alphen neck frequency I;, for Cu, related to the Fermi surface in the usual way by
When we turn to the virtual crystal results, we find an increase in the neck frequency for a 0.106 % alloy of 0.007 (4) in the above units, compared with the result of 0.013 in Table 6.1, which seems fairly encou- raging.
Secondly, we give in the second row of Table 6.1 the experimentally determined ratio of the belly to
>- -
H
F N = Akc - 2 z e
neck frequency. Whereas this ratio decreases by 0.12 as we go from pure Cu to the 0.106 % Cu-Zn alloy according to experiment, we find a decrease of 0.080 from the virtual crystal method, which again seems satisfactory.
Though, to our knowledge, there is no experimental
where A is the appropriate neck area, then their results correspond to a small increase in the neck frequency as shown in Table 6.1.
i check of our results, for the frequency of the dog's
-a.lrn a
olu
&Is a.ZWa.w
a Urn awu
o 6% D el* a 02-bone orbit we find a decrease of 0.003 7 from pure
TABLE 6.1
Frequencies from de Haas-van Alghen measurements of .Clzollet and Templeton on dilute Cu-Zn alloys
CuZn CuZn
Cu (0.04 %) (0.106 %)
- - -
~ ~ ( 1 0 ' G) 2.174 2.178 2.187
F ~ / F ~ 26.72 26.67 26.60
Cyclotron
mass 1 % Z n 1 0 % 2 0 %
- - - -
B(100) 1.24 1.23 1.15 1.09
D(110) 1.10 1.10 0.99 0.84
N(111) 0.40 0.41 0.47 0.54
It is interesting that the value 0.40 is near to the average mass between L and the Fermi surface neck obtained by Lindau and Walldin [7] of 0.37. Electron- phonon enhancement, as these workers suggest, brings these values into reasonable agreement with the value of 0.46 obtained by cyclotron resonance experiments (Koch, Stradling and Kip [8]). If the low concentration mass varies as little in the alloys as Table 6.2 suggests, it might be possible to get informa- tion from dilute Cu-Zn alloys on the variation of electron-phonon enhancement with electron-atom ratio.
The situation seems fairly satisfactory here, there- fore, the virtual crystal model being capable of predic- ting the features of the Chollet-Templeton measure- ments in at least a semi-quantitative way. The expla- nation may lie in the fact that we are discussing large orbits here and therefore the details of the field over a distance of the order of the size of a unit cell are not of central importance.
b) OPTICAL ABSORPTION. - The situation is less
satisfactory, however, when we turn to discuss optical
absorption in Cu-Zn alloys. Here the experimental
work of Biondi and Rayne [9] provides a good deal
of data. Unfortunately, however, much of the inter-
pretation involves the position of the Cu d-bands rela-
tive to the levels in the conduction band. It makes no
sense to use a virtual crystal method (i. e. an energy-
independent potential) to deal with these relatively
COPPER-ZINC ALLOYS AS TREATED BY VIRTUAL CRISTAL C3-263 narrow d-bands, which are intimately related to a TABLE 6.3
d-wave resonance in the Cu ion potential. Integrated Density of states
Thus, we shall confine our attention to the informa- density E in in tion that can be gained within the conduction band. of states Rydberg states/Rydberg
In the analysis made by Lettington [lo] based on earlier - - -
work of Philipp et al., the movement of the Fermi 1 .O 0.668 3.61 level relative to the bottom of the conduction band can 1.1 0.693 3.50
be detected. 1.2 0.726 3.15
Lettington finds a change in this of about 1 eV for 20 % Zn. However, the virtual crystal calculations yield 0.012 Ryd for this change with addition of 20 % Zn. This is far too small to agree with the Biondi- Rayne results.
On the other hand, a free electron result, which deals with the screening of the Zn atom carefully, making sure that the displaced charge satisfies the Frieded sum rule, gets the answer right. We show below that the virtual crystal method essentially does first-order perturbation theory, using Bloch waves, on the difference between V,, and Vzn. It seems better to calculate the movement of the bottom of the conduc- tion band relative to the Fermi level using first-order perturbation theory based on plane waves. It implies perhaps that the difference between Vcu and Vzn is too strong to be dealt with as a perturbation.
However, in connection with this point let us now consider the shift in the bottom of the conduction band relative to the Fermi level in the rigid band model.
We can obtain the predictions of this model from the work of Faulkner, Davis and Joy [5]. Their Table I1 gives the integrated density of states versus the energy
It is then clear that the shift in the Fermi energy in going to 20 % Zn is 0.057 Rydberg or 0.78 eV, which is in much better agreement with Lettington's result.
We wish to stress that this rigid band prediction though is based on a calculation with the pure Cu lattice spacing of 6.830 9 a. u.
This seems encouraging but we want to sound a note of caution here. If we examine a little further the possible electronic structure of these alloys, we can obtain some additional information from the eigen- values of Herman and Skillman [2] and also from the ordered brass calculation of Arlinghaus [ll]. Then, the d-resonance for Zn would appear somewhere near the bottom of the Cu band shown in figure 1. Our estimated position is marked by an arrow on this figure. From previous experience on more concen- trated alloys (*), one might suppose that the density of states for a 20 % Zn alloy would have a form like the dashed line we have sketched in on this figure.
If this is roughly the situation, it could raise problems when searching for the movement of the bottom of the band in an optical experiment.
above the bottom of the band. By interpolation, which c) ELECTRON~C SPECIFIC HEAT. - We also record is permissible because in the required energy range the the density of states at the Fermi level according to integrated density of states is a smooth function of the rigid band model in Table 6.4, and the correspond- energy E, the energies corresponding to electron to ing electronic specific heat coefficient y, together atom ratios of ela = 1.1 (10 % Zn) and ela = 1.2 with the results of our virtual crystal calculations are (20 % Zn) may be found and are shown in Table 6.3. shown in Table 6.4.
Electronic specific heat coeficient y in millijoule/gm atom deg2
Cu CuZn (I %)
- - 10 - % 20 %
-
Virtual crystal 0.659 0.646 0.624 0.590
Rigid band 0.626 - 0.608 0.586
(a = 6.830 9 a. u.)
Experiment 0.698 f 0.002 0.696 + 0.003 0.695
The most recent experimental work on the electro- nic contribution to the specific heat of Cu-Zn alloys appears to be that of Isaacs and Massalski [12], and their results are recorded in the final row of Table 6.4.
The experiments indicate a tiny decrease of y with increasing Zn concentration, whereas both virtual crystal and rigid band results show a pronounced decrease.
Certainly some enhancement in the band structure
(*) See G. M. Stocks, R. W . Williams and J. S. Faulkner, Phys. Rev., 1971, B 4,4390.
value of the density of states is to be expected due to electron-phonon interaction as has been discussed in connection with the rigid band model by Clune and Greene [13] and more recently by Grimvall (1141 in the press). The measured density of states at the Fermi level can then be usefully expressed as
N(Ef) = N(Ef) band structure $. I ] @-a
where [l + A] is the electron-phonon enhancement
factor. While A for many metals can be obtained either
from measurements on the superconducting state or
C3-264 N. H. MARCH, P. GIBBS, G. M. STOCKS AND J. S. FAULKNER by direct calculation, the noble metals are not super-
conductors and the theory is not yet developed for these metals. Fortunately, a realistic estimate can be found from the high temperature electrical resistivity (Grimvall [15] ; see also Young and Sham [16]).
Grimvall estimates that, for pure Cu, 1 = 0.14 which would bring the pure Cu value of Faulkner, Joy and Davis into quite good agreement with experiments namely y = 0.71. Of course, the value entered for pure Cu under virtual crystal' differs from this because of the different potential used by O'SulIivan, Switen- dick and Schirber. The corresponding value of y = 0.75 after enhancement by the electron-phonon interaction is too large. Though some dependence of
A on ela is to be expected, Grimvall argues that this is not sufficient to bring the rigid band model into agree-
ment with the measurements. However, taking 1 independent of ela would bring the 20 % Zn y up to 0.67, in reasonable agreement with experiment, though the enhancement needs to be larger for a consistent description, in agreement with Grimvall.
7. Comparison with other workers. - The virtual crystal approximation has been applied to Cu-Zn alloys by earlier workers and in particular by Amar, Johnson and Sommers [17] and by Pant and Joshi [18].
These calculations do not specifically give the Fermi energy, but a useful comparison of various energy level separations can be made.
Let. us consider for example LL - L,. We have collected in Table 7.1 below the various results
Separation L; - L, Energies in Rydbergs
Cu 10 % Z n 20 % Zn
- -
Amar et al. 0.44 0.38 0.38
Pant and Joshi
(their potential V,,) 0.38 0.36 0.35
Present work 0.35 0.32 0.29
At least the general trends here agree between different workers. Lettington finds, in sharp contrast, that this separation should increase with increasing Zn concentration. This suggests that reasonable changes in the potential will not avoid what appears to be a severe difficulty for the virtual-crystal approxi- mation.
In connection with the disagreement between Let- tington's deduction for L, - Li in Cu-Zn and the virtual crystal model, it may be relevant to remark that a decrease in L, - L; is found in AgIn alloys with increasing I n concentration in the photoemission measurements by Nilsson [19].
He also finds a substantial decrease in L, - E, in AgIn, of about 1.4 eV for 10 % of In. Our result is smaller for CuZn, about 0.5 eV for 10 % of Zn.
Of course, the fields will be so different that we cannot conclude with certainty from this data on AgIn any similarity with CuZn, though it seems reasonable to suppose some similar trends might occur. It would certainly be of interest to have virtual crystal calcula- tions of AgIn, though it would be necessary to consider relativistic corrections in this case.
In connection with Table 5.1 giving various energy levels, it is of interest to record that, for pure Cu, the photoemission measurements of Lindau and Wall- d i n (1971 ; private communication) locate L1 4.05 eV above the Fermi level. This agrees reasonably well with the separation 3.92 eV obtained from Table 5.1.
Again the separation L1 - L; is measured by these workers as 4.80 eV whereas we find 0,35 Ryd or
4.73 eV. Thus, L i lies 0.75 eV below the Fermi level from experiment and 0,81 eV from the theory. The agreement is quite satisfactory.
8. Relation between virtual crystal and coherent potential approximation. - We shall now show by a straightforward argument that the virtual crystal approximation is giving the same result as the coherent potential approximation (CPA) as the difference VA - V, becomes small. Furthermore the CPA goes over to the correct result as the concentration tends to zero.
The fact that the CPA does yield these limits correct- ly has been known from specific models (for example for one-dimensional models from the work of Faul- kner [20], for an extreme tight binding model from the work of Velicky et al. 1211 and for muffin-tin potentials from the work of Soven [22]). The proof is independent of any specific model, employing only basic definitions, which we take to be as follows :
v, A = potential operator for an A atom on n-th site, such that
with v: the same object for a B atom
COPPER-ZINC ALLOYS AS TREATED BY VIRTUAL CRISTAL C3-265 where on is the coherent potential for the site n and which can -. then be used in connection with the defini-
8 = Go + G O ~ o , l ] g
n
is the Green function describing an ordered array of coherent potential scatterers. Thus 3 is the scattering operator for an A atom inserted at site n in a system with o's on all the other sites.
a) RELATION TO VIRTUAL CRYSTAL. - The CPA condition that defines w may be written
"B
c;: + (1 - C ) tn = 0 . (8 - 1)
Inserting the definition of TA and ciB into this yields the result
A B ""A
on = cv, + (1 - c) vn + C ( V ~ - o n ) Gt, +
+ (1 - c ) (v; - o n ) 57," . (8 . 2 ) Letting
and using ( 8 . 1 ) again allows us to rewrite ( 8 . 2 ) as
""A
w n = < v n > + c ( u , A - v ~ ) ~ t n . ( 8 . 3 ) From this form it is clear that
on -+ < V , > as v t -+ vj:
or, in words, the CPA goes over to the virtual crystal model when the A and B potentials are almost the same.
b) LIMIT AS CONCENTRATION TENDS TO ZERO. -
To see what happens as c approaches zero it is conve- nient to write ( 8 . 3 ) in another form
on = vn B + c(v," - v:) (1 + 5;;) . ( 8 . 4 ) From this we see that
on = uj: + co:, + 0 ( c 2 ) (8 - 5 )
and it follows from the definition that
N
G = GB + cZ' + 0 ( c 2 ) (8 6 ) and
7; = t p + ~'2;: + 0 ( c 2 ) ( 8 . 7 ) where
is the Green's function for the pure host crystal and t,"" = (v; - v:) (1 + GB t t B ) ( 8 . 9 ) is the scattering operator for a single A atom inserted at site n in the host crystal.
Using these expansions in ( 8 . 4 ) leads to the result
o:, = t."" (8 .lo)
to show that
Thus, to first order in the concentration
The system with one A atom embedded in an other- wise pure B crystal has been investigated by Koster and Slater [23], Callaway [24], and others. This sys- tem is described by the Green's function
It is known that the energy spectrum for this system is the same no matter which site the impurity is placed upon. The spectrum can thus be found equally well from GzB for any n, or from the average of such func- tions
where N is the number of lattice sites. In the limit of one impurity, the concentration of impurities is just c = 1/N. Inserting this into ( 8 . 5 ) leads to
= < GAB > proving that the CPA goes to the right result in this limit. As long as the concentration is sufficiently small so that the interaction between impurities is negligible on the average we can easily generalize this result to the case where there are r impurity atoms. Since c = r / N we have
which is as it should be. The low concentration case is discussed by Lifschitz [25].
Thus, we see that the CPA approaches the proper limits as vA -+ vB and also as c + 0, whereas the virtual crystal method is correct only in the former limit, but incomplete as c
-t0 for general vA and vB.
c) AVERAGE t-MATRIX METHOD. - Interestingly
enough, the average t-matrix approximation is
slightly less good than the CPA even in the limit as
c -, 0. If the pure host crystal was taken as the refe-
rence state, the t-operator for scattering from an A
atom would be t$B and the average t-operator would
C3-266 N . H. MARCH, P. GIBBS, G. M. STOCKS AND J. S. FAULKNER be c t p . Then the approximate Green's function
obtained from this approach would be the same as in ( 8 . 5 ) to first order in c, but this is not the average t-approximation as proposed by Korringa and Beeby.
They suggested that the alloy should be described by a Green's function
where u, is a potential defined by the requirement that the scattering from that potential embedded in a vacuum should be the same on the average as the scattering caused by vA or vB embedded in a vacuum.
In operator language
where
r\ A
