The modification of electron energy levels by impurity atoms

ANNALS

OF PHYSICS:

14, %!bi(i

The Modification

Be//

Telephone

(1961)

of Electron Energy Impurity Atoms

Laboratories,

Murray

Hill,

Levels

New

by

Jersey

The influence of localized impurities in solids may be represented either by (i) the elect of randomly distributed classical scattering centers for which only averaged quantities are meaningful or, as proved herein, (ii) the effect of an eyuivalent quantum description which incorporates both the random character and t,he process of averaging. Adopting the lat,ter viewpoint,, we readily make contact with diagrammatic perturbation rnet)hods familiar in field theory and many-body studies. As an application of this formalism, various approsimstions, all of infinite order in the perturbation, are derived for the average density of states in an idealized solid for whirh the effects of the impurit,ies prevail and t.he periodic lattice may be neglerted. The results of these approsimations are compared with the exact results obtained hy Frisch and Lloyd for a special model in which the impurity potential may be represent.ed as an atractive one-dimensional delta-function. One of the approximations discussed is a direct analog of the Urueckner model, and for the delta-function potential example this approximation can be solved exactly. In the case of a high density of impurities, our solutions demonstrate the well-known value of propagator modification, while, for a low density of impurities, several solutions exhibit an impurity hand that is isolat,ed frorn the conduction band. Finally, we discuss the average wave function, a quantity that is open to some arbitrariness because a wide latitude exists in t,he choice of a rule by which a wave function is paired with a set of impurit,y sit.es. Tao distinrt “pairingrules” are analyzed in detail: one rule restricts the wave and its derivative to have specified initial values but, demands neither normalizahility nor singlevaluedness; the other rule requires that every acceptable wave be an eigerlvector whose phase, relative to the other waves, is selected in a definite manner. Differential equations are presented uhich are oheyed by the two differently defined average wave flmrtions in the case of a one-tlimcnsional delt:tfunction potential. I.

INTRODUCTION

The modification of electronic levels in solids by t,he presence of random impurit,ies is a problem well-suited, on t,hc one hand, for a complete quantjum mechanical investigation or, on the other hand, for a part quantum and part classical invest,igation. The correctness of both approaches provides, therefore, a model problem whose answer may be found (principally) by classical means 1.7

and to which various approximatiotts of an alternative, field-theoretic approach may he compared. Ott the prnct,ical side, of course, these two t,cchttiqucs provide a strong combination to st,udy the hand st,ructure of impure solids. In this paper an electSron is &died quant#um mec~hattirslly in t#he presence of a large numEer of impurity atoms represcnt#rd by fixt>d (infittit,e rffcctive mass) classical sratt,erittg cctttrrs whost positions are treated as random variables. For simplicity, we neglect, the influenre nf the periodic host lattice as well as electron-phonon or electron-electron ittt8crantions; these effects, as wrll as t,he effect of finite effective mass impuriCcs, could kc ittcludcd if desired. The oncelectron Hamilt,onian in the presence of the classical impurit,y sources is t#akett as H = g

+ 2 l!(X - y,), ,I =l

(1)

where p and x are momentum attd posit,iott operat,ors which obey [x, p] = il, expressed in units (which are used throughout) such that, fi, = 1. Each scnt#tcrer is assumed t,o interact with the elect,ron via the same potetttjial functiott, 11;y,, represents the c-number position of the nt,h impurity atom. In all, there are N impurities situated in a region of volume I’. Ultimately, we are interest,ed in thelimitN+ m, V+. ~0, and N/I’ = ‘11,n, being the (finite) densit,y of scatterers; the occasional appearance of N or V separat,ely may he int,erpreted formally. The positions of the N impurit#y atoms arr assumed to be stat.ist,ically independent random variahles, similar to t,hr model studied hy Iiohn and Lut,Gnger (1). The average of a qttantit’y Q which depends on t’hr set (y,,} of scattering sites is defined by (2) While there are several quarUies Q of interest, we mainly devote our nttention to the dens&y of stat,es, ~(3; yl , y2 , . . . ), a function of the energy E. The primary reason for our restricted at,t,entiott stems from the availability of comparison works by Lax and Phillips (2)’ artd Vrisch attd Lloyd (3), bot,h of whom have studied t#he densit,y of stat,es in a particular model and have obtained some exact, results. In the model studied in hot,h of the above papers, the rnt#ire problem is assumed t,o lit in a one-dimensional space rather than a thrce-dimensional space. In addition, the interaetJiott potential was chosen simply as an attrartive delta-functiort : U(Z) a - 6(z). In one dimension, the powerful SturmLouiville theorem (4) can he drawn upon to determine t,he density of st’ates from t,he number of nodes in various solutions. Such a technique was employed in the numerical &dies of Lax attd Phillips, as well as in the stochastic: analysis 1 References

to earlier

works

may

also

he found

in this

paper

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IMPURITIES

of Frisch and Lloyd. The latter aut’hors succeeded in reducing t’he density of states calculation to quadraCures, a rather remarkable accomplishment. However, the Sturm-Louiville approach is not useful in t,hrer dimensions and other methods to calculate t,he density of states are required. In this paper, we attack t)his questio!l with infinite-order perturbation m&hods, familiar from field theory and many-body studies ( 5). The predictions of several different approximations are investigated for the simple one-dimensional model and compared with the accurate results of Frisch and Lloyd, nugmcntcd at several points by numerical results of Las and Phillips. Our technique is based on the sp&ral t,heorem which relates the den&y of st,ates t,o transition amplitudes (6) .’ I;ormally, the densit.y of states is given by p(fl;

y,

) yt? )

‘.

) = NZ,,6(E

- h’,,) = IIf Tr 6CE’ - H),

(3)

where E, is an eigenvaluc of H which depends on the location of the scut8terers, M is a suitable normalization, and Tr represents the trace operat,ion. With the aid of t,he well-known identity iii+

= ijh

+ iqj = zy112)

when q is a small positive quantity, PW’:;

y1,

Y?,

‘.

- h-6(2),

(4)

we can put, .)

=

-q!!!Tr&H. 1

The resolvent operator, (E’ - H)-I, .1s the energy-represeutatioll propagation fun&on for a single particle in the presence of t’he external field due to t
=

(p(~;y1,y,,

*..))

= -__111Im . . . Tr [( E+ - H)-‘1 n ‘$ s n- s

(0)

.

In bhe following sections, the normalization for p is set by the formal relation J-’ = N, the number of particles contained in the system. We are indebted t,o Jr. Ambegaokar for pointing out some similarit,ics in our approach and that of Edwards (7). In this work of Edwards, however, only potentials are considered which simultaneously possess a vanishing first, third, etc., Born scattering, i.e., all odd orders vanish. lYnfortunatcly, these strict requirements rule out, the simple delt,a-function potential, as well as many others. * Spectral 16).

properties

of operators

are widely

used;

for esmnple,

see Martin

and

Schwinger

46

liLAUDEK

In Sect,ion II we develop a general perturbat,ion analysis to find the average densit,y of states. This analysis is based on a similarity between, on the one hand, a treatment of the impurity atoms as randomly-located classical scattering sources and, on the other hand, a complet,e (N + I )-body quantum mechanical treatment, of the N scatt’erers and t’he one electron. The equivalence of these two descriptions is shown in Section II. Translation of the pseudorlassical description to a field-theoret!k, caompletely quant,um description facilit,ates the introduction of pert’urbation and diagrammat,ic techniques of analysis familiar, for example, from t)he study of electron-electron correlations (8), or from ordinary quant,um electrodynamics (3). Rules are established whirh enable the contribut,ion to the propagator from any order to he readily writt,en down. Simplified rules are also given which apply to t’he test case of a one-dimensional delta-function potential. In Section III several approximat,ions are st,udied for the one-dimensional example, and those results are compared t,o the results previously obtained by Frisrh and Lloyd. The various approximations that WC study are all of infinite order in t)he perturbat~ion and are summarized in Fig. 6 as symbolic “integral equat’ions” for the one-body propagator. Each approximation involves t,he summat’ion of a restricted c~lsssof diagrams. One such class (which in the simple delta-futwtion example call be solved exact)ly) corresponds to that class which is used in trwlear studies to obtain the Brueckner equation (10). For most of the approximat,ions that we st,udy, the corresponding eqtu~tions are also presenkd in the case of an arbitrary t’hree-dimensional pot8ent,ial. The availability of highspeed computing techniques would seem to make some of these approximations pract’iral. In Section IV we discuss several equations obeyed by the ‘*average” wave function. Frisch and Lloyd, for example, found that, a very simple equation was satisfied by t,he average wave. However, tbeir equation is not obtained by au elementary and straightforward argument, as was first. pointed out by Lax (11’). We discuss t,his discrepancy and show that it arises because there are two mcatlingful definitions for the meaning of “average wave, ” one of which leads to the simple equation of motion of Frisch and Lloyd, while tbe other definition requires a much more involved equation of motion. Both definit,ions of “average” are extended to three dimensions where we show the connectjion of one of them (the definit,ion implicit, in the work of Lax and Phillips) t,o t,hc optical model averaging scheme employed in nuclear physics (,12 ) . II.

ALTERNATE SERIES

FORMULATION OF AND DIAGRAMMATIC

THE

PROBLEM: PERTURBATION REPRESETTATION

The formal statement of t,he impurity band problem presented in Section I was based on a quantum keatment of the elect.ron in the presence of a random

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IMPUHITIES

dist,ribution of classical scattering sources. But, the considerat,ion of an ensemble of classical systems for which only the average propert’ies are of interest is reminiscent of t#he usual statistical interpretation of a quant#um process. This similarit,y suggest,s t’hat there is an alternate formulat,ion of the impurit)y band calculation which is ent’irely quantum mechanical and which duplicates t,he combined quantum and classical approach of &&on I. If the impurit)ies could not he treatrd in an infinite effective mass approximation, then a fully quantum formulation would be required. By discussing the present examples in t,he same language we provide a formalism which can be readily adapt,ed to treat finite mass impurit,ies. The int,roduction of additional refinements such as clectronphonon and electron-elect)ron interact,ions may also be facilitated. It should be remarked, however, that’ t)he perturbat’ion and diagrammatic analysis we shall develop can be arrived at’ by a treatment which keeps the impurit,ies ‘Lclassical” (13). In this case it is natural to discuss diagrams eqlCvalent to ours but with a somewhat’ different int,erpretation. Let, us establish t,hat the alternate formulat)ion is based, as would be expected, on the (N + 1 J-body Hamiltonian .7(: = (p”/%l)

+ Z,v(x - X,,).

(7)

In this expression, x and p are conjugate operators for the rlect,ron and X, (II = 1, ... ) N) represent a set of impurity atom position operators which commute among t,hemselves (and with p or x), and obey [X, , PI] = i&J, where P[ is an impurity atom momentum operator. In order to obt’ain results consistent wit,h t,he ensemble averaging procedure, it is ncressary to study matrix elements of (N + I)-body operators between two states which specify zero moment,um for each of bhe N impurity atoms, and, in addition, remain arbitrary with respect to the electron coordinates. To prove t’he equivalence of t’he complet,e quantlum formulat,ion we proceed as follows: Let .f( X) be a function of the (N + 1 )-body Hamiltonian operator [e.g., f(x) = (Et - x)-‘1, and lat (pUPINP?N . . . 1f(x)

1p’P,‘P2’ . .)

be t’he matrix element hctween t)wo sets of momentum unit, operat’or,

s

(8) cigcnst,ates. Insert t,he

/ pX,‘X2’ . . . ) dp r~ dX,’ (pX1’X2’ . . . /,

(9)

expressed, in part, as a sum over position eigenstatrs of the impurity atoms. Since bhe subspace of S(X) hclonging t’o the operator X, is diagonal in the

48

KLATJDER

X-representation, s

(p” (f(H)

we find that (8) becomes ( p’)(P:P;

. . . 1X,‘X,’ . . .)rI dX,’ (XI’XZ’ . . . / P1’PZ’ . . . ),

(10)

wherein H is now an operator only in the electron variables since t,he impurity operators X, are replaced by their eigenvalues X,‘. To complete the proof we note that (P1”PZ” . . . / X1/X2’ . . .)(X,‘X,’

. . . ( Pl’P?’ . . .)

= V-” exp CSX,‘(P,’

-

P,“)

(11)

which is simply lieN if P,” = P,‘. For ronvenience, we choose the more restricted example P,,” = P,’ = 0 to discuss below.3 In either case, Eq. (10) becomes

(12) This is the same result that would have been predicted if t,he scatterers had been t’reated classically subject t’o an ensemble average; the equivalence of the two different’ viewpoints is t,herefore established. FIELD

FORMULATION

To facilitate the construction of the perturbation series it, is advantageous rewrite the Hamiltonian (7) in a second-quantized form (14) : X=

I#*

2

G fh + /y- +*ll/(xMx

-

Y)cp*cp(Y)

where $J*(x) and $J(x) are electron creat,ion and destruction obey iti(x),

#*(Y)l

to

(13)

dxdy,

operators which (14)

= 6(x - Y).

(As long as one electron is discussed at a time the part,icular commutation rules are not significant,.) The operat’ors q*(x) and cp(x), bot,h of which commut’e with 9(x) at, equal times, represent creation and deskuction operators for impurity atoms and obey

b(x),

(13)

(P*(Y)1 = 6(x - Y).

A pseudo- “vacuum” stat,e ) 0 ) is defined as t,hat st,ate which has N impurity atoms, all in the zero moment,um eigenst#ate. Thus, / 0 ) = (N!)-“2[a*(0)]x 3 Note that impurity distributions initial and an equal final state other

(16)

1vat ),

other than uniform than those discussed

are here.

described

by

choosing

an

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IMPURITIES

where a*(O) is the creation operat,or for an impurity atom of zero momentum, and where / vat ) is the truly empty state, defined by I)(X) 1 vat ) = p(x) 1 vat ) = 0. The pseudo-vacuum t,onian4

(17)

st,ate is not ouly an eigenstate of bhe “unperturbed”

Hamil-

where w=

I‘ P(X) dx, S’

(19)

but / 0 ) is also an eigenst’ate of the comp1et.e Hamiltonian sequently, the Green’s function (IS’) G(x”, 1”; x’, t’) = (0 1 T[&(x”,

t”)h*(x’,

X, Eq. ( 13). Cont’)]l 0),

(20)

in which Ic/H(x, t) obeys the Heisenberg equation of mot,ion, represents the true, time-dependent one-particle propagation function. As customary, T denotes t,he time-ordering operator. Because of the special choice for the “unperturbed” Hamiltonian the S matrix is defined by S = T {exp [-i

11 1: J/I*!h(X, t> ( dx

- y> - ; ) cp*p(y> dydxdt])

and permit’s a transformation between the Heisenberg and interaction #I(x, t), which are defined to obey

,

(21)

operators

i&(x,t)= (-~+nw)$I(x,t,.

(22)

On the ot,her hand, the interact,ion and Schriidinger representations coincide for both the impurity at’om creation and destruction operators. Equat’ion (20) for G assumes the following form when expressed in terms of interaction represent#ation operators (16). G(x”, t”; x’, t’) = (0 1 T[&h(x”,

t”)~//,*(x’, t’,]l 0).

(3.3)

This form depends, in t,he present, example, on t,hr invariance of the pseudovacuuni st,atc and t,hc absence of any “vacuum-to-vacuum” transitions. An expansion of S in a power series in t’he pot,ential generates the ordinary Feyn4 The analysis. constant

average potential w is included in the “unperturbed” Hamiltonian The density n arises from the integral of p*‘p over all space; of the motion, whose eigenvalue is proportional to the density.

this

to simplify operator

the is a

50

KLAUDER

0 ---*,

q Y---T,

0 /--+-p-q ‘f p

P’ y’, 4

I

2

3

1

-.

--x

‘\ ’ J

\ ,r-----------V I

---\

-\

\I I-f

(cl

1

___- A---

/---------‘\;---‘! : ; --d /. ----_--_/ ’ ‘_--

‘I \

FIG. 1. Symbolic diagrammatic (a) Second-order Born scattering ceeds from “position” 1 towards terer between “positions” 2 and purities. (c) Another process of process of the same order in the of a t,hird-order Born scattering

,----.

(b)

\,,.--I I

--A-

/--\

‘I j

,----,

\I vI

Cd)

,W

representation of various t,erms in the perturbation series. with a single impurity atom. The electron, solid line, pro4, suffering a temporary loss of moment.um q to the scat3. (b) Two second-order -xatters with two separate imthe same order in the potential as process (b). (d) A third potential as (b) and (c). (e) Diagrammatic representation with a single impurity atom.

man perturbat.ion theory (17) for which, according to Wick’s theorem ( 18), each term may be symbolically represented by a diagram. Ko diagram is present whose contribution is proportional to only the first, power of z](x) ; each such diagram is cancelled by another diagram that, involves the factor w/V. The cancellation of diagrams linear in U(X) exhausts the contribut’ion of the u~/V “renormalization” in (21). As is most often t.he case the labor involved in computation is greatly reduced when the problem is t.ranslat,ed into cnergy-momentum space. The Green’s function, expressed in energy-momentum variables,

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IMPIXITIES

is defined by

The represcnt,ation of a vertex inrvolves two electron lines and two impurity atom lines; at, each xrertes, three-momrntum is conserved. The contribution to G from t)hr term of second order ill P is diagramatically sketched in Fig. l(a). The electron line is represented by a solid line and the impurity atom by a dashed line and, following Brueckner (IO ), the impurity atom is shown “briefly” before and after scattering. The physical process to which the diagram rorresponds is the usual one: An electron ill an initial state of momentum p propagates from 1 to 2 where an impurity atom is cwount,ercd. In this scsttcring the impurity atom gains a momentum q, an amotult~ which the electron must lose. After propagating from 2 to 3 the electroll rcscattcrs from t,he same impurity atom and regains the momentum q which it originally lost. Finally the electron propagates from 3 to 4 with momrntum p’ (momentum conservation requires that, p’ = p ) . The n~unerical contribution to 6’ is correctly obtained if: (a) the propagator S(p) = ~_IT’+ -

T~u! (p2/2m)

-

(25)

ww

[w = J”u(x) dx I5 is associated with each electron line, and: ih) the factor

zj(k- s> w ’

(26)

where v(k) =

s

P-~~%(x) dx,

(27)

is associated at each vertex for which the impuritjy atom brings ill a momentum q and leaves with a momentum k. A delt,a-function is used t’o inslwe momentum conservation at each vrrtcs. The caontrihntion from each diagram is multiplied by t,he fact,or

(rw-‘[(2*)3n]-p, j The definition of S(p) in Eq. natural units. If w = 0, however, of (26) and (28) may he set, equal dimensions.

(25), being dimensionless, then 20 in the numerator to 1, or any convenient

(28j facilitates the introduction of (25), and in the denominator constant with the appropriate

of

52

KLAUDER

where p is the number of independent impurity momentum vectors in that. diagram. Thus, for example, the contribution to G from the process depicted in Fig. 1 (a) is @2) = 3(P’ - p) (2n)3n2w3

RP)

/

24 -q)S(p

-

qMq)

4;

over-all momentum conservation insures that this term as well as all contrihut,ions to G are proportional to 6(p’ - p). Three distinct diagrams t,hat, involve two impurity atmomsare illustrated in Fig. 1 (b)-(d). Since the impurity atoms are not creat#ed or absorbed at each vertex (like t,he photons in quantum electrodynamics), addit’ional diagrams arc necessary, corresponding t,o t>hird and higher Born seatterings, that, oecur with each impurity atom. The first Born approximation has, of course, already been accounted for by including w in the denominator in the definit~ion of N in EQ. (5). lqigure 1 (e) illust’rates a third Born scattering diagram for one impurit,y atom. If GC3’ denotes the contribut,ion to t-he complete C:reen’s function from this diagram, then G’s’ = f@’ - p> ___ x”(p) (27r)Ww4

I‘ v( -q)s(p

- q)o(q - k)S’(p - k)zl(k) dqdk.

(30)

All of the necessary diagrams are simple generalizations of those shown in Fig. 1. One simplification can be readily made by a partial summat,ion of the perturbation series, and a subsequent rest,rict,ion only t,o proper graphs: a proper graph being one for which the intermediat,e electron momentum is restricted not t,o have the initial electron momentum value [see Fig. I(b) for an improper -=

+ -I)

-=

+,-@-

(0) +

(b)

s-s’-

-

FIG. 2. Schematic representation of the “integral equation,” Eq. (31), which eFfectively sums the contributions of the improper diagrams: (a) The modified propagator S’ (heavy bar) equals S (light bar) plus S&S’, where the kernel Z is the contribution of the proper ” in (a). The first two terms arise if graphs; (b) Iteration solution to the “integral equat,ion t,he heavy bar is replaced by the light bar on the right-hand side, the third term arises when the resulting expression is substituted in the right side of the “integral equation” (a). Additional terms arise in a similar fashion.

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53

graph]. I’igure 2(a) illustrates the “int’egral equation” which sums up the additional contribution of improper graphs; Fig. 2(b) represents the iterative solution to t,his int#egral equat,ion. For, if /? denot’es the sum of all proper graphs, then t,he propagator S’ = s + SSS’ = s + szs

+ ssszs

+ ...

(31)

includes the contribution from every diagram. It is customary to call S’ t#he “dressed” electron propagator. In the section which follows, several approximate forms for 2 will he analyzed. As in Section I, t,he expression

p(#) = A!!

7rn

j WP,

P; fi)

$

=

-+$jkG

/ S’(P)

dp,

(32)

defines the densitjy of states for either the true or an approximat’e S’ when t,he positions of the impurit#y atoms are averaged out. When one-dimensional examples arc discussed, only a few modifications of the formulas presented in this section need he made. Resides t#he obvious transit,ion of all threefold integrat
then w=

a(:r) ds = V,,,

(34)

and

It’ follows from Eq. (26)) therefore, that the factor associated with each vertex is unity, independent of all momentum variables. For the delta-function potential then, only t,he propagator,

is a function of moment.um. The contribution to t.he complete one-dimensional Green’s function from the diagram shown in Fig. 1 (a) becomes (37)

54

KLAUDPJR

This correction to the propagator depends only on the energy and is independent of the initial elect,ron moment8um. The comparnCve simplicity of this result is indicat’ive of t’he fact that many clasnts of fairly complex processes ran he a~‘curately keated in this model. III. APPROXIMATE

SOLUTIONS

B17 RESTRICTI’II)

I>IAGRARI

SUMS

For an arbitrary potential t,he contrihut,ion to the Green’s fur&on from all diagrams camlot be exactly calculated. However, in certain limited cases, such as in a low or high density limit,, reasonably good approximate resulk can 1)~ found by summing selected subset,s of all diagrams. In principle, t.he subset selection is det’ermined by which set, of diagrams are expected t,o dominat,e t,hc perturbation series; t,oo often, in practice, t#he class of diagrams is rest,ricted to those which can be handled with reasonable facility. The one-dimensional deltafunction pot,ent,ial is a particularly at.tractive model t,o study because it lends ii-self to accurate treat,ment for several interesting restricted diagram sets. For example, the relative merit, of “dressing” the intermediate propagators or leaving t,hem “undressed” can be exactly studied in several (‘ascs. Throughout this section the result,s we obtain are compared for acbcuracsywith rcsult,s from t’he exact] treat,ment’ of Frisch and Lloyd. In most cases, equations are incl~~dcd for an arbitrary three-dimensional pot,ential t#hat would follow by studying the same class of diagfams. In our one-dimensional analysis, we employ several dimensionless variables used by Lax and l’hillips (2) and also by Frisch and Lloyd (.j). For an attracstive potential, V(J) = T’,J(.r), we plit T’G = -bIEo,lm < 0;

( :a I

E = Gtco2jm.

(39)

and for the energy we let The paramet,ers k and K of Refs. 2 and 3 are relat#ed to the energy parameter & by

(40) depending on whether E ? 0. The t#erm nw = ,nl’o is expressed with the aid of t,he dimensionless parameter t by -~zC’~, = -(nmV0,/~02)(~0”/m)

= e(~u2/,m).

(11)

In the high (low) densit,y limit’, it’ follows that c >> (<<) 1. When needed, momentum variables measured in units of IQ are denoted by X, p, Y, ct’c. In these natural units t,he electron propagat,or assumes the form S(X) = -E+--&-+-

/’

E;

(4%)

BAND

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55

the minus sign just) reflect’s the attractive sign of t’he potential. modifies the eontrihution of each diagram becomes

The fact,or which

(t)-‘(27Tp

(43)

when the p independent momentum integrations are carried out over X-like variables. In the natural units, the density of states p(&) is given by (44) wherein the normalization is set, by requiring p(&) d& = p(E) dE. In the following, one form of integral will be encountered often and we writ’e its form and its evaluation here: when 71 # 0, it can readily be shown that

s

[s(x>,” dX = -27ri( -c)“[a(n - 1)]!2l’” [(a - 1) !]“[4(& + e)]“-“2 ’

(45)

which is pure imaginary if E + t>Oandrealif&+t
0~

SEVERAL

XODELS

( 1) Virtual Crystal ilIode1 The most elementary model is, of course, one which assumes that’ all diagrams vanish t,hat, are higher than t)he zeroth order. The dressed propagator S’(h) is t,hus approximated simply by the “undressed” propagator S(X), Eq. (42), alld the dru&y of st,at,es is found tJo he pl(&) = (7~~’ Re [2(& + t)]-li2.

(46)

(A subscript is used to label each approximation; here the subscript is 1, this heing the first case studied.) This model would bc most accurate for a very weak potent,ial and a high density limit,. Lax and Phillips referred to this model as the “optical” model, hut, this is a misnomer, for it is not the optical model as the term is used in nuclear studies (see Section IV for further discussion of this point ). More accurately it should bc caalled the virt#ual crystal approximation since it is hased on the average pot,ent)ial. Figures 3-5 illustrat,e, for t)hree different c values, the integrated density N1(6) = J‘pl(&) d&, as well as several integrat.ed deusiGes for the remaining models to he &died. The curve labeled FL (Frisch and Lloyd) corresponds to t,he accurate results ohtained hy a st,orhastic analysis. For a t,hree-dimensional potential P(X), oue readily finds t,hat, p,(E)

= (27rZn)-‘m lie [21n(i7 -

)1.w)]1’2,

an expression which corresponds to the same approximation

as Eq. (46).

(47)

56

KLAUDER

k2 :: z : 0 s

'

is Fz 0 --1.3

-1.2

-1.1

-.P-mmtur -1.0 -0.9

-0.6

-0.7

-0.6

-0.5

-0.4

ENERGY,

-0.3

-0.2

-0.1

0

0.1

0.2

1 3

E

FIG. 3. Integrated density of states versus energy in dimensionless units for various approximate treatments of the one-dimensional delta-function potential when t’he density parameter t = 0.1. Curves l-5 represent 5 approximate expressions discussed in Section III; the ith curve is based on the density of states pi(&). The curve labeled FL is exact as found by Frisch and Lloyd, Ref. S.

e = 1.0

ENERGY,

&

FIG. 4. Integrated density of states versus energy when e = 1.0. Curves l-5 represent 5 approximate expressions p,(&) (i = l-5) discussed in Section III; the curve labeled FL is exact.

BAND

0

-16

MODIFICATIONS

-16

BY

-14

ENERGY, FIG.

approximate exact.

57

IMPURITIES

-12 &

-10

-6

-6

5. Integrated density of states versus energy when B = 10. Curves 1-5 represent expressions pi(&) (i = l-5) discussed in Section III; the curve labeled FL

5 is

(2, i3) Second Born Approximation Two cases are discussed together in this subsection because of their related charact,er. The first case may be symbolically represented by integral equation ( B 2) for X’(X) shown in Fig. 6(2). The second case, which involves a selfconsistency requirement, is symbolically represented in Fig. 6(3). The interpretation of the symbolic integral equations ( # 2) and ( $z!3) in Fig. 6 is typical of all t,he examples to be discussed and is perhaps worthy of further discussion at this point. The diagrams which integral equation ( 82) includes may be found by iteration as follows: Replace the heavy bar (A”) on the right-hand side by the light bar (S) as a first approximation; the new result for S’ is therefore, t’he first approximation, S, plus the diagram represented by Fig. 1 (a). To obtain the next diagram the present approximation for the heavy bar is substituted on the right side of the “integral equation.” The newest approximation for S’ t,hcn has three diagrams: a light bar (S), the diagram of Fig. 1 (a), and also the diagram shown in Fig. 1(b) . The additional diagrams which equation ( # 2) represents can be found by continually resubstituting the current approximation for the heavy bar on the right side. In the case of integral equat)ion ( # 3) the first approximation for S’ is t,he same as given by ( #! 2) : S plus Fig. 1 (a). Homever, when this approximation for S’ is resubstituted in equation ( #a), two additional diagrams appear: t,he diagram in Fig. 1 (b) , and t,he diagram in Fig. l(c), the latter being a direct result of the propagator modificahion indicated in integral equation ( # 3). As before, further diagrams that are included in t,his approximation for ‘S’ can be found by repeated iteration. Using t,he notion of proper diagram and Eq. (-1.5) as an evaluation for the

58

KLAUDER

(2)

(3)

-\ I- -, I-\ ,u K

-\ x---- /-

L-L+-+%-+

(4)

(5)

(6)

-\“K I-=A” -,--’ --\ *+q (6) ) i’ ‘I u_j‘- s/k\ _//4L _dJ’\ .s’‘.

FIG. 6. Symbolic representation of t,he various approximate “integral equations” discussed in Section III. [( # 1) (not illustrated) consists in replacing the modified propagator (heavy bar) simply by the unmodified propagator (light bar) .] ( B 2) An approximation which treats each scattering separately and only to second order. (13) An approximation which provides a “medium” in which an indefinite number of complet.ed scat,terings occur to second order. ( #4) An approximation, like ( # 2), which treats scatterers separately, but now to all orders in the potential. (,$5) An approximation, like (#3), which provides a “medium” in which complete scat,terings occur to all orders. This case treats all two-body interactions fully and is analogous to t,he Brueckner equation. ($6) An approximation, not analyzed numerically, which partially treats complete three-body interactions.

diagram of Fig. e(2), me are easily led t,o the propagator case : &‘(A)

= -ee[&+ -

(X2/2) + E + i2-““E/(G

corresponding to that + #2]-1.

(48)

IL’ote here that the proper part, has not contributed a term which depends on momentum (x). This simplicity permits P*(G) to be computed by an integral similar to that which appears in Eq. (G), and we find that p*(E) = (m)-’

Re 2-“‘[&

+ c + i2-“‘t/(&

+ t)“2]-“2;

(49)

BAA-I)

MOI)IFICATIOSS

BY

IMPURITIES

59

~2 vanishes for all energies lower than that energy for which the denominator of (49) het+omes zero. The curves labeled 2 in Figs. 3-3 represent int.egrated densities of states derived from p; . These curves show that pz extends to lower values of & t’han did p1 , before p? becomes identically zero. While pl(&) extends only down to 8 = -E, we find t,hat ps(&) ext,ends down as far as E = -c - 2-1’3z!3. It, appears generally characteristic of the pertSurhation approach that additional diagram sets extend the validity of the propagator to lower energy values. Consequently, it, may he expected that the density of stat’es is given most poorly for the lowest lying cl~ergy lcvrl~ at auy st.age of approximation. This model demonstrates an int,ercsting feat,ure of this order of c*alculation that was postulated by Lax and Phillipti and verified by Frisch and Lloyd: In the high-density limit, the number of states helow the “band edge” of the opt,ical model is proportional t’o cF”~, the band edge heing defined by 8 + c = 0. The number of st#ates helow the band edge is obtained hy integrating l&l. (49) from El = --E _ 2py, the value of & where it vanishes identically, to & = -E. --r

N, (below band edge) = & d 7rese, By a change of integration variables, I<([. (50) becomes exactly expressed in terms of the usual beta fun&on, Rip, q). (51) While the numerical coefficient is only approsimately correct, (Frisch and Lloyd find from their “diffusion approximat)ion” the asymptot#ic behavior N = 0.2532 E-“‘, a relation verified numerically by Lax and Phillips), t,he fun&ional dependence 011 t,he density is correctly given. The coefficient discrepancy is an indicat,ion that, the most pronounced errors are in t,he low-energy behavior of p. In t’hree dimensions, t,he restricted diagram sum reprcsent,ed hy Fig. C,(2) leads to the following expression for the propagator: s,'(p)

=

[s'(p)]-'

!

X?(E,

p)'

where sg(EI, p) = ____l

(2*)3nw”

s

L!( -q)S(p

- qh(q)

dq.

Actually this approximat’ion is similar to that employed by Gell-Mann and Brueckner (8) in their st,udy of electron correlation phenomena. The similarity arises hecause each case repeatedly treats a specified pair interaction and, in our

60

KLAUDEK

case, it is treated only to second order. In general, we find from (53) that Sz is a nontrivial function of p which prohibits giving an expression for pz(E) other than that provided by the defining integral, Eq. (32). The third model we shall discuss also pertains to a high density approximation. Figure 6(3) schematically represents the integral equation for S,‘(X) which depends on additional diagrams besides all the diagrams that S,‘(X) depended on. In the present example the internal electron line propagates with the dressed propagator S,‘(X) itself. The coupled integral equations this propagator satisfies are S,‘(X)

where

=

-

6

Gf -

(P/2)

+ E + E&(E)

9

X3(&) =&sx;(x) dX.

When Eq. (54) is substituted into (55) and the X-integration are led to an algebraic equation that determines &(&) , EZ3 = i2F2e(& + E + czp.

carried out we (56)

Since it follows from (55) and (44) that ,03(E) = (l/-J

Im S(E),

(57)

we are only interested in that solution of (56) for which & has a positive imaginary part. The necessity of a positive imaginary component of & has been extensively discussed in the general case by Van Hove (19). Solutions to the algebraic equation (56) are straightforward and we find that 112

P3(&) = ,Qf$@ 3

[(b + rY3 - (b - d1’31,

where

(59) and f = (26 -

l)“!

(60)

In contrast to the relation for pz(E) , the expression (58) is everywhere finite by virtue of the modified propagator. The integrated density of states for this self-consistent solution is given for three different values of E in Figs. 3-5. The current approximation, like the last, predicts a total number of states below the

BAND

MODIFICATIOSS

band edge which is proportional X3 (below band edge) = 2$,j . ([(I - g

BY

61

IMPURITIES

to EC*‘~.In the present, approximation, .g

12-“3 dx (61)

+ (1 - 2$)l’y

-

[(l - $) - (1 - 2~3)l’y)

w OS?!.

The coefficient, of eCLi31s much closer than before t,o the asymptotic result of Frisch and Lloyd. It’ is not, difficult to show t,hat their asymptotic result can be obtained by a summation which includes all diagrams wit,h interact,ions only up to the second order. For a general three-dimensional potential two coupled equations are found similar in form to (52) and (58) : *S,'(p)

1

= [s'(p)]-'

-

83(E,

(62)

P) ’

where Z3(E,

P) = (2s;3nu:” s 2,(-q)Ss'(p

- 4Mq)

dq.

(63)

The nonlinear integral equation for 83’ that the above equations represent is t,ypical of the type encountered in self-consistent treatment of nuclear problems (10). One of the next two examples to be studied provides a similar self-consist)ent, model applicable to low densities as well. (4, 5) Some Higher Order Processes The t,wo cases examined here are also schematically represented in Fig. 6. If an iteration of the K operator is made according to the integral equation of Fig. 6(4), then it, is apparent t,hat the scattering from each impurity atom is treated exactly in this approximation, but, t,hat’ no internal propagator modification takes place. For the one-dimensional delta-function potential, the iterat,ed solut,ion for &’ can easily be summed since

&‘(A) = S(X) + S(X,[Z, + ZJ2+ 543+ . ..lfh’(X) =-

(64) &+ -

(V/2)

+ : + E&,/(1 - Z,) ’

where

The density of states these expressions yield is given by t,he formula pa(E) = (m)-‘2-l’”

Re (& + 6 - c( 1 + i[2(& + t)]“z]-1)--1’4.

(66,

62

KLAUDER

Three integrated densities based on this expression are plotted in Figs. ~3-3. In the interval where G + E < 0, t’he density p4 vanishes unless both P

1 & + E jy2 -

1 > 0,

(67)

and E > ( & + t l(P”

/ & + E jl”? -

1).

(68)

Equations (67) and (68) lead to an isolat,ed energy band which extends from -52 t#o an c-dependent lower value determined by (68). The upper &+t= edge of t’he “impurit,y band” in this approximat8ion is related to the energy of t,he single bound state located, for a single scatt,erer, at G = -12 [i.e., E = --$;(IQ~/wL), which follows from (39)\. Even as the density is raised, t,he impurity band determined here remains ixolat’ed from the conducation band, although the scattering from each impurity is treated exact)ly. The general three-dimensional analogue in t,hc present approximation is expressed with the aid of a vertex function K4(E, p, q) which satisfies t,he integral equation s a($ -

q)S(p + s’)

(69)

.b(-q’)

+

In terms of the solution K4 , t’he dressed propagator, in Fig. B(4) determines, is defined by

k’,w,

p, q’)l dq’.

which the set of diagrams

(70)

,SJ(p) = [,$(p)]-1 1 -&(E, p) ’ wherein X4(E,

p) =

k’4(E,

p, 0).

(71)

The integral equation which determines K4 is reminiscent, of the Brueckner equation (10). But the usual Brueckner equation involves modified propagators as well, and is inst.ead integral equation ( A 5j represented by the symbolic equation shown in Fig. 6(S). It is a simple matt,er to transcribe the formulas obt,ained in discussing Fig. 6(A) to apply t#o the modified case of Fig. 6(S). The algebraic structure of t’he propagator is t’he same,

S,‘(h) =--

I+ -

(X2/2) + :+

e&/(1

-2,)

’

(72)

BASD

MODIFICATIONS

BY

IMPGRITIES

hut now we find that

S,(G) = & 1 S,‘(X)dX.

(73)

Agaiu, since Y$,depends only ou the energy, au integral over X of Eq. (72) may bc directly carried out, to yield au algebraic equation which determines & as a funrt,iou of G and, of course, t. The density p5(G) is then determined from the equation (Im zs)/‘( ~6). The algebraic equat,ion that Sg obeys is 2&Ss3 - “(G + EjS5? + -? 45 = 1,

(Tlj

au equation which may bc solved by familiar techniques for cubits. The integrat,ed dcusity curves for three values of c are again shown in Figs. :3-S for approximation ( I 5). E’or clarity, these curves are not. extended to energy values as high as those illustrated for t,he other models. Compared to the ot,her approximations st)udies here the Brueckner-like approximatiou is relat’ively good in a low-density limit because of the accurate treatment of each int,eraction, and relatively good in a high-densiby limit because of the inclusion of propagat,ion through a medium by use of the modified propagators. The Brueckner approximatjion applied to a degenerate Fermi gas can, according to Hugenholtz (20 ), be viewed only as a low-density approximation because of the neglect of interactions wit#h, and among, holes in the Fermi sea. However, t,he conclusion of Hugenholtz does not apply to t,he impurit,y problem as treated here for the analog of the holes really have no int,er action. This lack of hole interaction may easily be seen, for example, if Goldstoue diagrams (+??I) are drawn for the processes we discuss. In t,hree dimensions, where of course the prercding remarks still remain true, the Brueckner approximatiou is formally st,at,ed by a simple extension of Eqs. (69)-( 71) in t,erms of the vertex function Ks(E, p, q), which obeys / ds’

- d&‘CP

+ 9’)

(75)

.[a( -9’)

+ &(E:, p, q’)] dq’.

This equation is coupled with the equation &‘(p)

1 =

[s(p)]-l

-

p) ’

E5(E,

(76)

where S5(E,

p)

=

K5(E,

p,

0).

(77j

The system (75) through (77) represents a nonlinear integral equation for h’s , and thus for &,’ , which is generally very difficult to solve.

64

KLAUDER

(6)

Partial

Pair

Approximation

To treat low-density cases wit’h a greater accuracy than provided by t,hc Brueckner analogue it is necessary to include t’hree-body diagrams, such as that, in Fig. l(d). While an integral equation for the complete three-body (one electron and two impurity atoms simultaneously) can he set up for the onedimensional model, it, leads t#o a Z which is now a function of momentjum as well as energy. This behavior is present even for a 2 which is determined by the proper graph in Fig. l(d) alone. lcor simplicity we sketch a partial pair approximation which obeys the symbolic integral equation represented in Fig. 6(6). The vertex operat#or K is the same two-body operator which appears in Fig. 6(S). Note that, the particular three-body diagram of Fig. l(d) is not, included in the set of diagrams represented by Fig. C,(6). The propagator in this approximation obeys S,,(X)

= -

e &f -

where it is found t,hat & = &(&) &,

=

p -

(X2/2)

is determined cp3

-

E/3 111

(78)

+ E + t& ’

by t,he following

equations:

(1 - p’), (79)

I $ p = p(& + e z+ E.&,)]“~* A pair of equations which determine t,he densit,y of st’ates is then given by dE)

= (l/m)

Ini iP/l

+ B),

(80)

where 0 is found from a solution of the t,ranscendent,al equation

exp &+&+(&+c+i)$ff-/+]= [

1 -P’.

(81)

No analysis of these equations is made and their predict,ions are not presented in Figs. 3-5. These equations do make it apparent, however, t’hat as the class of diagrams included in the perturbation series is increased, the equation t,o find 2 rapidly becomes increasingly complicated, even when Z depends only on G. The involved equations applicable to a general three-dimensional pot,entJial in this approximation arc not, included here. &3X-SSION

Figures 3-S demonstrate clearly the superiority of propagator modification as embodied in approximations ( # 3) and ( # .5). In bot,h cases a weak singularity in the density of states is removed at the lowest, energy, and a much smoother and more realistic density of states results. As the density paramet#er E is increased

BAND

MODIFICATIOiVS

BT

IMPURITIES

65

(Figs. 4 and 5), a propagat’or modification up to second order ( #3) is more important in introducing lower energy states than treating the interaction exactly ( 14). These results are, of course, just’ what would be expected with a high density of scatt,ers because t’he clect,ron is simultaneously under t’he influence of a large number of cent,ers-an effect qoalit’atively and quantitatively described by propagation in a “medium.” Approximation ( # 4) is interestingly affected by propagator modification. illthough this case nominally involves no propagator changes, it does include the influence of the Jirst Born scatt,erings to all orders. &4sa consequence, the integrated density of states curve that is illustrated is shifted down in normalized energy units by t from a similar curve which would arise if t)he influence of the first Born scatterings were not, included in propagator modification. This explains the apparent improvement in approximation ( $4) in Fig. 3 if the curve were shifted by 6 = 0.1 units to the right. However, as the scat,terer densihy increases (Fig. 4 and particularly Fig. 5) the minor propagator change from the average potential becomes an asset rather than a liabilit,y. Both approximations ( ~4) and ( d5) t’ake account of the presence of t,he bound state by t’reating each int’eraction exactly. Whenever 6 < 36, approximation (d 5) will lead to an isolated impurity band separated from the conduction band by a gap which extends from I = 0 down to an energy such that the integrated density of stat#es reaches unit,y. Thus, approximations ( L 4) and ( A 5) should show a correspondingly improved behavior at very low densities where the bound state is the dominant contributor to the density of shates. As is well known, a perturbation analysis carried out only t’o a finihe order in the potential would not reveal the bound stat’e. In addition, the Brueckner-like approximation ( B 5) also becomes fairly accurate for reasons discussed earlier as the impurity density is increased (see Fig. 5). On the whole, it appears that approximation (# 5) provides a scheme quit,e adequat’e for most practical problems, especially in view of the much greater complexity introduced when higher order processes are included, for example, in approximation ( 16). For all values of 6 t,he asymptotic solutions of Frisch and Lloyd exhibit a long “tail” extending to arbitrarily low energy values. This behavior is not adequately t’reated in any of our approximat,ion schemes for the following reason: Very low energy values appear only when impurities are closely clustered. A proper treatment of clusters involves, in turn, an accurate t,reatment of t,hree, four, . . .-body collisions in the same manner that ( # 5) accurately treats twobody collisions. From the brief discussion of the partial three-body diagrams in ( N!(i) it is clear that large clusters are not easily handled in this formalism. An alternate approach is necessary, perhaps relat#ed to the “local density model” discussed by Lax and Phillips. ilt any rate, it appears that the qualit,at,ive error in t’he long tail (e.g., finite lower houlld) is more significant than the quantita-

66

KLAUDER

tive error when the exact, results are compared, for example, to approximation (85). IV.

EQUATIONS

OBEYED

BY

THE

iZVERAGE

WAVE

In the st,udy made by Frisch and Lloyd f.3) of the one-dimensional delt#afunction potential they found an interesting and very simple relation to be obeyed by the average wave function. Their equat’ion and another equation that is also related t,o the average wave are the subject’ of the present section. The equat,ion whirh det,ermines t,he behavior of t’he complete wave function, of course, is simply

expressed treats the determines and Lloyd

in ronvent.ional variables (Vo < 0) and in the representation which impurit,y atoms as classical s&tering cent,ers. The equation which the average wave, which we denote by (Ic/E(.c)), is found by Frisch to be --- 1 d2 (J/E) + ~V&E) 2m d.r2

= E&r),

which involves only t,he acerage potential, like t,he virtual crystal model approximation. The above exact) result is surprising since it is in apparent contradiction with t,he answer obt,ained by simply taking an ensemble average of Eq. (82)) namely 1 d2 - _-2m d.c2 ME) + I’oW(x

- XI)&(X))

= E(J/e).

(84)

In order for (83) to hold, the average of the product, term in (84) must equal the product of the averages, an equality which is generally not valid. Quite generally, we can define ($J~)by t’he following symbolic integral: (h)

= s, Mx;

Xl ) xt ) . . . ,n

q

= s, $da;

X)6X.

(85)

In (85 j we have used t,he shorthand 6X to represent the normalized multiple differential II( dX,/V) and X to denot#e the sequence X, , XZ , . . . . The symbol R designates some “rule,” as yet unspecified, whereby the desired value of $E is correlated with t,he locat,ion of all t,he scatterers. There are many possible “rules” that can he studied, but, the two we shall discuss shortly are among the more interesting. If we continue to keep t,he “rule” R arbitrary for the moment, then the term in (84) proportional to VOhecomes ir

F

s,

$b(.r;

Xl

) . . . x1-1

, .I-, x2+1,

. . . ) (,g;)

d+

(86)

BAND

Since the potential written as

MODIFICATIONS

BY

IMPURITIES

67

associated with each scatterer is identical,

Eq. (86) may he

where 6’X symbolizes II’ (dX,,/I’), the normalized differential for N - 1 \wahles. The first “rule” me shall discuss is that, employed by Frisch and Lloyd. Since their ‘LruIc” involves fised boundary conditions we shall refer to this model bar adding the subscript fbc. In order t,o associate a value of JJ~(.v;Xi with the collective scatt,erer locat’ions X, the values of tiE and dfiE/d.r = #El are preselected at, an arhit,rary .L‘point, say at, .r = 0. Hence, in the fhc model nil fields fiB(5; X) sat#isfy t,hc suhsidiary conditions tiE(O; X)

= GO and

fiE’(O; X) = &‘,

(88)

both of which can he romplex, and may if desired be chosen differently for difftrent energy values. Since the equation of motion (82) is a second-order equation in .r alone the boundary condit#ions in Eq. (88) uniquely determine a solution. In general, this solution is neither an cigenfunctiou nor is it, well-behaved at, infinity. In fact, Frisch and Lloyd shorn t,hat for t’hc fbc solutions the mean square wave diverges with probability one. Of course, these remarks in no way limit the usefullness of the fhr solutions in t,hc dcfinitjion of the average wave. There exists a very interesting propert)y of the fbc solutions which, when combined with t’he delt,a-funct,ion potential, accounts for the validity of the simple equation of motion (8.3 j found by Vriwh and Lloyd. Consider the value of #E(.L.; .P,X? ) . . .) as det,ermined by fbc and let, us in\-estigat,e the dependence of this expression on t,he secmd x, i.e., the vector which denotes t,he impurity location. Since the range of the delta-function impurity potent.ial is infinitesimal and also since #B must, be continuous on either side of a scatterer, the value of l)E(.r; .r, x.’ ) . . .) at the point n’ as determined by fixrd houndary conditions is independent of the existence or hhe nonexistence of an impurit,y atom at t,ht point, .I’. Therefore, !h(.r; 2’, x2 ) . . . ) = l&(.c; x2 , . . . ),

(89)

when both functions satisfy identical boundary conditions on value and derioat’ive ($ and #‘) at the origin. In view of (g(3), the specific average in (87) with the subscript R = fbc becomes

11 rl-, x2 ,... s 4M:r; fbc

)S’X

=

c

&&ix;

[,

c P

5,

) . . .)8X

irIo)

where t’he last equality follows because an infinite number of scatterers is being considered. The equation of mot’ion (83)) which tbe average wave

obeys, readily follows from (84) and bhe result of Eq. (90). Of more long-range interest than the simple equation of motion for the Frisch and Lloyd average wave is the characteristic class of solutions over which their average is carried out. To discuss this class of waves it is useful t,o introduce a probability density Pfbo(lC/, $‘; x) which represents the probability density at the point :c that the fbc average wave and its derivative will have the values # and $‘, respectively. This probability density must obey the usual constraints: Pfbc(J/, $‘; X) 2 0

and s PfhO(~, J/‘; 2) c&w

= 1.

(92)

Since the equation of motion has real coefficients, $J and #’ may eit,her be restricted to be real variables, as was done by Frisch and Lloyd, or taken as complex variables. In the latter case we interpret dJ/ as QT dJli and d#’ as d&. G!J/~’, where T and i label the real and imaginary parts. Most of the analysis of Frisch and Lloyd can be readily ext’ended to complex waves. Formally, we may define

Pn,(#, 6’; 2) = Jl,, SM- \1/EkX)lW - Jh’k m1m

(93)

where the delta functions are one- or two-dimensional depending on whether 9 and $’ are taken as real or complex variables. The functions fie(~; X) and #E’(x; X) are uniquely determined by their values at .1: = 0 and, in an obvious notation, they may be expressed as $d.r;

z

=

g1$0

+

g2N

,

=

g&o

+

g4dJo1 )

(94) b’(.c;

X)

where each gn is a function of J, E, and ,X. At z = 0, it follows that g1 = g4 = 1 and g% = g3 = 0, values which arc independent of X. ConsequenUy, t,he two delta functions may he brought outside the integral (which simply becomes unity) so that I’fhc(lC/,

It’;

0)

=

6($

-

Go)6(#’

-

90’)

(95)

as demanded by the fixed boundary conditions. The average wave at t’he point .C is recovered directly from the first moment of the distribution, i.e., ($ar))

=

1

WfldJ/,

$‘;

11:1 %w

(96) = s

fbc kk

XNX,

BAXD

IMODIFICSTIO~X

BY

IMPUIZITlES

69

just as in Eq. (91). A wealth of additional information is also contained in the probability density Pfl,C . In facst, it is from a study of Pr,,c(#, #‘; -c) that E’risch and Lloyd were led to their solut,ion, in the form of quadrat,ures, of the particular density of states problem that we have discussed in Se&ion III. A different t,ype of average wave function is obtained when t,he ensemble of functions differs from those with fixed boundary conditions. WC shall now briefly discuss the case where the “rule” R restrkt’s the field fiE(s; X) to be an eigenfunction, or t,heir familiar extension in t,he manner of Dirac in t,he case of a continuous spectrum. A fuller and more precise discussion of this important example is given shortly when we discuss average waves in t,hrre dimensions. Let, us symbolically define, therefore, the average ware over the class of eigenfunctions (cif) as

(97) Two comment,s regarding this definit,ion need t’o be made: First, for certain energy values, there may be no eigenfunction for a specific collective scatterer configuration; in this case zero is the appropriate value for 1+5~ . (It may be necessary to rescale (+) by a multiplicative f&or, but this is not germane to our discussion. ) Second, we must determine what rule governs the choice of the relative phases of the various eigenfunctions tig(.r; X) as the variable X is changed. We shall show in the discussion of the t,hree-dimensional analog that (+(z)) can be uniquely defined in the sense of (97) up to a single over-all phase factor. Our present purpose is only t’o point out that the eif average wave will not obey t,he same differential equat,ion as does the fbc average wave. There is one term in t,he equation of motion (84) which will appear differently in an eif average, namely,

(98) The assert3ion proved in the fbc case was that tiE(:c; 2, X2 , . ) was independent of the second Z; in t,he eif case this is no longer t#rue. While the dependence on the second s in t’he eif case is physically obvious, let us prove it by assuming t,he converse, i.e., that tiE(.r; s, X:: , . . . ) = IcE(x.; X2 , . . . ), and both waves are eigenfunctions. It follows from the differential equation that the derivative of Ic/E(s; X? , . . . ) is contimlous at .r, while the derivative of &(x; 9, X2 , . . . ) defined by

has a finit’e discont’inuity at .c. Consequently, the \~nluc and derivative of s+bfi(.r;x2 ) . . . ) cannot match those of qE(.r; .L’, X:! , . . ) 011 both sides of the

70

KLAUDER

point .r. Wit,h the same value of tig but a different, value of tiE’ it follows from a theorem of Frisch and Lloyd (3) that at least, on one side of the point .c t,he function GE(.r; X2 , . . ) will blow up exponentially fast’ wit’h probabilit,y one, and can thus not bc an rigenfunct’ion. We have thus proved the incompat,ahilit,y of removing one scatt,erer and still retaining the eigenfunctjion property with the previous field value. Hence, I
The quantity Tt’ a& essentially as an effective, nonlocal pot,cntial for the eif average wave (#,J in (9’7 ), since it, follows that)

From translational symmetry IV is restricted to be a function of .I’ - y, and a11 important result brought8 out in the three-dimensional analysis is that W(s - !I) is proportional t)o t,he Fourier transform of the sum of proper graphs S(X) discussed in Sect8ion III. The preceding analysis shows that the specific equation of motion which the average wave obeys is clearly connected with the meaning of the phrase “average.” The simple eyuatJion of motion (83) or Frisch and Lloyd is a consequence of t#heir choice of average coupled with special properties of the delta-func%ion model. The same simple differential equation was obt’ained in the paper by Lax and Phillips (2)) but t,heir derivation is incorrect .6 The class of functions which Lax and Phillips caonsidered were all of the form rsp (kc) which, due to the translational symmetry of the problem, are all equivalent to eigenfunctions and do not cover the larger function class studied by Frisrh and Lloyd. The type of average implicit in the analysis of Lax and Phillips is the eif average rather t,han t,he fbr average. THREE

DIMESSI~~S

Both the fixed boundary condition average wave and t,he eigenfunct8ion average wave can he extended t#o three dimensions. We consider first’ the extension of 1’ Ill0 , the fixed boundary condition probability density. In the one-dimensional case it, sufficed to give as initial data two constants, Go and $Q’, corresponding 6 M. LAX (private communication) has pointed out the following footnot,e 9 of Lax and Phillips, Ref. 2, is incorrect, u-hich implies not follow from their Eqs. (1.1) and (4.2).

error that

in their derivation : their Eq. (1.3) does

BAA-U

MODIFICATIONS

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71

IMPURITIES

to the value and derivative at’ the point x = 0. The initial or conditional data required in three dimensions is given by specifying $ and Ic/’on a surface of one less dimension than the full space, i.e., on a two-dimensional surface. That this is the appropriate analogue may be seen by the following argument. The class of wave functions over which Frisch and Lloyd average includes nearly all nomlormalizahle functions. It is only when such functione are accept,ed that J/,, and Jl,,’ can be selected arbitrarily wit)h no restriction imposed on B. A similar feat,urc is expected for most functions in the class over which the fbc average is made in three dimensions. By permitt’in g singulari& and multi~raluedness to occur, it, is possible to satisfy the time-independent, three-dimensional SchrGdingrr cquat,ion for any value of E in a neighborhood of the two-surface on which 9 and #’ = &b,/dn are specified. As a simple example of the kind of solutions we are discussing, let, us investigate Dhe Helmholtz equat,ion 02+ + l&

= 0.

(101)

The general solutions to this equation involve both spherical Bessel functions and spherical harmonics. If we restrict our attent’ion to spherically symmetric solut8ions t,hen l)(r)

= OLjO(k?-)+ /ho(kr).

A solution of (101) which obeys #(Q)

(102)

= $0 and #‘(TO) = Jlo’ is oht,aincd if

a = (kro)2[no’(kr”)$o

- no(li-ro)k-‘~“‘], (103)

/3 = (kr~)2[-j~‘(Icr0)~~

+ j”(kr,)li-‘~J~~‘].

This solution exists for any value of k, but in general it is a singular solution. The present example illustrates that a unique solution exist’s for any energy in the neighborhood of a two-dimensional surface if 1c,and #’ are bot)h specified thereon; we assume this result to be generally valid. Analogous t,o the onedimensional analysis, w-e can, for example, detint /‘fhr($; x), the probability dcnsit,y at x that, J/(x) = I,!J,by means of the symbolic int,egral ~fl,c(I;

xl

=

l,>,

6M

-

!Mx;

x,

441)

Ic/o’)l 6X,

(10-l)

in which the dependence of tiE(x) on the conditional data $Q and $“’ has been indicated. The wave function #E(~) depends on the init,ial value data in a linear mamier just as in the one-dimensional case. While the class of functions can be established, there does not appear t’o be any method analogous to that of Frisch and Lloyd by which the definit,ion of Pfhc( fi; x) or even P,,,,(+, v#; x), say, can be used to find the differential equation obeyed by the average wave. However, in view of the many exact results Frisch and Lloyd obtained by pursuing

72

KLAUDER

an equation analogous to (104), there is some merit, in drawing this t,hrecdimensional form to the reader’s atkntion. The cxt’ension to three dimensions for the eigeufunction averaging process can be symbolically writt,en down immediately simply as

W)) = Iif \~h;xj 6x.

(105)

We shall show that the averaging process defined by (105) corresponds to the usual “optical model” average of Watson (12), widely used in t,he study of nucleon-nuclei scattering. To facilitate the analogy with t’he nuclear problem, we can interpret the N impurit#y atoms to represent, the target nuclei, and t,he electron to represent the projectile. The entire problem is translated into a many-particle quant,um picture, as discussed in Section II, for which the ground state of the syst#em is the pseudo-vacuum state ( 0), defined by Ey. (16). We denote by ( ON) the ground st’ate of the “nuclei,” i.e., the N-particle system alone. The one-particle wave matrix w is defined by t’he expectat’ion of the (N + I)-particle wave mat,rx 9 in the N-particle ground state, w =

(ON

/ fd / ON).

(106)

The one-particle wave vect,or 1J/) is then defined by I #> = tJJI CF)= (ON I w,

(,107)

where 1\E) is an exact (N + I)-particle eigenvector and 1cp) is an eigenstate of the unperturbed one-particle syst#em. The x-representation of the wave vector / #) is just t,he inner product (x ( Ic/), and it follows directly from (107) that (x \ +) E (XON ( \E), where / xON) denotes the (N + I)-particle state formed from the direct product of the N-particle ground state and the position eigenvector of t,he extra particle. Analogous to t)he method of investigation pursued in Section II, we now make use of the unit operator, expressed by /

j x’, Xl’, X0’, . . . ) dx’rl dxn’(x’, Xl’, X2’, . . . 1 )

so that (x I#) becomes (x / I/,) = j- (xON ( x’, X,‘, Xz’, . . . ) dx’ rI dXnl(x’, X,‘, X,‘, . . . I !I”) (108) zx ( vy2

/*,7(x,

X)rI dx, )

where, as before, X is short for X1 , Xp , . . and \ks(x, X) = (x, Xl, x2, * . * I %,

(109)

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MODIFICATIONG

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IMPURITIES

73

and where we have used the fact that, / 0,) is a moment,um eigenstate with eigenvalue zero for each impurity atom. We now wish to show that the funct~ions\kE(x, X) have all the desired properties that are required of t,he eigenfunctions fig(x; X) needed in the eif average defined symbolically in (105). First, if a part*icular configuration X1 , X2 , . . . is inconsistent with the presence of an energy cigenfunction with energy A’, then the projection of j ‘k) ont>o the stat’e 1x, X1 , X2 , s . .) would be identically zero, and the contribut(ion to the average waire would likewise be zero. Second, the relative phases of t’he various N-particle vectors 1Xl , Xp , . . .) are fixed a priori by the choice of their inner product with the N-particle ground state 1 ON), namely, I,-N/2. Consequently, bhere remains only one arbitrary over-all phase factor in \ks(x, X), which can be associated w&h t,he eigenve&or 1\k). Finally, to prove that (x / $) is proportional to the eif average wave it is necessary to show that !PE(x, X) and \k8!(x, X) are energy eigenfunctions orthogonal for different energy values over the one-particle subspace of the coordinate L alone. But this special ort8hogonality follows immediately when it, is remembered that t’he manyparticle Hamiltonian is diagonal in the impurity-atom subspace in the impurityatom position represent’ation. Thus \kE(x, X) and \kE,(x, X) may be viewed as two solubions of the same one-particle Hamiltouian in which X1 , X, , . . * appear as c-number paramet,ers; as such, these two functions are orthogonal for inequivalent, E‘ values over just the one-particle subspace. An elementary analogue of this ort’hogonality in a proper subspace is provided by the following example of the three-dimensional free-particle Hamiltonian Hf = p2/2m. Consider t>wo arbitrary energy eigenfunctions I E) and / K), and assume t’hem to be represented in terms of the mixed x-coordinate and p, , p,-moment~um eigenvectors, I zp,pz). It follows that. both (Pi = are solutions of the same Hermitian (sp,p, I E’) and at(~) = (~p,p, I E”) differential equation in the single variable .r, an equation in which p, and p, appear simply as parameters. Hence, when E # I?, it’ follows quite generally that JP~*(.c)P~~(x) d.~ = 0, a result easily verified directly since qe(x:) = exp (i.rpz), where p, = f (2mE - p,” - p:)“‘. The basis of t,he orthogonality which occurs in the impurit,y problem is simply an ext,ension of that which occurs in the present example. We have established, therefore, that the functions !PB(x, X) are orthogonal in t,he one-part,icle subspace for different energy values. This verifies their desired eigenfunction property, and we hereafter identify up to a scale factor the average over eigenfunctions ( 105) with the wave function (x 1Ic/)in Eq. (108). We now turn our attention to a formal discussion of the differential equation that is obeyed by the eif average wave. Several formal derivat,ions already appear in the literature which determine t’he “optical potential” for t’he one-body SchrGdinger equat,ion from the complete (N + 1 )-body problem (12, 22, 23). For the impurity band applical;ion as dis-

74

KLAUDEH.

cussed here no antisymmetrization is required between t,he “target” and “projectile.” Thus, the simplified optical pot,ential derivation of Feshbach is directly applicable ($3). The ordinary one-body optical model equation is

(&g-w ) $=O,

(110)

where the c-number energy of the N-part,& ground stat,e vanishes in the present case, and U, the optical potential operator, is generally bot,h non-Hermitian and nonlocal. The customary equation which determines w is the following:

In t’he present model, the operator V = Z~U(X - Xl) is the interaction in which x and every Xl denote operators, and

potential

(112) Here Q is t’he projection operator state j a) is one of t,he eigenstat’es 10,) represent#s the ground state. atoms, therefore, we may represent

onto t’he subspace orthogonal to 1ON), The of the isolat,ed N-body system among which In the present treatment of the impurity the st,ates / CC)in coordinate space simply by

(Xl , x* , . .

, a) = (v)-s!spa,.X,,

(113)

where (~1is int,crpreted as the momentum eigenvalue for the Zth impurity atom; (Y,like X, is t’herefore a label covering all N scatterers. In order to show the formal connection of the optical pot’ential ‘u to quantities discussed in Sections II and III, the following alt’ernate definition is useful:

'0 = (ON/ V / ON)+ (ON1R-' 1OX)- (ON1R 1ON)-' = E' - ($/am) - (0.v/ R ( ON)-',

(114

where R is defined similar to R' in (112) but without any Q operators. Equation ( 114) can be derived from ( 111) and ( 112)) and has also been obt’ained by Namiki and Suzuki ($3). The inverse of the last term, (ON 1R / ON),is t,he same quantity that was discussed in Section II. In particular, we find the relation (p’ \ (ON 1R(X)

IO,) ( p) = “@-$@

S’(p; E),

(ii5

j

in terms of the mat.rix elements of the one-body operator (0~ j R j ON), and where t.he dimensionless form for S’, like Eq. (25), has been used. From ( 115), it

B-4&-I)

MODIFICSTIONS

BY

75

IMPlYRITIES

follows that (P’ I iCONI NE)

I OAW’ 1 P)

=

h+

0’

-

I~inally, if the operator w is exprewed in the p-rrpresentjation dc1t.a function is cancelled), we find that v(p)

= E - &

=

S(E’,

P).

(116)

(and a common

- nw . S’(p; E)

or when expressed in terms of S, the contribution U(P)

p) .

(lli)

from t,he proper graphs, (118)

According to ( 118) the optical model potential w is the same operator as Z itself7 the sum of the proper graphs discussed in Section II. As a result,, the average wave defined by the eif averaging process is an exact wave fun&ion that, describes a st.at,e of t,he average syxt,em and is one of the solutions of the homogeneous Green’s function equation. i-1 similar connect,ion between modified propagators and modified wave functions has been discussed in elemcntnry particle physics by Schwinger (15) and especially by Dcser et nl. (a/,). I:inally, let, us point out, once more, the fact that the optical model potent’inl (118) is t.he exact effective potent.ial rather than the average potential accounts for the unsuccrssflll at tempt of Lax and I’hillips lo recover simply the average potential obtained by J”risch and Lloyd. It is possible that an analog of the Vrisch and Lloyd procedure may be connected with the &Iandelstam representat,ion of nonrelativistic dispersion theory. E’or, as pointed out by Blankenbccler et al. (2%), under suitable conditions the scattering amplitude (or one-body propagnt,or) is determined complet,eIy by the Mandelst,am representsCon, unitarity, and the first Born approximation. Clearly, for the simple one-dimensional drlta function potential the first Born approximation and the average potential arc identical. ,kGNOWLEDGYENTS

It is a pleasure to thank Dr. M. Las for suggesting this problem, for his continued terest, and for numerous stimulating discussions. The aut,hor also enjoyed conversations with Drs. P. W. Anderson, J. G. Fletcher, H. L. Frisch, S. P. Lloyd, and P. A. Wolff subjects related to this paper. Some of the numerical calculations xere carried out with assistance of Miss B. Cetlin, to whom thanks are given.

RECEIVED: October 21, 1960

inon the

76

KLAUDER

REFERENCES W. M. 3. H. 4. P.

Rev. 108, 590 (1957). 110, 41 (1958). FRISCH AND S. LLOYD, Phys. Rev. 120, 1175 (1960). M. MORSE AND H. FESHBACH, “Methods of Theoretical Physics.” McGraw-Hill, New York, 1953, Chapter 6. 5. F. J. DYSON, P/q/s. Rea. 76, 486 (1949); R. P. FEYNMAN, Ph.ys. Rev. 80, 440 (1950); J. HUBBARD, Proc. Izoy. Sot. A240, 539 (1957); I). F. I)UBOIS,~,nn& of Physics 7, 174 (1959). 6. H. LEHMANN, SOUUO cin&ento 11, 342 (1954); P. C. MARTIN AND J. SCHWINGER, Z’hys. Rev. 116, 1342 (1959). 7. S. F. EDWARDS, Phil. IMay. [S], 3, 1020 (1958). 8. M. GELL-MANN AND I(. BRUECKSER, Phys. Rev. 106, 364 (1957); see also Dubois, Ref. 5. 9. F. J. tI)YSoN, “Advanced Quant~um Mechanics,” Cornell Lecture Notes (1951) ; J. M. JAUCH AND F. ROHRI,ICH, “The Theory of Photons and Electrons,” AddisonWesley Publishing Co., Cambridge, 1955. 10. Ii. A. BRUECKNER, Theory of Nuclear Structure, it1 “The Many-Body Prohlem.” Wiley, New York, 1959, and additional references therein. 11. M. LAS (private communication) (1959). Id. N. C. FRANCIS ANI) I<. M. WATSOX, Z’h,ys. Rev. 92, 291 (1953); G. TAKEDA AND B. M. WATSON, Phys. Rev. 97, 1336 (1955); K. M. WATSON, Phys. ZSev. 106, 1388 (1957). 13. P. W. ANDERSOX. Phys. Rev. 109, 1492 (1958). 14. See, for example, I,. I. Shift’, “Quantum Mechanics, ” 2nd ed., Chapter XIII. McGrawHill, New York, 1955. 15. J. SCHWINGER, Pm-. Nat/. Amd. fki. 37, 452, 455 (1951). 16. SCIIWEBER, BETHE, .~NL) DE HOFFMAN, “Mesons and Fields,” Volume 1. Row, Peterson and Co., Evanston, 195ti. f 7. It. P. FEYSMAN, Phys. Rev. 76, 769 (1949). 18. C:. C. WICK, Phys. Rev. 80, 2G8 (1950). 19. L. VAN Hove, Physicu 21, 901 (1955). 20. N. M. HUGENHOI~TZ, Physica 23, 533 (1957). 21. J. (:oI.DsToNE,P~o(..K~)~.SO(.. A239.267 (1957). 22. M. LAX, Phlys. Rev. 79, 2OOiA) (1950); M. NAMIKI ASD Y. ST-ZITRI, Progr. Theoret. Phys. Japan 9, 223 (1953) and several later articles; I’. A. WOI,FF iunpublished) (1956) ; J. S. BELL AND E. J. RQCIRES, Phys. Rel’. /,etters 3,96 (19%). 23. H. FESHBACII, in “Annual Review of Nuclear Srience,” Vol. 8, p. 49. Annual Reviews, Inc., Stanford, 1958; .4n,na/s of Physics 6, 357 (1958). 24. S. DESER, W.E. TEIIRRING, AND M. L. GOLUBERGER, Phys.Rev.94,711 (1954).See also M. Gell-Mann and F. I,ow, Phys. Rev. 84, 350 (1951). 25. R. BLANKENBECLER, ill. L. C;OI,I)BERGER, N. N. KITI-RI, AK‘D S. B. TREIMAN, .~1nnaZs of Physics 10, 62 (1960). 1. 2.

KUHN AND J. M. LUTTINGER, Phys. LAS AND J. C. PHILLIPS, Phys. Rev.