- Email: [email protected]
Bound state of two plasmon excitations
Solid State Communications,
Vol. 25, pp. 485-488,
Pergamon Press.
1978.
Printed in Great Britain
BOUND STATE OF TWO PLASMON EXCITATIONS P. Deo Rama Devi Women’s College, Bhubaneswar,
Orissa, India 75 1007
and D. N. Tripathy Institute
of Physics, A/l05
Saheed Nagar, Bhubaneswar,
Orissa, India 75 1007
(Received 7 July 1977 by A.R. Verma)
A theory for the bound state of two plasmons in an electron gas is developed with the four-particle density response function invoking only the electron-electron interactions. The plasmon-plasmon interaction in our theory arises due to the exchange of four one-particle Green’s functions for the electrons. This interaction is found to be attractive, and under certain conditions is reduced to the exchange of an electron-hole pair. However, this is not strong enough to bind two plasmons in the range of metallic densities, which is in contrast to the conclusions derived by Ruvalds et al, constant required to split a bound state off the two plasmon continuum, is determined. Comparing g”4, with the value of g4 calculated from the process arising out of the exchange of an electron-hole pair, one finds that the ratio
RECENTLY there has been a great interest in the study of the bound state of two plasmons in an electron gas. The first attempt to investigate this problem theoretically has been made by Ruvalds, Rajagopal, Carballo, and Grest [ 1] . These authors have used a second quantized version of the approach due to Bohm and Pines [2] according to which an interacting electron gas is represented by a system of two interacting fields, one due to the electrons and the other due to the plasmons. The interaction between the electrons and plasmons is brought out through terms that are linear and bilinear in the plasmon co-ordinates. By accounting for the effect of the bilinear interaction term they have written down the Bethe-Salpeter equation for the scattering of two plasmons. In this theory the dominant part of the effective interaction between the plasmons results from the exchange of an electron-hole pair and this is found to be attractive. The solution of the Bethe-Salpeter equation for the two plasmons obtained by Ruvalds et al. [ 1 ] is written in the form
F(Q, w) =
F"'(Q o) 1 -g4F'Oi(Q,
where F(O) is the is the strength of (1) it follows that of a plasmon pair 1 -gaF”‘(Q,
w)’
g4/gcq
(1)
bare two-plasmon propagator, and g4 the plasmon-plasmon coupling. From the criterion for forming a bound state is given as
w) = 0.
(2)
Solving (2) for the momentum Q = 0 and threshold energy o = 2w,r, g$, the critical value of the coupling 485
=
O.O9rz,
(3)
where r, is the well known parameter for the electron gas. In order that the bound state of two plasmons would be possible this ratio g,/g$ has to be greater than unity. From (3) it follows that the metallic densities (1.8 < r, < 5.6) such as r, 2 4, would be favourable for the formation of a bound state. The expression for F(‘)(Q, w) as obtained by Ruvalds et aZ. [I] seems to be unphysical as it is unsymmetric with respect to the interchange w -+ - w. This defect is not there in the subsequent work by Rajagopal et al. [3] . As can be seen from this work, it is very unlikely that there would be any momentum dependence in the expression for F’O’(Q, w). This is because the bare one-plasmon propagator Dy(k, E) = [E’ - w$(k)]-’ would remain momentum independent if any of the effect of the electron-plasmon interaction is not incorporated. One can of course give a momentum dependence to Dy(k, E) if one formally writes o&k) = mpr + oJc2, where o,r is the classical plasmon frequency and OLis the dispersion coefficient. It would then mean that a plasmon has already undergone dispersion at least in the random phase approximation (RPA), i.e. the effect of terms linear in electron-plasmon coupling has been taken into the considerations. Once this has happened one would expect the one plasmon propagator
486
BOUND STATE OF TWO PLASMON EXCITATIONS 102
F (1234)
Ic)----2
Vol. 25, No. 7
I~---~---~2
=
I
o---
+ 30---
I
o---
+ 3Q---
.
+.
+ 3 *4+3*---*4
.
3~---~---~4
, o__
---e2
cl
u--_
___~--~2
El
___e4+3e__*___
___u---
---02
cl
El
___~___
, o-_-u___ + 3Q___O_-_
+... Fig. 1. Feynman
* *
___~--_~4+*
---a4
---~___U__
_
63
____~___~2
Et
-__~___~--_
___Q___*4+'
l
.
. . . diagram describing the scattering of two plasmons.
to be associated with a coupling constant. This is, however, not evident from the expression given by Rajagopal er al. [3] Furthermore, if one assumes that the plasmon has undergone dispersion in the RPA fashion it would be difficult to justify the presence of a width F in the expression for the two plasmon propagator given by Ruvalds et al. [I] . From all these considerations we have tried to make a direct attack on the problem of the bound state of two plasmons in an electron gas by invoking only the electron-electron interactions. Our investigation is based on the study of the four particle density response function, which, in the Boson representation leads us to have the propagation of two collective modes each being restricted to the range of momentum less than a critical value k,. From the Feynman diagram (shown in Fig. 1) which represents the scattering of a pair of plasmons in our theory we identify the interaction between the two plasmons that is responsible for the scattering. Evaluating this interaction at the poles of the plasmon propagators for the momentum Q = 0 and frequency w = 20,~ we obtain the expression for the effective plasmon-plasmon coupling g4, which is found to be attractive. For forward scattering and values of the plasmon momenta around k, the expression for our g4 is simplified so as to involve a free electron-hole pair propagator. In other words, the effective plasmonplasmon coupling happens to be due to the exchange of an electron-hole pair. The Bethe-Salpeter equation for the scattering of two plasmons which has been represented by the Feynman diagram shown in Fig. 1 is solved without any difficulty by introducing a suitable decoupling. By this
its solution is cast into the form given in (1). Using our expression for the two plasmon propagator we have calculated gz , the critical coupling constant. From the ratio g4/gs we find that the bound state of two plasmons is not possible for any of the metallic densities. We start with the four particle density response function which is defined as F(l) 2,334)
= i (I T]&l)p(2)p(3)p(4)1
I),
(4)
where p denotes the density operator in the Heisenberg representation, I ) is the exact ground state of the manyelectron system, and each of the numbers 1,2 . . . etc. represent the space, time, and spin variables. We work in the interaction representation to write down a perturbative expansion for F. We go to many orders in the perturbation theory to obtain the terms that we need to account for the propagation of two collective modes in the system. Out of the large number of terms we come across we select only a particular set of terms which describe the coherent propagation of these modes. When this is done F is represented by the Feynman diagrams shown in Fig. 1. In this figure the bubbles denote the electron-hole pairs and the dotted lines bare Coulomb interactions. After contracting the co-ordinate variables 3 -+ 1 and 4 + 2, we make the Fourier transformation to write down F in (Q, o) space. With this the perturbation series for F is written as F(Q, a> = F”‘(Q, x
R2(pl>
wWz(p1
WI+ g
+
i2
Q,~I
1 dp, da1 s dp, da2 + a>
BOUND STATE OF TWO PLASMON EXCITATIONS
Vol. 25, No. 7
xi(;n)4 . -~(PMPI
X
5
dp3
da3
Go(p2
+
Go(P~,u~)Go(P~
x GO(PI + p3 +
x
+ QMPzMP~
+p3
+
Q, WI + Q, ~2
~3
+a3
+
+
Q)
Q, ~3
+ a)
a>
+w) I
x R2(pz,oz)R2(~2
+
Q, 02
+a)+.
..3
(5)
where J7'"'(Q,o)
= L!.(W4
x R2(~1
+
dp, dw 1‘
Rz(plw)
Q, WI + ~1.
(6)
In the above equation R2 represents the density-density response function, Go, the one-particle Green’s function, and v(p) the Fourier transform of the bare Coulomb potential. It is known that the spectrum of R2 consists of two parts [4], one due to the quasi particles and the other due to plasmons. In the RPA (pair approximation) the plasmons represent well defined collective excitations of the system for momenta less than k,. Since in the present case we are interested in the propagation of two collective modes in the system, we confine ourselves to the Boson part of the spectra [4] present in R2. That is, we write R~(P,u)=
Q(P) 2v(p)
-
$F~+
p
2fi(P) [iqp) - irJ2 - cd2I ’
(7)
where a(p) = w,r{ 1 + 3p2 vi /lo&}, v, being the Fermi velocity. The multiplying factor outside the bracket in (7) denotes the electron-plasmon coupling. With R2 as given above the expression for F(O) is given by F(O)(Q, o)
=
lim
(2n)-3
X
s pi
seen that the expression within the curly bracket represents g4. When this expression is replaced by an averaged value the decoupling is possible and the solution of (5) (with the inclusion of higher order terms) can be cast into the form shown in (1). For Q = 0 and w = 2w,r we have evaluated g4 at the poles of the plasmon propagators. This is the only value of g4 which is of interest to us, because this leads to the coherent propagation of the two plasmons. The purpose of choosing such a particular value of g4 is the following: firstly, like Ruvalds et al. [l] we also assume that Q # 0 situation would not appreciably alter our conclusion for Q = 0; secondly, since both g4 and & calculated by Ruvalds et al. [l ] are available for Q = 0 and w = 2w,r we will be able to compare our results with those of theirs if we evaluate them under these conditions. The resulting expression for g4 is further simplified for the case of forward scattering and for values of plasmon momenta pi and p2 around k,, where k, in metal is generally of the order of the Fermi momentum kF. After performing the spin summations for the electrons our calculated value ofg, is given as (9) the value of xo(2kF, 0) being (- mkF/2n2). It is interesting to note that the effective coupling g4 is attractive and under the approximations stated above the generalized expression of our g4 is reduced to the case where it is due to the exchange of an electron-hole pair. This leads to the same situation as considered by Ruvalds etal. [l] . From (1) we calculate f4 by setting 1 -g4Fto)(Q = 0, w = 2wpI) = 0. Following (S)f4 is found to be
(10) where p = (5upl/6EF)m. we obtain
q-0+
dp
IPI+Ql
-(u+WPI)+WPI
g4 f&i
1
+Q)-W’).
(8)
It may be seen from above that F(‘)(Q, o) is now symmetric with respect to the interchange w + - w. Here it is associated with a coupling constant arising due to electron-plasmon coupling. We now make use of (7) in the second term of (5) perform the o3 ,integration, and then try to express this as a product of F(O) times F(O) and a factor which could be identified as the effective coupling g, between the two plasmons. From (5) it is
487
=
O.O656r,.
With the help of (9) and (10)
(11)
It can be seen from the above that this ratio exceeds unity for r, > 15.25 which means that the binding of the two plasmons is only possible at densities far below the metallic density range. This is in contrast with the conclusion derived by Ruvalds et al. [ 1] . It may be noticed that the ratio g4/gC in our case depends linearly on r, whereas in their case a quadratic dependence on r, is observed. We conclude that the theory developed by us to study the bound state of two plasmons is a first principle approach which is based on the many body theory of an
488
BOUND STATE OF TWO PLASMON EXCITATIONS
electron gas invoking only the electron-electron interactions. Our primary calculations of the effective coupling indicates an attractive form of the interaction between the two plasmons. However, the coupling is not strong enough to bind the two plasmons in the range of metallic densities. We only expect a resonance in the two-plasmon spectrum for such densities. From the present work it is clear that there should not be any width present in the bare two-plasmon propagator if the individual plasmons are to be taken in the RPA. Assignment of a width in the two-plasmon propagator can only be justified if one goes beyond the RPA, in which case it would not be proper to use the RPA type expression for the one-plasmon propagator. We think that when the non-RPA corrections would be accounted for it perhaps will not drastically alter the conclusion derived in this letter.
Vol. 25, No. 7
Experimental measurements [S, 61 of the twoplasmon density of states in systems like graphite and quasi one-dimensional metals like TTF-TCNQ are expected to give us quantitative information about the specific nature of the two plasmon interactions. Once such information is available and the conclusion goes in favour of the formation of the two plasmon bound states, then only it would be proper to look into the effect of Q # 0 and of the non-RPA contributions.
Acknowledgements - The authors thank Dr. S.S. Mandal for many helpful discussions on the problem and to Dr. B.K. Rao for critical reading of the manuscript. One of us (P.D.) thanks Professor T. Pradhan, Director of the Institute of Physics, Bhubaneswar, for giving the opportunity to work at the Institute.
REFERENCES 1.
RUVALDS J., RAJAGOPAL
A.K., CARBALLO J. & CREST G.S., Whys. Rev. Lett. 36,274
(1976).
2.
BOHM D. & PINES D., Whys. Rev. 92,609
3.
RAJAGOPAL A.K., GREST G.S. & RUVALDS J.,Phys. Rev. B14,67 (1976).
4.
SCHULTZ T.D., Quantum Field Theory And K%eMany Body Problem, p. 103. Gordon and Breach, New York, (1964).
5.
BRADSHAW A., CEDERBAUM S., DOMCKE W. & KRAUSE U., J. Phys. C7,4503
6.
RITSKO J.J., LIPARI N.O., GIBBONS P.C., SCHNATTERLY Lett. 36,210 (1976).
(1953).
(1974).
S.E., FIELDS J.R. & DEVATY R., Phys. Rev.
Vol. 25, pp. 485-488,
Pergamon Press.
1978.
Printed in Great Britain
BOUND STATE OF TWO PLASMON EXCITATIONS P. Deo Rama Devi Women’s College, Bhubaneswar,
Orissa, India 75 1007
and D. N. Tripathy Institute
of Physics, A/l05
Saheed Nagar, Bhubaneswar,
Orissa, India 75 1007
(Received 7 July 1977 by A.R. Verma)
A theory for the bound state of two plasmons in an electron gas is developed with the four-particle density response function invoking only the electron-electron interactions. The plasmon-plasmon interaction in our theory arises due to the exchange of four one-particle Green’s functions for the electrons. This interaction is found to be attractive, and under certain conditions is reduced to the exchange of an electron-hole pair. However, this is not strong enough to bind two plasmons in the range of metallic densities, which is in contrast to the conclusions derived by Ruvalds et al, constant required to split a bound state off the two plasmon continuum, is determined. Comparing g”4, with the value of g4 calculated from the process arising out of the exchange of an electron-hole pair, one finds that the ratio
RECENTLY there has been a great interest in the study of the bound state of two plasmons in an electron gas. The first attempt to investigate this problem theoretically has been made by Ruvalds, Rajagopal, Carballo, and Grest [ 1] . These authors have used a second quantized version of the approach due to Bohm and Pines [2] according to which an interacting electron gas is represented by a system of two interacting fields, one due to the electrons and the other due to the plasmons. The interaction between the electrons and plasmons is brought out through terms that are linear and bilinear in the plasmon co-ordinates. By accounting for the effect of the bilinear interaction term they have written down the Bethe-Salpeter equation for the scattering of two plasmons. In this theory the dominant part of the effective interaction between the plasmons results from the exchange of an electron-hole pair and this is found to be attractive. The solution of the Bethe-Salpeter equation for the two plasmons obtained by Ruvalds et al. [ 1 ] is written in the form
F(Q, w) =
F"'(Q o) 1 -g4F'Oi(Q,
where F(O) is the is the strength of (1) it follows that of a plasmon pair 1 -gaF”‘(Q,
w)’
g4/gcq
(1)
bare two-plasmon propagator, and g4 the plasmon-plasmon coupling. From the criterion for forming a bound state is given as
w) = 0.
(2)
Solving (2) for the momentum Q = 0 and threshold energy o = 2w,r, g$, the critical value of the coupling 485
=
O.O9rz,
(3)
where r, is the well known parameter for the electron gas. In order that the bound state of two plasmons would be possible this ratio g,/g$ has to be greater than unity. From (3) it follows that the metallic densities (1.8 < r, < 5.6) such as r, 2 4, would be favourable for the formation of a bound state. The expression for F(‘)(Q, w) as obtained by Ruvalds et aZ. [I] seems to be unphysical as it is unsymmetric with respect to the interchange w -+ - w. This defect is not there in the subsequent work by Rajagopal et al. [3] . As can be seen from this work, it is very unlikely that there would be any momentum dependence in the expression for F’O’(Q, w). This is because the bare one-plasmon propagator Dy(k, E) = [E’ - w$(k)]-’ would remain momentum independent if any of the effect of the electron-plasmon interaction is not incorporated. One can of course give a momentum dependence to Dy(k, E) if one formally writes o&k) = mpr + oJc2, where o,r is the classical plasmon frequency and OLis the dispersion coefficient. It would then mean that a plasmon has already undergone dispersion at least in the random phase approximation (RPA), i.e. the effect of terms linear in electron-plasmon coupling has been taken into the considerations. Once this has happened one would expect the one plasmon propagator
486
BOUND STATE OF TWO PLASMON EXCITATIONS 102
F (1234)
Ic)----2
Vol. 25, No. 7
I~---~---~2
=
I
o---
+ 30---
I
o---
+ 3Q---
.
+.
+ 3 *4+3*---*4
.
3~---~---~4
, o__
---e2
cl
u--_
___~--~2
El
___e4+3e__*___
___u---
---02
cl
El
___~___
, o-_-u___ + 3Q___O_-_
+... Fig. 1. Feynman
* *
___~--_~4+*
---a4
---~___U__
_
63
____~___~2
Et
-__~___~--_
___Q___*4+'
l
.
. . . diagram describing the scattering of two plasmons.
to be associated with a coupling constant. This is, however, not evident from the expression given by Rajagopal er al. [3] Furthermore, if one assumes that the plasmon has undergone dispersion in the RPA fashion it would be difficult to justify the presence of a width F in the expression for the two plasmon propagator given by Ruvalds et al. [I] . From all these considerations we have tried to make a direct attack on the problem of the bound state of two plasmons in an electron gas by invoking only the electron-electron interactions. Our investigation is based on the study of the four particle density response function, which, in the Boson representation leads us to have the propagation of two collective modes each being restricted to the range of momentum less than a critical value k,. From the Feynman diagram (shown in Fig. 1) which represents the scattering of a pair of plasmons in our theory we identify the interaction between the two plasmons that is responsible for the scattering. Evaluating this interaction at the poles of the plasmon propagators for the momentum Q = 0 and frequency w = 20,~ we obtain the expression for the effective plasmon-plasmon coupling g4, which is found to be attractive. For forward scattering and values of the plasmon momenta around k, the expression for our g4 is simplified so as to involve a free electron-hole pair propagator. In other words, the effective plasmonplasmon coupling happens to be due to the exchange of an electron-hole pair. The Bethe-Salpeter equation for the scattering of two plasmons which has been represented by the Feynman diagram shown in Fig. 1 is solved without any difficulty by introducing a suitable decoupling. By this
its solution is cast into the form given in (1). Using our expression for the two plasmon propagator we have calculated gz , the critical coupling constant. From the ratio g4/gs we find that the bound state of two plasmons is not possible for any of the metallic densities. We start with the four particle density response function which is defined as F(l) 2,334)
= i (I T]&l)p(2)p(3)p(4)1
I),
(4)
where p denotes the density operator in the Heisenberg representation, I ) is the exact ground state of the manyelectron system, and each of the numbers 1,2 . . . etc. represent the space, time, and spin variables. We work in the interaction representation to write down a perturbative expansion for F. We go to many orders in the perturbation theory to obtain the terms that we need to account for the propagation of two collective modes in the system. Out of the large number of terms we come across we select only a particular set of terms which describe the coherent propagation of these modes. When this is done F is represented by the Feynman diagrams shown in Fig. 1. In this figure the bubbles denote the electron-hole pairs and the dotted lines bare Coulomb interactions. After contracting the co-ordinate variables 3 -+ 1 and 4 + 2, we make the Fourier transformation to write down F in (Q, o) space. With this the perturbation series for F is written as F(Q, a> = F”‘(Q, x
R2(pl>
wWz(p1
WI+ g
+
i2
Q,~I
1 dp, da1 s dp, da2 + a>
BOUND STATE OF TWO PLASMON EXCITATIONS
Vol. 25, No. 7
xi(;n)4 . -~(PMPI
X
5
dp3
da3
Go(p2
+
Go(P~,u~)Go(P~
x GO(PI + p3 +
x
+ QMPzMP~
+p3
+
Q, WI + Q, ~2
~3
+a3
+
+
Q)
Q, ~3
+ a)
a>
+w) I
x R2(pz,oz)R2(~2
+
Q, 02
+a)+.
..3
(5)
where J7'"'(Q,o)
= L!.(W4
x R2(~1
+
dp, dw 1‘
Rz(plw)
Q, WI + ~1.
(6)
In the above equation R2 represents the density-density response function, Go, the one-particle Green’s function, and v(p) the Fourier transform of the bare Coulomb potential. It is known that the spectrum of R2 consists of two parts [4], one due to the quasi particles and the other due to plasmons. In the RPA (pair approximation) the plasmons represent well defined collective excitations of the system for momenta less than k,. Since in the present case we are interested in the propagation of two collective modes in the system, we confine ourselves to the Boson part of the spectra [4] present in R2. That is, we write R~(P,u)=
Q(P) 2v(p)
-
$F~+
p
2fi(P) [iqp) - irJ2 - cd2I ’
(7)
where a(p) = w,r{ 1 + 3p2 vi /lo&}, v, being the Fermi velocity. The multiplying factor outside the bracket in (7) denotes the electron-plasmon coupling. With R2 as given above the expression for F(O) is given by F(O)(Q, o)
=
lim
(2n)-3
X
s pi
seen that the expression within the curly bracket represents g4. When this expression is replaced by an averaged value the decoupling is possible and the solution of (5) (with the inclusion of higher order terms) can be cast into the form shown in (1). For Q = 0 and w = 2w,r we have evaluated g4 at the poles of the plasmon propagators. This is the only value of g4 which is of interest to us, because this leads to the coherent propagation of the two plasmons. The purpose of choosing such a particular value of g4 is the following: firstly, like Ruvalds et al. [l] we also assume that Q # 0 situation would not appreciably alter our conclusion for Q = 0; secondly, since both g4 and & calculated by Ruvalds et al. [l ] are available for Q = 0 and w = 2w,r we will be able to compare our results with those of theirs if we evaluate them under these conditions. The resulting expression for g4 is further simplified for the case of forward scattering and for values of plasmon momenta pi and p2 around k,, where k, in metal is generally of the order of the Fermi momentum kF. After performing the spin summations for the electrons our calculated value ofg, is given as (9) the value of xo(2kF, 0) being (- mkF/2n2). It is interesting to note that the effective coupling g4 is attractive and under the approximations stated above the generalized expression of our g4 is reduced to the case where it is due to the exchange of an electron-hole pair. This leads to the same situation as considered by Ruvalds etal. [l] . From (1) we calculate f4 by setting 1 -g4Fto)(Q = 0, w = 2wpI) = 0. Following (S)f4 is found to be
(10) where p = (5upl/6EF)m. we obtain
q-0+
dp
IPI+Ql
-(u+WPI)+WPI
g4 f&i
1
+Q)-W’).
(8)
It may be seen from above that F(‘)(Q, o) is now symmetric with respect to the interchange w + - w. Here it is associated with a coupling constant arising due to electron-plasmon coupling. We now make use of (7) in the second term of (5) perform the o3 ,integration, and then try to express this as a product of F(O) times F(O) and a factor which could be identified as the effective coupling g, between the two plasmons. From (5) it is
487
=
O.O656r,.
With the help of (9) and (10)
(11)
It can be seen from the above that this ratio exceeds unity for r, > 15.25 which means that the binding of the two plasmons is only possible at densities far below the metallic density range. This is in contrast with the conclusion derived by Ruvalds et al. [ 1] . It may be noticed that the ratio g4/gC in our case depends linearly on r, whereas in their case a quadratic dependence on r, is observed. We conclude that the theory developed by us to study the bound state of two plasmons is a first principle approach which is based on the many body theory of an
488
BOUND STATE OF TWO PLASMON EXCITATIONS
electron gas invoking only the electron-electron interactions. Our primary calculations of the effective coupling indicates an attractive form of the interaction between the two plasmons. However, the coupling is not strong enough to bind the two plasmons in the range of metallic densities. We only expect a resonance in the two-plasmon spectrum for such densities. From the present work it is clear that there should not be any width present in the bare two-plasmon propagator if the individual plasmons are to be taken in the RPA. Assignment of a width in the two-plasmon propagator can only be justified if one goes beyond the RPA, in which case it would not be proper to use the RPA type expression for the one-plasmon propagator. We think that when the non-RPA corrections would be accounted for it perhaps will not drastically alter the conclusion derived in this letter.
Vol. 25, No. 7
Experimental measurements [S, 61 of the twoplasmon density of states in systems like graphite and quasi one-dimensional metals like TTF-TCNQ are expected to give us quantitative information about the specific nature of the two plasmon interactions. Once such information is available and the conclusion goes in favour of the formation of the two plasmon bound states, then only it would be proper to look into the effect of Q # 0 and of the non-RPA contributions.
Acknowledgements - The authors thank Dr. S.S. Mandal for many helpful discussions on the problem and to Dr. B.K. Rao for critical reading of the manuscript. One of us (P.D.) thanks Professor T. Pradhan, Director of the Institute of Physics, Bhubaneswar, for giving the opportunity to work at the Institute.
REFERENCES 1.
RUVALDS J., RAJAGOPAL
A.K., CARBALLO J. & CREST G.S., Whys. Rev. Lett. 36,274
(1976).
2.
BOHM D. & PINES D., Whys. Rev. 92,609
3.
RAJAGOPAL A.K., GREST G.S. & RUVALDS J.,Phys. Rev. B14,67 (1976).
4.
SCHULTZ T.D., Quantum Field Theory And K%eMany Body Problem, p. 103. Gordon and Breach, New York, (1964).
5.
BRADSHAW A., CEDERBAUM S., DOMCKE W. & KRAUSE U., J. Phys. C7,4503
6.
RITSKO J.J., LIPARI N.O., GIBBONS P.C., SCHNATTERLY Lett. 36,210 (1976).
(1953).
(1974).
S.E., FIELDS J.R. & DEVATY R., Phys. Rev.