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Study of the dynamic structure factor of an electron liquid
Volume 71A, number 5,6
PHYSICS LETTERS
28 May 1979
STUDY OF THE DYNAMIC STRUCTURE FACTOR OF AN ELECTRON LIQUID B.K. RAO
1
S.S. MANDAL and D.N. TRIPATHY
Institute of Physics, Bhubaneswar 751007, India Received 21 February 1979 Revised manuscript received 26 March 1979
A rigorous calculation of the dynamic structure factor S(k, w) of an electron liquid is made by taking into account the lifetime effects in the electron and hole parts ofthe RPA polarizability function. Our result shows a complete wipe out of the structure in S(k, w)reported by Mukhopadhyay et al. in the region k > kc. We notice that the effect of the lifetime only gives rise to a long tail of S(k, w)but not any structure.
Recently, there has been great interest in explaining the structure in the dynamic structure factor S(k, w) observed in the inelastic X-ray [1 —31 and electron [4,5] scattering experiments from various metals such as Li, Be and Al. What one observes in all these experiments is that there is a double-peak structure in S(k, w) in the momentum transfer region kc 2kF) this is entirely given by the partide—hole excitation spectrum. Therefore, to explain the structure in the intermediate momentum transfer region (kc
In an attempt to explain the structure of S(k, w) theoretically it is shown [6] that the random phase approximation (RPA) fails to reproduce the structure. Besides, it is also known [7] that in the RPA the plasmon excitation spectrum does not persist beyond kc. Attempts have also been made to explain the structure of S(k, w) using the dielectric function of Vashishta and Singwi (VS) [8]. This theory, which has been found to be quite satisfactory in calculating various static properties of the electron liquid fails to explain the structure. It was later realized by Mukhopadhyay et al. [9J that even though the VS theory goes beyond the RPA and accounts for the short-range correlations among the electrons properly, the electrons and the holes in this theory have infinite lifetime. To take care of this deficiency they modified the VS theory by adding finite lifetimes to the free particle energy appearing in the energy denominator of the Lindhard polarizability function, in an approximate way. With this, they were able to reproduce more or less the structure of S(k, w) in aluminium. I~ookingat the plots of S(k, w) versus w from the work of Mukhopadhyay et al. [9] for the RPA and VS theory one finds that the spectra in both cases are timilar to each other, the only difference being that the peak positions of S(k, w) are located at different positions in w-space. Since the VS theory with the introduction of lifetime effects gives rise to the structure in S(k, w), it therefore follows that the structure of 447
Volume 71A, number 5,6
PHYSICS LETTERS
S(k, w) is a consequence of the finite lifetime effects
of the electrons only. The effect of the short-range correlations among the electrons which is brought into the VS theory through the static local field factor, is only to properly adjust the peak positions of S(k,w). This point has been discussed by Mukhopadhyay et al. in their paper [9]. The lifetime added to the free particle energy in the VS theory can be seen to be a function of momenturn only. Therefore, it is not clear how this can give rise to any structure in S(k, w) when plotted in wspace for fixed k-values, because this factor does not change with frequency. What one actually needs is some equivalent frequency-dependent function, in which case one may expect some kind of structure in S(k, w)when plotted versus w. Since the local field factor of the VS theory is independent of frequency it is understandable why S(k, w) cannot reproduce any new structure different from that of the RPA where the local field factor is identically zero, when viewed in w-space. A recent calculation by Mukhopadhyay and Sjolander [10] indicates that the structure in S(k, w) very much depends on how the lifetime effect is introduced. Assuming that it is the lifetime effect that brings the structure in S(k, w), in this paper we proceed to calculate S(k, w) by introducing finite lifetimes to the electron and hole parts of the Lindhard polarizability function, in a rigorous way. Strictly speaking, we should include the correction due to the associated vertex to the Lindhard polarizability function, while we are considering the lifetime effects to the free electron propagators of this function. This is essential in order to explain the structure in S(k, w) correctly. Since here our aim is to study the electron—hole lifetime effects on S(k, w) and compare our result with those of Mukhopadhyay et al. [9] we do not take into consideration the effects due to the vertex correction. The value of the lifetime as a function of k is estimated by looking at the imaginary part of the self-energy operator following the work of Lundquist [11] We do not take into consideration the effect of the real part of self-energy operator in our the calculation it is known that this does modify strength although of the .
one-pair excitation spectrum in the presence of damping. We justify the neglect of the real part of the selfenergy byof saying thatThis its role is only shift the positions S(k, w). is the same toattitude as peak 448
28 May 1979
adopted by Mukhopadhyay et al. [9] in their work where the authors do not include the effect of the real part of the self-energy following the argument that the presence of the static local field factor in the VS theory takes care of this effect in some averaged way. Since we have said that a static local field factor displaces only the peak positions of S(k, w), the real part of the self-energy would do a similar job. Here, being concerned only with the structure of S(k, w) and not with the positions where these structures should appear, we are justified in not taking into account the effect of the real part of the self-energy. The response function ~(k,w) in the RPA is given by ~ (k, w) ~(k,w) = (1) I (4ire2Ik2)xo(k, w) where x 0 (k, w) is the usual Lindhard polarizability. This should now be modified by introducing the selfenergy insertions. After this is done, x0 (k, w) assumes the form: -—-----—-——-———-———---, —
~{n~0
X~(k, w) =
~,
+
—
—
i
~p+k, ~}
0
k)
+ ~(JJ)
Im [M(p + k, ~(,p + k))
—
M(p ~(p))]
}1•
(2)
The self-energy M(k, w) has been evaluated at the free particle energy ~ (k). The expression for its imaginary part is given by Mk ~ k m ~k+ ~2
~ dq ~ i 2ii* q ~ ~
2 dx{_Imc~p~(q,k—x)}
—
—‘
0 ~., ,~
~ 1~e~
,~
q ~2 2
—
~2 2 ---1
+iIme RPA (qk —x)~ }
2 x) 0(0.25 —x)] (3) X [0(k where ~ = (4/9~)1I3,r 5 is the usual dimensionless density parameter and ~RPA (k, w) is the RPA dielectric function. Here the momentum is measured of 2kF and the energy in units of 4CF. S(k, oi) inis units related to the imaginary part of x(k, w) as —
—
Volume 71A, numter 5,6
PHYSICS LETTERS
S(k,~,)=ir4Im~(k,w),
_________________________
~.,>0,
w<0.
(4)
It can be seen that the numerical calculations for S(k, w), although tedious, are straightforward. We have calculated S(k, w) for Al (r 5 =2kF. 2.0) The for several results values of k in the region kc
,,.3,.~ /7 /~ \\ 7_~.s:~
.0
~
28 May 1979
,c~-~-~\
1.0
k=I.8kF
0.8 07
o.~ 3, ~ 0.5 04
0.2 0.1
w/e F Fig. 2. S(k, w) versus ~ for k/kF = 1.8 for Al. Peak heights of all curves have been normalized to unity. Curve 1, RPA; curve 2, modified RPA; curve 3, modified VS theory.
K =l.6k~ I.0
k~2.Ok~
~ ‘a.0.6
Jf/j// [I
0.4 0.3
0.1
f
0.4 0.3
/
0.2 0.1
________________________________—
I
2
3
4
w/e~
5
6
7
8
Fig. 1. S(k, w) versus w for k/kF = 1.6 for Al. Peak heights of all curves have been normalized to unity. Curve 1, RPA; curve 2, modified RPA; curve 3, modified VS theory. The dashed curve represents experimental data of Platzman and Eisenberger [2].
____________________________ I 2 3 4 5 6 7
8
Fig. 3. S(k, w) versus ~ for k/kF = 2.0 for Al. Peak heights of all curves have been normalized to unity. Curve 1, RPA; curve 2, modified RPA; curve 3, modified VS theory.
449
Volume 71A, number 5,6
PHYSICS LETTERS
served in the plot obtained by the VS theory. This is shown in the figures of ref. [9] and it is argued by Mukhopadhyay et a]. that this shift is entirely due to the static local field correction factor. However, we find here that this shift is very clearly present even without the local field factor. (3) A long tail appears in our curve extending to large values of w which exists in the modified VS theory (curve 3) too. Our calculation thus confirms that the introduction of lifetime is not responsible for obtaining the structure in S(k, w). It seems that the effect of the lifetime is only to give rise to a long tail behaviour to S(k, w) in w-space. Although we have ignored the effects due to the vertex correction while renormalizing the electron—hole propagators of the RPA, we do not think that inclusion of such a correction will affect the conclusions of this paper. For it is rather well known that the vertex corrections sometimes cancel most of the self-energy corrections. As a result the electron—hole lifetime effect on S(k, w) will be further reduced, which would go more in favour of our conclusion. A more recent calculation by Raedt and Raedt (RR) [121 shows that the authors have succeeded in explaining the structure in S(k, w) using Mon’s memory function formalism which is known to account for the fast decaying processes. However, to obtain a fit with the experimental data these authors have to introduce a few parameters into their theory whose justification is lagging from a first-principles point of view. Some of these parameters are even found to vary drastically in going to large wave vectors. Besides the deficiencies mentioned above, it can be trivially seen that the dielectric function obtained by the above authors has the following drawbacks: (i) It violates the compressibility sum rule and (ii) it does not go to unity in the
450
28 May 1979
of large w, rather it is seen that 6RR (k 0, w oc) = (1 7r)~.From all these considerations the rather good agreement of the results of these authors with some experimental data (besides the several parameters in their theory) should not be regarded as a proof of the usefulness of their approach. We, therefore, conclude that the structure of S(k, w) in the range k~
limit
—~
—
One of the authors (SSM) is thankful to the University Grants Commission, India for financial support. References [11 P. Eisenberger, P.M. Platzman and K.C. Pandey, Phys. Rev. Lett. 31(1973) 311. [2] P.M. Platzman and P. Eisenberger, Phys. Rev. Lett. 33 (1974) 152. [3] P. Eisenberger, P.M. Platzman and P. Schmidt, Phys. Rev. Left. 34 (1975) 18. [4] P. Zacharias, J. Phys. C7 (1974) L26. [51 P.E. Batson, CU. Chen and .1. Silcox, Phys. Rev. Left. 37 (1976) 937. [6] R.K. Kalia 1243. and G. Mukhopadhyay, Solid State Commun. 15 (1974) [7] D. Pines and P. Nozières, The theory of quantum liquids (Benjamin, New York, 1966) Vol. I. [8] P. Vashishta and KS. Singwi, Phys. Rev. B6 (1972) 875. [9] G. Rev.Mukhopadhyay, Lett. 34 (1975)R.K. 950.Kalia and KS. Singwi, Phys. [10] G. Mukhopadhyay and A. SjOlander, Phys. Rev. Bl7 (1978) 3589. [II] B.I. Lundquist, Phys. Kond. Mater. 7 (1968) 117. [12] H. De Raedt and B. De Raedt, Phys. Rev. B18 (1978) 2039.
PHYSICS LETTERS
28 May 1979
STUDY OF THE DYNAMIC STRUCTURE FACTOR OF AN ELECTRON LIQUID B.K. RAO
1
S.S. MANDAL and D.N. TRIPATHY
Institute of Physics, Bhubaneswar 751007, India Received 21 February 1979 Revised manuscript received 26 March 1979
A rigorous calculation of the dynamic structure factor S(k, w) of an electron liquid is made by taking into account the lifetime effects in the electron and hole parts ofthe RPA polarizability function. Our result shows a complete wipe out of the structure in S(k, w)reported by Mukhopadhyay et al. in the region k > kc. We notice that the effect of the lifetime only gives rise to a long tail of S(k, w)but not any structure.
Recently, there has been great interest in explaining the structure in the dynamic structure factor S(k, w) observed in the inelastic X-ray [1 —31 and electron [4,5] scattering experiments from various metals such as Li, Be and Al. What one observes in all these experiments is that there is a double-peak structure in S(k, w) in the momentum transfer region kc
In an attempt to explain the structure of S(k, w) theoretically it is shown [6] that the random phase approximation (RPA) fails to reproduce the structure. Besides, it is also known [7] that in the RPA the plasmon excitation spectrum does not persist beyond kc. Attempts have also been made to explain the structure of S(k, w) using the dielectric function of Vashishta and Singwi (VS) [8]. This theory, which has been found to be quite satisfactory in calculating various static properties of the electron liquid fails to explain the structure. It was later realized by Mukhopadhyay et al. [9J that even though the VS theory goes beyond the RPA and accounts for the short-range correlations among the electrons properly, the electrons and the holes in this theory have infinite lifetime. To take care of this deficiency they modified the VS theory by adding finite lifetimes to the free particle energy appearing in the energy denominator of the Lindhard polarizability function, in an approximate way. With this, they were able to reproduce more or less the structure of S(k, w) in aluminium. I~ookingat the plots of S(k, w) versus w from the work of Mukhopadhyay et al. [9] for the RPA and VS theory one finds that the spectra in both cases are timilar to each other, the only difference being that the peak positions of S(k, w) are located at different positions in w-space. Since the VS theory with the introduction of lifetime effects gives rise to the structure in S(k, w), it therefore follows that the structure of 447
Volume 71A, number 5,6
PHYSICS LETTERS
S(k, w) is a consequence of the finite lifetime effects
of the electrons only. The effect of the short-range correlations among the electrons which is brought into the VS theory through the static local field factor, is only to properly adjust the peak positions of S(k,w). This point has been discussed by Mukhopadhyay et al. in their paper [9]. The lifetime added to the free particle energy in the VS theory can be seen to be a function of momenturn only. Therefore, it is not clear how this can give rise to any structure in S(k, w) when plotted in wspace for fixed k-values, because this factor does not change with frequency. What one actually needs is some equivalent frequency-dependent function, in which case one may expect some kind of structure in S(k, w)when plotted versus w. Since the local field factor of the VS theory is independent of frequency it is understandable why S(k, w) cannot reproduce any new structure different from that of the RPA where the local field factor is identically zero, when viewed in w-space. A recent calculation by Mukhopadhyay and Sjolander [10] indicates that the structure in S(k, w) very much depends on how the lifetime effect is introduced. Assuming that it is the lifetime effect that brings the structure in S(k, w), in this paper we proceed to calculate S(k, w) by introducing finite lifetimes to the electron and hole parts of the Lindhard polarizability function, in a rigorous way. Strictly speaking, we should include the correction due to the associated vertex to the Lindhard polarizability function, while we are considering the lifetime effects to the free electron propagators of this function. This is essential in order to explain the structure in S(k, w) correctly. Since here our aim is to study the electron—hole lifetime effects on S(k, w) and compare our result with those of Mukhopadhyay et al. [9] we do not take into consideration the effects due to the vertex correction. The value of the lifetime as a function of k is estimated by looking at the imaginary part of the self-energy operator following the work of Lundquist [11] We do not take into consideration the effect of the real part of self-energy operator in our the calculation it is known that this does modify strength although of the .
one-pair excitation spectrum in the presence of damping. We justify the neglect of the real part of the selfenergy byof saying thatThis its role is only shift the positions S(k, w). is the same toattitude as peak 448
28 May 1979
adopted by Mukhopadhyay et al. [9] in their work where the authors do not include the effect of the real part of the self-energy following the argument that the presence of the static local field factor in the VS theory takes care of this effect in some averaged way. Since we have said that a static local field factor displaces only the peak positions of S(k, w), the real part of the self-energy would do a similar job. Here, being concerned only with the structure of S(k, w) and not with the positions where these structures should appear, we are justified in not taking into account the effect of the real part of the self-energy. The response function ~(k,w) in the RPA is given by ~ (k, w) ~(k,w) = (1) I (4ire2Ik2)xo(k, w) where x 0 (k, w) is the usual Lindhard polarizability. This should now be modified by introducing the selfenergy insertions. After this is done, x0 (k, w) assumes the form: -—-----—-——-———-———---, —
~{n~0
X~(k, w) =
~,
+
—
—
i
~p+k, ~}
0
k)
+ ~(JJ)
Im [M(p + k, ~(,p + k))
—
M(p ~(p))]
}1•
(2)
The self-energy M(k, w) has been evaluated at the free particle energy ~ (k). The expression for its imaginary part is given by Mk ~ k m ~k+ ~2
~ dq ~ i 2ii* q ~ ~
2 dx{_Imc~p~(q,k—x)}
—
—‘
0 ~., ,~
~ 1~e~
,~
q ~2 2
—
~2 2 ---1
+iIme RPA (qk —x)~ }
2 x) 0(0.25 —x)] (3) X [0(k where ~ = (4/9~)1I3,r 5 is the usual dimensionless density parameter and ~RPA (k, w) is the RPA dielectric function. Here the momentum is measured of 2kF and the energy in units of 4CF. S(k, oi) inis units related to the imaginary part of x(k, w) as —
—
Volume 71A, numter 5,6
PHYSICS LETTERS
S(k,~,)=ir4Im~(k,w),
_________________________
~.,>0,
w<0.
(4)
It can be seen that the numerical calculations for S(k, w), although tedious, are straightforward. We have calculated S(k, w) for Al (r 5 =2kF. 2.0) The for several results values of k in the region kc
,,.3,.~ /7 /~ \\ 7_~.s:~
.0
~
28 May 1979
,c~-~-~\
1.0
k=I.8kF
0.8 07
o.~ 3, ~ 0.5 04
0.2 0.1
w/e F Fig. 2. S(k, w) versus ~ for k/kF = 1.8 for Al. Peak heights of all curves have been normalized to unity. Curve 1, RPA; curve 2, modified RPA; curve 3, modified VS theory.
K =l.6k~ I.0
k~2.Ok~
~ ‘a.0.6
Jf/j// [I
0.4 0.3
0.1
f
0.4 0.3
/
0.2 0.1
________________________________—
I
2
3
4
w/e~
5
6
7
8
Fig. 1. S(k, w) versus w for k/kF = 1.6 for Al. Peak heights of all curves have been normalized to unity. Curve 1, RPA; curve 2, modified RPA; curve 3, modified VS theory. The dashed curve represents experimental data of Platzman and Eisenberger [2].
____________________________ I 2 3 4 5 6 7
8
Fig. 3. S(k, w) versus ~ for k/kF = 2.0 for Al. Peak heights of all curves have been normalized to unity. Curve 1, RPA; curve 2, modified RPA; curve 3, modified VS theory.
449
Volume 71A, number 5,6
PHYSICS LETTERS
served in the plot obtained by the VS theory. This is shown in the figures of ref. [9] and it is argued by Mukhopadhyay et a]. that this shift is entirely due to the static local field correction factor. However, we find here that this shift is very clearly present even without the local field factor. (3) A long tail appears in our curve extending to large values of w which exists in the modified VS theory (curve 3) too. Our calculation thus confirms that the introduction of lifetime is not responsible for obtaining the structure in S(k, w). It seems that the effect of the lifetime is only to give rise to a long tail behaviour to S(k, w) in w-space. Although we have ignored the effects due to the vertex correction while renormalizing the electron—hole propagators of the RPA, we do not think that inclusion of such a correction will affect the conclusions of this paper. For it is rather well known that the vertex corrections sometimes cancel most of the self-energy corrections. As a result the electron—hole lifetime effect on S(k, w) will be further reduced, which would go more in favour of our conclusion. A more recent calculation by Raedt and Raedt (RR) [121 shows that the authors have succeeded in explaining the structure in S(k, w) using Mon’s memory function formalism which is known to account for the fast decaying processes. However, to obtain a fit with the experimental data these authors have to introduce a few parameters into their theory whose justification is lagging from a first-principles point of view. Some of these parameters are even found to vary drastically in going to large wave vectors. Besides the deficiencies mentioned above, it can be trivially seen that the dielectric function obtained by the above authors has the following drawbacks: (i) It violates the compressibility sum rule and (ii) it does not go to unity in the
450
28 May 1979
of large w, rather it is seen that 6RR (k 0, w oc) = (1 7r)~.From all these considerations the rather good agreement of the results of these authors with some experimental data (besides the several parameters in their theory) should not be regarded as a proof of the usefulness of their approach. We, therefore, conclude that the structure of S(k, w) in the range k~
limit
—~
—
One of the authors (SSM) is thankful to the University Grants Commission, India for financial support. References [11 P. Eisenberger, P.M. Platzman and K.C. Pandey, Phys. Rev. Lett. 31(1973) 311. [2] P.M. Platzman and P. Eisenberger, Phys. Rev. Lett. 33 (1974) 152. [3] P. Eisenberger, P.M. Platzman and P. Schmidt, Phys. Rev. Left. 34 (1975) 18. [4] P. Zacharias, J. Phys. C7 (1974) L26. [51 P.E. Batson, CU. Chen and .1. Silcox, Phys. Rev. Left. 37 (1976) 937. [6] R.K. Kalia 1243. and G. Mukhopadhyay, Solid State Commun. 15 (1974) [7] D. Pines and P. Nozières, The theory of quantum liquids (Benjamin, New York, 1966) Vol. I. [8] P. Vashishta and KS. Singwi, Phys. Rev. B6 (1972) 875. [9] G. Rev.Mukhopadhyay, Lett. 34 (1975)R.K. 950.Kalia and KS. Singwi, Phys. [10] G. Mukhopadhyay and A. SjOlander, Phys. Rev. Bl7 (1978) 3589. [II] B.I. Lundquist, Phys. Kond. Mater. 7 (1968) 117. [12] H. De Raedt and B. De Raedt, Phys. Rev. B18 (1978) 2039.