A. DI E L E C T R I C RE S P O N S E FU N C T I O N S
We consider now the properties of the longitudinal response functions of electron systems. For simplicity we consider only electron plasmas, that is, systems of electrons moving in a uniform background of positive charge. We shall begin with the quantum plasma at Τ = 0, and then later discuss the case of finite temperature plasmas. One of our principle aims will be the derivation of exact expressions for the screening length and the plasma oscillation frequency in the limit of very long wave
lengths.
1. Response to an External Field: \/e(q,co)
Let us suppose an external longitudinal electric field, «^(r, t), is ap
plied to an electron system. The vector 2 is the "dielectric displace
ment" defined in elementary electrostatics. It satisfies Poissons's equa
tion
div B(r, t) = Anzqe(r, t), (3.1)
where zge(r, t) is the density of "external charge" introduced into the gas at a point r. The external field will act to polarize the electron system.
The induced charge fluctuations may be regarded as producing a space charge field, the electric field inside the system is thus equal to
§T(r, 0 = 9(r9 t) + gT(r, 0 . (3.2)
According to the usual laws of electrostatics, S£(r, t) may be related to the polarization charge density by the equation
div 8^(r, t) = Ane <ρ(Γ, f)> · (3.3) When dealing with dielectrics, one must modify (3.3) in order to take
into account a surface term, due to the accumulation of charge at the boundary of the sample. Such a term does not enter in the present case, because we assume the system to be closed on itself (for instance, by means of an external "perfectly conducting" wire).
We may combine (3-1)—(3.3) to write
div ST(r, t) = 4π {ζρ€(ι\ t) + e <ρ(ι·, *)>} (3.4) as the equation which relates the electric field to the external and in
duced charge fluctuations. We next take the Fourier transform in space and time of (3.1) and (3.4):
iq-®(q, ω) = 4nzqe(q, ω) (3.5)
/q ·§*(<?,(ω) = 4n{zge(q, ω) + e <Q(qt ω)}} . (3.6) These equations have been obtained with reference to the macroscopic laws of electrostatics; we now extend them to a microscopic level by considering them as applicable for all wave vectors q and frequency ω ,
DENSITY F L U C T U A T I O N EXCITATIONS 65 corresponding to fields which vary arbitrarily rapidly in both space and time. (3.6) thus serves us a definition of the electric field ^(q, ω).
We now make a key assumption: that the dielectric response of the electrons, <ρ>, is proportional to the applied field 3>. This will be the case if the external field is sufficiently weak; we thus assume that in computing the system response one can neglect coupling terms propor
tional to «^2, etc. It follows at once that will be proportional to Since both 2 and are purely longitudinal fields, we may write
&{q, ω) = , (3.7)
where s(q, ω) is the frequency and wave vector dependent dielectric constant. We see in (3.7) that e(q9 ω) furnishes a direct measure of the dielectric response of the electron system; it tells us the extent to which the external field, ρΡ), is screened by the electronic polarization which it induces.
Thus e(q9 ω) furnishes a natural description of the important screening action of the electron gas. It likewise furnishes a description of the col
lective modes of the system, the plasmons. The plasmon dispersion re
lation is simply the condition that one have a nonvanishing charge fluctuation of the electrons in the absence of an external charge. That condition is, according to (3.5) and (3.6),
s(q, ω) = 0 .
If we substitute (3.7) into (3.6), and divide the resulting equation by (3.5), we find:
1 e <p(q, ω)}
= ι + ; / (3.8)
e(q, ω) ZQe(q> ω)
on comparing (3.8) with (2.4), and recalling that 5 ^ = 4ne2/q2 for the coulomb interaction between the electrons and the external charge den
sity, we see that
1 4π<?2
1 + - - - , — ω ) . (3.9)
e(q, ω) q
The various properties of the dielectric response function then follow
directly from those we have derived earlier for %{q9 co). Let us write
e(q, co) = ex(q, co) + is2(q, co) (3.10) From (3.9) it is then straightforward to establish the following results
for spectral representation
1 4ττ£?2 f30 1 1
= 1 + - — dco' S(q9 co') — , . . (3.11) q2 Jo «> — ο +ιη ω + οϊ + ιη
£(q, ω )
and for sum rule
f.
0
e
2(q, co) π 2n
2Ne
2dco co , = — ω 2 = (3.12)
\e(q,co)\2 2 p m ' '
where ω2 is, as before, the square of the electron plasma frequency, ωρ. The response of an electron system to an applied electric field is frequently specified in terms of the conductivity, which is the ratio of the current induced in the system, e(J}, to the effective electric field, 8s\ For the present case of fields which vary in space and time, we may write
e <J(<7, ω)} = a(q, ω) &(q, ω) . (3.13) Here e </(</ , ω ) > is the Fourier transform of the induced current, while
σ(<7, ω) is the scalar longitudinal conductivity. We are considering only a longitudinal external field, and have made our customary assumption that the system behavior is isotropic. There is a simple relation between σ and ε. To obtain it, we make use of the current conservation equation for the charge and current induced in the electron gas. Thus one has
div <J> + ^ ^ - = 0 (3.14)
dt and
q<i-{q9toy> = io(q(q9<oy> . (3.15) On taking the divergence of (3.13), and making use of Eqs. (3.5-3.7),
we find at once
4nia(q, co)
e(q,co)= 1 + ( 3.1 6 )
DENSITY F L U C T U A T I O N EXCITATIONS 67 2. Response to a Screened Field: xs(q, ω)
There is another quantity of interest for an electron system, one which we may call a screened response function, xs(q, co). According to (2.4), X(q, co) may be defined (for the case of an external charge in interaction with the electron system), as
(p(q, co)}
X(q9a>)= + , J\ (3.17)
where 9?ext(<7> w) *s the scalar field potential produced by the external charge. We define xs(q, ω) as the response of the system to the scalar potential, qo(q, ω), so that
Xs(q9 ω) = + — — — . (3.18)
ecp(q, co)
We call x£q, ω) the screened response function, because it represents the system response to the screened external probe; according to 3.7 (p(q, ω) = cpe(q, co)/e(q9 co) so that we can in fact write
x*(q,ω) = *(q, «>)x(q,ω) · ( 3 ·1 9) The utility of xs(q,(o) becomes clear if we combine (3.19) with (3.9)
to write
Ane2
e(q9 ω)=1 — x9(q9 ω) . (3.20) Xs(q9 s) is simply related to the longitudinal conductivity, o(q, ω). On
comparing (3.20) and (3.16) we see that icoe2
<r(q>ω) = —o ~ x*(a>ω) · (3·21) q2
3. Analytic Behavior of e(q, ω)
At first sight one would expect that e(q, co) is analytic in the upper half-plane, this being the usual property of causal response functions, and more specifically, a property of both %(q, ω) and l/s(q, co). Closer investigation shows that there does not exist a rigorous mathematical demonstration that e(q, ω) is indeed analytic in the upper half of the
complex ω plane. What one can show is that e(q, ω) will be analytic there, provided (30).
e(q, 0) > 0 . (3.22)