Longitudinal dielectric permeability of the quantum degenerate collisional plasmas

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by  Anatoly V. Latyshev

Longitudinal dielectric permeability of the quantum degenerate collisional plasmas
Anatoly V. Latyshev and Alexander Yushkanov
Department of Mathematical Analysis and Department of Theoretical Physics,
Moscow State Regional University, 105005, Moscow, Radio st., 10–A
(Dated: January 22, 2010)
Dielectric permeability of the degenerate electronic gas for the collisional plasmas is found. The
kineticequation ofWigner —Vlasov — Boltzmann with integralof collisions in relaxation form in
coordinate space is used. We will notice that dielectric permeability with using of the relaxation
equation in themomentum space has been received byMermin.
Keywords: Degenerate Electron Gas, Dielectric Permeability and Conductivity, Colli-
0
sion Integral, Lindhard Function, Kohn’s Singularities.
1
0
2 PACSnumbers: 50,52.25.DgPlasmakineticequations,52.25.-bPlasmaproperties

n
a I. INTRODUCTION optical properties of metal [25], [26].
J
Dielectric permeability of quantum plasma is widely
2
used also for studying the screening of the electric field
2 In the present work formulas for conductivity and for
dielectric permeability of quantum electronic plasma are and Friedel oscillations (see, for example, [19] - [21]). In

] deduced. the work [27] screening of the Coulomb fields in magne-
h
tised electronic gas has been is studied.
p Dielectric permeability is one of the major plasma
Intheoryofquantumplasmathereexisttwoessentially
- characteristics. Thisquantityisnecessaryfordescription
h various possibilities of construction of the relaxation ki-
of process of propagation and attenuation of the plasma
t
a oscillations, skin effect, the mechanism of electromagne- neticequationin τ –approximation: inthespaceofim-
m pulses (in the space of Fourier images of the distribution
tic wavespenetrationinplasma[1]–[3],andforanalysis
function)andinthespaceofcoordinates. Onthebasisof
[ of other problems in plasma physics.
Dielectric permeability in the collisionless quantum the relaxation kinetic equations in the space of momen-
1
tum Mermin[23] has carriedoutconsistentderivationof
v gaseous plasma was studied by many authors (see, for
thedielectricpermeabilityforquantumcollisionalplasma
7 example, [4]-[9]). In work [7], where the one-dimensional
3 case of the quantum plasma is investigated, importance in 1970 for the first time.
9 In the present work expressionfor the longitudinal di-
of derivation of dielectric permeability with use of the
3 electricpermeabilitywithuseoftherelaxationequations
quantum kinetic equation with integral collisions in the
.
1 in space of coordinates is deduced. If in the received
form of BGK - models (Bhatnagar, Gross, Krook) [10]
0 expression we make Planck constant converge to zero
wasmarked. Thepresentworkisdevotedtoperformance
0 (~ 0),wewillreceiveexactlyclassicalexpressionofdi-
1 of this problem. →
electric permeability of degenerate plasma. Various lim-
: Inthepresentworkforaderivationofdielectricperme-
v itingcasesofthedielectricpermeabilityareinvestigated.
ability quantumkinetic Wigner— Vlasov—Boltzmann
i
X equation (WVB–equation) with collision integral in the Comparison with Mermin’s result is carried out also.
r form of τ –models is applied. Such collision integral is
a
named BGK–collisionintegral.
II. SOLUTION OF THE KINETIC EQUATION
The WVB–equation is written for Wigner function,
which is analogue of distribution function of electrons
We consider the kinetic Wigner — Vlasov — Boltz-
for quantum plasma (see [11] and [12]).
mann equation [28]:
Themostwidespreadmethodofinvestigationofquan-
∂f ∂f ie
tum plasmas is the method of Hartree — Fock or a +v = W[U,f]+B[f,f].
method equivalentto it, namely, the method of Random ∂t ∂r −~
PhaseApproximation[17],[18]. Inwork[22]thismethod Here e is the charge of electron, W[U,f] is the func-
has been applied to receive expression for dielectric per- tional of Wigner — Vlasov for the scalar potential U ,
meability of quantum plasma in τ –approach. However, 1 ~b ~b
inwork[24]itisshown,thatexpressionreceivedin[22]is W[U,f]= U r+ ,t U r ,t
(2π)3 2 − − 2 ×
noncorrect,asdoes notturnintoclassicalexpressionun- Z h (cid:16) (cid:17) (cid:16) (cid:17)i
der a condition, when quantum amendments can be ne-
f(r,p′,t)exp(ib(p′ p))d3bd3p′, (1.1)
glected. Thus in work [24] empirically corrected expres- × −
sion for dielectric permeability of quantum plasma, free b = b ,b ,b is the vector, f = f(r,p,t) is the
x y z
fromthespecifiedlackhasbeenoffered. Bymeansofthis Wigner{function}for electrons, ~ is the Planck constant,
expressionauthorsinvestigatedquantumamendmentsto B[f,f] is the collision integral.
2
Collision integral for quantum plasma in general case f(x,p′,t)exp(ib(p′ p))d3bd3p′. (1.4)
× −
can have rather complex form. In particular, it can be
non-local by coordinates as well. A limiting case of such We will carry out linearization of the equations (1.3)
quantumnon–localityisconsideredin[23]. Inthepresent and (1.4). Unperturbed absolute Fermi — Dirac distri-
work the case when it is possible to present collision in- bution function for degenerate plasma has the form
tegral in a local form is considered. Particularly, we
f (p)=Θ(E E),
will consider collision integral representation in a form F F −
of standart model BGK–collisionintegral(Bhatnagar— where E is the kinetic energy of electron on the Fermi
F
Gross — Krook) [29]. Then the previous equation will
surface,
be written in the following form:
mv2 p2
∂f ∂f ie EF(p)= F = F .
+v = W[U,f]+ 2 2m
∂t ∂r −~
In linear approximation in expression (1.4) instead of
f itisnecessarytotakethe absoluteFermi—Diracdis-
+ν feq(r,p,t)−f(r,p,t) . (1.2) tribution function fF . Our linearization of the Wigner
h i function for electrons and the equilibrium distribution
Thisequationdescribesbehaviourofthecollisionalde- function leads to equalities:
generate quantum plasma.
Here ν is the effective scattering frequency of elec- f =fF(p)+U0ei(kx−ωt)f1(p), (1.5)
trons (in particular, on impurities), f is the equilib-
eq
rium Fermi — Dirac distribution function of electrons. f =f (p)+δ(E E)δE (x,t), (1.6)
eq F F F
Further we will considerthe case of degeneratequantum −
plasma. Then the equilibrium distribution function can where f (p) is a new unknown function, δ(x) is the
1
be expressed in terms of Heaviside function Dirac delta–function,
f =Θ(E (r,t) E), δE (x,t)=E (x,t) E ,
eq eq F F F
− −
the function Θ(x) is the function of Heaviside, U0 is the potential amplitude. We assume that has the
form of the traveling wave
1, x>0,
Θ(x)= 0, x<0, U(x,t)=U0ei(kx−ωt). (1.7)
(cid:26)
mv2 p2 The quantity δEF(x,t) describes local change of
E is the kinetic energy of electrons, E= = , Fermi’s energy of the electronic gas, caused by change
2 2m
of its density. Presence of this term in collisions integral
mv2(r,t) p2(r,t) provides realization of the particle number conservation
E (r,t)= F = F
F 2 2m law for electrons.
Let’s substitute (1.5) and (1.6) in the equation (1.3).
is the perturbed Fermi energy of the electrons, p = mv
We receive the following equation:
is the momentum of the electron, p is the momentum
F
of the electron on the Fermi surface. We assume that f (p) ν iω+ikv ) U ei(kx−ωt) =
1 x 0
Fermi surface is spherical. −
h i
Let’s consider, that distribution electron function de-
pends on one spatial coordinate x, time t and momen- ie
= W[U,f ]+νδ(E E)(δE (x,t)). (1.8)
tum p,andtheelectricpotentialdependsononespatial −~ F F − F
coordinate x and time t. Then the equations (1.1) and
The Wigner — Vlasov functional has the following
(1.2) can be written in a form:
form in linear approximation:
∂f ∂f
∂t +vx∂x = W[U,fF]= (2π1)3 U x+ ~2bx,t −U x− ~2bx,t ×
Z h (cid:16) (cid:17) (cid:16) (cid:17)i
ie
=−~W[U,f]+ν feq(x,p,t)−f(x,p,t) , (1.3) ×fF(p′)exp(ib(p′−p))d3bd3p′, (1.9)
h i
We derive following expression for potential:
W[U,f]= 1 U x+ ~bx,t U x ~bx,t U(x+ ~bx,t) U(x ~bx,t)=
(2π)3 2 − − 2 × 2 − − 2
Z h (cid:16) (cid:17) (cid:16) (cid:17)i
3
=U0ei(kx−ωt) exp(ik~2bx)−exp(−ik~2bx) . (1.10) =U0ei(kx−ωt) Θ+(v)−Θ−(v) ,
h i h i
where
We will integrate in (1.9) by d3b. Considering (1.10),
we deduce: ~k 2
Θ (v)=Θ v2 v (v2+v2) ,
1 U x+ ~bx U x ~bx eib(p′−p)d3b= + h F −(cid:16) x− 2m(cid:17) − y z i
(2π)3 2 − − 2
Z h (cid:16) (cid:17) (cid:16) (cid:17)i ~k 2
Θ (v)=Θ v2 v + (v2+v2) ,
− F − x 2m − y z
= U(x,t) exp(ik~bx) exp(ik~bx) e−ib(p′−p)d3b= So, the Wignerh— Vla(cid:16)sov’s func(cid:17)tional is equalito:
(2π)3 2 − 2
Z h i
W[U,f ]=U ei(kx−ωt) Θ (v) Θ (v) . (1.11)
M 0 + −
=U(x,t)δ(p′ p )δ(p′ p ) −
y − y z − z × The quantity δE (x,t) whe will find fromithe conser-
F
vation law of particle number:
~k ~k
δ p′ p + δ p′ p =
× x− x 2 − x− x− 2
ν(f f)dΩ =0,
h (cid:16) (cid:17) (cid:16) (cid:17)i eq − F
Z
=U(x,t)δ(py −p′y)δ(pz −p′z)× where
2d3p
δ p p′ ~k δ p p′ + ~k . dΩF = (2π~)3. (1.12)
× x− x− 2 − x− x 2
h (cid:16) (cid:17) (cid:16) (cid:17)i According to the equality (1.12) we get the following
It is necessary to integrate by momentums: equation:
W[U,fF]=U(x,t) δ(py−p′y)δ(pz −p′z)× ν(feq −f)d3v =
Z Z
~k ~k
δ p p′ δ p p′ + = δE (x,t)δ(E E) U ei(kx−ωt)f (p) d3v =0.
× x− x− 2 − x− x 2 × F F − − 0 1
h (cid:16) (cid:17) (cid:16) (cid:17)i Z h i
From this equality we obtain:
p2 p′2
Θ F dp′dp′dp′.
× 2m − 2m x y z f (p)d3v
1
As a result of in(cid:16)tegration b(cid:17)y momentums we obtain: δEF(x,t)=U0ei(kx−ωt) Z . (1.13)
δ(E E)d3v
F
−
W[U,f ]=U ei(kx−ωt) Z
F 0
×
The denominator of the expression (1.13) is equal to
the following:
Θ p2F p2y+p2z px− ~2k 2 1 4πv
× 2m − 2m − (cid:0) 2m (cid:1) − δ(EF E)d3v = δ(vF v)d3v = F.
h (cid:16) (cid:17) − mv − m
F
Z Z
Θ p2F p2y+p2z px+ ~2k 2 , According to the equality (1.13) we have:
− (cid:16)2m − 2m − (cid:0) 2m (cid:1) (cid:17)i δE (x,t)=U ei(kx−ωt) m f (p)d3v. (1.14)
or F 0 4πv 1
F
Z
W[U,f ]=U ei(kx−ωt) Substituting the expression (1.14) in the equation
F 0
×
(1.8), we obtain:
× Θ vF2 − vx− 2~mk 2−(vy2+vz2) − U0ei(kx−ωt)f1(p) ν−iω+ikvx) =−i~eW[U,fF]+
n h (cid:0) (cid:1) i (cid:16) (cid:17)
−Θ vF2 − vx+ 2~mk 2−(vy2+vz2) = +U0ei(kx−ωt)mνδ4(πEvFF−E) f1(p)d3v. (1.15)
Z
h (cid:0) (cid:1) io
4
Let’srewritetheequation(1.15)withthehelpof(1.11) Now from the equation (2.2) we obtain a relationship:
in a form:
ie 1 Θ (v) Θ (v)
f1(p) ν−iω+ikvx =−i~e Θ+(v)−Θ−(v) + A=−~ · 1−g0(ω,k,ν)Z ν++i(k−vx−−ω) d3v. (2.3)
(cid:16) (cid:17) h i Let’s consider the integral from (2.3)
+mνδ(EF −E) f1(p)d3v. (1.16) J(ω,k,ν)= Θ+(v)−Θ−(v)d3v.
4πvF Z Z ν+i(kvx−ω)
Taking into account the following equality According to definition of Heaviside function we have:
δ(EF −E)= m1vFδ(vF −v) Θ (v)= 1, vx∓ 2~mk 2+vy2+vz2 6vF2,
from the equation (1.16) we obtain: ±  0, (cid:16)vx∓ 2~mk(cid:17)2+vy2+vz2 >vF2.
(cid:16) (cid:17)
ieΘ (v) Θ (v) Hence, thisintegral is equal to the following:
f (p)= + − − +
1 −~ ν+i(kv ω)
x− dvxdvydvz dvxdvydvz
J(ω,k,ν)= .
ν+i(kv ω) − ν+i(kv ω)
x x
Aν δ(vF v) SZ+3 − SZ−3 −
+ − . (1.17)
4πvF2 ν+i(kvx−ω) or
J(ω,k,ν)=J+(ω,k,ν) J−(ω,k,ν),
−
where
III. LONGITUDINAL PERMEABILITY AND dv dv dv
J±(ω,k,ν)= x y z .
CONDUCTIVITY
ν+i(kv ω)
x
SZ3 −
±
Let’s designate:
Here S3 is a sphere with the centre in the point
±
A= f1(p)d3v. (2.1) (∓2~mk,0,0). Theradiusofthissphereisequaltoelectron
velocity on Fermi’s surface,
Z
Substituting (1.17) in the relationship (2.1), we get: ~k 2
S3 = (v ,v ,v ): v +v2+v2 6v2 .
± x y z x± 2m y z F
A= ie Θ+(v)−Θ−(v)d3v+ (cid:26) (cid:16) (cid:17) (cid:27)
−~ Z ν+i(kvx−ω) After obviousreplacementofa variable vx±2~mk →vx
we receive for integrals J±:
Aν δ(v v)d3v
F
+ − . (2.2)
4πvF2 Z ν+i(kvx−ω) J±(ω,k,ν)= ν+ikdvvxdvy~dkvz iω,
The last integral in (2.2) is easily calculated with the S3Z(0) x± 2m −
(cid:0) (cid:1)
use of spherical coordinates:
where S3 is the Fermi’s sphere with the centre in the
1 2π ∞ beginning of coordinates,
δ(v v)d3v v2δ(v v)dµdχdv
F F
− = − =
ν+i(kvx ω) ν+i(kvµ ω) S3(0)=S3(0,0,0)= (vx,vy,vz): vx2+vy2+vz2 6vF2 .
Z − −Z1 Z0 Z0 −
Fermi’s sphere S3(0(cid:8)) we will present in the form: (cid:9)
1 vx=vF
=2πvF2 ν+i(kdvµµ ω) =2πvF2 kvi lnωω++iiνν+kkvvF. S3(0)= Sv2F2−vx2(0,0).
−Z1 F − F − F vx=[−vF
Here S2 (0,0) there is a circle of the following
Let’s designate further: vF2−vx2
form:
iν ω+iν+kv
F
g0(ω,k,ν)= 2kvF lnω+iν kvF. Sv2F2−vx2(0,0)= (vy,vz): vy2+vz2 <vF2 −vx2 .
−
(cid:8) (cid:9)
5
Now we will calculate integrals J± as repeated: J(ω,k,ν) δ(v v)
F
+ − . (2.7)
vF dv 4πvF2(1−g0(ω,k,ν)) · ν+i(kvx−ω)#
J±(ω,k,ν)= x
ν+ik v ~k iω× Let’s consider a relationshipbetween electric field and
−ZvF x± 2m − potential
(cid:0) (cid:1)
vF E(x,t)= gradU(x,t),
(v2 v2)dv −
dv dv =π F − x x .
× y z ν+ik v ~k iω or
S2 ZZ(0,0) −ZvF x± 2m −
vF2−vx2 (cid:0) (cid:1) ∂U(x,t)
E(x,t)= ,0,0 ,
Now these integrals can be calculated easily: − ∂x
n o
2iπνv and the equation of a continuity for current and charge
J±(ω,k,ν)= k2 F(ω∓+iν)− densities:
∂ρ ∂j
x
+ =0.
iπν (ω +iν)2 k2v2 lnω∓+iν+kvF. ∂t ∂x
− k3 ∓ − F ω +iν kv
h i ∓ − F Here according to definition of dielectric conductivity
Here, as earlier, we may represent the current density in the form:
~k2 ∂U
ω± =ω± 2m. jx =σlEx =−σl∂x =
The difference of integrals J+ and J− is equal to:
= σU ikei(kx−ωt) = σikU(x,t).
2iπνv ~ − l 0 − l
F
J(ω,k,ν)= +
− m Hence,
∂j
+ikπ3ν (ω++iν)2−k2vF2 lnωω+++iiνν+kkvvF− ∂xx =σlk2U(x,t).
+ F
h i − Takingintoaccountobviousequalityforchargedensity
iπν ω +iν+kv
− k3 (ω−+iν)2−k2vF2 lnω−−+iν kvFF. (2.4) ρ=e fdΩF =e [f0(E)+U0ei(kx−ωt)f1]dΩF,
h i − Z Z
Let’s present a difference (2.4) in the form:
we obtain:
J(ω,k,ν)= ∂ρ
= iωeU(x,t) f dΩ .
1 F
∂t −
Z
2iπνv ~
= F 1 g(ω ,k,ν)+g(ω ,k,ν) , (2.5) Substituting lasttwoequalitiesinthe continuityequa-
+ −
− m −
tion, we find the generalformula for calculationoflongi-
h i
where tudinal conductivity:
g(ω±,k,ν)= σ = ieω f dΩ = 2ieωm3 f d3v. (2.8)
l k2 1 F (2π~)3k2 1
Z Z
= m (ω±+iν)2−k2vF2 lnω±+iν+kvF. (2.6) Substituting (2.7) in (2.8), we get:
2~k3v ω +iν kv
(cid:2) F (cid:3) ± − F
2e2τωm3 (Θ (v) Θ (v))d3v
Thus, the quantity A according to (2.3) and (2.5) is σ = + − − +
equal to: l (2π~)3k2~" ν+i(kvx ω)
Z −
ie J(ω,k,ν)
A= .
−~ 1−g0(ω,k,ν) + J(ω,k,ν) δ(vF −v)d3v .
4πv2(1 g (ω,k,ν)) ν+i(kv ω)
Hence, according to both (1.17) and (2.5) function F − 0 Z x− #
f (p) is constructed also:
1 and, using formulas
f (p)= ie Θ+(v)−Θ−(v)+ (Θ+(v) Θ−(v))d3v
1 −~" ν+i(kvx ω) ν+i−kv iω =J(ω,k,ν),
− x
Z −
6
δ(vF −v)d3v =4πv2g (ω,k,ν), Here
ν+ikv iω F 0
Z x− iy x+iy+1
g (x,y)= ln , z =x+iy,
we derive: 0 2 x+iy 1
−
iπe2ωm3v J(ω,k,ν)
F
σ = .
l 2π3~3 mk2 · 1 g (ω,k,ν) (z q/2)2 1 z q/2+1
− 0 g(z, q)= ± − ln ± .
± 2q z q/2 1
With the help of the formula for numerical electron ± −
density of degenerate plasma Let ν = 0, i.e. plasma is collisionless; then from the
expression (2.10) the following classical formula for the
mv 3
F =3π2N, dielectric permeability follows:
~
following expression(cid:16)for c(cid:17)alculation of longitudinal con- 3ωp2
ε (ω,k)=1+ 1 g(ω ,k)+g(ω ,k) , (2.12)
ductivity is obtained: l 2k2v − − +
F
h i
3ie2Nω 1 g(ω ,k,ν)+g(ω ,k,ν) where
+ −
σ = − , (2.9)
l −2mk2vF2 · 1−g0(ω,k,ν) g(ω ,k)= m(ω±2 −k2vF2)lnω±+kvF.
e2N ± 2~k3vF ω± kvF
or,withtheuseofclassicalconductivity σ = ,this −
0
mν
This formula is called (see, for example, [19] - [21])
formula will be written in the form:
dielectric Lindhard’s function [22] in the literature. It is
3i ων
deducedbythemethodofrandomphasesapproximation.
σ =σ
l 0· − 2 · (kvF)2· In dimensionless variables dielectric Lindhard’s func-
(cid:16) (cid:17)
tion has the following form:
1 g(ω ,k,ν)+g(ω ,k,ν)
· − 1+−g0(ω,k,ν)− . (2.9′) εl(x,q)=1+ 32x2p[1−g(x,+q)+g(x,−q)], (2.13)
Using definition of dielectric permeability where
4πi (x q/2)2 1 x q/2+1
εl =1+ σl, g(x, q)= ± − ln ± .
ω ± 2q x q/2 1
± −
with the help of (2.9) we will get the following represen-
tation for longitudinal dielectric permeability of plasma:
ε (ω,k,ν)=
l |d ldq|
1.8
3ω2 1 g(ω ,k,ν)+g(ω ,k,ν) 1
p + −
=1+ − , (2.10)
2k2vF2 · 1−g0(ω,k,ν) 2
3
where ω is the electron plasma frequency, 1.2
p
4πe2N
ω = .
p m 1.95 2.00 2.05 q

Let’s enter dimensionless parameters:
ω ν FIG. 1: Kohn’s singularity, xp = 1, x = 0; curves of 1,2,3
z =x+iy, x= , y = , correspond to parameters which are values of dimensionless
kv kv
F F collision frequencies y=0, 0.0050.01.
ω k
p
xp = kv , q = k , On Fig. 1 Kohn’s singularity in a case, when xp =
F F
1, x = 0 is represented; curves of 1,2,3 correspond
mv
where k = F is the Fermi wave number. to parameterswhicharevalues ofdimensionless collision
F ~
frequencies y =0, 0.0050.01.
The dimensionlessdielectric function ofthese parame-
ters has the the following form:
On Fig. 2 Kohn’s singularities in a case, when x =
p
3 1 g(z,+q)+g(z, q) 10, x = 0 are represented, curves of 1,2,3 correspond
ε (x,y,q)=1+ x2 − − . (2.11)
l 2 p 1 g (x,y) to the values of parameter y = 0; 0.01; 0.02. On
0
−
7
(z+q2/2)2 q2 z+q+q2/2
160 g (z,q)= − ln ,
|dl/dq| + 2q3 z q+q2/2
1 −
2
120
3 (z q2/2)2 q2 z+q q2/2
g (z,q)= − − ln − .
− 2q3 z q q2/2
− −
80
1.95 2.00 2.05 q Kohn’s singularities are determined from four equa-
2- /kFvF 2+ /kFvF
tions:
FIG.2: Kohn’ssingularities, xp =10, x=0,curvesof 1,2,3 q2 2q 2z =0. (2.14)
correspond to thevalues of parameter y=0; 0.01; 0.02. ± ±
These equations at y = 0;(ν = 0) define four Kohn’s
singularities,twoofwhichat ω =0 layinneighbourhood
6
of point q =2:
|d l/dq|
1 q =1 √1 2x,
150 1 1,2 ± ±
2
2 and two others lay in point neighbourhood q = 2:
3 −
3
q = 1 √1 2x.
120 3,4 − − ±
In the case of infinitesimal x we have from these for-
1.96 1.98 2.00 2.02 2.04 mulas:
2- /kFvF 2+ /kFvF q
ω ω
q 2 x 2 , q 2 x 2 .
1,2 3,4
≈ ± ≈ ±k v ≈− ± ≈− ±k v
FIG.3: Kohn’ssingularities, xp =10, x=0,curvesof 1,2,3 F F F F
correspond to values of theparameter y=0; 0.002; 0.004.
IntermsofdimensionalFermiwavenumbertheKohn’s
singularities are determined by equalities:
k ω
Fcuigrv.es3otfhe1,c2a,se3, cwohrernespxopnd=t1o0v,axlue=s o0f itsherepparersaemnetetedr, k1,2 =kF +rkF2 ±2 vFF ,
y =0; 0.002; 0.004.
and
In the formula (2.11) we will write new dimensionless
variables: k ω
k = k k2 2 F .
ω+iν ω 3,4 − F −r F ± vF
z = =x+iy, x= ,
kFvF kFvF Besides that, these formulas may be rewritten in a
form:
ν k
y = , q = . mv ~ω p ~ω
kFvF kF k1,2 = ~F 1+s1±2mvF2 = ~F 1+r1± EF
At such replacementof variableswe obtainthe follow- (cid:16) (cid:17) (cid:16) (cid:17)
ing formula for the longitudinal dielectric permeability: and
εl(x,y,q)=1+ 32xq22p1−g1+−(zg,0q()x+,yg,−q)(z,q), (2.13) k3,4 = m~vF −1−s1±2m~vωF2 = p~F 1+r1± E~ωF .
(cid:16) (cid:17) (cid:16) (cid:17)
where
Here E is the electron kinetic energy on a Fermi’s
F
ω2 surface
x2 = p ,
p kFvF mv2
E = F.
F
2
iy x+iy+q Thus,intocollisionlessplasma(ν =0)at ω =0 there
g (x,y,q)= ln , 6
0 2q x+iy q is a splitting of Kohn’s singularities.
−
8
III. COMPARISON WITH MERMIN’S RESULT Withthe helpof(3.2)–(3.4)Merminformula(3.1)will
be written in our designations in the following form:
Mermin (see Mermin N.D. [23]) has obtaned the fol-
3ω2
lowing expression of dielectric function: εM =1+ p
2k2v2 ×
F
(ω+iν) ε◦(ω+iν,k) 1
εM(ω,k)=1+ − . (3.1)
hε◦(ω+iν,k) 1 i (ω+iν)[1 g(ω++iν,k)+g(ω−+iν,k)]
ω+iν − − . (3.5)
ε◦(0,k) 1 1 g(ω++iν,k)+g(ω−+iν,k)
− ω+iν −
1 g(0 ,k)+g(0 ,k)
+ −
Theformula(3.1)isobtainedonthebasisofthekinetic −
equationforone–partialdensitymatrix ρ inmomentum From formulas (2.10) and (3.5) one can see, that in
space. the case ν 0 the formula deduced in this work and
→
In the formula (3.1) the function ε◦(ω,k) is the so- Mermin formula turn into the same expression for di-
called Lindhard’s dielectric function, i.e. the dielectric electric function for quantum collisionless plasma that is
function obtained for collisionless plasma, and defined Lindhard dielectric function (2.12).
by the equality (2.12):
3ω2 IV. CONCLUSION
ε◦(ω,k) ε (ω,k)=1+ p 1 g(ω ,k)+g(ω ,k) .
≡ l 2k2v − + −
F
h i It is interesting to notice, that in the case of low-
Expression ε◦(ω+iν,k) means that arguments of di- frequency limit, i.e. at ω = 0 Mermin dielectric func-
electric Lindhard function ω± are replaced formally on tion does not depend on collision frequency ν. Indeed,
ω±+iν, i.e. assuming ω =0 in the formula (3.1), we obtain
ε◦(ω+iν,k) 1= εM(0,k,ν)=ε◦(0,k)=
l − l
3ω2 3ω2
p p
= 1 g(ω +iν,k)+g(ω +iν,k) . (3.2) =1+ 1 g(0 ,k)+g(0 ,k) , (4.1)
2k2v2 − + − 2k2v2 − + −
F F
h i h i
So the function ε◦(0,k) has the form or, in dimensionless variables,
ε◦l(0,k)−1= 2k3ω2vp22 1−g(0+,k)+g(0−,k) . (3.3) εMl (0,k,ν)=1+ 32x2p 1− w22w−1lnww+11 . (4.2)
Fh i h − i
From the formula obtained in the this work (2.10) we
Here
have another formula in a low–frequency limit:
g(0 ,k)= m 4~mk42 −k2vF2 ln±~2km2 +kvF =
± (cid:16) 2~k3vF (cid:17) ±~2km2 −kvF εl(0,y,w)=1+ 3x2p 1 iy lniy+1 −1
2 − 2 iy 1 ×
h − i
= m(cid:16)4~2mk~22kv−FvF2(cid:17)ln±±22~~mmkk +−vvFF. ×h1− (iy+4ww)2−1lniiyy++ww−+11+
From these formulas follows that these two functions
differ only in sign: (iy w)2 1 iy w+1
+ − − ln − . (4.3)
4w iy w 1
g(0 ,k)= g(0 ,k). − − i
− +
−
From formulas (3.2) and (3.3) one can see that
The formula (4.3)transformsinto the formula(4.2)at
ε◦(ω+iν,k) 1 y =0.
− = So, in the present work analytical expression for the
ε◦(0,k) 1
− longitudinal quantum dielectric permeability of degener-
ateelectronplasmaisderived. KineticWigner—Vlasov
1 g(ω +iν)+g(ω +iν) —Boltzmannequationwithcollisionintegralintheform
+ −
= − . (3.4)
1 g(0 ,k)+g(0 ,k) of relaxation τ – model in coordinate space is used.
+ −
−
9
It is shown, that in a limiting case, when Planck con- [10] Bhatnagar P. L., Gross E. P., and Krook M. // Phys.
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