IL NUOVO CIMENTO VOL. XXV, N. 2 16 Luglio 1962

On the Relativistic Degenerate Electron Gas (').

B. JA~COVICI

Laboratoire de Physique Thdorique et Hautes Energies - Orsay

(ricevuto il 9 Aprile 1962)

R 6 s u m ~ . - - On 6tudie les propri6t6s d 'un gaz d'61ectrons ~ temp6rature z6ro et h densit6 assez forte pour que l '6nergie de Fermi soit relativiste (de te]les densit6s et des tempdratures re la t ivement faibles se rencontrent au sein des naines blanches). Le probl~me est 6tudi6 dans la formulation di61ectrique de l ' approximat ion des quasi-bosons: les paires ~lectron- trou qui apparaissent ~ cause des interactions coulombiennes ou des interactions avec le champ des photons transverses sont trait6es comme des bosons, et les processus autres que la cr6ation et l 'annihilat ion de telles paires sont n6glig6s ; les positrons sont inclus parmi les trous possibles. On 6crit et on diagonalise des hamiltoniens modSles d6crivant le syst~me

cette approximation. On calcule des constantes di61ectriques longi- tudinale et transversale, d6pendant du nombre d 'onde et de la fr~quence, et d6crivant la r6ponse du syst~me '£ des excitations ext6rieures; la renor- malisation d e charge n~cessaire appara l t tr~s naturellement. On 6tudie la propagation des ondes longitudinales et transversales; on en calcule la loi de dispersion. On calcule enfin l '6nergie de l '6tat fondamental jusqu 'au terme d 'ordre e 4 log e 2.

1 . - I n t r o d u c t i o n .

The p r e s e n t p a p e r dea ls w i th t he p r o p e r t i e s of a r e l a t i v i s t i c d e g e n e r a t e

e l ec t ron gas. R e l a t i v i s t i c m e a n s t h a t t he d e n s i t y is h igh enough for t h e F e r m i

m o m e n t u m ] not to be neg l ig ib le in f r o n t of t h e e l ec t ron mass m (**) ( / ~ m

co r r e sponds to a d e n s i t y of 6 .10 ~9 e lec t rons /cm3) . D e g e n e r a t e m e a n s t h a t t h e

t e m p e r a t u r e T is low enough for t h e c o r r e s p o n d i n g e n e r g y k T to be neg l ig ib le

(') Sponsored in par t by Air Force Office of Scientific Research, OAR, through the European Office, Aerospace Research, United States Air Force.

(**) We adopt the units h = e = l .

ON TI tE R E L A T I V I S T I C D E G E N E R A T E ELECTRON GAS 4 2 9

in front of the Fermi kinetic energy Ts = V/] ~ + m 2 - - m (for the above density, k T = 10-~Ts corresponds to a ((low ~) temperature of 2.5.10 ~ °K).

Densities and temperatures of these orders of magnitude actually occur

in the interior of the white dwarf stars (~). The aim of the present paper

however is definitely not to s tudy the interior of white dwarfs, but it is to

investigate the properties of a simple model: a high density electron gas, at

zero temperature, in a uniform positive background. This model still might

have applications to astrophysics, and we also believe that it might have some intrinsic interest.

The model is the relativistic generalization of the well-known nonrelativistic

high-density electron gas (2-4). For the sake of completeness, we shall how-

ever reformulate the whole argument. The problem will be studied here in

the dielectric formulation (5) of the quasi-boson approximat ion (e) (which is

equivalent to the random phase approximation).

The model is defined in Section 2, and the hamiltonians which describe

it within the quasi-boson approximation are settled and diagonalized in

Section 3. The momen tum and frequency-dependent longitudinal and trans-

verse dielectric constants are defined and computed in Section 4. F rom them,

the screening of a Coulomb field is computed in Section 5, the dispersion

laws for the propagation of waves are established in Section 6, and the ground state energy of the gas is computed in Section 7.

Some of our results have also been obtained by quan tum field theory techniques (7).

2. - D e s c r i p t i o n of the s y s t e m .

We consider here the idealized case in which the electrons move through

a uniform positive background which ensures total electric neutrality. If

there were no interactions, the ground state ((~ vacuum state ~)) of the system

of N electrons in a volume f2 would be the N electrons occupying in mo- mentum space a Fermi sphere of radius

(1) ] = (3~2N/Q)~.

(1) See e.g.M. SCHWARZ$CI t ILD : Structure and Evolution o] the Stars (Princeton, 1958). (2) M. GELL-MANN and K. A. BRUECKNER: Phys. Rev., 106, 364 (1957). (3) K. SAWA~)A: Phys. Rev., 106, 372 (1957). (4) K. SAWADA, K. A. BRUECKNER, N. FUKUDA and R. BROUT: Phys. Rev., 108,

507 (1957). (5) p. NOZI~RES and D. PINES: Nuovo Cimento, 9, 470 (1958). (6) G. WENTZEL: Phys. Rev., 108, 1593 (1957). (7) I. i . AKHIEZER and S. V. PELETMINSKII: SOY. Phys. JETP, 11, 1316 (1960).

42~0 :B. JANCOVICI

The relativist ic case is the one in which t h e density 2¢/s9 is so high t ha t the Fermi m o m e n t u m f becomes comparable to the electron mass m.

Because of the interactions, an electron m a y jump f rom a s ta te inside

the Fermi sphere to an unoccupied s ta te ; a hole, t ha t we shall call a ~ Fermi

hole ~>, is left behind. In the relat ivist ic ease, two new features appear . First ,

another type of exci tat ion also becomes impor t an t : the creation of a positron-

electron pair, which m a y be thought of as the jump of an electron f rom a negat ive energy s ta te to an unoccupied posit ive energy s ta te ; the posi t ron is a hole t ha t we shall call a (~ Dirae hole ~>. Secondly, the in teract ion is no longer a purely Coulomb one; photons are exchanged.

A complete basis for describing the sys tem is provided by a set of unper-

tu rbed states which mus t allow for positrons and photons as well as for

electrons. An unper tu rbed s tate therefore mus t be defined by the enumera t ion of the particles (electrons outside of the Fermi sea), the <~ Fe rmi holes ~, the

~ Dir~c holes ~, the photons. We find here convenient to describe the inter- actions in the <( old-fashioned ~ Coulomb gauge: only t ransverse photons exist

and in terac t with electrons, and in addit ion there is an unquant ized Coulomb potent ia l between the electrons (*).

3. - The quas i -boson h a m i l t o n i a n s .

I f we t ry to compute , let us say, the ground s ta te energy b y the per tur-

ba t ion theory, we mus t consider all possible v a c u u m to v a c u u m connected graphs (8). Most of these graphs however appear to diverge in the region of

a) b)

Fig. 1. - Ring-diagrams: a) is u Coulomb ring-diagram ; b) isa photon ring-diagram.

10w m o m e n t u m transfers, thus mak ing

simple per tu rba t ion theory unappli- cable. I n the nonrelat ivist ic case, this well-known difficulty m a y be circum- ven ted by the r e summat ion of the

class of the mos t divergent graphs, the so-called r ing-diagrams. A general ring-

d iagram is drawn on Fig. l a ; a ver-

t ical arrowed line represents an electron

or a hole, as usual, and a horizontal

do t ted line represents a Coulomb inter-

action. The obvious general izat ion in the relat ivist ic case is to consider in

addit ion r ing-diagrams with photons, as the one which is drawn on Fig. lb ;

(*) Since the positive background does provide a privileged frame of reference, an explicitly eovariant description with the introduction of longitudinal and temporM photons looses here much of its interest.

(s) J. GOLDSTONE: Proc. Roy. Sot., A 239, 267 (1957).

ON T H E R E L A T I V I S T I C D E G E N E R A T E E L E C T R O N GAS 431

a vert ieal dot ted line represents a photon. Also, in all diagrams, hole lines m a y represent ei ther Fermi or Dirac holes. I t m a y be noted t ha t there is no contr ibut ion f rom a r ing-diagram with both Coulomb and photon lines, beeause a photon carries a t ransverse polarizat ion and a. Coulomb line does not ; also, in a given photon ring-dis.gram, aI1 photon lines car ry the same

l inear t ransverse polarization.

The approx imat ion will therefore be to keep only the r ing-diagrams in

the Goldstone expansion for the ground state energy. The lowest order

neglected graphs are exchange graphs of order e 4, and therefore the ground

s ta te energy will be computed to within this order only. Terms of order e 4log e 2 will however appear and be taken into account.

This neglect of te rms of e 4 and higher order is justified only if the inter- action pa rame te r is small enough. In the non-relat ivist ic ease, the veloci ty

of light mus t not come into the problem, and the only re levant dimensionless pa rame te r which can be built with e 2 is me2~/. The approx imat ion therefore is a high densi ty one which is valid as soon as me"-//<< 1. At the still higher

densities of the extrenle relativistic ease m/] << 1, the electron mass becomes

un impor t an t and the re lewmt pa rame te r is the small quan t i ty e 2 ~ 1/137

itself. The approx imat ion therefore is valid for all (nonrelativistie or relati- vistie) densities high enough for making me"-~/<< 1.

A procedure whi('h is equiwflent to the summat ion of the r ing-diagrams is the in t roduct ion of soluble model hamil tonians chosen in such a way tha t they exact ly ffenerate the r ing-diagrams. Since, in a r ing-diagram, electrons

and holes are always created or annihi lated by pairs, model hamil tonians can be built with pair creation and annihilation operators, and of course photon operators.

Let us call bq~ the operator which annihilates an electron-hole pair of total

m o m e n t u m q; ~ is the set of the other q u a n t m n numbers which define the pair : the mon len tnm k of the hole state (the e lec t rum m o m e n t u m then is

k + q), the spins a~ and a._, of the hole and electron states, and finally the kind ~ of the hole (=o is +1 for a Fermi hole and - - 1 for a l)irac hole);

( - - q , - ~) are the indices for a pair with reversed m o m e n t a and spins. The (!oulomb r ing-diagrams are generated by the model hamil tonian

(2)

(3) 7y{. ~ + '.,=e~ , + = b _ q _ . ) G , , + b + q j , ) _ x ] l

O)q~ = V'(k + q)2 + m ~ _ o~/k2 + m2

is the kinetic energy of the pair (% ~).

{4) .Q~ vq ~ Vq.V = ~ 2 ( uk+q.~+ [,za~e} (uk,o;e, ] ua',q4+}

432 B. JANCOVICI

is a Coulomb matr ix element (Co is the electron charge, Uko 0 is the four- component spinor for a state of momen tum k, spin a, and energy sign ~).

is the total number of occupied (positive or negative energy) states: this te rm W appears when the one-particle creation and annihilation operators

are reordered before the approximations which lead to (2). Let us call aq~ the annihilation operator for a photon of m o m en tu m q and

polarization 2 (2 runs on two values of t ransverse polarization only). The photon ring-diagrams with photon t ransverse polarization 2 are generated by

the model hamiltonian:

(5)

(6)

= o~q~,bq~, bq~, + qaq~aq~.- eo [ q ~ . wq~,a(b~,-- b_q_~,) (aq~+ a_q~) .

is an electron-hole-photon ver tex (a~ is the usual Dirac matr ix along the direction). In (2) and (5), t ime-reversal properties have been used:

(7)

(S)

;uk+~o:+tuko,~> = (u j,_~,~lu_~,_~_o:+),

The hamiltonians (2) and (5) are supplemented with exact boson commu- ta t ion relations for the pair and photon operators: the only nonzero commu-

tators are :

(9) [bq~, bq,~,] = 6q~.a..,, [aqa , +

If the fictitious system defined by the quasi-boson hamil tonian (2) (with the commuta t ion relations (9)) is studied by the per turba t ion theory, its ground

state energy is found as being exact ly given by the Coulomb ring-diagram sum of the electron gas system (*); similarly, if the fictitious system defined by the quasi-boson hamil tonian (5) (with the commuta t ion relations (9)) is studied by the per turbat ion theory, its ground state energy is found as being exact ly

given by the 2-polarization photon ring-diagram sum of the electron gas system. I t is therefore interesting to s tudy the quasi-boson hamiltonians; we shall use them for an approximate calculation of not only the ground state correlation energy, but also of other quantities such as the dielectric constants.

(') The fact that the true pair operators do not exactly obey boson commutation relations is irrelevant. We here define a fictitious problem which is equivalent to the ring-diagram approximation, and this is all that is wanted.

O N T H E R E L A T I V I S T I C D E G E N E R A T E E L E C T R O N G A S 4 3 3

The next step is to find equations for the eigenvalues and eigenstates of the hamil tonians (2) and (5). They are coupled harmonic oscillator hamil- tonians, and it is convenient to reexpress them as functions of canonical con-

jugate (< position )> and (( m o m e n t u m >> variables. We put

i -

1 b + , } y ( b q ~ - b_q_,) = ~q~ ; ( 1 0 ) - (bq~ 4- -q-x) = 7~q~ i /o~q~ +

%/2wq,,

these operators obey the commuta t ion relations

(11) [~q~, =q,~.] = i ( ~ q q . ~ , , .

The Coulomb hamil tonian (2) then becomes

(12) H ((', ½ ~ [ ~ + --~oq~) n- (~"gq~7~q~ ~ 2 + O~q ~ ~)q ~ ~q.,

There is no coupling between different values of q. The contr ibut ion of each value of q to (12) is the quant ized form of a classical hamil tonian for a sys tem

of coupled oscillators. For each value of q (we m a y sometimes drop the index q) there is a hermi t ian f requency matr ix , the elements of which are

, S :~e~ , , , (13) M~,,,, = e )~5~ , ~ f2q: v~v~, ~/o)~ e),, .

Let us call v~ the eigenvalues and c~ the eigenvectors of this mat r ix :

(la) M~, ,ci~, = yTei,x,

~nd let us choose these eigenvectors as or thonormal :

( 1 5 ) =

If we then c~rry out the canonical t rans format ion

i i

(it is easy to see t ha t (16) ~ctually preserves the commuta t ion relations),

434

(12) takes the diagonal form

B . J A N C O V I C I

The corresponding system therefore behaves like an ~ssembly of uncoupled oscillators of frequencies vq~, which represent the possible longitudinal quasi- particle excitations. I t is easy, with a little algebra, to actually solve (14).

which are different f rom all the t,9~ are the roots of Those eigenvalues ~,~

(18)

where

1 + doFf(q, ~,) = O,

(19) F t ( q , ~o) = 87~ ~o~lv~l 2 8:z _ ~ O q k ~ T r [ A + ( k ÷ q)l lo(k)~ ]

O,)qk 0 is a notat ion for O~q~ which emphasizes its non dependence on the spin indices in ~;

(20) .4~(k) _ ~ . a +tim + ~ V k ~ ~

is a projector on the ~ sign energy states. The eigenvector which corresponds to a root ~,~ of (18) is

v~VG 2~ (21) ei~ = N , o)~ - - v i

with the normalization condition

Whenever n frequencies ~%1, (o~:, ...,¢o~, are equal, in addition to the roots of (18), the eigenfrequency v ~ = oJ~2 occurs with the multiplicity, n - - i ; the corresponding eigenveetors obey the condition

(23) ~ ~ / ~ . c,~ = 0;

it is still possible to define through (22) a quant i ty A'~ which is then zero. The transverse hamil tonian (5) can be made diagona b by a very similar

ON THE RELATIVISTIC DEGENERATE ELECTRON GAS

procedure. Instead of ( lo) , however, we rather put

With these conjugate variables, the transverse hamiltonian (5) becomes

In addition to the rows and columns referring to the pair states a, the fre- quency nlatrix has nonT a row and a column referring to the photon state 0 ; the matrix elements are (we may sometimes drop the indices q and A ) :

The eigenvalues of this matrix are again called I(; the eigenvectors have :L

component c,, in addition t o the components el , . Under the c~anonieal trims- formation

the transverse hamiltonian (12) takes the di;qonal form

These uncoupled oscillators of frequencies yYz represent the possible transverse quasi-perticle excitations of the system. Those eigenvalues v: of (25) which are different from all the cu: again are the roots of an equation like (IS),

436 B. JANCOVICI

where now however F z is replaced by

(29) Ft (q , ¢o) - - 8~ xf ]w~'l 2°~ _

9 ( q 2 - co2) 7" ~ - - ~o~

_. 8~ ~ q k ~ Tr[A+(k ÷ q)~aA~(k)~a] 2 ~0 2 ~9(q ~ - ~o~) hQ ~Oqk~-

2 of (18) is The eigenvector which corresponds to a root v~

1/s~ w:¢< (30) c~, = - - ieo - - _ ~ - ~ z C~o ' • [ 9 o ~ . - - v~ '

with the normalizat ion condition

8Jzeo2 Iw~, 12¢0~ ] - '

Whenever n frequencies ¢%,, o ~ , . . . ,o )~ are equal, in addi t ion to the roots of (18), the eigenfrequency v 2 = w 2 occurs with the mul t ip l ic i ty n - - 1 ; the

~t

corresponding eigenvectors obey the conditions

(32) C~o - - Y . w ~ V ~ c ~ = o .

The ground s ta te energies of the hamil tonians (17) and (28) are

(33)

and

(33')

These expressions could be used for obtaining the electron gas correlation energy. We shall however use an a l ternat ive method, af ter having inves t igated the

dielectric propert ies of the gas.

4 . - The dielectric constants.

4"1. - The dielectric constants describe the linear response of the gas to external charges and currents. Apar t f rom their interest for the actual cal-

culation of these responses, they are of interest for s tudying intrinsic prop-

ON TftE RELATIVISTIC I )EGENERATE ELECTRON GAS 4 3 7

crties such as wave propagation in the gas or the correlation energy. We

sham therefore compute the dielectric constants. Let us recMI their definition (9). The gas is supposed to be submit ted to

a weak external space and t ime-dependent test charge-current distribution.

This distribution is first assumed to be a longitudinal wave: the charge is

(34) (rq~ exp [ i q ' r - - iv)t] + rq~, exp [-- i q ' r + io)t]) exp [at],

where d is a small real positive quant i ty which ensures the adiabatic intro-

duction of the test distribution: the current is along q and determined by the

conservation condition. A eha.rge is induced in the electron gas, and is pro-

portional to the external charge as long as the latter is weak enough. The

total (external plus induced) (q(o) Fourier component of the charge is found

to be

(35) Rq~. exp [ i q . r --i(,)t + bt]

and the longitudinal dielectri(, constant e t is defined as

( 3 6 ) 1 = 1?,:,,, e~(q, (o) r,~,,,

Since there is no transverse external current, the toni)ling is only between

the external charge and the system described by the Coulomb hamiltonian (2); the coupling term is

(37) -l 7/~C o ~,q.,(bq~ @ b =t ~) exp [-- i(ot]

.* b + 7qw ~ '* *q~(bq. 5 -q ,) e x p [imt]] exp [6t]

'i S nce the external charge is arbitrarily weak, the induced one can be con> puted by perturbat ion theory. One then finds

1 (38)

d(q, o)) 8 ~ e 2 o O)no

- -1 @ f2qZ~]/O]~t,q~(bqi,,-]--b_,~ ~)in>[2 ...... " , ((,) + i 6 ) ~ - - ~ O ~ o '

where ]0} is tile ground state of the ha.miltonia.n (2), I~> its /~th excited

state, o&0 the excitation energy of tile state ]n>.

Let us now assume for the external distribution a transverse wave: no

(9) j . LINDHARD: Kgl. Danske Mat. Fys. Medd., 28, n. 8 (1954).

28 - I1 Nuovo Cimento.

438 ]3. JANCOWCI

charge, and a current distribution

(jq exp [ i q . r - - icot] 4- J,l,O exp [-- iq . r 4- icot) exp [dt] , (39)

where

(40) q "~(o = 0 .

The total (external plus induced) (qco) Fourier component of the current is found to be

(41) Jq~ exp [ i q . r - - icot 4- St],

with Jqo~ parallel to jq~ because the gas ground state is invariant under rota- tions, and the transverse dielectric constant s t is defined as (*)

1 Jq~ (42) e~(q, ~) -- ?,lo,

The coupling is now only between the external current and the photons of the system described by the transverse hamil tonian (5); the coupling t e rm is

(43) - - ~ [jq~(aq~ 4- a2q~) exp [-- i~t] 4- j~o~(a~ 4- a_q~) exp [/cot]] exp [~t ] ,

where ~ is the direction of jq~. One finds:

(44) st(q, co) -- 1 - - eo ~-~\Ol~wq~,x (bq+--b_q_~, ) !n} •

(DnO

• <nlaq~ 4- a+q~lO> (co 4- i5) ~ - coco"

4"2. - The sums on the excited states of a hamiltonian, which appea r in the expressions (38) and (44) for the dielectric constants, will now be expli-

citly computed. In the longitudinal case, one has from (10), (16), (21) or (23),. and (18)

9q~ (45) <o + - T

(') This definition differs from LINDHARD'S (9). In the present paper

q2--0)2gLindhard E t = q 2 0)2

ON THE RELATIVISTIC DEGENERATE ELECTRON GAS 439

The matr ix element (0[~5+]n,} is non-vanishing only when [n} is a one-i quasi-particle state, in which case In} may be called I i} and one has

1 (46) <0[¢~+Ti) V2~,~; o , ~ 0 = ~ .

With the help of (45), (46), and (22), (38) becomes

(47) et(q, ~o) = 1 - [- . [ ( (u+ iS)2--v~]8:~egDq i ~. (o)~,--v~) ~ h 1-1 •

Those u~ which are the roots of (18) are functions of co2; taking the derivative of (18) with respect to e~, one obtains

2 d ( ~ ) [ ' S ~ O 2 IV a 12(L)~ 1 - 1

(48) eo d(eo2) [f2q ~ ~---~ . . . . . . .2~ ; - ( ~ - ~,) ]

this equation is also valid for those v~ which are equal to a (,92. Therefore:

1 d(~)/d(eo 2) ~2"

On the other hand, the following ident i ty holds, for any value of the variable o :

(50) ] _L e~oF(q, co) 1 ÷ 8ne~ ~2 [v~]2(~ ,I~. (~°2- ~) - - z - ~ . . . . . . . . . . " ~-~ " 2 ~ " ~gq ~ , (~)~ - - co" l-[(o, - - o),)

2¢

Taking the logarithmic derivative with respect, to e~ of (50) and using it in (49), one finds

(51) eZ(q, co) = 1 q- eoFZ(q, (,) + i 6 ) ,

where F ~ is defined by (19). In the transverse case, one obtains from (44) in a very similar way

• ( ~ - ~ ) J

(49) is also found v~l id for ~, and one f inal ly finds

(53) ~'(q, c~) = ] + eoF'(q, ~o + i b ) ,

where F t is defined by (29).

44;t B . J A N C O V I C I

\Ve shall of ten in the fol lowing omi t the explicit n lent ion of ib in (51)

or (53)~ r emember ing however tha.t (o m u s t be t h o u g h t of as ha.r ing a small

posi t ive ima.gina.ry part .

4'3. C h a r g e r e n o r m a l i z a t i o n . - The dielectric consta.nts ob ta ined in (51)

or (53) depend on the u n k n o w n b~re e lectron cha.rge co, and F is a. d ivergen t

sum on k. F in i te results ca.n howeve r be ob t a ined in terms of the k n o w n renor-

realized electron cha.rge e. This (.ha.rge renorma.liza.tion can be (tarried ou t b y usintz

our results themeselves. The dielectric cons tan t s depend on the gas dens i ty

because the s u m m a t i o n domain on k depends on the Fermi Iuomen tun l ],

therefore on the dens i ty : for an electron-hole pair one m u s t ha.re ]k + q l > ]

if 0 = - - 1 , I k + q i > f and k < ] if ~2 ~ ~ 1. I n the specia.1 e~se of zero

dens i ty (] = 0), the dielectric eonsta.nt of the va.cuum is obta.ined:

(54) ~o(q, o ) ) = l + e,~/%(q, ~o).

W h e n two eha.rges co a.re e m b e d d e d in a. dielectric m e d i u m of dielectric con-

s t an t So, the effective in te rac t ion be tween t h e m is nmdified by the m e d i u m

pola.rization and is obta.ined by repla.eing" e"~o by e~/eo. The renorma.lized eha.rge

e m u s t be precisely defined in such a wa y t h a t the repla.eement of c o by ~

describes the effects of the w w u u m i)olarizat ion for low frequencies and mo-

men ta . Therefore , in the f r anwwork of our a.pproxima.tions:

,2 2 ( 0 (~o

(~51 e ~ = : ~ . _ . .

~:,,(0, 0) 1 + ~:~Fo(o, o)

and (51) or (53) m a y be wr i t t en :

1 4- e"-[F(q, o ~ ) - F o ( O , 0 ) ] (56) s(q' ")) - I - - e- ~ , / ( , - ~ ' ' ~ ~ - - > )

Aetua.lly, tlle gas dielectric eons tan t is a.lways measured re la t ively to the

va.cuum dMect r ie cons tant . The m e a s u r e m e n t process can be t h o u g h t of a.s

follows. Firs t , test charges a.re me~sm'ed in w~euum throua 'h their low Four ie r

c o m p o n e n t effects (for instance, a point ch~H'ge a.t rest is exa.miucd f r o m a

long distmlce); the resul t of such ~ mea.sm'ement is the ha.re tes t cha.rge d ivided

b y the v a c u u m dielectric eonsta.nt So(0, 0) (*). The ba.re test charges ~re then pu t in to mot ion m:d used to build a churge-cur ren t d is t r ibut ion like (34) or

(39), which is b r o u g h t into the gas, and the tot'~l eha.rge-current is rne:~sured.

l (') The l~orentz invarbmee of the wwuum requires that %(q, o~)::~'o~(q, ~).

O N T H E R ] ~ ; L A T I V I S T I C D E ( ; E N E R A T E E L E C T I U ) N C-AS 4.41

The r a t i o of these m e a s u r e m e n t s is, for the (.hard'e,

(5:) r~./~,o(O, o) = ~(q, (,)) [fqo~ co(O, ())'

a. s imi la r e q u a t i o n ho lds for the cu r ren t s . The r e l e v a n t q u a n t i t y (*) the re -

fore is

(58) ~"(q, o~) c(q, (,)) :~ ~o(O, o) - I + e~ IF(q , ,,,) - Yo(O, o ) ] .

This r e l a t i v e d ie lec t r ic c o n s t a n t is a f ini te q u a n t i t y , which ( leseribes t he phys i -

cMly m e a s u r a b l e response .

4"4. - The r e l a t i v e l o n g i t u d i n a l and t r a n s v e r s e d ie lec t r i c c ons t a n t s ,

def ined b y (58), wil l now be e x p l i c i t l y c o m p u t e d . F is a sum on k, def ined

b y (19) or (29), a n d has poles on the real axis in t he o) ~ c o m p l e x p l a n e ;

when the n o r m a l i z a t i o n v o l u m e .O becomes inf ini te , t he sum t u r n s in to an

i n t e g r a l ; a~nd t h e poles in to (.uts on the real axis .

I t is c o n v e n i e n t to wr i t e (58) as

(59) s'(q, (~,) 1 ~ e~[F(q, ,,~) - Fo(q, ( , ) ] - ~ [ t ' o (q , ~,,) - Fo(O, o ) ] .

The second b r a c k e t in (59) is the we l l -known s e c o n d - o r d e r v a c u u m pola -

r i za t i on , which m u s t be t h e s ame in the l o n g i t u d i n a l a n d in the t r a n s v e r s e

ease, s inee t he v a e u u m is L o r e n t z invar ia .n t . One has , f r om (19), t a k i n g

z 2 = (Eik+q I @ E a ) 2 q2 as one of t he i n t e g r a t i o n v a r i a b l e s ,

(~0)

co /" ~2 i

-[bo(q, (,)) l,'0((), 0)] : = - - e~(q~ c')~) d(z~) ~m! = 3 ~ z~(z ~ + q ~ - .2,,) z ~

l m e

e 2 4m ~ ,) q2 o)"- - 2m ~ q- (,)- 4- 4m- sinh * 4ma / 3 ~ q ~ (~l ~ " " q 2 o)2 p' q~__ (,)2 /

(*) ()he might a.s well define an a.lternative measurement process, m which the test charge or current distribution wlmld be first mea.sm'ed in vaeuum, directly at momentum a.nd frequeney va.lues q a.nd co. Tile releva.nt qnant i ty weuhl then be

~'(q, (:,9) t : e~[k ' (q, o~) Fo(O, 0)]

~o(q, o9) I • e2[Fo(q, co) 1"o(0, 0)]

('*) [;sing (29) instead of (19), one wouhl obtain the same l"o(q, eg) only through a proper definition of this a.mbiguous infinite integral. See for instance: \V. IIFITLE~: The Qu(ttl;l{t))t~ Theory o/ Radiat ion, 3rd edition (Oxford, 1954), p. 321.

442 B. JANCOVICI

Because of the smallness of e2~ 137 -x, (60) m a y be neglected compared to 1 in (59); (60) would become appreciable, behaving like

1 L

only for fantast ical ly high values of [qZ--(o~l which are not considered here.

I n the longitudinal case, one finds f rom (19) for the first b racke t t e rm in (59):

(6~)

-0.5

e~ [~ ' (q , co) - F o ( q , o9)] - -

e 2 (El~+q[ A- Ek) - - q Elk+ql - - Ek - - d3k . . . . . . . . . . . . . . . . . . . . . . . ~ 2

7~2q 2 2EIk+qlEk ( E I k + q l - EI~) - - 09 2 < ¢

Ik+ql>f

+ td3k (Elk+ql-Ek)~-q~ EI,+qI+E,~ } • 2EIk+~IEI~ (Elk+ql ~- E~) 2 - o) i. '

lk+q]<f

~qZ 6u f2~+~+m~

-l.0L Fig. 2. - The variation of ~"~- 1, when it is

real, as a function of w.

+

where

(62) Ek ~ ~v/k °"-- m ~ .

The explicit expression for (61), which is ra ther complicated, is given in the Appendix. An impor- t an t case however is the l imit q<<J , m < < E I , in which there are simplifications : the second integral in (61), describing the electron-positron pairs, becomes negligible compared to the first

integral describing the electron-

Fernfi hole pairs; the first integral itself has a simple leading term, and

the real and imaginary par ts of (59) become:

[ c°~/]2+m~ ~'~l] i+m2+]q] (63) Ree'~(q,e))=l+ e 2 4 ] ~ / ] 2 ~ 1 - - log - -

e22~°(]2 + m'~) if co < ]q • q3 V ] 2 + m;-'

(63') Im #l(q, e ) ) = ]q 0 if ( o > ~ / ] 2 + . m 2"

I n the w region where s ' z - - 1 is real, it is p lo t ted on Fig. 2 as a funct ion

of ~o. The approx imate forms (63) however are not valid a round o=fq/~/]2 +m 2

ON T I I E R E I , A T I V I S T 1 C D E G E N E R A T E E L E C T I ¢ O N GAS 4 4 3

where a bet ter approximat ion is (again assuming q << 1):

( , 1 nq~ - m~q

In the t ransverse case, one finds f rom (29) for the first bracket t e rm in (59):

(65) e'~[ F*(q, o)) - - Fo(q, (o)] . . . . . . 7e"-(q2--¢o2)

• 3k EIk+,~I . /Q,--( k - q /q )_~ k .q - - m" Etk+ql - Ek . . . . . 1 . . . . . .

EIk+ql ~/~ (Etk+ql - - E~) ~ - (o2 / c<f

Ik+ql>f

t Elk+,~l E,,, + (k.q/q)~ 4- l t .q + m ~ El~+q ! +E, , : /

d~k • EI~+ ~1 El,, (EI~+~ I + E~) ~ - o?]"

jk+q] <S

The complicated explicit expression for (65) is given in the Appendix. In the l imit q <</, oJ << Ej , simplifications again appear and the real and imag- inary par ts of (59) become:

(66) Re e'~(q, ~o) = ] + n(q ~ - o.,~)V~/'~ + m"

](o2,]zq rn:) , o~ /p - -~ m'~ l ~,?(/2 ÷ m~)] ~%/f~+ m Z+ f q }

q(q2~ ,,)~) [ 12q2 l j if , o < ~ / ] . ~ + m , ; (66') Jill e't(q, co) =

lq 0 if ~,) > %112 + m"-

In the (o region where e ' t - . 1 is real, it is p lot ted on Fig. 3 as :t funct ion of o). The behuviour a round the pole ~o = q is correct ly described by (66), even when q is large; a round the pole:

(66") e'f(q, ~o),-,., 1 +

~"2Iv? ~--+ m'- [1 " ~

~(q~-,,'~-) [ - E l ~ + :m~

Fig. 3. - - The variation of e l - 1, when it is real, as a function of (o, for the ease ] -= m.

3.0

2.0 . . . . .

1.0 i

0 I

-1 0 !

-2.[

-3.1

2e?f 3

/

co- f

O3

I

Y

444 B. JANCOVICI

5 . - Screening.

Tile to ta l potent ia l around a point test charge Q m a y be easily computed : it is the inverse Fourier t rans form of 4nQ/[q~sZ(q, 0)]. The large distance space

behaviour of tlfis potent ia l is given by the small q behaviour of s'~(q, 0), which is f rom (63):

(67) s"(q, 0) -- I + . ~q2

Therefore, far away from the charge Q, the potent ia l is screened and behaves like Q exp [--r/,~]/r, where the screening length ,~ is

(68) ), = 4e2]~/12+ m 2

instead of the non-relat ivist ic value

( 6 8 ' ) ;~ = \ 4 e ~ f . d "

6 . - Wave propagation.

F r o m the dielectric constants, informat ion is easily obta ined abou t the propaga t ion of waves in the gas. (36) and (42) tell tha t , for a charge-current

distr ibution to exist in the gas in the absence of an external one, the dielectric

constant mus t be zero. Therefore, tile equat ion

(69) ~,(q, o~) = 0 ,

or equivalent ly

(69') s'(q, (~) 0 ,

defines the dispersion relat ion between the f requency and the wave number of a wave which freely propagates in the gas. Longi tudinal and t ransverse

waves can propagate .

ON TIlE HEI,ATIVISTIC DE(IFNEIIATF ELECTI{ON ('AS 44,5

6'1. L o n g i t u d i n a l u,aees (p lasmons ) . - F o r q <</ a nd (, << E: , a"z is ~'iven

by (63). Fig'. 1 shows t h a t (69') lugs one root o), which is p lo t t ed 'ts a fun( , t ion

of q on Fig. ,i; this is the d ispers ion law for a l ong i tud ina l wave or p l a smon .

For q - - O , t he f re( tueney is

( 4f,/:, f, (70) , , . = \;3~//~ ! m2

in s te~d of the non-relat ivist i ( , c lassical p l a s m a f r e q u e n c y

( 7 0 ' ) ~'J°~ = \ 3 . ~ m ] "

W h e n q inereases, (,) (.omes (.lose to/q/~/fa+m='-.

(63) ceases then to be wdid , and, when q

/ e t s above the va lue q ...... for which (64)

vanishes , (69') no l<)ng'er has a root . There-

fore, f ( ) r q q ..... , <,) does rea('h the vahle

]ql~/fa 4 m "z, and for )figher wdues of q,

2~- 6o/O9o

I

l:ig. 4. The dispersion law fire- q II(}IICV (0 F,% ~V&VO lllIIIl bOl' ([)

i.r a hmgitudinal wave (plasmon). The a vro~ in(licates the (.u(:-off

for the case j - m .

p l a smons do not prol)ag'ate, q~,,. is easily seen l.o be of the or(ler of ee/V/2 ~ m ~,

a n d thus q~,,,~ :< f'- (since we a s sumed the dens i ty at least hi~'h enoug'h for

me"-//<< 1 to be va l id) : the root o) a lways does obey, when it exists,

the condi t ion (o <<< E: . The small q and ,) a p p r o x i m a t i o n is ther(,fore

j ustified.

6"2. Tra,s~:erse tc,~e.s,. - For q <~ / and e) .<:i E : , e 't is ~a'iw, n by (66). Fig. 2

shows tha4, (69:) a lways has one root o), which is plot.ted as a fun(.tio)~ ()f q

on Fi~'. 5; this is the dispersion

3 ?

law for a t r ansve r se wave. For q - - o , Ihe

frequen(.y is ~-0, g iven l) 3 ' (70), as f()r a

IonFi tud ina l wave: the eon(l i t ion (o,, < E, is

indeed satisfied. W h e n q is lar~'e enou~'h, o)

behaves like q, as for an eleetronm~'neti( '

wave in va ( .uum: the use of (66) is there-

fore just if ied, even for a laro'e q. There is

no hi~h-mom(,ntun~ ('ut-off for th(' t ransve) 'se

w a v e g .

Fig. 5. The (lispersion law (frequency so r.~. num- ber q) for a 1)'a.nsverse wave. in the ease ] m.

446 g. JANCOVICI

7. - The ground state energy.

The ground state energy Eo of the gas will now be computed. Within our approximat ions , this energy m a y be wri t ten as the sum

(71) Eo : T ~- AE(C) 4- AE (~),

where T is the kinetic energy of ~n ideal gas wi thout interactions, AE (c~ the

energy shift genera ted by the Coulomb forces and AE (t~) the energy shift gene-

ra ted b y the interact ions with t ransverse photons.

7"1. Kinet ic energy o] the ideal gas. - This is the quan t i ty

Y

?9 (72) T = J 4 ~ d k k q ~ / k 2 4 - m 2 - - m ) :

(2~)3j o

I n the nonrelat ivist ic ease ] << m,

1 1 m~ sinh ~ ] . + m~)~/1~ + ~ - ~ mt ~ -

(73) T ~ sc2J5 - - N 3 (3u2N~ ~ l o m ~ i ~ \ - ~ ] ;

in the ex t reme relativist ic case />> m,

(74) :T "Q]~ N 3 ( 3 ~ N ~ ~ 4 ~ = ~ ] "

7"2. Coulomb energy shi]t. - The Coulomb hami l tonian (2) m a y be wri t ten as

~75) H(°~= K + e~V (°) .

Since the ground state expecta t ion value of this hami l tonian is s ta t ionary,

i t is readily shown tha t the ground s ta te energy E (c) obeys

dE(C~ (76) e°2 d(e~) -- (I0 le~V !0} .

This expression can be re la ted to the longitudinal dielectric cons tant (38):

O N T I I E I ¢ I ~ ] I . ~ . T I V I S ' I ' I C D E G E N E R A T E E L E ( V I ? R O N G A S 4~7

in tegrat ing (38) with respect to o) along the inmginary axis, one finds

(77) "°!~°v'~'i°~ ~ / -4~2 l~(q,i,,,i-~/- ,:2q~-j"

T(C) Similar equat ions hold for the va(~uum energy E o . The difference is

T + A E `c' E (~:' - - E ~ ) ,

and therefore

co

(78) AE<"~>= -eT ~1 - -~ . I [e'(q, io,)--;o(q,i~o) ~ I 0 - -co

Owing to the simple dependen<.e (51.) of e on e~, the integmt.ion on e~ is easily performed, with the result :

(79)

co

]]-- f d sZ(q, i(o) 2~e~N[

. - c o

Within terms of order e ~, e~ m a y be replaced by e ~ in the second te rm of (79). The par t of AE ~') which is of order e ~ is (.Mled Coulomb exchange energy

AE(~C~h, and the pa r t which is of higher order is called ( 'oulomb corre |at ion

ener~'y AE((I~r. Using (58), one tinds:

(s0) A E ( I , ) ~(C) t / (C)

whel.( ~

(81)

~nd

c ~

. . . . . . . = ~ 14~ '"[':"(q' ':"')-';(q' ~(')J- ~2q~ ]' - c o

(82) . . . . . . . : q~ ~ d(o lo~z d~(q' i(,~) ' i(,)] ~'o(q, i(;') - D " ( q , i , , , ) - ~o(q, • - -co

Using (58) and (19), one finds for the ex(,hgnge Coulomb energy (81 :

(83) /x~(,,) ~ 4:~e"]l,2q.~ I "~ I . . . . . " -- W~ ~, Tr[A+(k+q)A~(k)+ ~ T,'[A+(k+q)A (k)j . k<j Ik+qt .i

k+ql < l

448 B. JANCOVICI

The second t e rm in (83) describes the exchange ()outomb in te rac t ion be tween

the electrons in the posi t ive energy s ta tes on one h a n d and the electrons

in the comple te ly filled nega t ive energy s ta tes on the o ther hand. This inter-

ac t ion is pa r t of the effects which lead to a mere unobse rvab le mass renor-

mffiization, and the second t e r m of (83) m u s t be discarded. Keeping" only

the first t e rm in (83), one finds:

(84) ]" 2=e " f (E[k+ul + E ~ : ) 2 - - q 2

k<y Ik+q[<i

~ e ~ [ -- - - (m' s inh ' --] - - # / ] ~ + .rrS) '-- 4 ] ' ~/f2 ~ m.~ sinh 1 i

(2~) ' [ m 5 m '

2 . . / ~ + m ~ 5/~ 2 ] + 5 (i-'+ "'~)'~ log ..... m ~ --~ ~ m " - i ~ .

To evalm~te the Coulomb corre la t ion energy (82), it is enough to consider

the m a i n con t r ibu t ion to (82), which comes f r o m small values of q and ~,~,

since gt(q, ion) diverges for small q; e'Z(q, i~o) m a y then be rep laced by its

a p p r o x i m a t e fo rm (66), and e' o by ] . Tak ing q and r ~ ) / q - - u as in t eg ra t ion

var iables in (82) and in t eg ra t ing first on q, which m a y be kep t small, one

finds, neglecting" te rms of order e~:

(85)

co

= ".- d u 1 - - t?.'- 1 =

X2 = 3= 4 (1 - log 2)p V/2 + .~e~ log e~.

7"3. T r a n s v e r s e e n e r g y s h i ] t . - The t ransverse energy shift m a y be com-

pu t ed along similar lines, f rom the t ransverse hami l ton i an (5); one now finds:

(86)

wh ere

(87)

and

(88)

/~E (tr) _._jr d(j0to0 ~ 1/-`'r'tl # d(o ~t(q,1 i0)) - - ~0(q, ~(.0) . . . . . . . 11 i ~- . . . . . . . ,

0 -ao

co

. . . . . ~ (,) [e"(q, "ioQ - - ~'o(q, ho)] ,

-co

co

AEoo,,,, = log ,, 2=t d ~'0(q, io))

--co

-- [a"'(q, i~ , ) - - e/o(q, ho)]}.

ON THE REI~ATIVISTIC I)EGENERATE EI.ECTR()N GAS 449

e L, q h ex<,ha, nge t ransverse ener~'v (87) m a y be wr i t t en :

(89) A•(t•) 4ste2 / Tr I A, (k q)~aA+(k)oo.] ..... , , = ~ ~; . - +

'1 "Q / ,c<'-~ q~ - - (E ik l -g l " E,,) ~ -I- I k I -q l - s

+ ~ ( T r A . (k + q)~A~_(k)=x Tr A (k ÷ q),9:,A+(k)=~i~

i , . y L q(Ez: - - El~+ql q) ~ v? ~ " - " - - q(E,,:- b;ik+~l + q) IJ

The second k suln iil (89 ) describes the t ransverse self-mass of the electrons in posi t ive energy states, and must be discarded af ter mass- renormal iza t ion . The, first t e rm gives:

(90) ar,<t~) __ D 4:nc ~ aq da k E [ k . F q ] ~ l - - ( k . ~ / ' q ) 2 k " q --- m 2 - - ~*"ex<~ ( 2 ~ ) s . " E i k ~ _ q ! E~,[q~ (E lk+q l __ E~)2 ] - -

k<f 1k4.77 <f

- - i~j.gi3 2 (,),2 s J n h 1 _ f ~ , / f 2 ~_ ,,~2) 2 T 3 / 3 ~ ' / f S ~ ,]¢~2 s i I ~ h - ' f

" ] '2 12 @ me 1 /., 3 m2]2 5 (/~ -~ "~'~)~ lo~z - - - * . rrH 3

NegleetinR terms ene rgy (88):

(91)

of order e ~, one finds for the

co

A/?(t~, ) = ~(2/"e ~ log e ~ / ' (1 du ...... 4=,~(p + m ~) . u~) ~

• {-- u'~(/'-' P+ m~) + uVf'~]+ m~ I 1

t r a n s v e r s e ( , o r r e l a t i o l t

-t- u~(t2/~ @ m!)] t~ -1 • . / 4 ' -A "

The integTal h~s been per for ined only in the l imit ing non-re la t iv is t ic a, nd ex t r eme relat ivist ic eases:

(92) if ] < < m , . . . . . . . . .

1 8 ~ ~m

ar~(~,) "q]* (8 loft :2 - - 5) c a log e ~

7"4. - W h e n tile componen t s of tile total ground state energy (71) are pu t together , one finds, b e y o n d tile u n p e r t u r b e d kinetic energy (72), a to ta l exchange energy

~,.~;<t~) ~(2e'-' 3 " )t [ (93) AE,~,.h = . . . . . A,d,c)~ ~ . . . . . ,~ = (2=) ~ (m- sinh -a --m - - ] V P + m~) u - 2] a[ ,

450 ]~. JANCOVICI

which cha.nges f rom negat ive to posit ive sign with increasing ]; the to taI correlation energy AE .... is ~lways positive.

I n the l imit ing non-relat ivist ic case ] << m, the ground s ta te energy is

5 m e 2 10(1 - - l og 2 ) / m e 2 \ ~ m e '~ ] (94) E o _ 107~2m'Q/s 1----2zr} ~ if- 3~ ~ ~ ) log" ] q- (9(e*) ;]

only the (~oulomb inteructions contr ibute.

In the l imiting ext reme relat ivist ic case, f >> m, the ground state energy is

e 2 e ~ log e 2 ] (95) Eo tg]~ 1 ÷ .. . . + e(e 4) ;

= 4 ~ + ~ - 2 ~

the exchange energy divides itself into

(96) ~hA~(c) __ ~]4e°- 2 q- 4 log 2 (27r) 3 3

and

(97) A.~,~) ~]4e2 5 q- 4 log 2

~oxoh = ~2~)3 3

and the correlation energy divides itself into

(98) Ar~(c) aO/4e ' log e 2 . . . . . . . = 24~4 - ( 8 - - 8 1 o g 2 )

and

(99) /,w(t~ ~]4e4 log e ~ ...... - - 247r 4 (8 tog 2 - - 5) .

In this ex t reme relativist ic case, the smallness of e 2 makes the ground

s ta te energy very close to the one of an ideal gas, as it could have been a

prioi expected. I t m a y also be noticed thut , in this case, the pressure

( l o o ) (~TE,,~ /3~ ~ \ ~ / N ~ / e 2 e4 log e 2 )

will be propor t ional to (N/[2)~, and tha t the more or less precise t r e a t m e n t

of the interact ions only slightly modifies the numerical propor t iona l i ty con-

s tan t ; this also could have been a priori expected on the basis of a simple

dimensional a rgument .

ON TtIE RELATIVISTIC DEGENERATE ELECTRON GAS 4 5 ]

A P P E N D I X

The full dielectric constants.

The full expression for the longitudinal relative dielectric constant is ob- tained by carrying out the integrations in (61). Since the integrand of the first integral changes sign under the transformation k --> -- (k + q), the integration domain k < ] and I k + q l > ] may be extended to the whole domain k < ] ; since the integrand of the second integral is i.nvariant under the same tran- sformation, the integration domain I k -- q I< / may be replaced by the domain k < / also. The integration may be performed after the two integrands have been summed together.

One finds for the real t)~rt of the longitudinal relative dielectric constant

, 1 {~ 2q , s i nh_ l ] + (A.I) Re e' t(q, co) = Re co(q, co) ÷ =q~ ]~/f2 + m 2_ ~ _m

q~(q~-- co~-- 2m ~) | / 'q2- co, ~ 4m :, + 6(q2 - io~ V q '=~oo~ "

[(q=-- ,~2_)V~ f ' +ira, + ]Y/ (q ! - io~)i(q,- oi~ + /l;m~)]'= 4m,c o~ • log [ ( q ' - co,)v?, + m 2 - l v ' (~ ,o6 (q2 ,o-- ~ :~ ,~ - f i ] ' - 4msco-" +

+~(q_~ co' p _ " (2q f÷2co~/ f ,~m2) ,_(q ,_co~)2 q \4 le m ' ) l o g (2~q-]_2oo~f12~fn2)2 -iq2 ~,), +

+ q ~- ¢ ] % : ~ - (], + ,n~) - ~ ( q , - co,) log: (eq/ - q~ + co~)~- 4co,(/~ + ~,)

where e'o(q, co) is defined through (58) and (60). l~emembering tha t co has a small positive imaginary part ÷ic~, and assuming co > 0, one finds an imag- inary part for the dielectric constant, with a non-zero value only in the fol- lowing cases :

(A.I')

- - For any value of q

if IEf_q-- E~]< co < E,+~-- E~,

1 im e't(q, w) = ~2qa [(2E~ + o)) 3 - 3q2(2E~ ~- co)] +

q2~ (_,?_ 2m21/q2..__w~-+ 4ra~. + 6(q ~ -eo ' ) [ q ' - - w ' '

if E1-q _L EI < co < E~+~ + Ei,

452

(a. l ' )

B . J A N C O V I C I

1 I m dr(q, (,,) -= ~2q3 [ ( 2 E f - - o)) a - 3 q ~ ( 2 E ~ - o))] +

q~-- (°2-- 2 m 2 i - - m2 + 4m2 6(q ~ - (o:) . /qe q 2 (,}2

if o}~- E j~o+ El ,

I m e"(q, {o) q 2 - - o ? - - 2 m " I'q'a'(,.)'a47 4-m'-' 3 ( q 2 _ _ o , 2 ) l / q 2 ~ o ? "

- - For q < 2]

if o ) < E f - - E ~,

l m e'~(q, {o) --- e) (12Ej + o) ' a - 3q~-). 6q a

For q > 2]

if x/q'-' @ 4m'-'< (o < E~_a q- E,,

l t n 6:'~(q, (o) = q z - - (O2-- 2<1a2 l//q'~- °,2 + (im'- 3 ( q " - - (,?) /' q 2 .)~ "

The small q and m l imi t m a y be ta, ken in (A.1). I t is howeve r s imple r to keep on ly the ill'st inte~'ral in (61), and to t ake the sma.]l q l imi t of the integ ' rand. One ob ta ins :

e ~ f 2q .k/Ej. (A.'2) ~"~(q, o)) = 1 q n=q~ dak {q'k/1;,.:)"--,' " o} ~"

k < f Ik+ql >s

In the smal l q l imit , the in teg ' ra t ion d o m a i n is such Chat, cal l ing I(1,% q . k ) the in tegrand , one has :

(A.3)

1

Ik~qt >*

I n this way , (63) is ob ta ined . This p rocedu re howeve r is no t in'eeise enough in the ne ig 'hbourhood of m - q]/Es, because tile i n t e g r a n d m a y b e c o m e infi- nite. A m o r e precise t r e a t m e n t is to t a k e the exac t va lue (A.1) a t the po in t a) = E ~ - - E s _ ~ where the in tegra l becomes rea.1, and to t ake the smal l q l imi t a f t e rwards . In this w a y (64) is ob ta ined .

The full express ion for the t r a n s v e r s e re la t ive die lect r ic c o n s t a n t is o b t a i n e d

ON THE RELATIVISTIC DEGENERATE ELECTRON GAS

f r o m (65) in u v e r y s imi lar way. One finds a real p a r t

453

1 .(A.4) R e s't(q, co) = Re s'o(q, co) +

n ( q : - - cod

2 2 . I 2 ( q ! ~ 20~!! IV/f> 4- m 2 _ 3 ( q __co2)sinh-1 / 3q 2

( [ ~ _ c o 2 ) V + ~ 4- f~ / [q2 - -co2) (q2_ o# 4- 4m2)] 2 - 4m40~ 2

- og m;_j q2 co2iiq2 co2+-- .2)]2_4m co2 CO 2

+ 24q 3 [(q _ cos)(3q2 + co2 4- 12/2 4- ] 2 m 2) _ 12m2q2].

- log (2q/ + 2co~fi-? m2) 2 - (q2 co2),~ 4- ('~qf -- '2coVfi U m~ )2-- (q~_co2)2

+~1 ~¢ / f~ m~ _ (q2_ (03)(]5 + m S) + 7 ( - q4 + co4 _~_~ 4m2q2) .

- log ( 2 q / + q 2 _ co2)2_ 4co2(p + m 2) I (2qt - - q~ + cod 2 - 4co2(12 + m 9 ] "

+

The i m a g i n a r y p a r t is nonzero in the fol lowing cases:

(A.4 ' )

- - F o r a n y va lue of q

if I E,_q- - Es I < co < E,+~-- El,

_ 1 [(2E, + co)3 I m e't(q, co) = 24 [ qa +

if El_q÷ E l < co < E/+~+ EI,

± [(2~,- co)~ I m e't(q, co) : - - 24 / q3 +

if co > E1+q + Er

3(q2__ co2_q(q24m2)~o2)(2Ef + co!] ÷

+ --6(q2_co'2) V q ~ - <0 2 ;

3(q2 _ co2 _ 4m 2) (2EI _ co)

~(q2_~ co D ÷

q2_ co2_ 2,n2 l /q2 - - ~ + 4 ~ + 6(q ~ -coD V q 2 ~ ;

q 2 _ c o 2 _ 2m 21/q2 (o3 + 4m 2 I m e't(q, co) = 3 ( q 2 _ w 2 ) [ q2~co2 •

- - F o r q < 2/

29 - I! Nuovo Cimen~o.

454 B. JANCOVICI

(A.4')

if o J < E I - - E t q

I m e ' t (q , e)) = ~ -

- - F o r q > 2]

if ~/q2 _}_ 4 m 2 < ~ < EI q + E~,

12E~ + ¢o 2 + 3q 2 12m -~ ] +

q ~ ] / .2 l m s'~(q, oJ) - - ° )~-- 2m~ - - 3(q 2 - - 092) /q q:092 +~)~.4m~

The sma l l q a n d o9 l i m i t m a y be o b t a i n e d b y t r e a t i n g the f irst i n t e g r a I of (65) in the same m a n n e r as in t h e l o n g i t u d i n a l case ; t he second i n t e g r a l of (65) h o w e v e r is no t neg l ig ib le (*), a l t h o u g h i t s i n t e g r a n d m a y be a p p r o x - i m a t e d b y i ts v a l u e for q = 0} = 0. One o b t a i n s

e~ { f k=--(k'q/q)~ q'k/E~ (A.5) s ' t(q, o)) = 1 - - . ~ 2 ( q 2 {,)2) dak E~ (q. k/Ek) ~- (o ~-

k < f ]k+ul >I

+ k2/3 + m!} O ,3 ~'

k < f

which g ives (66). A r o u n d the pole v~} = q, bo th t he e x a c t exp res s ion (A.4) ~md the a p p r o x i m a t e one (66) b e h a v e l ike (66'%

Because of t he fo rm of (60), (61) a n d (65), t h e r e is a d i spe r s i on r e l a t i o n b e t w e e n the rea l and the i m a g i n a r y p a r t of each d i e l e c t r i c c o n s t u n t ; t h e s e r e l a t i o n s m ~ y be used as checks on (A.1) a n d (A.4). B o t h l o n g i t u d i n a l and t r a n s v e r s e d i e l ec t r i c c o n s t a n t s o b e y :

(A.6)

co

' I t"Im[d(q't)--s~(q't)]d(t~) ( o ) , t > ( ) ) . Re [e'(q, ~o) - - so(q, o))] ,~ J ~ ÷ - - - t (~,) /{~)T

0

The w~cuum d i e l ec t r i c c o n s t a n t obeys :

co

(A.7) Re S'o(q, ~0) = 1 (q2_ {~2) ; I m so(q,: ~ /q2+-z2) d(z 2) . (oJ > 0).

4 m 2

(') Tile second integral in (65) arises because tile vacuum polarization current induced by an external vector potent ial A is modified in the presence of tile electron gas, since there are excitations of the vacuum which become forbidden by the exclusion principle. This process is not negligible even in the non-relativistic limit q, w, ] << m, but i t rather accounts for tile term (e2/2m.)~f*~A in the current of a nonrelativist ic electron with a wave function y~. The author is indebted to Professor A. KLEIN" for clarifying this point to him.

ON T H E R E L A T I V I S T I C D ~ G E N E R A T E E L E C T R O N GAS 455

R I A S S U N T O (*)

Si s t ud i ano le p rop r i e t~ di u n gas di e l e t t ron i a t e m p e r a t u r a zero e di dens i t~ a b b a - s t a n z a e l e v a t a perch6 l ' ene rg ia di F e r m i sia r e l a t iv i s t i ca (densit,~ simili e t e m p e r a t u r e r e l a t i v a m e n t e basse si r i s con t r ano en t ro le n a n e b ianche) . Si s tud ia il p r o b l e m a nel la f o rmu laz ione d ie l e t t r i ca de l l ' app ross imaz ione di quas i -boson i ; si t r a t t a n o come boson i le coppie e l e t t r o n e - b u c a che compa iono a causa delle in te raz ion i c o u l o m b i a n e o delle i n t e r az ion i con il c a m p o dei fo ton i t rasvers i , e si t r a s c u r a n o t u t t i i processi che n o n siano la creazione e l ' ann i ch i l az ione di ta l i coppie ; t r a le possibi l i b u c h e si c o m p r e n d o n o a n c h e i pos i t ron i . Si scr ivono e si d iagona l izzano gli h a m i l t o n i a n i model lo che desc r ivono il s i s t ema in q u e s t a appross imaz ione . Si calcolano le cos t an t i d ie le t t r i che l o n g i t u d i n a l e e t rasversa le , ehe d i p e n d o n o dal n u m e r o d ' o n d a e da l la f r e q u e n z a e ehe desc r ivono la r i spos t a del s i s t ema alle ecci tazioni e s t e rne ; la necessar ia r ino rmal i zzaz ione del la car ica c o m p a r e in modo mol to n a t u r a l e . Si s tud ia la p ropagaz ione delle onde longi- t u d i n a l e e t r a s v e r s a e se ne calcola la legge di dispers ione. Si calcola inf ine l ' ene rg ia dello s t a to f o n d a m e n t a l e sino al t e r m i n e de l l 'o rd ine e 4 log e 2.

(*) T r a d u z i o n e a cura della Redazione.