Covariant Wigner function approach to the relativistic charged gas in a strong magnetic field

Physica I13A (1982) 477--490 North-Holland Publishing Co.

COVARIANT WIGNER FUNCTION APPROACH TO THE R E L A T I V I S T I C C H A R G E D GAS IN A S T R O N G M A G N E T I C F I E L D I. ELECTRON GAS IN THERMAL EQUILIBRIUM

R. DOMINGUEZ TENREIRO and R6mi HAKIMt Groupe d'Astrophysique Relativiste, Observatoire, 92190 Meudon, France

Received 9 February 1982

The relativistic quantum electron gas embedded in a strong magnetic field is studied by calculating its covariant Wigner function in thermal equilibrium. Previous results obtained earlier by Canuto and Chiu are then recovered in a unified way. The polarization tensor is calculated with the use of a covariant quantum BGK equation. Also the lifetime of the neutron in such a medium is calculated for the sake of illustration of the usefulness of the covariant Wigner function.

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

The discovery of pulsars in 1967 has enhanced the interest shown for matter e m b e d d e d in strong magnetic fields (see e.g. the excellent review by Canuto and Ventura~)): a conservation-of-flux-type argument 1) (and also the observation of an X-ray line of the Crab pulsar, identified with synchrotron radiation2)) leads to values of the order of 1012 Gauss for the magnetic field. F u r t h e r m o r e , models of white d e v a r f s - w h e t h e r magnetic or not (see the review by Shulov3)) - provide central magnetic fields with intensities as high as 1011--1013 Gauss4). Besides this astrophysical context, the study of matter e m b e d d e d in strong magnetic fields (under the f o r m of a plasma in the case of the magnetosphere of a pulsar or under the f o r m of a "magnetic solid ''1) in the crust of a neutron star) deserves an interest of its own and it raises m a n y exciting theoretical problems. A m o n g those systems of physical interest is the relativistic quantum electron gas in a strong magnetic field, e m b e d d e d in a neutralizing uniform background of positively charged ions. The study of such a plasma is quite difficult not only because of some theoretical problems such as the structure of the v a c u u m and/or renormalization but, more particularly, owing to the f Dept. de Math6matiques, U.S.T.L., 34060 Montpellier C6dex, France. 0378-4371/82/0000-0000/$02.75 (~) 1982 North-Holland

478

R. DOMINGUEZ TENREIRO AND R. HAKIM

extremely complex calculations involved in the obtention of some physical results of interest, such as the proper modes of oscillation of the plasmaS'6), etc. This calculational difficulty renders necessary a check of the results claimed, by an independent calculation and, preferably, using different techniques. For instance, in ref. 5, Green's functions techniques have been used and, consequently, other techniques such as those involved in the use of covariant Wigner functions7'8), should not be considered as a superfluous luxury In this paper, we have bound ourselves to the calculation of the simplest possible case; i.e., the case of the ideal relativistic electron gas in thermal equilibrium whose knowledge of the covariant Wigner function is an utter necessity in any approximation scheme to be considered. After having calculated the thermal Wigner function (in the non-relativistic limit, it tends to the Wigner function obtained earlier by Kellyg)) of such a system, it is subsequently used to rederive, in a unified way, the equations of state first obtained by Canuto and Chiu ~'1°) (see also results by VisvanathanH)) in their pioneering works. Also, the lifetime of the neutron is calculated in a much simpler way as was done earlier12). Finally, with the help of a BathnagarG r o s s - K r o o k ~3) kinetic equation, the polarization tensor of the system is evaluated. Finally, we would like to mention that simpler although approximate results were obtained elsewhere~4): in the approximation where "spin was neglected"~4), the thermal Wigner function was calculated and the transport coefficients of the relativistic quantum plasma in a strong magnetic field obtained (see also ref. 15). N o t a t i o n s . In this paper we use the c o m m o n conventions: the metric g,~ has signature + - - - ; we also use a system of units where c (velocity of light) = h (Planck's constant/27r)= 1. e , ~ denotes the completely antisymmetric tensor with 0123= +1. Indices between brackets indicate an antisymmetrization; for instance: At~B~ 1=- A~B~ - A~B~.

2. The basic tools: covariant Wigner Function and relativistic quantum Liouville equations In refs. 7 and 8 some techniques involved by the use of covariant Wigner functions have been given and hence, in this section, we only mention a few properties which are to be used in the sequel (see also refs. 14 and 15). We begin with the definition of the covariant Wigner function F ( x , p) = ~

1 I

d4R exp[-i~- • Rl(t~(x + R/2) (~ qt(x - R/2)),

(2.1)

RELATIVISTIC CHARGED GAS IN A MAGNETIC FIELD I

479

where 7r = p + eA, A being the quadripotential of the external electromagnetic field; where $ is the electron field, obeying the usual Dirac's equations [iv" ( ~ - i e A ) -

m ] . ~k = 0 (2.2)

(J . "(iv • (0 + ieA) + m] = 0; and where the brackets denote a quantum statistical mechanical average (...) - Tr[p...],

(2.3)

where p is the density operator describing the statistical state of the system under consideration. The knowledge of F(x, p) provides the average four-current J"(x) of the electrons through

J'Xx) - (e~(x)'y~'~(x)) = e Sp f d4p~/"F(x, p)

(2.4)

and its average momentum-energy tensor

T ~ ( x ) - (~ J/(x)'y~OV~(x) I = Sp f d4pv~p VF(x, P)

(2.5)

as a simple calculation shows7'8'14). In most cases the data of T ~ and J " is amply sufficient for the problems at hand, such as the determination of the equations of state or the transport coefficients. The covariant Wigner function F(x, p) can easily be shown to satisfy the following relativistic quantum Liouville equations14):

[i,y . O + 2('y . p-m)-ie'y~,F~'V + i

F(x,p)=O,

F ( x , p ) i i ~ / • 0-2(3, • p - m ) - " le3,~,F~'~+1 = 0.

(2.6.a)

(2.6.b)

Here we should notice that F "" is the external (constant) magnetic field and, more particularly that, in the derivation of eqs. (2.6) use has been made of the Lorentz gauge condition

A"(x)

1,-,,~

= ~r vx .

(2.7)

This system will be studied in detail elsewhere~6). Decomposing now F(z,p) on the basis of the sixteen matrices of the

480

R. DOMINGUEZ TENREIRO AND R. HAKIM

1 . D i r a c ' s a l g e b r a (i.e. on: I, 7", ~r"~ =_ 2[3' , 3'"],

3,5

. o = 13' 3' I 3' 2 3' 3 , 3'"3'5) as

F = ~{fI + f . 3 ' " + ~.,,~r"" + fs3'5 + fs.v"3'5},

(2.8)

fa =- Sp [3'aF],

(2.9)

with A = 1.2 . . . . .

16,

o n e o b t a i n s the f o l l o w i n g s y s t e m : 0 . 0 ~ " - 2ipof" + 2imf + e F ~ 0--p-7f = 0,

(2.10a)

0 . O"f" + Z i p . f " - 2imf + e F [ 0 - ~ f = 0;

(2.10b)

- O.f + 2 i p . f +

eF[ ~ O f - 2imf. + O~f~ + 2i p o f .'~ + eF,,~ 0 ~O f . ,, = 0, (2.1 la)

0 0 ,, 0 . f + 2 i p . f + eF[ ~ p ~ f - 2imf. + 0of[ - 2ip~f~ + eF,~ o ~ f . = 0;

(2.11.b)

(O.f~ - O~f.) - 2 i ( p . f , - p J . ) + 2imf.,, +e

F~p~f,,-

~ 0- -p ~I"

+iE.A..

0 T-2ip'+eF

"~ 0 -0~

f~ = 0,

(2.12.a)

(O.f,, - O~f.) - 2 i ( p . f , , - p J . ) - 2imf.,, -e(F[

O _ F ~~-T-po1.)+i~.~.,,[O* O . \ ~p~[. + 2ip ~ + e F ~ " ~ ]0f ,

A=0; (2.12.b)

- ia~*f,,

"~ ~

- 2P ~ * f , , - i e F

+ 20"f5 - 4ip"fs + 2eF'~ + - i0T*f~.

+ 2p

f~, - ieF '~T

- 2a"f~ - 4 i p , f s - 2eFt, a"f~ - 2ip"f~ + eF~ -u ~p

*.L, f~ - 4imfs = 0, 0 *fT.

f5 + 4imf5 = 0;

f~ - 2imfs = 0,

O"f~ + 2ip"fs" + eF"~ ~ p ~ l

(2.13.a)

- 2 i m f , = 0;

(2.13.b) (2.14.a) (2.14.b)

w h i c h c o r r e s p o n d r e s p e c t i v e l y to the coefficients of 1, y", ~r "~, 3' " 3'5, 3;5- * f ~ is the dual t e n s o r of f,,~, i.e. * f ~ = E~o~f "~.

(2.15)

RELATIVISTIC CHARGED GAS IN A MAGNETIC FIELD I

481

3. The equilibrium Wigner function

A priori the equilibrium Wigner function F~q(p) [the invariance of the equilibrium state under spacetime translations implies the independance of Feq on x] is a 4 × 4 matrix. However, the use of the Liouville system (2.10a)(2.14.b) makes it possible to express the 16 fA's in terms of only two of them, namely, f and fJ2. To show this we first rewrite the above system by taking respectively the sum and the difference of equations a and b. One finds F~" ~ f~ = 0, 3p"

(3. l.a)

p j " - mf = 0;

(3.1.b)

p , f - m f , + ~e F ~ ~ 0 f,, = 0,

(3.2.a)

• o f~. = O; eF~oa_b_0 f - 21p

(3.2.b)

0 2 p J v ] - e~0~Y "~ 0-~ f~ = 0,

(3.3.a)

2E0~,vP f5 + 2 i m f ~ + eF'~l,~

f~l = 0;

4p j 5 + e e , , ~ F '~° 0 -O~ f~x = 0, e~,x,p~/~x + 2im/5, - eF~ ~ p~f~ = 0 eF~ - ~

3

(3.3.b) (3.4.a)

[5 = 0;

(3.4.b) (3.5.a)

f5 - 2im/5 = 0.

(3.5.b)

We note that eq. (3.2.a) is nothing but the Wigner function version of the usual Gordon's decomposition (see e.g. ref. 17 for details), its last term corresponding to the spin contribution (in another paper ~4) this term was neglected). Eq. (3.4.a) yields ep ~

f5 = - ' ~

e~x~F ~ ~

0

.,,~,

l

,

(3.6)

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R. D O M I N G U E Z T E N R E I R O A N D R. H A K I M

while eq. (3.2.a) provides f~ = p z f + e F,,~ O m 2i & f~"

(3.7)

Finally, inserting eq. (3.6) into eq. (3.4.b), one is led to 2im~"

~-F~ ~

F~

+p

(3.8)

Therefore, our goal has been achieved: all the f a ' s have been expressed in terms of f and f... It should be noticed that one can go a step further by remarking that (i) with the tensors and the pseudo-tensors at our disposal (i.e. F"% e~.~, g~., p.) one cannot construct a pseudo-scalar, and hence f~ - 0

(3.9)

(a direct calculation also confirms this result); and (ii) that the only antisymmetric tensor at our disposal is F "~ and hence, /~aF~.

(3.10)

Consequently, choosing the magnetic field parallel to the z-axis, the only calculation to be made is that of f,2. In fact, the calculation of f has already been performed elsewhere '4) (ref. 14 contains a misprint and the correct expression is the one given below) and was found to be 2m ~ 1 6(P ° - Eo) /(p) = ~ e x p ( - W 2) X (g0 exp[~-~-E~o7- ~r)] + 1 + n=l

(-1)" En

6(p ° - E,) [ L , ( + Z W 2) - L, ,(+2W2)]} exp[~--~,-?-~3r)]+l

(3.11)

with /3-= (kT) '; where el is the chemical potential and where the following notations have been used: E , = [ m Z + p ~ + 2 n e H ] ~/z, W 2 = p~/eH

n=0,1,2 .....

(3.12) (3.13)

(H: intensity of the magnetic field). In eq. (3.11) L, is a Laguerre polynomial of order n. In eqs (3.13) and (3.14) Pll and Pi denote the parallel and the perpendicular momentum of an electron with respect to the magnetic field. E, is the energy of the n th Landau level. Let us now pass to the calculation of /,2(P). Essentially, we have to calculate the Fourier transform (eq. (2.1)) of (4t(X + R/2)O'l~ql(X -- R/2)).

RELATIVISTIC CHARGED GAS IN A MAGNETIC FIELD I

483

This last quantity is easily calculated with the one-particle equilibrium density matrix

p(x, x') = ~'. ~r(X) ~ ~r(x')/exp[/3(Er - ~)] + 1,

(3.14)

r

where r designates the set of all quantum numbers necessary for the complete specification of an electron state [i.e. n; s = - 1 ; PlI; a, the degeneracy index linked to the position of the orbits in the (x, y)-planetS)]; where the ~r's are the well-known Johnson-Lippmann tS) spinors and where Er is the associated energy E~ = [ m 2 + p ~ + eH(2n + s + 1)] 1/2.

(3.15)

The calculation of In is in fact straightforward and of the same level of difficulty as the one for f, and is simply the Fourier transform of

~,(RI2)cr,2~,(-RI2)

~ ~ ~ - ~ ¥ i "

(3.16)

The result is

2m

f,2(P) = ~

/ 1

8(p ° - E°~

e x p ( - W 2) t/~o exp[-~-Eo-Z ~f)] + 1

3(p°-En) } .=1 E. exp[[3(E.-el)]+ l [Ln-l(2W2)+ L~(2W2)] "

+ ~ (-1)"

(3.17)

In order to calculate J" or T "v we also need f", which quantity is provided by eq. (3.7). Explicitly, one has

l~(p) = ~ f ( p ) +

e F21

a . o -~l,(p)--~12(p)

;

(3.18)

or _p~

f'(p)--~f(p),

/x = 0,3,

[ 0

(3.19)

,

0

-~f,(P)-~II~(P)

, i= 1,2.

(3.20)

From f~ one gets the normalization of F, i.e. the relation between the numerical density of the electrons, say neq, and the chemical potential e¢: neq ~ j 0 =

f d"pf°(p) = I d4p ~ - f ( p ) .

(3.21)

This last expression is identical with the one already obtained when the "spin was neglected"t4), i.e. it yields the results first obtained by Canuto and Chiu t°)

484

R. DOMINGUEZ TENREIRO AND R. HAKIM

as indicated in ref. 14. The explanation of this fact is now clear and is to be found in eqs. (3.19)-(3.20). Similarly, either the energy density

E =- T°°= f d4pp°f°(p)= f d4p (P~ f(p),

(3.22)

or the parallel pressure (with p3 _ P0 PII--- T33

d4pp3f3(p) =

=

drip ~ - - f ( p ) ,

(3.23)

have expressions identical with the ones already obtained when "spin was neglected"'4), identical themselves with those first given by Canuto and Chiu'°). It remains to check that the perpendicular pressure Pa --- T " = T 2~

(3.24)

calculated with our covariant Wigner function f " provides the same result as the one given by Canuto and Chiu. Let us calculate TI':

T 11= f d4pplfl(P) P' ~pl = f d p"p '{mf(p)_~F2 0 2,(p)} 4

i pl

e

21

Let us calculate briefly the second term; the first one is computed analogously.

f d4pf2,(p) = ~2mH f × {~(p~o E0) +

( _ 1 ) .+1

""

n=l

E.

dpO dp I dp2 dp 3 e x p ( - W :)

-1 exp[/3(Eo- Es)] + 1

6(p °- E,)

[L, ,(2W z) + L,(2W2)]~. k exp[~--~.S-'~-)t)]+ 1

(3.26)

The integral over pO is easily performed while the integral over p, and p2 [remember that d p l d p 2 = ~r dW z] or W. involves integrals of the following type'9):

f dt exp[-t/2] L.(t) 0

= (-1)"/2.

(3.27)

RELATIVISTIC CHARGED GAS IN A MAGNETIC FIELD I

485

Finally, one gets f d4pf21(p) = meH f 1 1 J dPll E00exp[/3(E0 - el)] + 1'

(3.28)

so that

T,I:T2:

(eH)2~,nfdpll

=~

,=t

1

exp[/3(E, - e¢)] + 1'

(3.29)

which is exactly the result of Canuto and Chiu.

4. The polarization tensor In this section, which serves as an illustration of the covariant Wigner function method, we want to calculate the polarization properties of the system due to the scattering of the electrons. This scattering is taken into account via a relaxation time approximation, i.e. via a B G K 13) kinetic equation: all the physics of the various scattering process is involved in the relaxation time T, which is allowed to depend on p. Therefore, we want to calculate the response of the four-current ~J~ of the system to an external electromagnetic perturbation ~A ", i.e. the polarization tensor P ~ defined through

~J~ = P~SA~;

(4.1)

of course, P ~ satisfies the constraint

k~P ~v = 0,

(4.2)

occuring from k ~ J ~= O. It should also be remarked that, in order for the external electromagnetic perturbation 8F, v associated with 8A ~ to be of electric character (i.e. such that there exists a reference frame where 8F "~ reduces to its electric components only) it must satisfy the following relations 2°) 8*F ~. 8F~ =0,

8F ~. ~F~>0.

(4.3)

We shall use a covariant generalization of the usual B G K equation which has been studied elsewhere~4). Its collision terms have been chosen so as (i) to be consistent, (ii) to generalize the relativistic B G K equation given by Anderson and Witting 2t) (which yields more precision than the Marie's 22) version as far as transport coefficients are concerned) and (iii) to lead to the so-called Landau and Lifshitz form 22) of relativistic hydrodynamics [for some

486

R. DOMINGUEZ TENREIRO AND R. HAKIM

details and the role of m a t c h i n g conditions 23) see refs. 24]. It reads

f

3' • O - 2i(3' • p - m ) + e3""F~

F = - 3' • u - -

(4.4.a)

, T

F[3".o+2i(3".p_m)+e3",F;OT_p, J= - F-Fo, r 3' • u,

(4.4.b)

w h e r e u is a unit timelike f o u r - v e c t o r parallel to the f o u r - c u r r e n t of the e l e c t r o n s and w h e r e ~- is a relaxation time which, a priori, d e p e n d s on the m a c r o s c o p i c quantities at our disposal (i.e. T, n~, or e¢, F,~) and p o s s i b l y on p. L e t us n o w switch on the external p e r t u r b a t i o n 8A"; then, in the left-hand side of eq. (4.4.a), we m u s t add the following term'-5): + ~ 2ei

f d4x ' d4p ' e x p [ - i p ' • (x - x')]3' • 8 A ( x ' ) F ( x ,

(4.5)

p - p'/2),

and, of c o u r s e , a similar term 2s) to eq. (4.4.b). B e f o r e calculating 8 J " we first need the solution of the B G K s y s t e m (4.4), w h i c h is o b t a i n e d a p p p r o x i m a t e l y via a C h a p m a n - E n s k o g e x p a n s i o n in the relaxation time r, F(x, p) = ~, r"F~(x,

(4.6)

p),

n=0

where F{0)(x, p) -= Feq(p).

(4.7)

Inserting n o w eq. (4.6) into eqs. (4.4), retaining the t e r m s of o r d e r one in r and multiplying by 3' • u [with (y • u ) : = 1], one gets F,)(x,p)=

-3" . u ×

{

3" . O - 2i(3" . p - m ) +

0}

e3""F~ ~ p ,

2ei

Feq + ~ - ~

f d4x ' d4p ' e x p [ - i p ' • ( x - x')]3" • 8 A ( x ' ) F e q ( p - p ' / 2 ) .

3" . u

(4.8)

N o t i c e that Feq is n o w x - d e p e n d a n t (local equilibrium) on a h y d r o d y n a m i c a l scale via the m a c r o s c o p i c quantities involved. T h e c o n t r i b u t i o n of Fm to the c u r r e n t is J}'l) = re Sp f d4p3"UFm(x, p).

(4.9)

N o w 8 J " is o b t a i n e d f o r m J~l b y selecting the term p r o p o r t i o n a l to 6 A ; i.e. 2~-e2i 8J ~' - (2.a.3z Sp f d4p d4x ' d4p ' x e x p [ - i p ' • (x - x')] 3"~3" • u y ~ S A ~ ( x ' ) F e q ( p - p ' / 2 ) .

(4.10)

RELATIVISTICCHARGEDGAS IN A MAGNETICFIELD I

487

Integrating this last equation over x' and p', using the definition (4.1) for P"~ and the following relation between Dirac's matrices:

3,.~/~.

= [g~t~g,l~+ g ~

g~]V~ + iE"~°'/5"/~,

(4.11)

P"~= 2re2i{n,qU~'u~-i,"W~'u,,~ d'pIs~(p)},

(4.12)

one finds

when r is independant of p and, in the opposite case: P"~ = 2re2i I

d4pr(p){u"ff(p)

* p,v~t~

-- IE

Utrfsa(P)}.

(4.13)

5. Neutron lifetime in magnetized matter

This problem has already been studied elsewhere ~z) and therefore, in order to illustrate the usefulness of the covariant Wigner function, we rederive shortly the expression for the neutron lifetime in magnetized matter. A quite similar technique can be used in the calculation of neutronization in magnetized matter26), an important problem in view of the internal structure and properties of white dwarfs. In this section, the notations of ref. 12 are generally adopted. The inverse of the neutron lifetime ~ is given by

(1/0

=!E 2 sr~o,,

(5.1)

where i is the initial state (a neutron with four-momentum Q); where f is the final state (a proton with four-momentum P, an electron in the quantum state r and a neutrino with four-momentum q)., In terms of the interaction Hamiltonian Hint 27) G~ Hint = ~ f d3x[OpV,(1 + Avs)UNI[O,V"(1 + vs)Uv],

(5.2)

(Up, UN, U,, U, are the spinors representing the state of the proton, of the neutron, of the electron and of the neutrino respectively), the S-matrix element Sti is given by Sfi = -

iI dt Hint.

= - i G ~ f d3x exp[ - ix • (Q - P - q)]]F, J

(5.3) (5.4)

488

R. D O M I N G U E Z

TENREIRO

A N D R. H A K I M

where F = [O(P)y,(l +

A'ys)U(Q)][(]e(X)'yu(1 + 3'5)U(q)].

(5.5)

In eq. (5.5) use has been made of Up(x) = e x p [ - i P • x ] U ( P ) ,

UN(X) = e x p [ - i Q • x] U ( Q ) ,

U,.(x)

= e x p l - i q • x] U ( Q ) ,

(5.6)

where U(P) is an eigenspinor of the free Dirac's equation. The neutron lifetime ~ is given by 27)

C-'

1 =2~f

T

Of,

(5.7)

where pf is the density of final states (the factor ~ in eq. (5.7) c o m e s from the average on the initial spin of the neutron). The phase space of the electron and of the proton is affected by the magnetic field but, for intensities of the order of those reached in neutron stars (1013 gauss) the proton can still be considered as a usual free particle: its critical field is of the order of 1016 gauss. This is the reason why we used eq. (5.6) for the proton, as in ref. 12. Finally, using eqs. (5.4)-(5.7) one obtains - 2T 2 (2~r~

.

d3p d3q dPll da d4x •

d4x '

ns

x e x p [ - i ( x - x ' ) . (Q - P - q)](l - o3r)It[ 2,

(5.8)

where (1-o3,) is the statistical weight of the unoccupied electron states; where the final states have been explicitly summed. The term Esp~o~trl2 is given by (ref. 28 eq. (161)) 2

E Irl 2-

spins

P

E N electron [Ue(x)y"(1 + vs) Ue(x')]Zu(Q, p, q),

(5.9)

spin

with

Z~(Q, P, q) = [(1 + A)2Q qP~ + (l - A)2P qQ. - (1 •

•

-

A 2)mpmNq~] (5.10)

so that, finally, 2

~-~ -

2

2~ G~f d3p d3q G'~(Q - P - q ) Z . ( Q , P, q), EN

Ep E~

(5.11)

where the function G ~ can be expressed from the equilibrium Wigner function of the unoccupied electron states F(k):

G"(k) = f ~d4R e x p [ - i k • R](:~(x + R/2)y"(1 + y~)qJ(x - R/2):),

(5.12)

RELATIVISTIC CHARGED GAS IN A MAGNETIC FIELD I

489

or

G~(k) =

S p { v " F ( k ) } - Sp{vsV"P(k)}.

(5.13)

F ( k ) h a s e x a c t l y t h e s a m e e x p r e s s i o n as F(k) except t h a t t~r h a s to b e r e p l a c e d b y (1 - ~3r). Eq. (5.12) c o m e s f r o m the f a c t t h a t Or(X)3~"(1 + vs)U,(x')[1 - t~r] = (:t~(X)3'"(1 + 3'5)~b(X'):).

(5.14)

r

T h e final e x p r e s s i o n (5.11) p r e s e n t s the f o l l o w i n g a d v a n t a g e s on p r e v i o u s c a l c u l a t i o n s : (i) c o m p a c i t y a n d (ii) o n c e t h e W i g n e r f u n c t i o n o f t h e e l e c t r o n s ( w h a t e v e r t h e i r s t a t i s t i c a l s t a t e , t h e r m a l e q u i l i b r i u m o r not) is k n o w n t h e s u m o v e r the final s t a t e s o f t h e e l e c t r o n s is a u t o m a t i c a l l y t a k e n into a c c o u n t a n d (5.9) h a s n o t to b e c a l c u l a t e d . O f c o u r s e , in t h e s i m p l e c a s e w h e r e t h e initial n e u t r o n is at r e s t a n d the p r o t o n is n o n r e l a t i v i s t i c , eq. (5.11) r e d u c e s to t h e r e s u l t g i v e n in ref. 12. In t h e c a s e o f neutronization 26) of p r o t o n s in a s t r o n g m a g n e t i c field it is d i r e c t l y the W i g n e r f u n c t i o n of t h e e l e c t r o n s t h a t c o m e s into p l a y r a t h e r t h a n the W i g n e r f u n c t i o n o f t h e u n o c c u p i e d e l e c t r o n s t a t e s .

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490 20) 21) 22) 23) 24)

25) 26) 27) 28)

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