P H Y S I C A L R E V I E W V O L U M E 17 3 , N U M B E R 5 2 5 S E P T E M B E R 1 9 6 8

Thermodynamic Properties of a Magnetized Fermi Gas VITTORIO CANUTO* AND HONG-YEE CHiuf

Institute for Space Studies, Goddard Space Flight Center, National Aeronautics and Space Administration, New York, New York 10025

(Received 26 April 1968)

In a previous paper, we calculated the equation of state of a Fermi gas at arbitrary temperature in a uniform and constant magnetic field of arbitrary strength. (Such a gas is now referred to as a magnetized Fermi gas.) In the present paper, we have studied the properties of the equation of state and have obtained simplified expressions for the normal stresses and the energy and particle densities. These expressions can be used in astrophysical applications. We have found that under suitable conditions of degeneracy (the temperature approaching 0), a magnetized Fermi gas behaves exactly as a one-dimensional gas, which has been studied extensively as a theoretical problem. We have also obtained expressions for the grand partition function.

I. INTRODUCTION

IN a previous paper/ we calculated the equation of state of a noninteracting Fermi gas in a constant

and uniform magnetic field of arbitrary strengths and at arbitrary gas temperatures. This kind of gas will be referred to hereafter as a magnetized noninteracting Fermi gas or a magnetized Fermi gas. The temperature and the chemical potential are well-defined quantities as well as being isotropic, as expected from thermo­dynamic considerations. We obtained the macroscopic energy-momentum tensor of a magnetized Fermi gas as a function of the chemical potential # and the temperature T. Detailed calculations of the equation of state have been discussed in Paper I. In this paper, we shall deal mainly with the thermodynamic properties of a magnetized Fermi gas.

One of the most fundamental properties of a magnet­ized Fermi gas is exhibited in the anisotropy of the normal stresses. (The normal stresses become the pressure in the case of an isotropic medium.) In our case, the normal stresses (the diagonal spatial element of the energy-momentum tensor) are different in the directions parallel and perpendicular to the magnetic field. This anisotropy is directly associated with the quantization of energy levels by the presence of a magnetic field. Classically, the electron orbits are helixes or circles with axes parallel to the field. The motions in the plane perpendicular to the field are circles of constant angular velocity, and can be de­composed into the motions of two correlated simple harmonic oscillators. Let the direction of the field be taken in the z direction. When these simple harmonic oscillators are quantized, the energy levels are2

r /pz\2~ H 1 1 / 2

E(pM,n%r) = ±m<Al+(—1+2—(n'+r-l) \ , (1) L \mc/ Hc J

* On leave of absence from Department of Physics, C.I.E.A., Ap. Post. 14740, Mexico 14, D.F.

fAlso at Department of Physics and Department of Earth and Space Sciences, State University of New York, Stony Brook, N. Y.

1 V. Canuto and H. Y. Chiu, preceding paper, Phys. Rev. 173, 1210 (1968), hereafter referred to as Paper I.

2 M. H. Johnson and B. A. Lippmann, Phys. Rev. 76, 828 (1949); H. Robl, Acta Phys. Austriaca 6, 105 (1952).

173

where pz is the momentum in the z direction, H is the magnetic field strength, m is the electron mass, c is the velocity of light, and He= mV/W*=4.414X1013 G. The + and — signs refer to electrons and positrons, respectively, r = l , 2 and n'—0, 1, 2, • • •. r and nf are two quantum numbers characterizing the spin and orbits of the electron. The energy levels are thus strongly quantized when pjrnc is small compared to 2H/HC, In the limit of large n', the correspondence between rtf and the x and y linear momentum is

\mc/ \mc/

H = 2—(n '+r-l).

He (2)

We have shown that in this limit (large nf) the gas becomes an ordinary Fermi gas. In other words, the parameter that characterizes a classical noninteracting Fermi gas is

{ = -(mc)2'

H 2—

Hc

The magnetic properties of a gas are quantized if £< 1, and classical if £2>1. This condition gives the fol­lowing criteria for a magnetized Fermi gas (see Paper I) :

k T/mc2 < 2H/HC, [ (nonrelativistic, nondegenerate) r«5.9X109 OK]

(kT/mc2)2<2H/HC, [(relativistic, nondegenerate) r»5.9X109 OK]

(3) f (eF/mc2) « (p/W)W<2H/Hc, (nonrelativis tic, degenerate)

l(ev/nu?)*** (P /107)2 / 3<2#/#c , (relativistic, degenerate)

where eF is the Fermi energy [_€^—mc2{^— 1) in the limit $—>0~|. The temperature encountered during and before gravitational collapse is in the range T9

= 1->100 [ r 9 - r / 1 0 9 O K ] . Hence the field strength of interest is of the order of 1013 G and beyond. Such a strong field may be present during gravitational collapse or in gravitationally collapsed objects (e.g., 1220

173 T H E R M O D Y N A M I C P R O P E R T I E S OF M A G N E T I Z E D F E R M I G A S 1221

the collapsed star in the center of Crab Nebula). Figure 1 shows the regions of a magnetized gas.

The nonrelativistic properties of a magnetized Fermi gas have been extensively discussed in conjunction with solid-state physics. The most important properties of a nonrelativistic magnetized Fermi gas are associated with the Landau diamagnetism and the Pauli para­magnetism. In addition to these nonrelativistic proper­ties, in the relativistic case we are also interested in the thermodynamic behavior (the relation between temper­ature and energy density, etc.). In the following, we shall study various thermodynamic properties of a magnetized Fermi gas.

II. EQUATIONS OF STATE OF A MAGNETIZED FERMI GAS

The equations of state of a magnetized Fermi gas, as given by Eqs. (88)-(91) of Paper I, can be rewritten in a more convenient form if we first perform the sum­mation over the index r, and then shift the summation index from n to n+1, and the integration on x (from — oo to +oo) to a new one (from 0 to <»). This last step gives only a factor of 2. The normalization volume can be taken equal to 1. The result is

p = p = — ( J- xx •*• yy 1

1/H\2mc2 » /•«

w\H,

\2OTC2 » / • * dx

• ) — ! > / F(x,n) , (4) / X„3 n-i Jo E(x,n,H)

_,__ v..j*rl f P«=-i — \-A- J F(x,0)

X (MX

E(xfi,H)

oo r°° x2dx "1 + E / F(*,n)- , (5)

» - i / o E(x,n,H)J

U=—(—)— - / F(xfi)E(x,0,H)dx T\HJXC

SL2JO

+ E / F{x,n)E(x,n,H)dx , (6) «-i Jo J

l / # \ l r l r 9 1 = - ( — ) — - / F(x,0)dx

TT\HJXA-2 JO

+ E [ F(x,n)dx~\, (7) «-iJo J

/ E(x,n,H)-n\ ;,») = ( ! + e x p J

where £(*>» l£D-/i\-1

1+exp ) (8)

and

E(x,n,H) = ll+xi+2(H/Hc)nJii> » = 0 , 1 , 2, • • •

x=pz/mc, <^>=ifer/mc2=r/5.903X109OK) (9)

X„= h/mc=3.86XlO-n cm.

Here U is the energy density and 91 is the particle energy density, and ix is the chemical potential plus the rest

11

10

9

8

7

_ 6

z 5

©• A

o 4

3

2

1

n

1-

-

- F

M F

_ F

M F „

— i i r r 1 1 •-• i T•"-•" i

F log(H/Hc)«0 MF

F log (H/Hc)»-2 MF

log (H/Hc)--0

s s y

/ s

lo<f(H/Hc)»-6 * s

M F

I

F

. ,.,! .„ 1 ,1.

M F

F

1 1 1

•

F

1 t

1—

M F

1

— 1 ' " I " I \

F J

1 I I - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10

log ip/fiB) (g/cm3)

FIG. 1. Approximate regions of a magnetized Fermi gas for several field strengths, as indicated by the solid lines, marked by '*MF." #C = 4.414X1013 G. The logarithms are to the base 10.

energy (in units of mc2).8 The macroscopic energy-momentum tensor TM„ is related to Pxx, Pyy, and Pez

as follows:

T =

* XX

0 0

I 0

0 Pyy

0 0

0 0

•L ZZ

0

o i 0 0

Tu,

(10)

Equation (10) is valid only in the frame of reference comoving with the gas. The energy-momentum tensor components are related to each other and to the temperature T and density 91 through Eqs. (5)-(8). The equations of state are therefore expressed as infinite series which generally cannot be approximated by an integration over n. When we approximate the sum over n by an integral over dn, we obtain the expression of the equations of state of classical non-magnetized Fermi gas, as we have shown in Paper I.

Introducing the following transformation in the integrands:

x / H X1'2

v=-—, an=ll+2—n) , an \ Hc /

(11) / H Y'2

f l + r H - 2 — n ) =an(l+v2)w, \ Hc /

the integrals in Eqs. (5)-(8) can be expressed in terms of the following functions:

• , M ) ^ [

Jo

Jo

dv

(12) ( l + W l + e x p K C l + e * ) 1 " - , , ) / ^ }

v2dv

(l + j2)l/2{1 + e x p r - ( ( 1 + 2)2)l/2_M)/(^-J}

- C . f a / O - C f a / O , (13) 8 H. Y. Chiu, Stellar Physics (Blaisdell Publishing Co.. Waltham,

Mass., 1968), Vol. I, Chap. III.

1222 V . C A N U T O A N D H . Y . C H I U 173

TABLE I. Values of C& (<£,#), listed in the order [Ci(<£>/*), C2(<£,£*), Cs(<f>}^), C4(<£,M)D, for a number of values of <£ and p,. <f>^kT/mc2=T/(5.903XIQ9 °K), and fx is the (chemical potential+rest mass)/W2.

1.0

1.5

2.0

3.0

4.0

5.0

6.0

7.0

8.0

10.0

Cdfal*)^

£ 4 ( 0 , ^

0.1

0.23617 0.03157 0.26774 0.25074 0.94123 0.38003 1.32126 1.10315 1.31029 1.09265 2.40294 1.72862 1.76054 3.37872 5.13926 2.82768 2.06230 6.73124 8.79354 3.87270 2.29173 11.11802 13.40975 4.89884 2.47741 16.52598 19,00339 5.91600 2.63357

22.94837 25.58194 6.92815 2.76840

30.28127 22.14966 7.93772 2.99306

48.26929 51.26235 9.94986

0.2

0.32946 0.09265 0.42210 0.36927 0.89063 0.44944 1.34007 1.07314 1.28645 1.15112 2.43756 1.71548 1.75353 3.43123 5.18477 2.82521 2.05880 6.78225 8.84105 3.87180 2.28960 11.16840 13.45800 4.89841 2.47597 16.57603 19.05200 5.91576 2.63252

22.99824 25.63076 6.92800 2.76760

30.43101 33.19861 7.93712 2.99255

48.31889 51.31144 9.94981

f°° (l+l?)lIWv

/o l+expKCl+^^-j

f L l+exp[«

dv

l+tfli/*-,

0.5

0.50279 0.40242 0.90520 0.64714 0.84340 0.84332 1.68672 1.13567 1.18230 1.54949 2.73179 1.69364 1.69453 3.80365 5.49818 2.80509 2.02978 7.14127 9.17105 3.86351 2.27302 11.52180 13.79482 4.89469 2.46518 16.92673 19.39190 5.91381 2.62488

23.34743 25.97231 6.92685 2.76187

30.77927 33.54114 7.93638 2.98897

48.66610 51.65507 9.94944

•*)/*] '

*)/*]•

0.8

0.61738 0.88018 1.49756 0.89004 0.87085 1.42454 2.29538 1.30050 1.13651 2.19135 3.32786 1.77487 1.61250 4.47762 6.09012 2.80677 1.97074 7.80872 9.77946 3.85378 2.23587 12.18046 14.41633 4.88737 2.44134 17.57959 20.02093 5.90937 2.60865

23.99682 26.60547 6.92419 2.75014

31.42655 34.17668 7.93471 2.98194

49.31111 52.29305 9.94868

(14)

(15)

1.0

0.67824 1.28702 1.96525 1.04446 0.89778 1.90207 2.79985 1.42638 1.12924 2.72269 3.85194 1.86393 1.56765 5.06590 6.63355 2.83736 1.92502 8.41276 10.33777 3.89178 2.20100 12.78546 14.98646 4.88562 2.41685 18.18209 20.59893 5.90659 2.59154

24.59674 27.18828 6.92190 2.73787

32.02451 34.76238 7.93307 2.97489

49.90674 52.88163 9.94788

r

2.0

0.89346 4.34605 5.23951 1.78164 1.03252 5.31564 6.34816 2.10203 1.17749 6.45322 7.63071 2.45321 1.47335 9.28964 10.76299 3.23601 1.75942

12.95316 14.71258 4.10259 2.02100 17.51365 19.53465 5.02584 2.25176

23.01654 25.26830 5.98442 2.45156

29.48934 31.94090 6.96375 2.62355

36.94787 39.57142 7.95471 2.90128

54.85283 57.75410 9.95240

5.0

1.23319 23.53205 24.76525 3.90504 1.30710

25.55397 26.86107 4.18472 1.38242

27.71887 29.10129 4.47694 1.53627

32.50237 34.03864 5.09811 1.69256

37.93076 39.62332 5.76626 1.84914

44.04953 45.89867 6.47832 2.00400

50.90085 52.90485 7.23072 2.15536

58.52311 60.67847 8.01953 2.30177

66.95069 69.25247 8.84070 2.57563

86.33941 88.91504 10.56443

E(x}n,H)dx

Jo l+exp£(E(x,n

f— dx

Jo l+exp|[(£(ff,».

fi)-v)/4>l

fl)-M*l

8.0

1.42745 57.59191 59.01936 6.00065 1.48009

60.65912 62.13922 6.26952 1.53336

63.86273 65.39610 6.54622 1.64147

70.69472 72.33619 7.12289 1.75117

78.11869 79.86986 7.73008 1.86182

86.16480 88.02661 8.36700 1.97281

94.86227 96.83508 9.03264 2.08353

104.23926 106.32279 9.72583 2.19341

114.32269 116.51609 10.44528 2.40858

136.70959 139.11817 11.95720

= an2Cd-,

= anCA —, -\an 1

10.0

1.52348 88.53946 90.06294 7.39328 1.56813

92.30214 93.87026 7.65846 1.61319

96.19897 97.81216 7.92991 1.70441

104.40760 106.11202 8.49148 1.79681

113.19015 114.98696 9.07769 1.89003

122.57105 124.46108 9.68811 1.98373

132.57426 134.55800 10.32222 2.07758

143.22318 145.30075 10.97939 2.17124

154.54049 156.71173 11.65891 2.35674

179.26736 181.62410 13.08176

-)• (19) In'

We find that the use of the CV functions gives

dx

I. P ^P = _ / \ J- xx •* yy I I

The equations of state can now be written as

1 /H\2mc2 ^

ITKHJ *„•

1 /H\mc2[

• 00 /<f> fM\

(20)

(\6) p»=^ — f^&*(*#)+£<^cj(-,-X\, (21) (16) **\HJWL —1 \an ajl

E{x,n,H){l+tvpi{.E(x,n,H)-ti/<t>-]}

= C 1 ( - - ) )

\an an/

J. Ei^HHl+e^Ei^H)-,)/^} U=^\TMiCs(<l"fl)+ £a"C\7:7J1' (22)

= an*cJ- - ) , (17) %=-(—)—hCi($,n)+ £ a»c/-, -)]. (23) \an aj ir2\Hc/Xc

sL »-i \an a J A

173 T H E R M O D Y N A M I C P R O P E R T I E S O F M A G N E T I Z E D F E R M I G A S 1223

oo /mix\ fm\ C I ( * , A O = E exp( — k 0 ( - ) .

The Ch functions are expectation values of quantities Therefore, relevant to a one-dimensional gas. Ci(0,ju) is the expectation value of (E')~l, where E'= (1+v2)112 is _ the energy of a one-dimensional particle; C%($,ix) is ™ $ $ the expectation value of wfo/dE, and is the pressure T h i s s e r i e s converges for all values if M < 1 . exerted by a one-dimensional particle; and C3 and C4 Similarly, using the transformation (26), we have are the expectation values of energy E' and particle number, respectively. We can therefore say that a Cz(<j>,y) magnetized Fermi gas has properties similar to a combination of one-dimensional gases. Later we shall show that in one limiting case—degenerate, high field, and low density—a magnetized gas does behave as a one-dimensional gas.

The equations of state thus can be written as a sum over the same function with different arguments. Table I lists Ck(<t>,y) for a number of values of fx and <j>. (Note that in the above sum the ratio of the two arguments of the Ck functions is a constant.)

(28)

-I. {\+&yiHv

l+expC((l+* ,)v ,-/»)/*]

(mix\ r™ (

= Eexp[ /mix\ r°° / m(l+v2)^2\ 1 * ' (1+v8)1'2 expf 1 \dv

m cosh0\ )cosh20d0. (29)

III. PROPERTIES OF THE Ck FUNCTIONS

A. y<l In the case / I<1> we can expand the denominator in

the definition of Ck [Eqs. (12)-(15)] into a geometrical series and we can express Ck(<t>,y<) as a sum over the McDonald functions Kn(z), which are defined by4

Kn(z)-

Wefind

CI(0,M)

Jo

-f— Jo ( l + i

coshnd exp (— z cosh0) d$. (24)

dv

(l+^)i/2 i+exp[((l+z)2)1/2-M)/0]

/•°° 1 °o /mjj\ - / S exp(—) Jo (l+i>2)1/2 —i \ < £ /

Xexpf—

oo /m/i\ r°° 1 = £ e x p — /

( •

0 /

Using the relation cosh20=|(cosh20+l), and the definition of Kn(z) [Eq. (24)], C3(<£,ju) becomes

oo /mjjL\r /m\ /m\n C«(*,M) = I E exp(^— | J T J - J + i U - J J . (30)

We also have

Ci (0,/*) = Cs (0,/*)—Ci (0,/x)

00 /mfjL\f fm\ lm\-\

C4(0,/*)=Eexp — k i - J. (32)

Xexpl

^(l+z;2)1

,2)1/2

7w(l + fl2)1/2\

dz;

The following recurring formulas for Kn(z) may be useful4:

Kn-i(z)+K n+i(z) = - 2KV(s), (33)

Kn+iW-Kn-^^ (2n/z)Kn(z). (34)

The behavior of Kn(z) at large and small values of z are

The transformation

)dv. (25) <t> /

fl=sinh0 (26)

reduces the integral in the last part of Eq. (25) to the McDonald function K0(m/<l>):

r00 / m(\+v2yi2\ dv

Jo \ <t> /(l+v2)1'2

K-HTP{I+ 4w2-l

(4rc2-l)(4^2-32)

2(8z)2 • ] • (35)

*->0: iT„(z) = J (w- l ) ! / ( iz ) n , K0(z)->lnz. (36)

For <£<3C1 we have, therefore,

» /^\i/2 / (1-MW C I ( ^ ) = ( 4 T ) V » E ( - ) exp( ) , (37) r00 / m \ fm\

= expf cosh6)dd=Ko[—). (27) JO \ <l> / V 0 / ^ /0V3/2 /

to ̂ e Sfcaty of Stellar C 2 (<£,/*) = ( i k ) 1 / 2 E ( — ) expf -York, 1957), Chap. X. m=i\mJ \

4 S. Chandrasekhar, An Introduction Structure (Dover Publications, Inc., New

0

(1—n)m

) • (38)

1224 V. CANUTO AND H. Y. CHIU 173

M

1 1.25 1.5 1.75 2.00 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0

10.0

C8(0,M) =

CM 0 0.69315 0.96242 1.15881 1.31696 1.56680 1.76275 1.92485 2.06344 2.29243 2.47789 2.63392 2.76866 2.99322

= (wi/,f;(

TABLE II. CM.

CM 0 0.12218 0.35731 0.67722 1.07357 2.08071 3.36127 4.90726 6.71425

11.10123 16.50930 22.93175 30.36469 48.25276

- e xp -(i

CM 0 0.81532 1.31974 1.83603 2.39053 3.64509 5.12401 6.83210 8.77769

13.39366 18.98718 25.56567 33.13335 51.24598

— y)m\

Ci(jx)

0 0.75000 1.11803 1.43614 1.73205 2.29129 2.82843 3.35410 3.87298 4.89898 5.91608 6.92820 7.93725 9.94987

we find that

CiG0= / Jo

- ( M 2 - ! ) 1 / 2

C 3(M)= /

Jo - ( M 2_l) l /2

C*(M)= /

Jo C 2 ( M ) = C3(M)—C]

when /-£>>!, we fir

dv = ln[ ju+(M2-l)1 '2], (46)

(l + z,2)l/2

(1 + ^2)1/2^= 1 l n [ M + ( M 2 - 1)1/2]

+§ M (M 2 - I ) 1 / 2 ,

p=a*a—i) 1/2

(47)

(48)

- 1 ) 1 / 2

•Aln|>+(M2-l)1/2]; (49)

N«f> 0

= Ci(0,/x).

C4(^,/i) = C'i(0,/l).

(39)

(40)

If (1-—/x)/#2>l, we only need to take the first term, and thereby have

C i C ^ - ^ ^ ^ V ^ e x p i

C f o / O - ^ i r W e x p .

C3 (<A,M) = C4 (<£,/x) = Ci (0,/*).

(41)

(42)

(43)

There is no convenient reduction in the general case /x>l.

B. The Limit («i-l)2>(*

This case occurs when the gas becomes degenerate. (The consequence of degeneracy is slightly different from the usual one, as will be discussed later.) In this case, the Fermi distribution function can be replaced by a step function

1+exp-(1+w2)1'2-/*

<t> T1

J = 1 , *<(M2 •1) 1/2

= 0, v>(n2-iyi* (44)

as is usually done for an ordinary Fermi gas. This is equivalent to replacing the upper limit of integration in Eqs. (12)-(15) by (M 2 -1) I / 2 . Then Ck(4>,fi) becomes a function of ju only. Let

(45)

Ci(/i) = ln2/x,

C3(/x) = C2(M) = iM2J

c40i)=/x;

when ^—1<<C1, we find (M= l + £)

CiOi) = ( 2 ? + a i / 2 D - ^ + ( 2 / 1 5 ) f ] ,

C 2 ( M ) - ^ K 2 ? + ^ 2 C f - ( l / 1 5 ) a ,

C3(M) -+ ( 2 { + r ) ^ C l + i f + ( l / 1 5 ) ? ] ,

C 4 ( M ) - * ( 2 H - £ 2 ) 1 / 2 .

Table II lists C&Ou) as a function of /*.

(50)

(51)

(52)

(53)

(54)

(55)

(56)

IV. THERMODYNAMIC PROPERTIES OF A NONDEGENERATE MAGNETIC GAS

As we have shown in Paper I, if the sum in the equa­tions of state (20)-(23) is contributed mainly by terms with large values of n, magnetized Fermi gas approaches a classical Fermi gas. Therefore, we only need to con­sider cases with small values of n. As an example, let us consider the case of the lowest term in n, namely, n=l. In this limit we have

1 (H\ J~ i . -I—Cil ( - • - ) •

1/H\mc2r /<j> M \ 1

""bMic*(**)+"H^)\-(*-)]• \ai a\> J

1 / H\mc2r =-(—)—UC3(^M)+«I2C3,

91=-f—V-("iC4(^)+«ic/- -)~| .

(57)

(58)

(59)

(60)

The Ck functions are decreasing functions of #. If we consider only terms small in n, we must have

l<<Cai<<Ca2<^- • - « a n , (61)

173 T H E R M O D Y N A M I C P R O P E R T I E S OF M A G N E T I Z E D F E R M I G A S 1225

direction is given by5 Therefore, we can also drop Ck(<t>/ai, M A I ) with respect to Ck(<l>,ii) in Eqs. (58)-(60). In the nondegenerate approximation, the factor 1 in the denominator in the Fermi distribution function may be neglected. This is equivalent to taking the first term in the series for Ck(4>,n) [Eqs. (28) and (30)-(32)]. We have

1 / H \2mc2 / /A /aA

'•"•"iirJwW? (62)

OH;)-1 / # y w c 2

2 7 r 2 \ # c A c Pzz=—(—)—exp

1 /H\mc*

9 1 = — ( — ) — e x p ( - V / - ) .

We find, from Eqs. (63) and (65), that

P^=9lwc2<£=3l£2\

(63)

(64)

(65)

(66)

This means that Boyle's law is valid along the z axis, i.e., the gas behaves as a normal gas along the % axis.

The anisotropy factor T may be defined as

Pxx 2H KB(ai/<j>)

'P„~*Ht K^lfr) (67)

From Eqs. (35) and (36) we find that, when <£»1 and

2H/T <£\1/2

Hc\2 aj (T)

-@®<T>- (68)

(dP„\» (

\du / \ : 2 + (1/202) ln0, )'"' (71)

which approaches the velocity of light as <t> —* <*>. In a nonmagnetized Fermi gas the limiting velocity of sound is c/V3 only.

For the nonrelativistic case $<3C1, and from Eqs. (35) and (36) we find that (<£«1)

2 ^ ( 1 / 0 )

U iT 2 ( l /^ )+ i^o( l /0 ) -><£E

kT

mcz (72)

However, U also includes the rest energy nic2, so that in the limit kT/tnc2y>\ Eq. (72) is identical with Eq. (66). We are really interested in the ratio of Pzz/iU—mc2), which is the thermodynamic energy. We find

1 /H\mc2r / 1 \ U- dlmc2=— — ] — p : 2 - 1

27r2\#c/Xc3L \<j>/

+iK0-K(i>Md (73)

1 /Hxmc*^ / x N1'2 / M ~ l \

"2VKH0)\/ \2/J C \ <i> ) ' which gives

P z , / ( £ / - 9 l w c 2 ) - > 2 . (74)

Equation (74) again differs from the classical result that P/(U-dlmc2) -> f by a factor of 3. The velocity of sound is accordingly greater by a factor of vX

The velocity of sound in the direction perpendicular to the field is

1/2 fd{rPzzW'2 /dPzz\

\ dU J \dU/ (75)

[The case <j£>>ai corresponds to a classical Fermi gas. See Eq. (3).] For the case $<<C1 (nonrelativistic case), we have

The ratio U/dl or (U—'Jltnc2)/'3l in the nonrelativistic case is

H

Hc (fry/ai

2 r - ( a i - l ) -exp

r - l a i -3

L <j>

(69)

U K2(l/<t>)+Ko(l/<i>) —= \mc2

31 KxilU) >nic2<t>=kT, (76)

The relation between Pzz and U is worked out as follows: For the relativistic case #2>1, we find, from Eqs. (35) and (36),

and using Eq. (73), we find

(U- 9lwc2)/9l= ^mc2<j)= \kT, (77)

2 ^ ( 1 / * ) 202

- > 1 -ln0

(70) U i £ 2 ( l / 0 ) + Z o ( l / 0 ) 202+ln<£ 2<p

but Pez/U< 1 if <j)< oo. The velocity of sound in the z

which are to be compared with the corresponding value of a nonmagnetized gas of 3kT (0—> <*>) and \kT

6 The sound velocity is not the Alfven velocity. The Alfven velocity is the velocity of transverse electromagnetic waves. To compute the Alfven velocity one needs to know the dielectric constant.

1226 V . C A N U T O A N D H . Y . C H I U 173

FIG. 2. The dependence of U, Pxx, Pzz, U—Nmc2, and N as functions of /* in the degenerate limit of the case H/Hc— 1.

(</>—> 0), respectively.6 We can thus conclude that only one out of the three degrees of freedom is excited in a strongly magnetized gas.

When terms other than n=0 are considered in Pzz, U, 91, and PXXf respectively, the anisotropy decreases and eventually U/Vl-*3kT, or (U-Vlnic2)/3l->%kT in the limit of n —> <*>. The speed of sound also decreases. The gas eventually becomes a three-dimensional gas.

In reality, when <j> is not zero, states other than the lowest one are always excited to some extent. Hence considerations made here are only of limited interest. The general case can be studied by resorting to more complicated formulas (20)-(23) and applying numerical methods.

V. DEGENERATE MAGNETIZED FERMI GAS. (V-l)/#»l

In this limit the equations of state become

1 / H \2mc2 3 / fx \

P**= i V = - ( — ) — £ nCil - ,

fic2(/i)+i; ^2c/-)l L n=i \an/J

1 / H \mc2\ Pzz=-[ — J—\iC2(fx)+E

<jr2\Hc/Xcs

(78)

, (79)

1 / H \mc2\

7r2\Hchc"\

UctW+ZanvI-X], (80) L n-i \ a„ / J

[iC40«)+Ea-c/-)"|. (81)

It can very easily be shown that if y./an< 1,

l imC»(^ ) -»0 , * = l , 2 , 3 , 4 .

Thus the sum over n does not extend to infinity but terminates at s such that

0s<M<<M-i> (82)

where s is an integer. 91, U, Pxz, and Pzz have now discontinuous deriva­

tives with respect to ju. The curves of 91, U, Pxx, and Pzz versus JJL and 91 will contain kinks, as is shown in Figs. 2 and 3. However, these kinks disappear at large values of s and also when one regards U as a function of 91, etc. These kinks signify the fact that the energy states are one by one excited as ju increases.

These kinks, maxima, and minima in the thermo­dynamic variables can be understood in terms of the

i.o

0.6

0.5

u

-•• p; U

for a Fermi Gos

for a Fermi Gos

for a Fermi Gas

6 L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962), 2nd ed., pp. 87-99.

FIG. 3. Functional dependence of PXx/Pzz, Pzz/N, Pxx/N, and Pzz/N or N for the degenerate case with H/Hc = i. The corre­sponding functions for a Fermi gas are also shown for comparison.

173 T H E R M O D Y N A M I C P R O P E R T I E S O F M A G N E T I Z E D F E R M I G A S 1227

behavior of the density of states rj(E). rj(E) is given by

dN(E) dtofa) ij(£) = = = 3 1 ' ( M ) ,

dE dfi

where N(E) is the number of states with energy below energy E. N(E) coincides with $l(jit) if we set E=ix. Figure 4 shows rj (E) as a function of E. y\ (E) shows sharp peaks as each magnetic state is excited.7

As before, we now consider the case s=Q. This case represents a physical example of a one-dimensional gas which has been discussed extensively in the literature.8

-t xx •* yy u j

1 /H\mc2

P„=—( —V—CiGO,

1 /H\mc2

u=— — — c 3 ( M ) , 2IAHJ\C

3

(83)

(84)

(85)

1 H mc2

Pzz= n^U^n2

4x2 Ec Xc3

1 H 1 91= fi^fi,

2,r2 HC *„3

FIG. 4. The density of states 11(E) as a function of E for the nonrelativistic case (from Reference 7).

At the high-density limit 0*—l)» l , we then have

(90)

(91)

91 1 /H\l

2TTA HJX*

P„cc3l*. (92)

(86)

In this case there is no lateral stress. A gas with no lateral stress is unstable against collapse. However, a small amount of pressure will always be present due to the finiteness of the temperature. Since

Other properties of this gas resemble a noninteracting one-dimensional Fermi gas. All relevant quantities can be calculated from the general equation of state. For example, the specific heat of a degenerate magnetized Fermi gas can be calculated, using the following formula4:

C i ( ^ ) - > ^ o ( l / * ) e x p O * / * ) ,

KQ(1/4>) -> W V " e x p ( - 0 ) (<*>-» 0 ) ,

we find the residual lateral pressure to be

H \2mc2/ <(> s1'2

Hc) xA(l+2H/Hc)vV

x e n : — ) •

(87) r00 [_d<p{u)/du}du

J0 e(M" t to)+l

where

P =P ~ L - ( -

= <p(u0)+2[_C2<p"(uo)

+Ci<pW(uo)+••••], (93)

C = E • (94)

(88)

This residual pressure can, under suitable conditions, prevent collapse in the direction perpendicular to the field.

Since C 2(M), CZ(V)> CAQX) are proportional to the

pressure, energy, and densities of a one-dimensional gas, we conclude that a degenerate electron gas in an intense magnetic field can behave almost exactly as a one-dimensional gas. The critical density corresponding to the one-dimensional behavior is such that

^{l+2H/HcyiK (89)

At H/Hc— 1, Eq. (89) gives a critical density of the order of 106 g/cm3 for a composition of helium.

7 D. C. Mattis, The Theory of Magnetism (Harper and Row, New York, 1965).

8 E. H. Lieb and D. C. Mattis, Mathematical Physics of One Dimension (Academic Press Inc., New York, 1966).

Equation (93) is accurate to the order e~uo.

VI. PAIR-CREATION EQUILIBRIUM

One might think that since classically the spin energy of an electron in a magnetic field is of the order of —HBH, where JJLB=eh/2mc, when H>2HC then the spin energy will be greater than 2mc2; it might be possible to create an electron pair with suitable spin directions at the expense of the field energy. However, because of the form of energy states [Eq. (1)], the separation between positive and negative energy states is always at least 2mc2. This means that pair creation will take place, not at the expense of the field energy, but at the expense of other forms of energy (e.g., thermodynamic energy). This conclusion is valid only for the case of a uniform and constant magnetic field. The question of whether pair creation will take place in a nonuniform or time-dependent magnetic field can only be answered by a study of the quantum origin of magnetic fields.

1228 V. C A N U TO A N D H. Y . C H I U 173

The chemical reaction of pair creation is

y <=>€-+e+9 (95)

where y stands for one or more photons. Equation (95) leads to the equation

fxy=lx-.+fi++2mc2f (96)

where ju7, /x_, and /x+ refer to the chemical potential of the photon, electron, and positrons, respectively. (Note that }j, — jjL-.-\-mc2 is the chemical potential for the electron used in previous discussions in this paper.) Since fiy=0, we find that

ju_-f tnc2=n= — (M++WC2) . (97)

The charge-conservation law requires that

dl„-dl+=dl, (98)

where 9l_ (91+) refers to the electron (positron) number densities and 31 is the number density of excessive electrons. Equations (97) and (98) are identical to those equations for pair creation in the absence of a magnetic field. Equations (97) and (98) together with Eqs. (20)-(23) can be solved to give 3Z_ and 9l+ as functions of 31 and T.

When the anomalous magnetic moment is taken into account, the lowest-energy eigenvalue of Dirac equa­tion for an electron in a magnetic field vanishes for values of H equal to 4nrorlHe, and therefore pair crea­tion phenomena can take place. The problem is fully discussed in Ref. 9.

VII. GRAND PARTITION FUNCTION

The grand partition function d is given by

ln3= In Tr e x p ( - # f c ) , (99)

where Tr means trace, /?= (kT)~l, and 3C is the Hamil-

(100)

tonian [defined in Eq. (80) of Paper I ]

3C^]L En,r<ir (n)ar(n), n,r

where

En>r=±nic2\ l + ( — + 2 — ( n + r - 1 ) (101) L \mcl Hc J

and arf(n) and ar(n) are the creation operators for the

state n. Equation (99) is easily reduced to (see Paper I for techniques used)

ln3 = ln I I [ l + X e x p ( - £ £ n , r ) ] w » . ' n,r,Pz

= I > n , i l n p + A e x p ( - ; g £ r e i l ) ] n,pz

+ E «..* l n [ l + X exp(-PEn,t)2, (102) n,pz

where cow,r is the degeneracy factor discussed in Paper I :

a>n,r=tt2f3eFI/2<irhc (103)

X=exp(ftu). (104) and

9 H. Y. Chiu, V. Canuto, and L. Fassio-Canuto, Phys. Rev. (to be published).

VIII. SUMMARY

We have calculated the thermodynamic properties of a magnetized Fermi gas (a Fermi gas of arbitrary temperature in a magnetic field of arbitrary strength) and have obtained convenient expressions for the energy and normal stresses. We have found that in the limit of low quantum numbers a magnetized gas behaves as a one-dimensional gas. In the limit of zero temperature, the gas behaves exactly as a one-dimen­sional gas for certain density ranges.

We have also considered the pair-creation phenomena and calculated the grand partition function for a magnetized gas.

ACKNOWLEDGMENTS

One of us (V.C.) acknowledges the support of an NAS-NRC resident research associateship.