P H Y S I C A L R E V I E W V O L U M E 1 7 3 , N U M B E R 5 25 S E P T E M B E R 1 9 6 8

Magnetic Moment of a Magnetized Fermi Gas VITTORIO CANUTO* AND HONG-YEE Cmuf

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

(Received 26 April 1968)

In this paper we have calculated the exact expression for the magnetic moment of a Fermi gas in a mag­netic field (magnetized Fermi gas). The susceptibility obtained from our result becomes the classical Curie-Langevin law in the weak-field limit for a nondegenerate and nonrelativistic gas. The induced magnetic moment 9H was found to be proportional to the square of the electron density, but at higher densities 3E passes through a maximum and eventually becomes negative; hence a magnetized gas is first paramagnetic but later becomes diamagnetic. We have also shown that spontaneous magnetization cannot take place in a noninteracting electron gas.

I. INTRODUCTION

IN two previous papers,1'2 we derived a general ex­pression for the equation of state for a system of non-

interacting Fermi particles in a magnetic field (a mag­netized Fermi gas), and studied the thermodynamic properties of such a gas. We found that in the limit of large quantum numbers, a magnetized Fermi gas re­duces to an ordinary Fermi gas, whose properties have been described elsewhere.3*4 In the case of small quan­tum numbers, the properties of a magnetized gas are quite different from a Fermi gas. The lowest magnetized state is characterized by the alignment of all electron spins with the field, and in this limit a magnetized gas behaves exactly as a one-dimensional gas, which has been studied extensively elsewhere as a theoretical problem.5 As the density or temperature increases, higher magnetized states are excited. The gas behaves more like an ordinary Fermi gas and eventually ap­proaches it in the classical limit. In previous papers,1,2

criteria were given for a gas to be considered as a mag­netized gas, and will not be discussed here.

In this paper we are interested in the magnetic prop­erties of a magnetized Fermi gas.6 The magnetic prop­erties of a system are intimately related to the macro­scopic magnetic moment of the system, which we will calculate in the sections that follow. We are also inter­ested in an important question: Can magnetization

arise spontaneously in an electron gas? Answers to this question will be found through a study of the total mag­netic moment of the system.

II. GRAND PARTITION FUNCTION

The grand partition function % is defined by7

ln&=In Tr exp[&-£(-jatf)] , (1)

where /3= (kT)*1, 3C is the Hamiltonian operator for the system, p is the chemical potential plus the rest energy in cgs units, and N is the particle occupation-number operator. From Eq. (1) we have derived2 the following expression for d:

n,pz

+ln[ l+exp( -£ (E 2 B - /0 ) ]} ) (2)

where o>n is the statistical weight7 for states of quantum number n for a system with volume 0:

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

f Also 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, second preceding paper, Phys. Rev. 172, 1210 (1968), hereafter referred to as Paper I.

2 V. Canuto and H. Y. Chiu, first preceding paper, Phys. Rev. 172, 1220 (1968), hereafter referred to as Paper I I .

3 E. Schatzman, White Dwarfs (North-Holland Publishing Co., Amsterdam, 1958), Chap. 4.

4 R . Kubo, Statistical Mechanics (North-Holland Publishing Co., Amsterdam, 1965).

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

6 The magnetic properties of the electron in solids and in rarefied gases have been so extensively studied that a comprehensive list is impractical. See, for example, D. C. Mattis, The Theory of Magnetism (Harper and Row, New York, 1965); R. E. Peierls, Quantum Theory of Solids (Oxford University Press, New York, 1955).

173

and

Wn~®2l3eH/2Thc

H Enr^±mA 1+1 — ) +2—(»+r-l) ,

L \mcf H0 J

(3)

(4)

r = l , 2, »=0, 1,2, ••*, oo.

pz is the % component of the momentum of the electron, H is the magnetic field which is in the z direction, Enr

is the quantized energy of the electron in the magnetic field, and

£rc=M2c3M=4.414X1013 G. (5)

The symbols e, ft, m, c} and k all have their usual meaning.

Carrying out the sum over pz in the usual mariner, we have the following expression for ln£ (see Paper II for

7 K. Huang, Statistical Mechanics (John Wiley & Sons. Inc., New York, 1963), Chap. 11, pp. 237-243.

1229

1230 V . C A N U T O A N D H . Y . C H I U 173

details): Using Eq. (6), we find that, explicitly, SflZ is given by

^{m[l+exp(-/3(£ l n- ju))] ^ = ~ — U * / l n 1 + < * P ( j kfe

+ m [ l + e x p ( - £ ( £ 2 n - j u ) ) ] } , * co /-00 r / E(x,n,H)-ny\

which can be written as +<t> Z / In l+exp( J \dx

1 eH/1 /-+00

- l n a = — - / dx ln{l+exp[-/9(£(*,0^)- | , )]} ff - /~ Hcn=l J0

+ l£dXln{l-i-exPL-KE(X)n,H)-fl)1}), (6) X [ l + e x p ( - £ ( ^ g ) " > t ) } , > (13)

/ ? = ^ 2 , ix=u/mc\ %=pz/rnc, w h e r e 4>=kT/™?> fxB=eh/2mc is the Bohr magneton, , % (60 and Xc=#/W is the Compton wavelength of the

E ( * , n ^ ) = ( l + ^ + 2 « H / J ? 0 ) ^ . V ' e l e c t r o n <

Equation (13) is the most general expression of the III. MAGNETIC MOMENT magnetic moment of a noninteracting Fermi gas in an

The magnetic moment operator Sfll is defined by8 external magnetic field. One can use the exact wave A __ *>* /ATT n\ function (discussed in Paper II) to evaluate (12) and

VK- — d3C/dH, (7) tQ g h o w t h a t^ a f t e r t a y n g statistical averages, Eq. (13) where 5C is the Hamiltonian for a Dirac particle (see be obtained from Eq. (8) using the Hamiltonian of the Paper I). The statistical average of 9TI then gives the Dirac equation as described in Paper I. total magnetic moment of the system, 9TC: We shall now study the classical properties of 9fTC as

, v * , , , given in Eq. (13). 9Tl=Tr(p)2fll/Trp, (8) 6

where p is the density matrix, in the grand canonical ^ CLASSICAL LIMIT-CURIE-LANGEVIN LAW ensemble In Paper I, the classical limit was defined as the

p= exp[-0(3C-jui\O]• (9) limit in which the sum over n can be replaced by an in-We now have tegration over n. We shall now show that in the clas-

^ >v . sical limit the nonrelativistic and nondegenerate expres-Tr{-(d3C/dH) expi-piW-pN)]} s i o n of E q , ( 1 3 ) g i v e s t h e Curie-Langevin law for the

Tr{exp[-£(5C-/xA0]} ' magnetic susceptibility. The nondegenerate and nonrelativistic case is given

If we keep the chemical potential ft constant with re- by the following condition: spect to the magnetic field,7 then Eq. (10) becomes „,

1 d exp( ) » 1 , E(x,n,H)->l+ix2+n—, (14) 9TC=-—lnTrexpC-^OC-juiV)], \ <t> / Hc

9m:= in&, L ^ <t> /A S dH

/ E(x,n,H)-ix\ where d is the grand partition function defined in Eq. ~-exPl ~ J • (15) (1) and explicitly given in Eq. (6). Equation (11) is a generalization to the nonzero-temperature case of the E ( l u a t l o n (13) becomes Pauli-Feynman theorem,8'9 which states that 2 w r l /u—1\ <» /u—l\

. 2^= -\ -4 expf )+<£ £ exp( J < n | 3 f e | n H - A n | & | n > (12) ^ M V * ' *=1 V * 7

dH / nH\ H oo / ^ X

when \v) is the eigenstate of 3C. e x p \ ^ c / ^ n - i ^ ^ V 0 / 8 C. Kittel, Elementary Statistical Mechanics (John Wiley & _

Sons, Inc., New York, 1958). / n&\l , x 9 W. Pauli, in Handbuch der Physik, edited by St Flugge (Sprin- X expf 1 \I0, (16) ger-Verlag, Berlin, 1958); see also Refs. 7 and 8, ' * \ <j)Hc/ J

173

where

Let

M A G N E T I C M O M E N T OF M A G N E T I Z E D F E R M I G A S

solutely convergent), we have

/ o = / exp{ ]dx.

1231

r00 / x2\

X = e x p [ O * - l ) / 0 ] (17)

^ = - — ( \ J l + e x p ( ) 1 £ e x p f - — )

E 00

2X— X) w expl

and, after rearranging terms in the sum (which is al­lowed because these series are easily seen to be ab- The following results are apparent

/*°° / x2 \ / o= / expf W = ( 2 T T 0 ) 1 / 2 ,

M / nH\ d » / nH\ d r «o / nH\~\ E n expl U-<t>Hc— £ exp \=-^Hc—\ 1 + 2 £ expf )

= _ i ^ f f Z expf — - ) + £ e x p f - — - ) = - ^ # c — l + e x p ( - ) £ expf )

(18)

(19)

(20)

' 2 # # < 3 l+exp( - f f /*£T. )

~dH l-exp(-H/<l>He)

d /(iBH\

-i*ff.—cothl-^J, (21) d# \ &r /

where

We therefore find

911=

»BH/kT=%H/4>Hc

A simplified expression for ln3 can be obtained from (22) Eqs. (6), (14), (15), and (19)-(21). We have

/ 6

I cothij+ij— cothr; ) •

Q X /lirkiy12

ln3 = ( ) •

Hence

9^= (y cotht;),

(23)

7T2XC drj

where t\—HBH/kT. Expanding rj coth?; and retaining terms of the lowest

order, we find

1 1 X /lirkiy12

0 4 T T 3 X C3 W C 2 /

Substituting Eq. (29) into Eq. (26), we obtain

(28)

(29)

(30)

—(?? coth^) ~> f 77, 7 ? - > 0 . drj

The magnetic susceptibility x is defined by4

X=lim(m/H).

which is the Curie-Langevin law for a classical gas at high temperature and low density.4'7

(24) This classical result includes both the diamagnetic part (due to particle motions in a magnetic field) and the paramagnetic part (due to the spin or the intrinsic magnetic moment of the particles). One can show that

(25) the paramagnetic contribution to X is ^B^l/kT and the diamagnetic contribution is — ̂ of the paramagnetic contribution, so that the resultant is that given in Eq. (30).

I t is difficult, however, to separate the general expres-(26) sion (13) into the corresponding diamagnetic and para­

magnetic parts. Nevertheless, we can still decompose10

TTT t - n r • *. \ r™ j r ... 14-u 4-- i (Gordon decomposition) the four-current j» We shall now eliminate X. The definition of the particle v r / JP density 91 is yM = iefarf^jp(1) + i M

(2)

1 d 31= —X— l n # . (27) i° J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley

ft d\ Publishing Co., Inc., Reading, Mass., 1967), pp. 107-110.

From Eqs. (23)-(25) we obtain

L47r3\

2fXB2

3kT

2TkT\s'2 1

2 /

1232 V. C A N U T O A N D H . Y. C H I U 173

such that

becomes the Landau Hamiltonian giving rise to diamag-netism in the nonrelativistic limit.

The other part,

y / 2 ) ^ = — eli 1 dAp _ 1 dA

2 dxu 2mcL2 dxi

becomes the Pauli Hamiltonian giving rise to spin paramagnetism in the nonrelativistic limit.

V. REDUCTIONS OF THE EXPRESSION FOR m

We can apply the same procedure as in Paper II to reduce 9TZ into a sum over the same function over differ­ent arguments. The following transformation, which has been explained previously, is introduced:

V~x/dn9

an~il+2{H/Hc)nJi\

and the following functions are defined:

(31)

(32)

Ci(4>,n) ,M)= [ Jo

a+v") - 1 / 2

X 1+exp /(l+*2)1/2-~/A

r r / (i+f2)l/2-A C2(0,/i) = <t> / In 1+expf J

By partial integration, we find that

, (33)

\dv. (34)

C2(4>; Jo (1+*-2) 1/2

X l+expf J dv. (35)

Equation (13) can be written as

wi= -c2(^)+E ^2c2( —,~) 7r2 Xc

3I2 »=i \an a J

End--)]. (36) Hc n=*l \an <*>n' 1

Numerically,

2fEo^(2/7r2)M^Ao3=3.2637Xl010 G.

0 —> 0, we have

d O i ) - lim d ( < M = M > + (M 2 - 1)1/2], (38)

C2(M)^limC2(^iu)

4{/z(M2-l) l / 2-ln[M+(M2-l)1 /2]}, (39)

for pL> 1, and

Ci0t) = C 2W = O, M < 1 . (40)

VI. DEGENERATE EXPRESSIONS FOR THE MAGNETIC MOMENT

As discussed in Paper II , the condition of degeneracy is expressed by

M-1>0, 0->O. (41)

From Eq. (40) it can be concluded that the sum over n in the expression for 31Z [Eq. (36)] terminates at s, where 5 satisfies the condition

#s<M<#s+ l - (42)

We therefore find

m=~ 2 fXB\

-GGO+L ] an2c2(~) i \aj

ZnCil-) • (43) Hcn=i \an/ J

(37)

The properties of Ci(<f>,n) and C2($,/x) have been exten­sively discussed in Paper II. In the degenerate limit

The first two terms are positive and the third term is negative. When $=0} the only nonvanishing term is the first term. As discussed in Paper II, this corresponds to the one-dimensional gas limit in which the spin of all electrons is aligned with the magnetic field. The result­ing magnetic moment is therefore the magnetic moment of all electrons. As is well known in nonrelativistic quantum mechanics, the electron spin gives rise to pa­ramagnetism, and the electron orbital motions give rise to diamagnetism. Hence, at the lowest-energy state there is only paramagnetism.

When quantum states other than the n~0 states are excited, the gas gradually becomes diamagnetic and 2m then becomes negative. The nonrelativistic and non-degenerate limit of large quantum numbers n has been discussed in Sec. IV.

A. The Case 5=0 . Ground State of a Magnetized Gas

When the density is such that

l < M < « i = ( l + 2 # / # c ) 1 / 2 , (44)

the only non vanishing term is the first term |C 2 (M)-This corresponds to the one-dimensional gas limit, as discussed in Paper I I . We shall obtain the magnetic moment 9TC as a function of particle density dl for this case.

173 M A G N E T I C M O M E N T OF M A G N E T I Z E D F E R M I G A S 1233

The relation between the density 91 and n is [Eq. (81), Paper I I ] , for the case given by (44),

l f l l H 31= |C4(M) = 3lo—C«0B), (45)

C4(M) = ( M 2 - 1 ) 1 / 2 , 9lo= . (46) 2xH c

3

From Eqs. (46) and (39), we find the relation between C2(M) and C4(M) :

C20x)=!c4G*)[C42G0+i]m

- i ln{C4(M)+[C42(M)+l]1 / 2} (47)

= i£(£2+l)1/2-§ InR+teH-D1'*], where

{=C«(M) = 9l/(3loff/H.). (48) Therefore,

1 W . / l n [ f + ( f H - l ) 1 / s >

1 MB 31

2x2V9X0fl-/JH',

£(£2+l) 2d \ 2 -l l /2

D1/2]\ 1/2 /

[(—M LxSJUH/HJ J

where

XSFif ) , \yioH/HJ

JFi(*) = l-

\<3l0H/Hc

ln[»+(x2+l)1/2]

(49)

and »(»2+l)i /

(SO)

(51)

(52)

From Eq. (49) it may be concluded that 201 increases as 9l2 at a given field strength. However, when 91 is too large, higher magnetic states with s>0 will enter, causing 9E to decrease. Hence, there is a combination of the field strength and the density such that 9E is a maximum. This maximum value of 911 can be estimated as follows:

The value of fx should be close to the upper limit given in Eq. (44), and for large values of H/He it is

^(2H/Hcyi\ (53)

This gives the following relation between 31 and H/Hc

for the maximum value of 911:

9l**i/2vio(H/He) 3/2 (54)

Substituting Eq. (54) into Eq. (49), we obtain the maximum value of 9E as a function of H for 9l]»9lo XH/HC:

\ fJLB H

9rc~ « io~m. (55)

B. The Case $>Q

In the case s>0, the higher magnetic states are oc­cupied and the gas gradually becomes diamagnetic. At a given field strength, 9TI increases with density until reaching a maximum given by Eq. (55), then decreases with density, eventually becoming negative. Figures 1 and 2 show the relation between 97Z and H/HCJ and 9H and 91. The undulating behavior of 91Z is due to the peculiar behavior of the density of states shown in Fig. 4 of Paper II.

VII. ABSENCE OF PERMANENT MAGNETISM IN A DENSE ELECTRON GAS

In solids, it has been discovered that the magnetic susceptibility X is given by the Curie-Weiss Law11

X = C / ( r - 0 ) , (56)

where C is a constant, and 8 is the transition tempera­ture, below which there is ferromagnetism and above which there is paramagnetism. This may suggest that an electron gas will also possess a transition tempera­ture so that it will become ferromagnetic at low tem­peratures. This is not the case in the absence of large-scale ordered phenomena, as will be shown.

We can now answer the question of whether a dense electron gas can be self-magnetized. This question has been raised many times with respect to gravitational-collapse problems.

The relation between the resulting field B, the im­pressed field II, and the induced magnetic moment 911 is

B=fi0(H+VK), (57)

where ju0 is a constant which takes care of dimensions. Consider a small cavity in a magnetized body (Fig. 3). The field felt by a sample in the cavity is B; hence if

FIG. 1. The relation between 3TC and N/N0 for H/Hc=l. M=m/mh where 3TC0= (2/ir*)m/\c* and #0= (lAr2)(l/Xc

3). 11 A. H. Morrish, Physical Principles of Magnetism (John Wiley

: Sons, Inc., New York, 1965).

1234 V. C A N U T O A N D H . Y . C H I U 173

*4 .6 .7

FIG. 2. The relation between M^M/MQ and H/H€.

the value of induced M matches B, the body may be­come self-magnetized through the interaction among electrons, and permanent magnetism may result. In order that this may occur, it is necessary that the in­duced magnetic field be greater than the impressed field, that is,

m>H, B^pfflC. (58)

Equation (55) shows that M<^iH. Hence, there is no possibility for permanent magnetism to exist in a dense electron gas:

Higher-order electromagnetic corrections will change the value ot 91Z computed here, but the magnitude of these corrections will be 1/137 of the computed value of 2fFl, unless some collective phenomena take place. These collective phenomena may include the presence

of an electric current or solid structure at high density and low temperature.

VIIL SUMMARY AND CONCLUSION

In this paper, we have computed the magnetic mo­ment of a magnetized Fermi gas. A general expression for the magnetic moment was obtained. The properties of the magnetic moment in a degenerate magnetized gas were also studied. We obtained the following results:

(i) At a given field strength, the magnetization 3fE (magnetic moment per unit volume) increases more than linearly with particle density, and for high field strengths (M may even increase quadratically with the particle density 91. However, if the density is too high, 9TC will decrease with the excitation of higher magnetic states. The maximum value of Wit at a given field strength is roughly

K /̂!

Covffy

. v r - - A Field strength

1 HBH (59)

(ii) Permanent magnetism will take place when the following condition isfulfilled:

H=*p$ll, (60)

EXTERNAL FIELD H « 0

FIG. 3. The condition for permanent magnetism,

where #o= 1 in Gaussian units (used in this paper). From Eq. (59), this condition cannot be fulfilled for an elec­tron gas. Thus there is no permanent magnetism in a noninteracting electron gas.

173 M A G N E T I C M O M E N T O F M A G N E T I Z E D F E R M I G A S 1235

ACKNOWLEDGMENTS ] paper. One of us (V.C., an NAS-NRC Resident Re­

search Associate) would like to thank Dr. Robert We would like to thank Dr. Laura Fassio-Canuto, J a s t row for the hospitality of the Institute for Space

with whom we collaborated for the first part of this Studies.

P H Y S I C A L R E V I E W V O L U M E 1 7 3 , N U M B E R 5 25 S E P T E M B E R 1 9 6 8

Self-Consistent Homogeneous Isotropic Cosmological Models. I. Generalities

R J M I HAKIM

Laboratoire de Physique TMorique et Hautes Energies* Faculte* des Sciences', 91-Orsay, France (Received 13 March 1968)

Self-consistent models of uniform universes are provided by coupling Einstein's equations to the one-particle Liouville equation. Correlations between the "particles" of the cosmic gas are thus neglected. As a conse­quence, an equation of state is not needed in the theory but is, rather, provided from these statistical con­siderations. I t is shown that an expanding self-consistent uniform universe behaves asymptotically as the relativistic polytrope p~(j,Vz

} and finally, as an expanding Friedmann universe. Near the possible singularity R=0 (JR=scale factor), self-consistent models are hot models. Friedmann models are shown to be a particular case of self-consistent models.

1. INTRODUCTION

IN a preceding paper we studied some simple proper­ties of the self-gravitating relativistic gas.1 Essen­

tially, we used a Vlasov approximation. In other words, the relativistic one-particle Liouville equation

d u»dlffl(x

p}u')-T*0'l(xp)uaufi ffl(x*,up) = 0 (1.1)

was coupled to Einstein's equations

RfiV—^RgfiV—\gfip=XTfiV (1.2)

through the definition of the momentum-energy tensor

f dzu T^(xp) = m(\/g) / — y i i x ^ u ^ u W . (1.3)

J u°

In the following the metric tensor g»p is of signature (H ) and g= | det(g/iI') | . We also use c= 1, where c is the velocity of light in a vacuum. In Eq. (1.2), R^ is the Ricci tensor, R is the scalar curvature, and X is the cosmological constant. X denotes the gravitational constant. The F ^ ' s are the well-known Christoffel symbols of the second kind. In Eqs. (1.1) and (1.3), yi(xp,up) is the invariant distribution function de­scribing the gas. In Eq. (1.3), m is the mass of a typical particle and the integral is extended to the hyperboloid

gnw(xi)uW= 1, u°>0 (1.4)

except when dealing with particles of vanishing mass. In

* Laboratoire associ^ au Centre National de la Recherche Scientifique.

1 Ph. Droz-Vincent and R. Hakim, Ann. Inst. H. Poincare* (to be published).

this latter case the integral is extended to the light cone

gpwix^uW =09 u°^0. (1.40

Finally, we will use the Einstein summation convention, Greek indices running from 0 to 3 and Latin ones from 1 to 3.

Since no explicit solution of Einstein's equations is known in terms of an arbitrary momentum-energy tensor TM„, it was of course impossible to get a Vlasov equation as is commonly done in the case of electro­magnetic interactions2 (i.e., the electromagnetic field is expressed as a functional of the distribution function and next eliminated in the one-particle Liouville equa­tion). Accordingly, we obtained a linearized kinetic equation by considering only small deviations of given "background quantities" (i.e., g^ and 91). It then follows that our previous paper can be mainly applied to problems of stability. We also stressed that the only known relativistic self-gravitating system where collective effects are dominant is constituted by the universe as a whole.3

In this paper we deal with homogeneous, isotropic cosmological models. It is indeed well known that, in this particular case, Einstein's equations reduce to two differential equations4 for the pressure p, the mass

2 See, e.g., S. Gartenhaus, Elements of Plasma Kinetic Theory (Holt, Rinehart and Winston, Inc., New York, 1964).

3 This is not completely true since the first stages of the gravi­tational collapse of a massive star is another such example. How­ever, as the star collapses, the neglect of correlations is less and less valid.

4 See, e.g., H. Bondi, Cosmology (Cambridge University Press, Cambridge, 1961), 2nd ed.; G. C. McVittie, General Relativity and Cosmology (Chapman and Hall Ltd., London, 1965); R. C. Tolman, Relativityf Thermodynamics and Cosmology (Clarendon Press, Oxford, 1958).