ISSN 10637761, Journal of Experimental and Theoretical Physics, 2012, Vol. 115, No. 3, pp. 420–426. © Pleiades Publishing, Inc., 2012. Original Russian Text © V.V. Skobelev, 2012, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2012, Vol. 142, No. 3, pp. 472–479.

NUCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS

Magnetism of a Relativistic Degenerate Electron Gas in a Strong Magnetic Field V. V. Skobelev Moscow State Industrial University, Moscow, 115280 Russia email: [email protected] Received February 20, 2012

Abstract—The magnetization and magnetic susceptibility of a degenerate electron gas in a strong magnetic field in which electrons are located on the ground Landau level and the electron gas has the properties of a nonlinear paramagnet have been calculated. The paradoxical properties of the electron gas under these con ditions—a decrease in the magnetization with the field and an increase in the magnetization with the tem perature—have been revealed. It has been shown that matter under the corresponding conditions of neutron stars is a paramagnet with a magnetic susceptibility of χ ~ 0.001. DOI: 10.1134/S106377611208016X

1. INTRODUCTION The study of a nonrelativistic electron gas in metals (at the elemental level, see, e.g., [1]), where electrons can be considered as free and an electron gas is para magnetic, was topical in early stages of investigations. The exhaustive quantumstatistical consideration of the problem was given in [2]. However, the electron gas in some astrophysical objects (white dwarfs, neutron stars) is significantly degenerate because of Pauli degeneracy and this property determines its contribu tion to the total pressure [3, 4] in white dwarfs or to the suppression of the β decay of a neutron in neutron stars [5]. Thus, in view of possible astrophysical appli cations, the study of the properties of the relativistic electron gas is of current interest. One of the first attempts was made in [6], where the corresponding mathematical technique was applied. The mentioned astrophysical objects have strong magnetic fields (up to 109 G in white dwarfs [7] and up to 1017 G in neu tron stars [8]). For this reason, it is of interest to exam ine the magnetic properties of the substance of white dwarfs and neutron stars, i.e., to calculate the magne tization J and magnetic susceptibility χ. The magnetic field of white dwarfs is weak B  B0 = m2/e = 4.41 × 1013 G; in this respect, the magnetic properties of the relativistic electron gas were quite comprehensively studied in [9], where closed expressions were obtained for this case. Discussing neutron stars, it is noteworthy that the contribution of a nonrelativistic degenerate neutron gas owing to the interaction of the anomalous mag netic moment of neutrons with the magnetic field leads to Pauli paramagnetism [2]; the condition of the weak field for neutrons has the form B  Bmax ≈ 5.6 × 1018 G [10] and this inequality is apparently sat isfied in neutron stars. It is interesting to determine the

contribution of electrons (and protons) to the quanti ties J and χ, which characterize the substance of a neu tron star at the strongest possible field on the order of 1017 G. The macroscopic effects of the significant influence of the electron gas on, e.g., the size of a mag netar can exist in such fields [5]. In view of this circumstance, it is noteworthy that the degenerate electron gas at the field [5] B 3 2/3 2 B > B cr = 0 ( 4π n e λ C ) , 2

where

1 λ C = , m

(1)

the degenerate electron gas is on the ground Landau level, i.e., becomes effectively onedimensional; at a typical electron density in the neutron star ne ~ 1035 cm–3, the field is Bcr ≈ 8 × 1016 G. For this reason, this should be taken into account when considering magnetars. This case was not considered in [9], because interpolation formulas used in that work are valid for the contribution of all Landau levels. In this work, the J and χ values are calculated for the case of the strong magnetic field satisfying condition (1), and the results are discussed in application to magnetars. The calculations are based on the following general relations. The quantity J = M/V, where M is the total mag netic moment of the electron gas, can be calculated using the generalized differential relation for the Ω potential [2] dΩ = – SdT – PdV – Ndμ – MdB

(2)

and the equality

420

˜ ∂Ω J = – ⎛ ⎞ ⎝ ∂B ⎠ T, V, μ

(3)

MAGNETISM OF A RELATIVISTIC DEGENERATE ELECTRON GAS

˜ = Ω/V is the poten following from Eq. (2), where Ω tial of the unit volume. Then [2], Ω = –T

μ – Ek

⎞ , ∑ ln ⎛⎝ 1 + exp  T ⎠

(4)

k

where the summation is performed over all quantum states with the set of quantum numbers k. In the con sidered case of static uniform magnetic field, the energy Ek is given by the expression 2

2

(5) Ek E n = m + p 3 + 2γn . Here, γ = eB, where e is the elementary charge; p3 is the momentum along the field; and n = 0, 1, 2, … are the ordinary numbers of the Landau levels. As is known [2] (see also [5]), the sum over quan tum states in the magnetic field has the form ∞

∞

γV 2 Γn dp 3 , ( 2π ) n = 0 –∞

∑

∑ ∫

k

(6)

where Γn = 2 and 1 for n ≠ 0 and n = 0, respectively, is the spin statistical weight. The specific potential can be represented in the form ∞

⎫ 1 1 ⎧ ˜ = –  Ω F ( n ) +  F ( 0 ) ⎬, 2⎨ 2 π ⎩n = 1 ⎭

∑

(7)

where ∞

μ–E F ( n ) = γT dp 3 ln ⎛ 1 + exp n⎞ . ⎝ T ⎠

∫

In the case of the completely degenerate electron gas (T = 0), function (7a) is nonzero only at En < EF ≡ μ|T = 0 and μ–E EF – En ln ⎛ 1 + exp n⎞ = , ⎝ T ⎠ T

∫

p max

dp 3

0

2 EF

∫

dp 3 ,

0

2 mn ,

2

In Eq. (8), y = ( E F – m2)/2γ and E(y) is the largest integer part; this notation is the same as in [4]. It is noteworthy that Eqs. (8) and (8a) are in agreement with the usual expression Ω = –PV and with Eq. (12a) from [4] for the pressure of the degenerate electron gas (a factor of 1/2 in the second term in the curly brackets in Eq. (8) was omitted in Eq. (12a) from [4] for conve nient calculations; this factor did not affect the calcu lation of the pressure of the electron gas in white dwarfs because the second term in the curly brackets in Eq. (8) is small for the characteristic magnetic fields in these astrophysical objects). These relations are sufficient for the repetition of the calculations performed in [9]. However, it seems to be more convenient to determine J in terms of the Ω potential rather than in terms of the free energy F, as was done in [9], because the derivative in Eq. (3) is cal culated at a constant μ value; this circumstance sim plifies the calculations as compared to [9], where the dependence μ(B) should be taken into account in dif ferentiation with respect to the field (because μ does not belong to the variables in terms of which dF is the exact differential). However, for completeness, the 3 field correction to the Fermi energy at (3π2ne λ C )2/3  ε (ne is the electron density and λC is the Compton wavelength) is calculated in Appendix A, and the tem peraturesquare correction to the chemical potential is calculated in Appendix B.

(7a)

0

∞

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2. MAGNETIZATION AND MAGNETIC SUSCEPTIBILITY OF THE COMPLETELY DEGENERATE ELECTRON GAS IN THE STRONG MAGNETIC FIELD According to the expression for y represented in the form 2

( E F /m ) – 1 y =  , 2ε on the ground Landau level (n = 0),

2

p max = – m n = m + 2γn . The elementary integration with respect to p3 and pas sage to dimensionless variables give E(y) ˜ ⎧ ⎫ Ω 1 ˜ dim ≡  (8) Ω  2ε F˜ ( n ) +  F˜ ( 0 ) ⎬, = – ⎨ 4 2 2 m /2π ⎩n = 1 ⎭

∑

where ε = B/B0 and F˜ is the following dimensionless function of the integer argument n: 1 F˜ ( n ) =  2ε ( 1 + 2εy ) ( y – n ) – ( 1 + 2εn )  2 (8a) 1 + 2εy + 2ε y – n × ln  . 1 + 2εn

(9)

2 2εy = p˜ F ,

p˜ F ≡ p F /m,

3

2

(9a)

and [4] 2

2π ( n e λ C ) ⎛ 2π n (10)  p F = e⎞ . p˜ F =  ⎝ ε γ ⎠ Since the contribution comes only from the second term in brackets in Eq. (8) and taking into account Eq. (8a), we have ˜ dim = – ε [ p˜ 1 + p˜ 2 – ln ( p˜ + 1 + p˜ 2 ) ], (11) Ω F F F F 2 and Eq. (3) has the form 2 ˜ dim m e ⎛ ∂Ω (12)  ⎞ , J = –  2 ⎝ ∂ε ⎠ p˜ F 2π

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i.e., 2

2 2 em J = 2 [ p˜ F 1 + p˜ F – ln ( p˜ F + 1 + p˜ F ) ] 4π or, in the equivalent form, α J =  2 4π 2

(13)

1 2 exp ⎛ –  x γ⎞ . ⎝ 2 ⎠ (13a)

2

× [ p˜ F 1 + p˜ F – ln ( p˜ F + 1 + p˜ F ) ]B 0 , where 2

α = e = 1/137. In view of the condition of the reduction to one dimensional form (1) rewritten in the form 3

2

3/2

2π ( n e λ C ) < ε , (14) and Eq. (10), it seems to be reasonable to consider the nonrelativistic case p˜ F  1; in this case, Eq. (13) can be reduced to the form 2

em  p˜ 3 . (15) J =  2 F 12π In view of Eq. (10), this expression can be rewritten in the form 3 3

2π e ( n e λ C ) J =    B. 4 3 ε 4 2

(15a)

The magnetic susceptibility χ defining through the relation (16) J = χB in the nonrelativistic case is given by the expression 3 3

4 2 e ( ne λC ) χ = 2π .   4 3 ε

(17)

As will be seen below, the ultrarelativistic case where p˜ F  1 is possible in neutron stars; in this case, it fol lows from Eqs. (13) and (16) that 3 2

2 2 2 ( ne λC ) em ˜ 2 J = p = π e   B, 2 F 3 4π ε 3 2 2 2 ( ne λC )

where the quasimomentum p2 determines the position of the center of the wavepacket on the x axis. For the simple case p2 = 0, this factor has the form

(18)

χ = π e  . (18a) 3 ε According to Eqs. (13)–(17), the field depen dences of J and χ are strongly nonlinear and the mag netization and magnetic susceptibility decrease with an increase in the field. This quite unusual result can be explained as follows. The wavefunction of the elec tron on the ground Landau level includes the expo nential factor (see, e.g., [4]) p 2 1 exp –  ⎛ x γ – 2⎞ , 2⎝ γ⎠

This means that the electron is localized inside the region with a width of Δx ~ 1/ γ ; i.e., the localization region decreases with an increase in the field. Further, the approximation of the ideal Fermi gas under con sideration can be implemented as for the classical gas in the case of either low densities or negligibly small “sizes” of particles (their localization regions). In both cases, a diluted ideal gas is obtained. Therefore, these two variants of “obtaining” the ideal electron gas in the strong magnetic field specified by Eqs. (1) and (14) are equivalent to each other. Formulas (10), (15), and (18) satisfy this requirement. In particular, according to Eqs. (15) and (18), the magnetization decreases either with a decrease in the density, which is quite clear, or with an increase in the field as follows from the mentioned equivalence of the variants. 3. TEMPERATURE CORRECTIONS TO THE MAGNETIZATION AND MAGNETIC SUSCEPTIBILITY IN THE STRONG MAGNETIC FIELD As is known [11], the temperature even in “old” neutron stars is T ~ 1 MeV, so that the electron gas in the real situation in collapsed astrophysical objects (white dwarf and neutron stars) is only partially degenerate and it is of interest to determine temperature corrections to J, χ, and, hence, the Ω potential given by Eq. (7). To this end, Eq. (7a) is transformed as follows. 2

(i) In terms of mn = m + 2γn introduced above, the total energy is represented in the form 2

2

En = p3 + mn . (19) (ii) The integration variable in Eq. (7a) is changed as (20) p3 Kn = En – mn (“kinetic energy”). After that, Eq. (7a) becomes ∞

Kn + mn  F ( n ) = γT dK n  2 K + 2m K n n n 0

∫

(21)

( μ – m n ) – K n⎞ × ln ⎛ 1 + exp   . ⎝ ⎠ T The integration by parts gives ∞

2

dK n K n + 2m n K n F ( n ) = γ  . Kn – ( μ – mn ) +1 0 exp  T The asymptotic expansion in temperature [2]

∫

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μ

∫

∫

2

dεf ( ε )  = f ( ε ) dε + π 2 f ' ( μ ) + … (23)   T ε–μ 6 exp   + 1 0 0 T will be used below. In the case under consideration,

2

T) π T 1 , ˜ (dim ˜2 E ˜ (F0 ) +  Ω = – ε  (28) (0) ˜ 6p˜ F EF or, in terms of the dimensionless Fermi momentum given by Eq. (10), 2

T) π ˜2 ˜ (dim Ω = – ε  T 6p˜ F

2

ε Kn , f K n + 2m n K n , μ μ – mn . Consequently, taking into account the expansion of the chemical potential in powers of the temperature μ = μ0 + μ2 + …, Eq. (22) can be represented in the form (24) F ( n ) = F0 ( n ) + FT ( n ) + … . Here, F0(n) is the F(n) value at zero temperature and makes a contribution to J that is given by Eq. (13) at n = 0 and the temperature contribution FT(n) has the form

EF (26) ˜ = T, E ˜ F =  T , m m ˜n = m n = 1 + 2εn . m m These general relations allow the determination of the temperature correction to χ obtained in [9]; in the case under consideration, the expression in the square brackets does not contribute to the derivative in Eq. (3). However, only the contribution of the ground Landau level is of interest and only the second term in the curly brackets in Eq. (8) should be taken into account. The corresponding contribution to the dimensionless specific Ω potential has the form ˜T T) Ω ˜ (dim Ω =  2 = – F˜ T ( 0 ) 4 m /2π (27) 2 ˜ (F0 ) 2 ( 0 ) ˜ ( 0 )2 E π ˜ ˜ 2 EF – 1 , = – ε  T  + μ 2 6 (0) ˜ EF – 1 where (0)

EF 2 ˜ (F0 ) =  E  = 1 + p˜ F m is the dimensionless Fermi energy corresponding to the dimensionless Fermi momentum given by Eq. (10) ˜ (20 ) is the dimensionless temperaturesquare cor and μ rection to the chemical potential on the ground Lan dau level n = 0, which is calculated in Appendix B. Using this result, we arrive at the expression

2 1 1 + p˜ F +  . 2 1 + p˜ F

(28a)

In view of Eqs. (11), (28a), and the relation Ω = ⎯PV, the pressure of the electron gas is given by the expression 4 2 2 m ⎧1 P = ε 2 ⎨  [ p˜ F 1 + p˜ F – ln ( p˜ F + 1 + p˜ F ) ] 2 2π ⎩ 2

π ˜2 +  T 6p˜ F

2

EF 2 2 2 (25)  + μ 2 E F – m n γ, FT ( n ) = π  γT  2 2 6 EF – mn where μ2 μ0 ≡ EF is set with the same quadratic accuracy in T. In terms of the dimensionless variables, Eq. (25) is written in the form 2 ˜F E π ˜ 2  ˜ 2F – m ˜ 2n ,  + μ˜ 2 E F˜ T ( n ) = ε  T 6 ˜ 2F – m ˜ 2n E

423

⎫ 2 1 1 + p˜ F +  ⎬ 2 1 + p˜ F ⎭

with constraint (B.6) for the temperature and Fermi ˜ = 0 coincides, as is momentum. This expression for T expected, with Eq. (25) from [4]. Under the extreme conditions of neutron stars considered in Section 4, p˜ F  1; hence, 2

T) π ˜2 ˜ (dim (28b) Ω ≈ – ε  T , 6 which, together with Eq. (11), gives the expression 2 2 ˜ 2 εp˜ F π T ˜ dim = –  Ω (29)  1 +  ⎛ ⎞ . 2 3 ⎝ p˜ F ⎠

Using Eq. (12), we obtain 3 2 2 ˜ 2 2 2 ( ne λC ) π ⎛  T⎞ B, J = π e   1 +   3 ⎝ ⎠ ˜ 3 p F ε

(30a)

and 3 2 2 ˜ 2 2 2 ( ne λC ) π ⎛ T χ = π e   1 +  ⎞ .    (30b) 3 ⎝ ⎠ ˜ 3 p F ε In contrast to the case of weak fields [9], the positive sign of μ2 (μ2 > 0) and, as a consequence, an increase in J and χ with the temperature have the same origin as the aforementioned decrease in J and χ with an increase in the field in the considered case of strong magnetic field satisfying conditions (1) and (14). It is noteworthy that, taking into account Eq. (13) for the Ω potential at zero 0) ˜ dim ≡ Ω ˜ (dim , this follows from the coinci temperature Ω T) ˜ (dim dence of the signs of Ω given by Eqs. (28a) and (28b) (0) 0) ˜ ˜ (dim and Ω dim . However, the sign of Ω is fixed by spin

paramagnetism because Landau diamagnetism is absent when transverse excitations are suppressed, T) ˜ (dim whereas the sing of Ω is determined by the sign of μ , 2

which is generally explained in Appendix B (second term in Eq. (27)), and by the same sign of the first term in Eq. (27), which is in turn fixed by the sign of the

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entropy (only this term contributes to the derivative because the derivative is calculated at a constant μ value and the second term contains μ2): 3 ˜ ˜ dim m  ⎛ ∂Ω S˜ = – ⎛ ∂Ω ⎞ = –   ⎞ 2 ⎝ ∂T ⎠ μ, B ⎝ ˜ 2π ∂T ⎠ μ ( p˜ ), ε F

˜ 2 ε   T =   1 + p˜ F > 0. 3p ˜ 6λ C F I could not find a more conclusive explanation for this phenomenon. Other features of the behavior of a spin 1/2 particle in the twodimensional spacetime and a change in the mathematical technique were men tioned, e.g., in [11, 12]. However, temperature correc tions under the conditions of magnetic neutron stars ˜ ≈ 2) considered in Section 4 are anyway ( p˜ F ≈ 50, T negligibly small according to Eqs. (30a) and (30b). 4. DISCUSSION The contribution of the electron gas to the magne tization of old neutron stars with the temperature T ~ 1 MeV [13] is first evaluated when temperature correc tions can be neglected as was mentioned above. At a probable density of ne ~ 1035 cm–3, conditions (1) and (14) for the reduction to the onedimensional form “in the limit” are satisfied at the strongest possible field B ~ 1017 G [8] with p˜ F ≈ 50. Then, using Eq. (18a), we obtain –4

(31) χ ≡ χ e ≈ 2 × 10 . At the same time, according to [14], at the correspond ing neutron density nn ~ 1038 cm–3 and at the same field, the magnetic susceptibility of the neutron gas owing to 1

the anomalous magnetic moment of the neutron is

from the respective formulas and estimates for χe with the change me mp. Since (p) (e) m –2 (33) p˜ F = p˜ F e ≈ 2.7 × 10 , mp the proton at the accepted density and field is strongly nonrelativistic and Eq. (17) should be used to estimate χp. As can be seen, ( nr ) ( nr ) m (34) χ p ≡ χ p = χ e e , mp ( nr )

where χ e is given by Eq. (17). The elementary cal culation gives χp ~ 10–6; consequently, the contribu tion of the proton component to the total magnetic ( st ) susceptibility in the strong field χ tot under the typical conditions of neutron stars can be neglected (the inclusion of the anomalous magnetic moment of the proton in the case of spin paramagnetism obviously does not change this conclusion). Thus, the substance of a neutron star in a strong field of about 1017 G is nonlinear (dχ/dB ≠ 0) paramagnet (J > 0) with the magnetic susceptibility ( st )

–3

χ tot ≈ χ e + χ n ∼ 10 .

(35)

( st )

Such a small χ tot value hardly affects the evolution of such stars. It is noteworthy that condition (1) of the one dimensional case could be satisfied not only under extreme conditions of neutron stars, but also in a usual situation. For example, for univalent metals, ne ~ 1023 cm–3 [1] and the corresponding critical field if Bcr ~ 109 G. Unfortunately, such fields are yet unat tainable in laboratories and it is currently impossible to test the anomalous properties of the electron gas.

–4

χ ≡ χ n ≈ 2.3 × 10 , (32) so that the contributions of the electron and neutron gases are of the same order of magnitude although the electron density is much lower. This certainly con cerns only fields of about 1017 G when it can be assumed that most electrons are on the ground level and the temperature corrections to χn [12] and χe (Section 3) can be ignored. The contribution of the proton component to the magnetic susceptibility χp in the strong field will also be estimated below. First, it is noteworthy that, taking into account Eq. (10), condition (1) of the “one dimensional case” is independent of the mass and, therefore, at equal densities np = ne, the corresponding formulas and estimates for χp disregarding the anoma lous magnetic moment of the proton are obtained (0) EF

≈ 40 MeV [14] should be taken in the corresponding formula; some remarks concern ing inaccuracies made in [10] were given in [14].

1 In this case, the more accurate value

APPENDIX A The total number of electrons can be determined by calculating the sum of the distribution function over all quantum states: N =

1

. ∑  E –μ

(A.1)

k exp  +1 T Transformation (6) gives k

∞

⎫ 1 ⎧ ne ≡ N  =  F ( n ) + 1 F ( 0 ) ⎬, ⎨ 2 V 2 π ⎩n = 1 ⎭

∑

(A.2)

where ∞

dp 3 F ( n ) = γ  . En – μ 0 exp  + 1 T In the zeroth approximation in temperature,

∫

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This gives the dimensionless relativistic Fermi energy with the same accuracy in the form

E(y)

⎧ ⎫ 1 1 n e =   × 2ε ⎨ F˜ ( n ) +  F˜ ( 0 ) ⎬, 2 3 2 2π λ C ⎩n = 1 ⎭

∑

(A.4)

F˜ ( n ) = 2εy – 2εn , and E(y) is the largest integer part of y [4]. Then, the following modification of the Euler– Maclaurin formula presented in [2] should be used: ∞

∑

n=1

F˜ ( n ) + 1 F˜ ( 0 ) = 2

∞

1˜

˜

∫ F ( x ) dx – 12F ' ( 0 ).

(A.5)

0

The condition of the applicability of this approxi mation, i.e., the condition under which discreteness can be neglected is the smallness of the quantity F˜ ( n + 1 ) – F˜ ( n ) /F˜ ( n ) for all n values in the range of summation in the expression for ne. As can be seen in the structure of this expression, the condition 2εy  (0) 2ε, i.e., z max  ε should be satisfied (see Eq. (A.7)). Furthermore, it is implied in the preceding formula that the sum and integral converge at the upper limit; if convergence is absent, the upper limit should be suf ficiently large in order to neglect discreteness in the transition from summation to integration; the latter is inherent in the case under consideration. Further, when using Eq. (9) for y, it is noteworthy that the dis crete variable n (corresponding to the integration vari able x on the righthand side in Eq. (A.5)) enters into the expression for F˜ only in the combination 2εn(2εx), so that it is convenient to introduce the new variable z = 2εx in the integral with respect to x; the ˜ 2F – 1, E ˜F = = E corresponding upper limit is z max

EF/m. Consequently, the expression for ne becomes 1  ⎧ 1 n e =  2 3 ⎨2 π λC ⎩

z max

∫ dz

z max – z

0

(A.6)

2

d z – z – ε  max 6 dz

⎫ ⎬, z=0⎭

or 2 3 1 ⎧ 1 z 3/2 +  ε ⎫ λ C n e =  ⎬ max 2⎨ 12 z max ⎭ π ⎩3 3/2 2 z max ⎛ ε ⎞ =  2 ⎜ 1 +  ⎟ . 2 3π ⎝ 4z max⎠

(A.6a)

This expression determines the zmax value with an accuracy of ε2 as (0) 1 ε ⎞2 z max = z max 1 –  ⎛   , 0) ⎠ 6 ⎝ z (max (A.7) (0)

2

3 2/3

z max = ( 3π n e λ C )

425

2

ε ˜F ≈ E ˜ (F0 ) 1 –   , E 2 ( 0 ) (0) ˜ ˜ 12E F ( E F – 1 )

(A.8)

where (0)

EF 3 2/3 2 ˜ (F0 ) ≡  E  = 1 + ( 3π n e λ C ) m is the usual dimensionless relativistic Fermi energy in the absence of the field. It is noteworthy that this expression confirms the conclusion made in [4, 5] that the Fermi energy decreases with an increase in the field on the basis of numerical calculations. APPENDIX B The fieldsquare correction, μ2 ~ T 2, to the chemical potential in the strong magnetic field satisfying condi tion (1) is evaluated as follows. The change of the vari able p3 Kn = En – mn modifies Eq. (A.3) to the form ∞

2

dK n ( K n + m n )/ K n + 2m n K n F ( n ) = γ  (B.1) . Kn – ( μ – mn ) exp  + 1 0 T Then, asymptotic expansion in temperature (23) is used for Kn + mn ε Kn , f  , μ μ – mn . 2 K n + 2m n K n To determine the μ2 value precise in field, the expres sion for F (n) should be substituted into Eq. (A.2) and, after summation, the μ2 value should be determined from the condition that ne is independent of the tem perature. However, for the aims of this work, only the contribution of the second term in Eq. (A.2) is taken into account; i.e., n = 0 is accepted. Then, taking into account the power expansion of the chemical poten tial in temperature μ = μ0 + μ2 + …, Eq. (B.1) can be represented in the form (B.2) F ( 0 ) = F 0 ( 0 ) + F T ( 0 ), where

∫

2 2 ⎧ EF μ2 ⎫ 2 m F T ( 0 ) = γ ⎨   ⎬.  – π T  3/2 2 2 ⎩ E 2F – m 2 6 ( E F – m ) ⎭

(B.3)

The density should be independent of the tempera ˜ 2 ≡ μ2/m is ture, i.e., F T (0) = 0; from this condition, μ determined as

 ε.

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where p˜ F is given by Eq. (10). Thus, 2 ˜2 2 π ⎛T ˜ ≡ μ = E F 1 +  μ 2 ⎞ , ˜ F⎠ m 6p˜ F ⎝ E

n

(B.5)

3 1/3 2 p˜ F = ( 3π n e λ C ) . As was mentioned in [2], Eq. (23) is an asymptotic, rather than convergent series. Thus, the temperature ˜ square correction is correct only when the series in T is convergent; this condition, at least in the quadratic approximation, imposes the constraints on the param eters in Eq. (B.5): 2 ˜ 2 π T (B.6) ⎞ < 1.  ⎛  ˜ p˜ ⎠ 6 ⎝E F F

The following general explanation of the positive sign of μ2 can be given. The strong magnetic field sat isfying condition (1) is responsible for the effective onedimensional motion. In view of this circum stance, μ2 in the ndimensional space is determined, where the dimension of space is specified by the same symbol as the Landau level. The total number of “electrons” in the ndimensional space is obviously ∞

n

n–1

Li ⎞ p dp ⎛  N = Ωn Γn (B.7)   , ⎝ 2π⎠ E–μ i=1 0 exp  + 1 T where Γn is the spin statistical weight; Ωn is the solid angle in space of n ≥ 2 measurements (see, e.g., [15]); by definition, Ω1 = 1; Li are the normalization length

∏

∫

2

2

in the axes; and E = m + p . Defining the “volume” of space as n

V =

∏L , i

i=1

we obtain the density Ωn Γn n e =  n I n , ( 2π ) ∞

(B.8) n–1 p dp I n =  . E–μ 0 exp  + 1 T K = E – m provides The change of the variable p

∫

∞

( n – 2 )/2

2

dK ( K + m ) ( K + 2mK ) (B.9)  . K – ( μ – m) + 1 exp  0 T Using expansion (23) in temperature, we obtain the following expression with the quadratic accuracy: In =

∫

(0)

(T)

In = In + In ,

p (0) I n = F . (B.10a) n Taking into account Eq. (B.8), we determine the Fermi momentum in the ndimensional space as n

1/n

n ( ne λC ) (B.10b) p F = 2π   m, Ωn Γn and the temperaturesquare part has the form 2

(T) n–2 π 2 I n = E F p F μ 2 +  T 6

×

n–4 pF [ ( n

–

2 1 )p F

(B.10c) 2

+ ( n – 2 )m ]. (T)

From the condition I n expression

= 0, we arrive at the

2

2

2 μ2 = – π  ⎞ ( n – 1 )E F – m  .  ⎛ T ⎝ ⎠ 6 pF EF

(B.11)

It can be seen that μ2 for n = 1 is positive and coin cides with Eq. (B.4), whereas μ2 in highdimensional space is negative. For the case n = 3, the result coin cide with the formula that was presented in [5] and is obtained from relativistic relations from [3] using the mentioned condition that the density is independent of the temperature (see [9]). REFERENCES 1. S. V. Vonsovskii, Magnetism (Wiley, New York, 1974; Nauka, Moscow, 1984). 2. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics: Part 1 (Butterworth– Heinemann, Oxford, 2000; Fizmatlit, Moscow, 2005). 3. G. S. BisnovatyiKogan, Physical Problems in the Theory of Stellar Evolution (Nauka, Moscow, 1989) [in Russian]. 4. V. V. Skobelev, JETP 113 (5), 791 (2011). 5. V. V. Skobelev, JETP 115 (2012) (in press). 6. D. K. Nadezhin, Nauchn. Inform. Astron. Sov. Akad. Nauk SSSR, No. 32, 3 (1974). 7. A. T. Potter and C. A. Tout, arXiv:0911.3657 [astro ph.SR]. 8. R. C. Duncan and C. Tompson, Astron. J. 392, L9 (1992). 9. L. Homorodean, Int. J. Mod. Phys. B 13, 3133 (1999). 10. V. V. Skobelev, JETP 111 (6), 962 (2010). 11. V. V. Skobelev, JETP 93 (4), 685 (2001). 12. S. Hamieh and H. Abbas, J. Mod. Phys. 3, 184 (2012). 13. Debades Bandyopadhyay, S. Chakrabarty, P. Dey, and S. Pal, Phys. Rev. D: Part. Fields 58, 121301 (1998). 14. V. V. Skobelev, JETP 112 (5), 910 (2011). 15. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (Nauka, Moscow, 1965; Ameri can Mathematical Society, Providence, Rhode Island, United States, 1968).

(B.10)

where

Translated by R. Tyapaev JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 115

No. 3

2012