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On the equation of state for an electron gas in an intense magnetic field
Volume 62B, number 3
PHYSICS LETTERS
ON THE EQUATION
7 June 1976
OF STATE FOR AN ELECTRON
INTENSE MAGNETIC
GAS IN AN
FIELD
V. CANUTO and E TSIANG
Institute for Space Studies, NASA, New York 10025, USA Received 14 April 1976 In this paper we derive the equatmn of state for a relativistic electron gas Imbedded m a static homogeneous magnetic field of arbitrary strength. The denvatmn is based on the evaluation of the energy-momentum tensor and the use of Dlrac's equatmn for such a problem Contrary to a derwatlon presented several years ago, the present derivation is completely gauge-mvariant. We also show how to recover, m an exact manner, the perfect gas law for the case of weak magnetic fields.
Introduction. Seven years ago, one of the authors [1 ] derived a relativistic equation o f state for an electron gas under the influence of a static uniform magnetic field of arbitrary strength, for any temperature and density. The exact solution of Dirac equations employed in (1) are infimtely degenerate with respect to the parameter characterizing the location o f the electron orbit m the plane perpendicular to the magnetic field. Such parameter does not enter in the eigenvalue formula since the electron energy IS independent of the location of the grading center. The equation of state p = p(p, T, H) cannot depend upon such a degeneracy parameter and the sum over the guiding centers was a major technical difficulty in the derivation presented in ref. [1]. The equation of state was evaluated by computing the &agonal terms of the energy-momentum tensor for the electrons, expressed in terms of the single particle wavefunctlons solutions o f Dirac's equations. Such wavefunctions are however gauge-dependent and a specific gauge had to be used. Even though there IS nothing incorrect with such a commlttment, it would be by far better if one could produce the same result in a gauge independent manner. Another problem with the derivation given in ref. [1 ] was that the electron Tuv was not gauge-mvariant. In this paper we shall present the derivatmn of the equation of state in a gauge invariant fashion, without using any specific form of Dirac's wavefunctmns. We shall show how the non-invarlant looking form of Tuv o f ref. [ 1] did not invalidate the results even though the 366
derwation was certainly cumbersome. In fact the results derived here coincides with those of ref. [ 1 ]. With this paper we intend to answer the critiosm of Korneev and Starostln [2] concerning the gaugeinvanance problem. At every stage of the present derivation we shall use only gauge lnvarlant expressions. Finally we shall show in which limit and how one should expect to recover the perfect gas law. All the necessary steps will be performed using exact integrations. This was also accbmplished in ref. [1] but in a much less transparent fashion. The energy-momentum tensor. The Lagrangian describing a system o f non-interacting electrons imbedded in an external field described b y a vector potential A u is known to be
2=-4
, (~Au ~Av)2 \ ~X~Xv ~xu/ -~(Tu~u+ m - ie'ruAu)~ "
(1)
The Lagranglan is clearly lnvariant under the gaugetransformation qJ(x) -~ ¢ ( x ) exp [ i a ( x ) ] , Au(x ) -+ Au(X ) + e - 1 0 a / a x u. Eqs. ( 2 ) - ( 3 ) and (4) o f ref. [1] were not written in a gauge invariant manner. The total energy momentum tensor ~uv is evaluated to be (Fv0 =Ao, v
-Av,p)
bA° Fvp - g6uvF~3 ~uv = ~TvO,u~ +.~x; ' 2 ,
(2)
upon remembering that the spinor ~ satisfies Dirac's equation
(Tu~u+rn-leTuAu)~O-(Tu~u+m)~ =0.
(3)
Because of the first two terms eq. (2) is not gauge m-
Volume 62B, number 3
PHYSICS LETTERS
variant. In ref. [1], the equations of state was identified with the diagonal terms of the first term of (2) and that gave rise to the criticism of reL [2]. In order to restore the gauge-invariance nature in (2), we shall recall that to any energy momentum tensor one can always add a term of the form ruvp, p where ruv o is an arbitrary tensor antisymmetric in the indices u, P. If we choose ruv o = -AuF~o so that
O Oxv r u . o -
OAu Fro - A u ~ F Ox° Ox°
vo'
(4)
the first term o f ( 4 ) is exactly what is needed to transform the second term o f ( 2 ) into the gauge invariant quantity FuoFvo. The second term of (4) can be transformed by using Maxwell equations Fvp ' o =/w Finally, we obtain
cjuv= ~(7vOu_leTvAu) ~ + FupFvo
1 uvF~t23. (5) ~6
This expression is now gauge-invariant. The last two terms are easily recognizable as the energy momentum tensor for the electromagnetic field alone. We shall therefore identify the first term of (5) with the matter energy-momentum tensor, out of which we shall compute the equation of state. The equation of state. From the work of Rabi [3], it is known that the eigenvalues of Dirac's equation (3) are
E2=p2+m2+2m2nB/Bq,
n = 0 , 1 , 2 .... ~ .
(6)
The quantum number n represents the harmonic motion in the x-y plane perpendmular to the magnetic field. The motion along the z-axis is unaltered by the presence of the field and this accounts for the presence o f p z2. Only the perpendicular motion, i.e., p2 + p2 has been changed into 2m2(B/Bq)n. The magnetic field B is measured m units of Bq = m2c3/eh = 4.4 X 1013 gauss. Clearly, when B ~ 0, n must go to + ~ m order to have a finite transverse momentum. Let us now compute the diagonal components of the energy momentum tensor for matter. Using Dirac's equation, we have
coupled with the motion in the x and y direction, the wavefunction ~ must be of the form expOpz" z) t~(x,y), so that we must have
p2 Txx + Try= - m f J ~ - ~ -
2n B ~ + ~ .
(9)
An analogous computation shows that
pz q~+~ 7", =~-
(10)
The quantities Tl and Tll are still operators m the occupation number space, where the ~b's are understood to be written. Using standard second quantizanon procedures, the product ~b+~Oturns out to be proportional to a s+ a,,, i.e. to the occupation number in the state a. We shall have
Txx = Tyy = T± = ~q
-~ ] ( ) ,
Tii =
-~ f ( x ) ,
(11) where ] ( a ) is the Fermi distribunon and a represents the quantum numbers necessary to characterize the state, i.e. n and Pz" As explained elsewhere [1] +~
~= (a)
1 ~(~qq)f dpl, 27r2 n _**
so that T±
= --
2rr2
=
n
"; -=
e(n,x)
(12)
f(e) (13)
PO
since ~ 3 = 03" In fact B x = By = O, B z 4=O, implies A3=0. Since the motion in the z-direction is totally un-
(8)
Txx+ T y y - E Bq
T~l = 2n 2 (7)
t~+~ +Et~74~ .
The factor pz/E comes from 73. In the non-relativistic limit i74"),3 = a 3 ~ 03, which is again P3" The structure of the spinor ~ must be such that 473 ~ = (P3/E)~+~, since the xg' motion is unrelated to the z-motion. By simple manipulanon of Dirac's equation, one has ~ = (m/E)~+¢ so that, using the mgenvalues formula, we finally obtain
Txx + Tyy = ~JTxC-l)x ~ + ~TyCJgy~ = - m t~¢ - ~730 3 ¢ - ~")'404 t~
7 June 1976
B
f
__~
e(n,x) f(e)
where
PO =me:~/X3c , eZ(n,x) = 1 +mZnB/Bq .
(14)
With a similar, but simpler computation one can show 367
Volume 62B, number 3
PHYSICS LETTERS
that the energy and particle densities are expressable as
7 June 1976
References
oo
= -27r2
d x e(n, x) f ( e ) , n=0 _ oa
N ] ~ = ~3 21r2
+~
(15)
n=0 - ~
Relations (13) and (15) constitute the sought equation of state and t h e y are valid for any temperature, density and strength o f the magnetic field. T h e y have been derwed in a gauge invariant w a y and they are the same as the ones derived in ref. [1].
368
[1] V Canuto and H Y Chin, Phys. Rev. 173 (1968) 1210; 1220, 1229 [2] V.V. Korneev and A.N. Starostw, Sov. Phys. JETP 36 (1973) 487. [3] I.I. Rabl, Z. f. Phys. 49 (1928) 507. [4] V. Canuto, m. The role of magneUc fields in Physics and Astrophysics, ed. V Canuto, The New York Academy of Sclences, Vol. 247 (1975) p. 108.
PHYSICS LETTERS
ON THE EQUATION
7 June 1976
OF STATE FOR AN ELECTRON
INTENSE MAGNETIC
GAS IN AN
FIELD
V. CANUTO and E TSIANG
Institute for Space Studies, NASA, New York 10025, USA Received 14 April 1976 In this paper we derive the equatmn of state for a relativistic electron gas Imbedded m a static homogeneous magnetic field of arbitrary strength. The denvatmn is based on the evaluation of the energy-momentum tensor and the use of Dlrac's equatmn for such a problem Contrary to a derwatlon presented several years ago, the present derivation is completely gauge-mvariant. We also show how to recover, m an exact manner, the perfect gas law for the case of weak magnetic fields.
Introduction. Seven years ago, one of the authors [1 ] derived a relativistic equation o f state for an electron gas under the influence of a static uniform magnetic field of arbitrary strength, for any temperature and density. The exact solution of Dirac equations employed in (1) are infimtely degenerate with respect to the parameter characterizing the location o f the electron orbit m the plane perpendicular to the magnetic field. Such parameter does not enter in the eigenvalue formula since the electron energy IS independent of the location of the grading center. The equation of state p = p(p, T, H) cannot depend upon such a degeneracy parameter and the sum over the guiding centers was a major technical difficulty in the derivation presented in ref. [1]. The equation of state was evaluated by computing the &agonal terms of the energy-momentum tensor for the electrons, expressed in terms of the single particle wavefunctlons solutions o f Dirac's equations. Such wavefunctions are however gauge-dependent and a specific gauge had to be used. Even though there IS nothing incorrect with such a commlttment, it would be by far better if one could produce the same result in a gauge independent manner. Another problem with the derivation given in ref. [1 ] was that the electron Tuv was not gauge-mvariant. In this paper we shall present the derivatmn of the equation of state in a gauge invariant fashion, without using any specific form of Dirac's wavefunctmns. We shall show how the non-invarlant looking form of Tuv o f ref. [ 1] did not invalidate the results even though the 366
derwation was certainly cumbersome. In fact the results derived here coincides with those of ref. [ 1 ]. With this paper we intend to answer the critiosm of Korneev and Starostln [2] concerning the gaugeinvanance problem. At every stage of the present derivation we shall use only gauge lnvarlant expressions. Finally we shall show in which limit and how one should expect to recover the perfect gas law. All the necessary steps will be performed using exact integrations. This was also accbmplished in ref. [1] but in a much less transparent fashion. The energy-momentum tensor. The Lagrangian describing a system o f non-interacting electrons imbedded in an external field described b y a vector potential A u is known to be
2=-4
, (~Au ~Av)2 \ ~X~Xv ~xu/ -~(Tu~u+ m - ie'ruAu)~ "
(1)
The Lagranglan is clearly lnvariant under the gaugetransformation qJ(x) -~ ¢ ( x ) exp [ i a ( x ) ] , Au(x ) -+ Au(X ) + e - 1 0 a / a x u. Eqs. ( 2 ) - ( 3 ) and (4) o f ref. [1] were not written in a gauge invariant manner. The total energy momentum tensor ~uv is evaluated to be (Fv0 =Ao, v
-Av,p)
bA° Fvp - g6uvF~3 ~uv = ~TvO,u~ +.~x; ' 2 ,
(2)
upon remembering that the spinor ~ satisfies Dirac's equation
(Tu~u+rn-leTuAu)~O-(Tu~u+m)~ =0.
(3)
Because of the first two terms eq. (2) is not gauge m-
Volume 62B, number 3
PHYSICS LETTERS
variant. In ref. [1], the equations of state was identified with the diagonal terms of the first term of (2) and that gave rise to the criticism of reL [2]. In order to restore the gauge-invariance nature in (2), we shall recall that to any energy momentum tensor one can always add a term of the form ruvp, p where ruv o is an arbitrary tensor antisymmetric in the indices u, P. If we choose ruv o = -AuF~o so that
O Oxv r u . o -
OAu Fro - A u ~ F Ox° Ox°
vo'
(4)
the first term o f ( 4 ) is exactly what is needed to transform the second term o f ( 2 ) into the gauge invariant quantity FuoFvo. The second term of (4) can be transformed by using Maxwell equations Fvp ' o =/w Finally, we obtain
cjuv= ~(7vOu_leTvAu) ~ + FupFvo
1 uvF~t23. (5) ~6
This expression is now gauge-invariant. The last two terms are easily recognizable as the energy momentum tensor for the electromagnetic field alone. We shall therefore identify the first term of (5) with the matter energy-momentum tensor, out of which we shall compute the equation of state. The equation of state. From the work of Rabi [3], it is known that the eigenvalues of Dirac's equation (3) are
E2=p2+m2+2m2nB/Bq,
n = 0 , 1 , 2 .... ~ .
(6)
The quantum number n represents the harmonic motion in the x-y plane perpendmular to the magnetic field. The motion along the z-axis is unaltered by the presence of the field and this accounts for the presence o f p z2. Only the perpendicular motion, i.e., p2 + p2 has been changed into 2m2(B/Bq)n. The magnetic field B is measured m units of Bq = m2c3/eh = 4.4 X 1013 gauss. Clearly, when B ~ 0, n must go to + ~ m order to have a finite transverse momentum. Let us now compute the diagonal components of the energy momentum tensor for matter. Using Dirac's equation, we have
coupled with the motion in the x and y direction, the wavefunction ~ must be of the form expOpz" z) t~(x,y), so that we must have
p2 Txx + Try= - m f J ~ - ~ -
2n B ~ + ~ .
(9)
An analogous computation shows that
pz q~+~ 7", =~-
(10)
The quantities Tl and Tll are still operators m the occupation number space, where the ~b's are understood to be written. Using standard second quantizanon procedures, the product ~b+~Oturns out to be proportional to a s+ a,,, i.e. to the occupation number in the state a. We shall have
Txx = Tyy = T± = ~q
-~ ] ( ) ,
Tii =
-~ f ( x ) ,
(11) where ] ( a ) is the Fermi distribunon and a represents the quantum numbers necessary to characterize the state, i.e. n and Pz" As explained elsewhere [1] +~
~= (a)
1 ~(~qq)f dpl, 27r2 n _**
so that T±
= --
2rr2
=
n
"; -=
e(n,x)
(12)
f(e) (13)
PO
since ~ 3 = 03" In fact B x = By = O, B z 4=O, implies A3=0. Since the motion in the z-direction is totally un-
(8)
Txx+ T y y - E Bq
T~l = 2n 2 (7)
t~+~ +Et~74~ .
The factor pz/E comes from 73. In the non-relativistic limit i74"),3 = a 3 ~ 03, which is again P3" The structure of the spinor ~ must be such that 473 ~ = (P3/E)~+~, since the xg' motion is unrelated to the z-motion. By simple manipulanon of Dirac's equation, one has ~ = (m/E)~+¢ so that, using the mgenvalues formula, we finally obtain
Txx + Tyy = ~JTxC-l)x ~ + ~TyCJgy~ = - m t~¢ - ~730 3 ¢ - ~")'404 t~
7 June 1976
B
f
__~
e(n,x) f(e)
where
PO =me:~/X3c , eZ(n,x) = 1 +mZnB/Bq .
(14)
With a similar, but simpler computation one can show 367
Volume 62B, number 3
PHYSICS LETTERS
that the energy and particle densities are expressable as
7 June 1976
References
oo
= -27r2
d x e(n, x) f ( e ) , n=0 _ oa
N ] ~ = ~3 21r2
+~
(15)
n=0 - ~
Relations (13) and (15) constitute the sought equation of state and t h e y are valid for any temperature, density and strength o f the magnetic field. T h e y have been derwed in a gauge invariant w a y and they are the same as the ones derived in ref. [1].
368
[1] V Canuto and H Y Chin, Phys. Rev. 173 (1968) 1210; 1220, 1229 [2] V.V. Korneev and A.N. Starostw, Sov. Phys. JETP 36 (1973) 487. [3] I.I. Rabl, Z. f. Phys. 49 (1928) 507. [4] V. Canuto, m. The role of magneUc fields in Physics and Astrophysics, ed. V Canuto, The New York Academy of Sclences, Vol. 247 (1975) p. 108.