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Equation of state of ideal neutrons in an intense magnetic field
Volume 67A, number 5,6
PHYSICS LEUERS
4 September 1978
EQUATIO$ OF STATh OF IDEAL NEUTRONS IN AN INTENSE MAGNETIC FIELD A.E. DEL~ANTE School of Thysics, University of Melbourne, Parkvile, Victoria 3052, Australia
and N.E. FRA1~iKEL School ofPl~ysics,University ofMelbourne, Parkvile, Victoria 3052, Australia and Department of TheoreticalPhysics, University of Oxford, Oxford, OXJ 3NP, UK Received 6 ~une 1978
The stati$tical mechanics of a gas of non-interacting, ultra-degenerate relativistic neutrons in a uniform intense magnetic field is presented. Exact results for the Fermi energy are obtained, and the equation of state in the high-field limit is given.
The interior o~fa neutron star is thought to consist mainly of dense egenerate neutrons, and it is generaliy agreed that m~gneticfields of the order of 1012 G exist at the surfa~e[1,2]; however, the size of the field in the interior is ~.inknownand could be larger. The equation of state~for relativistic, non-interacting neutrons was used ii~the work of Oppenheimer and Volkoff [3] to ca1ct~1atethe maximum mass of a neutron star. It is mterest~ngto see what effect an mtense magnetic field would~have on such an equation of state and hence on the crit~ca1mass of a neutron star. In this letter *e consider the statistical mechanics of a gas of non-inte~acting,relativistic neutrons in a urnform magnetic fi~ld.The Dirac equation for the system is [iflc~ +
—
mc2 j
=
o,
C
where B~= m2c3/eh, n = 0,1,2,” and ~ is the anomabus part of the magnetic moment of the fermion: if we let the charge of the particle go to zero, the magnetic moment becomes entirely anomalous. There are then no Landau levels for a neutral fermion. To recover eq. (1) from eq. (2)~wenote that the limit must be taken -,
such that (B/Bc)(2fl + s + 1) -+ p~/m2c2.We then obtain
which has been sblved [4] to yield the following energy levels: 2c2 + ~.s2B2+ m2c4 5= ±[p + 2pBs(p2c2s~n2O+ m2c4)~2]ii~
netic moment of the particle. We note that eq. (1) can also be obtained from the energy levels of a charged fermion with an anomalous magneticmoment in a magnetic field by the following process. The energy levels of such a fermion are [5]: 2 2 Emc [(p/mc) Z 2 21 2~ii~ ~ + [{1 + KB/B )(2n +s + 1)}1~’2+ s~B/mc
E~
E = mc2 [(~~/mc)2+ 1 + (pB/mc2)2 + (2iffls/mc2)(~~
(1)
where s = ±1is t~iespin quantum number, 0 is the angle between the mos~ientump and the magnetic field B, which we take td be in the z-direction, and ~.tis the mag-
+2 1)1/21 1/2, 1/mc) which directly recovers eq. (1) sincep~ p2sin2O, and ~1isL. The spectrum given in eq. (1)has some very interesting properties. Let E 2 and a = I~zB1, and consider = mc the spectrum for s =0+1, which corresponds to particles
435
Volume 67A, number 5,6
PHYSICS LETTERS
4 September 1978
identically so for our spectrum, eq. (1), with the interesting exception that the suppression of the spin up energy states occurs for all a > E0, which is not the case
a< a>E,
for eq. (2). To obtain the equation of state of the system we
e~r
first require the Fermi energy, CF, which is obtained from the number equation: Ia—E,I
a>E
e=~ N=En
En~1+En_1N++N_, p p
p,sp,s
_____________________________________________ ~ { 1. Single particle energy for a spin up particle as a func-
0 Fig.
tion ofits momentum p.
with spin up (p for a neutron is negative): E~÷
2c2 + a2 +E~ 1 [p 2a(p2c2sin2O +E~)112]1/2~
where N~is the number of spins up (down) and is taken to be the T’~0 K Fermi—Dirac distribution function. Hence, taking the sums over p to integrals, we have 3]ffp2 dp sinO do, N~= [2irV/(2irFi) where the appropriate limits on p and 0 are obtained from the constraint [p2c2+a2 +E~
=
—
‘
(3)
Eq. (3) is plotted in fig. 1 for various values of a and o as a function of p. We see that for a the spectrum is monotonically increasing. We note at this point that this is always the case for spin down, s = —1. However, and 0a * ir/2,atthere is a In smooth minimum fora>E0, which becomes cusp 0 = 1T/2. fact here at 0 = ~/2 the energy of the particle is zero forpc = (a2 2. Hence at zero temperature we expect the E~)~’to fill the levels starting at the lowest energy neutrons that is, zero. Thus the intense magnetic field suppresses the single-particle energies for spin up particles with maximum suppression occurring for particles moving at right angles to the field. The consequences of this peculiar behaviour for the Fermi energy will be —
—
+2a(p2c2sin2O +E~)1I2]1/2 This integral gives the following results: (a) For 0 ~ a ~ a 0: 3} N~= [27rV/(21T~lc) X [~(e~ (a—E 2)~’2(2e~ + (a—E 0) 0)(a+2E0)) a (4) + aeF(~w+ arsin( ~ / ~ —
—
N_ = [2irV/(2nhc)3] X
[~(4 (a +E
2)’/2(24 + (a +E
—
0)
quite marked as we will show. The fact that for some momenta the energy difference between the positive and negative energy states can be considerably less than 2E0, and in a particular case, zero, implies that pair production is possible. The possibility of pair production for charged fermions with an anomalous magnetic moment has also been discussed by O’Connell [6], and Chiu et al. [7]. They have shown that when pair production does occur, it does not do ~° at the expense of the magnetic field energy, but requires some thermal energy (however small). This is 436
+ ae
a + Eo) arsin( CF
0)(a 1
—
2E0))
Vi
~~)i’
(5)
where a0 is given by
[~ (
N = 2irV 3 (2~c)
112(3a~+ 5a
0E0)
0E0)
E0 vci +a0(a0 +E0)2 7r/2 +arsin( ~a0 +E0,i)} ~
—
(6)
Volume 67A, number 5,6
The equation for~~F is then obtained by adding N~and N_. We note that~at a = a0, CF = a0 + E0, andN÷= N, N_ = 0. (b) For a0 ~ ~ ~ a1, all spins are up and CF is given by eq. (4)but wi~hN~ replaced byN. The value ofa1 is given by 2
2, 2ira (2ithe)2 ~ —E0)
N V
(7)
1 is necessarily greater than E0, and at a = a1, a1 E0. (c) For a ~3p/2ir2a] a1, we have 1/2 (8) =
—
eF = [(2irllc) Note that for thi~range of B, 6F does not depend on E 0, and that asB-+ 00, e~-+0. In other words, referring to fig. 1, we see that for 2increasin~B thesoparticles E~)1~’2 and are conare trapped in the cusp at ~a range ofp and 0 values, frned to an incre~singlynarrow In fact CF is a mc*otonically decreasing function ofa for all a, and its value at a = 0 is [E~ + (3ir2p)213 .
—
x ~2~,2]
1/2,
The exact exptessions for the ground-state energy and pressure of t1~issystem are extremely complicated, involving integral~of elliptic integrals. Details of these will be given elseWhere. However, a great simplification can be obtained i~’weput E0 = 0. This corresponds to an ultra-relativisüc, large-B limit, since for large B, we see from fig. 1 th~tall the neutrons will have a momenturn of approxim~telypB/c (with 0 ir/2), which will dominate E0/c. 1~hisis also foreshadowed, perhaps, by the fact that for ~ > a1, CF is independent of E0. For the case E0 = 0, the equations for the Fermi energy are easily obtain~dfrom eqs. (4)—(7), and the total internal energy, U, is as follows: (a) For 0 ~a ~a0, 3~ U = [4ir V/(2ithc)~ where 2 2~1/2r2 1 2~ L~—2e,~CF_a ~ ~CF +~a ~, _!
~‘
2 ~
2
where 6F is again given by eq. (8). Note that forE0 = 0, a0 = a1 and the region between a0 and a1 disappears. In the course of this work we have found a very simple interesting relationship for the pressure, F, of an ideal Fermi system at T OK [81: FV=NCF U. Using this relationship we have immediately for the pressure when a ~ a0, P=~U/V. (10) —
This is an interesting result in that it is an example of
where a CF
4 September 1978
PHYSICS LETFERS
a non-interacting system which violates P ~ ~U/V, which was once thought to be a consequence of relativity. Zeldovich [9] has obtainedvia another counter ample, using particles interacting a vector mesonexfield. In our case the violation occurs due to the neutrons interacting with an external field. It would appear from eqs. (9) and (10) that the pressure vanishes asB 00~ since 0. However, thisexactly is strictly a T= result. In realCFsystems, Tis never zero, and OK to make the zero-temperature approximation we just require that CF/k T be very large that is, the gas is ultra-degenerate. In our case however,no matter how small kT is, CF will eventually be much smaller for a sufficiently large magnetic field. The gas wifi then become classical, with the pressure given by the classical ideal gas result P = pkT. Such fields will, of course, be huge, since a E 0 implies B ~‘ 1020 G. At these enormous fields pair production mechanisms must be considered, and also the Fermi energy will eventually be much smaller than the interaction energy of the particles, so that interparticle interactions must be taken into account in the equation of state. All these consideraticz~iswifi readily modify these model conclusions. For a typical neutron star a 4E0, so the above-mentioned effects will be small, and thus it would be of interest to study the effect ofsucha strong magnetic field on the configuration of a neutron star. The importance of these magnetic field effects are currently being investigated. —~
—
~‘
One of us (NEF) expresses his gratitude to D. ter Haar for his hospitality and for the generous frnancial support of an S.R.C. Grant; one of us (AED) acknowledges the support of a C.S.I.R.O. studentship.
1/2
f•’C~ ~F
+~ae~,arsin(.!_I+j~a4lnI_+(___l ~ “az
References
(b) For a> a 0~ 2
(9)
El] V.L. Ginzburg, Soy. Phys. Usp. 14 (1971) 83. E21 D. Ter Haar, Phys. Rep. 3C (1972) 57. 437
Volume 67A, number 5,6
PHYSICS LETTERS
[3] J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55 (1939) 374. [4] N.E. Frankel, G~I.Opat and JJ. Spitzer, Phys. Lett. 25A (1967) 716. [5] I.M. Ternov, V.G. Bagrov and V.Ch. Zhukovskil, Moscow Univ. Bull 21(1966)21.
438
4September 1978
[6] R.F. O’Connell, Phys. Rev. Lett. 21(1968)397. [7] H.Y. Chiu, V. Canuto and L. Fassio-Canuto, Phys. Rev. 176 (1968) 1438. [81 A.E. Delsante and N.E. Frankel, to be published. [9] Ya.B. Zeldovich and I.D. Novikov, Relativistic Astrophysics Vol. 1 (Univ. of Chicago Press, 1971).
PHYSICS LEUERS
4 September 1978
EQUATIO$ OF STATh OF IDEAL NEUTRONS IN AN INTENSE MAGNETIC FIELD A.E. DEL~ANTE School of Thysics, University of Melbourne, Parkvile, Victoria 3052, Australia
and N.E. FRA1~iKEL School ofPl~ysics,University ofMelbourne, Parkvile, Victoria 3052, Australia and Department of TheoreticalPhysics, University of Oxford, Oxford, OXJ 3NP, UK Received 6 ~une 1978
The stati$tical mechanics of a gas of non-interacting, ultra-degenerate relativistic neutrons in a uniform intense magnetic field is presented. Exact results for the Fermi energy are obtained, and the equation of state in the high-field limit is given.
The interior o~fa neutron star is thought to consist mainly of dense egenerate neutrons, and it is generaliy agreed that m~gneticfields of the order of 1012 G exist at the surfa~e[1,2]; however, the size of the field in the interior is ~.inknownand could be larger. The equation of state~for relativistic, non-interacting neutrons was used ii~the work of Oppenheimer and Volkoff [3] to ca1ct~1atethe maximum mass of a neutron star. It is mterest~ngto see what effect an mtense magnetic field would~have on such an equation of state and hence on the crit~ca1mass of a neutron star. In this letter *e consider the statistical mechanics of a gas of non-inte~acting,relativistic neutrons in a urnform magnetic fi~ld.The Dirac equation for the system is [iflc~ +
—
mc2 j
=
o,
C
where B~= m2c3/eh, n = 0,1,2,” and ~ is the anomabus part of the magnetic moment of the fermion: if we let the charge of the particle go to zero, the magnetic moment becomes entirely anomalous. There are then no Landau levels for a neutral fermion. To recover eq. (1) from eq. (2)~wenote that the limit must be taken -,
such that (B/Bc)(2fl + s + 1) -+ p~/m2c2.We then obtain
which has been sblved [4] to yield the following energy levels: 2c2 + ~.s2B2+ m2c4 5= ±[p + 2pBs(p2c2s~n2O+ m2c4)~2]ii~
netic moment of the particle. We note that eq. (1) can also be obtained from the energy levels of a charged fermion with an anomalous magneticmoment in a magnetic field by the following process. The energy levels of such a fermion are [5]: 2 2 Emc [(p/mc) Z 2 21 2~ii~ ~ + [{1 + KB/B )(2n +s + 1)}1~’2+ s~B/mc
E~
E = mc2 [(~~/mc)2+ 1 + (pB/mc2)2 + (2iffls/mc2)(~~
(1)
where s = ±1is t~iespin quantum number, 0 is the angle between the mos~ientump and the magnetic field B, which we take td be in the z-direction, and ~.tis the mag-
+2 1)1/21 1/2, 1/mc) which directly recovers eq. (1) sincep~ p2sin2O, and ~1isL. The spectrum given in eq. (1)has some very interesting properties. Let E 2 and a = I~zB1, and consider = mc the spectrum for s =0+1, which corresponds to particles
435
Volume 67A, number 5,6
PHYSICS LETTERS
4 September 1978
identically so for our spectrum, eq. (1), with the interesting exception that the suppression of the spin up energy states occurs for all a > E0, which is not the case
a< a>E,
for eq. (2). To obtain the equation of state of the system we
e~r
first require the Fermi energy, CF, which is obtained from the number equation: Ia—E,I
a>E
e=~ N=En
En~1+En_1N++N_, p p
p,sp,s
_____________________________________________ ~ { 1. Single particle energy for a spin up particle as a func-
0 Fig.
tion ofits momentum p.
with spin up (p for a neutron is negative): E~÷
2c2 + a2 +E~ 1 [p 2a(p2c2sin2O +E~)112]1/2~
where N~is the number of spins up (down) and is taken to be the T’~0 K Fermi—Dirac distribution function. Hence, taking the sums over p to integrals, we have 3]ffp2 dp sinO do, N~= [2irV/(2irFi) where the appropriate limits on p and 0 are obtained from the constraint [p2c2+a2 +E~
=
—
‘
(3)
Eq. (3) is plotted in fig. 1 for various values of a and o as a function of p. We see that for a the spectrum is monotonically increasing. We note at this point that this is always the case for spin down, s = —1. However, and 0a * ir/2,atthere is a In smooth minimum fora>E0, which becomes cusp 0 = 1T/2. fact here at 0 = ~/2 the energy of the particle is zero forpc = (a2 2. Hence at zero temperature we expect the E~)~’to fill the levels starting at the lowest energy neutrons that is, zero. Thus the intense magnetic field suppresses the single-particle energies for spin up particles with maximum suppression occurring for particles moving at right angles to the field. The consequences of this peculiar behaviour for the Fermi energy will be —
—
+2a(p2c2sin2O +E~)1I2]1/2 This integral gives the following results: (a) For 0 ~ a ~ a 0: 3} N~= [27rV/(21T~lc) X [~(e~ (a—E 2)~’2(2e~ + (a—E 0) 0)(a+2E0)) a (4) + aeF(~w+ arsin( ~ / ~ —
—
N_ = [2irV/(2nhc)3] X
[~(4 (a +E
2)’/2(24 + (a +E
—
0)
quite marked as we will show. The fact that for some momenta the energy difference between the positive and negative energy states can be considerably less than 2E0, and in a particular case, zero, implies that pair production is possible. The possibility of pair production for charged fermions with an anomalous magnetic moment has also been discussed by O’Connell [6], and Chiu et al. [7]. They have shown that when pair production does occur, it does not do ~° at the expense of the magnetic field energy, but requires some thermal energy (however small). This is 436
+ ae
a + Eo) arsin( CF
0)(a 1
—
2E0))
Vi
~~)i’
(5)
where a0 is given by
[~ (
N = 2irV 3 (2~c)
112(3a~+ 5a
0E0)
0E0)
E0 vci +a0(a0 +E0)2 7r/2 +arsin( ~a0 +E0,i)} ~
—
(6)
Volume 67A, number 5,6
The equation for~~F is then obtained by adding N~and N_. We note that~at a = a0, CF = a0 + E0, andN÷= N, N_ = 0. (b) For a0 ~ ~ ~ a1, all spins are up and CF is given by eq. (4)but wi~hN~ replaced byN. The value ofa1 is given by 2
2, 2ira (2ithe)2 ~ —E0)
N V
(7)
1 is necessarily greater than E0, and at a = a1, a1 E0. (c) For a ~3p/2ir2a] a1, we have 1/2 (8) =
—
eF = [(2irllc) Note that for thi~range of B, 6F does not depend on E 0, and that asB-+ 00, e~-+0. In other words, referring to fig. 1, we see that for 2increasin~B thesoparticles E~)1~’2 and are conare trapped in the cusp at ~a range ofp and 0 values, frned to an incre~singlynarrow In fact CF is a mc*otonically decreasing function ofa for all a, and its value at a = 0 is [E~ + (3ir2p)213 .
—
x ~2~,2]
1/2,
The exact exptessions for the ground-state energy and pressure of t1~issystem are extremely complicated, involving integral~of elliptic integrals. Details of these will be given elseWhere. However, a great simplification can be obtained i~’weput E0 = 0. This corresponds to an ultra-relativisüc, large-B limit, since for large B, we see from fig. 1 th~tall the neutrons will have a momenturn of approxim~telypB/c (with 0 ir/2), which will dominate E0/c. 1~hisis also foreshadowed, perhaps, by the fact that for ~ > a1, CF is independent of E0. For the case E0 = 0, the equations for the Fermi energy are easily obtain~dfrom eqs. (4)—(7), and the total internal energy, U, is as follows: (a) For 0 ~a ~a0, 3~ U = [4ir V/(2ithc)~ where 2 2~1/2r2 1 2~ L~—2e,~CF_a ~ ~CF +~a ~, _!
~‘
2 ~
2
where 6F is again given by eq. (8). Note that forE0 = 0, a0 = a1 and the region between a0 and a1 disappears. In the course of this work we have found a very simple interesting relationship for the pressure, F, of an ideal Fermi system at T OK [81: FV=NCF U. Using this relationship we have immediately for the pressure when a ~ a0, P=~U/V. (10) —
This is an interesting result in that it is an example of
where a CF
4 September 1978
PHYSICS LETFERS
a non-interacting system which violates P ~ ~U/V, which was once thought to be a consequence of relativity. Zeldovich [9] has obtainedvia another counter ample, using particles interacting a vector mesonexfield. In our case the violation occurs due to the neutrons interacting with an external field. It would appear from eqs. (9) and (10) that the pressure vanishes asB 00~ since 0. However, thisexactly is strictly a T= result. In realCFsystems, Tis never zero, and OK to make the zero-temperature approximation we just require that CF/k T be very large that is, the gas is ultra-degenerate. In our case however,no matter how small kT is, CF will eventually be much smaller for a sufficiently large magnetic field. The gas wifi then become classical, with the pressure given by the classical ideal gas result P = pkT. Such fields will, of course, be huge, since a E 0 implies B ~‘ 1020 G. At these enormous fields pair production mechanisms must be considered, and also the Fermi energy will eventually be much smaller than the interaction energy of the particles, so that interparticle interactions must be taken into account in the equation of state. All these consideraticz~iswifi readily modify these model conclusions. For a typical neutron star a 4E0, so the above-mentioned effects will be small, and thus it would be of interest to study the effect ofsucha strong magnetic field on the configuration of a neutron star. The importance of these magnetic field effects are currently being investigated. —~
—
~‘
One of us (NEF) expresses his gratitude to D. ter Haar for his hospitality and for the generous frnancial support of an S.R.C. Grant; one of us (AED) acknowledges the support of a C.S.I.R.O. studentship.
1/2
f•’C~ ~F
+~ae~,arsin(.!_I+j~a4lnI_+(___l ~ “az
References
(b) For a> a 0~ 2
(9)
El] V.L. Ginzburg, Soy. Phys. Usp. 14 (1971) 83. E21 D. Ter Haar, Phys. Rep. 3C (1972) 57. 437
Volume 67A, number 5,6
PHYSICS LETTERS
[3] J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55 (1939) 374. [4] N.E. Frankel, G~I.Opat and JJ. Spitzer, Phys. Lett. 25A (1967) 716. [5] I.M. Ternov, V.G. Bagrov and V.Ch. Zhukovskil, Moscow Univ. Bull 21(1966)21.
438
4September 1978
[6] R.F. O’Connell, Phys. Rev. Lett. 21(1968)397. [7] H.Y. Chiu, V. Canuto and L. Fassio-Canuto, Phys. Rev. 176 (1968) 1438. [81 A.E. Delsante and N.E. Frankel, to be published. [9] Ya.B. Zeldovich and I.D. Novikov, Relativistic Astrophysics Vol. 1 (Univ. of Chicago Press, 1971).