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Neutron matter
Nuclear Physics A154 (1970) 202 --224; ~ ) North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
N E U T R O N MATTER Binding energy and magnetic susceptibility E. O S T G A A R D t
Institute for Pure and Applied Physical Sciences, University of California, San Diego, La dolla, California 92037 ** Received 9 March 1970 (Revised 18 June 1970) Abstract: The binding energy and the magnetic susceptibility of neutron matter is caiculated by means of Brueckner theory. A modified Brueckner-Gammel method is used to solve the BetheGoldstone equation, calculate the reaction matrix or G-matrix, and obtain the interaction energy contribution from two-body correlations. For simplicity, the approximation of a reference energy spectrum with an effective mass and quadratic m o m e n t u m dependence is used for the input energy spectrum, which in principle should be fitted to self-consistent single-particle energies. The intermediate-state potential energies are, however, c h o s e n to be equal to zero. Hence, the three-body and possibly higher-order energy contributions should be estimated by separate calculations. The Bethe-Goldstone equation is solved numerically by iteration to give the perturbed two-body wave functions. Afterwards, the G-matrix elements are calculated by numerical integration. The binding energy is calculated as a function of the Fermi m o m e n t u m and the pure neutron-matter system seems to be u n b o u n d at any density, even at very low densities. In addition to the binding energy, also the magnetic susceptibility is estimated. To this end, the spin symmetry energy is calculated in terms of the derivatives of the G-matrix with respect to the Fermi m o m e n t u m . A ferromagnetic transition is found to occur at a density m u c h higher than the typical nuclear-matter density. Our method is, however, probably too simple to be reliable at such a high density.
1. Introduction A body with a large mass under gravitational attraction, may exist in two possible equilibrium states. If the mass is sufficiently large, a more compressed state with neutronic configuration would be more favorable than the less condensed electronnuclear state. Such a condensed state might exist as a neutron star, or at least as a neutron core formed by a supernova explosion at the last stage of the stellar evolution o f a sufficiently massive star. At the end of the nuclear evolution, the star is very hot and dense at the center, and the stellar core becomes unstable against gravitational collapse. This triggers a supernova explosion, which under appropriate conditions leaves behind a superdense core. The temperature is possibly sufficiently low that thermal energies are small compared with nuclear energies and with the Fermi energy o f the electrons. One then has an almost pure neutron gas, i.e., neutron matter. It is a * This research was supported by the United States Atomic Energy Commission under Contract A T ( 1 1 - 1 ) I G E N - 1 0 , P. A. 11 : Report N o . : UCSD-10P11-103. tt Present address: Institut Max y o n Laue-Paul Langevin, 8046 Garching bei Miinchen, Germany. 202
NEUTRON MA'I-EER
203
neutral nuclear matter, which is under strong hydrostatic pressure from gravitational forces. In principle, however, a superdense star is more complicated, with possibly a hyperon core, a neutron-dominant layer around that core, and an outer envelope of ordinary matter, i.e., electrons, protons and nuclei. The total energy of the neutron star may be written as the sum of its gravitational energy, rest-mass energy, nuclear energy, and the relativistic kinetic energy of the electron gas. Most mass is possibly concentrated in the superdense core. The neutron matter around consists of neutrons in equilibrium with a small amount of protons and electrons. This neutron layer is essentially isothermal because of high conductivity. At the surface of the star, there is possibly a crystalline ionic lattice or a solid mantle. Several theoretical models have been constructed, but the equilibrium configurations obtained for neutron stars are rather dependent on specific assumptions about the nuclear interactions, or, more directly, the equation of state for high-density matter. Chiu 1) discussed both the dynamic and static problems of neutron stars, making rough estimates of temperature and cooling time. He concluded that it should be possible, in principle, to detect neutron stars by their radiation of X-rays. This observation is prevented on the earth because the atmosphere is strongly opaque to radiation in the X-ray region. However, the interstellar medium is practically transparent to X-rays, so neutron stars should be detectable above the earth's atmosphere. Soon Xray sources outside the solar system were discovered from rockets and reported, their identification with neutron stars was proposed and their properties investigated. In a similar way, radio emission from a neutron star with a strong magnetic field can account for the radio emission of pulsars. A number of these pulsating radio sources were discovered, and theories were proposed. The regularity of the emissions lead to the idea that they are related to density oscillations of condensed matter. The nature of the radio sources is, however, an open question. It is possible that they are caused by oscillations of neutron stars, although there may be some difficulties with the assumption. A better understanding of neutron matter based on the nuclear interactions involved, could give better estimates of frequencies and lifetimes for X-ray or radio emission by such stars, and it is also of interest to find out whether the neutronneutron force alone can give binding to a pure neutron system. It is then important to treat neutron matter by sophisticated many-body methods. Brueckner et al. 2) applied the methods of Brueckner and Gammel 3), developed earlier for the nuclear-matter problem, to calculate the energy of a pure neutron gas or neutron matter as a function of density. They found that a neutron gas is not bound at any density. Salpeter 4) and Sood and Moszkowski s) looked at the lowdensity limit by calculating the interaction energy, using the S-state phase shifts in the phase-shift approximation 6-s), and Levinger and Simmons 9) made some work on the neutron-neutron potential near normal nuclear-matter densities. In addition to later Brueckner-type calculations by Nemeth et al. 10, ix), Wang 12) used a Jastrow method including only two-body contributions, i.e., a correlated basis
204
E. liJSTGAARD
function to represent the ground state wave function of the system, and Binder et aL ~3) and Pearson and Saunier 14) have also calculated the average energy per particle as a function of density. None of the calculations made of the ground state energy of the neutron system, gives any indication of a bound state due to the nuclear neutron-neutron force alone. The most promising models of pulsars involve rapidly spinning neutron stars with intense magnetic fields. As discussed by Canuto and Chiu 15, 16), neutron stars may have very intense magnetic fields and can emit radio energy with a rate and spectrum similar to those of pulsars. The possible existence of a ferromagnetic state in neutron stars would be of considerable interest, with influence on the equilibrium composition and evolution of the star. It thus seems appropriate to investigate the magnetic properties of pure neutron matter. Brownell and Callaway 17) discussed the possibility of a ferromagnetic state for neutron matter at densities corresponding to theoretically stable neutron stars. Neglecting the attractive part of the nuclear interaction, assuming only a state-independent hard core, they replaced reaction-matrix elements between pairs of particles by constants assumed to be their average values. It was concluded that as the density of superdense stars increases to more than the equilibrium density of normal nuclear matter, the ground state might experience a ferromagnetic transition 18, ~9). However, the effect of the attractive part of the nuclear interaction may cancel this conclusion. Also, the exchange character, i.e., the spin-dependence of the two-nucleon potential, will oppose ferromagnetism. If the ground state of neutron matter ever turns ferromagnetic, the transition will occur at such high density that current nuclear many-body theories are probably not applicable 20, 21). Migdal 22), Ginzburg and Kirzhnits 23), and Wolf 24) have argued that a neutron star or at least the central region may be superfluid. This follows from the assumption that the residual interaction between neutrons with antiparallel spins is attractive for the relevant momentum region in high-density neutron matter. The idea was later supported by Ginzburg 25) and Baym et al. 26, 27). Strong interactions could make neutron superfluidity possible at the general density expected in the neutron star interior and below some critical temperature. At high densities, however, there is little known generally about the properties of matter. And the density in the central region may be too high for neutron superfluidity, at least for S-state pairing-type superfluidity, except in very light stars. 2. G e n e r a l t h e o r y
A neutron star is a mechanical system in a quasi-hydrostatic equilibrium, which must have a negative total energy. This star is under rotation and vibration, and emits radiation. The radiation consists of neutrinos, X-rays, long-wavelength electromagnetic waves, and gravitational waves. A neutron star has a typical radius of about 10 km and a mass of the order of the solar mass. To do calculations on the structure and the evolution, we should know the equation of state, i.e., the pressure as function of the density up to ordinary nuclear densities and somewhat higher.
NEUTRON MATTER
205
Neutron star matter is an idealized infinite system of neutrons, protons and electrons which is electrically neutral. It consists mainly of neutrons, and the number of protons and electrons is very small because of the very large relativistic kinetic energy of the electrons at nuclear-matter densities. The neutrons form a degenerate Fermi sea, and interact by very strong short-range nuclear forces which induce strong twobody correlations between the particles. Three-body contributions to the energy will become more and more important as the density increases, and should then be taken into account together with other corrections. A pure neutron system seems not to be bound, even at very low densities, and there is no relative minimum in the energy as function of density. As the density increases, the lack of binding increases, probably causing a greater density of protons. Although ferromagnetism is not expected at nuclear densities, and ordinary nuclear matter is not ferromagnetic, it can possibly be expected in a neutron star at higher densities when the repulsive potential core is more dominant. For a gas of fermions with only short-range repulsive interactions at a sufficiently high density, it should be energetically favorable for the particles to align their spins. This would reduce the potential energy because of the Pauli exclusion principle acting to keep the particles from approaching each other. There is, however, also an increased kinetic energy, since in the ferromagnetic state only one particle can occupy a given momentum state, but it is not correspondingly great. The particles can avoid each other anyway, when the density is low, but at high densities they would possibly align their spins. We then would get a ferromagnetic instability. However, an important influence on the criterion for ferromagnetism in real neutron matter is the inclusion of the more long-range attractive interaction. The effect of the attractive component of the two-nucleon potential is to shift the critical density for onset of ferromagnetism to considerably higher values. With realistic forces included, there is no ferromagnetic instability of the normal ground state of the system for mass densities of the order of ordinary nuclear-matter density. With the attractive part of the nuclear interaction restored, there is a deep attraction in singlet states just outside the repulsive core, which opposes the tendency towards spin alignment. Because of this attraction, a pair of particles should approach each other closely with spins antiparallel. Any triplet attraction will favor ferromagnetism, but the odd-state attractive interactions are much weaker than the even-state attractive interactions. For neutron densities at nuclear-matter densities and somewhat higher, there will be cancelling effects due to repulsion and attraction in the two-neutron force, but a ferromagnetic instability is not expected to occur until at significantly higher densities. The neutron matter problem is basically simpler than that of nuclear matter, since the Pauli principle prohibits all triplet-even and singlet-odd state interactions. The neutron-neutron interactions occur in triplet states of two-body isospin and the potential can be separated in singlet-even and triplet-odd components involving only spin projection operators. Thus, the strong triplet-even tensor, the triplet-even spin-
206
E. OSTGAARD
orbit, and triplet-even and singlet-odd quadratic spin-orbit components do not appear. The only non-central components of the two-nucleon potential which are there, in principle, are the triplet-odd tensor and spin-orbit components, and the triplet-odd and singlet-even quadratic spin-orbit components. The extra-core oddstate interactions are, however, quite weak, and are in any case not important at low densities and low relative momenta. As the spin-orbit components also are quite weak, one can probably neglect noncentral components and give a reasonable description of neutron matter in terms of a purely central potential, at least if the density is not too high. Brueckner and others have developed a method to handle the strong repulsion in the two-body potential. Starting from a perturbation expansion for the energy, the expansion in terms of the large matrix elements of the potential is rearranged and replaced by an expansion in reaction matrix elements. This reaction matrix, or Gmatrix, is obtained by solution of a two-body problem in the medium. To include effects of the Pauli exclusion principle and the single-particle energy spectrum arising from the average interaction of each particle with the medium, Brueckner and Gammel a) use a Green function method to obtain a G-matrix propagator in coordinate space. In the integral equation for the G-matrix, single-particle potentials given by sums of diagonal G-matrix elements are included in the energy denominator, which introduces a problem of self-consistency. The method can, however, be modified and simplified, and supplemented by calculations of the contribution to the binding energy from three-body correlations. In their nuclear-matter calculations, Brueckner and Gammel treated the hard core in the two-nucleon potential in a special way. They assumed that the integrated product of the potential and the perturbed wave function has a finite value at the core boundary, but vanishes inside the core; i.e., they assumed a 6-function at the core boundary for the repulsive part of this product. In the Brueckner-Gammel method, the energy is calculated from a reaction matrix similar to the transition matrix for scattering in free space. This reaction matrix, or G-matrix, is defined by the integral equation G = v-v(Q/e)G
(2.1)
in operator form. Here v is the two-body potential, which is assumed to be the same as in free space, and the Pauli exclusion operator Q prevents scattering into occupied intermediate states. The energy operator e includes kinetic and potential single-particle energies, and is written e = ea+eb--em--en. (2.2) The single-particle energies e,, and e, are self-consistent energies for particles moving in the Fermi sea, and 8a and eb are energies of virtual excitations above the Fermi surface.
N E U T R O N MATTER
207
The energy of an unexcited neutron is
~(k,,) = T(kn)+ U(k,,),
(2.3)
and the total ground state energy is
E = • [T(k,,)+½U(k,,)],
(2.4)
n
where T is the kinetic energy
T(kn) = - - -h12 k.2
(2.5)
2M and M is the neutron mass. The single-particle potential U is given by the diagonal G-matrix elements by the relation
U(k,) = ~ [- ],
(2.6)
/rl
with summation over occupied states. The second term in eq. (2.6) comes from exchange of spin and m o m e n t u m coordinates. The Fermi momentum, PF = hkF, is related to the density by
where N is the total number of particles in a large volume f2. The normal density can then be determined from the minimum of the binding energy as a function of the number density N _ 1 k~ P--£2 3 ~2'
(2.8)
if such a minimum exists. We also define a mean interparticle spacing r 0 by p--1
--
"~ - - 4 g y 0 3 "
(2.9)
N Then
kr =
(9),
/ro = 1.92/r0.
(2.1o)
For a given Fermi momentum, assuming zero spin polarization, the density of symmetrical nuclear matter is twice that of pure neutron matter. We introduce c.m. and relative coordinates, and if we assume that the G-matrix depends on the total two-body m o m e n t u m only through its magnitude 2P, we can write a partial wave expansion with matrix elements GL(P, k,,,), where GL is a function of c.m. and relative momenta. The properties of the system can then be determined from the G-matrix. For central forces only, eq. (2.3) for the energy of a particle
208
I;'. I~STGAARD
moving on the energy shell with momentum kn becomes
~(k~) = ~ h2 k] + 2 . 2. Z [¼ Y, (2L+l)
m
even
L
+~ Y~ (2L+ l)(k.nIG,~Ikm)],
(230
oddL
with the sum over m taken over the Fermi sea. The first factor of 2 in the interaction term comes from two spin states per momentum state, and the second factor of 2 comes from the exchange term. The factors ¼ and ¼ give the relative weights of the singlet-even and triplet-odd states. We will later include all these factors in our definition of the Gr~ matrix elements. 3. Two-body terms According to general scattering theory, the two-body wave function is defined by c ¢ --
v~,
(3.1)
where ~b is the unperturbed free-particle wave function and ~ is the perturbed one. The distortion of the wave function because of the potential is written as = gb-~,.
(3.2)
Due to the hard or at least strongly repulsive core, the momentum-space matrix elements of the potential v are infinite or at least very large. The reaction matrix equation (2.1) is then reformulated and converted into coordinate space, and dividing on the left by v and multiplying from the right by ~b, we get the Bethe-Goldstone equation 2s) = ~b- --Q v@.
(3.3)
e
The propagator Q/e is a rather complicated nonlocal integral operator. We write eq. (3.3) in coordinate space, and after an expansion and separation into partial waves, we get the Bethe-Goldstone integral equation in the form
ur.(ko, r) = ~¢r.(ko r ) +
fo ' FL(r, r')vn(r')uz(ko,
r')dr',
(3.4)
where we have introduced
Jn(ko r) = ko rjL(ko r), uL(k o , r) = ko r~/n(ko, r), XL(ko, ~) = J,(ko ~)-- Udko, ~). 7[hen J L ( k o r)lcorresponds to the free-particle wave function.
(3.5)
N E U T R O N MATTER
209
If k o is the initial and k the intermediate-state relative momentum of the two interacting particles, the energy denominator e is defined by a reference energy spectrum 29)
e(k) = h2 ~2+ k2 M
m*
(3.6) '
where
2 = 2Ak2_k 2
(3.7)
for propagation on the energy shell and k o < kF. The basic parameters of the energy spectrum are m*, which is the dimensionless effective mass defining both the hole spectrum and the intermediate-state particle spectrum in our approximation, and A, which is a measure of the gap between the occupied and the intermediate-state energy spectra. We will later set m* = 1. Then the gap A or the parameter ?2 just defines the hole spectrum, which will be our real input parameter. The Green functions FL(r, r') are then defined by
rL(r,
r') =
-
m*
dk JL(kr)JL(kr')Q(e, k) ~2+k 2 ,
~
(3.8)
with propagation off the energy shell in the intermediate states. The Pauli exclusion operator Q(P, k) is approximated by its average over angles of the average momentum P, i.e., 0
for k < ( k ~ - p 2 ) ~
Q(P, k) = (p2+k:-k~)/EPk for ( k 2 - p 2 ) ~r < k < kF+P, ~1 for k > kF+P.
(3.9)
To obtain the integral (3.8) as accurately as possible, using the property of the Pauli function (3.9), we write eq. (3.8) in the form ~, (k2F -- p 2 ) "!"
r,(r, r') = r~°)(r, ~') + ~2m*
Jo
dk JL(k~)J,(kr')/(~ 2 + k')
+ - m* | 7~
dk JL(kr)JL(kr' )
,
(3.10)
./(k2F-P2)~
where
F[°)(r, r') = -- (2/tOm* f ~dk JL(kr) JL(kr') j o 72 + k2
(3.11)
corresponds to a Green function in which the exclusion principle is not included. For this latter function analytic expressions can be obtained, and for the lowest partial
210
E. e s ' r G ~
waves we get the solutions m*
r~o°)(,, r') = ~
{exp E-~(r+~')]-exp [-~1~-"1]},
---
a+
x+
exp[-~(,+,')] + [14- yl--r~IlYr ~r,l exp [ - T l r - r'.]} ,
r?'(,.
,3
=
1+
-
+
1+
-
yr'
7r
+
(3.12)
.
yr3--;+ ( ~ ?
xexp[-y(r+r')]-[l+3+~l[1T-yr
exp [-ylr-r"]} ~
and so on. In the expressions above, the upper sign is for r > r', the lower sign is for r
W e are also interested in the volume integral
~ = p
: I~(ko, r)12d~ = 4rcp :ooI~(ko, r)[ 2r2dr.
(3.13),
This quantity is proportional to the probability of finding a particle in an excited state. rather than in the Fermi sea and gives information about the saturation propertie~ of the system. We write x = 4rrp[½ ~
+½ ~ ](2L+ 1 ) f ~ (ZL~2dr = 2~rp ~ XL,
©venL
oddL
JO
\ko/
L
(3.14)'
which will be our definition of XL, i.e., we include appropriate statistical factors. The integral in eq. (3.14) ,which gives the volume of the correlation hole or the size of the wound in the wave function, is calculated as (ZL~ dr
\ko/
= ~2 ~o ~F~(k°' k)dk,
(3.15),
because an integration in momentum space seems more accurate here than an integration in coordinate space. The Fourier transforms F~.(ko, k) are given by
FL(ko, k)
ko ~
JL(kr)zL(ko, r)dr,
(3.16),
or
FL(ko, k) - Q(P' k)m* ko, f: ~2 + k2
j~(kr)v,(~)u~(ko,r)dr,
(3.17)
NEUTRON MATTER
211
because
zr(ko, r) = Q vLuL(ko, r).
(3.18)
e
Eq. (3.17) seems more accurate than eq. (3.16) because the values for zz(k o, r) will be more uncertain and have a greater relative error than the values for the wave function uz(ko, r). Also, the exclusion operator Q(P, k) gives a more correct behavior at k ~ kF. We will later set P = 0 in our calculations. Potentials and energies are usually expressed in units of MeV. But in the following sections, energies are also expressed in units of fm -2, and the conversion factor is
h2/M
--- 41.47 MeV • fm 2.
(3.19)
4. Binding energy We have now simplified the original Brueckner-Gammel method and made it more convenient in two ways. The first is to express the total Green function (3.8) as a reference spectrum Green function (3.11) which has an analytic form, plus a correction term which takes the exclusion principle into account, and in which the range of integration in momentum space is only of order k F. The second way is to use the reference spectrum idea for the energy denominator in the Bethe-Goldstone equation. It is very useful to have a reference spectrum or an effective-mass approximation for this denominator, and the release of the energy denominator from the loop of selfconsistency through the G-matrix elements and the single-particle potentials is a great advantage in the calculations. The use of single-particle energies calculated from G-matrix elements would involve extensive computing time. The average binding energy per particle is
EB_ 3
h2
10 M
k2 + ~fi:Fk 2 ( 1 _ 3 ko + 1 ka]dk, 2 kr 2 k~]
(4.1)
.or
kUo
M
where the single-particle potential (2.6) is given by zl
U(k)
/*~(kF-k)
k2dk°
= 5Jo 2
½(kF+k) 2
The first integral vanishes for k >
~o
(for k < -]
kF)
(4.3)
dk°"
k v. The diagonal
G-matrix elements are calculated
212
E. eSTGAARD
as
= • r =
--
2
4r~ r E ko2
even L
oo
+ 3 E ]( L + l ) f ,J 0
oddL
a¢'L(kor)vL(r)ur(ko, r)dr,
(4.4)
and we take this as our new definition of GL, i.e., we include appropriate statistical factors. The calculations are done for a simple central potential with hard core, i.e., the Moszkowski-Scott potential 3o), which is defined for even L as ~+oo
VL(r)
=
t-- 11"oexp [ - # ( r - - c ) ]
for r < c, for r > c,
(4.5)
V[MeV 200
I00
r[f°] I
I
3.0
-I00
-200
Fig. 1. The Moszkowski-Scott potential defined by eqs. (4.5) and (4.6).
where c -- 0.4 fm, /~ = 2.083 fm, Vo = 260 MeV = 6.27 fm -2.
(4.6)
The potential is shown in fig. 1. It is assumed to vanish in odd angular m o m e n t u m states. More realistic and " m o d e r n " potentials include tensor-force components, and it has been shown that a finite or soft core in the potential can increase the binding energy in nuclear matter by several MeV [refs. 31, 32)]. However, for the neutron gas
NEUTRON MATTER
213
we have no T = 0 or T = I, T3 = 0 interactions, and the extra-core triplet-odd-state nucleon-nucleon interactions are known to be significantly weaker than the extra-core singlet-even-state interactions. It is also more convenient to use a hard-core rather than a soft-core or velocity-dependent repulsion in the potential. The strongly repulsive core is necessarily central. In the Brueckner-Gammel method, the hard core in the potential is then treated in the following way. We separate the integral on the right-hand side of eq. (3.4) into two parts at the core boundary, and make the replacement
VL(r)UL(ko, r) = 2Lr(r--c )
for r < c,
(4.7)
where 2r is determined by the condition that the wave function vanishes for r = c. Then
2L = -- JL(ko C)+
FL(c, r)vL(r)uL(ko,
FL(c, c),
r)d
(4.8)
and eq. (3.4) becomes
ur.(ko, r) = SL(ko, r)+
f; fgz(r, r')VL(/)uL(ko, r')dr',
(4.9)
where
sL(ko,
r) =
eCL(ko r)-- aCr.(koc) ~
r) c)'
(4.10)
and
fCdr, r') = rL(r, r')- rL(r, c)Fdc, r')
r,.(c, c)
(4.11)
The above treatment of the hard core is equivalent to assuming a "hollow-hardcore" potential 33), neglecting contributions from the region inside the core. The method seems to be a very good approximation 34), at least for calculations on the energy shell and for relative momentum k o < kr, at not too high densities. And the results should not be very different from results obtained by a more "exact" treatment of the hard core in the two-nucleon potential. The assumption (4.7) must be replaced by a more complicated version when dealing with the coupled equations that arise in the presence of tensor forces. But it is sufficient in our case, because we consider a simple potential with central forces only. The radial integral in the G-matrix elements (4.4) is finally calculated as
fo JL(ko0V,(.)uL(ko,.)d.
=
J~(koc) + rde, c)
f? ~(ko, 0v~(r)u,(ko, 0d~:
(4.12)
The wave functions uL(ko, r) are determined by numerical integration and iteration of eq. (4.9). Afterwards, the G-matrix elements are calculated by numerical integration.
214
E. OSTOAARD TABLE 1 Converg ence p a r a m e t e r Z defined by eqs. (3.13) a n d (3.14). A = 0.55. m* = 1.0
x]
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
1.0
1.36
1.5
2.0
2.5
3.0
3.5
0.017 0.017 0.017 0.017 0.018 0.018 0.018 0.019 0.021
0.026 0.026 0.026 0.026 0.026 0.026 0.027 0.028 0.031
0.033 0.033 0.033 0.033 0.033 0.033 0.034 0.036 0.040
0.076 0.075 0.075 0.074 0.074 0.074 0.075 0.079 0.085
0.147 0.146 0.145 0.143 0.140 0.138 0.136 0.137 0.142
0.250 0.248 0.245 0.239 0.232 0.223 0.215 0.209 0.207
0.390 0.388 0.380 0.368 0.351 0.333 0.313 0.295 0.280
TABLE 2 D i a g o n a l GL m a t r i x elements in [fm] for even L, weigh t of (2L-k1) i n c l u d e d ; A = 0.55, m* = 1.0 kF[fm - 1 ]
ko/kF
L = 0
L ~ 2
L = 4
Total
0.5
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--35.3 --35.2 --34.8 --34.3 --33.6 --32.8 --32.0 --31.3 --30.9
0 0 0 0 --0.1 --0.1 --0.3 --0.5 --0.7
0 0 0 0 0 0 0 0 0
--35.3 --35.2 --34.8 --34.3 --33.6 --32.9 --32.2 --31.7 --31.7
1.5
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--24.9 --23.7 --20.4 --16.1 --11.7 --7.8 --4.5 -- 1.9 0
0 0 --0.3 --1.0 --2.3 -- 3.9 --5.6 --7.0 --8.3
0 0 0 0 --0.1 --0.2 --0.4 --0.8 --1.3
--24.9 --23.7 --20.7 --17.2 --14.1 -- 11.9 --10.5 --9.8 --9.5
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875
--14.1 --8.4 1.3 7.9 11.0 11.8 11.3 10.0
0 --0.4 --3.3 -- 6.6 -- 8.4 --8.7 -- 8.1 --7.0
0 0 --0.1 --0.8 --2.0 --3.2 --4.2 --4.8
--14.1 --8.8 --2.1 0.4 0.6 --0.1 --0.9 -- 1.8
-- 5.6
-- 5.2
--2.5
3.5
1.00
8.3
215
NEUTRON MATTER TABLE 3
Binding energy for n e u t r o n matter
kr. [ f m - 1] 0.25 0.5 1.0 1.36 1.5 2.0 2.5 3.0 3.5
r0 [fm]
p [fm - 3]
7.68 3.84 1.92 1.41 1.28 0.96 0.77 0.64 0.55
0.001 0.004 0.034 0.085 0.114 0.270 0.528 0.912 1.448
W2 [MeV]
TF [MeV]
En [MeV ]
--0.3 --1.5 --7.0 --13.9 --17.3 -- 31.6
0.8 3.1 12.4 23.0 28.0 49.8 77.8 112.0 152.4
0.5 1.6 5.5 9.1 10.6 18.1 33.4 68.5 142.8
--44.4 --43.5 --9.6
Only two-body terms are included. W2 is the two-body contribution to the interaction energy, TF is the average kinetic energy per particle, and EB is the binding energy, d = 0.55. m* = 1.0.
ko/kF I
- - - - o . 5 ~
I
~.,o
I
-I0
1
i
-20
-30
Fig. 2. Diagonal GL matrix elements for even L. Weight o f ( 2 L + l ) included, kr = 1.0 fm -1. A = 0.55, m* = 1.0.
216
E. ~ $ T G A A R D
GL[fo]
I
10t-
I
\[L.o] ,
I
!
iko/~
ZIL'4]
1.0 v
J -10
-20
Fig. 3. Diagonal GL matrix elements for even L. Weight of (2L-t-l) included, kr /I
=
=
3.0 fm -1.
0.55, m * = 1.0.
,EB[MeV] [NSB,PS]~ 20
_
[,,o,<]~
/
.//"-~//
..I' / ' /
[wl IO
0
~
"
J."'"'""
I......................... I 0.5 1.0
~rsl I L .I 1.5
I
,.~, -,~
r'FLVmJ
2.0
Fig. 4. Binding energy for neutron matter as a function of the Fermi momentum. Only two-body terms are included, d = 0.55, m* = 1.0. Comparison is made with other results given by Brueckner et al. 2) (BGK), Salpeter 4) (S), Levinger and Simmons 9) (LS), Nemeth et al. lo) (NSB), Nemeth and Sprung 11) (NS), Wang 12) (W), Binder et al. 13) (BPR), and Pearson and Saunier 14) (PS).
NEUTRONMATTER
217
The following mesh was used for the radial integration in the calculations: r, r' = (0.1)0.4(0.05)0.6(0.1)1.2(0.2)3.6(0.4)6.0 fro.
(4.13)
The integrations are carried out to a distance beyond which the exponential behavior of the potential makes a further contribution quite negligible. First we calculate and store the Green functions (3.10) and (4.11) for all distinct pairs of points. The F[°)(r,r ') are evaluated directly from eq. (3.11) and the correction integrals FL(r, r')-F~°)(r, r') are done numerically with the following mesh:
k/k e = 0(0.05)1.0
for k < kp = ~/k~-+P 2.
(4.14)
The radial wave equation (4.9) is solved by iteration, starting at r = c and proceeding outwards. The convergence is improved somewhat by using the results of the last iteration for the points r' < r in each cycle. The integrations were done by Simpson's rule. However, it was necessary to use an improved Simpson's method in the integration ofeq. (4.9). Because of the shape of the Green functions FL(r, r') and ffr.(r, r') with a peak at r = r', this point should always be at the end of a three-point Simpson interval. Results from the calculations are shown in tables 1-3 and in figs. 2-4. Table 1 gives values for the convergence parameter (3.14) as a function of relative momentum and density. Table 2 and figs. 2 and 3 give G-matrix elements in dimensions [fm], these results can be translated to [ M e V - f m s] according to eq. (3.19). Table 3 gives the binding energy together with some corresponding parameter values, and fig. 4 shows the binding energy as a function of the density, or, rather, the Fermi momentum. Comparison can be made with other results given by Brueckner et al. 2), Salpeter 4), Levinger and Simmons 9), Nemeth et aLlo, 11), Wang lz), Binder et al. 13), and Pearson and Saunier 14). Sood and Moszkowski s) have also given results for very low densities. 5. Magnetic susceptibility
The magnetic susceptibility is a measure of the energy required to produce a net spin alignment in a given direction. It is inversely proportional to the energy required to polarize the spins. This energy has been given for liquid 3He in the limit of small net spin alignment a s). Following the same procedure here, we get an expression for the energy of spin alignment, given in units of the spin polarization energy of the corresponding ideal Fermi gas. If we apply a magnetic field to the system, the two spin populations will no longer be equal in the ground state, and the total energy will be a function of the spin polarization parameter
N(+) N -) s - N(+)+N(_ ) ,
(5.1)
218
E. OSTGAARD
where N (+) and N (-) are the numbers of particles with spin up and spin down with respect to the applied field. The total magnetization ~ is = y ( N ( + ) - N (-)) =
ysN,
(5.2)
where 7 is the magnetic moment per particle, and N is the total number of particles. The magnetic field is then given by ~E
(5.3)
and the total magnetic susceptibility is defined as _-
z =
~-~
.,.
\a~1.,,,
/~as2t.,NIs=o"
(5.4)
Only the susceptibility for a small field is considered, s o (~2E/OS2)p.Nis evaluated for s = 0, and the susceptibility is determined by the coefficient of the s 2 term in the expansion of the energy E(s) about s = 0. We write
~as z]
~ as~ ] '
where E r is the energy of the noninteracting ideal Fermi gas, which has the susceptibility XF = ~2M kF h 2 ~2"
(5.6)
The Fermi momentum k(F±) of the two spin populations containing N (+) and N (-) particles, are given by the density relation p(±) _ N (±) I k(F:I:)3 I2 - 6 z~2 ,
(5.7)
or, in terms of the spin polarization parameter s, we have N ( ± ) = ½N(I __.s),
k(~+) = kF(l +s) ~.
(5.8)
To calculate the spin symmetry energy, we expand the total energy E(s) in powers of s, neglecting terms of higher order than s 2. We calculate the second derivative of E(s) with respect to s for s = 0. The first derivative of the energy vanishes. Following the same procedure s 5) as for liquid SHe, the magnetic susceptibility is finally obtained
NEUTRON MATTER
219
from eq. (5.5) as = M*
2 h2
kF
+ 2 ~22 ~ f f
[ae(kF, 3 - 1
(41ZkF)
kF)a,,.-ao(kv, kv)~.]
dk, kF O a o (ki' kF) ~k F
+
kf (4nkg)-'dkff(4rck~)-~dkjkv [k F ( ~ ) - - 2
(~kv)? [a.(k,, kj)
%
(5.9)
+ 3ao(k,, kj)]/•!
The first term depending on the effective mass m* comes from the change in the density of the single-particle levels due to the interaction, or arises from the momentum dependence of the single-particle potential. The second term comes directly from the spin dependence of the two-body interaction, or arises from the change in the interaction as spins are shifted from antiparallel to parallel alignment. The third term comes from the rearrangement terms in the single-particle energies, or arises from the variation of the reaction matrix with the Fermi momentum because of the spin polarization. The contribution from even and odd angular-momentum states is given by
ae(k,, kj) = Z (k,ilGL(k(F+), k(v-))lk,~>, eve. L ao(ki, k.i) = oddZ L(k,jlGL(k(v+), k(F-))lk,s>•
(5.10)
The statistical factor of 3 for odd L is here omitted from the Gr matrix elements, while it has been included in our earlier definition (4.4) of Gr. The factor of (2L+ 1) is still included as before. However, we need only a e and the corresponding derivatives because of our simple potential (4.5). After transforming the integrals in eq. (5,9) to
f dk,, = 16~zf~Fk2 (1-- ~FF)dko, (5.11)
f dki f
dk i
128 ~Z2kF3 = -T
(
ko2 1 - ~
~
+½
k~!
dko,
the third term in eq. (5.9) can be written as
1 - M ---230 kF 7t ~lx2 I(~-x+½xa) (k2 -~v --2kF --Oa~ OkF Ok~] + ( 4 - 2 x - Z x 3 )
~kv]A
(5.12)
220
E. I~ISTO,~J~,RD
where (5.13)
x = k o / k F.
The magnetic susceptibility is then obtained from eq. (5.9) after numerical estimates of the derivatives of ae. The G-matrix elements are calculated as functions of the relative momentum k0. The density dependence of the G-matrix, or the derivatives with respect to the Fermi momentum, are evaluated numerically, and the magnetic susceptibility is estimated. The differentiation of the G-matrix is carried out in the numerical solution of the Bethe-Goldstone equation by making a variation or finite shift in the population of the Fermi gas at the chosen momentum. Then the derivatives are determined by the finite shift in the G-matrix. We make a small variation of the Fermi momentum in the Bethe-Goldstone equation by making the change k F ~ k~, and resulting G-matrix elements are given in table 4. TABLE 4 D i a g o n a l G - m a t r i x e l e m e n t s i n [ f m ] , d = 0,55, m* = 1.0
k "F/kF kr [ f m -
1.36
3.0
1]
ko/kv
0.96
1.00
1.04
0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--24.35 --21.78 - - 18.59 --15.61 - - 13.26 --11.65 - - 10.67 - - 10.21
--24.21 --21.64 - - 18.45 --15.47 - - 13.13 --11.51 - - 10.53 - - 10.06
--24.08 --21.51 - - 18.32 --15.34 - - 13.00 --11.38 - - 10.39 --9.91
0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--15.32 --8.83 --5.22 --4.03 - - 3.98 --4.35 --4.87 - - 5.31
--14.44 --7.97 --4.40 --3.26 --3.24 --3.64 --4.17 --4.74
--13.49 --7.04 --3.51 --2.42 --2.45 --2.88 --3.45 --4.03
To obtain an impression of the behavior of eq. (5.9) as a function of the Fermi momentum, we rewrite the three terms as -
M* +D+R,
(5.14)
where D is the second or "direct" term, and R is the third or rearrangement term. With M * / M = 1.0, the final results for eq. (5.14) are given in table 5. Fig. 5 shows
221
NEUTRON MATTER
TABL~ 5 Magnetic susceptibility for neutron matter kv [fm- I ] 0.25 0.5 1.0 1.36 1.5 2.0 2.5 3.0 3.5 4.0
M/M *
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
D
R
0.67 0.76 0.83 0.88 0.90 0.94 0.86 0.62 0.18 --0.48
(Z/Ze) -
--0.16 --0.16 --0.10 --0.05 --0.04 --0.01 0 --0.03 --0.10 --0.34
Z/ZF
I
1.50 1.60 •.74 1.83 1.86 1.93 1.86 1.60 1.07 0.18
0.67 0.63. 0.58 0.55 0.54 0.52 0.54 0.63 0.93 5.48
D is the second or direct term and R is the third or rearrangement term in the expression (5.9), i.e. eq. (5.14); zl = 0.55, m* = 1.0.
XF IX 2.0
1.0
I
I
I
1.0
2.0
3.0
kF[fm "l]
4.0
Fig. 5. Magnetic susceptibility for neutron matter as a function of the Fermi momentum. Rearrangement terms included, d = 0.55, m* = 1.0. Comparison is made with other results given by Clark 21) (C). The dotted line indicates ferromagnetic transition. t h e m a g n e t i c s u s c e p t i b i l i t y as a f u n c t i o n o f t h e d e n s i t y , i.e., t h e F e r m i m o m e n t u m . C o m p a r i s o n c a n b e m a d e w i t h o t h e r r e s u l t s g i v e n b y C l a r k 21).
6. Discussion and summary F r o m t h e v a r i o u s t a b l e s a n d d i a g r a m s w e c a n see h o w t h e r e s u l t s c h a n g e a s p a r a m e t e r s a r e v a r i e d . O n e fixed p a r a m e t e r i n o u r c a l c u l a t i o n s is t h e effective m a s s m * = 1.
222
E. OSTGAARD
It should be a fairly good approximation to choose the intermediate-state singleparticle potentials to be essentially zero as we have done, i.e., set m* = 1. But then it is, in principle, necessary to calculate separately the three-body cluster energy. Also, the average momentum P or the c.m. momentum 2P is set equal to zero. Inclusion of P has been shown not to be important in nuclear-matter calculations for central forces only 36), and it should be a fair approximation to neglect it. It would be more important with tensor forces included, and possibly at very high densities. The effective mass for the hole spectrum is also chosen to be m* = 1. From table 5 we see that a possible deviation from unity is not very important, because the direct term D is generally fairly large and positive, while the rearrangement correction R is fairly small. The density or the Fermi momentum has been varied to cover probable values. However, the high-density results may not be very relevant because of pollution of pure neutron matter by protons and hyperons. According to Langer and Rosen 37), hyperon creation sets in at about ordinary nuclear density, and at k F = 3 f m - 1 there are already more strange particles than neutrons. The importance of the neglect of three-body contributions is rather difficult to estimate. The linked-cluster perturbation series of Goldstone 3s) provides a theoretical expression for the ground state energy of the system, but it is difficult to make any statements about the convergence of this series, or even of the resulting compactcluster expansion 39). Neutron star matter may be a "denser" system than, for instance, nuclear matter, but the Pauli exclusion principle helps to suppress the energy contribution from higher n-body terms. The increasing anti-symmetry of the n-body wave functions with increasing n helps to suppress the energy contribution from higher clusters. At densities equal to nuclear densities, the nuclear forces are fairly well known. If we assume charge independence, the neutron-neutron potentials must fit rather accurately determined phase shifts. However, there is no really firm evidence as to whether the two-body potential is static or velocity-dependent. One can hope that the properties of the many-body system are rather insensitive to the shape of the )otential as long as it reproduces the phase shifts at moderate energies, but that is probably not true. We have not studied the sensitivity of our results to method or ~otential, and the chosen potential may have some influence on the results of the calculations. Potentials with different forms will generally give different results or predictions, and the values obtained for the binding energy and the magnetic susceptibility should be affected by changes in the various parts of the potential. This is because the binding energy, for instance, is actually a rather small difference between large repulsive and attractive terms. In particular, the strongly repulsive short-range part of the potential is still not well known, and this may cause some errors in the calculations. The attractive long-range part is better determined, although the various contributions from central and tensor forces may be somewhat uncertain. We have used a simple central hard-core potential only, which is not a realistic potential. However, we do not expect non-central effects to be very important for our results. Also,
NEUTRON
MATTER
223
we do not include any odd-state effects, but these effects arise mainly from the oddstate core, so that their inclusion would only strengthen our conclusions regarding the absence of ferromagnetism. There has been some speculation that the ground state of neutron matter might experience a ferromagnetic transition. Disregarding the attractive component of the nuclear interaction and assuming only a state-independent hard core, Brownell and Callaway 17) first predicted the onset of ferromagnetism in a density region corresponding to normal nuclear matter. However, we find that the transition occurs at a much higher density; the density is in fact so high, that our many-body theory is probably, in principle, not applicable. Also, the way the rearrangement term R, i.e., the third term in the expression (5.9) or eq. (5.14) is evaluated, may be criticized 40). However, the general smallness of R reduces the importance of this objection. Our main results or conclusions in regard to the ferromagnetism of neutron matter are, otherwise, the same as the results of Clark and Chao 2 o, 21). However, the two studies also support one another, and are in considerable degree complementary because of certain differences in approach, In addition, Pearson and Saunier 14) have concluded that there is no ferromagnetism in neutron star matter for kv < 2.3 fm-1 As discussed by Clark and Chao z0,21), comparison with results for liquid aHe also suggests that the ground state of pure neutron matter will remain antiferromagnetic up to densities much higher than the normal nuclear-matter density. According to the arguments used by Brownell and Callaway 17), liquid 3He should, contrary to experiments, be expected to be ferromagnetic just like neutron matter. Ferromagnetism is generally favored by repulsive singlet-even and attractive tripletodd G-matrix elements. And with the attraction included, liquid aHe should be even more favorable to ferromagnetism than neutron matter, since the triplet-odd attraction which helps to align the spins is just as strong as the singlet-even attraction which inhibits alignment of spins. Due to the strongly attractive triplet-even interactions, we would also expect nuclear matter to favor a ferromagnetic instability more than the pure neutron matter system does. According to our two-body calculations, the neutron system is not bound at any density, and there is no relative minimum in the energy as a function of density. The system is properly regarded as a gas. A constrained neutron gas could, however, possibly show superfluidity, and Baym et al. 26,zv) and others have suggested the importance of investigating the role of superfluidity or superconductivity in neutron star matter. References
1) 2) 3) 4) 5)
H.-Y. Chiu, Ann. of Phys. 26 (1964) 364 K. A. Brueckner, J. L. Gammel and J. T. Kubis, Phys. Rev. 118 (1960) 1095 K. A. Brueckner and J. L. Gammel, Phys. Rev. 105 (1957) 1679; 109 (1958) 1023 E. E. Salpeter, Ann. of Phys. U (1960) 393; Astrophys. J. 134 (1961) 669 P. C. Sood and S. A. Moszkowski, Nucl. Phys. 21 (1960) 582
224 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40)
E. ~STGAAma N. Fukuda and R. Newton, Phys. Rev. 103 (1956) 1558 B. de Witt, Phys. Rev. 103 (1956) 1565 W. B. Riesenfeld and K. N. Watson, Phys. Rev. 104 (1956) 492 J. S. Levinger and L. M. Simmons, Phys. Rev. 124 (1961) 916 J. N6meth, D. W. L. Sprung and P. C. Bhargava, Phys. Lett. 24B (1967) 137 J. N~meth and D. W. L. Sprung, Phys. Rev. 176 (1968) 1496 C. T. P. Wang, Phys. Rev. 177 (1969) 1452 M. Binder, R. H. Pierce and M. Razavy, Can. J. Phys. 47 (1969) 2101 J. M. Pearson and G. Sannier, Phys. Rev. Lett. 24 (1970) 325 V. Canuto and H.-Y. Chin, Phys. Rev. 173 (1968) 1210, 1220, 1229 H.-Y. Chin and V. Canuto, Phys. Rev. Lett. 21 (1968) 110; Astrophys. J. 153 (1968) L157 D. H. Brownell, Jr. and J. Callaway, Nuovo Cim. 60B (1969) 169 S. D. Silverstein, Phys. Rev. Lett. 23 (1969) 139 M. J. Rice, Phys. Lett. 29A (1969) 637 J. W. Clark and N.-C. Chao, Nuovo Cim. Lett. 2 (1969) 185 J. W. Clark, Phys. Rev. Lett. 23 (1969) 1463 A. B. Migdal, JETP (Soy. Phys.) 10 (1960) 176 V. L. Ginzburg and D. A. Kirzhnits, JETP (Sov. Phys.) 20 (1965) 1346; Nature 220 (1968) 148 R. A. Wolf, Astrophys. J. 145 (1966) 834 V. L. Ginzburg, J. Stat. Phys. 1 (1969) 3; Comm. Astrophys. Space Phys. 1 (1969) 81 G. Baym, C. Pethick and D. Pines, Nature 224 (1969) 673; 224 (1969) 674 G. Baym, C. Pethick, D. Pines and M. A. Ruderman, Nature 224 (1969) 872 H. A. Bethe and J. Goldstone, Proc. Roy. Soc. A238 (1957) 551 H. A. Bethe, B. H. Brandow and A. G. Petschek, Phys. Rev. 129 (1963) 225 S. A. Moszkowski and B. L. Scott, Ann. of Phys. 11 (1960) 65 C. W. Wong, Nucl. Phys. 56 (1964) 213; 71 (1965) 385 P. C. Bhargava and D. W. L. Sprung, Ann. of Phys. 42 (1967) 222 S. A. Coon and J. Dabrowski, Phys. Rev. 140 (1965) B287 E. Ostgaard, Phys. Rev. 168 (1968) 1139 E. Ostgaard, Phys. Rev. 176 (1968) 351; 180 (1969) 263 G. Dahll, E. Ostgaard and B. H. Brandow, Nucl. Phys. A124 (1969) 481 W. D. Langer and L. C. Rosen, Astrophys. Space Sci. 6 (1970) 217 J. Goldstone, Proc. Roy. Soc. A239 (1957) 267 B. H. Brandow, Phys. Rev. 152 (1966) 863 S.-O. B~ckman, Phys. Lett. 30A (1969) 544
N E U T R O N MATTER Binding energy and magnetic susceptibility E. O S T G A A R D t
Institute for Pure and Applied Physical Sciences, University of California, San Diego, La dolla, California 92037 ** Received 9 March 1970 (Revised 18 June 1970) Abstract: The binding energy and the magnetic susceptibility of neutron matter is caiculated by means of Brueckner theory. A modified Brueckner-Gammel method is used to solve the BetheGoldstone equation, calculate the reaction matrix or G-matrix, and obtain the interaction energy contribution from two-body correlations. For simplicity, the approximation of a reference energy spectrum with an effective mass and quadratic m o m e n t u m dependence is used for the input energy spectrum, which in principle should be fitted to self-consistent single-particle energies. The intermediate-state potential energies are, however, c h o s e n to be equal to zero. Hence, the three-body and possibly higher-order energy contributions should be estimated by separate calculations. The Bethe-Goldstone equation is solved numerically by iteration to give the perturbed two-body wave functions. Afterwards, the G-matrix elements are calculated by numerical integration. The binding energy is calculated as a function of the Fermi m o m e n t u m and the pure neutron-matter system seems to be u n b o u n d at any density, even at very low densities. In addition to the binding energy, also the magnetic susceptibility is estimated. To this end, the spin symmetry energy is calculated in terms of the derivatives of the G-matrix with respect to the Fermi m o m e n t u m . A ferromagnetic transition is found to occur at a density m u c h higher than the typical nuclear-matter density. Our method is, however, probably too simple to be reliable at such a high density.
1. Introduction A body with a large mass under gravitational attraction, may exist in two possible equilibrium states. If the mass is sufficiently large, a more compressed state with neutronic configuration would be more favorable than the less condensed electronnuclear state. Such a condensed state might exist as a neutron star, or at least as a neutron core formed by a supernova explosion at the last stage of the stellar evolution o f a sufficiently massive star. At the end of the nuclear evolution, the star is very hot and dense at the center, and the stellar core becomes unstable against gravitational collapse. This triggers a supernova explosion, which under appropriate conditions leaves behind a superdense core. The temperature is possibly sufficiently low that thermal energies are small compared with nuclear energies and with the Fermi energy o f the electrons. One then has an almost pure neutron gas, i.e., neutron matter. It is a * This research was supported by the United States Atomic Energy Commission under Contract A T ( 1 1 - 1 ) I G E N - 1 0 , P. A. 11 : Report N o . : UCSD-10P11-103. tt Present address: Institut Max y o n Laue-Paul Langevin, 8046 Garching bei Miinchen, Germany. 202
NEUTRON MA'I-EER
203
neutral nuclear matter, which is under strong hydrostatic pressure from gravitational forces. In principle, however, a superdense star is more complicated, with possibly a hyperon core, a neutron-dominant layer around that core, and an outer envelope of ordinary matter, i.e., electrons, protons and nuclei. The total energy of the neutron star may be written as the sum of its gravitational energy, rest-mass energy, nuclear energy, and the relativistic kinetic energy of the electron gas. Most mass is possibly concentrated in the superdense core. The neutron matter around consists of neutrons in equilibrium with a small amount of protons and electrons. This neutron layer is essentially isothermal because of high conductivity. At the surface of the star, there is possibly a crystalline ionic lattice or a solid mantle. Several theoretical models have been constructed, but the equilibrium configurations obtained for neutron stars are rather dependent on specific assumptions about the nuclear interactions, or, more directly, the equation of state for high-density matter. Chiu 1) discussed both the dynamic and static problems of neutron stars, making rough estimates of temperature and cooling time. He concluded that it should be possible, in principle, to detect neutron stars by their radiation of X-rays. This observation is prevented on the earth because the atmosphere is strongly opaque to radiation in the X-ray region. However, the interstellar medium is practically transparent to X-rays, so neutron stars should be detectable above the earth's atmosphere. Soon Xray sources outside the solar system were discovered from rockets and reported, their identification with neutron stars was proposed and their properties investigated. In a similar way, radio emission from a neutron star with a strong magnetic field can account for the radio emission of pulsars. A number of these pulsating radio sources were discovered, and theories were proposed. The regularity of the emissions lead to the idea that they are related to density oscillations of condensed matter. The nature of the radio sources is, however, an open question. It is possible that they are caused by oscillations of neutron stars, although there may be some difficulties with the assumption. A better understanding of neutron matter based on the nuclear interactions involved, could give better estimates of frequencies and lifetimes for X-ray or radio emission by such stars, and it is also of interest to find out whether the neutronneutron force alone can give binding to a pure neutron system. It is then important to treat neutron matter by sophisticated many-body methods. Brueckner et al. 2) applied the methods of Brueckner and Gammel 3), developed earlier for the nuclear-matter problem, to calculate the energy of a pure neutron gas or neutron matter as a function of density. They found that a neutron gas is not bound at any density. Salpeter 4) and Sood and Moszkowski s) looked at the lowdensity limit by calculating the interaction energy, using the S-state phase shifts in the phase-shift approximation 6-s), and Levinger and Simmons 9) made some work on the neutron-neutron potential near normal nuclear-matter densities. In addition to later Brueckner-type calculations by Nemeth et al. 10, ix), Wang 12) used a Jastrow method including only two-body contributions, i.e., a correlated basis
204
E. liJSTGAARD
function to represent the ground state wave function of the system, and Binder et aL ~3) and Pearson and Saunier 14) have also calculated the average energy per particle as a function of density. None of the calculations made of the ground state energy of the neutron system, gives any indication of a bound state due to the nuclear neutron-neutron force alone. The most promising models of pulsars involve rapidly spinning neutron stars with intense magnetic fields. As discussed by Canuto and Chiu 15, 16), neutron stars may have very intense magnetic fields and can emit radio energy with a rate and spectrum similar to those of pulsars. The possible existence of a ferromagnetic state in neutron stars would be of considerable interest, with influence on the equilibrium composition and evolution of the star. It thus seems appropriate to investigate the magnetic properties of pure neutron matter. Brownell and Callaway 17) discussed the possibility of a ferromagnetic state for neutron matter at densities corresponding to theoretically stable neutron stars. Neglecting the attractive part of the nuclear interaction, assuming only a state-independent hard core, they replaced reaction-matrix elements between pairs of particles by constants assumed to be their average values. It was concluded that as the density of superdense stars increases to more than the equilibrium density of normal nuclear matter, the ground state might experience a ferromagnetic transition 18, ~9). However, the effect of the attractive part of the nuclear interaction may cancel this conclusion. Also, the exchange character, i.e., the spin-dependence of the two-nucleon potential, will oppose ferromagnetism. If the ground state of neutron matter ever turns ferromagnetic, the transition will occur at such high density that current nuclear many-body theories are probably not applicable 20, 21). Migdal 22), Ginzburg and Kirzhnits 23), and Wolf 24) have argued that a neutron star or at least the central region may be superfluid. This follows from the assumption that the residual interaction between neutrons with antiparallel spins is attractive for the relevant momentum region in high-density neutron matter. The idea was later supported by Ginzburg 25) and Baym et al. 26, 27). Strong interactions could make neutron superfluidity possible at the general density expected in the neutron star interior and below some critical temperature. At high densities, however, there is little known generally about the properties of matter. And the density in the central region may be too high for neutron superfluidity, at least for S-state pairing-type superfluidity, except in very light stars. 2. G e n e r a l t h e o r y
A neutron star is a mechanical system in a quasi-hydrostatic equilibrium, which must have a negative total energy. This star is under rotation and vibration, and emits radiation. The radiation consists of neutrinos, X-rays, long-wavelength electromagnetic waves, and gravitational waves. A neutron star has a typical radius of about 10 km and a mass of the order of the solar mass. To do calculations on the structure and the evolution, we should know the equation of state, i.e., the pressure as function of the density up to ordinary nuclear densities and somewhat higher.
NEUTRON MATTER
205
Neutron star matter is an idealized infinite system of neutrons, protons and electrons which is electrically neutral. It consists mainly of neutrons, and the number of protons and electrons is very small because of the very large relativistic kinetic energy of the electrons at nuclear-matter densities. The neutrons form a degenerate Fermi sea, and interact by very strong short-range nuclear forces which induce strong twobody correlations between the particles. Three-body contributions to the energy will become more and more important as the density increases, and should then be taken into account together with other corrections. A pure neutron system seems not to be bound, even at very low densities, and there is no relative minimum in the energy as function of density. As the density increases, the lack of binding increases, probably causing a greater density of protons. Although ferromagnetism is not expected at nuclear densities, and ordinary nuclear matter is not ferromagnetic, it can possibly be expected in a neutron star at higher densities when the repulsive potential core is more dominant. For a gas of fermions with only short-range repulsive interactions at a sufficiently high density, it should be energetically favorable for the particles to align their spins. This would reduce the potential energy because of the Pauli exclusion principle acting to keep the particles from approaching each other. There is, however, also an increased kinetic energy, since in the ferromagnetic state only one particle can occupy a given momentum state, but it is not correspondingly great. The particles can avoid each other anyway, when the density is low, but at high densities they would possibly align their spins. We then would get a ferromagnetic instability. However, an important influence on the criterion for ferromagnetism in real neutron matter is the inclusion of the more long-range attractive interaction. The effect of the attractive component of the two-nucleon potential is to shift the critical density for onset of ferromagnetism to considerably higher values. With realistic forces included, there is no ferromagnetic instability of the normal ground state of the system for mass densities of the order of ordinary nuclear-matter density. With the attractive part of the nuclear interaction restored, there is a deep attraction in singlet states just outside the repulsive core, which opposes the tendency towards spin alignment. Because of this attraction, a pair of particles should approach each other closely with spins antiparallel. Any triplet attraction will favor ferromagnetism, but the odd-state attractive interactions are much weaker than the even-state attractive interactions. For neutron densities at nuclear-matter densities and somewhat higher, there will be cancelling effects due to repulsion and attraction in the two-neutron force, but a ferromagnetic instability is not expected to occur until at significantly higher densities. The neutron matter problem is basically simpler than that of nuclear matter, since the Pauli principle prohibits all triplet-even and singlet-odd state interactions. The neutron-neutron interactions occur in triplet states of two-body isospin and the potential can be separated in singlet-even and triplet-odd components involving only spin projection operators. Thus, the strong triplet-even tensor, the triplet-even spin-
206
E. OSTGAARD
orbit, and triplet-even and singlet-odd quadratic spin-orbit components do not appear. The only non-central components of the two-nucleon potential which are there, in principle, are the triplet-odd tensor and spin-orbit components, and the triplet-odd and singlet-even quadratic spin-orbit components. The extra-core oddstate interactions are, however, quite weak, and are in any case not important at low densities and low relative momenta. As the spin-orbit components also are quite weak, one can probably neglect noncentral components and give a reasonable description of neutron matter in terms of a purely central potential, at least if the density is not too high. Brueckner and others have developed a method to handle the strong repulsion in the two-body potential. Starting from a perturbation expansion for the energy, the expansion in terms of the large matrix elements of the potential is rearranged and replaced by an expansion in reaction matrix elements. This reaction matrix, or Gmatrix, is obtained by solution of a two-body problem in the medium. To include effects of the Pauli exclusion principle and the single-particle energy spectrum arising from the average interaction of each particle with the medium, Brueckner and Gammel a) use a Green function method to obtain a G-matrix propagator in coordinate space. In the integral equation for the G-matrix, single-particle potentials given by sums of diagonal G-matrix elements are included in the energy denominator, which introduces a problem of self-consistency. The method can, however, be modified and simplified, and supplemented by calculations of the contribution to the binding energy from three-body correlations. In their nuclear-matter calculations, Brueckner and Gammel treated the hard core in the two-nucleon potential in a special way. They assumed that the integrated product of the potential and the perturbed wave function has a finite value at the core boundary, but vanishes inside the core; i.e., they assumed a 6-function at the core boundary for the repulsive part of this product. In the Brueckner-Gammel method, the energy is calculated from a reaction matrix similar to the transition matrix for scattering in free space. This reaction matrix, or G-matrix, is defined by the integral equation G = v-v(Q/e)G
(2.1)
in operator form. Here v is the two-body potential, which is assumed to be the same as in free space, and the Pauli exclusion operator Q prevents scattering into occupied intermediate states. The energy operator e includes kinetic and potential single-particle energies, and is written e = ea+eb--em--en. (2.2) The single-particle energies e,, and e, are self-consistent energies for particles moving in the Fermi sea, and 8a and eb are energies of virtual excitations above the Fermi surface.
N E U T R O N MATTER
207
The energy of an unexcited neutron is
~(k,,) = T(kn)+ U(k,,),
(2.3)
and the total ground state energy is
E = • [T(k,,)+½U(k,,)],
(2.4)
n
where T is the kinetic energy
T(kn) = - - -h12 k.2
(2.5)
2M and M is the neutron mass. The single-particle potential U is given by the diagonal G-matrix elements by the relation
U(k,) = ~ [
(2.6)
/rl
with summation over occupied states. The second term in eq. (2.6) comes from exchange of spin and m o m e n t u m coordinates. The Fermi momentum, PF = hkF, is related to the density by
where N is the total number of particles in a large volume f2. The normal density can then be determined from the minimum of the binding energy as a function of the number density N _ 1 k~ P--£2 3 ~2'
(2.8)
if such a minimum exists. We also define a mean interparticle spacing r 0 by p--1
--
"~ - - 4 g y 0 3 "
(2.9)
N Then
kr =
(9),
/ro = 1.92/r0.
(2.1o)
For a given Fermi momentum, assuming zero spin polarization, the density of symmetrical nuclear matter is twice that of pure neutron matter. We introduce c.m. and relative coordinates, and if we assume that the G-matrix depends on the total two-body m o m e n t u m only through its magnitude 2P, we can write a partial wave expansion with matrix elements GL(P, k,,,), where GL is a function of c.m. and relative momenta. The properties of the system can then be determined from the G-matrix. For central forces only, eq. (2.3) for the energy of a particle
208
I;'. I~STGAARD
moving on the energy shell with momentum kn becomes
~(k~) = ~ h2 k] + 2 . 2. Z [¼ Y, (2L+l)
m
even
L
+~ Y~ (2L+ l)(k.nIG,~Ikm)],
(230
oddL
with the sum over m taken over the Fermi sea. The first factor of 2 in the interaction term comes from two spin states per momentum state, and the second factor of 2 comes from the exchange term. The factors ¼ and ¼ give the relative weights of the singlet-even and triplet-odd states. We will later include all these factors in our definition of the Gr~ matrix elements. 3. Two-body terms According to general scattering theory, the two-body wave function is defined by c ¢ --
v~,
(3.1)
where ~b is the unperturbed free-particle wave function and ~ is the perturbed one. The distortion of the wave function because of the potential is written as = gb-~,.
(3.2)
Due to the hard or at least strongly repulsive core, the momentum-space matrix elements of the potential v are infinite or at least very large. The reaction matrix equation (2.1) is then reformulated and converted into coordinate space, and dividing on the left by v and multiplying from the right by ~b, we get the Bethe-Goldstone equation 2s) = ~b- --Q v@.
(3.3)
e
The propagator Q/e is a rather complicated nonlocal integral operator. We write eq. (3.3) in coordinate space, and after an expansion and separation into partial waves, we get the Bethe-Goldstone integral equation in the form
ur.(ko, r) = ~¢r.(ko r ) +
fo ' FL(r, r')vn(r')uz(ko,
r')dr',
(3.4)
where we have introduced
Jn(ko r) = ko rjL(ko r), uL(k o , r) = ko r~/n(ko, r), XL(ko, ~) = J,(ko ~)-- Udko, ~). 7[hen J L ( k o r)lcorresponds to the free-particle wave function.
(3.5)
N E U T R O N MATTER
209
If k o is the initial and k the intermediate-state relative momentum of the two interacting particles, the energy denominator e is defined by a reference energy spectrum 29)
e(k) = h2 ~2+ k2 M
m*
(3.6) '
where
2 = 2Ak2_k 2
(3.7)
for propagation on the energy shell and k o < kF. The basic parameters of the energy spectrum are m*, which is the dimensionless effective mass defining both the hole spectrum and the intermediate-state particle spectrum in our approximation, and A, which is a measure of the gap between the occupied and the intermediate-state energy spectra. We will later set m* = 1. Then the gap A or the parameter ?2 just defines the hole spectrum, which will be our real input parameter. The Green functions FL(r, r') are then defined by
rL(r,
r') =
-
m*
dk JL(kr)JL(kr')Q(e, k) ~2+k 2 ,
~
(3.8)
with propagation off the energy shell in the intermediate states. The Pauli exclusion operator Q(P, k) is approximated by its average over angles of the average momentum P, i.e., 0
for k < ( k ~ - p 2 ) ~
Q(P, k) = (p2+k:-k~)/EPk for ( k 2 - p 2 ) ~r < k < kF+P, ~1 for k > kF+P.
(3.9)
To obtain the integral (3.8) as accurately as possible, using the property of the Pauli function (3.9), we write eq. (3.8) in the form ~, (k2F -- p 2 ) "!"
r,(r, r') = r~°)(r, ~') + ~2m*
Jo
dk JL(k~)J,(kr')/(~ 2 + k')
+ - m* | 7~
dk JL(kr)JL(kr' )
,
(3.10)
./(k2F-P2)~
where
F[°)(r, r') = -- (2/tOm* f ~dk JL(kr) JL(kr') j o 72 + k2
(3.11)
corresponds to a Green function in which the exclusion principle is not included. For this latter function analytic expressions can be obtained, and for the lowest partial
210
E. e s ' r G ~
waves we get the solutions m*
r~o°)(,, r') = ~
{exp E-~(r+~')]-exp [-~1~-"1]},
---
a+
x+
exp[-~(,+,')] + [14- yl--r~IlYr ~r,l exp [ - T l r - r'.]} ,
r?'(,.
,3
=
1+
-
+
1+
-
yr'
7r
+
(3.12)
.
yr3--;+ ( ~ ?
xexp[-y(r+r')]-[l+3+~l[1T-yr
exp [-ylr-r"]} ~
and so on. In the expressions above, the upper sign is for r > r', the lower sign is for r
W e are also interested in the volume integral
~ = p
: I~(ko, r)12d~ = 4rcp :ooI~(ko, r)[ 2r2dr.
(3.13),
This quantity is proportional to the probability of finding a particle in an excited state. rather than in the Fermi sea and gives information about the saturation propertie~ of the system. We write x = 4rrp[½ ~
+½ ~ ](2L+ 1 ) f ~ (ZL~2dr = 2~rp ~ XL,
©venL
oddL
JO
\ko/
L
(3.14)'
which will be our definition of XL, i.e., we include appropriate statistical factors. The integral in eq. (3.14) ,which gives the volume of the correlation hole or the size of the wound in the wave function, is calculated as (ZL~ dr
\ko/
= ~2 ~o ~F~(k°' k)dk,
(3.15),
because an integration in momentum space seems more accurate here than an integration in coordinate space. The Fourier transforms F~.(ko, k) are given by
FL(ko, k)
ko ~
JL(kr)zL(ko, r)dr,
(3.16),
or
FL(ko, k) - Q(P' k)m* ko, f: ~2 + k2
j~(kr)v,(~)u~(ko,r)dr,
(3.17)
NEUTRON MATTER
211
because
zr(ko, r) = Q vLuL(ko, r).
(3.18)
e
Eq. (3.17) seems more accurate than eq. (3.16) because the values for zz(k o, r) will be more uncertain and have a greater relative error than the values for the wave function uz(ko, r). Also, the exclusion operator Q(P, k) gives a more correct behavior at k ~ kF. We will later set P = 0 in our calculations. Potentials and energies are usually expressed in units of MeV. But in the following sections, energies are also expressed in units of fm -2, and the conversion factor is
h2/M
--- 41.47 MeV • fm 2.
(3.19)
4. Binding energy We have now simplified the original Brueckner-Gammel method and made it more convenient in two ways. The first is to express the total Green function (3.8) as a reference spectrum Green function (3.11) which has an analytic form, plus a correction term which takes the exclusion principle into account, and in which the range of integration in momentum space is only of order k F. The second way is to use the reference spectrum idea for the energy denominator in the Bethe-Goldstone equation. It is very useful to have a reference spectrum or an effective-mass approximation for this denominator, and the release of the energy denominator from the loop of selfconsistency through the G-matrix elements and the single-particle potentials is a great advantage in the calculations. The use of single-particle energies calculated from G-matrix elements would involve extensive computing time. The average binding energy per particle is
EB_ 3
h2
10 M
k2 + ~fi:Fk 2 ( 1 _ 3 ko + 1 ka]
(4.1)
.or
kUo
M
where the single-particle potential (2.6) is given by zl
U(k)
/*~(kF-k)
k2
= 5Jo 2
½(kF+k) 2
The first integral vanishes for k >
~o
(for k < -]
kF)
(4.3)
k v. The diagonal
G-matrix elements are calculated
212
E. eSTGAARD
as
--
2
4r~ r E ko2
even L
oo
+ 3 E ]( L + l ) f ,J 0
oddL
a¢'L(kor)vL(r)ur(ko, r)dr,
(4.4)
and we take this as our new definition of GL, i.e., we include appropriate statistical factors. The calculations are done for a simple central potential with hard core, i.e., the Moszkowski-Scott potential 3o), which is defined for even L as ~+oo
VL(r)
=
t-- 11"oexp [ - # ( r - - c ) ]
for r < c, for r > c,
(4.5)
V[MeV 200
I00
r[f°] I
I
3.0
-I00
-200
Fig. 1. The Moszkowski-Scott potential defined by eqs. (4.5) and (4.6).
where c -- 0.4 fm, /~ = 2.083 fm, Vo = 260 MeV = 6.27 fm -2.
(4.6)
The potential is shown in fig. 1. It is assumed to vanish in odd angular m o m e n t u m states. More realistic and " m o d e r n " potentials include tensor-force components, and it has been shown that a finite or soft core in the potential can increase the binding energy in nuclear matter by several MeV [refs. 31, 32)]. However, for the neutron gas
NEUTRON MATTER
213
we have no T = 0 or T = I, T3 = 0 interactions, and the extra-core triplet-odd-state nucleon-nucleon interactions are known to be significantly weaker than the extra-core singlet-even-state interactions. It is also more convenient to use a hard-core rather than a soft-core or velocity-dependent repulsion in the potential. The strongly repulsive core is necessarily central. In the Brueckner-Gammel method, the hard core in the potential is then treated in the following way. We separate the integral on the right-hand side of eq. (3.4) into two parts at the core boundary, and make the replacement
VL(r)UL(ko, r) = 2Lr(r--c )
for r < c,
(4.7)
where 2r is determined by the condition that the wave function vanishes for r = c. Then
2L = -- JL(ko C)+
FL(c, r)vL(r)uL(ko,
FL(c, c),
r)d
(4.8)
and eq. (3.4) becomes
ur.(ko, r) = SL(ko, r)+
f; fgz(r, r')VL(/)uL(ko, r')dr',
(4.9)
where
sL(ko,
r) =
eCL(ko r)-- aCr.(koc) ~
r) c)'
(4.10)
and
fCdr, r') = rL(r, r')- rL(r, c)Fdc, r')
r,.(c, c)
(4.11)
The above treatment of the hard core is equivalent to assuming a "hollow-hardcore" potential 33), neglecting contributions from the region inside the core. The method seems to be a very good approximation 34), at least for calculations on the energy shell and for relative momentum k o < kr, at not too high densities. And the results should not be very different from results obtained by a more "exact" treatment of the hard core in the two-nucleon potential. The assumption (4.7) must be replaced by a more complicated version when dealing with the coupled equations that arise in the presence of tensor forces. But it is sufficient in our case, because we consider a simple potential with central forces only. The radial integral in the G-matrix elements (4.4) is finally calculated as
fo JL(ko0V,(.)uL(ko,.)d.
=
J~(koc) + rde, c)
f? ~(ko, 0v~(r)u,(ko, 0d~:
(4.12)
The wave functions uL(ko, r) are determined by numerical integration and iteration of eq. (4.9). Afterwards, the G-matrix elements are calculated by numerical integration.
214
E. OSTOAARD TABLE 1 Converg ence p a r a m e t e r Z defined by eqs. (3.13) a n d (3.14). A = 0.55. m* = 1.0
x]
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
1.0
1.36
1.5
2.0
2.5
3.0
3.5
0.017 0.017 0.017 0.017 0.018 0.018 0.018 0.019 0.021
0.026 0.026 0.026 0.026 0.026 0.026 0.027 0.028 0.031
0.033 0.033 0.033 0.033 0.033 0.033 0.034 0.036 0.040
0.076 0.075 0.075 0.074 0.074 0.074 0.075 0.079 0.085
0.147 0.146 0.145 0.143 0.140 0.138 0.136 0.137 0.142
0.250 0.248 0.245 0.239 0.232 0.223 0.215 0.209 0.207
0.390 0.388 0.380 0.368 0.351 0.333 0.313 0.295 0.280
TABLE 2 D i a g o n a l GL m a t r i x elements in [fm] for even L, weigh t of (2L-k1) i n c l u d e d ; A = 0.55, m* = 1.0 kF[fm - 1 ]
ko/kF
L = 0
L ~ 2
L = 4
Total
0.5
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--35.3 --35.2 --34.8 --34.3 --33.6 --32.8 --32.0 --31.3 --30.9
0 0 0 0 --0.1 --0.1 --0.3 --0.5 --0.7
0 0 0 0 0 0 0 0 0
--35.3 --35.2 --34.8 --34.3 --33.6 --32.9 --32.2 --31.7 --31.7
1.5
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--24.9 --23.7 --20.4 --16.1 --11.7 --7.8 --4.5 -- 1.9 0
0 0 --0.3 --1.0 --2.3 -- 3.9 --5.6 --7.0 --8.3
0 0 0 0 --0.1 --0.2 --0.4 --0.8 --1.3
--24.9 --23.7 --20.7 --17.2 --14.1 -- 11.9 --10.5 --9.8 --9.5
0.001 0.125 0.25 0.375 0.50 0.625 0.75 0.875
--14.1 --8.4 1.3 7.9 11.0 11.8 11.3 10.0
0 --0.4 --3.3 -- 6.6 -- 8.4 --8.7 -- 8.1 --7.0
0 0 --0.1 --0.8 --2.0 --3.2 --4.2 --4.8
--14.1 --8.8 --2.1 0.4 0.6 --0.1 --0.9 -- 1.8
-- 5.6
-- 5.2
--2.5
3.5
1.00
8.3
215
NEUTRON MATTER TABLE 3
Binding energy for n e u t r o n matter
kr. [ f m - 1] 0.25 0.5 1.0 1.36 1.5 2.0 2.5 3.0 3.5
r0 [fm]
p [fm - 3]
7.68 3.84 1.92 1.41 1.28 0.96 0.77 0.64 0.55
0.001 0.004 0.034 0.085 0.114 0.270 0.528 0.912 1.448
W2 [MeV]
TF [MeV]
En [MeV ]
--0.3 --1.5 --7.0 --13.9 --17.3 -- 31.6
0.8 3.1 12.4 23.0 28.0 49.8 77.8 112.0 152.4
0.5 1.6 5.5 9.1 10.6 18.1 33.4 68.5 142.8
--44.4 --43.5 --9.6
Only two-body terms are included. W2 is the two-body contribution to the interaction energy, TF is the average kinetic energy per particle, and EB is the binding energy, d = 0.55. m* = 1.0.
ko/kF I
- - - - o . 5 ~
I
~.,o
I
-I0
1
i
-20
-30
Fig. 2. Diagonal GL matrix elements for even L. Weight o f ( 2 L + l ) included, kr = 1.0 fm -1. A = 0.55, m* = 1.0.
216
E. ~ $ T G A A R D
GL[fo]
I
10t-
I
\[L.o] ,
I
!
iko/~
ZIL'4]
1.0 v
J -10
-20
Fig. 3. Diagonal GL matrix elements for even L. Weight of (2L-t-l) included, kr /I
=
=
3.0 fm -1.
0.55, m * = 1.0.
,EB[MeV] [NSB,PS]~ 20
_
[,,o,<]~
/
.//"-~//
..I' / ' /
[wl IO
0
~
"
J."'"'""
I......................... I 0.5 1.0
~rsl I L .I 1.5
I
,.~, -,~
r'FLVmJ
2.0
Fig. 4. Binding energy for neutron matter as a function of the Fermi momentum. Only two-body terms are included, d = 0.55, m* = 1.0. Comparison is made with other results given by Brueckner et al. 2) (BGK), Salpeter 4) (S), Levinger and Simmons 9) (LS), Nemeth et al. lo) (NSB), Nemeth and Sprung 11) (NS), Wang 12) (W), Binder et al. 13) (BPR), and Pearson and Saunier 14) (PS).
NEUTRONMATTER
217
The following mesh was used for the radial integration in the calculations: r, r' = (0.1)0.4(0.05)0.6(0.1)1.2(0.2)3.6(0.4)6.0 fro.
(4.13)
The integrations are carried out to a distance beyond which the exponential behavior of the potential makes a further contribution quite negligible. First we calculate and store the Green functions (3.10) and (4.11) for all distinct pairs of points. The F[°)(r,r ') are evaluated directly from eq. (3.11) and the correction integrals FL(r, r')-F~°)(r, r') are done numerically with the following mesh:
k/k e = 0(0.05)1.0
for k < kp = ~/k~-+P 2.
(4.14)
The radial wave equation (4.9) is solved by iteration, starting at r = c and proceeding outwards. The convergence is improved somewhat by using the results of the last iteration for the points r' < r in each cycle. The integrations were done by Simpson's rule. However, it was necessary to use an improved Simpson's method in the integration ofeq. (4.9). Because of the shape of the Green functions FL(r, r') and ffr.(r, r') with a peak at r = r', this point should always be at the end of a three-point Simpson interval. Results from the calculations are shown in tables 1-3 and in figs. 2-4. Table 1 gives values for the convergence parameter (3.14) as a function of relative momentum and density. Table 2 and figs. 2 and 3 give G-matrix elements in dimensions [fm], these results can be translated to [ M e V - f m s] according to eq. (3.19). Table 3 gives the binding energy together with some corresponding parameter values, and fig. 4 shows the binding energy as a function of the density, or, rather, the Fermi momentum. Comparison can be made with other results given by Brueckner et al. 2), Salpeter 4), Levinger and Simmons 9), Nemeth et aLlo, 11), Wang lz), Binder et al. 13), and Pearson and Saunier 14). Sood and Moszkowski s) have also given results for very low densities. 5. Magnetic susceptibility
The magnetic susceptibility is a measure of the energy required to produce a net spin alignment in a given direction. It is inversely proportional to the energy required to polarize the spins. This energy has been given for liquid 3He in the limit of small net spin alignment a s). Following the same procedure here, we get an expression for the energy of spin alignment, given in units of the spin polarization energy of the corresponding ideal Fermi gas. If we apply a magnetic field to the system, the two spin populations will no longer be equal in the ground state, and the total energy will be a function of the spin polarization parameter
N(+) N -) s - N(+)+N(_ ) ,
(5.1)
218
E. OSTGAARD
where N (+) and N (-) are the numbers of particles with spin up and spin down with respect to the applied field. The total magnetization ~ is = y ( N ( + ) - N (-)) =
ysN,
(5.2)
where 7 is the magnetic moment per particle, and N is the total number of particles. The magnetic field is then given by ~E
(5.3)
and the total magnetic susceptibility is defined as _-
z =
~-~
.,.
\a~1.,,,
/~as2t.,NIs=o"
(5.4)
Only the susceptibility for a small field is considered, s o (~2E/OS2)p.Nis evaluated for s = 0, and the susceptibility is determined by the coefficient of the s 2 term in the expansion of the energy E(s) about s = 0. We write
~as z]
~ as~ ] '
where E r is the energy of the noninteracting ideal Fermi gas, which has the susceptibility XF = ~2M kF h 2 ~2"
(5.6)
The Fermi momentum k(F±) of the two spin populations containing N (+) and N (-) particles, are given by the density relation p(±) _ N (±) I k(F:I:)3 I2 - 6 z~2 ,
(5.7)
or, in terms of the spin polarization parameter s, we have N ( ± ) = ½N(I __.s),
k(~+) = kF(l +s) ~.
(5.8)
To calculate the spin symmetry energy, we expand the total energy E(s) in powers of s, neglecting terms of higher order than s 2. We calculate the second derivative of E(s) with respect to s for s = 0. The first derivative of the energy vanishes. Following the same procedure s 5) as for liquid SHe, the magnetic susceptibility is finally obtained
NEUTRON MATTER
219
from eq. (5.5) as = M*
2 h2
kF
+ 2 ~22 ~ f f
[ae(kF, 3 - 1
(41ZkF)
kF)a,,.-ao(kv, kv)~.]
dk, kF O a o (ki' kF) ~k F
+
kf (4nkg)-'dkff(4rck~)-~dkjkv [k F ( ~ ) - - 2
(~kv)? [a.(k,, kj)
%
(5.9)
+ 3ao(k,, kj)]/•!
The first term depending on the effective mass m* comes from the change in the density of the single-particle levels due to the interaction, or arises from the momentum dependence of the single-particle potential. The second term comes directly from the spin dependence of the two-body interaction, or arises from the change in the interaction as spins are shifted from antiparallel to parallel alignment. The third term comes from the rearrangement terms in the single-particle energies, or arises from the variation of the reaction matrix with the Fermi momentum because of the spin polarization. The contribution from even and odd angular-momentum states is given by
ae(k,, kj) = Z (k,ilGL(k(F+), k(v-))lk,~>, eve. L ao(ki, k.i) = oddZ L(k,jlGL(k(v+), k(F-))lk,s>•
(5.10)
The statistical factor of 3 for odd L is here omitted from the Gr matrix elements, while it has been included in our earlier definition (4.4) of Gr. The factor of (2L+ 1) is still included as before. However, we need only a e and the corresponding derivatives because of our simple potential (4.5). After transforming the integrals in eq. (5,9) to
f dk,, = 16~zf~Fk2 (1-- ~FF)dko, (5.11)
f dki f
dk i
128 ~Z2kF3 = -T
(
ko2 1 - ~
~
+½
k~!
dko,
the third term in eq. (5.9) can be written as
1 - M ---230 kF 7t ~lx2 I(~-x+½xa) (k2 -~v --2kF --Oa~ OkF Ok~] + ( 4 - 2 x - Z x 3 )
~kv]A
(5.12)
220
E. I~ISTO,~J~,RD
where (5.13)
x = k o / k F.
The magnetic susceptibility is then obtained from eq. (5.9) after numerical estimates of the derivatives of ae. The G-matrix elements are calculated as functions of the relative momentum k0. The density dependence of the G-matrix, or the derivatives with respect to the Fermi momentum, are evaluated numerically, and the magnetic susceptibility is estimated. The differentiation of the G-matrix is carried out in the numerical solution of the Bethe-Goldstone equation by making a variation or finite shift in the population of the Fermi gas at the chosen momentum. Then the derivatives are determined by the finite shift in the G-matrix. We make a small variation of the Fermi momentum in the Bethe-Goldstone equation by making the change k F ~ k~, and resulting G-matrix elements are given in table 4. TABLE 4 D i a g o n a l G - m a t r i x e l e m e n t s i n [ f m ] , d = 0,55, m* = 1.0
k "F/kF kr [ f m -
1.36
3.0
1]
ko/kv
0.96
1.00
1.04
0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--24.35 --21.78 - - 18.59 --15.61 - - 13.26 --11.65 - - 10.67 - - 10.21
--24.21 --21.64 - - 18.45 --15.47 - - 13.13 --11.51 - - 10.53 - - 10.06
--24.08 --21.51 - - 18.32 --15.34 - - 13.00 --11.38 - - 10.39 --9.91
0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
--15.32 --8.83 --5.22 --4.03 - - 3.98 --4.35 --4.87 - - 5.31
--14.44 --7.97 --4.40 --3.26 --3.24 --3.64 --4.17 --4.74
--13.49 --7.04 --3.51 --2.42 --2.45 --2.88 --3.45 --4.03
To obtain an impression of the behavior of eq. (5.9) as a function of the Fermi momentum, we rewrite the three terms as -
M* +D+R,
(5.14)
where D is the second or "direct" term, and R is the third or rearrangement term. With M * / M = 1.0, the final results for eq. (5.14) are given in table 5. Fig. 5 shows
221
NEUTRON MATTER
TABL~ 5 Magnetic susceptibility for neutron matter kv [fm- I ] 0.25 0.5 1.0 1.36 1.5 2.0 2.5 3.0 3.5 4.0
M/M *
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
D
R
0.67 0.76 0.83 0.88 0.90 0.94 0.86 0.62 0.18 --0.48
(Z/Ze) -
--0.16 --0.16 --0.10 --0.05 --0.04 --0.01 0 --0.03 --0.10 --0.34
Z/ZF
I
1.50 1.60 •.74 1.83 1.86 1.93 1.86 1.60 1.07 0.18
0.67 0.63. 0.58 0.55 0.54 0.52 0.54 0.63 0.93 5.48
D is the second or direct term and R is the third or rearrangement term in the expression (5.9), i.e. eq. (5.14); zl = 0.55, m* = 1.0.
XF IX 2.0
1.0
I
I
I
1.0
2.0
3.0
kF[fm "l]
4.0
Fig. 5. Magnetic susceptibility for neutron matter as a function of the Fermi momentum. Rearrangement terms included, d = 0.55, m* = 1.0. Comparison is made with other results given by Clark 21) (C). The dotted line indicates ferromagnetic transition. t h e m a g n e t i c s u s c e p t i b i l i t y as a f u n c t i o n o f t h e d e n s i t y , i.e., t h e F e r m i m o m e n t u m . C o m p a r i s o n c a n b e m a d e w i t h o t h e r r e s u l t s g i v e n b y C l a r k 21).
6. Discussion and summary F r o m t h e v a r i o u s t a b l e s a n d d i a g r a m s w e c a n see h o w t h e r e s u l t s c h a n g e a s p a r a m e t e r s a r e v a r i e d . O n e fixed p a r a m e t e r i n o u r c a l c u l a t i o n s is t h e effective m a s s m * = 1.
222
E. OSTGAARD
It should be a fairly good approximation to choose the intermediate-state singleparticle potentials to be essentially zero as we have done, i.e., set m* = 1. But then it is, in principle, necessary to calculate separately the three-body cluster energy. Also, the average momentum P or the c.m. momentum 2P is set equal to zero. Inclusion of P has been shown not to be important in nuclear-matter calculations for central forces only 36), and it should be a fair approximation to neglect it. It would be more important with tensor forces included, and possibly at very high densities. The effective mass for the hole spectrum is also chosen to be m* = 1. From table 5 we see that a possible deviation from unity is not very important, because the direct term D is generally fairly large and positive, while the rearrangement correction R is fairly small. The density or the Fermi momentum has been varied to cover probable values. However, the high-density results may not be very relevant because of pollution of pure neutron matter by protons and hyperons. According to Langer and Rosen 37), hyperon creation sets in at about ordinary nuclear density, and at k F = 3 f m - 1 there are already more strange particles than neutrons. The importance of the neglect of three-body contributions is rather difficult to estimate. The linked-cluster perturbation series of Goldstone 3s) provides a theoretical expression for the ground state energy of the system, but it is difficult to make any statements about the convergence of this series, or even of the resulting compactcluster expansion 39). Neutron star matter may be a "denser" system than, for instance, nuclear matter, but the Pauli exclusion principle helps to suppress the energy contribution from higher n-body terms. The increasing anti-symmetry of the n-body wave functions with increasing n helps to suppress the energy contribution from higher clusters. At densities equal to nuclear densities, the nuclear forces are fairly well known. If we assume charge independence, the neutron-neutron potentials must fit rather accurately determined phase shifts. However, there is no really firm evidence as to whether the two-body potential is static or velocity-dependent. One can hope that the properties of the many-body system are rather insensitive to the shape of the )otential as long as it reproduces the phase shifts at moderate energies, but that is probably not true. We have not studied the sensitivity of our results to method or ~otential, and the chosen potential may have some influence on the results of the calculations. Potentials with different forms will generally give different results or predictions, and the values obtained for the binding energy and the magnetic susceptibility should be affected by changes in the various parts of the potential. This is because the binding energy, for instance, is actually a rather small difference between large repulsive and attractive terms. In particular, the strongly repulsive short-range part of the potential is still not well known, and this may cause some errors in the calculations. The attractive long-range part is better determined, although the various contributions from central and tensor forces may be somewhat uncertain. We have used a simple central hard-core potential only, which is not a realistic potential. However, we do not expect non-central effects to be very important for our results. Also,
NEUTRON
MATTER
223
we do not include any odd-state effects, but these effects arise mainly from the oddstate core, so that their inclusion would only strengthen our conclusions regarding the absence of ferromagnetism. There has been some speculation that the ground state of neutron matter might experience a ferromagnetic transition. Disregarding the attractive component of the nuclear interaction and assuming only a state-independent hard core, Brownell and Callaway 17) first predicted the onset of ferromagnetism in a density region corresponding to normal nuclear matter. However, we find that the transition occurs at a much higher density; the density is in fact so high, that our many-body theory is probably, in principle, not applicable. Also, the way the rearrangement term R, i.e., the third term in the expression (5.9) or eq. (5.14) is evaluated, may be criticized 40). However, the general smallness of R reduces the importance of this objection. Our main results or conclusions in regard to the ferromagnetism of neutron matter are, otherwise, the same as the results of Clark and Chao 2 o, 21). However, the two studies also support one another, and are in considerable degree complementary because of certain differences in approach, In addition, Pearson and Saunier 14) have concluded that there is no ferromagnetism in neutron star matter for kv < 2.3 fm-1 As discussed by Clark and Chao z0,21), comparison with results for liquid aHe also suggests that the ground state of pure neutron matter will remain antiferromagnetic up to densities much higher than the normal nuclear-matter density. According to the arguments used by Brownell and Callaway 17), liquid 3He should, contrary to experiments, be expected to be ferromagnetic just like neutron matter. Ferromagnetism is generally favored by repulsive singlet-even and attractive tripletodd G-matrix elements. And with the attraction included, liquid aHe should be even more favorable to ferromagnetism than neutron matter, since the triplet-odd attraction which helps to align the spins is just as strong as the singlet-even attraction which inhibits alignment of spins. Due to the strongly attractive triplet-even interactions, we would also expect nuclear matter to favor a ferromagnetic instability more than the pure neutron matter system does. According to our two-body calculations, the neutron system is not bound at any density, and there is no relative minimum in the energy as a function of density. The system is properly regarded as a gas. A constrained neutron gas could, however, possibly show superfluidity, and Baym et al. 26,zv) and others have suggested the importance of investigating the role of superfluidity or superconductivity in neutron star matter. References
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