- Email: [email protected]
Quasiparticle interactions in neutron matter for applications in neutron stars
NuclearPhysicsA555 North-Holland
(1993)
NUCLEAR PHYSICS A
128-150
Quasiparticle interactions in neutron matter for applications in neutron stars+ J. Warnbach’,
T.L. Ainsworth’
and D. Pines
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, II 61801, USA
1110 West Green Street,
Received 4 May 1992 (Revised 16 July 1992)
A microscopic model for the quasiparticle interaction in neutron matter is presented. Both particle-particle (pp) and particle-hole (ph) correlations are included. The pp correlations are treated in a semi-empirical way, while ph correlations are incorporated by solving coupled two-body equations for the particle-hole interaction and the scattering amplitude on the Fermi sphere. The resulting integral equations self-consistently sum the ph reducible diagrams. Antisymmetry is kept at all stages and hence the forward-scattering sum rules for the scattering amplitude are obeyed. Results for Landau parameters and transport coefficients in a density regime representing the crust of a neutron star are presented. We also estimate the ‘So gap parameter for neutron superfluidity and comment briefly on neutron-star implications.
Abstract:
1. Introduction We dedicate this article to Arkady B. Migdal, whose recent death deprived the physics community of one of its leading members. Kadya was a theoretical physicist of the first magnitude; his remarkable personality and breadth of interests made an indelible impression on all of us who were privileged to know him. As we note, the behavior of neutron matter in neutron stars was a topic dear to him, the more so because his pioneering contributions to neutron-matter superfluidity, the Fermiliquid description of nuclei and nuclear matter, and to the possibility of pion condensation
in neutron
stars, laid the ground
work for so much subsequent
work
on the problems we consider here. Neutron matter, bound under gravitational pressure, is a significant component of neutron stars. It is found over a wide density range, p,/lOOO < 2p,, where pO= 2.6 x 1014 g/cm3 is the saturation density of nuclear matter. The equation of state of neutron matter at densities >pO, the approximate location of the boundary between the solid outer crust and the quantum liquid core of the star, determines the stellar radius. For most currently accepted equations of state that means that + This paper is dedicated
to the memory of A.B. Migdal. Correspondence to: Prof. Dr. J. Wambach, Dept. of Physics, 1110 West Green Street, Urbana, 11 61801, USA. ’ Present address: Institut fiir Kernphysik, Forschungszentrum Jiilich, D-5170 Jiilich, Germany. * Present address: Department of Physics, Texas A&M University, College Station, TX 77843, USA. 03759474/93/$06.00
@ 1993 - Elsevier
Science
Publishers
B.V. All rights reserved
J. Wambach et al. / Quasiparticle
interactions
129
neutron matter is found between a few meters and a few kilometers below the surface of the star. In the crust, the neutrons coexist with a lattice of neutron-rich nuclei and a sea of relativistic electrons; in the quantum liquid core, they coexist with protons (whose density is -5-10% that of the neutrons) and relativistic electrons. Both the constituents and equation of state at densities >2p,, are uncertain; pion condensation, first suggested by Migdal I), kaon condensation *) or, as some believe, a quark-gluon plasma are some of the interesting possibilities. The present work has two motivations: to derive the Fermi-liquid parameters and transport coefficients for the normal phase of neutron matter, and to estimate the temperature at which neutron matter becomes superfluid. More than 30 years ago, Migdal suggested ‘) that since the outermost nucleons in nuclei exhibit superfluid properties, it was quite likely that if neutron stars existed, neutron matter inside these stars would be superfluid and this would in turn lead to interesting macroscopic phenomena. With the discovery of pulsars, and their identification as rotating neutron stars, that has turned out to be the case. Throughout most of the inner crust of the star, for all but newly formed pulsars, the stellar temperature is such that neutron matter found there can be shown to be in a ‘S,, pairing state. Because pulsars rotate, vortices form in this superfluid. Since the coherence length of the superfluid is comparable to the size of crustal nuclei with which the superfluid coexists, the vortices are pinned to these nuclei. This pinning neutron fluid plays a significant role in the spin down of the pulsar since catastrophic unpinning of vortices is believed to be associated with the giant glitches observed in the Vela and other pulsars. Furthermore the vortex creep theory of the postglitch behavior provides strong evidence for the presence of a crustal superfluid “). A recent detailed analysis of the behavior of the Vela pulsar following the eight glitches observed during the twenty-year period, March, 1969, to January, 1989, based on vortex creep theory, provides interesting constraints on the parameters relevant for pinning: the pinning energy, EP, and the coherence length, 5. Since both depend sensitively on the magnitude of the neutron superfluid gap, A&, one finds a set of mutual constraints which link the pinning parameters and the density-dependent energy gap ‘). Neutron superfluidity and proton superconductivity are also believed to play an important role in the evolution of neutron stars. At early times the cooling of the star is dominated by neutrino emission involving the modified Urea process as well as neutron-neutron and neutron-proton bremsstrahlung in the liquid interior “). It is well established that the existence of pairing gaps suppresses the neutrino processes by reducing the available phase space in the final state. At the same time the total heat capacity of the star may be drastically lowered. Finally, the time evolution of the magnetic field may be influenced by the interaction between neutron vortex lines and superfluid protons. Superfluid properties of neutron matter have been studied for a long time. While all calculations yield neutron ‘S,, pairing for k, less than 1.3-1.5 fm-’ which corresponds to matter densities between 0.5~~ and 0.7p, a prediction of the maximum
J. Wambach et al. / Quasiparticle
130
gap, AiF, as well as the pair condensation The predictions problem
for the maximum
originates
(qp) scattering
from the exponential
amplitude,
E,, has been notoriously
difficult.
gap vary by almost an order of magnitude
pairing interaction. In Fermi-liquid theory the pairing title
energy,
interactions
dependence
matrix
element
&, near the Fermi
of the superfluid is obtained surface
‘). This
gap in the
from the quasipar-
which
also determines
the transport properties of the neutron liquid. It is the aim of the present paper to obtain a reliable estimate for Op by exploiting its relationship the particle-hole interaction, 9 [ref. “)I. We have earlier presented a model ‘) for the particle-holeirreducible interaction, 9, which drives the coupled equations for 9 and ~9 (subsect. 2.1). The model is microscopic in the sense that the starting point is the bare two-body interaction between nucleons. Following a very successful approach for the He liquids lo) the short-distance part of the bare interaction is renormalized phenomenologically through a density-dependent effective core height which is adjusted to the equation of state. We will show that, in the density range 0 s p s po, this effective interaction is very close to the Brueckner G-matrix (subsect. 2.2). Given 9, the ph interaction and the scattering amplitude at the Fermi surface are determined and Landau parameters, transport coefficients, etc. can be deduced (subsects. 3.1 and 3.2). Also the pairing matrix element at the Fermi surface can be calculated
and the density
dependence
of the gap parameter
inferred.
2. Theory In Fermi-liquid theory a strongly interacting system of fermions is represented by dilute gas of weakly interacting quasiparticles. These “elementary excitations” exist in the vicinity of the Fermi surface and have the same statistics as the bare particles. They differ, however, through an effective mass, m*, and a finite lifetime, 7. A finite lifetime is caused by quasiparticle collisions and it sets a limit on the applicability susceptibility,
of the theory. Bulk properties such as the compressibility, the magnetic the specific heat, as well as the transport properties are described in
terms of the quasiparticle interaction. This interaction is characterized by a few “Landau parameters” which are directly related to the macroscopic equilibrium properties of matter. They also signal the onset of various kinds of instabilities. Given the complications of a strongly interacting many-body system, Landau suggested a phenomenological treatment of the quasiparticle interaction using empirical bulk information, conservation laws and symmetry principles. One is restricted, however, to small momenta, the so-called “Landau limit.” This is sufficient for many bulk phenomena but, some of the non-equilibrium properties as well as the superfluid behavior require a much wider range of momenta. To describe such processes one has to make contact with the momentum structure of the bare interaction. This is complicated by the fact that the bare interaction has a hard core which has to be dealt with in some non-perturbative way.
J. Wambach et al. / Quasiparticle
interactions
131
In our approach we are motivated by the polarization potential model (PPM) of Aldrich and Pines lo) which starts from the bare fermion interaction and includes medium effects such as Pauli correlations, polarization phenomena, etc. via phenomenological modification of its short-distance behavior. With relatively few parameters, adjusted to the bulk behavior, the PPM has been applied successfully in liquid 3He as well as in 3He-4He mixtures. Straightforward application to nuclear or neutron matter, however, faces some difficulties: (i) since the bulk properties can only be inferred from finite nuclei they are much less certain; (ii) in the density regime of interest, the role of the Pauli principle is quite different from that in liquid helium; (iii) the presence of non-central components in the bare interaction complicates the phenomenological treatment of medium effects. In a series of papers rzV9) we have therefore extended the PPM so that it can be applied to nucleonic matter. The formalism is reviewed below.
2.1. THE TWO-BODY
EQUATIONS
In determining the quasiparticle interaction two quantities are of interest, the particle-hole interaction, 9, and the scattering amplitude d (fig. 1). The particlehole interaction determines equilibrium properties, collective modes, etc. while the scattering amplitude yields the transport coefficients through damping phenomena induced by quasiparticle collisions. In a homogeneous system the qp interaction can be labeled by three independent momenta. In the following we choose the variables p = $(pr +p3), p’ = f(p2 +p4) and q = p3 -pl. There is an additional spin dependence which arises from the exchange character of the fermion interaction and, in the case of neutron matter, from meson exchange. Meson exchange also adds non-central and spin-orbit components which complicate the interaction. In the following we shall ignore these components since we are interested in densities up to po, where they are unimportant. With central components only .9 and d decompose into a spin-independent and a spin-dependent interaction,
(1)
Fig. 1. The ph interaction
9 and the scattering amplitude
s2.
132
.I. Wambach et al. / Quasiparticle
where p(a) is the symmetric neutron
pair (pa
(antisymmetric)
= t( @’ f @‘)),
are only meaningful
combination
of a spin-t?
and likewise for .vY. By definition,
in the vicinity
of the Fermi
the momenta to the Fermi sphere, IpI = /$I= is given by an angle 0, = O,,., the so-called expand
interactions
surface
and a spin-tJ qp interactions
and we therefore
confine
kF. Then the dependence on p and p’ “Landau angle”, and it is natural to
9 and JB in @, iv(O)F”
= c Fy(q)P,(cos I
N(O)&“=C
A;‘“(q)P,(cos
0,)
)
0,).
(2)
To scale out a trivial density dependence we have multiplied by N(0) = k,m*/d, the density of states at the Fermi surface. Due to the Fermi-surface restriction the magnitude of the momentum transfer q is limited by 0~ /q1s2kF. Therefore the q-dependence can be represented by a second angle 0, as q = 2k,( 1 - cos 0,). In microscopic approaches one tries to relate the quasiparticle interactions to the bare interparticle potential. In a diagrammatic sense, phenomenological modifications of the bare potential represent summations of pp diagrams (ladders) as well as ph diagrams (bubbles). We argue that both diagram sets effectively decouple and that the short-ranged pp correlations can be treated rather crudely. Care must be taken, however, in determining the ph correlations, particularly to ensure the Pauli principle. To set up a system of scattering equations for 9 and d we exploit the diagrammatic relationship between them (fig. 2). This diagram set is driven by a ph irreducible interaction, 9, (fig. 2b) and approximately sums particlehole diagrams in all momentum channels. The physical idea of separating the ph
Pl
l
p4 q
(b)
‘H=)i$ Pl
Fig. 2. Coupled
p4
Ii 3;
T q’
a4
equations for evaluation of ph interaction 9 and the scattering amplitude Fermi sphere. These equations are driven by 9, the “direct interaction.”
~4 on the
J. Wambach et al. / Quasiparticle
interactions
133
interaction, 9 into a “direct” part which incorporates short-range correlations and an “induced” part which includes exchange of virtual collective modes is quite similar to the Babu-Brown model (BBM) 15). The BBM was initially proposed for liquid 3He but later also applied to nucleonic matter 16), in a somewhat rudimentary form. The original intent was to calculate Landau parameters. Then the scattering amplitude is required to be antisymmetric only in the forward direction (q = 0). If one is interested in transport coefficients, superfluidity, etc., as we are here, non-zero momentum transfer is required and the BBM model is incomplete. Appropriate extensions to all possible values of q on the Fermi sphere are possible, however, and have been given in ref. ‘l). In ref. ‘) we have applied the extended theory to neutron matter and it has also been used successfully for liquid 3He and for the electron gas 13). In contrast to the q + 0 limit, the domain of the BBM, the resulting equations are no longer algebraic but constitute a set of coupled non-linear integral equations for the Landau components F?(q) and As”(q) (2). One obtains explicitly F;(q) = JX(q) + %(q, 4’)
x
C [f~"m(q')x~(s')Asl(q')+~~a,(s')x~"(q')A",(91)1 l’,m,n
1 ,
F;(q) = G(q)+ %(a 4’) ,,E n [3F”m(q’>x~“(q’)A”n(q’) -tF”,(s’)xF”(q’)A”.(s’)l , 1
. .
(3)
and
A;(q) = F;(q) - C Cn(q)x;ln(qMXq)
,
A;(q) = F;(q) - C CXq)xY”(q)KXq)
,
m,n
m,n
(4)
where q’ denotes the magnitude of the momentum in the exchange particle-hole channel (see fig. 2) and BI is an integral operator which projects onto the direct particle-hole channel momentum, q. Because of the Fermi-surface restriction, q and q’ are related. The direct interaction enters through the spin components D?(q) and ,$“(q) are angular projected ph propagators which are discussed in detail in ref. ‘l). They include the density and current response as well as higher tensor correlations. Since both, 9 and &, depend on two angular variables, 0, and O,, we can cast the integral equations (3) and (4) into a set of non-linear matrix equations for the coefficients F;$ and As: in a double-moment expansion, N(O)?T N(O).V’=
= 1 F;:P,(cos Lm
2 A;:P,(cos
1.m
@)P,(cos
f3,) ,
O,)P,,,(cos 0,).
(5)
Given a direct interaction, these matrix equations are solved iteratively in a truncated
134
J. Wambach et al. / Quasiparticle
set of coefficients
(l, m) which has to be chosen
interactions
sufficiently
large to ensure
gence. The convergence properties are governed by the momentum direct interaction and typically I,,,,, = mmax = 10 suffices.
2.2. THE
DIRECT
structure
converof the
INTERACTION
In our approach, the direct interaction, 9, completely specifies the particle-hole interaction 9 and the scattering amplitude &. In general, 9 represents the sum of all ph-irreducible diagrams. Therefore any microscopic calculation of the phirreducible interaction any double counting.
can drive the two-body equations (3) and (4) without causing In practice, however, such calculations are difficult. Fortu-
nately, screening corrections from the induced interaction render the quasiparticle interactions rather insensitive to the input 9 [refs. 99’“)]. In the present work we shall pursue two routes in constructing an appropriate direct interaction: (1) We approximate 9 by a Brueckner G-matrix as was done in the work of Sjiiberg 14). The G-matrix incorporates the most important short-range correlations and will be particularly well suited for low densities. This approach does not involve any adjustable parameters. (2) We represent 9 by a “pseudopotential” based on a local bare interaction, incorporating the effects of short-range correlations through a modification of the core height. In neutron matter, the regularization of the core heights introduces two parameters utf and at’ [ref. “)I. Due to the Pauli principle the results are insensitive to atT [ref . ‘)I , however. As we shall discuss below the two approaches give very similar results in the density regime of interest. 2.2.1. The G-matrix. For our purposes
it is convenient
to have a parameterization
of the neutron-matter G-matrix in terms of a local spin-dependent potential. Such parameterizations are available in the literature. In the following we utilize the most recent work by Nishizaki et al. I’) which fits a state-dependent five-range gaussian interaction ( ?) to the G-matrix elements of the Reid soft-core (RSC) potential over a wide range of densities and for arbitrary proton fraction [o = (p,-&/p]. In neutron-star applications this is particularly useful. The resulting local neutronneutron
potential
p(r)
has the following V(r)
= i
CT’(p,
form: Cl)
exp [-(r/hi)‘]
,
i=l
FL(r) = i cT’(p,a)
exp [-(r/hi)*],
i=l
cj(p, a) = aj(a)+bj(a)k~*.
(6)
The ranges hi and the strength parameters ai and Vi are listed in table 1 for the case of pure neutron matter (a = 1). The density dependence of the interaction is
J. Wambach et al. / Quasiparticle interactions
135
TABLE 1 Parameterization
of the G-matrix of Nishizaki et al. I’) for neutron matter. The ranges, Ai, are in fm whilst the a’s and b’s are in MeV
A
at1
att
btl
bff
0.50
l.l93E+03 -3.799E+02 6.049E+OO -7.078E-01 -1.878E-02
2.600E + 02 -9.396E+Ol -1.278E+OO 5.424E - 01 -2.104E-03
-6.164E+02 2.959E + 02 -6.089E + 00 -4.946E - 01 2.247E - 02
2.037E + 02 l.lOlE+02 -2.4068+00 -2.604E - 01 2.247E - 02
0.95 1.70 2.85 5.00
indicated
in fig. 3 which shows the radial form of the two spin-components
pTt and
prL for densities p/p0 = 0.03 (dashed lines) and p/p0 = 1.0 (solid lines). At short distances both interactions are repulsive reflecting the core of the bare interaction. At intermediate distances ?tr gives an attraction which is strong enough to cause pairing in the ‘So channel for densities up to bpO. With increasing p the G-matrix become increasingly repulsive as can be verified from the volume integrals of e(r). This is quite natural since, as the mean interparticle spacing decreases, the neutrons sample more of the core. This mechanism is the physical reason for the disappearance of s-wave superfluidity at higher density (see below). To gauge the quality of the G-matrix approximation to the direct interaction, 9, we compare the resulting equation of state (EOS) with more sophisticated many-body 700 ,
I
I
I
600 506 400 300 200
100 0 -100 ’
0.0
I 0.5
I
I
I
1.0
1.5
2.0
r (fm) Fig. 3. The neutron-matter pseudopotentials v?(r) and PA(r), derived from the G-matrix parameterization of ref. I’), for two densities p/p0 = 0.03 (dashed lines) and p/p,, = 1.0 (solid lines).
J. Wambach et al. / Quasiparticle
136
calculations
r8). Given the parameterization
32 -
5 C sj C (krAi)3c!(p, i=l
(v)=4J;;j=1
with k, = ($r*~)“~,
(6) it is straightforward
to calculate
the
E/A = $( kk/2m) + ( c’>, where
total energy/particle, I
interactions
k”= k/ kF and f(k)
L2dk”f(C)(l*exp[-(kAi)2]),
1)
= $ - 22’3<+ $i3. Spin degeneracy
(7) is accoun-
ted for by the factor Sj. From the total energy, the pressure is easily found as p(p) = p’d(E/A)/dp. The results are plotted in fig. 4 which gives a comparison with
1.5
-
1.0 -
0.5
-
0.0 0.00
0.10
0.05
P
0.15
W-0
Fig. 4. The neutron-matter equation of state from the G-matrix (full lines) used in the present work. The upper part displays the energy/particle as a function of density while the lower part gives the pressure. The circles denote results from the Friedman-Pandharipande equation of state I’).
et al. / Quasiparticle interactions
J. Wambach
137
the realistic Friedman-Pandharipande EOS la) for cold neutron matter (circles). We find that up to p. the agreement is very satisfactory which gives us confidence in the G-matrix approach for the determination of the quasiparticle interaction. Given the r-space pseudopotentials crt and ?‘, derived from the G-matrix, the Landau moments, D;“(q), of the direct interaction are now obtained by Fourier transformation and proper antisymmetrization as
G(q) = %(a q’)[Wq, 471 , n(s) = %(q, q’)[Wq, 471 ,
gscq, 4’) = +%I)- W”(s’)+w(s')l W(q,
,
q’)= P(q)-[$F(qy-#(q~)], 03
W(q)
= 47r I
dr r2 jo( qr) p*“(r)
,
(8)
0
where the integral operator 9,(q, q’) projects out the Ith Landau moment of the explicitly antisymmetric 9(q, q’). Results for I = 0 and 1= 1 are displayed in fig. 5. They indicate that the momentum dependence becomes stronger as density increases. This is easily understood. With growing density the Fermi momentum kF increases as does the momentum transfer q on the Fermi sphere. At low kF, q is small and one obtains essentially the volume integrals of the antisymmetrized direct interaction. At larger kF, on the other hand, the momentum transfer becomes appreciable and the spatial structure of the G-matrix is sampled. Antisymmetry of the direct interaction under exchange implies the following general sum rule: W(q=q’)+W(q=q’)=O,
and, as a special case, the forward-scattering
(9)
sum rule obtains when q = q’ = 0,
; [D;(q=O)+D;(q=o)]=O. I=0
(10)
This is a useful check of the convergence in the Landau expansion as well as the numerical accuracy of the exchange integral. At all densities of interest, the sum rule is exhausted by a few terms with the 1= 0 moments giving the largest contribution. This rapid convergence is due to the short-range nature of the G-matrix. For the same reason the symmetric and antisymmetric components, of given 1, sum to zero individually, to a high degree of accuracy. 22.2. Pseudopotentiab A less microscopic way to construct the direct interaction is based on the PPM. It is particularly useful in cases where the G-matrix fails to be a good starting point, such as the high-density 3He liquid. In the PPM one starts
138
J. Wambach et al. / Quasiparticle 1.o
I
0.4
-
0.2
-
interactions t
I
0.0 -0.2 -0.4 __..I.-______._--_ _.-- _ -1.0
0.8
-
0.6
-
I
I
I
I
1
I
-0.4 -0.6
r
Fig. 5. Momentum dependence of the first two Landau moments D;,;(q) of the direct interaction on the Fermi sphere. Results are shown at three densities: p/p0 = 0.03 (dashed lines), p/p0 = 0.2 (dash-dotted lines) and p/p0 = 1.0 (solid lines).
with a local bare interaction Vbare(r) and the magnitude of the repulsive core is modified in a density-dependent fashion. For neutron matter, suitable choices of V bareare the V, form of the RSC potential 19), or the more recent Argonne ANVi4 potential *‘). Ignoring tensor and spin-orbit components we then have Vbare(r) = vS( r) + Va( r)a * u’ ,
(11)
where Pa is the spin-symmetric (-antisymmetric) part of Vbare.At distances above -0.8 fm, which is somewhat larger than the core radius, these interactions are
139
J. Wambach et al. / Quasiparticle interactions
remarkably similar to the r-space G-matrix at low density. It is then argued the main effects of short-range correlations can be incorporated by fixing maximum value the cores can attain. For the relevant momentum ranges we implement this renormalization by using a rather crude cut-off prescription. define the “pseudopotentials” p(r) as pTt( r) = min {at?, V”(r)}
that the can We
,
V&(r) = min {afL, VT1(r)} ,
(12)
in which, below a certain distance, the bare potentials are set equal to constant “core heights” utt or utl. These core heights are parameters which generally depend on the matter density and, according to the PPM, should be determined from empirical I=0 Landau parameters. While this is possible in 3He, only limited information is available for nuclear matter with small proton fraction. Previously we have found ‘) that the calculated Landau parameters are largely insensitive to a”, due to the Pauli principle, so that at’ remains as the only parameter. Realistic values quoted in ref. ‘) range between 100 and 250 MeV. It is instructive to compare the Fourier transforms of the pseudopotentials (12) with those of the G-matrix. Such a comparison is given in fig. 6 at p/p,, = 0.2. Here
I
0.0
I
2.0
4.0
I
I
6.0
6.0
I
10.0
Fig. 6. Momentum dependence of the “pseudopotentials” P’( 4) which determine the “direct interaction” in the two-body equations (3) and (4). The solid lines denote the G-matrix results while the dashed lines give the PPM potentials derived from ANV,, with core heights adjusted to give the same volume integrals as the G-matrix interaction.
140
the pseudopotentials
J. Wambach et al. 1 Quasiparticle interactions
were obtained
from the ANVi4 potential
heights to give the same volume
integrals
and at’ = 170 MeV, well within
the ranges
as the G-matrix. quoted
adjusting
the core
This yields uTt = 115 MeV
in ref. ‘). The momentum
depen-
dence of the two interactions is strikingly similar over a very wide range. We can thus conclude that the basic physical effect of the Brueckner correlations in neutron matter is to regularize the core and simple prescriptions in terms core height are well justified over the momentum range of interest.
of an effective
3. Results 3.1. THE PARTICLE-HOLE
INTERACTION
In our microscopic model the ph interaction, 9, is determined from the solutions of the coupled two-body equations (3) and (4) which are driven by the direct interaction, 9. To lowest order 9 = 9. Higher-order corrections are built in through ph correlations. To assess their relative importance it is useful to examine eqs. (3) and (4) in second order. Since IDS(q)1 = [D?(q)\ (see fig. 5) there will then be no correction to F?(q) due to cancellation of spin-independent and spin-dependent correlations. On the other hand there is an additive contribution to F;(q) with the spin-dependent term being These conclusions basically
three times more important due to statistical weight. survive the non-linear solutions, as we shall see below.
Let us first discuss the q-dependence of the 1= 0 and 1= 1 moments of the ph interaction. In the density range of interest, those two terms are the dominant contributions to the Landau expansion. Using the G-matrix of subsect. 2.2.1 as the direct interaction we display the results in fig. 7 for three different densities (p/pO= 0.03, 0.20, 1.00). A comparison with fig. 6 shows that the q-dependence of F?(q) follows that of Ds;“( q) very closely. In fact, the antisymmetric parts F:(q) and Di( q) are almost identical, as suggested by the second-order arguments The spin-independent part FE(q), on the other hand, is substantially
given above. weaker than
Dg( q) over the entire momentum range. Closer inspection shows that this weakening is almost independent of momentum transfer indicating that the exchange ph correlations are rather short range, as already noted in ref. ‘). The lower part of fig. 7 gives results for the I= 1 moments. At low density they are small, reaching values of around 0.3 as p. is approached. The momentum dependence is rather flat (except near 2k,) indicating the short-range nature of the current-current interaction. It should also be mentioned there is no Pauli-principle requirement for 9 and no sum-rule equivalent to eq. (9) exists. Therefore the symmetric and antisymmetric components can be influenced rather differently by ph correlations. The lowest-order Landau parameters F;,:(O) characterize the equilibrium properties of a normal Fermi system. In particular, the effective mass m* is given by (13)
J. Wambach 1.o
et al. / Quasiparticle
I
interactions
I
141
I
I
0.6 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.6 -1.0
0.8 t 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6
Fig. 7. Momentum dependence of the first two Landau moments F:.:(q)of the particle-hole interaction on the Fermi sphere. Results are shown at three densities: p/po=0.03 (dashed lines), p/pa= 0.2 (dash-dotted lines) and p/p = 1.0 (solid lines).
and determines
the specific heat CV at constant C, = fm*kFT.
volume
as
(14)
Its density dependence is therefore easily calculated from F;(O) which is displayed in fig. 8. At very low densities, Cv is essentially that for a free Fermi gas, since m*/m is very close to unity. With increasing density it drops below the free-gas value, however. Comparison of F;(O) with results from the direct interaction (dotted line) shows that ph correlations render the spin-symmetric current interaction more
J. Wambach et al. / Quasiparticle interactions
142 1.2 1.0 0.8 0.6 0.4 g
0.2
m.V L
0.0 -0.2 -0.4 -0.6 -0.8
k, (fm-‘) Fig. 8. Density dependence of the lowest-order (solid lines) in comparison with the corresponding
Landau parameters F$(O) from the full ph interaction values from the direct interaction alone (dashed lines).
repulsive, thus increasing the effective mass M*. This is the well-known enhancement near the Fermi surface 30). The I= 0 Landau parameters determine the bulk modulus b = Sp/Sp via the relation b=--
2 k; 3 2m”
and the static magnetic susceptibility xS.=
1+ cdO)l ,
(15)
as
m*kF @: 1+ F:(o)]-’
-
7T2
)
(16)
where p denotes the magnetic moment of the neutron. Again, the density dependence of these quantities can be easily obtained from fig. 8. A comparison of Fy(O) with the results from the direct interaction only (dashed lines) reveals that, over the entire range, the components Ft,, are affected very little by ph correlations, for reasons given above. The component J$, however, is screened considerably, mostly by spin fluctuations. The attraction is strongly reduced. The effect is particularly pronounced near kF = 1 fm-’ (P/P,, - 0.2) where the direct interaction yields values very close to -1. According to the general stability criterion *‘) FF(O) > -(21-r 1) )
(17)
J. Wambach et al. / Quasiparticle interactions
143
a value less than -1 would cause break up of the homogeneous phase. The latter is, indeed, encountered in symmetric nuclear matter around OSp,-0.7~~ [ref. 20)]. Finally, we should mention that our result for the lowest-order Landau parameters are in good agreement with the findings in ref. 22) where a variational approach has been used.
3.2. SCATTERING
AMPLITUDES
As for the ph interaction, the scattering amplitude, &, is completely specified by the direct interaction and no new parameters are introduced. We can therefore determine the Landau components As” of d from the self-consistent solutions of eqs. (3) and (4). Using the G-matrix of subsect. 2.2.1, results for A;,:(q) as a function of momentum transfer q are given in fig. 9 at three different densities p/p0 = 0.03, 0.20, 1.00 which correspond to the choices in figs. 5 and 7. As for the corresponding moments of 9 and 9, the momentum dependence of the spin-symmetric component A:,:(q) becomes strong as density increases, being quite similar to that of F:(q) (fig. 5). On the other hand, At(q) hardly depends on momentum, except near 2kF. It should be reiterated that the scattering amplitudes are antisymmetric over the entire Fermi sphere which leads to the sum rule &P( q = q’) + d”( q = q’) = 0 ) f
[A;(q=O)+A;(q=O)]=O,
(18)
I=0
in analogy to eq. (9) for the direct interaction. The convergence in I is, however, much slower, particularly at higher density, as can be seen from fig. 9. As is well known, the non-equilibrium behavior of a low-temperature Fermi liquid as well as its superfluid properties are determined by the qp scattering amplitude 23). The scattering amplitude enters the collision integral of the quasiparticle transport equation via the transition amplitude A1234= (121.%!134)where W = 27rlA,,,,12 gives the probability for quasiparticle scattering from states 1 and 2 to states 3 and 4 in a binary collision. At the Fermi surface, the momentum dependence is usually expressed in terms of two scattering angles 0 and 4, where 8 is the angle between incoming and outgoing relative momenta of the scattered quasiparticles and 4 the angle between the scattering planes. Evaluation of the transition probability, W, then involves a transformation of the angular variables @ and 0, to the scattering angles 8 and 4 which is determined by the relations I’) cos e=-l++(l+cos cos 4 =
o,)(l+cos
2cos O,--$(l+cos
2-;(1+cos
0,))
o,)(l+cos 0,) o,)(l+cos 0,) ’
(19)
144
J. Wambach et al. / Quasiparticle interactions
I
1.0
I
f
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 1.0 0.8
i-
-l
0.6 0.4 0.2 s 0.0 -0.2 -0.4 -0.6 -0.8 -
Fig. 9. Momentum dependence of the first two Landau moments A$y(,(q) of the scattering amplitude on the Fermi sphere. Results are shown at three densities: p/p,, = 0.03 (dashed lines), p/p = 0.2 (dash-dotted lines) and p/p0 = 1.0 (solid lines). with
angular range 0 s 0 s m, 0 s 4 s 27~. The normalized transition probability scattering of two quasiparticles of like (tt) or unlike (T&) spin is then defined
W’“(l9,4)=277 and the total transition
[A:;;;)]“,
rate,
W, is given by
~(8,
+)=fwT”(e,
4)++w”(e, 4).
for as
(20)
(21)
145
J. Wambach et al. / Quasiparricle interactions
After averaging over the Fermi surface,
(22) one then obtains the well-known expression for the quasiparticle TT==-
8~~ m*3( W) ’
lifetime (23)
Results are shown in the left panel of fig. 10. One finds that the quasiparticle lifetime increases with density. This behavior is qualitatively different from the quasiparticle lifetime in other quantum fluids, e.g. liquid 3He [ref. “)I. The density dependence arises from a combination of the drop in m* and a decrease in ( W) as the density increases. The effective mass enters the expression for 7 as the third power, and therefore its density dependence is amplified. As the density increases m*/ m drops below unity (see fig. 8). Furthermore ( W) is a strongly decreasing function of density for 0 < p G po. For liquid 3He the effective mass increases with increasing density, hence the qualitatively different density dependence of the quasiparticle lifetime. 50.0
1
I
r
Quasiparticle lifetime 40.0 -
"'
Spin-diffusion coefficient
0.8 0.6 0.4 -
0.2 -
k, (fm-‘) Fig. 10. Density dependence of the transport coefficients in neutron matter. The solid lines denote the full calculations while the dashed lines correspond to results from using the direct interaction only.
146
J. Wambach et al. / Quasiparticle interactions
3.2.1. Transport coeficients. Given the transition probability W( 0, 4) we can now calculate the transport coefficients for neutron matter. In the extreme lowtemperature limit T/F~<< 1 the thermal conductivity K, the viscosity q and the spin diffusion coefficient d are obtained by suitable angular averages of the transition probabilities WTT(0, $41)and WT1(B, 4) over the Fermi sphere 24). Details can be found in ref. 24) and we only quote the results. Our starting point is the AbrikosovKhalatnikov-Hone (AK, H) approximation to the collision integral which gives
dH =f(l+
F;?(O))&
TAu=fP*v:7( T2(;_A,)),
(24)
where the h, are given by
I
h,(W)=
cos
(Q.))
dR W’@,
b(W)= A,( w
dfl WC446)
-4*
47, i
(1+2 cos e, ’
~)[COS 6+(1-cos
0) cos (P]+jWt’(6,
4) 9
2 cos ($9)
=
(25)
The exact results 25) for the transport coefficients can then be expressed in terms of the AK, H expressions (24) as 4n+5 K =
KAK
.=,,(n+1)(2n+3)[(n+l)fZn+3)-h,]
I ’
4n+3 I ’ 4n+3
f ,=o(n+l)(2n+l)[(n+1)(2n+l)-~,1
1 (26) ’
which are easily evaluated from our scattering amplitudes. Results are displayed in fig. 10 (solid lines). In the region of interest a strong density dependence is found. This is a direct consequence of the density dependence of the lifetime 7 which enters the transport coefficients as a prefactor. Fig. 10 also gives a comparison with coefficients which result from using the direct interaction only (dashed lines). 3.22. S~per~uidity. BCS theory 32) forms the basis for evaluating superfluid properties. It is assumed that, to first approximation, the system can be described by a Slater determinant I+0) in which all single-particle momentum states k with
J, Wambach et al. / Quasiparticle interactions \kl c
147
kF are occupied. S-wave superfluidity
appears if the formation of Cooper pairs from particles with opposite momenta and spins leads to a lowering of the groundstate energy. At zero temperature this occurs when the s-wave gap equation
A;=-)CP&, k'
A;. [(ek'-eF)2+(Air)2]1'2
(27)
has
non-trivial solutions for the gap function Ai. The quantity Ek = is interpreted as the energy needed to create a quasiparticle of [(ek-eF)2+A2k]1’2 momentum k (and given spin projection) in the superfluid while ek denotes the normal-state single-particle energy. For s-waves the pairing matrix elements entering (27) assume the form POkk,=(k~-k’ll~(‘So)lkt-k’~),
(28)
where zZ(‘S,) is the ‘S,, scattering amplitude. Because of isotropy, they only depend on the magnitudes of the momenta k and k’. The solution of the gap equation (27) requires knowledge of the full pairing matrix element, Ptk’. In our scheme for evaluating the scattering amplitude we obtain this matrix element only at the Fermi surface as 1
dcos~[s8”(f3=7r,~)-30Pa(8=~,~)], (29) i -1 where a”( 8,+) and .&( 0,4) are the spin-symmetric and spin-antisymmetric scattering amplitudes in the scattering-angle representation (19). With this Fermi-surface restriction we cannot solve the full gap equation but have to rely on some approximate solutions. The simplest is the BCS weak-coupling formula N(0)POkFkF
=
i
where er= k:/2m* is the Fermi energy. For the maximum gap this approximation is expected to be accurate to within a factor of two 33*34). The results, given in fig. 11, show the importance of ph correlations. Their possible influence was first recognized by Pethick and Pines 35) although an enhancement, rather than a suppression was estimated. The suppression of the gap was first pointed out by Clark et al. 35)within the Babu-Brown framework and also found in refs. 33*9). The large gaps obtained from the G-matrix are drastically reduced and the maximum attainable gap is only about 0.9 MeV. This is slightly lower than the values obtained in ref. ‘) and is consistent with the most recent findings of the variational+ CBF calculations by Clark et al. 36) which are represented by the dashed lines in fig. 11. It should be mentioned, however, that the present evaluation does not exclude particle-particle reducible graphs from the scattering amplitudes and hence (as in gap calculations based on the G-matrix) involves some double counting. Thus the agreement with the other works may be somewhat fortuitous.
J. Wambach ei al. / Qunsiparticle interactions
148
1.0
0.0
0.0
0.5
1.0
1.5
kF (fm.‘)
Fig. II. Density dependence of the gap parameter A ’+ The solid lines denote our results while the dashed lines quote results from variational calculations using variational methods with and without CBF corrections 9.
Our results imply that the crust of the neutron star is superfluid up to densities p - p0/2 with maximum gaps of the order of 1 MeV. This is qualitatively consistent with the constraints from the analyses of glitch data in the vo~ex-pinning model 504*37). Given the approximate nature of the present calculations of A& one would not be surprised if improved calculations led to somewhat larger Ai, profiles.
We have calculated the ph interaction, S, and the qp scattering amplitude Sa, in pure neutron matter for densities 0 s p 6 p,, which are relevant for the inner crust of a neutron star. We make use of coupled equations for 9 and ~4 which are driven by a ph irreducible interaction, 9 (the direct interaction). Given 9 these equations are solved exactly on the Fermi surface and antisymmetry is kept at all stages. To determine the direct interaction we have made two different choices which both lead to similar results. One is to represent 9 by a Brueckner G-matrix. While this would not be a good choice in liquid 3He, it turns out to be rather satisfactory for neutron matter in the density range of interest. A second approach makes use of the PPM where one starts from a bare local interaction which is renormahzed by introducing effective core heights. This one-parameter model gives very similar results as the G-matrix.
J. Wambach et al. / Quasiparticle interactions
149
The 1= 0 and Z= 1 Landau parameters which we deduce from our model are in good agreement with the variational calculations of ref. ‘*). We find that ph correlations give repulsive screening corrections to Fi and F”, mediated predominantly by the exchange of spin modes. The spin-antisymmetric interactions are hardly affected, a result which is easily understood from the spin structure of the underlying equations. From the calculated scattering amplitudes in spin-symmetric and spin-antisymmetric two-particle states we have determined the quasiparticle lifetime as well as the transport coefficients. A strong density dependence is found which arises from a decrease in m* as well as the average transition probability ( W). Using the weak-coupling BCS formula we have also estimated the density dependence of the ‘So gap parameter A”,,. This parameter is very sensitive to ph correlations, as has been known for some time. The peak values, when these are neglected, are reduced by more than a factor of three by the inclusion ph correlations. Although it may be somewhat fortuitous, our findings are in qualitative agreement with the recent, more elaborate, calculations of ref. 36). As compared to values derived from a most recent analyses of Vela timing data 37) they lie somewhat below the bounds quoted. We thank M.A. Alpar, J.W. Clark and C.J. Pethick for helpful discussions and suggestions. This work was supported by NSF grants PHY89-21025, PHY88-06265, PHY86-00377, PHY84-15064, DMR85-211041, NASAgrant NSG-7653, NATO grant RG 85/093 and the Texas Advanced Research Program grant 010366-012. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107 D.B. Kaplan and A.E. Belson, Phys. Lett. B175 (1986) 57 A.B. Migdal, Sov. Phys. JETP 10 (1960) 176 D. Pines and M.A. Alpar, Nature 316 (1985) 27, and references therein M.A. Alpar, H.F. Chau, KS. Cheng and D. Pines, submitted to Astrophys. J. (1985) O.V. Maxwell, Astrophys. J. 231 (1979) 201 J. Wambach, T.L. Ainsworth and D. Pines, in Neutron stars: theory and observation, ed. J. Ventura and D. Pines (Kluwer, Dordrecht, 1991) p. 37 A.B. Migdal, Theory of finite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967) T.L. Ainsworth, J. Wambach and D. Pines, Phys. Lett. B222 (1989) 173 C.H. Aldrich III and D. Pines. J. Low Temp. Phys. 32 (1978) 689 T.L. Ainsworth and KS. Bedell, Phys. Rev. B35 (1987) 8425 D. Pines, K.F. Quader and J. Wambach, Nucl. Phys. A477 (1988) 365 T.L. Ainsworth, F. Green and D. Pines, Phys. Rev. B42 (1990) 9978 0. Sjiiberg, Ann. of Phys. 78 (1973) 39 S. Babu and G.E. Brown, Ann. of Phys. 78 (1973) 1 S.-O. Blckman, G.E. Brown and J.A. Niskanen, Phys. Reports 124 (1985) 1 S. Nishizaki, T. Takatsuka, N. Yahagi and J. Hiura, preprint (1991) B. Friedman and V.R. Pandharipande, Nucl. Phys. A361 (1981) 502 V.R. Pandaripande and R.B. Wiringa, Rev. Mod. Phys. 51 (1979) 821 J.M. Lattimer and D.G. Ravenhall, Astrophys. J. 223 (1978) 314
150
J. Wambach
et al. / Quasiparticle
interactions
21) I.Ja. Pomerantschuck, Sov. Phys. JETP 8 (1958) 361 22) A.D. Jackson, E. Krotscheck, D.E. Meltzer and R.A. Smith, Nucl. Phys. A386 (1982) 125 23) D. Pines and P. Noziere, The theory of quantum liquids (Benjamin, New York, 1966) 24) A.A. Abrikosov and I.M. Khalatnikov, Rep. Prog. Phys. 22 (1959) 329; D. Hone, Phys. Rev. 121 (1961) 669 25) GA. Brooker and J. Sykes, Phys. Rev. Lett. 21 (1968) 279; H.H. Jensen, H. Smith and J.W. Wilkins, Phys. Lett. A27 (1968) 532 26) D. Pines, in Highlights of condensed matter theory, Proc. 89th Int. School of Physics, Sot. Italiana di Fisica, Bologna, Italy (1985) p. 580 27) R.V. Reid, Ann. of Phys. 50 (1968) 411 28) R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29 (1984) 1207 29) J.A. Niskanen and J.A. Sauls, preprint (1981) 30) C. Mahaux, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Reports 120 (1985) 1 31) B.R. Patton and A. Zaringhalam, Phys. Lett. A55 (1975) 95 32) J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175 33) J.M.C. Chen, J.W. Clark, E. Krotscheck and R.A. Smith, Nucl. Phys. A451 (1986) 509 34) M. Baldo, J. Cugnon, A. Lejeune and U. Lombardo, Nucl. Phys. A515 (1990) 409 35) D. Pines, Proc. XII Int. Conf. on low temperature physics, ed. E. Kandu (Kligatu, Tokyo, 1971); J.W. Clark, C.-G. Kallman, C.H. Yang and D.A. Chakkalakal, Phys. Lett. B161 (1976) 331 36) J.W. Clark, R.D. Dave and J.M.C. Chen, to be published 37) D. Pines and M.A. Alpar, in Structure and evolution of neutron stars, ed. D. Pines, R. Tamagaki and S. Tsuruta, (Addison-Wesley, Reading, MA, 1992)
(1993)
NUCLEAR PHYSICS A
128-150
Quasiparticle interactions in neutron matter for applications in neutron stars+ J. Warnbach’,
T.L. Ainsworth’
and D. Pines
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, II 61801, USA
1110 West Green Street,
Received 4 May 1992 (Revised 16 July 1992)
A microscopic model for the quasiparticle interaction in neutron matter is presented. Both particle-particle (pp) and particle-hole (ph) correlations are included. The pp correlations are treated in a semi-empirical way, while ph correlations are incorporated by solving coupled two-body equations for the particle-hole interaction and the scattering amplitude on the Fermi sphere. The resulting integral equations self-consistently sum the ph reducible diagrams. Antisymmetry is kept at all stages and hence the forward-scattering sum rules for the scattering amplitude are obeyed. Results for Landau parameters and transport coefficients in a density regime representing the crust of a neutron star are presented. We also estimate the ‘So gap parameter for neutron superfluidity and comment briefly on neutron-star implications.
Abstract:
1. Introduction We dedicate this article to Arkady B. Migdal, whose recent death deprived the physics community of one of its leading members. Kadya was a theoretical physicist of the first magnitude; his remarkable personality and breadth of interests made an indelible impression on all of us who were privileged to know him. As we note, the behavior of neutron matter in neutron stars was a topic dear to him, the more so because his pioneering contributions to neutron-matter superfluidity, the Fermiliquid description of nuclei and nuclear matter, and to the possibility of pion condensation
in neutron
stars, laid the ground
work for so much subsequent
work
on the problems we consider here. Neutron matter, bound under gravitational pressure, is a significant component of neutron stars. It is found over a wide density range, p,/lOOO < 2p,, where pO= 2.6 x 1014 g/cm3 is the saturation density of nuclear matter. The equation of state of neutron matter at densities >pO, the approximate location of the boundary between the solid outer crust and the quantum liquid core of the star, determines the stellar radius. For most currently accepted equations of state that means that + This paper is dedicated
to the memory of A.B. Migdal. Correspondence to: Prof. Dr. J. Wambach, Dept. of Physics, 1110 West Green Street, Urbana, 11 61801, USA. ’ Present address: Institut fiir Kernphysik, Forschungszentrum Jiilich, D-5170 Jiilich, Germany. * Present address: Department of Physics, Texas A&M University, College Station, TX 77843, USA. 03759474/93/$06.00
@ 1993 - Elsevier
Science
Publishers
B.V. All rights reserved
J. Wambach et al. / Quasiparticle
interactions
129
neutron matter is found between a few meters and a few kilometers below the surface of the star. In the crust, the neutrons coexist with a lattice of neutron-rich nuclei and a sea of relativistic electrons; in the quantum liquid core, they coexist with protons (whose density is -5-10% that of the neutrons) and relativistic electrons. Both the constituents and equation of state at densities >2p,, are uncertain; pion condensation, first suggested by Migdal I), kaon condensation *) or, as some believe, a quark-gluon plasma are some of the interesting possibilities. The present work has two motivations: to derive the Fermi-liquid parameters and transport coefficients for the normal phase of neutron matter, and to estimate the temperature at which neutron matter becomes superfluid. More than 30 years ago, Migdal suggested ‘) that since the outermost nucleons in nuclei exhibit superfluid properties, it was quite likely that if neutron stars existed, neutron matter inside these stars would be superfluid and this would in turn lead to interesting macroscopic phenomena. With the discovery of pulsars, and their identification as rotating neutron stars, that has turned out to be the case. Throughout most of the inner crust of the star, for all but newly formed pulsars, the stellar temperature is such that neutron matter found there can be shown to be in a ‘S,, pairing state. Because pulsars rotate, vortices form in this superfluid. Since the coherence length of the superfluid is comparable to the size of crustal nuclei with which the superfluid coexists, the vortices are pinned to these nuclei. This pinning neutron fluid plays a significant role in the spin down of the pulsar since catastrophic unpinning of vortices is believed to be associated with the giant glitches observed in the Vela and other pulsars. Furthermore the vortex creep theory of the postglitch behavior provides strong evidence for the presence of a crustal superfluid “). A recent detailed analysis of the behavior of the Vela pulsar following the eight glitches observed during the twenty-year period, March, 1969, to January, 1989, based on vortex creep theory, provides interesting constraints on the parameters relevant for pinning: the pinning energy, EP, and the coherence length, 5. Since both depend sensitively on the magnitude of the neutron superfluid gap, A&, one finds a set of mutual constraints which link the pinning parameters and the density-dependent energy gap ‘). Neutron superfluidity and proton superconductivity are also believed to play an important role in the evolution of neutron stars. At early times the cooling of the star is dominated by neutrino emission involving the modified Urea process as well as neutron-neutron and neutron-proton bremsstrahlung in the liquid interior “). It is well established that the existence of pairing gaps suppresses the neutrino processes by reducing the available phase space in the final state. At the same time the total heat capacity of the star may be drastically lowered. Finally, the time evolution of the magnetic field may be influenced by the interaction between neutron vortex lines and superfluid protons. Superfluid properties of neutron matter have been studied for a long time. While all calculations yield neutron ‘S,, pairing for k, less than 1.3-1.5 fm-’ which corresponds to matter densities between 0.5~~ and 0.7p, a prediction of the maximum
J. Wambach et al. / Quasiparticle
130
gap, AiF, as well as the pair condensation The predictions problem
for the maximum
originates
(qp) scattering
from the exponential
amplitude,
E,, has been notoriously
difficult.
gap vary by almost an order of magnitude
pairing interaction. In Fermi-liquid theory the pairing title
energy,
interactions
dependence
matrix
element
&, near the Fermi
of the superfluid is obtained surface
‘). This
gap in the
from the quasipar-
which
also determines
the transport properties of the neutron liquid. It is the aim of the present paper to obtain a reliable estimate for Op by exploiting its relationship the particle-hole interaction, 9 [ref. “)I. We have earlier presented a model ‘) for the particle-holeirreducible interaction, 9, which drives the coupled equations for 9 and ~9 (subsect. 2.1). The model is microscopic in the sense that the starting point is the bare two-body interaction between nucleons. Following a very successful approach for the He liquids lo) the short-distance part of the bare interaction is renormalized phenomenologically through a density-dependent effective core height which is adjusted to the equation of state. We will show that, in the density range 0 s p s po, this effective interaction is very close to the Brueckner G-matrix (subsect. 2.2). Given 9, the ph interaction and the scattering amplitude at the Fermi surface are determined and Landau parameters, transport coefficients, etc. can be deduced (subsects. 3.1 and 3.2). Also the pairing matrix element at the Fermi surface can be calculated
and the density
dependence
of the gap parameter
inferred.
2. Theory In Fermi-liquid theory a strongly interacting system of fermions is represented by dilute gas of weakly interacting quasiparticles. These “elementary excitations” exist in the vicinity of the Fermi surface and have the same statistics as the bare particles. They differ, however, through an effective mass, m*, and a finite lifetime, 7. A finite lifetime is caused by quasiparticle collisions and it sets a limit on the applicability susceptibility,
of the theory. Bulk properties such as the compressibility, the magnetic the specific heat, as well as the transport properties are described in
terms of the quasiparticle interaction. This interaction is characterized by a few “Landau parameters” which are directly related to the macroscopic equilibrium properties of matter. They also signal the onset of various kinds of instabilities. Given the complications of a strongly interacting many-body system, Landau suggested a phenomenological treatment of the quasiparticle interaction using empirical bulk information, conservation laws and symmetry principles. One is restricted, however, to small momenta, the so-called “Landau limit.” This is sufficient for many bulk phenomena but, some of the non-equilibrium properties as well as the superfluid behavior require a much wider range of momenta. To describe such processes one has to make contact with the momentum structure of the bare interaction. This is complicated by the fact that the bare interaction has a hard core which has to be dealt with in some non-perturbative way.
J. Wambach et al. / Quasiparticle
interactions
131
In our approach we are motivated by the polarization potential model (PPM) of Aldrich and Pines lo) which starts from the bare fermion interaction and includes medium effects such as Pauli correlations, polarization phenomena, etc. via phenomenological modification of its short-distance behavior. With relatively few parameters, adjusted to the bulk behavior, the PPM has been applied successfully in liquid 3He as well as in 3He-4He mixtures. Straightforward application to nuclear or neutron matter, however, faces some difficulties: (i) since the bulk properties can only be inferred from finite nuclei they are much less certain; (ii) in the density regime of interest, the role of the Pauli principle is quite different from that in liquid helium; (iii) the presence of non-central components in the bare interaction complicates the phenomenological treatment of medium effects. In a series of papers rzV9) we have therefore extended the PPM so that it can be applied to nucleonic matter. The formalism is reviewed below.
2.1. THE TWO-BODY
EQUATIONS
In determining the quasiparticle interaction two quantities are of interest, the particle-hole interaction, 9, and the scattering amplitude d (fig. 1). The particlehole interaction determines equilibrium properties, collective modes, etc. while the scattering amplitude yields the transport coefficients through damping phenomena induced by quasiparticle collisions. In a homogeneous system the qp interaction can be labeled by three independent momenta. In the following we choose the variables p = $(pr +p3), p’ = f(p2 +p4) and q = p3 -pl. There is an additional spin dependence which arises from the exchange character of the fermion interaction and, in the case of neutron matter, from meson exchange. Meson exchange also adds non-central and spin-orbit components which complicate the interaction. In the following we shall ignore these components since we are interested in densities up to po, where they are unimportant. With central components only .9 and d decompose into a spin-independent and a spin-dependent interaction,
(1)
Fig. 1. The ph interaction
9 and the scattering amplitude
s2.
132
.I. Wambach et al. / Quasiparticle
where p(a) is the symmetric neutron
pair (pa
(antisymmetric)
= t( @’ f @‘)),
are only meaningful
combination
of a spin-t?
and likewise for .vY. By definition,
in the vicinity
of the Fermi
the momenta to the Fermi sphere, IpI = /$I= is given by an angle 0, = O,,., the so-called expand
interactions
surface
and a spin-tJ qp interactions
and we therefore
confine
kF. Then the dependence on p and p’ “Landau angle”, and it is natural to
9 and JB in @, iv(O)F”
= c Fy(q)P,(cos I
N(O)&“=C
A;‘“(q)P,(cos
0,)
)
0,).
(2)
To scale out a trivial density dependence we have multiplied by N(0) = k,m*/d, the density of states at the Fermi surface. Due to the Fermi-surface restriction the magnitude of the momentum transfer q is limited by 0~ /q1s2kF. Therefore the q-dependence can be represented by a second angle 0, as q = 2k,( 1 - cos 0,). In microscopic approaches one tries to relate the quasiparticle interactions to the bare interparticle potential. In a diagrammatic sense, phenomenological modifications of the bare potential represent summations of pp diagrams (ladders) as well as ph diagrams (bubbles). We argue that both diagram sets effectively decouple and that the short-ranged pp correlations can be treated rather crudely. Care must be taken, however, in determining the ph correlations, particularly to ensure the Pauli principle. To set up a system of scattering equations for 9 and d we exploit the diagrammatic relationship between them (fig. 2). This diagram set is driven by a ph irreducible interaction, 9, (fig. 2b) and approximately sums particlehole diagrams in all momentum channels. The physical idea of separating the ph
Pl
l
p4 q
(b)
‘H=)i$ Pl
Fig. 2. Coupled
p4
Ii 3;
T q’
a4
equations for evaluation of ph interaction 9 and the scattering amplitude Fermi sphere. These equations are driven by 9, the “direct interaction.”
~4 on the
J. Wambach et al. / Quasiparticle
interactions
133
interaction, 9 into a “direct” part which incorporates short-range correlations and an “induced” part which includes exchange of virtual collective modes is quite similar to the Babu-Brown model (BBM) 15). The BBM was initially proposed for liquid 3He but later also applied to nucleonic matter 16), in a somewhat rudimentary form. The original intent was to calculate Landau parameters. Then the scattering amplitude is required to be antisymmetric only in the forward direction (q = 0). If one is interested in transport coefficients, superfluidity, etc., as we are here, non-zero momentum transfer is required and the BBM model is incomplete. Appropriate extensions to all possible values of q on the Fermi sphere are possible, however, and have been given in ref. ‘l). In ref. ‘) we have applied the extended theory to neutron matter and it has also been used successfully for liquid 3He and for the electron gas 13). In contrast to the q + 0 limit, the domain of the BBM, the resulting equations are no longer algebraic but constitute a set of coupled non-linear integral equations for the Landau components F?(q) and As”(q) (2). One obtains explicitly F;(q) = JX(q) + %(q, 4’)
x
C [f~"m(q')x~(s')Asl(q')+~~a,(s')x~"(q')A",(91)1 l’,m,n
1 ,
F;(q) = G(q)+ %(a 4’) ,,E n [3F”m(q’>x~“(q’)A”n(q’) -tF”,(s’)xF”(q’)A”.(s’)l , 1
. .
(3)
and
A;(q) = F;(q) - C Cn(q)x;ln(qMXq)
,
A;(q) = F;(q) - C CXq)xY”(q)KXq)
,
m,n
m,n
(4)
where q’ denotes the magnitude of the momentum in the exchange particle-hole channel (see fig. 2) and BI is an integral operator which projects onto the direct particle-hole channel momentum, q. Because of the Fermi-surface restriction, q and q’ are related. The direct interaction enters through the spin components D?(q) and ,$“(q) are angular projected ph propagators which are discussed in detail in ref. ‘l). They include the density and current response as well as higher tensor correlations. Since both, 9 and &, depend on two angular variables, 0, and O,, we can cast the integral equations (3) and (4) into a set of non-linear matrix equations for the coefficients F;$ and As: in a double-moment expansion, N(O)?T N(O).V’=
= 1 F;:P,(cos Lm
2 A;:P,(cos
1.m
@)P,(cos
f3,) ,
O,)P,,,(cos 0,).
(5)
Given a direct interaction, these matrix equations are solved iteratively in a truncated
134
J. Wambach et al. / Quasiparticle
set of coefficients
(l, m) which has to be chosen
interactions
sufficiently
large to ensure
gence. The convergence properties are governed by the momentum direct interaction and typically I,,,,, = mmax = 10 suffices.
2.2. THE
DIRECT
structure
converof the
INTERACTION
In our approach, the direct interaction, 9, completely specifies the particle-hole interaction 9 and the scattering amplitude &. In general, 9 represents the sum of all ph-irreducible diagrams. Therefore any microscopic calculation of the phirreducible interaction any double counting.
can drive the two-body equations (3) and (4) without causing In practice, however, such calculations are difficult. Fortu-
nately, screening corrections from the induced interaction render the quasiparticle interactions rather insensitive to the input 9 [refs. 99’“)]. In the present work we shall pursue two routes in constructing an appropriate direct interaction: (1) We approximate 9 by a Brueckner G-matrix as was done in the work of Sjiiberg 14). The G-matrix incorporates the most important short-range correlations and will be particularly well suited for low densities. This approach does not involve any adjustable parameters. (2) We represent 9 by a “pseudopotential” based on a local bare interaction, incorporating the effects of short-range correlations through a modification of the core height. In neutron matter, the regularization of the core heights introduces two parameters utf and at’ [ref. “)I. Due to the Pauli principle the results are insensitive to atT [ref . ‘)I , however. As we shall discuss below the two approaches give very similar results in the density regime of interest. 2.2.1. The G-matrix. For our purposes
it is convenient
to have a parameterization
of the neutron-matter G-matrix in terms of a local spin-dependent potential. Such parameterizations are available in the literature. In the following we utilize the most recent work by Nishizaki et al. I’) which fits a state-dependent five-range gaussian interaction ( ?) to the G-matrix elements of the Reid soft-core (RSC) potential over a wide range of densities and for arbitrary proton fraction [o = (p,-&/p]. In neutron-star applications this is particularly useful. The resulting local neutronneutron
potential
p(r)
has the following V(r)
= i
CT’(p,
form: Cl)
exp [-(r/hi)‘]
,
i=l
FL(r) = i cT’(p,a)
exp [-(r/hi)*],
i=l
cj(p, a) = aj(a)+bj(a)k~*.
(6)
The ranges hi and the strength parameters ai and Vi are listed in table 1 for the case of pure neutron matter (a = 1). The density dependence of the interaction is
J. Wambach et al. / Quasiparticle interactions
135
TABLE 1 Parameterization
of the G-matrix of Nishizaki et al. I’) for neutron matter. The ranges, Ai, are in fm whilst the a’s and b’s are in MeV
A
at1
att
btl
bff
0.50
l.l93E+03 -3.799E+02 6.049E+OO -7.078E-01 -1.878E-02
2.600E + 02 -9.396E+Ol -1.278E+OO 5.424E - 01 -2.104E-03
-6.164E+02 2.959E + 02 -6.089E + 00 -4.946E - 01 2.247E - 02
2.037E + 02 l.lOlE+02 -2.4068+00 -2.604E - 01 2.247E - 02
0.95 1.70 2.85 5.00
indicated
in fig. 3 which shows the radial form of the two spin-components
pTt and
prL for densities p/p0 = 0.03 (dashed lines) and p/p0 = 1.0 (solid lines). At short distances both interactions are repulsive reflecting the core of the bare interaction. At intermediate distances ?tr gives an attraction which is strong enough to cause pairing in the ‘So channel for densities up to bpO. With increasing p the G-matrix become increasingly repulsive as can be verified from the volume integrals of e(r). This is quite natural since, as the mean interparticle spacing decreases, the neutrons sample more of the core. This mechanism is the physical reason for the disappearance of s-wave superfluidity at higher density (see below). To gauge the quality of the G-matrix approximation to the direct interaction, 9, we compare the resulting equation of state (EOS) with more sophisticated many-body 700 ,
I
I
I
600 506 400 300 200
100 0 -100 ’
0.0
I 0.5
I
I
I
1.0
1.5
2.0
r (fm) Fig. 3. The neutron-matter pseudopotentials v?(r) and PA(r), derived from the G-matrix parameterization of ref. I’), for two densities p/p0 = 0.03 (dashed lines) and p/p,, = 1.0 (solid lines).
J. Wambach et al. / Quasiparticle
136
calculations
r8). Given the parameterization
32 -
5 C sj C (krAi)3c!(p, i=l
(v)=4J;;j=1
with k, = ($r*~)“~,
(6) it is straightforward
to calculate
the
E/A = $( kk/2m) + ( c’>, where
total energy/particle, I
interactions
k”= k/ kF and f(k)
L2dk”f(C)(l*exp[-(kAi)2]),
1)
= $ - 22’3<+ $i3. Spin degeneracy
(7) is accoun-
ted for by the factor Sj. From the total energy, the pressure is easily found as p(p) = p’d(E/A)/dp. The results are plotted in fig. 4 which gives a comparison with
1.5
-
1.0 -
0.5
-
0.0 0.00
0.10
0.05
P
0.15
W-0
Fig. 4. The neutron-matter equation of state from the G-matrix (full lines) used in the present work. The upper part displays the energy/particle as a function of density while the lower part gives the pressure. The circles denote results from the Friedman-Pandharipande equation of state I’).
et al. / Quasiparticle interactions
J. Wambach
137
the realistic Friedman-Pandharipande EOS la) for cold neutron matter (circles). We find that up to p. the agreement is very satisfactory which gives us confidence in the G-matrix approach for the determination of the quasiparticle interaction. Given the r-space pseudopotentials crt and ?‘, derived from the G-matrix, the Landau moments, D;“(q), of the direct interaction are now obtained by Fourier transformation and proper antisymmetrization as
G(q) = %(a q’)[Wq, 471 , n(s) = %(q, q’)[Wq, 471 ,
gscq, 4’) = +%I)- W”(s’)+w(s')l W(q,
,
q’)= P(q)-[$F(qy-#(q~)], 03
W(q)
= 47r I
dr r2 jo( qr) p*“(r)
,
(8)
0
where the integral operator 9,(q, q’) projects out the Ith Landau moment of the explicitly antisymmetric 9(q, q’). Results for I = 0 and 1= 1 are displayed in fig. 5. They indicate that the momentum dependence becomes stronger as density increases. This is easily understood. With growing density the Fermi momentum kF increases as does the momentum transfer q on the Fermi sphere. At low kF, q is small and one obtains essentially the volume integrals of the antisymmetrized direct interaction. At larger kF, on the other hand, the momentum transfer becomes appreciable and the spatial structure of the G-matrix is sampled. Antisymmetry of the direct interaction under exchange implies the following general sum rule: W(q=q’)+W(q=q’)=O,
and, as a special case, the forward-scattering
(9)
sum rule obtains when q = q’ = 0,
; [D;(q=O)+D;(q=o)]=O. I=0
(10)
This is a useful check of the convergence in the Landau expansion as well as the numerical accuracy of the exchange integral. At all densities of interest, the sum rule is exhausted by a few terms with the 1= 0 moments giving the largest contribution. This rapid convergence is due to the short-range nature of the G-matrix. For the same reason the symmetric and antisymmetric components, of given 1, sum to zero individually, to a high degree of accuracy. 22.2. Pseudopotentiab A less microscopic way to construct the direct interaction is based on the PPM. It is particularly useful in cases where the G-matrix fails to be a good starting point, such as the high-density 3He liquid. In the PPM one starts
138
J. Wambach et al. / Quasiparticle 1.o
I
0.4
-
0.2
-
interactions t
I
0.0 -0.2 -0.4 __..I.-______._--_ _.-- _ -1.0
0.8
-
0.6
-
I
I
I
I
1
I
-0.4 -0.6
r
Fig. 5. Momentum dependence of the first two Landau moments D;,;(q) of the direct interaction on the Fermi sphere. Results are shown at three densities: p/p0 = 0.03 (dashed lines), p/p0 = 0.2 (dash-dotted lines) and p/p0 = 1.0 (solid lines).
with a local bare interaction Vbare(r) and the magnitude of the repulsive core is modified in a density-dependent fashion. For neutron matter, suitable choices of V bareare the V, form of the RSC potential 19), or the more recent Argonne ANVi4 potential *‘). Ignoring tensor and spin-orbit components we then have Vbare(r) = vS( r) + Va( r)a * u’ ,
(11)
where Pa is the spin-symmetric (-antisymmetric) part of Vbare.At distances above -0.8 fm, which is somewhat larger than the core radius, these interactions are
139
J. Wambach et al. / Quasiparticle interactions
remarkably similar to the r-space G-matrix at low density. It is then argued the main effects of short-range correlations can be incorporated by fixing maximum value the cores can attain. For the relevant momentum ranges we implement this renormalization by using a rather crude cut-off prescription. define the “pseudopotentials” p(r) as pTt( r) = min {at?, V”(r)}
that the can We
,
V&(r) = min {afL, VT1(r)} ,
(12)
in which, below a certain distance, the bare potentials are set equal to constant “core heights” utt or utl. These core heights are parameters which generally depend on the matter density and, according to the PPM, should be determined from empirical I=0 Landau parameters. While this is possible in 3He, only limited information is available for nuclear matter with small proton fraction. Previously we have found ‘) that the calculated Landau parameters are largely insensitive to a”, due to the Pauli principle, so that at’ remains as the only parameter. Realistic values quoted in ref. ‘) range between 100 and 250 MeV. It is instructive to compare the Fourier transforms of the pseudopotentials (12) with those of the G-matrix. Such a comparison is given in fig. 6 at p/p,, = 0.2. Here
I
0.0
I
2.0
4.0
I
I
6.0
6.0
I
10.0
Fig. 6. Momentum dependence of the “pseudopotentials” P’( 4) which determine the “direct interaction” in the two-body equations (3) and (4). The solid lines denote the G-matrix results while the dashed lines give the PPM potentials derived from ANV,, with core heights adjusted to give the same volume integrals as the G-matrix interaction.
140
the pseudopotentials
J. Wambach et al. 1 Quasiparticle interactions
were obtained
from the ANVi4 potential
heights to give the same volume
integrals
and at’ = 170 MeV, well within
the ranges
as the G-matrix. quoted
adjusting
the core
This yields uTt = 115 MeV
in ref. ‘). The momentum
depen-
dence of the two interactions is strikingly similar over a very wide range. We can thus conclude that the basic physical effect of the Brueckner correlations in neutron matter is to regularize the core and simple prescriptions in terms core height are well justified over the momentum range of interest.
of an effective
3. Results 3.1. THE PARTICLE-HOLE
INTERACTION
In our microscopic model the ph interaction, 9, is determined from the solutions of the coupled two-body equations (3) and (4) which are driven by the direct interaction, 9. To lowest order 9 = 9. Higher-order corrections are built in through ph correlations. To assess their relative importance it is useful to examine eqs. (3) and (4) in second order. Since IDS(q)1 = [D?(q)\ (see fig. 5) there will then be no correction to F?(q) due to cancellation of spin-independent and spin-dependent correlations. On the other hand there is an additive contribution to F;(q) with the spin-dependent term being These conclusions basically
three times more important due to statistical weight. survive the non-linear solutions, as we shall see below.
Let us first discuss the q-dependence of the 1= 0 and 1= 1 moments of the ph interaction. In the density range of interest, those two terms are the dominant contributions to the Landau expansion. Using the G-matrix of subsect. 2.2.1 as the direct interaction we display the results in fig. 7 for three different densities (p/pO= 0.03, 0.20, 1.00). A comparison with fig. 6 shows that the q-dependence of F?(q) follows that of Ds;“( q) very closely. In fact, the antisymmetric parts F:(q) and Di( q) are almost identical, as suggested by the second-order arguments The spin-independent part FE(q), on the other hand, is substantially
given above. weaker than
Dg( q) over the entire momentum range. Closer inspection shows that this weakening is almost independent of momentum transfer indicating that the exchange ph correlations are rather short range, as already noted in ref. ‘). The lower part of fig. 7 gives results for the I= 1 moments. At low density they are small, reaching values of around 0.3 as p. is approached. The momentum dependence is rather flat (except near 2k,) indicating the short-range nature of the current-current interaction. It should also be mentioned there is no Pauli-principle requirement for 9 and no sum-rule equivalent to eq. (9) exists. Therefore the symmetric and antisymmetric components can be influenced rather differently by ph correlations. The lowest-order Landau parameters F;,:(O) characterize the equilibrium properties of a normal Fermi system. In particular, the effective mass m* is given by (13)
J. Wambach 1.o
et al. / Quasiparticle
I
interactions
I
141
I
I
0.6 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.6 -1.0
0.8 t 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6
Fig. 7. Momentum dependence of the first two Landau moments F:.:(q)of the particle-hole interaction on the Fermi sphere. Results are shown at three densities: p/po=0.03 (dashed lines), p/pa= 0.2 (dash-dotted lines) and p/p = 1.0 (solid lines).
and determines
the specific heat CV at constant C, = fm*kFT.
volume
as
(14)
Its density dependence is therefore easily calculated from F;(O) which is displayed in fig. 8. At very low densities, Cv is essentially that for a free Fermi gas, since m*/m is very close to unity. With increasing density it drops below the free-gas value, however. Comparison of F;(O) with results from the direct interaction (dotted line) shows that ph correlations render the spin-symmetric current interaction more
J. Wambach et al. / Quasiparticle interactions
142 1.2 1.0 0.8 0.6 0.4 g
0.2
m.V L
0.0 -0.2 -0.4 -0.6 -0.8
k, (fm-‘) Fig. 8. Density dependence of the lowest-order (solid lines) in comparison with the corresponding
Landau parameters F$(O) from the full ph interaction values from the direct interaction alone (dashed lines).
repulsive, thus increasing the effective mass M*. This is the well-known enhancement near the Fermi surface 30). The I= 0 Landau parameters determine the bulk modulus b = Sp/Sp via the relation b=--
2 k; 3 2m”
and the static magnetic susceptibility xS.=
1+ cdO)l ,
(15)
as
m*kF @: 1+ F:(o)]-’
-
7T2
)
(16)
where p denotes the magnetic moment of the neutron. Again, the density dependence of these quantities can be easily obtained from fig. 8. A comparison of Fy(O) with the results from the direct interaction only (dashed lines) reveals that, over the entire range, the components Ft,, are affected very little by ph correlations, for reasons given above. The component J$, however, is screened considerably, mostly by spin fluctuations. The attraction is strongly reduced. The effect is particularly pronounced near kF = 1 fm-’ (P/P,, - 0.2) where the direct interaction yields values very close to -1. According to the general stability criterion *‘) FF(O) > -(21-r 1) )
(17)
J. Wambach et al. / Quasiparticle interactions
143
a value less than -1 would cause break up of the homogeneous phase. The latter is, indeed, encountered in symmetric nuclear matter around OSp,-0.7~~ [ref. 20)]. Finally, we should mention that our result for the lowest-order Landau parameters are in good agreement with the findings in ref. 22) where a variational approach has been used.
3.2. SCATTERING
AMPLITUDES
As for the ph interaction, the scattering amplitude, &, is completely specified by the direct interaction and no new parameters are introduced. We can therefore determine the Landau components As” of d from the self-consistent solutions of eqs. (3) and (4). Using the G-matrix of subsect. 2.2.1, results for A;,:(q) as a function of momentum transfer q are given in fig. 9 at three different densities p/p0 = 0.03, 0.20, 1.00 which correspond to the choices in figs. 5 and 7. As for the corresponding moments of 9 and 9, the momentum dependence of the spin-symmetric component A:,:(q) becomes strong as density increases, being quite similar to that of F:(q) (fig. 5). On the other hand, At(q) hardly depends on momentum, except near 2kF. It should be reiterated that the scattering amplitudes are antisymmetric over the entire Fermi sphere which leads to the sum rule &P( q = q’) + d”( q = q’) = 0 ) f
[A;(q=O)+A;(q=O)]=O,
(18)
I=0
in analogy to eq. (9) for the direct interaction. The convergence in I is, however, much slower, particularly at higher density, as can be seen from fig. 9. As is well known, the non-equilibrium behavior of a low-temperature Fermi liquid as well as its superfluid properties are determined by the qp scattering amplitude 23). The scattering amplitude enters the collision integral of the quasiparticle transport equation via the transition amplitude A1234= (121.%!134)where W = 27rlA,,,,12 gives the probability for quasiparticle scattering from states 1 and 2 to states 3 and 4 in a binary collision. At the Fermi surface, the momentum dependence is usually expressed in terms of two scattering angles 0 and 4, where 8 is the angle between incoming and outgoing relative momenta of the scattered quasiparticles and 4 the angle between the scattering planes. Evaluation of the transition probability, W, then involves a transformation of the angular variables @ and 0, to the scattering angles 8 and 4 which is determined by the relations I’) cos e=-l++(l+cos cos 4 =
o,)(l+cos
2cos O,--$(l+cos
2-;(1+cos
0,))
o,)(l+cos 0,) o,)(l+cos 0,) ’
(19)
144
J. Wambach et al. / Quasiparticle interactions
I
1.0
I
f
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 1.0 0.8
i-
-l
0.6 0.4 0.2 s 0.0 -0.2 -0.4 -0.6 -0.8 -
Fig. 9. Momentum dependence of the first two Landau moments A$y(,(q) of the scattering amplitude on the Fermi sphere. Results are shown at three densities: p/p,, = 0.03 (dashed lines), p/p = 0.2 (dash-dotted lines) and p/p0 = 1.0 (solid lines). with
angular range 0 s 0 s m, 0 s 4 s 27~. The normalized transition probability scattering of two quasiparticles of like (tt) or unlike (T&) spin is then defined
W’“(l9,4)=277 and the total transition
[A:;;;)]“,
rate,
W, is given by
~(8,
+)=fwT”(e,
4)++w”(e, 4).
for as
(20)
(21)
145
J. Wambach et al. / Quasiparricle interactions
After averaging over the Fermi surface,
(22) one then obtains the well-known expression for the quasiparticle TT==-
8~~ m*3( W) ’
lifetime (23)
Results are shown in the left panel of fig. 10. One finds that the quasiparticle lifetime increases with density. This behavior is qualitatively different from the quasiparticle lifetime in other quantum fluids, e.g. liquid 3He [ref. “)I. The density dependence arises from a combination of the drop in m* and a decrease in ( W) as the density increases. The effective mass enters the expression for 7 as the third power, and therefore its density dependence is amplified. As the density increases m*/ m drops below unity (see fig. 8). Furthermore ( W) is a strongly decreasing function of density for 0 < p G po. For liquid 3He the effective mass increases with increasing density, hence the qualitatively different density dependence of the quasiparticle lifetime. 50.0
1
I
r
Quasiparticle lifetime 40.0 -
"'
Spin-diffusion coefficient
0.8 0.6 0.4 -
0.2 -
k, (fm-‘) Fig. 10. Density dependence of the transport coefficients in neutron matter. The solid lines denote the full calculations while the dashed lines correspond to results from using the direct interaction only.
146
J. Wambach et al. / Quasiparticle interactions
3.2.1. Transport coeficients. Given the transition probability W( 0, 4) we can now calculate the transport coefficients for neutron matter. In the extreme lowtemperature limit T/F~<< 1 the thermal conductivity K, the viscosity q and the spin diffusion coefficient d are obtained by suitable angular averages of the transition probabilities WTT(0, $41)and WT1(B, 4) over the Fermi sphere 24). Details can be found in ref. 24) and we only quote the results. Our starting point is the AbrikosovKhalatnikov-Hone (AK, H) approximation to the collision integral which gives
dH =f(l+
F;?(O))&
TAu=fP*v:7( T2(;_A,)),
(24)
where the h, are given by
I
h,(W)=
cos
(Q.))
dR W’@,
b(W)= A,( w
dfl WC446)
-4*
47, i
(1+2 cos e, ’
~)[COS 6+(1-cos
0) cos (P]+jWt’(6,
4) 9
2 cos ($9)
=
(25)
The exact results 25) for the transport coefficients can then be expressed in terms of the AK, H expressions (24) as 4n+5 K =
KAK
.=,,(n+1)(2n+3)[(n+l)fZn+3)-h,]
I ’
4n+3 I ’ 4n+3
f ,=o(n+l)(2n+l)[(n+1)(2n+l)-~,1
1 (26) ’
which are easily evaluated from our scattering amplitudes. Results are displayed in fig. 10 (solid lines). In the region of interest a strong density dependence is found. This is a direct consequence of the density dependence of the lifetime 7 which enters the transport coefficients as a prefactor. Fig. 10 also gives a comparison with coefficients which result from using the direct interaction only (dashed lines). 3.22. S~per~uidity. BCS theory 32) forms the basis for evaluating superfluid properties. It is assumed that, to first approximation, the system can be described by a Slater determinant I+0) in which all single-particle momentum states k with
J, Wambach et al. / Quasiparticle interactions \kl c
147
kF are occupied. S-wave superfluidity
appears if the formation of Cooper pairs from particles with opposite momenta and spins leads to a lowering of the groundstate energy. At zero temperature this occurs when the s-wave gap equation
A;=-)CP&, k'
A;. [(ek'-eF)2+(Air)2]1'2
(27)
has
non-trivial solutions for the gap function Ai. The quantity Ek = is interpreted as the energy needed to create a quasiparticle of [(ek-eF)2+A2k]1’2 momentum k (and given spin projection) in the superfluid while ek denotes the normal-state single-particle energy. For s-waves the pairing matrix elements entering (27) assume the form POkk,=(k~-k’ll~(‘So)lkt-k’~),
(28)
where zZ(‘S,) is the ‘S,, scattering amplitude. Because of isotropy, they only depend on the magnitudes of the momenta k and k’. The solution of the gap equation (27) requires knowledge of the full pairing matrix element, Ptk’. In our scheme for evaluating the scattering amplitude we obtain this matrix element only at the Fermi surface as 1
dcos~[s8”(f3=7r,~)-30Pa(8=~,~)], (29) i -1 where a”( 8,+) and .&( 0,4) are the spin-symmetric and spin-antisymmetric scattering amplitudes in the scattering-angle representation (19). With this Fermi-surface restriction we cannot solve the full gap equation but have to rely on some approximate solutions. The simplest is the BCS weak-coupling formula N(0)POkFkF
=
i
where er= k:/2m* is the Fermi energy. For the maximum gap this approximation is expected to be accurate to within a factor of two 33*34). The results, given in fig. 11, show the importance of ph correlations. Their possible influence was first recognized by Pethick and Pines 35) although an enhancement, rather than a suppression was estimated. The suppression of the gap was first pointed out by Clark et al. 35)within the Babu-Brown framework and also found in refs. 33*9). The large gaps obtained from the G-matrix are drastically reduced and the maximum attainable gap is only about 0.9 MeV. This is slightly lower than the values obtained in ref. ‘) and is consistent with the most recent findings of the variational+ CBF calculations by Clark et al. 36) which are represented by the dashed lines in fig. 11. It should be mentioned, however, that the present evaluation does not exclude particle-particle reducible graphs from the scattering amplitudes and hence (as in gap calculations based on the G-matrix) involves some double counting. Thus the agreement with the other works may be somewhat fortuitous.
J. Wambach ei al. / Qunsiparticle interactions
148
1.0
0.0
0.0
0.5
1.0
1.5
kF (fm.‘)
Fig. II. Density dependence of the gap parameter A ’+ The solid lines denote our results while the dashed lines quote results from variational calculations using variational methods with and without CBF corrections 9.
Our results imply that the crust of the neutron star is superfluid up to densities p - p0/2 with maximum gaps of the order of 1 MeV. This is qualitatively consistent with the constraints from the analyses of glitch data in the vo~ex-pinning model 504*37). Given the approximate nature of the present calculations of A& one would not be surprised if improved calculations led to somewhat larger Ai, profiles.
We have calculated the ph interaction, S, and the qp scattering amplitude Sa, in pure neutron matter for densities 0 s p 6 p,, which are relevant for the inner crust of a neutron star. We make use of coupled equations for 9 and ~4 which are driven by a ph irreducible interaction, 9 (the direct interaction). Given 9 these equations are solved exactly on the Fermi surface and antisymmetry is kept at all stages. To determine the direct interaction we have made two different choices which both lead to similar results. One is to represent 9 by a Brueckner G-matrix. While this would not be a good choice in liquid 3He, it turns out to be rather satisfactory for neutron matter in the density range of interest. A second approach makes use of the PPM where one starts from a bare local interaction which is renormahzed by introducing effective core heights. This one-parameter model gives very similar results as the G-matrix.
J. Wambach et al. / Quasiparticle interactions
149
The 1= 0 and Z= 1 Landau parameters which we deduce from our model are in good agreement with the variational calculations of ref. ‘*). We find that ph correlations give repulsive screening corrections to Fi and F”, mediated predominantly by the exchange of spin modes. The spin-antisymmetric interactions are hardly affected, a result which is easily understood from the spin structure of the underlying equations. From the calculated scattering amplitudes in spin-symmetric and spin-antisymmetric two-particle states we have determined the quasiparticle lifetime as well as the transport coefficients. A strong density dependence is found which arises from a decrease in m* as well as the average transition probability ( W). Using the weak-coupling BCS formula we have also estimated the density dependence of the ‘So gap parameter A”,,. This parameter is very sensitive to ph correlations, as has been known for some time. The peak values, when these are neglected, are reduced by more than a factor of three by the inclusion ph correlations. Although it may be somewhat fortuitous, our findings are in qualitative agreement with the recent, more elaborate, calculations of ref. 36). As compared to values derived from a most recent analyses of Vela timing data 37) they lie somewhat below the bounds quoted. We thank M.A. Alpar, J.W. Clark and C.J. Pethick for helpful discussions and suggestions. This work was supported by NSF grants PHY89-21025, PHY88-06265, PHY86-00377, PHY84-15064, DMR85-211041, NASAgrant NSG-7653, NATO grant RG 85/093 and the Texas Advanced Research Program grant 010366-012. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107 D.B. Kaplan and A.E. Belson, Phys. Lett. B175 (1986) 57 A.B. Migdal, Sov. Phys. JETP 10 (1960) 176 D. Pines and M.A. Alpar, Nature 316 (1985) 27, and references therein M.A. Alpar, H.F. Chau, KS. Cheng and D. Pines, submitted to Astrophys. J. (1985) O.V. Maxwell, Astrophys. J. 231 (1979) 201 J. Wambach, T.L. Ainsworth and D. Pines, in Neutron stars: theory and observation, ed. J. Ventura and D. Pines (Kluwer, Dordrecht, 1991) p. 37 A.B. Migdal, Theory of finite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967) T.L. Ainsworth, J. Wambach and D. Pines, Phys. Lett. B222 (1989) 173 C.H. Aldrich III and D. Pines. J. Low Temp. Phys. 32 (1978) 689 T.L. Ainsworth and KS. Bedell, Phys. Rev. B35 (1987) 8425 D. Pines, K.F. Quader and J. Wambach, Nucl. Phys. A477 (1988) 365 T.L. Ainsworth, F. Green and D. Pines, Phys. Rev. B42 (1990) 9978 0. Sjiiberg, Ann. of Phys. 78 (1973) 39 S. Babu and G.E. Brown, Ann. of Phys. 78 (1973) 1 S.-O. Blckman, G.E. Brown and J.A. Niskanen, Phys. Reports 124 (1985) 1 S. Nishizaki, T. Takatsuka, N. Yahagi and J. Hiura, preprint (1991) B. Friedman and V.R. Pandharipande, Nucl. Phys. A361 (1981) 502 V.R. Pandaripande and R.B. Wiringa, Rev. Mod. Phys. 51 (1979) 821 J.M. Lattimer and D.G. Ravenhall, Astrophys. J. 223 (1978) 314
150
J. Wambach
et al. / Quasiparticle
interactions
21) I.Ja. Pomerantschuck, Sov. Phys. JETP 8 (1958) 361 22) A.D. Jackson, E. Krotscheck, D.E. Meltzer and R.A. Smith, Nucl. Phys. A386 (1982) 125 23) D. Pines and P. Noziere, The theory of quantum liquids (Benjamin, New York, 1966) 24) A.A. Abrikosov and I.M. Khalatnikov, Rep. Prog. Phys. 22 (1959) 329; D. Hone, Phys. Rev. 121 (1961) 669 25) GA. Brooker and J. Sykes, Phys. Rev. Lett. 21 (1968) 279; H.H. Jensen, H. Smith and J.W. Wilkins, Phys. Lett. A27 (1968) 532 26) D. Pines, in Highlights of condensed matter theory, Proc. 89th Int. School of Physics, Sot. Italiana di Fisica, Bologna, Italy (1985) p. 580 27) R.V. Reid, Ann. of Phys. 50 (1968) 411 28) R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29 (1984) 1207 29) J.A. Niskanen and J.A. Sauls, preprint (1981) 30) C. Mahaux, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Reports 120 (1985) 1 31) B.R. Patton and A. Zaringhalam, Phys. Lett. A55 (1975) 95 32) J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175 33) J.M.C. Chen, J.W. Clark, E. Krotscheck and R.A. Smith, Nucl. Phys. A451 (1986) 509 34) M. Baldo, J. Cugnon, A. Lejeune and U. Lombardo, Nucl. Phys. A515 (1990) 409 35) D. Pines, Proc. XII Int. Conf. on low temperature physics, ed. E. Kandu (Kligatu, Tokyo, 1971); J.W. Clark, C.-G. Kallman, C.H. Yang and D.A. Chakkalakal, Phys. Lett. B161 (1976) 331 36) J.W. Clark, R.D. Dave and J.M.C. Chen, to be published 37) D. Pines and M.A. Alpar, in Structure and evolution of neutron stars, ed. D. Pines, R. Tamagaki and S. Tsuruta, (Addison-Wesley, Reading, MA, 1992)