Quasiparticle interaction in neutron matter for application in neutron stars

PHYSICS REPORTS

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Physics Reports 242 (1994) 387—401

EL.SEVIER

______________________

Quasiparticle interaction in neutron matter for application in neutron starst J. Wambach’ Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Abstract A model for the quasiparticle interaction in neutron matter is presented which includes, both, particle—particle (pp) and particle—hole (ph) correlations. The pp correlations are treated in a semi-empirical way, while ph correlations are incorporated by solving coupled two-body equations for the ph interaction and the scattering amplitude on the Fermi sphere. 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. The ‘S 0 gap parameter for neutron superfluidity is estimated and we comment briefly on neutron star implications.

1. Introduction Neutron matter, bound under gravitational a significant of neutron stars. 3p~ ~ pressure, p ~ 2Po’ is where Po = 2.6component x 1014 g/em3 is the saturaIt exists over of a wide density range, tion density nuclear matter. For10 most currently accepted equations of state, that means that neutron matter is found between a few meters and a few kilometers below the surface of the star (Fig. 1). In the inner crust, whose boundary with the liquid core is roughly located at Po’ the neutrons coexist with a lattice of neutron-rich nuclei, various “noodle phases” [1] and a sea of relativistic electrons. In the core, on the other hand, they coexist with protons (whose density is 5—10% that of the neutrons) and relativistic electrons. The work presented here has two motivations: to calculate the Fermi liquid, parameters and transport coefficients in the normal phase for densities extending into the liquid core, and to estimate the temperature at which neutron matter becomes superfluid. More than 30 years ago, Migdal suggested [2] 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 _

tDedicated to Tom Kuo for his 60th birthday. ‘Also at: Institut für Kernphysik, Forschungszentrum JUlich, D-52425 JUlich, Germany. 0370-1573/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0370-1573(94)0003 1-W

388

f. Wambach/Physics Reports 242 (1994) 387—401 Outer Crust 1 \O.

~.

tIC eI Neutron-rich ec rOfiS Neutron fl Nuclei, flOe a EIectrons~ Pinned SuperfluId

~

\

S 0

r u

/

Superfluid Neutrons Superconducting Protons Elect rons

\ ~

Pion Condeosote?

Fig. 1. A theorist’s perception of a l.4M0 neutron star.

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. In a rotating superfluid vortices form. Since the coherence length is comparable to the size of crustal nuclear clusters with which the superfluid coexists, the vortices are pinned to these clusters. 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 [3]. Superfluid properties of neutron matter have been studied for a 5Po°~7Po~ long time. a While all calculations yieldgap, neutron ‘S~pairing forcondensation matter densities lessE than prediction of the maximum A°,as well as the pair energy, °. 0, has been notoriously difficult and estimates vary by almost an order of magnitude [4]. This problem originates from the exponential dependence of the superfluid gap on the pairing interaction.

2. Theory In Fermi liquid theory a strongly interacting system of fermions is represented by a 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

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mass, m*, and a finite lifetime, -r. Bulk properties such as the compressibility, the magnetic susceptibility, 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 the 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 quasipartiele 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. In our approach we are motivated by the Polarization Potential Model (PPM) of Aldrich and Pines [5]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: (1) since the bulk properties can only be inferred from finite nuclei they are much less certain; (2) in the density regime of interest, the role of the Pauli principle is quite different from that in liquid helium; (3) the presence of non-central components in the bare interaction complicates the phenomenologieal treatment of medium effects. In a series of papers [6, 7] we have therefore extended the PPM so that it can be applied to nueleonic matter. 2.1. The two-body equations

In determining the quasipartiele interaction two quantities are of interest, the particle—hole interaction, and the scattering amplitude d (Fig. 2). The particle—hole 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 variable p = (Pt + p3)/2~p= (P2 + p 4)/2 and q = (p~ p,). There is an additional spin dependence which arises from the exchange character of the fermion interaction and, in the ease 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, ~ and d decompose into a spin-independent and a spin-dependent interaction .~,

—

,~=

5+

~s

+

~

d

=

(1)

~jag.~f

d

where ,~s(a)is the symmetric (antisymmetric) combination of a spin- 11 and a spin- ~ neutron pair ~ = 1/2(~h1±.~11)), and likewise for dS.~. By definition, qp interactions are only meaningful in the vicinity of the Fermi surface and we therefore confine the momenta to the Fermi ~.

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~ -

Fig. 2. The coupled equations for evaluation of the ph interaction ~ and the scattering amplitude d. These equations are driven by the “direct interaction”.

sphere, P1 = p’I = kF. Then the dependence on p and p’ is given by an angle t9, so-called “Landau angle”, and it is natural to expand F and d in 9, N(0)F~’~ = ~,F~~a(q)P,(cos s,),

N(0)dS~a= ~A~’~(q)P,(eos 9,).

&,,,,, the

(2)

To scale out a trivial density dependence we have multiplied by N(0) = krm*/1t2, 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 I~I 2kv. Therefore, the q-dependence can be represented by a second angle °q as q = 2kF(l eos &q). 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 setup a system of scattering equations for F and d we exploit the diagrammatic relationship between them (Fig. 2). This diagram set is driven by a ph irreducible interaction, ~, and approximately sums particle— hole diagrams in all momentum channels. The physical idea of separating the ph interaction, F into a “direct” piece which incorporates short-range correlations and an “induced” piece which includes exchange of virtual collective modes is quite similar to the Babu—Brown model (BBM) [8]. The BBM Was initially proposed for liquid 3He but later also applied to nucleonic matter [9], in a somewhat rudimentary form. The original intent was to calculate Landau parameters. Then the scattering amplitude is required to be antisymmetric 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 —

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the Fermi sphere are possible; however, and have been given in Ref. [10]. In Ref. [7] we have applied the extended theory to neutron matter and it has also been used successfully for liquid 3He and for the electron gas [11]. 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~’1’and A~’5(2). For a single component Fermi liquid one obtains explicitly —~

F7(q) = D7(q) +

F7(q)

=

~,(q, q’)[ ~ (~F~(q’)x~(q’)A~(q’) +

D7(q) +

~ 1(q, q’)[

~

(~F~(q’)x7~(q’)A~(q’) ~F~(q’)x~(q’)A~(q’))] —

(3)

l’,m,n

and A7(q)

=

F7(q)

A~(q)= F7(q)

—

—

~

~ F~,(q)~~n(q)A~(q),

(4)

where ~, is an integral operator which projects onto the direct particle—hole channel momentum, q (Fig. 2). The direct interaction enters through the spin components D~’a and f” are angular projected ph propagators which are discussed in detail in Ref. [10]. They include the density and current response as well as higher tensor correlations. Since both, F and d, depend on two angular variables, 0, and °q’ we can cast the integral equations (3) and (4) into a set of non-linear matrix equations for the coefficients F~and A~’in a double-moment expansion 5’ a = ~ F7;~P,(eos0j)Pm(CO5 Oq), N(0)F =

~ A~,Pi(cos0,)Pm(cos6q).

(5)

1, m

Given a direct interaction, these matrix equations are solved iteratively in a truncated set of coefficients (1, m) which has to be chosen sufficiently large to ensure convergence. The convergence properties are governed by the momentum structure of the direct interaction and typically ‘max = 1~max= 10 suffices. 2.2. The direct interaction

In our approach, the direct interaction, ~, completely specifies the particle—hole interaction and the scattering amplitude. In general, ~ represents the sum of all ph-irreducible diagrams. Therefore, any microscopic calculation of the ph-irreducible interaction can drive the two-body Eqs. (3) and (4) without causing any double counting. In practice, however, such calculations are difficult.

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J. Wambach/Physics Reports 242 (1994) 387—401

Fortunately, screening corrections from the induced interaction render the quasiparticle interactions rather insensitive to the input ~ [7, 9]. In constructing an appropriate direct interaction we pursue two routes (which give very similar results in the density range of interest). We approximate ~ by a Brueckner G-matrix as was done in the work of Sjöberg [12]. 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. To be specific we use a parameterization from the recent work of Nishizaki et al. [13] 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, ct. The resulting local neutron— neutron potential V takes the form (r)

12

=

c~(p,c~)=

c~1 (p, ~)e~~

,

~

(r) =

c~ (p. ~)e2~12

ai(c~)+ bi(ca~)k~’2 .

(6)

At short distances both spin components are repulsive reflecting the core of the bare interaction. At intermediate distances ~ gives an attraction which is strong enough to cause pairing in the ‘S 2. With increasing p the interaction becomes increasingly 0-channel for isdensities up to As po/the mean interparticle spacing decreases, the neutrons sample repulsive which quite natural. more of the core. This mechanism is the physical reason for the disappearance of s-wave superfluidity at higher density. To gauge the quality of the G-matrix we compare the resulting equation of state (EOS) with more sophisticated many-body calculations [14]. Given the parameterization (6) it is straightforward to calculate the total energy/particle and the pressure. The results are plotted in Fig. 3 which gives a comparison with the realistic Friedman—Pandharipande EOS [14] for cold neutron matter (circles). Up to p~the agreement is very satisfactory. A less microscopic way to construct the direct interaction is based on the PPM. It is particularly useful in eases where the G-matrix fails to be a good starting point, such as the high-density 3He liquid. In the PPM one starts with a local bare interaction Vb~e(r) and the magnitude of the repulsive core is modified in a density-dependent fashion. For neutron matter, suitable choices of are the V 6 form of the RSC potential [15], or the more recent Argonne ANy14 potential [16]. Ignoring tensor and spin—orbit components we then have Vb~e(r)= Vs(r) + Va(r)~,.q~

(7)

where v’~ a is the spin-symmetric (-antisymmetric) part of Vb~e. At distances above 0.8 fm, which is somewhat larger than the core radius, these interactions are remarkably similar to the r-space G-matrix at low density. It is then argued that the main effects of short-range correlations can be incorporated by fixing the maximum value the cores can attain. For the relevant momentum ranges this renormalization can be implemented by using a rather crude cut-off prescription ‘~

1(r) ~

=

min{a1 1,

v1 1(r)}

,

J~1t(r) =

min{a1 1,

v1 ~(r)} ,

(8)

J. Wambach/Physics Reports 242 (1994) 38 7—401

0.0000

0.05

0.10

393

0.15

p (fm~) Fig. 3. The neutron matter EOS from the G-matrix used here. The upper part displays E/A, as a function of density while the lower part gives the pressure. The circles denote the results of the FP EOS [14].

in which, below a certain distance, the bare potentials are set equal to constant “core heights” a1 1 or a1 ~. These core heights are parameters which generally depend on the matter density and, according to the PPM, should be determined from empirical 1 = 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 [7] that the calculated Landau parameters are largely insensitive to a11, due to the Pauli principle, so that a1 1 remains as the only parameter. Realistic values quoted in Ref. [7] range between 100 and 250 MeV. It is instructive to compare the Fourier transforms of the pseudopotentials (8) with those of the G-matrix. The momentum dependence is strikingly similar over a very wide range which leads to the conclusion that, at low density, the basic many-body effect is to regularize the core. Simple prescriptions in terms of an effective core height are well justified over the momentum range of interest. Given the r-space potentials V1 I and P1 ~, derived either from the G-matrix of via effective core heights, the Landau moments, D7’ a(q), of the direct interaction are obtained by Fourier

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J. Wambach/Pkvsics Reports 242 (1994) 387—401

transformation and proper antisymmetrization as D7(q) = D~(q)=

5(q

~,(q

q’)]

q’)[~

~3 1(q

q~)[~a(qq’)]

~S(q)

=

~S(q

q’)

~a(q,

q’) =

—

(1/2 ~S(q~)+ 3/2 ~a(q~)) 75(q/) 1/2 ~a(qP)) (1/2 J —

~s. a(q)

=

~a(q)

4~

—

dr r2j 0(qr) ~

(9)

a(r)

where the integral operator ~1,projects out the lth Landau moment of the explicity antisymmetric ~(q, q’). The results 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 V is sampled. Antisymmetry of the direct interaction under exchange implies a general sum rule = q’) + 9,~a(q= q’) = 0, (10) ~S(q

which, as a special case, contains the forward scattering sum rule when q D~(q=0)+D~(q=0))=0.

= q’ =

0, (11)

This is a useful check of the convergence in the Landau expansion as wellas the numerical accuracy of the exchange integral. At all densities of interest, the sum rule is exhausted by a few terms with the I = 0 moments giving the largest contribution. This rapid convergence is due to the short-range nature of V. For the same reason the symmetric and antisymmetric components, of given 1, sum to zero individually, to a high degree of accuracy.

3. Results 3.]. The particle—hole interaction The ph interaction, F, is determined from the solutions of the coupled two-body equations (3) and (4) which are driven by the direct interaction, ~. To lowest order F = 2i~.Higher-order corrections are built in through the ph correlations. To assess their relative importance it is useful to examine Eqs. (3) and (4) in second order. Since ID~(q)I~ D~(q)~ 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 F7(q) with the spin-dependent term being three times more important due to statistical weight. These conclusions basically survive the non-linear solutions (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.

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395

It is found that the q dependence of F~a(q) follows that of D~a(q) very closely. In fact, the antisymmetric parts P0(q) and D~(q)are almost identical, as suggested by the second-order arguments given above. The spin-independent piece F~(q),on the other hand, is substantially weaker than D~(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. [7].It should also be mentioned there is no Pauli principle requirement for F and no sum-rule equivalent to (10) exists. Therefore, the symmetric and antisymmetric components can be influenced rather differently by the ph correlations. The lowest-order Landau parameters F~(0) characterize the equilibrium properties of a normal Fermi system. In particular, the effective mass m* is given by m*/m

=

(1 + ~F~(0))

(12)

and determines the specific heat C~at constant volume as =

(13)

(m*kF/3)T.

Its density dependence is therefore easily calculated from F~(0) which is displayed in Fig. 4. At very low densities, C,.~is essentially that for a free Fermi gas, since m*/m is very close to unity. However, with increasing density it drops below the free gas value. Comparison of F~(0) with results from the direct interaction (dotted line) shows that the ph correlations render the spin-symmetric current interaction more repulsive, thus increasing the effective mass m*. This is the well-known enhancement near the Fermi surface [17]. 5p via the relation The 1 = 0 Landau parameters determine the bulk modulus b = t5p/ b = 4(k~/2m*)(1+ F~,(0)) (14)

1.2

I

1OD

10~~

kF (fm)

Fig. 4. The density dependence of the lowest-order Landau parameters F~ 1(0) from the full ph interaction (solid lines) in comparison with those from the direct interaction (dashed lines).

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J. Wambach/ Physics Reports 242 (1994) 387—401

and the static magnetic susceptibility as =

(m*kF/1r2)p2(1 + F~(0))’ ,

(15)

where p denotes the magnetic moment of the neutron. Again, the density dependence of these quantities can be easily obtained from Fig. 4. A comparison of F’~~(0) with the results from the direct interaction only (dashed lines) reveals that, over the entire range, the components F~,~ are affected very little by the ph correlations, for reasons given above. The component Ft,, however, is screened considerably, mostly by spin fluctuations. The attraction is strongly reduced. The effect is particularly pronounced near kF = 1 fm 1 (P/Po 0.2) where the direct interaction yields values very close to 1. According to the general stability criterion [18] —

F~~a(0) >

—

(2/ + 1)

(16)

a value less that 1 would cause break up of the homogeneous phase. The latter is, indeed, encountered in symmetric nuclear matter around °.5Po—°~7Po [19]. Finally, we should mention that our results for the lowest-order Landau parameters are in good agreement with the findings in Ref. [20] where a variational approach has been used. —

3.2. Scattering amplitudes and transport coefficients

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 d [21]. It enters the collision integral of the quasiparticle transport equation via the transition amplitude A = 2 gives the probability for quasiparticle scattering from1234 states 1 and 2 to where W= 21tIA1234I states 3 and 4 in a binary collision. At the Fermi surface, the momentum dependence is usually expressed in terms of two scattering angles U and ~, where 0 is the angle between incoming and outgoing relative momenta of the scattered quasipartieles and 4t the angle between the scattering planes. Evaluation of the transition probability, W, then involves a transformation of the angular variables 0, and 0q to the scattering angles 0 and ~ which is determined by the relations [10] cos0= —1 +~(1+cosO,)(1 +cosOq), cosd=

2cosOq ~(1 + cos0,)(1 + e050q) 2—~(1+cos0,)(1 +cosOq) —

with angular range 0 0 two quasiparticles of like

w11(0, ~)

=

it,

(11)

2ir[A11(O,

0

17

The normalized transition probability for scattering of or unlike (1 .~)spin is then defined as 4

2it.

4)/N(0)]2 ,

Wu(0,

4,) =

2ir[AU(0, 4,)/N(0)]2

(18)

and the total transition rate, W, is given by W(0, 4,) = ~ WIt (0,4,) + ~W11(0, 4,).

(19)

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After averaging over the Fermi surface, ~

<~>=

j 4it

(20)

eos(0/2)

one then obtains the well-known expression for the quasipartiele lifetime iT2 = 8114/m*3 .

(21)

Results are shown in the left panel of Fig. 5. One finds that the quasiparticle lifetime increases with density. This behavior is qualitatively different from that in other quantum fluids, e.g., liquid 3He [10]. The density dependence arises from a combination of the drop in m* and a decrease in as the density increases. The effective mass enters the expression for t as the third power, and therefore its density dependence is amplified. As the density increases m*/m drops below unity (see Fig. 4). Furthermore, is a strongly decreasing function of density for 0 p Po~For liquid 3He the effective mass increases with increasing density, hence the qualitatively different density dependence of the quasiparticle lifetime. Given the transition probability W(0, 4,) we can now calculate the transport coefficients for neutron matter. In the extreme low-temperature limit T/8F ~ 1 the thermal conductivity ic, the

500

2500

•...

IParticle

1.0

~

~

ç

lifetime

Viscosity

Spin-diffusion coefficient

Thermal conductivity

~

O~5 I

k~(fm”)

‘II’

~°Q,O

‘

O~5

kF (fm’)

Fig. 5. The density dependence of the transport coefficients in pure neutron matter. The solid lines denote the full calculation while the dashed lines represent results from the direct interaction only.

)

—

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J. Wambach/Physics Reports 242 (1994) 387—401

viscosity i~and the spin diffusion coefficient d are obtained by suitable angular averages of the transition probabilities W1 1(0, 4,) and W1 1(0, 4,) over the Fermi sphere [22]. Details can be found in Ref. [22] and we only quote the results. Starting from the Abrikosov—Khalatnikov—Hone approximation to the collision integral one obtains 2

Cvv~r(

3

KAK =

=

~it~(i

(1 + F~(0))v~t ( 2 \\ 5 ~it~(1 id) —

pm*v~t( i/AK

2

5

=

\\

—

,~))‘

(22)

where the )~are angular averages of the transitions probabilities [10]. The exact results [23] can be expressed in terms of the AK coefficients and the A’s (22) and are easily evaluated from our scattering amplitudes. Results are displayed in Fig. 5 (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 t which enters the transport coefficients as a prefactor. Figure 5 also gives a comparison with coefficients which results from using the direct interaction only (dashed lines). Superfluidity The BCS theory [24] forms the basis for evaluating superfluid properties. It is assumed that, to first approximation, the system can be described by a Slater determinant I4,~>in which all single-particle momentum states k with 1k
I

k

‘v’ =

—

L~

AO

o

kk’ ((ek’

—

eF)

1-’k’ + (A~)2)1t2

23

-

has non-trivial solutions for the gap function A~.The quantity Ek = [(ek eF) + ~]1I2 is interpreted as the energy needed to create a quasiparticle of 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 (23) assume the form —

=


—

k’~Id(’So)lkt —k’~> ,

(24)

where d(’S 0) is the ‘S0 scattering amplitude. Because of isotropy, they only depend on the magnitudes of the momenta k and k’. The solution of the gap equation (23) requires knowledge of the full pairing matrix element, Pk°k’. In our scheme for evaluating the scattering amplitude we obtain this matrix element only at the Fermi surface as 5(0

N(0)P~,~ =

d cos 4, [d

= it,

4,)

—

3dtI(0 =

it,

4,)]

(25)

J. Wambach/Physics Reports 242 (1994) 387—401

399

where d’(O 4,) and da(0, 4,) are the spin-symmetric and spin-antisymmetrie scattering amplitudes in the scattering-angle representation (17). 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 =

2eFexP(N(OP0),

(26)

where eF = k~/2m*is the Fermi energy. For the maximum gap this approximation is expected to be accurate to within a factor of two [25, 26]. The results, given in Fig. 6, show the importance of the ph correlations. Their possible influence was first recognized by Pethiek and Pines [27] although an enhancement, rather than a suppression was estimated. The suppression of the gap was first pointed out by Clark et al. [28] within the Babu—Brown framework and also found in Refs. [7,25]. The large gaps obtained from the G-matrix are so drastically reduced that the maximum attainable gap is only about 0.9 MeV. This is slightly lower than the values obtained in Ref. [7] and is consistent with the most recent findings of the variational + CBF calculations by Clark et al. [29] which are represented by the dashed lines in Fig. 6. 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. 2 with Our results theofcrust of the star is superfluid to the densities p ,oo/ maximum gapsimply of thethat order 1 MeV. Thisneutron is qualitatively consistentup with constraints from the analyses of glitch data in the vortex-pinning model [3, 30]. Given the approximate nature of the present calculations of A~one would not be surprised if improved calculations led to somewhat larger A~profiles.

k, (1m)

Fig. 6. The density dependence of the s-wave gap parameter A 0r, The solid lines denote our results while the dashed lines quote results from variational calculations without and with 5CBF corrections.

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J. Wambach/ Physics Reports 242 (1994) 387—401

4. Summary Within a semi-microscopical model the ph interaction, and the qp scattering amplitude d, in pure neutron matter for densities 0 p Po which are relevant for the inner crust of a neutron star have been calculated. We make use of coupled equations for and d which are driven by a ph irreducible interaction, ~ (the direct interaction). Given ~ 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 ~ 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 renormalized by introducing effective core heights. This one-parameter model gives very similar results as the G-matrix. The 1 = 0 and 1 = 1 Landau parameters which we deduce from our model are in good agreement with the variational calculations of Ref. [20]. We find that the ph correlations give repulsive screening corrections to F~and Fs, 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 twoparticle states we have determined the quasiparticle lifetime as well the transport coefficients. A strong density dependence is found which arises from a decrease in m* as well as the average transition probability . Using the weak-coupling BCS formula we have also estimated the density dependence of the ‘S 0 gap parameter A~.This parameter is very sensitive to the 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. [29]. As compared to values derived from a most recent analyses of Vela timing data [30] they lie somewhat below the bounds quoted. ~,

.~

Acknowledgments This work has been done in collaboration with T. Ainsworth and D. Pines and was supported by NSF grants PHY91-08066, PHY88-06265, PHY86-00377, PHY84-15064, DMR85-211041, NASA grant NSG-7653, NATO grant RG 85/093 and the Texas Advanced Research Program grant 010366-012.

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