Superfluidity of neutron matter

Nuclear Physics @ North-Holland

A437 (1985) 487-508 Publishing Company

SUPERFLUIDITY OF NEUTRON (I). Singlet pairing L. AMUNDSEN Fysisk Institutt,

AVH,

and

Universitetet

MAlTER

E. 0STGAARD

i Trondheim,

7055 Dragvoll,

Norway

Received 15 June 1984 (Revised 2 November 1984) Abstract: The possibility of neutron or proton singlet pairing and superfluidity in neutron star matter is investigated, and the energy gap and corresponding critical temperature is calculated or estimated as a function of Fermi momentum or density. The calculations are performed for three different potentials: a “one-pion-exchange gaussian” (OPEC) potential, an “effective” OPEC potential, and an effective Reid soft-core potential obtained by a method of “lowest-order constrained variation”. The results indicate that neutron superfluidity, corresponding specifically to ‘&-state pairing, may exist in a low-density shell in the nuclear-matter region in neutron stars, i.e. for densities 4.6 x IO” g/cm3 < p < 1.6 x lOI g/cm3, and the maximum self-consistent energy gap is A( k;) = 2-5 MeV for a neutron Fermi momentum k;= 0.7-l .O fm-‘. Superfluidity or superconductivity corresponding to ‘$,-state pairing for the proton subsystem is quite likely at higher densities, i.e. for 2.4 x lOI g/cm3 < p (7.8 x lOI g/cm3, and the maximum energy gap for the OPEC potential is A( kfL) f 0.3-0.6 MeV for a proton Fermi momentum kF 5 0.7 fm-‘. The estimated critical temperatures seem to be higher than expected temperatures inside neutron stars.

1. Introduction Migdal ‘) first suggested the possibility of superfluid states in neutron star matter. The effective interaction between two neutrons (or protons) is a combination of very strong repulsive short-range interactions and weaker attractive long-range interactions. In neutron matter at low densities, when the interparticle distance is much larger than the range of the repulsive interactions, we should expect the neutrons to “condense” into a superfluid state because of an effective attractive pairing interaction. According to the BCS theory for S-wave pairing we get Cooper pairs ‘) and an energy gap A - 5 exp [-l/N(O)

U]

(1.1)

at the Fermi surface, where 5 defines an energy interval corresponding to the attractive neutron-neutron interaction, U corresponds to this effective BCS interaction, and N(0) is the density of states at the Fermi energy for a given spin system. In neutron stars we probably get N(O)U< and we should

expect good conditions

1)

A>lMeV,

for superfluid 487

(1.2) states in certain

density regions.

488

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E. Osrgaard

/ Superfluidity

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In addition to superfluid neutrons we should also expect superfluid protons. The proton density is probably approximately two magnitudes smaller than the neutron density, and we should expect superconductive protons in certain density regions. Superfluid electrons, however, are not likely, since the critical temperature for superconductivity according to BCS theory ‘) is

~c-pcexp[-l/N(0)Ul,

( 1.3)

where N(0) =p2/27r%‘c,

N(O)U-e’/hc,

u-e’l(plh)‘,

i.e. pc corresponds to the Fermi temperature for relativistic essentially equal to zero, and the electrons are “normal”.

electrons,

( 1.4)

T, becomes

2. General theory Bardeen et al. ‘) (BCS) created the concept electron states by introducing the pair creation p: = C&&,

Pk

of Cooper pairing 2, of one-body and annihilation operators =

C-kICkt ,

(2.1)

for particles having momenta k or -k and spins “up” (t) or “down” (J). In the BCS theory there is a probability amplitude uk(O< k
hamiltonian

flk

cull +

hi&lo),

= (010) = u: + u; = 1 .

is a reduced

hamiltonian,

(2.2) and the expectation

value

is obtained by choosing the uk and uk which give the minimum. The variational problem S(BCSIHIBCS)=O

(2.3)

can be solved by a Bogoliubov-Valantin (BV) transformation 4-6). If the Fermi sea becomes unstable to the creation of Cooper pairs because of pairing interactions, then the Fermi surface is not clearly defined, and we have to define some new operators: a,‘” = u,c,, - u&t,, akl

where

uk and uk are real numbers

=

ukc-kJ+

,

ukC:t,

(2.4)

1.

(2.5)

satisfying u:+u:=

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L. Amundsen, E. Ostgaard / Superfluidity (I)

In the Bogoliubov transformation we need a “compensation of dangerous graphs” since we mix particle states and hole states by the canonical transformation (2.4). We then start by assuming a general many-body hamiltonian for a system of neutrons

interacting

by two-body

forces, i.e. H=H,+H,,

(2.6)

where

(2.7)

Here, t is the single-particle energy and V( 1,2) is the two-body potential. The terms in (2.7) can be treated by a BV transformation, and the hamiltonian can finally be written in the form H=

U+H,,+&,+H’,

(2.8)

where the potential U is a “constant” independent of the quasi-particle operators and the terms Hi, and H,,, are quadratic in the operator (Y,i.e. Hi, includes terms proportional to cz+cz,and H,, includes terms proportional to a ‘at and (Y(Y. The last term H’ in (2.8) includes products of four a-operators, which give zero when operating on the quasiparticle vacuum, or can be neglected in the following. We then try to determine uk and v, in such a way that the hamiltonian H=

U+H,,+H*,,

(2.9)

is diagonal. When we diagonalize the new hamiltonian (2.9), the term H20 must vanish only U and H,, commute with the quasiparticle number operator N, =C (Q:OG+

(Y:,G)

since

(2.10)

.

k

The requirement (2.11)

Hzo=O gives the equation hkk, = [e,~(ukvR~+ vkuk,) - Ak-,Jukuk, where the one-body

matrix

Ed, and the pairing

E,‘,‘,=(klfjk’)+ ekkf = Ed! - AS VU.=4

- vkvk.)] = 0,

potential

(2.12)

Ak-lrr are given by

VU,, Mr,

C (ka, k”c+‘(1,2)~k’q

k”a’-k”a’,

k’a)&,

k”md

A,_,,=-;(kt,

-k’J(V(l,

2)lk”T, -k”J)uk+,.

(2.13)

490

L. Amundsen,

If we assume

a “pure”

pairing

E. @stgaard / Superfluidity

interaction,

(I)

i.e.

(k?, -k’J] V( 1,2)lk”T, -k”J)

= 0,

for k’# k,

(2.14)

then Ak-k.= AL-,Jkk~ = A,J~,~,

(2.15)

2eku,uk - Ak( u’, - vi) = 0.

(2.16)

and (2.12) may be written If we assume ukuk

# 0

(2.17)

,

and use (2.5), we get the solutions u:=5[l+ek/(e:~Ad)“‘], v:=f[l-ek/(e;+A2,)“*]. Inserted

into (2.16), this produces

(2.18)

the gap equation

or non-linear

integral

Ak=-f~(k~,-k~~V(1,2)]k’~,-k’~)Ak~/(e~~+A~.)”2.

equation (2.19)

k’

Replacing summation by integration, taking plane waves exp (ik, . ri)/t6 for the single-particle wave functions, and introducing centre-of-mass and relative coordinates, we obtain the matrix element in the gap equation (2.19) as (kT,--kJIV(l,2))k’T,-k’J)=R-’ where k, k’ and r are relative partial-wave expansion A(k) = --r-l

J

exp(-ik.r)V(r)exp(ik’.r)d3r,

I

coordinates.

Considering

(2.20)

‘SO pairing,

k’* dk’(k)V(‘S,)(k’)A(k’)/[e*(k’)+

we get from a

A*(k’)]“‘,

(2.21)

where (k(V(‘S,)lk’)=

The matrix

J j,(kr)V(r)j,(k’r)r*dr.

potential

VU, and the one-body

vu, =f c

(ka, k”dl V( 1,2)lk’a,

matrix k”o’-

(2.22)

&kk’now become k”a’, k’a)6,_,&

k”m’

=f

C ka, k”-alV(1,2)lk’a,

k”-a)v&

k”m

=fC*(k)V(‘S,,)(k)

J v’(k”)

ekk’= ekSti,=h2k2/2M+2Y’(klV(‘S,)lk)

d3k”,

J v*(k”)k”* dk”-A.

(2.23)

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E. Qstgaard

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491

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We also get the condition 00 k;=3

(2.24)

u*(k)k’dk, I0

and we have to solve eqs. (2.18), (2.21), (2.23) and (2.24) self-consistently. We will also try two approximations for the single-particle energy. In the first approximation we decouple the gap equation (2.21) from the amplitude vi in (2.18) by inserting v; =

1,

for ks kF,

1 0,

for k> kF,

A = E(kF), (2.25)

~(k)=hZk2/2M+f~-‘k:(k~V(1So)~k), in (2.23). In the second in (2.13) by

approximation

we replace

the single-particle

E(k) = h2k2/2M* in the effective-mass

approximation,

E(k)--A =h2k2/2M+;

1

energies

E(k) (2.26)

and

(ka, k’cr’lV(1,2)lka,

k’a’-k’a’,

/+.$-A

k’mr’

=ti2k2/2M+ The effective

U(k)-[h2k:/2M+

mass then is defined

U(k,)]-h2(k2-k;)/2M*.

(2.27)

by (2.28)

3. Correlation functions and potentials Neutron matter can be considered as an infinite homogeneous translationalinvariant system and the unperturbed single-particle wave functions are plane waves. If the two-body potential V( 1,2) has a strongly repulsive core, two-body matrix elements become infinite and perturbation theory does not converge. In Brueckner theory, therefore, this problem is solved by the introduction of a so-called reaction matrix G in analogy to scattering theory, defined by the finite matrix elements (@IGl@)=(@IVl%

(3.1)

where I@) is the unperturbed two-body wave function and I !P) is the correlated two-body wave function, and the Bethe-Goldstone equation G=

V- V(Q/e)G,

(3.2)

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/ Superjluidity (I)

where e is an energy denominator and Q is a Pauli exclusion-principle operator “projecting out” intermediate states outside the Fermi sea. Calculations by the Brueckner method indicate that the G-matrix in certain cases is almost diagonal in the relative coordinate r, and can be approximated as a function of r and the density p. We may then define a correlation function f( p, r) by ~((P,r,k)l~(r,k)=G(p,r)lV(r)=f(p,r), where

k is the relative

momentum.

(3.3)

We may also define an effective

t&r)

=

interaction

v(r)fz(r) .

by (3.4)

Pandharipande ‘) has used this to develop a “lowest-order constrained variational” (LOCV) method which is simpler than Brueckner theory, and a correlated wave function q is produced from a trial wave function @ in the Jastrow assumption

Cf;(rq)SlA

II.h(ri)

,

(3.5)

I

where I is the angular momentum quantum number for the correlated is a projection operator for the Ith partial wave, and A is an operator the correct antisymmetry. For large r we get

f(r)-+ 1,

Wr) + Q(r) ,

pair (ij), P producing

(3.6)

i.e. we may assume

.fXr) = 1 df,/dr=O

for r> d,

(3.7)

where the “healing distance” d is defined in a suitable way. Making a cluster expansion for the energy, but including only one-body and two-body terms, we can show that the final variational equation to be solved for the correlation function is co 6 I0 where

J:[V~~-(hZIM)(fif;+2fLf~JIIJ,)1dr=0, J,( kr) = krj,( kr) .

(3.8)

(3.9)

The LOCV method is not a pure variational method, however, since we have to introduce a parameter A, i.e. add an “outer potential” or Lagrange multiplier A to the two-body potential. This is justified by arguments which may be not too convincing. A is, however, determined by boundary conditions on the correlation function and definitions of the healing distance d. In the Bethe-Goldstone equation (3.2) we get “healing” when P=@, i.e. when both particles

for Q=O,

(3.10)

are in the Fermi sea, i.e. for states where k
(3.11)

493

L. Amundsen, E. Ostgaard / Superfluidity (I)

when K and k are centre-of-mass and relative momenta. We then get “healing” outside the first maximum of the wave functionj,( k,r), and for an average centre-ofmass momentum K the maximum ofj,(k,r) is close to $n-/fik,. For neutron ‘So pairing we therefore choose a healing distance dN = 0.5 &0.7 k, . (3.12) We will also calculate the correlation momentum k, i.e. an rms value

function

for an average value k. of the relative

k,,=J?==&?k,.

(3.13)

Including the boundary conditions (3.7), the variational ing correlation function f0 becomes

equation

for the correspond-

-(~'lM)[f;:+2(J~lJ,l~~l+[v(~)+~Ifo=O, j-;=[V(r)+A](M/h’)f,-2kcotg(kr)f;,

(3.14)

which may be solved numerically by, for instance, the Runge-Kutta-Nystrdm (RKN) method. We have, however, a problem of self-consistency since we do not know the parameter h or the healing distance d. We, therefore, start by choosing a value h for a certain density and healing distance dN, and calculate by iteration new values A(‘) and d(‘), i.e. we start by A(‘) = A A(l)=A(o)+C(Of(d(0)-dN)r where C(O) is an arbitrary and check if

constant

to be chosen.

(3.15)

For each iteration

we solve (3.14)

lAdil=(d’i’-d,J
solution

(3.16)

where

d(‘) = dN . If (3.16) is not satisfied,

we take a new constant

(3.17) given by

Cci) = _[A(;)_A(i-l)]/[‘Ad(i)_Ad(i~l)],

(3.18)

and go on with the iteration until (3.16) is satisfied. The most important input in the calculations is probably the two-body nucleonnucleon potential V( 1,2) which is determined phenomenologically from experimental scattering data and bound-state data. The Reid soft-core (RSC) potential “) is a superposition of Yukawa forms of different strength and range, but it is clearly pheoomenological since it cannot be written on the most general theoretical form from meson field theories. The ‘So potential is given by V(‘S,) = [-lo.463

exp (-X) - 1650.6 exp (-4x)+6484.2

exp (-7x)1/x,

(3.19)

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L. Arn~~dsen, E. 0sfgaard

/ Su~r~u~dify

(I)

in MeV, where x=pr,

k = 0.7 fm .

(3.20)

Pandharipande’s LOCV method can then produce an “effective” RSC interaction from this potential to be used in our neutron ‘&-gap calculations. In addition to the RSC potential we will try the “one-pion-exchange gaussian” (OPEG) potential with a very “soft” repulsive core. Here V(‘S,)=

V,(r)=fpc2(f2/47r)Y(1+a,Y+b,YZ)F4+-u,(r),

(3.21)

where 9, p is the pion mass, and F=l-exp[-(r/d)‘],

Y = exp (-x)/x,

,

c=2000.0MeV, f2/4%- = 0.08,

4. Calculations

b, = 19.06,

a,=7.6,

u,(r)=2),exp[-(r/n,)‘l, nc= 0.5 fm

x = r/(~/~c),

pc2=137MeV, d=0.5fm.

(3.22)

and results

The integrals are solved numerically by the Gauss-Legendre

method

dy=8b-a) jE, wif(Yi)+&, Ja’.ff~)

(4.1)

where yi=~(b-a)xi+~(b+a), wi = 2(1 -x:)-‘[P:,(xi)]2,

(4.2)

i.e. we use a 9dpoint Gauss integration in intervals (a, b), and the points Xi are zeros in the Legendre polynomial P”(x) of degree n = 96. The r-dependence in the matrix elements (2.22) are then integrated over for given momenta k, k’. The complete self-consistent problem then is to solve the equations ,.=[,

Jk2u2(t)dk]“3,

~(k)=~~2k2/~+(2/~)(klV{‘S~)lk)

A(k)=-v-’

J

J k”*u(k”)dk”,

k’* dk’(k’~V(‘S,)~k)A(k’)/{[~(k’)-A]*fA2(k’)}”*,

v2(k)=~[1-[~(k)-h]/{[~(k)-~]2+A2(k)}],

(4.3)

495

L. Amundsen, E. Ostgaard / Superfluidity (I)

where k, is defined

in analogy

to (2.25) by c(k,)

To solve the self-consistent problem, in the effective-mass approximation

= A.

(4.4)

we first approximate by

the single-particle

energies

m”=lO . . The gap equation in (4.3) is solved maximum difference IA’“‘(k)-A’“-“(k)l<

In the integral

(4.5)

in momentum

space

by iteration

0.001 MeV,

for A( kF) > 1.0 MeV ,

O.OOIA(kF),

for A(k,)<

until

the

(4.6)

l.OMeV.

we get E(k’)>A(k’),

for k’pk,,

(4.7)

i.e. A(k) -

k’s-k,

(k’/k)

-r-I

sin(k’r)V(r)sin(kr)dr/[E(k’)-A],

dk’A(k’) I

I

(4.8)

and the integrand

becomes

effectively

equal to zero for

k’= 10 fm-’ .

(4.9)

We also must take special care for k’= kF, where the integrand changes rapidly and has a sharp maximum. We, therefore, apply a 96-point Gauss integration in the intervals (0, k,+O.l fm-‘), (k,+O.l fm-‘, 5 fm-‘) and (5 fm-i, lOfm-‘). The set of equations (4.3) is solved self-consistently, starting with some decoupling values for I, A = E(kF) and A(k) to obtain values for u2(k). We then calculate a new Fermi energy A defined by (4.4), and new values until we get self-consistent solutions A(%)

which replaces LOCV method

potential

defined

values

are chosen 10m3fm,

of calculation

etc.,

by (3.4), i.e. (4.11)

= V(r)f2(r),

V(‘S,) in (4.3). The correlation function f(r) is obtained from (3.14), i.e. by a RKN method with a step length

r,=

The procedure

v2(k),

(4.10)

h = 10m3 fm , and our starting

A(k),

# 0.

An effective mass is finally obtained by (2.28). For the RSC potential we calculate an effective r&r)

for E(k),

by the

(4.12)

to be f( ro) = 10M5,

is shown

in fig. 1.

f’( rO) = 10e5 .

(4.13)

L. Amundsen, E. 0stgaard

496

/ Superfluidity (I)

0

START + Given

kF

t

I I

Find fin the decoupI.BDlWOX.

Find f and d

+

Find A(k the decoupl.approx.

I I

Find v*(k). k, E (k)and A salfcons.

I I

Find

Find

*

veff I

I k’lv,ttlk 1

c3 STOP

Fig. 1. Scheme

of calculation

for the ‘SO energy

gap and effective

mass in neutron

stars.

Results of the calculations for the OPEG potential are shown in figs. 2-4. The energy gap A(k,) or A( k,) is shown in fig. 2. For m* = 1, the AEMA( kF) is obtained in the effective-mass approximation when we replace the single-particle energies in (4.3) by (2.26). ADKA( kF) is obtained in the decoupling approximation (2.25). A&k,) is the self-consistent solution (4.3), and from the self-consistent single-particle energy (4.3) we also obtain an effective mass through (2.28), which then gives new selfconsistent single-particle energies and an AEMA( k,) in an effective-mass approximation for m* f 1. In general, we obtain an energy gap by all methods for 0.1 fm-’ < k, < 1.5 fm-‘, and the maximum SC energy gap is A(k,)

= 1.7 MeV,

for k F20.7

fm-’ .

(4.14)

We also see that a smaller effective mass m* produces a smaller energy gap and makes the energy gap disappear faster with increasing densities. Fig. 3 shows the

L. A~undsen,

0

0.1

497

E. @sigaard / Super~~jdi?y (I)

0.5

1.5

1.0 k,.

kF(fm-‘1

Fig. 2. Energy gap A calculated for the OPEG potential as a function of density, i.e. kr, k, and p. Results are given for the self-consistent (SC) solution, the decoupling approximation (DCA) and the effectivemass approximation (EMA).

Fig. 3. Energy-gap functions A(k) for the different single-particle models, calculated for the GPEC potential and k,= 0.8 fm-‘. Results are given for the self-consistent (SC) solution, the decoupling approximation (DCA) and the effective-mass approximation (EMA).

k,=l.l fm“

0.1

0.5

1.0

*

k (fm-‘1

Fig. 4. Occupation probabilities u”(k), calculated for the OPEG potential and .$=a.2 0.8 fm-’ and 1.1 fm-‘.

fm-‘, 0.5 fm-‘,

498

L. Amundsen,

Fig. 5. The effective

RSC potential

E. Ostgaard

/ Superfluidity

uFic (r) for k,=0.4

fm-‘,

(I)

1.0 fm-‘,

1.6 fm-’

and 2.0 fm-‘.

energy gap A(k) as function of k for k, = 0.8 fm-‘, and fig. 4 shows the occupation probabilities vi for different Fermi momenta. Results for the effective RSC potential uFzc are shown in figs. 5 and 6. The potential is shown in fig. 5, and the energy gap is shown in fig. 6. We see that the effective RSC potential produces a much larger energy gap than the bare OPEG potential, and we get a gap for a larger density region. The maximum SC energy gap is A ( kF) = 4.9 MeV , for k F= l.Ofm-‘ . (4.15) To study further the correlation effects, we also define an effective OPEG potential uzpG in the same way as the effective RSC potential. The potential is shown in fig. 7 for kF = 1.O fm-‘, and results for the energy gap are shown in fig. 8. We see that the decoupling approximation again agree with the self-consistent results, and we

L. Amundsen,

E. Ostgaard

/ Superfluidity

p

0.01

a

0.05

0.1

499

(I) (fmm3)

0.2

0.3

0.4

(MeV)

7.0 6.0 5.0 4.0 -

2.0

k, , k, Fig. 6. Energy gap A calculated are given for the self-consistent

(fm-’

)

for the RSC potential as a function of density, i.e. k,, k, and p. Results (SC) solution, the decoupling approximation (DCA) .and the effectivemass approximation (EMA).

“.I1

(MeV) 100

50

3.0 r(fm)

0

-50

-100 Fig. 7. The effective

potentials

up,“” and uFgEG for k,=

1.0 fm-’

500

L. Amundsen,

E. 0stgaard

/ Superfluidity

(I)

p ( fme3)

a

0.01

0.05

0.1

0.2

0.3

0.4

(Mew 5.0

-

4.0

.

3.0 2.0

-

1.00

0.5

1.0

1.5

2.0 k,.k&fm-‘)

Fig. 8. Energy gap A calculated for the effective k,, k, and p. Results are given for the self-consistent and the effective-mass

get a maximum

SC energy

gap of for kF= 1.0 fm-’ .

A(k,)=3.4MeV, Fig. 9 shows

how matrix

OPEG potential u~~G as a function of density, i.e. (SC) solution, the decoupling approximation (DCA) approximation (EMA).

elements

(k(uEzC(kF)

(4.16)

and (k(v~~EGlkF) vary with

(fm3%sV) 10

Fig. 9. Matrix

elements

(kluf$clkF)

and (k[op,P”“lkF) for k,=

1.0 fm-‘.

k for

L. Amundsen,

E. 0stgaard

:“I, , * Q2

0.4

06

(

0.8

/ Superfluidity

501

(I)

,

(

,

,

*

/

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 10. Effective mass obtained from the single-particle energy in the self-consistent solution for different potentials VoPEG (solid line), u:zeG (dashed line) and ug” (dotted line).

kF = 1.0 fm-‘, and effective masses and energy gaps for the different potentials are compared in figs. 10 and 11. At densities smaller than typical nuclear-matter densities, neutron star matter will consist mainly of neutrons, protons and electrons. The proton number is probably only a few percent of the neutron number, and the proton Fermi momentum will be correspondingly small compared to the neutron Fermi momentum. We then expect ‘S, pairing for protons at neutron matter densities which are too high for neutron ‘So pairing. A proton pair in neutron star matter will create polarization effects, i.e. polarize both the neutron system and the proton subsystem. We also get a many-body dispersion effect, which gives the proton a smaller effective mass than the neutron. Results for the energy gap are shown in fig. 12, and the maximum

P (fi3)

0.01

A

0.05

0.1

0.2

0.3

k,.

kF (fm”)

0.4

t

&t&f 5.0

-

4.0

-

3.0 2.0 1.0 0

Fig.

II.

0.5

1.0

1.5

2.0

Energy gap A from the complete self-consistent calcutations for the different potentials VoPEG, ~~“” and VFzc.

502

L. Amundsen, E. Dstgaard / Superfluidity (I) 2.0 a (MEW

/

F-w

\

dk:) \

/

\ 1’

\ \ \ \

10

1.5 ky.k: (fm-‘)

Fig. 12. Proton energy gap A( kpF) for different effective masses m*. The dotted line show! calculated for effective masses given by Takatsuka ‘I). The dashed line shows the corresponding

the gap neutron

‘SO gap At&).

energy

gap is A(kP,)=0.3

MeV,

(4.17)

for an effective mass of m* = 0.6. Proton effective masses have been estimated by Ikeuchi et al. lo) and Takatsuka ‘I) to be 0.5-0.7, and we have chosen corresponding values in our calculations, A critical temperature T, for the ‘So proton gap can be estimated by the BCS weak-coupling formula, kT,=0.57d(k$)=A(kP,)/1.76, and corresponding results are shown in fig. 13 for the OPEC the temperature should not prevent proton superconductivity

5. Summary

(4.18) potential. We see that in neutron stars.

and discussion

We have tried to calculate the energy gap as a function of density in “neutron star matter” to investigate the possibility of superfluidity in neutron stars. Superfluid states in neutron stars may explain observed dynamical facts in pulsars. The coupling between the “normal” outer shell and the inner superfluid neutron liquid is weak, and the observed dynamical behaviour after “speed-ups” in Crab and Vela pulsars, should prove that these pulsars include superfluid regions. Observed effects related to ~‘unpinning” of vortex lines may also indicate or prove pinning mechanisms and neutron superfluidity in the neutron star interior.

L. Amundsen, E. 0stgaard

/ Superjluidity (I)

503

Tc(lt) 1o’O -

109

-

0.2

0.4

0.6

0.8

1.0

1.2

1.4

kF(fm-‘)

Fig. 13. Critical

temperature T, for the superconducting proton system in neutron shows results for effective masses given by Takatsuka ‘I).

stars. The dotted

line

We see that singlet or ‘So pairing and superfluidity is possible for both neutrons and protons in neutron stars, but in different density regions of neutron star matter. The two effective potentials give larger energy gaps than the bare OPEG potential, and also superfluidity at larger densities. This is because the effective potentials have a much weaker repulsive core than the bare potential. In neutron star matter, there is probably a “neutron-drip” point at a density p=4.3x10”g/cm3,

(5.1)

i.e. at this density the matter should mainly consist of neutron-rich nuclei in a neutron “sea” (together with relativistic electrons). Negele and Vautherin I*) find that the density p of the neutron gas (liquid) is quite different from the baryon density &, in this region, i.e. p 5 6.7 X 10” g/cm3 &,=4.7x

10” g/cm3

p = 4.3 X 10” g/cm3 pb= 1.0x10’*g/cm3

for k r==O.ll

fm-’ ,

for k r=O.2Ofm-l.

(5.2)

Just above the neutron-drip density we, therefore, may expect superfluid neutrons both inside and outside the nuclei, but this is not taken into account in our calculations which are for pure neutron matter. The critical temperature as function of density for the calculated neutron and proton ‘So energy gaps can be estimated

so4

L. Amundsen, E. Osrgaard / Superfluidiry

(I)

according to the. BCS weak-coupling formula (4.14). The density regions for the gap vary, but indicate strongly the existence of superfluid and superconducting regions in neutron stars. The neutrons then are superfluid in the IS,, state for baryon densities of 4.6~10”g/cm~
(5.3)

above the “neutron-drip” point, i.e. we get superfluid neutrons both inside and outside neutron-rich nuclei. In a rotating neutron star, we should get a non-uniform superfluid with vortex lines parallel to the rotation axis. The protons in the interior are superconducting in a density region 1.6fm-‘
.

(5.4)

The protons should corotate with the electrons because of strong magnetic fields. The proton effective mass will be smaller than the neutron effective mass, which gives a smaller energy gap, but polarization of the neutron system may increase the effective mass and the energy gap for protons. The ‘So proton superconductivity will be destroyed for temperatures T,>2.4x10YK, and the ‘So neutron

superfluidity

will be correspondingly

(5.5) destroyed

for

~~1.2xlO’“K, but these are very unrealistic stars.

(improbable)

temperatures

(5.6) in the interior

of neutron

The energy gap for neutron ‘S, pairing has earlier been estimated in analogy with the BCS theory of superconductivity. Kennedy 13) showed that it probably is a good approximation to solve the gap equation with Hartree-Fock single-particle energies in the normal state. gstgaard 14) investigated numerically the possibility of superfluid states in the nuclear-matter region in neutron stars, using reaction-matrix elements from Brueckner theory and a Moszkowski-Scott hard-core potential. The results give a maximum energy gap of A(k,.)=4.5

MeV,

for k F= 0.75 fm-’ .

(5.7)

Yang and Clark “) constructed a variational wave function from Jastrow-type correlation functions and BCS wave functions, and solved the gap equation with an Ohmura potential in the decoupling approximation. The corresponding maximum value for the neutron energy gap is A( k,,) = 2.36 MeV ,

for k F = 0.72 fm-’ .

(5.8)

Takatsuka 16) tried to include the effect of the neutron single-particle potential from Brueckner-type calculations by lkeuchi et al. lo). and solved the gap equation by

L. Amundsen,

iteration

for the OPEG

potential.

E. 0stgaard

/ Superjluidity

He obtained

A ( kF) = 2.4 MeV ,

(I)

a maximum

505

gap value of

for kF= 0.96 fm-’ ,

(5.9)

and found that the ‘S,, gap disappears at somewhat higher densities than obtained by Yang and Clark 15). Clark et al. I’) and Niskanen and Sauls Is) have investigated the sensitivity of the energy gap to neutron interactions induced by particle- and spin-density fluctuations. Niskanen and Sauls, for instance, used the simple BCS weak-coupling formula A(k,)

= (h2k~/M*)

exp (-A-‘),

(5.10)

where A was taken to be the scattering amplitude for neutrons of zero total momentum at the Fermi surface. They separated A in a “direct” and an “induced” part, to get an impression of how the energy gap is modified by polarization effects on the effective interaction in the neutron matter, and found that density-fluctuations will increase the energy gap for 0.7fm-‘
(5.11)

These results are compared with our results for the bare OPEG potential in fig. 14, and we see that the results are in rough agreement, i.e. the ‘S, energy gap exists A

A (k& (r&V) 4.0 -

3.0 -

2.0 -

1.0

0

-

0.2

0.4

0.6

0.8

1.0

1.2

1.4

kF(fm“)

Fig. 14. The neutron ‘S, energy gap calculated by 0stgaard 14) (0), Yang and Clark “) (YC), Takatsuka 16) (T), and Niskanen and Sauls ‘s) (NS). The dashed line shows our results for the OPEC potential.

506

L. Amundsen,

E. Qstgaard

/ Superfluidity

(I)

A ( kF)’ ‘Mew 1.0 -

0

0.2

0.4

0.6

0.8

10.

1.2

-

kF(fme’)

Fig. 15. The proton ‘S, energy gap calculated by Takatsuka ‘I) (T), Niskanen and Sauls ‘*) (NS) and Chao et al. 19) (CCY). The dotted line shows our results for effective masses given by Takatsuka ‘I), and the dashed line shows our results for proton single-particle potentials given by Takatsuka ‘I).

for k,> 0.1 fm-’

and disappears

at a density

of

k F= 1.2- 1.4fm-‘ ,

p = lOI g/cm’.

(5.12)

The proton IS,, gap has also been estimated by the same authors “,‘8,‘9), and results are shown and compared in fig. 15. Chao et al. 19) obtained a maximum value of A (k:) = 0.8 MeV , i.e. for k F= 1.3 fm-‘,

and Takatsuka

for kF=0.6fm-‘,

‘I) obtained

A(kP,)=O.S MeV,

a maximum

for kg = 0.7 fm-’ .

(5.13) value of (5.14)

Niskanen and Sauls ‘*) find that the proton gap should be modified by neutron spin-density fluctuations, i.e. suppressed by spin fluctuations at higher densities, but be increased by density fluctuations at lower densities. We have used the same bare OPEG potential as Takatsuka, but obtained rather different results. This is mainly because of different proton single-particle energies EJ k) and different ways of solving the integral equation for A(k). Takatsuka takes proton single-particle potentials V,(k) from calculations by Ikeuchi et al. lo), in the single-particle energies E,(k)=hZk2/2M+ while we take the effective-mass

approximation

V’,(k),

(5.15)

(2.26), where

m*=[1+(M/fi*k,)(dV,/ak),+]-’

(5.16)

L. Amundsen,

The gaps disappear,

however, p = 8 X

E. @stgQQrd / Superfluidity

at approximately

the same density,

for k$= 1.1 fm-' .

1OL4g/cm3,

507

(I)

i.e. at (5.17)

We have neglected effects from the rapid rotation and the strong magnetic field, if we compare our calculations with the real situation in neutron stars. The rotation will problably create a lattice of corotating vortex lines, and the strong gravitational fields could influence on the geometry of the vortex lattice. Rothen “) has tried to estimate this effect. We have also assumed non-relativistic conditions, while general relativistic effects could be important for the dynamics and the structure of the neutron star. There are also questions whether pions can exist in dense neutron matter and “destroy” neutron superfluidity. Theoretical calculations indicate that the difference u, - up between the neutron and proton chemical potentials is too small to permit spontaneous creation of pions if they interact with S-wave nucleons in the density region for neutron pairing, i.e. (5.18) where AEZ is the pion energy because of the S-wave interaction pion self-energy. Otherwise, we would get n+p+r-,

for u,--uup=

u,> mar

which modifies

the

(5.19)

if the pion did not interact with the matter. We may, however, get a pion condensate in the density region for proton superconductivity, and the existence of superfluidity in a pion condensate will probably depend on the effective mass of the nucleons. It seems, anyway, that the effects of pion condensation can probably be neglected in our discussions. We assume in our calculations that we have no pairing between neutrons and protons. This should be a good approximation since the pairing takes place in a small region at the Fermi surface, and the Fermi energies of neutrons and protons are quite different in a certain density region. We have also neglected polarization’ effects, but neutron density fluctuations could increase the neutron critical temperature for low densities, and spin-density fluctuations could decrease the energy gap and critical temperature for higher densities I’). Krotscheck 2’) has also shown that the weak-coupling formula (4.18) should be a good approximation for neutron matter.

References 1) A.B. Migdal, JETP (Sov. Phys.) 10 (1960) 176 2) 3) 4) 5)

L.N. Cooper, Phys. Rev. 104 J. Bardeen, L.N. Cooper and N.N. Bogoliubov, JETP (Sov. J.G. Valantin, Nuovo Cim. 7

(1956) 1189 J.R. Schrieffer, Phys. Rev. 108 (1957) 1175 Phys.) 7 (1958) 41 (1958) 843

508 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

L. Amundsen, E. 0stgaard

/ Superfluidity (I)

K. Yosida, Phys. Rev. 111 (1958) 1255 V.R. Pandharipande, Nucl. Phys. Al74 (1971) 641; Al78 (1971) 123 R.V. Reid, Ann. of Phys. 50 (1968) 411 R. Tamagaki, Prog. Theor. Phys. 39 (1968) 91 S. Ikeuchi, S. Nagata, T. Mizutani and K. Nakazawa, Prog. Theor. Phys. 46 (1971) 95 T. Takatsuka, Prog. Theor. Phys. 50 (1973) 1754 J.W. Negele and D. Vautherin, Nucl. Phys. A207 (1973) 298 R.C. Kennedy, Nucl. Phys. A118 (1968) 189 E. Plstgaard, 2. Phys. 243 (1971) 79 C.-H. Yang and J.W. Clark, Nucl. Phys. Al74 (1971) 49 T. Takatsuka, Prog. Theor. Phys. 48 (1972) 1517 J.W. Clark, C.-G. Klllman, C.-H. Yang and D.A. Chakkalakal, Phys. Lett. 61B (1976) 331 J.A. Niskanen and J.A. Sauls, preprint (1981) N.-C. Chao, J.W. Clark and C.-H. Yang, Nucl. Phys. Al79 (1972) 320 F. Rothen, Astron. Astrophys. 98 (1981) 36 E. Krotscheck, Z. Phys. 251 (1972) 135