Research Report

Many-Particle Theory of Nuclear Systems withApplication to Neutron Star Matter

NASA Headquarters A -4

Washington, D C.

Contract Number 953549 r0I

Jet Propulsion LaboratoryPasadena, CaliforniaD. C.

o t

ti

Dr. Dennis A. Chakkalakal ,Principal Investigator

030

CoDr.Chia-H. Y an umber 953549

Principal Invest Propulsiatorn Laboratory

Departsadent of Physics

Southern 3 anch Post OfficeSouthern UniversityBaton Rouge, Louisiana 7083

00P'A 4

Baton Rouge, Louisiana 70813

https://ntrs.nasa.gov/search.jsp?R=19740004366 2018-05-29T20:32:30+00:00Z

TABLE OF CONTENTS

In troduction

PROJECT A

A-i Constraints on Variation 1

A-2 Numerical Procedure and Results 4

A-3 Discussion 8

PROJECT B 17

B-i Introduction 17

B-2 Polarization Effect 18

B-3 An Approximate Calculation of Fo 18

B-4 Effect of Polarization on Condensation Energy and

Superfluid State Energy 20

B-S Discussion 20

Many-Particle Theory of Nuclear Systems withApplication to Neutron Star Matter

This is a report on the research carried out on the following

projects since submitting the semi-annual status report on

April 5, 1973.

A. Calculation of an improved energy-density relation

for the normal state of neutron-star matter

B. Calculation of the effects of superfluidity and

polarization in neutron star matter

PROJECT A

The Calculation of an Improved Energy-Density Relation in Noutrol

Matter in the High-Density Region

A-1 Constraints on Variation

The theoretical formalism is outlined in section III.1 of the origi-!

proposal. In particular, the energy of the normal state of neutron mateL.

E , is developed in a cluster series

In this report, we shall use E,,to refer to our approximation for energy,

namely, (CE +r. E ) .

Reference is made in the original proposal to certain physically

motivated necessary conditions on the radial distribution function. These

in turn, constrain the variational parameters in the trial two-body

correlation factor f(r). There is also the so-called Pauli condition which

restricts f(r) directly. It arises from the effect of the Pauli Principle

which prevents particles in the fermi sea from scattering back to the

occupied states. The conditions which we use as constraints on our

variational procedure are listed below:

(I) I 0 (Pauli Condition)

(III) 1g . 1-243 (Coulomb Inequality)

(IV) aJ= 0

(V) 5Cc) = 0 (Structure-Factor Sum Rule)

where IB, S(K), Ic and aJfare defined as follows:

Ig = p Jdr (fr - .) ( - LF))

inK, = 3( Sm X~Z oS x

SIa = 1 + CO) 4 SC) ...

Sn Lkd - W

(0)

Sv) {Lr) ( -- £c

S(c~C7) -= f (3r>.') tr 0

. £4t>-a.~~w1= £LkF)

** + 5

+ ITP e r s #- st ks) t*r +

(oo

[ e(k( r) + e('tPS) + C rPt)

(')Tc + +e

Sa) r r Cr

34N = -(p-4w?3 5o e r

3

We note that

Thus condition (IV) is equivalent to S (0) = -1. Condition (V) means

S(o) = i S'(o) t 3'o)I +--- 0

we use it in the truncated form

+ + $(0kO) 0

The origins of conditions (II) and (III) are discussed in E. Feenberg,

Theory of quantum Fluids (Academic Press, New York, 1969).

The relative order of magnitude of the terms in the cluster series

for energy and all other associated cluster expansions is determined by the

"correlation parameter" 9 defined by

5 = qw J(f r) -I ) A r,

We note also that the five conditions are not completely independent of

each other because of the following relations: Condition (V) is related

to the k = 0 version of (II). The former can be satisfied by first

satisfying (IV) and then requiring also that S(1)(o) = 0.

In order to expedite the numerical calculation, we usually avoid im-

posing (II) and (III) directly. Often we find that (V) enables us to find

more easily the region of the parameter space where (II) and (III) are

satisfied. Similarly, (IV)(or (I))enables us to locate regions where j lis

small,

4

A-2 Numerical Procedure and Results

Several methods of calculation', each involving a different set of

criteria governing the choice of constraints in the variational procedul:,

have been pursued. As indicated in the proposal, we have used one-

parameter, two-parameter and three-parameter forms for the trial two-body

correlation factor f(r). Calculations using the various methods of

approximation have been carried out at the density corresponding to fermi

wave number, kF = 3.5 fm-1 in order to test the reliability of these method;

in the high-density region. The two-nucleon potential used for the

purpose is the hard-core potential of core radius 0.4 fm given by Ohmura,

Morita and Yamada (omy-4) (Progr. Theoret. Phys. 15 (1956) 222). This is

a central potential of the form4

v < 4= W ( z)

A 4 A 3 =A- 4

where i = 1, 2, 3, 4 denotes, respectively, the component appropriate to the

singlet-odd, singlet-even, triplet-even, and triplet-odd state of the two-

nucleon system. The Atare the corresponding projection operators. The

parameters are

-1V 2 = -235.41 MeV, 2 = 2.0344 fm-

-lV 3 = -475.04 MeV, $3 = 2.5214 fm

-

V V o01o 04 '

r" = 0.4 fm (in all States)c

These parameters are chosen to fit the following data characterizing th'Z

low-energy interaction of two nucleons in free space:

Binding energy of the deuteron = 2.226 MeV

-13

Triplet scattering length of the neutron-proton system = 5.378 x 10 cr13

-13

Singlet scattering length of the neutron-proton system =-23.69 x 10 cm

Singlet effective range of the neutron-proton system = 2.7 x10-13 cm

A two-nucleon state must be either singlet-even or triplet-odd

according to the Pauli Principle. Therefore, in our calculation, there is

no need for (Vo3, 3)'

Method 1

A trial two-body correlation factor,

has been used. According to the variational principle, the minimum of E,

obtained using any trial wave function provides an upper -ound to the true

energy E. Therefore, at each density we minimize (F EZ) with respect to

/1. If this minimum occurs at P = o , then E3 ( l) is calculated. The

corresponding approximation to the energy per neutron is EF +E z()+E 3(f~)

The results obtained by this method are given in Table A-1.

6

Method 2

The following two-parameter form for f(r) has been used in this method:

O

The additional parameter gives f(r) more flexibility in order to help satisfy

the constraints better. Several alternate approaches in imposing these

conditions have been used. These are described below:

(A) The parameter P is fixed at r= 2.0 fm-1 .y is varied and (E +4 ) is

found to have a minimum with respect to / at ( = ( . Then E((,4,) is

calculated.

(B) r is determined by condition (I). Then the minimization procedure of

method 1 is attempted.

(C) The same as (B), using condition (IV) instead of condition (I).

(D) The same as (B), but using condition (V) instead of (I), followed by an

attempt to use the minimization procedure of method 1.

In methods (B) and (C) neither (TF + 2) nor eF + S2 + 83) is found

to have a minimum with respect to P . Results for method 2A, 2B and 2C

presented in Table A-i are intended to illustrate the following fact: As

the value of W increases, energy gets larger and convergence of all cluster

expansions improves. This causes some ambiguity regarding the determination

of the minimum energy. However, the ambiguity can be removed when we impose

the conditions (II) and (III) on the En(y) curve. This is done in method

2D for which the complete results are given in Table A-2 for p values in the

7

range 2.0 ! V t 7.0 fm- . It should be pointed out here that there are

two values of I that satisfies S(0) = 0 in method 2D. Only the smaller of

these values of I is used for each t , since that corresponds to lower

energy and better convergence of all the cluster expansions.

Method 3

Here we adopt the following three-parameter form for f(r):

Several different procedures involving this f(r) have been attempted. Of

these, the ones that turned out to be most fruitful are described below.

(A) For given y and 3K, ' is determined by conditinn (I). Then it is

found that at eachi(, both (~F+ E) and 2 have a minimum with respect to

y at approximately the same value of = . Now we bring the results for

( ,, to better agreement with the constraints by changing Y to the lower

of the two values that will make S(0) = o. 3( is to be determined by a

further minimization of energy. The results are presented in Table A-3.

(B) In this method, the procedure in (A) is followed upto the point of

obtaining the U () results. Then Xis chosen as the value IC which gives

the smallest lit. Then (f,o) is recalculated and its minimum with

respect to j is sought. The results are given in Table A-4.

(C) In this procedure, again we let Y and X vary, but use condition (I) to

fix = o as a function of Pand (. Then at each trial value of/ ,y is

chosen as the value.?O for which tMAC is smallest. The resulting data for

(,Y.,) and other auxiliary quantities are shown in Table A-5. The lowest

8

energy subject to the constraints.is to be determined from this data.

(D) This procedure is the same as (C) except for the following changes:

Condition (IV) is used instead of (I) to fix Y ; C is chosen correspond-

ing to the smallest value of IIB. The results are given in Table A-6.

A-3 Discussion

The lowest energy obtained by each method is shown in Table A-7.

The criteria that are used to determine the lowest energy are the following:

(1) Whenever or() or + E() has a minimum with respect to 1 , we

choose that as the lowest value in spite of the violation of S(O) = o that

is usually associated with it (but, only in the high density region).

However, S(k) Z 0 is satisfied for k values not near k = 0. This can be

considered adequate because in the high-density system the larger k

values are much more significant. Besides, in a convergent cluster

expansion for S(k),

S(k) = 1 + S ( ) ( k) + S (k) + ...

the neglected higher order terms, though expected to be small, may never-

thless be sufficient to "repair" small violations of S(o) = 0 and

S(k) Z 0 for small k. In short, the results for energy obtained from a

calculation in which all the cluster expansions converge rapidly and all

the conditions are satisfied with the exception of S(k),>o at small k,

may be considered reliable. (2) Even when (EF + e2) or En has no minimum

with respect to p in an unconstrained variation, it is found that conditions

(II) and (III) cannot be satisfied when y is below a particular value .

9

Thus Et(jo) may be taken as the lowest energy consistent with the cons"ra:, l-

(3) Note that condition (II) is imposed in the modified sense discussed

above, except for methods 2D and 3A, where we have sought to satisfy (Ck

for all k. The results form these latter two methods, when compared with

results from other procedures, give us an estimate of the error we mai L

allowing through the violation of S(k) o for small k. (4) The reliabiliL:

of all our results depends on the size of Vf Iobtained; if l(is large,

adequate convergence of the cluster development is in doubt. Based on

all these criteria, so far the best results are obtained from method 3D.

We are in the process of testing four additional procedures involving

the three-parameter correlation factor. This is expected to'be completed

in about three weeks from now. Then we will adopt the most reliable of

the methods we have tested and carry out the complete calculation for

the entire density range 0.25!5 kF' 3.5 fm-1. It should be emphasized

here that the difficulties associated with imposing S(k) ? 0 for all k

do not arise in the intermediate - and low-density regions.

METHoD F ' 5 (0) E_ IE I . E- 2+ E z 3 E-Fm' )(e (Me Mev) QMev) (MW)

I -3. 0 4.3 o + ,6 -1-57 -oI -1-t) -b0.1 C922 oI (6o.

2A -1-30 20 1-20 -1-1 - l-o-o - 0o-7g -59-1 " 4 13-9 '

28 o. IL 2.0 1-30 -1-. 0-30 o 1.31 -57.2 q4., g764 (11.

0. 010 40o 1-53 -o0'4 oI 0 I-07 +1%7 1 G 1, G%, 29.

2c -o,),, 2-0 1t2G -2-0 o -o-03 I-zo -7.2 'c"4 4 19. (70.

2 c- -0 29 4. 0 140o -0- 0 -0 -O -oIf +12-7 I(obS, 72.0 231

TABLE A-i

Results from Methods 1, 2A, 2B and 2C

, 2 S )Co Sco> 57 o) aN -Z Te E SCk) >o . qo

(t' ev) n'Jw) Cr4.V) Cfnv>) _ot___

-i.o 2.0 I.,P -2-31 1-37 0-06( -0'65 -0'D 0-5. -59-3 '3.o 72-2 L . 3.of~ , k

-tof 3-o 1.201 -1-bG 0-b7 0.o0 -0-33 -0-Do 09,q -A1.1 it . 7', 1%"/. ,o 5',k _ T

-D074 14-o 1-2L49 -l'3 0-43 O-oo -0-21 -ID3 O 96-A4-1 1 14. 1r(C - 224. No-et

-5 5-0 1I3t1 -1-31 0-31 o-oo -0.15 - O3 0,9q +q3.8 ir9. T'1-5 21 4. Noe

-0-41 6.0 1-3'9 -1-Z3 O'Z3 0.o o -O0.1 -0'DZ O-q 103. 55. . No- 33e.

-0. -7, 1-4I 90 -1-t.r 0 17 o-oa --09 --P02 0-4 1-15. 327. S3,( I1. N ,.ne

TABLE A-2

Results from Method 2D

5 > 5 Co) SCo) AK T-3 I +EZ S(fk) 3 kio(fC) ( ') (Mw) C MCV) CCAv) (tAV) V tol atov%

-Zo 06 3*- 2.0 o.3 -- *S.q 1 -o00I -0o17 -0-D O-2% ~-59., 92-7 0-6 I.3. Z.7-- 7 ?

-1i21 4.o 3.0 0"52 -1.30 0-jI *t0,ol -0o4o -0.05 O 9 -4q.- 103, 7( - \7. 4 o - k. !!

-60b3 5.0 4.0 0.6% -I.A9 0.47 -ooZ -0-24 -0.Oq 0.96 -1S,2 134. 17,6 212. No ..

-0D*3 60o 50 o02 -1.34 0-34 o00oo -0-17 -0.03 0.'L4 +26.l 178 1'4 2S7. NovI.

-050 7 .0 6.0 04 -IZ1 2 025 0-oo -0-13 -o0-Dz 0I92 g3-1 3 5. 91.2 317. No,

-0,1 -0o 7-0 Io10 -d.1 0oa19 000oo -0 o -0-0, 0-89 15Z. 304. 94.7 399. Noi

TABLE A-3

Results from Method 3A

S r s sC sI() SCO) 4XJ r :, E2 4 . + S) o(AeV) (%AeY) C twcy) (M4CV) SbicsCtd for

Ot1' 2-5 3.2 2.(.Z -060 -0*99 -0-59 0.20 o 0 ~-o 3%. 1 3. 61-' 29 9 4.Z k '

0'09 3-0 3.2 Ib -0.60 -0'1 -D'59 0-20 o 1-15 -11-3 14k. o700 211. 14fI .

0, 3. 35 3.Z I-I6 -0-5% -1.-1( - -b 2 o - 24.1 -z.1. -14-2 2oz. i,. k<

O'0 4-0 3-. 0-%7 -0-"o -1-1 -0.72. 0-21 0 1.18 -21.0 125. -15.1 200. 1.G p<k ' "

O0-S 4.5 3-. 0o-6 -0'3;V -1.15 -0.11 o-z1 o 1I.1 -254 121. 74 . 202. u. 4,',k C.

0,04 5,0 3.2- O0 & -0,59 -1,0% -0-61 0-21 0 1-1-7 -21,9 130. 74 .5 205. I- Gf '. .k

TABLE A-4

Results from Method 3B

(X S> S0o3 Sco AJ-r T : -e E2 EF +E E c kAo

( (f) ) (Mrv) (Mev) (Mey) Sakis'' for

0-14 < k < I-t fr0-23 IA4 175 2-36 -0-3b -2-59 -. '14r 0-'3 0 oWl -45-9 io&-. 5-5 192. 2 - < k 3-'

0.( Lo 2-oo 2-3% -o-44 -2.00 --I.4 02. 0 O Dq -39.0 113. 80-2 I53. 1

204 k< 1'30.17 Is~ 2- 2-42 -06.9 -1-63 -1.12 O- ' 0 1-4 -200 I'4, 15-, 200. < 0

0-19 2.0 2-60 2-64 -053 -I-37 -0o 0-Z3 o I04 -1-27 151. 13-5 22 . 0~,>o7 f I

0 -l 2-2 2-80 2-55 --O-S6 -. 19 -0475 o0-2 o 1-1 3,6, 165. 61- 2z~ . k > o 1'~ '

0-17 . 3-2. 2-,b( -o06o -0.-9 -0-51 020o o 1-o7 34 . 197. %.1 Z'48. k o f '

0.( 2- 355 2-59 -o0*6 -o0 , -0 42 0o'4 o I-o, 52-1 2o0. 56'9 Z61. k - I FhJ

0 -17 3.0 3-4 2-bI -o-65 -0 -79 -0 z. o 1% o 1-03 1To. - 2Z3. 54.0 217. 1i- Iff,

0o17 3.2 4,0 2- -62 -06! -0-79 -031-g o-,0 o I-ol ~8-2 241. S1.7 2.92. k'.o kFu

OIS 3- 4-96 2-'73 -0-65 -0-6% -0.33 o0 - o-97 12z8. 281. L '3 327. k -9F;

O.I 3.% t.90 Z-7Z -0-66 -0-63 -p-1 0o17 o 0195 15. os. q'4. 352. k 70 - f ,(

TABLE A-5

Results from Method 3C

p F % 7 5(o s o SM) A ( 3, I E . +E L E NoS(rMv) (4ey) (Mjtv) (mev) SokCfct for

-0 o40 J*'4 80 2-53 -1-01 -1.77 -I-79 0 -O'. 06b9 -'.j Ill. 99-7 2oo.2*o < ,<4-2;%

0-7 k < 1.3f i-0,37 16 2.04. 2.41 -I.oo -1-32 -I.3L- 0 -0C2 1-13 -35- IG . 79,7 196.

2-0 4 k < '!

-0.3 4 f. 2.- 0 2.30 -oo-.o -0 oo -- oo O - '20 1-20 -Z.72. 127. 74.0 2o01.

-0'30 2-0 2-bo 2-G2 -1.oo -0-77 -0.77 o -0l-oz Ir -,-30 I-I. "97 2 . k> 1.0o

-0-7 2-2 2-9 2.b7 -I[oo -0o61 -0-bl -0-02 1-13 11.2 16. 6bJq 22-. k >i.I fv,,

-0lf 2., 3,25 2-64 -I.oo -0.44 -P- . o -0.z 1.10 30-6 189. Si i. 22. k > 1'3 fp-

-0-2. 2- 3.6 2-65 -1.oo00 -0-33 -0"33 o -6~'t 1-o5 55-'3 2o4, 5q9 262. k > 1I3f . ;

-0-21 3-0 3-70 2-67 --o00 -0-28 -o.Z oa 6 .ot I-oz 72-7 225. 52.5 279. k > 1,2 I

-0oIS 3.2 q-25 2-3 -I-oo -0-23 -0-23 o -o.-o 0-98 /lo. 2bo. '7-3 3o9. k > 1 If

-o0-r 3.5 . 2.14 --Ioo -o-ig -o-IS > -o-or 0- 6 136. 21. 4S-5 3314. > o.9 f,

-0-16 3. 5.0& 2-90 -1-oo -0-15 -00I5 o -Ooi 0-93 /17. 325, 43,S 3( , k > O'7f

TABLE A-6

Results from Method 3D

Cc) Sc., SCo) ~J B I( ' 2 3 -,.,METHOD C-FV yOvIev) 1MQ9) (Mev)

3' -0-30 2-0 2-Go 2.67 -1.oo 00 "D' -017 O -D-02. 1.15 -6-30 14'G. 9-7 216.

3 C 0,(o 2.2 2- 0 2-55 -0-56 -1-19 -0"15 0-2 O 1.12. 3.G 154s. Z.9 2.23.

313 0-og 4-.0 3 0o 7 -o052 -,1l1 -0-12. 0-z21 0 1-1 -29.0 IS5. 1 ,.1 2o00o.

3A -0-%3 5.0 4.0 0o-6% -1i9 0-47 -oo02 -D0.2 -0.D O-q46 -19-2 134. 7,1G 21z.

2_D -0-74 4-.o 4.0 1.249 -- 3 0-43 O-oo -*1z -O*D3 0- . -4-If I48. '1,o 224.

TABLE A-7

Comparison of Results from Different Methods

17

PROJECT B

Effects of Polarization on Neutron Star Structure

B-I Introduction

The invisible components in the so called "single-line spectro-

scopic binaries" in Hercules, Scorpius, etc., seen by the UHURU satel-

lite(1) are now commonly believed to be rotating neutron stars and in

some casesmay be even black hole revolving around the center of mass of

the system. It is possible to estimate the mass of the neutron star

component by an elaborate study of the intensity curves and of the

spectral class of the visible components. Hence theoretical determinat-

ions of masses, m9ments of inertia and radii of stable neutron stars have

become more important than ever.

Macroscopic neutron star properties have been calculated during the

past fifteen years using equations of state derived from different pheno

menological two-body interactions. One of the realistic effective inter-

actions between two neutrons is a combination of a strong short-range re-

pulsion and a long range attraction. It was first suggested by Migdal

and later proved by Yang and Clark (2) and others, that in a comparatively

low-density degenerate neutron liquid, for which the interparticle spacing

is large compared to the range of the repulsive forces (10- 13cm), the

attraction between pairs of neutrons of opposite spin and momentum would

lead to the formation of a condensate and the appearance of superfluidity.

However, our estimation of the pairing energy for "S" wave attraction

is on the low side, since the enhancement of the attractive interaction

between neutrons arising from the fact that they are embedded in a highly

polarizable medium (the other neutrons) were not taken into account by us.

(Yang and Clark),

18

B-2 Polarization Effect

According to Pethick and Pines( 3) , the additional term coming from

the polarizability of the medium is always attractive and is approximately

- VFsI/(1 + Fs) where V is the "bare" interaction in the "S" wave

channel and Fs is the Fermi liquid parameter which describes the spin-o

symmetric part of the interaction between two quasi-particles on the

neutron Fermi surface. The net effective interaction, hence, has the

simple form

-8I

Since Fs for the neutron liquid is negative and according to Pethick &o

Pines may be as negative as -0.7, such enhancement effects can be very

important. An exact estimation of FS' seems unattainable at present al-

though such a calculation is desirable and necessary in order to under-

stand exactly how the polarized medium affects the energy state and

therefore the mass-energy density inside the neutron-star matter.

B-3 An Approximate Calculation of Fs0

Applying the Landau technique of functional differentiation of the

energy with respect to quasiparticle occupation numbers(4), we estimate

the Fermi liquid parameter F by summing over both the direct and exchange

interactions between the interacting quasiparticle pair via the following

integral 2k F

Where No = m*kf/2T(211 is the familiar density of states at the Fermi surface,

and A1is the volume of the system.

19

For the interaction between a quasiparticle pair, we choose the simple.

Yamaguchi potential(5)

(k')Vlk) = -(h2/m) K g(k') g(k), (S-wave only) (s - 7)

2 k2with g(k) = ( + -1. For K = 0.18725 fm- 3 and B= 1.254 fm the singlet

-n ae = -23.75 fm and 9

Pcattering length and effective range are as = -23.75 fm and r's

The Fermi liquid parameter F is calculated for several densities. The

results are listed in Table B-1

Fs for .different dessitiesTable B-1 o

-kf(fm- 1) 0.456 0.60 0.72 0.96 1.20

Fs -0.081 -0.262 -0 .364 -0.552 -0.734

As can be seen from Table B-1, Fs depends on density quite strongly. As0

the density increases, the Fs becomes more and more negative. At the0

density kf = 1.20 fm-l1 Fo is equal to "0.73. These results together with

the prediction by Pethick and Pinds give a very strong indication that the

enhancement due to the polarization of the neutron medium may even be large

enough to give rise to a major mass-energy density discontinuity due to

neutron pairing effects. The existence of a concavity in the mass-energy

density vs. the number density curve is sufficient to give a first-order

phase-transition. Since there is no precise way of calculating the en'ergy

state for the neutron-star matter including the polarization effect, it is cf

interest to see how strong the enhancement will have to be in order to produce

a first-order phase-transition.

20

B-4 Effect of Polarization on Condensation Energy and Superfluid-State Energy

As has been mentioned in our original proposal we have all the nzecssnar

ingredients for the calculation of the normal state energy,n,. and the

condensatinn energy, cE, and also the superfluid state energy, Es= n--E ).

To simplify the calculation, we assume that the enhancement due to polai-

zation results in an increase in the attractive potential well-depth only.

Then all we have to do in the calculations of the enhanced normal state energy,

nenh , and the enhanced condensation energy, enh, is to substitute A by

(6)jA o G>) in the Ohmura potential

V(12) = c0 , r12 < cA2 exp (-Ar), S-wave only, r12 > c 8-4 )

Where c is the radius of the hard core. Cenh enh enhn Ec an s

enh( =n -- nh) are then calculated according to the procedures described in

enhour proposal for the c = 0.4 fm Ohmura potential. The results of enhn

enh, and nh for 8= 1.0, 1.30, 1.45, and 1.50 are listed in Table B-2.

(Note that aB = 1.50 is corresponding to a FS = -0.33.) For the purpose of0

easier reference, we show in Figure B-1 the plots of the enhanced normal state

energy per particle vs. kf for R= 1.0, 1.15, 1.30, 1.45, 1.50, 1.55, 1.60,

and 1.80. In Figure B-2, the enhanced superfluid state energy per i particle

enhEs , vs kf for = 1.30, 1.45, and 1.50 are plotted. Some interesting sets

of energies in Figure B-1 are plotted vs. the specific volume,tr, in Figure B-3.

B-5 Discussion

From Table B-2 and Figure B-2 we find that atf= 1.45, the superfluid state

energy, enh , turns negative at k = 0.5 fm-I while at = 1.50,~enh turnsS f 5

21

negative at kf = 0.65 fm- 1 resulting in a major mass-energy density dis-

12 13 - 3continuity around the density 10 -10 gm-cm

We conclude that the polarization effect indeed enhances the condensation

energy (and the gap) and there is a tendency of the neutron-star matter to under-

12 13 -3go a first-order phase-transition around the density of 10 -10 gm-cm

provided the effect is as strong as indicated (or stronger).

1. R. Giacconi, S. Murray, H. Gursky, E. Kellogg, E. Schrier and H. Tanen7a.:;Astrophys. J., 178 (72) 281

2. C. -H. Yang and J. W. Clark, Nucl. Phys. A174 (71) 49

J. W. Clark and C. -H. Yang, Nuovo Cimo Lett. 1(1970) 272, 2(1971) 579

3. D. Pines in Proc. XIIth Into Conf. Low. Temp. Phys., edited by E. Kandu

Tokyo, 1971

4. G. E. Brown, Rev. of Modern Phys. 43 (1971) 1

5. Y. Yamaguchi, Phys. Rev., 95 (1954) 1678

6. T. Ohmura, Progr. Theo. Phys. 41 (1969) 419

y

Table B-2 Condensation energy, Ec, normal state energy per particle, andsuperfluid state energy per particle for = 1.0 (unenhanced),1.30, 1.45, and 1.50 (enhanced)

8 kf(fm- 1) (MeV) c(MeV) s(MeV)

1.0 0.24 0.606 0.114 0.4920.48 2.050 0.294 1.7560.60 2.960 0.301 2.6590.72 3.990 0.220 3.7700.84 5.150 0.131 5.0191.08 7.935 0.012 7.9231.20 9.655 0.001 9.654

1.30 0.36 1.103 0.436 0.6670.48 1.705 0.773 0.9320.60 2.319 1.120 1.1990.72 2.925 1.030 1.8950.84 3.527 0.765 2.7620.96 4.155 0.504 3.6511.08 4.853 0.352 4.5011.20 5.674 0.107 5.5671.32 6.679 0.021 6.658

1.45 0.36 1.028 1.065 -0.0370.48 1.533 1.601 -0.0680.60 1.996 1.777 0.2190.72 2.390 1.685 0.7050.84 2.719 1.331 1.3880.96 3.012 0.933 2.0791.08 3.315 0.564 2.7511.20 3.686 0.277 3.4091.32 4.189 0.051 4.138

1.50 0.36 1.003 1.270 -0.2670.48 1.476 1.853 -0.3770.60 1.888 2.070 -0.1820.72 2.212 1.959 0.2530.84 2.450 1.554 0.8960.96 2.632 1.105 1.5271.08 2.803 0.686 2.1171.20 3.023 0.359 2.6641.32 3.359 0.068 3:291

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