REFERENCE W ' IC/75/18

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

SYMMETRY COEFFICIENT, ISOSPIN-SPIN DEPENDEKCE

OF THE SINGLE-PARTICLE POTENTIAL,

COMPRESSIBILITY AND REARRANGEMENT ENERGY

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

K.M. Khanna

D. Jairath

and

P.K. Barhai

1975 MIRAMARE-TRIESTE

I .

|r

i i , .1* S,AJL JI n

IC/T5/18

International Atomic Energy Agency

and

United Mations Educational Scientific and Cultural Organization

IIJTERHATIONAL CEHTKE FOE THEORETICAL PHYSICS

SYMMETRY COEFFICIENT, ISOSPIN-SFIN DEFEfTDEHCE

OF THE SINGLE-PARTICLE POTENTIAL,

COMPRESSIBILITY AHD REARRANGEMENT ENERGY •

K.K. Khanna ••

International Centre for Theoretical Physics, Trieste, Italy,

D. Jairath and P.K. Barhai

Department of Applied Physics, Birla Institute of Technology,Ranch!, Indie.,

MRAMAHE - TRIESTE

March* 1975

* To be submitted for publication.

•• On leave of absence from Department of Applied Physics, Birla Institute

of Technology, Ranchi, India.

ABSTRACT

The symmetry energy coefficient a has been calculated using first—T

order perturbation theory. In one set of calculations we have used the

theory developed by Brueckner and Eabrowski, while in the other set of

calculations we have made straightforward calculations of energy In terms Of

the neutron excess parameter a . We e e t different results In the two sets

of calculations. But the results agree favourably with those obtained by

otners. The Isospin-spin dependent part of the single-particle potential

U has also been estimated using the explicit expression due to Brueckner

and Dabrovski.

Using the formalism developed by us in this manuscript we nave

calculated compressibility in terms of a . Our result is roughly 10J( more

than the nuclear matter result. Similarly the rearrangement energy Uft

has been calculated in terms of a . The results compare reasonably well

with those obtained by others.

-1-

I . IHTRODUCTIOB

2)

Huclear matter properties for an equal number of neutrons and protons

have been determined by several authors using various approaches. Host of

the attempts to calculate the properties of nuclear matter are based on

cueleon-micleon (n-n) forces determined in free nucleon scattering experiments.

Because of the singular character of the n-n forces,the problem of nuclear

matter could not have been solved before the reaction matrix theory of

Brueckner was developed. The calculations of Brueckner and Gammel (BG)

gave remarkable agreement between the calculated and empirically determined

values for the various nuclear matter parameters including the symmetry co-

efficient aT .

The symmetry energy for small neutron excess was first calculated

by BG. This was later more accurately determined by Brueckner and

Dabrovski (BD) . The symmetry energy of large neutron excess is of

significant interest in order to predict the nuclear masses far from the

region of normal nuclei presently known. Such masses are rarely measurable

but are of considerable importance in astrophysics! problems and in the

theory of nuclear fission. The properties of the neutron gas have been

pact5)

determined by Bruecluier, Gammel and Kubis . The impact of symmetry energy

on astrophysical problems has been studied by nemeth

The value of the symmetry coefficient a_ is not well determined6)

even when a good fit to nuclear masses near the stable valley is obtained .

Therefore, even for the agreement achieved by BG one could only argue that

the value of symmetry coefficient obtained with the most realistic case1 \ ft \

of Gammel-Thaler (GT) nuclear forces IJ>°' lies within the range of different

empirical estimates of this parameter. Besides the difficulties encountered

in the calculations due to the singular nature of the interactions, the

investigations carried out with the Brueckner-Goldstone theory are confrontedq)

with the criticism of Brown et al. J' They remark that the theory and the

potentials used in these investigations are unable to give reliable values

for the nuclear matter parameters. Consequently, the use of effective

interactions in nuclear calculations has gained renewed importance in recent

years.

The possibility of using effective interactions for nuclear matter

calculations has already been discussed in our earlier papers /•HJ»12/<

Though the effective interactions are usually determined primarily by a fit

to the binding energy and density of nuclear matter, if the interaction has

to reproduce approximately these quantities for spherical nuclei, then it is

-2-

reason&ble to require that it also fits the symmetry coefficient , a_ .

Keeping this in view,-in this paper we have calculated the synmetry coefficient

using the two effective interactions derived in our earlier paper

whether they also fit

to see

In one set of calculations we have calculated

a^ using the approximations of BD ', while in another set of calculations

we have calculated the symmetry energy coefficient in a simple and explicit

formalism developed by us in this paper.

We have calculated the rearrangement energy U in terms of aR

using the formalism developed by us in this paper. Knowing ot for differentnuclei, one could find out the U for finite nuclei.

n

Since the isospin-spin dependence of the single-particle potential

is closely related to the symmetry coefficient a~, , we have also calculated

the isospin-spin dependent part of the single-particle potential U, using3)an explicit expression due to i D ' .

This paper is essentially divided into two parts. In one, some of

the properties are calculated using the explicit expressions due to BD .

In the other, some of the properties are calculated using the formalism

developed by us. The various results thus obtained have been compared with

each other and with uhat is known about them so far.

II. INTERACTION

For convenience and comparison, we briefly describe the interactions

derived in Kef.10. The general form of the interaction is given by

V =where

(2)

corresponding to the modified interaction of set 1 and

(3)

-3-

corresponding to tbe modified interaction of pet 2 of Ref.10.

form of VCr) ia

The

and VK is given by

= c (5)

From now onwards the interactions represented by Egs.(l), (2) and (5) and

(1), (3) and (5) will be referred to as set 1 and set 2, respectively. The

values of the parameters corresponding to set 1 are

V, = 8S-71

•; C =(6)

Set 2 has the same values for V,, , n , u as given by Eq. (6), whereas V

and c are given by

V,= C = (7)

III. SrMMETHY COEFFICIEHT

We consider here the case of nuclear matter vith a given neutron excess.

We also assume that there is no spin excess, i.e. every momentum state is

occupied by two neutrons with spin up and down and/or by tvo protons with spin up

end down, or otherwise the momentum state is empty. If k and k^ denote

and

down, or otherwise the momentum state is empty. If k and

the Ferni momenta for protons and for neutrons, respectively, then

k are related to and the neutron excess parameter a =N—Z

Vsa.s follows:

(8)

where k_ is defined by the relation

-It-

'Z3 "

The average binding energy per proton, c , i s given by

(9)

Cio)

There is a similar relation for the neutron. U(k ,-) is the potential energy

felt by a proton wi

particle potential

felt by a proton with momentum k and has the same form &s the usual single-

except that all kp values ere replaced by k . ) in

eludes sums over spin and isospin.

Expanding the quantity

also in-

Ain powers of a and retaining the leading terms one obtainB

A

(12)

In Eq..(12) the coefficient of a Is the symmetry coefficient i.e.

C l 3 )

-5-

where the quantity S is the isospic flip term vhich ia s. measure of the force

to resist a change In the isospin of the system. To a good approximation theIk)

quantity S can he written as

vhere U is the mean potential energy per particle due to Interactions in the

indicated states. The expression for U is given hy

(15)

Thus for the interaction of set 1 one obtains:

3.3 .. ,3 J

and for set 2 :

(16)

vhere

z

i o 7 r.&

(IT)

and

A C19)

I t is clear from Eqs.(l6) and (17) that the contribution from the

isospin amplitude part is independent of the contribution from the short-range

density dependent delta interaction part. This, hovever, does not mean that

-6-

the short-range part is Inactive. Xhat happens in this case is tlurt therepulsive and the attractive cor.trihutioEs are equal and consequently canceleach other.

The value of U(k ] for the interaction represented hy aet 1 isID

given ly 3

(20)

For small momenta, J_{kmr) can he expanded and,keeping terms up toobtains

(21)

In the same vay, for the interaction of set 2 one obtains

*=[-101-38 (22)

Thus Eqs.(l6), (1T)> (21) and (22) give the necessary expressions for the

calculation of a_ .

However, taking into account the intrinsic dependence of our inter-

action on kp , ve must also add to Eq.(13) the rearrangement part of the

symmetry coefficient i€R .

Following BD, ve vrite the rearrangement part of the symmetry

coefficient, A ^ (essentially Eqs.(2>0 and (62) of Ref.3),aa

(23)

vhere

(2M

-7-

corresponding to the modified Interaction of set 2 of Eef.lQ. The general

form of V(r) ia

VB is given "by

V 6 = c (5)

Prom now onwards the interactions represented by Eqs.(l), (2) and (5) and

(1), (3) and (5) will be referred to as set 1 and set 2, respectively. The

values of the parameters corresponding to set 1 are

* 8371

•; C = fa-(.6)

Set 2 has the same values for V£ , y » U2 a s given hy Eq.(6), whereas V

and c are given by

V, = c = 73-65"Mtv (7)

III. SYHMETEY COEFFICIENT

We consider here the case Df nuclear matter with a given neutron excess.

We also assume that there is no spin excess, i.e. every momentum state is

occupied by two neutrons with spin up and down and/or by two protons with spin up

and down, or otherwise the momentum state is empty. If k and k denote. p n

the Termi momenta for protons and for neutrons, respectively, then k andH-Z "k are related to k_ and the neutron excess parameter a as follows:

(6)

vhere k_ is defined by the relation

-3 T. K = C9)

The average binding energy per proton, c , is given by

(10)

There is a similar relation for the neutron. U(k ,-) is the potential energym

felt by a proton with momentum k and has the same form as the usual single-

particle potential

also in-except that all iy values are replaced by i . ^ ^ i n

cludes Bums over spin and isoepin. m

Expanding the quantity

Ain powers of a and retaining the leading terms one obtains

2)

(12)

In Eq.(l2) the coefficient of a is the symmetry coefficient s_ •, i.e.

, (13)

-5-

where the quantity S is the isospin flip term which Is a measure of the force

to resist a change in the isospin of the system. To a good approximation,the

quantity S can tie written as Ik)

where U is the mean potential energy per particle due to interactions in the

indicated states. The expression for U is given by

Thus for the interaction of Bet 1 one obtains;

3 3 3 .3

ana for set 2 :

Vvcp_3,.3

/0 7T'1

(16)

(17)

where

and

(19)

It is clear from Eqs.(l6) and (17) that the contribution from the

isospin amplitude part is independent of the contribution from the short-range

density dependent delta interaction part. This, hovever, does not mean that

-6-

the short-range part ia inactive. Khat happens in this case is that -the

repulsive and the attractive contributions are eijual and consequently cancel

each other.

The value of U(k ] for the interaction represented by set 1 is

given by 3

t20]

For small momenta, Jn(k r) can be expanded and,Keeping terms up to 1 ,one

obtains

(21)

In the same way, for the interaction of set 2 one obtains

) (22)

Thus Eqs.(l6), (17), (21) and (22) give the neceBBary expressions for thecalculation of a~ •

However, taking into account the intrinsic dependence of our inter-action on k^ , we must also add to Eq.(l3) the rearrangement part of thesymmetry coefficient Afip .

Following ED, ve write the rearrangement part of the symmetrycoefficient, A^ (essentially Eqs.(2U) and (62) of Hef .3), as

(23)

S Tin,.-7-

f

i f (V

•M

The symbol implies

k

(26)

It is to be notea that Eqs.{2li) and C25] differ lay a factor of ± from

Eqs.(2!t) and (6E) of BD . This factor j is necessary here since the equations

of BD contain a factor of 2 due to the contributions from the exchange term.

Terms with V. do not appear in Eqs.(2l+) and (25) since 7ft has no intrinsic

dependence on k_ .

The quantity AnC_ given by Eq.(2lt) has been calculated using the

relation between the usual rearrangement energy

arrangement energy Up is given by

and • The re-

Comparing Sqs.(2i4) and (27) one obtains

(27)

(26)

where, for the interaction given by set 1, Eq..(2T) yields

-8-

(291

Set 2 also yieldB the same form for

c corresponds here to Eq.(T).

as given by Eq.(29), except that

Substituting for V B in Eg.. (25) one obtains

(30)

for both the interactions. Eovever, the mmerical values vill differ due to

different values of c .

IV. ISOSPIH-SPIN DEPEHTEHCE OF THE SINGLE-PAHTICLE POTEHTIAI,

The s i ng l e -pa r t i c l e po t en t i a l U of a nucleon with momentum k inm

the case of nucleus with a given neutron excess is dependent on a . Ito a

linear approximation in a it can be expressed in the form ;

(31)

where +(-) refers to the case of neutron (proton). A typical value of

U, can be estimated from the relation between. a_ and U., at the Fermi3)

surface which is

(32)

I t should be noted that in Eq.(32) UQ(kffi) usually includes also the

rearrangement part UR . This i s evident from Ea..(56) of BD. However,

in our case, inclusion of UR in Eq.(32) will not make any difference as

U,, given by Eq..(29) is independent of k . Thus, i t permits us to use U(k )K m m

given by Eas.{2l) and (22) for \ ( \ ) in Eq. (32). Of course, i t does notapply to Eq.(3l) where TJL must be included to get the correct neutron(proton) single-particle potential.

-9-

V. RESULTS ASD JHSCUSSIOH

In all our calculations we have used k^ » 1.35 fm which

corresponds to the saturation density for the interactions- of Hef.10.

The kinetic energy part of a^ gives

For U(km) given by Eq.(21), one obtains .= 12- 3o

The value of S is given by Eq..(l6) and has tieen found to he

Eg«.{28) and (29) give

vbere UR given by Ea.(29) 1B 12.37 MeV.

The value of i ^ p given hy Eq. (30) is

Co)

Thus from Eq.(£3) one ohtains

3o

Prom Eqs.(33), t ^ ) , (35) and (38) we obtain

(33)

(310

(35)

(36)

(37)

(38)

(39)

vhere t^, given by Eq. (39) also includes A^ .

The value of the isospin-spin part of the single-particle potential,U^kj,) , for the interaction represented by set 1, has been obtained fromEq.(32) to be

-10-

The values of aj, and ^(kp) without rearrangement effects are:

(in)

and

Similarly, for set 2, using the appropriate equations and the values of the

parameters, one obtains;

6

= - S-

Z Mev

and

The corresponding values without rearrangenent effects are:

a

(US)

The value of the symmetry coefficient as has been obtained from the

empirical estimates is not unique . By considering a pure volume symmetry

-11-

i\

coefficient, Green 5 ! finSa s , n 23.52 JfeV, whereas considering also a

surface part of the symmetry coefficient .Green finds &_ = 30. ~5k MeY andprefeCameron finds a_ = 31.!*5 MeV. Dabrcwski refers to the experimental

value of &J = 28-32 MeV. Using GT nuclear forces, * ' BD find a^ = 32.0 MeV.

As one can see, our values of a^ are somewhat higher than those

referred to above. However, our values compare favourably with the value

of a , = It3 MeV obtained by Falk and Wilets ' , our value for set 2 being

a_ = 1)3-16 MeV. One should also cote that the values of a without re-

arrangement effect as given by Ec_s.(|tl) and (1*1)) are in close agreement vith

the values of Refs.3> 13 and 15-17-

BC find that the symmetry coefficient is very much sensitive to the

nature of the potential used. Employing the beet GT potential they obtained

a_ = 26 MeV. However, using another potential vhich contained no odd-state

interactions but which also reproduced acceptable values for mean binding

energy density and compressibility, they obtained an = U3 MeV. They

attributed the reduction of the coefficient from k3 MeV to 2& MeV to the

inclusion of odd-state forces, where both the repulsive singlet and the

attractive triplet components act to lower the symmetry energy.

It is significantly interesting to note that this fact is borne out

by our results also. It can easily be understood if one looks at the values

of e_, given by Eqs.(39) and (*43). The value of aT given by Eq..(i43)

corresponds to the interaction of set 2, where V acts on both even and

odd states due to the unequal Wigner and Majorana exchange mixtures ana

consequently lowers the value of a^ . In contrast to this, a_, given by

Eq.(39), i.e. corresponding to the interaction of set 1, where VA acts

only on even states due to the equal Wigner and Majorana exchange mixtures,

is higher.

We therefore anticipate that inclusion of further repulsion in odd

states by altering the exchange mixtures and then re-adjusting the parameters

to give correct binding energy and-density of nuclear matter might also

improve our values of a,,, . However, the symmetry coefficient being of

considerable importance in finite nuclei also, it vould be a reasonable

procedure to adjust the potential parameters to fit also the two-particle

data or some finite nuclei properties like the binding energy of He , 0 ,

etc. and then calculate such properties as the compressibility and symmetry

energy. This vould thus constitute a semi-phenoraenological approach to the

parametriaation of the interactions, which might give us a better interaction

for calculations of further nuclear properties.

-12-

In our present investigation,in yiey of the inaccuraci.es contaj.nsd. in

U and A^ , one cannot attribute the entire disagreement solely to theinteractions used. The expression for A<£ given by Eg.. (25) is an approximate

one only (see the discussion of Ref.3). Since the rearrangement port Ae

constitutes a large part of a , , i t s Importance shoula be suitably taken care

of.' Especially in the case of isospin-spin dependent part of the eittgle-

particle potential U , the rearrangement contribution constitutes i t s

major part and approximately doubles the value without rearrangement for both

the interactions.

The value of the isospiti-spln dependent part of the single-particle

potential U, is also not reliably known. As one can see from the in-

vestigations of Lane , Hodgson *' and Dabrowski 2 0 ' , the situation vith

U is even much worse. BD using the 01 potential have obtained IL (k-) = 126

MeV . The results of all the investigations of Eefs.3, 18-20 indicate a value

of l^tkj,} = 100 ± 50 MeV . He find that U^kj.) given by Eq.(ltO) is in close

agreement with the upper limit of ^(kj.) • The non-rearranged value

U1(kF)Qr = 76.08 MeV Is well within the range. Toe values of U,(k-) » 102.61*

MeV and ^ ( k ^ ^ , = 53.01 MeV for the interaction of set 2, are also very much

within the limits of U^kp) already mentioned.

An estimate of the difference between the neutron and proton single-

particle potentials due to symmetry effect for a large nucleus like Pb

can be made using the value of U1 1jp l Ckp) a = 8 KeV and

For °Fb we have a « 0.21 and

a = k MeV for the interaction of aet 1.

These values show that the IS single-particle energies for neutrons and

protons inP.OB should differ due to symmetry effect by about 16 MeV with

rearrangement and 8 MeV without rearrangement. This latter value Is comparable1PI 1

For the interaction Of setto the 10 MeV value calculated by MoEZkowsJci2, we get ^ Ui^kF ° ~ MeV * T b i s vsJ-ue i s i n excellent agreement viththat of Moszkowski 21K

We also mention that although our interactions determined by a. f i t

to binding energy and density of nuclear matter give correct values for

compressibility, the single-particle energy obtained from them does not

satisfy the HugenholtiS-Van Hove (HV) theorem: Single-particle energy .

at the Fermi surface equals the average binding energy per particle of

nuclear matter. The single-partiele energy W(lO is given by

= U(kF) + u 0 0

-13-

whereA;

Using UCXp) from Bqs.C2O) and (2ll and

Eq.(29), one can easily find that Wtk^) le much less than the average

binding energy per particle 15.5 MeV. This is because in the parametrization

of our interactions the rearrangement effect has not been taken into

account.

In our subsequent communications, ve plan to take this effect into

account and reconstruct the effective interactions to satisfy the EV theorem

end calculate nuclear matter parameters and then finite nuclei properties.

This might give us a better agreement for aT with the experimental values

3ince the rearrangement symmetry part Ae^ of a^ has been found to be an

important correction.

However, apart from the minor disagreements, the agreements attained

for a , and U (kj,) are very much impressive in view of the apparent

sensitivity of these quantities on the nature of the interactions used.

As our results already lend credibility to the theoretical results obtained

by others, we conclude that the disagreements in our investigations with the

experimental values could be narrowed down by a suitable modification of our

effective interaction with due attention paid to the rearrangement effects also.

VI, ALTERNATIVE METHOD TOE CALCULATING SYMMETRY COEFFICIENT

In the calculations presented in Sees.Ill and IV, expressions derived

in Kefs,2 and 3 were used to calculate various nuclear matter properties.

In what follows we calculate symmetry energy, compressibility and rearrangement

energy in a simple and explicit formalism developed by us in this section.

For nuclear matter with neutron excess (NMNE) the symmetry energy is the

coefficient of a in the expression for the average energy per particle

of NHNE:

(1*6)

where the total energy E(ci) is written in terms of kinetic and potential

parts as

E60-

We then have

N+Z

where

<- 0*9)

is the kinetic energy contribution to the symmetry energy.

Corresponding to the long-range and short-range parts of the inter-

VA and VB , respectively, the total potential energy can heactions

written as

•pot(50)

I t is convenient to further consider E,(a) endA

a) aa sums of neutron-5)

neutron, proton-proton and neutron-proton parts, as suggested by Seaeth ,i.e.

(51)

Then the total neutron-neutron, proton-proton and neutron-proton potential

«nergies would be given by

with the superscripts xy standing for nn, pp and np in the respective cases.

The explicit expressions for the terms on the right-hand Bide pf Eq.(5l)

are given by /

S

(-0•S+l

(53)

-15-

and

(-•)

S+T

where the primes on the summation symbols mean that the summations over

momenta are up to the appropriately filled momentum states only.

The evaluations of Eqs.(53) and (5k) yield the folloving expression

for the average potential energy per particle of plane-vave JJMNE due to the

long-range part of the effective interaction

(55)

where the D'B and X's refer, respectively, to the direct and exchange energy

terms. The direct energy term contributicn to E. is

(56)

vhere the first term is the contribution from

(57)

and the coefficient of Aa is the contribution to the symmetry energy

coefficient a- from this part of E .

-16-

The eichange integralB T?B' and X^ can be e-raluated to give

(58)

(59)

The evaluation of the exchange integral A ^ is Eonevhat complicated and.

we finally get

lrVi

where

(60)

(61)

It is to be noted that the evaluation of any of the exchange integrals involves

•integrations over infinite power series in k or k or both.and toe

relative co-ordinate r . In obtaining E4.(60) terms up to fl only were

considered in the expansions of kn and k .

In the same manner the sum (X1111 + x " ) is evaluated aa:

X

where

-X

t62l

(631

The exchange energy contribution to E, is then;

-17-

We now proceed to evaluate the short-range contribution E^{a) of

Eq.(_52). Before we do thia ve Irish to deviate slightly from our assumptions

about V . We had assumed that V£ acts on all states, both even and odd,

and i ts form and Justification was taiea from Bethe's work . However,

on after thought, i t seems that since V Is limited by the delta function

to be of zero range, i t is more reasonable to consider V as acting only

in even states (really in S states). This does not change the results of

Khanna and Barhai (KB) . We simply replace the factor c in Kef.10 by

— and suppose that the short-range repulsion acts only in the six even

states; the final results are then unchanged. However, as will be seen

below, the value of the symmetry coefficient depends In an important vay

on whether V is assumed to act in all states or only in even states. In

fact, the contributions to the symmetry coefficient in the two cases are of

opposite sign.

The fact that the results of KB remain unchanged if V acts only£

in even states can be physically justified as follows.

There is a near zero net force in odd states (Serber force). This2k)

is cited by Bethe as one of the factors which produce-saturation. In

addition we have the fact that the contribution to the potential energyoil) pc )

of all states vitb 1 ^ 1 is relatively small

the case of the interaction of set 1,

How, in

is a pure Serber

force and the odd states give zero contribution to the potential energy

as far as the V part of the interaction is concerned. It is then very

reasonable to consider V,, as acting only is even states too so that it is

only these states which contribute to the repulsion that produces saturation.

In the case of interaction of set 2, the odd states already contribute

repulsively to the potential energy as far as the V^ part of the inter-

action is concerned. If then V^ is considered as acting in odd states too,

the total repulsion in the odd states is increased so that the situation

then deviates even more from that described at the beginnning of this

paragraph. We conclude that in both cases it is eminently reasonable to

consider V», as acting only in even states. However, we shall considera

both the cases, i.e. when VB acts in

i) all states, whan the quantity c remains that of Ref.lO, and

ii) only in even states, when c is to be replaced by —^ .

We shall eoapare the results of the tvo cases, although caseii) is physically

more correct.

-18-

Sinee ve are considering a system ylth •unequal numbers of neutrons

and protons, the argument of the density function in V_ y i l l have to bea

modified in viev of the different Fermi momenta for neutrons and protons.

The logical forms that follow from Eq.(?) are:

V,6 = K

2— K Rp

Sir) > (65)

(66)

(67)

(68)

The relative merits of the choices given in Eqs.(6?) and (68) are given later.

The contributions to ~E^(a) nov become:

7lTT*fc6

e p r >

and when we use Eq.(68) we get

(69)

(76)

C71)

(72)

The usefulness of the weight factors W1"1 , WTP and tf"1* is quite obvious.

In case i) when Vo is assumed to act in all states, we have W1111 « \rp • U115W115 = 8 , K « c

states, ve have

In case ii), vhen

Wpp 1;

V^ is assumed to act only In even

k ; K - % .

-19-

Thus the contribution from the short—range repulsion tn case 1) 18'.

ait rz. Lr h^

"6and la case 11) is:

277T

S

Z7TT

(73)

6 SI7TI t is clear from Eqs.(73) and (71*) that the contributions to s^ are ofopposite sign in the two eases i) and i i ) .

When we use Eg. (72) we obtain

(T5)

(76)

Thus the form of V?p given by Eg,. {68) does not contribute to a^, when theahort-range repulsion is assumed to act only in even s tates . This in a wayargues against i t s choice.

Summing up the various contributions to the total energy E of oursystem with neutron excess, we have for the average per part icle, Eq.Ci6),from where we can write s , in three parts

a T (77)

where for

= 12-72 Mtv- (78)

-20-

22.93 MeY for interaettmi of set 3.

S9-92 MeV for interaction of set 2,

2 7 TT*

T

= 0

(79)

(80)

(81)

(82)

(83)

3The values for different B , and the corresponding a™*s for, -two sets ofinteractions are given in Table I .

The results inTsble I when compared with those obtained in See.Vindicate that a_ may l ie between 1*2-1*9 MeV. Wltb a' slight difference inmagnitude, the results obtained by tvo different methods are more or lesssimilar ia magnitude. But the value of Bj = 32.96 MeV in Table I comparesvery veil with the values obtained earlier 3>'"'ts1>m

Two points need special mention with reference to the results obtainedin this section. First , for both interactions a^ ia lower in case i i )than in case i ) , and closer to the empirical values. This supports theassumption of VB acting only in even states in further calculations.Second, the values of a_ obtained with the interaction of set 2 in thetwo cases are higher than those obtained with the interaction of set 1.

This is easily explained by an examination of Eq.(79) for s , . TheB

corresponding increase in a_even should also be specially noted, as ease I i )2)is the physically more correct one. Brueckner and Gammel , however, obtained

the opposite effect. Considering a potential without odd-state interactions,they obtained a value for SL, = 1*3 MeV which was higher than that obtainedby considering a potential with odd-state interactions (£6 MeV). Theyattributed the decrease to the inclusion of odd-6tate interactions, but inthe light of our results this explanation should be re-examined.

As an exercise,we calculated a^ with the interaction of Bet 1 andcase i i ) for various values of kj, , and plotted e{a) against Itj, fordifferent a . The binding energy curves thus obtained are similar tothose obtained by Brueckner, Coon and Dabrowski , except that the

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of war exarvei ahift consi stoutly towards, lower densities as a is increasedbeyond Q.fc, up to vfaich. value the shift is negligible. At a = 0-5 there is avery small shift (Q.Q1 tsT ) , while for a = 0.6 and more the minima suddenlyshifts quite rapidly. ITcv for U , a - 0.23 - I t is quite remarkable

that HMHE should saturate at the same density as HM up to more than tvice themaximum natural value of a , which safely oovers the range of superheavy nucleithat are terres t r ia l ly attainable or might be produced in astronomicalprocesses.

VII. COMPRESSIBILITY OF HMNE

At saturation we define the compressibility K(a) of BMHE by

vhere

and,up to a « 0.5 t this can he safely computed at

2a-(77) ve can vrite

(8U)

(85)

m

r )T

vhere

K

1.35 fm . From

(86)

(87)

(88)

-32-

6.i

the parameters of the interaction of set 1, ve get

K = 68- %\

How

so that

and

(89)

(90)

(91)

(92)

(93)

This latter result Bhows that even if the neutron excess is increased till

N * 3Z , the compressiDility of the system is increased by about 10J of

the nuclear matter value.

EEAERAHOEHEIIT ENERGY FOR BMffi

The rearrangement energy is given by

and it cas he evaluated to give

-ai-

(95)

(96")

If we put a • 0 , we get the rearrangement energy for nuclear matter 11)

Values of TL, fOT different values of a are given in Table I I . Uc

goes on increasing with a whereas u * v e n g o e s ou decreasing.

is always more than Up(a = 0) while U^611 is always less than(a » 0) . The increase in U^ with a is faster than the decrease

in ufVen with a . But the values of u^ven seem to be more reasonable2) 29)since they l i e between the 12-13 HeV value obtained by Brueekner et a l . *

We feel that Up should be less than the corresponding NM value for finitevalues of a < 1 . Such a resis t is given by U,, only. This furthersupportB the iaea that Vg should act only in even states. The correctnessof the alternative method (Sec.VI} for calculating a? is established by theexcellent agreement we get for U- with the XL values obtained byBruectaier et al . " > < = ; "

ACKNOWLEDGMENTS

The author would like to thank Professor Abaus Salam, theInternational Atomic Energy Agency and UHESCO for hospitality at theInternational Centre for Theoretical Physics, Trieste. Financial assistancefrom the Atomic Energy Commission, Bombay, India, is gratefully acknowledged.

-2k-

KEFERENCES

1) See, for instance, the ejrtenslve review by B. Day, Ber. Hoa. Phys.32, Ti9 (196T).

2) K.A. Brueckner and J .L . Gajmnel, Phys. ReT. 109, 1023 (1958).

3) K.A. Brueckner and J . Dabrovski, Fhys. Rev. 13U. B722 (1961»).

•i) K.A. Bruec icne r , G.L. Gammel and J . T , E u b i s , P h y s . EeT. l l f l , IO95

( I960) .

5) J . Nemeth.in Theory of Nuclear Structure (IAEA, Vienna 1970), P.9I1I.

6) D.S. Falk ana L. V i l e t s , Phys. Eev. 12lt_, 188T (196l ) .

7) J . Gammel and E.H. Thaler , Phys. Eev. lOJ, 291 (1957).

8) J . Garamel and E.H. Thaler , Phys. Rev. 10J_, 1337 (1957).

9) O.E. Brown, G.C. Schappert and C.¥. Wong, Hucl. Phys. ^6., 191 (196!*).

10) K.M. Khanna ana P.K. Barhai , Nucl. Phys. A215. 31*9 (1973).

11) K.M. Khanna and P.K. Barhai, Hucl. Phys.( in Press} .

12) K.M. Khanna and P.K. Barhai, Phys. Rev. CI1. 26k (1975).

13) V- Vautherin. in Theory of Nuclear Structure (IAEA. Vienna 1970),

p.767.

lU) D.W.L. Sprung and P.K. Banerjee, Hucl. Phys. A168. 273 (1971).

15) A.E.S. Green, Rev. Mod. Phys. £0, 569 (1958); Phys. Rev. 9_5_,

1006 (1951*).

16) A.G.W. Cameron. Can. J . Phys. £2> 1 ° 2 1 (1957).

17) J . Dabrovski,in Theory of Nuclear St ructure (IAEA, Vienna 1970),

p .131 .

18} A.M. Lane, Hucl. Fhys. 25_, 676 (1962).

19) P.E. Hodgson, Phys. Letters 2, 352 (1963).

20) J. Dabrowski, Phys. Letters £, 90 (19&*).

21) S.A. Moszkowski, Phya. Rev. 2,, 1(02 (1970).

22) H.M. Hugenholtz and L. Van Hove, Physic* 2 ^ 363 (1958).

23) H.A. Bethe, Phys. Hev. 167. 879 (1968).

2k) H.A. Bethe, International Nuclear Fhyaics Conference- Gatlingberg,

Tennessee (Academic Press, Hew York 1966), p.625.

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25) D . W J . Sprung, P.C. Bhargav& ana T.K. DaWblom, Phys. Le t te r s 2^ , 538

(1966).

26) J . Dabrewski ana P. Haensel, Phye. Bev. C7_, 9 l 6 (1973).

27) P . J . Siemesn, Hud. Phys. AIM, 225 (1970).

28) K.A. Bmeckner, S.A. Coon ana J . Dabrwski , . Phys. Rev. 1§8, 118t

(1968).

29) K.A. Bmeckner, J .L . Gaamel and J .T . Kubis, Phys. Rev. 1X8, 11*38 ( i960) .

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TABLE I

Interaction ofset 1

Interaction ofset 2

a£(SfeV)

^ ( M e V )

^(MeV)

ajCMeV)

V- :Case ( i )

8.25

1*3.96

U.96

1*7.66

Case (11)

-2.75

32.96

-1.65

1*1-05

case (1)

10.31

1*6.02

6.20

l«8.9O

case (11)

0

35.71

0

1*2.70

Variation of a^ vith

TABLE II

a

0.15

0.20

0.25

0.50

U^Ven(MeV)

12.53

12.53

12.1*7

12.20

uf^MeV)

12.70

12.83

12.91

lit.01*

Variation of UD with a ; U-(a = 0) = 12-57 MeV.

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