,...^i^a|

REFERENCEic/75/26

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

WHERE TO LOOK FOB LAEGE OFF-SHELL EFFECTS OF

THE MJCLEON-MJCLEOTC INTERACTION?

0 . Zohn i

1975 MIRAMARE-TRIESTE

It. Ir

\*

itir

) r

IC/75/26

I n t e r n a t i o n a l Atonic Energy Agency

and

United Nations Educational S c i e n t i f i c and Cul tura l Organization

INTERNATIONAL CEHTEE FOR THEORETICAL PHYSICS

WHERE TO LOOK FOE LARGE OFF-SHELL EFFECTS OF

THE NUCLEON-NUCLEON INTERACTION? *

0. Zohni **

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

Off-shell variation over a wide range of energies and momenta

is examined for a model nonlocal short-range nucleon-nucleon repulsion.

The results point out to large off-shell effects in regions of phase

space which characterize three-nucleort scattering processes,particularly

in the break-up region, and nuclear matter binding-energy calculations.

MIHAMARE - TRIESTE

March 1975

* To be submitted for publication.

** Present addresss Institut fur Theoretiscbe Phyaifc der Univarsitit

Frankfurt/Main, Fed. Rep. Germany.

- 2 -

I. Introduction

The" study of the off-shell behaviour of the nucleon-nucleon ("N)

interaction has recently aroused much interest (Sprung and Srivastava

1974). Particularly important in this respect is to look for the

phenomena and regions of phase space in which new information about

the short-range nuclear interaction is possibly obtainable (Erayshaw 1974;

Haftel and Peteraen 1974). The investigation of potentials havinp

different shapes for the repulsive core is especially needed to srain

such knowledge about the NN off—shell effects.

In the present work we aim at exploring the energy-momentum

dependence of the off-shell behaviour associated with a class of potentials

having a norri.ocal-square-well-core (NLSWC) repulsion which has proved

to allow large flexibility in varying the shape of the repulsive core,yet

satisfying experimental data pertinent to the NN system (Zohni 1973a),

The off-shell t matrix associated with this interaction was given

analytically and shown to provide a generalization over the hard-core (KC)

and the finite local-square-well-core (ISWC) interactions as well as

the separable one (Zohni 1975b).

The two-body interaction is discussed in Section II. In Sections

III and IV, we examine how far variations in the "core" parameters

affect the off—shell behaviour over a wide range of energies and

momenta and a discussion is presented of the implications this might

have in applications to nuclear systems. Conclusions are given

in Section V.

I I . The two-body interaction

We consider a family of NLSWC s-wave two-body interactions of

the form (Zohni,1975a,b) :r sinh ,5 ( r ' - r 0 ) f o r O ^ r ^2 U S

— °2* sinhysr

V0(r.r') = . ^ _

70(r,r') = fy

(r-rQ) sinner' for 0$ r'<^r£

for TQ£T^T

for rSr,

- 3 - - 4 -

where r 0 is the range of the repulsive NLSWC, B its strength (1

P is its nonioeality parameter and r and U are the range and depth

of the outer attractive 1SW,respectively.

A family of on—the—energy-shell equivalent potentials of this

form were determined (Zohni 1973a) by matching the low-energy NN data

and phase shifts up to 340 KeV laboratory energy . We quote in Table I

four of these potentials^Labellod as P1,P2,P5 and P4) which are employed

in the present calculations to study the off—shell behaviour. It should

be noticed that the potentials considered preserve the outer shape

vhile being different only in the core region which mainly contributes

to the off—shell behaviour.

The off—shell variation is calculated with an analytic off—shell

t matrix associated with the potential £2.1] (Zohni 1975b). Thie t matrix

is of particular interest in view of its explicit analytic structure

and of the NLSWC interaction used. Expressions pertinent to our present

calculations are given in Appendix E.

III. Off—shell behaviour at negative energies

The off-shell elements to(k',k;q ) are shown in Figs. 1 and 2

for two negative values of the energy parameter, namely q = -0,2 and

—2.2 fm" , respectively. The values chosen are typical of triton and

nuclear matter binding energy calculations. In both figures the curves ere

labelled as the corresponding potentials. For the sake of comparison,

the off—shell elements corresponding to an almost on—shell equivalent

HC square (Kim and Tubis 1970a,b) are also shown. Further,the results

are selected and given for the initial momentum k in the range 0 to 7

and a single value Of the final momentum k' Other values axe available

but do not give more information and lead to a trend similar to that

shown (Zohni 1971).

Tor both energies the off-shell elements P1,P2,P3 and P4 show

that the HLSWC potentials are less repulsive than the HC square well,

as indicated by the depths of the respective minima. Further, all such

minima are noticed to occur at k«»%5 fin" , illustrating that the

repulsion in these potentials is concentrated to radial distances of

the order of 0.5 fm,

<The low-energy HN parameters and s-wave phase shifts were calculatedby a standard procedure (Wu and 0hriura,1962) and are given in Appendix Afor convenience.

-1

At a2= -0.2 fm"2 (typical of bound triton calculations), the

NLSWC potentials show about 15% off-shell differences in the momentum

range O £ k $ 2 fm"1. The differences get larger (— 5<M } in the

range 2 ^ i ^ 5 fm"1 and become more pronounced ( »- an ordex of

magnitude) for larger k values.

.The relevant range of momenta dominating- the bound three-nucleon

system was shown (Lavine 1971;Kharchenko 197?) to lie between 0 and 2

Accordingly, the small off-shell differences observed \Ln Fig. 1 in this

range would indicate that the three-nucleon binding energy is slightly

affected by variations in the shape of the NK repulsive core.

Indeed, exact calculations {Zohni 197?>a) employing the present

potentials have demonstrated the insensltivity of the triton binding

energy to the associated off-shell differences. This result agrees also

with other authors (Bahethi and Fuda 1973jKharchenko et al, 1975;

Lavine and Stephenson,Jr. 1974) who used different KM repulsive shapes,

indicating the model-independence of the conclusion drawn.

? —2At q = -2.2 fm , which is typical of nuclear matter binding-energy

calculations, a similar off-shell behaviour is observed (see Pig, 2).

Thus, the off-shell differences are smallest ( ~ 4% ) in the range

fir.

2 fm , become larger {•»» J5^ ) in the range 2 .£ k 6 fm

more pronounced (~ an order of magnitude) for larger k values.

-1 and

Nuclear matter calculations are actually known (Sprung and

Srivastava 1969; Haftel and Tabakin 1971) to be sensitive to momenta as

large as 6 fm . This shows that far off-shell elements play a significant

role in nuclear matter calculations.

The results in Pig. 2 indicate that large off-shell differences

are associates with far off-shell elements. We conclude that nuclear

matter calculations would then be more sensitive to variations in the

shape of the short-range MN repulsion, and therefore deserve nore

considerations in this concern (see,e.(T. Coester et al. 1970;

Wong an~ Sawada 1972).

- 5 -

TV. Off—shell behaviour at positive energies

Three values of the energy parameter were arbitrarily selected

to represent the off—shell behaviour at positive energies, namely

q2= 0.01, 0.2 and 1.0 fm~3 (correspondins to 1, ZO and 100 Me? lab.

energy for the HK system,respectively). These values cover the energy

range arouna the break-Tip region as well as higher energies for the

nucleon-deuteron ( M ) scattering.

The calculated off-shell elements are shovn in Pigs. 3-5 for

both the real ani imaginary parts. The following remarks could be

made from the results obtained:

(i) Considering their momentum dependence, the off-shell elements

k <[ 2 fm~ at the

(ii)

show large differences in the interval

energies considered.

When looked at as a. function of energy, the off-shell differences

in the interval 0 4- k ^ 2 f^"1 are relatively larger for

values of the energy parameter around that in Fig. 4 . This

corresponds to a nueleon of about 10 KeV centre-of—mass energy

incident on a denteron target, i.e., an energy around the

deuteron break-up region. The off-shell differences are relatively

smaller for higher energies as well as for energies below the

deuteron break-up threshold.

(iii) The HC off-shell behaviour is significantly different from that

of the HLSWC, particularly in the interval O ^ k ^ 2 fnT1.

However, this difference becomes more pronounced at higher energies

(notice a difference in sign in Pig. 5 between the HC and the

NLSWC off—shell elements), as iB expected from a general consideration

of the HC effects.

Considering the Nd scattering processes, it is known (Aaron and

Amado 1966; Cahill and Sloan 1971) that low momenta up to s 3 fm~ are

relevant for the Nd break-up processes. Further, the doublet and quartet

lengths in Nd-ecatterinr have been reported (Kharehenko 1975) to be

sensitive to momenta around 1.8B fm~ .

>Off—shell elements,were calculated at other values of the energyparameter around q = 0*2 fm which iB shown here as an example ofthe trend found.

- 6 -

Bearing that in mind, our results imply therefore that three-

nucleon scattering calculations are quite sensitive to off-shell

variations arising froti different shapes of the short-ranre KN repulsion,

since large off-shell differences mainly show up in that momentum

interval relevant for these calculations. Tn particular, and as implied

•fay the energy dependence of the off-shell behaviour obtained, calculations

in the break-up region would be particularly useful in this respect,

since large off—shell differences are observed to occur there. Indications

to conclusions of a similar type were recently shown by other

authors (Kloet and Tjon 1975; Haftel and Petersen 1971). The study of

the off-shell effects in deuteron break-up deserve much consideration

and should aleo be done using different types of interactions, so as

to avoid model—dependent conclusions. Investigations In

this direction are now in progress by the author.

V, Conclusions

We studied wide variations in the off—shell behaviour produced

by changing the parameters of a model KLSW repulsive core, while

preserving the same outside 3STJJ attraction. The NT) interaction used is

interesting in view of the well separated short-range repulsion for

r £0.5 fm.

The implications of the off-shell behaviour obtained in actual

threei-nucleon and nuclear natter calculations depend on which monenta

predominate at a given energy. The results are indicative of large

off-shell effects in nuclear matter binding-energy calculations and in

three-nucleon scattering processes in the break-up region.

ACKNOWLEDGMEHTS

The author is grateful to Professor Abdus Salam, the International

Atomic Energy Arency and UNESCO for hospitality at the International

Centre for Theoretical Physics, Trieste.

- 7 -

Appendix Ai Expressions for the NM parameters

- 8 -

The scattering length a 0 ana the effective range rg defined by

the relation:

Here we give the expressions obtained for the scattering: length a ,

the effective ranfre rg and the e-vave phase shift S corresponding to

the IJLSWC potential in [2,l"l . Defining

2

L(d

V cosh l> r Q - f cosh t> r—-} ^ = ; ,

V sinh >> r 0 - tf sinh V To

the two-nucleon bound-state energy is determined by the roots of the

following transcendental equation:

tA.2}L(-«2) cot*, (r - r )

= —-1 1 2_Lf-ft; ) + * (r - r )

where Efl= -ti ot /2 is the two-Tjoay binding enercy, 0 , = (U - 2 ) J ' 2

!?2= ~K 2 in t*-1!-

Further, the exact e-wave phase shift for the potential t2'

is given hy:

and

can easily be shown on using [k.'i] to have the expression:

(-TV^cot (-Vx)l/2 (r - rQ) + L(0)

U 6 3 ^ ^ i-^)1?2 L(0) cot (-U^ - r0) + U,

sinh

with C*o = l/a0 and

Einh sinh sinh

tA.5]

where

[A.4]

- 7 tan

0 + q tan qrn

cot L(q?)

A.l<tf»

Di)i/2ri + *°

2 ^ 1 / 2

sinh r - sinh

-7 -

Appendix B: Expressions for to(kj kj <i )

The off-shell t matrix associated with the NLSW-core potential in

[.2.1} can be analytically expressed as:

" 1j k; q ) = e + D"1 (a

vhere:

LB.2]I."1-

tB.53A ok'k

[ sin k ' r 0X |^— - — ( COsbfr0 sin kr0 - k sinhfr0 cos krQ)

sinh Pr0 f °™ ("'- ^)^o

I L k ' - k

A ^ f Bin ( k ' - k ) ^ sin (k'+ k)r.

2k'k !• k ' - k k'+ 1c

sin (k'+ k)rQ

k'+ k

sin (k'~ k ) r 0 sin (k'

k'-ik k'+ k

[B.4Jsin k'r_

sinh

coBh<Ti_ sinhBr.)u 1 0

^ n JJ

k' + p '>+r0 sin k'r0 - k'Einh

cosh i To sin k'r0 - k'coeb 3OE k^T^-.) T

-10 -

IE.53i U,

sin k ' ^ - k'cos k'r.)

- e -1 ° (iC<1 sin k'rQ - k'cos k'rQ) ,

i U,

- i « r

sin k'rx - k'cos k'

sin k'r_ — k'cos t ' r 0 ) ,

IB.81 B+= » 0 (p , r ) (Ax- l )X 0(k« + )

A j - AQ)

LB.9] B~- - U 0

-11 -

+ +

In the above equations, }~ are given as in [A. l l i J0(kr) and hp( 0( r) are

Bessel and Hankel functions of the f i r s t and Eecend type,respectively.

Further, uQ( p,r)=sinhi>+r + BinhiTr and the factors X- 'B are given by

expressions of the form:

LE.IO]

where ~b^ denotes the differential d/dr| . The factors &1 and Of are

1 2 1 / 2 ^given throughout by « \= (q - t^) ' , 1*1,2. Also the coeffieients kQ

and A are given by:

[B.11Jk4

+ k2( B2 - q2) - I

k 2 - n,

Bef erer.ces

Aaron,E. and Amado.R.B. I966, PhyE.Hev.

Bahethi.O.P. and Fuda.K.G. 1972. Phys.Hev. C6.1956.1 WR 18^51973-

Cahill,E.T. and Sloan,I.H. 1971. Nuel.Phys. ftlgc,l6l.

Coester,T.,Cohen,S.,Day,B, and Vincent,C.ft. 1970. Phys.Hev. 01,7^9.

Haftel,i : .I . and Petersen.K.L. 1974. Phys.Rev.Lett. ^J_,1229.

Haftel.K.I. and Tabakin,r. 1971. Phys.Rev. £2,921.

Kharchenko,V.F.,Sha.dchin,S.A. and StoTozhenkotS.A. 1971. Phye.Lett. 57B.1

Kharehenko.V.F. 1975. Sov.J.Nucl.Phys. 16,173.

Kim.Y.E. and Tubis.A. 1970a. Phys.Rev. C1.414.1970b. Phys.Rev. C1.1627.

Kloet.W.K. and Tjon.J.A. 1973- Hucl.Fhys. A210.58O.

LaTine, J .P. 1971. "Theoretical studies in the three-nucleon system",Technical Report No.72-028, University of Maryland.

lavine.J .P. and Stephenson.G.J.,Jr. 1974. Phys.Rev. ££,2095.

Sprunr.D.W.L. and Srivastava.M.K. 1969. Mucl.Phys. A1^9,6O5.— 1974. Adv, in Nuel.Phys.

Vonr.C.W. and Sa.w&da,T. 1972. Ann.Phys.(K.Y.) 12,107.

Wu,T.y, and Ohmura,T. 19^9- Quantum Theory of Scattering? {Prentice-Hall,Mew Jersey) .

Zohni, 0, 1971. "Off-energy-she11 t matrix for non-local-core potentials",ICTP, Trieste, preprint IC/7l/59.

1973a. J. Math. Phys. 14_, 20J.19731J. Pliys. Rev. C6, 1164.

- 13 -

Table I. Parameters of the NLSW-core HU potentials.

Potential

PI

P2

P3

P4

(r8)

0.51

0.44

0.38

0.34

<M8V)

248.82

829-40

4105.5

2571140

: f 6

0.18

0.10

0.04

0.001

(fm)

1.68

1.68

1.68

1.66

(KeV)

-65.5

-63-5

-63.5

-63.5

FIGURE CAPTIONS

2 —2Figure 1. Off-shell t matrix elements at q. = -0.2 fm as computed

from liq. [B.l]. The solid curve (lie) corresponds to a

HCSV potential with a core radius of 0.4 fm and an outside

attraction similar to our HLSW-core potentials, namely an

outer radius of 1.757 fa and a depth of -63.85 MeT (Kim and Tubis

2 _2 )2. Off-shell t matrix elements at q = —2.2 fm computed from

Eq. [B.l]. The solid curve HC has the same meaning as in

Fig. 1.

Figure 5. Eea} and imaginary t matrix eleffients at q = 0.01 fai" (i.e.

1 KeV lab. energy)- ?he solid curves have the same meaning

as in Fig. 1.

? —?Fignre 4. Real and imaginary t matrix elements at q ™ 0.2 fm (i.e.

20 HeV lab. energy)• The solid curves have the same meaning

as in Pig. 1.

Figure 5. Heal and imaginary t matrix elements at q = 1 fm (i.e.

100 KeV lab. energy)- The solid curves haw the same meaning

as in Fig. 1,

-14-

Fir.l

-15-

Fie.2

-16-

I '. I !» 1I M i

n

Fif.3

t I

-17-

111!i ! J !1111

Oo

Cta

>• Qr O- O-

1111I i i !I i * i

l i

™ "

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