/ N 5 & • -' '?•, \ -£\ IC/71/130*:.•>• . .. , \ M INTERNAL REPORT?• : • '• • ' ; • ' ! 1, ( L i m i t e d d i s t r i b u t i o n )

.A.

1 International Atomic Energy Agency

andUnited Nations Educational Soientific and Cultural Organization

INTEHNATIONAL CENTRE FOR THEORETICAL PHYSICS

NUCLEON-DEUTERON SCATTERING*

T.H

M.A.

. Eihan

and

Sharaf

ABSTHACT

The nucleon-deuteron scattering amplitude ia investigated on

the basis of Feynman-diagram technique. A superposition of an exchange

pole graph and triangle graphs is considered. With the aid of some

reasonable approximations, the triangle amplitude is given in terms of

nuoleon-nucleon amplitude. Angular distributions up to 155 MeV proton

energy are calculated for p-d scattering using phase-shift analyses for

the nuoleon-nucleon amplitudes. Excellent agreement with the experi-

mental data up to 180° is obtained.

MIRAMARE - TRIESTE

October 1971

* To be submitted for publication.

** On leave of absence from Physios Department, Faculty of. Education,University of Libya, Tripoli, Libya.

*** Permanent address : Faculty of Science, Cairo University,Cairo, Egypt.

1. INTRODUCTION

Most of the investigations on the high-energy (l GeV and more)

nucleon-deuteron scattering are "based on the Glauber multiple scattering

theory , where the scattering amplitude ia considered as a simple super-

position of single- and double-scattering amplitudes of the incident

nucleon from the (quasi—free) target nuoleons. Usually, the internal

motion of the target nucleona is neglected in this approach and a" simple

(high-energy) pararaetrization of the nucleon-nucleon amplitude is adopted.

This picture of scattering appears to be quite reasonable for small mo-

mentum transfers. When the momentum transfer becomes large (or the

energy of the incident particle becomes less) the preceding picture of

the scattering process fails for, in this case, several effects that were

relatively small can no longer be neglected. These effects may include

the internal motion of the target nucleons, the multiple scattering

corrections, the parametrization of the nucleon-nucleon amplitude at

energies of interest and essentially the exchange backward scattering.2) 3)

The latter effect was the subject of many investigations ' and seems

to have an appreciable effect even at high energies. In the energy

range up to 700 MeV, there seems to be a pronounced backward rise in the

angular distribution of p-d scattering. The Glauber theory as well as

the simple impulse approximation (even with the inclusion of the off-

energy shell effects in the nucleon-nucleon amplitude) gives satis-

factory resultsk¥n the forward direction. Although the recently de-5) 6)veloped dispersion relation theory ' gives satisfactory results, it is

laborious as one needs to solve N/D equations. In this work, the

proton-deuteron scattering is investigated on the basis of the disper-

sion theory for nuclear reactions . An exchange pole diagram plus*)

two. triangle diagrams (see Pig.l) seem to be sufficient for the descrip-

tion of our process. Using a simple one-particle model for the 3_

The relative importance of the Feynman graphs in calculating the

amplitude depends oritioally on the location of their respective singu-

larities from the physical region. The singularity of our pole graph

lies at 4.46 MeV amu while that of triangle graphs is eight times

further. Therefore the contribution of other complicated graphs may safely

be neglected in this model.

-2-

ray vertices, the triangle amplitude (after some reasonable approxima-

tions) is shown to be faotorized into rrucleon-nuoleon amplitude times

the deuteron foTm-factor. We take, for the nucleon-nucleon amplitude,

the phase-shift analysis in the form given by Stapp ' for p-n scatter-

ing and the appropriate form for the n-n scattering. The phase-shifts

are taken from fief.9- Angular distributions for p-d scattering at

proton energies up to 155 MeV are then calculated.

In Sec.I we give the derivation of the p-d amplitude, while

in Seo.IEthe method of calculation is indicated and a discussion of

the results obtained is given.

II. THE SCATTERING AMPLITUDE

The most important Feynman graphs contributing to the p-d

scattering process are those shown in Fig.l.. Now, according to the

Feynman diagram rules, the exchange pole amplitude (Fig. la) has the form

/q = - ^ p v \i\p(%) tliF<<P/cr+X'lf) (i)

where q = -k^ + k , q1 =• k. + -^kf, k f k are the initial and final

proton wave vectors, m* is the reduced mass of the neutron and thenp

proton, M. is the binding energy wave number of the deuteron. V is anp K

normalization constant and M.. is the vertex amplitude whioh, in the*#) -1 J

one-particle model , may be given by:

where s , ^ are the spin of particle i and its z-projection, while Z is

the relative orbital momentum of the pair of particles i and j. The

reduced vertex part ys-« is related to the spectroscopic factor and $* (q)

is the Fourier transform of the bound state wave function. Further, the

triangle graph amplitude may be given in the same notation (e.g. for the

triangle diagram of Fig.lb) by:

*)Here we use the system of units in whioh n - c • 1.

#*)' Such a definition is in accordance with the corresponding Born term inthe full scattering amplitude.

-3-

Here MTP is the 4-ray vertex amplitude which may be expressed in termsp 10)

of the off-energy-shell nucleon-nuoleon scattering amplitude as

where f. .(p, ,p«)$z

(4)

is the nuoleon-nucleon scattering ampli-

tude taking into aocount the possibility of spin-flip.1, , 1, , 1,

, 2

z - •

2mpd13

32m. + m2)

and k,, k2>k, are the momentum of particles 1, 2 and 3 while E , E 2 and

E., are their corresponding energies, respectively. We thus see that in

Eq.(3) the integration over E.. can easily be performed. By using the

residue method of the same trivial kinematical relations, one obtains:

-\

So that, making use of a relation similar to Eq..(2),the amplitude

may be given by

(6)

where s is the total spin of the two protons.

& " Ei + h * it "ifef - 7^ +4is4m

-4-

m is the nuoleon mass while A • (k, - k^) is the momentum transfer;»v . »"1 *^f

$(q.) is the Fourier transform of the bound deuteron wave function.

Now, from E<j.(6), one can see that the calculation of the triangle

amplitude leads to the calculation of the off-energy-shell nucleon-

nucleon amplitude. However, the calculations will simplify greatly

if one notices that the product |^>(k) $(k-x)j reaches its maximum

value for a given value of the momentum k in the direction of the

vector x. ' Henceforth, the integral in Eq..(6) will be evaluated

under the assumption that the principal contribution to the scattering

occurs for those values of k where k = - x (in our case x = A ).

Further, in this case (i.e. with k « - A ) the energy S can be approxi-

mated without muoh error by P. /2m(P. » - k. + •& A), which means that

our nuoleon-nucleon amplitude can be fairly well approximated by its

on-energy-shell version. With these simplifications, the triangle

amplitude (6) may finally be given by

<S C{ A) £7, (ifrffn /I//) C {/P'iy^ I '/'V;

(7)

v

25 it - fa ~ H and 2 - £

while S(x) - |^(k) i(k-x)dk .

Similarly^for the triangle graph of Pig.lc we have:

f-Mp*> (8)and for the exchange pole graph

VNow, in actual calculation of the differential cross-section of the

p-d scattering, the nucleon-nuoleon amplitudes appearing in Bq,s.(7) and

(8) must be multiplied by the faotor

-5-

which aocounts for the difference between the cm. of the p + deuteron

and that of the proton + proton (and/or neutron) cm. systems.

It should also be noted that the nucleon-nucleon amplitudeslab

must be calculated at energies T.. which differ from the incident

(lab.) energy T according to the relation

(ID

Consequently, the scattering amplitude for the p-d scattering may

finally be given by

(12)

III. METHOD OP CALCULATION

It was shown in expressions (7) and (8) that the triangle

amplitudes are given through the nucleon-nucleon scattering amplitude

f , The calculation of such amplitudes is,in general, difficult ands

depends on the suitable choice of the nucleon—nuoleon interaction. The

most general forms of such potentials must include spin-dependent inter-

actions and tensor foroes plus hard core. However, most of these

potentials reproduce the correct phase-shifts only at certain ranges

of energy. Thus the use of a special model for nucleon-nucleon inter-

action will be avoided, and only phase-shift analyses will be considered.

For this purpose one notes that the nucleon-nucleon amplitude must be

considered as^4 x 4 spin matrix, and hence will be written in terms

of two Pauli-spin matrices or. and cr* for the interacting particles.

The most general symmetric form for the nucleon-nucleon amplitude may

be given by :

f %,n,, w - ft A° -+ c i (m

+ £«?•£) to i)] A1

where q •• k. - k«; p = k. + k_} n » k. X k. (the hat over the vector

means unit vector). A »A are the singlet and triplet projection

operators, respectively. The coefficients A', B, C, D and E may be

expressed in terms of the eigenphase shifts and mixing parameters.

-6-

10)Such a representation has been given by many authors ; however, it will

representations 8) , v

be more convenient to use the ^ derived by Stapp . Expression (13; is

adopted in Eq.(4)> where the total spin S of the two nucleons takes the

values 0, 1 and, accordingly, the projections^ and ft' take the values 0,

-1, 0, 1. Considering all these terms for both A?p and A?n in Eq.(l2),

one finally finds the differential cross-section for p-d scattering from

the equation:

IV. COMPARISON WITH THE EXPERIMENTAL EESULTS AND DISCUSSION

Eq.(l4) is applied to the proton-deuteron scattering1 for differ-

ent • energies up to 155 MeV. The spatial deuteron wave function isX the _o

taken to be of g'aussian form e with^extension parameter o* = I.l8fm

The contribution of the eigenphase shifts to the scattering amplitude

for both p-p and p-n scattering is considered up to the H-

state. The numerical values as well as the mixing parameters for thesephase shifts are taken from the analysis of the experimental data

q)of the nucleon-nucleon scattering of MacGregor et al.

In our numerical calculation the variation of these phase shifts, with

energy which changes with the scattering angle 0(Z * pi /2m)^is taken into

account.

Agreement with the experimental data is good for both the

angular distribution and the magnitude of the cross-section(Pig.^.

The present method gives a clear physical picture for the

nuclson-deuteron scattering without a complicated computation. Further,

it gives an agreement with the experimental results similar to that ofagreement

the complicated N/D method and even better^for relatively low energies.

ACKNOWLEDGMENTS

The authors would like to thank Professors Abdus Salam and

P. Budini, as well as the International Atomic Energy Agency and UNESCO,

for kind hospitality at the International Centre for Theoretical Physics,

Trieste.

-7-

REFERENCES

1) V. Franco and R.J. Glauber, Phys. flev. 142, 1195 (1966).

2) H. Kattler and K.L. Kowalsky, Phys. Rev. JL38_, B619 (1965).

3) K.L. Kowalski, Nuovo Cimento _30, 266 (1963).

4) K.L. Kowalski and D. Feldman, Phys. Rev. 1J0, 276 (1963).

5) Y. Avishai, W. Ebenhoh and A.S. Rinat-Reiner, Ann. Phys. (NY).

21, 341 (1969).

6) W. Ebenhoh, A.S. Rinat-Reiner and Y. Avishai (to be published).

7) I.S. Shapiro, in Proceedings of the Varenna Summer School,

Lecture 38,p21O (1966);

L. Taffara and V. Vanzani, Nuovo Cimento ^2_, 570 (1967).

8) H.P. Stapp, T.J. Ypsilantis and N. Metropolis, Phys. Hev. 105,

302 (1957).

9) M.H.KacOregor, R.A. Arndt and R.M. Wright, Phys. Rev. l82_, YJlk (1969).

10) H.L. Goldberger and K.M. Watson in "Collision Theory" (John Wiley

and Sons, Hew York 1964) p.384.

FIGURE CAPTIONS

Fig.l Most important Feynman diagrams contributing to the p-d

scattering process.

Differential oross-section for p-d scattering1 at

incident energies indicated in the figure. The

experimental data are taken from Ref. 6 , and the

theoretical curves are calculated from Eq.(l4).

-8-

P1

(b)

(c)

Fig. 1

- 3 -

100

u(0

100

100

30 60 90

Fig. 2

120 150 180

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