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/ 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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