VOLUME 30, NUMBER 4
PHYSICAL REVIEW C
Closed form
S matrix
OCTOBER 1984
of matter distributions and nucleon-nucleon
interaction for heavy ion scattering
in terms
Department
Y. K. Gambhir
of Physics, Indian Institute of Technology,
Bombay-400 076, India
C. S. Shastry
Physics Group, Birla Institute of Technology and Science,
Pilani (Rajasthan) 33303-1, India
(Received 20 March 1984)
We derive an approximate analytical expression for the S matrix in terms of the parameters of the nuclear matter distributions and nucleon-nucleon interaction in the framework of folding model for heavy ion
scattering. The numerical calculations carried out for
Ni scattering {E]»=60 MeV), a test case,
agree well with the corresponding results of the phenomenological optical model.
0+
Since last decade a great deal of activity in nuclear physics
has centered around heavy-ion scattering (HIS) both in
theory and experiment. The HIS data are usually analyzed
either in the conventional phenomenological
optical model
(OM) or in the parametrized S matrix (PSM) (also known
as closed formalism approach). The parameters of the optical potential are deduced by requiring the best fit to the observed cross sections using quantum mechanical or &KB
method. In the PSM, the 5 matrix itself is parametrized in
terms of several parameters so as to reproduce the scattering data and the associated physical features. In the OM
several attempts have since been made to calculate microscopically the optical potential for HIS starting from the
nucleon-nucleon
interaction. The most successful attempt
in this direction is in terms of the folding model. The folding model discussed by Satchler and Love, ' in spite of its
simplicity, has been remarkably successful in accounting for
majority of the observed HIS cross section data covering different projectiles incident on a variety of targets, up to an
average of 20 MeV/nucleon beam energies.
Unfortunately,
hardly any attempt has been devoted to
correlate the PSM parameters to that of OM. A recent attempt by Shastry2 succeeds in establishing a link between
PSM parameters and phenomenological OM parameters us%e in this Brief Report attempt to
ing %KB technique.
present an analytical expression for S matrix in terms of the
parameters of the nucleon-nucleon interaction and of matter
distributions using a similar approach. This task is achieved
here in two steps. In the first we make use of the recent
observation of Goldfarb and Gambhir which results in an
approximate analytical expression for the folding potential.
This expression which involves the parameters
of the
nucleon-nucleon interaction and the matter distributions has
been shown to yield the results within 5% of the exact folding model results. Starting from this expression we then
derive, using %KB technique, an approximate but closedform expression for nuclear phase shifts.
It is to be emphasized that we do not aim in this Brief
Report to achieve a quantitative fit to the data, instead we
demonstrate the feasibility that an analytical closed form expression for phase shifts (or 5 matrix) can be arrived at for
HIS involving the parameters of nucleon-nucleon
interaction and the matter distributions.
%e check the formalism
for a test case
by carrying out explicit numerical'calculations
of '
Ni scattering at E]ab=60 MeV.
The folding potential Vr(r) is given by
0+
fO
V
(r)=J J
dr~dr2pt(r~)p2(r2)
The nucleon-nucleon
sum
interaction
vNN
vNN(r
—r~ —r2)
is chosen'
of two Yukawa interactions so as to fit the
to be the
6 matrix
of
Reid potential. A simple zero range term was added' to account for knock on exchange.
Goldfarb and Gambhir
chose &NN as
v„„(r)= [G)exp( —N)r)/Ntr
+ G2exp( —N2r)/N2r],
MeV,
with
G)=5642. 8, N)=4 fm
62= —1907, N2= 2.5 fm
This vNN yields results within 1% of that obtained with M3Y
version of vNN. ' An analytical expression for V (r) can be
derived which is accurate within 5% by approximating the
matter distribution
P~(ri) =Pa
[i+exp[(r —R, )/a, ]}
as
I
f
= po'(bO(R, —r) + (exp[v;(R; —r) ] — exp[v, '(R, —r) ]}8(r—R, ) ) fm
b=0. 55, f =0 45, v;=a; ', v .=2.lvI, and po' [Eq. (3)] is determined through
p&(r)
with
be expressed as sum of ten terms out of which the contributions
five terms Vr(r) can be written as [Eq. (4.4) of Ref. 31
5
V'(r) = g
O.'I
r
+gI exp(
(4)
the normalization.
The V (r) can then
from five terms are negligible. Therefore, omitting these
a;r)—
30
1343
1984
The American Physical Society
BRIEF REPORTS
where
a„ f~, and
g&
30
are defined below.
A~ = &2= V2, &3=&4= VI',
A5= N2
r
exp[v2(R1+ R2) ]
41r G2
+2
f2= (G2N2
)
2v2
1
2
(1)
(2)
S2(],) +&2S2(1)
+2
X2 —u2 2
1
f3= (2~ 1)f2, f4= (2~ 1)f1, g2=
(R, + R, ) ]
exp [N'
S2, 4S2, 2
N22
IY2
= po
PO
G1N1) f1,
4m2G
Si(&)
—&2
&2%+2
2
v(R(vi
(bv/
—p) —(v/ —v()
4)r G2 exp[v2(R1+ R2) )
g1
~
(N
G)N1)g1, g3=
(G2N2
f
[R1(vi —v1) —p]+ (v/
—v1')
respective
fined as
(i)
J) J('O
k
b exchanges the index a to the index b.
The symbol a
The remaining five terms which contribute & 1'/o and are
involve
with
here
exponentials
arguments
omitted
v1(R1+ R2), v2(R1+ R2), and N1(R1+ R2). These terms
can be included in the present derivation without any problem. With this, the sum over i will then extend up to 10 instead of 5 in Eq. (4).
In HIS the partial waves I
Io are almost completely absorbed and hence will not contribute to the S matrix.
Therefore, the most important partial ~aves which contribute to the S matrix will have one turning point except in the
It is therefore
special cases where barrier is transparent.
reasonable to use %KB approximation as is usually done,
for the calculation of the phase shifts. The WKB expression
for the nuclear phase shift 5)1t(X) in the presence of
Coulomb interaction is given by
R
pR
«]
l
ao
r
[k2 —U~(r)
R
[k' —U,c (r) —U„(r) ]'/'
Here rI and r, are the approximate
51v(A. )
=—
'" 2pr
2k
g c
—U, (r) —U), (r) ]'
k2
=5 ")() )+5
integrand
=,
E
2JM
U
h
(
51'(l(. ) = lim
(2~ 1)g2,
g4=
(1)
2(1)
2
(2~ l)g1
—p]]
[R1(v& —v/)
I
J-
PO
2)
g5
and
u=(
2
U, (r)
"
= „k
U(r)
=,
(r)=
2p,
,
(1+iW) VF(r)
h
3 —r'/R, '
r
for
r
r
The other symbols are de-
vanishes.
)
= i+ T,
R, = r, (A,'/ + A
'/
&
'
~Rc
= zze2'
"", ~,
)
Z~, Z2 and A~, A2 are the atomic numbers and the mass
numbers of the projectile and the target, respectively, E is
the bombarding energy, and p, being the reduced mass. The
turning point r, is
r, =
1
—
[g+ (q2+A2)'/2]
provided r,
k
~ R,
In the surface region ~here the important
dr
partial waves be-
long
1
dr
Therefore, one can expand (k2 —U, —U)v —U), )'/2 in powers
of U~ in the region r & r, and in the leading order obtain
turning points where the
j
'
g
4
1
f(
+g( exp(
—0.1r)/
r
2qr
—
' 1/2
+
k'2
'
[k —U~—
(r) —U, (r) —U), (r)]'/
dr
4
(2)() ) .
(6)
—;
For the calculation of 5~(')(A. ) one needs the integrals of the type
I
(p) = Q
r
dr r
exp(
—pr)dr/
r2
—2n r
k
&/2
k'
for
m
=0.1
(7)
From the table of integral transforms4 and using the leading terms5 in the expansion for the K-type modified Bessel function for the argument x [x = (p/k) (A. 2+ n2)'/2]
0, one finally obtains
)
I
5„(')() ) =—ao
2k
g (exp( —a [q+ (v)'+)12)' 2]/k])
Here ao= 1.253 31414.
The calculation of 511 (2)(X) requires the knowledge of the
value of r~, the turning point. We note that r] is complex
and ~Imrt( ((Re(r)) and )r) —r,
0 as X becomes large.
These facts are readily revealed through the explicit calculation of r~ and r, for the partial waves near the surface re-
' (~2 + )( 2)
1/2
f +g [7)+ (g'+)1.')' ']
(g)
I
gion. This then leads to the following iterative procedure
for estimating reasonable value of r~. Setting rq =r„with
this generating the zero of the function
(r),
f
1
~
— „",
U(.),
(9)
BRIEF REPORTS
30
1345
the first iteration gives
1.0—
= r, —r, U~(r, )/(k
r, +A.
—e)
(10)
rC
where
0.5—
e
= rg
d
—
[r
Ujy(r) ]
f~fC
It has been shown that ri ' is quite close to ri for I & 29
for ' 0+ Ni system for optical model nuclear potential
U~(r). With the value of r~ '
rt so obtained for our nuclear potential U~(r), the 5~ '(k) can be simplified to
0
—
8
~2~(h.
) =0.5r, 3[ —U~(r, )]3l2/(r, 2k2+
—e),
A.
(11)
l
30
25
35
40
FIG. 2. The real part of the phase shifts Rebz(X) for 25 ~ I » 40
0 + Ni ( E]+b = 60 MeV) scattering. The curve OM is for optical model results with parameters of Ref. 7.
for
where ~ defined above reduces to
e= X r, 'exp( —n, r, ) f,
l i
Finally, the
S matrix
I rc
—1 +g, (2 —n;r, )
is obtained through
S (I) =exp{2i[5
%e
t'~(h.
)+5
~2~(X)]}
(12)
would like to mention that a better prescription for
evaluating r i for partial waves
I&
„can be incorporated. As the above treatment is not expected to give
good results for the more or less fully absorbed low partial
waves I
lo [lo is defined through qi = l&w(lo) = 0 &].
I;„&
I,
~
I
this three point WKB can be used for the calcula-
remedy
0 le
)i
1.0
tion of the phase shifts. Since the S matrix is almost zero
for most partial waves with I & lp, we refrain from giving
the three point WKB analysis in this report.
For the present we check the present formalism by carry' Ni
numerical
calculation
for '
ing out explicit
(E~,b=60 MeV) a typical example of HIS. The calculated
reflection coefficients q, Re5~, and Imb~ are plotted as a
function of I for 25
I 40 in Figs. 1, 2, and 3, respectively. The calculated lp- 29 for this case. It is to be emphasized
that this calculation does not involve any parameter except the
value
of IV in Uz we choose IV = 0.9 here. The
phenomenological
optical model results
marked OM are
also included in the respective figures. The parameters used
in OM are ag = ai = 0.5 fm, R& = RI = 7.92 fm, and
Vo+iWO= (90.1+ l42. 9) MeV, taken from Ref. 7. It is
clear from the figures that the calculated results without the
use of any parameter compare well with phenomenological
OM results. For I
is excellent.
The
lp the agreement
agreement for I around the value of Ip can be further im-
0+
~ ~
—
—
)
1.5—
ll)
II
0.5
II
C
0. 5—
0
I
25
30
I
35
I
$0
L
FIG. 1. The calculated reflection function q for 25~ I ~40 for
Ni (E[,b= 60 MeV) scattering with 8'=0.9. The corresponding OM results with parameters of Ref. 7, also shown are
25~ I ~40
marked OM.
model results
0+
30
25
40
part of the phase shifts Imh~(A. ) for
Ni (Ei,b=60 MeV) scattering. The optical
are marked OM,
FIG. 3. The imaginary
for
35
0+
BRIEF REPORTS
1346
30
proved using three point WKB. The calculated results deviate from the OM results for l
Io as expected. This can be
remedied by adopting a better prescription for calculating
the value of rj.
On the basis of our results we conclude that it is feasible
to write down a closed form expression for S matrix for the
HIS directly in terms of the parameters of the nucleonnucleon interaction and the matter distribution of the colliding nuclei. This correlates directly the "observable" cross
sections to the intrinsic parameters of the colliding system.
~G. R. Satchler and W. G. Love, Phys. Rep. 55, 183 (1979).
C. S. Shastry, J. Phys. G 8, 1431 (1982).
L. J. B. Goldfarb and Y. K. Gambhir, Nucl. Phys. A401, 557
and I. A. Stegun, Handbook of Mathematical Func(Dover, New York, 1964), p. 379.
6C. K. Chan, P. Suebka, and P. Lu, Phys. Rev. C 24, 2035 (1981).
7F. Videback, P. R. Christensen, O. Hansen, and K. Ulbak, Nucl.
Phys. A256, 301 (1976).
(
(1983).
E. Erdelyi, Tables of Integral
York, 1954), Vol. 1, p. 139.
4A.
Transforms
(McGraw-Hill,
New
5M. Abramowitz
tions