– UNITED STATES A TO Mi C E N E R G Y CO M M I SS I C N
AECU-1075
HIGH-ENERGY NEUTRON SCATTERING BY NUCLEI
By
SimOn Pasternack
Hartland S. Snyder
ENGR LIBRARY
JUN 1 o so,
UNIV. OF WACH
September 26, 1950
Brookhaven National Laboratory
Technic a l l n form a ti on Service, Oak Ridge, Te n n e s see
UNIVERSITY OF MICHIGAN
9 ||||||||||||||
3 9015 08647 8644
Wy

PHYSICS
AEC, Oak Ridge,Tenn., 5-1-51--555-W1424
HIGH-ENERGY NEUTRON SCATTERING BY NUCLEI
By Simon Pasternack and Hartland S. Snyder
The so-called “transparent model” of a nucleus seems to be a useful model in treating the
scattering of 90-Mev neutrons by nuclei, since the nuclear radii fitted from experiment by this model
agree well with an A1/3 law.2 However, the calculated differential scattering cross sections deviate
somewhat from the experimental observations, 3 the latter being 10 to 20 per cent higher at low scatter-
ing angles.
The differential scattering calculations for the transparent model were made by assuming the
nucleus to be representable by a square well with potential 30.8 Mev, plus absorption, leading in the
classical (WKB) approximation to a sphere with a complex constant index of refraction. The angular
distribution amplitude was found to be
R $º- º |-2–02 -.
f(0) = *ſ | - e' K+2ik1) WR p J0(kp sin 9)p dp
0
where k is the neutron wave number, K the absorption coefficien, and k1/k the real part of the index
of refraction. This integral was evaluated by converting it to a sum of approximately kP terms.
The integral for f(0) may be evaluated by means of a series representation due to Van de Hulst,”
namely
2|J1(2) P_iw, ..., Jo(z). 13 z J1(2) 13:5 z” J2(z)
--5 eT" (1 – iw) + - + = e e &
p2 p4 p6
10-kº-,-->
where p = (2k1 + iR)R; z = kR sin 9; and w = p? + 2*.
This series for f(0) is particularly useful for large mass nuclei and for high energies for which kR
(the approximate number of terms in the summation method) is large. Van de Hulst derived this
formula by a double series expansion of f(0) in powers of z and p. A simpler proof may be obtained by
writing
J1 (z) a r1/2 ;
f(0) = kR2 | 1 + iſ elp cos ? Joſz sin y) sin y dy
Z ôp 0
If we define
sin y dy
I T/2 Jn(z sin Y) ip cosy
Il (z sin y)*
0
then integration by parts yields
eip Jn(z) 222 + 1
-: - + *
2nnlip z*ip ap2*n-1
Repeated integrations by parts yield the series for f(0).
AECU-1075 1
2 AECU-1075
Since the complex constant index of refraction is equivalent to the assumption of a square well with
a complex constant potential, it was considered desirable to check the validity of the classical approxi-
mation by making an exact partial wave analysis for the complex square well, using the corresponding
values of the parameters. This was done for aluminum, the results being shown in Fig. 1. The circles
represent the experimental points of Bratenahl et al., the dotted line the values of 0 (9) calculated using
the classical approximation, and the solid line the values of 0 (9) calculated by means of the exact
partial wave analysis. It is seen that the apparent deviations from experiment are at least partly due
to the calculational method rather than to the use of the complex square-well model.
The calculated scattering and absorption cross sections differ somewhat from those obtained with
the classical approximation. The scattering cross section becomes 0.83 barn instead of 0.75, and the
absorption cross section is 0.45 barn instead of 0.36. To make a closer comparison of the calculated
differential scattering cross section with the experimental results it would be necessary first to adjust
the complex potential parameters to fit the experimentally determined scattering and absorption cross
sections.
The phase shifts calculated by the exact partial wave analysis deviate considerably from those
obtained by the WKB method.” The latter gives, for aluminum,
ô1 = (1.35 + 0.452i) | – (1 + 1/2%is...]” 1 s 8
= 0 1 > 8
The former method yields the following phase shifts for aluminum (for 1 = 0,1,2,...):
1.29 + 0.38i, 1.44 + 0.56i, 1.20 + 0.40i, 1.36 + 0.40i, 1.12 + 0.49i,
1.01 + 0.29i, 1.11 + 0.32i, 0.85 + 0.47i, 0.27+ 0.19i, 0.065 + 0.025i,
0.012 + 0.005i, 0.002 + 0.001i, etc. .*
We wish to express our appreciation to Richard J. Weiss for some helpful discussion and to
William Donoghue, Theresa Danielson, and Dale Meyer for performing the numerical work.
REFERENCES
1. R. Serber, Phys. Rev., 72: 1114 (1947).
2. S. Fernbach, R. Serber, and T. B. Taylor, Phys. Rev., 75: 1352 (1949).
3. A Bratenahl, S. Fernbach, R. H. Hildebrand, C. E. Leith, and B. J. Moyer, Phys. Rev., 77: 597
(1950). -
. H. C. Van de Hulst, Recherches Astronomiques de l'Observatoire d’Utrecht, Vol. XI, Part 1, 1946.
4
AECU-1075
| | | | f> *
4 8 42 46 2O 24
NEUTRON SCATTERING ANGLE, 6, DEG
Fig. 1- Comparison of experimental differential scattering cross sections
with those calculated from transparent model theory in units of barn per
steradian. G), experimental (Bratenahl et al.). ..., classical approximation.
—, partial wave analysis.
END OF DOCUMENT