CALCULATION IN THE IMPULSE APPROXIMATION
To calculate the total cross sections s
tot, the reaction cross sections s
r, and the integrated (s el)
and differential cross sections for elastic scattering, we have used the
algorithm and program presented in Refs. 21,
40, 41, and 47. The optical potential for the interaction of protons, pions,
and kaons with nuclei was obtained in Refs. 20, 21, 40, 4l, and 47-49 by
means of a generalization and development of the results of the investigations
of Refs. 24, 46, and 50-54 in the form
where the summation is taken over the proton and neutron densities, l
is the orbital angular momentum in the xA c.m.s., s
is the spin of the proton in pA scattering or the spin of the nucleus
in the meson-nucleus interaction, p is the momentum of the
incident particle x in the xA c.m.s., w
xN is the reduced energy in the xN c.m.s., r
N(r) is the density of the nucleons N inside the
nucleus, and g N is a kinematic factor
in the strong-coupling approximation, which has the form [41] g
N = A(sxN/sxA)1/2
MN/p, in which Mp = Z, Mn
= A - Z, sij- is the s invariant of the (ij) system,
A is the mass number, and Z is the charge of the nucleus. In the NA
potential in the calculation, we have introduced a factor (A-1)/A: UpA(r)=U(r)(A-1)/A
(see Refs. 55- 57), i.e., we have adopted the
KMT approach in summing the multiple-scattering series (this makes it possible
to obtain better agreement with experiment). For mesons, we have used Watson’s
approach [57]. Expressions for the coefficients Ai, and Bi,
were obtained in Refs. 21 and 40, and are given in Appendix
1 of the present paper. When they are calculated in terms of the amplitudes
for interaction of the incident particle x with the nucleons N inside
the nucleus, it is possible to include the S, P, D, F, G, H, I, and J waves.
In the actual calculations, we used the results of phase-shift analyses
[58-60]. The choice of the K+N amplitudes [58] was justified
in Ref. 40. We note that the algorithm for obtaining
the potential was constructed in Ref. 21 on the basis of an expansion of
the elementary amplitudes in powers of the momentum transfer q
in the xN c.m.s. In the calculations reported in the present
paper, we discarded the terms of order (q/p)6
or higher in the expansion of the xN amplitudes with L>2. In comparison
with experiment, this led to an overestimation of the calculated values
of the differential cross sections for p 12C elastic scattering
in the backward hemisphere at energies above 100 MeV (see Fig. 7). It should
be noted that, in view of the smallness of the calculated and experimental
cross sections in this kinematic region, the above-mentioned differences
do not affect the conclusions of the present work. In the future, it will
be possible to take into account all the terms in the expansion of the
elementary amplitudes in order to obtain agreement with experiment in this
region.
The parameters of the nucleon densities for the
nucleus 12C were taken to be the same as in Ref.
40 (in the calculations, we used a two-parameter Gaussian distribution).
For the nucleus 4He, the parameters of a three-parameter Fermi
distribution were taken from Ref. 23. For the deuteron, r
N (r) was taken in the Hulthen and Gartenhaus-Moravcsik
forms with the parameters given in Refs. 61 and 62, respectively, and also
in the Gaussian form with the parameter obtained from the rms charge radius
using the algorithm of Ref. 63 for taking into account the charge form
factors of the nucleons. The form and the parameters of r
N(r) are given in Appendix 2.
In the calculations of the particle - nucleus interaction, we used the
Klein - Gordon equation for spinless systems and the Schrodinger equation
with relativistic kinematics for systems with spin.
The experimental data were taken from Refs. 64-69
(see also the references in Refs. 1, 5, 9, 21,
and 44) for the nucleon - nucleus interaction, from Refs. 70-79 (see also
the references in Ref. 21) for pion - nucleus scattering, and from Refs.
43, 62, and 80 - 84 as well as the compilation of Ref. 85 for K+
- nucleus scattering. We also made usee of the compilation of Ref. 86.
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