PEI YSICAL REVI EW VOLUME 142, &UMBER 4 FEBRUARY 1966 High-Energy Deuteron Cross Sections* V. FRANcot Iaboratory for Nzzctear Sczence and DePartment of Physics, IVassactzzzsetts Instztzzte of Technology, Cambridge, Massachusetts AND R. J. GLAUBER Lyman Laboratory of I'hysics, Harvard University, Cambridge, Massachusetts (Received 9 August 1965) A theoretical analysis of the collisions of particles with deuterons is carried out in the high-energy approxi- mation. This approximation, which corresponds to a generalized form of diffraction theory, takes explicit account of double collision processes as well as single ones. It is used to express the amplitudes for elastic and inelastic scattering by deuterons in terms of the elastic-scattering amplitudes of the neutron and proton and the deuteron wave functions, The resulting expressions are used to evaluate the differential cross sec- tion for elastic scattering, and the angular distribution of inelastic scattering (i.e. , the differential cross section for deuteron breakup integrated over 6nal energies of the incident particle). The contributions to these cross sections of the various single and double scattering processes and the terms which represent their interference are exhibited individually. Expressions are derived for the total cross section of the deuteron and for its elastic and inelastic total scattering and absorption cross sections. The difference be- tween the various types of deuteron cross sections and the sum of the corresponding cross sections for the free neutron and proton is explained in some detail. Spin-dependent interactions are treated, and for incident particles of spin 2 an expression is given for the deuteron total cross section in terms of the general spin- dependent scattering amplitudes of the neutron and proton. The theory is applied to antiproton-deuteron collisions in the energy range from 0.13 to 17. 1 BeV. The results for the total and absorption cross sections which are calculated for a variety of models of the deuteron wave function are found to be in good agreement with the measurements. The magnitudes of such effects as double scattering and the interference of single- and double-scattering amplitudes are seen to be appreciable. I. INTRODUCTION XPERIMENTAL studies have been made in recent ~ years of the interactions of a wide variety of high- energy particle beams with deuterium targets. Several « these studies have been directed toward determining the internal structure of the deuteron. More frequently, however, deuterons have been used as collision targets in the hope of estimating, by a simple subtraction pro- cedure, the cross sections of stationary neutron targets. This method has been based on the presumption that at sufficiently high energies, i. e. , when the incident-particle wavelength is much smaller than the deuteron radius, any deuteron cross section should be approximately the sum of the corresponding free-neutron and free-proton cross sections. If that is so, then once the deuteron and proton cross sections are measured, the neutron cross section is given by their difference. It has become evident, however, through many ex- periments that the deuteron cross section may differ quite substantially, even at the highest available en- ergies from the sum of the free-neutron and proton cross sections. An elementary discussion of the origin of this lack of additivity has been given by Glauber' ' *Supported in part by the U. S. Air Iorce Ofhce of Scientific Research under Contract No. AF49(638)-1380 and by the Atomic Energy Commission under Contract No. AT(30 — 1) — 2098. $ Present address: Lawrence Radiation Laboratory, Berkeley, California. z R. J. Glauber, Phys. Rev. 100, 242 l1955l; in Proceedzngs of the Conference on Nuclear F~orces and the Pen Nucleon Problem, edited by T. C. GrifFith and E. A. Power (Pergamon Press, Inc. , London, 1960},Vol. I, p. 233. R. J. Glauber, in Lectures in Theoretical I'hysics, edited by W. E. Brittin et al. (Interscience Publishers, Inc. , New York, 1959), Vol. I, p. 315. who made theoretical estimates of its magnitude for the case of incident nucleons and x mesons. In subse- quent experimental work much the same effect has been observed with incident beams of antinucleons' ' and E mesons. ' ' In particular the deviation from additivity of the cross sections for the case of incident antinucleons has been found in some measurements' 4 to amount to 20 to 40% of the individual antinucleon-nucleon cross sections. Since a detailed understanding of this correc- tion is evidently of some importance in establishing fundamental cross sections we shall try to improve the earlier theoretical analysis in a number of respects and to extend its domain of applicability. At the same time we shall take the occasion to study in detail the various types of collision processes in which the deuteron may participate. ' In particular we shall separate the differen- tial cross sections for elastic and inelastic scattering into contributions coming from single and from double scattering and their various interference terms. %e then illustrate the magnitudes and angular dependences of these cross sections by calculating them explicitly for the case of antiproton-deuteron collisions. In each of the cases which has been studied experi- ' Q. Chamberlain, D. V. Keller, R. Mermod, E. Segre, H. M. Steiner, and T. Ypsilantis, Phys. Rev. 108, 1553 (1957). 4 T. Elioff, L. Agnew, O. Chamberlain, H. M. Steiner, C. Wie- gand, and T. Ypsilantis, Phys. Rev. 128, 869 (1962). '%. Galbraith, E. W. Jenkins, T. F. Kycia, B. A. Leontic, R. H. Phillips, A. L. Read, and R. Rubenstein, Phys. Rev. 138, B913 (1965). 6 V. Cook, B. Cork, T. F. Hoang, D. Keefe, L. T. Kerth, W. A. Wenzel, and T. F. Zipf, Phys. Rev. 123, 320 (1961. ). ' V. Franco, thesis, Harvard University, 1963 (unpublished). 1195