D.C. Dodder

Experiments involving polarization techniques have nowhere unearthed a richer structure of observed phenomena than in those scattering and reaction processes involving very few nucleons. This paper is concerned with a program, carried out with colleagues at Los Alamos, with its first goal as a phenomenological understanding of these processes. It is the belief of those of us working on this program that our approach is essentially dictated by the fact that the overwhelming amount of information already collected seems to be still only marginally sufficient to describe the complex behavior of these systems. This somewhat paradoxical situation, that an almost unmanageable quantity of data is still perhaps just on the threshold of encompassing all aspects of the behavior of such a system, seems to call for an approach with two main features. First, all channels and all complementary experiments should be described simultaneously. Second, an energy dependent parameterization should be used, if only because the data are not sufficiently complete at single energies. The simultaneous description of all channels allows us to make use of the very important principle of unitarity of the collision matrix S. The need for an energy dependent parameterization has led us to choose the R-matrix formalism. In this context that formalism is to be regarded as a practical tool. This paper does not discuss the role of the R-matrix in general theories of nuclear reactions. It may be, however, that the results of the present work will have a place in such discussions.

Phase-shifts in $S$ and $P$ waves are found that fit the observed scattering of protons by helium (4) within the experimental accuracy of the recent Minnesota results. The $S$ wave phase is negative and rather small. The ${P}_{\frac{1}{2}}$ and ${P}_{\frac{3}{2}}$ waves have different phase-shifts, both positive, but there are two sets of possible values, one corresponding to a normal doublet in ${\mathrm{Li}}^{5}$, the other to an inverted doublet. In both cases there is resonant scattering and strong polarization of the proton beam. Measurement of the polarization would decide between the doublets. The values of the phase-shifts presented are those that minimize the sum of squares of percent differences between observed and calculated differential cross sections.

Proton-alpha elastic scattering experiments at 5.81 and 9.48 Mev are analyzed in terms of phase shifts, and the results, together with earlier ones at lower energies, are used to compute the logarithmic derivatives of the wave functions at a radius of 2.9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm. These logarithmic derivaties, for ${P}_{\frac{1}{2}}$ and ${P}_{\frac{3}{2}}$ states, are found to be linear functions of the energy. From the behavior of the logarithmic derivatives the widths and positions of the ${P}_{\frac{1}{2}}$ and ${P}_{\frac{3}{2}}$ energy levels are determined, the most striking results being the large splitting of the two levels and the very broad width of the ${P}_{\frac{1}{2}}$ level. These results are compared with neutron-alpha experiments and are found to be in sufficiently good agreement to support the conclusions about the ${P}_{\frac{1}{2}}$ level which differ from those previously expected. Small negative $D$ wave phase shifts are found in the proton-alpha experiments which also show evidence of an inverted doublet spin-orbit splitting.

Accurate measurements of $p\ensuremath{-}\ensuremath{\alpha}$ elastic scattering cross sections were made at energies of 11.157, 12.040, 13.600, and 14.230 MeV. The average relative error is about 0.6% and the scale error is 0.37%. These data and all available cross section and spin-dependent measurements of $p\ensuremath{-}\ensuremath{\alpha}$ scattering between 0 and 18 MeV were collected and prepared for input to a general purpose $R$-matrix analysis program. Strict statistical criteria were used for the elimination of data. The resulting search on 1131 data produced a unique fit with a ${\ensuremath{\chi}}^{2}$ per degree of freedom of 1.001 which is within one standard deviation in ${\ensuremath{\chi}}^{2}$ space. Arbitrary normalizations to the data were not allowed; a normalization was treated as another datum restrained by a scale error obtained from the experimental information. The parameter space was made up of background contributions in $S$, $P$, $D$, and $F$ states with an additional level each in the ${P}_{\frac{3}{2}}$ and ${P}_{\frac{1}{2}}$ states. There were 14 free parameters. For the first time, the reduced widths of the $p$-wave resonance states come out almost equal. Comparisons are made to the $R$-matrix analysis of Stammbach and Walter and to the phase shift analysis of Arndt, Roper, and Shotwell.NUCLEAR REACTIONS $^{4}\mathrm{He}(p, p)$, $E=11\ensuremath{-}14$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$, $\ensuremath{\theta}(\mathrm{c}.\mathrm{m}.)=19\ensuremath{-}167\ifmmode^\circ\else\textdegree\fi{}$, $\ensuremath{\Delta}\ensuremath{\theta}=0.03\ifmmode^\circ\else\textdegree\fi{}$, $\ensuremath{\Delta}\ensuremath{\sigma}=0.06%$; calculated $R$-matrix parameters for all $^{4}\mathrm{He}(p, p)^{4}\mathrm{He}$ data, $E=0\ensuremath{-}17$ MeV.

An approximately Coulomb-corrected, charge-symmetric R-matrix prediction for n${\mathrm{\ensuremath{-}}}^{3}$H scattering from analyzing p${\mathrm{\ensuremath{-}}}^{3}$He data is described, which deviates at most by 2% from the n${\mathrm{\ensuremath{-}}}^{3}$H total cross section. The total cross section, together with a new measurement of the coherent scattering length now allows two possible sets of singlet and triplet scattering lengths ${\mathit{a}}_{\mathit{s}}$ and ${\mathit{a}}_{\mathit{t}}$. Our analysis agrees well with the new value of the coherent scattering length, and determines the set with ${\mathit{a}}_{\mathit{s}}$/${\mathit{a}}_{\mathit{t}}$>1 to be the correct one.