Phonon coupling effects in proton scattering from ${}^{40}\mathrm{Ca}$

Background: Formal optical model theory shows that coupling to vibrational nuclear states generates a nonlocal and $l$-dependent dynamical polarization potential (DPP). Little is established concerning the DPP, yet its properties are crucial for explaining the departures of optical model potentials (OMPs) from global behavior and for the rigorous extraction of spectroscopic information from direct reactions.

Purpose: To appraise the application of channel coupling followed by $S$-matrix inversion for the systematic exploration of the contribution of the coupling of collective states to the nucleon OMP and to identify properties of nuclear potentials indicative of $l$-dependence.

Methods: $S$-matrix to potential, $S_{lj}\to V\left(r\right)+\mathbf{l}·\mathbf{s}{V}_{\mathrm{SO}}\left(r\right)$, inversion provides local potentials that precisely reproduce the elastic channel $S$-matrix from coupled channel (CC) calculations. Subtracting the elastic channel uncoupled (bare) potential yields a local and $l$-independent representation of the DPP. The dependence of this local DPP upon the nature of the coupled states and upon other parameters can be studied.

Results: All components of the DPP arising from coupling to vibrational states are substantially undulatory with a point-by-point magnitude therefore disproportionate to their contribution to volume integrals. Information relating to dynamical nonlocality is found. The proton charge leads to a substantial difference between DPPs for protons and neutrons.

Conclusions: Undulatory features in potentials found in precision fits to elastic scattering data are significant, are a consequence of coupling to inelastic channels and must be allowed for in phenomenology; they are indirect evidence of $l$-dependence. Within the model, coupling to excited states magnifies the effect of the proton charge on the difference between proton-nucleus and neutron-nucleus interactions. Coupled channel plus inversion is a procedure of wide applicability, complementary to evaluation of the Feshbach formalism.

Figure 1

For 30.3 MeV protons scattering from ${}^{40}\mathrm{Ca}$, the DPP calculated with full (complex) coupling to the GQR. The dashed line is for the bare potential A and the solid line is for the global potential B. Panels (a)–(d) are respectively real and imaginary central and real and imaginary spin-orbit terms.

Figure 2

For 30.3 MeV protons scattering from ${}^{40}\mathrm{Ca}$, the DPP calculated with bare potential A. The DPP for full (complex) coupling to the GQR is shown as the solid line and the DPP with real coupling only is the dashed line. Note that the $x$-axis scale is different from that for Fig. 1, which includes the same complex coupling DPP. Panel labels are as for Fig. 1.

Figure 3

For 20 MeV protons scattering from ${}^{40}\mathrm{Ca}$, the DPP calculated with full (complex) coupling to the GQR. The dashed line is for the bare potential A and the solid line is for the global potential, B. Panel labels are as for Fig. 1.

Figure 4

For 30.3 MeV protons scattering from ${}^{40}\mathrm{Ca}$, the solid line is the DPP calculated with full (complex) coupling to the $3^{-}$ state at 3.737 MeV and $\delta =1.6$ fm. The bare potential was the Becchetti-Greenlees global potential. The dashed line is the DPP for $\delta =0.45$ fm multiplied by ${(1.6/0.45)}^2$. Panel labels are as for Fig. 1.

Figure 5

For 30.3 MeV protons scattering from ${}^{40}\mathrm{Ca}$, the solid line is the DPP for the GQR with full complex coupling. The dashed line is the DPP for excitation of the $3^{-}$ state at 3.737 MeV and $\delta =1.6$ fm. In each case the bare potential was the Becchetti-Greenlees global potential. Panel labels are as for Fig. 1.

Figure 6

For 30.3 MeV protons scattering from ${}^{40}\mathrm{Ca}$, the solid line is the DPP arising from the coupling to two identical $3^{-}$ states at 3.737 MeV with $\delta =1.6$ fm. The dashed line is a factor of 2 times the DPP for a single $3^{-}$ state at 3.737 MeV. Panel labels are as for Fig. 1.

Figure 7

For 30.3 MeV protons and neutrons scattering from ${}^{40}\mathrm{Ca}$, the DPPs arising from the coupling to ten phonons. The solid and dashed lines repesent DPPs for two alternative inversion solutions for protons, and the dotted and dot-dashed lines represent DPPs for two alternative inversion solutions for neutrons. Panel (a) gives the real part and panel (b) gives the imaginary part.