We treat the response in a finite nucleus to longitudinal spin-isospin probes. Since linear momentum is here not a good quantum number, we adopt a formalism suggested by Toki and Weise, which incorporates an expansion in partial waves of good nuclear total angular momentum. The basic quantity involved is the pion self-energy II, which is nonlocal in momentum space. This quantity peaks at diagonal momentum distribution only for momentum $q\approx k_c$, where $k_c$ is the critical momentum for pion condensation precursor features, $k_c\sim 2-3m_{\pi }$. To solve for the response function $\mathcal{R}$, we iterate II by solving an integral equation exactly (using numerical methods). The response $\mathcal{R}$ is singular for criticality (Migdal parameter $g^{\prime }\approx 0.4$ and momentum $q\approx k_c$). We use $\mathcal{R}$ to renormalize the matrix element of a longitudinal spin-isospin probe and find, as expected, a large effect near criticality as well as a very significant effect of nonlocality in general. We apply our formalism to levels with $J^P=0^{-},T=1\mathrm{}$ in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and $J^P=1^{+},T=1\mathrm{}$ in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$. The results are compared with currently used approximations. The local density approximation employed in this context gives semiquantitative agreement only. Using infinite nuclear matter, we find the best agreement for the equivalent constant density $\rho =0.12\mathrm{}$ ${\mathrm{fm}}^{-3\mathrm{}}$ for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$; this agreement remains only qualitative, however. We give a detailed test of the model interactions suggested by Meyer-ter-Vehn in which it emerges that the value of $k_c$ is not correctly produced. The approximations of Toki and Weise are satisfactory for $g^{\prime }\gtrsim 0.6$ and $k\approx k_c$. Approximations cannot be reliably used, however, for small $g^{\prime }$ or far from the critical momentum transfer. This is especially the case for the description of the nonlocality of II in momentum space through the use of a Bessel-function distribution, which is unreliable throughout the relevant range of parameters.

[NUCLEAR REACTIONS Pionlike discrete nuclear excitations in finite nuclei; spin-isospin strength distribution effects; pion self-energy and response function; ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}_{}^{*}{}_{}{}^{}(J^P=0^{-},T=1)$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}_{}^{*}{}_{}{}^{}(J^P=1^{+},T=1)$; comparison with various approximations.]

• Received 12 July 1983

DOI:https://doi.org/10.1103/PhysRevC.29.914