Figure 1

Riemann surface of energy-wave-number mapping $\rho \left(E\right)$ for massive particles in the semiclassical limit (2). The square root ${\rho }_c\left(E\right)=\pm {\rho }_0\sqrt{E-E_{T_c}}$ gives rise to two sheets: $\left\{E,+\right\}$ and $\left\{E,-\right\}$. Units such that ${\rho }_0=1$. Threshold set at zero: $E_{T_c}=0$.

Figure 2

Real and imaginary parts of the reduced logarithmic derivative of the outgoing wave function $L_{\ell }\left(E\right)$ for semiclassical massive neutral-particle (Table 1) energy-wave-number mapping (2) for different angular momenta $\ell \in [[1,4]]$. These real and imaginary parts were used by Lane and Thomas to define the shift and penetration functions, $S_{\ell }\left(E\right)$ and $P_{\ell }\left(E\right)$, as (41). This definition commands branch points from mapping (2) (cf. Fig. 1). $\mathrm{Re}\left[L_{\ell }\left(E\right)\right]$ presents subthreshold discontinuities (for odd angular momenta $\ell$) and nonmonotonic behavior (for even angular momenta $\ell$) below threshold on the $\{E,-\}$ sheet. Units such that ${\rho }_0=1$. Threshold set at zero: $E_{T_c}=0$. (a) Real part $\mathrm{Re}\left[L_{\ell }\left(E\right)\right]$, on both $\{E,\pm \}$ sheets and (b) Imaginary part $\mathrm{Im}\left[L_{\ell }\left(E\right)\right]$, on both $\{E,\pm \}$ sheets.

Figure 3

Analytic shift $S_{\ell }\left(E\right)$ and penetration $P_{\ell }\left(E\right)$ functions, definition (44), nonrelativistic neutral-particles (Table 3) $k_c\left(E\right)$ mapping (2), angular momenta $\ell \in [[0,4]]$, and zero threshold $E_{T_c}=0$. The shift function $S_c\left(E\right)$ has no $\{E,\pm \}$ branches (proof in lemma t3) and has discontinuities (odd $\ell$) and nonmonotonic behavior (even $\ell$) below threshold. $P_{\ell }\left(E\right)$ has $\{E,\pm \}$ branches and is purely real above threshold and purely imaginary below. Units such that ${\rho }_0=1$. (a) Analytic shift function $S_{\ell }\left(E\right)$, for both $\{E,\pm \}$ sheets, (b) Analytic penetration function real part $\mathrm{Re}\left[P_{\ell }\left(E\right)\right]$, and (c) Analytic penetration function imaginary part $\mathrm{Im}\left[P_{\ell }\left(E\right)\right]$.

Figure 4

Brune eigenproblem (54) for one-level one-channel $p$ wave: Comparison of real solutions from definitions (41) versus (44), for angular momentum ${\ell }_c=1$, neutral-particle energy-wave-number mapping (2), and resonance width ${\gamma }_{\lambda ,c}=1$, using $B_c=-{\ell }_c$ convention and zero threshold $E_{T_c}=0$. Units such that ${\rho }_0=1$. Since both have a real subthreshold poles, both will yield two solutions (crossing the $E=E$ diagonal), one above and one below the discontinuity. If at threshold energy $E_{T_c}$ the left-hand side of (54) is above the $E=E$ diagonal, then the above-threshold solutions from both definitions coincide. In any case, the subthreshold solutions differ. Behavior is analogous for all odd angular momenta ${\ell }_c\equiv 1\left(\mathrm{mod}2\right)$. (a) Positive resonance energy $E_{\lambda }=2$ and (b) Negative resonance energy $E_{\lambda }=-2$.

Figure 5

Brune eigenproblem (54) for one-level one-channel $d$ wave:

comparison of real solutions from definitions (41) versus (44), for angular momentum ${\ell }_c=2$, neutral-particle energy-wave-number mapping (2), and resonance width ${\gamma }_{\lambda ,c}=\sqrt{2}$, using $B_c=-{\ell }_c$ convention and zero threshold $E_{T_c}=0$. Units such that ${\rho }_0=1$. Since there are no real subthreshold poles, both can yield one, two, or three solutions (crossing the $E=E$ diagonal), depending on the values of the resonance parameters. If at threshold energy $E_{T_c}$ the left-hand side of (54) is above the $E=E$ diagonal, then the above-threshold solutions from both definitions coincide. In any case, the subthreshold solutions differ. Behavior is analogous for all even angular momenta ${\ell }_c\equiv 0\left(\mathrm{mod}2\right)$. (a) Positive resonance energy $E_{\lambda }=1$ and (b) Negative resonance energy $E_{\lambda }=-2$.

Figure 6

${}^{134}\mathrm{Xe}$ Reich-Moore cross sections for spin-parity group $J^{\pi }=1/2^{(-)}$ $p$-wave resonances: The cross sections are generated using the ENDF/B-VIII.0 resonance parameters (an MLBW evaluation) into the Reich-Moore formalism equations. Similarly, the $R$-matrix cross section is generated by setting all capture (including eliminated) widths to zero. For reference, the exact cross-section values at the resonance peaks are reported in Table 7. (a) First $p$-wave resonance and (b) Second $p$-wave resonance.