Consistent optical potential for incident and emitted low-energy $\alpha$ particles. II. $\alpha$ emission in fast-neutron-induced reactions on Zr isotopes

Background: Challenging questions of the $\alpha$-particle optical-model potential (OMP) are still pointed out by recent high-precision measurements of $\alpha$-induced reaction data below the Coulomb barrier. Moreover, the reliability of a previous OMP for $\alpha$ particles on nuclei within the mass number range $45\le A\le 209$ has been recently proved for emitted $\alpha$ particles as well, but only in the case of proton-induced reactions on Zn isotopes [Phys. Rev. C 91, 064611 (2015) (Paper I)].

Purpose: Analysis of most recent $(\alpha ,\gamma )$ reaction data for Ge and Zr isotopes, which provides an additional validation of the above-mentioned potential, is related to a further account of $\alpha$-particle emission in neutron-induced reactions on Zr isotopes, at the same time with a suitable description of all competitive processes.

Methods: A consistent parameter set, established or validated by independent analysis of recent various data, particularly $\gamma$-ray strength functions, has been involved within model calculation of the $(\alpha ,\gamma )$ as well as $(n,\alpha )$ reaction cross sections. The latter are part of the whole analysis of the neutron activation of Zr isotopes, in order to avoid any error compensation or latent ambiguity.

Results: The aforesaid potential provides a consistent description of recent $\alpha$-induced reaction data with no empirical rescaling factors of the $\gamma$ and/or nucleon widths. On the other hand, its use leads to underestimated predictions of the pre-equilibrium emission and statistical models for the $(n,\alpha )$ reaction cross sections.

Conclusions: An optical potential with a volume imaginary component seems to be needed to describe the low-energy $\alpha$-particle evaporation, while only surface absorption occurs in $\alpha$-induced reactions at similar energies.

Figure 1

Comparison of the measured cross sections [12] of the $(p,n)$ reaction on ${}^{88}\mathrm{Sr}$ (top) and ${}^{89}\mathrm{Y}$ (bottom) and calculated values corresponding to proton OMP parameters of either the global (dashed curves) or local set for ${}^{90}\mathrm{Zr}$ (dash-dotted curves) of Ref. [50], and that of Johnson et al. [52] (solid curves). The finally calculated cross sections of $(p,p^{\prime })$ reaction (short-dashed curves) and $(p,\gamma )$ reaction (short dash-dotted curves) are also shown.

Figure 2

Comparison of measured [32, 33, 34] and calculated sum of $\gamma$-ray strength functions of the $E1$ and $M1$ radiations for ${}^{89}\mathrm{Y}$ (top) and ${}^{90}\mathrm{Zr}$ (bottom), using the models SLO (dotted curves), GLO (short-dotted curves), and EGLO without (dashed curves) and with (solid curves) a SR contribution, for $E1$ radiations, and the SLO model for $M1$ radiations (short dash-dotted curves). There are shown also measured dipole $\gamma$-ray strength functions for ${}^{92,94}\mathrm{Mo}$ nuclei [34, 35] and $s$-wave average radiation widths ${\mathrm{\Gamma }}_{\gamma }$ (in meV) either deduced from systematics for ${}^{89}\mathrm{Y}$ and ${}^{92}\mathrm{Mo}$, or corresponding to $M1$ and each of above-mentioned $E1$ functions.

Figure 3

Comparison of cross sections measured for $(p,\gamma )$ reaction on ${}^{88}\mathrm{Sr}$ [15] (top) and ${}^{89}\mathrm{Y}$ [16, 17, 18] (bottom) and calculated by using the $E1$ radiation RSF models SLO (dotted curves), GLO (short-dotted curves), EGLO (dashed curves), and EGLO+SR (solid curves). The finally calculated cross sections of $(p,p^{\prime })$ reaction (short-dashed curves) and $(p,n)$ reaction (short dash-dotted curves) are also shown.

Figure 4

Left: As Fig. 2 but for ${}^{74,78}\mathrm{Se}$ and dipole $\gamma$-ray strength functions for ${}^{76}\mathrm{Ge}$ nucleus [34, 71]. Right: Comparison of measured cross sections of $(\alpha ,\gamma )$ reaction on ${}^{70,74}\mathrm{Ge}$ [19, 68] and calculated values using the $\alpha$-particle OMPs of Refs. [74] (dash-dotted curves) and [7] (solid curves), and alternate involvements for the latter OMP of either the GLO (short-dotted curves) and SLO (dotted curves) RSF models, vs $\alpha$-particle laboratory energy (bottom) and ratio of center-of-mass energy to Coulomb barrier (top). There are shown also the ${\sigma }_R$ values given by OMP [74] (dash-dot-dotted curves).

Figure 5

As Fig. 4 but for RSFs of the compound nuclei ${}^{94,95,96}\mathrm{Mo}$ [44], and $(\alpha ,\gamma )$ reaction on ${}^{90,91,92}\mathrm{Zr}$ [19, 69].

Figure 6

Comparison of neutron-capture cross sections for ${}^{89}\mathrm{Y}$ and ${}^{90-92,94,96}\mathrm{Zr}$, measured [12], evaluated within the TENDL-2015 library [28] (dash-dotted curves), and calculated using the code TALYS-1.8 [11] and its default parameters (dash-dot-dotted curves) as well as in the present work using the $E1$ radiations RSF models SLO (dotted curves), GLO (short-dotted curves), EGLO (dashed curves), and EGLO+SR (solid curves). There are also shown the $s$-wave average radiation widths ${\mathrm{\Gamma }}_{\gamma }$ (in meV) either measured [31] or corresponding to $M1$ and each of the above-mentioned $E1$ functions.

Figure 7

Comparison of measured [9, 12], evaluated (dash-dotted curves [28], and histogram [81]), and calculated using the code TALYS-1.8 [11] and its default parameters (dash-dot-dotted curves) as well as in the present work (solid curves) fast-neutron reaction cross sections for ${}^{90}\mathrm{Zr}$ target nucleus. There are also shown the currently calculated excitation functions of the reactions $(n,n^{\prime }p)$ (dotted curve) and $(n,pn)$ (dashed curve).

Figure 8

As Fig. 7 but for ${}^{91}\mathrm{Zr}$ [9, 12].

Figure 9

As Fig. 7 but for ${}^{92,94,96}\mathrm{Zr}$ [9, 12].

Figure 10

Comparison of $(n,\alpha )$ reaction cross sections for ${}^{90,92,94,96}\mathrm{Zr}$ target nuclei, measured [1, 9, 12], evaluated [28] (dash-dotted curves), and calculated using the code TALYS-1.8 [11] and its default parameters (dash-dot-dotted curves) as well as in the present work using the $\alpha$-particle global OMPs of (i) Ref. [7] for the HF and PE model calculations (dashed curves), (ii) Ref. [7] only for the PE model while Ref. [56] is used for HF (short-dotted curves), and (iii) Ref. [56] for both HF and PE model calculations (solid curves).

Figure 11

Comparison of ${}^{91}\mathrm{Zr}(n,\alpha n+n^{\prime }\alpha )\stackrel{m}{{}^{87}\mathrm{Sr}}$ reaction reaction cross sections measured [9], evaluated [28] (dash-dotted curves), and calculated using the code TALYS-1.8 [11] and its default parameters (dash-dot-dotted curves) as well as in the present work using the $\alpha$-particle global OMPs of (i) Ref. [7] for the HF and PE model calculations (solid curves), (ii) Ref. [7], only the PE model, while Ref. [56] is used for HF (short-dotted curves), and (iii) Ref. [56] for both HF and PE model calculations (dashed curves).