Nucleon-induced inelastic scattering with statistical strength functions and the ECIS direct reaction code

Abstract

Modern theoretical descriptions of inelastic scattering make use of multi-step direct reaction approaches together with transition potentials obtained from sophisticated nuclear structure models. Here we demonstrate how the complexity of such calculations can be reduced to permit simpler ones, also using the ECIS code, but providing an almost equally precise alternative to a much more detailed calculation. We have studied the transition form factors within the random phase approximation (RPA), where these are obtained as linear combinations of particle–hole states. At moderate to high excitation energies, where interference effects tend to disappear, we have proposed an independent particle–hole formalism in which particle–hole states are spread in energy with an appropriate strength function obtained from the RPA. The effects of more complex modes, such as 2p–2h ones, are simulated with widths calculated in a semi-classical context. Here, we verify the validity of our approximations for pre-equilibrium proton-induced reactions on $^{90}$Zr target. Our calculations provide a good description of the reaction data and point toward a simplification of the description of nucleon-induced reactions based on averages of microscopic details of the projectile–target interaction.

Appendix

The effects of transitions to incoherent 2p–2h modes can be estimated by the product of an averaged value of the squared transition matrix element times the density of available 2p–2h states $\omega (E)$, [24]

$$\begin{aligned} \gamma _{2p2h}(E) = 2\pi M^{2} \omega (E), \end{aligned}$$

(16)

where we use a semi-empirical parameterization of the residual interaction proposed by Koning and Duijvestijn [23] for the average squared transition matrix element,

$$\begin{aligned} M^{2}=\frac{1}{A^{3}}\left[ 6.8 + \frac{4.2 \times 10^{5}}{(E/2+10.7)^{3}}\right] \;\text{ MeV}^2\,, \end{aligned}$$

(17)

with E being the excitation energy and A the target mass. The density of available 2p–2h states is obtained, following Cline and Blann (with $p=2$ and $h=2$) [24, 25], as

$$\begin{aligned} \omega (E)= {\left\{ \begin{array}{ll} \frac{(gE-1)^{2}}{6E} &{} gE>1 \\ 0 &{} gE<1 , \end{array}\right. } \end{aligned}$$

(18)

where $g=A/15$ MeV$^{-1}$ is the average single-particle density of states [23]. We approximate the escape widths, $\gamma _{n}^{\uparrow }$ and $\gamma _{p}^{\uparrow }$, as

$$\begin{aligned} \gamma ^{\uparrow }_{n,p}\left( E\right) =\frac{(2s+1)m}{\pi ^2\hbar ^2}\pi R^2\frac{\left( E-B_{n,p}-V_B\right) ^2}{2gE}\;\text{ MeV } \end{aligned}$$

(19)

where m is the nucleon mass, $s=1/2$ is the nucleon spin, $B_n$ and $B_p$ are the neutron and proton separation energies, the nuclear radius is taken to be $R=1.2 A^{1/3}$ fm and the emission barrier is taken to be 0 for neutrons and $V_B = 1.44*Z/(1.25A^{1/3})$ MeV for protons.