A Continuum Theory of the Compound Nucleus

The nucleus is described by an absorption coefficient $\sigma$ which gives the probability per unit time that an incident particle becomes amalgamated with the nucleus (Eq. (1)). This absorption coefficient appears as an imaginary potential in the Schrödinger equation. It is shown that a gradual decrease of $\sigma$ at the nuclear boundary is essential for achieving agreement with experiments (§2). This model gives automatically unit sticking probability for fast neutrons, a cross section proportional to $\frac{1}{v}$ for slow neutrons, and no one-particle resonances for particles which have to penetrate a potential barrier (§3). Quantitative calculations are made with $\sigma$ varying as $e^{-\frac{\mathrm{}(r-R)\mathrm{}}{b}}$ outside the nucleus. For neutrons of zero orbital momentum, the formation probability of the compound nucleus is found to be $\zeta =1-e^{-2\mathrm{}\pi \mathrm{kb}}$ where $k$ is the wave number. It is significant that $\zeta$ depends on the diffuseness $b$ of the nuclear boundary rather than on the nuclear radius $R$. On the other hand, the factor $2\pi$ ensures that $\zeta$ is close to unity already for energies of about 1 Mev (§4). The total cross section in the region of overlapping levels, and the average level width in the region of separated levels are expressed in terms of the formation probability $\zeta$. The relation with the elastic scattering is discussed (§5). The case of slow neutrons is treated in detail. With an average spacing $D$ of 10 volts between levels of the same $J$, the average neutron width is about $2\times {10}^{-3\mathrm{}}{E}^{\frac{1}{2}}$ for a neutron energy $E$, in rough agreement with the meager experimental data. With these assumptions, the neutron width will become larger than the radiation width already for $E\approx {10}^3$ ev; experiments on the capture of "medium fast" neutrons ($\approx 2\times {10}^5$ ev) can be interpreted roughly on this basis. The elastic potential scattering of slow neutrons is shown to be equivalent to the scattering from a hard sphere whose radius $R^{\prime }$ is defined by the condition that $\sigma \left(R^{\prime }\right)=\left(\frac{\hslash }{2m{b}^2}\right)e^{-2C}$ where $C$ is Euler's constant 0.577... (§6). The case of particles which move in a non-nuclear potential $V$ (electrostatic or centrifugal) is treated in §4, 7, 8 for various relations between the energy $E$ of the incident particle and the height $V\left(R^{\prime }\right)$ of the potential barrier. If $E-V\left(R^{\prime }\right)$ is more than about 1 Mev, the formation probability is close to one, as for a fast neutron (§4). If $E$ is about equal to $V\left(R^{\prime }\right)$, $\zeta$ is still of the order of unity (§8). For $E<V\left(R^{\prime }\right)$, $\zeta$ contains the well-known penetrability of the potential barrier, $e^{-2G}$, aside from other factors which increase slowly with $|E-V(R^{\prime }\left)\right|$ (§7). The magnitude of $\sigma$ inside the nucleus is derived for the case of extremely high energies from the Born approximation and the variation of $\sigma$ with energy is shown to be slight in this case. Although quantitative conclusions on the case of moderate energies cannot be drawn, it seems likely that $\sigma$ is at least 20-40 Mev in that case (§9). Finally, it is shown that no appreciable change of results is caused by an attractive or repulsive nuclear potential added to the nuclear absorption potential (§10). In the main part of the paper, it has been assumed that the average interaction between nucleus and particle is zero.