An Attempt to Calculate the Number of Energy Levels of a Heavy Nucleus

Experiments on slow neutrons, and theoretical considerations of Bohr have shown that heavy nuclei possess an enormous number of energy levels which are very closely spaced if the nucleus is highly excited. A crude method is suggested for calculating the spacing between these levels. The method is statistical: The individual nuclear particles are supposed to move in a simple potential hole, and the energy of the complete nucleus is supposed to be the sum of the energies of the individual particles. A critical discussion of these assumptions is given in section 5. The problem then reduces itself to the calculation of the "entropy" of a Fermi gas containing a given number of particles $A$ and having a given excitation energy $Q$ above the zero point energy of the Fermi gas (cf. section 2 and 3). This calculation gives the total number of levels of the complete nucleus in a given energy interval irrespective of the angular momentum, which will, for most of the levels, be very large. For the theory of neutron capture, it is necessary to calculate the density of nuclear levels with a given angular momentum $I$ (section 4). The spacing of nuclear levels is found to depend on the product of the mass number $A$ and the excitation energy $Q$ of the nucleus, and to be roughly given by $\{\Delta =\frac{4.1·{10}^6{x}^4{e}^{-x}}{\mathrm{}(2I+1)\mathrm{}} \mathrm{volts}\}\{x,=\frac{{\mathrm{}\left(\mathrm{AQ}\right)\mathrm{}}^{\frac{1}{2}}}{2.20},\}$ $Q$ being expressed in MV and $I$ being the nuclear spin. For the capture of slow neutrons by nuclei of medium weight ($A$ around 100), $\Delta$ is of the order 50 to 500 volts. The spacing between adjacent levels decreases rapidly with increasing atomic weight. For given atomic weight, the spacing of the nuclear levels responsible for neutron capture is wider if the capture leads to the formation of a radioactive nucleus than if a stable nucleus is formed. This explains the experimental fact that only moderately large cross sections are found for the capture of thermal neutrons leading to radioactive nuclei while the very largest cross sections are all connected with the formation of stable nuclei. The dependence of the spacing on various factors is discussed (section 6); the results seem to be in qualitative agreement with experiment.