# The Theory of Nuclear Masses C. F. von Weizsäcker, Leipzig. With 5 illustrations. (Received on July 6, 1935.) § 1. Problem Statement. § 2. Extension of the Thomas-Fermi Method. § 3. Numerical Evaluation (jointly with F. S. Wang). § 4. The Preference for Even Particle Numbers. § 5. Semi-empirical Representation of Mass Defects. § 6. Summary. § 1. Problem Statement It is now very likely that protons and neutrons are the only elementary constituents of nuclei. Since the rest energies of these particles are large compared to the binding energies of nuclei, their motion in the nucleus can be described in the first approximation by non-relativistic quantum mechanics. If the forces between the elementary particles were known, it would be possible in principle to calculate the binding energies, i.e., the mass defects of all atomic nuclei. However, since attempts to determine these forces theoretically have not yet led to definitive results, we have to take the reverse approach: deriving the nuclear forces from the empirically known mass defects. The mass defects of the lightest nuclei are now very precisely known from the energy balances of disintegration processes. With slightly less accuracy, mass spectrometric measurements by Aston and Bainbridge cover elements throughout the periodic table. These data are supplemented by upper bounds for the binding energies, which can be obtained from the decay or non-existence of certain nuclei, e.g., the radioactive isotope N718 must have a smaller binding energy than C613, since it spontaneously decays into this nucleus. The empirical data show the following laws: 1. The mass defects of the lightest nuclei (H12, H13, He23, He24) increase very rapidly with the particle number. 2. The mass defects of all heavier nuclei increase approximately linearly with the particle number. 3. The packing fractions (mass defects per particle) of lighter nuclei (up to Fe) are not strictly constant but increase slowly. 4. The packing fractions of heavier nuclei decrease slowly after a nearly constant course. 5. Nuclei with even proton and neutron numbers are generally more tightly bound than those with odd numbers. The possibility of explaining the first fact through a specific choice of the force law was shown by Wigner. According to him, an increase in packing fraction is accompanied by an increase in particle density in the nucleus. However, the behavior of nuclear radii of heavier nuclei suggests a constant density. This, combined with the second fact, was first described phenomenologically by Gamow through his "drop model." According to Majorana, this behavior follows from the assumption of an exchange force between protons and neutrons, with the sign of the exchange potential chosen so that the forces cancel. The particles in the nucleus behave exactly like molecules in# The Theory of Nuclear Masses C. F. von Weizsäcker, Leipzig. With 5 illustrations. (Received on July 6, 1935.) § 1. Problem Statement. § 2. Extension of the Thomas-Fermi Method. § 3. Numerical Evaluation (jointly with F. S. Wang). § 4. The Preference for Even Particle Numbers. § 5. Semi-empirical Representation of Mass Defects. § 6. Summary. § 1. Problem Statement It is now very likely that protons and neutrons are the only elementary constituents of nuclei. Since the rest energies of these particles are large compared to the binding energies of nuclei, their motion in the nucleus can be described in the first approximation by non-relativistic quantum mechanics. If the forces between the elementary particles were known, it would be possible in principle to calculate the binding energies, i.e., the mass defects of all atomic nuclei. However, since attempts to determine these forces theoretically have not yet led to definitive results, we have to take the reverse approach: deriving the nuclear forces from the empirically known mass defects. The mass defects of the lightest nuclei are now very precisely known from the energy balances of disintegration processes. With slightly less accuracy, mass spectrometric measurements by Aston and Bainbridge cover elements throughout the periodic table. These data are supplemented by upper bounds for the binding energies, which can be obtained from the decay or non-existence of certain nuclei, e.g., the radioactive isotope N718 must have a smaller binding energy than C613, since it spontaneously decays into this nucleus. The empirical data show the following laws: 1. The mass defects of the lightest nuclei (H12, H13, He23, He24) increase very rapidly with the particle number. 2. The mass defects of all heavier nuclei increase approximately linearly with the particle number. 3. The packing fractions (mass defects per particle) of lighter nuclei (up to Fe) are not strictly constant but increase slowly. 4. The packing fractions of heavier nuclei decrease slowly after a nearly constant course. 5. Nuclei with even proton and neutron numbers are generally more tightly bound than those with odd numbers. The possibility of explaining the first fact through a specific choice of the force law was shown by Wigner. According to him, an increase in packing fraction is accompanied by an increase in particle density in the nucleus. However, the behavior of nuclear radii of heavier nuclei suggests a constant density. This, combined with the second fact, was first described phenomenologically by Gamow through his "drop model." According to Majorana, this behavior follows from the assumption of an exchange force between protons and neutrons, with the sign of the exchange potential chosen so that the forces cancel. The particles in the nucleus behave exactly like molecules in