ABSTRACT
The liquid drop model was historically the first model to describe nuclear properties. The idea came primarily from the observation that nuclear forces exhibit saturation properties. The binding energy per nucleon, BE(A, Z)/A is a clear indication of this observation (figure 7.1): a non-saturated force would lead to a binding energy given by the A(A – l)/2 nucleon 2-body interaction energy, in total contradiction with the observation of figure 7.1. Also, the nucleus presents a low compressibility and so a well defined nuclear surface. It is soon clear though, that the liquid drop model taken as a fully classical model, cannot be extrapolated too far in the atomic nucleus
In the inset in figure 7.1, spikes on BE (A, Z)/ A appear at values A = An (n = 1 4He, n = 2 8Be, n = 3 12C, n = 4 16O1,...) and present a favoured n of-particle-like structure reflecting aspects of the nucleon-nucleon force.
In a liquid, the average distance between two fluid ‘particles’ is about equal to that value where the potential interaction energy is a minimum, which, for the nuclear interaction is ≃0.7 fm (figure 7.2). The nucleons are, on average, much farther apart. An important reason is the fact that fermions are Fermi-Dirac particles so we are considering a Fermi liquid. The Pauli principle cuts out a large part of nucleon 2-body interactions since the nearby orbitals in the nuclear potential are occupied (up to the Fermi level). Scattering within the Fermi liquid is a rare process compared to collision phenomena in real, macroscopic fluid systems. Thus, the mean free path of a nucleon, travelling within the nucleus easily becomes as large as the nuclear radius (see figures 7.3(a) and (b)) and we are working mainly with a weakly interacting Fermi gas (see chapter 8).
Saturation of binding energy results in a value of BE (A, Z)/A ≃ 8 MeV independent of A and Z and represents charge independence of the nuclear interaction in the nucleus. Each nucleon interacts with a limited number of <target id="page_242" target-type="page">242</target>Binding energy per nucleon as a function of the atomic mass number A. The smooth curve represents a pure liquid drop model calculation. Deviations from the curve occur at various specific proton (Z) and neutron (<italic>N</italic>) numbers. In the inset, the very low mass region is presented in an enlarged figure. (Taken from <xref ref-type="bibr" rid="ref57">Valentin 1981</xref>.) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367806576/84aaad7b-3d0e-4214-8c25-1fbd41c15abe/content/fig7_1_B.tif"/> nucleons, a conclusion that can be derived by combining the Pauli principle with Heisenberg’s uncertainty principle, and the short-range character of the nucleon-nucleon force. The simple argument runs as follows: Δ E ? · ? Δ t ? ≃ ? ℏ ? ? ? ? ? ? ? ? ? d ? ≃ ? Δ t ? · ? c ? ≃ ? ℏ Δ E ? · ? c ? ≃ ? ℏ · ? m c 2 m c ? · ? Δ E . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367806576/84aaad7b-3d0e-4214-8c25-1fbd41c15abe/content/eqn7_1_B.tif"/> With ΔE for a 2-body interaction in the central zone of the nucleus, ΔE ≃ 200 MeV and d ≃ 1 fm. The total binding energy is now the subtle difference between the total kinetic and potential energy (figure 7.2): the kinetic energy rapidly increases with decreasing internucleon distance whereas the potential 243 Balancing effect between the repulsive kinetic and attractive potential energy contribution to the total nuclear binding energy, all expressed as MeV per nucleon. It is shown that the total binding energy is a rather small, negative value (≃8 MeV/nucleon) and results as a fine balance between the two above terms. The various energy contributions are given as a function of the internuclear distance. (Taken from Ring and Schuck 1980.) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367806576/84aaad7b-3d0e-4214-8c25-1fbd41c15abe/content/fig7_2_B.tif"/> energy becomes negative. For small distances (d ≲ 1 fm), the kinetic energy dominates giving a total positive energy. For large distances (d ≲ 3 fm) there is almost no nuclear interaction remaining. A weak minimum develops at (r 0 ≃ 1.2 fm; the equilibrium value. So, this explains the limited number of nucleons interacting through the nucleon-nucleon interaction in the nucleus.
