Weizsaecker Formula – Semi-empirical Mass Formula

One of the first models that could describe the behavior of the nuclear binding energies and therefore of nuclear masses was the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF) published in 1935 by German physicist Carl Friedrich von Weizsäcker. This theory is based on the liquid drop model proposed by George Gamow. The physical meaning of this equation can be discussed term by term.

Volume term

Volume term – a_V.A. The first two terms describe a spherical liquid drop of an incompressible fluid that contributes from the volume scaling with A and from the surface, scaling with A^2/3. The first positive term, a_V.A is known as the volume term, which is caused by attracting strong forces between the nucleons. The strong force has a very limited range, and a given nucleon may only interact with its direct neighbors. Therefore this term is proportional to A instead of A². The coefficient a_V is usually about ~ 16 MeV.

Surface term

Surface term – a_sf.A^2/3. The surface term is also based on the strong force. It is, in fact, a correction to the volume term. The point is that other particles do not completely surround particles at the surface of the nucleus. In the volume term, it is suggested that each nucleon interacts with a constant number of nucleons, independent of A. This assumption is very nearly true for nucleons deep within the nucleus but causes an overestimation of the binding energy on the surface. By analogy with a liquid drop, this effect is indicated as the surface tension effect. If the volume of the nucleus is proportional to A, then the geometrical radius should be proportional to A^1/3, the surface term must be proportional to the surface area, i.e., proportional to A^2/3.

Coulomb term

Coulomb term – a_C.Z².A^-⅓. This term describes the Coulomb repulsion between the uniformly distributed protons and is proportional to the number of proton pairs Z²/R, whereby R is proportional to A^1/3. This effect lowers the binding energy because of the repulsion between charges of the equal sign.

Asymmetry term

Asymmetry term – a_A.(A-2Z)²/A. This term cannot be described as ‘classically’ as the first three. This effect is not based on any of the fundamental forces. This effect is based only on the Pauli exclusion principle (no two fermions can occupy exactly the same quantum state in an atom). The heavier nuclei contain more neutrons than protons. These extra neutrons are necessary for the stability of the heavier nuclei. They provide (via the attractive forces between the neutrons and protons) some compensation for the repulsion between the protons. On the other hand, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy levels in the nucleus. This is the basis for a correction factor, the so-called symmetry term.

Pairing term

Pairing term – δ(A,Z). The last term is the pairing term δ(A, Z). This term captures the effect of spin-coupling. Nuclei with an even number of protons and an even number of neutrons are (due to the Pauli exclusion principle) very stable thanks to the occurrence of ‘paired spin’. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

Table of Calculated Binding Energies

With the aid of the Weizsaecker formula, the binding energy can be calculated very well for nearly all isotopes. This formula provides a good fit for heavier nuclei. For light nuclei, especially for ⁴He, it provides a poor fit. The main reason is the formula does not consider the internal shell structure of the nucleus.

The coefficients a_V, a_S, a_C, a_A, and a_P must be known to calculate the binding energy. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0, Wiley, 1994., the coefficients in the equation are the following:Using the Weizsaecker formula. Also, the mass of an atomic nucleus can be derived and is given by:m = Z.m_p +N.m_n -E_b/c²

where m_p and m_n are the rest mass of a proton and a neutron, respectively, and E_b is the nuclear binding energy of the nucleus.

From the nuclear binding energy curve and the table, it can be seen that, in the case of splitting a ²³⁵U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ²³⁵U nucleus.

According to the Weizsaecker formula, the total energy released for such a reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.

See also: Liquid Drop Model