With the aid of the Weizsaecker formula (the semi-empirical mass formula) the binding energy can be calculated very well for nearly all isotopes. Periodic Table
Weizsaecker Formula – Semi-empirical Mass Formula
This formula is called the Weizsaecker Formula (or the semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.
Volume term – aV.A. The first two terms describe a spherical liquid drop of an incompressible fluid with a contribution from the volume scaling with A and from the surface, scaling with A2/3. The first positive term aV.A is known as the volume term and it is caused by the attracting strong forces between the nucleons. The strong force has a very limited range and a given nucleon may only interact with its direct neighbours. Therefore this term is proportional to A, instead of A2. The coefficient aV is usually about ~ 16 MeV.
Surface term – asf.A2/3. The surface term is also based on the strong force, it is, in fact, a correction to the volume term. The point is that particles at the surface of the nucleus are not completely surrounded by other particles. 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 A1/3 and therefore the surface term must be proportional to the surface area i.e. proportional to A2/3.
Coulomb term – aC.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 A1/3. This effect lowers the binding energy because of the repulsion between charges of equal sign.
Asymmetry term – aA.(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 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 level in the nucleus. This is the basis for a correction factor, the so-called symmetry 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 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.
In order to calculate the binding energy, the coefficients aV, aS, aC, aA and aP must be known. 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 following:Using the Weizsaecker formula, also the mass of an atomic nucleus can be derived and is given by:m = Z.mp +N.mn -Eb/c2
where mp and mn are the rest mass of a proton and a neutron, respectively, and Eb is the nuclear binding energy of the nucleus.
From the nuclear binding energy curve and from 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 reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.
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