The Bethe–Weizsaecker Formula

Whenever it comes to Nuclear physics, the Bethe Weizsaecker formula is one of the most fundamental formulas that one comes across. Also known as the Semi-empirical binding energy or mass formula,

The Bethe Weizsaecker formula approximates the Nuclear Binding energy as a function of protons and neutrons contained in the nucleus.

Bethe–Weizsäcker formula
Detailed Bethe Weizsaecker formula
Courtesy: web2.ph.utexas.edu

Origin of Bethe Weizsaecker formula:

The term binding energy of a nucleus generally refers to the minimum energy required to separate an atomic nucleus completely into its constituent protons and neutrons. The B.W formula aims to calculate the same Binding Energy. Every formula has a logic behind it. Some are derived as a result of experimental data, some as a result of simulations, and some are based on different theoretical models. The Bethe Weizsaecker has been generated by considering the Liquid drop model of a nucleus. So, in order to understand the origin of B.W formula, first of all we need to have an idea about what the liquid drop model is.

Liquid drop model:

Time and again, various Nuclear models have been put forward to explain the structure and properties of a nucleus. Proposed by Niels Bohr and F. Kalcar, the Liquid drop model is one the oldest model in this regime. The liquid drop model in nuclear physics treats the nucleus as a drop of incompressible nuclear fluid of very high density. Thus, it is an Analogical model, developed by considering a nucleus analogous to a drop of liquid. The basic similarity between the two is the fact that both of them have a definite constant density and a definite surface. No doubt, it is just a crude model that does not explain all the properties of the nucleus, but is still helpful to explain the spherical shape of most of the nuclei along with some other properties.

Components of Bethe Weizsaecker formula:

In total, the binding energy given by B.W formula is a combination of five different terms. These include the contributions from Volume Energy, Surface Energy, Coulomb Energy, Pairing Energy and some anti symmetric contributions. To derive and understand the mathematics of all these terms, first of all we need to understand the relationship between mass number A and the Radius R of nucleus.

Relationship between R and A: We have assumed that the nuclear density is nearly constant. Since, Density=Mass/Volume, therefore, the nuclear mass A is directly proportional to the volume V of the nucleus, ie. { A \propto V} . Also, {V\propto R^{3}} . This means that {A\propto R^{3}} . This eventually gives us the relation that {R \propto A^{\frac{1}{3}}} Now, Let's discuss each of these energy contribution terms in detail:

Volume Energy:

The first term in the formula corresponds to the volume energy {E_{v}}. When a large number of nucleons (protons+neutrons) of the same size are packed together into a small volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume of all the nucleons surrounding a single nucleon. Therefore, {E_{v} \propto A}. This means that {E_{v} = a_{v} A}, where {a_{v}} is a constant of proportionality.

As the nucleons on the surface are only surrounded by nucleons below them, so this term is generally over estimated. Physically, the volume energy corresponds to the amount of heat energy required to transform a liquid to its vapour state being proportional to the mass of the liquid.

Surface energy:

As mentioned earlier, the nucleons on surface are only affected by the nucleons below them. So, to compensate for the excessive volume energy, the surface energy term {E_{s}} comes into picture. This effect is similar to the concept of surface tension in liquids. It results in reduction of the total binding energy due to the nucleons on the surface. The correction due to surface energy is proportional to the surface area of the nucleus. Hence, Mathematically, {E_{s} \propto R^{2}}. This gives {E_{s} = a_{s} A^{\frac{2}{3}}}, where {a_{s}} is a constant of proportionality. Since, this term leads to decrease in the value of binding energy, so it comes with a negative sign in the B.W formula.

Coulumb Energy:

The third term, the Coulomb energy {E_{c}} originates from the Coulumb electrostatic repulsion between the charged particles in the nucleus. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy. This energy term is directly proportional to the possible number of combinations for a given proton number Z, which is {\frac{Z (Z-1)}{2} } and is inversely proportional to the distance of separation. So the energy associated with Coulomb repulsion is : {E_{c} \propto \frac{Z (Z-1)}{R}}. This gives {E_{c} = a_{c} \frac{Z (Z-1)}{A^{\frac{1}{3}}}}, where {a_{c}} is a constant of proportionality.

Asymmetry energy:

The fourth term, ie. the Asymmetry energy {E_{a}} originates from the asymmetry between the number of protons and the number of neutrons in the nucleus. The Nuclear data for stable nuclei has indicated that for lighter nuclei, the number of protons is almost equal to that of neutrons, ie. N=Z. However, as A increases, the symmetry of proton and neutron number is lost. Then, the number of neutrons exceed that of protons to maintain the nuclear stability. This excess of neutrons over the protons, ie. N-Z represents the measure of asymmetry. It eventually decreases the stability or the Binding energy of the medium or heavy nuclei.
Therefore, Mathematically, {E_{a} \propto \frac{ (N-Z)^{2}}{A}}. Substituting,{ A=N+Z}, we get {E_{a} = a_{a} \frac{ (A-2Z)^{2}}{A}}, where {a_{a}} is a constant of proportionality for the antisymmetry energy part.

Pairing Energy:

The last term in the formula is that of pairing energy {E_{p}}. It is a purely corrective term. This term arises from the tendency of proton pairs and neutron pairs to occur. This term takes into account the affect of spin coupling. Due to the Pauli exclusion principle, the nucleus would have a lower energy if the number of protons with spin up and spin down are equal. This is also true for neutrons. Both protons and neutrons have equal numbers of spin up and spin down particles only if both N and Z are even. Mathematically, { E_{p} = ( \pm,0) \frac {\delta}{A^{\frac{3}{4}}}}. Here {\delta} has different values for different configuration of nucleons as given in the table below:

Bethe–Weizsäcker formula
Pairing energy for different configurations of Nucleons

The pairing energy term shows that a nucleus with even number of particles is more stable than the one with odd number.

Importance of Bethe–Weizsäcker Formula:

The Bethe Weizsaecker formula helps to calculate the binding energy very well for nearly all isotopes. The B.W formula gives the value of binding energy. Since, the atomic masses depend upon the binding energy values, So B.W formula is also intrumental in calculating atomic masses. This formula provides a descent fit for the binding energy curve of heavy nuclei. However, it provides a poor fit for light nuclei like He. It also helps to explain the stability of nuclei against {\beta} decay.

The semi-empirical mass formula, based on the liquid drop model, compared to the data
Courtesy: web2.ph.utexas.edu

Author's message:

On the whole, The Bethe Weizsaecker formula is of utmost importance in nuclear physics. It plays a major role in explaining the Binding energy curve. The values of constants mentioned here can be found using different techniques. No doubt that different modified versions of this formula with a wider accuracy are here these days, but the base of all the modified formulae remains the same. I hope this article gave you a basic overview about what the famous Semi Empirical Mass formula really is !

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