## Liquid Drop Model of Nucleus

One of the first models which could describe very well 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**), that was published in 1935 by German physicist **Carl Friedrich von Weizsäcker**. This theory is based on **the liquid drop model** proposed by **George Gamow**.

According to this model, the atomic nucleus behaves **like the molecules in a drop** of liquid. But in this nuclear scale, the fluid is made of nucleons (protons and neutrons), which are held together by **the strong nuclear force**. The liquid drop model of the nucleus takes into account the fact that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. The **interior nucleons are completely surrounded** by other attracting nucleons. Here is the analogy with the forces that form a drop of liquid.

In the ground state the nucleus is **spherical**. If the sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a **dumbbell shape** and then may be splitted into **two fragments**. Since these fragments are a more stable configuration, the splitting of such heavy nuclei must be accompanied by **energy release**. This model does not explain all the properties of the atomic nucleus, but does explain the predicted nuclear binding energies.

The nuclear binding energy as a function of the mass number A and the number ofprotons Z based on **the liquid drop model** can be written as: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.

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 ^{4}He, 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 a_{V}, a_{S}, a_{C}, a_{A} and a_{P} 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.m _{p} +N.m_{n} -E_{b}/c^{2}**

where **m _{p}** and

**m**are the rest mass of a proton and a neutron, respectively, and

_{n}**E**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

_{b}^{235}U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original

^{235}U nucleus.

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

**critical energy (E**or

_{crit})**threshold energy**.This table shows critical energies compared to binding energies of the last neutron of a number of nuclei.