Question

What down the semi-empirical mass formula for the liquid drop model? Explain all the terms.

Answer 1

The semi-empirical mass formula (also called the Bethe-Wiezsacker's mass formula) is an expression for binding energy of a nucleus assuming the liquid drop model.

The full formula takes the form,

`E_B = a_1A - a_2A^(2/3) - a_3(Z(Z-1))/A^(1/3) - a_4(A - 2Z)^2/A + delta (A,Z)`

1) The first term is called the volume energy term :

Since, volume energy, `E_V prop V` and `V prop R^3`

But, `R = R_0A^(1/3)`

This gives, `E_V prop A implies E_V = a_1A`

2) The second term is the surface energy term :

The surface term is a correction to the volume term due to the fact that nucleons on the surface of the nucleus interact only with nucleons on one side, unlike the interior nucleons which are attracted equally from all sides.

Now, `E_S prop S` and `S prop R^2`

But, `R prop A^(1/3)` and this gives,

`E_S = -a_2A^(2/3)` where the negative sign indicates that the surface energy contributes to decreasing the binding energy.

3) The third term is the Coulomb energy.

The Coulombic interaction between protons destabilizes the nucleus and hence contributes to a decrease in the binding energy.

Since, each proton is repelled by `(Z-1)` other protons, hence there are `(Z(Z-1))/2` repelling pairs. (the factor of 1/2 is to avoid double counting of pairs).

Also, since Coulombic potential energy is inverse proportional to `R prop A^(1/3)` and hence, '

`E_C prop (Z(Z-1))/A^(1/3) = -a_3(Z(Z-1))/A^(1/3)`

4) The fourth term is the asymmetry term. The maximum stability occurs for Z = N. Any departure from this introduces an symmetry energy which again tends to destabilize and hence reduce the binding energy (hence a negative sign).

The symmetry energy is directly proportional to the neutron excess `(N-Z)` and also, the fraction of nuclear volume in which it is contained.

Thus, `E_A prop (N-Z)` and `E_A prop (N - Z)/A`

Therefore, the symmetry term takes the form,

`E_A = -a_4(A - 2Z)^2/A` where, `N - Z = A - Z - Z = A - 2Z`

5) The fifth term is the pairing energy among nucleons and can take both negative and positive values depending on the particular case.

Hence, the binding energy expression takes the form as written.

The mass of the atomic nucleus can then be estimated as,

`m = Zm_p + Nm_n - E_B/c^2` where, symbols have their usual meanings,