Question 189: Atoms & Nuclei

Question

For a nucleus $${ }_{\mathrm{A}}^{\mathrm{A}} \mathrm{X}$$ having mass number $$\mathrm{A}$$ and atomic number $$\mathrm{Z}$$
A. The surface energy per nucleon $$\left(b_{\mathrm{s}}\right)=-a_{1} A^{2 / 3}$$.
B. The Coulomb contribution to the binding energy $$\mathrm{b}_{\mathrm{c}}=-a_{2} \frac{Z(Z-1)}{A^{4 / 3}}$$
C. The volume energy $$\mathrm{b}_{\mathrm{v}}=a_{3} A$$
D. Decrease in the binding energy is proportional to surface area.
E. While estimating the surface energy, it is assumed that each nucleon interacts with 12 nucleons. ( $$a_{1}, a_{2}$$ and $$a_{3}$$ are constants)
Choose the most appropriate answer from the options given below:

Accepted Answer

In the semi-empirical mass formula, the terms have the following forms:
The volume term (binding energy) is proportional to the volume of the nucleus, and therefore to the number of nucleons $A$.
The surface term takes into account that the nucleons at the surface of the nucleus have fewer nearest neighbors than those in the interior, and so the binding energy is less for surface nucleons. This term is usually proportional to $A^{2/3}$.
The Coulomb term arises because the protons in the nucleus repel each other. This term is usually proportional to $\frac{Z(Z-1)}{A^{1/3}}$.
The symmetry term arises because of the Pauli exclusion principle, but it's not mentioned here.
The pairing term arises because of the tendency of protons and neutrons to pair up, but it's not mentioned here.
Now, let's check the statements:
A: The surface energy per nucleon $b_{s}=-a_{1} A^{2 / 3}$ is incorrect. The correct form should be proportional to $A^{2/3}$, but it should not be per nucleon.
B: The Coulomb contribution to the binding energy $b_{c}=-a_{2} \frac{Z(Z-1)}{A^{4 / 3}}$ is incorrect. The correct form should be proportional to $\frac{Z(Z-1)}{A^{1/3}}$.
C: The volume energy $b_{v}=a_{3} A$ is correct.
D: Decrease in the binding energy is proportional to surface area is correct. This represents the surface term in the semi-empirical mass formula.
E: While estimating the surface energy, it is assumed that each nucleon interacts with 12 nucleons is incorrect. While the exact number can vary depending on the model used, it's generally agreed that each nucleon interacts with a few nearest neighbors, not necessarily 12.
So, the correct answer is C, D only.

Answer

C, D only

Answer

B, C, E only

Answer

B, C only

Answer

A, B, C, D only

Atoms & Nuclei

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