Question

The semiempirical binding energy formula has four terms. Suppose we have a nucleus with 18 protons and 22 neutrons, For each term in the formula, indicate (without calculation) whether adding one more proton would cause an increase or a decrease, and explain why it should have this effect. Focus on the underlying idea.

Short answer

Adding a proton would cause an increase in the volume and Coulomb terms, would not sigificantly affect the surface term and would reduce the asymmetry term.

Step-by-step solution

Analyze the Volume Term

The volume term considers the number of nucleons. Protons are a kind of nucleon, so adding one more would increase the volume term.

Analyze the Surface Term

The surface term considers the surface area. This would remain largely unchanged by the addition of a proton because the size of a nucleus does not significantly change by adding one nucleon.

Analyze the Coulomb Term

The Coulomb term considers the repulsion between protons. Adding a proton would cause an increase in the Coulomb term because it would increase the repulsion.

Analyze the Asymmetry Term

The asymmetry term measures the difference between the number of protons and neutrons. Adding a proton would reduce the asymmetry term if the number of neutrons is larger, which is the case here.

Key concepts

Volume Term

The volume term in the semi-empirical binding energy formula is crucial as it accounts for the binding energy contribution from all nucleons, including both protons and neutrons. Each nucleon adds to the cohesion of the nucleus, contributing almost equally to the binding energy. Think of it like packing marbles tightly in a bag, where each marble adds to the overall compactness and stability.

When you add another proton to the nucleus, you're indeed adding another proton-sized piece to this tightly packed structure. This immediately increases the total number of nucleons, thereby increasing the volume term.

• Each additional nucleon increases the overall binding energy, attributed to the fact that the cohesive forces within the nucleus are roughly proportional to the total number of particles.
• Despite the increase in binding energy from the volume term, consider that there are balancing factors from other terms, such as the Coulomb term we will discuss next.

Coulomb Term

The Coulomb term is another significant component in the binding energy formula, specifically dealing with the repulsive forces that exist between protons within the nucleus. Remember that protons have a positive charge, causing them to naturally repel one another.

By introducing an additional proton into the nucleus, you're essentially intensifying this repulsive force. Thus, increasing the Coulomb term leads to an increase in the repulsive energy that counteracts the cohesive forces binding the nucleus.

• The repulsion results from the fact that like charges repel, and since all protons are positively charged, more repulsions occur as the proton count increases.
• Hence, each additional proton contributes negatively towards the binding energy by adding extra repulsive energy.
• Understanding this concept highlights the delicate balance needed to keep nuclei stable.

Asymmetry Term

The asymmetry term reflects differences in the ratio of protons to neutrons within a nucleus. Ideally, a stable nucleus has a balance between these two types of particles. When there's a noticeable discrepancy, it can destabilize the nucleus.

In the context of adding a proton when there are already more neutrons than protons, adding this proton actually helps to balance the asymmetry.

• This can slightly reduce the asymmetry term's contribution, moving towards a more balanced state.
• Remember, the ideal stability is often found when the number of protons and neutrons are close, minimizing this term’s effect.
• This stabilization effect is only partial; other factors can influence overall nuclear stability more drastically.

Thus, adding a proton in a neutron-rich environment assists in equalizing the balance, but it won't completely eliminate the contribution of the asymmetry term to the overall binding energy computation.