Question

(a) For a nucleus of \(A=220\), estimate very roughly how (b) If the binding of an interior nucleons would be at the surface. if binding energy of an interior nucleon due to the internucleon attraction were \(E_{1}\) and if all nucleons were as tightly bound, the total binding energy would be \(A E_{1}\). Taking into account the different binding of surface nucleons, roughly what would be the total binding energy?

Short answer

The total binding energy considering surface nucleons is given by the equation \(E_t = (A- N_s)E_1\), where \(N_s = 4\pi A^\frac{2}{3}\) is the number of surface nucleons. Therefore, the total binding energy depends both on the total number of nucleons and on whether the nucleons are on the surface or in the interior of the nucleus.

Step-by-step solution

Calculate the number of surface nucleons

For a roughly spherical nucleus, the number of nucleons on the surface can be estimated from the surface area of the sphere. With the radius of the sphere given by \(R = r_0 A^\frac{1}{3}\) (with \(r_0\) being a constant and \(A\) being the total number of nucleons), we have the surface area calculation as \(4\pi R^2\). Considering one nucleon per \(r_0^2\) on the surface, we find the number of surface nucleons is \(N_s = \frac{4\pi R^2}{r_0^2}\). Substituting the equation for \(R\) we then have \(N_s = 4\pi(r_0 A^\frac{1}{3})^2/ r_0^2 = 4\pi A^\frac{2}{3}\)

Calculate the total binding energy with uniformly bound nucleons

If it was assumed that all nucleons were bound with the same energy \(E_1\), the total binding energy \(E_t\) would simply be the product of the total number of nucleons and the binding energy per nucleon. Thus, \(E_t = A E_1\)

Calculate the total binding energy considering surface nucleons

If we consider that nucleons on the surface are less tightly bound, this would reduce the total binding energy of the nucleus. If we take the binding energy of the surface nucleons to be half that of the interior ones, then we subtract from the total binding energy the binding energy of the surface nucleons counted twice, resulting in \(E_t = A E_1 - 2 N_s E_1/2 = A E_1 - N_s E_1 = (A- N_s)E_1\)

Key concepts

Nucleon Binding Calculations

The concept of nucleon binding energy is fundamental in nuclear physics. It is the energy required to break a nucleus into its constituent protons and neutrons (nucleons). To calculate this, one may start with imagining a scenario where each nucleon is bound to the nucleus with a uniform energy of \(E_{1}\). In such an ideal and simplified case, the total binding energy of a nucleus with \(A\) nucleons would be \(A \times E_{1}\). However, this does not account for variations in binding energies between nucleons at the surface of the nucleus and those in the interior, which leads us to consider them separately for a more accurate estimate.

In practical calculations, various models and formulas are used to estimate the binding energy of nucleons, factoring in not only the number of nucleons but also their arrangement and interactions within the nucleus. This is vital, as it reflects the stability of the nucleus and can explain phenomena such as radioactivity and nuclear reactions.

Surface Nucleons in Nuclei

Nucleons at the surface of a nucleus experience a different environment than those in the interior. Because surface nucleons have fewer neighboring nucleons to interact with, they are not as tightly bound. Consequently, their binding energy is less compared to the interior nucleons.

This distinction is important when considering the stability of isotopes and predicting nuclear reactions. The idea can be visualized by comparing the nucleons in a nucleus to droplets of water in a droplet; those at the surface are more likely to evaporate (be less bound) than those within the droplet. Nuclear models often make adjustments to account for surface nucleons to provide a more accurate depiction of the nucleus's total binding energy.

Nuclear Physics Estimates

Estimating physical quantities is a crucial aspect of nuclear physics that requires understanding both theoretical models and empirical data. When calculating binding energies, physicists must estimate the contribution from both interior and surface nucleons. They may use terms like 'mass defect' and 'binding energy per nucleon' to characterize the stability and energy characteristics of a nucleus.

For example, the semi-empirical mass formula (also known as the Weizsäcker formula) incorporates terms that account for volume, surface area, Coulomb repulsion, asymmetry, and pairing effects. Such estimates guide us in predicting whether a nucleus will undergo radioactive decay or participate in nuclear reactions, and they have applications ranging from energy production to medical diagnostics.

Spherical Nucleus Approximation

Most light and medium-weight nuclei are approximately spherical in shape, and this simplification is widely used in nuclear physics to estimate properties like the binding energy. The spherical nucleus approximation is derived from the liquid drop model, where the nucleus is treated similarly to a drop of an incompressible fluid, with its properties being coherent throughout.

The model is useful for demonstrating concepts such as nuclear binding energy calculations. For instance, by treating the nucleus as a sphere, the surface area can be calculated, allowing for an estimate of the number of surface nucleons, which are important in determining the nucleus's total binding energy. However, it's important to note that this is an approximation; real nuclei can have various shapes, especially heavier ones, which can be more ellipsoidal or even exhibit more complex deformations.