Atomic physics

In the extreme single particle model, all the properties of the nucleus are ascribed by the single unpaired nucleon moving in a potential due to the remaining nucleons.

A nucleon of mass \(M\) moving in a mean spherically symmetric potential \(V(r)\) are determined by the Schrödinger equation

\begin{equation} \biggl[\nabla^2 + \frac{2M}{\hbar^2} \bigl(E - V(r)\bigr)\biggr] \ket{\psi(r)} = 0 \end{equation}

where \(E\) is the energy eigenvalue. The solution is of the form

\begin{equation} \psi_{nlm}(r) = u_{nl}(r) Y_{lm}(\theta,\phi) \end{equation}

where \(Y_{lm}(\theta,\phi)\) are the spherical harmonics and \(u_{nl}\) is the radial function.

One of the objective is to find a potential model that reproduce the magic numbers.

def woods_saxon(r, V0=50.0, A=50, a=0.5, r0=1.25):
    R = r0 * (A ** (1/3))
    return -V0 / (1 + np.exp((r - R) / a))

The single particle model refine the extreme single particle model by taking into account the states of several particles. The nucleus is then consists of

1. filled shells which contain the maximum number of neutrons and protons allowed by the Pauli exclusion principle
2. unfilled shells containing loose particles

All the properties of the nucleus are then described by all the loose particles. [2, p. 238]

By assuming that the nucleons move independently and the only correlations between them are entirely due to the Pauli exclusion principle, the nuclear wave funciton is represented by the configuration of its loose particles and is given by the Slater determinant (\(k < A\))

\begin{equation} \ket{\Psi^v} = \sqrt{\frac{1}{k!}} \begin{vmatrix} \psi^{v_1(\mathbf{r}_1)} & \ldots & \psi^{v_1(\mathbf{r}_k)} \\ \vdots & & \vdots \\ \psi^{v_k(\mathbf{r}_1)} & \ldots & \psi^{v_k(\mathbf{r}_k)} \end{vmatrix} = (nlj)^k \end{equation}

All the different states of the \(k\) particles are then degenerate in energy.

The degeneracy is removed by considering the two-particle interaction.

The individual particle model, also referred to as the many particle shell model, all the particles in the nucleus are taken into account for the properties of the entire nucleus. [2, p. 248]

The wave function of the nucleus is the properly antisymmetrized Slater determinant of the single-particle wave function for all the \(A\) particles

\begin{equation} \ket{\Psi^v} = \sqrt{\frac{1}{A!}} \begin{vmatrix} \psi^{v_1(\mathbf{r}_1)} & \ldots & \psi^{v_1(\mathbf{r}_A)} \\ \vdots & & \vdots \\ \psi^{v_A(\mathbf{r}_1)} & \ldots & \psi^{v_A(\mathbf{r}_A)} \end{vmatrix} \end{equation}

and are solutions of the Schrödinger equation

\begin{equation} \hat{H}_1 \ket{\Psi^v} = E_1 \ket{\Psi^v} \end{equation}

where

\begin{equation} H_1 = \sum_{i=1}^A \biggl[-\frac{\hbar^2}{2m}\nabla_i^2 + V_1(\mathbf{r}_i)\biggr] \end{equation}

and \(V_1(\mathbf{r}_i)\) is a suitably chosen common single particle potential (most frequently taken to be the harmonic oscillator potential).