The paper title is “Binding states of individual nucleons in strongly deformed nuclei” by Sven Gösta Nilsson on 1955.

In the introduction, the total nuclear wave function

where is Wigner D-matrix for rotation, is the intrinsic motion of all nucleons. The vibration component is skipped. Because of imcompressible nuclear matter, a vibration needs a lot of energy.

The total wave function should preserve parity, so, a complete wave function should be

Below is a picture of the quantum number.

- In large deformation, is not a good quantum number, but is always a good one.
- At ground state, , so, the rotation angular momentum is perpendicular to the body frame axis.

The single nucleon potential is the usual.

The basis in Nilsson’s paper is the eigen state of harmonic oscillator in L-S representation. The Hamiltonian is

with

The original paper use . And the function solution is

where is the confluent hypergeometric function, and it can be expressed as Laguerre polynomial. So, the solution can be rewrite as

In contrast, in my calculation, I use the presentation, that, the connection is

The different is that, Nilsson needs additional transformation to calculate the coefficient, and the calculation of the and is a bit complicated, due to L-S is not a good quantum number when interaction was included. Thus, in Nilsson paper, he spent sometimes to talk about the calculation of and .

Next, Nilsson gives the calculation parameters of . And since he is using L-S representation, the Nilsson orbital is expressed in that basis. Here is a comparison between my calculation and Nilsson calculation.

Next, he explained that for , energy increase with increase due to “surface coupling”. It can be imagine like this:

In above picture, when is smallest, is perpendicular to the body axis, so the nucleon has large overlap with the whole nucleus, thus it is most bounded.

For small deformation, the deformation field is treated as a perturbation, so that is a good quantum number.

For large deformation, the spin-orbital interaction is treated as perturbation, and the good quantum number is , since , so is also a good quantum number.

In our previous notation, Nilsson orbital is notated as or , which is equivalent in Nilsson as

The wave function of many nucleons.

After established the single nucleon wave function. The receipt of the construction of many nucleons wave function is

- select a set of Nilsson orbitals and form the Slater determent.
- Minimum the total energy by adjusting the deformation parameter.

It is interesting that the total wave function is not a mixture of various Slater determents from different combination of Nilsson orbitals, but rather a single Slater determent.

Ground state spin

Since each Nilsson orbital is degenerated to , which are rotate oppositely. For even-even nucleus, the ground state spin must be zero. For even-odd nucleus, the ground state spin is equal to the of the last single nucleon.

For odd-odd nucleus, the ground state spin can be , the p-n interaction decide which is the ground state.

Decoupling parameter

For odd-A nuclei, the rotational energy is modified by a decoupling factor

And

Magnetic moments, EM transition probability, and ft-value for beta-decay are skipped.

Besides of the skipped material, it turns out the original Nilsson paper did not surprise me. And I still don’t really understand the “decoupling” thing.