Nilsson model

The Nilsson model is a nuclear shell model treating the atomic nucleus as a deformed sphere. In 1953, the first experimental examples were found of rotational bands in nuclei, with their energy levels following the same J(J+1) pattern of energies as in rotating molecules. Quantum mechanically, it is impossible to have a collective rotation of a sphere, so this implied that the shape of these nuclei was nonspherical. In principle, these rotational states could have been described as coherent superpositions of particle-hole excitations in the basis consisting of single-particle states of the spherical potential. But in reality, the description of these states in this manner is intractable, due to the large number of valence particles—and this intractability was even greater in the 1950s, when computing power was extremely rudimentary. For these reasons, Aage Bohr, Ben Mottelson, and Sven Gösta Nilsson constructed models in which the potential was deformed into an ellipsoidal shape. The first successful model of this type is the one now known as the Nilsson model. It is essentially a nuclear shell model using a harmonic oscillator potential, but with anisotropy added, so that the oscillator frequencies along the three Cartesian axes are not all the same. Typically the shape is a prolate ellipsoid, with the axis of symmetry taken to be z.

Hamiltonian

For an axially symmetric shape with the axis of symmetry being the z axis, the Hamiltonian is

H

=

1

2

m

ω

z

2

z

2

+

1

2

m

ω

⊥

2

(

x

2

+

y

2

)

−

c

1

ℓ

⋅

s

−

c

2

(

ℓ

2

−

⟨

ℓ

2

⟩

N

)

.