Collective Degrees of Freedom

Figure C.1
Collective rotation.

The shape of the ground state of a nucleus can be either spherical or deformed. For a quantum mechanical system with spherical symmetry (i.e. a spherically symmetric wave function), rotation is not physically observable. Therefore, the existence of nuclear rotation constitutes proof of nuclear deformation. Evidence for nuclear rotation abounds. This rotational behavior is termed "collective" since it is necessary for many individual nucleons to move coherently in order to rotate the nucleus as a whole. In close correspondence with classical rotational behavior, the excitation energy spectrum of a quantum mechanical rotor is given by

Figure C.2
Collective gamma vibration.

Here is the moment of inertia and the angular momentum I takes the values

I = I₀ , I₀ + 1, I₀ + 2 ... (I₀ ≠ 0)
I = 0, 2, 4, 6... (I₀ = 0)

There are many nuclei that exhibit such sequences of excited states – called rotational bands. Figure C.1 illustrates the rotation of an axially symmetric deformed nucleus with ground state spin I_GS = 0. It shows that in this case the axis of rotation is perpendicular to the symmetry axis of the nucleus. This is also approximately true for I_GS ≠ 0. provided that I » I_GS .

Shape vibrations in which the shape of a nucleus performs a vibration about an equilibrium shape are another form of collective excitations. There are two different forms called β- and γ- vibrations. In the former case an axially symmetric shape β₂- oscillates about an equilibrium deformation β₀; hence axial symmetry is maintained. Consequenly, for nuclei with even numbers of protons and neutrons for which I_GS = 0, a quantum of β₂ vibration has angular momentum and parity 0⁺. Figure C.3 demonstrates this vibration. A nucleus in this state can perform a collective rotation, thus creating a rotational band called β -band based on a 1-quantum β excitation.

Figure C.3
Collective beta vibration.

In γ vibrations the equilibrium shape is generally assumed to be axially symmetric (γ₀ = 0); figure C.2 illustrates this type of vibration. A quantum of γ-vibration has angular momentum ±2 along the symmetry axis. A γ-band is a rotational band based on a 1-quantum γ-excitation of the ground state. So the lowest state in a γ-band has spin and parity 2⁺. Figure C.4 is a partial level scheme of the deformed Nucleus Er illustrating a typical rotational ground state band as well as a β- and γ- band. The excitation energies of the excited states are given in keV. The arrows indicate observed gamma-ray transitions.

Click on the thumbnail to see the full image.

Figure C.4
A partial level scheme.