Notation for nuclear energy states

Somewhat parallel to the labeling of atomic energy states, the labels of nuclear states are determined by their angular momenta. For single particle states you know that the spin part is S=1/2 and then the orbital angular momentum is used to determine a letter designation according to the spectroscopic notation. If L=2, then the state is represented by "d" and then the total angular momentum could be either L-S=3/2 or L+S=5/2. The two states would then be labeled:

The number n preceding the letter designation is analogous to the principal quantum number in the hydrogen atom in that it indicates the order of the energy states. But this number n is not subject to the same limitations as in the atomic case - it is just an indexing parameter. So the lowest energies with L=2 are labeled 1d (1d_3/2,1d_5/2), but there are higher levels with L=2 with would be labeled 2d_3/2,2d_5/2,3d_3/2,3d_5/2, etc.

For light nuclei it is presumed that the nucleons will fill levels according to the sequence of the shell model. Those nucleons in closed shells will be presumed not to contribute to the total nuclear spin, so the configuration of the nucleus will be indicated by the symbols for the nucleons outside closed shells. For example, ¹⁷O has one neutron outside a closed shell (Z=N=8= magic number) and its ground state would be designated (1d_5/2)¹.

The parity of the nuclear state is also an important characteristic and often listed as a part of the notation for the state. By correlation with the single particle shell model, a value of L can be associated with a given odd nucleon, and the parity of the state is (-1)^L. For the ¹⁷O case, the associated L is 2 (d state), so the parity can be predicted to be even. For ¹⁵O, the odd neutron is in a p state (L=1) so the parity of that state is odd. The state of the ¹⁵O is sometimes written as 1p_1/2^- and that of ¹⁷O as 1d_5/2⁺ to indicate both the spin and parity of those nuclei in their ground states. The parity, like the nuclear spin, is taken to be a characteristic of the entire nucleus, but the parity would be even for closed shells so it is usually sufficient to consider the parity of the odd nucleon.