57Ni Nucleon Configuration

The isotope is specified as ${}^{57}_{28}\text{Ni}$.

• Number of protons (Z) = 28.
• Number of neutrons (N) = A - Z = 57 - 28 = 29.

The proton number Z = 28 corresponds to a closed shell (magic number), meaning the proton configuration is stable.

For neutrons, the number is N = 29. The $N=28$ neutron number also represents a closed shell. Thus, the first 28 neutrons completely fill the available shells up to the $1f_{7/2}$ orbital.

The 29th neutron will occupy the next single-particle energy level available after the $N=28$ closure.

Shell Model Level Ordering

The sequence of single-particle energy levels following the $N=28$ closure is important for determining the states.

The typical ordering of the lowest energy levels for neutrons beyond $N=28$ is:

• $2p_{3/2}$
• $1f_{5/2}$
• $2p_{1/2}$
• ...

Ground State Spin-Parity

The ground state configuration is determined by the placement of the 29th neutron in the lowest available energy level.

• The lowest energy level available is $2p_{3/2}$.
• For the $2p_{3/2}$ state:
  • The orbital angular momentum quantum number is $l=1$ (p-orbital).
  • The total angular momentum quantum number is $j=3/2$.
  • The parity is calculated as $P = (-1)^l = (-1)^1 = -1$.
• Therefore, the ground state spin-parity is $(3/2)^-$.

First Excited State Spin-Parity

The first excited state arises from the promotion of the valence neutron to the next higher energy level.

• The next higher energy level after $2p_{3/2}$ is $1f_{5/2}$.
• For the $1f_{5/2}$ state:
  • The orbital angular momentum quantum number is $l=3$ (f-orbital).
  • The total angular momentum quantum number is $j=5/2$.
  • The parity is $P = (-1)^l = (-1)^3 = -1$.
• Therefore, the first excited state spin-parity is $(5/2)^-$.

Conclusion

Based on the single-particle shell model predictions for the neutron configuration of ${}^{57}\text{Ni}$, the spin-parity assignments are $(3/2)^-$ for the ground state and $(5/2)^-$ for the first excited state.