Neutron–proton spin–spin correlations in the ground states of $N=Z$ nuclei

Abstract

We present expressions for the matrix elements of the spin–spin operator $\mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}$ in a variety of coupling schemes. These results are then applied to calculate the expectation value $\langle \mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}\rangle$ in eigenstates of a schematic Hamiltonian describing neutrons and protons interacting in a single-l shell through a Surface Delta Interaction. The model allows us to trace $\langle \mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}\rangle$ as a function of the competition between the isovector and isoscalar interaction strengths and the spin–orbit splitting of the $j=l\pm {{1}\big /{2}}$ shells. We find negative $\langle \mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}\rangle$ values in the ground state of all even–even $N=Z$ nuclei, contrary to what has been observed in hadronic inelastic scattering at medium energies. We discuss the possible origin of this discrepancy and indicate directions for future theoretical and experimental studies related to neutron–proton spin–spin correlations.

Appendix: SU(4) character of \(\mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}\)

The tensor character of $\mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}$ under SU(4) can be determined by noting that the generators (3) transform as $[211]\equiv (101)$ tensors, where the first refers to the notation $[f'_1f'_2f'_3]\equiv [f_1-f_4,f_2-f_4,f_3-f_4]$ in terms of the Young tableau $[f_1f_2f_3f_4]$ (for a discussion of the latter, see Ref. [29]) and the second is the notation of supermultiplets from Ref. [30], $(\lambda \mu \nu )\equiv (f_1-f_2,f_2-f_3,f_3-f_4)$. The tensor character of any product of the SU(4) generators must therefore belong to

$$\begin{aligned}&[211]\times [211]\nonumber \\ {}&\quad = [0]+[211]^2+[22]+[31]+[332]+[422], \nonumber \\&\quad =(000)+(101)^2+(020)+(210)+(012)+(202). \end{aligned}$$

(20)

Furthermore, since $\mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}$ is a scalar in spin and a combination of a scalar and a tensor in isospin, it can only belong to an SU(4) irreducible representation which contains the spin–isospin combinations $(ST)=(00)$ or (02). Given the $\mathrm{SU}(4)\supset \mathrm{SU}_S(2)\otimes \mathrm{SU}_T(2)$ reductions

$$\begin{aligned}{}[f'_1f'_2f'_3]={}&(\lambda \mu \nu )\rightarrow \textstyle {\sum _{ST}(ST)}, \nonumber \\ [0]={}&(000)\rightarrow \mathbf {(00)}, \nonumber \\ [211]={}&(101)\rightarrow (01)+(10)+(11), \nonumber \\ [22]={}&(020)\rightarrow \mathbf {(00)}+(11)+\mathbf {(02)}+(20), \nonumber \\ [31]={}&(210)\rightarrow (01)+(10)+(11)+(12)+(21), \nonumber \\ [332]={}&(012)\rightarrow (01)+(10)+(11)+(12)+(21), \nonumber \\ [422]={}&(202)\rightarrow \mathbf {(00)}+(11)^2+\mathbf {(02)}+(20)\nonumber \\&+(12)+(21)+(22), \end{aligned}$$

(21)

it follows that $\mathbf {S}_\mathrm{n}\cdot \mathbf {S}_\mathrm{p}$ must be a combination of (000), (020) and (202) SU(4) tensors.