Nuclear tensor force ($S_{12}=3({\mathbf{\sigma }}_1\cdot \hat{\mathbf{r}})({\mathbf{\sigma }}_2\cdot \hat{\mathbf{r}})-{\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2$)

The linear operator $V_T(r)S_{12}$ couples nucleon spin orientations to their separation direction $\hat{\mathbf{r}}$. Here ${\mathbf{\sigma }}_i$ are Pauli matrices. A scalar radial potential energy alone omits this directional dependence.

An isotropic pure $L=0$ spatial state has zero electric quadrupole moment. The nonzero deuteron quadrupole moment consequently requires anisotropic structure. With ground-state $J^P=1^{+}$, a nuclear tensor force can mix ${}^3{S}_1$ and ${}^3{D}_1$ waves while preserving total angular momentum and parity. A purely central spin-independent potential cannot produce this mixing. The inference of an orientation-dependent force was already explicit in 1939.^[ref]