One-pion exchange

Although the pion has spin zero, one-pion exchange depends on nucleon spin and isospin and includes a nuclear tensor force. Shorter-range and multiple-nucleon interactions also matter. The scalar Yukawa potential isolates a range mechanism rather than specifying the complete nuclear force.^[ref]

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]