# Why are non-central nuclear forces also called tensor forces?

Experiments suggest that nuclear forces are non-central. Sometimes this is called tensor forces. Why?

• Tensor forces account for the dipoles and quadrupole moments and higher order moments existing due to the spin affect of nucleons Nov 16, 2021 at 5:16
• @SohamPatil Why tensor? Nov 17, 2021 at 18:57
• Higher order terms in moment like dipoles and quadrupoles are described in tensor form charges are 1x1 tensor dipoles 3x1 and quadrupoles 3x3 tensors in electrostatics Nov 19, 2021 at 5:08
• Anything that is not scalar (like the internuclear distance r) is described by direction-dependent entities, thus tensors. Dec 2, 2021 at 21:50

The nuclear force also has a tensor component which depends on the interaction between the nucleon spins and the angular momentum of the nucleons, leading to deformation from a simple spherical shape.

• Kindly only read last 2 pages Dec 3, 2021 at 3:28

The potential energy of interaction of two nucleons depends on their distance and their spins, there are three vectors on which this energy can depend, the unit vector $$\,\mathbf{n}\,$$ following the radius vector from one nucleon to the other and their spins.

It is obvious that we can only form from these three vectors $$\mathbf{n}, \mathbf{s_{1}}, \mathbf{s_{2}} \;$$ two independent scalars: $$\mathbf{s_{1}} \mathbf{s_{2}}$$ and $$(\mathbf{n}\mathbf{s}_{1}) (\mathbf{n}\mathbf{s}_{2})$$ and $$\mathbf{n}\mathbf{s}$$ is not a real scalar but a pseudoscalar ($$\mathbf{n}$$ being a polar vector and $$\mathbf{s}$$ an axial vector).

The interaction potential of two nucleons can be put in the form :

$$U= U_{1}(r)+U_{2}(r)(\hat{\mathbf{s}}_{1}\hat{\mathbf{s}}_{2})+U_{3}(r)[3(\hat{\mathbf{s}}_{1}\mathbf{n}) (\hat{\mathbf{s}}_{2}\mathbf{n})-\hat{\mathbf{s}}_{1}\hat{\mathbf{s}}_{2}]\;\;\;\;$$ (1)

the 3rd term is called tonsorial forces because the term: $$S_{12}=3(\hat{\mathbf{s}}_{1}\mathbf{n}) (\hat{\mathbf{s}}_{2}\mathbf{n})-\hat{\mathbf{s}}_{1}\hat{\mathbf{s}}_{2}\;$$ can be written in the form $$\;S_{12}=\sum_{i,j=1}^{3}T_{ij}\mathbf{n}_{i}\mathbf{n}_{j}\;$$ where $$\;T_{ij}=\frac{3}{2}(\sigma_{1i}\sigma_{2j}+\sigma_{1j}\sigma_{2i})-\delta_{ij}\vec{\sigma}_{1}.\vec{\sigma}_{2}\;$$ is an irreducible tensor of 2nd order, with $$\;\vec{\mathbf{s}}_{1}=\frac{1}{2}\vec{\sigma}_{1};\;\vec{\mathbf{s}}_{2}=\frac{1}{2}\vec{\sigma}_{2} \;\;\;\;$$

(1) Quantum mechanics by l.d. landau and e.m. lifshitz (volume III)

and also (sorry it's in french) https://lphe.epfl.ch/oschneid/cours/cours4_2005-2006/polycopie_noyau_atomique.pdf

formulas (4,39) and (4,40)