Experiments suggest that nuclear forces are non-central. Sometimes this is called tensor forces. Why?
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asked Nov 16, 2021 at 5:11
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1$\begingroup$ Tensor forces account for the dipoles and quadrupole moments and higher order moments existing due to the spin affect of nucleons $\endgroup$– Soham PatilNov 16, 2021 at 5:16
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$\begingroup$ @SohamPatil Why tensor? $\endgroup$– SolidificationNov 17, 2021 at 18:57
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1$\begingroup$ 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 $\endgroup$– Soham PatilNov 19, 2021 at 5:08
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$\begingroup$ Anything that is not scalar (like the internuclear distance r) is described by direction-dependent entities, thus tensors. $\endgroup$– Cosmas ZachosDec 2, 2021 at 21:50
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2 Answers
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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.
This link may also help:- https://www.ancpatna.ac.in/departments/physics/lectures/physics_III_sem/M.Sc-III%20drarunkumar%20Physics%20%20%20Exchange%20Forces%20and%20Tensor%20Forces.pdf
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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)
