Tensor force

Brueckner, K. A., Phys. Rev. 96, 508, Nuclear saturation and two-body forces. II. Tensor forces/ ... 

The simplest bound nuclear system, the deuteron, consists of a neutron and a proton. The deuteron is known to have a quadmpole moment, 0.00286 barns, which tells us that the deuteron is not perfectly spherical and that the force between two nucleons is not spherically symmetric. Formally, we say the force between two nucleons has two components, a spherically symmetric central force and an asymmetric tensor force that depends on the angles between the spin axis of each nucleon and the line connecting them. 

Rubinow, S.I. (1955). Variational principle for scattering with tensor forces, Phys. Rev. 98, 183-187. 

The new force that replaced the central force was more complicated. A central force acting between two objects depends only on the magnitude of their separation. The presence of a quadrupole moment within the deuteron required a tensor force that depended not only on the separation between protons and neutrons, but also on the angles that their spins make with the line joining them. 

When we studied the radio-frequency spectrum of D2 we hit another surprise [5]. The separation of the spectral lines in D2 were greater than in H2 even though the nuclear spin-spin interaction and the nuclear spin molecular rotation interaction should be much less. We found a similar anomaly for HD. We finally interpreted this as due the deuterium nucleus having a quadrupole moment (being ellipsoidal in shape) which gave rise to a spin dependent electrical interaction. The existence of the quadrupole moment, in turn, implied the existence of a new elementary particle force called a tensor force. In this way, magnetic resonance made a fundamental contribution to particle physics. 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

To demonstrate the influence of on-shell differences on nuclear structure results, one needs to find a case, where two potentials are essentially identical off-shell (including the strength of the tensor force, see discussion below), but differ on-shell. It is not easy to isolate such a case, since, in general, different potentials show differences on- and off-shell. 

Let us consider an example. The T-matrix in the Si state is attractive below 300 MeV lab. energy. If a potential has a strong (weak) tensor force, then the integral term in Eq. (5) is large (small), and the negative Born term will be small (large) to yield the correct on-shell T-matrix element. 

MeV ( ,ab = 2q /M). The abscissa, k, is the variable over which the integration in Eq. (5) is performed. It is seen that, particularly for large off-shell momenta, the Bonn B potential is smaller than the Paris potential. However, notice also that at the on-shell point (q = k, solid dot in Fig. 12) both potentials are identical (both potentials predict the same Ei parameter). Thus, the Bonn B potential has a weaker off-shell tensor force than the Paris potential. Since the Bonn B and the Paris potential predict almost identical phase shifts, the Born term (central force) in the Si state will be more attractive for the Bonn B potential than for the Paris potential. 

A neat measure of the strength of the nuclear tensor force is the D-state probability of the deuteron, Pd- This quantity is defined as... 

From our previous discussion, we know already that a potential with a weak tensor force (small Pp) produces a smaller integral term than a strong tensor-force potential. Thus, the G-matrix resulting from a strong-tensor force potential will be subject to a larger quenching thus, the G-matrix will be less attractive than the one produced by a weak-tensor force potential. This explains why NN interactions with a weaker tensor force )deld more attractive results when applied to nuclear few- and many-body systems. 

We show now three examples which demonstrate clearly these differences in predictions for nuclear energies by weak tensor-force potentials versus strong tensor-force potentials. As examples we choose the binding energy of the triton (Fig. 13), the spectrum of a s-d shell nucleus ( Ne,... 

Fig. 14), and nuclear matter (Fig. 15). To obtain an idea of the strength of the tensor-force component of the various potentials applied, we list in Table 5 .

The three examples suggest that potentials with weak tensor force (as implied by relativistic meson theory) may be superior in explaining nuclear structure phenomena. 

The Bonn A potential that occurs in some of the figures is a variation of Bonn B with an even weaker tensor force, but otherwise very similar to Bonn B. Predictions by Bonn A are slightly more attractive as compared to Bonn B, but the difference is not substantial e.g. for the triton binding energy Bonn B predicts 8.1 MeV while Bonn A yields 8.3 MeV. 

Furthermore, we have investigated the influence that differences between different NN potentials have on nuclear structure predictions. It turns out that for potentials that fit the NN data reasonably well, on-shell differences have only a negligible effect. However, potentials that are essentially identical on-shell, may differ substantially off-shell. Such off-shell differences may lead to large differences in nuclear structure predictions. Relativistic, meson-theory based potentials (which are non-local) are in general weaker off-shell than their local counterparts. In particular, the weaker (off-shell) tensor force component (as quantified by a small deuteron D-state probability, Pd) leads to more binding in finite nuclei. For several examples shown, these predictions compare favourably with experiment. 

In this work we discuss various approaches to the effective interaction appropriate for finite nuclei. The methods we review are the folded-diagram method of Kuo and co-workers and the summation of folded diagrams as advocated by Lee and Suzuki. Examples of applications to sd-shell nuclei from previous works are discussed together with hitherto unpublished results for nuclei in the pf-shell. Since we find the method of Lee and Suzuki to yield the best converged results, we apply this method to calculate the effective interaction for nuclei in the pf-shell. To calculate this effective interaction we have used three recent versions of the Bonn meson-exchange potential model. These versions are fitted to the same set of data and differ only in the strength of the tensor force. The importance of the latter for finite nuclei is also discussed. 

To evaluate the OBE amplitudes we use in this work the Bonn potential as it is defined by the meson parameters in Table A.2 of Ref. [7]. For further discussion of this topic, see e.g. Machleidt s contribution in these proceedings. There are three sets of meson parameters which then define three potentials, referred to as the Bonn A, B and C potentials. These potentials differ in the strength of the tensor force, which is reflected in the probability of the D-state of the deuteron. The significance of the tensor force for both nuclear matter and finite nuclei will be discussed in Section 2.3. The coupling constants, cutoffs and masses of the various mesons of Table A.2 of [7] are redisplayed in Table 1. These meson parameters are obtained through a solution of the scattering equation for... 

Vq is the central part of the NN interaction while Vj is the tensor force. Thus, if the tensor force is weak (strong), a stronger (weaker) central force is needed to arrive at the same on-shell. -matrix. A similar mechanism is present when we evaluate the G-matrix for either nuclear matter or finite nuclei as well, though, anticipating the discussion in the next subsection, in these cases we must also account for medium effects such as the modification of the energy denominator i and the inclusion of the Pauli principle. 

It has been argued that the increased attraction provided by modern meson-exchange potentials like those of the Bonn group [7,12], could be ascribed to the fairly weak tensor force exhibited by these interactions. In order to quantitatively explain this increased binding, we show in this... 

For finite nuclei the situation is not as transparent as in nuclear matter. The density dependence of the Pauli operator is couched by the fact that one integrates over all relative momenta k and k to obtain a G-matrix in a HO basis. Moreover, the summation over partial waves in Eq. (9) may include several channels representing contributions from central, spin-orbit and tensor contributions. However, the quenching of the tensor force due to the energy denominator can easily be studied. 

The quenching mechanisms discussed hitherto, explain why a potential with a weak tensor force results in G-matrix elements which are more attractive compared to a potential with a stronger tensor force. However, it ought to be remarked that the modern potentials of the Bonn group, like... 

In the next section, we show that a potential with too weak a tensor force introduces too much attraction in the JT = 10 for nuclei in the pf-shell as well. 

The spectra for Sc show that the different strengths in the tensor force are already reflected at the level of the bare G-matrix, with potential A being the most attractive. The qualitative pattern for the first-order effective interaction pertains to second order as well. 

Finally, in connection with Fig. 12, we see that the potential with the weakest tensor force, Bonn A, results in too much binding energy for the ground state, in line with the observation we made for... 

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