Two-body forces

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

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

As noted above at higher densities the EoS is sensitive to 3NF contributions. Whereas the 3NF for low densities seems now well understood its contribution to nuclear matter densities remains unsettled. In practice in calculations of the symmetry energy in the BHF approach two types of 3NF have been used in calculations in ref.[4] the microscopic 3NF based upon meson exchange by Grange et al. was used, and in ref. [15] as well in most VCS calculations the Urbana interaction. The latter has in addition to an attractive microscopic two-pion exchange part a repulsive phenomenological part constructed in such a way that the empirical saturation point for SNM is reproduced. Also in practice in the BHF approach to simplify the computational efforts the 3NF is reduced to a density dependent two-body force by averaging over the position of the third particle. 

In spite of the satisfactory convergence, the saturation density misses the empirical value po = 0.17 fm-3 extracted from the nuclear mass tables. This confirms the belief that the concept of a many nucleon system interacting with only a two-body force is not adequate to describe nuclear matter, especially at high density. 

Since long it is well known that two-body forces are not enough to explain some nuclear properties, and TBF have to be introduced. Typical examples are the binding energy of light nuclei, the spin dynamics of nucleon-deuteron scattering, and the saturation point of nuclear matter. Phenomenological and microscopic TBF have been widely used to describe the above mentioned properties. 

In the framework of the Brueckner theory a rigorous treatment of TBF would require the solution of the Bethe-Faddeev equation, describing the dynamics of three bodies embedded in the nuclear matter. In practice a much simpler approach is employed, namely the TBF is reduced to an effective, density dependent, two-body force by averaging over the third nucleon in the medium,... 

Here = Aw (I — ty 13) (1 — ]% ) and is the free wave function of the third particle. This effective two-body force is added to the bare two-body force and recalculated at each step of the iterative procedure. 

The combined effect of these TBF is a remarkable improvement of the saturation properties of nuclear matter [12], Compared to the BHF prediction with only two-body forces, the saturation energy is shifted from —18 to —15 MeV, the saturation density from 0.26 to 0.19 fm-3, and the compression modulus from 230 to 210 MeV. The spin and isospin properties with TBF exhibit also quite satisfactory behavior [18],... 

After reducing this TBF to an effective, density dependent, two-body force by the averaging procedure described earlier, the resulting effective two-nucleon potential assumes a simple structure,... 

These values of A and U have been obtained by using the Argonne v i two-body force [20] both in the BHF and in the variational many-body theories. However, the required repulsive component ( U) is much weaker in the BHF approach, consistent with the observation that in the variational calculations usually heavier nuclei as well as nuclear matter are underbound. Indeed, less repulsive TBF became available recently [21] in order to address this problem. 

Fet us now confront the EOS predicted by the phenomenological TBF and the microscopic one. In both cases the BHF approximation has been adopted with same two-body force (Argonne uis). In the left panel of Fig. 4 we display the equation of state both for symmetric matter (lower curves) and pure neutron matter (upper curves). We show results obtained for several cases, i.e., i) only two-body forces are included (dotted lines), ii) TBF implemented within the phenomenological Urbana IX model (dashed lines), and iii) TBF treated within... 

The results are shown in Fig. 5. We notice that the EOS calculated with the microscopic TBF produces the largest gravitational masses, with the maximum mass of the order of 2.3 M , whereas the phenomenological TBF yields a maximum mass of about 1.8 M . In the latter case, neutron stars are characterized by smaller radii and larger central densities, i.e., the Urbana TBF produce more compact stellar objects. For completeness, we also show a sequence of stellar configurations obtained using only two-body forces. In this case the maximum mass is slightly above 1.6 M , with a radius of 9 km and a central density equal to 9 times the saturation value. 

These results confirm the observation of Mihailovic and Rosina that second-order methods cannot characterize the correlations induced by two-body forces. 

We consider a system made up of N fermions (nucleons or electrons) interacting through two-body forces and with an external field, with respective potentials >(r,r ) and V(r). We assume there exists a possibility of decomposing the one-particle density matrix into an averaged part and an oscillating part, i.e. ... 

There however remains a delicate point. Keeping only two-body forces wc a priori should write the interaction as... 

V. Efimov, Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33 (1970) 563. 

The calculation of the forces at each time step is one of the most demanding task. Since I), = —Ffor the two-body forces, and F, = —F— Ffor the... 

A famous example for the presence of a hidden symmetry is the Toda Hamiltonian (see, e.g., Lichtenberg and Lieberman (1983)). In its simplest form it describes the dynamics of three particles moving on a ring subjected to repulsive two-body forces according to... 

There are two reasons why so much is unknown. First, at high densities three (and even four) body forces are important. This is particularly so when chemically reactive atoms are present. Then, even for two-body forces, the strongly repulsive regime is not well understood and, in addition, close in, as one approaches the united atom limit, there is considerable promotion of molecular orbitals. This is a universal mechanism for electronic excitation which means a breakdown of the Born-Oppenheimer approximation for close collisions. 

Then if only two-body forces act between the atoms of an assembly, the energy of formation at zero temperature is simply... 

If the dielectric properties are known, the expression represents a complete solution to the interaction problem, provided Aat a liquid separating the two interacting solid surfaces is itself not perturbed in structure by the surfaces. These forces can be calculated and measured. We make the following remarks. The assumption of two-body forces is completely misleading and qualitatively erroneous for condensed media interactions. The sum in the 11 expression above includes contributions from all frequencies. 

Computational quantum mechanics continues to be a rapidly developing field, and its range of application, and especially the size of the molecules that can be studied, progresses with improvements in computer hardware. At present, ideal gas properties can be computed quite well, even for moderately sized molecules. Complete two-body force fields can also be developed from quantum mechanics, although generally only for small molecules, and this requires the study of pairs of molecules in a large number of separations and orientations. Once developed, such a force field can be used to compute the second virial coefficient, which can be used as a test of its accuracy, and in simulation to compute phase behavior, perhaps with corrections for multibody effects. However, this requires major computational effort and expert advice. At present, a much easier, more approximate method of obtaining condensed phase thermodynamic properties from quantum mechanics is by the use of polarizable continuum models based on COSMO calculations. 

---