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Register a new userOn a translationally invariant nuclear single particle picture
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If one assumes a translationally invariant motion of the nucleons relative to the c. m. position in single particle mean fields a correlated single particle picture of the nuclear wave function emerges. A single particle product ansatz leads for that Hamiltonian to nonlinear equations for the single particle wave functions. In contrast to a standard not translationally invariant shell model picture those single particle s-, p- etc states are coupled. The strength of the resulting coupling is an open question. The Schroedinger equation for that Hamiltonian can be solved by few- and many -body techniques, which will allow to check the validity or non-validity of a single particle product ansatz. Realistic nuclear wave functions exhibit repulsive 2-body short range correlations. Therefore a translationally invariant single particle picture -- if useful at all -- can only be expected beyond those ranges. Since exact A = 3 and 4 nucleon ground state wave functions and beyond based on modern nuclear forces are available, the translationally invariant shell model picture can be optimized by an adjustment to the exact wave function and its validity or non-validity decided.
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The need to enforce fermionic antisymmetry in the nuclear many-body problem commonly requires use of single-particle coordinates, defined relative to some fixed origin. To obtain physical operators which nonetheless act on the nuclear many-body system in a Galilean-invariant fashion, thereby avoiding spurious center-of-mass contributions to observables, it is necessary to express these operators with respect to the translational intrinsic frame. Several commonly-encountered operators in nuclear many-body calculations, including the magnetic dipole and electric quadrupole operators (in the impulse approximation) and generators of SU(3) and Sp(3,R) symmetry groups, are bilinear in the coordinates and momenta of the nucleons and, when expressed in intrinsic form, become two-body operators. To work with such operators in a second-quantized many-body calculation, it is necessary to relate three distinct forms: the defining intrinsic-frame expression, an explicitly two-body expression in terms of two-particle relative coordinates, and a decomposition into one-body and separable two-body parts. We establish the relations between these forms, for general (non-scalar and non-isoscalar) operators bilinear in coordinates and momenta.
Precision tests of the Standard Model and searches for beyond the Standard Model physics often require nuclear structure input. There has been a tremendous progress in the development of nuclear ab initio techniques capable of providing accurate nuclear wave functions. For the calculation of observables, matrix elements of complicated operators need to be evaluated. Typically, these matrix elements would contain spurious contributions from the center-of-mass (COM) motion. This could be problematic when precision results are sought. Here, we derive a transformation relying on properties of harmonic oscillator wave functions that allows an exact removal of the COM motion contamination applicable to any one-body operator depending on nucleon coordinates and momenta. Resulting many-nucleon matrix elements are translationally invariant provided that the nuclear eigenfunctions factorize as products of the intrinsic and COM components as is the case, e.g., in the no-core shell model approach. An application of the transformation has been recently demonstrated in calculations of the nuclear structure recoil corrections for the beta-decay of 6He.
Chiral expansions of the two-pion exchange components of both two- and three-nucleon forces are reviewed and a discussion is made of the predicted pattern of hierarchies. The strength of the scalar-isoscalar central potential is found to be too large and to defy expectations from the symmetry. The causes of this effect can be understood by studying the nucleon scalar form factor.
The single-nucleon potential in hot nuclear matter is investigated in the framework of the Brueckner theory by adopting the realistic Argonne V18 or Nijmegen 93 two-body nucleon-nucleon interaction supplemented by a microscopic three-body force. The rearrangement contribution to the single-particle potential induced by the ground state correlations is calculated in terms of the hole-line expansion of the mass operator and provides a significant repulsive contribution in the low-momentum region around and below the Fermi surface. Increasing temperature leads to a reduction of the effect, while increasing density makes it become stronger. The three-body force suppresses somewhat the ground state correlations due to its strong short-range repulsion, increasing with density. Inclusion of the three-body force contribution results in a quite different temperature dependence of the single-particle potential at high enough densities as compared to that adopting the pure two-body force. The effects of three-body force and ground state correlations on the nucleon effective mass are also discussed.
[Background:] It is well known that effective nuclear interactions are in general nonlocal. Thus if nuclear densities obtained from {it ab initio} no-core-shell-model (NCSM) calculations are to be used in reaction calculations, translationally invariant nonlocal densities must be available. [Purpose:] Though it is standard to extract translationally invariant one-body local densities from NCSM calculations to calculate local nuclear observables like radii and transition amplitudes, the corresponding nonlocal one-body densities have not been considered so far. A major reason for this is that the procedure for removing the center-of-mass component from NCSM wavefunctions up to now has only been developed for local densities. [Results:] A formulation for removing center-of-mass contributions from nonlocal one-body densities obtained from NCSM and symmetry-adapted NCSM (SA-NCSM) calculations is derived, and applied to the ground state densities of $^4$He, $^6$Li, $^{12}$C, and $^{16}$O. The nonlocality is studied as a function of angular momentum components in momentum as well as coordinate space [Conclusions:] We find that the nonlocality for the ground state densities of the nuclei under consideration increases as a function of the angular momentum. The relative magnitude of those contributions decreases with increasing angular momentum. In general, the nonlocal structure of the one-body density matrices we studied is given by the shell structure of the nucleus, and can not be described with simple functional forms.
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