Nouredine Zettili

On the construction of a new solvable model and validity of many-body approximation methods☆

Abstract This work deals both with the construction of a new analytically solvable model and with the quantitative test of the time-dependent Hartree-Fock (TDHF) method. First, we construct a new analytically solvable model, which serves as a testing ground for the various many-body approximation methods. The construction is based on two vector operators that are the generators of a Lie algebra. The model consists of a one-dimensional system of two distinguishable sets of fermions interacting via a schematic two-body force. The model has a simple analytic energy spectrum. Second, we use this model to test the validity of the TDHF approximation. Exact eigenvalues are compared with the corresponding solutions of the TDHF method. The TDHF approximation is shown to be reasonably accurate in the description of the systems eigenstates.

Validity of the nuclear Born-Oppenheimer method

Abstract The validity of the adiabatic nuclear Born-Oppenheimer (NBO) approximation method is investigated by means of an analytically solvable model. The NBO equation of collective motion derived, when this method is applied to the model, is shown to have the structure of a Schrodinger equation. The NBO energy spectrum is then obtained by numerical integration of this equation and compared with the analytic energy spectrum. We show that the NBO approximation is very accurate in the description of the systems eigenstates. The time-dependent Hartree-Fock (TDHF) results, obtained in a previous publication for the solvable model, are compared with their NBO counterparts. We find that, although both methods describe the systems states very well, the NBO approximation is more accurate in the adiabatic domain.

Nuclear Rotations and the Born‐Oppenheimer Approximation

We deal here with the application of the Nuclear Born Oppenheimer (NBO) method to the description of nuclear rotations. As an edifying illustration, we apply the NBO formalism to study the rotational motion of nuclei which are axially‐symmetric and even, but whose shells are not closed. We focus, in particular, on the derivation of expressions for the rotational energy and for the moment of inertia. Additionally, we examine the connection between the NBO method and the self‐consistent cranking (SCC) model. Finally, we compare the moment of inertia generated by the NBO method with the Thouless‐Valantin formula and hence establish a connection between the NBO method and the large body of experimental data.

MATRIX ELEMENTS OF A HYPERBOLIC VECTOR OPERATOR UNDER SO(2,1)

We deal here with the use of Wigner–Eckart type arguments to calculate the matrix elements of a hyperbolic vector operator by expressing them in terms of reduced matrix elements. In particular, we focus on calculating the matrix elements of this vector operator within the basis of the hyperbolic angular momentum whose components , , satisfy an SO(2,1) Lie algebra. We show that the commutation rules between the components of and can be inferred from the algebra of ordinary angular momentum. We then show that, by analogy to the Wigner–Eckart theorem, we can calculate the matrix elements of within a representation where and are jointly diagonal.

THE NUCLEAR BORN–OPPENHEIMER METHOD APPLIED TO NUCLEAR COLLECTIVE MOTION

We deal with the application of the nuclear Born–Oppenheimer (NBO) method to the study of nuclear collective motion. In particular, we look at the description of nuclear rotations and vibrations. The collective operators are specified within the NBO method only to the extent of identifying the type of collective degrees of freedom we intend to describe; the operators are then determined from the dynamics of the system. To separate the collective degrees of freedom into rotational and vibrational terms, we transform the collective tensor operator from the lab fixed frame of reference to the frame defined by the principal axes of the system; this transformation diagonalizes the tensor operator. We derive a general expression for the NBO mean energy and show that it contains internal, collective and coupling terms. Then, we specify the approximations that need to be made in order to establish a connection between Bohrs collective model and the NBO method. We show that Bohrs collective Hamiltonian can be recovered from the NBO Hamiltonian only after adopting some rather crude approximations. In addition, we try to understand, in light of the NBO approach, why Bohrs collective model gives the wrong inertial parameters. We show that this is due to two major reasons: the ad hoc selection of the collective degrees of freedom within the context of Bohrs collective model and the unwarranted neglect of several important terms from the Hamiltonian.