Hartree-Fock Tilted-Axis-Cranking calculations

So far, the TAC model has been described in the literature only in its PQTAC and SCTAC variants [15,17,16]. Therefore, in this Section we give several details that are specific for its self-consistent implementation. The discussion concerns mainly the way of iteratively solving the HF equations which is adopted in this work.

As far as non-rotating states are concerned, the HF method consists in minimizing the expectation value of the many-body Hamiltonian, $\hat{H}=\hat{T}+\hat{V}$, in the trial class of Slater determinants. Here, $\hat{T}$ is the kinetic-energy operator, and $\hat{V}$ is a two-body effective interaction. Equivalently, one can formulate the method in terms of the energy density functional $E\{\rho\}$, which is minimized with respect to the one-body density matrix $\rho$ on which it depends, and this latter representation is used in the present study. Since $\hat{H}$ or the density functional $E\{\rho\}$ are invariant under rotations in space, it is clear that the HF solution is defined only up to an arbitrary rotation. For each solution it is useful to introduce an intrinsic frame of reference, whose axes we define as principal axes of the tensor of the electric quadrupole moment of the mass distribution. Due to the mentioned arbitrariness, this frame can be rotated with respect to the frame originally used to solve the HF equations, which we refer to as the program (or computer-code) frame. The program frame is the one defined by the axes $x$, $y$ or $z$, used for solving the mean-field equations, e.g., in a computer code ¹.

To describe rotational excitations within the TAC approach, in the program frame one imposes a linear constraint on angular momentum and minimizes the expectation value of the Routhian,

$$\hat{H}'=\hat{H}-\vec {\omega}\cdot\hat{\vec {I}}~,$$ (1)

or the energy density in the rotating frame,

$$E'\{\rho\}=E\{\rho\}-\vec {\omega}\cdot (\hat{\vec {I}}\rho)~,$$ (2)

where $\hat{\vec {I}}$ is the total angular momentum operator, and vector $\vec {\omega}$ is composed of three Lagrange multipliers; it is called rotational frequency vector. Its components in the program frame are fixed as part of the definition of $\hat{H}'$ or $E'\{\rho\}$. Because of the rotational invariance, solutions obtained for the same length, but different directions of $\vec {\omega}$ differ only by their orientation in the program frame, so that only the length, $\omega=\vert\vec {\omega}\vert$, has physical meaning.

Within the HF procedure, one obtains that the sought Slater determinant or the one-body density are built of the eigenstates of the s.p. Routhian,

$$\hat{h}'=\hat{h}-\vec {\omega}\cdot\hat{\vec {I}}~,$$ (3)

where the mean-field Hamiltonian, $\hat{h}=\hat{T}+\hat{\Gamma}=\delta{E\{\rho\}}/\delta\rho$, is a sum of the kinetic energy, $\hat{T}$, and the mean field, $\hat{\Gamma}$. A non-selfconsistent approximation to the HF method consists in replacing $\hat{\Gamma}$ with a model potential, whose deformation dependence is usually parametrized in terms of the multipole deformations of the nuclear surface. Then, the expectation value of the Routhian, $\hat{h}'$, is minimized over the deformation parameters of the potential, which allows to achieve self-consistency with respect to the nuclear shape.

In the PQTAC/SCTAC implementations of this approach, a hybrid [26] of the Woods-Saxon [33] and Nilsson [34] s.p. potentials is taken. The Quadrupole-Quadrupole interaction of PQTAC just amounts to the deformation, while SCTAC takes into account the nuclear liquid-drop energy within the standard Strutinsky method [35,36].

Obviously, only the relative orientation of the angular momentum vector with respect to the nuclear body carries physical information. Common orientation of the angular momentum vector and nucleonic densities with respect to the program frame can thus be arbitrary. In the PQTAC/SCTAC method, where the orientation in space of the nuclear surface is under control via the multipole deformations $\alpha_{\lambda\mu}$, one can take advantage of this fact and fix $\alpha_{21}=\mathrm{Im~}\alpha_{22}=0$ so that the intrinsic and program frames coincide. Minimization of the expectation value of the Routhian, $\hat{h}'$, at a given magnitude of $\omega$ is then performed by varying the direction of $\vec {\omega}$ and all $\alpha_{\lambda\mu}$ except of $\alpha_{21}$ and $\mathrm{Im~}\alpha_{22}$.

In the HF method, however, the Euler angles defining the orientation of the intrinsic axes in the program frame are not free variational parameters, but complicated functions of densities, which in turn change from one HF iteration to another. The only possible way to make the two frames coincide is by imposing dynamical constraints on the off-diagonal components of the quadrupole tensor and requiring that they vanish. Then, one can vary the cranking frequency vector like in the PQTAC/SCTAC approach. This is actually the only way to proceed if the energy dependence on the intrinsic orientation of $\vec {\omega}$ is sought. However, constraints strong enough to confine the nucleus easily lead to divergencies, and adjusting their strengths properly may be a cumbersome task. If only the energy minimum is of interest, a more natural and incomparably faster way is to fix $\vec {\omega}$ in the program frame and let the mean potential reorient and conform to it self-consistently in the course of the iterations. The intrinsic axes now do become tilted with respect to the program frame.

The Kerman-Onishi theorem [37] states that in each self-consistent solution the total angular momentum vector, $\vec {I}=\langle\hat{\vec {I}}\rangle$, is parallel to $\vec {\omega}$. In calculations, the angle between those vectors converges to zero very slowly in terms of the HF iterations, because the whole nucleus must turn in the program frame in order to align its $\vec {I}$ with the fixed $\vec {\omega}$. Therefore, a much faster procedure is to explicitly reset $\vec {\omega}$ in each iteration to make it parallel to the current $\vec {I}$, while keeping its length, $\omega$, constant [30]. This purely heuristic procedure does not correspond to a minimization of any given Routhian. However, once the self-consistent solution is found, it is the Routhian for the final angular frequency that takes its minimum value.

Some quantities, like mean angular momenta and multipole moments, are easiest to discuss only when expressed in the intrinsic frame of the nucleus, but it is a natural way to calculate them first in the program frame. Since the two frames do not necessarily coincide, one has to find the axes of the intrinsic frame by diagonalization of the quadrupole deformation tensor and to transform the considered quantities into that frame by use of the Wigner matrices. (Such a procedure may fail when the solutions have vanishing quadrupole moments, but in the present paper such cases are of no interest and will not be discussed).

The HF TAC solutions are arbitrarily tilted with respect to the program frame, and, moreover, their orientation is not known a priori. Therefore, when solving the problem numerically one should ensure such conditions that the same solution be represented equally well in all orientations. In particular, the energy must not depend on the orientation. If the s.p. wave-functions are expanded onto a basis, this means that the choice of the s.p. basis and of the basis cut-off must not privilege any axis of the reference frame. In the case of the Cartesian harmonic-oscillator (HO) basis used in the present study, this by definition amounts to taking the three oscillator frequencies equal and including only the entire HO shells. To obtain a reasonable description of deformed nuclei in such non-deformed bases, the only way is to use sufficiently many HO shells.