Nuclear Surfaces Invariant Under Symmetry Point-Groups

We are going to consider the nuclear surface equation written down in the form of the expansion in terms of the spherical harmonics, cf. Eq. (3). We wish to write down the nuclear mean-field Hamiltonian with the deformed Woods-Saxon and spin-orbit potentials, Eqs. (1) and (2), that is invariant under all the operations $\hat{g}$ of a given symmetry point-group $\mathbf{G}$. Here $\hat{g}$ denotes any point-group symmetry operation such as finite-angle rotations, proper or improper², plane reflections, and possibly inversion. The condition of invariance implies, by definition, that under the action of any group element $\Sigma\stackrel{\hat{g}}{\to}\Sigma^{\,\prime}\equiv\Sigma$. The latter can be written down as

$$\sum_{\lambda=2}^{\lambda_{max}} \sum_{\mu=-\lambda}^{\lam... ...\alpha^\star_{\lambda\mu} Y_{\lambda\mu}(\vartheta,\varphi).$$ (5)

In what follows we will need a representation of the operators $\hat{g}\in\mathbf{G}$ adapted to the action on the spherical harmonics. Here we consider explicitly the inversion, $\hat{\mathcal{C}_i}$, and spatial rotations denoted $R(\Omega)$. In the latter expression $\Omega$ represents the set of three Euler angles. The plane reflections can be treated explicitly or, alternatively, with the help of the other two operations by employing the group multiplication properties. The simultaneous action of inversion and proper rotations as well as the individual actions of these two operations can be conveniently written down using an auxiliary parameter $\eta$ taking possibly the values 0 (no inversion involved) or 1 (inversion involved):

$$\hat{g} \to \hat{g}(\eta,\Omega)=(\hat{C}_i)^{\eta} R(\Omega).$$ (6)

With this notation

$$% latex2html id marker 183\mathrm{Eq.~(\ref{eqn05}):~l.h.... ...mbda_{\mu' \mu}(\Omega)\, Y_{\lambda\mu}(\vartheta,\varphi).$$ (7)

Introducing relation (7) into Eq. (5), and re-ordering terms, we find the following equation,

$$\sum_{\mu'=-\lambda}^\lambda \sum_{\lambda=2}^{\lambda_{ma... ..._{\lambda\mu'} \bigg]\, Y_{\lambda\mu}(\vartheta,\varphi),$$ (8)

that must hold for any pair of angles $\vartheta$ and $\varphi$. Since spherical harmonics are linearly independent, Eq. (8) splits into a system of linear algebraic equations,

$$\sum_{\mu=-\lambda}^{\lambda} \alpha^\star_{\lambda\mu} (-... ...bda_{\mu' \mu}(\Omega) - \alpha^\star_{\lambda\mu'} = 0.$$ (9)

This uniform system of equations can be re-written in a more compact way as

$$\sum_{\mu=-\lambda}^{\lambda} \bigg[ (-1)^{\eta \lambda} ... ...\delta_{\mu\mu'} \bigg]\, \alpha^\star_{\lambda\mu} = 0,$$ (10)

where $\Omega$ is a fixed set of Euler angles corresponding to a given rotation as defined by $\hat{g}$. For instance in the case of a four-fold $\mathcal{O}_z$-axis rotation this could imply $\Omega=\{\pi/2,0,0\}$. The form of the invariance condition in Eq. (10) suggests that solutions can be taken as eigen-vectors of the $(2\lambda+1)\times(2\lambda+1)$ matrix of the form $(-1)^{\eta \lambda}D^\lambda(\Omega)$ with the eigen-value equal +1, leading to the simplest solution. With the corresponding set of the solutions, say $\bar{\alpha}_{\lambda\mu}$, inserted into (10) we can go back to Eq. (5) to convince ourselves³ that the condition of invariance under $\hat{g}=\hat{g}(\eta,\Omega)$ is indeed satisfied.

Some remarks may be appropriate at this point. Firstly, solutions corresponding to other eigen-values can be equally acceptable, although the preference can be given to those involving the minimum of non-zero components in terms of $\bar{\alpha}_{\lambda\mu}$. Secondly, because the system of equations in (10) is uniform, multiplying the corresponding vector by a constant corresponds again to a solution. This allows to select, e.g., $\bar{\alpha}_{\lambda\mu=0}$ as an independent parameter, which uniquely fixes all the other non-zero components. By exploring all possible values, say, $\alpha_{\lambda0}^{min}\leq\bar{\alpha}_{\lambda0}\leq\alpha_{\lambda0}^{max}$ we explore all possible surfaces invariant under the symmetry element $\hat{g}$. Thirdly, all other eigen-solutions correspond to equivalent orientations of the surface under considerations. Fourthly, the number of non-null eigenvectors gives the number of possible orientations.

So far we have presented the solution of a limited problem, i.e., the one of invariance with respect to a single symmetry operation. Formally our problem consists in searching for the simultaneous invariance conditions with respect to all the symmetry elements $\hat{g}\in\mathbf{G}$. Suppose that there are $f$ elements in the group considered, in which case $\eta_k$ and $\Omega_k$, for $k=1,2,\dots,f$ enumerate the corresponding transformations. In this case, we obtain a system of $f\times\lambda(\lambda+1)$ equations of the form

$$\left. \begin{array}{ccc} \displaystyle \sum_{\mu=-\lamb... ...\, \alpha^\star_{\lambda\mu} &=& 0 \end{array} \right\}.$$

This system of equations is over-defined - it contains more equations than unknowns. The fact that solutions exist (sometimes) is a result of symmetry; this and related problems are a subject of a forthcoming publication. In a practical treatment of the problem we do not need to solve all of these equations. One can use the fact that all the elements of any point-group can be generated out of two, possibly three chosen elements called generators. The original system of $f\times(2\lambda+1)$ equations reduces to a system of much smaller number of them, yet the property of being over-defined remains for all the groups of interest.

Finally, let us observe the following mathematical subtlety. Suppose that an operator $\hat{S}$ belongs to the ensemble of the symmetry operations, and thus its action on the surface $\Sigma$ transforms this surface into itself. However, this operation influences also the rotation-axes of the rotational-symmetry elements of the group considered - the axes following the operation $\hat{S}$. As a consequence the original group $\mathbf{G}$ ceases being a group of symmetry of the considered surface and it is the new group, $\hat{S}\mathbf{G}\hat{S}^{-1}$, isomorphic with the previous one, that overtakes the invariance rules. This mathematical subtlety has no influence on the physical consequences neither does it influence the interpretation of the discussed symmetry relations, yet it may (and often does) lead to a non-unique description of geometrically/physically equivalent objects, occasionally implying some confusion related to the particular combinations of the three Euler angles and/or signs of some of them.

In the present context we may also arrive at yet another type of technical complications. It is often convenient to limit the parametrization of the surfaces in e.g. deformed Woods-Saxon potential and/or that of the multipole moments in the constrained Hartree-Fock method to real $\alpha_{\lambda\mu}$-parameters and/or real $Q_{\lambda\mu}$-moments. It may also be convenient to work with a particular spatial representation of the symmetry operators through selection of certain reference frames. Moreover, there exist several ways of selecting the generators of the groups in question, leading to totally equivalent realizations of the group considered, yet possibly differing on the level of constructing the operators within a preselected basis. For instance, consider surfaces generated by combination $(Y_{3,+2}+Y_{3,-2})$ that are invariant under the group realization, say $T^{\,\prime}_d$, with the generators selected as $\{C_i\circ C_{4z},\,C_{8z}\circ(C_i\circ C_{4y})\circ C_{8z}^{-1}\}$. Consider next the surfaces with octahedral symmetry generated by Eqs. (15-17), the corresponding group $O_h^{\,\prime}$ generated e.g. by $\{C_i,\,C_{8z}C_{4z}C_{8z}^{-1},\,C_{8z}C_{4y}C_{8z}^{-1}\}$ The latter group contains as its sub-group tetrahedral group $T_d^{\,\prime}$ introduced above. The generators of the latter (octahedral) group could have been selected differently, in which case relations (12-14) and (15-18) would have lost consistency.