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Description
The main modes of excitation of medium and heavy atomic nuclei are associated with collective forms of motion, including surface and elastic vibrations. In this case, the shape of the nuclear surface is represented in an expansion in five parameters corresponding to the spherical harmonics of the second order, which describes quadrupole vibrations. Using the Pauli procedure the quantization of the kinetic energy of a deformed nucleus in curvilinear coordinates is considered in detail. The corresponding Hamiltonian includes both kinetic and potential components in a certain set of variables, which allows one to describe the dynamics of vibrations of the nuclear surface. This Hamiltonian is known as the Bohr Hamiltonian.
After the discovery of the Bohr Hamiltonian, many solutions of the Schrodinger eigenvalue equation have been proposed, but in many cases limited to the quadrupole degree of freedom. The Hamiltonian describing quadrupole and octupole deformations was investigated, where the core surface is represented by an expansion in seven parameters, which correspond to third-order spherical harmonics. Strong octupole correlations leading to pear-shaped shapes can occur for certain numbers of protons and neutrons Z and N. A joint parameterization of quadrupole and octupole deformations is considered, where internal variables defined in the rest frame of the general moment of inertia tensor.
The collective model with quadrupole and octupole deformations was applied to deformed axially symmetric nuclei. The Schrodinger equation with the same collective Bohr Hamiltonian are analyzed in detail. In addition, it is an interest of analytical solutions of the Schrodinger equation with Bohr Hamiltonian, and to study their connections with the critical behavior of nuclei. Note, that full form of Bohr Hamiltonian with quadrupole and octupole deformations is not given. However, it should be taken into account that it is necessary to quantize the classical kinetic energy octupole vibrations of the surface of the nucleus in curvilinear coordinates. The present work is aimed at obtaining the complete form of the Bohr Hamiltonian, taking into account quadrupole and octupole deformations.
The kinetic energy of a deformed nucleus with octupole vibrations of the surface is firstly quantized and they are presented in curvilinear coordinates. This Hamiltonian is determined by seven dynamic variables: $\beta_2$, $\gamma$, $\beta_3$, $\eta$, $\theta_1$, $\theta_2$, $\theta_3$. The obtained Hamiltonian differs from the previously known expression for quadrupole vibrations only by the coefficients before the differentiation operators $\partial$/$\partial\gamma$ and $\partial$/$\partial\eta$. This is due to the difference in the components of the moment of inertia tensor of the nucleus for quadrupole and octupole modes. The importance of taking into account deformation asymmetry parameters are shown. As well as in triaxial nuclei the third component of the angular momentum is not a good quantum number (K-mixing).
