Model for collective motion

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  Introduction

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  Nuclear Shape Parameters

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  Nuclear Surface Oscillations

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  The Rotation-Vibration Model

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  Microscopic Derivation of the Collective Hamiltonian

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  Microscopic Collective Hamiltonian Based on Density Functional Theory

A medium-heavy or heavy atomic nucleus presents a typical example of a complex quantum system, in which different interactions between a relatively large number of constituent nucleons give rise to physical phenomena that are qualitatively different from those exhibited by few-nucleon systems. There are a number of features that characterize complex systems, but for the topic of the present chapter of particular interest is the emergence of collective structures and dynamics that do not occur in light nuclei composed of only a small number of nucleons.

Collective motion is the simplest manifestation of emergent phenomena in atomic nuclei. It can be interpreted as a kind of motion in which a large number of nucleons contribute coherently to produce a large amplitude oscillation of one or more electromagnetic multipole moments. Collective motion gives rise to excited states that are characterized by large electromagnetic transition rates to the ground state, that is, rates that correspond to many single-particle transitions Rowe1970 . This chapter will mainly focus on low-energy large-amplitude collective motion (LACM), such as surface vibrations, rotations, and fission. Theoretical studies of LACM started as early as in the 1930s. Based on the liquid drop model of the atomic nucleus Weizsacker1935 , Flügge applied Rayleigh’s normal modes Rayleigh1879 to a classical description of low-energy excitations of spherical nuclei Flugge1941 . The model of Flügge was quantized by Bohr Bohr1952 , who formulated a quantum model of surface oscillations of spherical nuclei, and also introduced the concept of intrinsic frame of reference for a quadrupole deformed nuclear surface characterized by the Euler angles and the shape parameters $\beta$ and $\gamma$ (nowadays often called the Bohr deformation parameters). Subsequently, Bohr and Mottelson Bohr1953 generalized the model to vibrations and rotations of deformed nuclei. A generalization of the Bohr Hamiltonian to describe large-amplitude collective quadrupole excitations of even-even nuclei of arbitrary shape was introduced by Belyaev Belyaev1965 and Kumar and Baranger Kumar1967 . Several specific forms of the collective Hamiltonian, designed to describe collective excitations in nuclei of particular shape were also considered Wilets1956 ; Davydov1958 ; Davydov1960 ; Greiner1962a ; Greiner1962b ; Faessler1965 .

In the past several decades, enormous progress has been made in developing microscopic many-body theories of nuclear systems. However, a description of collective phenomena starting from single-nucleon degrees of freedom still presents a considerable challenge. One of the methods that has been used to obtain such a description is the adiabatic time-dependent HFB theory (ATDHFB) Belyaev1965 ; Baranger1968 ; Baranger1978 which, in the case of quadrupole collective coordinates, leads to the Bohr Hamiltonian. Another approach to collective phenomena that is based on microscopic degrees of freedom, is the generator coordinate method (GCM) Hill1953 . In the Gaussian overlap approximation (GOA) Brink1968 ; Onishi1975 , the GCM also leads to the Bohr collective Hamiltonian Une1976 ; Gozdz1985 .

Excitation spectra of even-even nuclei in the energy range of up to $\approx 3$ MeV, exhibit characteristic band structures that are interpreted as vibrations and rotations of the nuclear surface in the geometric collective model, first introduced by Bohr and Mottelson Bohr1952 ; Bohr1953 , and further elaborated by Faessler and Greiner Greiner1962a ; Greiner1962b .

For low-energy excitations the compression mode is not relevant because of high incompressibility of nuclear matter, and the diffuseness of the nuclear surface layer can also be neglected to a good approximation. One therefore starts with the model of a nuclear liquid drop of constant density and sharp surface. With these assumptions, the time-dependent nuclear surface can, quite generally, be described by an expansion in spherical harmonics with shape parameters as coefficients:

where $R(\theta,\phi;t)$ denotes the nuclear radius in spherical coordinates $(\theta,\phi)$, and $R_{0}$ is the radius of a sphere with the same volume. The shape parameters $\alpha_{\lambda\mu}(t)$ play the role of collective dynamical variables, and their physical meaning will be discussed for increasing values of $\lambda$.

To lowest order, the dipole mode $\lambda=1$ corresponds to a translation of the nucleus as a whole and, therefore, is not considered for low-energy excitations. Dynamical quadrupole deformations, that is, the mode with $\lambda=2$, turn out to be the most relevant low-lying collective excitations. Most of the following discussion of collective models will focus on this case, so a more detailed description is included below.

Octupole dynamical deformations, $\lambda=3$, are the principal asymmetric modes of a nucleus associated with negative-parity bands. While there is no evidence for pure hexadecupole excitations in low-energy spectra, this mode plays an important role as admixture to quadrupole excitations, and for fission dynamics. Shape oscillations of higher multipoles are not relevant for low-energy excitations.

For the case of pure quadrupole deformation the nuclear surface is parameterized

where the time dependence is implicit for dynamical variables. If the shape of the nucleus is an ellipsoid, its three principal axes ($x$, $y$, $z$) are linked with the ($X$, $Y$, $Z$) axes of a Cartesian coordinate system in the laboratory frame. From the symmetry of the ellipsoid, it follows that $a_{1}=a_{-1}=0,a_{2}=a_{-2}$, where $a_{\mu}$ are the shape parameters in the principal-axis system. Obviously, the five coefficients $\alpha_{\mu}$ in the laboratory frame reduce to two real independent variables $a_{0}$ and $a_{2}$ in the principal-axis system, which, together with the three Euler angles, provide a complete parameterization of the nuclear surface. The details of the transformation between $\alpha_{\mu}$ and $a_{\mu}$ are included below. In the principal-axis system, the nuclear radius is given by

Two parameters $(a_{0},\ a_{2})$ are generally used to describe quadrupole deformations but, instead of $a_{0}$ and $a_{2}$, the polar coordinates $\beta$ and $\gamma$ are usually employed. They are defined as follows:

Using Eq. (3) and Eq. (4), we can express the increments of the three semi-axes in the principal-axis system:

where $\kappa=1,2,3$ correspond to $x,y,z$, respectively. The parameters $\beta$ and $\gamma$ only describe exactly ellipsoidal shapes in the limit of small $\beta$-values.

Figure 2 displays nuclear shapes in the $(\beta,\ \gamma)$ plane. At $\gamma=0$ the nucleus is elongated along the $z$ axis, with the $x$ and $y$ semi-axes being equal. This axially symmetric shape is somewhat reminiscent of a cigar and is called prolate. As $\gamma$ increases, the $x$ semi-axis lengthens with respect the $y$ and $z$ semi-axes through a region of triaxial shapes with three unequal semi-axes, until axial symmetry is again reached at $\gamma=60^{\circ}$, but now with the $z$ and $x$ semi-axes equal in length. These two are longer than the $y$ semi-axis. The flat, pancake-like shape, is called oblate. This pattern is repeated: every $60^{\circ}$ axial symmetry recurs and prolate and oblate shapes alternate, but with the semi-axes permuted in their relative length.

A simple framework that accurately describes the most prominent features of low-energy collective excitations is provided by the model of surface oscillations. Let the surface of the nucleus be parametrized by Eq. (1) and, if the coefficients $\alpha_{\lambda\mu}$ are small, the deformation potential energy and the associated kinetic energy take the form

respectively. The dot denotes a derivative with respect to time, and the quantities $B_{\lambda}$ and $C_{\lambda}$ depend on the assumed properties of nucleonic matter. For an incompressible nucleus of constant density $\varrho_{0}$, one finds $B_{\lambda}=\lambda^{-1}\varrho_{0}R_{0}^{5}$, assuming nuclear matter to exhibit irrotational flow. If, moreover, the charge of the nucleus $Ze$ is uniformly distributed over its volume, one obtains

where $S$ is the surface tension.

It is convenient to make a change of coordinates in such a way that the kinetic energy separates into a vibrational and a rotational part. This coordinate transformation will be considered in the following paragraphs for deformations of order $\lambda=2$. For convenience, $B_{2}$ and $C_{2}$ are simplified to $B$ and $C$.

Consider a coordinate system $K^{\prime}$ whose axes coincide with the principal axes of the ellipsoid. The coordinate transformation from the five $\alpha_{\mu}$ to the new coordinates $a_{0}$, $a_{2}$ and $\theta_{i}$ must be given by

where $D_{\mu\nu}^{2*}\left(\theta_{i}\right)$ is the Wigner function and $\theta_{i}\equiv(\theta_{1},\ \theta_{2},\ \theta_{3})$ are the three Euler angles which describe the orientation of the body-fixed axes in space. Using Eq. (4) and the unitary character of $D^{2}_{\mu\nu}$, one obtains

From equations (6) and (9), it follows $V=\frac{1}{2}C\beta^{2}$ for the potential energy of quadrupole deformation. To express the kinetic energy of the oscillating nucleus, from Eq. (8) one derives

If this expression is used in Eq. (6), the kinetic energy can be written as a sum of three terms. The first term is quadratic in $\dot{a}_{\nu}$, and represents shape vibrations of the ellipsoid that retains its orientation. The second term, quadratic in $\dot{\theta}_{i}$, represents a rotation of the ellipsoid without change of its shape. The third term, which contains mixed time derivatives $\dot{a}_{\nu}\dot{\theta}_{i}$, vanishes as can be shown from simple properties of the Wigner coefficients $D^{2}_{\mu\nu}$ and their derivatives.

Therefore, the kinetic energy reads

For the vibrational energy, one obtains

by means of Eq. (4). To derive a convenient form for $T_{\text{rot}}$, the most natural description is achieved if the principal axes of the nucleus determine the body-fixed system, because then the inertia tensor is diagonal. The classical rotational energy reads

where $\mathcal{J}_{i}$ is the moment of inertia about the $i$-th principal axis of the nucleus, and $\omega_{i}^{\prime}$ is the corresponding angular velocity of rotation.

The next step is the quantization of the classical Hamiltonian. As it is well known, there is no unique prescription for such a quantization in the general case. Usually one adopts the Pauli prescription, for which the Laplace operator is expressed in curvilinear coordinates. The final form of the Hamiltonian when expressed in $\beta$ and $\gamma$, as was done originally by Bohr Bohr1952 , reads

where $\hat{I_{i}}$ are operators of the projections of the nuclear angular momenta on the principal axes. However, the operators $\hat{I_{i}}$ cannot be identified with the standard angular-momentum operators in the laboratory frame. Their commutation relations look similar to those of the laboratory-fixed operators, but with a crucial change in sign:

Detailed expression for these operators can be found in Ref. Davydov1965 .

The stationary wave functions separate in the following way:

The solution proceeds through the familiar procedure of separation of variables and, thus, the normalized solution for the $\beta$ part of the wave function reads

where $n$ is the principal quantum number, while $l$ is the quantum number that corresponds to the solution of the eigenvalue equation of the Casimir operator of the $SO(5)$ group (also called $SO(5)$ seniority). The total energy $E$ of the five-dimensional quadrupole oscillator reads

The wave function $\Phi$ can be written in the form

where $I$, $M$, and $K$ are the quantum numbers for the total angular momentum, its projections on $Z$ axis in the laboratory frame, and on the third axis in the intrinsic frame, respectively. $\nu$ is a label that distinguishes between multiple sets of the same quantum numbers $I$, $M$ in a given $SO(5)$ irreducible representation. The functions $g$ are given explicitly in Ref. Corrigan1976 .

In addition, the above expression must be symmetrized because of the ambiguity in the choice of intrinsic axes. The total number of possible choices is 24, and they can be related by three basic transformations $\hat{R}_{1}$, $\hat{R}_{2}$, $\hat{R}_{3}$, shown in Fig. 3.

Under these transformation the collective wave functions must satisfy:

By applying $\hat{R}_{2}^{2}$ to the wave function (19), one obtains that $K$ must be even. $\hat{R}_{1}$ will transform $D_{MK}^{I*}(\mathbf{\theta})$ into $(-1)^{I}D_{M-K}^{I*}(\mathbf{\theta})$, and require that $g_{-Kl\nu}^{I}(\gamma)=(-1)^{I}g_{Kl\nu}^{I}(\gamma)$. Finally, the wave function is written to explicitly display the symmetry property which restricts the sum to $K\geq 0$:

where

are the properly symmetrized wave functions for the symmetric rotor. ${N}_{nl\nu}^{I}$ normalizes the $\gamma$ part of Eq. (21). Detailed expression for the wave function can be found in Ref. Corrigan1976 .

The most important special case of collective surface motion is that of well deformed nuclei, whose potential energy surfaces exhibit a deep axially deformed minimum like in the simple rotor model, but with the additional feature of small oscillations around that minimum in both $\beta$ and $\gamma$ degrees of freedom (see the illustration in Fig. 4).

As the harmonic vibrations are easier to deal with in Cartesian coordinates, the potential is expressed in the coordinates $(a_{0},a_{2})$ instead of $(\beta,\gamma)$. Assuming the potential minimum is located at $a_{0}=\beta_{0}$ and $a_{2}=0$, the small displacement from the equilibrium deformation can be written in the form

and it is easy to show that the Hamiltonian can be expressed in the form

Note the additional factor of $2$ in the $\eta$-dependent part of the potential, which takes into account that $\eta$ represents the two coordinates $a_{2}$ and $a_{-2}$.

In a quantum mechanical description, a nucleus cannot rotate around the symmetry axis. However, here the vibrations make such rotations possible dynamically, and they are coupled to the dynamic fluctuations from axial symmetry described by $\eta$. The moments of inertia are given by the lowest order expressions

and the simplest expression for the kinetic energy reads

Similarly, the quantized Hamiltonian can be written in the form

The term with $\eta^{-2}$ is similar to the centrifugal potential and can be treated accordingly. The eigenfunctions of the rotational operator in the Hamiltonian correspond to those of the rigid rotor, while the $\xi$-dependent and $\eta$-dependent parts can obviously be separated. Thus, a trial wave function takes the form

The next step is to complete the separation of the $\xi$ and $\eta$ dependence, and finally they correspond to typical eigenvalue problems with one-dimensional and three-dimensional harmonic oscillator potentials, respectively. The total energy reads

with

For the eigenfunctions that depend on the $\xi$ ($\beta$) coordinate, the one-dimensional harmonic oscillator eigenfunctions can be used directly, and they are simply written as $\langle\xi|n_{\beta}\rangle$. For the $\gamma$ coordinate the solution for the three-dimensional harmonic oscillator potential is used

with $\lambda=2B\omega_{\gamma}/\hbar$, $\l_{K}=(-1\pm K)/2$, and $N_{Kn_{\gamma}}$ is the normalization factor. After symmetrization, the wave functions read

and $\phi_{MK}^{I}\left(\theta_{i}\right)$ are the symmetrized eigenfunctions for the symmetric rotor of Eq. (22). The allowed values of the quantum numbers are:

The structure of a typical spectrum is shown in Fig. 5. The bands are characterized by a given set of $K,n_{\beta},n_{\gamma}$ and the excitation energies follow the $I(I+1)$ rule of a rotor. The principal bands are:

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  the ground-state band (g. s.), includes the states $|IM000\rangle$ with $I$ even. The excitation energies are given by $\hbar^{2}I(I+1)/2\mathcal{J}$.

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  the $\beta$ band, includes the states $|IM010\rangle$ with one quantum of vibration in the $\beta$ direction. It starts at $\hbar\omega_{\beta}$ above the ground state, and also contains only even angular momenta.

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  the $\gamma$ band corresponds to $K=2$. It is characterized by coupling between rotation (around the third principal axis) and $\gamma$-vibration, induced by the term $(\hat{I}_{3}^{2}-\hbar^{2})/16B\eta^{2}$ in the Hamiltonian. In the spectra it is easy to distinguish it from the $\beta$ band, since it starts with $2^{+}$ and contains both even and odd angular momenta.

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  the next higher bands are the $\gamma$ band with $n_{\gamma}=1$ and the one with $K=4$, the second $\beta$ band with $n_{\beta}=2$.

The simplest observables that can be calculated and compared to data, are the electric quadrupole moments and transition probabilities. The collective quadrupole operator in the intrinsic system (in lowest order of deformation parameters) is given by

The collective quadrupole moment is defined by

It does not depend on $n_{\beta}$ and $n_{\gamma}$, and its value reads

where the intrinsic quadrupole moment is defined

The reduced electric quadrupole transition probabilities are calculated using the expression

The so-called stretched $BE2$-values in a rotational band read

and, for $K=0$ bands, one obtains

The transition probabilities allow the determination of the intrinsic quadrupole moment and thus the deformation itself, whereas the energy spacings in the excitation spectra determine the moment of inertia.

The general form of the collective Bohr Hamiltonian has been derived starting from a microscopic Hamiltonian or from an energy density functional in two rather different ways: (i) using the generator coordinate method (GCM) Haff1973 ; Banerjee1973 ; Giraud1974 ; Onishi1975 ; Ring1980 , and (ii) applying the time-dependent Hartree-Fock (TDHF) theory Baranger1968 ; Baranger1978 ; Goeke1978 . Here, the former one is presented, taking into account that the derivation based on the GCM is fully quantum-mechanical. It relies on the validity of the Gaussian overlap approximation (GOA) for the overlaps between configurations with different deformations, and on the assumption that the collective velocities are small, i.e., that the expansion in the collective momenta up to second order is adequate. The validity of these approximations is demonstrated in a comparison with a full GCM calculation for a nucleus characterized by shape coexistence: ⁷⁶Kr Yao2014 .