11institutetext: Z. P. Li 22institutetext: School of Physical Science and
Technology, Southwest University, Chongqing 400715, China, 22email:
<EMAIL_ADDRESS>33institutetext: D. Vretenar 44institutetext: Department of
Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia;
State Key Laboratory of Nuclear Physics and Technology, School of Physics,
Peking University, Beijing 100871, China, 44email<EMAIL_ADDRESS>
# Model for collective motion

Z. P. Li and D. Vretenar<EMAIL_ADDRESS>
###### Abstract

Collective motion is a manifestation of emergent phenomena in medium-heavy and
heavy nuclei. A relatively large number of constituent nucleons contribute
coherently to nuclear excitations (vibrations, rotations) that are
characterized by large electromagnetic moments and transition rates. Basic
features of collective excitations are reviewed, and a simple model introduced
that describes large-amplitude quadrupole and octupole shape dynamics, as well
as the dynamics of induced fission. Modern implementations of the collective
Hamiltonian model are based on the microscopic framework of energy density
functionals, that provide an accurate global description of nuclear ground
states and collective excitations. Results of illustrative calculations are
discussed in comparison with available data.

##  1 Outline

  * •

Introduction

  * •

Nuclear Shape Parameters

  * •

Nuclear Surface Oscillations

  * •

The Rotation-Vibration Model

  * •

Microscopic Derivation of the Collective Hamiltonian

  * •

Microscopic Collective Hamiltonian Based on Density Functional Theory

##  2 Introduction

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 .

##  3 Nuclear Shape Parameters

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:

$R(\theta,\phi;t)=R_{0}\left(1+\sum_{\lambda=0}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}(t)^{*}Y_{\lambda\mu}(\theta,\phi)\right)$
(1)

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 $\lambda$.

Figure 1: Nuclear shapes with dipole ($\lambda=1$), quadrupole ($\lambda=2$),
octupole ($\lambda=3$), and hexadecupole ($\lambda=4$) deformations.

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

$R(\theta,\phi)=R_{0}\left(1+\sum_{\mu=-2}^{2}\alpha_{\mu}^{*}Y_{2\mu}(\theta,\phi)\right)$
(2)

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

$R(\theta^{\prime},\phi^{\prime})=R_{0}\left[1+a_{0}Y_{20}(\theta^{\prime},\phi^{\prime})+a_{2}Y_{22}(\theta^{\prime},\phi^{\prime})+a_{2}Y_{2-2}(\theta^{\prime},\phi^{\prime})\right].$
(3)

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:

$\displaystyle a_{0}$ $\displaystyle=\beta\cos\gamma,$ (4) $\displaystyle
a_{2}$ $\displaystyle=\frac{1}{\sqrt{2}}\beta\sin\gamma.$

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

$\delta
R_{\kappa}=R_{0}\sqrt{\frac{5}{4\pi}}\beta\cos\left(\gamma-\frac{2\pi}{3}\kappa\right),\quad\kappa=1,2,3$
(5)

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: Nuclear shapes in the $(\beta,\ \gamma)$ plane. Panel (a) is from
Ref. Fortunato2005 .

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.

##  4 Nuclear Surface Oscillations

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

$sV=\frac{1}{2}\sum_{\lambda\mu}C_{\lambda}\left|\alpha_{\lambda\mu}\right|^{2},\quad\
\
T=\frac{1}{2}\sum_{\lambda\mu}B_{\lambda}\left|\dot{\alpha}_{\lambda\mu}\right|^{2},$
(6)

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

$C_{\lambda}=(\lambda-1)(\lambda+2)R_{0}^{2}S-\frac{3}{2\pi}\frac{\lambda-1}{2\lambda+1}\frac{Z^{2}e^{2}}{R_{0}}$
(7)

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

$\alpha_{\mu}=\sum_{\nu}a_{\nu}D_{\mu\nu}^{2*}\left(\theta_{i}\right),$ (8)

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

$\sum_{\mu}\left|\alpha_{\mu}\right|^{2}=\sum_{\nu}a_{\nu}^{2}=a_{0}^{2}+2a_{2}^{2}=\beta^{2}.$
(9)

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

$\dot{\alpha}_{\mu}=\sum_{\nu}\dot{a}_{\nu}D_{\mu\nu}^{2*}\left(\theta_{i}\right)+\sum_{\nu
j}a_{\nu}\dot{\theta}_{j}\frac{\partial}{\partial\theta_{j}}D_{\mu\nu}^{2*}\left(\theta_{i}\right).$
(10)

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

$T=T_{\mathrm{vib}}+T_{\mathrm{rot}}.$ (11)

For the vibrational energy, one obtains

$T_{\mathrm{vib}}=\frac{1}{2}B\sum_{\nu}\left|\dot{a}_{\nu}\right|^{2}=\frac{1}{2}B\left(\dot{\beta}^{2}+\beta^{2}\dot{\gamma}^{2}\right)$
(12)

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

$\displaystyle T_{\mathrm{rot}}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{3}\mathcal{J}_{i}\omega_{i}^{\prime 2},$
(13) $\displaystyle\mathcal{J}_{i}$
$\displaystyle=4B\beta^{2}\sin^{2}(\gamma-i\frac{2\pi}{3}),\quad i=1,2,3$

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

$\hat{H}=-\frac{\hbar^{2}}{2B}\left[\frac{1}{\beta^{4}}\frac{\partial}{\partial\beta}\beta^{4}\frac{\partial}{\partial\beta}+\frac{1}{\beta^{2}}\frac{1}{\sin(3\gamma)}\frac{\partial}{\partial\gamma}\sin(3\gamma)\frac{\partial}{\partial\gamma}\right]+\sum_{i=1}^{3}\frac{\hat{I}_{i}^{2}}{2\mathcal{J}_{i}}+\frac{1}{2}C\beta^{2}$
(14)

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:

$\left[\hat{I}_{1},\hat{I}_{2}\right]=-\mathrm{i}\hbar\hat{I}_{3},\quad\left[\hat{I}_{2},\hat{I}_{3}\right]=-\mathrm{i}\hbar\hat{I}_{1},\quad\left[\hat{I}_{3},\hat{I}_{1}\right]=-\mathrm{i}\hbar\hat{I}_{2}.$
(15)

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

The stationary wave functions separate in the following way:

$\Psi\left(\beta,\gamma,\theta_{i}\right)=f(\beta)\Phi\left(\gamma,\theta_{i}\right).$
(16)

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

$f_{nl}(\beta)=(-1)^{n}\left[\frac{2(n!)^{3}}{\Gamma\left(n+l+\frac{5}{2}\right)}\right]^{1/2}\beta^{l}e^{-\beta^{2}/2}L_{n}^{l+3/2}\left(\beta^{2}\right),$
(17)

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

$E=(2n+l+\frac{5}{2})\hbar\omega=(N+\frac{5}{2})\hbar\omega,\quad
N=0,1,\cdots$ (18)

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

$\Phi_{Ml\nu}^{I}\left(\gamma,\theta_{i}\right)=\sum_{K=-I}^{I}g_{Kl\nu}^{I}(\gamma)D_{MK}^{I*}\left(\theta_{i}\right)$
(19)

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.

[width=3.5cm,height=3.5cm, autoplay, loop]1picture/fig3-transformations/R1-12

(a) $\hat{R}_{1}$

[width=3.5cm,height=3.5cm, autoplay, loop]1picture/fig3-transformations/R2-14

(b) $\hat{R}_{2}$

[width=3.5cm,height=3.5cm, autoplay,
loop]0.5picture/fig3-transformations/R3-13

(c) $\hat{R}_{3}$

Figure 3: Three basic transformations $\hat{R}_{1}$, $\hat{R}_{2}$ and
$\hat{R}_{3}$ that define the symmetry of the collective wave function.

Under these transformation the collective wave functions must satisfy:

$\displaystyle\hat{R}_{1}:\Psi\left(\beta,\gamma,\theta_{1},\theta_{2},\theta_{3}\right)=\Psi\left(\beta,\gamma,\theta_{1}+\pi,\pi-\theta_{2},-\theta_{3}\right),$
(20)
$\displaystyle\hat{R}_{2}:\Psi\left(\beta,\gamma,\theta_{1},\theta_{2},\theta_{3}\right)=\Psi\left(\beta,-\gamma,\theta_{1},\theta_{2},\theta_{3}+\frac{\pi}{2}\right),$
$\displaystyle\hat{R}_{3}:\Psi\left(\beta,\gamma,\theta_{1},\theta_{2},\theta_{3}\right)=\Psi\left(\beta,\gamma+\frac{2\pi}{3},\theta_{1},\theta_{2}+\frac{\pi}{2},\theta_{3}+\frac{\pi}{2}\right).$

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 $K\geq 0$:

$\displaystyle\Psi_{Mnl\nu}^{I}\left(\beta,\gamma,\theta_{i}\right)$
$\displaystyle={N}_{nl\nu}^{I}f_{nl}(\beta)\sum_{K\geq 0\atop\text{ even
}}g_{Kl\nu}^{I}(\gamma)\phi_{MK}^{I}\left(\theta_{i}\right),$ (21)

where

$\phi_{MK}^{I}\left(\theta_{i}\right)=\sqrt{\frac{2I+1}{16(1+\delta_{K0})\pi^{2}}}\left[D_{MK}^{I*}(\theta_{i})+(-1)^{I}D_{M,-K}^{I*}(\theta_{i})\right]$
(22)

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 .

##  5 The Rotation-Vibration Model

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).

Figure 4: Schematic plot of the collective potential $V(\beta,\ \gamma)$ for
$0\leq\gamma\leq\pi/3$, shape of global minimum, vibrations in $\beta$ and
$\gamma$ directions, and rotation.

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

$\xi=a_{0}-\beta_{0},\quad\eta=a_{2}=a_{-2}$ (23)

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

$H=T_{\text{vib}}+T_{\text{rot}}+V=\frac{1}{2}B\left(\dot{\xi}^{2}+2\dot{\eta}^{2}\right)+\frac{1}{2}\sum_{i=1}^{3}\mathcal{J}_{i}\omega_{i}^{\prime
2}+\frac{1}{2}C_{0}\xi^{2}+C_{2}\eta^{2}.$ (24)

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

$\mathcal{J}\equiv\mathcal{J}_{1}=\mathcal{J}_{2}=3B\beta_{0}^{2},\quad\mathcal{J}_{3}=8B\eta^{2},$
(25)

and the simplest expression for the kinetic energy reads

$T=\frac{1}{2}B\left(\dot{\xi}^{2}+2\dot{\eta}^{2}\right)+\frac{1}{2}\mathcal{J}\left(\omega_{1}^{\prime
2}+\omega_{2}^{\prime 2}\right)+4B\eta^{2}\omega_{3}^{\prime 2}.$ (26)

Similarly, the quantized Hamiltonian can be written in the form

$\hat{H}(\xi,\eta,\boldsymbol{\theta})=\frac{-\hbar^{2}}{2B}\left(\frac{\partial^{2}}{\partial\xi^{2}}+\frac{1}{2}\frac{\partial^{2}}{\partial\eta^{2}}\right)+\frac{\hat{I}_{1}^{2}+\hat{I}_{2}^{2}}{2\mathcal{J}}+\frac{\hat{I}_{3}^{2}-\hbar^{2}}{16B\eta^{2}}+\frac{1}{2}C_{0}\xi^{2}+C_{2}\eta^{2}.$
(27)

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

$\psi(\xi,\eta,\boldsymbol{\theta})=g(\xi)\chi(\eta){D}_{MK}^{I*}(\theta_{i}).$
(28)

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

$E_{n_{\beta}n_{\gamma}IK}=\hbar\omega_{\beta}\left(n_{\beta}+\frac{1}{2}\right)+\hbar\omega_{\gamma}\left(2n_{\gamma}+\frac{1}{2}|K|+1\right)+\frac{\hbar^{2}}{2\mathcal{J}}\left[I(I+1)-K^{2}\right],$
(29)

with

$\omega_{\beta}=\sqrt{\frac{C_{0}}{B}},\quad\omega_{\gamma}=\sqrt{\frac{C_{2}}{B}},\quad
n_{\beta}\ \text{or}\ n_{\gamma}=0,1,\cdots$ (30)

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

$\chi_{Kn_{\gamma}}(\eta)=N_{Kn_{\gamma}}\sqrt{|\eta|}\eta^{K/2}\mathrm{e}^{-\lambda\eta^{2}/2}{}_{1}F_{1}\left(-n_{\gamma},l_{K}+\frac{3}{2},\lambda\eta^{2}\right),$
(31)

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

$\psi^{I}_{MKn_{\beta}n_{\gamma}}(\xi,\eta,\boldsymbol{\theta})=\left\langle\xi\mid
n_{\beta}\right\rangle\chi_{Kn_{\gamma}}(\eta)\phi_{MK}^{I}\left(\theta_{i}\right),$
(32)

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:

$\displaystyle K$ $\displaystyle=0,2,4,\ldots$ (33) $\displaystyle I$
$\displaystyle=\begin{cases}K,K+1,K+2,\ldots\ \ \ \ \text{ for }K\neq 0\\\
0,2,4,\ldots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ for }K=0\end{cases}$
$\displaystyle M$ $\displaystyle=-I,-I+1,\ldots,+I.$

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:

Figure 5:  Structure of the excitation spectrum of the rotation-vibration
model. The probability density distributions in the $(\beta,\ \gamma)$ plane
for the band-heads are shown above each band. The notation and the
corresponding quantum numbers are indicated below the band-heads.

  * •

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

  * •

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.

  * •

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.

  * •

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

$\hat{Q}_{2\mu}=\frac{3Ze}{4\pi}R_{0}^{2}\left\\{{D}_{\mu
0}^{2*}(\theta_{i})(\beta_{0}+\xi)+\left({D}_{\mu
2}^{2*}(\theta_{i})+{D}_{\mu-2}^{2*}(\theta_{i})\right)\eta\right\\}.$ (34)

The collective quadrupole moment is defined by

$Q_{IKn_{\beta}n_{\gamma}}=\sqrt{\frac{16\pi}{5}}\left\langle
IM=IKn_{\beta}n_{\gamma}\left|\hat{Q}_{20}\right|IM=IKn_{\beta}n_{\gamma}\right\rangle.$
(35)

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

$Q_{IK}=Q_{0}\frac{3K^{2}-I(I+1)}{(I+1)(2I+3)},$ (36)

where the intrinsic quadrupole moment is defined

$Q_{0}=\frac{3ZeR_{0}^{2}}{\sqrt{5\pi}}\beta_{0}.$ (37)

The reduced electric quadrupole transition probabilities are calculated using
the expression

$B\left(E2;I_{i}\rightarrow I_{f}\right)=\frac{1}{2I_{i}+1}\left|\left\langle
I_{f}||\hat{Q}||I_{i}\right\rangle\right|^{2}.$ (38)

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

$B(E2;I+2\rightarrow I)=Q_{0}^{2}\frac{5}{16\pi}\left|C_{K\ \ \ \ 0\ K}^{I+2\
2\ I}\right|^{2}$ (39)

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

$B(E2,I+2\rightarrow
I)=Q_{0}^{2}\frac{5}{16\pi}\frac{3}{2}\frac{(I+1)(I+2)}{(2I+3)(2I+5)}.$ (40)

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.

##  6  Microscopic Derivation of the Collective Hamiltonian

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 .

###  6.1  General Concepts of the Generator Coordinate Method

In the generator coordinate method, the nuclear wave function $|\Phi\rangle$
can be expressed as a continuous superposition of generating functions
$|\varphi(\beta)\rangle$, that are labeled by an arbitrary number of real or
complex parameters $\\{\beta\\}=\beta_{1},\beta_{2},\ldots,\beta_{i}$, the so-
called generator coordinates,

$|\Phi\rangle=\int d\beta f(\beta)|\varphi(\beta)\rangle.$ (41)

The weight $f(\beta)$ is assumed to be a well behaved function of the
variables $\beta$, and the multidimensional integral includes all real and
imaginary parameters $\beta$. The set of generating wave functions
$\varphi(\beta)$ is often determined by a specific problem or geometry. In
many cases, the intrinsic states of a nucleus are chosen as the generating
states, and the corresponding shape deformation parameters as the generator
coordinates. See the schematic illustration in figure 6.

Figure 6:  The nuclear wave function $|\Phi\rangle$ as a continuous
superposition of generating functions $|\varphi(\beta)\rangle$, labeled by the
parameters $\beta$.

The variational principle implies

$\begin{split}\delta
E(\Phi)=\delta\frac{\langle\Phi|\hat{H}|\Phi\rangle}{\langle\Phi|\Phi\rangle}=\frac{1}{\langle\Phi|\Phi\rangle}\left(\langle\Phi|\hat{H}-E|\delta\Phi\rangle+\langle\delta\Phi|\hat{H}-E|\Phi\rangle\right)=0.\end{split}$
(42)

By inserting Eq.(41) into Eq.(42), and performing the variation with respect
to $f(\beta)$, an integral equation is obtained

$\displaystyle\int\langle\varphi(\beta)|\hat{H}|\varphi(\beta^{\prime})\rangle
f(\beta^{\prime})d\beta^{\prime}=E\int\langle\varphi(\beta)|\varphi(\beta^{\prime})\rangle
f(\beta^{\prime})d\beta^{\prime}.$ (43)

This is the _Hill-Wheeler equation_ and it can formally be written in the form

$Hf=ENf,$ (44)

with the overlap functions

$\displaystyle\begin{split}H(\beta,\beta^{\prime})&=\langle\varphi(\beta)|\hat{H}|\varphi(\beta^{\prime})\rangle~{}~{}~{}~{}~{}~{}The~{}energy~{}kernel\\\
N(\beta,\beta^{\prime})&=\langle\varphi(\beta)|\varphi(\beta^{\prime})\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}The~{}overlap~{}kernel\end{split}$
(45)

as integral kernels.

To solve this equation, a transformation is defined from the space of the
generating coordinates to the coordinate space in which the wave functions
specified by the true coordinates $\eta$ are orthogonal. The transformation
$\beta\rightarrow\eta$ is defined in such a way that
$N(\beta,\beta^{\prime})\rightarrow\delta(\eta-\eta^{\prime})$.

In the first step, the eigenvalue equation for the overlap kernel is solved

$\displaystyle\int
N(\beta,\beta^{\prime})\chi(s,\beta^{\prime})d\beta^{\prime}=\nu(s)\chi(s,\beta).$
(46)

The eigenfunctions specified by a new variable $s$ form a complete
orthonormalized set

$\displaystyle\int\chi^{*}(s,\beta)\chi(s^{\prime},\beta)d\beta=\delta(s-s^{\prime});\
\ \ \ \
\int\chi^{*}(s,\beta)\chi(s,\beta^{\prime})ds=\delta(\beta-\beta^{\prime}).$
(47)

Next, a transformation kernel is introduced

$\displaystyle
N^{\frac{1}{2}}(\beta,\eta)=\int\chi(s,\beta)\nu^{\frac{1}{2}}(s)\chi^{*}(s,\eta)ds,$
(48)

that represents the transition from $\delta(\eta-\eta^{\prime})$ to
$N(\beta,\beta^{\prime})$ through

$\displaystyle\int
N^{\frac{1}{2}*}(\beta,\eta)\delta(\eta-\eta^{\prime})N^{\frac{1}{2}}(\eta^{\prime},\beta^{\prime})d\eta
d\eta^{\prime}$ $\displaystyle=$
$\displaystyle\int\chi^{*}(s,\beta^{\prime})\nu(s)\chi(s,\beta)ds$ (49)
$\displaystyle=$ $\displaystyle\int
N(\beta,\beta^{\prime\prime})\delta(\beta^{\prime}-\beta^{\prime\prime})d\beta^{\prime\prime}$
$\displaystyle=$ $\displaystyle N(\beta,\beta^{\prime}).$

Obviously, the inverse $N^{-\frac{1}{2}}$ has to be determined to transform
$N(\beta,\beta^{\prime})$ to $\delta(\eta-\eta^{\prime})$. Accordingly, the
energy kernel should be transformed to the coordinate representation by using
the narrowing kernel

$\displaystyle L(\eta,\eta^{\prime})=\int
N^{-\frac{1}{2}*}(\beta,\eta)H(\beta,\beta^{\prime})N^{-\frac{1}{2}}(\beta^{\prime},\eta^{\prime})d\beta
d\beta^{\prime},$ (50)

and the new wave function $g(\eta)$ in coordinate representation is related to
the generating function $f(\beta)$

$\displaystyle g(\eta)=\int N^{\frac{1}{2}}(\beta,\eta)f(\beta)d\beta.$ (51)

By replacing each factor in Eq.(43) with the corresponding one after
transformation, the orthogonal Hill-Wheeler equation is obtained

$\displaystyle\int
L(\eta,\eta^{\prime})g(\eta^{\prime})d\eta^{\prime}=Eg(\eta),$ (52)

or in matrix form

$\displaystyle Lg=Eg.$ (53)

Eq. (52) can be considered as a stationary Schrödinger equation, where $L$ is
the collective Hamiltonian and $g$ the corresponding eigenfunction in the new
representation. To derive the collective Hamiltonian, one needs an explicit
expression for the transformation kernel $N^{\frac{1}{2}}$. This is achieved
by introducing the Gaussian overlap approximation.

###  6.2  The Gaussian Overlap Approximation

A simple and pedagogical derivation of the collective Hamiltonian can be
presented by using a single generator coordinate. A set of time-reversal
invariant generating functions $|\beta\rangle=|\varphi(\beta)\rangle$, which
depend on a single collective parameter $\beta$, can be obtained by
constrained self-consistent mean-field calculations. To derive the collective
Hamiltonian, the Gaussian overlap approximation (GOA) is based on two
assumptions: 1) generally, the overlap function
$N=\langle\beta|\beta^{\prime}\rangle$ rapidly becomes smaller with the
increase of the distance $|\beta-\beta^{\prime}|$; and 2) both the energy
kernel $H$ and overlap kernel $N$ are well behaved functions. Therefore, a
Gaussian function is assumed for the overlap kernel $N$

$\displaystyle
N(\beta,\beta^{\prime})=e^{-\frac{1}{2}\gamma_{0}(\beta-\beta^{\prime})^{2}}.$
(54)

Note that Eq.(54) is written in a homogeneous form, namely $\gamma_{0}$ is
constant.

The overlap kernel is a function that only depends on the difference
$(\beta-\beta^{\prime})$. In this case, the solutions of equation (46) are

$\displaystyle\nu(s)=\sqrt{\frac{2\pi}{\gamma_{0}}}e^{-\frac{s^{2}}{2\gamma_{0}}};\
\ \ \ \ \chi(s,\beta)=\frac{1}{\sqrt{2\pi}}e^{is\beta},$ (55)

for the eigenvalue and eigenfunction, respectively. Hence, the kernel
$N^{\frac{1}{2}}$ can be expressed as

$\displaystyle
N^{\frac{1}{2}}(\beta,\eta)=\int\chi^{*}(s,\beta)\nu^{\frac{1}{2}}(s)\chi(s,\eta)ds=(\frac{2\gamma_{0}}{\pi})^{\frac{1}{4}}e^{-\gamma_{0}(\beta-\eta)^{2}}.$
(56)

Starting from Eq. (50), and using the expressions above for $\chi(s,\beta)$,
$\nu(s)$, and $N^{-\frac{1}{2}}$, N. Onishi and T. Une derived the collective
Hamiltonian in Onishi1975 . Here, the approach proposed by P. Ring and P.
Schuck in their textbook The Nuclear Many-Body Problem Ring1980  will be
described.

One starts from the expectation of the Hamiltonian

$\displaystyle E=\langle\Phi|\hat{H}|\Phi\rangle=\int
f^{*}(\beta)H(\beta,\beta^{\prime})f(\beta^{\prime})d\beta d\beta^{\prime}$
(57)

and, to transform $f(\beta)$ and $H(\beta,\beta^{\prime})$ to $g(\eta)$ and
$L(\eta,\eta^{\prime})$ as in Eq. (52), the reduced energy kernel
$h(\beta,\beta^{\prime})=H(\beta,\beta^{\prime})/N(\beta,\beta^{\prime})$ is
introduced. The diagonal element of the reduced energy kernel is just the
expectation value of the Hamiltonian in the state $|\beta\rangle$. Then Eq.
(57) takes the form

$\displaystyle E=\int
f^{*}(\beta)N(\beta,\beta^{\prime})h(\beta,\beta^{\prime})f(\beta^{\prime})d\beta
d\beta^{\prime}.$ (58)

Inserting Eq. (49) into Eq. (58), the following expression is obtained

$\displaystyle E=\int
f^{*}(\beta)N^{\frac{1}{2}*}(\beta,\eta)\delta(\eta-\eta^{\prime})h(\beta,\beta^{\prime})N^{\frac{1}{2}}(\eta^{\prime},\beta^{\prime})f(\beta^{\prime})d\beta
d\beta^{\prime}d\eta d\eta^{\prime}.$ (59)

Next, $h(\beta,\beta^{\prime})$ is expanded to second order around the point
$\beta=\eta$, $\beta^{\prime}=\eta^{\prime}$

$\displaystyle\begin{split}h(\beta,\beta^{\prime})&=h+h_{\beta}(\beta-\eta)+h_{\beta^{\prime}}(\beta^{\prime}-\eta^{\prime})\\\
&\ \ \
+\frac{1}{2}\left[h_{\beta\beta}(\beta-\eta)^{2}+2h_{\beta\beta^{\prime}}(\beta-\eta)(\beta^{\prime}-\eta^{\prime})+h_{\beta^{\prime}\beta^{\prime}}(\beta^{\prime}-\eta^{\prime})^{2}\right].\end{split}$
(60)

To calculate the derivatives $h$, the Hamiltonian kernel and overlap kernel
are written in the form

$\displaystyle\begin{split}H(\beta,\beta^{\prime})&=\langle\eta|e^{-i(\beta-\eta)\hat{P}/\hbar}\hat{H}e^{i(\beta^{\prime}-\eta^{\prime})\hat{P}/\hbar}|\eta^{\prime}\rangle\\\
N(\beta,\beta^{\prime})&=\langle\eta|e^{-i(\beta-\eta)\hat{P}/\hbar}e^{i(\beta^{\prime}-\eta^{\prime})\hat{P}/\hbar}|\eta^{\prime}\rangle\end{split}$
(61)

where $\hat{P}=-i\hbar\frac{\partial}{\partial\beta}$ is the momentum
operator. Then the derivative $h$ can be expressed as

$\displaystyle
h_{\beta^{\prime}}=h_{\beta}=\left.\frac{\partial}{\partial\beta}\left(\frac{H(\beta,\beta^{\prime})}{N(\beta,\beta^{\prime})}\right)\right|_{\beta=\beta^{\prime}=\eta=\eta^{\prime}}=\left.-\frac{i}{\hbar}\langle\eta|\hat{P}\hat{H}|\eta^{\prime}\rangle\right|_{\eta=\eta^{\prime}}=0,$
(62)

where $\eta=\eta^{\prime}$ because of $\delta(\eta-\eta^{\prime})$ in Eq.
(59). Note that the generating functions $|\eta\rangle$ are time-reversal
invariant states, and $\hat{P}$ is a time-odd operator. Therefore, the first
derivatives vanish, while the second derivatives read

$\displaystyle\begin{split}h_{\beta^{\prime}\beta^{\prime}}&=h_{\beta\beta}=\left.-\frac{1}{\hbar^{2}}\left(\langle\eta|\hat{P}^{2}\hat{H}|\eta^{\prime}\rangle-\langle\eta|\hat{H}|\eta^{\prime}\rangle\langle\eta|\hat{P}^{2}|\eta^{\prime}\rangle\right)\right|_{\eta=\eta^{\prime}}\\\
h_{\beta\beta^{\prime}}&=\left.\frac{1}{\hbar^{2}}\left(\langle\eta|\hat{P}\hat{H}\hat{P}|\eta^{\prime}\rangle-\langle\eta|\hat{H}|\eta^{\prime}\rangle\langle\eta|\hat{P}^{2}|\eta^{\prime}\rangle\right)\right|_{\eta=\eta^{\prime}}.\end{split}$
(63)

In addition, the term
$\frac{\partial}{\partial\eta}h_{\beta}=h_{\beta\beta}+h_{\beta\beta^{\prime}}$
will be used when performing partial integration. Setting
$N_{1}=N^{\frac{1}{2}}(\beta,\eta)$,
$N_{2}=N^{\frac{1}{2}}(\beta^{\prime},\eta^{\prime})$, and inserting the
expansion of $h$ into Eq. (59), one obtains

$\displaystyle\begin{split}E&=\int d\beta d\beta^{\prime}d\eta
d\eta^{\prime}f^{*}(\beta)f(\beta^{\prime})N_{1}N_{2}\Big{\\{}h+h_{\beta}(\beta-\eta)+h_{\beta^{\prime}}(\beta^{\prime}-\eta^{\prime})\\\
&\ \ \ \ \
+\frac{1}{2}\big{[}h_{\beta\beta}(\beta-\eta)^{2}+2h_{\beta\beta^{\prime}}(\beta-\eta)(\beta^{\prime}-\eta^{\prime})+h_{\beta^{\prime}\beta^{\prime}}(\beta^{\prime}-\eta^{\prime})^{2}\big{]}\Big{\\}}.\end{split}$
(64)

From Eq.(56), $(\beta-\eta)N_{1}$ and $(\beta-\eta)^{2}N_{1}$ can be expressed
in the form

$\displaystyle\begin{split}(\beta-\eta)N_{1}&=\frac{1}{2\gamma_{0}}\frac{\partial}{\partial\eta}N_{1}\\\
(\beta-\eta)^{2}N_{1}&=(\frac{1}{4\gamma_{0}^{2}}\frac{\partial^{2}}{\partial\eta^{2}}+\frac{1}{2\gamma_{0}})N_{1}.\end{split}$
(65)

Obviously, $(\beta^{\prime}-\eta^{\prime})N_{2}$ and
$(\beta^{\prime}-\eta^{\prime})^{2}N_{2}$ have the same forms as in Eq. (65),
except that the derivatives are with respect to $\eta^{\prime}$. Then Eq. (64)
becomes

$\displaystyle E=$ $\displaystyle\int d\beta d\beta^{\prime}d\eta
d\eta^{\prime}\delta(\eta-\eta^{\prime})f^{*}(\beta)f(\beta^{\prime})\left[(h+\frac{h_{\beta\beta}+h_{\beta^{\prime}\beta^{\prime}}}{4\gamma_{0}})N_{1}N_{2}+\frac{1}{2\gamma_{0}}h_{\beta}N_{1}^{\prime}N_{2}\right.$
(66)
$\displaystyle\left.+\frac{1}{2\gamma_{0}}h_{\beta^{\prime}}N_{1}N_{2}^{\prime}+\frac{1}{8\gamma_{0}^{2}}h_{\beta\beta}N_{1}^{\prime\prime}N_{2}+\frac{1}{4\gamma_{0}^{2}}h_{\beta\beta^{\prime}}N_{1}^{\prime}N_{2}^{\prime}+\frac{1}{8\gamma_{0}^{2}}h_{\beta^{\prime}\beta^{\prime}}N_{1}N_{2}^{\prime\prime}\right].$

After partial integration, and inserting Eq. (51), the final result is
obtained neglecting higher-order derivatives

$\displaystyle E=\int
g^{*}\delta(\eta-\eta^{\prime})\left[h-\frac{h_{\beta\beta^{\prime}}}{2\gamma_{0}}+\frac{\partial}{\partial\eta}\frac{1}{4\gamma_{0}^{2}}(h_{\beta\beta}-h_{\beta\beta^{\prime}})\frac{\partial}{\partial\eta}\right]gd\eta
d\eta^{\prime}.$ (67)

Comparing to Eq. (52), the collective Hamiltonian $L(\eta,\eta^{\prime})$
finally reads

$\displaystyle\begin{split}L(\eta,\eta^{\prime})&=\delta(\eta-\eta^{\prime})\left[h-\frac{h_{\beta\beta^{\prime}}}{2\gamma_{0}}+\frac{\partial}{\partial\eta}\frac{1}{4\gamma_{0}^{2}}(h_{\beta\beta}-h_{\beta\beta^{\prime}})\frac{\partial}{\partial\eta}\right]\\\
&=\delta(\eta-\eta^{\prime})\left[-\frac{\partial}{\partial\eta}\frac{1}{2M(\eta)}\frac{\partial}{\partial\eta}+V(\eta)\right],\end{split}$
(68)

with the collective mass and potential

$\displaystyle\begin{split}\frac{1}{M(\eta)}&=\frac{-1}{2\gamma^{2}_{0}}(h_{\beta\beta}-h_{\beta\beta^{\prime}}),\\\
V(\eta)&=h-\frac{h_{\beta\beta^{\prime}}}{2\gamma_{0}},\end{split}$ (69)

respectively. In a more general case, namely for inhomogeneous GOA,
$\gamma_{0}$ in Eq. (54) depends on the collective coordinates. However, the
inhomogeneous GOA can always be brought into a homogeneous form by performing
a transformation from $\beta$ to a new coordinate
$\alpha=\int^{\beta}\sqrt{\gamma(\beta^{\prime})/\gamma_{0}}d\beta^{\prime}$
Ring1980 . Finally, to obtain the general form of the collective Hamiltonian,
a transformation back to the original coordinate is carried out by inserting
$d\eta=\sqrt{\gamma/\gamma_{0}}d\xi$ into Eq. (67), and using a new wave
function

$\displaystyle\tilde{g}=\gamma_{0}^{-1/4}g=\int(\frac{2}{\pi})^{\frac{1}{4}}e^{-\gamma(\beta-\xi)^{2}}f(\beta)d\beta.$
(70)

Then,

$\displaystyle
E=\int\sqrt{\gamma}\tilde{g}^{*}\delta(\xi-\xi^{\prime})\left[h-\frac{h_{\beta\beta^{\prime}}}{2\gamma}+\frac{1}{\sqrt{\gamma}}\frac{\partial}{\partial\xi}\sqrt{\gamma}\frac{1}{4\gamma^{2}}(h_{\beta\beta}-h_{\beta\beta^{\prime}})\frac{\partial}{\partial\xi}\right]\tilde{g}d\xi
d\xi^{\prime},$ (71)

Note that $h_{\beta\beta}$ and $h_{\beta\beta^{\prime}}$ will involve
$\gamma_{0}/\gamma$ when transforming back to the original coordinate. The
general form of the collective Hamiltonian reads

$\displaystyle
L(\xi,\xi^{\prime})=\delta(\xi-\xi^{\prime})\left[-\frac{1}{\sqrt{\gamma}}\frac{\partial}{\partial\xi}\sqrt{\gamma}\frac{1}{2M(\xi)}\frac{\partial}{\partial\xi}+V(\xi)\right],$
(72)

where the collective mass and collective potential have the same forms as in
Eq. (69), with $\gamma_{0}$ replaced by $\gamma$.

##  7  Microscopic Collective Hamiltonian Based on Density Functional Theory

In the previous section the collective Bohr Hamiltonian was derived starting
from a microscopic GCM framework. To illustrate possible applications of the
collective Hamiltonian to various low-energy structure and dynamical
phenomena, selected examples that use different collective coordinates will be
presented. The Hamiltonian that will be employed in the following examples is
constructed using covariant density functional theory (CDFT, details can be
found in Chapter 6) but, of course, it can equally well be built from any of a
number of successful non-relativistic density functionals. In the first
representative case, the quadrupole five-dimensional collective Hamiltonian
(5DCH) is applied to the phenomenon of shape coexistence in the neutron-
deficient nucleus 76Kr. The spectroscopy of pear-shaped nuclei can be
described using a quadrupole-octupole collective Hamiltonian. In the final
example, nuclear fission dynamics is modeled by the time-evolution of a
collective wave packet in the space of quadrupole and octupole axially
symmetric deformations.

###  7.1  The five-dimensional collective Hamiltonian

To describe complex phenomena that originate in collective quadrupole
excitations of nuclei, e.g., shape phase transitions, shape coexistence, and
super-deformed bands, the simple Bohr Hamiltonian of Eq. (14) has to be
extended to a general five-dimensional collective Hamiltonian (5DCH),

$\displaystyle\hat{H}(\beta,\gamma,\theta)$
$\displaystyle=\hat{T}_{\text{vib}}+\hat{T}_{\text{rot}}+V_{\text{coll}}$
$\displaystyle=-\frac{\hbar^{2}}{2\sqrt{wr}}\left\\{\frac{1}{\beta^{4}}\left[\frac{\partial}{\partial\beta}\sqrt{\frac{r}{w}}\beta^{4}B_{\gamma\gamma}\frac{\partial}{\partial\beta}-\frac{\partial}{\partial\beta}\sqrt{\frac{r}{w}}\beta^{3}B_{\beta\gamma}\frac{\partial}{\partial\gamma}\right]\right.$
$\displaystyle\ \ \
\left.+\frac{1}{\beta\sin{3\gamma}}\left[-\frac{\partial}{\partial\gamma}\sqrt{\frac{r}{w}}\sin{3\gamma}B_{\beta\gamma}\frac{\partial}{\partial\beta}+\frac{1}{\beta}\frac{\partial}{\partial\gamma}\sqrt{\frac{r}{w}}\sin{3\gamma}B_{\beta\beta}\frac{\partial}{\partial\gamma}\right]\right\\}$
$\displaystyle\ \ \
+\frac{1}{2}\sum_{k=1}^{3}{\frac{\hat{J}^{2}_{k}}{\mathcal{I}_{k}}}+V_{\text{coll}}.$
(73)

where $\hat{J}_{k}$ denotes the components of the angular momentum in the
body-fixed frame of a nucleus, and both the mass parameters $B_{\beta\beta}$,
$B_{\beta\gamma}$, $B_{\gamma\gamma}$, and the moments of inertia
$\mathcal{I}_{k}$, depend on the quadrupole deformation variables $\beta$ and
$\gamma$. $V_{\text{coll}}$ is the collective potential. Two additional
quantities that appear in $\hat{T}_{\text{vib}}$, namely $r=B_{1}B_{2}B_{3}$
($B_{k}$ are related to $\mathcal{I}_{k}$ Niksic2009 ), and
$w=B_{\beta\beta}B_{\gamma\gamma}-B_{\beta\gamma}^{2}$, determine the volume
element in the collective space. Details about the DFT-based 5DCH can be found
in Refs. Niksic2009 ; Libert1999 ; Prochniak2004 .

The 5DCH describes quadrupole vibrations, rotations, and the coupling of these
collective modes. The corresponding eigenvalue equation is solved by expanding
the eigenfunctions on a complete set of basis functions that depend on the
deformation variables $\beta$ and $\gamma$, and the Euler angles Prochniak2004
. The result are the excitation energy spectrum $E^{I}_{\alpha}$ and the
collective wave functions

$\displaystyle\Psi^{IM}_{\alpha}(\beta,\gamma,\theta)=\sum_{K\in\Delta
I}\psi^{I}_{\alpha K}(\beta,\gamma)\phi^{I}_{MK}(\theta),$ (74)

where $M$ and $K$ are the projections of angular momentum $I$ on the third
axis in the laboratory and intrinsic frames, respectively, and $\alpha$
denotes additional quantum numbers. The rotational function
$\phi^{I}_{MK}(\theta)$ is defined by Eq. (22). Using the collective wave
functions, various observables, such as $E2$ transition probabilities, can be
calculated,

$\displaystyle B(E2;\alpha
I\rightarrow\alpha^{\prime}I^{\prime})=\frac{1}{2I+1}|\langle\alpha^{\prime}I^{\prime}||\hat{M}(E2)||\alpha
I\rangle|^{2},$ (75)

where $\hat{M}(E2)$ is the electric quadrupole operator.

The next step is to calculate the collective inertia parameters $B$ and
$\mathcal{I}_{k}$, and the collective potential $V_{\rm coll}$, using
solutions of microscopic self-consistent deformation-constrained triaxial CDFT
calculations (c.f. Chapter 6). In principle, one can use results obtained with
the GCM+GOA framework. However, in practice, the resulting inertia parameters
are generally too small Baranger1978 ; Goeke1980 . The adiabatic time-
dependent Hartree-Fock (ATDHF) theory Baranger1978  provides an alternative
way to derive a classical collective Hamiltonian, and, after requantization, a
Bohr Hamiltonian of the same structure is obtained but with different
microscopic expressions for the inertia parameters Villars1977 . This method
has the advantage that the time-odd components of the microscopic wave
functions are also included and, in this sense, the full dynamics of a nuclear
system. In many applications a further simplification is thus introduced in
terms of cranking formulas for the inertia parameters and zero-point energy
corrections Girod1979 .

The entire map of the energy surface as a function of quadrupole deformations
is obtained by imposing constraints on the axial and triaxial mass quadrupole
moments:

$\langle H\rangle+\sum_{\mu=0,2}C_{2\mu}(\langle\hat{Q}_{2\mu}\rangle-
q_{2\mu})^{2},$ (76)

where $q_{2\mu}$ is the constrained value of the multipole moment and
$C_{2\mu}$ is the corresponding stiffness constant Ring1980 .
$\langle\hat{Q}_{2\mu}\rangle$ denotes the expectation value of the mass
quadrupole operator:

$\langle\hat{Q}_{20}\rangle=2z^{2}-x^{2}-y^{2}\ \ \text{and}\ \
\langle\hat{Q}_{22}\rangle=x^{2}-y^{2}.$ (77)

The single-nucleon wave functions, energies, and occupation factors, generated
from constrained triaxial CDFT calculations, provide the microscopic input for
the parameters of the collective Hamiltonian. The moments of inertia are
calculated according to the Inglis-Belyaev formula Inglis1956 ; Belyaev1961

$\mathcal{I}_{k}=\sum_{i,j}\frac{\left(u_{i}v_{j}-v_{i}u_{j}\right)^{2}}{E_{i}+E_{j}}\left|\left\langle
i\left|\hat{J}_{k}\right|j\right\rangle\right|^{2},\quad k=1,2,3,$ (78)

where $k$ denotes the axis of rotation, and the summation runs over the proton
and neutron quasiparticle states. $E_{i}$, $v_{i}$, and $|i\rangle$ are the
quasiparticle energies, occupation probabilities, and single-nucleon wave
functions, respectively. The mass parameters associated with the two
quadrupole collective coordinates
$q_{0}=\left\langle\hat{Q}_{20}\right\rangle$ and
$q_{2}=\left\langle\hat{Q}_{22}\right\rangle$ are also calculated in the
cranking approximation Ref.Girod1979

$B_{\mu\nu}\left(q_{0},q_{2}\right)=\hbar^{2}\left[\mathcal{M}_{(1)}^{-1}\mathcal{M}_{(3)}\mathcal{M}_{(1)}^{-1}\right]_{\mu\nu},$
(79)

with

$\mathcal{M}_{(n),\mu\nu}\left(q_{0},q_{2}\right)=\sum_{i,j}\frac{\left\langle
i\left|\hat{Q}_{2\mu}\right|j\right\rangle\left\langle
j\left|\hat{Q}_{2v}\right|i\right\rangle}{\left(E_{i}+E_{j}\right)^{n}}\left(u_{i}v_{j}+v_{i}u_{j}\right)^{2}.$
(80)

The collective energy surface includes the energy of zero-point motion, which
has to be subtracted. The collective zero-point energy (ZPE) corresponds to a
superposition of zero-point motion of individual nucleons in the single-
nucleon potential. In practice, ZPE corrections originating from the
vibrational and rotational kinetic energy are considered. These are given by
the following expressions Girod1979

$\Delta
V_{\mathrm{vib}}\left(q_{0},q_{2}\right)=\frac{1}{4}\operatorname{Tr}\left[\mathcal{M}_{(3)}^{-1}\mathcal{M}_{(2)}\right],$
(81)

and

$\Delta V_{\mathrm{rot}}\left(q_{0},q_{2}\right)=\Delta
V_{-2-2}\left(q_{0},q_{2}\right)+\Delta
V_{-1-1}\left(q_{0},q_{2}\right)+\Delta V_{11}\left(q_{0},q_{2}\right),$ (82)

with

$\Delta
V_{\mu\nu}\left(q_{0},q_{2}\right)=\frac{1}{4}\frac{\mathcal{M}_{(2),\mu\nu}\left(q_{0},q_{2}\right)}{\mathcal{M}_{(3),\mu\nu}\left(q_{0},q_{2}\right)}.$
(83)

Finally, the collective potential $V_{\rm coll}$ in Eq. (7.1) is obtained by
subtracting the ZPE corrections from the total mean-field energy:

$V_{\text{coll }}\left(q_{0},q_{2}\right)=E_{\text{tot
}}\left(q_{0},q_{2}\right)-\Delta V_{\text{vib
}}\left(q_{0},q_{2}\right)-\Delta V_{\text{rot }}\left(q_{0},q_{2}\right).$
(84)

###  7.2  Shape coexistence in 76Kr

The observation that an atomic nucleus with a particular combination of
neutron $N$, and proton $Z$ numbers, can exhibit states characterized by
different shapes, i.e., shape coexistence, appears to be a unique feature of
finite many-body quantum systems Heyde2011 . The low-lying states of neutron-
deficient even-even krypton isotopes are of particular interest due to their
rapid structural change with neutron number, and the presence of multiple
shape coexistence; i.e., several $0^{+}$ states with different intrinsic
shapes coexist at low excitation energy. The irregularities observed in the
ground-state bands at low spin in 74,76Kr were explained by shape coexistence
in Ref. Piercey1981 . As an example of the application of the quadrupole
collective Hamiltonian, here the CDFT-based 5DCH model is used to analyze
shape coexistence in 76Kr. A systematic study of shape evolution and shape
coexistence in Kr isotopes has been reported in Ref. Fu2013 .

Figure 7 displays the collective potential energy surface in the
$\beta$-$\gamma$ plane for the even-even nucleus 76Kr, with the ZPEs of
rotational and vibrational motion subtracted from the total deformation energy
[cf. Eq. (84)]. As shown in the panel on the right, after subtraction of the
ZPEs the prolate deformed minimum becomes deeper, and the energy with respect
to the spherical shape is reduced by $\approx 0.9$ MeV. This leads to a
coexistence picture of competing spherical and prolate minima on the energy
surface. When correlations related to restoration of broken symmetries are
taken into account in the 5DCH calculation, the ground state of 76Kr is
dominated by the prolate deformed configurations.

Figure 7: (a) Collective potential energy surface of 76Kr Fu2013  in the
$\beta$-$\gamma$ plane calculated using the constrained triaxial CDFT with the
PC-PK1 functional Zhao2010 . (b) Comparison of the total axial deformation
energy and the collective potential of76Kr. All energies are normalized to
that of the spherical shape.

The 5DCH low-spin excitation spectrum of the nucleus 76Kr is compared with
available data in Fig. 8. The main features are reproduced very well by the
model calculation, in particular for the ground-state band and the low-lying
$0^{+}_{2}$ state. The observed large $E2$ transition strength from the
$0^{+}_{2}$ state to the $2^{+}_{1}$ state is underestimated by about a factor
of four. This may imply that the mixing between the two calculated bands is
too weak. In contrast, the large $B(E2;0^{+}_{2}\to 2^{+}_{1})$ can be
reproduced by a similar 5DCH calculation that uses the Gogny D1S functional,
which predicts a rather large mixing between the two $0^{+}$ bands,
characterized by a very large electric monopole transition rate
$\rho(E0;0^{+}_{2}\to 0^{+}_{1})$ Clement2007 .

Figure 8:  (Color online) Low-spin excitation spectrum of 76Kr Fu2013 , in
comparison with available data Clement2007 ; NuclearData . The $E2$ transition
strengths are in units $e^{2}$ fm4.

The density distribution of the collective states, which takes the following
form,

$\displaystyle\rho_{I\alpha}(\beta,\gamma)=\sum_{K\in\Delta
I}|\psi^{I}_{\alpha K}(\beta,\gamma)|^{2}\beta^{3},$ (85)

with the normalization

$\displaystyle\int^{\infty}_{0}\beta
d\beta\int^{2\pi}_{0}\rho_{I\alpha}(\beta,\gamma)|\sin(3\gamma)|d\gamma=1,$
(86)

provides further insight into shape coexistence. Figure 9 displays
$\rho_{I\alpha}$ in the $\beta$-$\gamma$ plane for the first two $0^{+}$
states, and the $2^{+}_{1}$ state in 76Kr. Obviously, the dominant
configurations of the ground state band correspond to the large prolate
deformed shape at $(\beta\sim 0.50)$. The distribution of probability density
$\rho_{I\alpha}(\beta,\ \gamma)$ indicates a prolate-oblate mixed
configuration for the $0^{+}_{2}$ state. From the potential energy surface
shown in Fig. 4, one notices that 76Kr is rather soft with respect to $\gamma$
deformation, and this is reflected in the structure of the $0^{+}_{2}$ state.

Figure 9: Distribution of the probability density $\rho_{I\alpha}(\beta,\
\gamma)$ for the first two $0^{+}$ states, and the $2_{1}^{+}$ state in 76Kr
Fu2013 .

###  7.3  Quadrupole-octupole collective Hamiltonian for pear-shaped nuclei

Even though most deformed medium-heavy and heavy nuclei exhibit quadrupole,
reflection-symmetric equilibrium shapes, there are regions in the mass table
where octupole deformations (pear shapes) occur, in particular, nuclei with
neutron (proton) number $N(Z)\approx 34,56,88$ and $134$. Pear shapes are
characterized by the occurrence of low-lying negative-parity bands [c.f. Fig.
(10)], as well as pronounced electric octupole transitions
Gaffney2013Nature199 . The physics of octupole correlations was extensively
explored several decades ago (see the review of Ref. Butler1996 ), but there
has also been a strong revival of interest in pear shapes more recently, as
shown by a series of experimental studies Butler2019Nature ; Butler2020PRL ;
Chishti2020Nature .

Figure 10: The low-lying rotational spectrum of 224Ra, compared with that of
the H35Cl molecule Butler1996 . Schematic shapes for the two systems are also
shown.

The quadrupole-octupole collective Hamiltonian (QOCH), which can
simultaneously treat the axially symmetric quadrupole and octupole vibrational
and rotational excitations, is expressed in terms of two deformation
parameters $\beta_{2}$ and $\beta_{3}$, and the Euler angles $\theta$ that
define the orientation of the intrinsic principal axes in the laboratory
frame,

$\begin{split}{{\hat{H}}_{\text{coll}}}=&-\dfrac{\hbar^{2}}{2\sqrt{w\mathcal{I}}}\left[\dfrac{\partial}{\partial\beta_{2}}\sqrt{\dfrac{\mathcal{I}}{w}}B_{33}\dfrac{\partial}{\partial\beta_{2}}-\dfrac{\partial}{\partial\beta_{2}}\sqrt{\dfrac{\mathcal{I}}{w}}B_{23}\dfrac{\partial}{\partial\beta_{3}}\right.\\\
&\left.-\dfrac{\partial}{\partial\beta_{3}}\sqrt{\dfrac{\mathcal{I}}{w}}B_{23}\dfrac{\partial}{\partial\beta_{2}}+\dfrac{\partial}{\partial\beta_{3}}\sqrt{\dfrac{\mathcal{I}}{w}}B_{22}\dfrac{\partial}{\partial\beta_{3}}\right]\\\
&+\dfrac{\hat{J}^{2}}{2\mathcal{I}}+{{V}_{\text{coll}}}({{\beta}_{2}},{{\beta}_{3}}).\end{split}$
(87)

$\hat{J}$ denotes the component of angular momentum perpendicular to the
symmetry axis in the body-fixed frame of a nucleus. The mass parameters
$B_{22}$, $B_{23}$, and $B_{33}$, the moments of inertia $\mathcal{I}$, and
collective potential $V_{\text{coll}}$, depend on the quadrupole and octupole
deformation variables $\beta_{2}$ and $\beta_{3}$. The additional quantity
that appears in the vibrational kinetic energy,
$w=\sqrt{B_{22}B_{33}-B_{23}^{2}}$, determines the volume element in the
collective space. Just as in the case of the 5DCH model, all the collective
parameters are calculated from the self-consistent solutions of constrained
reflection-asymmetric CDFT, using cranking formulas Xia2017 .

Figure 11:  Deformation energy surfaces of the nuclei 222-228Ra Sun2019  in
the $\beta_{2}$-$\beta_{3}$ plane calculated with the CDFT, using the PC-PK1
functional Zhao2010 . For each nucleus the energies are normalized with
respect to the binding energy of the global minimum. Contour lines are
separated by 0.5 MeV (between neighboring solid curves) and 0.25 MeV (between
neighboring dashed and solid curves), respectively. The density distributions
of the global equlibrium minima of 224,226Ra are also shown in panels on the
right.

As an illustrative example, the CDFT-based QOCH is used to calculate the low-
energy excitation spectra and electromagnetic transitions of Ra isotopes
Sun2019 . In Fig. 11 it is shown that already at the mean-field level the
calculation predicts a very interesting structural evolution in Ra isotopic
chain. Quadrupole deformation increases starting from 222Ra, and one also
notices the emergence of octupole deformation with $\beta_{3}\sim 0.11$. For
224,226Ra the occurrence of a rather strongly marked octupole minimum is
predicted. The deepest octupole minimum is calculated in 226Ra and the
octupole deformation energy is $\sim 0.94$ MeV. In 228Ra the deformation
energy surface exhibits a softer minimum in the $\beta_{3}$ direction, and the
octupole deformation starts to decrease.

Figure 12: (Color online) The excitation spectrum, intraband $B(E2)$ (in
W.u.), and interband $B(E3)$ (in W.u.) values of 222-228Ra, calculated with
the QOCH based on the functional PC-PK1, and compared to experimental results
NuclearData ; Gaffney2013Nature199 ; Butler2020PRL .

In Fig. 12 the low-energy excitation spectra of positive- and negative-parity
states, the corresponding $B(E2)$ values for intraband transitions, and the
interband $B(E3)$ values, calculated with the QOCH based on the PC-PK1
functional, are compared with data for the nuclei 222-228Ra
Gaffney2013Nature199 . The level schemes show that the lowest negative-parity
bands are located close in energy to the corresponding ground-state positive-
parity bands. In fact, one notices that the lowest positive- and negative-
parity bands form a single, alternating-parity band, starting with angular
momentum $J\approx 5$. Except for the negative-parity band-heads in 222Ra and
228Ra that are calculated somewhat higher and lower than their corresponding
experimental counterparts, respectively, the overall structure agrees very
well with the available data. The calculated $E2$ and $E3$ transition rates
are also in reasonable agreement with the experimental values. The shape
transition is also confirmed by other characteristic collective observables,
such as the $B(E3;3^{-}_{1}\to 0^{+}_{1})$ and $B(E2;2^{+}_{1}\to 0^{+}_{1})$.
The $B(E2;2^{+}_{1}\to 0^{+}_{1})$ values increase from 222Ra to 228Ra,
indicating an enhancement of quadrupole collectivity. The theoretical
$B(E3;3^{-}_{1}\to 0^{+}_{1})$ values can be used as a measure of octupole
collectivity, and exhibit a maximum in 226Ra, together with $E3$ rates for
higher spin states.

Figure 13: Potential energy surface of $~{}^{228}$Th in the
$(\beta_{2},\beta_{3})$ plane, calculated using the reflection-asymmetric CDFT
with the PC-PK1 functional. The nucleon density distributions at
$(\beta_{2},\beta_{3})=(3.6,2.56)$ and $(5.36,0)$, are shown in the panels on
the right.

###  7.4  Time-dependent GCM+GOA for nuclear fission

As a final example, the time-dependent generator coordinate method plus
Gaussian overlap approximation (TDGCM+GOA) is applied to nuclear fission
dynamics Tao2017 ; Verriere2020 ; Schunck2016 . In the exothermic process of
fission decay, an atomic nucleus splits into two or more independent
fragments. A quasi-stationary initial state evolves in time through a sequence
of increasingly deformed shapes. The system reaches the outer saddle point on
the deformation energy surface, and continues to deform while a neck appears
that eventually becomes so thin that scission occurs. Fission can be described
as a slow adiabatic process determined by only a few collective degrees of
freedom, for example the quadrupole $\beta_{2}$ and octupole $\beta_{3}$
deformation parameters. In the TDGCM+GOA fission dynamics is modeled by a
local, time-dependent Schrödinger-like equation in the space of collective
coordinates

$i\hbar\frac{\partial g(\beta_{2},\beta_{3},t)}{\partial t}=\hat{H}_{\rm
coll}(\beta_{2},\beta_{3})g(\beta_{2},\beta_{3},t),$ (88)

with the collective Hamiltonian

$\hat{H}_{\rm
coll}(\beta_{2},\beta_{3})=-\frac{\hbar^{2}}{2}\sum_{kl}\frac{\partial}{\partial\beta_{k}}[B^{-1}(\beta_{2},\beta_{3})]_{kl}\frac{\partial}{\partial\beta_{l}}+V(\beta_{2},\beta_{3}).$
(89)

$g(\beta_{2},\beta_{3},t)$ is a time-dependent complex wave function in the
$(\beta_{2},\beta_{3})$ plane. Eq.(88) describes how an atomic nucleus
characterized by the collective mass $B_{kl}$ evolves in time in the
collective potential $V$. As an illustration, the left panel of Fig.13
displays the potential energy surface of $~{}^{228}$Th, calculated using CDFT
with the PC-PK1 functional Zhao2010 . From
$(\beta_{2},\beta_{3})\approx(0.9,0.0)$ to $\approx(3.6,2.56)$, an asymmetric
fission valley extends on the energy surface, with two saddle points located
at $(\beta_{2},\beta_{3})\approx(1.2,0.35)$ and $\approx(2.1,1.0)$. For
elongations $\beta_{2}>2.0$, a symmetric valley extends up to the scission
point at $\beta_{2}\approx 5.36$. The symmetric and asymmetric fission valleys
are separated by a ridge from $(\beta_{2},\beta_{3})\approx(1.6,0.0)$ to
$\approx(3.8,1.2)$.

Figure 14: Time evolution of the probability density $|g|^{2}$ in the
$(\beta_{2},\beta_{3})$ plane. The solid contour corresponds to the scission
hypersurface.

In the panels on the right of Fig. 13, the nucleon density distribution is
plotted at $(\beta_{2},\beta_{3})=(3.6,2.56)$ and $(5.36,0)$, two point on the
potential energy surface that are close to scission. The nascent fission
fragments can be clearly identified and, therefore, the number of particles in
the connecting neck can be defined

$Q=\int_{0}^{2\pi}d\varphi\int_{0}^{\infty}rdr\int_{-\infty}^{+\infty}\rho(r,\varphi,z)exp[-(z-z_{n})^{2}]dz$
(90)

where $\rho$ is the nucleon density distribution in cylindrical coordinates,
and $z_{n}$ is the point of minimum density on the z-axis. When the number of
nucleons in the neck region falls below a certain value, typically few
nucleons, scission occurs. Considering the entire PES, the scission points
define the scission hypersurface. Given an appropriate initial state, Eq. (88)
evolves the collective wave function from the inner region with a single
nuclear density distribution, to the external region that contains the two
fission fragments. (see the illustration in Fig. 14).

Figure 15: The charge yields (normalized to 200%) for induced fission of
$~{}^{228}$Th, calculated using the TDGCM+GOA, in comparison with data of
photo-induced fission at an excitation energy $\approx 11$ MeV Schmidt2001 .

The flux of the probability current through the scission hypersurface provides
a measure of the probability of observing a given pair of fragments at time
$t$. The expression for the probability current $J$ in the collective space
reads

$J_{k}(\beta_{2},\beta_{3},t)=\frac{\hbar}{2i}\sum_{l=2}^{3}B_{kl}(\beta_{2},\beta_{3})[g^{*}(\beta_{2},\beta_{3},t)\frac{\partial
g(\beta_{2},\beta_{3},t)}{\partial\beta_{l}}-g(\beta_{2},\beta_{3},t)\frac{\partial
g^{*}(\beta_{2},\beta_{3},t)}{\partial\beta_{l}}].$ (91)

The integrated flux for a given scission surface element associated with a
specific pair of fragments is computed from

$Y=\int_{0}^{\infty}dt\vec{J}\cdot\vec{ds}.$ (92)

By considering all points on the scission contour, the final charge or mass
distribution of fission fragments is obtained. Figure 15 displays the charge
yields (normalized to 200%) for induced fission of $~{}^{228}$Th, calculated
using the TDGCM+GOA, in comparison with data of photo-induced fission at an
excitation energy $\approx 11$ MeV Schmidt2001 . The calculation reproduces
the trend of the data except that, without particle number projection, the
model obviously cannot describe the odd-even staggering of the experimental
charge yields.

##  8  Further Reading

This chapter has focused on the basic features of large amplitude collective
motion in atomic nuclei. Various additional topics related to nuclear
collective motion are thoroughly covered in the following textbooks:

  * •

P. Ring and P. Schuck, The nuclear many-body problem, Springer-Verlag, 2004.

  * •

W. Greiner and J. A. Maruhn, Nuclear models, Springer-Verlag, 1996.

  * •

D. J. Rowe, Nuclear collective motion, World Scientific, 2010.

  * •

D. J. Rowe and J. L. Wood, Fundamentals of nuclear models, World Scientific,
2010.

  * •

W. Younes and W. D. Loveland, An introduction to nuclear fission, Springer-
Verlag, 2021.

Different aspects of quadrupole collective dynamics, and more detailed
derivations of specific implementations of the collective model, are discussed
in review articles:

  * •

L. Próchniak, and S. G. Rohoziński, Quadrupole collective states within the
Bohr collective Hamiltonian, J. Phys. G: Nucl. Part. Phys. 36, 123101 (2009).

  * •

T. Nikšić, D. Vretenar, and P. Ring, Relativistic nuclear energy density
functionals: Mean-field and beyond, Prog. Part. Nucl. Phys. 66, 519 (2011).

  * •

S. G. Rohoziński, Gaussian overlap approximation for the quadrupole collective
states, J. Phys. G: Nucl. Part. Phys 39, 095104 (2012).

  * •

S. G. Rohoziński, On the Gaussian overlap approximation for the collective
excitations of odd nuclei, J. Phys. G: Nucl. Part. Phys. 42, 025109 (2015).

  * •

K. Matsuyanagi, M. Matsuo, T. Nakatsukasa, K. Yoshida, N. Hinohara, and K.
Sato, Microscopic derivation of the quadrupole collective Hamiltonian for
shape coexistence/mixing dynamics, J. Phys. G: Nucl. Part. Phys 43, 024006
(2016).

  * •

J. L. Egido, State-of-the-art of beyond mean field theories with nuclear
density functionals, Phys. Scr. 91, 073003 (2016).

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§ ACRONYMS AND NOTATION

§.§ List of Acronyms

2SRAtwo-step random access
4SRAfour-step random access
5GNR5G New Radio

AWGNadditive white Gaussian noise

BPbelief propagation
BSbase station

CCScoded compressed sensing
c.c.u.complex channel use
CPcyclic prefix
CRDSAcontention resolution diversity slotted Aloha
CScompressed sensing
CSAcoded slotted Aloha
c.u.channel use

IDMAinterleaver division multiple access
IoTInternet of Things
IRSAirregular repetition slotted Aloha
ISIinter-symbol interference

LDPClow-density parity-check
LLRlog-likelihood ratio
LTELong Term Evolution

MACmultiple access
MMSEminimum mean square error
MPRmulti-packet reception
MTCmachine-type communication

NB-IoTNarrowband IoT
NRnew radio

OFDMorthogonal frequency-division modulation
OMPorthogonal matching pursuit

PAMpulse-amplitude modulation
PBCHphysical broadcast channel
PDSCHphysical downlink shared channel
POPUSCH occasion
PRACHphysical random access channel
PRBphysical resource block
PSSprimary synchronization signal
PUPEper-user probability of error
PUSCHphysical uplink shared channel

QPSKquadrature phase shift keying

RArandom access
RCUrandom coding union

SB-IDMAsparse block interleaver division multiple access
SCLsuccessive cancellation list
SCSsub-carrier spacing
SICsuccessive interference cancellation
SNRsignal-to-noise ratio
SPARCsparse regression code
SSSsecondary synchronization signal

TAtime advance

UTuser terminal
UMACunsourced multiple access

§.§ Notation Summary

$\code$     Error correcting code

$E_b$     Energy per information bit

$k$     Number of information bits per active user

$K$     Total number of users

$K_a$     Number of active users

$n$     Frame length

$n_s$     Number of segments (SB-IDMA)

$N$     Number of PUSCH occasions

$N_0$     Single-sided noise power spectral density

$n_c$     Block length of the error correcting code

$\POlength$     number of channel uses in a PO

$\preamblelength$     Preamble length in channel uses

$\NRX$     Number of antennas at the base station

$\PUPE$     Per-user probability of error