Bifurcations in quantum rotational spectra

REPORTS (Review Section of Physics Letters) 226, Nos. 4 & 5 (1993) 173—279. Iland

PHYSICS REPORTS

ifurcations in quantum rotational spectra I. M. Pavlichenkov1

Russian National Research Center ‘Kurchatov Institute”, Moscow, 123 182, Russia

Received August 1992; editor: E.W. McDaniel Contents: I. Introduction 2. Bjfurcatjons in an isolated rotational band ofnonaxial many-body systems 2.1. Effective rotational Hamiltonian and its symmetry 2.2. Types of symmetric bifurcations: classical analysis 3. Phenomenological theory of local bifurcations: quanturn approach 3.1. C 2 -type bifurcation 3.2. C3~-typebifurcation 3.3. C4 -type bifurcation 4. Experimental situation: molecules 4.1. Bifurcation in rotational spectra of symmetric three-atom molecules XY2 4.2. Spherical top molecules with high symmetry 5. Bifurcation in the rotational spectra of odd-A deformed nuclei

175 180 181 185 196 196 207 210 214 215 222 234

5.1. 5.2. 5.3. 5.4.

Effective Hamiltonian of the particle—rotor model Classical picture of bifurcation Quantum bifurcation Electromagnetic transitions and static moments in the lowest levels of multiplets 5.5. Comparison with experimental data 5.6. Nonaxial shape of odd-A deformed nuclei 6. Conclusion Appendix A. Proof of the universality of Hamiltonians

235 236 242

for local bifurcations Appendix B. Harmonic approximation for effective Harniltonians Appendix C. Calculation of matrix elements of electromagnetic transitions in the harmonic approximation Appendix D. Angular momentum operators in the partide—rotor model References

261

246 251 255 260

266

269 274 277

Abstract: The nonlinear effects in rotational spectra of molecules and atomic nuclei caused by the centrifugal and Coriolis forces are investigated for high values of angular-momentum quantum number I. It is shown that qualitative changes of the rotational-motion regime may occur in a rotational spectrum for the critical value I~due to the appearance or disappearance of a degeneracy in a rotational band. This critical phenomenon corresponds to the bifurcation in classical mechanics and manifests itself both in the rearrangement of rotational multiplet levels and in the qualitative change of electromagnetic transitions in the band. The classification of bifurcations for a purely rotational motion is given. The classification is based on the concept of a local symmetry group. There exist five types ofbifurcations in rotational spectra, the most interesting ofwhich are those analogous to the second-order phase transitions (local bifurcation). The difference between the local bifurcations in a finite many-body system and second-order phase transitions in macroscopic ones is discussed. It was shown that a universal effective rotational Hamiltonian exists in the neighborhood of I~,which makes it possible to develop a phenomenological theory of local bifurcations. The existence of bifurcations of various types in experimentally observed molecular and nuclear rotational spectra is established. The bifurcations in high-spin rotational states of odd-A nuclei, nonlinear symmetrical three-atomic molecules XY2, and tetrahedral molecules XY4 are investigated theoretically and compared with experimental data.

e-mail address: pavi~jbivn.kiae.su. 0370-1573/93/$24.00

©

1993

Elsevier Science Publishers B.V. All rights reserved.

BIFURCATIONS IN QUANTUM ROTATIO SPECTRA

I.M. PAVLICHENKOV Russian National Research Center “Kurchatov Institute”, Moscow 123 182, Russia

NORTH-HOLLAND

1. Introduction The motion of a finite many-body system becomes more intricate as the excitation energy increases. The nonlinear interaction between different degrees of freedom of a system complicates its energy spectrum so much that it becomes possible to speak about a dynamic chaos. But even before that (at much lower excitation energy) the nonlinear dynamics should lead to bifurcations, which qualitatively change the energy spectrum for one of the excitation branches of the system. We will consider a rotational branch of excitation. The quantum rotation is a specific type of excitation of a microscopic system such as hadrons, nuclei, molecules, atoms and clusters of atoms. The rotational excitations of molecules and nuclei have been studied in more detail. Electronic excitations are much higher than vibrational ones for most of the so-called “normal” molecules. Therefore they may be described adequately in the Born—Oppenheimer approximation. There is no analog to the Born—Oppenheimer approximation for atomic nuclei. Yet the occurrence of rotational bands with strong (nearly 100 single particles) E2 transitions between neighboring states shows the existence of the collective rotation. All nucleons participate cooperatively into this collective motion with the internal degrees of freedom being completely or partly frozen. The rotational excitations are grouped into rotational bands having states characterized, in the simplest case of an axially symmetric system, by the energy E

=

h21(I + 1)/2Z~

(1.1)

and quantum number I of the total angular momentum. Z is a moment of inertia. Over the last decade new methods in molecular and nuclear spectroscopy were developed, which allow one to investigate the rotational states with high-I values. Nuclei in states with very high angular momentum are produced by allowing two heavy nuclei to collide with energies above the Coulomb barrier [1]. The angular momentum of the relative motion of the nuclei may be more than lOOh, and if the two nuclei fuse, the compound system has the same angular momentum. A compound nucleus is initially in a state with excitation energy up to 200 MeY and will begin to cool by the evaporation of neutrons. Each neutron carries away a little angular momentum. After the evaporation of neutrons, therefore, the nucleus remains in an excited state with high angular momentum and energy of about 30 MeV. In the last stage of this heavy-ion collision, a nucleus is de-excited through three 7-ray cascades that contain spectroscopic information concerning the high-spin states of a nucleus produced in the (HI, xn) reaction. The first to occur is the statistical cascade of mainly El transitions, which take the nucleus to rotational states, usually called the yrast band. This band consists of levels with the lowest energy for a given spin. The statistical cascade carries a small amount of angular momentum, so that the yrast states have a maximal spin of about 40h and energy of about 10 MeV. Next to proceed is the yrast cascade, consisting of E2 transitions with 7-quanta between the levels of the yrast band. This cascade carries off both energy and angular momentum. At spins of about 20h and energies of about 5 MeV, the yrast band transforms into the ground-state band. The third cascade of 7-quanta contains information about E2 transitions in this band. The time interval between the formation of the compound nucleus and the beginning of the third cascade is approximately 10 Ps. 175

176

f.M. Pavlichenkor’, Bifurcations in quantum rotational spectra

We have outlined the most probable ‘y-decay mode of a compound nucleus. Other modes, having much smaller probability, take the nucleus to the levels of the ground-state band through the states of side bands including superdeformed bands discovered in 1986 [2]. The heavy-ion reaction generates a horribly complex spectrum of ‘y-quanta that contain cascades of E2 transitions in the band (1.1) with energy E~(I)= E(I)

—

E(I

—

2)

=

(h2/~)(2I

—

1).

(1.2)

To extract from the enormous background these signals, which give information on the discrete quantum states in the band, experimenters are using large arrays of high-resolution germanium detectors. Such a detector measures both the multiplicity and total energy of the ‘y-quanta, which is associated with the double or triple coincidence. As an example, fig. 1 ~1contains a remarkable spectrum of E2 transitions in the ‘52Dy superdeformed band, which was found by Twin and his collaborators [2]. The decisive element that has been exploited in extracting the peaks of ‘y-transitions from the noise is the approximate constancy of the spacing between them. The absorption spectrum of a gaseous molecule contains information concerning transitions, separately or in combination, between its rotational, vibration and electronic energy states. Rotational transitions correspond to the energies of photons in the microwave and far infrared regions. Recently infrared spectra with resolution from 0.05 to 0.01 cm’ have been obtained by Fourier-transform spectrometers. Due to the use of these devices, many spectral details of the polyatomic molecules are concealed, for the widths of the absorption peaks are much less than the resolution. A revolution in molecular spectroscopy began with the development of tunable lasers, which combine very narrow linewidth with tunability. The most useful for the investigation of the rovibrational transitions are semiconductor diode lasers. The rovibrational spectra are obtained by passing this laser emission through the sample and recording the intensity of the transmitted light

I

81-

8

0 Q6 U, I—. X 0 42 0

~

2830

34

38 40 42 44 46 48 50 52

1.5

E 7(MeV) 8Pd(4~Ca, 4n)152Dy in ref. [2]. The numbers above the gammaFig. indicate rays 1.1. Coincident the angular-momentum gamma-ray spectrum quantum obtained number by the I of reaction the emitting ‘° states in a superdeformed rotational band of ‘52Dy.

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

177

as the wavelength ofthe source is changed. The infrared diode lasers have resolution ofthe order of 10_5_10_6 cm1 and allow spectra to be resolved to the Doppler broadening limit, which is typically between 0.01 and iO~cm1. Next step is the sub-Doppler spectroscopy using the saturation effect of the transition being pumped. This technique allows one to eliminate the

Doppler broadening and reveal a wealth of additional spectral details. Figure 1.2 taken from ref. [3] shows the rich details revealed in the v

3 absorption spectra of the SF6 molecule as the resolution is increased. The Fourier-transform infrared spectrum has a small resolution and does not reveal the underlying structure. The second spectrum is a portion of the v3 band obtained with a tunable diode laser. The complexity of the band becomes apparent. The

(a) 944’

940

0

(b)

0. 0 ‘I)

I

I

941.?0

H~

I

I

I

I

I

I

9I~

(C) p.7q~o PRE~LJENCY

9’t~.743O

(cn~~)

Fig. 1.2. Infrared absorption spectra ofrovibrational transitions in a SF6 molecule for three resolutions. (a) Fourier-transform spectrum at a resolution of 0.06 cm 1; (b) Doppler-limited spectrum obtained with a tunable semiconductor diode laser at a resolution of 0.001 cm — 1; (c) sub-Doppler saturation spectrum recorded with a resolution of better than 10 ~cm — ~. The trace shows the first derivative of the absorption. The figure is taken from ref. [3].

178

f.M. Pavlichenkor, Bifurcations in quantum rotational spectra

sub-Doppler saturation spectrum resolves some rovibrational transitions inside the small range of frequencies possible for the lasing of a single line of a CO2 gas laser. The above described high-resolution laser spectroscopy gives us information concerning rotational-band energy levels encoded in the absorption spectrum of a molecule, containing up to tens of thousands of discrete transitions for heavy nonsymmetric polyatomic molecules. The next step consists in assigning each absorption feature to the transition between corresponding rovibrational levels. This assignment is carried out by the reduced effective Hamiltonian, the parameters of which have direct connection with molecular rotational dynamics. Thus, the rotational states with high I values and a regular sequence of energy levels may be extracted experimentally from the complex excitation spectra of these many-body systems by using nuclear yrast spectroscopy and high-resolution laser spectroscopy. Yet, from the viewpoint of the dynamic behavior of the finite many-body system, deviations from regularity are of primary interest because they provide a deeper insight into the internal structure of these systems. These deviations may be caused by the interaction of the rotation with other degrees of freedom. Centrifugal and Coriolis forces play an important role in such interaction for high I values. In the 70s irregularities in the rotational spectra of nuclei were studied in connection with the backbending phenomenon [1,4]. The essence of this phenomenon is that the ‘y-transition energies (1. 2) do not increase monotonically as I increases within the range 12—16, but remain constant or may even decrease, which is equivalent to a sharp increase in the nuclear moment of inertia in these states. The backbending was found to be induced by band crossing, i.e., the substitution of one rotational excitation branch by another branch preferable energetically in the yrast band. The primary branch does not disappear but goes over into a higher nuclear-excitation energy range. Another critical phenomenon related to band crossing is the Mottelson—Valatin effect [5]. The Coriolis force in a nucleus reduces the pairing correlations of nucleons. So, by increasing the nuclear spin I, i.e., by moving upward within a rotational band, one can eventually reach a level above which there is a no pairing correlation. The spin I~of this level corresponds to the crossing point between the ground-state band with pairing correlations and the band based on the normal state. It is the first-order phase transition point from the macroscopic point of view. The phase transition of first order in the yrast band of non axially deformed nuclei was considered also in ref. [6]. The critical point I~,here is the intersection point of the bands corresponding to the rotation around two mutually perpendicular axes. Therefore the transition through the critical point is accompanied by a 90°rotation of the angular-momentum vector I around the body-fixed frame (BFF). The nonlinear dynamics at high angular momenta can result in qualitatively new effects, which do not have relevance to the band crossing. The analog of the second-order phase transition in an isolated band of a tetrahedral molecule XY4 was investigated in ref. [7]. This transition results in the rearrangement of rotational multiplet*) levels near some critical value I~.Such effects in microsystems were called “critical phenomena” to distinguish them from the phase transitions in macroscopic systems. The notion of a critical phenomenon is taken from the Catastrophe Theory [8]. It could be shown (see, e.g., refs. [9, 10]) that each critical phenomenon is associated with a degenerate critical point on the classical energy surface of the system considered. Until now critical phenomena have been studied mainly for model systems. Usually, these phenomena are known as non-thermodynamic phase transitions or ground-state phase transitions, since the second derivative of the ground-state energy of the system as a function of some control *)

The rotational multiplet is a set of rotational levels with the same value of the quantum number I.

I. M. Pavlichenkov, B~furcationsin quantum rotational spectra

179

parameter ~ undergoes a discontinuity at the critical point The critical points of the classical energy surface are analyzed by this scheme. The latter can be obtained by averaging the algebraic Hamiltonian over a generalized coherent state of the same symmetry as the Hamiltonian. By using this method, the ground-state phase transitions were investigated for the SU (2) Lipkin—Meshkov—Glick model [9], the SU (6) interacting boson model [11], the SO(8) [12] and Sp(6) [13] fermion dynamical symmetry models. The coherent-state method has been proven [14] to be equivalent to the Hartree—Fock—Bogolyubov mean-field theory for algebraic Hamiltonians. The mean-field approach allows one to analyze the bifurcations Of the stationary points on the energy surface and topological changes in the system phase trajectories [15]. Still a fully quantum analysis of a critical phenomenon is the most adequate for microscopic systems. There are a few works in this direction. The quantum bifurcations were studied in refs. [16, 17] for the simplest model systems, in which their appearance was due to that of a double-well potential at a certain value of the control parameter. To find the spectrum of the system, the quasiclassical approximation was used for the Lipkin model in ref. [16]. This approximation turned out to be a very convenient method. It allowed the system spectrum to be determined within a wide range of the control parameter, except in a small neighborhood around the critical point. As a limiting case, the random-phase approximation and its domain of validity have been obtained from the quasiclassical treatment. Let us return to the quantum rotational problem. Here, the angular-momentum quantum number I may play the role of the control parameter i~.It means that the rotational dynamics of the system changes at the critical point I~,i.e., the system itself modifies its rotational motion. This phenomenon was considered for the first time in ref. [7] for the rotational band based on a nondegenerate vibrational state of a tetrahedral molecule XY4. The stationary points of the rigid ~.

non-axial top are known to coincide with the stationary rotation axes. Therefore bifurcation in the considered band results in the appearance or disappearance of equivalent rotational axes, or, in other words, in the rearrangement of the cluster structure of the rotational multiplet. An earlier observation of the clusters was made by Dorney and Watson in 1972 [18] when they calculated the rotational structure of the spherical-top molecule CH4 for I = 2—20. Dorney and Watson explained the six- and eight-fold near-degenerate states of the rotational multiplets in terms of two- and three-fold equivalent stationary axes. Six years later the cluster states were resolved in the SF6 molecule by using the laser diode technique [19, 20] and a precise computer analysis [21]. This result stimulated the development of a new theoretical approach for the description of the cluster structure. In a series of works by Harter, Patterson and their coworkers [22—31], a quantum model including simple symmetry analysis was produced for high-I states of spherical top molecules. The splitting of clusters, which had been considered usually as invariable objects,*) was predicted by this theory in terms of the angular-momentum vector I tunneling through barriers separating equivalent stationary axes. Thus, the critical phenomenon in rotational spectra represents the qualitative change (bifurcation) of the collective-motion dynamics of a microscopic system, occurring at a certain value of the angular-momentum quantum number I. The bifurcation is induced by nonlinear effects of the centrifugal or Coriolis forces. It manifests itself both in the rearrangement of rotational multiplet levels and in the abrupt change of the electromagnetic transition probabilities in a rotational band. The study of critical phenomena is a fundamental and yet unresolved problem in the physics of finite many-body systems. Four problems are encountered in the investigation of critical phenomena. 6)

The rearrangement of cluster levels with the variation of Hamiltonian parameters was considered, in refs. [26, 27].

180

f.M. Pavlichenkov, B(furcations in quantum rotational spectra

(i) The phenomenological or microscopical construction of the effective Hamiltonian for a particular rotational band. This Hamiltonian is represented by an infinite power-series expansion in angular-momentum operators. Such Hamiltonians are found in section 2 for the centrifugal force and in section 5 for the Coriolis force. (ii) The classification, i.e., determination of the possible types of critical phenomena for the rotational motion of the system. This problem is reduced to the analysis of the critical points on the rotational energy surface*) in the case of an isolated band, in the spirit of the Catastrophe Theory. The analog of a critical phenomenon in classical mechanics is bifurcation. An important point at this stage is the introduction of the idea of local symmetry group g, which characterizes a small region of the rotational-motion phase space. The critical phenomena are classified according to the group g. All types of critical phenomena can be divided into two groups: local ones occuring in a small region of the phase space and global ones, which are not so restricted. The classification problem is solved in section 2. (iii) The investigation of a rotational spectrum near the critical point. This problem, which may naturally be called the theory of quantum bifurcation, can be solved only for local critical phenomena for which the effective rotational Hamiltonian can be approximated near the critical point by a finite power-series expansion in angular-momentum operators. This Hamiltonian is universal, i.e., it depends on the internal structure of a system through expansion coefficients only. The solution of this problem, meaning the construction of a phenomenological theory of a local critical phenomenon, makes it possible to find the changes of the rotational spectrum and its electromagnetic properties while passing through the critical point. In a finite system these changes are the only attribute by which the bifurcation can be identified. The phenomenological theory of the bifurcations in rotational spectra is presented in the sections 3 and 5. (iv) The microscopic theory of a local critical phenomenon should answer the question as to whether bifurcation really exists for a given rotational branch in the system considered. If bifurcation exists, the theory must predict parameters of the phenomenological Hamiltonian. The results of the phenomenological theory are general and independent of the specific form of the system. That is why this review contains illustrations from molecular and nuclear physics.

2. Bifurcations in an isolated rotational band of nonaxial many-body systems In this section we shall consider the bifurcations in rotational spectra caused by the centrifugal forces. Centrifugal effects are common for both molecules and atomic nuclei. The centrifugal distortion in these systems plays an important role at sufficiently high angular momenta. Usually, the centrifugal distortion effects are treated phenomenologically. The model of a variable moment of inertia is used in nuclear physics [32, 33] and the so-called reduced effective rotational Hamiltonians are used in molecular physics [34, 35]. The latter is the power series of the angularmomentum operators with the minimum number of parameters required for the description of rotational states having angular-momentum quantum number I ‘max~Both models enable one to obtain a good description of a large number of rotational levels. For example, the rotational band of the ground vibrational state of the H2 S molecule requires 29 adjustable parameters to reproduce 426 experimentally observed transitions with I 22 [36]. It should be noted that for I ~ 20 the subsequent terms of the Hamiltonian are no smaller than the preceding ones. The 61The useful concept of the rotational-energy surface was introduced and graphically illustrated by Harter and Patterson in refs. [28. 29].

I.M. Pavlichenkov, Bifurcations in quantum rotational spectra

181

convergence of the power expansion is worse for the H 20 molecule, and the corresponding reduced Hamiltonian includes terms with higher powers of angular-momentum operators. Such a situation is typical for the ground rotational bands of many molecules, since the expansion near I = 0 cannot adequately describe high-I states for which the nonlinear features of the rotational dynamics are essential. Nonlinear effects of the centrifugal forces for high I result in a new phenomenon in a rotational band, viz, a bifurcation, which will be considered theoretically independent of a concrete type of system. 2.1. Effective rotational Hamiltonian and its symmetry It is convenient to study critical phenomena in rotational spectra using the effective rotational Hamiltonian H

=

h +~h5I4+ ~ h~I5I~ + ~ a

h~~I4I~I5 +

.

,

(2.1)

a,~

which is an infinite power series in I~,namely the projections of the total angular momentum operator on the BFF (body-fixed frame) axes ~ = 1, 2, 3. The coefficients of the series depend on the internal structure of the system. The Hamiltonian (2.1) may be obtained with the help of the generalized density-matrix approach, which is used for the description of collective excitations of atomic nuclei [37, 38]. For molecules we may use as a starting point the Wilson—Howard rovibrational Hamiltonian [39] for a nondegenerate ground electronic state H

~

[(Ia

—

x~)~(I~ lIp) —

_ap~]

+

+ U(q),

(2.2)

where I~are the projections of the total angular-momentum operator on the Eckart frame axes, ir~are the same components of the vibrational angular-momentum operator, and ~afl stands for the matrix elements of the inverse inertia tensor, which depends on the normal molecular coordinates q~whose conjugated momenta are Pk. U(q) is the molecular potential energy for a ground electronic state. The effective Hamiltonian (2.1) results from (2.2) by using the contact transformation or any type of operator perturbation theory [35, 40]. The rotation—vibration interaction is reduced to the centrifugal distortion effect due to the ~ dependence on q and the Coriolis interaction due to ira. If the vibrational state is not degnerate, the Coriolis effects are absent from the corresponding rotational band. In such a case h5 = = = 0 in (2.1) and H describes the centrifugal distortion effects only. We shall study these effects for an isolated rotational band whose coupling with other bands is negligible. For this band the coefficients h, h~,etc., are the real c-numbers due to hermiticity of the Hamiltonian. The effective rotational Hamiltonian for an isolated band is invariant with respect to time reversal and inversion of the BFF. Further restrictions on the coefficients of this Hamiltonian are due to the system point symmetry group. The set of point group operations and inversion form a symmetry group G of the effective rotational Hamiltonian. It contains such elements as symmetry planes, o~ n-fold symmetry axes, C~their combinations C~, C~h;and n-fold mirror symmetry axes S,. The symmetry axes C5h contribute nothing new to the properties of the effective Hamiltonian, since C5h is identical with the Ca-axis (for even n) or with C2n (for odd n). The axis Sn ..

182

I.M. Par’lichenkot’, Bifurcations in quantum rotational spectra

is identical with

Cflh

for even n and with C25h for odd n because the Hamiltonian is invariant with

respect to inversion. We write H for an isolated rotational band as an expansion, 2k

v~.

k0

(2.3)

mt~’mT2k,_m],

m0 ~ [tkmT2k.m+(l)

in terms of the irreducible spherical tensor operators Ti,mz(_l)mU,m(I2,I

2,I 3)I~,

Ti,_m

1m+Uim(1

3),

m0,

(2.4)

where 12 = I~+ I~+ I~,I~= I~±i12*), and u stands for the real function, the explicit form of which is given, for example, in ref. [41]. The t coefficients in (2.3) can be expressed in terms of the h ones. We shall find it convenient to regroup the terms of the sum (2.3) and write the effective rotational Hamiltonian for an isolated band in the form 2, H

=

where the

.t’m

13) +f

2, I3)I~],

(2.5)

m(1

[I~f~(I

2m0

functions satisfy the relationships

(_1)mf

2,

—

I3)I~= I’~f

m(1

2,13),

(—1)mI’~f~(I2, —13) =f~(I2,I3)I~,

(2.6)

m(I

and depend on the t coefficients and the symmetry group G. Besides the quantum number I, the levels of rotational multiplets are characterized by the irreducible representations of the G group. For example, the symmetry properties of a rigid rotor are described by the group D 2 = {1, ~ ~R2,913}, which contains the identity operator and three 180°rotations around the BBF axes, 9~~(it)=

exp(— iirl4),

G~=

1, 2, 3.

(2.7)

The irreducible representations of D2 are labeled by 1 and A1, A2, B1, B2 [42]. They correspond to the eigenvalues r4 = ±1 of the operators (2.7) according to table 2.1. The subscripts 1 and 2 label even-and odd-symmetry levels with respect to the 9~rotation, A and B label even and odd symmetry levels with respect to the ~R2rotation. Note that r1r2r3

=

1

(2.8)

for each of the four representations. To solve the Hamiltonian (2.5) we are going to use different methods. As a first step we will calculate the classical rotation energy E defined as a function in the system phase space. The phase space of rotational motion is formed by three Euler angles 4’, ~ i~fr and three conjugate momenta p4, p3, p~,.The absolute value of the angular momentum I and its projection I~= p,,~on the z-axis of *)J

is the raising operator and 1÷is the lowering one in the BFF.

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

183

Table 2.1 Irreducible representation of the D

2 point group

D2

A1

r1 r2 r3

+ + +

A2

B1

B2

—

—

+

—

+

—

+

—

—

the space-fixed frame are the integrals of motion. It is suitable to carry out a canonical transformation [43] to new conjugate variables I and q1, I~and q~, 13 and q. Since q1 and q~are cyclic variables, the phase space of rotational motion is two-dimensional. It is convenient to map it on the surface of the sphere with radius I and center at the origin of the BFF (the phase sphere). *) The point on the sphere with coordinates 0 and ‘p determines the orientation of the vector I in the BFF. The canonical transformation enables us to relate the conjugate variables 13 and q to the angles 0 and ‘p. For I~= q1 = 0 and arbitrary q, we have cos0

13/I,

=

‘p

=

—

q.

(2.9)

Thus, the trajectories of the end point of vector I on the phase sphere are classical trajectories of the system in its rotational phase space. From Hamiltonian (2.5) we find the rotational energy surface E(0,’p)

=

c0(I, 0) + 2m_i [c~(I, 0)eim~+ Cm(I, 0)e~m~]sinm0,

(2.10)

which determines the rotational energy as a function of an angular-momentum vector in the BFF [28, 29]. It allows the classical trajectories of rotational motion to be determined in the system phase space and relate them to the energy of quantum levels using Bohr’s quantization rule. Equation (2.10) will help us in subsection 2.2 to find the critical points and classify the critical phenomena. The next step consists in using the classical equations of motion, d15/dt

=

{H, I~},

(2.11)

for the projections of the vector Ion the BFF axes. In this equation H means the classical analog of the Hamiltonian (2.5) and ~ } is the Poisson bracket. Equations (2.11) resemble the Euler equations and for a rigid asymmetric top [44] but they have a more complicated form. Let us introduce the classical concept of stationary rotation axis defined in the BFF by the three equations {H, i~~} = 0.

(2.12)

The stationary states I~are identical with the fixed points of the energy surface (2.10). Each stable stationary axis is connected with a level group in a rotational multiplet corresponding to a precession of the vector I around this axis. The spacing between these levels is determined by the 6)

J~is called the Bloch sphere in quantum optics.

184

f.M. Pavlichenkov. B~furcationsin quantum rotational spectra

precessional frequency w calculated from the linearized set of equations (2.11). An unstable stationary axis is associated with a group of levels located in the transition region between states corresponding to the precession of the vector I around different stable axes. Thus, the gross structure of the rotational multiplet is defined by stationary axes and their stability. For the effective Hamiltonian (2.5), several equivalent stable or unstable stationary axes may exist due to the symmetry of the system. The precession around equivalent stable axes is independent and identical up to a symmetry transformation. This means that the classical approach gives us strongly degenerate cluster states. In quantum mechanics, the eigenfunctions of the Hamiltonian (2.5) can be written in the form

~IMv

=

K=-I aIKVDMK(4’,

~

(2.13)

~).

where M and K are the quantum numbers of the operators I~(the angular-momentum components along the z-axis of the space-fixed frame) and 13, respectively. The quantum number v = 1, 2, 3, . is the ordinal number of the level in a rotational multiplet as its energy increases. D ‘MK is the Wigner function depending on the Euler angles. The exact function (2.13) will be used for numerical diagonalization of the Hamiltonian. To clarify the physical picture of a critical phenomenon, we will use the harmonic approximation, which is described in appendices A and B. This approximation is based on the idea of sharply localized states widely used in mapping a complex nonlinear dynamics onto a simple system [45—49]. Let us consider a rigid asymmetric top with the Hamiltonian .

.

H

=

AI~+ Bfl + Cfl,

(2.14)

which is the simplest problem in rotational dynamics. The rotational constants A, B and C are each one-half of the principal moments of inertia ~ ~, ~2 and Z~, A=~i11=1/2~1,

B=~u22=1/2Z~2,

C=~J133=1/2Z~3.

(2.15)

The diagonalization of (2.14) with the functions (2.13) involves tridiagonal matrices or three-term difference equations for the a coefficients. Solving these equations for large I, we obtain 2/(2I + 1)], aIK~

c

—~

1,

(2.16)

exp[— cK

for states located in the lower part of the rotational multiplets if A
f.M.

Pavlichenkov. B~furcationsin quantum rotational spectra

185

2.2. Types of symmetric b~furcations:classical analysis We shall distinguish between regular and critical changes in the rotational multiplet level structure. In the former case the variation of the quantum number I changes only the orientation of the stationary axes, which leads to a monotonic dependence of the level energies on 1. Critical phenomena are accompanied by a change in the number of stationary axes and in the character of their stability. The appearance or vanishing of equivalent stable rotation axes leads to a change in the cluster structure of the rotational multiplet levels. To study such irregular behavior, we consider the effective Hamiltonian (2.5) near the selected direction namely near axis 3 of the BFF. Let us consider a subgroup g ~ G making axis 3 invariant. We will call g a local symmetry group. It determines the form of the Hamiltonian (2.5) near the selected direction. Let the axis 3 lie in a symmetry plane i (the group Cs), which coincides, for example, with the (13)-plane. The reflection in this plane is equivalent to the replacement of 1÷with —I and 13 with —13. Using eqs. (2.6) one can easily prove that the Hamiltonian (2.5) is invariant under the C6 transformation if f~ =fm. The invariance of the Hamiltonian H with respect to the Cn symmetry axis requires that the only nonzero fm functions are those with m = np, where p = 0, 1, 2 For the C5~-axisthe fm functions must satisfy simultaneously both requirements. The functions fm in eq. (2.5) are undoubtedly determined by the symmetry of the system as a whole, i.e., by the group G of the effective Hamiltonian. Even so, it is the group g whichis suitable for the description of a part of the rotational multipet levels and for the classification of the critical phenomena. The concept of a local symmetry group is closely related with the fundamental difference between the proposed theory of rotational spectra and other approaches based on the power-series expansion of the rotational Hamiltonian in terms of I~.The sum (2.5) is not a power series, and can be widely used, as will be shown below, to describe bifurcations in rotational spectra. We begin our consideration with the study of the rotational energy surface (2.10) near axis 3 for different g. 2.2.1. Bjfurcation for the local symmetry group C1 First of all we consider the case g = C1, when the axis 3 is a general position axis and all the function Cm in (2.10) differ from zero. We expand the rotation energy E(O,q,), assuming the angle 0 to be small, and write this expansion in terms of the Cartesian coordinates x

=

Ocos’p,

y

=

Osin’p

(2.17)

near the north pole of the phase sphere (the axes x and y are directed along the axes 1 and 2 of the BFF, respectively), 2+a 2 E(x, y) = E0(I) + a10x + a01y + a20x 11xy + a02y +a 3 +a 2y + a 2+a 3 + ~, (2.18) 30x

21x

12xy

03x

where E 0(I) is the energy of rotation around axis 3, and the coefficients ajj depend on I. Let axis 3 be at I = I~,the stationary rotation axis of the system, i.e., a10(I0) = a01(I0) = 0. Then we can restrict the expansion (2.18) to terms quadratic in x and y. If I is varied, the local behavior of the function E(x, y) does not change qualitatively. The system, as before, has a stationary axis of rotation, which can move with respect to its position at I = I~.This is the regular variation of the stationary axis.

I.M.

186

Pavlichenkov, B~/i~rcations in quantum rotational spectra

The critical point *) I~(if it exists) is determined by both the stationary condition and the equation uilr

\

~

—

2 ir \

A ii \ (1 \ -1a02~1C)a20kJC,

—

A —

where W is the Hessian of the energy surface E(x, y). Near the critical point it is necessary to take into account the cubic terms in expansion (2.18). A nonlinear transformation of the coordinates can reduce (2.18) to the canonical form of a catastrophe function of the “fold” type [8], which takes in our case the form E(x, y)

=

E0(I)

2+a —

ot(I

—

I~)x+ a02y

3,

(2.20)

30x

where the coefficients a 02 and a30 are specified at the point I~.The analysis of the surface (2.20) shows that the considered bifurcation is connected with the appearance of two stationary rotation axes, one stable and the other unstable. The energy of rotation around these axes is

3/27a E1,2(I)

=

E0(I) ~ ~

—

I~)

3o,

(2.21)

and the directions of these axes are close to that of axis 3. The stationary axes arise for I > I~if c~and a30 are of the same sign and for I 0 the axis with the minimal rotation energy E1 is stable and for a02 <0, the one with the maximal energy E2 is stable. In the critical point I~,a singularity arises in the second derivative of the rotational energy around the stable axis, =

E’,2(I)

—

E’~(I)=

~

~\/~~/3a3o(I

—

(2.22)

Ia).

Figure 2.1 shows the classical trajectories of the rotational motion on the left and on the right of the critical point I~in the part of the phase space where the stationary axes appear. For I I~the local trajectories correspond to the precession of the vector Iaround the stable axis if their energies satisfy the condition E1 < E < E0. The local trajectories are separated from the global ones by the separatrix S that passes through the saddle point of the surface (2.20) with energy E0. The separatrix is a global curve, and therefore this bifurcation is nonlocal. 2.2.2. B?furcations for the local symmetry groups C6, C2 and C2~ We begin with the C. symmetry group. Let us choose the symmetry plane o’ to be the (13)-plane of the BFF. Then the ~mfunctions in eq. (2.10) are real. Using the small-0 expansion of E(0, ‘p) and the coordinates (2.17), we obtain, with the required accuracy, 2 +a E(x, y)

=

E0(1) + a10x + a20x 4

2+a 02y

22

3+ a 30x

2 12xy

4

+a

40x + a22x y + a04y 6)

.

More accurately, the degenerate critical point according to the terminology of Catastrophe Theory [8].

f.M.

Pavlichenkov, Bifurcations in quantum rotational spectra

I
187

I>1c

Fig. 2.1. Classical trajectories close to the north pole of the phase sphere for the C

1-type bifurcation. The positive values of the

coefficients a, a02 and a30 are used in eq. (2.20).

If the coefficient a02 differs from zero for all values of I, the stationary axis changes its direction in a regular manner and remains in the plane a. Then the critical point can arise if both coefficients a10 and a20 become zero simultaneously. This bifurcation has been discussed in subsection 2.2.1. It is connected with the appearance of two stationary axes in the a-plane. The critical point I~,defined by aO2(I~)= 0, is a new one. Let us transform expression (2.23) for this case. We rotate the BFF to align axis 3 along the stationary one. Then, using a nonlinear coordinate transformation, one can transform eq. (2.23) near I~to the form 2+ a E(x, y)

=

E0(I)

—

cc(I

—

I~)y

2+a 20x

4,

(2.24)

04y

where the coefficients a 20 and a04 are taken at I~.It is not difficult to show that the critical phenomena for the local symmetry groups C2 and C2~are described by the same energy surface (2.24). Depending on the relative signs of the coefficients a20 and a04 in eq. (2.24), two types of bifurcations exist. If a20 and a04 have opposite signs, the bifurcation is reduced to a change in the character of the stability of the stationary rotation axis 3. At equal signs of the coefficients ~ and a20, this axis, which is stable for I < I~,becomes unstable for I> I~.The stability changes in the reverse direction if ~ and a20 have opposite signs. The appearance or disappearance of the stable rotation axis is accompanied by the simultaneous appearance or disappearance of two equivalent unstable axes lying in the (23)-plane symmetrically with respect to the axis 3. These axes correspond to the saddle points A and A’ on the surface (2.24) with energy 2/4a E1(I)

=

E0(I)

—

~2(I

—

Ij

04.

(2.25)

Classical trajectories and the separatrix S are shown in fig. 2.2, from which we see that this bifurcation is nonlocal, and is not accompanied by symmetry breaking. If the coefficients a20 and a04 have the same sign, the bifurcation results in a change of the direction and number of the stationary rotation- axes. If ~, a20 and a04 all are of the same sign, a rotation around axis 3, with energy E0(I), goes over at I > IC into rotation around either one of

f.M.

188

Pavlichenkov, B(furcations in quantum rotational spectra

S 9

S

-i

I
I>Ic

Fig. 2.2. Classical trajectories for the nonlocal C

2~-typebifurcation [a

9

>

0, a20

>

0, a04 <0 in eq. (2.24)].

9

__

1~~t

~~5X

I<1c

I>1c

Fig. 2.3. Classical trajectories for the local C2~-typebifurcation [a

>

0, a20

>

0, a04

>

0 in eq. (2.24)].

the two equivalent axes with energy E1(I) (2.25). If the sign of ~ is opposite to that of a20 and a04, the transition occurs in the reverse order. In both cases the symmetry of the classical solution with respect to the local C2~,group is violated. Another feature of the considered bifurcation is the discontinuity of the second derivative of the rotational energy at the critical point, 2/2ao AE”(I~)= E’(I~) E~(I~) = a~ 4. (2.26) —

—

The character of the critical phenomenon can be determined by considering the family of classical trajectories of rotational motion in phase space. Figure 2.3 shows these trajectories near

f.M.

Pavlichenkov, B~furcationsin quantum rotational spectra

189

the north pole of the phase sphere on the left and on the right of the critical point. The trajectories describe the precession of the vector I around axis 3 with energy E > E0 for I I, the precession with energy E1
2,

H = 2 n0

2~2,

I3) +f2fl(1

I 3)I~].

(2.27)

Ii),

(2.28)

In classical limit, we find from eqs. (2.6) 2,

13)

f2n(12,

—

13) =f2n(I2,

f25(1

and consequently H

~ n~ø(I~!’+ I~)f2n(I2,I

=

3).

(2.29)

Using the well known formula for the Poisson bracket of the angular-momentum components [see eq. (D.16)] {I~,I~}= ~

(2.30)

we obtain from eq. (2.11) a system of differential equations, 2+”1—I)f2fl(I2,I3)++I2~(I~+I~)f’2fl(I2,I3),

11

=

—iI3~n(I

12

=

13 ~ n(I~’ + I~.”’~’)f2n(I2, I3)

~

2,13),

~ (I?~’+ I2_’~)f’

(2.31)

2n(I —

13

—

1

V ~

(r2fl fl~I

+

—

y2r,~~(72 — )J2n~.’

I

r ,13

n

The quantity f~

5is the derivative of the f2n function with respect to 13. These equations describe

the motion of the I vector in the BFF. They have only one constant of motion, namely 12, and are

invariant under rotation by 180°around axis 3. The stationary state 110

=

120

=

0,

I3~= I

(2.32)

190

f.M. Pavlichenkov, B(furcations in quantum rotational spectra

is identical with the fixed point x = y = 0 on the energy surface (2.24). The linearized set of equations describing precession motion has for the small quantities I~and 12 the form

It

=

[2If2(I2, I) +f’o(I2, I)]12,

12

=

[21f2 (12,1)

f’o(12, I)]I~.

(2.33)

The vector I undergoes small-amplitude oscillation with frequency =

21{[f’o(I2,

f~(I2,I)}112

I2)]2

and its end point moves along the trajectories represented in fig. 2.3 for I determines the stability of the stationary state (2.32); it is stable if f~(I2,1)

<

(2.34) <

I~.Expression (2.34)

[f~(I2, I2)]2

(2.35)

The precession frequency (2.34) vanishes at the critical point IC, in which inequality (2.35) becomes an equality. It is important to note that the precessional motion around the symmetry axis does not depend on the n> 1 terms in the effective Hamiltonian (2.27). This circumstance (as it will be seen in section 3) allows us to construct the universal rotational Hamiltonian of the local bifurcation of the C 2~type. Higher local symmetry groups C,, or C,1~, with n 3 are appropriate for molecules and quasimolecular states of nuclei. There are two types of associated critical phenomena. For their classification it makes no difference whether axis 3 is the C, or the C,,~symmetry axis, because the difference between corresponding local energy surfaces are not essential. For simplicity, we shall consider the local symmetry groups ~ The rotational energy surface close to axis 3 can be obtained by the expansion of the function (2.10) in power series of the small value 0. The critical point exists if the term proportional to 02 vanishes at I = IC. The ‘p dependence of the energy surface determines only the number n of equivalent axes but not the type of critical phenomenon. 2.2.3. Bifurcation for the local symmetry groups C3 and C30 The rotational energy surface for the C3,, local symmetry group has the form 3cos3’p, E(0,’p)

=

E0(I)

—

c~(I

—

IC)02

(2.36)

+ 2bO

where E 0(I) is the energy of rotation around the stationary axis 3, ~ and b are constants. A critical phenomenon involves a change in the character of the fixed point 0 = 0. If c~> 0 and b is arbitrary, the stable axis 3 with the minimal rotation energy E0(I) for I IC. The transformation takes place in the reverse order for ~ <0. The change of a extremum is accompanied by the reorientation of three unstable rotation axes placed symmetrically around the axis 3. They are associated with three saddle points on the surface (2.36) with the energy 3/27b2. E1(I)

=

E0(I) + ~~(IC

—

I)

While going through the critical point, these axes rotate by 60°around the axis 3.

(2.37)

f.M.

Pavlichenkov, Bifurcations in quantum rotational spectra

191

~

I
I>Ic

trajectories for the C3~-typebifurcation for a >

0,

b

>

0 in

eq. (2.36).

The classical trajectories close to the north pole of the phase sphere are shown in fig. 2.4. The local trajectories are separated from the global ones by three separatrices S, being themselves global curves, passing through the saddle points of the surface (2.36) with energy E1 (2.37). It is interesting that the rotational energy surface (2.36) is identical with the potential energy of the Henon—Heiles system [50], which is widely used to study chaotic motion. Let us consider the precession motion of the I vector along the local trajectories of fig. 2.4. The effective Hamiltonian of the local symmetry group C3,, has the form ~jf

(7 72\

JØ~I~13)

I

+

L~3 ç(7 2’

7 ~ I +J3~’,13) + .lf(7 2J3~L,

\73

13)1

—

+

.

We can find linearized equations describing precession in the same way as was done in the previous subsection, 2,I2)I 2,I2)Ii. (2.39) I~=2If’~(I 2, 12= —2If’t,(I The precessional frequency CO

=

211f’o(12,12)I,

(2.40)

vanishes at I = I, because of f’o I I~.The local motion disappears at the critical point, since the barriers separating local trajectories from global ones disappear, and the separatrices pass through the north pole. Thus the considered critical phenomenon is nonlocal. —

2.2.4. B~furcationsfor the local symmetry groups C 4 and C4,, The rotational energy surface in the neighborhood of axis 3 has the form 4. E(0,q~)= E0(I)

—

ot(I

—

I~)O2+ (a + 2bcos4’p)0

(2.41)

192

f.M. Pavlichenkov, B(furcations in quantum rotational spectra

Here E0(I) is the energy of the rotation around the stationary axis 3 and ~, a, b are constants. In general, one may assume b > 0. The type of critical phenomenon depends on the relations between the coefficients a and b. If I a I <2b, the critical phenomenon is nonlocal and reduced, in analogy with the phenomenon for the symmetry group C3,,, to a change in the character of the fixed point 0 = 0. There are four saddle points, according to the four equivalent unstable rotational axes rotated by an angle of 90° relative to each other. Depending on the position of these points, their energy equals E1 or E2, where 2(I IC)2/4(a ~ 2b),

(2.42)

—

E1,2(I)

=

E0(I)

—

c~

and E 0 is the maximal or minimal rotational energy around axis 3. Whilegoing through the critical point I~,these axes rotate by 45°and their energy changes from E1 to E2 and vice versa. The critical phenomenon becomes more complicated if I a I > 2b. In this case, besides the stable rotation axis 3, there appear or vanish four stable equivalent axes that are close to it in direction and rotated by the angle of 90°relative to each other. Axis 3 remains stable but changes its extremal character. If a > 2b and ~ > 0 the minimal energy E0(I) of rotation around axis 3 for I
i

=

1,

. .

. ,

4;

0~ = c~(I Ij/2(a —

—

2b).

(2.43a,b)

The rotational energy corresponding to these axes is E1 (I) [see eq. (2.42)]. At ~ <0 the transition takes place in the reverse order with increasing I. If a 2b, the same bifurcation occurs with the small difference of the interchange of the minimal and maximal rotational energy E0(I) and substitution of E1 by E2. Four unstable equivalent axes arise simultaneously with the stable ones in all cases. The unstable axes are rotated by an angle of 45° relative to the stable ones. They correspond to the saddle points of the surface (2.41) with the energy E2 (I) for a> 2b and E1 (I) for a < 2b. The classical trajectories near the north pole of the phase sphere are shown in fig. 2.5 for the case a > 2b and c~> 0. The trajectories with energy E > E0 for I < I~and E2 IC correspond to the precession of the I vector around axis 3 (the point 0). The precession frequency is determined by eq. (2.40) if the effective rotational Hamiltonian for this local symmetry group has the form <

—

—

H =f0(I, fl) + ~I’~.f4(I, 13) + ~f4(I, I3)I’~i+

....

(2.44)

The precession around each of the four stable axes k, (2.43a) with the energy E1 I~together with the precession around axis 3. The separatrix S passing through the four saddle points of the surface (2.41) with energy E2 separates the trajectories of these two types of precessional motion. It also separates local precessional trajectories from the global ones. The pattern of the phase trajectories is typical of a local bifurcation. It is characterized by the symmetry violation of the classical rotation relative to the C4 group and by the discontinuity of the second

f.M.

Pavlichenkov. Bifurcations in quantum rotational spectra

1< Ic

193

I> I~

Fig. 2.5. Classical trajectories

for

the C

4~-typebifurcation

[a >

0, a/b

=

8.0

in eq. (2.41)].

derivative, with respect to I, of the minimal (E1) or the maximal(E2) rotation energy at the critical point I~, AE”(I~)= E’,2(I~) E’~(I~) = —

2/2ja ~ 2b1.

—

~

(2.45)

2.2.5. B~furcationsfor the local symmetry groups C,, and C,,,,, n 5 Local symmetries containing axes of higher order than the fourth are either rarely encountered or occur only in heavy molecules whose rotational structure cannot be resolved spectroscopically. However, we consider the critical phenomena for the local symmetry groups C,,,, (n 5) for completeness. The rotational energy surface near axis 3 can be written in the form E(0,’p)

=

E

4 + 2b05cosncp. 0(I)

—

cL(I

—

(2.46)

I~)02+ a0

The last term in this expression is responsible for the appearance of equivalent axes. It is small compared to a04. Thus there is only one type of critical phenomenon for the all the groups considered, namely the same as that for the C 4,, group in the case of I a > 2b. The difference is that there exist n stationary axes for the surface (2.46). Let us list all the considered bifurcations in purely rotational spectra. It is convenient to use for their classification the classical picture, in which a bifurcation is characterized by a local symmetry group g, by the form of the energy surface near axis 3, and by the singularity of the energy of rotation around the stable axis at the critical point ‘C• These characteristics are presented in table 2.2, which includes also the bifurcation diagram showing in (E, 1)-coordinates the dependence of the stationary energy near the critical point for positive values of the parameters ~, a, b, etc. specifying the energy surface. The solid lines correspond to the energy of rotation around the stable axis; the dashed lines, around the unstable one. The solid line is associated with the minimal rotation energy if it is below the dashed line in energy. If it lies above the dashed line, it corresponds to the maximal rotation energy. The integers in parentheses above the lines show the number of equivalent stationary rotational axes. Number 1 is not shown for the sake.of simplicity.

f.M.

194

Pavlichenkov, Bjfurcations in quantum rotational spectra

Table 2.2 Typesof bifurcations for a nondegenerate isolated band. The energy surface is determined by the polar coordinates (0, ~) of the I vector or by the Cartesian coordinates x = 0 cos çs, y = 6 sin q. The description of the bifurcation diagrams and the value of AE” see the text. Local symmetry

group g

C

1

Energy surface close to axis 3 of the body-fixed frame

Singularity at the

ciritical point E(l)

E(x,y) =

E0(1)

+

C,

Bifurcation diagram

E(x, y) =

a(I +a

I~)x 3 30x

—

E

—

a3I2/~12aso(I—

IC

2

0(I)

C

—2

a02y

a(l

I~)y + a20x + a,,,y’

2, C2~

—

—

/ /

~2)

2/2a —

C3, C3~

E(0,

~)

=

E0(I)

3cos3~ a(I — I~)62

—

+2b6

C,, C,~

E(0, ~s) =

2 +0(I) (a + — 2bcos4~)0’ a(I — 1j0

E

~-7’

a

0,

,t’/__ (3) L

/ (4)

—

a2/2(a

—

(n)

~

E(6, ~)

=

I

-

a2/2a

2b)

l~)

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

195

The above consideration is part of a more general approach involving the study of generic bifurcations in one-parameter Hamiltonian systems which are invariant under a compact Lie group. The word “generic” means that the type of bifurcation cannot be changed by a small invariant perturbation of the Hamiltonian. The work of Golubitsky and Stewart [51] contains as an example the generic bifurcations of a Hamiltonian system with one degree of freedom for the symmetry groups C1, C2, C3 and C4. The normal forms of these bifurcations coincide with ones for the effective rotational Hamiltonian (2.5). The mathematical analysis of bifurcations and symmetry breaking is given also in refs. [52, 53]. These critical phenomena can be called elementary in the sense that they involve the vanishing, at the critical point I~,of one coefficient of the local rotational Hamiltonian. More complicated cases are possible, when several or even many coefficients vanish. In the latter case a trough is produced on the energy surface, and the critical phenomenon, which leads to rotation of the I vector through a finite angle relative to the BFF, is analogous to the first-order phase transition [6]. The next evident generalization of the purely rotational motion is the rotational dynamics of a group of degenerate or quasidegenerate states. It involves Coriolis coupling between rotational and internal degrees of freedom. Following the work by Sadovskii and Zhilinskii [54, 55], let us consider rotational bands based on a finite number of close-lying internal states (vibrational, single particle, etc.), so that the rotational multiplets of these bands in general strongly overlap, and all other rotational levels are well separated in energy. For N internal states, the effective Hamiltonian has the form of the matrix operator,

H=

H11

H12

.

~

H:2!

H22

.•.

H2N

HN1

HN2

(2.47)

HNN

All the matrix elements H1~can be represented as power expansions in the angular-momentum operators 1~and L analogous to eq. (2.5). Symmetry requirements put some restrictions on which are similar to those for the pure rotational Hamiltonian (2.5). The classical limit for the Hamiltonian (2.47) is the N-dimensional matrix whose elements are functions of two variables 9, ‘p and the parameter I

H

=

H11(0,’p)

H12(0,’p)

H21(0,’p)

H22(0,’p)

H~l(O,’p) HN2(O,’p)

.

. .

H~N(O,’p) H2N(O,’p)

. . .

(2.48)

H~(0,’p)

The classical energy surfaces of the internal states can be obtained as the eigenvalues of the matrix (2.48). They may be visualized as N-fold or overlapping surfaces (see the excellent computer-made pictures obtained by Harter [31]) and provide the bifurcational analysis even when parts of them show a breakdown of the adiabatic approximation. Each rotational eigensurface has stable fixed points, regions surrounding them, and separatrices, which connect saddle points and separate different regions. A fixed point of each surface corresponds to a stationary rotation axis. Investigation of the stability of stationary axes allows one to find bifurcations and classify them according to the local symmetry group. This investigation shows that all critical phenQmena for an isolated

196

f.M.

Pavlichenkov, Bifurcations in quantum rotational spectra

rotational band may exist in each eigensurface (see in section 4 the bifurcational analysis of rotational structures based on quasidegenerate vibrational states of tetrahydrides). The rotational motion becomes very complicated when the state mixture evolves and precession is no longer a simple one-dimensional motion. The path of the end point of the I vector on the energy surface can change dramatically as the surface changes even slightly. As a result, there is a possibility for classically chaotic rotational motion. The simplest example of such a motion has been found in ref. [56] for two coupled rotators. In the nonadiabatic parts of the energy surface, a new type of singularity arises, which is caused by the degeneracy of the eigenvalues of the matrix (2.48). Such a singularity is known as a conical or diabolic point [57,58].

3. Phenomenological theory of local bifurcations: quantum approach There are two local bifurcations among the five types of the elementary critical phenomena described in the previous section. In the classical limit local bifurcations are characterized by degeneracy (equivalent rotational axes), breaking of symmetry with respect to a local group g, and a discontinuity of the second derivative of the rotational energy at the critical point. Thus, they resemble second-order phase transitions [59]. For a quantum system such as molecular and atomic nuclei, the classical picture is no more than an illustration. Yet the classical approach allows one to understand a local bifurcation taking place in a restricted region of the system phase space. We will use this feature in the present section to develop the phenomenological quantum theory of local bifurcations. 3.1. C~,,-typebifurcation Let us begin with quantum effective Hamiltonian (2.27) for the local symmetry group C2,,. The set of local trajectories shown in fig. 2.3 occupies that part of phase space which corresponds to the quantum states v satisfying the condition 2

I~IIMv>/I2<<1.

—

(3.1)


=

E

0(I) + a1(I)(I~

—

I

Ii,

rr2

72

T2U2L131

72

2+ + 1~)+ a 12)

+ b1(I

~ 721

2(I~

—

I

,1+m1_J+mC1~1+m

jj’4

~

y4

12)2

‘~

—1,

where the regular part of the Hamiltonian, 2, I) =f E0(I)

fo(1

2, 12), 0(I

(3.3)

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

197

is the energy of classical rotation around axis 3, and the coefficients ( j~ —

ç” ~ 2

2\

—

/

—

T

(7

-~j,

7 \

c”(

—

.1.

—

2J Ok c’

2

j2~

a 1~ij—J

b1

=

~JI~

I

~f2(I~, IC),

)Xt~I

—

L~)

a2

~.tij,

—

b2 =f’2(I~,I~)/4I,

c1

c~),

~f4(I~, Ia),

=

(3.4)

depend on the internal structure of the system. As is shown in appendix A, the Hamiltonian (3.2) is sufficient for the description of the lowest levels of rotational multiplets for which 1 I~I/I~<< 1. Thus, this Hamiltonian, depending on the six parameters I~,~, a2, b1, b2, c1, is universal for the description of the C2~-typebifurcation in different systems. Yet not all the parameters of the Hamiltonian (3.2) are independent*). It is possible, as was found by Watson [34], to reduce the number of parameters in the Hamiltonian (3.2). A unitary transformation —

U~H~2U,

U = exp(iS);

S

=

ia[13,

12+

—

I~]÷,

(3.Sa,b)

transforms (in the considered approximation) this Hamiltonian in an operator with the same terms but different parameters. We can choose the coefficient a so that the fifth or the sixth term in the new Hamiltonian vanishes. In the first case we obtain the A-reduction of the effective Hamiltonian, 2)+bl(I2+ +I~) Hc2ozEo(I)+al(I)(I~I iy2 -1-a2k13—1

,2~2

)

17. rr2 T’2U2L131

y2

72

,2 1

,l+-1-l_J+.

This choice is preferable because Hamiltonian (3.6) has a larger existence region of the bifurcation as is shown by a simple classical analysis. Besides, the eigenvalues of (3.6) are the same as those of the Hamiltonian (3.2) because U is a unitary transformation. Considering the Hamiltonian (3.6) as a starting point, we will discuss the quantum manifestation of a C 2,,-type bifurcation. To clarify the character of the changes in the system spectrum while going through the critical point, let us obtain first the solution of the Schrödinger equation, Hc2,WIM,,

=

EI,,WIM,,,

(3.7)

in the harmonic approximation. The general solution of eq. (3.7) in this approximation is obtained in appendix B. We follow the changes of this spectrum with the increase of the angular-momentum quantum number I in the band. 3.1.1. Multiplet levels For I < I~axis 3 is stable and the lowest levels of the Hamiltonian (3.6) correspond to the precession of the vector I around this axis. In quantum mechanics, the precession is described by small oscillations near the stationary state (2.32). The single minima of the potential V(k) (B.1 1) at the point k0 = 0 correspond to this state. Using formulas (B.16) and (B.17) we find the energy of the lowest levels of a multiplet Ej,,=Eo(I)+Ia1(I)+w<(n+1), 6)

n=0,1,2,...,

The author thanks Igor Kozin for drawing his attention to this circumstance.

(3.8)

198

Pavlichenkov. Bifurcations in quantum rotational spectra

f.M.

where the precession frequency is equal to =

2(21 + 1)..,,/otb!(IC

—

1).

(3.9)

The last expression is reduced to the classical formula (2.34) for large I. The oscillator quantum number n differs in the general case from the quantum number v, n = v 1. The value n = 0 corresponds to the lowest level of a multiplet. Its energy in the classical limit equals that of the rotation around axis 3. The expansion coefficients of the wave function (2.13) in the harmonic approximation have, according to eqs. (B.18), the form —

w<

(~ 2

aIK,,

=

=

2 ~lI(2r~

1))

1/2

___________ H,,(K~~/(2I + 1)) exp [—w< K2/(2I + 1)],

(3.10)

[(a 1 + 2b1)/(a1

2b1)] 1/2

—

(3.11)

where K is the quantum number of operator 11. Thus, the wave function of the lowest levels of multiplets with I < I~has the form of a sharply localized state (2.16). The eigenfunctions of the Hamiltonian (3.6) are simultaneously the eigenfunctions of the operators ~R6(2.7) of the D2 group. Consequently, the a coefficients should satisfy the relation a1, ~K,v = r3(— 1)’aJK,,, which in the harmonic approximation equals (—1)~= r3(1).

(3.12)

This symmetry property results in a doublet structure of multiplet states with energy (3.8). For even I, two degenerate states A1, A2 or B1, B2 (see table 2.1) have, respectively, even or odd quantum number n. The same is true for odd I except that states A are replaced by states B and vice versa. In particular, there is only one fully symmetric state 1 in the rotational spectra of deformed nuclei and molecules, containing nuclei with zero spins. In this case the lowest levels of the multiplets with even I have the oscillator quantum number n = 0, 2, 4,. , and those with odd I, n = 1, 3, 5 At I > I~the frequency (3.9) becomes imaginary, meaning instability of the precession around axis 3. As a result, two equivalent precession axes k and k’ appear in the (23)-plane. Two degenerate minima of the potential V(k) (B. 11) at points ±k0, where .

.

k~= (a1 + 2b1)/21(I + 1)(a2 + 2b2),

(3.13)

correspond to these axes. The angle Oo specifying 2 0~= the k~.orientation of the precession axes with respect to The axis energy 3 of theofBFF is determined by sin the lowest states of the multiplets according to eq. (B.16) is 3(a

E 1,,

=

E0(I) + Ia1(I)

—

I

2(I I~)2/4(a —

2

—

b2)

—

c~

2+ 2b2) + w>(n + ~), n = 0,1,2,

...

(3.14) (0>

(21 + 1){8a~(I I~)(1 k~)[b1 ~(2I+ 1)[8o~bl(I—1C)]112, =

—

—

—

2 112 1(1 + 1)b2k 0]} (3.15)

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

199

where w,. is the precession frequency. Equations (3.14) and (3.15) refer to precession motion in either of the two wells of the double-well potential, and represent the approximate eigenvalues of the Hamiltonian (3.6) with a broken C2,, symmetry. The a coefficients, describing precession motion in the right-hand well, have the form a1~(k k0)

=

—

4 (~I(2~

_____________

‘~)H,,((k 1/2

—

k0)~w>(2I+ 1))

2], xexp[—(I + ~)w>(k w>

=

(3.16)

k0)

—

.~,/(a 1+ 2b1)/2s(k0),

where the function s(k) is determined by eq. (B.10). The coefficients for the left-hand well differ from eq. (3.16) in the sign of k0. Thus, we obtain a solution with broken symmetry, which Peierls refers to as a “symmetry-breaking approximation” [60]. It is convenient to use the simple harmonic wavefunction to describe the sharply localized states involved in the C2,, -type bifurcation. To improve the method without going beyond the framework of the harmonic approximation, we may use for the coefficients the symmetrized expression 2[aIK,,(k =

[2(1 + r3J,,)]

—

k

“

0) + r3(—1)’~aIK,,(k+ k0)],

(3.17)

where J is the overlap integral for a(k k0) and a(k + k0). Having averaged Hc2~(3.6) over the function (2.13) with coefficients (3.17), we obtain an additional symmetry-dependent term in formula (3.14), —

‘‘r3(”~1)w> exp[—w>k~(2I + 1)].

(3.18)

This term appears due to the tunneling of the angular-momentum vector I across the potential barrier, which separates the two degenerate minima of the potential V(k) (B. 11). Thus, the splitting energy AE between the levels of A and B types with the same n decreases with increasing I I~ proportionally to —

exp[—s(I

—

I~)3/2],

s

‘-~

1.

(3.19)

The mutual approach of the multiplet levels as the quantum number I varies is a specific feature of quantum bifurcation, which we shall call the level clustering. The harmonic approximation becomes inappropriate near the critical point I~.An exact diagonalization of the Hamiltonian Hc2~(3.6) is required to reveal the level clustering in multiplets near I~.It is convenient to use the basis set IIMKr2>

=

21+1 (16E2(1 + ~K.0))

1/2

[D~K(4’,~, ~) + r2(—1)’~DL~,_K(4’, ~ ~)],

(3.20)

as the eigenfunctions of the operators 12, I~’13 and 9~2. The type of irreducible representation is determined by r2 and the parity of the quantum number K. The results of the numerical

200

f.M.

Pavlichenkov, Bjfurcations in quantum rotational spectra

:::

~

400-

~

~ ~

4’O

~

ANGULAR MOMENTUM QUANTUM NUMBER I

Fig. 3.1. Energies of the lowest levels of multiplets (taken relative to the lowest level with v = 1) in an isolated rotational band for the C 2,-type bifurcation. The parameters of the Hamiltonian (3.6) are: I, = 30, a = 0.25, b1 = 1.0, a2 = 0.005, b2 = — 0.001. The full lines represent the result of a numerical diagonalization, the dashed ones are the harmonic approximation.

diagonalization of Hc2~for a particular set of parameters is shown in fig. 3.1. It exhibits the energy dependence of the lowest levels on the angular-momentum quantum number I. At I < 4 these levels form a system of doublets A1 + A2 for the even and B1 + B2 for the odd values of K. The splitting of the doublets is of the inversion type and is too small to be shown in the figure. As I approaches 4, the doublets A1 + A2 and B1 + B2 tend to be approach each other and form at I > 4 a four-fold cluster A1 + A2 + B1 + B2, corresponding to delocalized precession around two equivalent axes. For the sake of convenience, the energies of the levels in a multiplet are taken relative to its lowest level with v = 1. The same figure shows for comparison the result of the harmonic approximation calculations with eqs. (3.8), (3.14). 3.1.2. Electromagnetic transition intensities The bifurcation manifests itself not only in the modification of the energy spectrum but also in the change of the level properties with respect to electromagnetic interaction. The latter change is more radical since the wave function is more sensitive to the variation of potential energy. In this section we investigate the change of El, E2 transitions and a static quadrupole moment on going through the critical point I~.It is convenient to express the probability of the electric transition between states Iv and l’v’,

(~)

8 (~+l) 1 AE I’v’) = ~[(2I + 1)!!]2 ~

P(E~,Iv

2A+1

B(E2; Iv

I’v’),

(3.21)

in terms of the reduced transition probability 2. B(EA; Iv

—*

I’v’)

=

p.~

I(WF’M.,,’.~t~(E2)!t’IM,,)I

(3.22)

f.M.

Pavlichenkov. Bjfurcations in quantum rotational spectra

201

Here AE is the transition energy, and the electric multipole operator, =

~ D~(4,~9,~i)~t~(E2),

(3.23)

is expressed in terms of its components ~ in the BFF. The method of obtaining expressions for the reduced probabilities in the harmonic approximation is described in appendix C. The analytical formulas allow one to understand better the physical interpretation of a phenomenon under consideration. We begin the discussion with the electric El transitions, which are observed in dipole-active molecules. It should be noted that the line strength S used in molecular spectroscopy [61] differs from the reduced probability, which is used mostly in nuclear physics [62]. The two quantities are connected with the relationship S(Iv

—~

I’v’)

~it(2I + l)B(E~.%;Iv

=

I’v’)

—~

(3.24)

.

It is reasonable for us to use the reduced probability because it depends on the initial and final functions only. Thus, this quantity may be used as the measure of the modification of a wave function due to a bifurcation. The El transitions exist if a system has an inherent dipole moment D0. The dipole-moment operator of a system having the C2,, symmetry is parallel to the symmetry axis. Let us choose the C2,, axis to coincide with one of the axes 1, 2 or 3 of the BFF. Table 3.1 gives the selection rules for the allowed El transitions between the states of the D2 group. We will investigate the transitions of the second type when the dipole moment becomes aligned along axis 2. It is of interest to compare transitions in the I < I~region with those in the I > I~ region. To this end, the corresponding formulas for the reduced probability are placed in sequence: first for I < I~and then for I > IC. Let us consider first the transitions inside a multiplet (the Q branch, Al = 0), B(El; I, v

I ~

— -

—*

I, v

—

1)

=

[3D~(2b1

=

(3D~/l6ir)k~, An

=

[3D~n/l6ir(2I + l)](l

—

a1)/l6irw<]n, =

0, I —

An >

=

—1,

—

<

I~

I~

k~)1(I),

An

2’ 2b1 —a1 +21(1+ l)a2k~ {8~(I I~)(l k~)[b1 1(1 + l)b2k~]}” —

1

—

Table 3.1 Allowed El transitions in the system with the

C 2~symmetry axis Direction of dipole moment

El transitions

axis 1 axis 2 axis 3

A,*-÷B2 A2o-~B, A1—B, A2i-~B2 A,i-oA2 B,i-~B2

=

—1,

I> I~,

(3.25)

(.326

202

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

where k~is determined by eq. (3.13). It is necessary to distinguish the transitions between the levels of a cluster (An = 0) from those between adjacent clusters (An = 1). In the harmonic approximation, at I > ‘C~ the latter transitions are approximately n/I times as weak as the former without changing the quantum number n. A characteristic quantity is the parameter (io/I)2, where i0 is —

equal to the precessional amplitude in the nth state given by 2, (3.27) = [nw
—*

B(El; I, v

—+

I

—

1, v)

=

(3D~/32ir)n,

An

=

(3D~/32i~)(1 ks), —

=

0,

—

I

An

—

< 4

=

0,

1

>

I~.

(3.28)

The exact dependence of this B(El)-value on I shows its drastic fall for I > 4, as k~approaches unity. The weaker transitions have reduced probabilities, which may be written in the form 3D~n(n—1)( B(E1;I,v—*I—1,v—2)=

12(1+

32it(I+~) \

2b 1—a1 (21+1)1, 2w< /

An= —2,

2,

1>4.

An= —1,

1<4, (3.29)

32~(2i±1),1(I)[1+~(I)] 0.02

0.00_2~’3b3~”4b” ANGULAR MOMENTUM QUANTUM NUMBER I

Fig. 3.2. Calculated B(El)-values for the Q-branch transitions between the lowest multiplet states A 1 and B1 in an isolated rotational band. The parameters of the Hamiltonian (3.6) are the same as those of fig. 3.1.

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

203

Examination of eqs. (3.29) shows that the quantity n/I appears with a different power. Thus, the probability of this transition increases while going through I~.The same is true for the transition with Av = 2, B(El;I,v_*I_1,v+2)=3D~)~2)(1

=

311)

_2~

[1

—

at(2I+ l))~

An

~(I)]2,

=

1, I

An=2,

>

I~.

1<4,

(3.30)

The other transitions in the P-branch have I Av I > 2. They are weaker than those considered above because they involve higher powers of the small quantity n/I. The E2 transitions are typical of nuclear rotational spectra, since deformed nuclei have an ellipsoidal shape and therefore a quadrupole moment. The nuclear shape is characterized by the two invariants depending on the axial parameter /3 and nonaxial one y [63]. So the operator (3.23) for E2 transitions between rotational states depends on two internal moments, 112Q =

(5/l6ir)

112Q 0cosy,

~#‘+2(E2)

=

(5/32ir)

0sin~,

(3.31)

where the internal electric quadrupole moment, Q0

=

(3eZRo2/~J~f3,

(3.32)

3 fm and the static deformation /3 of the nucleus, which has is by Z theprotons. radius R0 = l.1A” A determined nucleons and Only the full symmetric A 1 states exist in the nuclear rotational bands. According to eq. (3.20) they have different wave functions !PIM,, for multiplets with even and odd quantum number I. In the harmonic approximation, for I < 4, this difference results in different sequences (even and odd) of the oscillator quantum number n [see eq. (3.12)]. So another sign of bifurcation arises in the E2 transitions for such systems. We begin with intermultiplet transitions, for which B(E2) may be expressed as B(E2; I,v~I,v— 1)— l5Q~(2bi

An= —2, 1<4,

a1(n+i)sin(k~+Y))

4x (3.33) =

15Q~k~(l—k~)n(J).2(I + y),

An

=

—1,

1>4,

where the function ~(I) is determined by eq. (3.26). The I dependence of expression (3.33) for I < 4 involves that of w (I) and the change of the parity of the quantum number n when I changes by one unit. The former is a smooth dependencewith a derivative of the order 1/I. The latter results in a zigzag dependence of the reduced probability in the region of the aligned configuration. By contrast, for I > 4, the I dependence of B(E2) in eq. (3.33) is smooth because there is no alteration of the quantum number n in the region I > IC. Thus, the bifurcation manifests itself in

204

f.M.

Pavlichenkov, B~furcationsin quantum rotational spectra

a change of the dependence of B(E2)(3.33) on I, while going through the critical point 4. The same is true of the static quadrupole moment in a band, which is determined by 1 n)”2(Wj = ( 1,,~#0(E2)~1’j1,,) = Qo{cosy [~~/~(2n+ 1)/w<(2I + 1)]sin(~it+ y)}, —

Q0[cosy

=

—

~/~k~sin(~ic + y)],

I

> 4.

I <4, (3.34)

The B(E2) values of other E2 transitions inside a multiplet are proportional to higher powers of the small parameter n/I. We consider now the E2 transitions between levels of neighboring multiplets B(E2; I, v

—~

I

—

1, v) 2(n + 1),

=

An =

5Q~(2b~1 a1) [sin(~ + y) 8irw —

=

1,

‘~0’.

—

w< sin(~n y)] —

even I, I <4, —a 1 11[sin(~ir + y) + w< sin(~ir y)]2n, —

An

=

—1,

odd 1,1
oltW<

=

k~(l k~)sin2(sir + y), —

An

=

0, even or odd I, I

> IC,

(3.35)

where w< is determined by eq. (3.11). The difference in the expressions for even and odd quantum numbers I in the region I <4 results in the zigzag dependence of the reduced probability on I. A plot of B(E2) obtained by a numerical calculation is shown in fig. 3.3. The strong suppression of the reduced probability for even I leads, for certain values of the nonaxial parameter y, to large oscillations of B(E2) in the region I < I~.This effect is stronger than that of transition (3.33) and of the quadrupole moment (3.34), and involves a different physical reason.

0.04

0.00 20~3b’3~”4b ANGULAR MOMENTUM QUANTUM NUMBER I

Fig. 3.3. Calculated B(E2;

I

—~I —

1) values for the transitions between the full symmetric A

1 lowest states ofadjacent multiplets in an isolated rotational band. The parameters ofthe Hamiltonian (3.6) are the same as those of fig. 3.1. The nonaxial parameter y = 18°.

LM.

Pavlichenkov, Bj/’urcations in quantum rotational spectra

205

To understand this nontrivial zigzag dependence of B(E2) in (3.35), let us return to the classical description of the precessional motion. Equations (2.33) and (2.34) determine the motion of the angular-momentum vector I in the BFF, I!

=

i0 sinw< t,

I2

=

i0 [(a,

—

2b,)/(a, + 2b1)]

1/2 cosw< t,

(3.36)

where i0 is the amplitude of the precession given by eq. (3.27). To find the time dependence of the E2 moment (3.23), we need to know the absolute motion of the system determined by the Euler angles. Let the z-axis of the laboratory frame be parallel to the Ivector. Then the components of this vector in the BFF are given by 11

=

—Isin~9cos~/, 12

=

—Isin19sini/i,

I3

=

Icos9.

Using these formulas one obtains the t dependence of the angles ~9and 9=

—

(3.37)

~,

(iosinw~~t)/(IcosiIi), tan~/’= (cotw
(3.38)

The angle 4’ is determined by the projection of the angular velocity vector Q, =

4’ cos 9+ ~fr.

In the small-amplitude approximation, Q3

=

Q = dE/dI, and we have

cb+~/i=Qt.

(3.39)

Finally, substitution of the Euler angles (3.38), (3.39) and the amplitude i0 into eqs. (3.23) gives the classical value of the transition operator 112[w< sin(~x y)sinw
~..t_,(E2,

t) =

—

—

(3.40)

The Fourier coefficients of this value are equal to the quasiclassical matrix elements of the E2 transition operator with frequencies E(I)—E(I— 1)—w< ~Q—w<,

An= 1,

E(I)—E(I—l)+w<~Q+w<,

An= —1.

It is at this stage of the derivation that the difference in formulas for the reduced probability (3.35) in the region 1 <4 appears, because of different Fourier coefficients. Thus, the difference in the formulas for transitions with An = 1 and An = —1 can be attributed to the fact that these transitions take place between specific pairs of states. It should be emphasized that such combination of pairs is the consequence of full symmetric multiplet states. One should note also that this simple quasiclassical approach allows one to obtain all the reduced probabilities listed in this section.

206

f.M. Pavlichenkor, Bifurcations in quantum rotational spectra

Next we consider transitions I I 1 between states for which Av equals ~ 1. The former has the corresponding reduced probability —*

B(E2; I, v =

—+

I

—

[5Q~(2b,

—

1, v

—

—

1)

a,)/8mw<][sin(jm + y) + w< sin(~ir y)]2n, —

An=—1, evenI,I
An

An

odd I, I

—3,

=

[5Q~n/16itIt~(I)][sin(~ir =

—1,

—

y)

—

< IC,

k~sin(~ir + y)]2

even or odd 1,1>4.

(3.41)

The I dependence of this probability has again a specific zigzag pattern in the region I < I~and a monotonic one for I > I~.The same is true for the transition with Av = 1 except that the phase of the zigzag dependence is different, —

B(E2; I, v

-~

I

—

1, v + 1)

=

—~Q~(n/I)3,An

=

[5Q~(2b

=

even I, I

<

‘C~

2(n + 1), 1

An=1,

3,

=

—

a1)/8irw<] [sin(~1t + y)

—

w< sin(~1r y)] —

oddl,I<4,

5Q~(n+ 1) [sin(~ 16itI~(I)

—

y)

—

k~sin(~ + ~)]2

An

=

1,

even or odd 1,1>4.

(3.42)

The zigzag I dependence of the B quantities in eqs. (3.41) and (3.42) is explained again by the specific combination of initial and final states in full symmetric multiplets. The remaining transitions between neighboring multiplets are much weaker since they contain higher powers of the small quantity n/I. The transitions I I 2 are also classified according to the powers of the quantity n/I. The strongest transitions of this type are those involving no change in the quantum number v. The corresponding B(E2) quantity is B(E2; I, v

I

—+

—

—

2, v)

=

~

[sin~ 2b —

1

—

32m

(0<

(5Q~/32x)[siny

—

a1 (2n + 1)sin(~+ 2,

=

k~sin(4ir + y)]

)]

An = 0, I

< 4,

An

> IC.

=

0, 1

(3.43)

These formulas resemble the eqs. (3.34) for the static quadrupole moment except that sin y is included instead of cos y. This result can be accounted for by observing that the transition (3.43) is

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

207

induced by the rotation of the quadrupole moment around axis 3. The transitions with Av come next in this sequence. The expression for B for transitions with Av = 1 has the form B(E2; I, v

=

=

—~

I

—

2, v

—

~

±1

1)

[3w< siny + sin(~x+ 32irw~(2I±1)2 ~

=

[1 + ti(I)]2sin2(~ir+

~)]2

An

=

—2, 1

y),

An

=

—1,

<

I~,

I> I~.

(3.44)

Another B-value has a different form, B(E2; I, v

—~

I

—

2, v + 1)

5Q~(n+l)(n+2) =

32irw~(2l+

1)2

1

2

[3w< sin~’ sin(Tlt + ~‘)] ,

~

—

=

2, I

<

I~,

2 sin2(~ir+ y), An = 1, 1 > I~. (3.45) 1(I)] These transitions have a small odd—even I staggering in the region I < 4 due to the difference in the sequence of the oscillator quantum number n. Let us note that the transitions in this region are much weaker than those for I > 4. The transitions with I Av I 2 are proportional to higher powers of the small parameter n/I. Thus, the set of lowest multiplet levels may be classified according to El and E2 transitions between them. They form “subbands” (the group of states with the same quantum number v or n) with strong transitions inside each subband and weaker ones between them.*) The small quantity n/I, where n = v 1 is an oscillatory quantum number, plays a central role in such a classification. We will use this classification in section 5. =

1) [1

An

—

—

3.2. C 3~-typebjfurcation It is clear from fig. 2.4 that the motion near axis 3 is local if the system is far from the critical point This axis becomes unstable at the very critical point. Thus, it is necessary to include terms with higher powers of the operators I~in the Hamiltonian (2.38) to describe those states of multiplets for which the quantum number I is close to 4. Let us consider the quantum problem assuming that the transition region is small. In such a case the discrete character of the quantum number I enables us to exclude the essentially nonlocal region and to use the two first terms of the power expansion of H (2.38) in terms of the small quantity (I~ 12)/I 2~The thus obtained Hamiltonian has the form 4.

—

~

(3.46)

where E0(l) is determined by eq. (3.3), and —

6)

I~)= I2f~(I2, 12),

b

=

I~f3(I~, I~)/2.

The subband called track in the nuclear yrast spectroscopy [64J.

(3.47)

208

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

The Hamiltonian (3.46) has the symmetry properties of the total effective Hamioltonian (2.38). It means that the operators C~and =

exp(—i~irI3)

(3.48)

commute with (3.46). Thus, we can choose the basis functions with a different sequence of the quantum number K in the form t

IIMKr2>

=

]16~2(1~

~K,0)

3~DL~ -K(4’, ~

x [D~K(4’,9, ~) + r2e

~)],

(3.49)

for the A, and A 2 states (K

3k), and

=

2 D~K(4’, 9, II’) (3.50) ,.J(2I + 1)/8ir for the E, (K = 2k + 1) and E 2 (K = 2k + 2) states. The latter are degenerate because the C, operator transforms the function ~P~1into the function ~PE2 and vice versa. Let us use the boson representation (A.4) for the angular-momentum operators. Then, in the harmonic approximation, the last term of the Hamiltonian Hc3~vanishes and the operator of the boson number commutes with 13. So, the eigenfunction of the harmonic approximation is simultaneously that of the 13 operator. The energy of the multiplet “lowest levels” is equal to

I IMK>

ElK

=

=

E0(I)

—

(2x/I)(I

—

I~)(I IKI), —

K

=

±1,±(I

—

1)

(3.51)

These levels are placed in the lower part of multiplets with I < I~and in the upper one for I > IC. They are degenerate with respect to the sign of the quantum number K. The wave functions of the lowest levels in the harmonic approximation are determined by eqs. (3.49) and (3.50) with K = ±(I n), where n is the oscillator quantum number. Another characteristic of these levels is the type of irreducible representation of the C3~group. Accordingly, there exist three sequences of quantum numbers I = 3m, 3m + 1, 3m + 2 (m is a positive integer) with different orders for the lowest levels in the rotational multiplets (see table 3.2) The sequence of states in a multiplet repeats cyclically with period 3 for n 3. Figure 3.4 shows the I dependence of the energy ratio of the “lowest levels” to the precession frequency w = 2a~I IC I/I in eq. (3.51). The ratio is obtained by the numerical diagonalization of the Hc3~with the basis functions (3.49) and (3.50). The energies of the levels are displayed relative to the first (for I I~)level of the multiplet. The value ~/b is chosen large to narrow down the transition region. The levels form a system of doublets A1 + A2 and doubly degenerate levels E1 + E2 arranged in the same order as shown in table 3.2. The fine-level structure is not shown in fig. 3.4 for simplicity. The figure clearly reveals that the critical phenomenon manifests itself in the inversion of the multiplet levels while passing through the critical point. Let us consider the electromagnetic El transitions for the Hamiltonian (3.46). The symmetry of the problem under consideration suggests that the dipole moment D is parallel to axis 3. So the allowed transitions are —

—

A,÷-+A2,

E1.i—*E1,

E2+-÷E2,

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

209

Table 3.2 Classification of multiplet lowest levels of the Hc

3, Hamiltonian in the

harmonic approximation n

K

I=3m

I=3m+l

I=3m+2

O 0

1 —I 1—1

—1+1 1—2

E1 E2 A,,A2 A1,A2 E2 E1

E2 E1

1 2 2

A1,A2 A1,A2 E2 E1 E1 E2

—1+2

E2 A1,A2 A1,A2

5.

—3

4.

~20

‘2~’

3b’3’54’O

ANGULAR MOMENTUM QUANTUM NUMBER I

Fig. 3.4. Energies of the extreme levels of muliplets (in units of the precessional frequency w = 2a I I — I~I/I) in an isolated rotational band for the C3~-typebifurcation. The full lines connect the repeated sequence of states A—E—E. The parameters of the Hamiltonian (3.46) are I~= 30.5, a/b = 0.8. E,1, is the energy of the lower (for I < I~)or upper (for I > 1,) multiplet level.

and involve states having the same sequence of quantum number K. The Q-branch transitions I, v I, v 1, the P-branch transitions I, v I 1, v in the region I < I~,and R-branch transitions I, v I + 1, v + 1 for I > 4 are forbidden in the harmonic approximation because of the opposite direction of the I vector in the E1 and E2 states. By using the exact wave functions of H~3,,with the parameters of fig. 3.4 one can obtain for these “directionally forbidden transitions” (see below) the values B(E1)/D~ l0~°—l0~°. The transitions, allowed in the harmonic approximation, involve the ones in the P-branch for I <4 and in the R-branch for I > 4. The former transitions have the corresponding B-value —*

—÷

—

—

—~

3fl2 i~sr

B(E1;I, v—~I—1, v

1)=-~-~-_ 1(21 +n,n 1)’ -~~-‘~ IIh~1 —

—

where the oscillator quantum number n

=

v

—

(3.52)

1. The reduced probability of the latter is

3D~(21 n + l)(n + 1) B(E1;I~v—+I+1~v+1)=-j~-—(I+l)(2I+l) . —

(3.53)

210

f.M. Pavlichenkov. Bifurcations in quantum rotational spectra

Thus, the C3,, -type bifurcation results in the substitution of the P-branch by the Q-branch while passing through the critical point 4. 3.3. The C4~-type bifurcation The local character of the considered critical phenomenon makes it possible, as is shown in appendix A, to obtain a universal Hamiltonian for the description of the lowest levels. Using eq. (2.44) one finds 12

=

12

E0(I) + a,(I) ~

+ a2

( ~~) 12

12

2

+ c1

I~ I~ + +

(3.54)

-‘

2, 12), a 2f~(12, 12) = ot(I Ic), E0(I) =f0(I 1(I) = l a ....114f”(7272\1c )‘ C 1c J4k’c ‘1c 2 2 JO k’c ‘ 1 2 This Hamiltonian has the same symmetry properties as the initial one (2.44): it commutes with the operators C, and —

—

—

~R 3(ir/2)= exp(—i~irI3).

(3.56)

Thus, one can choose the basis functions with a different sequence of the quantum number K in the form (3.20) for the A and B states or (3.50) for the E1 and E2 degenerate states. Let us consider the quantum precession near the critical point I~,for the Hamiltonian parameters I a2 I > 2c, and ~ > 0. Using the boson representation (A.4) for the angular-momentum operators, we obtain for the energy of the levels corresponding to precession around the stable axis 3 the expression =

E0(I) + w(I)n,

n

0,1,2, ... ,

=

(3.57)

where the precession frequency equals w(I)

=

tx(I~ I)(21 —

—

(3.58)

l)/I2.

While going through the critical point, these levels invert relatively to the level with the quantum number n = 0. They are degenerate with repect to the sign of the quantum number K = I n and form a two-fold cluster structure. The wave functions corresponding to eigenvalues (3.57) are determined by eqs. (3.20) and (3.50) with K = ±(I n), since the boson number operator commutes with 13. Another characteristic of these levels is the type of irreducible representation of the C4,, group. There exist four sequences of the quantum numbers I = 4m, 4m + 1, 4m + 2, 4m + 3 (m is a positive integer) with different order of the levels (3.57) in the rotational multiplets, as is presented in table 3.3. The sequence of states in a multiplet repeats cyclically with the period 4 for n 4. The allowed electromagnetic El transitions for the direction of the dipole moment D parallel to axis 3 are given in table 3.1. They are —

—

A,+—i.A21

B1+—i.B2,

E1÷-+E,,

E2.*—+E2,

f.M. Pavlichenkov. Bifurcations in quantum rotational spectra

211

Table 3.3 Classification of multiplet lowest levels of the H~

4,Hamiltonian

in the harmonic approxi-

mation n

K

I=4m

I=4m+l

I=4m+2

I=4m+3

O O 1 1 2 2 3 3

1 —I

A1,A2

E1

A1,A2 E2 E1

E2 A1,A2 A1,A2

B1,B2

E2

B1,B2 E1 E2

E1 B1,B2 B1,B2

B1,B2 B1,B2 E1 E2 A1,A2 A1,A2 E2 E1

E2 E1 B1,B2 B1,B2 E1 E2 A1,A2 A1,A2

1—1 —1+1 1—2 —1+2 1—3 —1+3

and involve the states having the same sequence of the quantum number K. The allowed (in the harmonic approximation) transitions in the P-branch for I < 4 and in the R-branch for I > 4 have the reduced probabilities given by eqs. (3.53) and (3.54), respectively. Thus, for the multiplet states corresponding to the precession around axis 3, bifurcation results upon substitution of the P-branch by the Q-branch while passing through the critical point 4. Four equivalent precession axes k~(2.43a) appear in the region I > 4. Let us consider the precession around the k1(00, fir) axis. To realize the harmonic approximation let us express the Hamiltonian (3.54) in terms of the spherical tensor operators in the form (A.11) and perform the rotation ¶R(~ir,0~,0) (A.14) as described in appendix A. The angle 0~is determined by eq. (A.22), which leads in the classical limit to the value (2.43b). Using the boson representation (A.4) of the angular-momentum operators and retaining in the transformed Hamiltonian ~RHc4~ 91-1 the terms quadratic in the boson operators, one can obtain the energy of the lowest levels in the form 12’ 12’ (r\ ( 15 .I~~Ifl—L~1k1,+w>kn+2,,

(51 1 nU,1,L,...,

2/4(a E1(I)

=

E0(I)

—

,~2(I

—

4)

2

—

2c1),

(3.60)

where E, (I) is the energy of rotation around the k1 axis, and 2 =

(I

—

Ij(8c/I)[c1/(a2

—

(3.61)

2c1)]”

is the precession frequency. We have found again the states with the broken symmetry similar to those for the C 2~-typebifurcation. To improve this solution one has to symmetrize the harmonic wavefunction according to the C4~group. The thus obtained wave function describes a delocalized precession, which results in an eight-fold cluster structure. To follow the level clustering while going through the critical point let us use the result of a numerical diagonalization of the Hamiltonian (3.54) with the functions (3.20) (for A and B states) and (3.50) (for E states) as a basis set. Figure 3.5 shows the I dependence of the rotational-multiplet lower-level energy, divided by the precession frequency (3.58) for the fixed value I = 20. The level energies are displayed relative to the regular part E0 (I). At I
212

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

Iw(20)I

2~13IO!115bo1,

I

Fig. 3.5. Energies ofthe lowest levels ofmultiplets [in units ofthe precessional frequency (3.58) at I = 20] in an isolated rotational band for the C 4,-type bifurcation. The full lines connect the repeated sequence of two-fold cluster states A1 + A2 (0), B1 + B2 (S) and E1 + E2 (s). The parameters ofthe Hamiltonian (3.54) are: 1~= 30, a/c1 = 0.18, a2/ci = 8.0. E0 is the regularpart ofthe system energy.

lines with the repeated sequence of states A E B E. The splitting of the doublets is too small to be shown in the figure. As I increases these lines either come closer together to form the eight-fold clusters —

—

—

A1 +A2+B1+B2+E1+E2, or are regrouped into the initial structure of two-fold clusters. The transition region with freakish crossing and pseudo-crossing of the connected lines divides the two precession domains. Note that these lines run from the upper left-hand to the lower right-hand corner of fig. 3.5 since the four degenerate minima E1 4. The transition region is clearly visible in fig. 3.6 showing the I depedence of the reduced probability of El transitions between states connected by the lines c~and ~ in fig. 3.5. These states belong to the two-fold clusters for I 4. The maximum of the probability at I = 38 corresponds to the pseudo-crossing point of the lines ~ and ~. The initial and final states are strongly mixed in this point. Note that the reduced probability is changed by two orders of magnitude while going through the critical point. It is interesting to compare this result with fig. 3.2, which shows the same transitions for the C2,,-type bifurcation. The Hamiltonian (3.54) gives us a fine opportunity for studying “directionally forbidden transitions.” Let us consider transitions between the states of the rotational multiplet belonging to the different types of clusters. In the classical picture, the precession axes corresponding to these

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

“I

10~’

r u >~

I

ii.

—

I

io-~

213

~ io~

I I

/ 10_b.

10

6

20

~

30

40

50

ANGULAR MOMENTUM QUANTUM NUMBER I

Fig. 3.6. Calculated reduced probabilities (in units of the square of the dipole moment) of the Q-branch El transitions between the states connected by the solid lines a and J3 in fig. 3.5. The parameters of the Hamiltonian (3.54) are the same as those of fig. 3.5.

10

45

‘‘‘I’’’’

50

I

55

ANGULAR MOMENTUM QUANTUM NUMBER I

Fig. 3.7. Calculated reduced probabilities (in units of the square of the dipole moment) of the El “directionally forbidden transitions” between the levels of two- and eight-fold cluster states with the oscillator quantum number n = 0. The same fixed parameters as in figs. 3.5 and 3.6 are used.

states, and consequently the angular-momentum vectors, have different direction in the BFF. Thus, their wavefunctions have small overlap for high values of I [6, 7]. As a result, the electromagnetic transitions between the considered states will be hindered. Figure 3.7 shows the numerically calculated I dependence of B(E1) for the transition between the states with the quantum number n = 0 belonging to the two- and eight-fold clusters. The plot begins with I = 45, for which 00 = 28.3°,and ends with I = 56, 0~= 38.6°.In this interval of the quantum number I the reduced probability decreases approximately as exp(—sIOo2),

S =

0.64,

(3.62)

except at the pseudo-crossing points. The same exponential dependence, but with s = 0.49, is obtained if the hindrance factor of this transition is approximated by the overlap integral of the two harmonic functions with quantum number n = 0, as has been done in refs. [6, 7]. The above-considered quantum bifurcations in an isolated rotational band can be divided into two groups. The nonlocal bifurcations of C 3,, and C4,, types result in the inversion of multiplet levels and, consequently, in substitution of the P-branch of electromagnetic transitions by a Qbranch while going through the critical point. The local bifurcations of C2,, and C4,, types are more interesting. They result in the clustering of multiplet levels and in the radical change of electromagnetic transitions between them. In the classical limit these bifurcations are similar to second-order phase transitions and involve a breaking of the symmetry group g (C2,, or C4,,) due to the localized precession. The classical picture is only a useful illustration of the energy level clustering phenomenon, which includes the delocalized precession of the angular-momentum vector around equivalent axes. How to reconcile the quantum delocalization with classical localization?

214

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

While I I~increases, the intracluster splitting decreases exponentially, as follows from eq. (3.19). But the precession motion is not localized at one axis and the cluster states have a definite symmetry with respect to the group g. For the C2,,-type bifurcation, say, there exist two delocalized —

states: a symmetric A-state and an asymmetric B-state. The quantum delocalization may be viewed as a consequence of the possibility of tunneling of the angular-momentum vector through a potential barrier with finite height. Thus, at first glance the quantum bifurcations do not have relevance to the problem of spontaneously broken symmetry [65]. The Hamiltonian and its eigenstates remain invariant against the local symmetry group. Here lies the distinction between the quantum bifurcation and the phase transition in a macroscopic system, in which the broken symmetry makes its appearance directly after a critical point. For sufficiently large I I~,the semiclassical conditions arise: a high potential barrier and a small intracluster splitting. In such conditions the delocalized precession becomes unstable against an asymmetric perturbation comparable with the splitting. This perturbation produces a different effect on the angular momentum in the left- and right-hand well. As a result, the correct zero-order wave functions are those which localize the angular momentum in either well and thus have a broken symmetry. The localized state is not strictly an eigenstate of an unperturbed Hamiltonian and therefore it is nonstationary. Yet its decay time may be large enough (due to a very high potential barrier separating equivalent axes) for it to be treated as really existing. This is exactly the spontaneously broken symmetry of the first kind according to Peierls [60]. Recently the effect of the environment as a possible source of symmetry breaking in pyramidal molecules XY3 is discussed in ref. [66]. The inversion symmetry, which occurs in these molecules may be broken by small perturbations due to the system-environment coupling. The onedimensional motion along the inversion coordinate is similar to the delocalized precession considered above. The instability of a delocalized state is determined by the mass of the atoms X. A molecule with a sufficiently heavy atom gets localized by a fluctuation from neighboring molecules. The localization is stabilized by the random process of molecular collisions [67, 68] or by the cooperative effect of molecule-environment interaction [66]. The level clustering in molecular rotational spectra is as simple a process as inversion symmetry breaking. It allows one to investigate the localization of the angular momentum due to the molecule-environment interaction by varying the quantum number I, which leads to the instability of tunneling. This study has relevance to the role of environment effects in quantum measurement theory. —

4. Experimental situation: molecules Molecules represent an excellent object for the study of the quantum bifurcations. They have well developed ro-vibrational spectra with different types of rotational bands, in which levels up to quantum number I 100 can be extracted from their absorption or emission spectra. The threeatom nonlinear molecules are the simplest ones with different principal moments of inertia and with a well developed multiplet structure. These molecules can be treated on the microscopic level (ab initio) by solving the Schrödinger equation for the electronic motion to obtain the molecular potential energy or for the nuclear one to calculate the ro-vibrational spectra. The three-atom molecule XY2 with two identical atoms has a C2,, symmetry group. On the other hand, more complicated polyatomic molecules may have the higher symmetry axes C3,, and C4,,. The microscopic analysis of their rotational spectra meets with great difficulties. So, effective Hamiltonians are generally used while dealing with ro-vibrational spectra of these molecules. We begin our discussion with symmetric nonlinear three-atom molecules.

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

215

4.1. Bifurcation in rotational spectra of symmetric three-atom molecules X Y2 The H2O and and H2S molecules were the first microscopic objects, in which bifurcation in well known experimental rotational spectra has been found [69]. Later this phenomenon in a H2O molecule was investigated theoretically by Makarewicz [70, 71] with the self-consistent approach to the description of the ro-vibrational states. Normal coordinates used in the solution of the multidimensional small-oscillator problem are unsatisfactory in the description of large-amplitude motions, which are necessary for the investigation of the bifurcation problem. There are many possibilities for the choice of internal coordinates of a triatomic system as is described in refs. [72—74].We will use the bond-length—bond-angle coordinates R,, R2, c~(see fig. 4.1) for treating highly excited rotational states. The equilibrium shape of an XY2 molecule is determined by the valence angle ~o > 90° and the bond length R, = R2 = R0. Let us specify the orientation of the principal axes of the inertia tensor so as to coincide with the BFF axes: axes 1 and 3 lie in the molecular plane and axis 2 is perpendicular to it. Then the principal moments of inertia have the form

~

2(~x/2), =

~osin

=

2mR~,

/1

=

=

[~JI,/(l + j.t)]cos2(cx/2),

Z~

2= 3~+ ~

2m/M,

(4.1) (4.2)

where M and m are the masses of atoms X and Y, respectively. We will be interested in light hydride and fluoride molecules of elements1A,. of the fourth and sixth The experimental groups with the compensated spins in the ground electronic state X ro-vibrational spectra of the H 20 and H2S molecules and their isotopic species D2O, HDO, D2S, and HDS are investigated in more detail. The centrifugal effects for high I values in these molecules are described by the effective rotational Hamiltonian in the A-reduction of Watson [34] Heff°H2+H4+H6+ ..., H2

=

LI

—

114

—

—

=

H~I~+ H~1I~l

H

6

A1~+ BIf + Cfl, A

,4

‘—‘K’ 3

A —

1272

‘—~IK’3’

.s r,2 ~2 1 13,1 12J

74

A —

~-‘I’

—

1~ y2 ,2 ‘-~‘I’121

2 + HJKflI4 + H 6 + hK[I~,If

1I 2+ 2h

+ h~1[fl,If2]÷I

where If

+

~-‘KL

—

4,

2]~ (4.3)

1If2I

2 = If I~.The constants A, B, C, ... are determined by fitting the energies of the observed rotational transitions. The number of the H25 terms in the power series (4.3) depends on the maximal value of I and on the type of band in the molecule. For example, in an H2O molecule, it is necessary to use terms up to H10 for fitting the multiplet levels in the ground-state band with I 35 [75, 76]. The strong centrifugal distortion in this molecule leads one to include in the Hamiltonian (4.3) the terms with I ~2 and I ~. Thus, this effective Hamiltonian is inapplicable to floppy molecules since the expansion (4.3) is slowly convergent. We will use another approach to describe the centrifugal effects in triatomic molecules. Let us begin with the classical equations for the ro-vibrational motion. The threefundamental frequencies —

216

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

Fig. 4.1. Internal bond.length—bond-angle coordinates R

1, R2, a for a triatomic molecule XY2. The point 0 is the center ofmass ofthe

molcule.

Molecule H 160 D2160 T 2160 28SiH 2 32S 2 H232S D H2 80Se H2130Te 2

Table 4.1 Critical angular momenta of the ground-state rotational bands in triatomic hidride molecules 1) v 1) v 1) f~(kcal/mol) ~2 (kcal/mol) R0 (A) ao (deg.) a, (deg.) v~(cm 2 (cm~ 3 (cm 0.96 104 87 3657 1595 3756 118 101 0.97 104 84 2672 1178 2788 120 103 0.97 104 81 2235 996 2368 121 103 1.34 1.52 1.34 1.46 1.66

92 93 92 91 90.3

88 87 89 89.6

2032 2615 1910 2345 2070

1183 1008 858 1034 861

2628 2022 1922 2358 2070

91 80 92 78 67

83 70 84 74 63

I,

26 44 58 15 18 26 9 7

Table 4.2 Critical angular momenta of the ground-state rotational bands in triatomic fluoride molecules Molecule

R 0

28Si”F 32S19F 2 40Ca19F 2 74Ge”F 2 19F 2 208Pb19F “Sr 2 2

(A)

1.59 1.59 2.10 1.73 2.20 2.13

a0 (deg.)

a, (deg.)

v1 (cm

101 98 140 97 110 95

66 68 71 72 80 85

855 795 490 663 450 545

1)

v2 (cm~I)

v3 (cm

345 430 120 263 90 170

872 830 575 691 455 520

1)

~

164 92 139 124 133 103

(kcal/mol)

~2

(kcal/mol)

131 81 127 123 129 82

I,

396 338 577 332 337 198

of a symmetric triatonic molecule include the v2(A1) bending mode, and the v,(A1) and v3(B,) stretching modes. The vibrational frequency v2 is 2—3 times as low as other ones for the considered molecules (see tables 4.1 and 4.2). The former is connected with the variation of the angle cx, the latter with the bond lengths R, and R2. Thus, the rigid-bender model with forzen stretching vibrations v1 and v3 is suitable for the description of ro-vibrational dynamics. This model proposed first by Hougen et al. [77] is widely used for classical [78, 79], quasiclassical [80, 81] and quantum [70, 82, 83] calculations. The Hamiltonian of this model has the form H

2/~[l + ~tcos2(~cx)]+ Trot(cx) + U(cx),

=

2(1 + p)p

(4.4)

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

217

where p is the momentum conjugate to the valence angle cx, and the molecule rotational energy is given by Trot(cx)

=

If/2~3,(cx)+

fl/2~32(cx)+ fl/233(cx).

(4.5)

The bond-angle potential energy U(cx) has a minimal value at the point cx0 for a nonrotating molecule. The classical equations of ro-vibrational motion are (1 l~’r

1 11

—

l~1/~J3

1 —

/e’1~ ~ J

7

‘/~32)2’3~

I’ 12

—

(I l~’( k’J-.Jl

1 /~ ~1 —

L/..j3J1

1

1

113,

‘3

I l~’T \1

(1 /“t —

~

2sincx T’ ~ ~ ~(1 + jicos2(~cx)] ~z)p ~3~[1+ roucx,

/-..)2

—

(4.6)

—

cx

— —

4(1 + ~u)p ~Jo[1 + j.tcos2(~cx)]’

.

— ‘~‘—

I

~l-31J’ 1’2’

cx.

They allow us to find the stationary axes of the system and investigate their stability. There are four different stationary states of eq. (4.6). The three ones have stationary axes which coincide with the axes of the BFF. The rotation around axis 2 with the maximal moment of inertia is described by the stationary state If=I~=0,

fl=12,

p=O.

(4.7)

In addition, the following condition is also fulfilled: T 0~(cx~) + U’(cxe)

=

0.

(4.8)

This equation determines the equilibrium valence angle cx~> cx0 of a rotating molecule, which increases with growing I. The linearized equations describing small amplitude oscillations near the stationary state (4.7) have two eigenfrequencies. The frequency Q(cxe)

=

(~~[l±~cos2(~cxe)] [T~ot(cxe)+ U”(cxe)])

(4.9)

corresponds to the harmonic vibrational motion relative to the equilibrium value cxe. Another frequency, w2(cxe)

=

I { [3i

1(cxe)

—

~i

‘(cxi)]

~

1

(cx)

—

Z~i1(cxe)] } 1/2

(4.10)

describes the precession of the angular-momentum vector around axis 2 in the equilibrium configuration determined by eq. (4.8). The two frequencies differ in magnitude namely Q ~ ~2. The stationary rotation axis 2 is stable because the inequality 32

(cxc)

> 3i(~~e)> 33(cxe)

(4.11)

is fulfilled for all values of I. 2, I~= I~= 0, p = 0 corresponds to rotation around axis 1. The The stationary state Ifcx~increases = I equilibrium valence angle with increasing I according to eq. (4.8). So, inequality (4.11)

218

I.M. Pavlichenkot’, Bifurcations in quantum rotational spectra

is satisfied and the stationary axis 1 is unstable since it involves rotation around the axis with intermediate moment of inertia 3,. The molecule rotates around axis 3 in the stationary state If=If=0,

p=O.

I~=I2,

(4.12)

According to the supplementary condition (4.8), the equilibrium valence angle decreases with increasing I up to the value cx~ = arccos [m/(M + m)],

(4.13)

corresponding to the critical angular momentum I~=

—

23oU’(cx~)~~/TT~/(2 + i~),

for which the two principal moments of inertia become equal,

31(cxC)

(4.14)

= 33(cxC).

In the region 0
I{[3~’(cxe)

—

3i’(cx~)][3~’(cxe)

—

3~1(cxe)]}h/2

(4.15)

for the precession around axis 3. The last frequency vanishes in the critical point I~. For I > ‘C~ the stationary axis 3 becomes unstable and the new degenerate stationary state If

=

I2cos2~,

fl

I~= 0,

=

I2sin2~,

p

=

0

(4.16)

appears. The solution (4.16) satisfies the conditions (4.8) and (4.14). The stationary state is characterized by the two equivalent precessional axes placed in the molecular plane symmetrically with respect to axis 1 and formed with the latter the angle x~ cos2x

=

[m/2(M + m)](l

—

I~/I2),

which is obtained from eq. (4.8). On the other hand, the condition (4.14) determines the equilibrium valence angle cx~= cxC. The new ro-vibrational regime is characterized by the oscillatory frequency Q(cx~)(4.9), and the precessional one, w

=

[41(1 + ~i)(2 + jt)/3~Q(cx~)] \/I2

—

I~2,

(4.17)

the inequality U) ~ Q being fulfilled. Thus, the rotational dynamics of triatomic XY 2 changes at the critical value I~.The bifurcation is the consequence of the competition between the centrifugal force and the mutual repulsion of two

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

219

X atoms. As a result the precession around axis 3 becomes unstable and the delocalized precession around two equivalent axes arises in the quantum case. It is the orientation of these axes in BFF rather than the equilibrium molecular geometry (the valence angle cxc) that varies with the angular momentum in the region I > I~.In the laboratory frame, the bifurcation results in the rotation of a molecule relative to the direction of the 1 vector by the angle x. Such rotational regime is preferable to minimize the energy of a system. The reorientation of a molecule relative to the I vector may strongly change the cross section of a molecular reaction with polarized or oriented reagents. The critical value of the angular momentum ‘C can be obtained if the potential-energy surface for a triatomic XY2 is known [84]. The potential energy U is a function of the relative nuclear coordinates R,, R2 (for pairs X—Y) and R3 (for pair X—X). This function must give a good representation of the molecular dynamics over a range of nuclear displacements about the equilibrium. We begin our consideration with the simplest approximation of U by pairing potentials, U(R,, R2, R3)

=

U,(R1) + U,(R2) + U2(R3),

(4.18)

where U, and U2 are the interaction potentials between X lations will be performed using the Morse potentials UK(RI

—

R10)

=

U~0{l

—

exp [

QK(RI

—

—

R10)] }2

—

—

Y and X

—

X, respectively. Calcu-

U~,

(4.19)

where R~0are the equilibrium interatomic distances. The parameters U10 and U20 are determined by using the energies of the reactions: XY2 = XY + Y (6”, = U,0 + U20) and XY = X + Y (~2 = U,0). The parameters QK are connected with the force constant of the harmonic oscillations, if the nuclei undergo small amplitude motion. The force constant represents the derivatives of the potential at the equilibrium geometry of a molecule and are expressed in terms of the fundamental frequencies v,, v2 and v3. The final result can be written in the form 2(~cxo)]}”2, Qi

=

Q2 =

6.512

x l0~ {mi’~/U,0[l + ,usin

4.605x

2(~cx

(4.20)

2

\U 10 3(mvfv~[1 + ,usin 0)]~ 20(l + ,u)v3cos (~cx0)J

where m is in amu, UKO is in kcal/mol, vK is in cm’, and Q in A1. Using the above determined parameters we can find the critical value of the angular momentum by minimizing the effective potential energy of the rotating molecule in the symmetrical configuration (R1 = R2) 2/23 Ueff(Rl,

R3)

=

U(R,, R,, R3) + I

(4.21)

3(R,, R3),

with the supplementary condition 31(R,, R3)

=

33(R1, R3). The final expression has the form

2)(l + /t)Q I~= (m/h

2R~~ U20 {exp[Q2(R30

—

R3~] 1}exp[Q2(R30 —

—

R3C)],

(4.22)

220

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

where the equilibrium distance Q,U,o{l

—

exp[

—

1~1~ is determined

R

3C

=

from the equation

2R,C/\/~

Q,(R 1~ Ro)]}exp[ —.Q1(R,~ R0)] —

=

—

Q2U2o.~/~~{exp[Q2(R3o R3~)] 1}exp[Q2(R30 —

—

—

(4.23)

R3C)].

The thus determined critical values are presented in table 4.1 for hydride molecules. For a H2 160 molecule, another value of I~= 28 is obtained by using a more realistic potential-energy surface U of Murrell [85] in eq. (4.21). The latter may be considered as the testing of the pairing potential approximation in the determination of I,~.It is essential that the rotational energy for all the values of ‘C~listed in table 4.1, is still much smaller than that of the first electronic excitation. The last statement does not seem true for the heavier fluoride molecules shown in table 4.2. What changes should we see in rotational bands of triatomics molecules? Axis 3 corresponds to the smallest moment of inertia. So four-fold clusters should appear in the upper parts of the multiplets, and the lower parts have to remain unchanged. Let us analyze at first experimental data concerning the ground-state rotational bands of the H20 [75, 76], HDO [86, 87], D20 [88, 89], and H2S [36], HDS [90], D25 [90] molecules. We will be interested in three extreme doublets in the upper (with quantum numbers K,, = I, I 1, I 2) and lower (KC = I, I 1, 1 2) parts of the multiplets. These states correspond to precession of the angular-momentum vector around axes 2 and 3, respectively. It is convenient to subtract the regular parts E2 (I) or E3 (I), being equal to the energy of rotation around these axes, from the multiplet level energies. These values contain the main dependence of the rotational energy on the quantum number I. To find E2(I) and E3(I) we use the harmonic approximation of the effective Hamiltonian (4.3) with rotational constants specified in the above cited experimental works for the considered bands. Using the boson representation (A.4) for the angular momentum operators and eqs. (A.6) and (A.10), one can obtain the energies of the lower or upper multiplet levels ‘C

—

E~(I)= E,,(I) + w,,(I)(n + ~), n

=

v

—

—

1

=

0, 1,2,

—

...

,

—

(4.24)

where cx = 2, 3 is the number of the precession axis of the BFF and w,,(I) is the corresponding precession frequency. The thus obtained energy differences divided by the precessional frequencies ~2 (4.10) and (03 (4.15) for the equilibrium valence angle cx0 are plotted in fig. 4.2 as functions of I for water molecules. As seen from these graphs, the precession around axis 2 is stable for all the molecules. A tendency toward level clustering in the upper parts of the multiplets is observed in symmetric molecules only. An asymmetric HDO molecule has unequal principal moments of inertia, 3, and 3,, for any values of I. Therefore, the bifurcation is absent from this molecule. The regularities observed in water molecules exist also in hydrogen-sulfide molecules and in the v2 bands of both molecules. The evolution of the rotational surface of the H2O molecule while going through the critical point is shown in fig. 4.3, taken from ref. [71]. The four local maxima appear on the surface at I > I~instead of two old local ones, which convert into saddle points. Accordingly, four new stable precessional axes appear. The harmonic approximation gives us another possibility to find the critical value IC using the zero point of frequency (03(1) in eq. (4.24). We have IC = 20 and 31 for the moderately floppy molecules H2S and D2S, respectively. Another example of multiplet level clustering has been found recently in the ground-state rotational band of a H2Se molecule [91]. The four-fold clusters were traced experimentally up to

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

221

evvvvv~~ 90000

tO

000

0000

0.5

0.0 -0.5

(a)

00

000

-

-i

5

10

15

I

20

~ A

2.0

1.0

A

(b~

~

~

10

15

20

25

I

Fig. 4.2. Energies of three extreme doublets in the upper (a) and lower (b) parts of rotational multiplets as a function ofthe quantum 160 (0), HD160 (~),and D 160 (V) molecules. The solid lines denote the harmonic number I for the ground bands of H approximation (4.24) for the effective Hamiltonians 2 from refs. [75, 86, 89]. 2

an angular momentum I = 20 and interpreted as the bifurcation occurring at I~= 11 (let us compare it with I~,= 9 in table 4.1). The strong deviation of the line strength from the rigid-rotor prediction has been observed with the effective Hamiltonian obtained from fitting the experimental transitions. This deviation agrees with the predicted I dependence of the B(E1)-value shown in fig. 3.2. The critical phenomenon in the rotational spectra of triatomic molecules raises a question about the possibility of reproducing it in ab initio ro-vibrational calculations. The latter involve the two steps. The first one lies in obtaining the potential-energy surface U by using high-resolution spectroscopic data on ro-vibrational transitions with low angular momentum. Then this surface, fitted to obtain a set of vibrational band origins, can be used to compute rotational states with high value of I. In both steps, a computationally very expensive procedure is used. Such calculations have made significant progress over the last decade, but have been largely limited to low I values (see for example refs. [92—94])because the size of computations increases rapidly with I. Only recently calculations have been performed for obtaining the states of H 20 with I 10 [95] and H2D + with I ~ 20 [96]. These calculations aim at solving problems different from bifurcation. Thus, the question whether the existing variational methods for calculating the highly excited rotational states of triatomic molecules could describe the bifurcation in the upper part of rotational multiplets, remains open.

222

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

‘
I>Ic

Fig. 4.3. Rotational-energy surfaces of a water-like molecule on the left and on the right ofthe critical point I. from ref. [71]. The solid lines present the typical trajectories of the end point of the vector I around local minima (m) and maxima (M). The dotted lines are separatrices and the dashed ones show the tunneling path. Numbers indicate the axes of a body-fixed frame.

The universal nature of the considered bifurcation (the “centrifugal governor” effect) makes one think it should be observed in other more complicated molecules, too. No doubt, this phenomenon is bound to exist in nuclear rotational spectra, since some nuclei become nonaxial for I > 40. However, it is difficult to observe the phenomenon in the upper part of rotational multiplets, i.e., much higher than the yrast band, where the density of levels with a given spin is very high. Another difficulty consists in the absence of clusters in nuclear rotational multiplets since all nuclear rotational states belong to the A1-type symmetry. We will consider this problem later in section 5. 4.2. Spherical top molecules with high symmetry The spherical top molecules are referred to as the ones with equal principal moments of inertia in the ground state. This equality is a consequence of a symmetry, namely the cubic symmetry group, to which all the spherical top molecules belong. They are the cubic XY8, tetrahedral XY4, and octahedral XY6 molecules. These molecules represent the most complicated molecular objects allowing ro-vibrational motion to the studied and described in detail. A large number of vibrational modes combined with the compact high I rotational multiplets make the spherical top molecules attractive for the investigation of nonlinear effects in ro-vibrational motion. On the other hand, the highly symmetric cubic group assumes the presence of different symmetry elements, such as symmetry axes and symmetry planes, which according to section 2 play a crucial role in the critical phenomena. In this section we outline the results of a bifurcation analysis of the rotational bands based on the ground and low-excited vibrational states for several tetrahedral molecules such as CH4, CD4, SiH4, GeH4, and SnH4 [7, 55, 97]. The equilibrium shape of tetrahedral molecules represents

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

223

a tetrahedron with the Y atoms placed at the top and X at the center. These molecules belong to the Td symmetry group, which has three C2 axes, four C3 axes passing through the tops and the centers of opposite faces, three S4 axes passing through the centers of opposite edges, and the six symmetry planes a passing through the C3 axes and edges. The irreducible representations of the Td-group and its characters are presented in table 4.3. There are two one-dimensional A, two-dimensional E, and two three-dimensional F representations of this group. The four fundamental frequencies of tetahedral molecules include: the symmetric nondegenerate stretching oscillation v1 (A1); the two-fold degenerate bending oscillation v2 (E); the three-fold degenerate oscillations v3(F2) (stretching) and v4(F2) (bending). The lowest vibrational levels of the above-mentioned tetrahydrides form well defined polyads because the bending frequencies are approximately twice as small as the stretching ones. Theby v2/v4 dyad in five molecules, havingand the 1,is characterized a comparatively weak degeneracy frequencies in the range 700—1500 cm~ large spacing from the other vibrational states. The v 1/v3 dyad of tetrahydrides with a heavy central atom (Si, Ge, Sn) is strongly degenerate. The splitting of the v1 and v3 states is only 2 cm~, the energy of the symmetric vibration being greater than the energy of the triply degenerated F2 state. The v1/v3 dyad of stannane SnH4 is situated in another environment; there is a nearest group of vibrational states namely the tetrad formed by bending modes. The structure of a polyad and its environment is essential for the understanding of the rotational states based on the polyad levels. Due to quasidegeneracy of the latter, the description of the rotational states in terms of isolated vibrational bands is possible for limited values of I only. Thus, the effective Hamiltonian, depending on the rotational coordinates and on the internal variables of the polyad under consideration, is used for the treatment of experimental data. It effectively takes into account the admixtures of high-lying states with the same I. It is convenient to use for highly symmetric molecules the effective Hamiltonian in terms of the direct product of vibrational V and rotational R irreducible tensor operators [98—101], —

eff

Q(N.flrj..’Q(N,r)

tkm

L’km

f2(N,F)1A1 X

j

where ~ ~ are the spectroscopic par~tmetersfitted to the experimental data. The sum in (4.25) is taken over all the repeated indices: k, m are the indices of the boson creation and annihilation operators; N is the rank of the R operator with respect to the three-dimensional rotation group; Q = N, N + 2, is the power of the angular-momentum operators; F is the type of irreducible representation. The tensor V contains the product of vibrational boson creation and annihilation operators and has nonzero matrix elements only within the considered block of vibrational states. These operators can be divided into diagonal and nondiagonal ones. The former appears in the effective Hamiltonians for an isolated vibrational state. The simplest operator V = 1 is used for ...

Table 4.3 Irreducible representations of the Td group and its characters Representation

E

3C2

8C3

A1 A2 E F1 F2

1 1 2 3 3

1 1 2 —1 —t

1 1 —1 0 0

6S4 1 —l 0 1 —1

6a 1 —1 0 —1 1

f.M. Pavlichenkot,, B(furcations in quantum rotational spectra

224

a description of the rotational band based on a nondegenerate isolated vibrational state. In this case the Hamiltonian (4.25) contains, like eq. (4.3), only the angular-momentum operators. The effective Hamiltonians describing the polyad of degenerate and quasidegenerate vibrational states include both diagonal and nondiagonal operators V. The physical reason for the appearance of nondiagonal opetators is the Coriolis force coupling the rotational and vibrational motions. The rotational tensor operator R is constructed analogously to the spherical tensor operator (2.4) from the projections I~of the total angular-momentum operator on the BFF axes*) taking into account the molecular symmetry group. Usually the basis in the chains of groups SO(3) ~ 0 is used for tetrahedral molecules. The tensors R are specified by the total power Q of 12 operators and the rank N with respect to the SO(3) group. For example the simplest four-rank tensor operator ~

(4.26)

has been introducing by Hecht [98] for the description of the anisotropic distortion effects in the ground-state band of spherical-top molecules. The spectrum of this operator was studied in several works [18, 21, 23]. There are two systems of localized states in the rotational multiplets with high I values. They correspond to the C4 and C3 stationary axes. The quantum delocalized precession around these axes results in the system of six-fold and eight-fold clusters, respectively. The six unstable stationary axes C2 are associated with a group of levels in the transition region between localized states. The effective Hamiltonian (4.25) is obtained from the Wilson—Howard Hamiltonian (2.2) by using the perturbation theory based on the Born—Oppenheimer small parameters K = M h/4, where M is the average nuclear mass in amu. Accordingly the terms in (4.25) can be classified by utilizing 2thisThis parameter. Thus, scheme the arbitrary termThere in themay sumexist (4.25)significant has the order to magnitude classification is crude. deviations from it, K~~~m caused in particular by the quasidegeneracy of vibrational states. As a result, different approaches with a different basis of tensor operators, different notation of the spectroscopic parameters, and different terms in the expansion (4.25) are used. Another problem consists in the ambiguity of the effective Hamiltonian resulting in correlations between the spectroscopic parameters t. This problem is analogous to the ambiguity of the simpler Hamiltonian (3.2) considered in subsection 3.1. Such an ambiguity analyzed by Watson [34] exists because the power series in operators Ir. is invariant under unitary transformations in the space of these operators. In the last decade a detailed study of the ambiguities in the effective rotational Hamiltonian for degenerate and quasidegenerate vibrational states was performed in a series of papers [102—107].The determination of correlations between the spectroscopic parameters sometimes simplifies the fitting procedure, that is essential for high-order terms in the series (5.25). The Hamiltonian (4.25) is used as a starting point in the bifurcational analysis of the rotational spectra in tetrahedral molecules. 4.2.1. Ground-state band of tetrahedral molecules The nondegenerate ground-state band of the considered molecules is described by the simple Hamiltonian ~

=

B12 + t~4,A1)R4(4~1) + t~61)R6t6,A1).

*)The S 4 symmetry axes are used as the BFF axes for tetrahedral molecules.

(4.27)

f.M. Pavlichenkov, B~furca1ionsin quantum rotational spectra

225

The first term represents the unperturbed rotational energy of the rigid spherical top and the subsequent ones are responsible for the centrifugal distortion effects. As I increased, the contribution of the R6~6’Ai) tensor might be significant. As a consequence a nonlocal C2 bifurcation arises, which transforms the six unstable equivalent rotational axes into stable ones and twelve-fold clusters are formed in rotational multiplets. This phenomenon was been discovered by Harter and Patterson [27]. Unfortunately, for realistic parameters of the Hamiltonian (4.27) the critical value I~is too high to be observed. A more sophisticated molecular system has been suggested in ref. [7] for the critical phenomenon to be observed in the ground-state band of a tetrahedral molecule. Let us assume that tetrahedral symmetry is slightly broken by an isotopic substitution of one or two Y atoms in the XY4 molecule. Two isotopic species, XY~Y2 and XY*Y3, of this molecule are considered below. Let us assume that the following simple Hamiltonian is sufficient for the description of the rotational structure of the isolated vibrational state in these molecules 2 + Hasph + tR4(4~1). (4.28) Heff = B! The first term represents the rotational energy of the rigid spherical top. The second one describes deviations from the rigid spherical top due to isotopic substitution. Its form depends on an isotopic species. The third term takes into account the centrifugal distortion effects in the spherical molecule. The contribution of the lower-symmetry fourth-rank tensor operators is small due to the small asphericity of the molecule. The first two terms in eq. (4.28) have the form of an asymmetric top Hamiltonian B!2 +

Hasph

=

~ (A + C)12 + ~(A

—

C)(I~+ kI~ Ii),

(4.29)

—

for an XY~Y 2molecule. Its principal axes slightly differ from the ones for an undistorted tetrahedral XY4 molecule. Taking into account this difference one can obtain after straightforward calculations the rotational constants 2) (1 + Am/4m

—

Am2/2mM)

A

=

(3/16mr

B

=

(3/16mr2)(1 + i~m/2m)~,

C

=

(3/16mr2)(1 + 3Am/4m

—

1,

(4.30)

Am2/2mM)~,

where m and m* are the masses of the Y and ~* atoms (Am = m* m), M is the total mass of the molecule, and r is the distance between the X and Y or X and ~* atoms. The asphericity parameter —

A—C Am/ Am aAC4~l+22M)

Am2~’

(4.31)

,

and the asymmetry parameter

k=

2B

A C A—C —

—

=

Am ~

I

8m

4Am

4Am2\ / Am\ 1 M2 )~~,1+~—) ,

(4.32)

226

f.M. Pavlichenkov. B~furcationsin quantum rotational spectra

are proportional to the quantity Am/m supposed to be small. Therefore, the term [~k(A C)]I~in eq. (4.29) can be omitted as it is of second order in the small quantity. Finally the effective Hamiltonian for the ground-state band of the XY~Y2molecule has the form —

2 + ~(A

—

C)(T

Heff = ~(A + C)!

2,2 + T2,

-2

+ 4t~/~(T4,4+ T4,4 + ~/~T4o),

(4.33)

where the spherical tensor operators are determined by T2,±2=~I~, T2,0=(1/..J~)(3fl_!2),

T4~4=~I~’

T4,0 = (1/~/~ö)(35I~3012I~+ 3J4 + 25I~ —

—

(4.34)

6!2).

The Hamiltonian (4.33) is invariant with respect to the D2 group. We use the boson representation (A.4) of the angular-momentum operators and the harmonic approximation (A. 10) for analyzing the multiplet cluster-structure changes with the increase of I. It is convenient to use the parameter =

~

[16t/(A

—

C)] (I

—

1)(2I

—

3),

(4.35)

reflecting the structure of the Hamiltonian (4.33) with different rank tensors. The harmonic approximation yields accurate energies for multiplet levels with a nearly maximal projection 1K I on the stationary precession axis. 5)have the same rotational structure as an The rotational multiplets I (small ~ I results in new stable rotation axes and in asymmetric top. The increase with in thesmall quantum number a change of the stability of the old axes. Only axis 1 corresponding to the minimal moment of inertia remains unchanged. The upper multiplet levels (inversion doublets) are described in the harmonic approximation by ElK

~ (A + C)I(I + 1) + (A

=

~

—

C)(2I

—

1) [~ (I + 5)(5 + 5)

~ ~1,2

=

~ {3 ±...J2I/(2I

(4.36) —

1)}.

(4.37)

The intermediate axis 2, which is unstable for small I becomes stable for high values of I. Consequently the two-fold clusters with energy ElK

=~(A+ C)I(I + 1) + ~(A

—

C)(21

—

1)[~(I + 5)ö

—

~/c52

—

5~(I IKI + ~)], (4.38) —

appear in the middle for the multiplets. The critical value I~is determined from the equation =

~J2I~/(2I~ — 1).

(4.39)

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

227

The energies of the levels corresponding to axis 3 are ElK

=

1(A + C)I(I + 1) + ~(A ±~/(5~

—

~)(~2

—

~)(1

—

—

C)(21

—

1)[

—

~(I + 5)(5

—

5) + ~

IKI + i)],

(4.40)

Examination of the precession frequency in eq. (4.40) shows that the stability of this axis changes at the angular momenta corresponding to the values ~ = ~ and t5 = ö2. In the region 0 < 5 <5k, axis 3 is stable and the energy of the levels in the lower parts of the multiplets is described by eq. (4.40) with the upper sign. Then for ö~<~ <~2 this axis is unstable. It becomes stable again for ~ > ~2, the lower sign being used in eq. (4.40). Figure 4.4 shows the bifurcation diagram of the Hamiltonian (4.33), i.e., the dependence of the energy of the stationary axis, E, on the parameter ö (4.35). The energies of states with the maximal projection 1K I = I on the stable stationary axes coincide with the solid lines in fig. 4.4. These states form extreme clusters. The clusters with smaller 1K I are positioned below (axes 1, 2 for ~ > 5~axis 3 for ~ > ~) or above (axis 3 for 0 < 5 <5k; axis V2 for b1 <ö < 3; axis C3 for ö > 3) the extreme ones. The two equivalent stable precessional axes V2 placed in the (23)-plane arise at point A. This bifurcation is analogous to that considered above for triatomic molecules. It results in the appearance of four-fold clusters. For higher values of I, at point B, another C2~-typebifurcation converts the two V2 axes into four C3 axes and accordingly the four-fold clusters into eight-fold ones. The dashed lines in fig. 4.4 correspond to the energy of rotation around the unstable axes. The states with energy close to that of these lines form the transition region between the clusters of different types. Thus, six types of stable rotation axes may exist in the ground-state band of a XY~Y2molecule for different quantum numbers I. Accordingly, six types of clusters are present in the rotational multiplets of this band; the general tendency that arises is that clusters with higher degeneracy appear as the quantum number I increases.

ENERGY

vz

-l------~--=-------.--------

0

2

3

4

S

Fig. 4.4. Relative positions ofthe extreme clusters (solid lines) and the transition regions (dashed lines) in the ground-state band of an — 1(A + C) 1(1 + 1)]/(A — C) 1(21 + 1). Indices denote the axis of rotation (see text).

XY~Y2molecule. The rotational energy E is given as 4[E

1.M. Pal’Iichenkov, Bifurcations in quantum rotational spectra

228

It is convenient for the XY*Y3 molecule to choose the orientation of the molecular frame so that axis 3 coincides with axis C3. As a result, we obtain the following effective Hamiltonian 2 + (C A)I~+ tR4(4~1), Heff = A!

(4.41)

—

with the rotational constants A=B= 1632(1

2~

~

(4.42)

C= 16mr

The R tensor in the new coordinate system has the form R4(4,A1)

=

16,,/~4(T 43

—

T4,

±~=

T4, -~+ \/~ T40),

(4.43)

T (1/~/~)I~ (213 ±3).

(4.44)

The Hamiltonian (4.41) is invariant with respect to the C3~-group.A bifurcational analysis similar to that described above shows a C3~-typecritical phenomenon exists for the Hamiltonian (4.41). Let us estimate the critical moment I~,of some molecules using the equality r5 = 1. For example a CC14 molecule has the rotational constant B ~ 5.7 x 10-2 cm The contribution of the 37C1 35C1 aspherical terms for C 2 2 is two orders of magnitude smaller since ~(A C) ~ 8.1 x iO~cm j. A crude estimation of the spectroscopic constant gives t ~ iO~cm 1 for the ground state of CCLI.. Thus, one can obtain I~= 120. This value is too high to be observed in an experiment. Another molecule Os 1802160 may be more suitable from the point of view of experimental investigation but there are no highly degenerate clusters of rotational levels in this molecule because the nuclei 160 and 180 have zero nuclear spin. To realize the above considered idea in the experimental observation of bifurcations in the ground-state band of tetrahedral-like molecules other molecules need to be found. On the other hand the spectroscopic constant t has to be considerably larger for vibrational states due to the quasidegeneraty of these states as described above. Now we will consider this problem. ~.

—

4.2.2. B~furcationsin the rotational structure of the v2/v4 dyad of tetrahydrides The v2/v4 dyad of bending modes in tetrahedral molecules is an example of the most widely studied vibrational polyads. The separation between the v2 and v4 vibrational levels in CH4, CD4, SiH4, GeH4, and SnH4 is about 200 cm Thus, the description of the rotational structure based on these states in terms of isolated vibrational bands is possible for I < 10 only. For higher I values the utilization of the effective Hamiltonian (4.25) with nondiagonal operators V is necessary for the treatment of experimental data. Such models are successfully used now for the analysis of the v2/v4 bands in tetrahydrides. This rotational structure includes five well separated branches corresponding to five vibrational levels of the dyad. The v2(E) band consists of two branches: upper E - and lower E ~ which are formed due to the Coriolis interaction. For low I valus v4(E) show a three-branch structure: F~,F°,F caused by diagonal Coriolis interaction. For higher I values this interaction results in the redistribution of the energy levels between different branches: a six-fold cluster passes from the lower E~branch to the upper F~one, an eight-fold cluster passes from the upper F + branch to the middle F°one, and a six-fold cluster from ~.

229

f.M. Pavlichenkov, Bjfurcations in quantum rotational spectra

F°to F. The redistribution of levels can be easily understood by introducing the approximate quantum number of the vibrational angular momentum 1 = 2. For high I, when the Coriolis force exceeds the frequency difference v2 v1, 1 becomes a good quantum number. Then we can —

introduce the rotational quantum number R(1 = R + 1), which takes values I 2 for the E branch, I 1 for E~,I for F~,I + 1 for F°,and I + 2 for F. Accordingly the v2/v4 rotational structure is split into five rotational multiplets with 2R + 1 levels in each. This means the transition to a new coupling scheme of the rotational and vibrational angular momenta. This phenomenon was shown to be related in the classical limit to the formation of the conical intersection points 28SiHof different rotational-energy surfaces [31, 54, 58]. The example of the v2/v4 dyad in a 4 molecule [55] (see fig. 4.5) shows clearly this transition. In a CD4 molecule it takes place for higher I due to the large value of v2 v4. The regrouping of levels in tetrahidrides ends for I > 20. For smaller values of I, in the transition region, a modification of the cluster structure is likely to occur in the considered molecules. The bifurcation analysis of the v2/v4 rotational structures was performed in ref. [97]. The analysis is based on the effective Hamiltonian (4.25) with spectroscopic obtained by 2CH parameters28SiH fitting the experimental data associated with these structures in ‘ 4 [108] and 4 [109]. The thus obtained Hamiltonian is written as the matrix (2.47) whose elements depend on the operators I~.The Harter method [25, 31] is used for the calculation of the five-fold rotationalenergy surface E(9, tp). When this surface is found, the method of ref. [54] is applied for obtaining stationary axes (they coincide with the symmetry axes C2, C3 and C4 in tetrahedral molecules) and their stability. The last problem is reduced to the investigation of the fixed points on each —

—

—

28

....~

SIHA

R-I-2

_~!-

E

—

~

—=

992 ——

a E

R=i-l

~

~

_—~~=

(I)

RI

—

F-_ 892

R=I.t

—_:_::_~=E~::!: —

—

~...

——

——

R~I~2 =

5 10 15 20 ANGULAR MOMENTUM QUANTUM NUMBER I

28SiH

Fig. 4.5. Ro-vibrational energy levels for the dyad v2/v4 of B0 1(1 + 1) + .... R is the rotational quantum number.

4 from ref. [55]. The energy is given without a scalar term

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

230

eigensurface. Their stability is characterized by the Hessian (~2E \~2 ~2E~2E ~

(4.45)

.

When taken at a fixed point it depends on the quantum number I only. The Hesian is the determinant of the matrix of second derivatives. So it is equal to the product of its eigenvalues, which define the stability of a fixed point. The Hesian is positive for the maxima and minima and negative for the saddle point. The critical point corresponds to the zero of the Hessian. Its absolute value can be used as a measure of stability. Figure 4.6 shows the I dependence of the Hessian for the C 2 stationary axes of the v2 band in 28SiH 4. It is seen that the C2 axis for the lower E~branch becomes a stable axis at I = 11. So, the twelve-fold form at I > 11. The experimental values of the twelve-fold cluster energies for 2CH clusters 28SiH the ‘ 4 and 4 molecules are listed in ref. [55]. In ref. [54], a simple model of the v2 rotational structure was considered with the effective Hamiltonian including only two tnesor operators 2(2E)]A1 + t[(a~ a 3(3.k2)]A1, (4.46) H = [(at a2)ER 2)A2R W 2000

Energy,

.

1030

/

1000-

E7

/ 28S i 200

,//‘

~

100 -

-

/ ‘s....S.-

—50

Solo

/‘

/

50

0

1020

/ /

-

/

1\ i’~

__~‘

—

!

,/~

-

1’

1000

,/

\

,~

990

980

-1000

-2000

.

~::~—— ~ 2

4

8

B

10

12

14

16

I

I

1.8

20

Fig. 4.6. Hessian values (upper curves) for the stationay C 28SiH 2 rotation axes of the rotational energy surface corresponding to the lower (E ~)and upper (E ~)branches of the v2 band in 4 from ref. [55]. The lower curves are the energies ofthe fixed points with C2 local symmetry on the energy surface. The energy scale is the same as in fig. 4.5. The stable (unstable) axes are designated by solid (dashed) lines. The nonlinear scale is used for representing the Hessian behavior.

f.M. Pavlichenkov, Bjfurcations in quantum rotational spectra

231

where a~and a2 are the boson creation and annihilation operators of the v2 vibrational mode. The bifurcation analysis shows that the nonlocal C2-type critical phenomenon for the Hamiltonian (4.46) is identical with that for a more complicated effective Hamiltonian (4.25). For the CH4 and SiH4 molecules the bifurcations thus predicted in ref. [54] take place at I~ 10 20. As follows from fig. 4.6, the structural changes in the upper branch E have a different pattern. They are related to the changes of the C3 and C4 stationary axes. Let us consider these 120SnH changes for a 4 molecule, as an example. The rotational structure of the v2 state of this molecule is similar to that shown in fig. 4.5. Two branches are clearly seen in this structure: the upper E branch with quantum number R = I 2 and the lower E~one with R = I 1. The redistribution of the six-fold clusters between the E + branch and the v4 state takes place in this molecule at higher I values. Figure 4.7 shows the results of the bifurcational analysis of the v2(E) subband based on the experimental data obtained by Oshima et al. [110]. The energies of the stationary C2, C3 and C4 axes and the corresponding Hessian values are plotted as functions of I. At low I values the rotational multiplets of the E + and E — branches exhibit a cluster structure similar to that of the ground-state band; there are two stable rotation axes, C3 and C4, and an unstable axis, C2. Yet in the rotational multiplets one can see only the six-fold clusters because at low I-values the number of quantum states in a multiplet is too small to produce all kinds of clusters. —

-

—

—

1 Energy c~n

/7 // /I

/ /

60-

/

,~‘

/,‘ /,‘

~==~j// /‘11/~~~~ \

-150•

~

-200-

—250~

I

2

I

I

4

Fig. 4.7. Bifurcational analysis of the lower v

I

I

~

I

I ~

I

10

I 12

I

I

4

14

I 18

I 18

I

20

J

20SnH 2(E) subband of ‘ 4 from ref. [97]. The lower curves are the energies ofthe fixed points with C2 (•), C3 (ta) and C4 (0) local symmetry on the energy surface. The upper curves labeled in the same way are the Hessian values of these points. The remaining notations is the same as in fig. 4.6.

232

f.M. Pavlichenkov, B?furcations in quantum rotational spectra

As I increase the C2 rotation axis becomes stable beginning with 12 = 8 for the E branch and 4 branch. A nonlocal C with ‘2 = 10 for the E 2-type bifurcation takes palce at these momenta. As result, the saddle point corresponding to the C2 axis on the rotational-energy surface transforms into a maximum for the E + and into a minimum for the E subband. Simultaneously two new saddle points corresponding to the unstable axes C, arise. With these critical points, the evolution of the C2 stationary axes in the two subbands becomes completely different. The Hessian value for the C2 axis of the E + subband increases with I, and twelve-fold clusters appear in the upper part of this subband for I> 14 (cf. with the E + subband of S1H4 in fig. 4.6). The C~axes move toward the C3 axes and at 13 = 14 they coincide with the latter. At this momentum a C3~nonlocal bifurcation takes place. It transforms the local maximum into a minimum. The C4 stationary axis is characterized by high positive values of the Hessian and six-fold clusters are clearly seen in the lower part of 4) subband. theThe v2(Eupper v 20SnH 2(E) subband of ‘ 4 shows a clearly seen crossover phenomenon at I = 8 12 (see fig. 4.7). After the C3 bifurcation at 13 = 9, the C, axes continue to move toward the C4 ones and at 14 = 11 they coincide with the latter, causing a C4~nonlocal bifurcation. It corresponds to the transformation of the local maximum into the minimum. As a result, the six-fold C4 clusters are shifted in the lower part of the subband multiplets and the eight-fold C3 clusters in the upper. Consequently, the inversion of these clusters in the v2(E)subband is observed at I > 14. For higher I values, the axes C~return to the C2 axes and disappear in the C2,, nonlocal bifurcation point at “2 = 12. Thus, four successive bifurcations, -

—

C2,,

—~

C3,,

—~

C4,,

—+

C2,,,

(4.47)

take place in the considered subband of the v2/v4 rotational structure. In the crossover region the v2 (E ) energy surface is almost spherical and the rotational splitting of multiplets is anomalously small. The C2 minima on this surface exist for a very short interval of I values and the correspond20SnH ing rotational multiplets do not show the presence of twelve-fold clusters for ‘ 4. These 3CD clusters are much better pronounced for the v2(E) subband of ‘ 4 [111]. This molecule possesses the same type of crossover but it takes place at higher I values, and the bifurcation has a wider interval in I. The chain of bifurcations (4.47) was observed also for the v3(R ) subband of a CF4 molecule at I 22 [112, 113]. 4.2.3. B~furcationsin the rotational structure of the v1/v3 dyad of tetrahydrides A strongly degenerate v1/v3 dyad includes four vibrational states taking into account the degeneracy of the v3 vibration. The rotational structure based on these states undergoes considerable changes in a series of tetrahydrides SiH4, GeH4, SnH4 mostly because of the difference in the Coriolis interaction. The v3 vibration has angular momentum I = branches 1. Therefore v3 band 4. These havethe correspondpossesses the three branches usually denoted as F, F°,and F ingly the approximate rotational quantum number R = I 1, I, I + 1. The constant of the Coriolis interaction in the v 3 band has a small positive value for Sift1, and takes large negative values for GeH4 and SnH4. A high degree of accuracy is achieved in the treatment of the experimental data by including a resonance interaction between vi(A1) and v3(F2) into the effective Hamiltonian. 28S1H The spectroscopic parameters of the v1/v3 dyad in 4, obtained from the experimental data in ref. [97] made it possible to follow the rearrangement of a rotational structure. The interaction of the i’~ state with the v3 state results in small multiplet level splitting of the former. One succeeds in distinguishing the F°,F~branches in the v3 subband at 1> 5 only. Then the rotational structure —

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

233

changes drastically as I increases. Such a strong distortion can be explained, according to ref. [114], by an anomalously low value of the Coriolis constant. The investigation of the four rotational energy surfaces of the dyad shows that the v1 and F surfaces are only slightly deformed. 4 have deep maxima and minima. At I ~ 7 they form a conical intersection The surfaces point lying onF°and the C F 3 symmetry axis. The lowest part of the F°-branch corresponds to the region of six-fold clusters associated with rotation around the C4 axis. For large I values, eight-fold clusters form in the upper part of the F + branch. There are no bifurcations in the investigated part of the v3 28SiH subband. An experimental study of this rotational structure in a 4 molecule for higher I values is needed to obtain information in order to investigate critical phenomena further. 20SnH As is mentioned above, the v1/v3 dyad of a ‘ 4 molecule possesses some peculiarities, which are absent from lighter tetrahydrides. Its rotational-structure analysis based on the experimental study of Krivtsun et al. [115] was performed in ref. [97]. A set of spectroscopic parameters found from the experimental data reproduce the levels multiplets up interaction. to I = 16. The band hasofthree 4 split by a oflarge Coriolis Thev3 analysis the clearly visible branches F, F°,F rotational energy surface of the v 3 (F°)subband shows that the stationary points corresponding to the C4 axes have a deep minimum, and the Hessian in these points has a large positive value, as seen from fig. 4.8. Consequently six-fold clusters are clearly seen in the lower part of the F°branch. For low I values there are not localized states in the upper part of the F°multiplets because the C2 rotation axis is unstable, and the stable C3 axis has a Hessian with too small a positive value. As I, increases, the C2 axis becomes stable at I ~ 9. The corresponding maxima on the energy surface -

W

Energy,ce’

2000W

1000-

-

~ S 0

C3 C2 C4

.1907

500.

400-

-

200-

5040:1905

20-

10-1904 5-

2-

-

0-

_._,~

-1903 —2.,

I

I 4

I

I

~

I

I

I 10

I1

I

I

12

14

16

I 18

I

I

20SnH Fig. 4.8. Bifurcational analysis of the v3(F°)subband of ‘

4 from ref. [97]. See the captions to figs. 4.6. and 4.7.

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

234

grow with I (see

4.8). The C2,, nonlocal bifurcation is accomplished with the formation of the twelve-fold clusters in the upper part of the F°multiplets. The saddle points appearing near the C2 axes begin to move toward the C3 axes as I increases. At I ~ 11 they merge with the latter axes, and the C3~nonlocal bifurcation takes place. It results in the formation of a minimum in the C3 direction instead of a maximum. Thus, the critical phenomena are analogous to those in the v2 (E ~) subband of this molecule. The bifurcational analysis of the two upper branches of the v1/v3 structure involves rotational dynamics close to conical intersection points. The v1 subband is placed higher in energy than the v3(F) one for I <9. Both rotational surfaces posses minima and maxima in the C4 direction. Six-fold clusters are well pronounced in the upper parts of the F multiplets. As I increases, minima and maxima become close and at I ~ 9 the two energy surfaces form a two-sheet surface with six equivalent conical points in the C4 direction. The Hessian goes to infinity at these diabolic points. After the intersection point, the six-fold clusters pass from the lower surface to the upper one. The second intersection point of these surfaces is formed in the C3 direction at I ~ 14. After this point, the modified F - branch consists of 21 + 7 rotational levels (the rotational quantum number R = I + 3). This branch is characterized by a rich cluster structure, which includes twelve-fold clusters in the upper parts of the multiplets, and both six-fold and eight-fold ones in the lower parts. The v1 branch includes 21 5 rotational levels (R = I 3) after the intersection point. In contrast to the F - branch, it contains several delocalized quantum states with small positive Hessian values corresponding to the C2 and C4 stationary axes. Besides, the C4,, nonlocal bifurcation takes place at I = 13 and the C2 rotational axis becomes unstable again for I 25. Such an organization of bifurcations would complete the crossover phenomenon (4.47) in the v1 band similarly to that of v2(E) branch discussed above. We considered existing experimental facts concerning the bifurcations in rotational spectra of polyatomic molecules. A large amount of bifurcations was found after the phenomenological theory has been developed in ref. [10]. They are the C2,, local and nonlocal, and C3,, and C4,, nonlocal bifurcations according to the classification of Table 2.2. All the experimental facts have two different levels of reliability. Those observed in triatomic molecules were considered on the phenomenological and microscopic level. On the other hand, the bifurcations observed in tetrahedral molecules were treated only phenomenologically using the effective Hamiltonians, whose parameters were obtained by fitting the experimental data. Accordingly, the second part of this section has a descriptive and illustrative character. Its main purpose is to establish the existence of various types of bifurcations in experimentally observed rotational structures. As there remain many unanswered questions, the microscopic description of the observed phenomena is necessary. fig.

—

—

—

‘-~

5. Bifurcation in the rotational spectra of odd-A deformed nuclei The nonlinear effects in rotational spectra are produced both by centrifugal and Coriolis forces. The latter is responsible for the interaction between the rotational and quasiparticle degrees of freedom in nuclei. It manifests itself most markedly in rotational spectra of odd nuclei. The Coriolis force is proportional to the nucleon angular momentum j. So, nucleons (or quasiparticles) in the levels of the nuclear mean-field subshell with the maximalj near the Fermi surface interact with the rotation most strongly. These levels are distinguished from other states of the shell by their parity. Therefore, it follows that j is a good quantum number for the subshell states considered, since the admixture of states with other values of j corresponds to transitions to a neighboring shell. The

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

235

Coriolis force decouples the j vector from the nuclear axis of symmetry and tries to align it along the axis of rotation. As a result, there appears a decoupled band. Presently decoupled bands are observed in many odd-A nuclei. They have been studied most thoroughly in the rare-earth nuclei [116], in which an odd neutron or proton occupies the levels of subshells i1312 or h1112. In molecular physics, the Rydberg spectra of the two-atom molecules are similar to those observed in odd axially deformed nuclei. The spacing between Rydberg levels decreases with increasing electron energy. When this spacing becomes smaller than that between the rotational levels, the electron angular momentum aligns itself in the direction of the rotational momentum of nuclei [117]. It will be shown below that in rotational bands of odd axially deformed nuclei the quantum bifurcation accompanies the transition from the strong-coupling scheme to the decoupled band with increasing the nuclear angular-momentum quantum number I [118, 119]. 5.1. Effective Hamiltonian of the particle—rotor model From all the levels of an odd axially symmetric nucleus, let us separate the rotational band based on the odd nucleon state belonging to the anomalous-parity subshell with angular momentum j. The total spin of the nucleus is ! = R + j, where R is the rotational angular momentum of the spinless even core. Therefore, for a fixed I, there are several states forming the rotational multiplet. The considered band is composed of multiplets as in the case of a nonaxial system. We shall analyze an effective Hamiltonian in the space of multiplet states. In the BFF, it has the form (5.1) wherefm are arbitrary functions. The first termf0 stands for the rotational energy of the core, while the subsequent ones describe the rotation-dependent average axially symmetric field, in which the nucleon moves. Equation (5.1) is a most general Hamiltonian for the particle-rotor model [63] widely used in nuclear physics. This Hamiltonian is correct up to the first backbending only. For higher I values the three-quasiparticle configuration is essential. The direction of the symmetry axis of the core, n, in the laboratory frame is specified by the polar angles 6 and p. The BFF axes are chosen so that axis 3 is directed along the vector n. The thus defined body-fixed frame coincides with the spherical coordinate system. That is why the components of the total angular-momentum operator ! (D.8) in the BFF have unusual commutation rules (D.9). 2, the projection M of I on the The integrals ofaxis motion laboratory-frame z, and for R the Hamiltonian (5.1) are !2, j 3 13 —-f3 = 0. The last expression shows that there is no rotation around the symmetry axis. It means that the 13 andj3 operators have the same quantum number K. Thus, it is impossible to choose the representations of the I and j operators with entirely independent quantum numbers. That is why the components of! do not commute with those off. The commutation relations for these operators are2obtained in appendix is given by eq. (D.13).D [see eq. (D.10)]. The square of the core angular-momentum operator R The eigenfunctions of the Hamiltonian (5.1) have the form ~1IMv

=

>~alK,,DMK((p,O,0)IfK>,

(5.2)

236

f.M. Pavlichenkov, B~/iircationsin quantum rotational spectra

where IiK> are the eigenfunctions of the operatorsj2 and j3. The sum in eq. (5.2) is taken over the quantum number K from I to I for I j. The angular momentum quantum number R of the spinless core is restricted to even values only. Consequently, the wave function (5.2) is invariant under rotation [63], —

91

exp [

=

—

—

iic(12 —j2)]

(5.3)

and its coefficients should satisfy the condition al,K,,,

=

(

—

1)’~aJ,_K,,,

(5.4)

.

Thus, the function can be rewritten in the form 1(j) =

K= 1/2

afl(,. [D~K(q7,0, 0)11K> + (

—

1)I_~D~,.~(p,0,O)~j, K)’] —

.

(5.5)

The basis functions used in eq. (5.5) correspond to a strong-coupling scheme, for which the valence nucleon couples with the axially symmetric nuclear field. Condition (5.4) reduces the number of states in a multiplet down to j + ~ if I > j. It means also that states with a different parity of the number I j are described by a wave function with different symmetry properties. Only the lowest level of each multiplet is populated in the heavy-ion reactions used for excitation of high-spin rotational states in odd-A nuclei. These levels are usually split into two sequences: a favored band with even values of I I and an unfavored one with odd I j. The quantum-number signature, —

—

—

=

(

—

i)li

(5.6)

,

can be introduced for these sequences. 5.2. Classical picture of bifurcation In the classical approximation, the energy of a level in a multiplet is determined by two angles. Let us specify them as the polar angles 99 and 4 of thej vector in the BFF, as shown in fig. 5.1. The angle ~9between the vectors R and j is acute for the lower multiplet levels and obtuse for the upper ones. According to eq. (5.1) the rotational energy surface has the form E(8, 4) = m~o~m ([(J2

28)”2 —j cos

~2

99]2

sin

The fixed points of this surface corresponding to 0 [(12j2

A(99)

=

C(9) =

sin29)~2—jcos

99]2

A(99) +j(I2

=

~2

~2 sin29)”2 —jcos 9]2

mfm([(I2

.....~2sin299)’~2 —jcos

(5.7)

ir/2 are determined by the equation sin299)”2 C(~9)cos99] sin 99=0, (5.8)

X0~~~’2 9n0

j2) (j sin 99 sin 4,)2m

j2)(jsin

99)2m

(5.9) ,9]2

j2)(j sin

99)2m_2

f.M. Pavlichenkov. Bjfurcations in quantum rotational spectra

237

/~~~l3

Fig. 5.1. Angular-momentum coupling scheme in odd-A nuclei before backbending: the total angular momentum I, the core angular momentum R, and the odd-particle angular momentum j. The symmetry axis of the deformation is axis 3, whilethe axis of the collective rotation of the core is axis 1 of the body fixed frame.

wheref, is the derivative Offm with respect to R2. The fixed point ~9= 0 corresponds to an aligned configuration of the angular-momentum vectors

j(j, 0,0),

!(I, 0,0),

00

ir/2,

=

R2

=

R~= (I

J)2

,

(5.10)

with the minimal energy E

2).

(5.11)

0(I) =f0(R~,j The anti-aligned configuration

j( ~

1(1,0,0),

0,0),

R2

=

R~= (I +j)2

(5.12)

corresponds to the upper level of a multiplets. To investigate the energy surface near the aligned configuration, let us expand the function (5.7) into a series, assuming that the angle ~9is small, and rewrite this expansion in terms of the Cartesian coordinates x, y (2.17) near axis 1. One can obtain, after a little algebra, the result E(x, y)

=

2+a

E

a2o

2+a

02y

0(I) + a20x

4,

(5.13)

40y

=1 R~f~(R~,j2)/I, a

2)/I +j2f 2). 02 =j R~f~(R~, j 1(R~,j

(5.14)

If f~<0, the coefficient a

02 vanishes at the critical value of the total angular momentum,

I~=j(1

—

p

+

,/p2

—

2p),

p

=f1(I~—J)/2fb(I~—j).

(5.15)

The functions f’o and f1 are assumed to be smooth functions of the variable I near the critical point. Thus, we have a02 = tx(I I~)with ~x> 0 and eq. (5.13) coincides with the canonical form (2.24) of the C2,, type bifurcation. —

238

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

For I
(5.16) 2

j 0(jcos ~ 0, ±j sin

cot 6~= —130/Ito,

I9~),

R

=

R2 0

=

(I~~ .....J~)2

appears in the (13)-plane symmetrically with respect to axis 1. The stationary angle mined from eq. (5.8). The energy of the fixed points (5.16) is

7;’ (fl..... —

99~ is

deter-

V’ ((1)2 ~ 1330 ~ Jm~”0,f m0

We will refer to the state corresponding to this fixed point as a decoupled state since the vectors I and j are decoupled from the nuclear symmetry axis. It is not difficult to show that E’~(I~) E’~(I~) ~ 0. The phase diagram of the considered C 2,, -type critical phenomenon is presented in fig. 5.2. The parameter p (5.15) shows the competition between the two forces leading to the bifurcation: the Coriolis interaction, which aligns the angular momentum j of the valence nucleon along the rotation axis, and the interaction with the axially deformed mean field, which aligns j along the symmetry axis of a nucleus. As a result, the intermediate regime with decoupled angular momenta arises. The bifurcation is involved not only in the change of the precession motion, but also in the change of the space correlation of the angular momenta. This statement becomes more evident while studying the precessional motion of the three vectors !, R and j near the stationary states (5.10) and (5.16). Consider the classical equations of motion for_ji)2 the projections I andj on theofBFF 2 = (I~ + (12 ~j2) of the andvectors the Poisson bracket the axes. Using the Hamiltonian (5.1) with R —

I/J 0:

Fig. 5.2. Phase diagram of the C 2~-typebifurcation in the lower part of the multiplets in the rotational band of odd-A nuclei.

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

239

angular-momentum components calculated in appendix D, one finds =

2{

—

(I~—11)12 cot 0+ (1~_J2)

[(‘i

—Ji) cot 0

+~~]}

~2(Iiji)(Iicot0+j3)~f~(R2j2)j~m

=

=

=

=

~

2(j~12

—

1ij2) ~

—

2[(I~ —11)12 cot 0+ (12

—

2j2j3

m~1

12)f3] (5.18)

2 2)j~m2, mfm(R 1j

2(I~—ji)(Ji cot 0 +J3) ~

2,j2)j~m

+ 2j1j3

m~i

mfm(R2,j2)j~m2,

0~~

=

2(1~

32)

11—li 2

cit

—

sin0

~R2,j2)j~m, ~

R2

‘

m~otm(

~

)f3

This system has enough motion constants to be integrable. The general solution of the canonical form of the classical particle-rotor model has been obtained by Kamchatnov [120]. We will be interested in investigating the precessional solutions of the system (5.18) only. Its stationary states coincide with the stationary points of the energy surface (5.7). 5.2.1. Precessional motion near the aligned configuration Let us consider first the small-amplitude motion about the aligned state (5.10), in which the nuclear angular momentum is approximately aligned in the direction of the axis of rotation. The linearized set of equations (5.18) for the small quantities 12,12,13 =13 and c5 = 0 has the form —

12

=

Q(j

2), 3 + Jo),

12

=

Q(j3 +j6) + 2jj3f1(R~,j

j

2),

~=

3=2(jI2—Ij2)f~(R~,j Q

=

~E =

2), 2—j2)f~(R~,j

2),

0/~I

—2(I

(5.19)

2(1 —j )f’0 (R~,j

(5.20)

240

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

where Q is the core rotational frequency. Equations (5.19) describe the two normal vibrations. The vibration with frequency Q represents a zero energy mode, originating in the conservartion law {H, 13 —13 } = 0. In this mode, the vectors I and j are aligned along a straight line and oscillate in the (12)-plane normally to the symmetry axis n. A normal oscillation with frequency (5.21)

[Q2 + 4Ijf1(R~,j2)f’0(R~,j2)]1/2

=

remains upon separation of the zero mode. The corresponding precessional motion is described by i I2(t)

=

0,

32(t)

=

—

0w>cosw>t2)’ 2If~(R~,j

(5.22)

I 3(t) =j3(t)

=

i0 sin w> t,

0(t)

=

—

I3(t)/I,

where i0 is the precession amplitude. The time-dependent parts of the I~and Jj components are proportional to i~.The precession of the angular momenta is shown in fig. 5.3. The land R vectors oscillate about axis 1 in mutually perpendicular planes, while the j vector precesses around this axis. Therefore, the motion of the angular-momentum vectors is localized near axis 1. The last equation of system (5.18) allows one to find the time dependence of the azimuthal angle, = Qt, and so the motion of the symmetry axis n in the laboratory frame, —

n~=sin0cos q,~cosQt, =

cos 0 ~

—

(i0/I) sin w> t

n~=sin0sin p~ —sinQt, (5.23)

.

Thus, the symmetry axis rotates uniformly in the (xy)-plane with angular frequency Q, and simultaneously participates in the small-oscillation motion with frequency w> in a transverse direction.

I
1>lc

Fig. 5.3. Two precessional regimes of the angular-momentum vectors I, R and 3 of the particle—rotor model in a body-fixed frame.

f.M. Pavlichenkov. B~furcationsin quantum rotational spectra

241

5.2.2. Precession near the decoupled configuration The frequency w> decreases proportionally with (I I~)”2as I approaches the critical value. Accordingly, the oscillation amplitudes of the vectors I and j along axis 3 increase infinitely. The aligned state (5.10) becomes unstable for I
‘2

(2AI2/I~

=

0)(I~~ —j10)(j3

—130

+ I~~O),

2/I~o)j~o(I~o _j~~)O,j~= 2A(j 12

(2Z/j10I10)(j3

=

=

—130)

+ (2AI

1oI2 —11012),

(5.24)

2A(I2 —12) 2(1 4 Z = Aj 1o jio) + 2A’(j~0/I1o)(I1o Jto) —

+ 4C’jtoj~o(Ito—Ito)2

Cj~o(I

—

1o+jto) + 2Dj~0j~0I10

(5.25)

.

The coefficients A and C are determined by eq. (5.9) for the stationary value 99)~, while 2)j~2, D A’

C’

~

=

=

m>1

=

m2

m(m

—

mf~(R~,j (5.26)

The considered motion is described by the small quantities 13

I~—I~~ = —(j

f3Q~

3o/ilo)(j3

(13o/Ilo)(1313o),

1111o

—130),

(5.27)

Orzr0—0~.

The zero-energy mode of eqs. (5.24) has the frequency £2

2A(I

=

10 —j10)/sin 0~,

(5.28)

and corresponds to oscillatory motion of the nonparallel vectors I and j with time-independent projections on axes 1 and 3. The vibrational mode with frequency 2 + 2A’R~+ 4C’ R~I

{(4Aj~o/J10j10)[ C(110 +jio) is represented by the solutions =

1 1(t)

=

~

I2(t)

=

0,

—

i0(j30/110) sin co
I3(t) =j3(t) =j~o+ io sin w< t,

2, (5.29) 10j10+ 2DI~oj~0]”

—

io(j30/j10)sinco
j1(t) =Jio

—

j2(t) =

(i0w
—

0(t) = 0~ (i0/I10) sin w< t. —

(5.30)

242

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

It follows from these formulas that the vectors I and R oscillate in mutually perpendicular planes, respectively, near the stationary position l~and near axis 1. The vector j precesses around the stationary position Jo. Unlike in the region I> I~,the motion of these vectors is localized in the different space areas (see fig. 5.3). The stationary value 130 is small near the critical point I,. Therefore, one can find an analytical solution of eq. (5.8) for the quantity 99g. In this limit, the frequency w< (I~ 12)1/2 depends on five ~ andf2(I~)only. —

This feature, which is a consequence of the local property of the considered bifurcation, will be used in the next subsection. 5.3. Quantum b~furcation To obtain a quantum-mechanical description for the considered bifurcation, let us simplify the effective Hamiltonian (5.1) by recalling that the C25-type bifurcation occurs in a small part of phase space corresponding to the lowest multiplet states IIMv>, for which 2

—

R~ IMv>/12<<1

(5.31)

.


=

E

2

—

R~.)+ a

0(I) + a1(R —

a1 a2

(‘(1)2

~J0~”e,J

~

—

)‘

—

I (“(1)2

—

2J0I~1%c,) J’

2

—

R?)2 + b

2(R f

(1)2

~

~jl —J 1’.~’c,J ~,

~

(‘(1)2 •2~ —J ik”c,J ),

—

c1

f (1)2

—

c

—

R ~, we

R~,j~]++ c 1j~,

(5.32)

~ I,

~J2k”c,J

—

U2

2 1j~+ ~b2 [R

—

-

—

The terms that we have written out are sufficient to describe the bifurcation. This claim can be proved by the method described in appendix A. The regular term E0(I) is the energy (5.11) of the lowest multiplet state in the aligned part of a band for I>> 1. The remaining five parameters are determined by the structure of the nucleus as a whole, and can be obtained by using a microscopic theory. They have a definite physical meaning. The constants a1 and a2 are the rotational parameters in the expansion of the energy of an even-even core in terms of the square of the angular momentum. The parameters b1 and c1 define the level splitting in the j-subshell by quadrupole and hexadecapole deformation of the nucleus. Finally, the term proportional to the parameter b2 describes the interrelation of both effects, rotation and splitting. The five parameters of the Hamiltonian (5.32) are not independent because the Hamiltonian is invariant under the unitary transformation =

exp(is[j+ I... —j.I+,j3]+)

.

(5.34)

As was shown in section 3, the coefficient s can be chosen to eliminate the last or the next to last term in (5.32). The first choice is preferable for nuclei with a small hexadecapole deformation. To

f.M. Pavlichenkov. B(furcations in quantum rotational spectra

243

investigate the change in the level structure of multiplets and in the electromagnetic transitions in a band while going through the critical point, we will find the solution of the Schrodinger equation

HWIM,,

=

E1,,

1IMv

(5.35)

,

~‘

with the Hamiltonian H

E

=

2

—

R~)+ a

0(I) + a1(R

2 2(R

—

R~42+ b

2

—

R~,j~]÷ .

(5.36)

1j~+ ~b2 [R

At first, we use the harmonic approximation for the precessional motion near the stationary states (5.10) and (5.16). The general solution of eq. (5.35) for the lowest multiplet states is obtained in such an approximation in appendix B. Let us examine the above-mentioned changes as the angularmomentum quantum number I increases within the band in which the bifurcation occurs at the critical point I~.The lower multiplets of the band with I
~

.~

k

1~~

Fig. 5.4. Potential energy V ofthe Hamiltonian (5.36) as a function ofthe continuous variable k = K/.,./I(I + 1) (K is the projection of Ion axis 3) for different values ofthe quantum number I. The horizontal lines represent the two lowest eigenvalues of this Hamiltonian with parameters b1/a1 = — 1.0, a2/a1 = — 2.Ox l0~, b2/a1 = — 1.Ox i0~.The critical value I, = 18.,

244

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

correspond to this state. The k0 value is determined from the equation 2

{

x~/(1

k~)(p2—k~)+1

—

+2(4~_’3~~)k~]I(I+ 1)}

—

2k~ 2)k~+ 4k~]

2/a1) [/2(1 + + (b

2/a1) [

—(3 + 2it + 3p

~)2

2 + 3(1 +

~2)

k~ 4k~]}1(1 + 1) —

=

0,

(5.37)

2p

—

J~2/J(J+

=

+p2

{4(a

+

k2

+~-~.p

—2 +~-~+2[ —2~(1 +p)

1),

=i(i + 1)/I(I + 1)

~2

(5.38)

.

In the classical limit we have k 0 = p sin ~ and eq. (5.37) is reduced to eq. (5.8) in the considered approximation (5.31). Equation (5.37) has a solution under the condition 2 —(2— b 1 + p

1/a1)p <0,

(5.39)

which is fulfilled in the decoupled region. If the quantities a2/a1 and b2/a1 are small the solution of eq. (5.37) has the form (~2 —

=

p~)/(1 pt),

p~

—

=f(j + 1)/I~(I~ + 1)

(5.40)

.

This approximation is correct for well deformed nuclei with small nonadiabatic effects in the rotational band up to the first backbending. The energy of the lowest multiplet levels in this approximation is given by E1~= E0(I)

—

a1(I + ~)(j + ~)(I~

—

1)2 +

w<(n +

~), n

v

=

1

—

=

0,1,2,...,

(5.41)

where the precession frequency is =

[2a1/(I~ + ~)] [(Ia

—

I)(I~+ I + 1)(I~ j)(i~+1 +

In the other limiting case k~<<1,which is valid if(I

k~ 2p~(p ~ [i =

—

—

p~

+~

~

~: ((1

—

—

1)]1/2

(5.42)

.

I~)/I~<<1, eq. (5.37) has the solution

~)2

~ + p~

(IC + ~)2]

.

(5.43)

In this approximation the energy of the lowest levels of the multiplet is given by the expression E1~= E0(I) =

—

4a1[(I~

(a1 k~/2p~) (1 —

I)(IC

—

—J)(1C

p~)(I~ + ~)(I~ I) + w<(n + 1) , —

+1 + 1)/(2I~+

1)]t/2

.

(5.44) (5.45)

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

245

Formulas (5.41) and (5.44) refer to the motion in one well of the double-well potential V(k) shown in fig. 5.4. The corresponding eigenfunctions with a broken symmetry are determined by expression (5.2), in which the coefficients aIK,, in the harmonic approximation are given by eq. (B. 25) for the right-hand well. The coefficients for the left-hand well differ from eq. (B. 25) in the sign of k0. To proceed further, the symmetrized expression 2 [aJK~(k k —

ajne(k) = {2[1 + a(

—

1)~J~]}~

0) + a(

—

1)~aJKfl(k + k0)] ,

(5.46)

must be used for the a coefficients. The value J is an overlap integral for a(k0) and a( k0). Having calculated the expectation value of the Hamiltonian (5.36) with function (5.2) and coefficients (5.46), we obtain the signature-dependent term —

—

l)~1+1 exp

[—k~w
(5.47)

in the lowest level energy (5.41) or (5.44). Its origin lies in the possible tunneling of the angular momenta I and j across the potential barrier, which separates the two wells of the potential V(k). There is no level-clustering phenomenon in the considered band, since levels with different signature belong to adjacent multiplets. The bifurcation is revealed by mutual approaching of the curves Ene = + ~(I) and E5a ~(I), which symbolize the I dependence of the lowest levels of the multiplet with fixed signature a. We will refer to these functions as tracks according to ref. [64]. The gap between tracks with opposite signatures and the same n decreases with increasing I~, I proportionally with exp[ d(I~ 1)3/2], where d 1. The closeness of these tracks is a specific feature of the decoupled region. It manifests itself most markedly for tracks with n = 0. The states with n 1 are described less well by the harmonic approximation, since their energy is close to the top of the potential barrier. =

—

—

—

—

5.3.2. Aligned part of a band: I> I~, In this region, located above the bifurcation curve in fig. 5.2, the angular momentumj is aligned along the nuclear rotation axis. It is the so-called weak-coupling scheme. The energy of the lowest levels is determined by the formula E1~= E0(I) + w>(n + ~),

n

=

v

—

1

=

0,1,2,...,

(5.48)

where the precession frequency is 2 + I + I~ 2j)/(2I~+ —

=

2a1[(I

—

IC)(2IIC

—

(5.49)

1)]1~’2 .

2j

For large I and j this expression is reduced to the classical formula (5.21). The expansion coefficients of the wave function (5.2) in this region have the form aIK~=

[(1/Yn!) %,/2w/1v(2J + 1)]1’2 H~(K...J2w/(2I+ 1)) exp [

w

—

=

(1

—

wK2/(2I + 1)] ,

(5.50)

1/p)2 + b 1/a1p

(5.51)

.

The symmetry property (5.4) of these coefficients suggest that the parity of the quantum number n coincides with that of the number I j. Therefore, the lowest levels of the multiplet with positive —

246

f.M. Pavlichenkov. Bifurcations in quantum rotational spectra

2-

1-

-~ +

.... ~

.

...

.

~

IC w

~ -1 Lu

-2

.

-3. I 2

I 2

IIIII.~..

I

33 T

43 T

I

53 T

Fig. 5.5. Energies of the lowest levels of multiplets [in units of(21 + 1)a,] in the rotational band of an odd-A nucleus near the critical point I~= 18. The full lines (tracks) connect the states with positive (•) and negative (0) signature o~The dotted lines correspond to the harmonic approximation.The parameters ofthe Hamiltonian (5.36) are the same as in fig. 5.4. E,~is the regular part ofthe system energy (see text).

signature have n = 0, 2, 4,..., and those with negative, n = 1, 3, 5 So, the gap between the tracks E5,,, = + 1 (I) and E~,,= ~(I)is equal to w> in the considered approximation. The closeness of —

these tracks due to the bifurcation can be seen in fig. 5.5 where the results of the exact and approximate solutions of eq. (5.35) are presented. It should be noted that the precession motion (in the quantum and classical descriptions) in the considered region does not depend on the effects resulting from higher terms of the Hamiltonian (5.32) proportional to the parameters a2, b2 and c1. This circumstance makes it possible to compare the present results for the region I > I~,with those obtained in refs. [121, 122] for the canonical form of the particle-rotor model [63] with the Hamiltonian 2+b H = a1 R

1j~

(5.52)

.

Let us note that the components of the angular momentum operators I and j are assumed to commute in obtaining the equations of motion in these works.*) 5.4. Electromagnetic transitions and static moments in the lowest levels of multiplets The E2 and Ml transitions observed in low-lying bands of odd-A nuclei are very sensitive to the bifurcation considered above. The analytical formulas for the B(E2)- and B(M1)-values and for the static magnetic dipole and electric quadrupole moments are calculated in the harmonic approximation by using the quasiclassical method of appendix C. To compare the B-values for the *>Special method for the restoration of the conservation law [H, 13 obtain the proper result.

—33]

= 0 broken by this assumption is used in ref. [122] to

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

247

decoupled and aligned parts of a band we will write down the corresponding formulas in parallel first for a transition in the region I < I~and then in I > I,~.It is convenient to characterize each reduced probability by the transition frequency which is found from the time dependence of the electric and magnetic transition operators. In the classical limit this dependence is determined by the angles 0, ~ and eqs. (5.22) and (5.30). As in section 3 above, we will use the subband or track classification of electromagnetic transitions. A subband in an odd-A nucleus includes the states with definite signature: a favored band with a = + 1 and an unfavored one with a = 1. Each subband characterizes strong E2 transitions between its levels. These transitions, which do not change the oscillator quantum number n, have the form —

B(E2; I, v

—~

I

—

2, v)

=

(15Q~/128ir)(1

—

k~)2,

An

=

0,

1

<

I~

An=0,

I>I~,

(5.53)

where k 0 is the root of eq. (5.37). They are induced by the quadrupole-moment rotation around axis 1 with the frequencies 2Q (5.28) and 2Q (5.20). Let us note the small signature oscillations of B(E2) in the region I> I~. The E2 transitions between levels of adjacent subbands having differentsignatures correspond to the frequencies Q and £2 ±w>. They are described by the formulas B(E2; I, v —÷I

—

1, v)

=

(15Qo~/32ir)k~(1 ks),

An

—

32(2I±i)~+~~

=

0,

even or odd I, 1
An=+1,

evenJ,I>I~,

15Q~(2j + 1)a1 =

32x (21 + ~

n,

An

=

—

1,

odd I, I

>

(5.54)

‘C•

The B(E2)-value in the region I > ‘C is n/I times as small as that for I < ‘C~ There are no marked oscillations of this quantity, unlike the analogous transition (3.36) for the nonaxial system. We consider now E2 transitions between levels of adjacent subbands having the same signature. They are the transitions with frequency 2Q + w for Av = 1 —

B(E2; I, v —+1 An

=

—

1,

—

2, v

—

1)

=

l5QokoS0a1 n [U-_k~w<

even or odd I, I

or with frequency 2Q + w< for Av B(E2; I, v An

=

—+

+

I

—

=

<

I~,

+

1,

2, ~ + 1) = 15Q~k~soa1 (n + 1) [~a~~-~)

1, even or odd I, I

< IC’

(i

—

p.—

(~s/~P)kkO) +

112, (5.55)

(~—

~

—

i]

2

(5.56)

248

f.M. Pavlichenkov. Bifurcations in quantum rotational spectra

where the definition s0 = s(k0) is introduced. The B(E2)-values for these transitions with frequen2. ciesThe 2Q E2 ±2w> in the region I >=IC are1 of order (n/I) Q + w< between the levels of adjacent transitions with Av and frequency subbands having different signatures, are described by the formula —

B(E2;I,v—~I—1,v— 1)

=

An

l5Qgs 12w
[

=

—

p

1

‘\

—

2kg 12

—

—

~—(~s/~P)kko)

(1—

kg)h12j

1, even or odd I, I <1~,

while transitions with Av

=

(5.57)

+ 1 and frequency £2 + w< for Av

=

+ 1 have

B(E2;I,v—+J— 1,v+ 1) =

An

1’2w< l5Qgs0a1(n kg) 32~tw<+ 1)[kg(1s 0a1(21 + 1)

[

+ 1, even or odd I, I

=

(

p

‘\

1— 2kg 12

—

~

—

~

(~s/~/2)kko) + (1

—

kg) 112] (5.58)

< I~,

The 3.B(E2)-values of these transitions with frequencies ±3w> in the region I > with I,, areAvof= order1 Thus, the probabilities of E2 transitions between Q levels of adjacent subbands (n/I) and Av = 1 are different. Finally, the static quadrupole moment in a subband is determined by —

Qi,,

=

(W?

(1~L1~1/2

1,,.I/0(E2) ~P11,,)

=

—

=

—

3soa~[i + ~P_(as/ak)kkO] (2n + 1)}~ 1< I~, 21+3 {i — 3kg — [1Q 0/(2I + 3)] {1 [3(2] + 1)a1/(21 + 1)w>] (2n + 1)} I > IC .

—

(5.59)

This quantity has small signature oscillations in the region I > I~.The E2 transitions between the levels of more distant subbands with lAy I 2 have smaller B(E2) values and are not considered here. The magnetic Ml transitions between rotational states of odd-A nuclei contain rich information about their structure. The Ml operator has the form 2 pN[gR’~ + (g~ g~)j~] , (5.60) = (3/4x)” where PN is the nuclear magneton. The 9R is the collective g-factor, while —

g~=g,+(g,—g,)/(2j+l4l),

j=l±~

is the function of the nucleon orbital g, and spin g~factors.

(5.61)

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

249

The Ml transition occurs between states with different signatures. Thus, there exist interband transitions only. The B(M1) values for the transitions with frequencies £2 and £2 ±w> have the form B(M1; I,v—~I—l,v— l)=p?~(gJ—gR)2 x 1(1 + 1)

(~/i1117~ ~/~2

~

—

—

kg),

An

~ 3a 2(n+1)~ 1(2j+1) 8itw>(21 + 1) (I_J+~) 2a 1

—j

X 8irw>(21±l)(~

n,

—

0,

=

even or odd I, I

An= +1,

An

=

—

1,

<

‘CI

evenl,I>IC,

odd I, I> I~.

(5.62)

The bifurcation changes essentially the I dependence of the reduced probability of these transitions. It follows from the first of eqs. (5.62) that B(Ml) is a smooth function of I in the region I < ‘C~ For I> I~,we obtain from eqs. (5.20) and (5.49) ~ 2a1(I —j) =

(5.63)

if I is large. Therefore, according to the two last formulas of eq. (5.62), the Ml unfavored-favored transitions with frequency Q + w> are strongly suppressed compared with the favored-unfavored ones with frequency £2 w>. So, the specific zigzag I dependence of B(M1) in (5.62) should be observed above the critical point ‘C~ One can see the change of the I dependence while going through the critical point in fig. 5.6, which presents the results obtained by the exact diagonalization —

m

O~

I

13

I

I

I

I

I

23

7

I

I

I

I

I

I

I

I

I

I

I

I

I

I

~l 2

C

2

2

Fig. 5.6. Reduced probabilities of Ml transitions between the states offavored (•) and unfavoured (0) bands [inunits of~ (g~— g~ )2] in odd-A nucleus near the critical point 1. = 18. The parameters of the Hamiltonian (5.36) are the same as in fig. 5.4.

250

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

of the Hamiltonian (5.36). The almost vanishing B(M1)-values for the unfavored-favored transition can be seen in this figure. The obtained I dependence may be understood by using the classical picture of the precessional motion (5.22) in the region I > I~.According to eq. (5.60) Ml transitions are induced by the time-dependent nucleon angular momentum component J+1, which in the region I > has the form ‘C

j+1(t)

1w>)t

_____

=

1/2

[(1 +

—

(i

—

~)

(5.64)

e~~0>)t].

e

Thus the smallness of the oscillation amplitude with the frequency £2 + w> is the origin of the suppression of unfavored-favored Ml transitions. This phenomenon in the aligned part of the rotational band in odd-A nuclei was predicted by Hamomoto [123, 124]. The frequencies Q ±w> have Ml transitions between levels of adjacent subbands with Av = 1, —

B(Ml; I, v —*1

—

1, v

—

1)

= p~(g~

—

g~)2

1(1 + 1) (3s 0a1 n/8mw.<)

~

+

+ vP / 2 An

=

and with Av

—

=

—

0]

k2’

1) [kg(~Ti~

—

—

\/p2

—

kg) (1

—

f

(~s/~P)k=kO)

2—2kg 12 (1 1—2kg kg)”2 + (~2 p kg)1/2J —

—

1, even or odd I,

I

<

(5.65)

‘C

+ 1,

B(M1;I,v—~I—1,v+ l)=p~~(gJ—gR)2I(I+l)(3s 0a1(n+ l)/8irw<)

~{;; + ~P ~

An

=

—

+ 1) [kg(~JT~i~

1 0]

+

(11—2kg kg)112 —

-

—

Jp2

—

kg) (1

-

~- (~s/~P)kkO)

12

p2—2kg (~2 — kg)1/21

k2

1, even or odd 1,

I
The B(M1)-values of these transitions with frequencies Q ±3w> in the region I order (n/I)3.

(5.66) > IC

are of

f.M. Paulichenkov, Bifurcations in quantum rotational spectra

251

Finally, the static magnetic moment in a subband is determined by the formulas =

(4ir/3)112 (~~211,, I ~..é’0(M1)I

=pN{gR+(gj—gR)[ko+\/(l—ko)(p—ko)]}I,

‘
+-~i)(I—f) (2n+ l)]}~

~

‘>‘C~

(5.67) We have written down terms of order n/I in the second expression to show the small signature oscillations of the magnetic moment in the region I > A transfer from the smooth I dependence to the zigzag one for B(E2) and B(Ml) while going through the critical point is the general consequence of the bifurcation in the rotational spectra of the systems possessing only one totally symmetric state A1 (cf. subsection 3.1.2). There are two types of zigzag dependence. The first is a signature oscillation due to the dependence of the oscillator quantum number n on the signature in the aligned part of the band. It leads to a small (-.-~n/I)amplitude oscillations of B. The second [such as B(E2) in (3.35) or B(Ml) in (5.62)] depends not only on signature but also on the character of the precessional motion and has oscillation amplitude 1. Let us note that the precessional motion and the electromagnetic transition rates between adjacent subbands in the aligned part of the band are described by the lower terms (of second order in the angular-momentum operators) of the effective Hamiltonians. The corresponding formulas were first obtained by Marshalek [122]. ‘C~

‘~

5.5. Comparison with experimental data The harmonic approximation allows us to follow the qualitative changes, which accompany the bifurcation in the rotational spectra of odd-A nuclei. Yet the harmonic approximation is inapplicable to the transition region separating the decoupled and aligned parts of a band. This region resulting from the double-well potential with too low a potential barrier represents a distinctive feature of the quantum bifurcation, the greater the quantum number I, the smaller is the width of the transition region. To follow the modification of the rotational spectra and to compare the theory with the experiment an exact solution of the Schrödinger equation (5.35) is required. There are some difficulties in observing the bifurcation in the energy spectra of odd-A nuclei. Since the level-clustering phenomenon is absent for full symmetric A1 states, one may observe only the closeness of the tracks representing the I dependence of the favored and unfavored state energy. Yet the closeness of the tracks is disguised by their rapid change with I. To isolate the required effect, the regular part of the rotational energy containing the main dependence on I should be subtracted from the multiplet level energy. It is convenient to take this to be the energy E0(I) of the lowest multiplet level with a positive signature in the region ~ > ‘~.In this region, the difference E,,~(I) E0(I) increases proportionally with I according to eqs. (5.48), (5.49). Thus, the dimensionless quantity —

[Evg(I)

—

E0(I)]/a1(2I + 1)

(5.68)

is constant in the limit of large I. For I < ‘C~ this value increases with I according to eq. (B. 16) since V(k0), while negative, vanishes at the critical point. In this region, the bifurcation should lead to a specific dip of the quantity (5.68).

252

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

To illustrate the above-mentioned peculiarities let us discuss the results of the numerical diagonalization of the Hamiltonian (5.36) shown in fig. 5.5. The right-hand part of this graph presents a system of equidistant levels corresponding to the precession with the frequency w <. The dip in the left-hand part visualizes the double-well potential. The Hamiltonian (5.36) with parameters typical of well deformed nuclei of Dy, Er and Yb isotopes with quadrupole deformation /3 = 0.3 is used in numerical computations. The rotational parameters are a1 10 keV and a2 a1 l0~. The parameter b1 determines the level splitting due ‘~

to the mean quadrupole field of nucleus and subshell occupation. It is equal to b1

[3K/j(j + 1)] (u~ v~)

=

(5.69)

—

in the particle-rotor model for an isolated j subshell. In this formula, K —~ 2.0—2.5 MeV is the energy unit for a singlej shell [125], while the factor u~ v~,depending on the amplitudes u and v of the Bogolyubov transformation, determines the subshell occupation [126]. Using the above-quoted estimate, one can obtain for] = ~ the corresponding value of the bifurcational parameter (5.15) —

p =

b,/2a1

—~

5(u~ v~) —

(5.70)

.

Thus, the bifurcation for an isolated j-subshell exists only if the subshell is more than halffilled. The larger the absolute value of p, the larger is the critical momentum (see fig. 5.2) and the more noticeable are the bifurcation effects, i.e., the mutual approach of tracks and the dip formation. Let us follow the above-considered regularities in the rotational bands of four odd-neutron Yb isotopes, for which the experimental data were obtained in refs. [127—129].We shall analyze positive-parity levels in the I j angular-momentum region below the first backbending in the bands based on one-quasiparticle Nilsson orbits K~[Nn~A][130] in the unique-parity subshell i1312. According to eq. (5.68), the quantity ‘C

=

[E1ff(I)

—

E,,+1(j)

—

E0(I)]/(2I + l)A

(5.71)

is used, where Eie(I) stands for level energies of favored and unfavored bands, and 4 + (5.72) E0(I) = E1 + A(I _J)2 + B(I _J) is the energy of favored states in the aligned part of the band. The energy in eq. (5.71) is measured from the level with I = ~ for convenience of the graphical representation. Equation (5.72) is not suitable for the determination of the rotational constants A and B, since the angular momenta are not completely aligned in the bands under consideration. That is why we will use the expression E

2(I + 1)2 + , 0(I) = E1 + AI(I + 1) + BI which leads to the following formulas for the determination of these constants: ...

1

(12+71+13

(5.73)

12+31+3

A= 4(2J5)~ B = 8(21+ 5) ( AE(I)

=

21+3 —

AE(I)—

21+7

AE(I+2))1

1 21 ~ 3AE(I) + 21 + 7AE(I + 2))~

E1,+1(I + 2)— E,,+1(I)

,

(5.74)

(5.75)

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

253

Table 5.1 Rotational constants of aligned parts of positive parity bands in odd-A ytterbium isotopes Nucleus

Band head

I

A (key)

~[65l]

~

9.5 (14.8) 7.7 (10.9) 7.0 (10.5) 7.0~

B (eV)

63Yb

‘

93 67Yb ‘ 97 69Yb ‘ 99 71Yb ‘ 101

~[642]

~‘

~[633]

~

~[633]

—

—

1.3

( — 6.1) 0.06 (—0.7) 0.3 (— 0.7) 0.0 69Yb.

~ The value was not calculated but taken the same as that in ‘

where AE(I) is the level spacing. The thus determined values are presented in table 5.1, where the rotational constants obtained from eq. (5.73) are listed in brackets for comparison. As the neutron number N in the Yb isotopes increases, the levels in the i 1312 subshell with high values of the quantum number K become occupied. The Fermi level rises above the midpoint of the subshell and the parameter b, (5.69) becomes negative for N = 93—97. For greater N, thenuclei critical 69’171Yb as value increases and the decoupled region becomes more pronounced for the ‘ is shown in fig. 5.7. In the 171Yb nucleus the separation between the tracks is smaller that in 169Yb, though the band based on the same single particle state is considered in both nuclei. One can explain this effect by the increase of caused by the addition of two neutrons. Another factor acting in the same direction is the negative hexadecapole deformation ($~= 0 for 168Yb and 0.02 for ‘71Yb), which increases the splitting of the subshell levels. Note also the anomaly in the upper part of the 169Yb unfavored band (the experimental points are not connected by the solid line in fig. 5.7). Possibly this anomaly can be accounted for by the appearance of nonaxial quadrupole deformation. Let us apply the above-developed phenomenological theory of bifurcations to the description of the experimental data concerning level energies and electromagnetic transitions in the band based on the neutron state ~ [624] from the i 1312 subshell in 161 Dy [131]. The problem consists in the choice of four parameters of the effective Hamiltonian (5.32) for the best description of the signature splitting of the level energies, and B(Ml)- and B(E2)-values. The hexadecapole deformation is absent from this nucleus, therefore we will use the Hamiltonian (5.36). The spacing between the favored and unfavored tracks and the oscillation amplitude of B(M1) strongly depend on the ratio bj/a1. The first approximation for this ratio is found assuming the a2 and b2 parameters to be zero. Then the a, and a2 parameters are found by fitting the favored track*) ‘C

‘C

—

=

[E1,+1

2

—

a1(I _j)

—

a

4]/a 2(I —j)

1(21 + 1).

(5.76)

By using the thus determined ratio a2/a1, the values b1/a1 and b2/b1 are refined. The latter and b1/a1 are responsible for signature-dependent effects. The variation of the parameters a2 and b2 slightly changes the tracks and B(M 1), and manifests itself mainly in the transition region. The higher the energy of a level in a multiplet, the more noticeable is the influence the nonadiabatical *)The E, value in eq. (5.72) is obtained by the parallel displacement of the experimental and theoretical tracks.

254

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

t~Ybgg

0

i

7

167

70Yb97

Fig. 5.7. Appearance of bifurcation in the rotational bands of odd-A ytterbium isotopes. The experimental points corresponding to the ~ values (5.71) for favored (•) and unfavored (0) bands are connected by solid lines (tracks) to guide the eye.

parameter a2 resulting in the enhancement of the anharmonicity of the precessional spectra. Figure 5.8(c, d) presents the results of the experimentalt6tDy data fit to the for parameters a1, a2, and b2. nucleus fitting, since the b1I dependence Unfortunately, one cannot use B(E2) in the differs from that predicted by eqs. (5.53), (5.54). These’ differences seem to involve a nonaxial deformation (static or dynamic). The static one modifies B(E2; I —*1 1) in (5.54) by the factor (sin y 3 1/2 cos y)2 where y is the parameter of the nonaxial deformation. The decrease in B caused by this factor allows us to obtain agreement with experiment for -y ~ 30°,as it shown in fig. 5.8a. The nonaxial deformation also improves B(E2; I —~I 2). The solid line in fig. 5.8b was obtained for axially symmetric nuclei with quadrupole moment Qo (3.32). Using nonaxial transition moments (3.31) we find within the harmonic approximation in the region I > ‘C~ —

—

—

B(E2;I, n

I

—

2, n)

=

(l5Qg/l28it) [(1

—

kg) cos

y

+ (1 + kg) (sin

)/~~/~]2

(577)

f.M. Pavlichenkov, Bjfurcations in quantum rotational spectra

2b2

255

b

a

B(E2;NI-1)/e

B(E2;I.—I—2)/e2bt

I

0,14*

B(Mt;I—I-1~p~

I

13

7

I

I

I

I

I

23

7

I

I

33T

I

I

I

I

I

13 7

I

I

23

7

Fig. 5.8. Calculated and experimental signature dependence ofenergies and B-values along the positive-parity band based on the state ~ [642] from the i 61Dy. The experimental points of favored (•) and unfavored (0) bands are taken from ref. [131]. 1312 subshell ‘ The solid lines represent the incalculations with the Hamiltonian (5.36) having the parameters a 1 = 13.2 keV, b1/a1 = — 0.3, 2and wereexperimental found from the of the the Ml effect and E2 a2 /a1 = — 8.0 x 10 ~,b~Ia, = — 2.0 x 10 ~,obtained byQ~ fitting data.experimental The dashed B-values line represents of transitions a nonaxial deformation (see therespectively. text). The constants (g1 — g~) ~ and ‘~ -+ ~,

instead of eq. (5.53). The dashed line in fig. 5.8b obtained from eq. (5.77) shows that nonaxial deformation eliminates the decrease of B(E2) for small values of the spin

~.

5.6. Nonaxial shape of odd-A deformed nuclei The last estimate of the previous subsection show that nonaxiality can play an appreciable role in the rotational dynamics of odd deformed nuclei at high spins. There is no clear evidence of triaxiallity in the ground state of heavy stable nuclei. Yet the deviation of the nuclear shape from axial symmetry is expected at high angular momenta. The tendency of the high-f quasiparticle

256

f.M. Pavlic/,enkov, Bifurcations in quantum rotational spectra

orbits from the unique-parity subshell to drive the rotating nucleus towards a triaxial shape has been observed in ref. [132]. In several subsequent publications [133—135]the signature dependence of the energy, E2 and Ml transitions in a band were used for obtaining evidence for existence of a triaxial shape in high I states of odd-A nuclei. Now the available experimental data cannot produce a definite conclusion on this matter. On the other hand, as follows from the previous subsection, the fine bifurcational effects allow one to obtain information concerning the internal structure of a nucleus. It will be shown below that a nonaxial deformation changes drastically the intermediate regime of nuclear rotational dynamics. The rotational spectra of odd nonaxial nuclei was treated initially in the adiabatic approximation, which considered an odd nucleon to be in a definite single-particle state [136, 137]. This strong-coupling regime is not adequate for high-spin states. The model consisting of an even core and an external nucleon in the states of an isolated j subshell seems more appropriate. The first calculations with this model have been performed by Pashkevich and Sardaryan [138]. Then Mayer-Ter-Vehn [139] used this model but treated an odd nucleonas a quasiparticle in the pairing field. The model contains a transition from the strong-coupling scheme to the weak~coupling*)and interesting intermediate regimes. We will consider the Pashkevich—Sardaryan Hamiltonian H = ~ [A3(I3 _J)2 + Q3jfl , Qi

=

—

Q~=

cj0(sin

y)/...J~,

(5.78)

Q~

=

q,,

cos

y

(5.79)

,

where the Q, are the mean-field parameters depending on the subshell filling [q0 = b1, see eq. (5.69)] and on the nonaxial deformation parameter y, for which we use the Bohr—Mottelson definition, 0°
c~fl~’=1,2,3,

~33=~30(r~+r~),

(5.80)

2 MARg, ~h where ~ is the moment of inertia of a spherical nucleus with mass MA and radius R

(5.81)

=

0. The reduced principal semi-axes of the ellipsoid with a volume independent of the deformation are given by 2/l6ir. (5.82) r3 = R3/R0 = 1 + /3 cos (y ~ ir~x)+ 5/3 The rotational constants for the rigid-body moments of inertia satisfy the condition

~

—

A 2
(5.83)

.

On the other hand, the hydrodynamical moments of inertia [63], ~

~t2 =

J~rfl

2\211(2 —

2\

r~31.~rp+ r73,

~Q...L

~

,.-

p

_l1~t 1-

—

1,

4-,

*>In molecules Hund’s cases a (or b) and d are equivalent to the strong- and weak-coupling schemes, respectively.

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

257

result in the inequalities A1
for

0°
<

30°;

A1
for

30°
(5.84)

The y-dependence of the principal moments of inertia in the more realistic nuclear model with pair correlations of nucleons has been found in ref. [140]. This dependence is close to that for the hydrodynamical moments of inertia, but the shell effects resulting in a non-monotonic behaviour could play a significant role. By using the Poisson brackets for the angular momentum components I~andj3 (see appendix D), one can obtain the classical equation of motion 26 1y~ J~= 2~sfly[ A~(I~ jp) + Q,~j~]j~ , (5.85) = 3p~Afl(Ifl—jp) where c~~,is an antisymmetric tensor (repeated indices are summed). The stationary states Jo,Jo of system (5.85) are determined by six algebraic equations. For fixed values of I and j, there are four independent angles defining the orientation of these vectors in the BFF. Therefore, a stationary state with an arbitrary orientation of the vectors I~and J 0 does not exist. There are two types of stationary states of the system (5.85). The three aligned stationary states —

—

S3: I~= I, j3 =j,

~ = 1,2, 3,

(5.86)

with energy

2+Q 2 (5.87) E3(I) = A3(I j) 3J have parallel vectors l~andf 0 aligned along the ~ axis.*) The three plane stationary states 9sfi~ I~= I sin °ap~ ~ip= I sin ‘9sfl~ /3 = 1, 2, 3, S3p: I~= I cos ~ j3 = j cos ‘ (5.88) have vectors ~ andJ 0 placed in the (cxfJ)-plane. The angles °afl and ~ measured from the o~axis can be expressed in terms of the critical angular momenta in the region I > j, 3fl

.J1

2

—J~ + Qci—Qp 2A [(QrsQp”~ 3 + 2A3 —

[~

)

(Qa—Qp)Ap1112 —

A3(A3

—

A4]

589

(

The index ~ in eqs. (5.88), (5.89) denotes the axis, from which the angular momenta I~and Jo decouple, while /3 denotes the axis toward which they approach when I increases. Thus, both indeices ~ and /3 denote the plane in which these vectors move for ~
*)

=

1p I~3A3/[jA~+ (A~ Ap) 3J —

.

The anti-aligned states corresponding to the upper part of a multiplet are of no interest here.

(5.90)

258

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

The critical momenta T~determine bifurcation points in the region I < j. The expression for this quantity is the same as for ‘zfl except for a sign in front of the square bracket in eq. (5.89). We will consider below the region I > j only. The angles 0 and ~9are described in this region by 2,9 [jAn + (A 2(I2 I~) 20 [jAn + (A3—A~)1~3]2(I2 I~~) 3 A~)I~3] Sifl ~1’{[jA~ + (A 2 A~}12’ sin {[jA~ + (A _j2A~}~ 2 3 Aa)Ips] 3 A~~)Ifl3] ‘

—

-

—

—

—

—

—

—

—

—

(5.91) The energy of a state S 3~has the form 2 E3fl(I)

=

—

I~~)/I~~ .

(5.92)

E3(I3~)+ Afi(1fl3 _j)(J

It can be shown that the second derivative of the rotational energy has a discontinuity at I = This discontinuity is a distinguishing feature of a local bifurcation. If there is no rotation, the stationary position of the Jo vector in the state with minimum energy coincides with axis 2 for q 0 > 0 and with axis 3 for q0 <0. The strong-coupling scheme is associated with low values of I. There is no stationary state corresponding to this adiabatic regime. In the weak-coupling limit, for high angular momentum I, the stationary state with minimum energy corresponds to the aligned configuration S2 for rigid-body moments of inertia or S1 for hydrodynamic ones. In the first chase the equations of precessional motion have in the approximation I>>j the form 11

=

2I(A3

—

A2)I3

—

2A3Ij3,

i3

=

—

21(A1

—

A2)I1 + 2A,Ij1, (5.93)

=

—

J~ 2IA2j1.

2IA2j3,

=

Equations (5.93) resemble the system (5.19) describing precessional motion of the vectors I andj in an axially symmetric nucleus. However, there is an important difference. The normal mode with frequency w2 = 21A2, which is equal to the core angular velocity around axis 2, is not the zero-energy mode. It involves the uniform rotation of the vectors I and J around axis 2, I1(t) =J1(t)

=

i0 cos w1t,

13(t) =j3(t)

=

i0 sin w1t

(5.94)

.

Inspection of eqs. (5.94) shows that the core angular momentum R = I J does not participate in the precessional motion of this normal mode. The R vector points along axis 2. The zero-energy mode of eq. (5.19) is a consequence of dependence of the angular-momentum vectors namely {H, 13 j3} = 0, which is lacking in the case of a nonaxial core. The normal mode with frequency —

—

=

2I.~/(A1 A2)(A3 —

—

A2)

(5.95)

represents the precession of the considered vectors with different amplitudes. While the I vector circumscribes an elliptical cone about axis 2, I1(t)

=

i0(A3/A2

—

1)1/2

cos w2t,

I3(t)

=

—

io(A1/A2

—

1)1/2

sin w2t ,

(5.96)

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

259

with amplitude i0, the precessional amplitude of thej vector is f/I times that of I. The normal mode corresponds to the precession of vector R only. This precessional motion is exactly the same as for an asymmetric top described by Euler’s equations. Thus, in the weak-coupling regime, the y-dependence of the moments of inertia determines the precession axis, rather than a precession pattern. On the other hand, intermediate regimes, involving the plane stationary states S3fi, are determined mostly by the -y dependence of the moments of inertia. For a hydrodynamic dependence the intermediate regime involves the S2, state if q0 > 0 or the S31 one if q0 <0. In the last stationary state the vectors J~andjo, decoupled from axis 3, move in the (31)-plane for 13j 0; the only stationary state S2 with minimal energy exists for all angular momenta in this case. If q0 <0, the situation is more interesting. An intermediate regime depends on the relationship between the critical momenta (5.89). Under the conditions 113

> ‘12,

‘23 >

‘31 >

‘21,

(5.97)

‘32

the intermediate regime involves the stationary state sequence S3 —. S32 S2 similar to the case of the hydrodynamic moments of inertia. Another scenario takes place if inequalities (5.97) ate inverted. The stationary state S3 becomes unstable at I = 131. Then, asl increases, the sequence of states S31 S1 S12 —p 52 in the (31)-and (12)-planes leads the system to the final aligned configuration S2. This scenario, which satisfies 2 fl/(r 0> Zo~o > (45/l6ir)” 1 + r2)(r1 + r3)(r2 + r3) (45/l6ir)~2*13, (5.98) —~

—+

—

—*

—

is a consequence of a competition between the aligned stationary states S, and S2 with the energies (5.87). The differences in the critical momenta (5.89) are plotted as a function of ~j0q0 in fig. 5.9. There is a small region determined by inequality (5.98), in which the precession around axis 1 with

—0.10

—0.08

—0.06

—0.04

—0~02

—0.00

STRENGHT OF MEAN FiELD 8.q.

Fig. 5.9. Critical angular-momentum differences asa function ofthe mean quadrupole field strength and the i1312 subshell occupation in the particle—rotor model with nonaxial deformation. The Bohr—Mottelson definition [63]of the parameter y is used.

260

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

an intermediate moment of inertia exists in the region ‘13
12

=

2[jA1 + (A3

—

A1)I] 13 + 2A31j3,

—

—

2jA3I3 + 2[A11 + (A3

—

2[A1I

—

(A1

—

A2 + Qi

—

A1 + —

i3

Q~

—

Q2)]]12

2[jA1 —(A1

=

Qt)j]j3,

—

A2)I]

‘2

2A2Ij2,

j~= 2jA2I2 (5.99)

.

The normal mode frequencies are readily extracted as the roots of a biquadratic equation. The frequency of the soft mode may be written under the assumption I>>j and (A1 A2)/A1 <<1, —

2)/I]h/2 =

2 [(A1

—

A2)(A3

A1)(I

—

—

I13) (If2

—

(5.100)

.

1

The time dependence of the angular momentum components for this normal mode is defined by I 2(t)

=

i0(A3/A1

I3(t)

=

to [(xi

-

~/

—

1)1/2

cos w2t,

If2 —) ] A1 1(1—113) ~

—

—

,2

11/2

j2(t) = (j/I) I2(t), .

.

sin w2t,

.

j3(t)

(j/I) I3(t)

=

(5~0fl

Thus, the end points of vectors I and j move on elliptic orbits stretched along axis 2 if the angular momentum I is close to the critical value ‘12 or along axis 3 if I is close to ‘13. For another normal mode of frequency w1 = 2IA1, the time dependence of these components may be written as 12(t) =j2(t)

=

i0

cos

w,t,

13(t) =j3(t)

=

—

i0 sin w,t

.

(5.102)

This precession motion is similar to those described by eqs. (5.94).

6. Conclusion The bifurcations in quantum rotational spectra have been shown to exist in different systems. They are induced by different forces and involve different energy scales. The bifurcations can be classified according to the symmetry elements and divided into two groups. The nonlocal bifurcations of the C3,, and C4,, types result in the inversion of multiplet levels and consequently in the substitution of the P-branch by the Q-branch in the electrical El transitions between levels of multiplets. The local bifurcations of the types C2,,, C4,,, etc. are characterized by level clustering and an abrupt change in the electromagnetic transitions caused by the modification of the precessional motion while going through the critical point. Similarly to the second-order phase transitions, the local bifurcations display a universal character of rotational motion near the critical point. This universality is due to the fact that a bifurcation occurs in a limited region of the phase space of the *)Strictly speaking we would have to consider the precession around an axis placed in the (12)- or (13)-plane near axis 1 because the precession region around the latter, as is shown in fig. 5.9, is too small,

‘12

— ‘13

<

1.

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

261

system. Accordingly, it is described by an effective Hamiltonian, which depends on the internal structure of the system through a few parameters. The last circumstance, having made it possible to develop the phenomenological theory of local bifurcation, encourages one to create the microscopic theory of this phenomenon. The theoretical analysis of the experimental data on molecular and nuclear rotational spectra performed in this review shows that rotational excitations near the critical point are sensitive to the internal structure of the system; i.e., to the potential energy surface of a polyatomic molecule or to the parameters of the nuclear mean field. Therefore studying the bifurcation will allow us to gain deeper insight into these finite many-body systems. That is why the experimental search for bifurcations in quantum rotational spectra is of undisputed interest.

Acknowledgements I would like to thank Dr. Turgay Uzer for his suggestion of writing this article. The work was done in the Laboratory of Many Body Systems of the I.V. Kurchatov Atomic Energy Institute. I am grateful to my colleagues from this laboratory for fruitful discussions and interest in my work during the last seven years. I would also like to thank Dr. B.I. Zhilinskii from the Moscow State University for introducing me to molecular physics, which illuminates the idea of critical phenomena in rotational spectra. For editorial assitance I would like to thank Lada Gorbunova.

Appendix A. Proof of the universality of Hamiltonians for local bifurcations The effective Hamiltonian describing the lowest states of rotational multiplets near the critical point I~will be called universal if it depends on a finite number of parameters, which are determined by the internal structure of the system. The universal Hamiltonian is a power series in terms of the angular-momentum operators 1~.Universality is inherent in the Hamiltonian of local critical bifurcations only, since they take place in a restricted region of the system phase space. Let the effective Hamiltonian be written in the form of a series H

=

+ Ht4~+ H~6~ + E0(1) + H~2~

(A.1)

...

where H~contains the operators 13 in powers not exceeding k. For the local bifurcations of the C t2~and H~4~ terms are given by eqs. (3.2) and (3.54), respectively, and the 2,, and C4,,can types H H16~terms be the obtained by using the expansion method described in section 3. After some calculations, it is possible to obtain the ft~llowingresults: ~

=a

6)[(I~ 3[(1f +

_I2)/12]3

(c

6)[I~

—

..J2)2J~

+I~]+

+(b3/2I j2j~

+ I~i]÷ + (d,/I6)(I~+ I~),

(A.2)

2/2I ~

=

a

6)[I~ 3[(If

—

12)/J2]3

+ (c2/21

—

j2

I1 + I~_]+.

(A.3)

262

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

The parameters E0, a~,b, c1, etc., are expressed in terms of derivatives of the fm functions and 6~ and subsequent powersinofeq. I,~. Theyare have theinsame order of magnitude*). We will prove thethis H~we use the boson terms (A.1) small comparison with the H~2~ and H~4~ ones.that To do representation of the angular-momentum operators of the quantized rotator [141]. We shall consider the components of the angular-momentum operator in the BFF, 13

I

=

b~b,

—

1+

=

I, + i1 2

b~~/~T— b~b= I~,

=

(A.4)

which determine in the space of wave functions ‘

aKV

=

[(I

(b+)~K K)!] 1/2 10>,

(A.5)

—

where v is the quantum number of a state in a rotational multiplet. The state I 0>, corresponding to K = I, is a vacuum state of the boson creation and annihilation operators b ~, b. A reasonable way to prove the above-mentioned claim is to use the harmonic approximation for the effective Hamiltonian (A. 1). Let us consider first the precession motion around axis 3. In this case, the square-root operator in I + and I may be expanded in a Taylor series in the small quantity ti/I, where tI = b + b is the boson number operator. Thus, we transform this Hamiltonian to boson operators using eqs. (A.4) and the formulas of ref. [142] for rearranging them into normal form. The next step is the extraction of leading-order quadratic boson terms. The general form of this Hamiltonian in the harmonic approximation is -

H

=

E0(I) + Sb~b+ P(b~b+ bb),

(A.6)

where the coefficients E0, P, S, depending on the parameters a1, b1, c1, etc. and the quantum number I, can be written as an expansion E0=E~+E~+E~+E~+

~

~

~

(A.7)

t2>+pt4~+p~6~+ ..., p=p

so that ~ S~, and p(k) correspond to the term H~ of the series (A.l). Straightforward calculations yield the following formulas for the H~ 2,Hamiltonian (2.27): 2’= E~’~ =

=

0,

S~2~ =

—

a

E~ 6~=—a S~ 3[(2I—1)/12]3, 2) = (b ______ p( 1/,2),%/~J~, 1), —

6) p(

4~ = a 1(2I

=

4)

=

p(

—

—

1)/12,

S~

b 2[(I

—

1)/Ia]

2[(21

______ /~~j~j—

8b 3[(I

—

1)/13]2

~J2I(2I 1), —

IlIn this appendix we use for the Hamiltonian H~>the same definition coefficients as for He41.

—

1),

1)/12]2,

(A.8)

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

263

For the H~4Hamiltonian (2.44), all the E and S coefficients are the same as those for H~21but for p(k) = 0. The Hamiltonian (A.6) can be diagonalized by a canonical transformation 2—1v12=l, b=ufl+v/3~, 1u1 to new boson operators /3 and /3 ~ The energy of the lowest levels is given by

(A.9)

E

2(n + ~), n = 0, 1,2 (A.10) 1~= E0(I) — ~ S + (S/I SI)\/S2 4P It is clearly seen from the substitution of coefficients (A.8) into eq. (A. 10) that the small parameter 1/I appears in the harmonic approximation for the lowest states of the multiplet. Dropping anharmonic terms does not cause any serious difficulty. They are small for the lowest states in the region where the harmonic approximation works. In the transition region, where anharmo~nicityis essential, another small parameter (I~ I)/IC appears as will be seen below. To describe precessional motion around the axis k whose direction differs from that of the axis, 3, the transformation of the Hamiltonians (2.27) and (2.44) must be performed. It can be done most easily by expressing them in terms of the spherical tensor operators (2.4) in the form —

—

~

I

AjmT

~

(A.11)

1m, 10

m

—l

where according to definition (A.1) one has Aim = Ai,_m

=

A~?+ A~ffn~ +~

+

(A.12)

~

For the H~2Hamiltonian the A~coefficients are given by 2 2, A~Ø 0~= a1(2I l)/31, A~= a,\/~/3I2, A~= 2b,/I A~= (a 3)(2I 1)(412 + 1), A~ = (a 2/151 2\/~/2lI4)(812 —61 + 5), 4) (6,2 5) A~= (2b2/71 = 2a 4, 2~,/iö/35I~, A~= 2b2/,.,,/7I~, A~ = 4c1/I A~= (a 5)(21 1) (24I~+ 6,2 + 5), 3/105I A~= (a 3~/~/2lI6)(8I4 8I~+ 1012 101 + 7), 6)(16I4 4,3 1612 + 131 177), A~= (2b3/21I = (2a 3%,/7ö/385I6) (18,2 151 + 35), —

—

—

—

—,

—

—

—

—

—

—

—

—

—

—

—

=

—

A~=

—

(2b3/.~/~I6)(16,2 61 41), 6)(10I2 —1—38), A~ = 4a (4c2/111 3/,~/~i16,

A~=8b3/~/~I6,

—

—

A~=~/~c2/I6,

A~=8d1/I6,

(A.13)

264

f.M. Pavlichenkov, B(furcations in quantum rotational spectra

The Hamiltonian H~4 has the same A coefficients as Hc2, except those proportional to the vanishing parameters b1, b2, b3 and d1. Consider now the rotation of the BFF by the angles oc, /3 and y to align axis 3, along k. The rotation operator ~R(c~, /3, y)

exp(iyI3) exp (i/H2) exp (ictl3)

=

(A.l4)

transforms the Hamiltonian (A.ll) into the form ~RH9~’

E0(I) + ~

=

~

Ai,m’D~im’(

—

y,

13,

—

—

(A.l5)

C~)Tim= ~Hm,

10m,m’

m

where D,’nm’ is the Wigner function in the Edmonds definition [143]. The operators Hm can be expressed in terms of linear combinations of spherical tensors T2m, T4m, T6m, etc. The rotation of the BFF is equivalent in the boson representation to a translational transformation resulting in terms of the Hamiltonian (A.15) with an odd numbers of operators b~,b. In the harmonic approximation, one must eliminate terms proportional to b~or b in the expression~H1or H÷1. These conditions are used to determine the angles ~, /3, y. The H0 term results in the series E0 of eq. (A.7). The terms H2 and H -2 containing bilinear combinations of boson operators yield the coefficients S and P in the Hamiltonian (A.6). The others terms are neglected in the harmonic approximation. To study the convergence of the series (A.7) we calculate their terms up to sixth order. For the Hamiltonian H~,the results are 2) = —[(21— l)/21](a 2/3, Et 1 + 2b1) sin Sf21 p(2)

=

=

[(21

1)/212] [

—

2a

—

(A.l6)

2f3],

1 + 3(a1 + 2b1) sin

(1/412) [4b 1

(a1

—

2b1)

+

sin 2/3]

..,/21(21

—

1)

t4~= [(21— 1)/41~][2(21 — l)a E

2 + 8(1 2/3] x(a2 + 2b2

S~4~ = [(21

+

5(1

1)(2I

—

(1/4I~)x

—

8(1

—

2/3 —

—

5)[(4I

—

3)a2 + 6(1

—

1)b2] sin (A.l7)

4fl},

2b2 + 2c1)

+

sin

1)b 2 + 3(1

—

x (21 E~6~ = [(21

l)(21 —3)

2/3 —

3)(a

—

2 p14) =

—

l)a

—

2 + (21 —

l)b2 + (I

2c1)sin

1)/2I~]{2(21

—

—

sin2f3,

3)c1] sin 1)/8I~]{ 4(21 —

—

121 + 7~z2 + 2(1 1)(61 ll)b2 + 12(1 l)(21 3)(a 4/3} /~J~I 1), 2 + 2b2 + 2c1) sin 1)2a 3 641(1 2)b3 [(412 —

—

—

—

—

—

—

—

2/3 —

2(1

— (I

—

—

1)(2I

1)(I

—

—

3)[(61

2)(2I

—

—

5)a3

3)(21

—

+

2(61

—

l1)b3 + 8(1

—

2)c2]sin

5)(a

4/3} sin2 /3, 3 + 2b3 + 2c2

+

2d1)sin

—

1)

I.M. Pavlichenkov, B~furcationsin quantum rotational spectra ~(6)

[(21 1)/8,6].[

=

—

8(21

—

1)2a3

—

—

4[(48I~ 2f3

—

172,2 + 1841

—

2(1

—

265

—

63)a3

1)(21 —3)

+ 6(12I~ 60,2 + 891— 77)b3]sin —

x [(31

20)(61

—

7)a

—

4/3

3 + 4(812

—

661 + 95)b3 + 10(1

2)(21 — 13)c2] sin

—

+21(I—l)(I—2)(21—3)(21—5)(a

6I3},

(A.18)

3+2b3+2c2+2d1)sin p(6)

(l/816){64,(,

=

2)b

—

3 + 2[(241~_84I2 + 901— 31)a3 2f3

+ 2(36I~ 16412 + 2351

—

99)b3 + 48(1

+ 2(1

—

151 + 15)a

—

—

1)(21

3) [3(212

—

—

1)(I

—

2)(21

—

3)c2] sin

3 + (14,2 ~99I + 136)b3 4f3

+ 15(1

—

2)(21

—

15(1

—

1)(I

—

2)(21

—

3)(21

—

5)

5)d1] sin

—

x (4a 3

+ b3 + c2 + d1) sin6J3}~/2I(2I 1). —

The angle /3 in the rotation operator (A.14) is determined in the approximation considered by the equation 4(a 161

1 + 2b1)

—

—

1612(21

1)a2

—

6412(1

—

l)b2 + 16(21

1)(21

—

l)c2] sin

3)[21

—

—

2J3

8(1 x (12a

2 + 2b2 + 2c1) —(61— 5)a3

+ (I

—

1)(I

—

4f3

3 + b3 + 12c2

—

1)a3 + 2561(1

—

2(a

8(1

—

—

—

—

2)b3

2(61 + 11)b3

2)(21 — 1)(21 — 3) =

0.

(A.19)

12d1) sin

The two other angles are ~ = ~z/2,y = ir/2. The analysis of eqs. (A.16)—(A.19) shows that there are two small parameters j.1 and sin2/3 —~ (I IC)/IC in the series (A.7) for I > The leading terms with respect to these parameters are included in Ht2~and H14~only. Thus, for I > as well as for I
—

‘C~

‘C

2~ = —[(21— 1)/2I]a E~ p(2)

=

(l/412).,,/2,(2,

—

2fl, 1 sin 1)a, sin2fl,

~(2) =

[(21— 1)/212](

—

2a

2fl), 1 + 3a, sin (A.20)

266

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra = [(21 E~4~

—

S14~= [(21

—

1)/4I~][2(21

l)a

—

2/3] sin2f3,

2 + (I l)/2I~][2(21

5(1

—

l)(21

2 p(4)

=

(l/4I~)[(412

—

2c1) sin 2f3

—

5)(41

—

3)a2 sin

4/3],

3)(a

—

3)(a2

—

l)a

—

2 + (21 —

l)(21

—

(A.21)

2c1)sin

—

121 + 7)a

—

2

—

12(1

—

1)(21

—

3)c1 2f3,

3(1 l)(21 3)(a2 2ci)sin2f3]~,,/2I(21 l)sin where the angle /3 is determined by the equation —

—

—

—

—

12a

2 /3 1 —(21— 1)a2

—

(I

—

l)(21

—

3)(a2

—

=

0.

(A.22)

2c1) sin

Appendix B. Harmonic approximation for effective Hamiltonians The aim of this appendix is to develop an aproximate method for solving the Schrödinger equation with the Hamiltonians depending on the angular momentum operators. The method is based on the approximation of recurrence relations by a second-order differential equation for high I values [144, 145]. It has the advantage of being simpler than the boson expansion method in the calculation of electromagnetic transition matrix elements, which will be treated in appendix C. We begin our consideration with solving eq. (3.7) describing the C 2~type bifurcation. Let us rotate the BFF by angles x = 0, /3 = fir, y = ~it to change its axes in such a way that the axis of precession coincides with axis 1. The operatOr (A.14) of this rotation transforms the Hamiltonian (3.7) into the form 2±+ j2) H~2= E,(I) + A1I~+ A2I~+ B1(1 —

—

*(a 2 + 2b2)[If, I~+ I~]++ 1~~(a2 2b2)(I1 + I~i), —

E1(I)

=

E0(I)

—

~a11(1

+ *a2I(I

—

1) + b1I(1 + 1)

l)(312 + I + 2) ~b2(I 1)1(1 + l)(31 + 2), 2—6I+5)+~b 2—6I—5), Ai=—~ai—3b,+*a2(2I 2(6I B 2, A 1 = *(a1 2b1) (*a2)I(I 1) + ~b2I 2 = *(3a2 + lOb2). —

(B.l)

—

—

—

—

(B.2)

—

The eigenfunctions of the Hamiltonian (B.1) are sought in the form (2.13), in which K is the quantum number of the 13 operator in the new frame. For the coefficients aJKV, a five-term recurrence relation is obtained, PKaI,K4,V

+

QKaI,K_2,v

+

[WK

+ QK+2a1,K+2,v +

—

(E1,,

—

El)]aIKV

PK+4a1,K+4,,,

=

0,

(B.3)

267

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

~(a2

=

—

2b2) mUi [(I

K + m)(I + K

—

2 + (K QK =

{B1

WK

A

=

—

—

—

m + 1)11/2, [(I

2)2]}

K + m)(I + K

—

—

m + 1)]h/2,

(B.4)

*(a2 + 2b2)[K

2 +A 1K

4. 2K

By using the small parameter =

[1(1 +

(B.5)

1)]1/2

for large I values, let us introduce the continuous variable k

=

K/~/I(I+ 1).

(B.6)

We will treat the coefficients P, Q and W as smooth functions of this variable. As a result, the recurrence relation (B.3) may be rewritten in the form of the SchrOdinger equation 1’a 1,,(k) = e15a1,,(k),

(Ej,,

t~,,=

—

E1)/I(I + 1),

(B.7)

with the Hamiltonian h= [P(k + 45) + P(k)] cos4j3c5 + i[P(k + 45)

—

P(k)] sin 4~5

+ [Q(k + 25) + Q(k)] cos 2~5+ i[Q(k + 2ö)

Q(k)] sin 2~c5+ W(k),

—

(B.8)

where = id/dk is the canonically conjugate momentum to the coordinate k. Equations (B.7) and (B.8) are the generalization of similar expressions obtained in ref. [145] for a three-term recurrence relation. In the harmonic approximation, eq. (B.8) is reduced to a second-order To 52 inclusive.differential The resultequation. is obtain the latter, let us expand eq. (B.8) in terms of ö up to ~ ~

—

t~2s(k)d2a

2+ 52s’(k)da

15/dk + [~

1,,/dk

21(I + 1)](1 s(k)

=

[2b1

V(k) = *(ai

—

—

—

2s”(k)]a —

V(k) + i~5

1,,= 0,

(B.9)

k2),

(B.10)

a1 + 2a2k 2b

2 1)

—

+ 2b1)k

(a1

—

4]I(I + 1).

*[3(a 2

—

2b2)

—

(B.!!)

8(a2 + 2b2)k

It is convenient (see ref. [145]) to transform eq. (B.9) into the standard form of a one-dimensional Schrödinger equation, d2~/íj,,/dx2+ [c

—

V(x)]t/i 1,,

=

0,

(B.12)

268

f.M. Pavlichenkov, B~furcationsin quantum rotational spectra

for the wave function s114(k)a1,,(k),

=

(B.13)

which depends on the new variable (B.14)

Xrz~J~[__.

The potential energy of eq. (B.12), 2V”(xo) + ..., V(x) = V(xo) + ~(x x0) is obtained by the power-series expansion of the function V(B.11) near its local minimum x

(B.15)

—

0, which is different for two stationary states on the left and on the right of the critical value I,~.Within our approximation it is sufficient to confine ourselves to two terms in expansion (B.l5). Finally, we find the eigenvalue spectrum near the minimum of the potential energy, E1~= E1(I) + 1(1 + l)V(k0) + w(n + ~),

n

=

0, 1, 2,

(B.16)

... ,

where the oscillator quantum number n is connected with the quantum number v by the relation v = n + 1. The frequency of oscillation, w

221(1 + 1),

=

2

=

(B.17)

~[~sd2V/dk2]~!~0,

is determined by the k0 value at the point x0 = x0(k0) of the minimum. The eigenfunctions of eq. (B.l2) enable us to determine the expansion coefficients of wave function (2.13),

112] 1/2 H~((x

—

aIKfl

[(2/2~n!)(2/ms)

=

xo).~/5~) exp

[

—

~2(x

—

x 2], 0)

(B.18)

where H~is the Hermite polynomial. Let us turn to the particle—rotor model. We have the five-term recurrence relation AKaI,K2,,,

—

—

BKaI,Kl,,, + [WK

BK+la,,K+1,V

—

(E,,,

+ AK+2a,,K+2,,,

=

—

Eo)]aJK,,

0,

(B.19)

for the Schrodinger equation (5.35), with the effective Hamiltonian (5.36) describing the bifurcation in the rotational band of odd-A nuclei. Its coefficients have the form AK = a

2 mUi [(I

—

K + m)(I + K

2 K2 (K {ai + 2a2[J x [(I — K + l)(I + K)(j —

BK

=

—

—

—

—

m + l)(j

K + m)(j + K

2 + (K 2[K K + l)(j + K)]1’2, 1)2]

+ 4b

—

—

1)2]}

—

m+

1)]h/2,

I.M. Pavlichenkov, B~furcationsin quantum rotational spectra

WK

= —

a1J2

(2a

2+a 2 2K2)2 + 21(1 + 1)j(j + 1) 1 b1)K 2{(J 2K2[I(I + 1) +j(j + 1) 1] + 2K4} + b 2 2K2)K2, 2(J 1(1 + 1) +j(j + 1) (I —j)2. —

—

—

—

=

269

—

—

(B.20) (B.21)

By using the continuous variable k (B.6), we transform the recurrence relation (B.19) into the Scrodinger equation (B.7) with the Hamiltonian =

[A(k + 2ö) + A(k)] cos 2~5+ i[A(k + 15) —

[B(k + ö) + B(k)] cos

—

i[B(k + ö)

—

—

A(k)] sin 2j3ö B(k)] sin j~5+ W(k).

(B.22)

As described above, the last equation is reduced in the harmonic approximation to eq. (B.9) and then to (B.12), the s and V functions having the form s(k)

=

{a

2) + b 1 + [4a2(j.t

2]I(I + l)}\/(1

—

k2)(,u2

—

k2)

2k

2)(p2 k2)I(I + 1), k 2a 2) + b 2 1(~i k 1k

(B.23)

—

2(1

=

k

4a

—

V(k)

—

—

—

+ {4a

2

—

(1 + ~i)2k2 + 2k4] + 2b~(p k2)k2}I(I + 1) —

2[2~i —

2) + b

2{a 1 + [4a2(j~

—

k

2]I(I + 1)}~/(1 k2)(~2 k2), —

—

(B24)

2k

where ~ = ö .,,Jj (I + 1). The potential energy (B.24) is shown in fig. 5.4 for different values of the quantum number I. These expressions and the above-described formalism allow one to find the energy spectrum (B.16) and the a-coefficients of the Hamiltonian (5.36). The latter have the form 1

a

1

I

I

2w

\1/2

1~(k)= ~2”n’.\Jit(2I +1) 2], x H~((k ko)\/(I + ~)w) exp[ 2/5s(k 0), —

w

=

—

*w(21 + l)(k

—

(B.25)

k0) (B.26)

where 2 is determined by eq. (B.17).

Appendix C. Calculation of matrix elements of electromagnetic transitions in the harmonic approximation In this appendix some of the B(E2)- and B(M1)-valuôs of Sections 3 and 5 will be obtained by using the sharply localized orientation of the angular-momentum vector in the states described by the harmonic approximation. The calculation method is based on the utilization of analytic

270

f.M. Pavlichenkov, B~furcations in

quantum rotational spectra

formulas (B.l8) or (B.25) for the expansion coefficients in the wave functions of the corresponding effective Hamiltonians. The reduced electric and magnetic transition probabilities are expressed in terms of the matrix elements of the multipole operator A~,,

(W7M.,,.I.A~l~‘JM,,) It depends on coefficients

a7K~,,~aIK,,.

=

(C.1)

and a1~K • with small differences in the quantum numbers I and K. According to appendix B, they can be replaced by the functions a1~(k) and a1 ~~,,,( k + s5) (with i.~I<
-

‘~

a,~j,3(k+ s5)

=

a1~(k) Al ~a1~/5I + s5 äa1~/ôk. —

(C.2)

The relationship between the variables k and x, k

=

xs3~,,/~,

s0

=

s(0),

(C.3)

can be derived from eq. (B.14) within the same accuracy. The approximation (C.2) and (C.3) is sufficient for obtaining the leading2/2~5) terms in the matrix element (C. 1) since the function a15(k) H~(k/~/~) exp( k —

~

has a sharp maximum of the width at k = 0. Thus, in calculating the matrix elements, we can confine ourselves to lower powers of k in the integrand. The simplest matrix elements we treat are the diagonal ones for the electric quadrupole and magnetic dipole moments. The quadrupole moment in the particle-rotor model can be written as —

.rJ..~ \l/2j~* II 113’)\W k 5 ltJ k IJw..’~t0I1_~4-~ 11,

=

(I + 1)(21 + 3) K=-j [3K2

n

~1v

1

(21+3)ö

1

J

(3k2

—

1(1 + !)]ai~,,

—

!)a?~(k)dk.

(C.4)

Replacing the integration over k by that over x and using eqs. (5.50) and (C.3), we reduce the integral to the oscillator matrix element . The final result, Qiv

=

[QoI/(2I + 3)][ —1 + 3

2 + 1tö

leads immediately to eq. (5.59).

0(t52)],

(C.5)

f.M.

271

Pavlichenkov, B~furcationsin quantum rotational spectra

Due to nonaxial deformation, the quadrupole moment for the states of the H~2Hamiltonian can be expressed in the form Qiv

Qo[(I + l)(21 + 3)]

=

1[

—

EØ

(sir

cos

—

y)

+

+

~(~2

.~‘—2)

sin (~ir

—

y)],

(C.6)

where ~‘ois exactly the sum in eq. (C.4) and =

>J~aIKVaI,K±2,V[3(I

~J

± K + l)(I ±K + 2)(I ~ K

a

2 —

1)(I ~ K)]”

2 T k~)(l k2 T 3k~ 2i2)]”2dk. —

1~(k)a~(k ±15)[3(1

—

—

(C.7)

k

Now, using eq. (C.2), one can approximately reduce the last integral to the same oscillator matrix element and get

(C.8) With the substitution of the sums from eqs. (C.4) and (C.8) into eq. (C.6), it is a straightforward exercise to obtain formula (3.34). In a similar way, one calculates the magnetic moment in the particle—rotor model (W~I~o(M1)I W

= (~)hI2

115)

= PN

(SRI +

(g~

—

g~)~

~ma).

(c.9)

The first sum is calculated by using the same approximation as in eqs. (C.4) and (C.5): 2a/K,, ~ ~ m0

=

1

K=-j

+ 1) + O(~).

(C.!0)

K

The extra term a2a~~/3k2 should be taken into account in eq. (C.3) to calculate the sum 1

m~ 1= 2’I

~ +

~f

i

P ~

aIKVaIK± ~,~[(I

T K)(I ±K + 1)(j ~ K)(j ±K +

1)]1~~2

JK=-j

2)Qz2 a~(k)a1~(k ±~)[(1

—

—

k2)]”2 dk.

(C.11)

k

Having performed straightforward but rather cumbersome calculations, we obtain m+i=~I{~u_[52w>/4ai+(1+p2)ai/w>](n+~)+O(ö2)}.

(C.12)

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

272

In the next step we consider the matrix element of the E2 transition I, v H~2Hamiltonian, 2 (!P7_l,M.,,,LA’~(E2)IWIM,,)

=

—

l)(21

—~

I

—

1, v for the

—

[5Q~f/16irI(I x{E 0cos(~ir—y)+~(~2 +~.2)sin(~it—y)},

(C.l3)

2) =

~

aI.1KVaIKVK\/3(I

~J =

—

K

a

2) k dk, 11,~±1(k)a1~(k)~3(1 —

—

k

±>a1_1K±2,,,aIKV[(I ± K + l)(I T K

~

2)(I

—

~

K

—

1)(I ~ K)]”2

(C.14)

Ja

115+i(k±2o)ain(k)(1

k)dk,

—~

1jj~is the Clebsch—Gordan coefficient. The nondiagonal oscillator matrix elements in where Cji~ eq. (C.l4) are just a consequence of the harmonic approximation, in which the states of adjacent multiplets, having the same quantum number v, are described by the quantum numbers n differing by a unit. By using the approximations (C.2) and (C.3) we get ~J

2)[(6s 0(n-.÷n± 1)=(l/5

0/wj(2n+

1 ±1)]1/2,

E2(n—*n±l)=~.2(n—+n±1) 2)(l ~ w<ö/4s = (l/ö 0)[(2s0/w<)(2n + 1 ±1)]h/2.

(C.15)

The substitution of eqs. (C.l5) into the r.h.s. of eq. (C.l3), and the subsequent substitution of the thus obtained expression in eq. (3.22), yields formulas (3.35). The B(M1) quantity in (5.62) as well as other B expressions for the aligned part of a band are calculated similarly. For the decoupled part of a band we will use the symmetrized expansion coefficients of eq. (3.17) or (5.46), the ansatz (C.2), and the approximation 2(x — XO)2(aS/~k)k_k k — k0 ~ 5(x — xo)~~/~+ ~5 0, (C.l6) where s0 a15(k

—

=

s(k0). Neglecting further small exponential terms resulting from overlap integrals of

k0) and a1.~.(k+ k0), one can obtain F(k)a1.~.0.(k)aj~~(k) dk

~J

[F(k) + pp’F(

—

k)] a1•~•(k k0)a15(k —

—

k0) dk,

(C.17)

rotational spectra

f.M. Pavlichenkov, B~furcationsin quantum

273

where the quantum number p equals r3 ( 1)I or signature a (5.6). This expression allows the matrix element (C. 1) to be calculated using the expansion coefficients (B. 18) and (B.25). These functions have a sharp maximum of the width at k = k0. Therefore, evaluating the matrix element, we may expand the integrand in the r.h.s of eq. (C.17) in a power series of k k0. For example, eq. (C.4) is a starting point in calculating the quadrupole moment. As approxima2 asa afirst power series of tion,kthe integral equals 3k~ 1. To obtain other terms, one should express k k 0 and use eq. (C.l6). As a result, we obtain the final formula for Q~,,in eq. (5.59). As a more complicated example, let us consider the matrix elements of the E2 and Ml transitions I, v I 1, v ±1 in the particle—rotor model. For the E2 transition we have —

~

—

—

—

—‘

—

(!t’F._l,M,,+

1

IA’~~(E2)I ~‘1Mv)

2 =

—

l)(21

—

~

!)]1/2

0Cf~~f,

[5Q~/16EI(I

(C.18)

2) 2~=—~aI...l,KV±,aIKVK\/3(I

J

~

—

~

J

K

a

2)kdk 11,~÷1(k)a15(k)~3(1k —

a~

2) k dk, 1,~+ ,(k

—

k0)a13(k

—

k0)~3(1

—

(C.19)

k

and expansion coefficients in the last integral are determined by eq. (B.25). To evaluate this integral, we use the ansaz (C.2) for a 1 -1. K, ±1 and expand the integrand near the fixed point k0 retaining only the terms linear in k k0. The final result may be presented by —

2)(3a,so/2w<)”2 —+

n ±1)

=

—

(1/ö

~f+(l_~2w< —

soai(2I + 1)

~ (as\

~l [

—

i~

1

‘\aP)k=kOj

!—2k~ + (1

—

x~,/2n+1±1,

k~)”2+

~ (C.20)

which leads to eqs. (5.57), (5.58). The magnetic matrix element

(~‘7-1,M’ ~I.~#~(M1) I !PIM~) =

(3j~2(g

g)2)

1/2

C 1’~~~

1

m~

(C.21)

274

i.M. Pavlichenkov, Bifurcations in quantum rotational spectra

is expressed in terms of sums: m0 1 4-

aIKVaI1.K±1,V±1

~J —

—

l)(j T K)(j ±K +

l)]1/2

a1,,~ 1(k ±ó)ai~(k)f+1(k) dk,

(C.22)

±1.

(C.23)

2, (1

E0/~,/~ (C.19), and

K= -j

x [(I ~ K)(I ~ K

=

=

xk)~J,u2

—

t

=

k

The integral (C.22) is evaluated in the same approximation as that in eq. (C.19). The result is given by m,,(n

—+

n ±1) = (t/ö2)(a

1’2 1so/8w<)

{~

~<

[~

1~~’°<~ +

k

0

—

~

(~)kkO]

—J’(k0) + O(~5)}

x1.,/2n+1±l.

(C.24)

Equations (C.20) and (C.24) allow us to obtain B(Ml)-values of (5.65) and (5.66).

Appendix D. Angular-momentum operators in the particle—rotor model Let us begin with an axially symmetric rotor. The rotor symmetry axis n is determined in the laboratory fixed frame by the polar angles 0 and p, while the particle position, by the radius vector r with the components x, y, z. The total nuclear angular momentum is I = R + j, where R is the angular momentum of a rotor having the following components in the laboratory frame: R~= i(sinq~’/30 + cot0cosq~’/3cp), (D.l) R~= i(

—

cos q ~‘/0O+ cot 0 sin q, a’/~q3),

R~=

—

iô’/8p.

The components of the particle angular momentum are =

—

i(y3/~z z~/~y), j,, = —

—

i(z13/ax

—

x~/~z),

Jz

=

—

i(xa/~y y8/~x). —

(D.2)

The prime symbol in eq. (D. 1) denotes the partial derivative providing differentiation with fixed x, y, z. The spin part of the particle angular momentum is omitted in eqs. (D.2) for simplicity.

f.M.

Pavlichenkov, B~furcationsin quantum rotational spectra

275

To perform the transformation of eqs. (D.l) and (D.2) to the body fixed frame (BFF) with axis 3 directed along n, we use the transformation matrix x1 x2

cos 0 cos tp =

sin p

—

x3

sin 0 cos ip

cos 0 sin p

sin 0

x

0

y

cos 0

z

—

cos ~p sin 0 sin ‘p

(D.3)

,

and replace the differentiation with respect to 0, ‘p with fixed x, y, z by that with fixed x1, x2, x3. This differentiation will be denoted below by a partial derivative without prime. Simple calculations yield =

a~ao

—

ij2,

ä’/~p=

ô/~q,

—

ij3 cos0 +

ij1 sin 0,

(D.4)

where j3, ~ = 1, 2, 3 stands for the angular momentum components in the BFF. Equations (D.4) allow the lab frame components of the total angular momentum to be determined, I~= i[sinq,3/öO + cot0cos’p3/~p

—

(cosq,/sin0)j3], (D.5)

I~= i[

—

cos’pa/eO + cotO sinpa/~p + (sinq/sin0)j3],

I~=

—

Another Euler angle cit should be introduced to determine the orientation of a nonaxial rotor. The method used above allows one to obtain the total angular-momentum operators for the nonaxial variant of the particle—rotor model =

i[sin’pa/aO + cotocos’pô/a’p

—

(cos ‘p/sin (D.6)

=

i[

—

cos ‘p 3/~0+ cot 0 sin p o/ap

—

(sin ‘p/sin 0) ô/~],

I~=

—

ia/tip.

The operators (D.5) and (D.6) satisfy normal commutation relations of the form [Ii, Ik]

=

(D.7)

‘~ik1’i,

61k1 is an antisymmetric tensor. where Now we refer the components of the total angular-momentum operator to the BFF. For an axial rotor, its principal axes 1, 2 and 3 coincide, respectively, with the unit vectors e 0, e4, and e, of the polar coordinate system. Using the transformation (D.3) we obtain

I~=10=(i/sinO)ô/tip +j3cotO,

‘2

i4,_~a/a0,

13

1y’J3.

(D.8)

A straightforward calculation results in commutation rules for these components, [11,12] =—i(I,cot0+j3),

[11,13] =[I2,I3]

=0,

(D.9)

276

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra

and for the components of I and j, [j,,I1]

=

[J2,Ii]

=ij1cot0,

[j~,I~]=

—if2 cotO,

0

[Jl,I3]

=

=0,

[12,13]

=—ij,,

0,

[f~,13]

=

[f,,12]

=

[f2,12]

[13,

=

12]

0,

—

if2, (D.10)

0.

The j3 components have the same commutation rules as in eq. (D.7). In the classical limit the commutators reduce to the Poisson bracket ~A, B} = 2i[A, B]. of the I~(D.8) and j~operators is in terms starting point for obtaining an expression for R theThe ansatz R

=

e 9(11

11)

+ e42(I2

(D.ll)

—f2).

Squaring (D. 11) and applying the commutation relations, [Ii, e,]

=

ie42,

[12, er]

=

—

ie0, [13, er]

=

0,

[I,,e0] =icotOeq,,

[I2,e9] =ier,

[I3,e0] =0,

[I1,e4,]=—i(e,

[I2,e~,] =0,

[I3,e~,] =0,

+cot0e9),

(D.l2)

we obtain a well known expression used in nuclear physics and in the nonadiabatic theory of diatomic molecules (see for example ref. [146]) 2 = (Il _f)2 + (12 32) icot0(1 2 +j2 2jf j±I_ f_I+. (D.13) R 2 —12) = j —

—

Let us also write down the square of the total angular momentum,

ai’.

1 j2

a~

1

13

-

ok\81fbj_2~~1f3coto)

_~

(D.14)

+j~.

Comparing eqs. (D.8) and (D.14) we find that 13 commutes with j2 For a nonaxial rotor, the total angular-momentum components in the BFF have the form I, ~

.1

a

.

a

.1 ‘2 =

—

cosci/~

-

a

coscii8\

a

sini/i3\

sin0~)’ .3

(D.l5)

+ cot0s1ncii~—------~—-)~ 13=— 1~.

Consequently, the BFF components of the angular-momentum operators obey the commutation relations [13, I~]=

—

i8 3~,I~, [j~, I~]= 0.

(D.l6)

f.M. Pavlichenkov. B~furcationsin quantum rotational spectra

277

The difference in the commutation relations of the angular-momentum operators for axial and nonaxial particle—rotor models lies in the absence of rotation around the symmetry axis for an axial rotor and in the independence of the vectors I and j for the nonaxial case.

References [1] R.M. Lieder and H. Ryde, in: Advances in Nuclear Physics, vol. 10, eds M. Baranger and E. Fogt (Plenum, New York 1978) p. 1. [2] P.J. Twin, B.M. Nyakó, A.H. Nelson, J. Simpson, MA. Bentley, et al., Phys. Rev. Lett. 57 (1986) 811. [3] R.S. McDowell, C.W. Patterson and w.G. Harter, Los Alamos Science 3 (1991) 40. [4]I.M. Pavlichenkov, Usp. Fiz. Nauk. 133 (1981) 193 [Soy. Phys.—Usp. 24 (1981) 79]. [5] BR. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5 (1960) 511. [6]I.M. Pavlichenkov, Zh. Eksp. Teor. Fiz. 82 (1982) 9 [Soy.Phys.—JETP 55 (1982) 5]. [7] I.M. Pavlichenkov and B.I. Zhilinskii, Chem. Phys. 100 (1985) 339. [8] R. Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley—Interscience, New York, 1981). [9]R. Gilmore and D.H. Feng, NucI. Phys. A 301 (1978) 189. [10]TM. Pavlichenkov and B.I. Zhilinskii, Ann. of Phys. (NY) 184 (1988) 1. [11]D.H. Feng, R. Gilmore and S.R. Deans, Phys. Rev. C 23(1981)1254. [12] W.-M. Zhang, D.H. Feng and J.N. Ginocchio, Phys. Rev. C 37 (1988) 1281. [13] W.-M. Zhang, C.-L. Wu, D.H. Feng, J.N. Ginocchio and MW. Guidry, Phys. Rev. C 38 (1988) 1475. [14] W.-M. Zhang, D.H. Feng, C..L. wu, H. Wu and J.N. Ginocchio, NucI. Phys. A 505 (1989) 7. [15]C.E. Vignolo, D.M. Jezek and ES. Hernández, Phys. Rev. C 38 (1988) 506. [16] S.T. Belyaev and I.M. Pavlichenkov, NucI. Phys. A 388 (1982) 505. [17] R. Gilmore, S. Kais and R.D. Levine, Phys. Rev. A 34 (1986) 2442. [18]A.J. Dorney and J.K.G. watson, i. Mol. Spectrosc. 42 (1972) 135. [19]ED. Hinkley, T.C. Harman and C. Freed, AppI. Phys. Lett. 13 (1968) 49. [20] ED. Hinkley and C. Freed, Phys. Rev. Lett. 23 (1969) 277. [21] K. Fox, H.W. Galbraith, B.J. Krohn and iD. Louck, Phys. Rev. A 15 (1977) 1363. [22] W.G. Harter and C.W. Patterson, Phys. Rev. Lett. 38 (1977) 224. [23] W.G. Harter and C.W. Patterson, J. Chem. Phys. 66 (1977) 4872. [24]W.G. Harter and C.W. Patterson, J. Chem. Phys. 66 (1977) 4886. [25]W.G. Harter, H.W. Galbraith and C.W. Patterson, J. Chem. Phys. 69 (1978) 4888. [26]C.W. Patterson and H.W. Galbraith, J. Chem. Phys. 69 (1978) 4896. [27]W.G. Harter and C.W. Patterson, J. Math. Phys. 20 (1979) 1453. [28] W.G. Harter, Phys. Rev. A 24 (1981) 192. [29]W.G. Harter and C.W. Patterson, J. Chem. Phys. 80 (1984) 4241. [30]W.G. Harter, J. Stat. Phys. 36 (1984) 749. [31]W.G. Harter, Comput. Phys. Rep. 8 (1988) 319. [32]M.A.J. Mariscotti, G. Scharf-Goldhaber and B. Buck, Phys. Rev. 178 (1969) 1864. [33]G. Scharf-Goldhaber, NucI. Phys. A 347 (1980) 31. [34] J.K.G. Watson, in: Vibrational Spectra and Structure, Vol. 6, ed. JR. Dung (Elsevier, New York, 1977) p. 1. [35] MR. Aliev and J.K.G. Watson, in: Molecular Spectroscopy. Modem Research, Vol VII, ed. K. Narahari Rao (Academic Press, New York, 1985) p. 1. [36] J.-M. Flaud, C. Camy-Peyret and J.W.C. Johns, Can. J. Phys. 61(1983)1462. [37]S.T. Belyaev and V.G. Zelevinsky, Yad. Fiz. 17 (1973) 525 [Soy. J. NucI. Phys. 17 (1973) 269]. [38] A. Klein, in: Progr. Part. NucI. Phys. Vol. 10 (Oxford Univ. Press, Oxford, 1983) p. 39. [39]E.B. Wilson, J.C. Decius and P.C. Cross, Molecular Vibrations (McGraw.Hill, New York, 1955). [40]G. Amat, H.H. Nielsen and G. Tarrago, Rotation—Vibration of Polyatomic Molecules (Dekker, New York, 1971). [41]HA. Buckmaster, R. Chatterjee and Y.H. Shing, Phys. Status Solidi a 13 (1972) 9. [42]L.D. Landau and EM. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965). [43]S.D. Augustin and W.H. Miller, J. Chem. Phys. 61(1974) 3155. [44] H. Goldstein, Classical Mechanics, 2nd Ed. (Addison-Wesley, Reading, MA, 1980). [45] U. Fano, F. Robicheaux and A.R.P. Rau, Phys. Rev. A 37 (1988) 3655. [46] A.R.P. Rau, Phys. Rev. Lett. 63 (1989) 244. [47] T. Uzer, Physica D 36 (1989) 51. [48] T. Uzer, Phys. Rev. A 42 (1990) 5787. [49]A.R.P. Rau and Lijun Zhang, Phys. Rev. A 42 (1990) 6342. [50] M. Henon and C. Heiles, Astron. J. 69 (1964) 73.

278

f.M. Pavlichenkov, Bjfurcations in quantum rotational spectra

[51]M. Golubitsky and I. Stewart, Physica D 24 (1987) 391. [52]M. Michel, Rev. Mod. Phys. 52 (1980) 617. [53]G. Gaeta, Phys. Rep. 189 (1990) 1. [54] D.A. Sadovskii and B.I. Zhilinskii, Mol. Phys. 65 (1988) 109.[55] D.A. Sadovskii, B.I. Zhilinskii, J.P. Champion and G. Pierre, J. Chem. Phys. 92 (1990) 1523. [56] M. Feingold and A. Peres, Physica D 9 (1983) 433. [57] MV. Berry, Proc. R. Soc. London, Ser. A 392 (1984) 45. [58] yB. Paylov-Verevkin, D.A. Sadovskii and B.I. Zhilinskii, Europhys. Lett. 6 (1988) 573. [59]L.D. Landau and EM. Lifshitz, Statistical Physics (Pergamon, Oxford, 1965). [60] R. Peierls, J. Phys. A. 24 (1991) 5273. [61] C.H. Townes, AL. Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955). [62] A. Bohr and BR. Motte!son, Nuclear Structure, Vol. I (Benjamin, New York, 1969). [63] A. Bohr and BR. Mottelson, Nuclear Structure, Vol. II (Benjamin, New York, 1974). [64]R.J. Liotta and R.A. Sorensen, NucI. Phys. A 297 (1978) 136. [65] J. Goldstone, Nuovo Cimentol9 (1961) 154. [66]P. Claverie and G. Jona-Lasino, Phys. Rev. A 33 (1986) 2245. [67] R.A. Harris and L. Stodolsky, Phys. Lett. B 78 (1978) 313. [68] R.A. Harris and L. Stodolsky, J. Chem. Phys. 74 (1981) 2145. [69]B.!. Zhilinskii and I.M. Pavlichenkov, Opt. Spectrosk. 64 (1988) 688. [70] J. Makarewicz and Jan Pyka, Mol. Phys. 68 (1989) 107. [71] J. Makarewicz, Mo!. Phys. 69 (1990) 903. [72] B.T. Sutcliffe and J. Tennyson, in: Understanding Molecular Properties, eds J. Avery et a!. (Reidel, Dordrecht, 1987) p. 449. [73] J. Makarewicz, J. Phys. B 21(1988)1803. [74]B.T. Sutcliffe and J. Tennyson, mt. J. Quantum Chem. 39 (1991) 183.. [75] J.-M. Flaud, C. Camy-Peyret and J.P. Maillard, Mo!. Phys. 32 (1976) 499. [76]J.W.C. Johns, J. Opt. Soc. Am. B 2 (1985) 1340 [77] J.T. Hougen, P.R. Bunker and J.W.C. Johns, J. Mo!. Spectrosc. 34 (1970) 136. [78] J.H. Frederick and G.M. McCle!land, J. Chem. Phys. 84 (1986) 4347. [79]G.S. Ezra, Chem. Phys. Lett. 127 (1986) 492. [80] J.H. Frederick and G.M. McCle!land, J. Chem. Phys. 84 (1986) 876. [8!]J.H. Frederick, Chem. Phys. Lett. 131 (1986) 60. [82] J. Makarewicz, J. Mol. Spectrosc. 130 (1988) 316. [83] J. Makarewicz, J. Phys. B 21(1988) 3633. [84]!.M. Pavlichenkov, in: Nuclear Physics, Proc. XXIV Winter School ofthe Leningrad Nuclear Physics Institute, Vol. 2 (Leningrad, 1989) p. 67. [85]iN. Murrell and S. Carter, J. Phys. Chem. 88 (1986) 4887. [86] R.A. Toth, V.D. Gupta and J.W. Brault, App!. Opt. 21(1982) 3337. [87] J.K. Messer, F.C. De Lucia and P. He!minger, J. Mo!. Spectrosc. 105 (1984) 139. [88] G. Steenbeckeliers and J. Bellet, J. Mol. Spectrosc. 45 (1973) 10. [89] N. Papinenan, J.-M. F!aud and C. Camy-Peyret, J. Mo!. Spectrosc. 87 (1981) 219. [90]C. Camy-Peyret, J.-M. Flaud and L. Lechuga-Fossat, J. Mo!. Spectrosc. 109 (1985) 300. [91] IN. Kozin, S.P. Belov, O.L. Polyansky and M.Yu. Tretyakov, J. Mol. Spectrosc. 152 (1992) 13. [92] J. Tennyson and B.T. Sutcliffe, Mol. Phys. 51(1984) 887. [93] S. Carter and N.S. Handy, Mol. Phys. 52 (1984) 1367. [94]V. Spirko, P. Jensen and P.R. Bunker, J. Mo!. Spectrosc. 115 (1986) 269. [95] C.-L. Chen, B. Maessen and M. Wolfsberg, J. Chem. Phys. 83(1985)1795. [96]J. Tennyson and B.T. Sutcliffe, Mo!. Phys. 58 (1986) 1067. [97] V.M. Krivtsun, D.A. Sadovskii and B.I. Zhilinskii, J. Mol. Spectrosc. 139 (1990) 126. [98] K.T. Hecht, J. Mo!. Spectrosc. 5 (1960)355. [99]J.-P. Champion, Can. J. Phys. 55 (1977) 1802. [100] J.-P. Champion and G. Pierre, J. Mo!. Spectrosc. 79 (1980) 255. [10!] B.I. Zhilinskii, V.1. Perevalov and V.G. Tyuterev, Methods of Irreducible Tensor Operators in the Theory ofMolecular Spectra (Nauka, Novosibirsk, 1987). [102] VI. Pereva!ov, V.G. Tyuterev and B.I. Zhilinskii, J. Mo!. Spectrosc. 103 (!984) 147. [!03] V.G. Tyuterev, J.-P. Champion, G. Pierre and VI. Pereva!ov, J. Mol. Spectrosc. 105 (1984) 113. [104] V.!. Pereyalov, V.G. Tyuterev and B.!. Zhilinskii, J. Mol. Spectrosc. 111 (1985) 1. [105] D.A. Sadovskii and B.I. Zhilinskii, J. Mol. Spectrosc. 115 (1986) 235. [106] V.G. Tyuterev, G. Pierre, i-P. Champion, VI. Perevalov and B.I. Zhi!inskii, J. Mo!. Spectrosc. 117 (1986) 102. [107] V.G. Tyuterev, J.-P. Champion, G. Pierre and V.!. Perevalov, J. Mol. Spectrosc. 120 (1986) 49. [108] J.-P. Champion, J.C. Hilico, C. Wenger and L.R. Brown, J. Mol. Spectrosc. 133 (1989) 256.

f.M. Pavlichenkov, Bifurcations in quantum rotational spectra [109] G. Pierre, A. Va!entin and L. Henry, Can, J. Phys. 64 (1986) 341. [110] Y. Ohshima, Y. Matsumoto, M. Takami, S. Yamamoto and K. Kuchitsu, J. Chem. Phys. 87 (1987) 5141. [111] G. Pierre, G. Millot, A. Valentin, L. Henry, B. Foy and iT. Steinfeld, Can. J. Phys. 66 (1988) 622. [112] 0.!. Davarashvi!i, B.I. Zhilinskii, V.M. Knvtsun, D.A. Sadoyskii and E.P. Snegirev, JETP Lett. 51(1990)17. [113] L. Mortensen, S. Brodersen and V.M. Krivtsun, J. Mol. Spectrosc. 145 (1991) 352. [114] A. Cabana, DL. Gray, AG. Robiette and G. Pierre, Mol. Phys. 36 (1978) 1503. [115]V.M. Krivtsun, D.A. Sadovskii, E.P. Snegirev, A.P. Shotov and I.!. Zasavitskii, J. Mol. Spectrosc. 139 (1990)107. [116] F.S. Stephens, Rev. Mod. Phys. 47(1975)43. [117] U. Fano, Phys. Rev. A 2 (1970) 353. [118] I.M. Pavlichenkov, Europhys. Lett. 9(1989) 7. [119] !.M. Pavlichenkov, Zh. Eksp. Teor. Fiz. 96 (1989) 404 [Soy. Phys.—JETP 69 (1989) 227]. [120] AM. Kamchatnov, Zh. Eksp. Teor. Fiz. 98 (1990) 754. [121] A. Bohr and BR. Mottelson, Phys. Scr. 22 (1980) 461. [122] ER. Marshalek, Phys. Rev. C 26 (1982) 1678. [123] I. Hamamoto, Phys. Lett. B 102 (1981) 225. [124] I. Hamamoto, Phys. Lett. B 106 (1981) 281. [125] I. Hamamoto, NucI. Phys. A 271 (1976) 15. [126] N.N. Bogolyubov, Usp. Fiz. Nauk 67 (1959) 549 [Soy. Phys.—Usp. 67 (1959) 236]. [127] Th. Lindblad, H. Ryde and D. Barneoud, NucI. Phys. A 193 (1972) 155. [128] J. Kownacki, iD. Garrett, J.J. Gaardhøje, GB. Hagemann and B. Herskind, NucI. Phys. A 394 (1983) 269. [129]J.C. Bacelar, M. Diebel, C. Ellegaard, J.D. Garrett, G.B. Hagemann, B. Herskind, et al., NucI. Phys. A 442 (1985) 509. [130] S.G. Nilsson, K. Dan. Vidensk. Se!sk. Mat..Fys. Medd. 29 (1955) 1. [131] M. Oshima, E. Minehara, S. Ichaikawa, S. limura et al., Phys. Rev. C 37 (1988) 2578. [132] S. Frauendorf and FR. May, Phys. Lett. B 125 (1983) 245. [133] I. Hamamoto and B. Mottelson, Phys. Lett. B 132 (1983) 7. [134] I. Hamamoto and H. Sagawa, Phys. Lett. B 201 (1988) 415. [135] GB. Hagemann and I. Hamamoto, Phys. Rev. C 40 (1989) 2862. [136] K. Hecht and OR. Satchler, Nuc!. Phys. 32 (1962) 286. [137] LW. Pearson and J.O. Rasmussen, NucI. Phys. 36 (1962) 666. [138] V.V. Pashkeyich and R.A. Sardaryan, NucI. Phys. A 65 (1965) 401. [139] J. Meyer-Ter-Vehn, NucI. Phys. A 249 (1975) 111. [140] J. Helgesson and I. Hamamoto, Phys. Scr. 40 (1989) 595. [141] ER. Marshalek, Phys. Rev. C 11(1975)1426. [142] S.G. Pang, A. Klein and R.M. Dreizler, Ann. Phys. (NY) 49 (1968) 477. [143] A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press., Princeton, 1957). [144] K. Neergárd, P. Vogel and M. Radomski, NucI. Phys. A 238 (1975) 199. [145] PA. Braun, AM. Shirokov and Yu.F. Smirnov, Mo!. Phys. 56 (1985) 573. [146] W. Kolos and L. Wo!niewicz, Rev. Mod. Phys. 35(1963) 473.

279