Semiclassical description of a triaxial rigid rotor

PHYSICAL REVIEW C 76, 064309 (2007)

Semiclassical description of a triaxial rigid rotor

A. A. Raduta,1,2 R. Budaca,1 and C. M. Raduta1

1Institute of Physics and Nuclear Engineering, Bucharest, P. O. Box MG6, Romania2Department of Theoretical Physics and Mathematics, Bucharest University, P. O. Box MG11, Romania

(Received 14 September 2007; published 10 December 2007)

A triaxial rotor Hamiltonian is treated by a time-dependent variational principle, using a coherent state as atrial function. The critical points of the constant energy surface depend on the ordering relations satisfied by thethree moments of inertia. All three orderings are considered and to each of them a specific wobbling frequency isderived. The transition between two distinct phases is discussed in terms of a cranking Hamiltonian. Four distinctpairs of complex canonical phase-space coordinates are pointed out. Each of them yields, after quantization, adistinct boson expansion for the angular-momentum components. The separation of the potential energy is treatedfor one of the four phase-space bases. One of the semiclassical descriptions is used for the yrast state energies of158Er and an excellent agreement with experimental data is obtained up to very high angular momentum.

DOI: 10.1103/PhysRevC.76.064309 PACS number(s): 21.10.Re, 21.60.Ev, 27.70.+q

I. INTRODUCTION

Many collective properties of the low-lying states arerelated to the quadrupole collective coordinates. The simplestphenomenological scheme of describing them is the liquiddrop model (LD) proposed by Bohr and Mottelson [1]. Inthe intrinsic frame of reference the Schrodinger equationfor the five coordinates, β, γ , and �, can be separatedand an uncoupled equation for the variable β is obtained[2,3]. However, the rotational degrees of freedom, the Eulerangles describing the position of the intrinsic frame withrespect to the laboratory frame and the variable γ , andthe deviation from the axial symmetry are linked together.Under certain approximations [4] the equation describing thedynamic deformation γ is separated from that associated tothe rotational degrees of freedom. Recently many articleswere devoted to the study of the resulting equation for theγ variable [5–9].

Here we focus on the rotational degrees of freedom byconsidering a triaxial rotor with rigid moments of inertia. Thecoupling to other degrees of freedom, collective or individual,will be treated somewhere else. We attempt to describe, by asemiclassical procedure, the wobbling motion correspondingto various ordering relations for the moments of inertia. Wepresent four canonical conjugate variables to which correspondfour distinct boson representations for the angular momentum,respectively. These could be alternatively used for a bosondescription of the wobbling motion. We stress the fact that thesemiclassical description provides a better estimate of the zeropoint energy. Another advantage of the method presented hereover the boson descriptions consists in separating the potentialand kinetic energies.

The results are presented according to the followingplan. In Sec. II, the semiclassical formalism is presented.Four boson representations for the angular momentum areobtained in Sec. III, through a quantization procedure ofcanonical complex coordinates defining the classical phasespace. Wobbling frequencies, corresponding to three distinctorderings for the moments of inertia, are obtained in Sec. IVby expanding the energy function around the minimum points.

The case when the magnitudes of the three moments of inertiaare comparable with each other is treated in Sec. V by a specificcranking of the rotor Hamiltonian. The separation of potentialand kinetic energies is treated in Sec. VI. The application tothe case of 158Er is described in Sec. VII. The final conclusionsare summarized in Sec. VIII.

II. SEMICLASSICAL DESCRIPTION OF A TRIAXIALROTATOR

We consider a triaxial rigid rotor with the moments of inertiaIk, k = 1, 2, 3, corresponding to the axes of the laboratoryframe, described by the Hamiltonian:

HR = I 21

2I1+ I 2

2

2I2+ I 2

3

2I3. (2.1)

The angular-momentum components are denoted by Ik . Theysatisfy the following commutation relations:

[I1, I2] = iI3, [I2, I3] = iI1, [I3, I1] = iI2. (2.2)

The raising and lowering angular-momentum operators aredefined in the standard way, i.e., I± = I1 ± iI2. They satisfythe mutual commutation relations:

[I+, I−] = 2I3; [I+, I3] = −I+; [I−, I3] = I−. (2.3)

The case of rigid rotor in the intrinsic frame with the axestaken as principal axes of the inertia ellipsoid is formallyobtained by changing the sign of one component, say I2 →−I2, and replacing the moments of inertia with respect to axes1, 2, and 3 with those corresponding to the principal axes ofthe inertia ellipsoid.

This quantum mechanical object has been extensivelystudied in various contexts, including that of nuclear physics.Indeed, in Ref. [10], the authors noticed that there are somenuclei whose low-lying excitations might be described by theeigenvalues of a rotor Hamiltonian with suitable choice forthe moments of inertia. Since then, many extensions of therotor picture have been considered. We mention just a few of

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http://dx.doi.org/10.1103/PhysRevC.76.064309

A. A. RADUTA, R. BUDACA, AND C. M. RADUTA PHYSICAL REVIEW C 76, 064309 (2007)

them: the particle-rotor model [11], the two-rotors model [12]used for describing the scissors model, and the cranked triaxialrotor [13]. The extensions provide a simple description of thedata but also lead to new findings like scissors mode [12], finitemagnetic bands, and chiral symmetry [14].

In principle it is easy to find the eigenvalues of HR by usinga diagonalization procedure within a basis exhibiting the D2

symmetry. However, when we restrict the considerations tothe yrast band it is by far more convenient to use a closedexpression for the excitation energies.

An intuitive picture is obtained when two moments ofinertia, say those corresponding to axes 1 and 2, are close toeach other in magnitude and much smaller than the moment ofinertia of the third axis. The system will rotate around an axisthat lies close to the third axis. Because the third axis is almosta symmetry axis, this is conventionally called the quantizationaxis. Indeed, a basis having the angular momentum projectionon this axis as one of the quantum numbers is suitable fordescribing excitation energies and transition probabilities.Small deviations of angular momentum from the symmetryaxis can be quantized, which results in having a bosondescription of the wobbling motion. This quantization canbe performed in several distinct ways. The most popularone consists of choosing the Holstein-Primakoff (HP) bosonrepresentation for the angular-momentum components andtruncating the resulting boson Hamiltonian at the secondorder. However, the second-order expansion for the rotorHamiltonian is not sufficient to realistically describe thesystem rotating around an axis that makes a large angle withthe quantization axis. Actually, there is a critical angle wherethe results obtained by diagonalizing the expanded bosonHamiltonian is not converging. However, one knows from theliquid drop model that a prolate system in its ground staterotates around an axis that is perpendicular to the symmetryaxis. Clearly, such a picture corresponds to an angle betweenthe symmetry and rotation axes, equal to π/2 that is larger thanthe critical angle mentioned above. Therefore, this situationcannot be described with a boson representation of the HPtype. To treat the system exhibiting such a behavior onehas two options: (a) to change the quantization axis by arotation of an angle equal to π/2 and to proceed as beforein the rotated frame and (b) to keep the quantization axisbut change the HP representation with the Dyson (D) bosonexpansion.

Note that if we deal with the yrast states the zero-pointoscillation energy corresponding to the wobbling frequencycontributes to the yrast energies. There are experimental datathat cannot be described unless some anharmonic terms ofHR are taken into account. It should be mentioned thatanharmonicities may renormalize both the ground-state energyand the wobbling frequency.

In what follows we describe a simple semiclassical proce-dure where these two effects are obtained in a compact form.

We suppose that a certain class of properties of the Hamil-tonian HR can be obtained by solving the time-dependentequations provided by the variational principle:

δ

∫ t

0〈ψ(z)|H − i

∂

∂t ′|ψ(z)〉dt ′ = 0. (2.4)

If the trial function |ψ(z)〉 spans the whole Hilbert space of thewave functions describing the system, solving the equationsprovided by the variational principle is equivalent to solvingthe Schrodinger equation associated to HR . Because this task isbeyond the scope of the present work, we chose the variationalstate as:

|ψ(z)〉 = N ezI−|II 〉, (2.5)

where z is a complex number depending on time and |IM〉denotes the eigenstates of the angular-momentum operatorsI 2 and I3. N is a factor that assures that the function |ψ〉 isnormalized to unity. Its expression is given in Appendix. Thisfunction is a coherent state for the group SU (2) [15], generatedby the angular-momentum components and, therefore, issuitable for the description of the classical features of therotational degrees of freedom.

To make explicit the variational equations, we have to calcu-late the average values of HR and the time derivative operatoron the trial function ψ(z). The results for the expectation valuesof the terms involved in the rotor Hamiltonian are collected inAppendix.

The averages of HR and the time derivative operator havethe expressions:

〈H 〉 = I

4

(1

I1+ 1

I2

)+ I 2

2I3+ I (2I − 1)

2(1 + zz∗)2

×[

(z + z∗)2

2I1− (z − z∗)2

2I2− 2zz∗

I3

],⟨

∂

∂t

⟩= I (zz∗ − zz∗)

1 + zz∗ . (2.6)

Denoting the average of HR by H, the time-dependentvariational equation yields:

∂H∂z

= − 2iI z∗

(1 + zz∗)2,

∂H∂z∗ = 2iI z

(1 + zz∗)2. (2.7)

Using the polar coordinate representation of the complexvariables z = ρeiϕ , the equations of motion for the newvariables are:

∂H∂ρ

= − 4ρI ϕ

(1 + ρ2)2,

∂H∂ϕ

= 4Iρρ

(1 + ρ2)2. (2.8)

It is convenient to choose that pair of conjugate variablesthat brings the classical equations of motion in the canon-ical Hamilton form. This goal is reached by changing ρ

to

r = 2I

1 + ρ2, 0 � r � 2I. (2.9)

Indeed, in the new variables the equations of motion are:

∂H∂r

= ϕ,∂H∂ϕ

= −r . (2.10)

The minus sign from the second line of the above equa-tions suggests that ϕ and r play the role of generalizedcoordinate and momentum, respectively. In terms of thenew variables, the classical energy function acquires the

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SEMICLASSICAL DESCRIPTION OF A TRIAXIAL RIGID . . . PHYSICAL REVIEW C 76, 064309 (2007)

expression:

H(r, ϕ) = I

4

(1

I1+ 1

I2

)+ I 2

2I3+ (2I − 1)r(2I − r)

4I

×(

cos2 ϕ

I1+ sin2 ϕ

I2− 1

I3

). (2.11)

III. BOSON REPRESENTATION FORANGULAR-MOMENTUM COMPONENTS

In general, treating the classical equations is an easiertask than solving the time-dependent Scrodinger equation.For some particular cases analytical solutions are possible tobe found. Moreover, good approximation for solutions in acertain region of the coordinate space can be achieved. Theusefulness of this procedure is realized when from the classicalpicture we could come back to the initial quantum mechanicalproblem. This desire is accomplished by transcribing the aboveequations in the complex variables and then quantizing theconjugate complex coordinate. A recipe to find a pair ofcanonical complex variable is provided by Cartan’s theorem[16]. For the case to be treated here, we suggest four pairs ofcanonical complex coordinates.

To begin, let us consider the average of the angular-momentum components, expressed in terms of the (ϕ, r):

J cl+ ≡ 〈I+〉 =

√r(2I − r) · eiϕ,

J cl− ≡ 〈I−〉 =

√r(2I − r) · e−iϕ, (3.1)

J cl3 ≡ 〈I3〉 = I − (2I − r) = r − I.

The method of associating to a model Hamiltonian, a system ofclassical equations by means of a time-dependent variationalprinciple, is usually called a dequantization procedure. Ac-cordingly, the average of HR is the classical energy function,whereas 〈Ik〉 the k-th component describes the classicalangular momentum.

Let f and g be two complex functions defined on the phasespace spanned by the canonical conjugate variables (ϕ, r). ThePoisson bracket associated to these functions is defined by:

{f, g} = ∂f

∂ϕ

∂g

∂r− ∂f

∂r

∂g

∂ϕ. (3.2)

With this definition Eqs. (2.10) can be written as:

{r,H} = r , {ϕ,H} = ϕ, {ϕ, r} = 1. (3.3)

The classical angular-momentum components satisfy theequations:

{J cl+ , J cl

− } = −2iJ cl3 , {J cl±, J cl

3 } = ±iJ cl±. (3.4)

The functions J cl± , J cl

3 with the inner product defined by thePoisson brackets generate a classical algebra that will bedenoted by SUcl(2).

A. Holstein-Primakoff boson expansion

Let us consider the complex coordinate

C = √2I − r · eiϕ, (3.5)

and denote by C∗ the corresponding complex conjugatevariable. They obey the equations:

{C∗, C} = i, {C,H} = C, {C∗,H} = C∗. (3.6)

These equations suggest that the complex coordinates are ofcanonical type. To quantize the classical phase space meansachieving a homeomorphism between the algebra of the C, C∗with the multiplication operation {, } and the algebra of theoperators a, a† with the commutator as inner multiplier:

(C, C∗, {, }) −→ (a, a†,−i[, ]). (3.7)

A consequence of this homeomorphism is the boson characterof the operators a, a†, expressed through the relation:

[a, a†] = 1. (3.8)

The quantization of an arbitrary function f (C, C∗) is performedby replacing C and C∗ by the operators a and a†, respectively.Concerning the terms containing mixed product of C and C∗,the product must be symmetrized first and then the complexcoordinates must be replaced by the boson operators. Thesimplest example is the angular-momentum components thatafter quantization become the operators:

J+ =√

2I

(1 − a†a

2I

) 12

a,

J− =√

2I a†(

1 − a†a

2I

) 12

, (3.9)

J3 = I − a†a.

One can check that these boson operators obey the commuta-tion relations [Eq. (2.3)] and, consequently, generate an SU(2)algebra that hereafter will be denoted by SUb(2). The productof the two successive homeomorphisms:

SU(2) → SUcl(2) → SUb(2) (3.10)

is a homeomorphism SU(2) → SUb(2) that is in fact theboson representation of the angular-momentum algebra. Equa-tions (3.9) are known under the name of boson expansion ofthe angular-momentum components. They were found out longago by Holstein and Primakoff [17] by a different method. Weshall refer to it as to the HP boson expansion.

In what follows we shall briefly mention another three bo-son expansions for angular momentum, obtained by quantizingother pairs of canonical complex variables.

B. Dyson boson expansion

One canonical pair is:

C1 = 1√2I

√r(2I − r)eiϕ,

(3.11)

B∗1 =

√2I

√2I − r

re−iϕ.

Indeed, calculating their Poisson bracket, one obtains:

{B∗1, C1} = i. (3.12)

064309-3


Through the quantization

(C1,B∗1, {, }) −→ (b, b†,−i[, ]), (3.13)

one obtains the Dyson’s boson representation (D) of angularmomentum [18]:

J D+ =

√2Ib

J D− =

√2I

(b† − (b†)2b

2I

), (3.14)

J D3 = I − b†b.

Note that although the HP expansion preserves the hermiticityproperty, the D expansion does not have such a virtue. Indeed,the Hermitian conjugate of JD

+ is not equal to JD− , although

the classical component J cl− is the complex conjugate of J cl

+ .Also, the Hermitian conjugate of the boson operator b is b†,despite the fact that their classical counterparts are not relatedby a complex conjugation operation.

C. An exponential boson expansion

Consider now the complex coordinates:

C2 = 1√2

(r − iϕ), C∗2 = 1√

2(r + iϕ). (3.15)

These coordinates are also canonically conjugate because{C∗

2 , C2} = i. The isomorphism(C2, C∗

2 , {, }) −→ (a, a†,−i[, ]

)(3.16)

yields a third boson representation of angular momentum:

J R+ =

√1√2

(a† + a) e1√2

(a†−a)

√2I − 1√

2(a† + a),

J R− =

√2I − 1√

2(a† + a) e

− 1√2

(a†−a)

√1√2

(a† + a), (3.17)

I R3 = 1√

2(a† + a) − I.

This boson expansion has been derived by one of us (A.A.R.)in Ref. [19] for the generators of the quasispin algebra.Obviously, this expansion preserves the hermiticity.

D. A new boson expansion

The pair of coordinates:

C3 = r√2I

, D∗3 = i

√2Iϕ (3.18)

satisfies the equation

{D∗3, C3} = i. (3.19)

The quantization

(C3,D∗3, {, }) −→ (b, b†,−i[, ]) (3.20)

provides a new boson representation for the angular momen-tum:

J+ = 2I

√b√2I

eb†√2I

√1 − b√

2I,

J− = 2I

√1 − b√

2Ie− b†√

2I

√b√2I

, (3.21)

J3 =√

2Ib − I.

These boson expressions satisfy the commutation relations(2.3) and, therefore, represents a new boson expansion forthe angular-momentum components operators. This expansiondoes not preserve the hermiticity. Each of these four bosonexpansions can be used to study the wobbling motion associ-ated to the Hamiltonian HR . The achievements in the field ofboson expansion for both phenomenological and microscopicoperators were reviewed in Ref. [20].

IV. HARMONIC APPROXIMATION FOR THE ENERGYFUNCTION

Solving the classical equations of motion (2.10) one findsthe classical trajectories given by ϕ = ϕ(t), r = r(t). Dueto Eq. (2.10), one finds that the time derivative of H isvanishing. That means that the system energy is a constantof motion and, therefore, the trajectory lies on the surfaceH = const.. Another restriction for trajectory consists in thefact that the classical angular-momentum square is equal toI (I + 1). The intersection of the two surfaces, defined by thetwo constants of motion, defines the manifold on which thetrajectory characterizing the system is placed. According toEq. (2.10), the stationary points, where the time derivative arevanishing, can be found just by solving the equations:

∂H∂ϕ

= 0,∂H∂r

= 0. (4.1)

These equations are satisfied by two points of the phase space:(ϕ, r) = (0, I ), (π

2 , I ). Each of these stationary points mightbe minimum for the constant energy surface provided themoments of inertia are ordered in a specific way.

Studying the sign of the Hessian associated to H, oneobtains the following: (a) If I1 > I2 > I3, then (0, I ) is aminimum point for energy, whereas (π

2 , I ) is a maximum.This situation is illustrated in Fig. 1. Expanding the energy

function around minimum and truncating the resulting seriesat second order, one obtains:

H = I

4

(1

I2+ 1

I3

)+ I 2

2I1− 2I − 1

4I

(1

I1− 1

I3

)r ′2

+ I (2I − 1)

4

(1

I2− 1

I1

)ϕ2, (4.2)

where r ′ = r − I . This energy function describes a classicaloscillator characterized by the frequency:

ω =(

I − 1

2

) √(1

I3− 1

I1

) (1

I2− 1

I1

). (4.3)

064309-4


FIG. 1. (Color online) The classical energy, given in MeV, is rep-resented as function of the phase-space coordinates r (adimensional)and ϕ[rad]. The rotor is characterized by the following moments ofinertia:I1 = 125h2 MeV−1, I2 = 42h2 MeV−1, I3 = 31.4h2 MeV−1.

(b) If I2 > I1 > I3, then (0, I ) is a maximum pointfor energy while in (π

2 , I ) energy is minimum. For thissituation the function H given by Eq. (2.11) is plotted inFig. 2. Considering the second-order expansion for the energyfunction around the minimum point one obtains:

H = I

4

(1

I1+ 1

I3

)+ I 2

2I2− 2I − 1

4I

(1

I2− 1

I3

)r ′2

+ I (2I − 1)

4

(1

I1− 1

I2

)ϕ′2, (4.4)

where r ′ = r − I, ϕ′ = ϕ − π2 . Again, we have a Hamilton

function for a classical oscillator with the frequency:

ω =(

I − 1

2

) √(1

I3− 1

I2

)(1

I1− 1

I2

). (4.5)

(c) To treat the situation when I3 is the maximal momentof inertia we change the trial function to:

|�(z)〉 = N1ez ˆI−|I, I ), (4.6)

where |I, I ) is eigenstate of I 2 and I1. It is obtained byapplying a rotation of angle π/2 around the axis OY toψ(z):

|I, I ) = e−i π2 |I, I 〉. (4.7)

The new lowering operator corresponds to the new quantiza-tion axis:

ˆI± = I2 ± iI3. (4.8)

Following the same path as for the old trial function, one

obtains the equations of motion for the new classical variables.In polar coordinates the energy function is:

H(r, ϕ) = I

4

(1

I2+ 1

I3

)+ I 2

2I1+ (2I − 1)r(2I − r)

4I

×(

cos2 ϕ

I2+ sin2 ϕ

I3− 1

I1

). (4.9)

FIG. 2. (Color online) The classical energy, given in MeV, versusthe phase-space coordinates r (adimensional) and ϕ[rad]. The energycorresponds to the following moments of inertia: I1 = 42h2 MeV−1,I2 = 125h2 MeV−1, I3 = 31.4h2 MeV−1.

When I3 > I2 > I1 the system has a minimal energy in(ϕ, r) = (π

2 , I ). For a set of moments of inertia satisfyingthe restrictions mentioned above, H is represented in Fig. 3as function of (r, ϕ). Note that the graphs 2 and 3 looksimilar. However, the coordinates (ϕ, r) involved in Fig. 3are different from those shown in Fig. 2, as suggested byEqs. (2.5) and (2.6).

The second order expansion for H(r, ϕ) yields:

H(r, ϕ) = I

4

(1

I1+ 1

I2

)+ I 2

2I3+ 2I − 1

4I

(1

I1− 1

I3

)r ′2

+ (2I − 1)2I

4

(1

I2− 1

I3

)ϕ′2. (4.10)

The oscillator frequency is:

ω =(

I − 1

2

) √(1

I1− 1

I3

)(1

I2− 1

I3

). (4.11)

FIG. 3. (Color online) The classical energy H(r, ϕ), given byEq. (4.9), versus the phase-space coordinates r (adimensional) andϕ[rad]. Energy is determined by the following moments of inertia:I2 = 42h2 MeV−1, I1 = 31.4h2 MeV−1, I3 = 125h2 MeV−1. Therepresented energy is given in units of MeV.

064309-5


V. CRANKED ROTOR HAMILTONIAN

The second-order expansion, yielding the wobbling fre-quency obtained previously, is a reasonable approximation forthe situation when the angular momentum stays close to theaxis with maximal moment of inertia. This is not, however,the case when the three moments of inertia are comparable inmagnitude. To approach such a picture we consider a constraintfor the trial function |ψ(z)〉 to provide a certain average valuefor the angular momentum:

〈�n · �I 〉 ≡ 〈cos θ I3 + sin θ I1〉 = I. (5.1)

We repeat the procedure of the time-dependent descriptionby writing down the classical equations yielded by thevariational principle with the constraint (5.1), associated tothe Hamiltonian

H = HR − λ(I3 cos θ + I1 sin θ

)(5.2)

and the trial function |ψ(z)〉 defined by Eq. (2.5). The crankedHamilton function is:

H = I

4

(1

I1+ 1

I2

)

+ I 2

2I3+ (2I − 1)r(2I − r)

4I

(cos2 ϕ

I1+ sin2 ϕ

I2− 1

I3

)

− λ(r − I ) cos θ − λ√

r(2I − r) cos ϕ sin θ. (5.3)

The equation ∂H/∂ϕ = 0 has three solutions for ϕ. Thesolution that is interesting for us is ϕ = π . The constraintequation yields:

r = I (1 + cos θ ), (5.4)

whereas the remaining equation (∂H/∂r = 0) provides theLagrange multiplier:

λ = −2I − 1

4

(1

I1− 1

I3

). (5.5)

Expanding the cranked Hamiltonian around the point (ϕ, r) =[π, I (1 + cos θ )], up to the second order in the deviations(ϕ′, r ′), one obtains:

H(r, ϕ) = I

4

(1

I1+ 1

I2

)+ I 2

2I3

+ (2I − 1)I

4

(1

I1− 1

I3

)cos2 θ

+ 2I − 1

2I

(1

I1− 1

I3

)1 − 2 sin2 θ

2 sin2 θ· r ′2

2

+ I (2I − 1)

2

[1

I2− 1

2

(1

I1+ 1

I3

)]sin2 θ · ϕ′2

2.

(5.6)

Obviously, the oscillator frequency is:

ω =√(

1

I1− 1

I3

)[1

I2− 1

2

(1

I1+ 1

I3

)] (1

2− sin2 θ

).

(5.7)

The existence condition for (ϕ, r) = [π, I (1 + cos θ )] to bea minimum point for energy is:(

1

I1− 1

I3

)(1

2− sin2 θ

)> 0,

1

I2− 1

2

(1

I1+ 1

I3

)> 0.

(5.8)

It is worth noticing that for θ = π/4, the frequency isvanishing. This suggests that θ = π/4 is a separatrice oftwo distinct phases and ω = 0 plays the role of a Goldstonemode. The classical trajectories are periodic, their periodgoing to infinity when θ is approaching the critical value ofπ/4. Another attempt to correct the wobbling frequency byaccounting for some anharmonicity effects was reported inRef. [21,22].

VI. THE POTENTIAL ENERGY

In a previous section we described four distinct bosonexpansions for the angular momentum. Inserting a chosenboson representation of the angular momentum into thestarting Hamiltonian, we are faced with finding the eigenvaluesof the resulting boson operator. Alternatively, we couldexpress first the classical Hamilton function in terms ofthe canonical complex coordinates and then, in virtue ofthe quantization rules, the complex coordinates are replacedwith the corresponding bosons. The two procedures yieldboson Hamiltonians which differ by terms multiplied withcoefficients like 1/2I . This suggests that the two procedurecoincide in the limit of large angular momentum.

The classical energy function comprises mixed terms ofcoordinate and conjugate momentum. Therefore, it is desirableto prescribe a procedure to separate the potential and kineticenergies. As a matter of fact this is the goal of the presentsection. We consider the pair of complex coordinates (C1,B∗

1).In terms of these coordinates the classical energy functionlooks like:

H = I

4

(1

I1+ 1

I2

)+ I 2

2I3+ 2I − 1

8I

(1

I1+ 1

I 2− 2

I3

)

×[r(2I − r) + k

4I

(r2B∗2

1 + 4I 2C21

)]. (6.1)

where:

r = 2I − B∗1C1, 2I − r = B∗

1C1,(6.2)

k =1I1

− 1I2

1I1

+ 1I2

− 2I3

.

The components of classical angular momentum have theexpressions:

〈I+〉 =√

2I

(B∗

1 − B∗21 C1

2I

),

〈I−〉 =√

2IC1, (6.3)

〈I3〉 = I − B∗1C1.

Here we consider the classical rotor Hamilton function asbeing obtained by replacing the operators Ik by the classical

064309-6


components expressed in terms of the complex coordinates B∗1

and C1.

H = I

4

(1

I1+ 1

I2

)

+ I 2

2I3− 1

4

(1

I1+ 1

I2− 2

I3

)H(B∗

1, C1). (6.4)

Here H denotes the term depending on the complex co-ordinates. Its quantization is performed by the followingcorrespondence:

B∗1 → x, C1 → d

dx. (6.5)

By this association, to H it corresponds a second-order differ-ential operator whose eigenvalues are obtained by solving theequation:[(

− k

4Ix4 + x2 − kI

)d2

dx2+ (2I − 1)

(k

2Ix3 − x

)

× d

dx− k

(I − 1

2

)x2

]G = E′G. (6.6)

Performing now the change of function and variable:

G =(

k

4Ix4 − x2 + kI

)I/2

F,

(6.7)

t =∫ x

√2I

dy√k

4Iy4 − y2 + kI

,

Equation (6.6) is transformed into a second-order differentialSchrodinger equation:

− d2F

dt2+ V (t)F = E′F, (6.8)

with

V (t) = I (I + 1)

4

(kIx3 − 2x

)2

k4I

x4 − x2 + kI− k(I + 1)x2 + I. (6.9)

We consider an ordering for the moments of inertia such thatk > 1. Under this circumstance the potential V (t) has twominima for x = ±√

2I , and a maximum for x = 0. For a setof moments of inertia that satisfies the restriction mentionedabove, the potential is illustrated in Fig. 4 for few angularmomenta.

The minimum value for the potential energy is:

Vmin = −kI (I + 1) − I 2. (6.10)

Note that the potential is symmetric in the variable x. Due tothis feature the potential behavior around the two minima areidentical. To illustrate the potential behavior around its minimawe make the option for the minimum x = √

2I . To this valueof x it corresponds t = 0. Expanding V (t) around t = 0 andtruncating the expansion at second order we obtain:

V (t) = −kI (I + 1) − I 2 + 2k(k + 1)I (I + 1)t2. (6.11)

FIG. 4. (Color online) The potential energy involved in Eq. (6.8),associated to the Hamiltonian HR and determined by the moments ofinertia I1 = 125h2 MeV−1, I2 = 42h2 MeV−1, I3 = 31.4h2 MeV−1,is plotted as function of the adimensional variable x, defined in thetext. The defining equation (6.9) has been used.

Inserting this expansion into Eq. (6.8), one arrives at aSchrodinger equation for an oscillator. The eigenvalues are

E′n = −kI (I + 1) − I 2 + [2k(k + 1)I (I + 1)]1/2 (2n + 1).

(6.12)

The quantized Hamiltonian associated to H has an eigenvaluethat is obtained from the above expression. The final result is:

En = I (I + 1)

2I1+

[(1

I2− 1

I1

) (1

I3− 1

I1

)I (I + 1)

]1/2

×(

n + 1

2

). (6.13)

VII. NUMERICAL APPLICATION

In the previous sections we provided several expressions forthe wobbling frequency corresponding to different orderingrelations for the moments of inertia. Also, the frequency forthe cranked triaxial rotor has been derived. Here we attempt toprove that these results are useful for describing realisticallythe yrast energies. The application refers to 158Er, where dataup to very high angular momentum are available [23]. Weconsider the case where the maximum moment of inertiacorresponds to the axis OX. In our description the yrast stateenergies are, therefore, given by

EI = I

4

(1

I2+ 1

I3

)+ I 2

2I1+ ωI

2. (7.1)

The last term in the above expression is caused by thezero-point energy of the wobbling oscillation. The wobblingfrequency was derived previously, with the result:

ωI =(

I − 1

2

) √(1

I3− 1

I1

) (1

I2− 1

I1

). (7.2)

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FIG. 5. (Color online) The excitation energies of yrast states,calculated with Eq. (7.1), are compared with the results obtained inRef. [24] by a different method as well as with the experimentaldata from Ref. [23]. The moments of inertia were fixed by a least-squares procedure with the results: I1 = 100.168h2 MeV−1. Insertingthe value of κ provided by the fitting procedure one obtains thefollowing equation relating the moments of inertia I2 and I3: 1/I2 =0.576837 + 1/I3 ± 1.519

√1/I3 − 0.00998318.

We applied a least-squares procedure to fix the momentsof inertia. Because the derivatives of the χ2 function withrespect to 1/I2 and 1/I3, respectively, are identical, the appliedprocedure provides only two variables, 1/I1 and

κ = 1

2

(1

I2+ 1

I2

)+

√(1

I3− 1

I1

)(1

I2− 1

I1

). (7.3)

The mentioned variables are the coefficients of I 2/2 and I/2in the expression of the yrast energies normalized to the stateof vanishing angular momentum.

The results of our calculations are shown in Fig. 5 where, forcomparison, the experimental data and the results obtained inRef. [24] by a different method are also plotted. As shownin Fig. 5, the agreement between the calculated and theexperimental energies is very good. Also the figure suggeststhat our description of the yrast states in the considered nucleusis better than that reported in Ref. [24].

VIII. CONCLUSION

In the present work a triaxial rigid rotor considered in thelaboratory frame has been studied semiclassically through atime-dependent variational principle using a coherent stateas a trial function. The trial function depends on a complexparameter that plays the role of a phase-space coordinate. Asuitable change of coordinate in the classical equations ofmotion is performed such that these are brought to a canonicalHamiltonian form. The stationary points for the classicalenergy function were analyzed in terms of the orderingrelations satisfied by the moments of inertia. When one ofI1 or I2 is the largest moment of inertia, the energy functionexhibits a minimum point. Expanding the energy function upto the second order around the minimum point, one obtains

a classical oscillator energy whose frequency describes theharmonic wobbling motion. The two minima describe twodistinct phases for the classical Hamiltonian function. Thisis at best seen by considering a cranked rotor Hamiltoniansubjected to the restriction that the rotation axis has a givenorientation. It turns out that starting from the reference picturedetermined by a given minimum, the rotation axis can makean angle with the axis of largest moment of inertia that variesfrom 0 to π/4, where the period of the classical trajectorybecomes infinity. The case where the moment of inertia withrespect to the axis 3 is the largest one is obtained by applyingto the former trial function a rotation of an angle π/2 aroundthe axis 2.

Therefore, we obtained the wobbling frequency for threedistinct situations when the largest moment of inertia corre-sponds alternatively to axes 1, 2, and 3. The situation whenthe three moments of inertia are comparable in magnitude istreated by a cranked rotor Hamiltonian. It is worth mentioningthe advantage of the classical description over the bosonmethod, consisting of the fact that the zero-point energy isbetter treated.

Concerning the boson description we obtained four dis-tinct boson pictures corresponding to four distinct complexcanonical coordinates obtained by exploiting the Cartan’stheorem. Applied to the angular-momentum quantization, fourdistinct boson expansions for the angular momentum havebeen derived: two of them are the well-known expansions ofHolstein-Primakoff and Dyson, one is that obtained by one ofus (A.A.R) by treating the quasispin algebra, and the last oneis a by-product of the present work. Two of them preserve thehermiticity property, whereas the other two do not.

Another feature that pleads in favor of the semiclassicaltreatment refers to the separation of the potential energy. Aprescription for getting the potential energy is presented inconnection with the complex canonical coordinates that yield,after quantization, the Dyson boson expansion.

To convince ourselves about the usefulness of our descrip-tion, we calculated the yrast state energies by using one ofthe scenarios from this work for 158Er, where data up to veryhigh angular momenta are available. The agreement with theexperimental data is quite impressive. Also, one concludes thatthe description presented here is superior to that of Ref. [24].

Our plan for the near future is to apply the present formalismto the description of the excited wobbling bands in the even-even nuclei. Also, the procedure will be extended to the caseof odd nuclei where this issue has been investigated from bothexperimental [25–27] and theoretical points of view [28,29].

APPENDIX

Here we give a list with the intermediate results concerningthe matrix elements involved in the variational equation.

The norm of the trial function is:

N = (1 + |z|2)−I . (A1)

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Some useful matrix elements are obtained by performingpartial derivatives for N−2:

〈I−〉 = N 2 ∂

∂z(N−2) = 2Iz∗

1 + zz∗ ,

〈I+〉 = N 2 ∂

∂z∗ (N−2) = 2Iz

1 + zz∗ ,

〈I+I−〉 = N 2 ∂2

∂z∂z∗ (N−2) = 2I

1 + zz∗ + 2I (2I − 1)zz∗

(1 + zz∗)2,

〈I 2+〉 = N 2 ∂2

∂z∗2 (N−2) = 2I (2I − 1)z2

(1 + zz∗)2,

〈I 2−〉 = N 2 ∂2

∂z2(N−2) = 2I (2I − 1)z∗2

(1 + zz∗)2. (A2)

Here we denoted by 〈〉 the expectation value of the involvedoperator corresponding to the trial function ψ . Using theequation

eSAe−S = A + 1

1![S, A] + 1

2![S, [S, A]] + · · · , (A3)

which holds for any operator S and A, one obtains:

〈I3〉 = I − z〈I−〉 = I − 2Izz∗

1 + zz∗ . (A4)

Using Eq. (A3) we get:

〈I 23 〉 = 〈(I3 − zI−)(I − zI−)〉 = I 2 − 2I (2I − 1)zz∗

(1 + zz∗)2. (A5)

The results for the averages of the squared components 1 and2 are obtained by combining the expressions listed above:

〈I 21 〉 = 1

4

[2I + 2I (2I − 1)

(1 + zz∗)2(z + z∗)2

],

(A6)

〈I 22 〉 = −1

4

[−2I + 2I (2I − 1)

(1 + zz∗)2(z − z∗)2

].

It is worth mentioning that the sum of the averages of the Ik

operators is I (I + 1)⟨I 2

1

⟩ + ⟨I 2

2

⟩ + ⟨I 2

3

⟩ = I (I + 1). (A7)

This is a reflexion of the fact that ψ(z) is an eigenfunction ofI 2.

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Semiclassical description of a triaxial rigid rotor