First order Coriolis-coupling for rotational spectrum of a tetrahedrally-deformed core plus one-particle system

First order Coriolis-coupling for rotational spectrum of a tetrahedrally-deformed core plus one-particle system

Abstract

The possible existence of shape-coexisting nuclear configurations with tetrahedral symmetry is receiving an increasing attention due to unprecedented nuclear structure properties, in particular in terms of the exotic 4-fold nucleonic level degeneracies and the expected long lifetimes which may become a new decisive argument in the exotic nuclei research programs. The present article addresses the rotational structure properties of the tetrahedrally-symmetric even-even core configurations coupled with a single valence nucleon. We focus on the properties of the associated Coriolis-coupling Hamiltonian proposing the solutions based on the explicit construction of the bases of the irreducible representations of the tetrahedral point-group on the one-hand side and the microscopic angular-momentum and parity projection nuclear mean-field approach on the other. It is shown that for one-particle occupying an orbital belonging to the or irreducible representation, the rotational spectrum splits into two sequences, the structures analogous to those of the rotational bands in the axially symmetric nuclei. Although the spectrum is generally more complicated for one-particle occupying a 4-fold degenerate orbital belonging to the representation, an appearance of the correlated double-sequence structures persists. The spectra of the doubly-magic tetrahedral core plus one-particle systems can be well interpreted using the analytical solutions of the first order Coriolis-coupling Hamiltonian. We introduce the notion of the generalized decoupling parameters, which determine the size of the energy-splitting between the double-sequence structures.

pacs:
21.10.Re, 21.60.Ev, 23.20.Lv

I Introduction

A great majority of atomic nuclei are non-spherical both in their ground-, and in the excited-states. This implies that their orientation in space can be defined and thus the corresponding systems may rotate collectively forming what is referred to as rotational bands. It turns out that the structure of the rotational bands and the related collective electromagnetic transitions depend on the geometrical symmetries of the nuclei in question and can be used for testing of the presence of certain point-group symmetries in nuclei.

The studies of the geometrical forms of nuclei found in the literature focus primarily on the quadrupole axial, in particular prolate and oblate shapes and their possible coexistence, and quadrupole triaxial ones; less frequently, on the octupole (pear-shape) deformations. The idea that the nuclear matter density in atomic nuclei may acquire more exotic symmetries resembling those of certain molecules was put forward already in the 30’s of the previous century in Ref. JAW37 (). It is natural to expect that nuclei in which tightly packed alpha-, and/or other light-clusters can coexist, may take more exotic symmetries and thus nearly at the same time, the alpha-cluster structures accompanied by a single-nucleon particle (hole) states have been discussed in Ref. LRH38 (). In particular, the structures composed of 4-, or 6 tightly-packed alphas become the prototypes of quantum systems, whose symmetry properties are governed by tetrahedral and octahedral point groups, and their associated the so-called double point-group realizations. At the same time the corresponding collective wave functions transform according to the irreducible representations of the point-groups in question. We return to the group-theory aspects in the more general context of non-alpha cluster nuclei in some detail later in this article.

Numerous studies of the nuclear alpha-cluster tetrahedral-symmetry prototype nucleus, O, have been undertaken in the past, cf. early Refs. DMD54 (); SLK56 () – and in many articles which followed. Specific efforts were undertaken later on to develop algebraic methods capable of describing the nuclear cluster structures, cf. e.g. Ref. RBI00 () focussing on the unitary groups and Ref. RBI02 () discussing in particular the -symmetry. Interested reader may consult e.g. Refs. RBi15 (); RBi16 (); RBi17 () and references therein, where the algebraic methods are applied in the context of various properties and observables in nuclei described within nuclear cluster structures. The most recent applications of these techniques in the context of the identification of the tetrahedral symmetry in O can be found in Refs. RBi14 (); RBi17a (), cf. also references therein.

Whereas proposing geometrical symmetries of the nuclear objects on the basis of the alpha (or for that matter any other light clusters) can be seen as a direct conceptual analogy with the molecular structures, finding such symmetries on the basis of the many-body (e.g. mean-field) Hamiltonians is a totally different matter. Among early studies addressing the microscopic origin of the alpha-structures in nuclei by beginning the description with the one-particle (single-nucleon) wave functions such as the ones generated by a mean-field Hamiltonian while taking into account the model nucleon-nucleon interactions one finds Ref. KWK58 (). Our approach is relatively close to the nuclear mean-field description, which introduces explicitly the issue of the single-particle levels of the nucleonic spectra in the tetrahedral-symmetric mean-fields leading to a number of exotic nuclear structure properties. To give an example of such exotic properties let us recall that the tetrahedral-symmetry double point group, , applicable to the mean-field Hamiltonians is characterized by two two-dimensional irreducible representations and one 4-dimensional one. This implies that certain nucleonic levels in the tetrahedral symmetry nuclei should produce a very exotic, so far unprecedented feature: some of the levels may be occupied by up to 4 nucleons of the same isospin. Thus nuclei obeying tetrahedral symmetry exactly may, among other exotic features, manifest the presence of the 16-fold degenerate particle hole excitations. One of the early predictions focusing on the 4-fold degeneracies in realistic mean-field calculations for heavy nuclei can be found in Ref. XLD94 (). The mechanisms involving the presence of highly degenerate excited states propagate in an interesting manner to the rotational properties of the systems composed of particles coupled to the collective rotors and the underlying so-called Coriolis-coupling mechanism. This mechanism will be addressed explicitly in the present article in the case of the tetrahedral-symmetry quantum rotors.

Let us mention in passing another of those exotic symmetry properties which makes the whole matter particularly interesting for the international programs of the exotic nuclei research. Indeed it can be shown that nuclei with the exact tetrahedral symmetry produce neither collective -, nor -transitions, the corresponding multipole dipole and quadrupole moments vanishing due to symmetry hindrance. Such a hindrance is expected to lead to an increase in the lifetimes of such exotic states by several orders of magnitude making them particularly attractive in the research of the exotic nuclei in which tetrahedral symmetry isomers may live significantly longer than e.g. the nuclear ground states. All these features attracted particular attention within the nuclear mean-field community. In particular, one of the moderately heavy (non-alpha-cluster) nuclei in which the presence of tetrahedral symmetry has been predicted by independent teams of researchers working with the self-consistent Hartree-Fock-Bogolyubov method is Zr nucleus as early as towards the end of the previous century, Refs. STK98 (); KMY01 () and later on, Refs. KZb06 (); TSD15 (). Later on several quantum mechanisms and their description pertinent to studying the point-group symmetries in nuclei have been developed. This concerns in particular: constructing the nuclear mean-field Hamiltonians with a predefined point group symmetry, relating systematically the Hamiltonian-symmetry groups and nuclear stability, constructing quantum rotor collective model-Hamiltonians of predefined point-group symmetry, multi-dimensional deformation spaces involving in particular the so-called isotropy groups and orbits, detailed analysis of the transformations between the laboratory and rotating frames and the associated symmetrization group, and several others. The reader interested in these issues can consult an overview article Ref. JDu13 (), cf. also Ref. DGM10 ().

The most recent discussion of a new approach to examining the experimental evidence for the presence of the tetrahedral and octahedral symmetries in nuclei focussed on the realistic example of Sm can be found in Ref. DCD18 ().

The nuclear tetrahedral symmetry invokes an extra stability leading to the so-called tetrahedral magic numbers. We have performed the angular-momentum and parity projection calculation from the tetrahedrally deformed mean-field states TSD13 (); TSD15 (), and found that the characteristic spectra suggested by the group theory naturally come out for even-even closed tetrahedral-shell nuclei by such a microscopic approach. In the present work, we extend this type of research for nuclei with a valence nucleon on top of a doubly closed tetrahedral-shell configuration at the asymptotic limit of very large tetrahedral deformations.

For the axially-symmetric quadrupole-deformed nuclei, the effect of an odd nucleon on the collective rotation is well-known and described in terms the quantum analogue of the Coriolis interaction, see e.g. Ref. BM75 (); RS80 (). In the present article we choose to follow, in analogy, the first-order Coriolis-coupling description for the strongly-deformed systems with the tetrahedral point-group symmetry by employing the techniques of the group representation theory, see below. For this purpose, the wave function of the so-called strong-coupling type BM75 (), which is suitable for large deformation, is introduced. It is found that the matrix elements of the first-order Coriolis-coupling can be diagonalized analytically and formula for the rotational excitation-energy spectrum can be derived. We present the results of the microscopic projection calculations and show that they can be interpreted in terms of the generalized decoupling parameter(s) in analogy to the axially-symmetric quadrupole deformation.

The paper is organized as follows. We present how the Coriolis coupling can be calculated in Sec. II, where the necessary mathematical ingredients are included with the help of group theory. In Sec. III we present the results of energy spectra for the typical core plus one-particle system in Zr nucleus, where the microscopic angular-momentum projection method is employed with the Hamiltonian composed of the Woods-Saxon mean-field and the schematic interactions TS12 (). The results are investigated in relation to the energy expression obtained by the calculation of the Coriolis coupling. Sec. IV is devoted to the summary and conclusions. Some mathematical details are discussed in Appendices. Preliminary results were already published in Ref. TSF14 ().

Ii First order Coriolis-coupling for tetrahedrally-deformed systems

In the present work, we formulate the generalized decoupling parameter technique known from the traditional literature describing the coupling of an odd particle with a quadrupole-deformed second-order quantum-rotor. A discussion of the structure of the Hamiltonian of such systems, in the form of the so-called particle-rotor model, can be found for instance in Sec. 4-2 of Ref. BM75 () or in Sec. 3.3 of Ref. RS80 (). The Hamiltonian in question has the general form

(1)

where the first term is a deformed nuclear mean-field Hamiltonian and the second one describes the collective rotation of the system.

The generalization considered in this article consists in obtaining a mathematically similar decoupled-band picture for the systems with tetrahedral symmetry rather than triaxial or axial ellipsoids. To introduce the framework of the presentation we first discuss in some detail the structure of the Hamiltonian applied here.

The first term in Eq. (1) represents the mean-field Hamiltonian, which is assumed to be invariant under the symmetry elements of the tetrahedral point-group, thus in general breaking the symmetry under inversion. In the following we will work under the approximation of no residual interaction included in the Hamiltonian, thus in particular ignoring the nuclear pairing. Such an approximation is partially justified by the fact that tetrahedral-symmetry nuclear-configurations are due to relatively large tetrahedral shell-gaps, the mechanism known to weaken the pairing interactions represented by the so called BCS- gap-parameter. Moreover, the presence of an odd nucleon weakens the pairing interactions even more due to the well known blocking mechanism.

The second term represents the quantum-rotor Hamiltonian. In the present work we choose a quadratic form involving the three components of the collective angular momentum operator ,

(2)

but interested reader may consult alternative formulations which can be found in the literature, cf. Ref. JDG01 (). Here and in what follows we use the body-fixed coordinate frame, and the quantities, , , , are the moment of inertia around the three principal axes. The total angular-momentum operator is composed of the rotor collective angular momentum and of the valence-particle angular momentum  contributions:

(3)

and it follows that the rotor Hamiltonian can be written down as

(4)

with

(5)

Above, describes the collective rotational energy, and the second term, , represents the so-called recoil energy of the valence particle. Some authors use the argument that this latter term, which depends only on the intrinsic degrees of freedom, can be absorbed in the mean field part of the Hamiltonian and, assuming that the corresponding modifications of the mean field are small, its presence is neglected. Other authors, arguing that the most often used mean fields do not contain the necessary framework allowing to include the recoil-term, and calculate the corresponding impact explicitly using alternative approaches, cf. e.g. Ref. ORG75 (), or deepen the detailed description involving the two-body mechanisms of the corresponding over-all effect as e.g. Ref. REO79 () and/or employ the links with other excitation modes as e.g. scissor-mode, cf. Ref. SI92 (). These early studies were followed by more recent ones but since in the present article we neglect this term as an approximation, we do not address these issues anymore. The last, so-called Coriolis-coupling term between the total system and the valence particle, , will be explicitly treated in the present work.

In the following we use the unit if not stated otherwise.

ii.1 The case of axial symmetry: splitting of rotational bands

Let us begin by recalling the axially symmetric case with the (or 3-rd) axis chosen as the symmetry axis in the body-fixed frame. The eigenvalue of angular momentum , which coincides with the eigenvalue of , is a good quantum number; see, e.g., Secs. 4-2 and 4-3 of Ref. BM75 () or Sec. 3.3.1 of Ref. RS80 (). With the requirement of the -invariance (here the signature is the operation of rotation through about the -axis in the body-fixed frame, see Sec. II.4 for details), the Coriolis-coupling effect for such a system can be easily calculated; the leading-order expression for the rotational excitation-energy spectrum is given by Eq. (4-61) in Sec. 4-3a of Ref. BM75 (), i.e.,

(6)

where . Thus, for rotational band, the spectrum splits into two sequences because of the oscillations of the second term

(7)

The size of splitting is determined by the so-called decoupling parameter

(8)

where is the axially deformed single-nucleon wave function of a valence particle. Note that the splitting of rotational energy spectrum appears only for the band in the axially-symmetric rotor (for the band with the Coriolis-coupling effect is of higher order).

In the following, we will see that in the case of tetrahedral deformation there is always -mixing and the Coriolis coupling is effective for all the rotational bands in the core plus one-particle systems. It will be further shown that the similar energy expression and the splitting to two rotational sequences are obtained by the Coriolis coupling with slightly different definition of the “decoupling parameter(s)”.

ii.2 Strong-coupling limit for the wave functions in the presence of a point-group symmetry

The eigenstates of the axially-symmetric collective-rotor Hamiltonian involve , the eigenvalues of angular-momentum and its third projection in the laboratory frame, and , the eigenvalue of in the body-fixed reference frame. These eigenstates can be taken as Wigner -functions, , depending on the Euler angles . We follow the convention of Ref. Ed57 () for the angular-momentum algebra in the present work. When analyzing systems with point-group symmetries, however, a complication arises since the constructed wave functions should transform as irreducible representations of the considered point group – in our case tetrahedral. We say that each wave-function belongs to an irreducible representation of .

Irreducible representations of the tetrahedral group will be labelled with symbol ; each irreducible representation is characterized by its dimension, . We introduce an extra quantum number to distinguish between various basis states belonging to the same representation  (anticipating the results of the discussion below a convenient choice of the quantum number  in the tetrahedral symmetry case will the so-called -doublex quantum number defined in Sec. II.4). Collective wave-functions respecting the discussed point-group symmetry can be written down as

(9)

where we also introduced the parity quantum number , and is an additional quantum number necessary to specify the point-group symmetric state with angular-momentum and parity , whose occurrence numbers, can be found in the literature, cf. e.g., Table VI and VIII in Appendix of Ref. TSD13 (). The expansion coefficients are for the moment unknown and will be specified later.

For the core plus one-particle systems, the intrinsic single-nucleon states are described by the eigenstates of the deformed mean-field Hamiltonian. These eigenstates will be denoted as since they should transform according to the irreducible representations () of the same point-group. For the sake of the following discussion it will be possible to omit other quantum numbers characterizing the single-nucleon properties. For sufficiently large deformations, the following “strong-coupling” wave function structure

(10)

is expected to be a good approximation BM75 ().

The collective and the intrinsic wave functions, and , should have consistent transformation properties in the sense that whereas the collective part transforms according to the representation here denoted as

(11)

the intrinsic (single-nucleon) wave functions transform according to representation ,

(12)

for an arbitrary symmetry-group element . Operators and are the group-representation operators acting in the spaces of intrinsic and collective wave-functions, respectively, and is the common unitary matrix for each group element in the irreducible representation (observe different orders of the indices and in Eqs. (11) and (12)). In other words, transforms the collective states in the same way as transforms the intrinsic single particle states, where is the complex conjugate representation of , cf. Ref. Ham62 ().

The transformation operators for are given explicitly by

(13)

and

(14)

Above, , and are Euler angles corresponding to the discrete rotations represented by , and , the operator of inversion in the intrinsic reference frame if contains inversion, alternatively . Operator is defined in full analogy but for the collective degrees of freedom. Note that the rotation operators for the collective and intrinsic degrees of freedom are formally different since the angles and are interchanged. This is a consequence of the fact that the components of obey the usual commutation relations of the form etc, whereas the components of satisfy the analogous commutation relation but with opposite signs on the right-hand sides. It follows that and in the following we omit the subscript “” or “” as long as there is no risk of confusion.

For the transformations of the rotor-associated functions we introduce operators identical to since the components of satisfy the same commutation relations as those of . We may straightforwardly verify that using and one obtains

(15)

and it follows that the wave function in Eq. (10) is invariant under ,

(16)

Alternatively,

(17)

We say that the results of transformations of the collective wave functions and those the intrinsic variables are conjugated, which is indeed the required symmetry property with respect to the point-group (see e.g. Sec. 4-2c of Ref. BM75 ()).

ii.3 Coriolis coupling for tetrahedrally-deformed core plus one-particle system

To discuss the spectra for the even-even core plus one-particle systems generated by the tetrahedral-symmetric Hamiltonian, we will introduce three irreducible representations of the group known in literature, cf. Secs. 9-6 and 9-7 of Hamermesh, Ref. Ham62 (). We use here the notation as in Table VIII, Appendix of Ref. TSD13 () according to which we set , and for the representations denoted as , and in the above textbook. The and orbitals are 2-fold degenerate, while the orbital is 4-fold degenerate. The irreducible representations appropriate for the boson-like tetrahedral -symmetric even-even systems are denoted according to the same references as , , , and . In the ground-state of an even-even core nucleus, all the 2- and 4-fold degenerate single-particle orbitals are fully occupied forming an configuration. Such a state may belong exclusively to the irreducible representation. It then follows that the single-particle state of the odd, valence nucleon coupled to the ground-state, determines uniquely the representation of the total odd- system.

Since the classical tetrahedral symmetric bodies have all the three principal-axis moments of inertia equal, , we impose this result in the rotor Hamiltonian in Eq. (2). Then, the total rotational energy described by in Eq. (5) is given by the usual quadratic spin dependence, . In order to obtain the spectra for the core plus one-particle systems, one has to diagonalize the first-order Coriolis-coupling Hamiltonian,

(18)

In the following we discuss diagonalization of this coupling matrix analytically by suitably constructed basis states, which can be performed exactly for and and approximately for .

Because the quantum number does not play any dynamical role for the energy spectra, we omit it to simplify the notation.

ii.4 Doublex symmetry and the corresponding good quantum number

In the following, we consider the tetrahedral group, , and the tetrahedral double group, . We will begin by specifying the body-fixed coordinate frame. For this purpose we will introduce the nuclear surface parameterization in terms of spherical harmonics

(19)

and use coordinate system for which the lowest order tetrahedral-deformed shapes are described by , see e.g. Ref. DDD07 ().

In analyzing rotational properties of nuclei whose shapes are described in terms of the spherical harmonics the discrete symmetries referred to as -signature and -simplex turned out to be very practical. They are defined in a body-fixed reference frame as the operations of rotation through the angle of about -axis, and a combination of the latter with the operation of inversion, respectively. In analogy one may introduce another useful discrete operation referred to as doublex, cf. e.g. Refs. SDF05 (); JDu13 () by In what follows it will be more practical to work with the -doublex, Here and below we use as generic symbols representing angular-momentum operators with the following correspondence

(20)

For even systems of fermions we have

(21)

and following Ref. SDF05 () the eigenvalues of can be written down as where the fourth-order roots can be parametrized with the help of . Any value of differing from the above values by an integer multiple of will be equivalent to one of the above. In what follows we will be using the doublex exponent (an analogue of the signature exponent) denoted as ; we have the correspondence and because of the presence of the factor of in the exponential in the definition of doublex operation, the physically significant values of can be determined modulo 4.

Irreducible representations will be used for examining the properties of either the collective or the intrinsic wave functions. It will be instructive to introduce certain formal properties of the basis states of the representation . Since for and groups, doublex operation associated with the -axis is among the symmetry elements, it will be possible to choose the quantum number in Eq. (9) for parametrizing its eigenvalues as follows

(22)

Thus, for the general angular-momentum eigenstate in a body-fixed reference frame, where represents the 3rd (or )-component of the angular-momentum and the parity (distinction should be made between two different roles of the symbol in the following expression), the -doublex(-exponent) is given by

(23)

where represents the doublex eigenvalue.

It follows that (mod 4), for boson systems (), and (mod 4), for fermion systems (), and it is easy to find the appropriate values of -doublex exponent in each representation. They are collected in Table 1 (see Appendix A for details).

Table 1: The dimension of the irreducible representations in the and groups and the values of -doublex exponent associated with them.

Let us notice that the -simplex operation introduced above, which is a group element of both and , satisfies

(24)

and it follows that the operation of changes the -doublex from to (mod 4). Therefore,

(25)

and for the states with and we arrive at an extra symmetry, for which . Consequently

(26)

The signs of simplex, i.e., symmetry or anti-symmetry with respect to , for all possible representations having and are summarized in Table 2 (see Appendix A for details).

Table 2: The symmetry or antisymmetry indices for the representations having and .

ii.5 Properties of the wave functions in the presence of tetrahedral-symmetry

The expansion coefficients in Eq. (9) can be represented as

(27)

and satisfy the orthonormality condition

(28)

They can be constructed in various ways. As an example, one can obtain the coefficients in Eq. (27) according to the group theory considerations and the angular-momentum coupling. Consider the representation: Its lowest possible state is . Considering the value of the -doublex and the symmetry in the previous section, one obtains and zero otherwise (the additional quantum number is unnecessary in this case since as it can be seen from Table VI in Ref. TSD13 () there is only one state in the representation). Then the coupling gives the coefficients for the and states. This process can be continued to obtain all the expansion coefficients for the representation: Those of are easily obtained because is the parity conjugate to . Others can be obtained by coupling the states and the lowest possible state of other representations because . Although all the expansion coefficients can be obtained in principle in this way, it is tedious to perform such calculations for high-spin states.

An alternative way of obtaining these coefficients is via numerical diagonalization of the projection operator onto the representation ,

(29)

within the space of . Here is the number of group elements, is the character of for the representation , cf. Ref.  Ham62 (), and is a group representation, cf. Eq. (14). This is a general way to construct basis states for an arbitrary representation of the point group. With the help of projection operator in Eq. (29) the occurrence number in Eq. (9) can be calculated as

(30)

from which follows because .

Below we will explicitly construct the tetrahedral-symmetric basis states for the core plus one-particle systems with of by coupling the even systems belonging to irreducible representation of to the lowest spin system with . In the same way, those with and are constructed by coupling and , respectively, to the lowest spin system with . The underlying coupling properties follow from the direct-product properties, , and , respectively. With this construction the Coriolis-coupling matrix elements in Eq. (18) can be diagonalized analytically for the and representations (see Appendices B and C for details). In this way one obtains the rotational energy expressions for the and representations. For the case of we are not able to obtain energy expression analytically with this construction; only an approximate expression is obtained. In the general case of representation the expansion coefficients obtained numerically from the projection operator in Eq. (29) were employed.

Without loss of generality, we choose the same phase convention for the coefficients in Eq. (27) as that of the angular-momentum state , see e.g. Ref. BM75 (); i.e., the action of the simplex operator and of the time-reversal operators on the wave function in Eq. (9) are the same:

(31)

which leads to

(32)

namely, the expansion coefficients are real for and pure imaginary for . The same phase convention is employed for the single-particle states.

ii.6 Coriolis coupling for the representation

The representation is two dimensional with -doublex . Because of the -simplex symmetry in Eq. (25), we choose

(33)

for both the collective and single-particle wave-functions. Then, for , taking into account Eq. (20), the strong-coupling wave function can be written as

(34)

and the Coriolis-coupling matrix element is given by

(35)

where the relations and the similar one for have been used. Note that () decreases (increases) by one unit (mod 4). It can be seen that the wave function in Eq. (34) has essentially the same form as the -invariant wave function for the axially symmetric rotational band in Sec. 4-2c of Ref. BM75 () (in fact, the signature operation should be replaced by the simplex operation).

As already mentioned we construct a specific collective basis wave-function with by coupling the basis states with that of the smallest spin positive-parity state of , . In fact, it is possible because , and it is enough because ; i.e., all the basis states are generated in this way (in obtaining these relations the information contained in Tables VI and VIII of Ref. TSD13 () has been used). Thus,

(36)

where is normalization constant and with and denotes the additional quantum number for the basis states of . Although we are not able to prove it generally, we have confirmed that operators and of appearing in the Coriolis coupling in Eq. (35) are diagonal within these specific basis states . If the numerically calculated basis states by diagonalizing the projection operator in Eq. (29) are employed, the matrix elements of are not diagonal and it turns out that the eigenvalues are and , corresponding to Eq. (38).

The diagonal matrix-elements in Eq. (35) with the basis state in Eq. (36) can be evaluated by using the identities of the expansion coefficients of . The details are presented in Appendices B and C, whereas the result of interest reads:

(37)

where the function is defined by the following generic expression with ;

(38)

with representing the allowed values of angular-momentum within -representation. Then the energy spectrum is given by one parameter, here denoted as ;

(39)

defined by

(40)

Consequently, the spectrum splits into two parabolas according to , and the amount of splitting is determined by the generalized decoupling parameter, . This result is structurally similar to the one valid in the case of the axial symmetry, cf. Sec. II.1, Eqs. (6) and (8).

ii.7 Coriolis coupling for the representation

The representation is parity-conjugate of and has -doublex . The basis state can be constructed by coupling the basis states with because (or equivalently, one can construct it by coupling the basis states with the smallest spin-parity state of , , because ). Again, this gives all the basis states because . Note that the corresponding -doublex exponent of the resulting wave function satisfies (mod 4), and consequently,

(41)

whereas the simplex-conjugate state is defined by

(42)

One shows that the structure of the wave functions in the representation is analogous to the one in Eq. (34) with . Here, similar calculations can be performed as in the case of , with the only difference that has opposite symmetry to as shown in Table 2 (see Appendices B and C for details). The matrix elements for the representation are then given by

(43)

where is defined by Eq. (38) with . The corresponding spectrum is given by

(44)

where the generalized decoupling parameter is defined as

(45)

This result is similar to the case with the axial symmetry, cf. Eqs. (6) and (8) in Sec. II.1.

ii.8 Coriolis coupling for the representation

It can be demonstrated that the covariant wave function for the representation has four components with the -doublex exponent :