Thouless–Valatin Rotational Moment of Inertia from the Linear Response Theory

Kristian Petrık∗

Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-40014 Helsinki, Finland andDepartment of Physics, University of Jyvaskyla, P.O. Box 35, FI-40014 Jyvaskyla, Finland

Markus Kortelainen†

Department of Physics, University of Jyvaskyla, P.O. Box 35, FI-40014 Jyvaskyla, Finland andHelsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-40014 Helsinki, Finland

(Dated: August 27, 2018)

Spontaneous breaking of continuous symmetries of a nuclear many–body system results in ap-pearance of zero–energy restoration modes. Such modes introduce a non–physical contributions tothe physical excitations called spurious Nambu–Goldstone modes. Since they represent a specialcase of collective motion, they are sources of important information about the Thouless–Valatininertia. The main purpose of this work is to study the Thouless–Valatin rotational moment of iner-tia as extracted from the Nambu–Goldstone restoration mode that results from the zero–frequencyresponse to the total angular momentum operator. We examine the role and effects of the pair-ing correlations on the rotational characteristics of heavy deformed nuclei in order to extend ourunderstanding of superfluidity in general. We use the finite amplitude method of the quasiparticlerandom phase approximation on top of the Skyrme energy density functional framework with theHartree–Fock–Bogoliubov theory. We have successfully extended this formalism and established apractical method for extracting the Thouless–Valatin rotational moment of inertia from the strengthfunction calculated in the symmetry restoration regime. Our results reveal the relation between thepairing correlations and the moment of inertia of axially deformed nuclei of rare–earth and actinideregions of the nuclear chart. We have also demonstrated the feasibility of the method for obtainingthe moment of inertia for collective Hamiltonian models. We conclude that from the numericaland theoretical perspective, the finite amplitude method can be widely used to effectively studyrotational properties of deformed nuclei within modern density functional approaches.

PACS numbers: 21.60.Jz, 21.60.Ev, 67.85.De

I. INTRODUCTION

Collective modes or excitations of many–fermion sys-tems are excellent sources of vital information about theproperties of the effective interaction among the con-stituent particles and correlations that govern the dy-namics of the many–body system. It is a major objectivefor the nuclear theory to extensively describe all types ofcollective nuclear motion. Studying the low–lying exci-tations brings better understanding of the properties ofpairing correlations, shell structure or nuclear deforma-tion, while the higher–lying giant nuclear resonances re-veal essential details about nuclear photoabsorption reac-tions or nuclear matter symmetry energy and compress-ibility [1–4]. One can analyze a response of a nucleus to atime–dependent external field to gain important knowl-edge about such excited states.

There exist several groups of models dealing withthe many–body structure and low–energy dynamics ofatomic nucleus, for example interacting shell–model ap-proaches, ab–initio methods, or the density functionaltheory (DFT) formalism. The nuclear DFT, belongingto the family of the self–consistent mean–field (SCMF)approaches, determines the nuclear one–body mean–fieldby using an energy density functional (EDF) adjusted

∗ kristian.k.petrik@jyu.fi† markus.kortelainen@jyu.fi

for a given task [5]. Withing the SCMF approach, thenuclear mean–field is obtained by solving Hartree–Fock(HF) or Hartree–Fock–Bogoliubov (HFB) equations self–consistently. Currently, the nuclear DFT can rathersuccessfully describe the nuclear ground state propertiesthroughout the whole nuclear chart [6–8].

An important ingredient of the nuclear DFT approachis the concept of a spontaneous symmetry breaking.A mean–field wave function, obtained self–consistently,usually breaks some of the symmetries of the full Hamil-tonian. In principle, an exact ground state wave func-tion of a nucleus does not break symmetries of the un-derlying many–body Hamiltonian, however, such kind ofwave–function is often computationally out of reach andsimplifying approximations have to be used. Effectively,the spontaneous symmetry breaking allows to introducediverse short–range and long–range correlations to thedeformed wave function.

In order to access various excitation modes, one needsto go beyond the static mean–field. One of the mostutilized methods is the linear response theory, that is,the random–phase approximation (RPA) theory. Excel-lent progress has been achieved with the application ofthe RPA and the quasiparticle random–phase approxi-mation (QRPA), the superfluid extension of the RPA,to the excited collective states of nuclei within the nu-clear DFT framework. Modern approaches usually em-ploy the self–consistent RPA or QRPA calculations to-gether with well–established EDFs [5, 9–12]. However,

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traditional approaches, like the very successful matrixformulation of the quasiparticle random–phase approx-imation (MQRPA), suffer from the fact that they arecomputationally heavy, especially when spherical sym-metry becomes broken. This is the main reason why theQRPA formalism has been able to treat the deformednuclei only recently.

To deal with various drawbacks of standard meth-ods for collective excitations, a very efficient formal-ism has been proposed. The so–called finite amplitudemethod (FAM) was introduced, first to calculate the RPAstrength functions [13] and later extended to sphericallysymmetric QRPA (FAM–QRPA) [14]. The FAM–QRPApromptly became a powerful tool to effectively handle abroad range of nuclear phenomena. For example, it wassuccessfully applied to study deformed axially symmet-ric nuclear systems within the HFB–Skyrme framework,individual QRPA modes, beta decay modes, collectivemoments of inertia, quadrupole and octupole strengthsor the giant dipole resonance (GDR) [15–22]. Due tomany distinct advantages, we chose the FAM–QRPA asthe main theoretical framework for the purposes of thiswork.

The properties and dynamics of heavy nuclear systemscan provide helpful insights for the physics of confinedatomic fermions, especially when superfluid characteris-tics are of interest [23–29]. Although the length and en-ergy scales of confined atomic systems are quite differentto those of nuclear scale, the rotating superfluid Fermiliquid drop picture of a nucleus covers the essential con-cepts to build up a connection between these two fields.One of the connecting ingredients is the moment of iner-tia, which is believed to provide an unambiguous signa-ture of superfluidity in general. Since the similarities be-tween nuclear systems and trapped fermions are strong,studying the rotational moment of inertia of heavy de-formed nuclei by the best available theoretical methodswill offer crucial information about the phenomenon ofsuperfluidity. This paper is thus aimed to present resultsthat will be valuable not only for the nuclear physics, butfor the physics of trapped fermionic gases as well. Suchuniversality can also help to study the transition betweenmacroscopic and microscopic fermionic systems.

It is well know that the experimental moments of in-ertia of nuclei are usually notably smaller than the cor-responding rigid body values [30]. When deriving theexpression for the moment of inertia, the simplest caseleads to the Inglis moment of inertia, which representsthe response of free gas. The moment of inertia resultingfrom this basic Inglis formula is typically very close to therigid body values. In order to improve the description ofthe data, one has to take into account the pairing cor-relations, which lead, within the BCS formalism, to theso–called Belyaev formula. This superfluid expansion ofthe Inglis inertia lowers the theoretical values towards theexperimental results, which manifests the importance ofthe correlations of the pairing type.

More generally, one can obtain the moment of inertiafrom the self–consistent cranking theory or the linear re-

sponse theory as derived by Thouless and Valatin [31].In this context, the inertia takes into account also thenuclear response, i.e. the effects of induced fields thatrepresent a reaction of the system to an external pertur-bation, determined by a given operator. In this way, themoment of inertia is defined by a generalized expressionthat contains, as special cases, both Inglis and Belyaevformulas.

Although the moment of inertia of superfluid nuclei issignificantly smaller than the rigid body inertia, it is stilllarger than that of a strictly irrotational motion. Fromthis observation, we can conclude that the currents insuperfluid rotating nuclei have a two–component charac-ter; the total current consists of rotational and irrota-tional components and the moment of inertia is the keyobservable to determine the effects of pairing correlationson the collective motion of nuclear systems. Analogousbehavior is found to be present in the trapped superfluidFermi gases at zero and non–zero temperatures, wherethe temperature dependence is shown to have crucial ef-fects on the moment of inertia as well as the qualitativebehavior of the currents [24, 25].

The main objective of this work is to apply the FAM–QRPA approach to study the Thouless–Valatin (TV) ro-tational moment of inertia in various nuclear systems andinspect the significance of the pairing correlations withrespect to the rotational characteristics of studied nuclei.The TV inertia can be obtained from the spurious zero–frequency (zero–energy) mode, which is a consequenceof a broken symmetry. We propose an efficient and nu-merically accessible method for extracting this quantityfrom the total angular momentum operator that acts ina role of an external perturbation. This method can beeasily used to describe collective modes of a wide varietyof deformed nuclei.

As a proof of concept, we will also demonstrate theusefulness of our method in providing local QRPA cal-culations, specifically constrained calculations of the ro-tational moment of inertia as a function of density. Inthis way, FAM–QRPA could be used to easily access col-lective mass parameters that represent a key input formicroscopic collective Hamiltonian models.

Our article is organized in the following way. In Sec.II, an overview of the theoretical framework is presented,in which the time–dependent HFB theory and FAM–QRPA approach are discussed in detail. Additionally, webriefly analyze the spontaneous symmetry breaking phe-nomenon and its connection to the collective TV inertia.In Sec. III, the numerical setup and main FAM–QRPAresults are covered. Final conclusions and prospects aregiven in Sec. IV.

II. THEORETICAL FRAMEWORK

In this section we recapitulate all necessary theoreticaldetails and go through the essential expressions that wereused as a foundation for our calculations. Main ideas andassets of the FAM–QRPA will be discussed.

3

II.1. Finite amplitude method

A static mean–field approach without pairing cor-relations (Hartree–Fock) or with pairing correlations(Hartree–Fock–Bogoliubov) can basically provide the nu-clear binding energy and other ground–state bulk prop-erties. In order to access excited states and dynamicsof the system, one has to go beyond the mean–field for-malism. One approach is to apply time–dependent ex-tensions to the stationary mean–field models. In the fol-lowing, we will use the quasiparticle random–phase ap-proximation as a theoretical method that represents asmall–amplitude limit of the time–dependent superfluidHFB (TD–HFB) theory.

The HFB ground state |Φ〉 is obtained by the mini-mization of the total energy defined through the energydensity E(ρ, κ, κ∗). In general, expectation values of anoperator with an HFB state can be expressed with one–body densities. In the quasiparticle picture, the energydensity E(ρ, κ, κ∗) introduces the one–body density ma-trix ρ and the pairing tensor κ,

ρij = 〈Φ|c+j ci|Φ〉 = (V ∗V T )ij = ρ∗ji,

κij = 〈Φ|cj ci|Φ〉 = (V ∗UT )ij = −κji, (1)

where V and U are the Bogoliubov transformation matri-ces that define the linear relation between the Bogoliubovquasiparticles a+µ , aµ and the bare particles c+k , ck,

a+µ =∑k

[Ukµc

+k + Vkµck

],

aµ =∑k

[U∗kµc

+k + V ∗kµck

]. (2)

The single–particle Hamiltonian h and the pairing po-tential ∆ follow from the variation of the energy densityfunctional E with respect to ρ and κ∗,

hij(ρ, κ, κ∗) =

∂E∂ρij

, ∆ij(ρ, κ, κ∗) =

∂E∂κ∗ij

. (3)

By minimizing the total Routhian, one can derive theHFB equations in terms of the transformation matricesV and U ,(

h− λ ∆−∆∗ −h∗ + λ

)(UµVµ

)= Eµ

(UµVµ

), (4)

where Eµ are the quasiparticle energies and λ is thechemical potential introduced in order to fix the averageparticle number.

The TD–HFB can be used to describe the time evolu-tion of the quasiparticles under a one–body external per-turbation that induces a polarization on the HFB groundstate. Such perturbation can be expressed as a time–dependent field in a conjugated form,

F (t) = η[F (ω)e−iωt + F †(ω)eiωt

], (5)

where η is a small real parameter introduced for the pur-pose of the small amplitude approximation, used in theFAM–QRPA.

The time evolution of a quasiparticle operator underthe influence of the external field that forces the oscilla-tions can be expressed as the TD–HFB equation,

i~∂

∂taµ(t) =

[H(t) + F (t), aµ(t)

], (6)

where the quasiparticle oscillations are given as

aµ(t) = [aµ + δaµ(t)] eiEµt,

δaµ(t) = η∑ν

a+ν[Xνµ(ω)e−iωt + Y ∗νµ(ω)eiωt

], (7)

where Xνµ and Yνµ are the FAM–QRPA amplitudes andEµ is the one–quasiparticle energy.

The time–independent part F (ω) of the one–body ex-

ternal perturbation F (t) can be now expressed (underthe linear approximation) in the quasiparticle space as

F (ω) =∑µ<ν

[F 20µν(ω)a+µ a

+ν + F 02

µν(ω)aµaν

]. (8)

The external perturbation induces oscillations of thedensity atop the static HFB solution, thus the self–consistent Hamiltonian will contain also the induced partas H(t) = HHFB+δH(t), where the oscillating part δH(t)is defined similarly as the external field in Eq. (5).

The response of the self–consistent Hamiltonian, givenby the induced time–independent matrices δH20

µν and

δH02µν , can be expressed with the HFB matrices U and

V and induced fields δh, δ∆ and δ∆ as

δH20µν(ω) =

[U†δh(ω)V ∗ − V †δh(ω)TU∗

− V †δ∆(ω)∗V ∗ + U†δ∆(ω)U∗]µν,

δH02µν(ω) =

[UT δh(ω)TV − V T δh(ω)U

− V T δ∆(ω)V + UT δ∆(ω)∗U]µν. (9)

Here, the fields δh, δ∆ and δ∆ were obtained throughan explicit linearization of the Hamiltonian and can bethus expressed through fields linearized with respect toperturbed densities, namely h′ and ∆′,

δh(ω) = h′ [ρf , κf , κf ] ,

δ∆(ω) = ∆′ [ρf , κf ] ,

δ∆(ω) = ∆′ [ρf , κf ] , (10)

where the non–Hermitian density matrices depend on theexternal perturbation through the FAM–QRPA ampli-tudes,

ρf(ω) = +UX(ω)V T + V ∗Y (ω)TU†,

ρf(ω) = +V ∗X(ω)†U† + UY (ω)∗V T ,

κf(ω) = −UX(ω)TUT − V ∗Y (ω)V †,

κf(ω) = −V ∗X(ω)∗V † − UY (ω)†UT . (11)

4

To access a transition strength function one needs toobtain the FAM–QRPA amplitudes Xµν(ω) and Yµν(ω).By using the linear approximation (i.e. linear response)one can derive the FAM–QRPA equations, given as

Xµν(ω) = −δH20

µν(ω)− F 20µν(ω)

Eµ + Eν − ω,

Yµν(ω) = −δH02

µν(ω)− F 02µν(ω)

Eµ + Eν + ω.

(12)

Since induced matrices δH20µν and δH02

µν depend on Xµν

and Yµν amplitudes, one has to employ an iterative, self–consistent scheme to solve the FAM–QRPA equations.

Formally, the FAM–QRPA system (12) is equivalent tothe linear response theory,

R(ω)−1(X(ω)Y (ω)

)= −

(F 20

F 02

), (13)

where the response function R(ω) can be expressed as

R(ω)−1 =

[(A BB∗ A∗

)− ω

(1 00 −1

)], (14)

where A and B are the well–known QRPA matrices. Ifthe external field is set to zero, the linear response equa-tion (13) will transform to the standard matrix QRPAequation (

A BB∗ A∗

)(X(ω)Y (ω)

)= ω

(X(ω)−Y (ω)

). (15)

Since matrices A and B have typically very large dimen-sions in the deformed case, solving the above equation iscomputationally rather demanding. The essential assetof the FAM–QRPA lies in the fact that instead of a verytime–consuming construction and diagonalization of thelarge QRPA matrix within the standard MQRPA proce-dure, one calculates the FAM–QRPA amplitudes itera-tively with respect to the response of the self–consistentHamiltonian, i.e. induced fields δH20 and δH02, to theexternal field. This significantly reduces the computa-tional cost in comparison to the MQRPA.

The transition strength function of the operator F atthe frequency ω is defined as

dB(F ;ω)

dω= − 1

πImS(F ;ω), (16)

where the FAM–QRPA strength function S(F ;ω) can beexpressed as

S(F ;ω) =∑µ<ν

[F 20∗µν Xµν(ω) + F 02∗

µν Yµν(ω)]. (17)

In order to obtain a finite value for the FAM–QRPA tran-sition strength, a small imaginary part, ω → ωγ = ω+iγ,is added to the excitation energy. This leads to aLorentzian smearing of transition strength function witha width of Γ = 2γ. However, in order to access a Nambu–Goldstone mode, one needs to evaluate strength function(17) at the vanishing frequency ωγ = ω = 0. We willdiscuss this relation in the next subsection.

II.2. Collective Thouless–Valatin inertia andNambu–Goldstone modes

When some form of a mean–field approximation isintroduced, a spontaneous symmetry breaking phe-nomenon that contains information about correlationsto the one–body approximation can occur. Continuoussymmetries conserved by the exact many–body Hamilto-nian, which may be broken spontaneously, are the trans-lational, rotational and particle number gauge symme-try [1]. In addition, the isospin symmetry is broken ex-plicitly by the presence of the Coulomb interaction andspontaneously due to the mean–field [32].

As a result, spontaneous breaking of any of the abovecontinuous symmetries leads to the appearance of anew zero–energy mode that restores the given symmetry.Such zero–energy restoration modes are called Nambu–Goldstone (NG) modes and are a consequence of thesymmetry–broken mean–field and represent a special caseof collective motion [1, 33, 34]. They are also associ-ated with the infinitesimal transformation of the frameof reference and since they do not represent real physicalexcitations in the intrinsic frame, they are often calledspurious modes.

It is well known that breaking of the translational sym-metry introduces the center of mass NG mode, rotationalsymmetry the rotational NG mode and particle numbergauge symmetry the pairing–rotational NG mode. In theintrinsic frame, each NG mode can be associated with acollective inertia that has experimental correspondenceto an actual spectroscopy measurement in the laboratoryframe. In the QRPA framework, such collective inertiacoupled to the NG mode is called the Thouless–Valatininertia [19, 31]. For the translational (center of mass)NG mode the TV inertia is simply the total mass ofthe nucleus and is easily obtainable since the coordinateand momentum QRPA phonon operators, needed for theestimation, are known in advance. In the case of the(pairing–)rotational NG mode the TV inertia representsthe nuclear (pairing–)rotational moment of inertia of thenucleus.

With the exception of the center of mass NG mode,in order to obtain the TV inertia one has to fully solvethe QRPA equations. Although all necessary expres-sions are known in the standard matrix QRPA formal-ism, the full evaluation of the QRPA A and B matricesis computationally heavy, mostly due to their large di-mensions. Other equivalent approaches have been there-fore employed to obtain this quantity. The most impor-tant ones are the perturbation expansion procedure inthe adiabatic time–dependent HFB theory, the crankedmean–field calculations within the HFB theory and mostrecently, the FAM–QRPA approach developed for the nu-clear density functional theory.

The FAM–QRPA has been already successfully formu-lated for calculating the symmetry restoring NG modesof the translational and particle number gauge symme-tries [19, 35]. It has been shown that the TV inertiacalculated in this way provides crucial information about

5

FIG. 1. Thouless–Valatin rotational moment of inertia in erbium (a) and ytterbium (b) isotopic chains. Full lines indicatethe volume pairing and vanishing pairing options; in the latter case, the values of the TV inertia are divided by 2 in order tomake a comparison with other results. The dashed lines show the experimental data from the rotational bands (2+ states).For erbium isotopes, the volume pairing was adjusted to the experimental proton and neutron pairing gaps of 166Er, while forytterbium isotopes, the pairing gaps of 168Yb were chosen.

the ground state correlations to the relevant broken sym-metry and that the FAM–QRPA represents a very preciseand effective method to study this quantity. Our presentwork continues in this direction and extends the FAM–QRPA also to the case of the rotational NG mode andrelated rotational TV moment of inertia.

To obtain information about the given NG mode onecan use a relevant one–body operator as the external fieldand evaluate the value of the strength function at thezero frequency. In the case of the translational symmetry,which introduces the center of mass NG mode, both thecoordinate Q and momentum operators P are known andthe TV inertia is simply the total mass of the studiednucleus, MNG = Am, where A is the atomic mass numberand m mass of the nucleon. This is the only examplewhere it is not necessary to solve QRPA equations toobtain the TV inertia.

The aim of this work is to study the rotationalThouless–Valatin moment of inertia of the rotational NGmode, where the one–body operator of the interest isthe total angular momentum. Depending on the sym-metries chosen, a specific component Ji of the angularmomentum operator has to be selected. Such a compo-nent, when used as the external perturbation, leads tothe rotational TV inertia that can be extracted from thestrength function at zero frequency as

S(Ji; 0)NG =∑µ<ν

[(J20∗i )µνXµν(0) + (J02∗

i )µνYµν(0)]

= −MJiNG . (18)

Here, MJiNG is the TV moment of inertia along the i-axis.

III. NUMERICAL RESULTS

Our calculations were carried out using the FAM–QRPA code on top of the HFB code, specifically the com-

puter program HFBTHO [36], which is an HFB solver pro-viding axially symmetric solutions using the harmonic os-cillator basis with the Skyrme energy density functional.

For the HFB calculations, and subsequent FAM-QRPAcalculations, we have employed the Skyrme SkM* param-eter set [37] at the particle–hole channel. This parameterset is know to be stable to the linear response in infi-nite nuclear matter [38]. The standard definition of theSkyrme EDF and associated densities can be found e.g.in Ref. [5]. For the pairing part, the particle–particlechannel interaction was taken to be a simple contactinteraction without a density dependence, yielding thepairing energy density commonly called the volume pair-ing. In all studied cases, this type of pairing providedgenerally better reproduction of the data over the den-sity dependent mixed pairing. A specific nucleus in agiven isotopic chain was chosen to adjust the pairingstrengths separately for neutrons and protons in orderto reproduce the empirical pairing gaps. Due to the usedzero–range pairing interaction, it was necessary to intro-duce a pairing window to prevent divergent energy in thepairing channel. In this work, the quasiparticle cut–offenergy was set to 60 MeV in all considered cases. Thedeformed HFB ground state obtained through this setupbreaks the translational, rotational and particle numbersymmetries.

One of the main assets of the FAM–QRPA formal-ism is that no additional truncation or cut–offs of thetwo–quasiparticle space are imposed. This guaranteesthe full self–consistency between the QRPA solution andthe HFB ground state. Even though the underlying HFBcalculations have conserved the time–reversal symmetry,the full time–odd part of the EDF was used at the FAM–QRPA level. The FAM module implementation followedthat of Ref. [20] and the FAM equations were solved inthe presence of the simplex–y symmetry [39].

To ensure a satisfactory convergence, we have selected20 major oscillator shells for the HO basis in all our cal-

6

FIG. 2. The same as Fig. 1 but for uranium (a) and plutonium (b) isotopic chains. For uranium isotopes, the volume pairingwas adjusted to the experimental proton and neutron pairing gaps of 238U, while for plutonium isotopes, the pairing gaps of240Pu were chosen.

culations and confirmed that no relevant improvementcould be achieved from employing more shells. TheFAM–QRPA iterative solution was obtained using thevery efficient modified Broyden method [40–42] that pro-vides stable and fast convergence, especially when multi-processor tasks are considered. Values of the spatial in-tegrals were numerically approximated using the Gauss–Hermite (mesh–point number set to NGH = 80) andGauss–Laguerre (mesh–point number set to NGL = 40)quadratures.

In this work, we focus on two areas of the nuclearchart, the heavy rare–earth isotopes and heavy actinideisotopes. Both regions offer plenty of information aboutcollective properties and rotational characteristics of ax-ially deformed nuclei.

III.1. Rotational Thouless–Valatin inertia

To obtain the TV collective inertia we need to cal-culate the strength function of the symmetry restor-ing NG mode and take the value at zero energy (zerofrequency). Since we are interested in the rotationalTV inertia under the assumption of the axial symme-try and simplex–y symmetry at the HFB level, we focuson the response to the y–component of the total angularmomentum operator, which is a combination of the y–components of the orbital and spin angular momentumoperators Jy = Ly + Sy. In order to investigate the roleof the pairing with respect to the TV inertia, we comparethe calculations where pairing correlations are taken intoaccount with those calculations of vanishing pairing.

First, we have studied the rare–earth region of thenuclear chart, specifically the erbium (even–even nu-clides from 160Er to 172Er) and ytterbium (even–evennuclides from 164Yb to 174Yb) isotopic chains. In thecase of erbiums, the pairing strength parameters (V p0 =−211.20 MeV fm3 and V n0 = −178.83 MeV fm3) were ad-justed separately to 166Er proton and neutron pairinggaps of 1.20 MeV and 1.02 MeV, respectively, obtained

from the three–point mass difference formula [43]. Forytterbiums, we adjusted the pairing strengths (V p0 =−212.40 MeV fm3 and V n0 = −180.60 MeV fm3) to 168Ybproton and neutron pairing gaps of 1.26 MeV and1.09 MeV, respectively.

Consequently, we have calculated the TV inertia us-ing the same input parameters, but with the vanishingpairing option chosen (pairing strengths were selected tobe negligibly small to induce a collapse of the pairing)and compared both sets of results to the experimentalmoments of inertia extracted from the 2+ state energyof the ground–state rotational bands of studied isotopes[44]. This comparison can be found in Fig. 1.

When the volume pairing is considered, the reproduc-tion of the experimental moment of inertia by the TVrotational inertia from FAM–QRPA is good for both er-bium and ytterbium isotopic chains. It can be easilydemonstrated that by adjusting the pairing strengths ofall isotopes to the pairing gaps of a different nucleus fromthe chain, the values of the TV inertia shift up or down,basically retaining the overall pattern. Therefore, onlypartial improvements can be achieved by changing thepairing strengths from case to case with no significantimpact on the descriptive or predictive capability.

When the nuclear pairing is not taken into account,the calculated TV inertia values are markedly higher (inFig. 1, values without pairing are divided by 2 for the sakeof comparison). We can conclude that the pairing cor-relations have crucial effects on the rotational collectivemotion and the TV inertia can be several times higher incontrast to the superfluid calculations.

In the next step, we focused on the actinide region ofthe nuclear chart, where we picked the uranium (even–even nuclides from 232U to 242U) and plutonium (even–even nuclides from 236Pu to 246Pu) isotopic chains. Herewe again studied the effects of the pairing correlationson the rotational TV inertia and compared cases withand without pairing. The pairing strengths (V p0 =−216.18 MeV fm3 and V n0 = −170.95 MeV fm3) for ura-niums were adjusted to 238U proton and neutron pairing

7

FIG. 3. The flow (lines with arrows) of the induced isoscalar current density ~j0 and its magnitude (color contours and thicknessof the flow line) for 166Er with volume pairing setup are shown in the x–z plane. The full cross–section (a) shows the totalcurrent density, while the half–plots (b)–(e) indicate the partial induced currents as given by the specific QRPA blocks. Theblack dashed contour line indicates the nuclear surface at matter density of ρ0 = 0.08 fm−3. See the text for further details.

gaps of 1.11 MeV and 0.67 MeV, and pairing strengths(V p0 = −213.20 MeV fm3 and V n0 = −170.80 MeV fm3)for plutoniums to 240Pu proton and neutron gaps of0.99 MeV and 0.64 MeV, respectively. Results are shownin Fig. 2.

Once again, we obtain rather good agreement betweenthe experiment and the rotational inertia obtained bythe FAM–QRPA approach. One can see especially pre-cise, although presumably coincidental reproduction inthe case of 238U and 240Pu, which were the isotopes cho-sen for the pairing strength adjustments. The TV inertiavalues for collapsed pairing (values divided by 2 in theplots) are again notably higher, indicating the impor-tance of the pairing.

In order to test the precision and validity of our FAM–QRPA approach, we carried out several independentcalculations using a different HFB solver (the HFODDcode [45]) and extracted the moment of inertia by usingthe cranking formalism. Since the HFB cranking calcu-lations are rather time–consuming, we chose 12, 14, and16 major oscillator shells for our setup and erbium iso-tope 166Er as our test nucleus. These calculations werecompared to the 12–, 14–, and 16–shell FAM–QRPA re-sults. In all cases the same SkM∗ parametrization wasused together with identical pairing strengths.

The TV moments of inertia from the cranking formulawere obtained by extrapolation to the zero cranking fre-quency based on the behavior of the inertia for non–zerofrequencies. We have obtained an excellent agreementin all studied cases, for example, in the case of 16–shellcalculation, the FAM–QRPA TV inertia yields a valueof 34.511 MeV−1, while the cranking moment of inertiaequals 34.533 MeV−1. See supplementary material formore detailed results.

This excellent correspondence between two calcula-tions underlines the efficiency and usefulness of the FAM–QRPA formalism in calculating the rotational TV inertia

and we can confirm that our approach is very convenientand computationally significantly faster than standardcranking HFB calculations.

III.2. Induced current density

In this section, we investigate the collective rotationalbehavior originating from the total angular momentumoperator more closely. As discussed previously, the NGmode, related to the spontaneous symmetry breaking ofthe rotational symmetry, leads to the TV moment of in-ertia connected to the strength function of the angularmomentum operator at zero frequency. Application ofthis operator as an external perturbation gives rise tothe induced densities and currents of various types. Herewe want to have a look on the induced isoscalar currentdensity, ~j0 ≡ ~jn +~jp, which can give us some hints aboutthe movement of protons and neutrons in the studied iso-topes. We have selected the 166Er nucleus as an exampleand studied the cases with and without pairing present.

In Fig. 3, one can see the x–z plane cross–sectionof the nucleus (panel (a)), where the total magnitude(color contours and line thickness) along with the di-rections (lines with arrows) of the total induced currentdensity are shown. The right side of Fig. 3 (panels (b)–(e)) shows the partial contributions coming from the firstseven QRPA blocks with values normalized to the samemaximum value to make comparisons possible. Theseblocks give the largest contribution to the total TV mo-ment of inertia. When two block numbers are indicated(e.g. block 2–3), there is only one plot shown since thecontribution is identical from both QRPA blocks. Specifi-cally, the block 1 corresponds to quasiparticle transitionsbetween |Ω| = 1

2 quasiparticle states. Blocks 2 and 3

correspond to transitions between |Ω| = 12 and |Ω| = 3

2quasiparticle states, blocks 4 and 5 correspond to tran-

8

FIG. 4. The same as Fig. 3, but with vanishing pairing.

sitions between |Ω| = 32 and |Ω| = 5

2 states, and so on.The Ω quantum number denotes the total angular mo-mentum projection of the quasiparticle state along thez–axis. In this way, one can understand the importanceof quasiparticle states of a given Ω and their impact onthe total induced current. This decomposition is ratheruseful when the comparison between the cases with andwithout pairing is made. In order to examine this, wehave calculated the induced current density and matterdensity under the vanishing pairing setup as well. Theycan be found in Fig. 4.

We already know that the presence of the pairing cor-relations significantly lowers the value of the rotationalTV inertia and thus one would expect to observe this ef-fect also in the induced current density. Comparing thefull plots in Figs. 3 and 4, we clearly see that the flowin the case with pairing resembles the irrotational flowbehavior, whereas the case without pairing is noticeablycloser to the rigid body rotation. Similar kind of flowpatters also emerge with the rotation of trapped atomicgases [24], where the temperature has a critical impacton the moment of inertia and the current flow.

In order to track down the source of this effect, weanalyzed the contributions from the QRPA blocks sep-arately. Looking at the panels (b)–(e) of Figs. 3 and 4,we see that although the first couple of QRPA blocks arebasically similar, a strong effect can be seen in the blocks6 and 7 in the case without pairing. Here, we observe thecontribution that explains the major difference discussedabove. The vanishing pairing enhances the rotationalcomponent, which shifts the irrotational flow structuretowards the rigid body flow pattern. This shows thatthe role of high–Ω states becomes more prominent withvanishing pairing when compared to the superfluid case.Similar kind of behavior occurs in the rotational flowof 240Pu and also in the flow of the oblate minimum of166Er, see the supplementary material. We did not seeany large qualitative differences between the proton andneutron flow apart from the fact that the neutron flowwas stronger in general due to higher abundance of neu-

FIG. 5. Calculated Thouless–Valatin rotational moment ofinertia and the HFB binding energy of 166Er, both as functionof the deformation parameter.

trons.

III.3. Deformation dependence of TV inertia

Rare–earth isotopes described in the present work aredeformed nuclei with a moderate prolate deformation.An example of such prolate ground state can be easilyseen from the plot of the HFB binding energy versus de-formation dependence of 166Er, as shown in Fig. 5. Herewe can see that in addition to the prolate minimum, thereexists also a higher–lying oblate minimum, therefore, it isinteresting to examine the TV rotational inertia depen-dence on the deformation as well. To obtain the resultsfor 166Er seen in Fig. 5, we carried out constrained HFBcalculations and scanned deformation from −0.5 to +0.5.For each deformation point we calculated the TV inertiain the FAM–QRPA framework. As expected, the TV in-ertia vanishes in the spherical case, when the deformationis set to zero, however, once the deformation is present,it increases smoothly with the increasing deformation inboth, prolate and oblate cases.

One can observe small dips in the moment of inertia,

9

which appear when the deformation is increased fartheraway from both minima. These dips are a direct conse-quence of the behaviour of pairing gaps and pairing en-ergy with changing deformation. They are linked to thelocal increase of the pairing gaps and related increase ofthe pairing energy in the region we are focusing on. Sucha feature supports the general picture and conclusionsabout the role of the superfluidity in the properties ofthe rotational TV inertia.

These calculations are of great interest, since the rota-tional moment of inertia is one of the vital inputs for themicroscopic collective Hamiltonian model [46, 47] thatrequires such local QRPA results. Our work proves thefeasibility of the FAM–QRPA for calculating collectivemass parameters of this type. The FAM–QRPA frame-work is thus an excellent candidate for furnishing neces-sary quantities for this kind of models.

IV. CONCLUSION

In the present work, we have studied the small–amplitude collective nuclear motion related to the rota-tional moment of inertia, which is a consequence of aresponse to the total angular momentum operator act-ing in the role of the external field. We have success-fully extended the FAM–QRPA approach to the case ofthe rotational NG mode and demonstrated the numeri-cal efficiency in calculating important nuclear properties.This work also paves a way for the removal of the spu-rious component from the transition strength function,appearing with Kπ = 1+ type of operators when defor-mation is present.

Our calculations covered several axially deformed nu-clei from the rare–earth and heavy actinide regions. Theanalysis of the rotational Thouless–Valatin moment ofinertia has shown the importance of the pairing correla-tions in the description of the rotational attributes of su-perfluid nuclei. When the pairing strength was adjustedto the experimental pairing gaps, the rotational momentof inertia for most of the isotopes was reproduced verywell. The high–Ω quasiparticle states were found to havea major role in increasing the moment of inertia value inthe case with no pairing considered.

Lastly, we demonstrated that FAM–QRPA can be usedto extract the rotational moment of inertia within theconstrained HFB calculations. Such kind of local QRPAcalculations are essential when constructing a micro-scopic collective Hamiltonian model. As a future avenue,we therefore plan to employ the FAM–QRPA approachto provide such microscopic input for a collective Hamil-tonian, with a purpose to apply it to the large amplitudecollective motion in nuclei.

ACKNOWLEDGMENTS

We thank Jacek Dobaczewski, Karim Bennaceur andNobuo Hinohara for valuable discussions and helpful ob-servations. This work has been supported by the Uni-versity of Jyvaskyla and Academy of Finland underthe Centre of Excellence Program 2012–2017 (Nuclearand Accelerator–Based Physics Program at JYFL) andFIDIPRO programme. We acknowledge CSC-IT Centerfor Science Ltd., Finland, for the allocation of computa-tional resources.

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