peterolivius - lth2. peter olivius, ragnar bengtsson, peter m¨oller, cranking the folded yukawa...

Extending the nuclear cranking modelto tilted axis rotation and alternative mean

field potentials

PETER OLIVIUS

PhD ThesisLund, Sweden, 2004

sid1 04-01-22, 16.031

Lund-MPh-04/01

ISBN 91-628-5966-8

Division of Mathematical Physics

Lund Institute of Technology

Box 118, SE - 221 00 Lund, Sweden

To be defended Friday 20 of Februari 2004, at 1.15 pm in lecture room F, Solvegatan14A, Lund.

c© Peter Olivius,

Tryck: Wallin & Dalholm Boktryckeri AB, 2004

Contents

Contents 3

1 Preface 51.1 About the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Abstract 7

3 Popularvetenskaplig sammanfattning pa svenska 9

4 Introduction 114.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 My contribution to the cranking model and its applications . . . . . 13

5 The model and the program 155.1 Basis states and basis transformation . . . . . . . . . . . . . . . . . . 155.2 The Hamiltonian and energy units . . . . . . . . . . . . . . . . . . . 265.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Calculation of matrix elements . . . . . . . . . . . . . . . . . . . . . 305.5 The volume conservation . . . . . . . . . . . . . . . . . . . . . . . . . 375.6 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.7 Diagonalization of the Hamiltonian matrix . . . . . . . . . . . . . . . 415.8 Angular momentum expectation values . . . . . . . . . . . . . . . . . 455.9 Particle hole excitations . . . . . . . . . . . . . . . . . . . . . . . . . 465.10 Strutinsky averaging of energy and spin . . . . . . . . . . . . . . . . 515.11 The Liquid drop model . . . . . . . . . . . . . . . . . . . . . . . . . . 545.12 Calculation of total quantities . . . . . . . . . . . . . . . . . . . . . . 585.13 The low energy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.14 Removing states with weakly interacting single-particle states . . . . 595.15 The Yrast band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.16 Spin interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.17 Errors in total quantities in each meshpoint . . . . . . . . . . . . . . 645.18 Mesh calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3

4 CONTENTS

5.19 Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.20 The use of alternative potentials . . . . . . . . . . . . . . . . . . . . 68

Bibliography 69

6 PAPER 1: On the interpretation of single-particle level dia-grams in tilted rotation of the atomic nucleus 71

7 PAPER 2: Moment of inertia properties of the rotated Folded-Yukawa potential 73

8 PAPER 3: Investigation of tilted rotation of triaxial superde-formed shapes in 166Hf 75

Chapter 1

Preface

1.1 About the thesis

This thesis consists of a popular scientific summary in Swedish, an introductionconsisting of the background and my contribution to the field, a description of themodel and the computer program used for the computations, and the followingpapers

1. Peter Olivius, Ragnar Bengtsson, On the interpretation of single-particle leveldiagrams in tilted rotation of the atomic nucleus. Accepted for publication inPhys.Rev.C

2. Peter Olivius, Ragnar Bengtsson, Peter Moller, Cranking the folded Yukawapotential

3. Peter Olivius, Ragnar Bengtsson, Investigation of tilted rotation at triaxialsuperdeformed shapes in 166Hf

1.2 Acknowledgments

I would like to thank my supervisor, Ragnar Bengtsson, for his patience and en-couragement, and for always being helpful whenever I asked him.

I would also like to thank the members of the division of Mathematical Physicsfor being friendly and helpful. A special thank to Dr. Stephanie Reimann andalso Dr. Thomas Guhr, Prof. Ikuko Hamamoto and Prof. Ingemar Ragnarssonfor lending me computer resources. My thanks also go to Dr. Peter Moller forhis hospitality and helpfulness during my visiting time at Los Alamos NationalLaboratory.

And of course I whish to thank you all, my dear family and friends, for all yoursupport and encouragement.

S.D.G.

5

Chapter 2

Abstract

The nuclear model for three dimensional cranking is extended to include a general-ization of the Strutinski renormalization procedure from principal axis cranking tocranking about an tilted axis. A new computer code has been written for realistic,full-scale calculations within the three dimensional cranking model. The code isdeveloped for calculations to produce output of high precision to allow resolutionof the often weak energy variations with the tilting angles of the direction of thecranking vector. The code also allows eigenvalues and angular momentum expec-tation values from another mean-field potential, to be input for the calculation oftotal quantities such as total energy and spin. The papers contain results from thefirst applications of the program.

7

Chapter 3

Popularvetenskapligsammanfattning pa svenska

Den foreliggande doktorsavhandlingen handlar om en karnfysikalisk modell, kalladcranking-modellen, for berakningar av egenskaper hos snabbt roterande atomkarnor.I modellen tvingas ett deformerat (icke-klotformat) kraftfalt att rotera kring enaxel genom masscentrum, samtidigt som partiklarna i atomkarnan, protonernaoch neutronerna, ror sig i sina s.k. banor inuti den begransningsyta, eller form,som kraftfaltet definierar. Modellen kombinerar klassisk fysik for rotationen avbegransningsytan, och kvantmekanik for de banor som partiklarna inuti kan rrasig i. Modellen har funnits i ungefar 50 ar, men har i de allra flesta fall anvantsunder antagandet att kraftfaltet roterar kring en s.k. troghetsaxel, da det gar attargumentera att for en klassisk fast kropp som roterar ar det mest gynnsamt (gerlagst energi) att rotera kring den kortaste av troghetsaxlarna.

Men cranking-modellen gar att generalisera till rotation kring en godtyckligaxel genom masscentrum, vilket kallas tre-dimensionell cranking. Det har visats,att under vissa omstandigheter av kvantmekanisk natur, kan det vara gynnsamtmed rotation kring en axel som inte sammanfaller med en troghetsaxel, s.k. tiltadrotation. I cranking-modellens bild av atomkarnan kan protonerna och neutronernanamligen placeras i olika banor, och for vissa val av banor kan tiltad rotation bliextra gynnsam. Arbetet har till stor del bestatt i att skriva ett nytt datorprogramfor storskaliga berakningar inom modellen for tre-dimensionell cranking, men ocksaav att vidareutvecka delar av den vanliga cranking-modellen till att galla for tre-dimensionell cranking. Det nyskrivna programmet har anvants for berakningar paframforallt 166Hf (Hafnium 166) som har 72 protoner och 94 neutroner, i sokandetefter gynnsam tiltad rotation.

Avsnittet ”The model and the program” innehaller en beskrivning av den tre-dimensionella cranking-modellen och de tillagg som gjorts, samt en beskrivning avdet datorprogram som utvecklats.

I artikel 1 har sarskilda egenskaper for tiltad rotation pavisats angaende oversattnin-

9

10CHAPTER 3. POPULARVETENSKAPLIG SAMMANFATTNING PA

SVENSKA

gen av energier fran ett referenssystem som foljer med i karnans rotation till ettreferenssystem som ser karnan rotera (laboratoriesystemet). For tiltad rotation is.k. icke-sjalvkonsistenta punkter behover partiklar ofta placeras i hoga banor settinifran karnan, for att generera den energi for hela karnan som ar lagst i laborato-riesystemet.

I artikel 2 har ett alternativt kraftfalt for partiklarna, givet av den s.k. folded-Yukawa potentialen, generaliserats till s.k. triaxial deformation, och roterats inomcranking-modellen. Banornas energier i folded-Yukawa potentialen har jamfortsmed motsvarande energier i det kraftfalt som ar standard i programmet, givet avden s.k. modifierade oscillator-potentialen. Likasa har det s.k. trohetsmomentetberaknats och jamforelser gjorts mellan de tva potentialerna.

I artikel 3 redovisas berakningar pa 166Hf med den tre-dimensionella cranking-modellen. For denna atomkarna hittades tiltade rotationer, men de ar allra lagst ienergi bara for mycket snabb rotation De banor for enskilda partiklar som orsakartiltade rotationer ar identifierade i tre analyserade exempel. Nagot andrade val avantalet protoner och/eller antalet neutroner, skulle kunna ge de tiltade rotationernaden allra lagsta energin for mindre snabb rotation.

Chapter 4

Introduction

4.1 Background

The atomic nucleus consists of particles we call protons and neutrons. The protonsare repelling each other by terms of the Coloumb interaction, while all the nucleonsare interacting with the strong interaction, holding together the nucleons to a com-pound object, if it is not decaying by fission or emission of one or several nucleons.By the properties of these forces, a distribution of the nucleons is determined, andthe mass or charge distribution defines a shape of the nucleus.

Apart from the intrinsic motion of the individual nucleons, there is the possi-bility for the nucleus to perform collective motion, i.e. a coherent motion of manynucleons, such as translation, shape vibration or rotation.

Some nuclei have in experiments shown properties consistent with nuclear mod-els including a spherical shape of the mass distribution. But in large regions of thenuclear chart, nuclei have in measurements shown sequencies of energies consistentwith rotational bands in models for deformed shapes, i.e. non-spherical shapes. In1951 it was shown that the rotational motion of nuclei follows by necessity fromthe existence of strongly deformed shapes [1]. In collective rotational motion thereare many small contributions to the angular momentum from many particles [3].Several models exist for collective rotational motion of nuclei.

In the particle-rotor model a rotor Hamiltonian, for an energy increasing withthe square of the angular momentum similar to the classical expression, is cou-pled to a one-particle Hamiltonian. It is a phenomenological model, and gives anapproximate description of low-lying bands in odd-A nuclei [4],[5].

In 1954, the cranking model was described by Inglis to determine magneticmoments of rotating molecules [2]. Usually in the cranking model, the nucleonsare moving independently of each other in a deformed mean-field generated by thenucleons themselves. The field, a static deformed field and, if included, a pairingfield, is forced to rotate uniformly about an axis through the center of mass. Thecranking model combines a description of rotational bands and of single-particle

11

12 CHAPTER 4. INTRODUCTION

states. The model applies particularly well to regions of middle and high angularmomentum, and has been the most useful model in comparison with experimentalhigh spin data [6]

The principal axes are the eigenvectors of the moment of inertia tensor [11]Normally the rotation in the cranking model is assumed to take place about one ofthe principal axes. One motivation is that the states of lowest energy for a rotorcorrespond to rotation about the axis with the largest moment of inertia, which isone of the principal axes. However, the cranking Hamiltonian is easily generalizedto rotation about an arbitrary axis. The question that naturally arises is if thereexist stable cranking solutions for rotation about a non-principal axis.

For free rotation of a rigid body in classical mechanics, the rotation is stable,with respect to small fluctuations in the angular velocity vector for rotation abouteither the short or the long principal axes. For the Riemann ellipsoid, a classicalsystem for a rotating ellipsoidal fluid shells, it can be proven that there does notexist any equilibria points where the angular momentum vector does not lie in aprincipal plane and not along a principal axis [12], or equivalently if there is anequilibrium then the angular momentum vector must lie in a principal plane oralong a principal axis.

The possibilities for nuclear collective rotation about non-principal axes, or ofthe total angular momentum vector pointing in non-principal axis directions, havebeen investigated in many works. A few results of other theoretical works done inthe field will here be mentioned.

In ref. [5] high-K bands are treated. They appear in axially symmetric nuclei,where a collective angular momentum about a principal axis perpendicular to thesymmetry axis is coupled to an angular momentum of one or several particleswith large components along the symmetry axis, which causes the total angularmomentum to lie in a principal plane. Another motion dealing with total angularmomentum not coinciding with a principal axis is wobbling [5]. In wobbling theprincipal axes of a triaxial shape are performing a precessional motion with respectto the angular momentum vector. It is rather different from tilted axis crankingwhere the shape is rotating uniformly about a tilted axis. Using time-dependentvariations, semi-classical equations for general nuclear rotation were derived in [8].Time Dependent Hartree Fock solutions to a cranked QQ-potential that is uniformlyrotating about a tilted axis, were investigated in [7]. Within a tilted axis crankingmodel, but simplified for investigation of a single-j shell, it was found that rotationabout a non-principal axis is energetically favored under certain conditions [10]. Fora three dimensional cranking model of a rotating triaxial harmonic oscillator, energyminima were found for tilted rotation for triaxial, but close to oblate deformedshapes [9].

4.2. MY CONTRIBUTION TO THE CRANKING MODEL AND ITSAPPLICATIONS 13

4.2 My contribution to the cranking model and itsapplications

A computer program was written for realistic, full-scale calculations for three-dimensional cranking of the modified harmonic oscillator including hexadecapoledeformation, in a multi dimensional mesh spanned by deformation and tilting an-gle parameters. In the beginning, attempts were made to generalize an older code,but it turned out to be rather complicated. Then a decision was made to write anew code from scratch.

The first aim for the design was to create a program that produce single particleeigenstates and eigenvalues, total energies and angular momenta of high precision,i.e. with small errors within the model compared to what would have been theresult in the ideal case of using an infinite number of basis states, infinite many anddense rotational frequencies, and an infinite number of close-lying mesh points indeformation and tilting space. As was realized during the project, the variation ofthe energies with the direction of the rotational axis was in many cases very weakin regions were minima were expected to be found, high precision was crucial toget reliable results.

Second, the program needed to be fast, to bring down the calculation timesfor large scale mesh calculations in the five dimensional space of deformation andtilting parameters. For the use of available computer resources, there was a constantstruggle between the need for high precision and the need for large meshes, thatboth cost time. Compromises had to be made to be able to get a minimal requiredprecision, and a sufficiently large mesh space of deformations and tilting anglescovered, under the constraint of a reasonable total computation time.

Third, the program is designed to be flexible, in the meaning of both the possi-bility to change many physical parameters and various limits, by only adjusting theinput file, rather than the program itself, but also the possibility to use eigenvaluesand angular momentum expectation values from alternative mean field potentialsas input for the program. The alternative input possibility was used for the Folded-Yukawa potential in the calculations of paper 2.

As a minor biproduct from the calculation of the hexadecapole matrix elements,the program also contains code for transformation of eigenstates between cylindricaland spherical harmonic oscillator bases.

The program package contains about 14000 lines, not including the comments.The development of the program and extensions made to the three-dimensional

cranking model was the major part of the work that this thesis is based upon.Invention of algorithms was required for fast calculation of very high-lying con-

figurations, one for the particle-hole excitations for the protons and the neutronsseparately, and one for the combination of these to total configurations. In order tobring down the calculation times to practical levels, the algorithms were needed forthe investigation reported in paper 1 and in the large-scale calculations to paper 3for mesh-points far away from self-consistency.

14 CHAPTER 4. INTRODUCTION

Also so-called rotated particle-hole excitations were constructed to optimize thecalculation of configurations. The procedure of the Strutinski renormalization ofthe energy was extended to the three dimensional cranking model, to a form thatsmoothly pass over in the principal axis cranking expressions in the limit when therotational axis coincide with a principal axis.

The computer program developed was used in the investigations reported in thepaper 1 and 3, and together with a folded-Yukawa program [13] for paper 2. Thepapers are briefly summarized below.

• Paper 1 deals with the interpretation of single particle levels in tilted axisrotation. The need for calculating a in the rotating frame highly excitedconfiguration in order to construct the yrast band (lowest in the lab frame)for three-dimensional cranking far away from self-consistent mesh points isexplained. The explanation involves to the component of the total angularmomentum vector perpendicular to the cranking axis and implies that single-particle energy diagrams (with respect to the cranking frequency) can not beused alone to construct the yrast band, since they lack information of theperpendicular component of the angular momentum vector.

• In paper 2 an alternative mean field, the folded-Yukawa potential, is gener-alized to triaxial shapes, and cranked about the principal axes. Properties ofthe single-particle energies and of the Strutinski smoothed moment of inertiaare investigated. Comparisons are made to the properties of the cranked Nils-son potential. The effect of the Coulomb term in the folded-Yukawa potentialon the moment of inertia is investigated.

• In paper 3 results from tilted axis cranking calculations at triaxial super-deformed shapes in 166Hf are reported. Self-consistent tilted axis crankingenergy minima of three types are found. First, for rotation about an axislying in the plane spanned by the short and intermediate principal axes.Second, for rotation about an axis lying in the plane spanned by the shortand long principal axes. Third, about an axis tilted out from all principalplanes. The single-particle levels causing the tilted energy minima to appearare shown to be members of high-j subshells. Examples of shape coexistencesare shown. The triaxial super-deformed yrast band is found to be of principalaxis cranking type for for spin ≤ 69~, but from spin 70~ and above the yrastband is tilted. Several other tilted bands are found, although they are excitedabout 500 keV to more than 1 MeV above the yrast line. Other proton andneutron numbers are discussed, which from the single-particle level diagramswith respect to the tilting angles seem likely to Cavour tilted bands more andbring them closer to the yrast line.

Chapter 5

The model and the program

5.1 Basis states and basis transformation

Among nuclear models, in the mean field approximation, the nuclear single-bodyforce potential is to the lowest order a harmonic oscillator potential.

The solutions to the time-independent Schrodinger equation for the harmonicoscillator, the harmonic oscillator eigenstates, can be used as a basis for diagonaliz-ing Hamiltonians that contain more realistic nuclear potentials where various termsare added as corrections to the harmonic oscillator.

Stretched cylindrical harmonic oscillator basis |NnzΛΩ〉The harmonic oscillator potential, for a general quadrupole deformation, is definedas

Vho(x, y, z) =12M(ω2

xx2 + ω2yy2 + ω2

zz2) (5.1)

where M is the nucleon mass, and

ωx = ω0(1− 23ε cos(γ + 2π

3 ))

ωy = ω0(1− 23ε cos(γ − 2π

3 ))

ωz = ω0(1− 23ε cos γ)

(5.2)

are the harmonic oscillator angular frequencies, defined as in the Lund convention[3], [14]. Note that we are omitting the isospin index τ for separation of protonsand neutrons, when not needed explicitly to differ proton and neutron quantities.

By adding the kinetic energy operator

T = − ~2

2M∆ = − ~2

2M(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2) (5.3)

15

16 CHAPTER 5. THE MODEL AND THE PROGRAM

one gets the harmonic oscillator Hamiltonian

Hho = T + Vho = − ~2

2M(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2) +

12M(ω2

xx2 + ω2yy2 + ω2

zz2). (5.4)

The Hamiltonian is traditionally split up in three terms

Hho = Hx + Hy + Hz

where

Hz = − ~2

2M

∂2

∂z2+

12Mω2

zz2 (5.5)

and analogously for Hx and Hy.In the case of symmetry about the z-axis, ωx = ωy = ω⊥, and the x and y-terms

are added to get

H⊥ = − ~2

2M(

∂2

∂x2+

∂2

∂y2) +

12Mω2

⊥(x2 + y2). (5.6)

Define stretched coordinates [15], which are dimensionless and obtained via a lineartransformation from the Cartesian coordinates x, y, z as

ξ =√

Mωx

~ x

η =√

Mωy

~ y

ζ =√

Mωz

~ z

(5.7)

The stretched radius vector is denoted

~rt = (ξ, η, ζ) (5.8)

The stretched spherical coordinates rt, θt, φt are defined by

ξ = rt sin θt cos φt

η = rt sin θt sin φt

ζ = rt cos θt

(5.9)

The oscillator angular frequencies are, as well as the mass, considered as constants,so the stretched partial derivatives are

∂∂ξ =

√~

Mωx

∂∂x

∂∂η =

√~

Mωy

∂∂y

∂∂ζ =

√~

Mωz

∂∂z

(5.10)

5.1. BASIS STATES AND BASIS TRANSFORMATION 17

Now define the operators

ax = 1√2(ξ + ∂

∂ξ )

ay = 1√2(η + ∂

∂η )

az = 1√2(ζ + ∂

∂ζ )

(5.11)

The stretched linear momentum operator is

(ptx, pt

y, ptz) = −i~(

∂

∂ξ,

∂

∂η,

∂

∂ζ) (5.12)

which is Hermitian. One can then express ax = 1√2(ξ + i

~ptx), so the Hermitian

adjoint operator is

a†x =1√2(ξ − i

~pt

x) =1√2(ξ − ∂

∂ξ), (5.13)

and analogously for a†y, a†z. For them, the following commutator relations hold true

[ai, a†i ] = 1, i = x, y, z

[ai, aj ] = [ai, a†j ] = 0, i, j = x, y, z i 6= j.

(5.14)

Satisfying such an algebra, the a†i :s are called creation operators and the ai:s anni-hilation operators, see [16]. Define the linear combinations

R = 1√2(ax − iay) S = 1√

2(ax + iay) (5.15)

with Hermitian adjoint operators

R† = 1√2(a†x + ia†y) S† = 1√

2(a†x − ia†y). (5.16)

They satisfy the commutator relations

[R, R†] = [S, S†] = 1, [R, S] = [R, S†] = 0 (5.17)

The stretched coordinates and their derivatives can be expressed in terms of theoperators as

ξ = 12 (R† + R + S + S†) ∂

∂ξ = 12 (R + S − S† −R†)

η = i2 (R† + S† − S −R†) ∂

∂η = i2 (R† + R− S − S†)

ζ = 1√2(a†z + az) ∂

∂ζ = 1√2(az − a†z)

(5.18)

Define the stretched orbital angular momentum operator

~lt = −i~(η

∂

∂ζ− ζ

∂

∂η, ζ

∂

∂ξ− ξ

∂

∂ζ, ξ

∂

∂η− η

∂

∂ξ

)(5.19)


and the stretched total angular momentum operator

~jt = ~lt + ~s (5.20)

where ~s is the intrinsic spin operator. The z-components are

ltz = −i~(ξ ∂∂η − η ∂

∂ξ )

jtz = ltz + sz

(5.21)

Also define the ladder operators

lt± = ltx ± ilty

s± = sx ± isy

(5.22)

By transforming the oscillator Hamiltonian terms and the stretched angular mo-mentum operators to the stretched coordinates, and substitute by operators as ineq. (5.18), one gets

H⊥ = ~ω⊥(R†R + S†S + 1)

Hz = ~ωz(a†zaz + 12 )

ltz = ~(R†R− S†S)

jtz = ~(R†R− S†S) + sz.

(5.23)

The harmonic oscillator Hamiltonian eq. (5.4) can also be expressed as

Hho =∑

i=x,y,z

~ωi(a†iai +

12) (5.24)

which always apply (also if no z-axial symmetry). Now define the stretched cylin-drical harmonic oscillator basis states, also called the Nilsson basis states, or asymp-totic basis states [3], as

|rsnzΣ〉 = Crsnz (R†)r(S†)s(a†z)nz |0〉 |Σ〉 (5.25)

where the normalization coefficient is

Crsnz= (r!s!nz!)−1/2 (5.26)

and |Σ〉 are the usual spin 1/2 eigenstates. The ket |0〉 is the ground state of theHarmonic oscillator, which is defined to satisfy

ai |0〉 = 0 , i = x, y, z (5.27)

and is given by its wave function representation

〈~rt | 0〉 = π−34 e−

12 r2

t (5.28)


which is normalized in stretched space. Note that the phase of |rsnzΣ〉 is unity.The basis states |rsnzΣ〉 are orthogonal and normalized to unity, and they aresimultaneous eigenstates of H⊥, Hz, ltz and jt

z with the eigenvalues

~ω⊥(n⊥ + 1), ~ωz(nz + 12 ), ~Λ and ~Ω (5.29)

respectively, where the quantum numbers n⊥, Λ and Ω are defined as

n⊥ = r + s, Λ = r − s, Ω = Λ + Σ (5.30)

Note that |rsnzΣ〉 is also an eigenstate of s2

s2 |rsnzΣ〉 = ~2 34|rsnzΣ〉 (5.31)

because of the factorization |rsnzΣ〉 = |rsnz〉 |Σ〉 and that |Σ〉 is an eigenstate ofs2 (and of sz). However, the basis states |rsnzΣ〉 are eigenstates to Hho only inthe z-axial symmetric case (when ωx = ωy). With the principal quantum number

N = n⊥ + nz (5.32)

we have, among other combinations, the following identical basis states

|NnzΛΩ〉 ≡ |n⊥nzΛΣ〉 ≡ |rsnzΣ〉 (5.33)

Note also that these quantum numbers are all referring to a stretched basis. Theindex t, for stretched, is omitted for them. The following quantum number rulesapply for the stretched Nilsson basis, derived from the condition that r, s, and nz

should be integers greater than or equal to zero.

N ∈ 0, 1, 2 . . .nz ∈ 0, 1, 2, . . . , NΛ ∈ −n⊥,−n⊥ + 2, . . . , n⊥ − 2, n⊥Σ ∈ − 1

2 , 12

(5.34)

and then Ω is given by eq. (5.30). For each N -shell, there are (N + 1)(N + 2) basisstates (for both spin projections). The number of basis states, for both spins, withprincipal quantum numbers N ≤ Nmax is

Nmax∑

N=0

(N + 1)(N + 2) =13(Nmax + 1)(Nmax + 2)(Nmax + 3) (5.35)

When calculating the matrix elements of the Hamiltonian with respect to the Nils-son basis, the commutators (5.14) and (5.15) are used to deduce the “derivatives”

R(R†)k = k(R†)k−1 + (R†)kR

S(S†)k = k(S†)k−1 + (S†)kS

az(a†z)k = k(a†z)k−1 + (a†z)kaz

(5.36)


and thereby the following terms of the first order

R |n⊥nzΛΩ〉 = 1√2

√n⊥ + Λ |n⊥ − 1, nz,Λ− 1, Ω− 1〉

R† |n⊥nzΛΩ〉 = 1√2

√n⊥ + Λ + 2 |n⊥ + 1, nz, Λ + 1,Ω + 1〉

S |n⊥nzΛΩ〉 = 1√2

√n⊥ − Λ |n⊥ − 1, nz,Λ + 1, Ω + 1〉

S† |n⊥nzΛΩ〉 = 1√2

√n⊥ − Λ + 2 |n⊥ + 1, nz, Λ− 1,Ω− 1〉

az |n⊥nzΛΩ〉 =√

nz |n⊥, nz − 1, Λ, Ω〉a†z |n⊥nzΛΩ〉 =

√nz + 1 |n⊥, nz + 1, Λ,Ω〉

(5.37)

For the spin ladder operators s+, s−, and the spin projection operator sz, we have

s+ |n⊥nzΛΩ〉 = ~√

( 12 − Σ)(3

2 + Σ) |n⊥nzΛ, Ω + 1〉s− |n⊥nzΛΩ〉 = ~

√( 12 + Σ)( 3

2 − Σ) |n⊥nzΛ, Ω− 1〉sz |n⊥nzΛΩ〉 = ~Σ |n⊥nzΛΩ〉

(5.38)

From the eqs. (5.36) and (5.27), one deduces the matrix element

〈0 |ajz(a

†z)

k| 0〉 =

k! , j = k

0 , j 6= k(5.39)

and analogously for ax, ay, R and S.

The natural harmonic oscillator basis |nxnynzΣ〉The natural basis for a three dimensional harmonic oscillator is a generalization ofthe one dimensional harmonic oscillator basis in terms of oscillator-quanta creationoperators, derived in textbooks in quantum mechanics, e.g. [16]. In the annihilationand creation operator notation, the harmonic oscillator terms 5.5, are given by

Hi = ~ωi(a†iai + 1

2 ), i = x, y, z (5.40)

The natural harmonic oscillator basis state is a simultaneous eigenket of Hx, Hy,Hz, with eigenvalues such that

Hi |nxnynzΣ〉 = ~ωi(ni + 12 ) |nxnynzΣ〉, i = x, y, z (5.41)

and of the spin projection operator sz such that

sz |nxnynzΣ〉 = ~Σ |nxnynzΣ〉 (5.42)


The normalized solutions are the factorized states

|nxnynzΣ〉 = |nxnynz〉 |Σ〉 =1√

nx!ny!nz!(a†x)nx(a†y)ny (a†z)

nz |0〉 |Σ〉 (5.43)

which are orthogonal to each other. The quantum number rules are

ni ∈ 0, 1, 2, . . ., i = x, y, z

Σ ∈ − 12 , 1

2(5.44)

Note that the phase is also here chosen to be unity, as traditionally.

The stretched spherical harmonic oscillator basis |NlΛΩ〉For calculation of the hexadecapole matrix elements, the stretched spherical har-monic oscillator basis |NlΛΩ〉 was used. The unstretched analogue of the basis isderived in [17]. To collect information on the various harmonic oscillator bases,the derivation in the stretched case is be presented here. We search a separated ket

|NlΛΣ〉 = |NlΛ〉 |Σ〉 (5.45)

which is a simultaneous eigenket of the spherical harmonic oscillator HamiltonianH0, the square of the stretched orbital angular momentum lt

2, the projection of thestretched orbital angular momentum ltz, and the spin projection sz, with eigenvaluessuch that

H0 |NlΛ〉 = ~ω0(N + 32 ) |NlΛ〉

lt2 |NlΛ〉 = ~2l(l + 1) |NlΛ〉

ltz |NlΛ〉 = ~Λ |NlΛ〉sz |NlΛΣ〉 = ~Σ |NlΛΣ〉

(5.46)

The operators H0 and lt2 are given by

H0 = 12~ω0(−∇2

t + r2t )

lt2 = ltx

2 + lty2 + ltz

2(5.47)

where ∇t = ( ∂∂ξ , ∂

∂η , ∂∂ζ ). In terms of the creation and annihilation operators,

eq. (5.11), we have

H0 = ~ω0(~a† · ~a + 32 )

lt2 = ~2

(~a† · ~a + (~a† · ~a)2 − (~a†)2(~a)2

)

ltz = −i~(a†xay − a†yax)

(5.48)


where the annihilation and creation vectors are defined as

~a = (ax, ay, az), ~a† = (a†x, a†y, a†z). (5.49)

Because of the analogy between the two commutators

[ai, a†j ] = δij

[ ∂∂xi

, xj ] = δij

(5.50)

and the identity (5.27) for the Harmonic oscillator ground state |0〉, the problemgiven by eq. (5.46) has an analogous problem in finding a polynomial P = P (~rt)such that

~rt · ∇tP = NP

lt2P = ~2l(l + 1)P

ltzP = ~ΛP

(5.51)

A homogenous function P of degree N satisfies by definition P (t~x) = tNP (~x) forany scalar t. It follows that ~x · ∇P = NP . To be an eigenfunction of lt

2 and ltzthe solution to 5.51 also contains spherical harmonics factors. From the identityYlm(π − θ, φ + π) = (−1)lYlm(θ, φ), the function

P (~rt) = rNt YlΛ(θt, φt) (5.52)

is easily shown to be homogeneous of degree N , if N − l is even. As usual, wedemand that l is an integer and ≥ 0, and that Λ is an integer and −l ≤ Λ ≤ l. Thesolid spherical harmonic function is defined as

YlΛ(~rt) = rltYlΛ(θt, φt) (5.53)

It is a polynomial in ξ, η, ζ, and can be expressed as the sum [18]

YlΛ(~rt) = blΛ

∑

k

(−ξ − iη)k+Λ(ξ − iη)kζl−2k−Λ

22k+Λ(k + Λ)!k!(l − Λ− 2k)!(5.54)

where the coefficient

blΛ =

√2l + 1

4π(l + Λ)!(l − Λ)! (5.55)

and the sum is taken for all integer k such that the arguments to the factorialexpressions are ≥ 0, that is

max(0,−Λ) ≤ k ≤ b(l − Λ)/2c (5.56)

Define the quantum number

n =N − l

2(5.57)


which is integer for N − l even. Eq. (5.52) can now be written as

P (~rt) = (~rt · ~rt)nYlΛ(~rt) (5.58)

The condition that P must be a polynomial is satisfied if n ≥ 0. The theory ofharmonic oscillators gives that N ≥ 0. The original problem (5.46) is thus solvedby

|NlΛ〉 = AnlΛ(~a† · ~a†)nYlΛ(~a†) |0〉 (5.59)

where AnlΛ is a normalization coefficient. The ground state, which as usual satisfyai |0〉 = 0, is identical to the one of the stretched cylindrical harmonic oscillatorbasis, with wavefunction given by eq. (5.28). A somewhat lengthy calculation showsthat the normalization coefficient is independent of Λ and that it has the value

Anl = (−1)n

√4π

(2n)!!(2n + 2l + 1)!!(5.60)

where the phase is chosen to (−1)n for consistency with the standard phase ofLaguerre polynomials, according to [17]. To conclude, the stretched spherical har-monic oscillator basis state is

|NlΛΩ〉 = Anl(~a† · ~a†)nYlΛ(~a†) |0〉 |Σ〉 (5.61)

They are normalized and orthogonal to each other, with the quantum number rules

N ∈ 0, 1, 2, . . .l ∈ N, N − 2, . . . , 0 or 1Λ ∈ −l,−l + 1, . . . , l − 1, lΣ ∈ − 1

2 , 12

(5.62)

Note that, for example the following basis states are identical

|NlΛΩ〉 ≡ |nlΛΣ〉 (5.63)

The stretched second spherical harmonic oscillator basis |NljΩ〉The stretched second spherical harmonic oscillator basis consists of states, in whicheach is a simultaneous eigenket of the spherical harmonic oscillator Hamiltonian H0

( 5.47), the square of the stretched orbital angular momentum lt2, ( 5.47), the square

of the total stretched angular momentum jt2 (the square of ~jt in equation 5.20),and of the projection operator jt

z (equation 5.21), which eigenvalues such that

H0 |NljΩ〉 = ~ω0(N + 32 ) |NljΩ〉

lt2 |NljΩ〉 = ~2l(l + 1) |NljΩ〉

jt2 |NljΩ〉 = ~2j(j + 1) |NljΩ〉jtz |NljΩ〉 = ~Ω |NljΩ〉

(5.64)


The problem is easily solved by the basis states |NlΛΩ〉, derived in section 5.1, andthe use of Clebsch- Gordan coefficients [16].

|Nl, j = l − 12 , Ω〉 =

√l+Ω+1/2

2l+1 |Nl, Λ = Ω + 12 , Ω〉 −

√l−Ω+1/2

2l+1 |Nl, Λ = Ω− 12 , Ω〉

|Nl, j = l + 12 , Ω〉 =

√l−Ω+1/2

2l+1 |Nl, Λ = Ω + 12 ,Ω〉+

√l+Ω+1/2

2l+1 |Nl, Λ = Ω− 12 , Ω〉

(5.65)The quantum number rules for this basis are

N ∈ 0, 1, 2, . . .l ∈ N, N − 2, , . . . , 0 or 1j ∈ l − 1

2 , l + 12 for l > 0, j = 1

2 for l = 0

Ω ∈ −j,−j + 1, . . . , j − 1, j

(5.66)

Note that the phase is the same as that of |NlΛΩ〉, that is (−1)n, and that thestates are also eigenstates of s2 but not of sz nor of ltz.

Transformation between |NnzΛΩ〉 and |NlΛΩ〉The transformation brackets between the stretched cylindrical harmonic oscillatorbasis, and the stretched spherical harmonic oscillator basis, are needed to the hex-adecapole matrix elements calculation, for the method chosen. I calculated thetransformation brackets in the following way. To avoid confusion, the quantumnumbers n and Λ will here only refer to the spherical basis. From the ket expres-sions (5.25) and (5.61) for the bases, we first notice that the spin projections mustbe the same to get a non-vanishing bracket

〈rsnzΣ′ |nlΛΣ〉 = 〈rsnz |nlΛ〉δΣ′Σ (5.67)

For the space-part of the kets, we get the expression

〈rsnz |nlΛ〉 = CrsnzAnl〈0 |RrSsanzz (~a† · ~a†)nYlΛ(~a†)| 0〉 (5.68)

The creation operator part in the matrix element is written in terms of the com-muting operators R†, S† and a†z. The nth power of the dot product

~a† · ~a† = 2S†R† + (a†z)2 (5.69)

is written as

(~a† · ~a†)n =n∑

p=0

(n

p

)2p(S†)p(R†)p(a†z)

2n−2p (5.70)

by the binomial theorem. The solid spherical harmonic function (5.54) is in operatorspace written as

YlΛ(~a†) = blΛ

∑

k

(−1)k+Λ(R†)k+Λ(S†)k(a†z)l−Λ−2k

2k+Λ/2(k + Λ)!k!(l − Λ− 2k)!(5.71)


which gives the product

(~a†·~a†)nYlΛ(~a†) =blΛ

2Λ/2

n∑p=0

∑

k

(n

p

)(−1)k+Λ2p−k(R†)p+k+Λ(S†)p+k(a†z)

2n−2p+l−Λ−2k

(k + Λ)!k!(l − Λ− 2k)!

(5.72)To get the matrix element (5.68) one needs to calculate the matrix element consist-ing of all the operator powers, enclosed by the ground state. By use of eqs. (5.27)and (5.36), and the fact that R, S and az are commuting, one gets

〈0 |RrSsanzz (R†)p+k+Λ(S†)p+k(a†z)

2n−2p+l−Λ−2k| 0〉 =

=

r!s!nz! , for p + k + Λ = r , p + k = s , 2n− 2p + l − Λ− 2k = nz

0 , otherwise(5.73)

Thus, the space-part factor of the transformation matrix element is

〈rsnz |nlΛ〉=(r!s!nz!)12 Anl

blΛ

2Λ/2

n∑p=0

∑

k

(n

p

)(−)k+Λ2p+kδ2n−2p+l−Λ−2k,nzδp+k,sδp+k+Λ,r

(k + Λ)!k!(l − Λ− 2k)!

(5.74)where the limits of the k-sum are given by eq. (5.56). From the linear system ofequations formed by the arguments to the Kronecker δ’s, one deduces that if 2n+ l(= N in the spherical basis) 6= r + s + nz (= N in the cylindrical basis), or if Λ (inthe spherical basis) 6= r − s (= Λ in the cylindrical basis) then 〈rsnz |nlΛ〉 = 0. Italso follows that for each p there can be at most one k = s− p.

Transformation between |NlΛΩ〉 and |NljΩ〉

The transformation brackets between the two spherical harmonic oscillator basesin this paper are simply the Clebsch-Gordan coefficients in eq. 5.65, where |NljΩ〉are expressed in terms of |NlΛΩ〉.

〈Nl, Λ = Ω + 12 , Ω |Nl, j = l − 1

2 ,Ω〉 =√

l+Ω+1/22l+1

〈Nl, Λ = Ω + 12 , Ω |Nl, j = l + 1

2 ,Ω〉 =√

l−Ω+1/22l+1

〈Nl, Λ = Ω− 12 , Ω |Nl, j = l − 1

2 ,Ω〉 = −√

l−Ω+1/22l+1

〈Nl, Λ = Ω− 12 , Ω |Nl, j = l + 1

2 ,Ω〉 =√

l+Ω+1/22l+1

(5.75)

All other transformation brackets between kets from these two bases are zero.


5.2 The Hamiltonian and energy units

The oscillator potential

The oscillator potential is in stretched coordinates, eq. (5.7), defined as

V τosc = 1

2~ωτ0 r2

t

[1 + α20Y20 + α22(Y22 + Y2−2)

+α40Y40 + α42(Y42 + Y4−2) + α44(Y44 + Y4−4)],

(5.76)

where τ = p for protons, n for neutrons. The spherical harmonics are evaluated inthe stretched angles (θt, φt), and the α-coefficients are

α20 = − 23ε

√4π5 cos γ

α22 = 23ε

√4π5

sin γ√2

α40 = 2ε4

√4π9 (

√712 cos δ4 +

√512 sin δ4 cos γ4)

α42 = −2ε4

√4π9

sin δ4√2

sin γ4

α44 = 2ε4

√4π9 (

√524 cos δ4 −

√724 sin δ4 cos γ4).

(5.77)

The α-coefficients were obtained from [19]. Note that we are here using the Lundconvention for the sign of γ. In [19] the opposite sign is used. The oscillatorpotential is a function of the position (rt, θt, φt) in stretched space and has fivedeformation parameters: (ε, γ) for the quadrupole deformation, and (ε4, γ4, δ4) forthe hexadecapole deformation. The quadrupole part of the oscillator potential canbe expressed in the familiar way

Vosc(ε4 = 0) = Vho (5.78)

where Vho is given by eq. (5.1).

The non-rotating single particle Hamiltonian

The non-rotating single particle Hamiltonian consists of the modified oscillator withhexadecapole terms, and is written as

Hτ0 = T + V τ

osc − ~ω

τ

0 κτ[2~lt · ~s~2

+ µτ( lt

2 − 〈lt2〉N~2

)](5.79)

where coordinates and operators refer to the body-fixed coordinate system Oxyz.The spin-orbit term ~lt · ~s is of a strong-force origin, and the parameter κ is thecoupling-strength [3]. The lt

2 − 〈lt2〉N term modifies the oscillator shape to givea flatter bottom and a steeper wall (between the harmonic oscillator and a squarewell), resulting in a shape closer to the one of the Woods-Saxon potential which isa more realistic mean field potential [3].

5.2. THE HAMILTONIAN AND ENERGY UNITS 27

The cranked single particle Hamiltonian

The cranked single particle Hamiltonian is

Hτ = Hτ0 − ~ω ·~j (5.80)

where τ = p for protons, n for neutrons, and

~ω ·~j = ωxjx + ωyjy + ωzjz (5.81)

is the 3D cranking term, and

~ω = (ωx, ωy, ωz) (5.82)

is the cranking vector. (The components are ofcourse not to be confused with theharmonic oscillator frequencies, although the same symbols). The tilting angles θ, φare by spherical coordinates defined as

ωx = ω sin θ cos φ

ωy = ω sin θ sin φ

ωz = ω cos θ.

(5.83)

A cranking unity vector, in the direction of the cranking vector, is also defined as

~n =1ω

~ω = (sin θ cos φ, sin θ sin φ, cos θ). (5.84)

The body-fixed (intrinsic) frame of reference is a positively oriented Cartesiancoordinate system Oxyz which follows the shape of the deformed nucleus, i.e. x, y, zaxes keep parallel to the semi axes Ax, Ay, Az, respectively, of the quadrupole de-formed equipotential surface Vosc(ε4 = 0) = V ′

osc. It is rotating in the positivedirection about the cranking vector (i.e. counter-clockwise seen from the point of~ω) with respect to the laboratory frame of reference OXY Z. The two systems canbe assumed to coincide at time zero.

The cranking Hamiltonian can be derived from a transformation of the time-dependent wavefunction Ψlab(t) from the lab frame to the body-fixed frame. Inref. [20] the derivation is given in the principal axis cranking case. The generaliza-tion to the three-dimensional cranking case is straightforward.

The time-independent Schrodinger equation in the body-fixed frame of referenceis

Hτψ = Eτψ, (5.85)

which is the eigenvalue problem that is solved.


Energy units

The deformed harmonic oscillator angular frequency ωτ0 is related to the spherical

harmonic oscillator angular frequencyω

τ

0 by

~ωτ0 = ~

ωτ

0

ω0ω0

(5.86)

where the fraction is given by the condition of volume conservation, section 5.5.The quantity

ω

τ

0 is independent of the deformation. Along the β-stability line is,except for light nuclei, the number of neutrons somewhat higher than the numberof protons. For the nuclear radius to be approximately the same for protons andneutrons, the neutron potential must be more narrow than for the protons. Thatis obtained by defining the oscillator frequency to be higher for the neutrons thanfor the protons, as

~ ω

n

0= (1 + N−Z3A )~

ω0

~ ω

p

0= (1− N−Z3A )~

ω0

(5.87)

where~

ω0= 41A−1/3 MeV (5.88)

to reproduce the experimental nuclear radii [21]. The matrix elements of the crank-ing Hamiltonian are calculated in units of ~ωτ

0 , and the program is actually diago-nalizing the dimensionless Hamiltonian

Hτ

~ωτ0

=T + V τ

osc

~ωτ0

+ω0

ω0κτ

(2~lt · ~s~2

+ µτ (lt

2 − 〈lt2〉N~2

))− ~ω

ωτ0

·~j

~(5.89)

The input cranking vector is in MeV-units

~~ωMeV

. (5.90)

It is converted to ωτ0 -units as

~ω

ωτ0

=~~ω

MeVMeV~ωτ

0

(5.91)

by the eqs. (5.86), (5.87) and (5.88), which assembled give

MeV~ωτ

0

=A1/3

ω0 /ω0

41(1± N−Z3A )

(5.92)

with plus sign for neutrons, minus sign for protons.

5.3. SYMMETRIES 29

5.3 Symmetries

In the general case for the cranking Hamiltonian, eq. (5.80) all symmetries arebroken, except the symmetry of parity due to

V (−~r) = V (~r) (5.93)

for the potential. Additionally, in the model the equipotential surface has a coordi-nate plane reflexion symmetry, that is the distance from the center of mass (origin)to the surface does not alter if the radius vector is mirrored through any of thecoordinate planes. Consider a spherical harmonics expansion of a general shape

R(θ, φ) = R0(1 +∞∑

λ=0

λ∑

µ=−λ

aλµYλµ(θ, φ)) (5.94)

The coordinate plane reflexion symmetry now implies for reflexions in the yz-planethat R(θ, φ) = R(θ, π − φ) which gives that

aλµ = aλ−µ (5.95)

and for reflexions in the xz-plane that R(θ, φ) = R(θ,−φ) which gives that

aλµ = 0, µ odd (5.96)

and in the xy-plane that R(θ, φ) = R(π − θ, φ) giving the condition

aλµ = 0, λ odd (5.97)

These conditions explain the form of the oscillator potential up to hexadecapoledeformation in eq. (5.76). In the program Ultimate Cranker [22], commonly usedfor principal axis cranking calculations, only one hexadecapole parameter, ε4 isused, but the hexadecapole deformation is restricted such that α4µ are functions of(ε4, γ) according to

α40 = ε4

√4π9

13 (5 cos2 γ + 1)

α42 = −ε4

√4π9

√306 sin 2γ

α44 = ε4

√4π9

√706 sin2 γ

(5.98)

which follows from the substitution

δ4 = arctan

√57≈ 40.203, γ4 = 2γ (5.99)

The restriction makes the hexadecapole deformation z-axis symmetric for γ = 0,just as the quadrupole deformation.


5.4 Calculation of matrix elements

The matrix elements of all operators in the cranked single particle Hamiltonianwere calculated with respect to the stretched cylindrical harmonic oscillator basis|NnzΛΩ〉. But for the hexadecapole terms, the matrix elements were first calculatedwith respect to the stretched spherical harmonic oscillator basis |NlΛΩ〉, and thentransformed to |NnzΛΩ〉 by the transformation in section 5.1.

The general method is the following. Terms in the Hamiltonian (except thehexadecapole part) are first transformed to stretched coordinates, by eqs. (5.7)and (5.10), then expressed in terms of the operators R, R†, S, S†, az, a

†z, eq. (5.18).

Finally, the matrix elements are calculated by using the terms in eq. (5.37).The parity symmetry of the Hamiltonian implies that with respect to a stretched

harmonic oscillator basis, all matrix elements between basis states for which ∆Nis odd, are zero. The Hamiltonian matrix is not split in any smaller sub matri-ces. Compared to the principal axis cranking case, where the signature symmetrygives one further matrix split, the tilted axis cranking case gives larger matrices todiagonalize, requiring more computing power or time, for meshes of similar size.

Note that different N -shells are not coupled by the matrix elements of T +Vosc(ε4 = 0), ~lt · ~s and of lt

2. But for jx, jy, jz and the hexadecapole terms coupledifferent N -shells.

Matrix elements of T + Vosc(ε4 = 0)

The harmonic oscillator part of the non-rotating single particle Hamiltonian, Hho =T + Vosc(ε4 = 0) is defined in eq. (5.4), but given in terms of the operators a†iai

by eq. (5.24). By inversion of the definitions (5.15), to express ax, a†x, ay, a†y inR,R†, S, S† one gets

Hho =~2(ωx+ωy)(R†R+S†S+1)+~ωz(a†zaz+

12)+~2(ωx−ωy)(R†S+S†R) (5.100)

The first two terms are diagonal with respect to |n⊥nzΛΩ〉, and the third is off-diagonal. The non-vanishing matrix elements are

〈n⊥nzΛΩ |Hho|n⊥nzΛΩ〉 = ~2 [(ωx + ωy)(n⊥ + 1) + ωz(1 + 2nz)]

〈n⊥nz, Λ− 2, Ω− 2 |Hho|n⊥nzΛΩ〉 = ~4 (ωx − ωy)

√(n⊥ + Λ)(n⊥ − Λ + 2)

〈n⊥nz, Λ + 2, Ω + 2 |Hho|n⊥nzΛΩ〉 = ~4 (ωx − ωy)

√(n⊥ − Λ)(n⊥ + Λ + 2)

(5.101)

Matrix elements of r2t Y4µ

No way of expressing the hexadecapole terms of the oscillator potential as a poly-nomial of annihilation and creation operators was found. The reason is occurencesof r2

t in the denominators of the Cartesian expressions for r2t Y4µ. The hexadecapole

5.4. CALCULATION OF MATRIX ELEMENTS 31

matrix elements are naturally calculated with respect to the spherical harmonicoscillator basis |c〉 ≡ |NlΛΩ〉. Denoting |b〉 ≡ |NnzΛΩ〉, the matrix elements withrespect to the Nilsson basis is the sum

〈bi |r2t Y4µ| bj〉 =

∑

k

∑

l

〈bi | ck〉〈ck |r2t Y4µ| cl〉〈cl | bj〉 (5.102)

where the transformations brackets are calculated in section 5.1.Because of the factorization given in eq. (5.45), and because the hexadecapole

operator terms have no spin operator factors, the spin projections must be the sameto get a non-vanishing matrix element of f(rt)Yλµ with respect to |NlΛΣ〉.

〈N ′l′Λ′Σ′ |f(rt)Yλµ|NlΛΣ〉 = 〈N ′l′Λ′ |f(rt)Yλµ|NlΛ〉δΣ′Σ (5.103)

In [17] the matrix elements of f(r)Yλµ(θ, φ) with respect to the spherical harmonicoscillator basis |Nlm〉, both unstretched, are calculated to be

〈Nl′m′ |f(r)Yλµ(θ, φ)|Nlm〉 =

(−1)m′√

(2l+1)(2l′+1)(2λ+1)4π

l λ l′

m µ −m′

l λ l′

0 0 0

∑

p B(n′l′nlp)Ip(f)

(5.104)where the parenthesis are Wigner-coefficients, the B-function is defined to

B(n′l′nlp) =12Γ(p + 3/2)

∑

k

anlkan′,l′,p−k− 12 (l+l′) (5.105)

where

anlk =(−1)k

k!

√2(n!)

Γ(n + l + 3/2)Γ(n + l + 3/2)

(n− k)!Γ(k + l + 3/2)(5.106)

and the summation variable runs over all integers in

max(0, p− 12(l + l′)− n′) ≤ k ≤ min(n, p− 1

2(l + l′)) (5.107)

To continue,

Ip(f) =2

Γ(p + 3/2)

∫ ∞

0

r2p+2f(r)e−r2dr (5.108)

is the Talmi integral. If the multipole order λ is even, then p is integer, otherwisep is half-integer. In both cases

12(l + l′) ≤ p ≤ 1

2(l + l′) + n + n′ (5.109)


The result in stretched coordinates is analogous. By the use of Clebsch-Gordan co-efficients, the notation is somewhat simplified. In stretched coordinates the generalmatrix element expression is

〈Nl′Λ′ |f(rt)Yλµ(θt, φt)|NlΛ〉 =√(2l+1)(2λ+1)

4π(2l′+1) Clλl′ΛµΛ′C

lλl′000

∑p B(n′l′nlp)Ip(f)

(5.110)

For the Clebsch-Gordan coefficients, the Raca’s first form [18] was used

Cj1j2jm1m2m = δ(m1 + m2,m)·√

(2j+1)(j1+j2−j)!(j1−m1)!(j2−m2)!(j−m)!(j+m)!(j1+j2+j+1)(j+j1−j2)!(j+j2−j1)!(j1+m1)!(j2+m2)!

·∑

t(−1)j1−m1+t (j1+m1+t)!(j+j2−m1−t)!t!(j−m−t)!(j1−m1−t)!(j2−j+m1+t)!

(5.111)

for which the summation index runs in the interval

max(0, j − j2 −m1) ≤ t ≤ min(j + j2 −m1, j −m, j1 −m1) (5.112)

For the second Clebsch-Gordan factor in 5.110 the following simpler form was used.

Cj1j2j000 =

0, j1 + j2 + j odd√

(2j+1)(j1+j2−j)!(j1−j2+j)!(−j1+j2+j)!(j1+j2+j+1)!

(−1)J−jJ!(J−j1)!(J−j2)!(J−j)! , j1 + j2 + j = 2J even

(5.113)We are interested in the case f(rt) = r2

t , for which

Ip(f = r2t ) = p + 3/2 (5.114)

and the sequence of multipoles

λ = 4 µ ∈ −4,−2, 0, 2, 4 (5.115)

The hexadecapole matrix elements will couple different oscillator shells of equalparity, i.e. give non-vanishing matrix elements for ∆N = 0,±2,±4, . . .. However,they will decay with |∆N |.

Matrix elements of ~lt · ~sThe stretched spin-orbit term in the non-rotating single particle Hamiltonian isexpressed in terms of ladder operators, eq. (5.22), as

~lt · ~s =12(lt−s+ + lt+s−) + ltzsz (5.116)

In terms of the operators R, S, az, and their adjoints, one gets

lt+ = ~√

2(Sa†z −R†az)

lt− = ~√

2(S†az −Ra†z)(5.117)


The operator expression for ltz is given by eq. (5.23). The non-vanishing matrixelements are

〈n⊥nzΛΩ |~lt · ~s|n⊥nzΛΩ〉 = ~2ΛΣ = ~2Λ(Ω− Λ)

〈n⊥ + 1, nz − 1, Λ− 1, Ω |~lt · ~s|n⊥nzΛΩ〉 = ~22

√nz(n⊥ − Λ + 2)

〈n⊥ − 1, nz + 1, Λ− 1, Ω |~lt · ~s|n⊥nzΛΩ〉 = −~22√

(nz + 1)(n⊥ + Λ)

〈n⊥ − 1, nz + 1, Λ + 1, Ω |~lt · ~s|n⊥nzΛΩ〉 = ~22

√(nz + 1)(n⊥ − Λ)

〈n⊥ + 1, nz − 1, Λ + 1, Ω |~lt · ~s|n⊥nzΛΩ〉 = −~22√

nz(n⊥ + Λ + 2)

(5.118)

Note that they all have ∆N = 0, which is an advantage of using stretched or-bital angular momentum operator in the spin-orbit term. Historically, it was veryimportant to avoid coupling between different N -shells, but also with modern com-putation resources, it is still an advantage as the stretched operators help reducingthe basis truncation error, see section 5.7.

Matrix elements of lt2

The operator lt2 = ltx

2 + lty2 + ltz

2 is calculated from the definition of ~lt (equa-tion 5.19), and the operators 5.11, 5.15 and their adjoints. For the two first terms,

ltx2 + lty

2 = 2~2(a†zaz + (R†R + S†S)/2 + S†Sa†zaz + R†Ra†zaz︸︷︷︸

diagonal

+

−RS(a†z)2 −R†S†(az)2︸︷︷︸

off−diagonal

) (5.119)

and for the thirdltz

2 |n⊥nzΛΩ〉 = ~2Λ2 |n⊥nzΛΩ〉 (5.120)

which result in the following non-vanishing matrix elements

〈n⊥nzΛΩ |lt2|n⊥nzΛΩ〉 = ~2(N + (1 + 2N)nz − 2n2

z + Λ2)

〈n⊥ + 2, nz − 2, ΛΩ |lt2|n⊥nzΛΩ〉 = −~2√

nz(nz − 1)(n⊥ − Λ + 2)(n⊥ + Λ + 2)

〈n⊥ − 2, nz + 2, ΛΩ |lt2|n⊥nzΛΩ〉 = −~2√

(nz + 1)(nz + 2)(n⊥ − Λ)(n⊥ + Λ)(5.121)

Note that also here there are only ∆N = 0 matrix elements. The mean value in aN -shell of the square of the stretched orbital angular momentum can be shown tobe

〈lt2〉N =12~2N(N + 3) (5.122)

and is subtracted only for the diagonal matrix elements of the Hamiltonian. Thematrix elements of lt

2 alone would give a compression of shells, which is avoidedby the subtraction [3].


Matrix elements of jx, jy and jz

The unstretched angular momentum operator is

~j = ~l + ~s (5.123)

where ~l is the unstretched orbital angular momentum operator, and ~s is the intrinsicspin operator. The orbital angular momentum operator is

~l = −i~(y

∂

∂z− z

∂

∂y, z

∂

∂x− x

∂

∂z, x

∂

∂y− y

∂

∂x

)(5.124)

with coordinates in the body-fixed frame Oxyz. Thus, also ~j refer to the body-fixedframe. The z-component of the spin operator has the well known property

sz |Σ〉 = ~Σ |Σ〉. (5.125)

The x and y-components are expressed in ladder operators as

sx = 12 (s+ + s−)

sy = 12i (s+ − s−)

(5.126)

from eq. (5.22), and the ladder identities

s+ |Σ〉 = ~√

(12 − Σ)( 3

2 + Σ) |Σ + 1〉s− |Σ〉 = ~

√( 12 + Σ)( 3

2 − Σ) |Σ− 1〉(5.127)

are used [16]. The matrix elements resulting from the spin components are alwaysof the ∆N = 0 type. By the stretched coordinates, eq. (5.7), and their derivatives,eq. (5.10), the x-component of the orbital angular momentum operator is

lx = −i~(√

ωz

ωyη

∂

∂ζ−

√ωy

ωzζ

∂

∂η

). (5.128)

By inserting expressions for the stretched coordinates and their derivatives in termsof the oscillator quanta creation and annihilation operators, eq. (5.18), one gets

lx = ~2√

2

(√ωz

ωy−

√ωy

ωz

)(Raz − S†a†z − Saz + R†a†z)︸︷︷︸

∆N=±2

+

+(√

ωz

ωy+

√ωy

ωz

)(−Ra†z + S†az + Sa†z −R†az)︸︷︷︸

∆N=0

(5.129)


The non-vanishing matrix elements of jx with respect to the Nilsson basis are

〈n⊥ − 1, nz − 1, Λ− 1, Ω− 1 |jx|n⊥nzΛΩ〉 = ~4 (

√ωz

ωy−

√ωy

ωz)√

nz(n⊥ + Λ)

〈n⊥ + 1, nz + 1, Λ− 1, Ω− 1 |jx|n⊥nzΛΩ〉 = −~4 (√

ωz

ωy−

√ωy

ωz)√

(nz + 1)(n⊥ − Λ + 2)

〈n⊥ − 1, nz − 1, Λ + 1, Ω + 1 |jx|n⊥nzΛΩ〉 = −~4 (√

ωz

ωy−

√ωy

ωz)√

nz(n⊥ − Λ)

〈n⊥ + 1, nz + 1, Λ + 1, Ω + 1 |jx|n⊥nzΛΩ〉 = ~4 (

√ωz

ωy−

√ωy

ωz)√

(nz + 1)(n⊥ + Λ + 2)

〈n⊥ − 1, nz + 1, Λ− 1, Ω− 1 |jx|n⊥nzΛΩ〉 = −~4 (√

ωz

ωy+

√ωy

ωz)√

(nz + 1)(n⊥ + Λ)

〈n⊥ + 1, nz − 1, Λ− 1, Ω− 1 |jx|n⊥nzΛΩ〉 = ~4 (

√ωz

ωy+

√ωy

ωz)√

nz(n⊥ − Λ + 2)

〈n⊥ − 1, nz + 1, Λ + 1, Ω + 1 |jx|n⊥nzΛΩ〉 = ~4 (

√ωz

ωy+

√ωy

ωz)√

(nz + 1)(n⊥ − Λ)

〈n⊥ + 1, nz − 1, Λ + 1, Ω + 1 |jx|n⊥nzΛΩ〉 = −~4 (√

ωz

ωy+

√ωy

ωz)√

nz(n⊥ + Λ + 2)

〈n⊥nzΛ, Ω + 1 |jx|n⊥nzΛΩ〉 = ~2

√( 12 − Σ)(3

2 + Σ)

〈n⊥nzΛ, Ω− 1 |jx|n⊥nzΛΩ〉 = ~2

√( 12 + Σ)( 3

2 − Σ)(5.130)

Note that there are no diagonal elements of jx with respect to this basis. In thespecial case of x-axis symmetry, ωy = ωz and only the ∆N = 0 matrix elementsremain.

Analogously, the y-component of the orbital angular momentum operator is bystretched coordinates written as

ly = −i~(√

ωx

ωzζ

∂

∂ξ−

√ωz

ωxξ

∂

∂ζ

)(5.131)

and in terms of quanta creation and annihilation operators as

ly = i~2√

2

(√ωx

ωz−

√ωz

ωx

)(−Raz + S†a†z − Saz + R†a†z)︸︷︷︸

∆N=±2

+

+(√

ωx

ωz+

√ωz

ωx

)(−Ra†z + S†az − Sa†z + R†az)︸︷︷︸

∆N=0

(5.132)


By adding sy, eq.( 5.126), the resulting matrix elements for jy are

〈n⊥ − 1, nz − 1,Λ− 1, Ω− 1 |jy|n⊥nzΛΩ〉 = − i~4 (

√ωx

ωz−

√ωz

ωx)√

nz(n⊥ + Λ)

〈n⊥ + 1, nz + 1,Λ− 1, Ω− 1 |jy|n⊥nzΛΩ〉 = i~4 (

√ωx

ωz−

√ωz

ωx)√

(nz + 1)(n⊥ − Λ + 2)

〈n⊥ − 1, nz − 1,Λ + 1, Ω + 1 |jy|n⊥nzΛΩ〉 = − i~4 (

√ωx

ωz−

√ωz

ωx)√

nz(n⊥ − Λ)

〈n⊥ + 1, nz + 1,Λ + 1, Ω + 1 |jy|n⊥nzΛΩ〉 = i~4 (

√ωx

ωz−

√ωz

ωx)√

(nz + 1)(n⊥ + Λ + 2)

〈n⊥ − 1, nz + 1,Λ− 1, Ω− 1 |jy|n⊥nzΛΩ〉 = − i~4 (

√ωx

ωz+

√ωz

ωx)√

(nz + 1)(n⊥ + Λ)

〈n⊥ + 1, nz − 1,Λ− 1, Ω− 1 |jy|n⊥nzΛΩ〉 = i~4 (

√ωx

ωz+

√ωz

ωx)√

nz(n⊥ − Λ + 2)

〈n⊥ − 1, nz + 1,Λ + 1, Ω + 1 |jy|n⊥nzΛΩ〉 = − i~4 (

√ωx

ωz+

√ωz

ωx)√

(nz + 1)(n⊥ − Λ)

〈n⊥ + 1, nz − 1,Λ + 1, Ω + 1 |jy|n⊥nzΛΩ〉 = i~4 (

√ωx

ωz+

√ωz

ωx)√

nz(n⊥ + Λ + 2)

〈n⊥nzΛ, Ω + 1 |jy|n⊥nzΛΩ〉 = − i~2

√(12 − Σ)( 3

2 + Σ)

〈n⊥nzΛ, Ω− 1 |jy|n⊥nzΛΩ〉 = i~2

√( 12 + Σ)( 3

2 − Σ)(5.133)

Note that there are no diagonal elements and that all matrix elements are imaginary.In the special case of y-axis symmetry, ωx = ωz and only ∆N = 0 elements remain.

The z-component of the orbital angular momentum operator is by stretchedcoordinates

lz = −i~(√

ωy

ωxξ

∂

∂η−

√ωx

ωyη

∂

∂ξ) (5.134)

and in terms of the operators R,S, az, and their adjoints, as

lz =~4

(√ωy

ωx−

√ωx

ωy

)(R†

2+ R2 − S†

2 − S2

︸︷︷︸off−diagonal

) +~2

(√ωy

ωx+

√ωx

ωy

)(R†R− S†S︸︷︷︸

diagonal

)

(5.135)By adding sz, which gives a diagonal matrix element term according to eq. (5.125),one gets the non-vanishing matrix elements of jz as

〈n⊥nzΛΩ |jz|n⊥nzΛΩ〉 = ~

[ 12 (√

ωy

ωx+

√ωx

ωy)− 1]Λ + Ω

〈n⊥ + 2, nz, Λ + 2, Ω + 2 |jz|n⊥nzΛΩ〉 = ~8 (

√ωy

ωx−

√ωx

ωy)√

(n⊥ + Λ + 2)(n⊥ + Λ + 4)

〈n⊥ − 2, nz, Λ− 2, Ω− 2 |jz|n⊥nzΛΩ〉 = ~8 (

√ωy

ωx−

√ωx

ωy)√

(n⊥ + Λ)(n⊥ + Λ− 2)

〈n⊥ + 2, nz, Λ− 2, Ω− 2 |jz|n⊥nzΛΩ〉 = −~8 (√

ωy

ωx−

√ωx

ωy)√

(n⊥ − Λ + 2)(n⊥ − Λ + 4)

〈n⊥ − 2, nz, Λ + 2, Ω + 2 |jz|n⊥nzΛΩ〉 = −~8 (√

ωy

ωx−

√ωx

ωy)√

(n⊥ − Λ)(n⊥ − Λ− 2)

(5.136)In the z-axis symmetric case γ = 0, ωx = ωy and only the diagonal element of jz

remains.

5.5. THE VOLUME CONSERVATION 37

5.5 The volume conservation

5.6 Motivation

Following a path in deformation space, the volume of a given nucleus should beheld constant, due to the experimental result that the nuclear density is almostconstant.

Therefore, the oscillator angular frequency ω0 is made a function of deforma-tion, to ensure that the volume inside an equipotential surface is the same for alldeformations.

Transformation to unstretched space

For the purpose of the volume conservation condition calculation, the volume ele-ment is transformed to the unstretched system. However, the oscillator potential,which in eq. (5.76) is expressed in stretched coordinates, is here transformed to theunstretched (physical) space, because it turns out to give simpler expressions whensome of the liquid drop integrals are calculated.

By unstretched spherical coordinates (r, θ, φ), defined as

x = r sin θ cosφ

y = r sin θ sinφ

z = r cos θ,

(5.137)

the squares of the stretched coordinates, eq. (5.7), are by the expressed as

ξ2 = Mω0~ r2Ωx sin2 θ cos2 φ

η2 = Mω0~ r2Ωy sin2 θ sin2 φ

ζ2 = Mω0~ r2Ωz cos2 θ

(5.138)

where the Ωi:s are the dimensionless oscillator angular frequencies

Ωx = 1− 23ε cos(γ + 2π/3)

Ωy = 1− 23ε cos(γ − 2π/3)

Ωz = 1− 23ε cos γ

(5.139)

The square of the radius vector in stretched coordinates is then

r2t ≡ ξ2 + η2 + ζ2 =

Mω0

~r2f00(θ, φ) (5.140)

where the function

f00(θ, φ) = (sin2 θ(Ωx cos2 φ + Ωy sin2 φ) + Ωz cos2 θ) (5.141)


The first quadrupole term in the potential is transformed to unstretched coordinatesas this

r2t Y20(θt, φt) = r2

t

√5

16π(3 cos2 θt − 1) =

√5

16π(2ζ2 − ξ2 − η2) =

=

√5

16π

Mω0

~r2f20(θ, φ) (5.142)

wheref20(θ, φ) = 2Ωz cos2 θ − sin2 θ(Ωx cos2 φ + Ωy sin2 φ) (5.143)

and the second quadrupole term in the potential as this

r2t (Y22(θt, φt) + Y2−2(θt, φt)) = r2

t

√1532π

2 sin2 θt cos(2φt) =√

1158π

(ξ2 − η2) =

√158π

Mω0

~r2f22(θ, φ) (5.144)

wheref22(θ, φ) = sin2 θ(Ωx cos2 φ− Ωy sin2 φ) (5.145)

The hexadecapole terms are transformed in a similar manner

r2t Y40(θt, φt) =

158√

4π

Mω0

~r2f40(θ, φ)r2

t (Y42(θt, φt) + Y4−2(θt, φt)) =

34

√104π

Mω0

~r2f42(θ, φ)r2

t (Y44(θt, φt) + Y4−4(θt, φt)) =

38

√704π

Mω0

~r2f44(θ, φ) (5.146)

If one substitute cos θ = t, then the collection of f functions are expressed as

f00(t, φ) = (1− t2)(Ωx cos2 φ + Ωy sin2 φ) + Ωzt2

f20(t, φ) = 2Ωzt2 − (1− t2)(Ωx cos2 φ + Ωy sin2 φ)

f22(t, φ) = (1− t2)(Ωx cos2 φ− Ωy sin2 φ)

f40(t, φ) = 7Ωzt4

(1−t2)(Ωxcos2φ+Ωy sin2 φ)+Ωzt2− 6Ωzt

2+35 ((1− t2)(Ωx cos2 φ + Ωy sin2 φ) + Ωzt

2)

f42(t, φ) = (1−t2)(Ωx cos2 φ−Ωy sin2 φ)(6Ωzt2−(1−t2)(Ωx cos2 φ+Ωy sin2 φ))(1−t2)(Ωx cos2 φ+Ωy sin2 φ)+Ωzt2

f44(t, φ) = (1−t2)2(Ω2x cos4 φ−6ΩxΩy cos2 φ sin2 φ+Ω2

y sin4 φ)

(1−t2)(Ωx cos2 φ+Ωy sin2 φ)+Ωzt2

(5.147)

The expressions above for the monopole, quadrupole and hexadecapole terms areinserted in the oscillator potential and r2, that is the square of the radius vector inthe unstretched space, is factored out resulting in

Vosc =12Mω2

0r2(f00(t, φ)− ε2d2(t, φ) + ε4d4(t, φ)

)(5.148)

5.6. MOTIVATION 39

where the d-functions are

d2(t, φ) = 13 cos γf20(t, φ)− 1√

3sin γf22(t, φ)

d4(t, φ) = 54 (

√712 cos δ4 +

√512 sin δ4 cos γ4)f40(t, φ)−

√5

2 sin δ4 sin γ4f42(t, φ)+√

704 (

√524 cos δ4 −

√724 sin δ4 cos γ4)f44(t, φ)

(5.149)

The volume conservation condition

Consider an equipotential surface

Vosc(r, t, φ) = V ′osc (5.150)

where V ′osc is a positive constant energy. The cube of the distance from the center

(the origin) to the equipotential surface is

r3(t, φ) =(2V ′

osc

M

)3/2 1ω3

0

1(f00(t, φ)− ε2d2(t, φ) + ε4d4(t, φ)

)3/2(5.151)

One can then compute the volume inside the equipotential surface as a definiteintegral over all space angles (θ, φ), i.e. (t, φ). The volume of an infinitesimallynarrow cone with length r and surface element d~S is

dυ =13~r · d~S =

13r3 sin θdθdφ (5.152)

The entire volume is the definite integral

υ =13

∫ 1

t=−1

∫ 2π

φ=0

r3(t, φ)dtdφ (5.153)

By the coordinate plane reflexion symmetries of the oscillator potential, see sec-tion 5.3, which imply that

r(−t, φ) = r(t, φ)r(t, φ) = r(t, π − φ) (5.154)r(t, φ) = r(t,−φ)

and thus reduce the integral domain to one octant to get the volume expression

υ =83

∫ 1

t=0

∫ π/2

φ=0

r3(t, φ)dtdφ (5.155)

For a sphere, the oscillator frequency is defined to

ω0≡ ω0(ε2 = 0, ε4 = 0) (5.156)


For a sphere f00 = 1 , and therefore

r3sphere =

(2V ′osc

M

)3/2 1ω

3

0

(5.157)

Then the volume of a sphere is

υsphere =4π

3

(2V ′osc

M

)3/2 1ω

3

0

(5.158)

By setting this constant equal to the general expression for the volume, the volumeconservation condition is simplified to

(ω0ω0

)3

=2π

∫ 1

t=0

∫ π/2

φ=0

dφdt(f00(t, φ)− ε2d2(t, φ) + ε4d4(t, φ)

)3/2(5.159)

where the constants M and V ′osc are canceled.

In the special case of no hexadecapole deformation, the volume conservationratio can be calculated analytically. If ε4 = 0 the oscillator potential can be writtenas in eq. (5.1). The oscillator potential expression is rewritten to

x2

2V ′osc

Mω2x

+y2

2V ′osc

Mω2y

+z2

2V ′osc

Mω2z

= 1 (5.160)

that is, the equipotential surface is in this case an ellipsoid with semi axes

Ax =(2V ′

osc

M

)1/2 1ω0

Ω−1x

Ay =(2V ′

osc

M

)1/2 1ω0

Ω−1y (5.161)

Az =(2V ′

osc

M

)1/2 1ω0

Ω−1z

Thus, the volume of the ellipsoid is

υ =4π

3AxAyAz =

(2V ′osc

M

)3/2 1ω3

0

Ω−1x Ω−1

y Ω−1z (5.162)

The volume conservation ratio in the case ε4 = 0 is simplified to

ω0ω0

= (ΩxΩyΩz)−1/3 (5.163)

In the ε4 6= 0 case, the volume conservation ratio is numerically calculatedby Gaussian double quadrature using Legendre polynomials as basis [23]. Thequadrature order N = 25 was used, for which the error was estimated to ∆(ω0/

ω0

) . 10−15, for deformation ε ≤ 0.6 and ε4 ≤ 0.1.

5.7. DIAGONALIZATION OF THE HAMILTONIAN MATRIX 41

5.7 Diagonalization of the Hamiltonian matrix

The matrix elements 〈bi |Hτ | bj〉 of the cranked single particle Hamiltonian eq.(5.80)are calculated with respect to all combinations i, j from a subset B = |b〉i =|N, nz, Λ, Ω〉i, i = 1, . . . , n of the stretched cylindrical harmonic oscillator basisstates (Nilsson basis) eq.(5.25). They are collected in a matrix

H =

〈b1 |Hτ | b1〉〈b1 |Hτ | b2〉 . . .

〈b2 |Hτ | b1〉〈b2 |Hτ | b2〉 . . ....

.... . .

(5.164)

called the Hamiltonian matrix, which is always hermitian.Actual calculations are needed only for i ≤ j, as

〈bj |Hτ | bi〉 = 〈bi |Hτ | bj〉∗ (5.165)

where ∗ denotes complex conjugate. Also, because of the parity symmetry, thesubset B can be split in two sets B+ for positive parity, and B− for negative parity.Matrix elements for for basis states from different parity are always zero, and sothe Hamiltonian matrix can be split in two submatrices

H =

H+ 0

0 H−

(5.166)

Thus, for each mesh point and cranking frequency, there are four diagonalizationsto be carried out: For the protons one for positive and one for negative parity, andlikewise for the neutrons.

If cranking is taking place about an axis lying in the xz-plane, there is no ycomponent in the cranking vector, giving no contributions from the imaginary 〈jy〉matrix elements. In that case the Hamiltonian matrix becomes a real, symmetricmatrix.

Accuracy of diagonalization routines

For the diagonalization of the Hamiltonian matrix, we need in this model a nu-merical diagonalization routine of both high accuracy and speed. Accuracy, to getthe possibility to find also very shallow energy minima, as the spectrum can varyonly slowly with the tilting angles. Speed, to be able to do calculations in largemeshes. The deformation-tilting space has five dimensions, and so the total numberof meshpoints can be very large due to the combinatorial explosion.

Generally, the numerical algorithms for diagonalization of hermitian or sym-metric matrices are of a high accuracy. Nevertheless, a set of them were tested andcompared for their accuracy.


Let (λk, zk), k=1, . . . , n be the numerically calculated eigenvalues and eigen-vectors of a Hermitian test matrix A. As measures of the quality of the differentdiagonalizers, the following quantities were calculated

max1≤k≤n ‖(A− λkI)zk‖ (eigenvalue eq)

max1≤k,j≤n |〈zk | zj〉| (orthogonality)

max1≤k≤n |〈zk | zk〉 − 1| (normality)

(5.167)

The closer to zero each of these quantities are, the better the quality of the eigen-values and eigenvectors returned by the numerical routine. Of special importanceis the orthogonality measure (2nd line in eq. above) in cases with high degeneracy.

The following diagonalizers were tested for complex, hermitian matrices

Diagonalizer From package

ch Eispack [24]

zheev LAPACK [25]

F08* and F02HAF NAG [26]

In the real, symmetric case, the tested diagonalizers were

Diagonalizer From package

rs Eispack [24]

sdiag Napack [27]

bigmax see ref. [28]

dsyev LAPACK [25]

F08* and F02FAF NAG [26]

These diagonalizers operate in complex and real double precision, respectively,(round off unit about 10−16) which is used for all floating point calculations inthe cranking program. When taking both speed and accuracy into account, thebest diagonalizer in the investigations turned out to be zheev in the complex case,and dsyev in the real case, which therefore were used in the cranking calculations.

As all the matrix elements for the cranking Hamiltonian with respect to theNilsson basis are algebraically computed (no quadrature) the numerical error inthe matrix elements themselves are very small and can be neglected.

For estimates of the actual errors in the computed eigenvalues, a method basedon Geshgorin’s circle theorem was used[29].

The resulting error estimates for the errors in the eigenvalues returned by zheevand dsyev were in the order 10−11, for cranking Hamiltonian matrices. That is avery small error which can be neglected in comparison to errors due to the trunca-tion of the infinite basis space discussed in the following section.

5.7. DIAGONALIZATION OF THE HAMILTONIAN MATRIX 43

Truncation of basis space - eigenvalue convergence

The harmonic oscillator basis has an infinite number of members. That is, for anexact expression in terms of a Fourier series of an arbitrary state |ψ〉 in the Hilbertspace,

|ψ〉 =∞∑

i=1

ci |bi〉, (5.168)

where ci = 〈bi |ψ〉, an infinite number of basis states |bi〉 are needed. No matter howa truncated basis B is chosen, there will be missing matrix elements because of cou-plings to higher lying oscillator shells, as soon as there is any cranking term or hex-adecapole term involved which have non-vanishing matrix elements∆N = ±2 and∆N = ±2,±4, . . ., respectively. Thus, the eigenvalues and eigenvectors of eq.(5.164)for a truncated (and thus finite) basis B are in general only approximations to theexact solution. However, for low lying eigenvalues (and their eigenvectors) the trun-cation error becomes small if B is sufficiently large, and so the approximation to theexact spectrum becomes better the larger the truncated basis is made. This processwhere the basis truncation error diminishes by the use of an increasing number ofbasis states, we call eigenvalue convergence. Note that in the special case of nocranking and no hexadecapole deformation, if B is made up of full oscillator shell,an exact solution will be obtained although B is finite.

In the program there are two main options for how the truncated base B is tobe chosen. In the first option, B is made up of full oscillator shells for the mainquantum numbers in the interval 0 ≤ N ≤ Nmax. In the second option, the diagonalmatrix elements of the hamiltonian are used, which are first order approximations tothe eigenvalues. The diagonal elements di = 〈bi |Hτ | bi〉 are calculated for all basisstates with N ≤ Nmax, and sorted in ascending order. The diagonal element dN ,where N is the number of particles for the present isospin, is a rough approximationof the Fermi level. Starting from dN , a number of higher sorted diagonal elementsdi, i = N + 1,N + 2, . . ., are included until the unequality di ≥ dN + sm + em issatisfied. Let n denote this i-value.

The truncated basis is chosen as B = |b1〉, . . . , |bn〉 corresponding to thesorted diagonal elements d1, . . . , dn. Thus, the number n is the number of basisstates, and becomes the number of eigenstates and corresponding eigenvalues aswell.

The parameter sm > 0 is called the Strutinsky marginal and the idea is thateigenvalues up to an energy corresponding to the Fermi level plus the Strutinskymarginal (that is with index j = 1, 2, . . . , jmax, such that dj ≤ dN + sm), shallhave a small basis truncation error, as they are used for the single-particle energysums and for Strutinsky averaged quantities, to give a sufficiently small error inthe total energy and spin. See section 5.17. The other parameter em > 0 is calledthe eigenvalue convergence marginal, which can be varied to obtain different sizesof the basis truncation error.


Let ei, |ψi〉 be the eigenvalues and eigenvectors of a truncated basis B, ande∞i , |ψi〉∞ be the eigenvalues and eigenvectors of an infinite basis B∞ (always exactsolution). The basis truncation error for the eigenvalues ei can be defined as

∆ei = ei(B)− ek(B∞), (5.169)

where k is chosen to maximize the overlap |〈ψi |ψk〉∞|, that is |ψi〉 approximatethe exact eigenstate |ψk〉∞. However, for numerical estimates the second term isapproximated with ek(Blarge), where the large basis can be chosen by using fulloscillator shells to Nmax + n, where n > 0 is some number of extra shells, or byusing a large em value, in both cases such that a further increase of the basis sizewill give neglectable changes in the eigenvalues ei, i ≤ jmax, or the single-particleenergy sums.

The ∆N 6= 0 couplings enter in the matrix elements of the angular momentumoperators jx, jy, jz, i.e. in the cranking term, and in the matrix elements of thehexadecapole terms r2

t Y4µ in the potential. The matrix elements of jx, jy, jz havethe following mesh parameter dependent factors

fx =√

ωz

ωy−

√ωy

ωz= 2ε[cos(γ−120)−cos γ]

3√

(1− 23 ε cos(γ−120))(1− 2

3 ε cos γ)

fy =√

ωx

ωz−

√ωz

ωx= 2ε[cos γ−cos(γ+120)]

3√

(1− 23 ε cos γ)(1− 2

3 ε cos(γ+120))

fz =√

ωy

ωx−

√ωx

ωy= 2ε[cos(γ+120)−cos(γ−120)]

3√

(1− 23 ε cos(γ+120))(1− 2

3 ε cos(γ−120))

(5.170)

(compare eqs. (5.130), (5.133), (5.136)). The absolute values of the fi factors in-crease with ε (or are constant and equal to zero, e.g. fx = 0 for γ = 60 or −120

i.e. in the x-axial symmetric case).The matrix elements of the hexadecapole terms are multiplied by a factor ε4

in the potential eq. (5.76). Thus, larger ε and/or ε4 will produce larger ”missed”matrix elements, and demand a higher em to keep the basis truncation error belowsome limit.

For the lab energy and total angular momentum, introduced in section 5.12,errors emerging from the basis truncation error are discussed in section 5.17. How-ever, a formula was developed to get a maximum basis truncation error in theeigenvalues up to the Fermi level of the order 10−6~ω0, in order to reach an error inthe order of 10−3 MeV = 1 keV in the lab energy, which is the typical accuracy ofexperimental data. But due to limited computer resources, and experiences of theactual need of an error limit in total energy and spin in 3D cranking calculationsfor 166Hf, as reported in article 3, the use of that formula was abandoned. Insteadan error limit of 10 keV was used for the lab energy.

In practice a compromise often has to be made between the prestanda of avail-able computer system, that is to time, and to the required maximum error limit.For large survey calculations a rather small em value is suitable as it will allowlarger meshes for a given amount of time, compared to what a large em valuewould. For zoom in calculations in a neighborhood of interesting energy minima,

5.8. ANGULAR MOMENTUM EXPECTATION VALUES 45

an em value large enough to hold the basis truncation error below some limit isdesirable.

5.8 Angular momentum expectation values

For calculation of the total angular momentum vector (section 5.12), the expecta-tion values of the angular momentum operators jx, jy and jz, with respect to thesingle particle eigenstates of the cranked Hamiltonian, are needed.

An eigenstate of Hτ can be expanded in terms of basis states |bk〉 as

|ψi〉 =∑

k

ck |bk〉 (5.171)

where ck = 〈bk |ψi〉. In the calculations, ck is (approximated by) the kth componentin the ith eigenvector of the Hamiltonian matrix H. The expectation value of theangular momentum operator jµ, µ = x, y or z, with respect to the eigenstate |ψi〉is

〈ψi |jµ|ψi〉 =∑

l

∑

k

c∗l ck〈bl |jµ| bk〉. (5.172)

As a first optimization, using the fact that the angular momentum operators arehermitian, the sum is rewritten to

〈ψi |jµ|ψi〉 =∑

k

|ck|2〈bk |jµ| bk〉+∑

k

∑

l>k

2Re(c∗l ck〈bl |jµ| bk〉), (5.173)

which reduce the number of terms by roughly a factor 2.In the actual computation of the expectation values, only the eigenvector coef-

ficients with absolute value |ck| > limit contribute to the sum, which will diminishthe number of iterations somewhat further. Because the signature symmetry isbroken as soon as the rotational axis is tilted, the number of basis states, as wellas the number of eigenvectors, are in principle each twice as many as for a pure 1Dcranking code which splits the two signatures in two separate calculations. Also theuse of high eigenvalue convergence marginals for high precision, gives many termsin the summation. In the triaxial case, when all three of the oscillator angularfrequencies ωx, ωy, ωz are different from each other, all ∆N ±2 couplings in the an-gular momentum matrix elements with respect to the basis states, contribute, andthere are many more non-zero eigenvector coefficients than in the axially symmetriccase.

The total time for calculation of expectation values with respect to the eigen-states can in such cases be larger than that for the diagonalization, and large scalemesh calculations are hard to realize. The time complexity of the straightforwardsummation is, for all eigenstates, of O(n3).

The expectation values for µ = x, y and z are combined into an expectationvalue vector

〈~j〉i = (〈jx〉i, 〈jy〉i, 〈jz〉i). (5.174)


The slope of a single-particle energy level, with respect to the cranking angularfrequency, is

∂ei

∂ω= −〈~j〉i · ~n (5.175)

where ~n is the cranking unity vector from eq. (5.84).It should also be mentioned that the expectation values are calculated for the

proton and the neutron states separately.

Basis state loop optimization

As a second optimization of the calculation of the expectation values, an algorithmwas developed that, for each basis state |bk〉, runs a loop over only the basis states|bl〉 that couple to the former, that is 〈bl |jµ| bk〉 6= 0, as follows from the non-vanishing matrix elements in eqs. (5.130), (5.133) and (5.136). The coupling basisstates |bl〉 are collected in a set B′. Thus time is not spent to calculate any of allthe angular momentum matrix elements that anyway are zero.

The time complexity is reduced to O(n2), as the number of elements in B′ islimited and does not increase with n. In test runs the total time for calculation ofthe expectation values was reduced by more than a factor 10.

5.9 Particle hole excitations

Definitions and motivation

Consider the proton and neutron single particle states (eigenstates of the crankingHamiltonian). Among the proton states, exactly Z states are occupied, and amongthe neutron states, exactly N states. The occupied states are called particles, theunoccupied holes. The single particle energies (the corresponding eigenvalues) aresummed for the particles to get

E = Ep + En =∑

i occ

epi +∑

i occ

eni (5.176)

called the total Routhian, which is a term in the total energy in the model, seesection 5.12, in can be interpreted as the total microscopic energy in the rotatingframe. The lowest total Routhian is obtained by occupying the Z and N energeti-cally lowest single particle states, respectively. By moving one particle to a higher,unoccupied single particle state (of the same isospin), larger total Routhians are ob-tained. Such an operation is called a particle-hole excitation. Several particle- holeexcitations can be combined. Each way in which Z proton states and N neutronstates can be occupied, we call a configuration.

For a given configuration, the total angular momentum vector (spin vector) isgiven by

~I = ~Ip + ~In =∑

i occ

〈ψpi |~j|ψpi〉+∑

i occ

〈ψni |~j|ψni〉 (5.177)

5.9. PARTICLE HOLE EXCITATIONS 47

The total Routhian and the total angular momentum vector are combined into

Elab = E + ~ω · ~I, (5.178)

which we call the lab energy, and that can be understood as the total microscopicenergy in the lab frame.

When taking the angular momentum contribution into account, the lab en-ergy closest to the yrast band, may not necessarily correspond to the lowest totalRouthian, although in principal axis cranking the lowest total Routhian is alwaysmapped on the yrast band. In paper 1 it is shown that in tilted cranking the yrastband can correspond to highly excited total Routhians. Thus, in a 3 dimensionalcranking program, the ability to calculate highly particle-hole excited configurationsis very important.

The construction of different configurations are broken down in two steps. First,particle-hole excitations are calculated for protons and neutrons separately. Second,the proton and neutron configurations are combined to obtain total configurations.

A fast particle-hole excitation algorithm

For each isospin, different particle-hole excitations (giving different configurations)are calculated, according to the following specification: For the Routhian

Eτ =∑

i occ

eτi (5.179)

determine the energetically lowest configuration (i.e. that gives the lowest single-particle energy sum), the next-to-lowest configuration, and so on.

Denote the lowest configuration σ = 1, the next-to-lowest σ = 2, etc. Theσ = 1 configuration is trivial and corresponds to a summation of the N lowestsingle-particle energies. It is called the vacuum configuration. The configurationsfor a few of the lowest σ values are easily constructed, but higher up the problemgets more complicated.

A method that tests all(

nN

)ways to pick N single-particle states out of n > N

totally, and sorts the resulting sums, is very inefficient, because N is of the order102 and n of the order a few hundreds, which makes the number of configurationsenormous. Of course, only the configurations with particle-hole excitations suffi-ciently ”close” to the Fermi surface need to be calculated, but if the problem isto be solved up to a high σ, the limit may be difficult to set. A better iterativeapproach was tested, where for each number of particles to excite above the N thlevel, the particles and holes are moved forward in a systematic way from a startconfiguration. However, as mentioned above in tilted cranking configurations ofhigh σ are needed, but the iterative approach was not very efficient for calculatinghigh configurations.

A much more efficient way to solve the problem is based on the following con-cept. Starting from the vacuum configuration, the σ = 2 configuration must be the


one where a term as small as possible is added (the new particle) and a term asbig as possible is subtracted (the new hole), i.e. the N th particle is lifted to levelN +1. Such a particle-hole excitation is called a minimal increase operation (MIO).All other ph-excitations, or combinations of them, will result in a larger sum Eτ .From the σ = 2 configuration, two MIOs are possible, either the N − 1 particleis excited to level N , or the N + 1 particle is excited to level N + 2. The σ = 3configuration must be one of these two candidates, because all other ph-excitationswill give a larger sum.

A configuration can be denoted by an n-bit binary number with exactly N ones(for particles), and n − N zeros (for holes). Then e.g. the vacuum configurationis denoted by the number (1 . . . 10 . . . 0), and a sub sequence (10) admits a MIO tomap it on the sub sequence (01). All configurations of N particles distributed overn single-particle states, can be constructed from combinations of MIOs startingfrom the vacuum configuration. Therefore, no configurations are missed by MIO-combinations.

The MIO-based algorithm is schematically given here in pseudo-code

Let C(1) = vacuum configurationsigma=0loop1: sigma=sigma+1

Let conf = C(sigma) ! is the sigma’th smallest config in Cif break condition satisfied

then exit loop1loop2: for each (10) subsequence in conf

newconf = MIO on confif newconf is not a double to an existing element in C

then add newconf to C sorted after increasing sumend2

end1

The parity of a configuration is the product of the parities of the occupied singleparticle states, which can be written as

πτ =∏

i occ

πτi, (5.180)

so that if an even number of single particle states of odd parity are occupied, theconfiguration parity is even (+1), otherwise odd (-1).

In the program, the break condition in the algorithm above is that the numberof even and odd configurations each are greater than or equal to some limit sethigh enough to admit solving the combination problem (described in the followingsection) for ord ≤ maxord.

In the test runs, for σmax & 103 the MIO-based algorithm was more than 100times faster than the iterative approach.

5.9. PARTICLE HOLE EXCITATIONS 49

A fast combination algorithm

The resulting configurations from the MIO-algorithm, described in the previoussection, are splitted in two sequences, one for configuration parity πτ = + and onefor πτ = −. The proton configurations cp

σ=1, cp2, ... and the neutron configurations

cnσ=1, cn

2 , ..., can be combined to elements in four parity groups (πp, πn) = (+,+),(+,-), (-,+) or (-,-).

For each of the parity combinations, total configurations are calculated to getthe total Routhian

E(cp, cn) = Ep(cp) + En(cn) (5.181)

with the lowest energy, next-to-lowest energy, and so on. The combination of oneproton and one neutron configuration that gives the lowest total Routhian, withinthe present parity combination, is denoted ord = 1. The combination that gives thenext-to-lowest total Routhian for the given parity combination, is denoted ord = 2,and so on.

A straightforward method, where all combinations are calculated, up to someexcitation energy cut off, and sorted with respect to increasing energy, was tested.But in the tilted case, high order configurations of ord & 103 may be needed forthe yrast line. For such high orders, the method was found to be very inefficient.

However, there is also here a better solution, which is quite similar to the fastparticle-hole excitation algorithm in the previous section. Let (p, n) denote a com-bination of the proton configuration with the pth lowest energy (for the presentconfiguration parity πp), and the neutron configuration with the nth lowest energy(for πn). The ord = 1 total configuration is of course the combination (1, 1). Theord = 2 combination must be a member of the set D2 = (2, 1), (1, 2), because allcombinations (p, n), where p, n ≥ 2, have higher energy (or equal if there are degen-erated configurations 1). Such operations, where either the proton or the neutronconfiguration index is increased by unity, I call combination-minimal-increasement(CMI). The ord = 3 combination is either the member of D2 that was not picked asthe ord = 2 combination, or it is a member of D3 that consists of the two combina-tions by CMIs on the ord = 2 combination. All combinations can be generated bysequences of CMIs. A fast combination algorithm based on CMIs, that calculatesa solution to the combination problem for ord = 1, . . . , maxord, is schematicallygiven below

comb(1) = (1,1)loop1: for k = 1,maxord

c = comb(k)if k < maxord

cp = proton CMI on c ! one higher proton configif cp not a double to element in comb

insert cp sorted in comb

1In the case of degenerated configurations, the solution is not unique, but different solutionswill only differ in the sorting within each subset of degenerated combinations


cn = neutron CMI on c ! one higher neutron configif cn not a double to element in comb

insert cn sorted in combend1

For each ord ∈ 1, . . . , maxord, the combination of order ord is available incomb(ord), after the termination of the loop.

Rotated ph-excitations

As mentioned above, and as described in paper 1, in tilted axis cranking the yrastband (in the lab frame) may correspond to highly excited bands in the rotatingframe.

Before the fast algorithms for particle-hole excitations and combinations weredeveloped, the computation time was critical when calculating highly excited con-figurations with ord above the order 103. Then the following idea helped to reducethe combination order (ord) for the total configurations close to the yrast band.

On average, the lab energy and spin, for different cranking frequencies and totalconfigurations, follow the Strutinsky averaged lab energy, eq.( 5.201), versus theStrutinsky averaged spin, eq.( 5.200). The total configurations which are close toyrast in total energy, eq.( 5.229), are also close to yrast in lab energy, eq.( 5.178).

By rotating, for each isospin, the lab energy counter-clockwise in the energy-spinplane, so that the energy is calculated along an axis perpendicular to the Eτ

lab(Iτ )

curve, and combining the ”rotated” proton and neutron configurations, the ”ro-tated” lowest total Routhian (ordα = 1) would come at or close to the yrast bandin the lab frame.

If the energy is counted in MeV, and the spin is mapped on MeV by multipli-cation with a fix angular frequency ωf , a rotation angle (one for protons, one forneutrons) can be defined as the slope of a line segment along the Eτ

lab(Iτ )-curve,

as follows

tanατ =[Eτ

lab(ωk)− Eτlab(ωk−1)]/ MeV

[Iτ (ωk)− Iτ (ωk−1)]ωf/ MeV(5.182)

The rotation operator is linear, implying that a rotation of the lab energy is equiv-alent to the sum of rotations of each term eτi + ~ω · 〈~j〉τi. But the size I of the totalspin vector, eq. (5.177), is in general not the sum of the sizes of the terms 〈~j〉τi.However, as discussed in section 5.19, close to self-consistency the total spin vector~I is close to parallel to the tilting vector ~n, eq. (5.84), so that

∑

i occ

~n · 〈~j〉i = ~n · ~I ≈ I, (5.183)

and the rotation of the spin size I is approximately the sum of the rotations of theterms ~n · 〈~j〉i. The terms

xτi = ~n · 〈~j〉τiωf/ MeV, yτi = (eτi + ~ω · 〈~j〉τi)/ MeV (5.184)

5.10. STRUTINSKY AVERAGING OF ENERGY AND SPIN 51

are rotated counter-clockwise the angle ατ , to get a ”rotated” lab energy term

yατi = [− sin α~n · 〈~j〉τiωf + cos α (eτi + ~ω · 〈~j〉τi)]/ MeV. (5.185)

The sequence of ”rotated” lab energy terms are sorted to get

eατi = sort(yα

τ1, . . . , yατn)i, (5.186)

which is sent as input to the fast particle-hole excitation routine. The index vec-tor for the sorting is saved, so that the ”unrotated” Routhians and spins can becalculated.

The ”rotated” proton and neutron lab energies are the input to the fast combi-nation routine. For each ”rotated” combination, the real lab energy and spin arecalculated.

The ”rotated” combination order is denoted ordα and is normally not equal toord, but as designed to be, smaller.

But close to self-consistency, the ord = 1 band will anyway be mapped on theyrast band, so one may ask why rotate the energies there when a fairly low maxordis sufficient to get the yrast band anyway? On the other hand, tests showed that ina mesh point far from self-consistency, the gain in combination order was roughlya factor 2, which gave a significant reduction in computation time.

5.10 Strutinsky averaging of energy and spin

The sum of the occupied single-particle states can not alone be used as total energyfor the nucleus, since it does not correspond to experimental energies. By the Shellcorrection method, where the sum is renormalized to the liquid drop model energy,a good agreement is achieved [30]. Also the sum of angular momenta expectationvalues of the single particle states can be renormalized in an analogue manner [31].

The procedure of Strutinsky renormalization of energy and spin is described indetail in [32], from which the code for the renormalization was derived. The mostimportant definitions and some explicit formulas for calculations will be given here.

The level density is defined as

gτ (E) =∞∑

i=1

δ(E − eτi), (5.187)

implying that∫ b

agτ (E)dE is the number of single particle states between energy a

and b for isospin τ , where τ = p for protons, n for neutrons. Now, the level densitycan be written as a sum of a smooth part and an oscillating part, as

gτ (E) = gτ0 (E) + δgτ (E). (5.188)

The Strutinsky averaged level density is defined as

gτ (E) = SMgτ (E) =1γs

∞∑

i=1

M∑µ=0

a2µγ2µs (

d

dE)2µf(

E − eτi

γs) (5.189)


where SM is the Strutinsky averaging operator and f is the smoothing function.SM has the property that when it acts on any polynomial Pn of degree n ≤ 2M +1,the polynomial is unchanged, i.e. only oscillations corresponding to monoms oforder ≥ 2M + 2 are affected. The idea of the smoothing function f is to replaceeach delta spike by an appropriate 2 curve with maximum at the eigenvalue, andwith decaying tails in both directions. The coefficients a2µ are determined by f .The parameter γs is called the smearing parameter. The larger the value of γs, thewider the distribution curve in the averaged level density around each eigenvalue.The integer 2M ≥ 0 is called the curvature correction order. For more details,see [32].

A natural choice for the smoothing function is a Gaussian f(x) = e−x2/√

π. Forthat choice, the Strutinsky averaged level density can be expressed in a form, usefulfor computations, as

gτ (E) =1

γs√

π

∞∑

i=1

M∑µ=0

(−1)µ

22µµ!p2µ(xi)e−x2

i , (5.190)

where the variable xi = (E − eτi)/γs is used, and pn(x) is a Hermite polynomialof degree n. The averaged Fermi energy λτ is determined such that the averagednumber of particles up to λτ is equal to the actual number of particles N τ , i.e. thesolution to the integral equation

N τ =∫ eλτ

−∞gτ (E)dE, (5.191)

which can be written as

N τ =∞∑

i=1

[12(1 + erf(xi)) +

e−ex2i√

π

M∑µ=1

(−1)µ

22µµ!p2µ−1(xi)], (5.192)

where xi = (λτ − eτi)/γs. The Strutinsky averaged single particle energy sum isdefined as

Eτ =∫ eλτ

−∞E gτ (E)dE, (5.193)

which can be expressed in a useful form as

Eτ =∞∑

i=1

[eωτi

12(1+erf(xi))+

e−ex2i

2√

π

(−γs +

M∑µ=1

(−1)µ

22µ−1µ!λτp2µ−1(xi)−γsp2µ−2(xi)

)].

(5.194)

2Suitable properties are that f(x) is analytical, positive,R∞−∞ f(x)dx = 1, symmetric and with

maximum at x=0.

5.10. STRUTINSKY AVERAGING OF ENERGY AND SPIN 53

To continue with the averaging of the spin sums, the spin density is defined as

gτ,jk(E) =∞∑

i=1

〈jk〉τiδ(E − eωτi). (5.195)

for each of the angular momentum operator for jk, k=x, y, z. A Strutinsky averagedspin density can, similarly to eq. (5.190), be written as

gτ,jk(E) =1

γs√

π

∞∑

i=1

〈jk〉τi

M∑µ=0

(−1)µ


i . (5.196)

The Strutinsky averaged spin sum is defined as

Iτk =

∫ eλτ

−∞gτ,jk(E)dE, (5.197)

and can be written in a useful form as

Iτk =

∞∑

i=1

〈jk〉τi

[12(1 + erf(xi)) +

e−ex2i√

π

M∑µ=1

(−1)µ

22µµ!p2µ−1(xi)

]. (5.198)

The proton and neutron contributions are added to get

E = Ep + En, (5.199)

~I = (Ipx + In

x , Ipy + In

y , Ipz + In

z ). (5.200)

The Strutinsky averaged lab energy is defined as

Elab = E + ~ω · ~I. (5.201)

which can be broken down in proton and neutron terms if needed.Because of the limits erf(xi) → −1 and e−ex2

i → 0, as eτi →∞, the contributions,in the i summation to the averaged energy and spin, decay towards zero above theFermi surface. Still, the i series need to be truncated appropriately. In the program,an energy difference cut off above the Fermi surface is used, which can be variedvia input parameters. In the calculations, these input parameters were chosen togive a truncation error . 1 keV for the averaged energy sum and . 10−3~ for theaveraged spin sum.

The Strutinsky averaged energy sum varies somewhat with the parameters γs

and M . The averaged energy sum flattens out in a region of values of γs & ~ω0

and of M , see [32]. On the other hand, M must not be made too big, otherwisethe averaged level density would contain too fast oscillations and there would be


essentially no smoothing. For a given mesh point, ideally, the parameters shouldbe chosen in a local plateau, i.e.

∂E

∂γs= 0,

∆E

∆M= 0. (5.202)

But in mesh calculations, I have used fixed Strutinsky parameter values, so thatwhen comparing quantities between neighboring mesh points, differences are notdue to local behavior of the averaging calculations.

5.11 The Liquid drop model

The bulk contribution to the total energy and spin, as defined in section 5.12, comesfrom the liquid drop model. The shell effects are smaller variations added to theliquid drop energy.

The liquid drop energy used in the present calculation is

Eld = (Bs − 1)[c1A

2/3 + c2(N − Z)2

A4/3

]+ (Bc − 1)c3

Z2

A1/3, (5.203)

see [33]. The first term corresponds to the surface energy, the second is the symme-try term, and the third corresponds to the Coulomb energy, renormalized to zerofor spherical shape.

The deformation dependent surface integral Bs and Coulomb integral Bc are de-fined in sections 5.11 and 5.11, respectively. The coefficients c1, c2, c3 are constants,for which the values

c1 = 17.9439MeV, c2 = −31.9868 MeV, c3 = 0.70531MeV, (5.204)

of ref. [33], are used in the calculations.The zero level of any energy quantity can be freely chosen, and in this case the

liquid drop energy is defined such that it is zero for a sphere, i.e. when ε = 0, ε4 = 0.For calculating the rotational energy the rigid body moment of inertias are used,

see section 5.11. For the matrix expressions below, it is convenient to collect themin an inertia matrix J , eq. (5.226) below. The angular momentum of the rigidbody, at cranking angular frequency ω, is given by the column vector

~Ild = J ~ω =

Jxωx

Jyωy

Jzωz

. (5.205)

For principal axis cranking, a liquid drop angular frequency for total spin I can bedefined as ωld = I/J . But a generalization to a vector ~ωld = J−1~I would give a

5.11. THE LIQUID DROP MODEL 55

norm |~ωld| that would not vary linearly with the spin size I = |~I|, unless in thespecial case of a spherical shape. To get that linear dependence, the vector

~Ir =I

Ild

~Ild (5.206)

is defined, which is the rigid body angular momentum rescaled to a length equal tothat of the total spin ~I, and the liquid drop angular frequency vector is set to

~ωld = J−1~Ir. (5.207)

The rotational energy for spin ~Ir is then set to

Erotld =

12~Itr J−1~Ir =

12(I2rx

Jx+

I2ry

Jy+

I2rz

Jz) =

I2

2

(Jxn2

x + Jyn2y + Jzn

2z

J 2x n2

x + J 2y n2

y + J 2z n2

z

), (5.208)

which is thus calculated for a spin size I, just like the configuration for the mi-croscopic energy has spin size I. Note that there is no explicit dependence of thecranking angular frequency ω in the rotational energy above. The explicit spin de-pendence enters in the factor I2, the deformation dependence in Ji, and the tiltingangle depenence in the cranking unity vector components ni, i=x, y, z, eq. (5.84).

In the case of principal axis cranking, the rotational energy is reduced to thefamiliar form

Erotld =

I2

2J (1D only) (5.209)

Note that the rotational energy is configuration dependent, as I is.

Surface energy

Let S′ be the equipotential surface given by

Vosc = V ′osc = const, (5.210)

and R be the radius of a sphere of equal volume (as contained in S′). For thesurface energy term in eq. (5.203) above, the surface integral is

Bs =1

4πR2

∮

S′dS (5.211)

In the spherical case, Bs = 1. By eq. (5.148), the distance from the origin to thesurface in the direction specified by the spherical angles θ, φ is given by

r =(

2V ′osc

Mω20

)1/2 1(f00 − εd2 − ε4d4)1/2

(5.212)


where f00, d2 and d4 are functions of θ, φ given in sec. 5.6. From the radius vector~r = r~er, the surface element vector can be calculated as d~S = ∂~r

∂θ × ∂~r∂φdθdφ with

length

dS =

[r2 + (∂r

∂θ)2] sin2 θ + (

∂r

∂φ)2

1/2

r dθdφ (5.213)

Differentiation of r with respect to θ and φ, respectively, gives an algebraic expres-sion for dS, although too long to list here.

The reflexion plane symmetry can be used to reduce integration space to θ, φ ∈[0, π/2] (starting from [0, π] for θ, and [0, 2π] for φ), that is by a factor 2 · 4 = 8.

For computation of Bs Gauss-Legendre quadrature was used, of order 35 forwhich the numerical estimate of the error is . 10−14 for deformations ε ≤ 0.6 andε4 ≤ 0.1.

The Coulomb energy

For a homogenous charge density within a volume V , the electrical potential is

W (~r) =∫

V

dV ′

|~r′ − ~r| , (5.214)

where constant factors that contains, among others, permitivitty and charge, areomitted, as they would anyway have be canceled in the expressions below. TheCoulomb energy for the charge is then

B′c =

∫

V

W (~r)dV, (5.215)

which is a six-fold integral. For a sphere of radius R it has the value B′c(sphere) =

3215π2R5. Let V be the volume contained within the equipotential surface given byeq. (5.210), with radius according to eq. (5.212). The Coulomb energy relative tothat of a sphere is

Bc =B′

c

B′c(sphere)

, (5.216)

which is used in the liquid drop energy defined above. Using Gauss’ theorem twice,the Coulomb energy integral is rewritten to

B′c =

∮

S

∮

S′

16

(~r − ~r′) · d~S(~r′ − ~r) · d~S′

|~r − ~r′| , (5.217)

and is thus reduced to a four-fold integral. The surface element vector is

d~S =(

r sin θ ~er − sin θ∂r

∂θ~eθ − ∂r

∂φ~eφ

)rdθdφ, (5.218)

and analogously for d~S′, by which the integrand is algebraically expressed.

5.11. THE LIQUID DROP MODEL 57

By the reflexion plane symmetry the integration space is reduced to θ, θ′, φ, φ′ ∈[0, π/2], that is by a factor 64.

For the computation of Bc Gauss-Legendre quadrature was used of order N =16, for which the estimated numerical error was . 10−7.

The rigid body moments of inertia

For a homogenous rigid body of mass M and volume V the moment of inertia aboutthe, e.g., z-axis is

Jz =M

V

∫

V

(x2 + y2)dV. (5.219)

Let V be the volume contained in the equipotential surface eq. (5.210), for whichthe distance from the center to the surface is given by eq. (5.212). The ratio betweenthe moment of inertia for any deformation, and the moment of inertia for a sphere(i.e. ε = ε4 = 0), about the z-axis is

Tz =3π

( ω0

ω0

)5 ∫ 1

t=0

∫ π/2

φ=0

(1− t2)(f00 − εd2 − ε4d4)5/2

dtdφ (5.220)

where t = cos θ, the integration space is reduced, and the functions f00, d2 and d4

are defined in sec. 5.6. Similarly, the moment of inertia ratios for the x and y axescan be expressed as

Tx =3π

( ω0

ω0

)5 ∫ 1

t=0

∫ π/2

φ=0

(t2 + (1− t2) sin2 φ)(f00 − εd2 − ε4d4)5/2

dtdφ (5.221)

Ty =3π

( ω0

ω0

)5 ∫ 1

t=0

∫ π/2

φ=0

(t2 + (1− t2) cos2 φ)(f00 − εd2 − ε4d4)5/2

dtdφ (5.222)

In the ellipsoidal case (ε4 = 0), the moment of inertia ratios can be analyticallycalculated to

Tx =12(ΩxΩyΩz)2/3(Ω−2

y + Ω−2z ),

Ty =12(ΩxΩyΩz)2/3(Ω−2

z + Ω−2x ), (5.223)

Tz =12(ΩxΩyΩz)2/3(Ω−2

x + Ω−2y )

where the relation eq. (5.161) between the ellipsoid semiaxes Ax, Ay, Az and theharmonic oscillator angular frequencies Ωx, Ωy, Ωz is used.

In a very simplified view of the atomic nucleus, consider it as a homogenoussphere of mass M = mA, where m is the mass of a nucleon (only about 0.1%difference between neutron and proton masses). The radius for a spherical nucleus


can be taken as R = r0A1/3. Then the moment of inertia, about any axis through

the center, is

Jsph =25mAR2 ≈ 1

72A5/3 ~2

MeV(5.224)

The rigid body moment of inertia is then defined as

Jx = Tx Jsph, Jy = Ty Jsph, Jz = Tz Jsph (5.225)

The moment of inertias are collected in the inertia matrix

J =

Jx 0 0

0 Jy 0

0 0 Jz

, (5.226)

where the off-diagonal elements, the inertia products Jxy,Jyz and Jzx are all zero,due to the coordinate plane reflexion symmetry, described in section 5.3.

For the computation of the moment of inertia ratios when ε4 6= 0, Gauss-Legendre quadrature was used of order N = 30, with an estimated numerical error. 10−14 for deformations ε ≤ 0.6 and ε4 ≤ 0.1.

5.12 Calculation of total quantities

The shell energy is defined as

Eshell = Elab − Elab(I2 =I2). (5.227)

The first term, on the right hand side, is the lab energy, eq. (5.178). The secondterm is the Strutinsky averaged lab energy from eq. (5.201) interpolated to squaredspin I2 = ~I · ~I, so that the shell energy is defined for a given value of I and not ofω. The averaged lab energy varies approximately as along a straight line with thesquare of the averaged spin, see [31]. Therefore, the interpolation is carried out as

Elab(I2 =I2) = Elab(ωj) +Elab(ωk)− Elab(ωj)

I2(ωk)− I2(ωj)

[I2 − I2(ωj)

]. (5.228)

From the sequence of cranking angular frequencies, used in the calculation, ωj andωk are chosen such that, either j = 1 and k = max, or j is chosen as the largestindex such that I2(ωj) ≤ I2, and k as the smallest index such that I2 < I2(ωk).

The total energy of the nucleus in the 3 dimensional cranking model is definedas

Etot = Eld + Erotld + Eshell, (5.229)

where the liquid drop terms are given in sec. 5.11. Note that all cranking dependentterms are taken for a common value of the spin I.

The total spin vector ~I is given by eq. (5.177).For each deformation, tilting, cranking angular frequency and configuration,

values of the total energy and of the spin vector are defined in the model.

5.13. THE LOW ENERGY SETS 59

Figure 5.1: Figure from ”On the use of spin adiabatic energy surfaces” Nucl PhysA569, p 475

5.13 The low energy sets

For a meshpoint, there are for each parity combination (πp, πn) and cranking an-gular frequency ωi, configurations calculated for ord = 1, 2, . . . , maxord. In orderto calculate the yrast band in an efficient way, the configurations are sorted in thefollowing structure. A configuration with spin I is inserted in a sequence of indexk = bIc sorted with respect to increasing lab energy Elab,

The collection of sequences (one sequence for each integer spin k) is called alow energy set. Thus, for a meshpoint, there are four low energy sets, one for eachparity combination. As only the configurations closest to the yrast line are needed,a cut off is used for the number of elements for each integer spin k, such that amongall the calculated configurations only the mth lowest are saved.

5.14 Removing states with weakly interactingsingle-particle states

”States” with a configuration that has a particle-hole pair in weakly interactingsingle-particle states are called spurious. They occur close to band crossings. Seethe schematic illustration in figure 5.1 of, among other things, spurious states whichare found between a1 and a2 in the lower-left corner sub figure. For more details,see [34].

The spurious states should be removed. Otherwise, the yrast line may get toosmall energy close to the band crossing, which may result in a false local minimumin the total energy. Note that it is not the configuration in itself, nor all theconfigurations for a fixed cranking angular frequency, but a point (I, Etot) coming


from a certain combination of a meshpoint, a cranking angular frequency and aconfiguration, that should be removed. Figure xxx shows an example of spuriousstates in calculations, and the resulting yrast lines when the spurious states areremoved, and if they would not have been removed.

If a ”state” satisfy all of the following conditions it is considered as spurious,and not included in the low energy set.

1. Its configuration has a hole in a single-particle state ψh, and the closest neigh-bor among the single-particle states ψp of the same parity is a particle.

2. The hole and the particle are sufficiently close in energy |eh − ep| < elimit

3. The absolute difference in slope is not too large (non-interacting), nor toosmall (may be a signature pair which should be kept) limsignpair < |〈~j〉h ·~n− 〈~j〉p · ~n| < limj

4. The cranking angular frequency is sufficiently large to give a clear separationof non-interacting states ω > omlim.

An advantage of this method for detecting spurious states is that a test, whetherthe conditions above are satisfied, can be done for each value of ω independent ofothers. Thus, different single-particle levels need not be tracked, which gives asimple implementation.

A disadvantage is all the parameters for the conditions above. It is difficult tofind a optimal set of values for them. If they are too ”narrow”, there is a large riskof missing one spurious state that could cause much harm. If the parameters aretoo ”wide”, unnecessary many ”states” may be removed.

However, the parameters were matched for 166Hf and the results for removal ofspurious states in the tests are good.

5.15 The Yrast band

A rotational band can in the cranking model be defined as a sequence of states,with increasing angular momentum, and which have the same configuration,

It is necessary that the nucleus is deformed, or if it is deformed with axialsymmetry that it is not rotating about the symmetry axis, for a rotational band tobe built.

By definition, the yrast band is the rotational band, or sequence of bands, thathas for each spin the minimal energy in the lab frame.

For each parity combination (πp, πn), the low energy set is used to determinethe yrast band. Thus, for each meshpoint (a given deformation and tilting), fourindependent yrast bands are calculated, one for each of the parity combinations(+, +), (+,−), (−, +) and (−,−).

In principal axis cranking, the lowest total routhian (ord = 1) is always mappedon the yrast band. Close to a band crossing, a few excited configurations, with

5.15. THE YRAST BAND 61

ord . 10, will contribute. Thus, the entire yrast band can be determined from onlya few of the lowest configurations.

In paper 1 it is shown how in the case of tilted axis cranking for high spin, theyrast band can contain high configurations (i.e. high ord in the rotating frame),in non self-consistent (sec. 5.19) mesh points. This is possible because for a non-self consistent meshpoint (which is always tilted) the total spin vector ~I and thecranking vector ~ω are not parallel. In the lab energy there is term ~ω ·~I = ωIω, whereIω is the projection of ~I on ~ω. For some excited configurations this projection termcan be extra small, although the spin vector length I is extra large, which pushesthe point (I, Etot) close to the yrast band.

However, for self-consistent meshpoints the configurations of ord close to 1, (i.e.lowest in the rotating frame) will build the yrast line.

As described in section 5.19 below, for each parity combination and spin, thetotal energy is minimized with respect to the meshpoints (i.e. to the deformationsand tilting angles in the calculation). Then, in principle, the yrast lines for thedifferent meshpoints are compared at the present spin. As shown in paper 3, thetotal energy may vary very weakly with one, or both, of the tilting angles, a variationthat sometimes come close to the error limits within the model. Then it is crucialthat the yrast lines are calculated in a good way, i.e. with high precision andconsistent from one meshpoint to another.

In particular for the tilted meshpoints away from self-consistency, calculation ofhighly excited configurations are needed to generate the appropriate yrast band. Ifonly low order configurations are used, then for high spins the energy will be to highat these meshpoints (higher than for the true yrast), and as how much too high canvary from one meshpoint to another, that can produce false local minima, as wasdiscovered in early applications of the program, before the need for including highlyexcited configurations was realized. Such false local minima could be ruinous asnot only in the global minima (that often are principal axis cranked states) are ofinterest, but also the higher local minima (that sometimes are tilted).

An yrast band that meets these requirements, is realized by on one hand calcu-lation with small steps in the cranking angular frequency, to get sufficiently smallsteps in total energy along the yrast band, and on the other hand by the smallestslope and rotational band tracking algorithms, described in section 5.15, that in areliable way determine the ”states” along the yrast band.

Along a band the total energy can be approximated as E ≈ E0 + Jbω2/2.

Differentiation gives

∆ω ≈ ∆E

Jbω. (5.230)

For spins & 10~ for mass number in the order of 160, the band moment of inertiais roughly of the order Jb ≈ 102~2/ MeV. For a constant step in total energyalong the band, of the order of ∆E ≈ 0.1MeV, the step in the cranking angularfrequency decreases as 10−3/ω MeV2/~2. The step length is chosen as a constantfor ω ≤ 0.1 MeV/~ such that the step length function remains continuous. The


step length for the 166Hf investigation in paper 3 was chosen to

∆ω =

0.01, ω ≤ 0.1

0.001/ω, ω > 0.1(5.231)

in units of MeV/~. This keeps the error in the spin interpolated energies small andof the same order along the whole yrast band, see section 5.16, but the number offrequencies can be fairly large. For ωmax = 0.6MeV/~ the number of frequencies isabout 180, for ωmax = 0.7MeV/~ the number is 250.

The method of smallest slope

A simple algorithm to determine the yrast band from the low energy set for agiven (πp, πn), assuming no band crossings, is the following. (To handle the bandcrossings, the rotational band tracking alghorithm, described in section 5.15, isused).

The starting point in the yrast band is assigned the point (I1, E1) with minimaltotal energy E =Etot, among all points in the low energy set for the present paritycombination. The next point is chosen as the point (I2, E2), which gives the smallestslope

µ(1, 2) =E2 − E1

I2 − I1(5.232)

for a line segment that begins in (I1, E1). The search is repeated from the secondpoint, and so on. The procedure is continued along the yrast band to the largestspin available in the calculation, unless, for the spin Iss found by the method ofsmallest slope, the condition

Iss − Ik ≥ β(Ik − Ik−1) (5.233)

is satisfied, in which case a rotational band crossing is expected to occur somewherein the interval [Ik,Iss]. The value of the parameter β was chosen to 2, which workedwell in the calculations for 166Hf carried out for paper 1 and 3.

Rotational band tracking

If the condition (5.233) is satisfied, then a phase of rotational band tracking isperformed in the interval [Ik,Iss]. Along a rotational band the moment of inertiais roughly a constant, so that the total spin and energy vary as

I ≈ Jbω, E − E0 ≈ I(I + 1)2Jb

. (5.234)

Differentiation gives∆I ≈ Jb∆ω, ∆E ≈ I∆ω, (5.235)

5.15. THE YRAST BAND 63

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70

Eto

t (M

eV)

Ι (−h)

++ yrast example

Etot(Ι)14

15

16

17

18

19

43 44 45 46 47 48 49 50 51 52

Ι (−h)

++ yrast example

k

k-1

ss

Figure 5.2: To the left, an yrast line for X = 0.375, Y = 0.125, ε4 = 0.03, θ = 90

and φ = 9 calculated with the method of smallest slope and rotational bandtracking. To the right, a zoom-in around a band crossing, an example where therotational band tracking is performed.

for a step ∆ω in the cranking angular frequency. From (5.234), the band momentof inertia can be assigned

Jb = IkIk − Ik−1

Ek − Ek−1. (5.236)

Starting from (Ik, Ek) the next point along the band is expected to be close tos ≡ (Ik + ∆I, Ek + ∆E), according to eq. (5.235). A point P is searched fromamong the points in the low energy set, within a box centered around s, and withsmallest slope µ(k, P ). P is added to the calculated yrast band, unless the slopeµ(k, P ) is much larger than µ(k−1, k) (in which case a jump to an excited band hasoccurred, which must here be avoided as we are looking for the yrast band only).It can be expressed in terms of a condition on the band moment of inertia ratio

Jb(P )/Jb(k) < lim < 1, (5.237)

where lim was chosen to 0.8 in the 166Hf calculations. If the condition above issatisfied, P is rejected, and the method of smallest slope is resumed. If not, then kis assigned to P and the band tracking is continued. However, the band tracking isalso interrupted if no point P is found, or if the spin of P becomes larger than Iss.

See fig. (5.2) for an illustration.


5.16 Spin interpolation

In order to compare yrast bands for different meshpoints, an interpolation betweenthe calculated states, i.e. points (Ik, Ek), with respect to spin along each yrastband is performed.

As the signature symmetry is broken for tilted axis cranking, the energy isinterpolated to spin values in steps of 1~, instead in steps of 2~ as in principalaxis cranking. The algorithms for calculation of the yrast band gives a sequence ofpoints with strictly increasing spin, assuming a deformed shape. Then it is alwayspossible to interpolate along the yrast band with respect to spin.

Note that the energy is interpolated with respect to the length I of total angularmomentum vector, i.e. I = ‖~I‖2. Also vector quantities are interpolated withrespect to I, but separately for the x, y and z-components.

For the spin interpolation calculations, weighted quadratic polynomial interpo-lation was used.

The total energy can then be minimized with respect to the meshpoints (i.e.deformation and tilting), for each parity combination and given integer step spinvalue.

5.17 Errors in total quantities in each meshpoint

As the calculations have shown, the total energy can vary very weakly with thetilting angles. Then it became important to estimate the errors in the total energy,eq. (5.229), and spin, eq. (5.177), within the model, to assure that the errors aresmaller than the depth of energy minima. The errors are due to the basis truncationerror, uncompleted Strutinsky averaged quantities convergence, spin interpolationand uncompleted convergence in quadratures for integrals.

Note that errors defined as discrepancy between model and experiment, and theerrors due to interpolation between different meshpoints, are not taken into accounthere.

For the total spin and for each of the terms in the total energy, the errors wereestimated. Actual values apply for the 166Hf calculations in paper 3.

The error in the total spin is due to the basis truncation error for the expecta-tion values of the angular momentum. The error can be estimated by calculationsfor successively higher eigenvalue convergence marginals, and for a fixed state com-paring the spin for a em value (see section 5.7) with the converged spin (for a veryhigh em value when changes in the spin are relatively negligible), which gave

∆I . 0.01~ (5.238)

for em = 5 sm = 2 (in units of oscillator shells), which also was the error limitaimed for in those calculations.

The error in the liquid drop energy, eq. (5.203), are due to small errors in thecalculated surface and Coulomb energy integrals, see sections 5.11 and 5.11. The

5.18. MESH CALCULATIONS 65

error in Bs is negligible compared to the error in Bc. The error estimate is

∆Eld ≈ 10−7 Z2

A1/3MeV, (5.239)

giving a small error of 0.1 keV.The error in the rigid body rotational energy, eq. (5.208), comes from the basis

truncation error for the spin, eq. (5.238), and from small errors in the moment ofinertia ratio integrals. Differentiation with respect to the quantities that containerrors give

∆Erotld =

∂Erotld

∂I∆I +

∂Erotld

∂Tx∆Tx +

∂Erotld

∂Ty∆Ty +

∂Erotld

∂Tz∆Tz. (5.240)

The three latter terms are negligible compared to the first. For the 166Hf calcula-tions, the error coming from (5.238) is of the order ∆Erot

ld . 10 keV.The error in the shell energy, eq. (5.227), is due to the basis truncation error and

to uncompleted Strutinsky convergence , and was estimated to the order ∆Eshell .10 keV, for em = 5, sm = 2 and smextra = 3.

The estimated error in the total energy is then

∆Etot . 20 keV. (5.241)

However, the error terms are not likely fully additive, which would bring down theerror estimate above somewhat, but still it should be of the order 10 keV.

Also, the errors in neighboring meshpoints are likely to go in the same direction,such that when they are compared in the minimization, the fluctuations are evensmaller than the order 10 keV.

5.18 Mesh calculations

A mesh is a discrete set of points in a multi-dimensional space for deformation,tilting angles and particle numbers.

In the program the parameters that may be varied in a mesh are given intable (5.1).

For the calculations in each of the three papers, the particle numbers were keptfixed. Thus, the mesh space is 5-dimensional, and the total number of mesh pointscan be large although the number of discrete values in each dimension is fairlysmall.

As for each mesh point the program loops through a sequence of cranking angu-lar frequencies ω=ω1, . . . , ωn, the total number of iterations (meshpoint-frequencycombinations) can be very large.

Due to the high time complexity for a six dimensional mesh-frequency space,and the need of both small errors, requiring large matrices to diagonalize, and ofcalculations of highly excited configurations, it was necessary to use fast algorithms


Name Explanation

ε Quadrupole deform. size

γ Triaxiality angle

ε4 Hexadecapole deform. size

θ Polar tilting angle

φ Azimuthal tilting angle

N,Z Neutron and proton numbers

Table 5.1: The mesh parameters that can be varied in the calculations.

to bring down the total computation times to reasonable orders. Still, for theproduction calculations for paper 3, the total time were in the order of a week,distributed over about 20 computers. That is for using 100% of the CPU-time. Ona facility where CPU-time has be shared with other users, the total time is evenlarger.

In the actual calculations the Cartesian quadrupole coordinates

X = ε cos(γ)

Y = ε sin(γ)(5.242)

were used to vary the quadrupole deformation.

5.19 Minimization

For each proton and neutron parity combination, and integer step spin I, the min-imization of total energy Etot was carried out in the following way.

Strict local mesh minima

The mesh point with minimal Etot, among all mesh points for the present paritycombination and spin, is determined. Let us denote that mesh point m. The numberof mesh parameters (that is deformation-tilting variables varied in the calculation)is denoted by N. A N-dimensional sub mesh with 3N mesh points, with m in themiddle, is set up. See fig. (5.3).

If m is along the border of the grand mesh, then the center of the sub meshis moved in one step into the mesh. If a loop parameter along the border of thesub mesh is a mirroring value pm for that parameter p, i.e. Etot is symmetric withrespect to mirroring around pm,

Etot(pm − p) = Etot(pm + p), (5.243)

5.19. MINIMIZATION 67

p

p

2

1

Figure 5.3: A sample sub mesh of 9 points (circles) in the case N=2. The minimalEtot point m is marked with a filled circle.

the mesh is moved one step towards the mirroring plane and mirrored about thatplane. For example in the case N=2 and one border along φm = 0, for which thereis a symmetry Etot(−φ) = Etot(φ).

Newton minimization of multidimensional interpolatingpolynomials

The total energy is interpolated in the sub mesh by a multi-polynomial L(~p) ofdegree 2N (2nd order polynomials for each parameter), and minimized within thesub mesh by Newton-Raphson iterations

~pn+1 = ~pn −H(~pn)−1∇L(~pn) (5.244)

where H is the Hessian matrix holding the 2nd order derivatives of L, and ∇Lis the gradient vector with the 1st order derivatives of L. The iteration does notalways converge towards a local minimum of the multi-polynomial surface. If oneiteration goes outside the sub mesh, or if a stationary point is found which is not alocal minimum, or if the number of iterations gets larger than some limit withoutany stationary point being found, then the output minimum is chosen to m.

By this method, the total energy minimum is calculated by simultaneous mini-mization with respect to all mesh parameters in the calculation.

Properties at energy minimum, self-consistency condition

As shown in [8] for a self-consistent energy minima the spin vector, eq. (5.177), andthe cranking angular frequency vector, eq. (5.82), must be parallel, i.e.

~I ‖ ~ω. (5.245)


Define spherical angles (θI , φI) for the spin vector ~I = (Ix, Iy, Iz) by

Ix = I sin θI cos φI

Iy = I sin θI sin φI

Iz = I cos θ

(5.246)

To measure the deviation from self-consistency, define

cθ = θI − θ, cφ = φI − φ, (5.247)

which are called self-consistency measures. A self-consistent energy minimum has,ideally cθ = cφ = 0. But in practical calculations, energy minima with numericalself-consistency measure close to zero are considered approximations to (exact) self-consistent minima. For 2Dφ minima, the angle between the spin vector and thecranking vector is simply [~I, ~ω] = |cφ|. For 2Dθ minima, the angle is [~I, ~ω] = |cθ|.

5.20 The use of alternative potentials

The program allows alternative mean-field potential single-particle eigenvalues andangular momentum operator expectation values as input for the calculation of totalquantities such as total renormalized energy and total angular momentum. Thisgives a possibility to compare other mean-field potentials with the default modifiedoscillator potential built into the program package. In paper 2, the folded-Yukawasingle particle eigenvalues and expectation values of the angular momentum oper-ators, calculated by another program, ref. [13], entered before the Strutinski renor-malization calculation.

Bibliography

[1] Bohr A., Phys.Rev, 81 134

[2] Inglis, D. R., Phys. Rev. 96 (1954) p 1059, and 97 p 701.

[3] S.G. Nilsson, I. Ragnarsson, Shapes and shells in nuclear structure, Cambridge1995

[4] Rowe, D.J.,Nuclear collective motion Models and Theory, Methuen 1970

[5] Bohr A.,Mottelson B.R., Nuclear Structure, vol II, W.A. Benjamin, 1975

[6] S. Frauendorf,Rev. Mod. Phys. 73 (2001), 463

[7] S. Frauendorf, Nucl. Phys., A557(1993), 259.

[8] A.K. Kerman, N. Onishi, Nucl. Phys., A361(1981), 179.

[9] W.D. Heiss, R.G. Nazmitdinov, Phys. Lett., B397(1997).

[10] R. Bengtsson, Nucl. Phys., A557(1993), 277c

[11] Fowles, Cassiday, Analytical Mechanics,Suanders 1993.

[12] Riemann, B., Abhandlungen der mathematischen Classe der kniglichenGesellschaft der Wissenschaften zu Gttingen. Neunter Band. 1860. Gttingen,1860.

[13] Computer program for the folded-Yukawa potential by Peter Moller.

[14] S.E. Larsson, Physica Scripta, Vol.8(1973)17

[15] S.G. Nilsson, Mat. Fys. Medd. Dan Vid. Selsk., 29, no 16

[16] J.J. Sakurai Modern quantum mechanics, Addison-Wesley, 1994.

[17] M. Moshinsky, Y.F. Smirnov The harmonic oscillator in modern physics, Har-wood academic publishers, 1996.

[18] L.C. Biedenharn, J.D. Louck Encyclopedia of mathematics and its applications,volume 8 : Angular momentum in quantum physics, Addison-Wesley, 1981.

69

70 BIBLIOGRAPHY

[19] Rohozinski, Sobiezewski, Acta Physica Polonica, vol B12(1981).

[20] R. Bengtsson, J.D. Garrett, Lund-Mph-84/18

[21] S.G. Nilsson et al., Nucl. Phys., A131(1969)1

[22] Ultimate Cranker, http://www.matfys.lth.se/Ragnar.Bengtsson/ultimate.html

[23] W.H. Press, S.A. Teukolsky et.al. Numerical recipes in FORTRAN. The art ofscientific computing, Cambridge, 2nd edition, 1992.

[24] EISPACK http://www.netlib.org/eispack/

[25] LAPACK http://www.netlib.org/lapack/

[26] NAG The Numerical Algorithms Group Ltd, http://www.nag.co.uk

[27] NAPACK http://www.netlib.org/napack/

[28] BIGMAX M.Elson, R.E.Fundnrcit

[29] S. Spanne, Forelasningar i matristeori, Dept of Mathematics, Lund Instituteof Technology, 1994, p 157.

[30] V.M. Strutinsky, Nucl. Phys. A95(1967), 420.

[31] Anderson et. al., Nucl. Phys. A268(1976), 205

[32] M. Brack, The Shell correction method, International Workshop Nuclear Struc-ture Models, 1992

[33] W.D. Myers, W.J. Swiatecki, Nucl. Phys. 81 (1966) 1

[34] R. Bengtsson et. al., Nucl. Phys. A569 (1994) 469

PAPER I

TUMGREPP 04-01-22, 14.231

TUMGREPP 04-01-22, 16.106

On the interpretation of single-particle level diagrams in tilted rotation of the atomicnucleus

Peter A. Olivius∗ and Ragnar BengtssonDepartment of Mathematical Physics, Lund Institute of Technology, Lund, Sweden

(Dated: January 22, 2004)

In the generalization of the cranking model from rotation about a principal axis (1D) to tiltedrotation (2D and 3D), the mapping of the lowest total routhian on the yrast band is broken. Intilted rotation the single particle energy diagrams cannot alone be used to construct the yrast band,because they lack information of the spin component perpendicular to the rotation axis. In fact, intilted rotation the yrast band can for high spin correspond to high particle-hole excitations in therotating frame.

A rotating rigid body has stationary states, i.e. theenergy has an extremum, only when it rotates about itsprincipal axes. An atomic nucleus does not behave likea rigid body, since it consists of particles (protons andneutrons) moving in individual orbitals inside a common,usually deformed, potential well. As the potential well,and thus the nucleus, is forced to rotate, the individ-ual particles are bound to collectively follow this motion,creating an angular momentum along the rotational axes.However, particles moving inside the rotating potentialwell may generate an angular momentum pointing in adifferent direction than that of the enforced collective ro-tational axis.

Typical examples of the above situation are rotationalbands built on high-K bandheads in axially symmetricnuclei.The collective rotation takes place about an axisperpendicular to the symmetry axis, where as one or afew particles generate a sizable angular momentum com-ponent along the symmetry axis. The total angular mo-mentum will then point in a direction, which deviatesfrom the principal axes of the nuclear potential. Thissituation is referred to as tilted rotation.

Tilted axis rotation can be studied theoretically bymeans of two- or three-dimensional cranking. In the firstcase, the cranking axis (i.e. the enforced rotational axis)is restricted to lie in a principal plane, whereas in thelatter case it does not belong to any principal plane. Byminimizing the energy with respect to the direction of thecranking axis, selfconsistent solutions can be found.Thecranking axis and the total angular momentum are par-allel. For high-K bands of the kind mentioned above,the configuration and bandhead deformation are known.It is then feasible to fix the configuration and study itsevolution with respect to the rotational frequency andthe tilting angle. Note that only one tilting angle (theangle between the cranking axis and the symmetry axis)is required. However, in more general situations, e.g.when studying strongly deformed triaxial nuclei at highangular momentum, neither the exact deformation, northe configurations that may result in tilted rotation areknown a priori. Furthermore, two tilting angles must beconsidered, since the nucleus lacks rotational symmetry.

To find selfconsistent tilted rotational states, the total en-ergy typically has to be minimized in a five-dimensionalspace, spanned by three deformation parameters and twotilting angles. It may require that the total energy is cal-culated in as many as about 103 to 104 mesh points ormore. Only after the energy minimization is done, theconfiguration of the selfconsistent tilted rotational statescan be determined. Consequently, it is not possible tobase calculations on tracing a few pre-selected configura-tions.

Different strategies can be used for finding selfconsis-tent solutions. A very appealing approach is to minimizethe total energy for a fixed value of the total angular mo-mentum by making calculations in a multidimensionalmesh, since this also produces a picture of the energylandscape surrounding the energy minima. Thus, onecan e.g. determine whether an energy minimum is soft orstiff and draw conclusions about whether a stable tiltedaxis rotation is to be expected or whether a wobblingmotion is more likely. A complication in such mesh cal-culations is that most of the mesh points lie far from theselfconsistent points and that the configurations that arerepresented in the lowest energy surfaces away from theselfconsistent points may correspond to highly excitedconfigurations in the rotating system. We shall discussthis problem here, explaining why it appears and illumi-nate the consequences it gives rise to in practical meshcalculations.

The cranking model [1] is designed for calculating ro-tational states of the atomic nucleus. The generalizedmodel Hamiltonian can be written

Hτω = Hτ

0 − ~ω ·~j, (1)

where Hτ0 is the Hamiltonian in the non-rotating (lab)

frame, ~ω is the cranking angular frequency vector and ~jthe angular momentum operator. Hτ

ω has eigenvalues eωτi

and eigenfunctions Ψωτi, where τ = p for protons, n for

neutrons. The total energy in the rotating frame is

EωTot =

∑

i occ

eωpi +

∑

i occ

eωni, (2)

where the sums are to be taken over the occupied pro-ton and neutron single particle states, respectively. Since

2

we are only interested in the single-particle properties weomit in this paper the renormalization of the total energy,which is otherwise required [2]. By calculating Eω

Tot, usu-ally referred to as the total routhian [2], for a set of nu-clear deformations (shapes), arranged in a regular mesh,total routhian surfaces (TRS), showing Eω

Tot for a fixedcranking vector, ~ω, can be constructed. TRS have beenwidely used as a tool for displaying results of principalaxis (or 1D) cranking calculations. In 1D cranking, ~ωpoints along one of the principal axes of the (deformed)nucleus. In 2D cranking, ~ω lies in a principal plane, andin 3D cranking it does not belong to any principal plane,see fig. 1 and ref. [4].

In the rotating frame, the lowest total energy is ob-tained by filling all single particle levels below the Fermienergy. In this paper we are only going to consider stateswith positive parity for both protons and neutrons. Thelowest total routhian in this subset of states will be de-noted ord=1, the excited but energetically next-to-lowestrouthian ord=2, and so on.

In order to compare with experimental data, total en-ergies must be given in the non-rotating (lab) frame forfixed (integer or half-integer) values of the angular mo-mentum. The total angular momentum vector is calcu-lated from the eigenfunctions of (1) as

~I =∑

i occ

〈Ψωpi |~j|Ψω

pi〉+∑

i occ

〈Ψωni |~j|Ψω

ni〉. (3)

The corresponding total energy in the lab frame is definedas

Elab =∑

τ

∑

i occ

〈Ψωτi |Hτ

0 |Ψωτi〉 = Eω

Tot + ~ω · ~I, (4)

using the notations of eqs. 1–3. In order to construct theexcitation energy spectrum at a given angular momen-tum I = |~I|, an interpolation between calculated pointsElab(I) is required.

In 1-D cranking ~I ‖ ~ω and there is a simple relationbetween Eω

Tot(ω) and Elab(I): The lowest sequence of

ωz

y

long

axi

s

intermediate

axis

φ

θ

x

short axis

FIG. 1: The cranking vector ~ω, the cranking angles θ and φ(spherical coordinates), and the directions of the three prin-cipal axes of the deformed nucleus.

states (band) in the rotating frame (ord=1) is mappedon the lowest sequence of states in the lab frame (yrastband). Furthermore, the lowest excited bands in the labframe are generated from the lowest excited bands in therotating frame. This is why the TRS can be successfullyapplied to the investigation of rotating nuclei in the 1D(principal axis) cranking limit. However, in the 3D and2D cases (tilted cranking) the simple relation betweenexcitation energy spectra in the rotating frame and thelab frame breaks down. We shall illustrate the situationby a couple of examples taken from an investigation ofthe triaxially superdeformed (TSD [3]) nucleus 166Hf.

First consider principal axis cranking. The relevantsingle-particle energy diagrams for protons and neutronsare shown in fig. 2 for cranking about the shortest axisat a deformation close to the TSD energy minimum. Fill-ing all levels below the Fermi energy (long-dashed line)generates the energetically lowest band in the rotatingframe. Each point that belongs to this band correspondsto a point in the yrast line in the lab frame, as illustratedby the -points (ord=1) in fig. 3. Some states are labeledwith their ord number and, within parenthesis, the valueof the rotational angular frequency ω. Two situationsshould be particularly observed. The first is when anoccupied level is interacting weakly with a down-slopingunoccupied level above the Fermi level. Then extra an-gular momentum is gained over a short frequency rangedue to the alignment of an occupied level. Such an in-teraction (marked P1 i fig. 2) is found in the protonsystem at ~ω ≈ 0.32 MeV. The effect in the lab frameis seen in the spin range 30 – 35 ~ in fig. 3, where thedistance between the calculated points is larger than itis for both lower and higher spin, although the frequencymesh points are equidistant in this case (∆~ω = 25 keV).By exciting a proton from level -a to a, the dot-dashedcurve at P1 in fig. 3 is generated.

The second situation is a sharp level crossing (no in-teraction) like the one marked P2 in fig. 2 at ~ω ≈ 0.50MeV in the neutron system, where the levels A and –Across. When the lowest routhian is mapped onto the labframe, a gap, ∆I, is produced in the yrast line, see fig.3. The width of the gap is determined by the differencein slope of the two crossing levels

∆I = 〈Ψωni |jx|Ψω

ni〉 − 〈Ψωnj |jx|Ψω

nj〉, (5)

where in this case i = A and j = –A. The gap in theyrast line is filled by considering particle-hole excitations,namely the one in which –A is filled and A is emptyabove the crossing as well as the one in which A is filledand –A is empty below the crossing marked P2 in fig. 2.These particle-hole excitations generate the two bands atP2 in fig. 3.

Let us now turn to tilted axis cranking [4–7](and the references therein), in which case ~ω =ω(sin θ cosφ, sin θ sin φ, cos θ), see fig. 1. It must be ob-served that in tilted axis cranking, the direction and

3

B CA

A

−A−A

P2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8hω (MeV)

49.5

50

50.5

51

51.5

52

e i (M

eV)

94 NEUTRONS ε=0.39 γ=14.93 o ε4=0.03 θ=90

o φ=0

o

a

a

−a

−a

P1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8hω (MeV)

42

42.5

43

43.5

44

e i (M

eV)

72 PROTONS ε=0.39 γ=14.93 o ε4=0.03 θ=90

o φ=0

o

FIG. 2: Single-proton (top) and single-neutron (bottom) en-ergy levels for θ = 90 and φ = 0 (principal axis rotation).Levels for single-particle states with positive parity are solid,and those for negative parity are dot-dashed.

length of the angular momentum vector, ~I, is differentin different configurations and at different deformationseven if the cranking vector ~ω is the same. Only for a self-consistent solution, i.e. when the energy has a minimumwith respect to θ and φ and the deformation simulta-neously, the vectors ~I and ~ω will be parallel, as shownin [8].

To illustrate the effects of 3D cranking, we choose theangles θ = 40 and φ = 40. The corresponding single-particle energy levels are shown in fig. 4.

As previously, the lowest total routhian (ord=1) is cal-culated by filling all single-particle levels below the Fermienergy. Structural changes appear when down-slopinglevels are crossing the Fermi surface. The first crossingof this kind (marked T1 in fig. 4) appears in the neu-tron system at ~ω ≈ 0.28 MeV, where the levels A and–A cross at the Fermi surface without interaction. Thestructure of the lowest total routhian (denoted ord=1)changes at this point. Below the crossing level –A is oc-cupied and level A is empty, whereas above the crossinglevel A is occupied and level –A is empty. For the othertotal routhian involved in the crossing (denoted ord=2),the occupation numbers are reversed. However, whenthe corresponding rotational bands are mapped on to thelab frame and plotted as energy (eq.4) versus angular mo-mentum I, the structure of the yrast line does not change(figs. 5 and 6). The rotational band in which level A is

5942

5944

5946

5948

5950

5952

5954

5956

5958

5960

10 15 20 25 30 35 40 45 50 55

Ela

b(M

eV)

Ι( −h )

Elab(Ι) for θ=90°,φ=0°,step(−hω)=0.025 MeV, πp=+,πn=+

1(0.225)

1(0.325)

2(0.450)

1(0.475)

2(0.525)

2(0.500)

1(0.550)

∆ Ι

P1

P2

yrast, ord=1yrast, ord>1

FIG. 3: The yrast line structure in principal axis rotation forθ = 90 and φ = 0. Some states are labeled with their ordnumber and ω-values within parenthesis.

occupied and level –A is empty, does not reach the yrastline above the crossing, as one would expect based on theexperience from principal axis cranking. The explanationis simple. The lab energy (eq. 4) is calculated by addingthe term ωIω = ~ω · ~I to the total routhian, whereas thetotal angular momentum is calculated as I =

√I2ω + I2

⊥,where I⊥ is the angular momentum component perpen-dicular to the cranking vector.

For the ord=1 configuration, the change in the angularmomentum vector at the neutron level crossing T1 infig. 4 is given by

∆~I = 〈Ψωni |~j|Ψω

ni〉 − 〈Ψωnj |~j|Ψω

nj〉, (6)

with i = A and j = –A and the change of the length ofthe angular momentum vector can be written as ∆I =∆Iω + ∆IS where IS = I − Iω =

√I2ω + I2

⊥ − Iω. Thus,∆I depends on the perpendicular component I⊥. Thechange of I at the crossing can be broken down in twosteps. First, I increases with an amount given by theterm ∆Iω, corresponding to the difference in slope be-tween the crossing single-particle levels. At the sametime the lab energy increases with ω∆Iω. This results ina point X near the yrast line as shown in fig. 6. Second, Iis decreased by |∆IS |, as ∆IS is negative. Therefore, therotational band, corresponding to the ord=1 routhian,

4

−A BB

−B

T1 T3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8hω (MeV)

A

A

A

−A−B

49.5

50

50.5

51

51.5

52

e i (M

eV)

94 NEUTRONS ε=0.39 γ=14.93 o ε4=0.03 θ=40

o φ=40

o

−a aa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

T2

hω (MeV)

42

42.5

43

43.5

44

e i (M

eV)

72 PROTONS ε=0.39 γ=14.93 o ε4= 0.03 θ=40

o φ=40

o

FIG. 4: Single-proton (top) and single-neutron (bottom) en-ergy levels for θ = 40 and φ = 40 (tilted axis rotation).Positive parity levels are solid, negative dot-dashed. Holes(circles) below the Fermi surface, and particles (filled circles)above the Fermi surface, are marked for the yrast configura-tion ord=3693, ω = 0.540. That configuration is also markedin fig. 5.

will not appear along the yrast line above the crossing,but at a relatively high excitation energy (about 1 MeVin the case shown in fig. 6).

Following the yrast configuration towards higher fre-quencies, additional crossings are encountered. First asharp proton crossing at ~ω ≈ 0.38 MeV (marked T2 infig.4) and then a neutron crossing with a weak interactionat ~ω ≈ 0.40 MeV (marked T3 in fig. 4).

The proton crossing has similar properties as the neu-tron crossing at ~ω ≈ 0.28 MeV, i.e. the yrast configura-tion in the lab frame does not change, since the config-uration with the down-sloping level marked a occupieddoes not reach the yrast line, that is conf IV below T2and conf I above T2 as defined in legend to fig. 6.

At the neutron crossing T3 at ~ω ≈ 0.40 MeV theweak interaction between the two crossing routhians Band –B results in a gradual mixing, which in the labframe leads to the smooth continuous exchange of prop-erties between the bands denoted conf III and conf IV infig. 6. The precise shape of the mixed bands in the labframe depends strongly on the value of ∆I = ∆Iω +∆IS .For ∆IS = 0 we have the same kind of band crossingsas in principal axis cranking. On the other hand, when∆I < 0, i.e. when the absolute value of ∆IS is larger than

5944

5946

5948

5950

5952

5954

5956

10 15 20 25 30 35 40 45 50

Ela

b(M

eV)

Ι( −h )

Elab(Ι) for θ=40°, φ=40°, πp=+, πn=+

1 (0.274)

22 (0.376)

1743 (0.502)

837 (0.492)

3693 (0.540)

15438(0.585)

Elab<Elab(ord=1)ord=1yrast,ord>1

FIG. 5: The near-yrast line structure in tilted axis rotationfor θ = 40 and φ = 40. The lowest total routhian is mappedon the -curve (ord=1). The -curve shows the part of theyrast line for which ord is >1. The grey dots mark the stateswith ord>1 for which the lab energy is less than for ord=1.Some states along the yrast line are labeled with their ordnumber and ω-values within parenthesis.

∆Iω, the rotational band with lowest energy below thecrossing bends back as it loses angular momentum at thecrossing. When the interaction is weak, it is preferableto construct non-interacting single-particle levels throughthe crossing, creating the same situation as at crossingT1 with smoothly behaving rotational bands. The resultof removing the interaction at crossing T3 is schemati-cally shown by a dashed line along the yrast line betweenI ≈ 26 and 28~ in fig. 6.

If the energies in the lab frame are plotted versus Iω, in-stead of I, the lowest routhian forms the yrast line in thelab frame, in the same way as in principal axis rotation.However, when the perpendicular component I⊥ is takeninto account, the rotational bands in the lab frame areshifted towards higher angular momentum by an amountIS =

√I2ω + I2

⊥ − Iω as illustrated for the two bands infig. 6 that have the configurations conf II and conf III. Itis obvious that a large perpendicular component I⊥ andthus a large IS is favorable, bringing such bands closerto the yrast line. For the lowest total routhian (ord=1)IS is comparatively small, which explains why the corre-sponding band has a high excitation energy above the T1

5

5945

5946

5947

5948

5949

5950

20 22 24 26 28 30

Ela

b(M

eV)

Ι( −h )

Elab(Ι) for θ=40°, φ=40°, πp=+, πn=+

1 (0.274)

22 (0.376)

(0.585)

2 (0.312)

2 (0.259)

1 (0.292)

X

∆Ιω

ω ∆Ιω

∆ΙS

T1

T3

T2

Ιω ΙS

Ιω ΙS

Elab<Elab(ord=1)yrast, not conf I-IVconf Iconf II (ord=1)conf IIIconf IV

FIG. 6: Zoom-in of the marked region in fig.5. Let [h,p] de-note that the single particle state h is a hole and p is a particle.Around the T1 crossing in fig. 4: Conf I has the neutron con-figuration [–A,A] below T1, but [A,–A] above T1, giving the-conf I curve in this figure. Conf II has opposite occupationof the A and –A levels, has ord=1, and gives the -curve inthis figure. The discontinuous jump in spin and lab energy forthe two configurations when they pass the crossing is markedwith the same symbol T1 in the present figure.Around the T2 crossing: Conf I has the proton configuration[a,–a] below T2 but [–a,a] above T2. Conf IV has the protonconfiguration [–a,a] below T2, but [a,–a] above T2.Around the T3 crossing conf III has the neutron configura-tion [–B,B] both below and above the crossing, see fig.4.The grey dots mark the states with ord>1 for which the labenergy is less than for ord=1.For the illustrations of the component Iω and the shift IS ,the states along the ord=1 curve and along conf III are cho-sen such that both have the same value of Iω.

crossing in figs. 5 and 6. In fact, above T1 a large num-ber of rotational bands lie between the yrast line and theband corresponding to the ord=1 routhian as indicatedby the grey dots in fig. 5 and 6. In the tilted case, theyrast band in the lab frame may at high angular momen-tum correspond to highly excited states in the rotatingframe, which is illustrated by the ord numbers for somestates along the yrast line in fig. 5.

From the examples discussed above, a few importantconclusions can be drawn.

1. In the tilted case, the single-particle level diagramscannot alone be used to predict the configuration of thelowest rotational bands in the lab frame, since they giveno information about the angular momentum componentperpendicular to the rotational axis.

2. The lowest total routhians do, in the tilted case, ingeneral not correspond to the lowest rotational bands inthe lab frame. In fact, the lowest routhians may corre-spond to highly excited rotational bands.

3. The yrast band and other low-lying bands in the labframe may correspond to particle-hole excitations withhigh energy in the rotating frame.

4. At high angular momentum (of the order 50~) theremay be thousand or more total routhians with a lowerenergy in the rotating frame, than the total routhianscorresponding to the bands which build up the yrast bandin the lab frame.

∗ Electronic address: [email protected]

[1] G. Andersson et al., Nucl. Phys. A268 (1976) 205[2] A. Granderath et al., Nucl. Phys. A597 (1996) 427[3] H. Schnack-Petersen et al., Nucl. Phys. A594 (1995) 175[4] R. Bengtsson, Nucl. Phys. A557 (1993) 277[5] S. Frauendorf, J. Meng, Nucl. Phys. A617 (1997) 131[6] S. Frauendorf, Nucl. Phys. A677 (2000) 115[7] S. Frauendorf, Rev. Mod. Phys. 73 (2001) 463[8] A.K. Kerman, N. Onishi, Nucl. Phys. A361 (1981) 179

TUMGREPP 04-01-22, 16.102

PAPER II

TUMGREPP 04-01-22, 16.103

TUMGREPP 04-01-22, 16.104

DRAFT

Cranking the folded-Yukawa potential

Peter Olivius and Ragnar Bengtsson

Department of Mathematical Physics, Lund Institute of Technology, Lund, Sweden

Peter Moller

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545

Abstract

The folded-Yukawa potential is extended to describe triaxial shapes and a crankingterm is included. Single-particle energy spectra and moment of inertia propertiesare investigated and compared to those of the Nilsson potential.

Preprint submitted to Elsevier Preprint 22 January 2004

Contents

1 Introduction 4

2 The potential 4

3 The nuclear shape parameterization 5

3.1 Quadrupole deformations 6

3.2 Higher-multipole deformations 9

3.3 Shape symmetries 11

4 Cylindrical coordinates and basis functions 12

5 Choosing appropriate sectors of the (ε,γ) plane 13

6 The spin-orbit term 15

7 The cranking term 17

7.1 Cranking about the z axis 17

7.2 Cranking about the x axis 17

8 Comparisons with the modified harmonic oscillator potential 18

8.1 The single-particle energies 18

8.2 The moment of inertia 20

9 Conclusions 24

References 26

A Matrix elements of jx 27

A.1 The jx operator 27

A.2 Calculation of the matrix elements of lx 27

A.3 Coupling rules and algebraic expressions for the matrixelements of lx 31

A.4 Matrix elements of sx 32

2

A.5 Matrix elements of jx 32

B Symmetries 33

B.1 Parity 33

B.2 Signature for rotation about the z axis 33

B.3 Signature for rotation about the x axis 34

B.4 The α-kets 35

B.5 Matrix elements of jx with respect to the α-kets 35

C Some orthogonal polynomial relations 37

C.1 Hermite polynomials formulas 37

C.2 Laguerre polynomials formulas 37

3

1 Introduction

Calculations of nuclear properties at high spin primarily make use of the crankingmodel. There was a major break-through in this field [1,2] in the mid seventies asthe result of a collaboration between research groups at the universities of Lundand Warsaw. The basic ideas have since then been developed considerably, e.g. bythe inclusion of pairing correlations [3,4]. Today, two versions of the cranking modeldominate high-spin physics. They are based on the modified-harmonic-oscillator(Nilsson) potential and on the Woods-Saxon potential, respectively. However, forboth models when compared to experimental data, the average moment of inertia ofthe yrast line is not correctly reproduced. In the oscillator model, the reason is thesomewhat unphysical l2 term that occurs in the potential. In the Woods-Saxon po-tential the commonly used universal parameter set has an unphysically large valuefor the radius parameter. The problem is overcome by a Strutinsky-like renormal-ization of the moment of inertia. The renormalization is necessary and works well atvery high angular momentum. At low spin (I < 20~) no renormalization is required.In fact, a better agreement with data is obtained without it. The reason is that atlow spin the angular momentum is built from the mixing of a few single-particlelevels near the Fermi surface, whose energies and relative position have been fittedto experimental data.

At high spin (and in particular if combined with a large deformation) high-j levelsreach the Fermi surface from above. They bring in large amounts of angular mo-mentum. Due to the deficiencies mentioned above of the currently used mean fieldpotentials, this happens at a too low rotational frequency, resulting in a too largeeffective moment of inertia. A renormalization of the moment of inertia is thereforerequired. A consistent treatment of the high- and low-spin regions is not possible,which is a major drawback of the currently used models.

In a mean field potential, where the high-j levels appear at more appropriate po-sitions, the above mentioned difficulties should not occur, resulting in a major im-provement of cranking calculations. This may be the situation in the folded-Yukawamodel [5]. To investigate this possibility we have extended the model to axiallyasymmetric shapes and included cranking terms in the single-particle potential anduse the enhanced model to investigate if properties such as the effective moment ofintertia are better described than in the Nilsson and Woods-Saxon potentials.

2 The potential

The single-particle potential felt by a nucleon is given by

V = V1 + VC + Vs.o. (1)

4

The first term is the spin-independent nuclear part of the potential, which is calcu-lated in terms of the folded-Yukawa potential as

V1(r) = − V0

4πapot3

∫

VG

e−|r−r′|/apot

|r− r′|/apotd3r′ (2)

where the integration is over the volume, VG, of the generating shape, whose vol-ume is held fixed at 4

3πRpot3 as the shape is deformed. The potential radius Rpot,

potential depth V0 and the potential diffuseness apot are defined in ref. [5]. Rpot andV0 are functions of the proton and neutron numbers and V0 has different values forprotons and neutrons. The potential diffuseness apot has the constant value 0.8 fm.More details about the Folded-Yukawa potential are found in ref. [6].

The second term in Eq. (1) is the Coulomb potential for protons which is given by

VC(r) =qpρc

4πεo

∫

VG

d3r′

|r− r′| (3)

where the charge density ρc is given by

ρc =Zqp

43πAr0

3(4)

For the nuclear radius constant r0 the value 1.16 fm is used [5], and qp is the chargeof the proton, and εo is the permittivity of free space.

The generating volume VG is defined by r(θ, φ) which is the distance from the centerof the volume to its surface, expressed in terms of the angles θ and φ of Eq. (8).r(θ, φ) here defined by the ε-parameterization, described in detail in section 3.

The last term in Eq. (1) is the spin-orbit term Vs.o.. It depends on the generatingvolume in a more complicated way than the two other terms in Eq. (1) and istherefore described in more detail in section 6.

3 The nuclear shape parameterization

The phrase “shape parameterization” is actually a misnomer, because the param-eterizations that are required to define nuclear single-particle models primarilyparameterize the nuclear single-particle potential. We are here mainly concernedwith the Nilsson perturbed-spheroid parameterization [7] and its extensions, theso-called ε parameterization. It was originally introduced by Nilsson to define thesingle-particle potential in such a way that matrix of the full Hamiltonian containedno matrix elements between basis functions with different main oscillator quantumnumber N , when the basis functions were defined in the so-called “stretched” repre-sentation. Although this parameterization is quite cumbersome, it is still very muchused today, so that current results are directly comparable with the wealth of resultsobtained in this parameterization over the past 50 or so years.

5

Once the eigen-functions of the single-particle Hamiltonian have been calculated, anuclear density distribution can be obtained from these by use of obvious expres-sions. However, the concept of nuclear shape is also useful, although it is less exactthan density distribution.

In the oscillator model the nuclear shape is somewhat arbitrarily postulated to bea particular potential-energy surface of the single-particle potential, such that ithas the same volume as a spherical nucleus, namely 4πr3

0A/3. In the folded-Yukawamodel it is the generating volume for the single particle potential that is definedby the parameterization and the “nuclear shape” is postulated to be given by thesurface shape of this volume. In both models the “shapes” defined in this wayallow the Coulomb-energy, surface-energy, and additional macroscopic terms to becalculated.

Below we will introduce the ε-parameterization by tracing its historical emergenceas the parameterization defining the Nilsson single-particle potential. We then carryover the parameterization to the folded-Yukawa model by defining the surface of itsgenerating volume VG in terms of the expression that is obtained for the single-particle equipotential energy surface in the Nilsson model.

With this background we will feel free below to follow customary practice and usethe term “nuclear shape” somewhat loosely.

3.1 Quadrupole deformations

In the Nilsson model the quadrupole degrees of freedom are introduced by definingthree oscillator angular frequencies

ωx = ω0[1− 23ε cos(γ + 2

3π)]

ωy = ω0[1− 23ε cos(γ − 2

3π)]

ωz = ω0[1− 23ε cos(γ)]

(5)

The corresponding triaxially deformed oscillator potential is

Vosc =m

2(ω2

xx2 + ω2yy

2 + ω2zz

2). (6)

The shape of the nucleus is considered to be given by the shape of the equipotentialsurface obtained by setting Vosc = C = constant. Equation (6) then describes anellipsoid with the length of the three semi-axes given by

ax =1ωx

√2C/m , ay =

1ωy

√2C/m , az =

1ωz

√2C/m (7)

Thus, the lengths of the semi axes are inversely proportional to the oscillator fre-quencies.

6

180Pt N=102 [ε=0.25]

-180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180γ (degrees)

7.5

8

8.5

9

9.5

10

Osc

illat

or fr

eque

ncy

(MeV

)

Sector 3

Sector 2

Sector 1

Sector 4

Sector 5

Sector 6ωy

ωx

ωz

Fig. 1. The oscillator frequencies ωx, ωy and ωz in the Lund convention for ε and γ.The frequencies are shown as functions of γ for a fixed value of ε. A base frequencyω0 appropriate for the neutron system of 180Pt is used. Note that the length of thesemi axes of the potential are inversely proportional to the oscillator frequencies,cf. Eq. (7).

The over-all magnitude of the quadrupole deformation is given by ε, where ε = 0corresponds to spherical shape. The triaxiality parameter γ determines throughEq. (5) the oscillator frequencies which define the shape of the nucleus. The oscillatorfrequencies ωx, ωy and ωz are shown in Fig. 1 for a full period −180 ≤ γ ≤ 180.The same set of frequencies (and thus also nuclear shapes) are repeated in each60 sector, which therefore all describe the same family of nuclear shapes. The sixsectors correspond to the six permutations of the coordinate axes with respect tothe principal axes of the nucleus.

To define r(θ, φ) we introduce the spherical coordinates rs, θ and φ, defined by therelations

x = rs sin θ cosφ , y = rs sin θ sinφ , z = rs cos θ (8)

Inserting (5) and (8) into (6), putting Vosc = C, gives

r2(θ, φ) =2C

mω20

·[

11 + ε

3 cos γ(1− 3 cos2 θ) + ε√3sin γ sin2 θ cos 2φ

]·

·[

11 + ε

3 cos γ(1− 3 cos2 θt) + ε√3sin γ(1− cos2 θt) cos 2φt

](9)

where θt and φt are introduced through

cos θt =

[1− 2

3ε cos γ

1 + ε3 cos γ(1− 3 cos2 θ) + ε√

3sin γ sin2 θ cos 2φ

]1/2

cos θ (10)

7

and

cos 2φt =(1 + ε

3 cos γ) cos 2φ + ε√3sin γ

1 + ε3 cos γ + ε√

3sin γ cos 2φ

. (11)

By inserting ε = 0 (spherical shape), r(θ, φ) = r0 and ω0 =ω0 in the formulas above,

the value of C can be expressed as

C =m

ω

2

0

2r20 (12)

where r0 andω0 are the radius and the oscillator angular frequency, respectively,

for the spherical nucleus.

We now insert the above expression for C into the formula for r2(θ, φ) and simul-taneously replace sin2 θt with 1− cos2 θt. This gives

r2(θ, φ) = r20

( ω0

ω0

)2 [1

1 + ε3 cos γ(1− 3 cos2 θ) + ε√


]·

·[

11 + ε

3 cos γ(1− 3 cos2 θt) + ε√3sin γ(1− cos2 θt) cos 2φt

](13)

The value of ω0 is determined by the volume conservation condition

4π

3axayaz =

4π

3ax(ε = 0)ay(ε = 0)az(ε = 0) (14)

that is the volume of the ellipsoidal equipotential surface at any quadrupole defor-mation equals the volume of the spherical equipotential surface. By Eq. (7) it issimplified to

ωxωyωz =ω

3

0 (15)

As is shown in Fig. 2, a nuclear shape as given by Eq. (13) is associated with eachpoint in the (ε, γ) plane. Physical quantities for the nucleus are often presented interms of contour diagrams versus ε and γ. A typical layout is that the center ofthe figure corresponds to ε = 0 (spherical shape) and the distance from the centeris equal to ε. The angular variable γ varies from −180 to +180, as shown in thefigure. The six 60 sectors correspond to the six sectors in Fig. 1. Thus, the nucleushas a prolate shape (the two shorter semi axes have the same length) at γ = 0

and at γ = ±120. It has an oblate shape (the two longer semi axes have the samelength) at γ = ±60 and at γ = ±180. The same nuclear shapes appear in each60 sector. Therefore, only one sector is needed to describe all shapes that can begenerated by the shape parameters ε and γ. Usually sector 1, with 0 ≤ γ ≤ 60, ischosen and the γ = 0 axis is then drawn horizontally.

In one-dimensional cranking the nucleus is studied when it rotates about any ofits three principal axes (which coincide with the semi axes). It would be enough to

8

The Lund convention

Sector 1Short axis rotation

Sector 2Intermediate axis rotation

Sector 3Long axis rotation


Sector 5Intermediate axis rotation


γ = 120o

(prolate)

γ = 60 (oblate)o

γ = 0o

(prolate)

γ = -60o

(oblate)

γ = -120 (prolate)o

γ = 180 o

(oblate)

-1 -0.5 0 0.5 1ε cos(γ + 30o)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ε sin

( γ+

30

o)

Fig. 2. The Lund convention for the (ε, γ) plane, and the 1 dimensional crankingconvention.

use sector 1 for this purpose if we study separately cranking about the x-axis (theshort axis in this sector), the y-axis (intermediate axis) and the z-axis (long axis).However, in the 1 dimensional cranking-model one usually only considers rotationabout the x-axis. Then, if one uses the sectors 1, 2 and 3 of the (ε, γ) plane, rotationabout all three principal axes will be covered. A comparison with Fig. 1 then showsthat sector 1 corresponds to rotation about the short axis, sector 2 to rotation aboutthe intermediate axis and sector 3 to rotation about the longest axis. At γ = 0 theshort and intermediate axes have the same length, which gives a prolate shape, andat γ = −60 the long and intermediate axes have the same length, which gives anoblate shape. This guarantees continuity of the cranking model description whenpassing from one sector of the (ε, γ) plane to the neighboring sector at γ = 0 andγ = −60. It should also be observed that the (ε, γ) plane, when used for displayingresults of cranking calculations, is reflection symmetric in the line ε cos(γ +30) = 0as shown in Fig. 2.

3.2 Higher-multipole deformations

The quadrupole deformations introduced previously were originally introduced [7]in a way which gives computational advantages in the MHO model. However, thefunction r(θ, φ) becomes complicated. We shall here give a brief description of howhigher multipoles were subsequently introduced [8–13] in the ε-parameterizationand present the corresponding expression for r(θ, φ).

9

First stretched Cartesian coordinates are introduced:

ξ =√

mωx/~ x , η =√

mωy/~ y , ζ =√

mωz/~ z (16)

They are used to define stretched spherical coordinates ρt, θt and φt through therelations

ξ = ρt sin θt cosφt , η = ρt sin θt sinφt , ζ = ρt cos θt (17)

where θt and φt are identical to those of Eqs. (10) and (11), respectively.

Rewriting the oscillator potential (6) using Eqs. (16), (17) and (5) gives

Vosc =~ω0

2ρ2t

[1 +

ε

3cos γ (1− 3 cos2 θt) +

ε√3

sin γ cos 2φt sin2 θt

](18)

In the next step the potential is rewritten in terms of spherical harmonics, Y m` ,

which can be expressed as

Y m` (θt, φt) = (−)m

[(2` + 1)(`−m)!

4π(` + m)!

]1/2

Pm` (cos θt)eimφt , 0 ≤ m ≤ l (19)

where Pm` are Legendre functions defined as

Pm` (u) =

(1− u2)m/2

2``!d`+m

du`+m(u2 − 1)` ,

` = 0, 1, 2, ..., ∞ and m = 0, 1, 2, ..., ` (20)

Whereas the associated Legendre functions only are defined for m ≥ 0, the sphericalharmonics are also defined for m < 0. The spherical harmonics with m < 0 aredefined as

Y −m` (θt, φt) = (−)mY m

`∗(θt, φt) (21)

Using the above expressions of the spherical harmonics, the MHO potential (18)can be written as

Vosc =~ω0

2ρ2t

1− 4ε

3

√π

5

[cos γ Y 0

2 (θt, φt)−sin γ1√2

(Y 2

2 (θt, φt)+Y −22 (θt, φt)

)]

(22)

Higher multipole deformations are now introduced by adding additional terms tothe potential according to the following prescription:

Vosc =~ω0

2ρ2t

1− 4ε

3

√π

5

[cos γ Y 0

2 (θt, φt)−sin γ1√2

(Y 2

2 (θt, φt)+Y −22 (θt, φt)

)]

+∑

`>2

∑

m=−`

a`mY m` (θt, φt)

(23)

10

which can be written on the more compact form

Vosc =~ω0

2ρ2t

[1 +

∑

`

∑

m=−`

a`mY m` (θt, φt)

](24)

Solving for the radius in the same way as in Eq. (13) gives the result

r2(θ, φ) = r20

( ω0

ω0

)2 [1

1 + ε3 cos γ(1− 3 cos2 θ) + ε√


]·

·[

1

1 +∑

`

∑`m=−` a`mY m

` (θt, φt)

](25)

The quadrupole coefficients a2m can be directly determined by comparing Eq. (23)with Eq. (24), which gives

a20 = −4ε3

√π5 cos γ

a22 = a2,−2 = 4ε3

√π10 sin γ

a21 = a2,−1 = 0

(26)

The hexadecapole coefficients a4m are given by

a40 = 2ε49

√π(5 cos2 γ + 1)

a42 = a4−2 = − ε49

√30π sin 2γ

a44 = a4,−4 = ε49

√70π sin2 γ

a41 = a4,−1 = a43 = a4,−3 = 0

(27)

which are determined by requiring that the shape satisfies reflection symmetry withrespect to any coordinate-plane and that the hexadecapole deformation is z axissymmetric for γ = 0.

3.3 Shape symmetries

We shall in this paper only consider quadrupole and hexadecapole deformations.This implies certain symmetries for the nuclear shape, which can be deduced fromthe expression for r(θ, φ) given in Eq. (25). For the folded-Yukawa model the shapeof the generating volume VG is also subject to these symmetries, which are therefore

11

also present in the single-particle potential V , defined in Eq. (1). Thus V has goodparity, i.e. V (r) = V (−r). Furthermore, V is symmetric under rotation an angle πabout any of the principal axes. This symmetry is in the cranking model called thesignature. The parity remains a good quantum number when the nucleus rotatesand so does the signature, provided that it rotates about one of its principal axes. Agood single-particle basis for the rotating nucleus should therefore consist of basisstates with good parity and good signature.

4 Cylindrical coordinates and basis functions

In the folded-Yukwawa model we use a set of deformed oscillator basis functionsexpressed in cylindrical coordinates ρ, φ and z, which are defined by the relations

x = ρ cosφ , y = ρ sinφ , z = z (28)

The basis functions, which can be labeled by the four quantum numbers nρ, nz, Λand Σ, are separable in the variables φ, z and ρ and a spin factor. We refer to thisbasis as the αz-basis. Thus

Ψnρ,nz ,Λ,Σ(ρ, z, φ) = ψ|Λ|nρ(ρ)ψnz(z)ψΛ(φ)χΣ (29)

whereψΛ(φ) = 1√

2πeiΛφ

ψnz(z) = Nnz

(Mωz~

)1/4e−ζ2/2Hnz(ζ)

ψΛnρ

(ρ) = NΛnρ

(2Mω⊥~

)1/2u|Λ|/2e−u/2L

|Λ|nρ (u)

(30)

with Hnz being Hermite polynomials and L|Λ|nρ Laguerre polynomials, see appendix C.

The dimensionless coordinates (u, ζ) are defined as

u = Mω⊥~ ρ2

ζ =√

Mωz~ z(31)

The normalization constants in Eq. (30) are given by

Nnz = 1√2nz nz !

√π

NΛnρ

=√

nρ!(nρ+|Λ|)!

(32)

The three functions in Eq. (30) are, respectively, eigenfunctions of the operators

lz = ~(x ∂∂y − y ∂

∂x) = −i~ ∂∂φ

Hz = − ~22M

∂2

∂z2 + 12Mω2

zz2

H⊥ = − ~22M ( ∂2

∂x2 + ∂2

∂y2 ) + 12Mω2

⊥(x2 + y2)

(33)

12

with eigenvalues ~Λ, ~ωz(nz +1/2) and ~ω⊥(2nρ + |Λ|+1). In addition to the threefunctions of Eq. (30), the basis functions (Eq. (29)) contain the spin function χΣ,which is eigenfunction to the operator sz with eigenvalues ~Σ.

The quantum number nz takes integer values ≥ 0. The quantum number nρ, asLaguerre polynomial order, takes integer values ≥ 0. It should not be mixed upwith the quantum number

n⊥ = 2nρ + |Λ| (34)

which implies that n⊥ and Λ must have equal parity, that is either both are odd orboth are even. Also, nρ = (n⊥ − |Λ|)/2 ≥ 0 implies that |Λ| ≤ n⊥. The quantumnumber rule for Λ is therefore

Λ ∈ −n⊥,−n⊥ + 2, . . . , n⊥ − 2, n⊥ (35)

When giving labels to the basis functions, the quantum number N is usually usedinstead of n⊥ or nρ, where

N = n⊥ + nz (36)

and Ω instead of Σ, whereΩ = Λ + Σ (37)

It should be observed that the basis functions defined in Eq. (29) preserve parityas well as signature for rotation about the z-axis. We refer to Appendix B for moredetails.

5 Choosing appropriate sectors of the (ε,γ) plane

The definition of the nuclear shape involves three angular frequency parameters,ωx, ωy and ωz, whereas the basis functions only involves two angular frequencyparameters, ω⊥ and ωz. The angular frequency parameters determine the spatialextension of the basis functions. In order to get an optimal set of basis functions, theangular frequencies must depend on the shape of the nucleus in a suitable way. Inthe z direction the choice is simple, since there is a single unique angular frequencyappearing both in the definition of the nuclear shape and in the basis functions.However, in the plane perpendicular to the z axis, the shape is defined by means oftwo angular frequencies, ωx and ωy, whereas the basis functions contain only oneangular frequency parameter, ω⊥.

Only for a pure prolate deformation at γ = 0 and for a pure oblate deformationat γ = ±180 do we have ωx = ωy = ω⊥. (We can choose ω⊥ = ωx). However,for deformations with γ close to 0 (near prolate shapes) ωx ≈ ωy ≈ ω⊥(γ = 0).Similarly, for γ close to ±180 (near oblate shapes) ωx ≈ ωy ≈ ω⊥(γ = ±180). Itis therefore an advantage to use only sectors of the (ε, γ) plane close to γ = 0 andγ = ±180. It turns out that all nuclear shapes, that can be described by the (ε, γ)parametrization, can be generated by using only two 30 sectors of the (ε, γ) plane,namely 0 < γ < 30 and −180 < γ < −150. The maximum deviation from either

13

180Pt N=102 [ε=0.25]

-180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180γ (degrees)

7.5

8

8.5

9

9.5

10

Osc

illat

or fr

eque

ncy

(MeV

)

Sector 3

Sector 2

Sector 1

Sector 4

Sector 5

Sector 6

ωy

ωx

ωz

B CA D

Fig. 3. The frequencies in the Lund convention for ε and γ. Choice of optimal sectors.

γ = 0 or γ = ±180 is then 30. An alternative choice would be to use the sectors−30 < γ < 0 and 150 < γ < 180. The four optimal 30 sectors are denoted A,B, C and D in Figs. 3 and 4.

In cranking calculations, one wants to describe rotation about any of the threeprincipal axes. If the x axis is used as the cranking axis, which is normally the case,one must use three full 60 sectors of the (ε, γ) plane. Usually the sectors denoted 1,2 and 3 in Fig. 4 are chosen, i.e. −120 < γ < 60. Alternatively, sectors 4, 5 and 6could be used. However, if one allows for cranking about both the x axis and the zaxis, the full (ε, γ) plane can be covered by performing calculations only in the fouroptimal 30 sectors A, B, C and D. From an input γ in the interval [−120, 60],one can transform to a γ(calc), to be used in the calculations, such that it lies inone of the sectors A, B, C or D, and then crank about either the z or the x axis,according to table 1.

For γ in calculate for γ(calc) = sector cranking about

[−120,−90] γ + 120 C (or B *) z axis

]− 90,−60] −γ + 120 A x axis

[−60,−30[ −γ + 120 D x axis

[−30, 0] γ B x axis

[0, 30[ γ C x axis

[30, 60] γ + 120 D (or A *) z axis

Table 1Optimal choice of sectors and cranking axes. * In [−120,−90] or in [30, 60], onecan alternatively perform the transformation γ(calc) = −γ − 120.

14

C

B

A

D

The Lund convention


Sec

tor

2

Inte

rmed

iate

axi

s ro

tatio

n


Sector 4

Sec

tor

5

Inte

rmed

iate

axi

s ro

tatio

n

Sector 6

γ = 120 o(prolate)

γ = 6

0 (

obla

te)

o

γ = 0o (prolate)

γ = -60 o(oblate)

γ = -

120

(pr

olat

e)o

γ = 180o (oblate)

Cx

Bx

Dx

Ax

Az or

Dz

Bz

or Cz

-1 -0.5 0 0.5 1ε cos(γ + 30o)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ε sin

( γ+

30

o)

Fig. 4. The Lund convention for (ε, γ) plane. Choice of optimal sectors and crankingaxes.

6 The spin-orbit term

The spin-orbit potential is given by the expression

Vs.o. = −λ

(~

2mnucc

)2 σ · ∇V1 × p

~(38)

where λ is the spin-orbit interaction strength, mnuc is the nucleon mass, σ representsthe Pauli spin matrices, p is the nucleon momentum. For an early discussion on thespin-orbit interaction we refer to [14]. V1 is the potential defined in Eq. (2). We canthen write the spin-orbit operator as

Vs.o. = −1iλ

(~

2mnucc

)2

[σx, σy, σz] · ∇V1 ×∇ (39)

or alternatively by using the step operators σ+ = σx + iσy and σ− = σx − iσy as

Vs.o. =i

2λ

(~

2mnucc

)2

[σ+ + σ−,−i(σ+ − σ−), 2σz] · ∇V1 ×∇ (40)

By expressing the nabla operator by means of cylindrical coordinates Eq. (28) wecan write

15

Vs.o. =i

2λ

(~

2mnucc

)2

[σ+,−iσ+, 0] + [σ−, iσ−, 0] + [0, 0, 2σz] ·

[(cosφ

∂V1

∂ρ− sinφ

ρ

∂V1

∂φ

)x +

(sinφ

∂V1

∂ρ+

cosφ

ρ

∂V1

∂φ

)y +

(∂V1

∂z

)z]

(41)

×[(

cosφ∂

∂ρ− sinφ

ρ

∂

∂φ

)x +

(sinφ

∂

∂ρ+

cosφ

ρ

∂

∂φ

)y +

(∂

∂z

)z]

Note that in the above equation we write σ as a sum of three vectors, where eachof the three vectors contains one and only one of the three operators σ+, σ− andσz.

The above expression is simplified to

Vs.o. =−λ

2

(~

2mnucc

)2 σ+eiφ

[∂V1

∂ρ

∂

∂z− ∂V1

∂z

(∂

∂ρ− i

ρ

∂

∂φ

)+

i

ρ

∂V1

∂φ

∂

∂z

]

+σ−e−iφ

[−∂V1

∂ρ

∂

∂z+

∂V1

∂z

(∂

∂ρ+

i

ρ

∂

∂φ

)+

i

ρ

∂V1

∂φ

∂

∂z

](42)

−2σzi

ρ

(∂V1

∂ρ

∂

∂φ+

∂V1

∂φ

∂

∂ρ

)

This operator consists of 10 terms, which are all of the same type. They containeach, from right to left, the following factors.

1. A partial derivation operator, where the derivation is with respect to one of thecylindrical coordinates ρ, z or φ.

2. A partial derivative with respect to ρ, z or φ of the potential V1(r) defined inEq. (2).

3. A factor, which is a simple function of φ and/or ρ.

4. A spin operator, which is one of σ+, σ− or σz.

The spin-orbit potential depends on the nuclear shape through the partial deriva-tives ∂V1

∂ρ , ∂V1∂z and ∂V1

∂φ , where V1 is the nuclear potential of Eq. (2). For triaxialshapes, V1 depends on the angle φ and thus ∂V1

∂φ 6= 0. This is the only significant dif-ference compared to rotation-symmetric shapes. For more details on the spin-orbitterm and its matrix elements we refer to [15].

16

7 The cranking term

The cranking Hamiltonian for the folded Yukawa potential can be written

Hω = T + V − ωjk (43)

where T is the kinetic energy term, V is the potential of Eq. (1) and ωjk is thecranking term. The cranking term is often thought of as describing a rotation aboutthe x-axis, in which case jk = jx. However, in section 5 it was concluded that incertain sectors of the (ε, γ) plane, a rotation about the z-axis is to be preferred,implying that jk = jz. We shall therefore discuss cranking about the z axis andabout the x axis in this section.

7.1 Cranking about the z axis

Cranking about the z-axis is the easiest type of rotation to implement, since thebasis functions of the αz-basis defined by Eqs. (29) and (30) are eigenfunctions ofboth lz and sz and since

jz = lz + sz (44)

it follows directly that

jz Ψnρ,nz ,Λ,Σ = (Λ + Σ)~ Ψnρ,nz ,Λ,Σ = Ω~ Ψnρ,nz ,Λ,Σ (45)

Thus, when solving the eigenvalue problem in the basis defined in Eqs. (29) and(30), the cranking term only contributes with diagonal matrix elements.

7.2 Cranking about the x axis

The quadrupole and hexadecapole shapes defined in section 3 are symmetric withrespect to a rotation 180 about the x axis. However, the basis functions defined inEqs. (29) and (30) are not eigenfunctions to the rotation operator

Rx(π) = e−iπjx/~ (46)

and do therefore not have good signature with respect to rotation about the x axis.This is shown in Appendix B. Since it is desirable to have a set of basis functionswith good signature, a new set of basis functions will be introduced for describingrotation about the x axis.We start by calculating the matrix elements of jx = lx + sx between the αz basisfunctions of Eqs. (29) and (30). We denote these basis functions |n⊥, nz,Λ,Σ〉 andobserve that

lx =12(l+ + l−) sx =

12(s+ + s−) (47)

17

It is then clear that lx will only give non-zero matrix elements between basis statesdiffering with one unit in Λ, and sx only between basis states differing with oneunit in Σ. A detailed calculation, which is included in Appendix A, results in thefollowing non-zero matrix elements for jx.

〈n⊥ + 1, nz + 1, Λ± 1, Σ |jx|n⊥nzΛΣ〉 =~4 (

√ωzω⊥−

√ω⊥ωz

)√

(n⊥ + ΛSΛ′−Λ + 2)(nz + 1)SΛ′+Λ−1

〈n⊥ + 1, nz − 1, Λ± 1,Σ |jx|n⊥nzΛΣ〉 =

− ~4 (

√ωzω⊥

+√

ω⊥ωz

)√

(n⊥ + ΛSΛ′−Λ + 2)nz SΛ′+Λ−1

〈n⊥ − 1, nz + 1, Λ± 1, Σ |jx|n⊥nzΛΣ〉 =

− ~4 (

√ωzω⊥

+√

ω⊥ωz

)√

(n⊥ − ΛSΛ′−Λ)(nz + 1)SΛ′+Λ−1

〈n⊥ − 1, nz − 1,Λ± 1,Σ |jx|n⊥nzΛΣ〉 =~4 (

√ωzω⊥−

√ω⊥ωz

)√

(n⊥ − ΛSΛ′−Λ)nz SΛ′+Λ−1

〈n⊥nzΛΣ′ |jx|n⊥nzΛΣ〉 = ~2δΣ′,−Σ

(48)

where Sx = sign(x) = 1 if x ≥ 0, and −1 if x < 0. Equation (48) implies that jx willonly have non-zero matrix elements between basis states differing with one unit inΩ, where Ω = Λ + Σ.

It is shown in Appendix B that one can form simple linear combinations of theαz basis functions of Eq. (29), which have good signature with respect to rotationabout the x axis. Thus, for α = 1/2 and α = −1/2

|n⊥nzΛΩα〉 =1√2( |n⊥nzΛΩ〉+ (−1)nz+ 1

2−α |n⊥nz,−Λ,−Ω〉) (49)

are eigenfunctions to Rx(π). For α = 1/2 the eigenvalue of Rx(π) is −i and forα = −1/2 it is +i. Thus, α is the normal signature quantum number. We refer tothis basis as the αx-basis. It is assumed in Eq. (49) that Ω = Λ + Σ > 0, whichimplies that Λ ≥ 0 and that Σ > 0 when Λ = 0. All matrix elements of jx betweenstates with different signatures are zero. Thus, jx conserves the signature. For moredetails, see Appendix B.

8 Comparisons with the modified harmonic oscillator potential

8.1 The single-particle energies

As a first exploratory study of the cranked folded-Yukawa potential, we have chosento calculate single-particle energies as functions of the rotational frequency for anumber of different shapes and directions of the rotational axis and compared them

18

Cranked Nilsson potential

66

60

78

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55hω(MeV)

a1 a2

a3

b2

b1

b3

c2c3

c4

c5

c6c7,8

c9,10

[413 5/2]

[411 3/2]

[411 1/2]

[404 7/2] [402 5/2]

38.5

39

39.5

40

40.5

41

41.5

42

42.5

43

43.5

44

44.5

e i(MeV

)

68 PROTONS ε=0.25 γ=0.0 ε4=0.0 θ=90 φ=0Cranked Nilsson potential

76

98

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55hω(MeV)

A1

A2A3

A4

A5A6

A7

B1

B2

[521 3/2][523 5/2]

[521 1/2]

[512 5/2]

[514 9/2]

[505 11/2]

47

47.5

48

48.5

49

49.5

50

50.5

51

51.5

52

52.5

53

53.5

e i(MeV

)

96 NEUTRONS ε=0.25 γ=0.0 ε4=0.0 θ=90 φ=0

Fig. 5. Cranked Nilsson single-proton (left) and single-neutron (right) energy levels,plotted as functions of the cranking frequency for a deformation close to the groundstate deformation of 164Er. Levels for single-particle states with positive parity aresolid, and those for negative parity are dot-dashed. The dashed line marks theposition of the Fermi energy. Som levels near the Fermi surface are labeled by theirdominating asymptotic quantum numbers [N nz Λ Ω]. Proton levels originating fromspherical i13/2 subshell are labeled a, where a1 is the lowest level, a2 the secondlowest level and so on. Similarly, levels from the mixed h9/2 and f7/2 subshells arelabeled b and levels from the h11/2 subshell are labeled c. The neutron levels arelabeled in the same way, except that upper case letters A, B and C are used.

to the corresponding single-particle energies calculated with the modified harmonicoscillator potential. We chose nucleon numbers corresponding to 164Er, since thisnucleus is well studied experimentally, and for which the Nilsson parameters arewell determined. One result of such a comparison is shown in Figs. 5 and 6. It canbe observed that the energy levels, which lie in the vicinity of the Fermi surfaceat ω = 0 have similar relative positions and respond in a very similar fashion tothe rotation. Also the pronounced gaps e.g. at proton numbers 60 and 66 appearin both potentials. For the neutrons significant gaps at 76 and 98 occur in bothmodels. Such similarities are to be expected, since the parameters of both modelshave been fitted to roughly equivalent experimental data.

There are, however, a few striking differences between the single-particle energyspectra generated in the two models. Thus, in the proton system, the two lowestlevels from the mixed h9/2 and f7/2 subshells (b1 and b2) lie much lower in thefolded-Yukawa potential than in the Nilsson potential, opening up a significantenergy gap at low rotational frequencies at proton number 78. It can also be observedthat the lowest members of the high-j subshells, in particular the levels a1 and b1are descending faster with increasing rotational frequency in the Nilsson model thanin the folded-Yukawa model.

In the neutron system, the most noticeable difference between the two models is thehigher position of the negative parity levels [521 3/2], [523 5/2], [521 1/2] and [5125/2] relative to the i13/2 levels (labeled A) in the folded-Yukawa potential. As a

19

60

66

78

164Er λp = 32.10, ap = 0.80 fm

Cranked folded-Yukawa potential ε = 0.25 γ = 0.0 ε4 = 0.0 θ = 90 φ = 0

- 0.0 0.1 0.2 0.3 0.4 0.5 0.6

hω (MeV)

− 10

− 9

− 8

− 7

− 6

− 5

− 4

− 3

− 2 S

ingl

e-P

roto

n E

nerg

y (M

eV)

a1 a2

a3

b1 b2

b3

c2c3

c4

c5c6

c7,8

c9,10

[413 5/2]

[411 3/2]

[411 1/2]

[402 5/2][404 7/2]

76

98

164Er λn = 34.58, an = 0.80 fm

Cranked folded-Yukawa potential ε = 0.25 γ = 0.0 ε4 = 0.0 θ = 90 φ = 0

- 0.0 0.1 0.2 0.3 0.4 0.5 0.6

hω (MeV)

− 13

− 12

− 11

− 10

− 9

− 8

− 7

− 6

− 5

Sin

gle-

Neu

tron

Ene

rgy

(MeV

)

A1

A2A3 A4

A5A6

A7

B1B2

[514 9/2]

[505 11/2]

[521 3/2][523 5/2]

[521 1/2]

[512 5/2]

Fig. 6. Cranked folded-Yukawa potential single-proton (left) and single-neutron lev-els. The same line types and labels as in Fig. 5 have been used.

result, the energy gap at neutron number 98 gets considerably larger in the folded-Yukawa model than in the Nilsson model. On the other hand, the higher membersom the h11/2 subshell (the levels [514 9/2] and [505 11/2]) lie at nearly the sameposition relative to the i13/2 levels in the two potentials.

8.2 The moment of inertia

The Strutinsky smoothed moment of inertia for protons and neutrons are definedas

Jp =Ip

ω, Jn =

In

ω, (50)

where Ip, In are the Strutinsky smoothed total spins for protons and neutrons,respectively [16]. The total smoothed moment of inertia is

J = Jp + Jn. (51)

From the rigid body moment of inertia JLD [2], the proton and neutron rigid bodymoments of inertia are defined as

JpLD = JLD

Z

A, Jn

LD = JLDN

A. (52)

When solving the cranked folded-Yukawa eigenvalue problem (43), we choose aharmonic oscillator basis with good signature with respect to the rotational axis.

20

Thus, for example, if we consider an oblate nucleus rotating about its symmetry axiswe can, according to the Lund convention, choose γ = 60, that is an oblate shapewith respect to the x-axis (x-oblate), and let the nucleus rotate about the x axis,using the αx-basis, eq. (49). However, that is not an optimal choice, since the basisfunctions are constructed under the assumption that the z-axis is the symmetryaxis. A better choice is to use the αz-basis, eq. (29), let γ = 180, that is an oblateshape with respect to the z axis (z-oblate), and let the nucleus rotate about thez-axis, see fig. 4. The basis can then be truncated to a smaller size.

As a second example, consider a prolate nucleus rotating about an axis perpendicu-lar to the symmetry axis. In the Lund convention, we can in this case choose γ = 0(z-prolate), let the nucleus rotate about the x-axis, and use the αx-basis. Then, thez-axis is the symmetry axis of the potential, and the basis functions, which havecylindrical symmetry about the z-axis, fit well. However, the equivalent physicalsituation can be described by choosing γ = −120 (x-prolate), let the nucleus ro-tate about the z-axis, and use the αz-basis. The x-axis is then the symmetry axisand the basis functions are not optimal, and a larger number of basis functions areneeded.

We have studied cases like the two described above, in order to determine how muchthe basis space can be reduced by making the optimal choice of γ and rotationalaxis. For large deformations (large ε) the reduction is very significant. This is illus-trated for the prolate case in fig. 7. The curves marked x-CFY show the Strutinskysmoothed moment of inertia calculated with x-axis cranking for γ = 0, using anαx-basis space consisting of 12 and 16 full oscillator shells, respectively. It is clearthat 12 shells is sufficient for getting nearly full convergence even for the largest val-ues of ε. The curves marked z-CFY in fig. 7 are calculated with the less favourablechoice of z-axis cranking at γ = −120. At small deformations the same resultsare obtained as for x-CFY, but for large deformations (ε & 0.3) the curves divergerapidly. Even an αz-basis space of 16 oscillator shells is too small in order to reachconvergence at the largest deformations in fig. 7.

As is apparent from Fig. 7, the use of optimal basis functions is crusial, implyingthat appropriate sectors of the (ε,γ) plane and appropriate cranking axes must bechosen. The principles for this choice are described in section 5. Thus, for each 30o

sector of the (ε, γ)-plane in the interval −120 ≤ γ ≤ 60, a mapping is performedfrom the input γ of the Lund convention to a γ(calc) to be used in the calculations,so that the basis fits the shape of the potential in the best possible way, see table 1and Fig. 4. For z axis cranking, the αz-basis basis is used and for x axis cranking,the αx-basis is used. In this way we can perform cranking of the folded-Yukawapotential for all triaxial shapes with a cylindrical symmetric bases, using only jx

or jz matrix elements. Thus there is no need to include the imaginary jy matrixelements.

In Fig. 8, the γ-dependence of the moment of inertia is shown for ε = 0.25, which isclose to its ground-state value. The cranked folded-Yukawa moment of inertia lies inbetween the cranked Nilsson and the rigid-body moments of inertia, all with similarγ-dependence.

21

70

80

90

100

110

120

130

0 0.1 0.2 0.3 0.4 0.5 0.6

J (

− h2 /MeV

)

ε

Moment of inertia(ε), 164Er

JLD

CN ∼J

x-CFY ∼J

z-CFY∼J

CN ∼Jx-CFY ∼J Nmax=12

x-CFY ∼J Nmax=16

z-CFY ∼J Nmax=12

z-CFY ∼J Nmax=16JLD

Fig. 7. For a prolate shape cranked about an axis perpendicular to the symme-try axis, the figure shows the advantage of x cranking a z-prolate shape of thefolded-Yukawa potential (γ = 0), compared to z cranking an x-prolate shape(γ = −120), as the basis is z-prolate. The bottom curve is the liquid drop mo-ment of inertia, and the upper one is the Strutinsky smoothed moment of inertiafor the cranked Nilsson potential with identical deformation.

The ε and γ dependence of the separate proton and neutron Strutinsky smoothedmoments of inertia, relative to the rigid body moments of inertia, are shown inFig. 9. The ratio between the proton smoothed moment of inertia and the protonrigid body moment of inertia will be denoted the proton ratio, and similarly for theneutrons. For 164Er, at the line of β-stability, the cranked folded-Yukawa potentialproton ratio (on average about 1.08) is smaller than the neutron ratio (about 1.13).However, the relation between the proton and neutron ratios vary with particlenumber along an isotope chain. Thus, for the neutron rich isotope 184Er the crankedfolded-Yukawa proton ratio is only about 1.04, which is much smaller than theneutron ratio, which has increased slightly to about 1.14. However, for the neutrondeficient isotope 144Er the proton ratio is bigger (about 1.13) than the neutronratio (about 1.09). The results can be understood in terms of changing radius andpotential depth for protons and neutrons along an isotope chain.

In the cranked Nilsson potential, the Strutinsky smoothed moment of inertia isessentially determined by the strength of the l2 term. In the absence of this term,both the proton and neutron ratios are close to 1. In the rare earth region, thestrength of the l2 is about 50 % larger for protons than for neutrons. As a result,

22

55

60

65

70

75

80

85

90

95

100

105

-120 -90 -60 -30 0 30 60

J (

− h2 /MeV

)

γ

Moments of inertia(γ), ε=0.25, 164Er

CN

CFY

LD

CN ∼JCFY ∼J Nmax=16JLD(x,r0=1.2fm)

Fig. 8. The cranked folded-Yukawa moment of inertia for ε = 0.25, and−120 ≤ γ ≤ 60, plotted together with the cranked Nilsson and the rigid bodymoments of inertia for comparison. The small discontinuity in the CFY moment ofinertia at γ = 30 is due to the sudden change of the orientation of the potentialfrom one with the z axis being the long semi-axis, to one with the z axis being theshortest semi-axis, see table 1 and Fig. 4. Similar discontinuities occur at γ = −90

and γ = −30, but they can hardly be seen in this figure.

the proton ratio is much larger than the neutron ratio as can be seen in fig. 9. Ifthe strength of the l2 term does not change, the proton and neutron ratios remainessentially constant along an isotope chain.

The Strutinsky smoothed moment of inertia of the cranked folded-Yukawa potentiallies always closer to the rigid body moment of inertia, than the Strutinsky smoothedmoment of inertia of the cranked Nilsson potential does. The difference is particu-larly large for the protons. The average deviation from the rigid body value is about10 % (protons plus neutrons combined) for the folded-Yukawa potential comparedto about 25 % for the cranked Nilsson potential. It can be noticed that the momentof inertia scales with the square of the radius parameter. Thus, a 5 % decrease ofthe radius parameter in the folded-Yukawa potential would give a moment of inertiavery close to the rigid body value.

To investigate how the Coulomb term influences the moment of inertia for protonsin the cranked folded-Yukawa model, calculations for 164Er at γ = 0 were alsoperformed when the Coulomb term in the hamiltonian was removed. The resultis shown in Fig. 10. When the Coulomb term is included the ratio between theproton Strutinsky smoothed moment of inertia and the proton rigid body momentof inertia, lies between 1.06 and 1.09 and increases with the deformation. When the

23

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6

∼ J/J L

D

ε

Moments of inertia ratios(ε), γ=0

CN ∼J(p)/JLD(p)CN ∼J(n)/JLD(n)

CFY ∼J(p)/JLD(p) Nmax=16CFY ∼J(n)/JLD(n) Nmax=16

0.9

1

1.1

1.2

1.3

1.4

-120 -90 -60 -30 0 30 60

∼ J/J L

D

γ

Moments of inertia ratios(γ), ε=0.25

CN ∼J(p)/JLD(p)CN ∼J(n)/JLD(n)

CFY ∼J(p)/JLD(p) Nmax=16CFY ∼J(n)/JLD(n) Nmax=16

Fig. 9. Proton Strutinsky smoothed moment of inertia relative to the proton rigidbody moment of inertia, and similar for the neutrons, plotted for the crankedfolded-Yukawa potential and, for comparison, also for the cranked Nilsson potential.The discontinuities in CFY neutron and proton ratios are explained in the captionof Fig. 8.

Coulomb term is removed, the ratio lies between 0.98 and 1. Thus, the effect of theCoulomb term is to increase the proton smoothed moment of inertia by about 8 to10 %. In spite of this, the proton ratio is usually smaller than the neutron ratio.

In the Nilsson model, the Coulomb interaction is taken into account by using alarger strength of the l2 term for protons than for neutrons. This increasis theproton ratio which is consistent with how the Coulomb term in the folded-Yukawapotential changes the moment of inertia. However, it also implies that the protonratio will always be larger than the neutron ratio, which is inconsistent with thefolded-Yukawa results.

9 Conclusions

The single-particle energy levels generated by the Nilsson potential and the folded-Yukawa potential show great similarities in the vicinity of the Fermi surface in thenon rotating nucleus. These levels also respond to the rotation in a very similarway. However, the position of certain intruder levels, originating from higher-lyingsubshells, differs. Since such levels carry a large amount of angular momentum, theygenerate differences in the moment of inertia.

The Strutinsky smoothed moment of inertia in the cranked folded-Yukawa modelis shown to be somewhat larger than the rigid body moment of inertia, but muchsmaller than smoothed moment of inertia in the cranked Nilsson model.

24

0.9

0.95

1

1.05

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6

∼ J/J L

D

ε

Proton moments of inertia ratios(ε), γ=0

CFY ∼J(prot,Coul)/JLD(prot) Nmax=16CFY ∼J(prot,No Coul)/JLD(prot) Nmax=16

Fig. 10. Effect of the Coulomb term for the proton moment of inertia.

The effect of the Coulomb term in the folded-Yukawa hamiltonian is to increase themoment of inertia. This is expected since the Coulomb interaction repel the protonsfrom each other. This increases the radius in the proton mass distribution and as aconsequence, the moment of inertia increases.

Although the Coulomb term increases the Strutinsky smoothed proton moment ofinertia, the Strutinsky smoothed neutron moment of inertia (relative to the rigidbody moment of inertia) is larger, except for very neutron deficient nuclei.

For the cranked Nilsson model the opposite holds true, the smoothed moment ofinertia (relative to the rigid body moment of inertia) is greater for protons than forneutrons. This can be explained by the the fact that the parameter µ (strength ofthe l2 term) is larger for protons than for neutrons. The proton potential thereforedeviates more from the pure harmonic oscillator potential which has the rigid bodymoment of inertia.

It is generally accepted that nuclei at high angular momentum, where the pairingcorrelations can be neglected, should have an average moment of inertia close tothat of a rigid rotor. The folded-Yukawa moment of inertia is only about 10 %larger than the rigid body value and can be reduced to about this value by mod-erate changes of the model parameters. It should therefore be feasible to performcranking calculations based on the folded-Yukawa potential without renormalizingthe moment of inertia. For this reason, we expect the folded-Yukawa potential tobecome an interesting alternative to the commonly used Woods-Saxon and Nilssonpotentials for describing rapidly rotating nuclei.

25

References

[1] R. Bengtsson, S. E. Larsson, G. Leander, P. Moller, S. G. Nilsson, S. Aberg,and Z. Szymanski, Phys. Lett. 57B (1975) 301.

[2] G. Andersson, S. E. Larsson, G. Leander, P. Moller, S. G. Nilsson, I. Ragnarsson,and S. Aberg, Nucl. Phys. A268 (1976) 205.

[3] R. Bengtsson and S. Frauendort, Nucl. Phys. A314 (1979) 27.

[4] R. Bengtsson and S. Frauendort, Nucl. Phys. A327 (1979) 139.

[5] P. Moller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, Atomic Data Nucl.Data Tables 59 (1995) 185.

[6] M. Bolsterli, E. O. Fiset, J. R. Nix, and J. L. Norton, Phys. Rev. C 5 (1972)1050.

[7] S. G. Nilsson, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd. 29:No. 16(1955).

[8] C. Gustafson, I.L. Lamm, B. Nilsson, and S. G. Nilsson, Proc. Int. Symp. onwhy and how to investigate nuclides far off the stability line, Lysekil, 1966, Ark.Fysik 36 (1967) 613.

[9] S. G. Nilsson, C. F. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C.Gustafson, I.-L. Lamm, P. Moller, and B. Nilsson, Nucl. Phys. A131 (1969)1.

[10] B. S. Nilsson, Nucl. Phys. A129 (1969) 445.

[11] P. Moller, S. G. Nilsson, A. Sobiczewski, Z. Szymanski, and S. Wycech, Phys.Lett. 30B (1969) 223.

[12] P. Moller and B. Nilsson, Phys. Lett. 31B (1970) 171.

[13] P. Moller and S. G. Nilsson, Phys. Lett. 31B (1970) 283.

[14] J. Damgaard, H. C. Pauli, V. V. Pashkevich, and V. M. Strutinsky, Nucl. Phys.A135 (1969) 432.

[15] H. C. Pauli, Physics Reports, 7C (1973) 35.

[16] M. Brack, International Workshop: Nuclear Structure Models, Oak Ridge, 1992,Eds. R. Bengtsson, J. Draayer and W. Nazarewicz (World Scientific, Singapore,1992) p. 165.

[17] J.J. Sakurai, Modern Quantum mechanics, section 3.2, Addison-Wesley 1995.

[18] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, Doverpublications, N. Y., (1972).

26

Appendices

A Matrix elements of jx

A.1 The jx operator

The angular momentum operator jx is the sum of the orbital angular momentumoperator lx and the spin operator sx. From the definition ~l = ~r × ~p one gets

lx = −i~(y∂

∂z− z

∂

∂y) (A.1)

By the chain rule, lx is transformed to the cylindrical coordinates of Eq. (28):

lx = −i~[ρ sinφ(

∂

∂z− z

ρ

∂

∂ρ)− z cosφ

ρ

∂

∂φ

](A.2)

The basis functions defined in Eq. (30) were written in terms of the dimensionlesscylindrical coordinates u and ζ as of Eq. (31), in which lx is expressed as

lx = −i~[√

ωz

ω⊥u1/2 sinφ

∂

∂ζ−

√ω⊥ωz

ζ u−1/2(2 sinφu∂

∂u+ cosφ

∂

∂φ)]

(A.3)

A.2 Calculation of the matrix elements of lx

In the equations below, the notations m = |Λ|, m′ = |Λ′| are used. The u and ζdependence of the polynomials is not written explicitly, when not necessary.

By the recurrence relations Eqa. (C.3) and (C.9) for Hermite and Laguerre polyno-mials, respectively, the action of lx, defined in Eq. (A.3), on ψnρ,nz ,Λ of Eq. (29) iscalculated to be

lx/(−i~)ψnρ,nz ,Λ =√

ωzω⊥

u1/2 sinφψmnρ

ψΛ(−√

nz+12 ψnz+1 +

√nz2 ψnz−1)+

−√

ω⊥ωz

ζu−1/22 sin φψΛψnz [(nρ + m−u2 )ψm

nρ−√

nρ(nρ + m)ψmnρ−1]+

cosφψmnρ

ψnz iΛΨΛ

(A.4)

The integrand in the matrix element integral is therefore

ψ∗n′ρ,n′z ,Λ′lx~ ψnρ,nz ,Λ =

√ωzω⊥

u1/2ψmnρ

ψm′n′ρ

(√

nz+12 ψnz+1 −

√nz2 ψnz−1)ψn′z+

2√

ω⊥ωz

u−1/2[(nρ + m−u2 )ψm

nρ−√

nρ(nρ + m)ψmnρ−1]ψ

m′n′ρ

ζψnzψn′z

ψΛψ∗Λ′i sinφ+

−√

ω⊥ωz

u−1/2ψmnρ

ψm′n′ρ

ζψnzψn′zΛcos φ, ψΛψ∗Λ′(A.5)

27

where it is written such that the φ-dependence is factored out to one sinφ -termand one cosφ -term. The volume element is

dxdydz = ρdρdφdz =~

2Mω⊥

√~

Mωzdudφdζ (A.6)

The matrix element triple integral can be written as

〈ψn′ρn′zΛ′ | lx~ |ψnρnzΛ〉 =

= 12( ~M )3/2 1

ω⊥ω1/2z

[√ωzω⊥

I1(n′ρ,m′, nρ m)I3(n′z, nz)+

2√

ω⊥ωz

I4(n′ρ, m′, nρ m)I2(n′z, nz)]i∫ 2π0 sinφψΛψ∗Λ′dφ+

−Λ√

ω⊥ωz

I5(n′ρ,m′, nρ m)I2(n′z, nz)∫ 2π0 cosφψΛψ∗Λ′dφ

(A.7)

where the integrals I1 to I5 are defined as

I1(n′ρ, m′, nρ m) =∫∞0 u1/2ψm

nρψm′

n′ρdu

I2(n′z, nz) =∫∞−∞ ζψnzψn′zdζ

I3(n′z, nz) =∫∞−∞

(√nz+1

2 ψnz+1 −√

nz2 ψn′z−1

)ψn′zdζ

I4(n′ρ, m′, nρ,m)=∫∞0 u−1/2

[(nρ + m−u

2 )ψmnρ−√

nρ(nρ + m)ψm′nρ−1

]ψm′

n′ρdu

I5(n′ρ, m′, nρ m) =∫∞0 u−1/2ψm

nρψm′

n′ρdu

(A.8)

By Euler’s formulas for sin(φ) and cos(φ) the φ-integrals are calculated to be

i∫ 2π0 sinφψΛψ∗Λ′dφ = 1

2SΛ′−Λδ|Λ−Λ′|,1∫ 2π0 cosφψΛψ∗Λ′dφ = 1

2δ|Λ−Λ′|,1(A.9)

where Sx = sign(x) = 1 if x ≥ 0, and −1 if x < 0. Note that the integrals are zeroif Λ′ 6= Λ± 1.To get rid of the factors containing M , ωz, ω⊥ and ~, which appear in the definitionof the basis functions, see Eq. (30), new integrals Jk are introduced through therelations

I1 = 2Mω⊥~ J1

I2 = (Mωz~ )1/2J2

I3 = (Mωz~ )1/2J3

I4 = 2Mω⊥~ J4

I5 = 2Mω⊥~ J5

(A.10)

For the calculation of J2 we first integrate by parts

J2 =∫∞−∞ ζψnzψn′zdζ = NnzNn′z

∫∞−∞(ζe−ζ2

)(HnzHn′z)dζ

= NnzNn′z12

∫∞−∞ e−ζ2 d

dζ (HnzHn′z)dζ(A.11)

28

then use the recurrence relation (C.3) to get

J2 = NnzNn′z(nz

∫ ∞

−∞Hnz−1Hn′ze

−ζ2dζ + n′z

∫ ∞

−∞HnzHn′z−1e

−ζ2dζ) (A.12)

By the orthogonality relation C.2 for Hermite polynomials, the expression for J2

can be simplified to

J2 = NnzNn′z [nz√

π2nz−1(nz − 1)!δnz−1,n′z + n′z√

π2nznz!δnz ,n′z−1]

=√

nz2 δn′z ,nz−1 +

√nz+1

2 δn′z ,nz+1

(A.13)

In a similar way, the integral J3 is calculated as

J3 =√

nz+12 Nnz+1Nn′z

∫∞−∞Hnz+1Hn′ze

−ζ2dζ−

√nz2 Nnz−1Nn′z

∫∞−∞Hnz−1Hn′ze

−ζ2dζ

=√

nz+12 Nnz+1Nn′z

√π2nz+1(nz + 1)!δn′z ,nz+1−

√nz2 Nnz−1Nn′z

√π2nz−1(nz − 1)!δn′z ,nz−1

=√

nz+12 δn′z ,nz+1 −

√nz2 δn′z ,nz−1

(A.14)

Note that both J2 and J3 are zero for n′z 6= nz ± 1. It is possible to express J4 interms of J1 and J5 as

J4(n′ρ,m′, nρ,m) = (nρ + m2 )J5(n′ρ,m′, nρ,m)

−12J1(n′ρ,m′, nρ, m)−√

nρ(nρ + m)J5(n′ρ, m′, nρ − 1,m)(A.15)

Also define J32 as

J32 =√

ωzω⊥

J3 −√

ω⊥ωz

J2 =(√

ωzω⊥−

√ω⊥ωz

)√nz+1

2 δn′z ,nz+1 −(√

ωzω⊥

+√

ω⊥ωz

)√nz2 δn′z ,nz−1

(A.16)

Then collect separately terms containing J1 and J5 in the matrix element to writeit as

〈ψn′ρn′zΛ′ | lx~ |ψnρnzΛ〉 =

12SΛ′−ΛJ32(n′z, nz)J1(n′ρ, m′, nρ,m) + ω⊥

ωz·[

[SΛ′−Λ(2nρ+m)−Λ]12J5(n′ρ,m

′, nρ,m)−SΛ′−Λ

√nρ(nρ + m)J5(n′ρ,m

′, nρ−1, m)]

︸︷︷︸J5p

·J2(n′z, nz) δ|Λ′−Λ|,1(A.17)

The selection rule for Λ gives the following implication

Λ′ = Λ± 1 ⇒ m′ = m± 1 (A.18)

Because the matrix element of lx is zero whenever Λ′ 6= Λ ± 1 it is sufficient tocalculate the integrals for m′ = m ± 1. The technique is to use the recurrence

29

relations for the Laguerre polynomials to map the integral on to a sum of integralswith the same m-value for each integral. Then one can use the orthogonality relationEq. (C.6) for Laguerre polynomials to express each integral as a Kronecker deltamultiplied by some function of the quantum numbers.Assume m′ = m− 1. By the recurrence relation Eq. (C.7) for Laguerre polynomialsone gets

J1(n′ρ,m′ = m− 1, nρ,m) = Nmnρ

Nm′n′ρ

∫∞0 Lm

nρLn′ρm

′u(m+m′+1)/2e−udu =

= Nmnρ

Nm−1n′ρ

(∫∞0 Lm

n′ρLm

nρume−udu− ∫∞

0 Lmn′ρ−1L

mnρ

ume−udu) =

= Nmnρ

Nm−1nρ

(nρ+m)!nρ! δn′ρ,nρ −Nm

nρNm−1

nρ+1(nρ+m)!

nρ! δn′ρ,nρ+1 =

=√

nρ + mδn′ρnρ −√

nρ + 1 δn′ρ,nρ+1

(A.19)

Now assume m′ = m+1. This is equivalent to m = m′−1. By interchanging primedand unprimed symbols, one can use the latter expressions for m′ = m− 1 to get

J1(n′ρ,m′ = m + 1, nρ,m) =

√nρ + m + 1 δn′ρnρ −

√nρ δn′ρ,nρ−1 (A.20)

By use of a sum expression of the Laguerre polynomials Eq. (C.5) and term-wiseintegration for J5, one can construct a double-sum expression for J5. This expressionhas been used to show numerically that J5 does not obey selection rules that are assimple as those for J1 (n′ρ = nρ ± 1 or 0). But the J5p parenthesis in (A.17) does.There are four combinations of ∆Λ and SΛ for which we need calculate J5p.Assume Λ′ = Λ + 1 and Λ ≥ 0. Then SΛ′−Λ = 1, m = Λ, and m′ = m + 1.

J5p(n′ρ,m′=m+1, nρ, m; Λ′ = Λ + 1) =

= nρJ5(n′ρ,m′=m+1, nρ, m)−√nρ(nρ+m)J5(n′ρ, m′=m+1, nρ−1, m) =

=√

nρ!n′ρ!

(nρ+m)!(n′ρ+m+1)! ·

·[nρ

∫∞0 Lm

nρLm+1

n′ρume−udu− (nρ + m)

∫∞0 Lm

nρ−1Lm+1n′ρ

ume−udu]

=

=√

nρ!n′ρ!

(nρ+m)!(n′ρ+m+1)!

∫∞0

1u

[nρL

mnρ− (nρ + m)Lm

nρ−1

]Lm+1

n′ρum+1e−udu

(A.21)Now we use the recurrence relation Eq. (C.8) for the expression inside the squarebracket in the last line, and the orthogonality relation (C.6) to get

J5p(n′ρ, m′ = m + 1, nρ,m; Λ′ = Λ + 1) =

= −√

nρ!n′ρ!

(nρ+m)!(n′ρ+m+1)!

∫∞0 Lm+1

nρ−1Lm+1n′ρ

um+1e−udu =

= −√nρδn′ρ,nρ−1

(A.22)

In a similar way one gets for Λ′ = Λ + 1 and Λ ≤ −1

J5p(n′ρ,m′ = m− 1, nρ,m; Λ′ = Λ + 1) =

√nρ + mδn′ρ,nρ (A.23)

and for Λ′ = Λ− 1 and Λ ≥ 1

J5p(n′ρ,m′ = m− 1, nρ,m; Λ′ = Λ− 1) = −√

nρ + mδn′ρ,nρ (A.24)

30

and finally, for Λ′ = Λ− 1 and Λ ≤ 0

J5p(n′ρ,m′ = m + 1, nρ, m; Λ′ = Λ− 1) =

√nρ δn′ρ,nρ (A.25)

The four different expressions for J5p can be summarized in the following compactexpression:

J5p(n′ρ,m′, nρ,m; Λ′ = Λ± 1) =

(√

nρ + mδn′ρnρδm′,m−1 −√nρδn′ρnρ−1δm′,m+1)SΛ′−Λ

(A.26)

We tested the correctness of the algebraic expressions for J1, J2, J3 and J5p bycomparing these to explicit calculations of their original definitions in terms ofintegrals by use of Maple for several sets of quantum numbers.

The algebraic expressions are added together according to (A.17) in the two casesm′ = m + 1, m′ = m− 1. The result is found in the following section.

A.3 Coupling rules and algebraic expressions for the matrix elements of lx

The φ-part of the matrix element integral gives that for non-vanishing matrix ele-ments it is necessary that Λ′ = Λ± 1. The z-part gives a selection rule n′z = nz ± 1.The ρ-part gives a selection rule n′ρ = nρ ± 1 or 0. Since lx does not affect the spinpart, Σ′ = Σ for non-vanishing matrix elements when including spin.

Omitting the spin factor we obtain for the non-vanishing matrix elements of thespacial part of the wave function:

〈nρ + 1, nz + 1, Λ± 1 |lx|nρnzΛ〉 =

− ~2√

2(√

ωzω⊥−

√ω⊥ωz

)√

(nρ + 1)(nz + 1) SΛ′−Λδm′,m−1

〈nρ + 1, nz − 1, Λ± 1 |lx|nρnzΛ〉 =~

2√

2(√

ωzω⊥

+√

ω⊥ωz

)√

(nρ + 1)nz SΛ′−Λδm′,m−1

〈nρ − 1, nz + 1, Λ± 1 |lx|nρnzΛ〉 =

− ~2√

2(√

ωzω⊥

+√

ω⊥ωz

)√

nρ(nz + 1)SΛ′−Λδm′,m+1

〈nρ − 1, nz − 1, Λ± 1 |lx|nρnzΛ〉 =~

2√

2(√

ωzω⊥−

√ω⊥ωz

)√nρnz SΛ′−Λδm′,m+1

〈nρ, nz + 1, Λ± 1 |lx|nρnzΛ〉 =~

2√

2(√

ωzω⊥− Sm′−m

√ω⊥ωz

)√

(nρ + m + δm′,m+1)(nz + 1)SΛ′−Λ

〈nρ, nz − 1,Λ± 1 |lx|nρnzΛ〉 =

− ~2√

2(√

ωzω⊥

+ Sm′−m

√ω⊥ωz

)√

(nρ + m + δm′,m+1)nz SΛ′−Λ

(A.27)

31

If nρ is expressed in terms of n⊥ and Λ, according to (34), the non-vanishing matrixelements can be simplified to

〈n⊥ + 1, nz + 1, Λ± 1 |lx|n⊥nzΛ〉 =~4 (

√ωzω⊥−

√ω⊥ωz

)√

(n⊥ + ΛSΛ′−Λ + 2)(nz + 1)SΛ′+Λ−1

〈n⊥ + 1, nz − 1, Λ± 1 |lx|n⊥nzΛ〉 =

−~4 (√

ωzω⊥

+√

ω⊥ωz

)√

(n⊥ + ΛSΛ′−Λ + 2)nz SΛ′+Λ−1

〈n⊥ − 1, nz + 1, Λ± 1 |lx|n⊥nzΛ〉 =

−~4 (√

ωzω⊥

+√

ω⊥ωz

)√

(n⊥ − ΛSΛ′−Λ)(nz + 1)SΛ′+Λ−1

〈n⊥ − 1, nz − 1, Λ± 1 |lx|n⊥nzΛ〉 =~4 (

√ωzω⊥−

√ω⊥ωz

)√


(A.28)

A.4 Matrix elements of sx

The spin operator sx = 12(s+ + s−) only affects the spin factor of the basis kets:

〈n′ρn′zΛ′Σ′ |sx|nρnzΛΣ〉 = 〈n′ρn′zΛ′ |nρnzΛ〉〈Σ′ |sx|Σ〉 (A.29)

The ladder operators give non-vanishing matrix elements only for anti-parallel spin

〈Σ′ |sx|Σ〉 =~2δΣ′,−Σ (A.30)

A.5 Matrix elements of jx

The matrix elements of jx are sums of matrix elements of lx and sx. Thus,

〈n⊥ + 1, nz + 1, Λ± 1, Σ |jx|n⊥nzΛΣ〉 =~4 (

√ωzω⊥−

√ω⊥ωz

)√

(n⊥ + ΛSΛ′−Λ + 2)(nz + 1)SΛ′+Λ−1

〈n⊥ + 1, nz − 1, Λ± 1,Σ |jx|n⊥nzΛΣ〉 =

−~4 (√

ωzω⊥

+√

ω⊥ωz

)√

(n⊥ + ΛSΛ′−Λ + 2)nz SΛ′+Λ−1

〈n⊥ − 1, nz + 1, Λ± 1,Σ |jx|n⊥nzΛΣ〉 =

−~4 (√

ωzω⊥

+√

ω⊥ωz

)√

(n⊥ − ΛSΛ′−Λ)(nz + 1)SΛ′+Λ−1

〈n⊥ − 1, nz − 1, Λ± 1,Σ |jx|n⊥nzΛΣ〉 =~4 (

√ωzω⊥−

√ω⊥ωz

)√


〈n⊥nzΛΣ′ |jx|n⊥nzΛΣ〉 = ~2δΣ′,−Σ

(A.31)

32

For Λ′ 6= Λ, the expression ΛSΛ′−Λ, occurring in the jx matrix elements in (A.28),are invariant when signs of Λ and Λ′ are both changed. But for Λ′ = ±Λ ± 1, thefunction SΛ′+Λ−1 changes sign when both Λ and Λ′ change signs, that is S−Λ′−Λ−1 =−SΛ′+Λ−1. Therefore, when all of Λ′, Σ′, Λ,Σ change signs, then the matrix elementssimply change their signs

〈n′⊥n′z,−Λ′,−Σ′ |jx|n⊥nz,−Λ,−Σ〉 =

−〈n′⊥n′zΛ′Σ′ |jx|n⊥nzΛΣ〉, if Λ′ = ±Λ± 1(A.32)

B Symmetries

B.1 Parity

The parity symmetry operator changes ~r to −~r. In the cylindrical coordinates de-fined in Eq. (28), this corresponds to the transformation

ρ 7→ ρ, φ 7→ φ + π, z 7→ −z, (B.1)

which will transform the basis wavefunction factors as

ψΛ 7→ (−1)ΛψΛ, ψnz 7→ (−1)nzψnz , ψ|Λ|nρunchanged (B.2)

where the property Eq. (C.4) was used for the Hermite polynomial in ψnz . By useof Eqs. (35) and (36) one gets (−1)Λ = (−1)n⊥ so that (−1)Λ(−1)nz = (−1)N .Therefore, a basis wavefunction at ~r is transformed to

ψn⊥nzΛ(−~r) = (−1)Nψn⊥nzΛ(~r). (B.3)

Thus, the basis states have positive parity for N even, and negative parity for Nodd.

B.2 Signature for rotation about the z axis

The signature operator is the operator for rotation 180 about the z axis

Rz(π) = e−iπjz/~ = e−iπlz/~e−iπsz/~ = R(L)z (π)R(S)

z (π). (B.4)

because lz and sz commute. The rotation operator in spin space is derived in [17],and is here

R(S)z (π) = −2isz/~, (B.5)

The rotated spin state isR(S)

z (π) |Σ〉 = −2iΣ |Σ〉. (B.6)

33

For the rotation in ordinary space it is easier to act upon the coordinates of thenon-rotated wavefunction, rather than letting R

(L)z (π) act upon the state. The wave-

function for the rotated state is

ψrot(~r) = ψ( R(L)z (−π)~r ). (B.7)

The coordinate transformation R(L)z (−π) acting on ~r is

x 7→ −x, y 7→ −y, z 7→ z, (B.8)

and in cylindrical coordinates

ρ 7→ ρ, φ 7→ φ− π, z 7→ z. (B.9)

The basis wavefunction factors are transformed as

ψΛ 7→ (−1)ΛψΛ, ψnz unchanged, ψ|Λ|nρunchanged, (B.10)

so the space part of the total wavefunction for the z-rotated state is

ψrotn⊥nzΛ(~r) = (−1)Λψn⊥nzΛ(~r). (B.11)

Thus, a basis state is for rotation 180 about the z axis transformed to

Rz(π) |n⊥nzΛΣ〉 = −2i Σ(−1)Λ |n⊥nzΛΣ〉, (B.12)

implying that the basis states have good signature for rotation about the z axis.

B.3 Signature for rotation about the x axis

As in the z-case, the operator for rotation 180 about the x axis is factored as

Rx(π) = e−iπjx/~ = e−iπlx/~e−iπsx/~ = R(L)x (π)R(S)

x (π), (B.13)

The rotation operator in spin space is in this case

R(S)x (π) = −2isx/~ = −i(s+ + s−)/~. (B.14)

which gives a x rotated spin state

R(S)x (π) |Σ〉 = −i | − Σ〉. (B.15)

For the rotation in ordinary space, a coordinate transformation in the wavefunctionargument is used, just as in the z-case. The wavefunction for the rotated state is

ψrot(~r) = ψ( R(L)x (−π)~r ). (B.16)

Here, the coordinate transformation R(L)x (−π) acting on ~r is

x 7→ x, y 7→ −y, z 7→ −z, (B.17)

34

and in cylindrical coordinates

ρ 7→ ρ, φ 7→ −φ, z 7→ −z. (B.18)

The basis wavefunction factors are transformed as

ψΛ 7→ ψ−Λ, ψnz 7→ (−1)nzψnz , ψ|Λ|nρunchanged, (B.19)

so the space part of the total wavefunction for the rotated state is

ψrotn⊥nzΛ(~r) = (−1)nzψn⊥nz ,−Λ(~r). (B.20)

To conclude, a basis state is for rotation 180 about the x axis transformed to

Rx(π) |n⊥nzΛΣ〉 = −i(−1)nz |n⊥nz,−Λ,−Σ〉, (B.21)

demonstrating that the basis states do not have good signature for rotation aboutthe x axis.

B.4 The α-kets

For α ∈ −12 ,+1

2 define the kets

|n⊥nzΛΩα〉 =1√2( |n⊥nzΛΩ〉+ (−1)nz+ 1

2−α |n⊥nz,−Λ,−Ω〉) (B.22)

where Λ and Ω here are taken to be positive (or zero for Λ). Below the kets arereferred to as + for α = +1

2 and – for α = −12 . From Eq. (B.21) it follows that they

are eigenkets of the signature operator in Eq. (B.13):

Rx(π) |n⊥nzΛΩα〉 = −i(−1)12−α |n⊥nzΛΩα〉 (B.23)

and thus, they have good signature for rotation about the x axis, (but not aboutthe z axis). For α = −1

2 the eigenvalue is +i, for α = +12 the eigenvalue is −i, in

accordance with the standard definition of the signature quantum number α.

Note that though n⊥ and nz are also good quantum numbers, Λ and Ω are notin this case. In fact, the α ket is instead an eigenket of |lz|, |sz| and of |jz|, witheigenvalues m = |Λ|, |Σ| and |Ω|, respectively.

B.5 Matrix elements of jx with respect to the α-kets

The α kets can be combined in a matrix element as ++,−−, +− or −+. By thelinearity of the scalar product, the matrix element of the α kets can be written asa sum of four matrix elements in terms of the original basis.

In the following, for the α kets (left hand side of equations), α = ± means α = ±12 .

35

• In the ++ case, one gets

M(++) def= 〈n′⊥n′zΛ′Ω′, α′ = + |jx|n⊥nzΛΩ, α = +〉 =

= 12 [〈n′⊥n′zΛ′Ω′ |jx|n⊥nzΛΩ〉+ (−1)nz〈n′⊥n′zΛ′Ω′ |jx|n⊥nz,−Λ,−Ω〉+

(−1)n′z〈n′⊥n′z,−Λ′,−Ω′ |jx|n⊥nzΛ,Ω〉+(−1)n′z+nz〈n′⊥n′z,−Λ′,−Ω′ |jx|n⊥nz,−Λ,−Ω〉

](B.24)

By use of the sign change rule Eq. (A.32), the matrix elements can be pairedtogether, giving

M(++) = 12

[1− (−1)n′z+nz ]〈n′⊥n′zΛ′, Ω′ |jx|n⊥nzΛ,Ω〉+

[(−1)nz − (−1)n′z ]〈n′⊥n′zΛ′, Ω′ |jx|n⊥nz,−Λ,−Ω〉 (B.25)

If n′z 6= nz ± 1, then by the selection rule for nz the matrix elements on the righthand side are zero. But if n′z = nz ± 1, the powers of −1 can be simplified to give

M(++) = 〈n′⊥n′zΛ′Ω′ |jx|n⊥nzΛΩ〉+ (−1)nz〈n′⊥n′zΛ

′Ω′ |jx|n⊥nz,−Λ,−Ω〉(B.26)

where the jx matrix elements in the right hand side are obtained from Eq. (A.31).

• In the −− case one gets similarly

M(−−) = 〈n′⊥n′zΛ′Ω′ |jx|n⊥nzΛΩ〉 − (−1)nz〈n′⊥n′zΛ

′Ω′ |jx|n⊥nz,−Λ,−Ω〉(B.27)

Note that the matrix elements in the ++ and −− cases differ only in the signbefore the second term.

• In the +− case, the linearity and the sign change rule give

〈n′⊥n′zΛ′Ω′, α′ = + |jx|n⊥nzΛΩ, α = −〉 =12

[1 + (−1)n′z+nz ]〈n′⊥n′zΛ′Ω′ |jx|n⊥nzΛΩ〉−

[(−1)nz + (−1)n′z ]〈n′⊥n′zΛ′Ω′ |jx|n⊥nz,−Λ,−Ω〉

(B.28)

If n′z 6= nz±1, the matrix elements in the right hand side are zero. If n′z = nz±1,the factors in front of the matrix elements are simplified to

1 + (−1)n′z+nz = 1 + (−1)2nz±1 = 0

(−1)nz + (−1)n′z = (−1)nz [1 + (−1)±1] = 0(B.29)

so that all +− matrix elements of jx are zero. Because jx is Hermitian, it followsthat all −+ matrix elements are also zero.

〈n′⊥n′zΛ′Ω′, α′ = + |jx|n⊥nzΛΩ, α = −〉 = 0

〈n′⊥n′zΛ′Ω′, α′ = − |jx|n⊥nzΛΩ, α = +〉 = 0(B.30)

Thus, jx does not couple α kets of different x signature.

36

C Some orthogonal polynomial relations

In for example Ref. [18] many of the relations for orthogonal polynomials used inthis paper are found. We present some of the more useful ones below and add a fewadditional relations.

C.1 Hermite polynomials formulas

Rodrigues’ formula for Hermite polynomials

Hn(x) = (−1)nex2 dn

dxn(e−x2

) (C.1)

Orthogonality relation∫ ∞

−∞Hn(x)Hm(x)e−x2

dx = δnm

√π2nn! (C.2)

Recurrence formulae for derivatives

d

dxHn(x) = 2nHn−1(x) (C.3)

The Hermite polynomials are even for even order, and odd for odd order

Hn(−x) = (−1)nHn(x) (C.4)

C.2 Laguerre polynomials formulas

Summation formula

Lmn (x) =

n∑

k=0

(−1)k

(n + m

n− k

)xk

k!(C.5)

Orthogonality relation for m integer∫ ∞

0Lm

n1(x)Lm

n2(x)xme−xdx = δn1n2

(m + n1)!n1!

(C.6)

m-recurrenceLm−1

n+1 = Lmn+1 − Lm

n (C.7)

Recurrence with x multiplication

xLm+1n (x) = (n + m + 1)Lm

n (x)− (n + 1)Lmn+1(x) (C.8)

Recurrence formula two steps back

nLmn (x) = (2n + m− 1− x)Lm

n−1(x)− (n + m− 1)Lmn−2(x) (C.9)

37

TUMGREPP 04-01-22, 16.104

PAPER III

TUMGREPP 04-01-22, 16.105

TUMGREPP 04-01-22, 16.106

Investigation of tilted axis rotation at triaxial superdeformed shapes in 166Hf

Peter Olivius and Ragnar BengtssonDepartment of Mathematical Physics, Lund Institute of Technology, Lund, Sweden

(Dated: January 23, 2004)

Tilted axis cranking calculations for 166Hf have been performed starting from the γ±20 principalaxis cranking energy minima at spin around 30~. Self-consistent tilted axis cranking energy minimafor triaxial superdeformed shapes have been found. The single-particle energy levels causing thetilting are investigated. The energy varies weakly with the tilting angles around the minima, andthe gain in energy compared to principal axis cranking is in the order 10 to 200 keV.

INTRODUCTION

Triaxial superdeformed (TSD) states are theoretically predicted to appear in a group of nuclei centered around166Hf. The TSD minima, which are found in nearly all low-lying configurations, are the result of pronounced shellstructures. Calculations based on the Nilsson [1] and Woods-Saxon [2, 3] potentials give very similar results. In thehigh spin regime of several nuclei, TSD energy minima appear pairwise in the total energy surface with deformationcoordinates such that the quadrupole deformation ε is approximately the same for both minima, whereas the triaxialityparameter γ has nearly the same absolute value but with opposite sign. This implies that the nucleus has the sameshape in the two minima, but rotates about different axes: For γ > 0, the nucleus rotates about its shortest principalaxis and for γ < 0 about its intermediate axis. In the (ε, γ) deformation plane the TSD minima are often separatedfrom each other and from the energy minimum at normal deformation by energy barriers of the order 1 MeV or more,as illustrated in fig. 1.

Since the shape of the nucleus is nearly the same in the two TSD minima, they might be connected without changingthe deformation, namely by gradually changing the direction of the rotational axis from that of the short axis of thenucleus to that of its intermediate axis. It is not obvious that there has to be an energy barrier between the twoTSD minima if the rotational axis is allowed to tilt from one minimum to the other. It might also be that there isan absolute energy minimum for a direction of the rotational axis which does not coincide with one of the principalaxes. This is referred to as tilted axis rotation.

Tilted axis rotation can be described by the cranking model, either as two-dimensional cranking, in which case therotational axis is restricted to lie in a principal plane, or as three-dimensional cranking, in which case the rotationalaxis is allowed to point in any direction, see e.g. [7]. In this paper we shall report the results of a study of the stabilityof TSD minima in 166Hf against a tilting of the rotational axis and the appearance of energy minima correspondingto tilted axis rotation.

An extended version of the three-dimensional cranking model was developed to allow for realistic full-scale calcu-lations in a multi-dimensional deformation space. In the calculations, we have allowed three deformation parameters(ε, γ and ε4) and two tilting angles (θ and φ) to vary simultaneously. Pairing correlations are not included, since theyare not considered to be important at the high spin values for which the TSD states appear on or near the yrast line.This is illustrated in fig. 2. It should be observed that there are no TSD minima for very low angular momenta andif the paring correlations are included, additional angular momentum is needed before the TSD minima develop. Thepairing correlations have a tendency to smear out the barrier between the TSD minima and the energy minimum atnormal deformation, in particular at low spin where the pairing correlations still are important.

We have chosen the nucleus 166Hf, which has proton number 72 and neutron number 94, for the present study sincethere are major energy gaps in the single-particle level spectrum at TSD shapes for these particle numbers, whichcan be seen from figs. 3 and 4. The gaps stabilize the location of the TSD energy minima in deformation space. Thepresent calculations could therefore be restricted to a limited part of the (ε,γ) plane surrounding the TSD energyminima, see fig. 1. Since the tilting angles are allowed to take all values between 0 and 90, rotation about thelongest principal axis will automatically be included. It corresponds to tilting angle θ = 0 (see eq. (14) and fig. 5)and appears in the (ε, γ) plane for values of γ between −60 and −120 [5]. This part of the (ε, γ) plane is notincluded in fig. 1 since rotation about the long axis results in a large rotational energy for TSD shapes in nuclei near166Hf. The energy may for I & 30~ be several MeV higher than for rotation about any of the two shorter axes.

2

166Hf N=94 π=1 α=0 I=36 h

0.1 0.2 0.3 0.4 0.5ε cos γ

−0.2

−0.1

0

0.1

0.2

ε sin

γ

Step=0.50 MeV, Min=8.57 MeV, Entry=ETot

FIG. 1: Potential energy surface with pairing for I = 36 ~. The regions around the TSD energy minima, which have beeninvestigated with respect to tilted axis rotation are marked by shading. The upper shaded region corresponds to the tiltingangles θ = 90 and φ = 0 (principal axis rotation about the short axis) and the lower shaded region to θ = 90 and φ = 90

(principle axis rotation about the intermediate axis). The absolute energy minimum appears at normal deformation and ismarked by a filled circle.

THE HAMILTONIAN FOR 3D CRANKING

For the potentials described below we need some harmonic oscillator quantities and coordinate definitions. In termsof the size of the quadrupole deformation ε, and the triaxiality parameter γ [5], dimensionless harmonic oscillatorfrequencies are defined as

Ωx = 1− 23ε cos(γ + 2π

3 )Ωy = 1− 2

3ε cos(γ − 2π3 )

Ωz = 1− 23ε cos γ,

(1)

Note that the Lund convention is used for the sign of γ [5]. An isospin dependent oscillator parameter is set to

~ ω0= 41A−1/3(1± N − Z

A)MeV, (2)

where + sign is used for the neutrons and - sign for the protons, as in [6]. The oscillator spacing is

~ω0 = ~ ω0 rvc (3)

where the factor rvc is the volume conservation condition ratio which is a function of the deformation [5], [6]. A setof stretched coordinates are introduced as

ξ =√

MΩxω0~ x

η =√

MΩyω0~ y

ζ =√

MΩzω0~ z,

(4)

From them, stretched spherical coordinates are defined by

ξ = rt sin θt cosφt

η = rt sin θt sin φt

ζ = rt cos θt,(5)

3

166Hf

0 10 20 30 40 50I ( h)

TSD, calculated w

ithout pairing

TSD, calculated with pairing

Normal deformation, with pairing

−4

−3

−2

−1

0

1

2

3

Eto

t−0.

0080

I(I+

1) (

MeV

)

FIG. 2: Total energy versus angular momentum at TSD shape with γ > 0, calculated with (open circles) and without (filledcircles) pairing. The total energy calculated with pairing at normal deformation is also included. The calculations with pairingare performed as described in ref. [4]. The data shown in the figure refer to the lowest configuration with positive parity forboth protons and neutrons and signature α = 0 for both kind of nucleons.

that are not to be confused with the tilting angles eq.( 14). The single particle Hamiltonian in the lab frame is thesum of the kinetic energy operator and the modified oscillator potential,

H0τ = T + Vτ , (6)

where the isospin index τ =p for protons and n for neutrons. The potential can be written as

Vτ =Vosc,τ − ~ ω0 κτ

(2~lt ·~s~2

+ µτ (lt

2−〈lt2〉N~2

)). (7)

Note that the coupling parameters κτ and µτ , (and also ~ ω0) are different for protons and neutrons. For values ofthe coupling parameters see ref. [6]. The orbital angular momentum operator, used in the spin-orbit term and in thelt

2 term, is defined in the stretched coordinates eq. (4) as

~lt = −i~(η

∂

∂ζ− ζ

∂

∂η, ζ

∂

∂ξ− ξ

∂

∂ζ, ξ

∂

∂η− η

∂

∂ξ

). (8)

The oscillator takes into account quadrupole and hexadecapole deformation. In terms of spherical harmonics evaluatedin the stretched spherical coordinate angles θt, φt eq. (5), it is expressed as

Vosc,τ = 12~ω0r

2t

(1 + α20Y20 + α22(Y22 + Y2,−2)

+α40Y40 + α42(Y42 + Y4,−2) + α44(Y44 + Y4,−4)) (9)

The quadrupole and the hexadecapole deformations both have a coordinate plane reflexion symmetry, which reducethe number of parameters to two for the quadrupole deformation, and three for the hexadecapole deformation. The

4

7270

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hω (MeV)

41

42

43

44

e i (M

eV)

PROTONS ε=0.40 γ=20.0o ε4=0.04 θ=90.0o φ=90.0o

7270

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hω (MeV)

41

42

43

44

e i (M

eV)

PROTONS ε=0.40 γ=20.0o ε4=0.04 θ=90.0o φ=0.0o

FIG. 3: Single-proton energy levels for principle axis rotation at TSD shape. The top figure corresponds to rotation about theshort axis (θ = 90, φ = 0) and the bottom figure to rotation about the intermediate axis (θ = 90, φ = 90).

two quadrupole coefficients α20, α22 are expressed in the parameter ε and γ as

α20 = − 4ε3

√π5 cos γ

α22 = 4ε3

√π10 sin γ.

(10)

Additionally, the hexadecapole deformation is restricted to be z-axis symmetric for γ = 0. The hexadecapole coeffi-cients can, in terms of ε4 and γ, be expressed as

α40 = 2ε49

√π(5 cos2 γ + 1)

α42 = − ε49

√30π sin 2γ

α44 = ε49

√70π sin2 γ.

(11)

The cranked single particle Hamiltonian is

Hωτ = H0

τ − ~ω ·~j (12)

where

~ω ·~j = ωxjx + ωyjy + ωzjz (13)

is the 3D cranking term, and ~ω=(ωx, ωy, ωz) is the cranking vector, that is the angular velocity vector for rotation ofthe potential, and the intrinsic frame of reference, about a fixed lab frame. For the cranking vector, we define tilting

5

9497

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hω (MeV)

49

50

51

52

53

e i (M

eV)

NEUTRONS ε=0.40 γ=20.0o ε4=0.04 θ=90.0o φ=90.0o

94 97

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hω (MeV)

49

50

51

52

53

e i (

MeV

)

NEUTRONS ε=0.40 γ=20.0o ε4=0.04 θ=90.0o φ=0.0o

FIG. 4: Single-neutron energy levels for principle axis rotation at TSD shape. The top figure corresponds to rotation aboutthe short axis (θ = 90, φ = 0) and the bottom figure to rotation about the intermediate axis (θ = 90, φ = 90).

angles θ, φ by spherical coordinates

ωx = ω sin θ cos φωy = ω sin θ sin φωz = ω cos θ

(14)

In 1D cranking, ~ω points along one of the principal axes of the (deformed) nucleus. In 2D cranking, ~ω lies in aprincipal plane, and in 3D cranking it does not belong to any principal plane, see fig. 5 and ref. [7].

The time-independent Schrodinger equation in the intrinsic frame of reference is

Hωτ Ψ = eω

τ Ψ, (15)

which is the problem that is solved.

THE STRUTINSKY RENORMALIZATION

The sum of the energies of the single particle states do not correspond to experimental total energies. By theShell correction method, where the sum is renormalized to the liquid drop model, a good agreement is achieved [8].Also the sum of angular momenta expectation values of the single particle states can be renormalized in an analoguemanner [9]. The procedure of Strutinsky renormalization of energy and spin is described in detail in [10], from whichthe code for the renormalization was derived. Let for isospin τ the eigenvalues in eq. (15) be eω

τi. The Strutinsky

6

ω

long

axi

s

intermediate

axis

φ

θ

short axis

Y

X

Z

FIG. 5: The cranking vector ~ω, the tilting angles θ, φ, and the directions of the three principal axes of the deformed nucleus.

averaged level density is defined as

gτ (E) = SMgτ (E) (16)

where gτ (E) =∑∞

i=1 δ(E − eωτi) is the level density, and SM is the Strutinsky averaging operator [10]. A natural

choice for achieving a smooth level density is to replace each delta spike by a Gaussian e−x2/√

π, centered about theeigenvalue. For that choice, the Strutinsky averaged level density can be expressed in a useful form for computationsas

gτ (E) =1

γs√

π

∞∑

i=1

M∑µ=0

(−1)µ


i , (17)

where the variable xi = (E − eωτi)/γs is used (where the index τ is omitted), and pn(x) is a Hermite polynomial of

degree n. The averaged Fermi energy λτ is determined such that the averaged number of particles up to λτ is equalto the actual number of particles N , that is the solution to the integral equation

N =∫ eλτ

−∞gτ (E)dE, (18)

which can be written as

N =∞∑

i=1

[12(1 + erf(xi)) +

e−ex2i√

π

M∑µ=1

(−1)µ

22µµ!p2µ−1(xi)], (19)

where xi =(λτ − eωτi)/γs. The Strutinsky averaged single particle energy sum is defined as (for protons and neutrons

separately)

Eτ =∫ eλτ

−∞E gτ (E)dE, (20)

which can be expressed as

Eτ =∞∑

i=1

[eωτi

12(1 + erf(xi)) +

e−ex2i

2√

π

(−γs +

M∑µ=1

(−1)µ

22µ−1µ!λτp2µ−1(xi)− γsp2µ−2(xi)

)]. (21)

7

Let jk be the angular momentum operator for k=x, y, z. For the exact spin density

gτ,jk(E) =∞∑

i=1

〈jk〉τiδ(E − eωτi), (22)

a Strutinsky averaged spin density can, similarly to eq. (17), be written as

gτ,jk(E) =1

γs√

π

∞∑

i=1

〈jk〉τi

M∑µ=0

(−1)µ


i . (23)

The Strutinsky averaged spin sum is defined as

Iτk =

∫ eλτ

−∞gτ,jk(E)dE, (24)

and can be written in a useful form as

Iτk =

∞∑

i=1

〈jk〉τi

[12(1 + erf(xi)) +

e−ex2i√

π

M∑µ=1

(−1)µ

22µµ!p2µ−1(xi)

]. (25)

The proton and neutron contributions are summed up to get

E = Ep + En, (26)

~I = (Ipx + In

x , Ipy + In

y , Ipz + In

z ). (27)

In the calculations, we have used the Strutinsky parameter values γs = 1.2 and M = 3.

TOTAL QUANTITIES AND SYMMETRIES

By occupying exactly Z proton single particle states, and N neutron single particle states (eigenstates Ψpi and Ψni

of the cranked Hamiltonian in eq. (12), a total Routhian energy is obtained

Eω =∑

i occ

eωpi +

∑

i occ

eωni. (28)

The Hamiltonian in eq. (12) conserves parity (it commutes with the space inversion operator Π that changes ~r to−~r). The eigenvalue problem eq. (15) is solved for protons and neutrons separately, and the single particle eigenstateseach either have positive or negative parity, i.e.

ΠΨ = ±Ψ. (29)

A configuration where an odd number of negative parity single particle states are occupied, results in a negativeparity state for the isospin, but for an even number a positive parity state. Thus, there are four parity combinations(πp, πn) = (+, +), (+,−), (−, +) and (−,−).

The lowest total Routhian, for a parity combination, we denote ord=1. The particle-hole excited but energeticallynext-to-lowest total Routhian we denote ord=2, the configuration with the third lowest total Routhian ord=3, andso on. This results in sequences ord=1, 2, . . . of configurations, for each of the four parity combinations.

The signature symmetry, that is invariance under rotation 180 about the cranking axis, holds for 1D cranking.Then the single particle eigenstates of the Hamiltonian eq. (12) are also eigenstates of the operator for rotation 180

about the cranking axis, i.e.

R~n(π)Ψ = −i(−1)12−α Ψ (1D only), (30)

8

where ~n = ~ω/ω is a direction unity vector along the cranking axis, and α =± 12 is the signature quantum number.

However, the signature symmetry is broken for 2D and 3D cranking.The total angular momentum vector of a configuration is calculated from the eigenstates as

~I =∑

i occ

〈Ψωpi |~j|Ψω

pi〉+∑

i occ

〈Ψωni |~j|Ψω

ni〉, (31)

which is usually different to both length and direction for different configurations (ord). The corresponding energy inthe lab frame is defined as

Elab =∑

τ

∑

i occ

〈Ψωτi |H0

τ |Ψωτi〉 = Eω + ~ω · ~I. (32)

A Strutinsky averaged lab energy is similarly defined as

Elab = E + ~ω · ~I, (33)

using eqs. (26) and (27). The shell correction energy is calculated as

Eshell = Elab − Elab(I2 =I2), (34)

where the second term is the Strutinsky averaged lab energy from eq. (33), but interpolated to squared spin I2 = ~I · ~I,so that the shell energy is defined for a constant I and not a constant ω.

The liquid drop model energy is taken as

Eld = (Bs − 1)(c1A2/3 + c2

(A−2Z)2

A4/3 )+(Bc − 1)c3

Z2

A1/3 .(35)

where deformation dependence enters through the surface integral BS and the Coulomb energy integral BC , definedin [11]. The parameters c1, c2, c3 are constants, see [11]. For the rotational energy the matrix holding the rigid bodymoment of inertias

J =

Jx 0 00 Jy 00 0 Jz

(36)

are used [11]. The angular momentum of the rigid body is given by

~Ild = J ~ω =

Jxωx

Jyωy

Jzωz

, (37)

from which we define the vector

~Ir =I

Ild

~Ild, (38)

which is the rigid body angular momentum rescaled to a length equal to that of the total spin ~I. The liquid droprotational energy for spin ~Ir is then defined as

Erotld =

12~Itr J−1~Ir =

12(I2rx

Jx+

I2ry

Jy+

I2rz

Jz). (39)

The renormalized total energy for spin I is defined as

Etot = Erotld + Eld + Eshell. (40)

The total spin is that of eq. (31), that is the sum of the angular momenta expectation values of the occupied singleparticle states as is, and it was not renormalized.

Instead we renormalized the cranking angular frequency vector.

9

7

8

9

10

11

12

0 10 20 30 40 50 60 70 80 90

0102030405060708090

(MeV

)

φ (deg)

Eldrot for A=166, spin 36 −h, X=0.38,Y=0.13,ε4=0.03

θ (deg)

(θ=90,φ=0) Short axis rot

Interm. axis rot(θ=90,φ=90)(θ=90,φ=90)

Long axis rot(θ=0,φ arb.)

θ varies; φ=90 fix

θ varies; φ=0 fix

φ varies; θ=90 fix

FIG. 6: The liquid drop rotational energy term for fixed spin 36~, and fixed deformation X = 0.38, Y = 0.13, ε4 = 0.03 (roughlythe same as for the γ > 0 minimum in fig. 1). The solid line shows how Erot

ld increases from principal axis rotation about theshort axis to rotation about the long axis. The dashed curve shows the variation from short axis rotation to intermediate axisrotation, the dot-dashed curve shows the variation from intermediate axis rotation to long axis rotation. This would be thevariation of the total energy (omitting the constant Eld) with the tilting angles if the shell correction energy would have beenzero. Note that the two points in the graph denoted (θ = 90, φ = 90) correspond to the same point in the tilting angle space(intermediate axis rotation).

YRAST DETERMINATION

For each ω and parity combination (πp, πn), particle-hole excitations are calculated up to ord=maxord, which givesa sequence of points (I, Etot), ord=1, 2, . . . , maxord. For each (πp, πn), these sets are used to determine the yrastband, i.e. the energetically lowest band in the lab frame for that combination of parities. Thus, we calculate fourindependent yrast bands, one for each of the parity combinations (+,+), (+,−), (−, +) and (−,−).

For 1D cranking, the lowest total Routhian (ord=1), or when encountering a band-crossing, a few low lying config-urations (ord . 10) are sufficient to generate the yrast band.

As we described in [12], in tilted axis cranking for high spin, it can be necessary to perform high excitations in therotating frame (high ord), to generate the yrast band in mesh points where the self-consistency condition, eq. (49)is not fulfilled. In fact for angular momentum of the order 50~ a maxord in the order of 103 to 104 may be neededin mesh points far away from any self-consistent point. However, for self-consistent deformation-tilting meshpointsconfigurations with ord=1, or of ord close to 1, are sufficient for calculating the yrast line. But for the minimizationof the total energy with respect to deformation and tilting (page 12) it is necessary to calculate high particle-holeexcitations to generate a correct yrast line also for the meshpoints away from self-consistency. Otherwise, the energywill be to high at those meshpoints, and as how much too high can vary from meshpoint to meshpoint, that wouldproduce false local minima, which would be ruinous as we are interested not only in the global minima, for a given spinand parity combination, (that often is a 1D cranked state), but also in the higher order local minima (that sometimesare tilted).

Our method to calculate the yrast band for a given (πp, πn) will be schematically described here. The startingpoint is assigned the point (I1, E1) with lowest total energy E = Etot, among all ω and ord in the calculation. The

10

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70

Eto

t (M

eV)

Ι (−h)

++ yrast example

Etot(Ι)14

15

16

17

18

19

43 44 45 46 47 48 49 50 51 52

Ι (−h)

++ yrast example

k

k-1

ss

FIG. 7: To the left, an yrast line for X = 0.375, Y = 0.125, ε4 = 0.03, θ = 90 and φ = 9 calculated with the method ofsmallest slope and rotational band tracking. To the right, a zoom-in around a band crossing, an example where the rotationalband tracking is performed. Starting from (Ik, Ek) a point P within a box, centered around (Ik + ∆I, Ek + ∆E), eq. (44), andwith minimal slope µ(k, P ) is searched. P is added to the yrast line, unless the slope µ(k, P ) is much bigger than µ(k−1, k)which is interpreted as a sudden jump to an excited band. That can be expressed in terms of a condition on the band momentof inertia ratio Jb(P )/Jb(k) < lim < 1 where lim=0.8 in our calculations. If the condition is satisfied, P is rejected, and themethod of smallest slope is continued. The rotational band tracking is also interrupted if the spin of P becomes larger thanIss.

next point is chosen as the point (I2, E2), among all ω and ord in the calculation, which gives the smallest slope

µ(1, 2) =E2 − E1

I2 − I1(41)

for a line segment that begins in (I1, E1). The search is repeated from point 2, and so on. This simple algorithm wecall the method of smallest slope. It continues along the yrast band to the largest spin available in the calculation,unless, for the spin Iss found by the method of smallest slope, the condition

Iss − Ik ≥ 2(Ik − Ik−1) (42)

is satisfied, in which case a rotational band crossing is expected to occur somewhere in [Ik,Iss]. Then a phaseof rotational band tracking is performed in the same interval: Along a rotational band the spin and energy varyapproximately as

I ≈ Jbω, E − E0 ≈ I(I + 1)2Jb

, (43)

which give

∆I ≈ Jb∆ω, ∆E ≈ I∆ω, (44)

for a step ∆ω in the cranking angular frequency. The moment of inertia for the present band is taken as

Jb = IkIk − Ik−1

Ek − Ek−1. (45)

11

Name Explanation

ε Quadrupole deform. size

γ Triaxiality angle

ε4 Hexadecapole deform. size

θ Polar tilting angle

φ Azimuthal tilting angle

N, Z Neutron and proton numbers

TABLE I: The mesh parameters that can be varied in the calculations.

In the algorithm of rotational band tracking, yrast points are searched in directions in the energy-spin plane basedon these equations. See fig. (7) for an illustration and more details.

In order to minimize the total energy with respect to deformation and tilting for a given spin, an interpolationbetween calculated points (I, Etot) is required. As the signature symmetry is broken for tilted cranking, the totalenergy is interpolated to steps of 1~ in spin, not 2~ as is the case in principal axis cranking.

Suppose that for a given constant spin a minimum of the renormalized total energy is found, with respect to varia-tions of the deformation and tilting angles. For a constant cranking angular frequency, which at the energy minimumgenerates the given spin, the lowest total Routhian (i.e. for ord=1) is also a function Eω(~p) of the deformation andtilting angle parameters. Note that the direction of the cranking vector varies with the tilting angles, but the sizeis fixed at the value at the minimum. Then, the lowest total Routhian function also has a minimum, at the samepoint in the deformation and tilting angle space, though, however, the Strutinski renormalization will generate a smalldeviation. This connection will be used when the shell structure causing tilted energy minima is analyzed.

PARAMETERIZING THE DEFORMATION AND TILTING SPACE

A mesh is a discrete set of points in a multi-dimensional space for deformation, tilting angles and particle numbers.In the computer program the parameters that may be varied in a mesh are given in table (I). However, in this paperwe have kept the particle numbers fixed. Thus, the mesh space is 5-dimensional, and the total number of mesh pointscan be large although the number of discrete values in each dimension is fairly small. As for each mesh point theprogram loops through a sequence of cranking angular frequencies ω = ω1, . . . , ωn, the total number of iterations(meshpoint-frequency combinations) can be very large. In the calculations that this paper is based upon, the totalnumber of iterations is in the order of 106. The step length for ω decreases, as ω increases, in order to get a step intotal energy of the order ∆Etot =0.1MeV for a step in ω along the band, and thus reach a sufficiently high precisionin the spin interpolation, also in the high spin region. See page 13 for the results of an error analysis.

In the actual calculations the Cartesian quadrupole coordinates

X = ε cos(γ)Y = ε sin(γ)

(46)

were used to vary the quadrupole deformation, not to be mixed up with the lab frame space coordinates.The step length for the tilting angles θ, φ have been varying from the order of 10−25 for survey calculations, down

to 3 for the most detailed zoom-in calculations in regions with interesting energy minima. For the deformation, thestep length in X and Y was varied from 0.025 in the survey, where the rough location of the minima was determined,to 0.02 in the two zoom-in calculations for which the LO and HI deformation regions defined in fig. 9 were explored.In ε4 the step length was 0.03. That is still unsatisfactory large steps lengths, but for the available computer resourcesthey yet resulted in calculation times of the order a week, due to the high time-complexity in five dimensional space,the dense cranking angular frequencies, and the high demands on small basis truncation errors giving large matricesto diagonalize.

As there is no pairing included in the calculations, solutions in the low spin region, i.e. for spins . 30~ are, witha few exceptions, not included. The upper limit for the cranking angular frequency was ωmax = 0.7MeV/~ in thecalculations, generating states along the yrast band up to about spin 70 to 80~ for the present deformation regionand nucleus.

12

p

p

2

1

FIG. 8: A sample sub mesh of 9 points (circles) in the case N=2. The minimal Etot point m is marked with a filled circle.

THE MINIMIZATION PROCEDURE

For each proton and neutron parity combination, the total energy Etot was minimized for integer values of the spinI [17] in the following way.

Among all mesh points with the same parity combination and spin, the program searches for all mesh points, suchthat each of them has a lower energy than all its neighbors (in the deformation and tilting angle space). Such a meshpoint is called a strict local mesh minimum. Note that there may exist several strict local mesh minima, e.g. one forone deformation and principal axis rotation, and a second for another deformation with tilted axis rotation.

Let us denote a strict local mesh minimum point m. The number of deformation parameters and tilting anglesvaried in the calculation is denoted by N. There may, in the N-dimensional mesh space be several strict local meshminima, with different deformations and/or tilting angles, as e.g. in fig. 10. An N-dimensional submesh containing3N mesh points, with m in the middle, is set up. See fig. (8). If m lies along a border of the full mesh, with no mirrorsymmetry, then the center of the sub mesh is moved one step into the mesh. If m lies at a border of the full meshwith mirror symmetry with respect to a mesh parameter pk, that is Etot is symmetric with respect to pk for mirroringaround its value pk(m) in m, i.e.

Etot(pk(m)− x) = Etot(pk(m) + x), (47)

for any x, the submesh will include mirror points lying outside the original full mesh. Examples when mirroring isperformed are, if m is along the border in the φ-direction and φ(m) = 0, because of the symmetry Etot(−φ) = Etot(φ),or if m is along the border in the θ-direction and θ(m) = 90, because of the symmetry Etot(90− θ) = Etot(90+ θ).

The total energy is interpolated in the sub mesh by a multi-polynomial L(~p) of degree 2N (2nd order polynomialsfor each parameter), and minimized within the sub mesh by Newton-Raphson iterations

~p (n+1) = ~p (n) −H(~p (n))−1∇L(~p (n)) (48)

where H is the Hessian matrix holding the 2nd order derivatives of L, and ∇L is the gradient vector with the 1storder derivatives of L, and ~p = [p1, . . . , pN]t is the vector collecting the parameters pk. However, the iteration doesnot always converge towards a local minimum of the multi-polynomial surface. In the N = 5 case (three deformationand two tilting parameters) it is frequent that after some iterations, one of the parameters cross the border of thesubmesh. But if the iterations converge towards a stationary point (the norm of ∇L is zero) where the Hessian ispositively definite (all N eigenvalues are strictly positive), then the algorithm has ended in a strict local minimum ofthe interpolating polynomial.

By this method, the total energy local minima are calculated by simultaneous minimization with respect to allmesh parameters in the calculation.

THE SELF CONSISTENCY MEASURES AND ERROR ESTIMATES

As shown in [13] for a self-consistent energy minima the total angular momentum vector, eq. (31), and the crankingangular frequency vector, (see page 4 and fig. 5) must be parallel, i.e.

~I ‖ ~ω. (49)

13

0.1

0.125

0.15

0.175

0.2

0.325 0.35 0.375 0.4 0.425 0.45

Y =

ε s

in γ

X = ε cos γ

(X,Y) distribution of TSD self-consistent minima in 166Hf

LO

HI

1D LO1D HI

2Dθ LO2Dθ HI

2Dφ LO2Dφ HI

3D

FIG. 9: Quadrupole deformation distribution of tilted minima with good self-consistency measure (page 12). 2Dφ denote theφ-tilted minima (~ω in principal plane spanned by the short and middle semi axes). 2Dθ denote the θ-tilted minima (~ω inprincipal plane spanned by the short and long semi axes). 3D denote the minima tilted both in φ and θ. The areas in whichthe zoom calculations were performed are marked with rectangles.

Introduce spherical coordinates (I, θI , φI) for the total angular momentum vector, as

Ix = I sin θI cos φI

Iy = I sin θI sin φI

Iz = I cos θI .

(50)

To check the results of the energy minimization for self-consistency, the following measures have been used

cθ = θI − θ, cφ = φI − φ, (51)

where θ, φ are the spherical angles for the cranking angular frequency vector as defined in eq. (14). A self-consistentsolution has, in the ideal case, cθ = cφ = 0. In practice, how close to zero the numerical values of the self-consistencymeasures can be expected to lie has to be considered. An upper limit of 1− 2 was used for the zoom-in calculations.Furthermore, it is required from the tilting angle variation around the minimum that the self-consistency measureintersects zero and does it for a tilting angle close to that of the energy minimum, see e.g. fig. 14(b) for the φ-variationaround a 2Dφ-tilted solution, where cφ intersects zero at φ ≈ 9 and the tilted energy minimum also occurs at φ ≈ 9.Another example is shown in fig. 17(b) for a 2Dθ-tilted solution, where cθ intersects zero at θ ≈ 75 and the energyminimum is at θ ≈ 76. If the two conditions on the self-consistency measures, about the size of the absolute valueless than some limit, and about the intersection with zero at a tilting angle close to the one giving minimum totalenergy, are satisfied, as well as the condition that the energy minimum is a stationary point with a positively definiteHessian (see page 12) then the energy minimum is considered a self-consistent solution - a ”good” energy minimum.

As the calculations have shown, the total energy can vary very weakly with the tilting angles. Then it is importantto estimate the errors in the total energy, eq. (40), and spin, eq. (31), within the model, to assure that the errors aresmaller than the depth of energy minima. The errors are due to the basis truncation error, incomplete Strutinskyaveraged quantities convergence, spin interpolation and incomplete convergence in quadratures for integrals. Note thaterrors defined as discrepancy between model and experiment, and the errors due to interpolation between differentmeshpoints, are not taken into account here.

14

166Hf Z= 72 N= 94 πp= 1 πn= 1 Spin= 45h

0.34 0.36 0.38 0.4 0.42 0.44 0.46X = ε cos γ

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

Y =

ε s

in γ

step=100 keV min=14.92 entry=Etot

(a)

166Hf Z= 72 N= 94 πp=−1 πn= 1 Spin= 53 h

0.34 0.36 0.38 0.4 0.42 0.44 0.46X = ε cos γ

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

Y =

ε s

in γ


(b)

FIG. 10: (a) Shape coexistence between two principal axis cranking minima in the low (denoted LO) and high deformation(denoted HI) regions, respectively. The parity combination is πp =+, πn =+, and the spin I = 45~.(b) Shape coexistence between a principal axis cranking minima in the high-deformation (HI) region, and a 2Dθ-tilted (θ ≈ 86)cranking minima in the low-deformation (LO) region. The parity combination is πp =−, πn =+, and the spin I = 53~.

The basis truncation error for the spin is, for the size of the basis used in the calculations, estimated to ∆I . 0.01~.The error in the liquid drop energy, eq. (35), is of the order ∆Eld ≈ 0.1 keV. For the shell energy, eq. (34), the errorcomes from the basis truncation error and from incomplete Strutinsky convergence, and was estimated to the order∆Eshell . 10 keV. The estimated maximal error in the total energy is then

∆Etot . 20 keV. (52)

However, the error terms are likely not fully additive, which would bring down the estimated error somewhat. Alsothe error terms are expected to be similar in neighboring meshpoints, so that a difference in energy between them,smaller than the error limit above, is still relevant. But as a rough upper limit, results within the order 10 keV canbe considered fully reliable for the calculations in the model.

PRINCIPAL AND TILTED AXIS ROTATION SOLUTIONS

The self-consistent solutions are located in two regions in the quadrupole deformation space, see fig. 9. One region,which we denote LO, has relative to the other region, a somewhat lower deformation with ε about 0.3 to 0.4. Theother region, which we denote HI has a higher deformation with ε about 0.4 to 0.5. Both regions correspond totriaxial shapes with positive γ-values. After localizing the minima to these two regions, the zoom-in calculations wereperformed in each of them. The ranges for the quadrupole deformation of the self-consistent solutions are in polarcoordinates ε ∈ [0.36, 0.42], γ ∈ [14, 23] in the LO region, and ε ∈ [0.43, 0.49], γ ∈ [21, 26] in the HI region. Thehexadecapole deformation, ε4, is for most of the self-consistent solutions about 0.02 to 0.04.

In both regions there are self-consistent solutions of both principal axis rotation and of tilted axis rotation.A spin sequence of self-consistent solutions with the same parity combination and configuration, and with similar

deformation and tilting angles, we call a miniband.The minibands are for each parity combination illustrated in figs. 11(a), 11(b), 12(a), and 12(b), with their energies

and tilting angles plotted versus spin. The energies are shown relative to that of a rigid rotor of fixed moment of

15

3

3.5

4

4.5

5

5.5

20 30 40 50 60 70 80

Eto

t-Erig

.rot

(M

eV)

1D LO

1D HI2Dθ HI

πp = +, πn = + minima overview

+ + 1D LO deform+ + 1D HI deform+ + 2Dθ HI

90

87

84

81

78

75

θ (d

eg)


0369

121518

20 30 40 50 60 70 80

φ (d

eg)

Ι (−h)


(a)

3

3.5

4

4.5

5

5.5

20 30 40 50 60 70 80

Eto

t-Erig

.rot

(M

eV)

1D LO2Dθ LO

2Dφ LO

3D LO

1D HI2Dθ HI

TSD total yrast

πp = +, πn = - minima overview

1D LO2Dθ LO2Dφ LO3D LO1D HI2Dθ HI

90

87

84

81

78

75

θ (d

eg)


0369

121518

20 30 40 50 60 70 80

φ (d

eg)

Ι (−h)


(b)

FIG. 11: (a) Minibands total energies, relative to a rigid rotor, and tilting angles for the minima in the πp =+, πn =+ group.The rigid rotor energy is Erig.rot = I2/182.951MeV/~2, calculated for a fixed deformation and a fixed rotational axis, so thatdifferent bands energies can be compared in the figure. Note that the θ-axis decreases upwards.(b) Minibands total energies, relative to a rigid rotor, and tilting angles for the minima in the πp =+, πn =− group. The rigidrotor energy is Erig.rot = I2/182.951MeV/~2, calculated for a fixed deformation and a fixed rotational axis, so that differentbands energies can be compared in the figure. The gap in the 1D HI band between 51 and 54~ is because ε4 during theminimization ended outside the border 0.06 of the submesh, which was also the upper limit of ε4 in the calculation.

inertia.Four kinds of minibands can be distinguished.1. Principal axis solutions, denoted 1D, which correspond to rotation about the shortest principal axis (θ = 90

and φ = 0).2. Tilted solutions denoted, 2Dφ, with tilting in the φ-direction i.e. the rotational axis is tilted from the short

principal axis towards the intermediate principal axis lying in the principal plane spanned by these axes. For thesolutions of the 2Dφ type the values of φ lies between 4 and 13, on average about 9.

3. Tilted solutions, denoted 2Dθ, with tilting in the θ-direction i.e. the rotational axis is tilted from the shortestprincipal axis towards the longest principal axis and lying in the plane spanned by these two axes. The tilting angleθ of the 2Dθ solutions have values between 76 and 88, on average 85. This is about 5 from the principal axiscranking limit θ = 90.

4. Tilted solutions, denoted 3D, which are tilted in both the φ and θ directions, i.e. the rotational axis does notlie in any principal plane. The tilting angles for the 3D solutions have values of θ between 82 and 86, i.e. 4 to 8

in the direction towards the longest principal axis, and values of φ between 9 and 15.For each of the parity combinations, the energetically lowest self-consistent solutions are for most spins of principal

16

3

3.5

4

4.5

5

5.5

20 30 40 50 60 70 80

Eto

t-Erig

.rot

(M

eV)

1D LO

2Dθ LO

1D HI

2Dφ HI

TSD total yrast

πp = -, πn = + minima overview

1D LO2Dθ LO1D HI2Dφ HI

90

87

84

81

78

75

θ (d

eg)


0369

121518

20 30 40 50 60 70 80

φ (d

eg)

Ι (−h)


(a)

3

3.5

4

4.5

5

5.5

20 30 40 50 60 70 80

Eto

t-Erig

.rot

(M

eV)

2Dθ LO

1D HI

2Dφ HI

2Dθ HI

TSD total yrast

πp = -, πn = - minima overview

2Dθ LO3D LO1D HI2Dθ HI2Dφ HI

90

87

84

81

78

75

θ (d

eg)


0369

121518

20 30 40 50 60 70 80

φ (d

eg)

Ι (−h)


(b)

FIG. 12: (a) Minibands total energies, relative to a rigid rotor, and tilting angles for the minima in the πp =−, πn =+ group.The rigid rotor energy is Erig.rot = I2/182.951MeV/~2, calculated for a fixed deformation and a fixed rotational axis, so thatdifferent bands energies can be compared in the figure. The gap in the 2Dθ LO band above spin 60~ is because X, Y or φended outside the submesh during minimization. The gap in the 1D LO band above spin 20~ is because of Y ended outsidethe submesh, or the stationary point was not positive definite. The gap between spin 45 and 48~ is because of φ ended outsidethe submesh border, or the stationary point was not positive definite. The gap between spin 61 and 68~ is because of X, Y orφ ended outside the submesh border in the minimization.(b) Minibands total energies, relative to a rigid rotor, and tilting angles for the minima in the πp =−, πn =− group. The rigidrotor energy is Erig.rot = I2/182.951MeV/~2, calculated for a fixed deformation and a fixed rotational axis, so that differentbands energies can be compared in the figure.

axis cranking type (about the shortest semi axis). In some spin intervals, the lowest solutions of a given paritycombination are tilted, e.g. - - 2Dφ for spins 39 to 56~ see fig. 12(b). In some other spin intervals, there arecoexistence between energetically lower principal axis cranking solutions and higher principal axis or tilted crankingsolutions, e.g. for the + + parity combination for spins 43 to 46~ between principal axis cranking solutions in theLO and HI regions, respectively, see fig. 11(a). Thus, in that interval there are shape-coexistences between LO andHI region deformed minima, see fig. 10(a). Also, e.g. in - + for spins about 50 to 60~ there are shape-coexistencesbetween energetically lower HI deformation region principal axis solutions and energetically higher LO region 2Dθtilted solutions, see figs. 12(a) and 10(b).

Among all parity combinations, the πp = +, πn = + minibands are TSD yrast for all spins. The TSD yrast bandconsists of three minibands. For spins ≤ 46~ the TSD yrast band is of principal axis type with deformation ε ≈ 0.39,and γ ≈ 17, i.e. in the LO region, and ε4 = 0.03 to 0.04. For spins between 47 and 69~ another principal axis bandis yrast. It has a higher deformation with ε ≈ 0.47 and triaxiality γ ≈ 23, i.e. in the HI region, and ε4 ≈ 0.04.

17

166Hf Z= 72 N= 94 πp= 1 πn= 1 Spin= 36h

80 82 84 86 88 90 92 94 96 98 100θ (deg)

−15

−10

−5

0

5

10

15

φ (d

eg)


(a)

10.3

10.4

10.5

10.6

10.7

10.8

10.9

11

11.1

0 9 18 27 36 45 54 63 72 81 90

Eto

t (M

eV)

φ (deg)

φ varied from ++ Ι=36 1D min: X=0.38,Y=0.13,ε4=0.03

Etot(φ), fix θ=90

local max

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 9 18 27 36 45 54 63 72 81 90

φ Ι-φ

(de

g)

φ (deg)

(b)

FIG. 13: (a) Total energy contours around the + + principal axis cranking solution at spin 36~. The tilting angles θ and φ arevaried, but the deformation is fixed at the mesh point X = 0.38, Y = 0.13, ε4 = 0.03 closest to the interpolated minimum.Theenergy difference between neighboring contour levels is 5 keV.(b) Variation of φ around the + + 1D minimum at spin 36~, from short axis rotation (φ = 0) to intermediate axis rotation(φ = 90). In the upper part, the total energy variation is plotted. Note the local maximum at φ = 78. As Etot is symmetricabout φ = 90, the local maximum also implies a local minimum in φ = 90, although very shallow (about 8 keV). The lowerpart shows the self-consistency measure cφ versus φ. Note how it is zero (self-consistency) at both the absolute minimum andthe local maximum and local minimum.

Finally, for very high spins from 70~ and above, the TSD yrast is the 2Dθ tilted miniband of deformation ε ≈ 0.48,γ ≈ 24 (also in the HI deformation region), and ε4 ≈ 0.03.

PROPERTIES OF THE SOLUTIONS

As a background and for comparison to the tilted axis cranking solutions (self-consistent tilted energy minima),the properties of a principal axis cranking solution will be shown here. The principal axis cranking solutions haveenergy minimum with respect to variations in both deformation and tilting around the minimum meshpoint. Theyare always self-consistent as for rotation about a principal axis the spin vector will be parallel that axis.

.As an example, the + + 1D LO solution at spin 36~, is chosen. For this solution, the deformation is X = 0.38,

Y = 0.13 (that is ε = 0.40, γ = 19) and ε4 = 0.04. The solution has total parity +, and is in fact the upperTSD minimum in fig. 1 obtained by principal axis cranking calculations of ref. [14]. If the tilting angles θ, φ arevaried around (90, 0) the yrast energies, for the same parity combination and spin, increase smoothly in both tilting

18

166Hf Z= 72 N= 94 πp=−1 πn=−1 Spin= 52 h

80 82 84 86 88 90 92 94 96 98 100θ (deg)

−15

−10

−5

0

5

10

15

φ (d

eg)


(a)

20

20.5

21

21.5

22

22.5

23

0 9 18 27 36 45 54 63 72 81 90

Eto

t (M

eV)

φ (deg)

φ varied from - - Ι=52 2Dφ min: X=0.44,Y=0.19,ε4=0.06,θ=90

20.08

20.09

20.1

20.11

20.12

0 3 6 9 12 15 18

-8

-7

-6

-5

-4

-3

-2

-1

0

1

0 9 18 27 36 45 54 63 72 81 90

φ Ι-φ

(de

g)

φ (deg)

(b)

FIG. 14: (a) Total energy contours around the - - 2Dφ solution at spin 52~ (tilted about φ = 9). The tilting angles θ and φ arevaried, but the deformation is fixed at the mesh point X = 0.44, Y = 0.19, ε4 = 0.06 closest to the interpolated minimum.Theminima at φ ≈ ±9 are marked with dots. The energy difference between neighboring contour levels is 5 keV. The tiltingenergy is about 20 keV.(b) At the deformation of the the - - 2Dφ solution at spin 52~, tilted about φ = 9, the tilting angle φ is varied and the - -I = 52~ yrast energies are plotted in the upper part. In the lower part, the self-consistency measure cφ is plotted versus φ,which shows how it smoothly passes zero (self-consistency) where the energy has its minimum.

directions, see fig. 13(a). To follow the tilting all the way from rotation about the short principal axis to rotationabout the intermediate principal axis, the tilting angle φ can be varied from 0 to 90 while the deformation is fixedat the short axis cranking minimum. The resulting total energy variation is shown in fig. 13(b). The energy increasessmoothly towards the value for intermediate axis rotation, corresponding to the lower minimum in fig. 1. The variationalso gives, in this example, a local maximum (with respect to φ) of the total energy at φ = 78, which implies a localminimum at φ = 90 as the total energy is symmetric with respect to reflection in φ = 90. Note that the barrier inthe φ direction, from the intermediate axis local minimum towards the short axis minimum, is only 8 keV, comparedto about 1MeV in deformation space as shown in fig. 1. The self-consistency measure cφ deviates from zero as thetilting increases, but at about φ ≈ 50 it bends back and intersects zero at the local maximum, and bends back againto intersect zero again at φ = 90. A similar variation, from the same + + 1D LO solution at spin 36~, of the tiltingangle θ from short axis rotation (θ = 90, φ = 0) to long axis rotation (θ = 0) results in a strictly increasing totalenergy, and the self-consistency measure cθ intersects zero only at θ = 90 and θ = 0.

For the description of the tilted axis cranking minima, the following quantity is used. As a measure of the gainin energy that the nucleus gets due to tilted axis cranking compared to principal axis cranking, we define a tiltingenergy as

Etilt = Etot(PAC)−Etot(min). (53)

19

72 PROTONS for 2Dφ min

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85φ (deg)

40

40.5

41

41.5

42

42.5

43

43.5

e i(MeV

)

ε=0.479 γ=23.356 X=0.44 Y=0.19 ε4=0.06 ω=0.454 θ=90

(a)

94 NEUTRONS for 2Dφ min

10 20 30 40 50 60 70 80φ (deg)

49.2

49.4

49.6

49.8

50

50.2

50.4

50.6

50.8

51

51.2

51.4

51.6

51.8

e i(MeV

)

ε=0.479 γ=23.356 X=0.44 Y=0.19 ε4=0.06 ω=0.454 θ=90

(b)

FIG. 15: (a) Single-proton energies versus φ, at the deformation for the - - 2Dφ solution at spin 52~ (generated at ~ω =0.454MeV), and θ = 90. The levels for positive parity states are solid, and for negative parity states dot-dashed. The Fermilevel (between particle numbers 72 and 73), is marked with a long-dashed line. The configuration for the tilted solution ismarked by a circle for the g7/2 hole below the Fermi surface, and a filled circle for the h9/2 particle above the Fermi surface.(b) Single-neutron energies versus φ, at the deformation for the - - 2Dφ solution at spin 52~ (generated at ~ω = 0.454MeV),and θ = 90. The levels for positive parity states are solid, and for negative parity states dot-dashed. The Fermi level is markedwith a long-dashed line. The configuration is marked by a circle for the i11/2 hole below the Fermi surface, and a filled circlefor the h9/2 particle above the Fermi surface.

The second term on the right-hand side is the yrast energy for a the deformation-tilting mesh point that, for a givenspin and parity combination has a strict local mesh minimum of the total energy. The first term is the correspondingprincipal axis cranking yrast energy for the same parity combination and spin. [18]

As an example from the 2Dφ tilted self-consistent solutions, the properties of the - - 2Dφ spin 52~ tilted solutionare described here. It is tilted to about φ = 9 (but θ = 90), and has deformation about X = 0.44, Y = 0.19(that is ε = 0.48, γ = 23), and ε4 = 0.05 in the interpolated minimum. It thus belongs to the HI region andthe - - 2Dφ HI miniband, see figs. 9 and 12(b). A variation of the tilting angles around the minimum, gives theenergy landscape shown in fig. 14(a). Due to the reflection symmetry in φ, there is a minimum also at φ = −9. Inthe short axis cranking point θ = 90, φ = 0 there is a saddle point; the energy decreases in the φ-directions, butincreases in the θ-directions. By fixing the deformation and θ (=90) at those of the minimum, and varying the tiltingangle φ from 0 (short axis rotation) to 90 (intermediate axis rotation), the yrast energy and the self-consistencymeasure variation in fig. 14(b) is obtained. The total energy has a smooth minimum at φ = 9, but increases allthe way to 90. The self-consistency measure intersects zero both at the minimum, and at φ = 0 and φ = 90. Ataround φ ≈ 70 there is a change in configuration for the protons causing an unsmooth and faster variation in theself-consistency measure. The tilting energy is about 20 keV. The explanation for the tilted minimum is found inthe variation of the single-particle energies with respect to φ for the occupied orbitals. In figs. 15(a) and 15(b), theproton and neutron single-particle energy levels are plotted versus φ, for a cranking frequency generating the presentspin around the minimum. Note, that cranking frequency is increasing a little as φ grows large, as the moment ofinertia decreases somewhat, so that the single-particle energy diagrams do not for high φ correspond exactly to thespin used in fig. 14(b). However, between φ = 0 and roughly 40, i.e. in a large surrounding of the minimum, thefigures match well the actual single-particle levels for the spin. From φ = 0 up to about 70 the proton configuration

20

-0.05

0.00

0.05

0.10

0.15

0.20

0 3 6 9 12 15 18 21 24 27 30

(MeV

)

φ (deg)

φ varied from - - Ι=52 2Dφ min: X=0.44,Y=0.19,ε4=0.06,θ=90

Ep

En

Ep+En

Total proton Routhian - refTotal neutron Routhian - refTotal Routhian - ref

FIG. 16: Total Routhians for protons and neutrons, and their sum, versus φ, at the deformation and θ for the - - 2Dφ solutionat spin 52~ and the cranking angular frequency ~ω = 0.45MeV generating spin 52~ at φ = 9. All curves are renormalized toenergy zero at φ = 0.

has a particle in the down-sloping level, just above the Fermi surface, which is an h9/2 orbital, and a hole in a g7/2

orbital, see the markings in the figure. Note that the proton configuration in this case needs at least one particle-holeexcitation in order to give odd total proton parity, as there are an even number of single-parity levels of odd paritybelow the Fermi surface. The vacuum configuration is the lowest but it is of even total proton parity. For the neutronconfiguration (of odd total neutron parity), see the markings in the neutron fig.15(b). The h9/2 proton particle causethe total proton Routhian to initially decrease with φ, which create a minimum in Ep at 12, but is then increasingwith φ, see fig. 16. One may ask how it is possible for one single particle to cause the minimum, when there are manyoccupied orbitals of which many give increasing terms with φ, and many give decreasing terms, to the total protonRouthian. However, the h9/2 proton decreases about 100 keV from φ = 0 to φ = 12, while the total proton Routhianat its minimum have decreased only about 35 keV. If the configuration would not have included the h9/2 proton (butinstead would have the last proton in e.g. the horizontal g7/2 level), then the total proton Routhian would have beenstrictly increasing with φ.

The total neutron Routhian is strictly increasing with φ, but initially not as fast as the total proton Routhiandecreases. The sum, i.e. the total Routhian, forms a minimum, but it is slightly shifted towards a smaller tiltingangle φ = 9, see fig. 16. The single-particle level responsible for the tilting towards the intermediate principal axisis thus a high-j level. It lies in the middle of the lower half of the proton h9/2 subshell.

Among the 2Dθ tilted self-consistent solutions, the + - 2Dθ solution at spin 33~ will be discussed here. It is tiltedabout 14 into the plane spanned by the short and the long principal axes to θ ≈ 76, and φ = 0. The deformationin the interpolated minimum is X = 0.34, Y = 0.12 (that is ε = 0.36 and γ = 19), and ε4 = 0.03, and thusbelongs to the + - 2Dθ LO miniband, see fig. 11(b). If the tilting angles are varied around the minimum, but thedeformation is fixed at that of the minimum, the energy landscape shown in fig. 17(a) is obtained. It shows minimaat θ ≈ 76, φ = 0 and at the point θ ≈ 104, φ = 0, mirrored in θ = 90. A saddle point is formed at θ = 90, φ = 0,from which the energy is decreasing in the θ-direction, but increasing in the φ-direction. A variation of θ from shortaxis rotation (θ = 90) to long axis rotation (θ = 0), results in the yrast energies and self-consistency measure shownin fig. 17(b). The total energy curve varies smoothly around the minimum, and is strictly increasing all the way tolong axis rotation. The unsmooth parts of the curve come where the configuration is changed. The self-consistencymeasure falls of below θ = 90, and is then for a wide range increasing, and intersects zero at about θ = 75.5. It turns

21

166Hf Z= 72 N= 94 πp= 1 πn=−1 Spin= 33 h

75 80 85 90 95 100 105θ (deg)

−25

−20

−15

−10

−5

0

5

10

15

20

25

φ (d

eg)


(a)

9.5

10.0

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

14.5

09182736455463728190

Eto

t (M

eV)

θ (deg)

θ varied from + - Ι=33 2Dθmin X=0.35,Y=0.115,ε4=0.03,φ=0

min

9.95

10.00

10.05

10.10

10.15

10.20

66707478828690

-10

-5

0

5

10

15

20

25

30

35

40

09182736455463728190

θ Ι-θ

(de

g)

θ (deg)

(b)

FIG. 17: (a) Total energy contours around the + - 2Dθ solution at spin 33~. The tilting angles θ and φ are varied, but thedeformation is fixed at the mesh point X = 0.44, Y = 0.19, ε4 = 0.06 closest to the interpolated minimum.The energy differencebetween neighboring contour levels is 10 keV. The energies for θ = 90 are approximated by energies for θ = 89.9.(b) Around the + - 2Dθ minimum at spin 33~, the tilting angle θ is varied and the yrast energies are plotted in the upperpart. In the zoom plot the smooth variation around the minimum is shown, as well as the rapid energy change very close toθ = 90 (closest point θ = 89.9) due to a particle-hole pair in a signature pair, as described at page 21, The lower part showsthe self-consistency measure versus θ. Note that the θ-axis decreases in the right hand direction.

back towards zero below θ ≈ 30, when the configuration changes. A configuration change is in the self-consistencymeasure in fact seen even clearer, than in the total energy curve.

The total energy changes rapidly as soon as the rotational axis is tilted out from θ = 90. Already at θ = 89.9 ithas decreased about 130 keV, see the zoom plot in fig. 17(b). This is due to a signature pair of two h11/2 single-neutronstates at 51.5MeV at θ = 90 (see fig. 18(b)), which very fast bend away from each other as soon as the θ tiltingstarts (but they are not degenerated at θ = 90 although very close). The lower level is occupied, but the upper isunoccupied. This leads to an initial gain ∆Iz = − ∂ei

∂(ω cos θ) in the z-component of the spin vector to Iz = 5.28~ atθ = 89.9 (from zero at θ = 90), and the self-consistency measure cθ = θI−θ falls to about −9. When the rotationalaxis is tilted out more, it approaches the spin vector again, and at the minimum they are parallel. The change in thetotal energy can be expressed as the change in the projection term ~ω · ~I, as the other terms vary smoothly with thetilting angles.

Etot(θ = 90, φ = 0)− Etot(θ = 89.9, φ = 0) ≈ ωI −[ω sin θ I

√1− I2

z (89.9)/I2 + ω cos θ Iz(89.9)]≈

≈ ω2I I2

z (89.9) = 12J I2

z (89.9),(54)

using a MacLaurin approximation of the square root expression. The total energy in θ = 89.9 is an approximationof the total energy in θ = 90 if the interaction in the signature pair was removed, i.e. the levels were continued

22

72 PROTONS for 2Dθ min

10 20 30 40 50 60 70 80θ (deg)

41

41.2

41.4

41.6

41.8

42

42.2

42.4

42.6

42.8

43

43.2

43.4

43.6

e i(MeV

)

ε=0.368 γ=18.189 X=0.35 Y=0.115 ε4=0.03 ω=0.32 φ=0

(a)

94 NEUTRONS for 2Dθ min

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85θ (deg)

49.5

50

50.5

51

51.5

52

52.5

53

e i(MeV

)

ε=0.368 γ=18.189 X=0.35 Y=0.115 ε4=0.03 ω=0.32 φ=0

(b)

FIG. 18: (a) Single-proton energies versus θ, at the deformation for the + - 2Dθ solution at spin 33~ (generated at ~ω =0.32MeV), and φ = 0. The levels for positive parity states are solid, and for negative parity states dot-dashed. The Fermi levelis marked with a long-dashed line. The configuration is vacuum for the protons.(b) Single-neutron energies versus θ, at the deformation for the + - 2Dθ solution at spin 33~ (generated at ~ω = 0.32MeV),and φ = 0. The levels for positive parity states are solid, and for negative parity states dot-dashed. The Fermi level is markedwith a long-dashed line. Around the minimum at θ ≈ 76, the configuration has a i13/2 hole (marked with a circle) below theFermi surface, and a h11/2 particle (filled circle) above the Fermi surface.

diabatically through θ = 90. The tilting energy is about 230 keV when compared to ordinary principal axis cranking,but when compared to the diabatic energy the gain in total energy is about 95 keV.

The tilted minimum is explained by the variation of the occupied single-particle energies with the tilting angle θ.In figs. 18(a) and 18(b) the proton and neutron single-particle energy levels are shown for a fixed cranking frequency,chosen as the one generating spin 33~ around the minimum. Note that also here, similarly to the φ-tilted example,the cranking frequency for a given spin increases, as the moment of inertia around the rotational axis decreases, whenmoving towards long principal axis rotation. But between θ = 90 and θ ≈ 50 the energy levels are consistent withspin 33~. In that interval the neutron configuration has a particle in the same down-sloping h11/2 level as above,and a hole in an i13/2 level, see the markings in fig. 18(b), while the protons have the vacuum configuration. Thedown-sloping h11/2 neutron causes the total neutron Routhian to decrease to a minimum at θ ≈ 50, see fig. 19 Asin the 2Dφ case, for a configuration without a particle in the h11/2 level (e.g. the vacuum configuration), the totalneutron Routhian would not have a tilted energy minimum. But the total proton Routhian is strictly increasing, asθ decreases. The sum, i.e. the total Routhian, gets a minimum at θ ≈ 75. The particle responsible for the rotationabout an axis tilted towards the long axis is a neutron at the top of the h11/2 subshell, consistent with [7].

As an example of the 3D tilted self-consistent solutions the + - 3D solution at spin 45~ is discussed. The interpolatedminimum is tilted about 7 in the θ direction to θ ≈ 83, and about 15 in the φ direction φ ≈ 15. The deformationis X = 0.34, Y = 0.15 (i.e. ε = 0.37 and γ = 24), and ε4 ≈ 0.03. Thus it belongs to the LO deform region infig. 9, and to the + - 3D LO miniband in fig. 11(b). If the tilting angles are varied around the minimum, the energylandscape shown in fig. 20 is obtained. It shows the minimum at θ ≈ 83, φ ≈ 15, and the corresponding tree minimaobtained by reflections in θ = 90 and φ = 0. At θ = 90, φ = 0 there is in this case a local maximum, and in thefour points in the middle between each pair of minima there are saddle points, e.g. in θ ≈ 83, φ = 0. Chirality canbe assigned to each of these four mirror minima [15], [16]. For the minimum in the tilting angle region used in thecalculations, that is θ ≈ 83, φ ≈ 15, the chirality is right-handed, because the short-intermediate and long axes canbe seen counter-clockwise from the point of the spin vector (or equivalent from the point of the cranking vector as

23

-0.5

0

0.5

1

1.5

2

2.5

09182736455463728190

(MeV

)

θ (deg)

θ varied from +- Ι=33 2Dθ min: X=0.35,Y=0.115,ε4=0.03,φ=0

Ep

En

Ep+En

Total proton Routhian - refTotal neutron Routhian - refTotal Routhian - ref

FIG. 19: Total Routhians for protons and neutrons, and their sum, versus θ, at the deformation and φ for the + - 2Dθ solutionat spin 33~ and the cranking angular frequency ~ω = 0.32MeV generating spin 33~ at the total energy minimum. The curvesare normalized to energy zero at θ = 90. The unsmoothness in the total proton curve is due to a configuration shift for theprotons at θ ≈ 40.

166Hf Z= 72 N= 94 πp= 1 πn=−1 Spin= 45 h

L

R L

R

80 82 84 86 88 90 92 94 96 98 100θ (deg)

−20

−15

−10

−5

0

5

10

15

20

φ (d

eg)


FIG. 20: Total energy contours around the + - 3D solution at spin 45~. The tilting angles θ and φ are varied, but thedeformation is fixed at the mesh point X = 0.34, Y = 0.15, ε4 = 0.03 closest to the interpolated minimum. The level differencebetween neighboring contour levels is 5 keV. The energies for θ = 90 are approximated by energies for θ = 89.9. The chiralityfor the original and mirrored solutions are marked by an R for right-handed and L for left-handed chirality.

24

15.4

15.6

15.8

16.0

16.2

16.4

16.6

0 9 18 27 36 45 54 63 72 81 90

Eto

t (M

eV)

φ (deg)

φ varied from +- Ι=45 3Dmin X=0.34,Y=0.15,ε4=0.03,θ=81,φ=15

min

local max15.42

15.43

0 4 8 12 16 20

-5

-4

-3

-2

-1

0

1

0 9 18 27 36 45 54 63 72 81 90

φ Ι-φ

(de

g)

φ (deg)

15.0

16.0

17.0

18.0

19.0

20.0

21.0

22.0

23.0

09182736455463728190

Eto

t (M

eV)

θ (deg)

θ varied from +- Ι=45 3Dmin X=0.34,Y=0.15,ε4=0.03,θ=81,φ=15

min

15.45

15.50

15.55

15.60

707478828690

-10

-5

0

5

10

15

20

25

30

09182736455463728190

θ Ι-θ

(de

g)

θ (deg)

FIG. 21: (a) Around the + - 3D minimum at spin 45~, the tilting angle φ is varied and the yrast energies are plotted in theupper part. The zoom plot shows the smooth variation around the minimum. Note the local maximum at φ = 78, whichimplies a local minimum when mirrored around 90. The lower part shows the self-consistency measure versus φ, and how itpasses zero in the stationary points.(b) Around the + - 3D minimum at spin 45~, the tilting angle θ is varied and the yrast energies are plotted in the upperpart. The zoom plot shows the smooth variation around the minimum, and the sudden decrease in total energy after a smalltilting (0.1) out from θ = 90, due to a particle-hole pair in a signature pair that splits fast with θ. The lower part shows theself-consistency measure versus θ.

they are parallel in a self-consistent solution). After a reflection in φ = 0, the short-intermediate and long axes areseen clockwise and the chirality is therefore left-handed. Similarly, starting from the original minimum and makinga reflection in θ = 90 gives a left-handed chirality. A reflection in both φ = 0 and in θ = 90 gives a right-handedchirality again. If the deformation and θ are fixed at the minimum and φ is varied from 0 to 90 (note that this is notin a principal plane), the energy and self-consistency measure curves in fig. 21(a) are obtained. There is a shallowand smooth minimum at φ ≈ 15, and for larger φ the energy increases up to a local maximum at φ ≈ 78 about13 keV above the local minimum at φ = 90. The self-consistency curve varies smoothly all the way since there isno change in configuration as φ varies. The curve intersects zero at all stationary points. If the deformation and φare fixed at the minimum, and θ is varied from 90 to 0 (neither in a principal plane), the curves in fig. 21(b) areobtained. By the same mechanism as in the 2Dθ case described above, the energy decreases rapidly as soon as θdeviates from 90, to an about 130 keV lower value already at θ = 89.9. The self-consistency curve is unsmooth forsmall θ, since the configuration changes, but intersects zero at the minimum. The diabatic tilting energy is about110 keV (compared to a tilting energy of 230 keV when the minimum energy is compared to pure short principal axiscranking, i.e. θ = 90, φ = 0). The appearance of the 3D minimum is caused by down-sloping levels of occupiedsingle-particle states in both the θ and the φ directions. The variations with φ of the single-proton and single-neutronlevels are shown in figs. 22(a) and 22(b). Note that, as before, the single-particle levels in the figures do not correspondexactly to spin 45~, but around the minimum they are consistent with the present spin. For the neutrons, which have

25

72 PROTONS for 3D min

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85φ (deg)

41

41.2

41.4

41.6

41.8

42

42.2

42.4

42.6

42.8

43

43.2

e i(MeV

)

ε=0.372 γ=23.806 X=0.34 Y=0.15 ε4=0.03 ω=0.436 θ=81

(a)

94 NEUTRONS for 3D min

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85φ (deg)

49.6

49.8

50

50.2

50.4

50.6

50.8

51

51.2

51.4

51.6

e i(MeV

)

ε=0.372 γ=23.806 X=0.34 Y=0.15 ε4=0.03 ω=0.436 θ=81

(b)

72 PROTONS for 3D min

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85θ (deg)

41

41.2

41.4

41.6

41.8

42

42.2

42.4

42.6

42.8

43

43.2

e i(MeV

)

ε=0.372 γ=23.806 X=0.34 Y=0.15 ε4=0.03 ω=0.436 φ=15

(c)

94 NEUTRONS for 3D min

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85θ (deg)

49.6

49.8

50

50.2

50.4

50.6

50.8

51

51.2

51.4

51.6

51.8

e i(MeV

)

ε=0.372 γ=23.806 X=0.34 Y=0.15 ε4=0.03 ω=0.436 φ=15

(d)

FIG. 22: (a) Single-proton energies versus φ, varied from the + - 3D solution at spin 45~ (generated at ~ω = 0.436MeV).The levels for positive parity states are solid, and for negative parity states dot-dashed. The Fermi level is marked with along-dashed line. The configuration is vacuum.(b) Single-neutron energies versus φ, varied from the + - 3D minimum at spin 45~ (generated at ~ω = 0.436MeV). Theconfiguration is vacuum, but two down-sloping i13/2 particles are marked by filled circles.(c) Single-proton energies versus θ, varied from the + - 3D minimum at spin 45~ (generated at ~ω = 0.436MeV). Theconfiguration is vacuum.(d) Single-neutron energies versus θ, varied from the + - 3D minimum at spin 45~ (generated at ~ω = 0.436MeV). Theconfiguration has a h11/2 particle which is marked by a filled circle, and an i13/2 hole marked by a circle.

26

the vacuum configuration for all φ, there are two down-sloping i13/2 neutrons, see the markings in fig. 22(b). Theycause the total neutron Routhian to decrease with φ. The protons also have the vacuum configuration, but the totalproton Routhian is increasing with φ, although initially somewhat slower than the total neutron Routhian decreases,so that their sum, the total Routhian gets a minimum at φ > 0. The i13/2 neutrons responsible for the tilting inthe direction towards the intermediate axis are in the lower half of the subshell, about a third from the its bottom.The variation of the single-particle levels with θ is shown in figs. 22(c) and 22(d). The neutron configuration hasa down-sloping h11/2 particle, and an i13/2 hole, see the markings in fig. 22(d). The h11/2 neutron cause the totalneutron Routhian to initially decrease as θ decreases. The proton configuration is from θ = 90 to θ roughly 50 thevacuum configuration. The total proton Routhian is increasing as θ decreases, but initially not as fast as the totalneutron Routhian decreases, and the total Routhian gets a minimum at θ < 90. The particle responsible for thetilting towards the long axis is an h11/2 neutron at the top of the subshell.

From the single particle levels obtained in the calculation one can suggest slightly different proton and neutronnumbers that may favor tilting more and push the tilted solutions closer to the yrast line. For the - - 2Dφ solutionat spin 52~, changing to 73 protons would bring the Fermi surface to lie just above the tilting driving h9/2 level,and the odd parity configuration would be the lowest configuration, instead of lying about 1 MeV above the yrastline, which is the case for 72 protons, see fig. 15(a). By also changing to 92 neutrons, the two up-sloping levels (justbelow the Fermi surface for 94 neutrons) would be unoccupied for the vacuum configuration, which would make thetotal neutron Routhian less resistant to tilting. Thus, 165Ta seems to be a good candidate for having low-lying tiltedrotational bands. For the + - 2Dθ solution at spin 33~, changing to 69 protons would put the last proton in thedown-sloping level 69, and emptying the up-sloping level 70, see fig. 15(b). Changing to 95 neutrons would bring theFermi surface to lie just above the tilting-driving h11/2 level. Thus, 164Tm seems to be a good candidate. For the +- 3D solution at spin 45~, one can argue in a similar way that the proton numbers 69 or 70 may be more favorableto tilting, but the neutron number 94 is already a good choice, see figs. 22(a) to 22(d). Thus, 163Tm and 164Yb mayalso have low-lying tilted rotational bands.

SUMMARY

Within the model of three-dimensional cranking of the modified harmonic oscillator with hexadecapole deformation,self-consistent tilted axis cranking solutions have been found in the triaxial superdeformed (TSD) region for 166Hf.

The tilted solutions are of three kinds. First, solutions for tilted rotation about an axis lying in the plane spannedby the short and intermediate principal axes, on average about 9 from the short axis. The gain in energy is due totilting on average about 30 keV. Second, solutions for tilted rotation about an axis lying in the plane spanned by theshort and long principal axes, typically about 5 from the short axis. The gain in energy compared to principal axiscranking about the short axis is on average about 150 keV. Third, solutions for tilted rotation about an axis that isnot lying in any of the principal planes, on average tilted out about 10 in the direction towards the intermediate axisand about 6 in the direction towards the long principal axis. The gain in energy compared to principal axis crankingis for this kind of tilted solutions about 200 keV.

The TSD yrast band consists of two principal axis rotational bands extending up to spin 69~, and for spin 70~ andhigher a tilted band of the second kind, i.e. tilted in the plane spanned by the short and long principal axes.

The tilted solutions studied are caused by high-j orbitals that are down-sloping as the tilting from the short axisincreases. The ones that cause tilting towards the intermediate axis lie in the lower half of the subshell, but not atthe bottom. The ones that cause tilting towards the long axis are at the top of the subshell.

The barrier between the principle axis cranking TSD minima at γ < 0 and γ > 0, is essentially disappearing if theyare connected with tilting in the plane spanned by the short and intermediate principal axes, than it is in deformationspace. See figs. 13(b) and 1. This implies that when including the tilting degree of freedom, the γ < 0 minimum isvery unstable, and is not likely to be observed for this nucleus.

The TSD yrast band, which is also the global yrast band for spin above 50~ (see fig. 2), is found to be a tilted axisrotational band for spin above 69~. Several other bands with tilted rotation are found at excitation energies varyingfrom about 500 keV to more than 1MeV above the TSD yrast line.

For the single-particle levels obtained in the calculations, the variation with the tilting angles suggest that changingthe proton number and/or the neutron number a few units up or down, according to values given at page 26, mayfavor tilted rotation even more by pushing the tilted bands closer to, or on the TSD total yrast line, also for lowerspin.

27

[1] H. Schnack-Petersen, et al., Nucl. Phys. A594(1995)175.[2] W. Schmitz, et al.Nucl. Phys. A539(1992)112.[3] W. Schmitz, et al.Phys. Lett. B303(1993)230.[4] R. Bengtsson, et al. Nucl. Phys. A569(1994)469.[5] S.E. Larsson, Physica ScriptaVol.8(1973)17[6] S.G. Nilsson et al., Nucl. Phys. A131(1969)[7] R. Bengtsson, Nucl. Phys. A557(1993)277c[8] V.M. Strutinsky, , Nucl. Phys. A95(1967), 420[9] G. Anderson et al., Nucl. Phys. A268(1976), 205

[10] M. Brack, International Workshop Nuclear Structure Models, 1992[11] P. Olivius, PhD. Thesis, ISBN 91-628-5966-8 (2004)[12] P. Olivius, R. Bengtsson, accepted for publication in Phys.Rev.C[13] A.K. Kerman, N. Onishi, Nucl. Phys. A 361(1981)179[14] R. Bengtsson, H. Ryde, to be published.[15] S. Frauendorf, J. Meng, Nucl. Phys. A 617(1997),131[16] S. Frauendorf, Rev. Mod. Phys. 73(2001),463[17] as 166Hf is an even A nuclei. For odd A nuclei the minimization is carried out for each half-integer spin value[18] For practical reasons, the mesh point energies are used here, rather than the interpolated minimum energies, as the latter

is for the principal axis cranking not immediately available from the calculation, if the minimum is tilted, because theprincipal axis cranking energy is not calculated for that interpolated deformation.

TUMGREPP 04-01-22, 16.106

peterolivius - lth2. peter olivius, ragnar bengtsson, peter m¨oller, cranking the folded yukawa...

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