IV. Nuclear Structure · IV. Nuclear Structure Topics to be covered include: Collective rotational...

1

IV. Nuclear Structure

Topics to be covered include:

Collective rotational model

Collective vibrational model

Shell model

Advanced shell models

Hartree-Fock

General References:

1) deShalit and Fesbach, Theoretical Nuclear Physics, Volume I:

Nuclear Structure, Chapter VI

2

Collective rotational model:

rotor Rigid small)(very MeV 10 to10 ; MeV 0.02 to01.0

)1()1()( :fit Empirical

56

22

BA

JBJJAJJE

Previously I presented the excitation spectra for nuclei which display level spacings

characteristic of a rigid rotor. These rotational bands occur in most nuclei between closed

shells where the electric quadrupole moments are large. We also observe rotational energy

level spacings built on states with non-zero angular momentum. The driving force for the

intrinsic deformation away from a spherical shape is the non-central tensor force in the N-N

interaction. Below are examples for even-even and even-odd nuclei:

3


Qualitative restriction for rotational collective

motion: the motion of the nucleons which define

the deformed shape should be rapid compared

with the rotation of the overall shape.

cv

KE

nucleon

nucleon

2.0

MeV 20

2.0MeV 940

MeV 40

Rel.)-(Non MeV 20 22

1

c

v

mvKEnucleon

MeV 0.01~ are energies typical whereMeV, 8.2 ,180For

MeV 16

10

motion)nucleon of (frequency fm, 1.2 ,2

2.0

shape deformed intrinsic, theoffrequency rotational Overall

3/13/1

0

03/1

0

A

AAr

c

rAr

c

5


Consider the Hamiltonian for a many-body system with overall rotation:

x

xy

y

0 frame Lab theinhowever

0 symmetry, From

cossin

sincos

cossin

sincos

,2,1 particlesFor

1

1

A

i

ii

A

i

ii

iii

iii

iii

iii

yx

yx

yxy

yxx

yxy

yxx

Ai

n.Hamiltonia in the separated becan scoordiante

internal and C.M. the wherecase analogous in the unlike appears, termcross that theNote

,,,)(,

form thehavemust generalin but

,)(,

where, coordiante external theand scoordinate )(intrinsic internal into separated

becannot frame Lab in then Hamiltonia theand 2

arctan2

1 general,In

,2,13,2,121,2,1

,2,121,2,1

22

AiiAiiiAiiiLab

AiiiAiiiLab

ii

ii

yxHyxHHyxH

yxHHyxH

yx

yx

Lab

6


Born-Oppenheimer Approximation:

If the body rotational frequency is small compared with the internal nucleon

frequencies, then the orientation can be treated as a parameter, rather than

dynamical, and the wave function for internal coordinates can be approximated by

system. coordinate fixed-body in the axes principal

wrt theinteria of moments and ,, where222

scoordinate fixed-body on the acts which operator momentumangular theDefine

,

n toHamiltonia thegsimplifyin thuson,perturbati

a as treatedbemay andmotion collective and intrinsic couples ,

angles.Euler e.g. Lab, in the

system fixed-body theof angles 3D theare ,

and ,

system, theof eigenvalueenergy theis where

and scoordinate

intrinsic 3D therepresents where,,

22

22

22

int21

3

321

1

32

zyxz

z

y

y

x

x

rot

k

krotiikki

i

kki

ii

iiiii

RRRT

R

TrHrHHrH

rH

r

ErHrHHH

E

EEH

rrErrHrH

JJJJJJ

7


z z

.0 where affect t doesn'

, ˆ axisarbitrary about by scoordinate lab ofRotation

particles. affect thenot doesbut frame fixed-body on the acts which operator and

, scoordinate lab on the acts which operator momentumangular , scoordinate

nucleon intrinsic on the acts which operator momentumangular theDefine

ˆ

ii

ii

Ini

ii

rIr

rren

R

rIr

J

Rotate body-fixed

axis by is equivalent to

rotating the

particles by

JRIrJR

rere

ki

ki

Jni

ki

Rni

0 gives which

ˆ)(ˆ

8


)ˆ()ˆ( harmoinc) (spherical where

)1(

:matrices-Drotation Wigner by the drepresente

topsymmetric-3D a ofly that specifical , of eigenstatean is and

of eigenvaluean isfunction waveintrinsic symmetric,axially thewhere

222

2

22222

:are inertia of moments The however).not are nuclei (some symmetricaxially is system

coordinate fixed-body in theon distributi particle intrinsic that theassume also

willI discussion For this .unphysical are and operators ;discussion in this

operators momentumangular physicalonly theare and , Operators

*

**

**

**2

222

222

222

22

22

222

2

rYDrY

KDDI

MDDI

DIIDI

T

rrJ

J

JIJIJI

RRRRRT

JR

III

IK

K

I

MKIM

k

I

MKk

I

MKz

k

I

MKk

I

MKz

k

I

MKk

I

MK

rotk

iiz

z

zz

z

z

z

z

z

yxrot

yx

z

JJJ

JJJJJ

JJJ

9


ik

I

MK

zz

z

zz

yxyx

couple

zz

z

zz

rDI

JIJIIHH

iJJJiIII

JIJIH

JIJIIJHH

*

2

222

22

int0

2

222

22

22

int

8

12

bygiven ionseigenfunct normalizedwith

222

2

isn Hamiltoniaorder -lowest The

on.perturbati small a as treatedbecan termsCoriolis and coupling theand

2

1 ,

2

1 where

222

22

:becomesn Hamiltonia The

JJJ

J

JJJJ

10


The angular momentum vector operators and their z-projections are represented as

z

z

I

J R

M

K

rotation. fixed

-body overall theoft independen ismotion

nucleon internal that themodel rotational

theoft requiremen theembodies This

thereforeand

requires which ,0 Therefore

momentum.angular ofcomponent -z

itsor function waveinternal thechange

cannot scoordinatebody theofRotation

K

zRRz

11


z

y

x

.by axis

about the rotating s,coordinatebody on the acts where

thatrequiressymmetry This axes. or

about the rotations 180 or wrt to plane, , about the

sreflection wrt symmetric isfunction waveintrinsic The

o

y

R

e

yx

yx

y

Ri y

i

JJ

J

Ji

J

J

J

Ji

i

J

J

J

Ji

i

Ji

i

Ri

k

I

KM

KI

k

I

MK

Ii

kyk

I

MK

JIi

k

I

MK

Ri

rarae

raerere

DDe

JDeDe

y

yyy

y

yyy

)1(

gives This .introduced is

function venucleon wa internal theofexpansion multipole a steplast in the where

scoordinate intrinsic for theresult out theThen work

)()1()(

on operatenot does where, )()(

:first scoordinate fixed-body for theresult out the work and scoordinate lab

and intrinsic ofn combinatio ith theoperator w fixed-body theReplace

*

,

*

*)(*

12


.0 where

)()1()(16

12)(

in resulting added, bemust functions wave

reflected and normal hesymmetry t reflection ensure To multipole.each for

)()1(8

12

gives by ofrotation or ,reflection The

. using , )1()1()1()1(

tosimplified be orecan theref phase The

integer.even an is )(2 thereforeand integer,-halfor integer either

are and en integer whan is that notingby phase heSimplify t

)()1(8

12

get weoperations theseCombining

*

,

*

2

*

,2

)(2

*

,2

K

rDrDaI

IKM

rDI

K

J

JJ

rDaI

e

J

i

J

Kk

I

KM

JI

i

J

Kk

I

MKJ

i

J

k

I

KM

JI

JIJKIJJKIJKI

i

J

k

I

KM

JKI

J

J

Ri y

13


MeV 013.02/

parameter onewith

fittedroughly are levelsenergy theall and ,,for appear bands

rotationalLu For . states internal excited different, toingcorrespond

of valuesseveralfor appear may bands rotational nuclei odd-evenFor

parameter.fit only theis inertia ofmoment the where,)1(2/by

described wellis spectrum excitation theand ,0 nuclei,even -evenFor

state intrinsic on thebuilt is spectrum rotational state excited thewhere

,2 ,1 , like goes spectrum then the, 0 state intrinsic theIf

0)(

where

2)1(2

is spectrumenergy The

exist. 4 2, 0, momentumangular totaleven, with statesOnly

)1(116

12),0,(

is w.f. theand ,0 then ,0 when case special heConsider t

2

25

29

27

177

2

22

22

int

0

0

*

02

J

JJ

J

K

K

IIE

K

KKKIK

KJIJIJI

KIIEE

I

rDI

MI

KJ

J

K

K

zzzzzz

IKM

ik

I

M

I

14


Three rotational bands in 177Lu built on three intrinsic states with K = 7/2+, 9/2, 5/2+

15


Next we will consider estimates of the moments of inertia and the relation between

the shape deformation and the electric quadrupole moments discussed in Chapter 2.

In the body-fixed reference frame the radius parameter may be expressed as

)(R

z

946.04

5

2

3 isparameter n deformatio the

16

51)2/( ,

4

51),0(

1

toreduces hissymmetry t axial with and

,1

0

00

200

220

R

R

RRRR

YRR

YRR

sphererig

NNrig

Q

ZRQ

A

MARM

JJ

J

0

0

2

00

2

052

but ,0 as 0 that Note

16.015

3

ismoment quadrupole electric ingcorrespond The number.nucleon is and

massnucleon is where31.01

density uniform and shape spheroidal thisrotor with rigid aFor

16


large toois inertia ofmoment rotor rigid that themeanswhich

KeV 5.152

is valueempirical the whereKeV 32

gives inertia ofmoment resulting The

fm 2.1 assuming 3.0cm 101.8 HfFor

exp

22

3/1

0

224

0

178

JJ

rig

ARQ

Another limiting model is the irrotational flow model where a spherical “core”

does not rotate and only the outer “bulge” flows around the core.

models. limiting esebetween th are nuclei

small; toois model in this and

KeV 602

HfFor

8

9

2178

22

0

irrot

irrot

Nirrot ARM

J

J

J

For rare earth nuclei

17


Single-particle model in a deformed binding potential – a 3D axially symmetric

harmonic oscillator:

2

20

22

0

22

0

22

0

6/1327

163

40

322

0

223

42

0

2

22222222

int

2

2222222

int

)(53

4

2

1

2

isNilsson by defined potential binding deformedn with Hamiltonia particle,-single The

)(.1

constraint thegives olumeconstant v requiring and

1 ,1

where

22

(1955)) No.16 29, Medd. Fys. Mat. (Dan.n HamiltoniaNilsson the

isresult The .splittings level ),(proper theensures potentialorbit -spin a

and shape surfaceSaxon - Woods theof effects themimicks potential an terms;

additional requireslimit 0 in the levelsenergy model-shell thereproduce To

22

DsCYrMrMM

H

const

DsCzyxM

MH

js

zyxM

MH

yxz

i

i

i

ii

i

iziix

i

i

i

iziix

i

i

18


,

2

2

023

0

22

0

22

0

,

, fixed withstatesfor dconstructe are expansions ionEigenfunct

. eigenvalue with

wherean HamiltoniNilsson full the withcommutescomponent -z

momentumangular totalThe neglected. are )( q.n. principaldifferent

between Couplings equations. coupled involves solution theand states

basis thesecouple components andorbit -spin spherical,-non The

ions.eigenfunct full theof expansionsfor states basis as serve willand

)1(

seigenvalue following thehave

2

1

2

an Hamiltoni H.O.ninteractio-central spherical, theof ionsEigenfunct

Na

N

sj

N

D

NNs

NN

NN

NNNH

rMM

H

N

zzz

z

z

19


Diagonalizing the coupled-equations

gives the single-particle energy levels

to the right and on the next slide for

nuclei with 8 < (Z,N) < 20 as a

function of deformation where the

x-axis parameter h is defined by

Positive (negative) values of h

correspond to prolate (oblate)

quadrupole deformations.

Deformations alter the energy level

ordering and hence the ground-state

spin-parities and the excited-state

spectrum of the intrinsic state upon

which rotational bands are built.

C/)(2 0 h

Nilsson states

20

Collective

rotational

model:

Nilsson states

21


A better understanding of the nuclear moments of inertia is attained by including residual pairing

interactions via the so-called “cranking” model.

A. Bohr and B. Mottelson [Mat. Fys. Medd. Dan. Vid. Selsk. 30, no.1 (1955)], using a simple

two-particle model, demonstrated that residual pairing interactions would reduce the “rigid

rotor” moment of inertia.

D. R. Inglis [Phys. Rev. 96, 1059 (1954); 97, 701 (1955)] worked out the problem of a self-

consistent, mean field potential, which determines single-particle orbitals, being externally

“cranked,” either rotationally or vibrationally.

The rotational energy is calculated as the extra energy the nucleons require to follow the slowly

rotating self-consistent potential field.

Nilsson and Prior [Mat. Fys. Medd. Dan. Vid. Selsk. 32, no.16 (1961)] applied this model to

rotational spectra and calculated nuclear moments of inertia, showing that pairing

interactions are quite capable of explaining these moments.

Aage Niels Bohr Ben Roy Mottelson

Nobel Prize in

Physics - 1975

22


splitting. massnuclear even -odd thefittingby determined is

states) unoccupied to levelenergy Fermi (from , energy gap thewhere

, 12

1 , 1

2

1

where

2

is inertia ofmoment the

,1 ,

ation transformparticle-quasiValatin -Bogoliubov theusing formalism BCS In the

form theof is nsinteractio pairingn with Hamiltonia The

operator.creation particle-single a is where0

states) H.O. ofion superposit (e.g.

:moment Quad. electric by the determined parameter n deformatiowith

potential deformed ain states particle single define Prior, andNilsson

2222

,

2

2

2

22

,

VUG

EE

VE

U

UVVUEE

j

VUVUa

VUa

aaaaGaaaaH

aa

c

x

j

j

Kj

J

2

V 2

U

23


From Nilsson and Prior:

(cases A and B correspond to

minor adjustments in the

single-particle basis states)

rigidJ

rigidJ

24

Collective vibrational model:

The other, major type of collective nuclear excitation is vibrational modes

corresponding to quantized phonon excitations in the nucleon motion. The dominant

modes are the = 0, 1, 2, and 3 multipoles corresponding to monopole (expansion –

contraction) or the so-called “breathing” mode, dipole oscillations between protons

and neutrons, quadrupole oscillations and octupole oscillations. Each is illustrated

below:

25


= 0 monopole vibration or “breathing”

mode; these states tend to be at higher

excitation energies due to the large,

nuclear compressibility.

= 1 dipole vibration between protons

and neutrons – Giant Dipole state

= 2 quadrupole vibrations, both

isoscalar (I = 0) and isovector (I = 1)

or the Giant Quadrupole, corresponding

to protons and neutrons oscillating out-of-

phase.

Vibrational excitations of deformed nuclei

also occur, where the protons and neutrons

oscillate out-of-phase in a so-called “scissors

mode” as shown to the right.

26


Vibrational excitations of deformed nuclei

also occur, where rotational bands build on

top of vibrating, deformed nuclei

vibration g vibration

27

Collective vibrational model:We invoke the same assumptions used to model rotational motion wherein the vibrational

frequencies are much smaller than the characteristic nucleon crossing frequencies, i.e. the

Born – Oppenheimer approximation.

I will only discuss the most common isoscalar quadrupole and octupole vibrations of

approximately spherical nuclei. The model was proposed by Aage Bohr (Niels’son) in

1952.

BC

theoremadditionYYY

YR

CBH

YRR

vib

/

frequency with oscillator harmonic a of that isn Hamiltonia classical above The

)1(

and , theusingscalar a ,

where, as rotationsunder transform that require wereal is order thatIn

2

1

2

1

wherems,energy ter potential and kinetic containsmotion lvibrationa

,collective theofn Hamiltonia )(classical The ry).(oscillatodependent - timeis where

),(1

time,and angleon depend willradius potential binding that theidea thestart with We

:nHamiltoniaBohr

,

*

0

*

*

22

0

28


5 to1 are states thes vibrationoctupole & quadrupolephonon -2 combinedFor

6,4,2,0 are statesphonon -2 thes vibrationoctupolefor and

4,2,0 have statesphonon -2 thes vibrationquadrupoleFor only.-even is that find we

|)1(|

tscoefficienGordon -Clebsch

forsymmetry theFrom sign. thechangenot does 21 labels inginterchang statistics BoseFor

0|,

issymmetry -Boseproper theand , momentumangular with 0 statephonon -2 The

,11,

,1,

statephonon -1 a is ,1 where,10

state uumphonon vac-no theis 0 where00

where,)1(parity and ,component -z and , momentumangular with phononsfor

ˆ

isoperator number phonon The states.boson )(spin integer for symmetry theexpressinglatter the

0,, , ,

where , operatorson annihilati andcreation phonon with replaces Quantizing

1221

,

212

21

21

21

P

P

P

I

phonon

vib

I

I

II

IMIM

aaIMMI

MIaa

nnna

nnna

a

a

aan

aaaaaa

aaH

29


phononsn

n

IM

nnE

n

theallfor degenerate are which

2

12ˆ

is multipole of phonons for levelsenergy The

21

= 2

quadrupole

= 3

octupole

0+,2+,4+

2+

0+gs

0+,2+,4+,6+

3

0+gs

30


Examples of =2

1- and 2-phonon

states

Anharmonic interactions

break the 0+,2+,4+

degeneracy

The picture for =3

1- and 2-phonon

states is messy.

The next slide shows the

excited state spectrum for

208Pb which has a

very strong 3 state.

31


1st excited states is a 3

at 2.6 MeV

208Pb – the messy real world!

Several 2,3,4,5 (but

no 1) states which may

have = 2&3 2-phonon

mixtures

Several 0+,2+,4+,6+ states at about

twice the energy of the 3

(MeV)

32

p n

1d3/2

2s1/2

1d5/2

1p1/2

1p 3/2

1s1/2

18O

If the 2 neutrons do not

interact, then they both

go into the lowest

energy 1d5/2 state.

p n

1d3/2

2s1/2

1d5/2

1p1/2

1p 3/2

1s1/2

18O

p n

1d3/2

2s1/2

1d5/2

1p1/2

1p 3/2

1s1/2

18O

ORetc.

If they interact (scatter)

then all 6 combinations of

2 neutrons in the 3 levels

of the “s-d” shell may be

populated with some

probability.

Shell Model Calculations

In conventional shell model calculations

the lower, closed shell states (core) are

treated as inert and valence nucleons

populate the higher energy levels in

multiple configurations as determined by

the two-body effective interaction. Each

configuration contributes to the nuclear

J state. We will calculate basis states

with which to expand the full wave

function, then diagonalize to find the

shell-model states and energies.

33


b

i

vali

a

i

corei

b

ji

valij

a

i

b

j

valcoreij

a

ji

coreij

b

i

vali

a

i

corei

b

i

vali

a

i

corei

A

i

i

ji

ij

A

i

i

A

i

i

ji

ijij

jigsC

gsCgs

iji

C

ji

ij

A

i

i

UUvvvUUTTH

baAba

UvUTH

gijijV

g

VH

viT

VHvTH

1

,

1

,

1

,

1 1

,

1

,

1

,

1

,

1

,

1

,

111

0

0

1

)( nucleons valenceand core , withons,contributi nucleon valence"" and core"" theseparate Next,

states. basis theof an Hamiltoni thein termpotential thisinclude and nucleons, 1)-(A

other theand nucleon one between ninteractio field-mean effective theestimates which

an Hamiltoni the topotentialbody -one effective, anct add/subtra will we thisachieve To

.components ofnumber reasonablea withconverge expansions function wavethat the

order-in practical as states real thesimilar to as be should states basis model-shell The

matrix- by the edapproximat is PEg.s. the3)(Chapter theory Bruecknerin that Recall

state. groundnuclear theof PEand KE total theare ,

potential,N -N theis , nucleon of KE theis where

by given is nucleus theof an Hamiltoni totalThe

AB,G

34


elements.matrix state lying-low other,for validalso is

result, theory Brueckner that theassume wewhere

is an Hamiltonimodel-shell The effects. theseinclude nscalculatio model shell realisticbut

termsninteractio valence core residual

eneglect th willI discussion For thisnucleons. )( valenceand )( core ofnumber

the withscale whichparts twointo separated are nucleons valencefor the potential

body-one theand into collected are nucleons core only the involving termswhere

1

,

1

,

1

,

1

,

1

,

1

,

1

,

1

,

1

,

1

,

1

,

1

,

1 1

,

1

,

1

,

1

,

1 1

,

1

,

1

,

ji

ij

gsji

ij

valcore

b

ji

eff

valij

b

i

vali

b

i

valicore

b

i

vali

b

ji

valij

b

i

vali

b

i

valicore

b

i

vali

b

ji

valij

b

i

vali

b

i

valicoreSM

b

i

vali

a

i

b

j

valcoreij

core

b

i

vali

b

i

vali

b

ji

valij

a

i

b

j

valcoreij

b

i

vali

b

i

valicore

gv

HH

vUTHUgUTH

UvUTHH

Uv

UU

H

UUvvUTHH

35


022/312/33022/112/12022/512/51

1618

11 , 22 , 11

:2 & 1 neutronsfor ,0 tocouple which nscombinatio only those be tostates

basis2n therequires which state ground 0 theCalculate shell. d-s in the levels three theofany

occupy toneutrons valence2 theallow and core Oinert an with example Ohe Consider t

part valencefor the solvemust weand

where

as written bemay function wave thensinteractio valencecore ofneglect With the

ddssdd

EH

EEEHH

EH

EHHH

valvalvalval

valcorecorevalvalcorecorevalSM

corecorecorecore

valcorevalcorevalcoreSM

valcore

p n

1d3/2

2s1/2

1d5/2

1p1/2

1p 3/2

1s1/2

18O

p n

1d3/2

2s1/2

1d5/2

1p1/2

1p 3/2

1s1/2

18O

p n

1d3/2

2s1/2

1d5/2

1p1/2

1p 3/2

1s1/2

18O

+ +

36


ts.coefficien expansion of sets three toingcorrespond

seigenstate and seigenvalue 0 for three solved is

00

00

00

equationmatrix The

element matrix thedefine and

obtain and state basisarbitrary Project

thenis nucleons valencefor the r Eq.Schrodinge The

as expanded is function wave total theand

by determined are states basis theof seigenvalue The

3

2

1

3

2

1

3

2

1

333231

232221

131211

2

1

,

2

1

,

2

1

,

02

1

,

0

0

,2,1,2,1

c

c

c

E

E

E

c

c

c

vvv

vvv

vvv

vvcEvcc

cEcvHvH

c

HUUTT

J

J

J

effeffeff

effeffeff

effeffeff

k

ji

eff

valijm

eff

mkmJk

k

ji

eff

valijmkmm

m

k

kkJk

kk

ji

eff

valijvalJji

eff

valijval

i

iiJ

iiivalivalvalvalval

1

2

3

0+

0+

0+

37


State-of-the-art shell model calculations attempt to improve the effective

interactions, the mean potential used to determine the basis states, the

estimates of core-valence interactions, and to minimize the size of the core,

including “no-core” shell models, and to expand the set of basis states.

Modern calculations use harmonic oscillator basis states in order to greatly

speed up the calculation of the effective interaction matrix elements.

However, the H.O. states are rather poor estimates of the nominal Woods-

Saxon potential eigenstate basis, so many H.O. levels are required to obtain

realistic results.

Including sufficient H.O. levels and all the accompanying total, orbital, and z-

component angular momentum states, plus the total and z-component iso-spin

states, and then including all the n-body valence nucleon configurations

which may contribute to the set of nuclear J states to be described requires of

order 109 basis states and must be done on dedicated super-computers.

38

Shell Model

Calculations

Vintage shell-model

calculations:

Glaudemans et al. Nucl. Phys.

56, 529 and 548 (1964).

39


Includes up to 10 quasi-particle basis states

40

Shell Model

Calculations

41

Advanced Shell Model Calculations (as used in 2 and 0 decay transitions)

The calculation of nuclear transition matrix elements for use in 2-neutrino and 0-neutrino

double-beta decay is a topic of present day research. See for example: Horoi and Neacsu,

Phys. Rev. C 93, 024308 (2016). Because the relevant nuclear isotopes for these studies

are neither light, nor closed-shell, nor ideal collective nuclei which rotate or vibrate, the

nuclear structure aspects of these transitions is complex. Multiple approaches are given in

the literature. The above paper lists the following general categories with several

references for each. These include:

Interacting Shell Model (ISM)

Quasi-particle Random Phase Approximation (QRPA)

Interacting Boson Model (IBM-2)

Projected Hartree-Fock Bogoliubov (PHFB)

Energy Density Functional (EDF) – not discussed

Relativistic Energy Density Functional (REDF) – not discussed

The ISM calculations seem to be regular shell model calculations as described in the

previous slides but with varying cores, shell configurations and effective interactions.

For the other models I will only describe those approaches in general terms.

42

Advanced Shell Model Calculations

A basic difference in the nuclear transition matrix elements for 0 and 2

involves the role of intermediate nuclear states. For 0 the “closure” approximation

(see below) suffices [see, Mustonen & Engel, PRC 87, 064302 (2013) and Pantis and

Vergados, Phys. Lett. B 242, 1 (1990)] while for 2 intermediate nuclear states must

be summed explicitly. The importance of intermediate states in 2 prevents using

measured 2 rates to determine the NME for 0.

if

nn

ba

MME

innf

ba baba

closure

n

baba

n fin

b bba aa

,

0

,,

2

tosimplifiesnumerator theand ion,approximat closure In the

., nucleonsfor operatorsin spin/isosp are and

ns, transitiodominated GTfor 2/

M

M

1,

0

Intermediate

states in 150Pm

populated in

2

This difference

applies to all

methods.

43

Advanced Shell Model Calculations

ISM: Interacting Shell Model

Retamosa et al. PRC 51, 371 (1995)48Ca to 48Ti double-beta decay

Kuo & Brown effective interaction

Richter (RVJB) effective interaction

40Ca core

8 neutrons in

1f7/2 – 2p1/2 valence levels

44

Advanced Shell Model Calculations ISM:

Caurier et al. PRL 100, 052503 (2008)

A = 76, 82 with {2p3/2, 1f5/2, 2p1/2, 1g9/2}

valence basis states; 56Ni core.

G-matrix effective interaction;

veff tuned to fit the low-lying spectrum in

each isotope

A = 124, 128, 130, 136 with

{1g7/2, 2d5/2, 2d3/2, 3s1/2, 1h11/2} basis

Of order 1010 dimensional matrix!

56Ni core

4 valence

orbitals

5 valence

orbitals

100Sn core

45

Advanced Shell Model Calculations – Quasi-particle Random Phase Approximation

In the random phase approximation (RPA) 2p-2h matrix elements are simplified;

simultaneous 2p-2h excitations are neglected and are reduced to sums of 1p-1h

“active” excitations times “inert” ground-state averages of the other particle-hole

excitation in the pair. An arbitrary 2p-2h excitation is driven by effective

interactions containing the operator which excites 2 nucleons from

orbitals r,s to p,q as in the diagram. The essence of the RPA is to write this

operator as:

srqp aaaa

elements.matrix state-ground are etc. , thewhere

rq

rqspsqrprpsqsprqsrqp

aa

aaaaaaaaaaaaaaaaaaaa

p n

p

q

r

s

p n

p

q

r

s

Mustonen and Engel, PRC 87, 064302 (2013) is a deformed, self-consistent, Skyrme QRPA

applied to 76Ge, 130Te, 136Xe, 150Nd: deformed HF basis, Skyrme contact veff

46

Advanced Shell Model Calculations – Interacting Boson Model (IBM)

In the IBM valence protons and neutrons in even-even nuclei are paired (pp and nn) into

integer spin objects (bosons) which are, in turn, allowed to interact and excite into boson

plus boson-hole states. Lower energy levels are treated as an inert core as in the SM. For

example, 128Xe (Z = 54, N = 74) is near closed shells 50 and 82 with 4 valence protons

forming 2 pp bosons and 8 valence neutron holes forming 4 nn bosons. The bosons are

assumed to be in either L = 0 (s) or 2 (d) states only.

The IBM Hamiltonian using s-boson and d-boson creation and annihilation operators

contains 9 terms with corresponding fitting parameters and is given by

.1

~ where

2

~

~~

2

~~~

2

~~12

~

0)0()0(

00

)2()2(

2

0)0()0()0()0(

0

0)2()2()2()2(

2

0

4,2,0

21

dd

ssssU

sdsdU

ddssssddV

ddsdsdddV

ddddCLddssHL

LL

Lds

With a fixed set of parameters the solutions, as a function nuclear Z,N, range from

rotational to vibrational.

47

Advanced Shell Model Calculations – Interacting Boson Model (IBM)

For example, Barea and Iachello, PRC 79, 044301 (2009) apply the IBM to

0 NME for 76Ge, 82Se, 100Mo, 128Te, 130Te, 136Xe, 150Nd, 154Sm

48

Advanced Shell Model Calculations -- Projected Hartree-Fock Bogoliubov

See e.g. Rath et al., PRC 88, 064322 (2013); PRC 82, 064310 (2010) applied to 94,96Zr, 98,100Mo, 104Ru, 110Pd, 128,130Te, and 150Nd

Hartree-Fock models will be discussed later. In a nutshell HF solutions are self-

consistent solutions of the many-body Hamiltonian with effective interactions in

which the potentials are determined by the effective two-body interactions

folded with the single-particle states, and the states are determined by solving

the Sch.Eq. with those potentials. HF requires an iterative solution. Brueckner

HF uses the g-matrix interaction; HF Bogoliubov uses quasi-particle basis

states. The PHFB uses deformed basis states with = 0, 2, 4 multipole

components of the effective two-body interaction. The PHFB Hamiltonian is

),(

andparameter coupling a is where4,2for

)1()(

are sexcitation2h -2pfor nsinteractio multipole The

potentials particle-single and KE theincludes particle)-(single

,

g

g

g

Yrq

C

aaaaqqCV

UH

VVVHH

isp

lehexadecupoquadpairingsp

49

Hartree-Fock Models

The basic idea is to find a method to optimize the single particle states and single-particle

binding potential such that the effects of the two-body interactions on the nuclear

eigenstates are accounted for with a simple, single particle model. Start with a simple,

product form for the A-body wave function where anti-symmetrization is ignored. This

will give the Hartree potential.

0)()()()1(2

1)()1(

:constraint

ion normalizat with condition, al variation the write tosmultiplier Lagrange Use. statein of

eigenvalue the toscorrespondelement matrix above theachieved isty stationari

Whents.requiremenion normalizat above thegmaintainin whilein variations

respect to with stationary is such that ofset best thefind and

constructThen

. statebody -many toingcorrespond orbitalslowest theup fill states particle single and

systembody -A theof eigenstatean denotes ,1 where)()2()1()2,1(

1

3*3

1

3

1

**

1

3*

11

21

A

k

kkkkA

ji

ij

A

i

i

ji

ij

A

i

i

rdrrrdrdAvTA

H

H

vTH

rdAA

kkAA

i

A

50

Hartree-Fock Models

slide.next on the isfactor this where,0 require we where0)( :form theof iswhich

0)()(

)()1()1()()1()1()()(

)()()1()1(2

1)1()1(

0)()(

)()1()1(2

1)()1()1(

zero. befactor tivemultiplica its that requires )( variationsarbitrary for validbe toeqn. above For the

. vary weif obtained are results same The t.independen considered are and of variations

and constraint multiplier Lagrange one requiresonly function -state single a of variationwhere

0)()(

)()1()1(2

1)()1()1(

toreduces eqn. preceding then the,for that state, specific one ofariation Consider v

3*

3*

1

1

3

1

3***3*

3*

1

3

1

31

1

1

1

**

3*

3

1

31

1

1

1

1

1

***

*

*

3*

3

1

3

1

***

1111

1111

1111

1111

XXrdr

rdrr

rdrdAAvAArdATA

rdAArdrdAvTA

rdrr

rdrdAAvvTTAA

A

rdrr

rdrdAAvTAA

Ai

AA

AAAA

A

j

AAjAA

AA

A

ji

ij

A

i

i

AAAA

A

A

j

Aj

A

ji

ijA

A

i

i

AAAA

A

ji

ij

A

i

i

A

AA

AAAAAA

AAAA

AA

AAAA

A

AAA

AA

AAAA

51

Hartree-Fock Models

restored. is

ityorthogonal that find and functions wavericantisymmetconsider will weNext, solution. Hartree theis

energy, andfunction body wave-A resulting The

.orthogonalnot are states n theseHamiltoniadifferent a of eigenstatean is state particle

singleeach Since converge. results theuntilon so and new a then and potential, new a calculate

,for eq. Sch. thesolve ,)( potential particle single theguess tois followed procedure The

nucleons. )1(other all with ginteractin

nucleon by seen that is )( potential The nucleons. A}{1,2 allfor results eq. Sch. same The

0)(

0)()1()1()1()1(

in resulting , with combined

andconstant aby replaced becan andA #nucleon on dependnot does integral second The

0)()1()1(2

1)1()1(

)1()1()1()1(

1

1

3

1

31

1

**

1

3

1

31

1

1

1

**

1

3

1

31

1

**

1111

1111

1111

A

i

i

ii

ii

AAAA

AAA

A

j

AjA

A

AA

A

ji

ij

A

i

i

AA

A

j

AjA

E

rU

A

irU

rUT

rrdrdAvAT

rrdrdAvTA

rdrdAvAT

i

i

A

AAA

AAA

AA

52

Hartree-Fock Models

For a many-body wave function of identical fermions, antisymmetrization

requirements can be satisfied with a Slater determinant form given by

)()2()1(

)()2()1(

)()2()1(

!

1),2,1(

where

0),2,1(2

1)2,1(

bygiven is orbitalarbitrary toappliedcondition al variationThe labels.nucleon are

A1,2 and ,,,, e.g. numbers, quantum allrepresent subscripts where

)()2()1(

)()2()1(

)()2()1(

!

1),2,1(

0

3*3

1

3

0

1

*

0

3

0

A

A

A

AA

rdrdrdAvTA

mjN

A

A

A

AA

A

ji

ij

A

i

i

j

53

Hartree-Fock Models

equal.similarly are 4th terms and 3rd The above. first term theis which )2()1()2()1(

)1()2()1()2()1()2()1()2()2()1()2()1(

obtain and , symmetric, isn interactio that thenote

andenergy theofsign thechangenot does which term,second in the 21 labels Exchange

)2()1()2()1()2()1()2()1(

)2()1()2()1()2()1()2()1(2

1

)2()1()2()1()2()1()2()1(!2

1)2,1()2,1(

example 2A with thestarting n terminteractio out the work Similarly,

)2()2()1()1()1()1(

is this2AFor

arbitrary. are labelsnucleon where)2()2()1()1()1()1(

)2()1()2()1()2()1()2()1(!2

1)2,1()2,1(

examplean as case 2A thedoing term,KE out theWork

12

122112

2112

1212

1212

120120

,

210

1

0

21

210210

v

vvv

vv

vv

vv

vv

TTT

TT

TTTT

A

i

i

54

Hartree-Fock Models

ry.antisymmet of econsequenc a potential, exchange"" theis 3rd and potential direct"" theis term2nd The

)1()1()2()2()1()2()2()1(

equationr Schrodinge channels-coupled in the results zero to)1(on actsch factor whi

tivemultiplica theSetting before. as multiplier Lagrange with thecombined andconstant awith

replaced becan and function wave varied theinvolvenot does termsecond in thesummation The

0)1()1()2()1()2()1()2()1()2()1(

)2()2()1()1()1()1(

gives termsall Combing

)2()1()2()1()2()1()2()1(

in resulting n,restrictio

theremoves which )2()1()2()1( term, ct theadd/subtramay We

)2()1()2()1()2()1()2()1(

get wegeneralin and

)2()1()2()1()2()1()2()1()2,1()2,1(

Therefore

12121

1212

,

21

121200

12

,

121200

12120120

vvT

vv

TT

vvv

v

vvv

vvv

ji

ij

ji

ij

55

Hartree-Fock Models

A self-consistent solution is obtained as above where an initial guess for the direct

and exchange potentials and solution of the coupled-equations gives the initial,

trial set of single particle states. Then new potentials are calculated from the

matrix elements, a new set of states is obtained and so on.

Typical, effective interactions

for the direct and exchange

terms are shown here, from

Negele, Phys. Rev. C1, 1260

(1970).

56

Hartree-Fock Models

Example Hartree-Fock charge densities calculated by Negele, PRC 1, 1260 (1970)

are shown below.

Data

Hartree-Fock

57

This concludes Chapter IV:

Nuclear Structure

IV. Nuclear Structure · IV. Nuclear Structure Topics to be covered include: Collective rotational...

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Transcript of IV. Nuclear Structure · IV. Nuclear Structure Topics to be covered include: Collective rotational...