nuclear structure collective models...

Nuclear Structure�(II) Collective models�

P. Van Isacker, GANIL, France �

NSDD Workshop, Trieste, March 2014

TALENT school�TALENT (Training in Advanced Low-Energy Nuclear

Theory, see http://www.nucleartalent.org).�Course 5: Theory for exploring nuclear structure

experiments.�Lecturers: M.A. Caprio, R.F. Casten, J. Jolie, A.O.

Macchiavelli, N. Pietralla and P. Van Isacker. �Audience: Master of Science and PhD students,

and early post-doctoral students.�Place: GANIL. Dates: 11-29 August 2014.�

Collective nuclear models�Rotational models�Vibrational models�Liquid-drop model of vibrations and rotations�Interacting boson model�

Rotation of a symmetric top �Energy spectrum: ��Large deformation ⇒ large ℑ ⇒ low Ex(2+).�R42 energy ratio: �

€

E rot I( ) =2

2ℑI I +1( )

≡ A I I +1( ), I = 0,2,4,…

€

E rot 4+( ) /E rot 2

+( ) = 3.333…

Evolution of Ex(2+)�

J.L. Wood, private communication �

The ratio R42 �

Rotation of an asymmetric top �Energy spectrum: ��Reflection symmetry only allows even I with positive parity π.��

3

33

• At low spins 222Ra has an octupole vibrational spectrum; the I+3 negative parity states are one ht higher in energy than the even parity states with spin I.

• At "high-spins" the spectrum for 222Ra shows a 6I=1 sequence

with alternating parity; consistent with a stable octupole deformation, and similar to the reflection- asymmetric rotor, HCl.

Reflection AsymmetricShapes in Nuclei

3

33





Erot Iπ( ) =

2

2ℑI I +1( )

I π = 0+,1−, 2+,3−, 4+,…

Nuclear shapes�Shapes are characterized by the variables αλµ in

the surface parameterization: ��

λ=0: compression (high energy)�λ=1: translation (not an intrinsic deformation)�λ=2: quadrupole deformation �λ=3: octupole deformation ��

€

R θ,ϕ( ) = R0 1+ αλµYλµ* θ,ϕ( )

µ=−λ

+λ

∑λ

∑(

) * *

+

, - -

Quadrupole shapes�Since the surface R(θ,φ) is real: �è Five independent quadrupole variables (λ=2).�Equivalent to three Euler angles and two intrinsic

variables β and γ:

αλµ( )*= −1( )µαλ−µ

α2µ = a2νDµν2 Ω( )

ν

∑ , a21 = a2−1 = 0, a22 = a2−2

a20 = β cosγ, a22 =12β sinγ

The (β,γ) plane�

Modes of nuclear vibration �Nucleus is considered as a droplet of nuclear

matter with an equilibrium shape. Vibrations are modes of excitation around that shape.�

Character of vibrations depends on symmetry of equilibrium shape. Two important cases in nuclei: �Spherical equilibrium shape�Spheroidal equilibrium shape�

Vibrations about a spherical shape�Vibrations are characterized by λ in the surface

parameterization: �λ=0: compression (high energy)�λ=1: translation (not an intrinsic excitation)�λ=2: quadrupole vibration ��λ=3: octupole vibration ��

⇔ ⇔

Spherical quadrupole vibrations�Energy spectrum: ��R42 energy ratio: ��E2 transitions: �

€

Evib n( ) = n + 52( )ω, n = 0,1…

€

Evib 4+( ) /Evib 2

+( ) = 2

€

B E2;21+ → 01

+( ) =α 2

B E2;22+ → 01

+( ) = 0

B E2;n = 2→ n =1( ) = 2α 2

Example of 112Cd�

Possible vibrational nuclei from R42�

Spheroidal quadrupole vibrations �The vibration of a shape

with axial symmetry is characterized by aλν.�

Quadrupole oscillations: �ν=0: along the axis of

symmetry (β)�ν=±1: spurious rotation �ν=±2: perpendicular to axis

of symmetry (γ)�

βγ⇔

γ⇔

β

Spectrum of spheroidal vibrations�

Example of 166Er�

Quadrupole-octupole shapes�It is difficult to define an intrinsic frame for a pure octupole shape.��Quadrupole-octupole: use quadrupole frame è two quadrupole and seven octupole intrinsic variables α3µ.�Most important case: β3=α30 (axial symmetry).��

formed) are discussed below in Sec. II.A (isoscalar mo-ments) and Sec. II.B (isovector electric dipole moment).

A. Reflection-asymmetric shapes

In many applications, the nuclear shape is param-etrized in terms of a spherical harmonic (multipole) ex-pansion. The spheroidal nuclear surface is defined bymeans of standard deformation parameters a

lm

describ-ing the length of the radius vector pointing from theorigin to the surface (Bohr, 1952; Hill and Wheeler,1953):

R~

V

!

5c~

a

!

R0F11(

l52

lmax

(

m52l

1l

a

lm

Ylm

*~

V

!G , (1)

with c(a) being determined from the volume-conservation condition and R0=r0A1/3. The requirementthat the radius be real imposes the condition

~

a

lm

!

*5~

2!

m

a

l2m

. (2)

The three dipole deformations, a161 and a10 , aregiven by the constraint that fixes the center of mass(c.m.) at the origin of the body-fixed frame:

EV

rd3r50, (3)

where V is the total volume enclosed by the surface de-fined in Eq. (1). For shapes axially symmetric with re-spect to the z axis, all deformation parameters with m

fi 0 vanish. The remaining deformation parameters a

l0are usually called b

l

:

b

l

[a

l0 . (4)

For a well-defined axial octupole minimum in the to-tal energy, the intrinsic charge octupole moment is

Q 30,c[E 2r3P30rc~r!

d3r, (5)

where rc(r) is the charge density. Assuming rc=constinside the sharp surface of Eq. (1), Q 30,c can be relatedto the deformations b

l

by (Leander and Chen, 1988)

Q 30,c53

A7p

ZR03b3 , (6)

where

b35b315

A4p

S 4A515

b2b31611

b3b4

160A7

91A11b4b51••• D . (7)

The mass octupole moment and corresponding defor-mations are defined in a similar way.

There exist several parametrizations of reflection-asymmetric shapes other than the a and b parametriza-tions discussed above. A compilation of other param-etrizations is contained in the Appendix.

The number of deformation parameters that appear inthe multipole expansion Eq. (1) grows rapidly with l .For instance, the general quadrupole-plus-octupoleshape is described by two quadrupole deformations(a20 and a22 , or b2 and g) and seven independent de-formations a3m

. Figure 2 displays four shapes resultingfrom the superposition of axial quadrupole and octupoledeformations with m=0, 1, 2, and 3.

A general parametrization of the combinedquadrupole-octupole field, covering all possible shapeswithout double counting, was proposed by Rohozinski(1990). The basic requirements are that the parametri-zation obeys simple transformation rules under O h (agroup of 48 transformations changing the names and ar-rows of the axes), and have simple ranges for the param-eters. After introducing the seven real Cartesian compo-nents a3m

and b3m

(Rohozinski et al., 1982; Rohozinski,1988),

a305a30 , a36m

5~

61!

m

A2~

a3m

6ib3m

!

~

m51,2,3!

,

(8)

FIG. 2. Quadrupole-octupole shapes represented by multipoleexpansion, Eq. (1). In all cases, the same axial quadrupole de-formation a20=b2=0.6 is assumed. The four shapes correspondto octupole deformations with m=0, 1, 2, and 3

@

a30=b30 ;a32m

=(21)m

a3m

=b3m

/2; b3m

=0.35]. (Courtesy of T. Misu.)

351Butler and Nazarewicz: Intrinsic reflection asymmetry

Rev. Mod. Phys., Vol. 68, No. 2, April 1996

formed) are discussed below in Sec. II.A (isoscalar mo-ments) and Sec. II.B (isovector electric dipole moment).

A. Reflection-asymmetric shapes

In many applications, the nuclear shape is param-etrized in terms of a spherical harmonic (multipole) ex-pansion. The spheroidal nuclear surface is defined bymeans of standard deformation parameters a

lm

describ-ing the length of the radius vector pointing from theorigin to the surface (Bohr, 1952; Hill and Wheeler,1953):

R~

V

!

5c~

a

!

R0F11(

l52

lmax

(

m52l

1l

a

lm

Ylm

*~

V

!G , (1)

with c(a) being determined from the volume-conservation condition and R0=r0A1/3. The requirementthat the radius be real imposes the condition

~

a

lm

!

*5~

2!

m

a

l2m

. (2)

The three dipole deformations, a161 and a10 , aregiven by the constraint that fixes the center of mass(c.m.) at the origin of the body-fixed frame:

EV

rd3r50, (3)

where V is the total volume enclosed by the surface de-fined in Eq. (1). For shapes axially symmetric with re-spect to the z axis, all deformation parameters with m

fi 0 vanish. The remaining deformation parameters a

l0are usually called b

l

:

b

l

[a

l0 . (4)

For a well-defined axial octupole minimum in the to-tal energy, the intrinsic charge octupole moment is

Q 30,c[E 2r3P30rc~r!

d3r, (5)

where rc(r) is the charge density. Assuming rc=constinside the sharp surface of Eq. (1), Q 30,c can be relatedto the deformations b

l

by (Leander and Chen, 1988)

Q 30,c53

A7p

ZR03b3 , (6)

where

b35b315

A4p

S 4A515

b2b31611

b3b4

160A7

91A11b4b51••• D . (7)

The mass octupole moment and corresponding defor-mations are defined in a similar way.

There exist several parametrizations of reflection-asymmetric shapes other than the a and b parametriza-tions discussed above. A compilation of other param-etrizations is contained in the Appendix.

The number of deformation parameters that appear inthe multipole expansion Eq. (1) grows rapidly with l .For instance, the general quadrupole-plus-octupoleshape is described by two quadrupole deformations(a20 and a22 , or b2 and g) and seven independent de-formations a3m

. Figure 2 displays four shapes resultingfrom the superposition of axial quadrupole and octupoledeformations with m=0, 1, 2, and 3.

A general parametrization of the combinedquadrupole-octupole field, covering all possible shapeswithout double counting, was proposed by Rohozinski(1990). The basic requirements are that the parametri-zation obeys simple transformation rules under O h (agroup of 48 transformations changing the names and ar-rows of the axes), and have simple ranges for the param-eters. After introducing the seven real Cartesian compo-nents a3m

and b3m

(Rohozinski et al., 1982; Rohozinski,1988),

a305a30 , a36m

5~

61!

m

A2~

a3m

6ib3m

!

~

m51,2,3!

,

(8)

FIG. 2. Quadrupole-octupole shapes represented by multipoleexpansion, Eq. (1). In all cases, the same axial quadrupole de-formation a20=b2=0.6 is assumed. The four shapes correspondto octupole deformations with m=0, 1, 2, and 3

@

a30=b30 ;a32m

=(21)m

a3m

=b3m

/2; b3m

=0.35]. (Courtesy of T. Misu.)

351Butler and Nazarewicz: Intrinsic reflection asymmetry

Rev. Mod. Phys., Vol. 68, No. 2, April 1996

Quadrupole-octupole vibrations�

Octupole rotation-vibrations�

3

33





3

33





3

33





3

33





Octupole rotation-vibrations�

3

33





3

33





Example: 222Ra �

3

33





Example: 220Rn and 224Ra �

experimental features: shell-corrected liquid-drop models10,11, mean-fieldapproaches using various interactions12–16, models that assume a-particle clustering in the nucleus17,18, algebraic models19 and other semi-phenomenological approaches20. A broad overview of the experimentaland theoretical evidence for octupole correlations is given in ref. 21.

In order to determine the shape of nuclei, the rotational model canbe used to connect the intrinsic deformation, which is not directly

observable, to the electric charge moments that arise from the non-spherical charge distribution. For quadrupole deformed nuclei, thetypical experimental observables are the electric-quadrupole (E2)transition moments that are related to the matrix elements connectingdiffering members of rotational bands in these nuclei, and E2 staticmoments that are related to diagonal matrix elements for a single state.If the nucleus does not change its shape under rotation, both typesof moments will vary with angular momentum but can be related to aconstant ‘intrinsic’ moment that characterizes the shape of the nucleus.For pear-shaped nuclei, there will be additionally E1 and electric-octupole (E3) transition moments that connect rotational states havingopposite parity. The E1 transitions can be enhanced because of the sepa-ration of the centre-of-mass and centre-of-charge. The absolute valuesof the E1 moments are, however, small (,1022 single particle units)and are dominated by single-particle and cancellation effects9. In con-trast, the E3 transition moment is collective in behaviour (.10 singleparticle units) and is insensitive to single-particle effects, as it is gene-rated by coherent contributions arising from the quadrupole-octupoleshape. The E3 moment is therefore an observable that should providedirect evidence for enhanced octupole correlations and, for deformednuclei, can be related to the intrinsic octupole deformation parameters22.Until the present measurements, E3 transition moments have beendetermined for only one nucleus in the Z < 88, N < 134 region, 226Ra(ref. 23), so that theoretical calculations of E3 moments in reflection-asymmetric nuclei have not yet been subject to detailed scrutiny.

Experiments and discussionCoulomb excitation is an important tool for exploring the collectivebehaviour of deformed nuclei that gives rise to strong enhancementof the probability of transitions between states. Traditionally, thistechnique has been employed by exciting targets of stable nuclei withaccelerated ion beams of stable nuclei at energies below the Coulombbarrier, ensuring that the interaction is purely electromagnetic incharacter. Whereas E2, E1 and magnetic dipole (M1) transition pro-babilities dominate in the electromagnetic decay of nuclear states, andhence can be determined from measurements of the lifetimes of thestates, E2 and E3 transition moments dominate the Coulomb excita-tion process allowing these moments to be determined from measure-ment of the cross-sections of the states, often inferred from the c-raysthat de-excite these levels. In exceptional cases, the Coulomb excita-tion technique has been applied to radioactive targets like 226Ra,which is sufficiently long-lived (half-life 1,600 yr) to produce a mac-roscopic sample. It is only comparatively recently that the techniquehas been extended to the use of accelerated beams of radioactive nucleisuch as those from the Radioactive beam EXperimental facility atISOLDE, CERN (REX-ISOLDE24). In the experiments described here,

Energy (keV)

Cou

nts

per k

eV

10–1

1

10

102

103

104

105

106

10–4

10–3

10–2

10–1

1

10

102

103

104

105

X-rays

2+ →

0+

4+ →

2+

6+ →

4+

1– →

0+

1– →

2+

3– →

2+

5– →

4+

2 γ+ →

2+

Pile

up o

f 2+ →

0+

and

4+ →

2+

2 γ+ →

0+

a

*

Energy (keV)

Cou

nts

per k

eV

1

10

102

103

104

105

10–3

10–2

10–1

1

10

102

103X-rays

b

2+ →

0+

4+ →

2+

1– →

2+

5– →

4+

3– →

2+

1– →

0+ 6+ →

4+

8+ →

6+

Counts per keV

100 200 300 400 500 600 700 800 900 1,000

Counts per keV

50 100 150 200 250 300 350

Figure 1 | Representative c-ray spectra following the bombardment of2 mg cm22 60Ni and 120Sn targets by 220Rn and 224Ra. a, 60Ni (blue) and 120Sn(red) bombarded by 220Rn; b, targets as a but with bombardment by 224Ra. Thedifferences in excitation cross-section for the targets with different Z areapparent for the higher spin states. The c-rays are corrected for Doppler shiftassuming that they are emitted from the scattered projectile. The asterisk ina marks an unassigned, 836(2) keV transition. A state at 937.8(8) keV isassigned Ip 5 21 on the basis of its excitation and decay properties; it isassumed to be the bandhead of the c-band in 220Rn.

84.4

131.6216.0

166.4

74.4205.9

142.6182.3228.4

161.7 207.7

113.8

275.7

151.5265.3

312.6

965.5

241.0

292.7404.2

645.4

422.0

188.8318.3340.2

254.3 276.2370.4

937.8

0

2+ 84

4+ 251

6+ 479

8+ 755

10+ 1067

1–216

3–290

5–433

7–641

9–9062+ 966

0+ 0

2+ 241

4+ 534

6+ 874

8+ 1244

645

3–663

5–852

7–1128

2+ 938

881.1

a b220Rn

1–696.9

224Ra

0+

Figure 2 | Partial level-schemes for 220Rn and 224Ra, showing the excitedstates of interest for this work. a, 220Rn; b, 224Ra. Arrows indicate knownc-raytransitions. All energies are in keV (upright font refers to transition energies,

italic font refers to state energies) and spins in units of B. Note that the level at938 keV in 220Rn is observed for the first time in this work.

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experimental features: shell-corrected liquid-drop models10,11, mean-fieldapproaches using various interactions12–16, models that assume a-particle clustering in the nucleus17,18, algebraic models19 and other semi-phenomenological approaches20. A broad overview of the experimentaland theoretical evidence for octupole correlations is given in ref. 21.

In order to determine the shape of nuclei, the rotational model canbe used to connect the intrinsic deformation, which is not directly

observable, to the electric charge moments that arise from the non-spherical charge distribution. For quadrupole deformed nuclei, thetypical experimental observables are the electric-quadrupole (E2)transition moments that are related to the matrix elements connectingdiffering members of rotational bands in these nuclei, and E2 staticmoments that are related to diagonal matrix elements for a single state.If the nucleus does not change its shape under rotation, both typesof moments will vary with angular momentum but can be related to aconstant ‘intrinsic’ moment that characterizes the shape of the nucleus.For pear-shaped nuclei, there will be additionally E1 and electric-octupole (E3) transition moments that connect rotational states havingopposite parity. The E1 transitions can be enhanced because of the sepa-ration of the centre-of-mass and centre-of-charge. The absolute valuesof the E1 moments are, however, small (,1022 single particle units)and are dominated by single-particle and cancellation effects9. In con-trast, the E3 transition moment is collective in behaviour (.10 singleparticle units) and is insensitive to single-particle effects, as it is gene-rated by coherent contributions arising from the quadrupole-octupoleshape. The E3 moment is therefore an observable that should providedirect evidence for enhanced octupole correlations and, for deformednuclei, can be related to the intrinsic octupole deformation parameters22.Until the present measurements, E3 transition moments have beendetermined for only one nucleus in the Z < 88, N < 134 region, 226Ra(ref. 23), so that theoretical calculations of E3 moments in reflection-asymmetric nuclei have not yet been subject to detailed scrutiny.

Experiments and discussionCoulomb excitation is an important tool for exploring the collectivebehaviour of deformed nuclei that gives rise to strong enhancementof the probability of transitions between states. Traditionally, thistechnique has been employed by exciting targets of stable nuclei withaccelerated ion beams of stable nuclei at energies below the Coulombbarrier, ensuring that the interaction is purely electromagnetic incharacter. Whereas E2, E1 and magnetic dipole (M1) transition pro-babilities dominate in the electromagnetic decay of nuclear states, andhence can be determined from measurements of the lifetimes of thestates, E2 and E3 transition moments dominate the Coulomb excita-tion process allowing these moments to be determined from measure-ment of the cross-sections of the states, often inferred from the c-raysthat de-excite these levels. In exceptional cases, the Coulomb excita-tion technique has been applied to radioactive targets like 226Ra,which is sufficiently long-lived (half-life 1,600 yr) to produce a mac-roscopic sample. It is only comparatively recently that the techniquehas been extended to the use of accelerated beams of radioactive nucleisuch as those from the Radioactive beam EXperimental facility atISOLDE, CERN (REX-ISOLDE24). In the experiments described here,

Energy (keV)

Cou

nts

per k

eV

10–1

1

10

102

103

104

105

106

10–4

10–3

10–2

10–1

1

10

102

103

104

105

X-rays

2+ →

0+

4+ →

2+

6+ →

4+

1– →

0+

1– →

2+

3– →

2+

5– →

4+

2 γ+ →

2+

Pile

up o

f 2+ →

0+

and

4+ →

2+

2 γ+ →

0+

a

*

Energy (keV)

Cou

nts

per k

eV

1

10

102

103

104

105

10–3

10–2

10–1

1

10

102

103X-rays

b

2+ →

0+

4+ →

2+

1– →

2+

5– →

4+

3– →

2+

1– →

0+ 6+ →

4+

8+ →

6+

Counts per keV

100 200 300 400 500 600 700 800 900 1,000

Counts per keV

50 100 150 200 250 300 350

Figure 1 | Representative c-ray spectra following the bombardment of2 mg cm22 60Ni and 120Sn targets by 220Rn and 224Ra. a, 60Ni (blue) and 120Sn(red) bombarded by 220Rn; b, targets as a but with bombardment by 224Ra. Thedifferences in excitation cross-section for the targets with different Z areapparent for the higher spin states. The c-rays are corrected for Doppler shiftassuming that they are emitted from the scattered projectile. The asterisk ina marks an unassigned, 836(2) keV transition. A state at 937.8(8) keV isassigned Ip 5 21 on the basis of its excitation and decay properties; it isassumed to be the bandhead of the c-band in 220Rn.

84.4

131.6216.0

166.4

74.4205.9

142.6182.3228.4

161.7 207.7

113.8

275.7

151.5265.3

312.6

965.5

241.0

292.7404.2

645.4

422.0

188.8318.3340.2

254.3 276.2370.4

937.8

0

2+ 84

4+ 251

6+ 479

8+ 755

10+ 1067

1–216

3–290

5–433

7–641

9–9062+ 966

0+ 0

2+ 241

4+ 534

6+ 874

8+ 1244

645

3–663

5–852

7–1128

2+ 938

881.1

a b220Rn

1–696.9

224Ra

0+

Figure 2 | Partial level-schemes for 220Rn and 224Ra, showing the excitedstates of interest for this work. a, 220Rn; b, 224Ra. Arrows indicate knownc-raytransitions. All energies are in keV (upright font refers to transition energies,

italic font refers to state energies) and spins in units of B. Note that the level at938 keV in 220Rn is observed for the first time in this work.

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L.P. Gaffney et al., Nature 497 (2013) 199 �

Discrete nuclear symmetries�Tetrahedral symmetry: ��Octahedral symmetry: ��Experimental evidence?�

6 TetraOcta08 printed on 6th October 2006

Above, ! ! {!!,µ; " = 2, 3, . . . "max; µ = "", "" + 1, . . . + "}, R0 is thenuclear radius parameter,, and c(!), a function whose role is to insure thatthe nuclear volume remains constant, independent of the deformation.

3.1. Octahedral Deformations

We can demonstrate that there exist special combinations of sphericalharmonics that can be used as a basis for surfaces with octahedral symme-try. The lowest order of the octahedral deformation, called by conventionthe first, is characterized by the fourth rang spherical harmonics. By intro-ducing a single parameter o1 we must have in this case

!40 ! +o1; !4,±4 ! +

!

5

14· o1 , (8)

i.e. three hexadecapole deformation parameters must contribute simulta-neously and with proportions

"

5/14 fixed by the octahedral symmetry

Figure 1. Comparison of two octahedrally deformed nuclei. Left: octahedral de-formation of the first order, o1=0.10; right: octahedral deformation of the secondorder, o2=0.04 .

requirement. No deformation with " = 5 is allowed and the next possibleare deformations with spherical harmonics of " = 6. Similarly, we introduceone single parameter, o2, with the help of which the next allowed octahe-dral deformation, called of the second order, and depending on the 6th rangspherical harmonics can be defined. We must have:

!60 ! +o2; !6,±4 ! "

!

7

2· o2 . (9)

The third order octahedral deformation is characterized by the 8th rangspherical harmonics and we can demonstrate that it can be defined with

TetraOcta08 printed on 6th October 2006 7

the help of a single parameter, o3, where:

!80 ! +o3; !8,±4 !

!

28

198· o3; !8,±8 !

!

65

198· o3 . (10)

Of course the basis of the octahedrally deformed surfaces is infinite, butthe increasing order of the octahedral deformations implies immediately atwice as fast increase in the rang of the underlying multipole deformations;the possibility of having such a situation in real nuclei is unlikely and theexpansion series can be cut o! quickly.

3.2. Tetrahedral Deformations

In a similar fashion, the tetrahedral deformation basis can be introducedin terms of the standard spherical harmonics. The first order tetrahedraldeformation, t1, is characterized by a single octupole deformation with " = 3and µ = 2 and we have

!3,±2 ! t1 (11)

The second order tetrahedral deformation, t2, is characterized by multipo-

Figure 2. Comparison of two tetrahedrally deformed nuclei. Left: tetrahedraldeformation of the first order, t1=0.15; right: tetrahedral deformation of the secondorder, t2=0.05 .

larity " = 7 (observe that the multipoles with " = 4, 5 and 6 are not allowedat all by the symmetry studied) and we have

!7,±2 ! t2 ; !7,±6 ! "

!

11

13· t2 . (12)

The third order tetrahedral deformation, t3, is characterized by " = 9 andby definition we must have:

!9,±2 ! t3; !9,±6 ! +

!

13

3· t3 . (13)

α3±2 ≠ 0

α40 =145α4±2 ≠ 0

J. Dudek et al., Phys. Rev. Lett. 88 (2002) 252502 �

Rigid rotor model�Hamiltonian of quantum-mechanical rotor in terms

of ‘rotational’ angular momentum R: ��Nuclei have an additional intrinsic part Hintr with

‘intrinsic’ angular momentum J.�The total angular momentum is I=R+J.�

€

ˆ H rot =2

2R1

2

ℑ1

+R2

2

ℑ2

+R3

2

ℑ3

#

$ %

&

' ( =2

2Ri

2

ℑ ii=1

3

∑

Rigid axially symmetric rotor�For ℑ1=ℑ2=ℑ ≠ ℑ3 the rotor hamiltonian is��Eigenvalues of Hrot: ��Eigenvectors ⎟KIM〉 of Hrot satisfy: �

€

ˆ H rot =2

2ℑi

Ii2

i=1

3

∑ =2

2ℑIi

2

i=1

3

∑ +2

21ℑ3

−1ℑ

%

& '

(

) * I3

2

€

EKI =2

2ℑI I +1( ) +

2

21ℑ3

−1ℑ

$

% &

'

( ) K 2

€

I2 KIM = I I +1( )KIM ,Iz KIM = M KIM , I3 KIM = K KIM

Ground-state band of axial rotor�The ground-state spin of even-even nuclei is I=0. Hence K=0 for ground-state band: �

€

EI =2

2ℑI I +1( )

E2 properties of rotational nuclei�Intra-band E2 transitions: ��E2 moments: ��Q0(K) is the ‘intrinsic’ quadrupole moment: �

€

B E2;KIi →KIf( ) =516π

IiK 20 IfK2e2Q0 K( )2

€

Q KI( ) =3K 2 − I I +1( )I +1( ) 2I + 3( )

Q0 K( )

€

e ˆ Q 0 ≡ ρ $ r ( )∫ r2 3cos2 $ θ −1( )d $ r , Q0 K( ) = K ˆ Q 0 K

E2 properties of gs bands �For the ground state (K=I):��For the gsb in even-even nuclei (K=0): �

€

Q K = I( ) =I 2I −1( )

I +1( ) 2I + 3( )Q0 K( )

€

B E2;I→ I − 2( ) =1532π

I I −1( )2I −1( ) 2I +1( )

e2Q02

Q I( ) = −I

2I + 3Q0

⇒ eQ 21+( ) =

2716π ⋅ B E2;21

+ → 01+( )

Generalized intensity relations�Mixing of K arises from�

Dependence of Q0 on I (stretching)�Coriolis interaction �Triaxiality�

Generalized intra- and inter-band matrix elements (eg E2): �

€

B E2;K iIi →K fIf( )IiK i 2K f −K i IfK f

= M0 + M1Δ + M2Δ2 +

with Δ = If If +1( ) − Ii Ii +1( )

Inter-band E2 transitions�Example of γ→g transitions in 166Er: �

€

B E2;Iγ →Ig( )Iγ2 2 − 2 Ig0

= M0 + M1Δ + M2Δ2 +

Δ = Ig Ig +1( ) − Iγ Iγ +1( )

W.D. Kulp et al., Phys. Rev. C 73 (2006) 014308 �

Rigid triaxial rotor�Triaxial rotor hamiltonian ℑ1 ≠ ℑ2 ≠ ℑ3 : ��H'mix non-diagonal in axial basis ⏐KIM〉 ⇒ K is not

a conserved quantum number�

€

ˆ " H rot =2

2ℑ i

Ii2

i=1

3

∑ =2

2ℑI2 +

2

2ℑ f

I32

ˆ " H axial

+2

2ℑg

I+2 + I−

2( )ˆ " H mix

1ℑ

=12

1ℑ1

+1ℑ2

&

' (

)

* + ,

1ℑ f

=1ℑ3

−1ℑ

, 1ℑg

=14

1ℑ1

−1ℑ2

&

' (

)

* +

Rigid triaxial rotor spectra�

€

γ = 30

€

γ =15

Tri-partite classification of nuclei�Empirical evidence for seniority-type, vibrational-

and rotational-like nuclei. �

N.V. Zamfir et al., Phys. Rev. Lett. 72 (1994) 3480 �

Interacting boson model�Describe the nucleus as a system of N interacting

s and d bosons. Hamiltonian: ��Justification from�

Shell model: s and d bosons are associated with S and D fermion (Cooper) pairs.�

Geometric model: for large boson number the IBM reduces to a liquid-drop hamiltonian.�

€

ˆ H IBM = εiˆ b i

+ ˆ b ii=1

6

∑ + υ i1i2i3i4ˆ b i1

+ ˆ b i2+ ˆ b i3

ˆ b i4i1i2i3i4 =1

6

∑

Dimensions�Assume Ω available 1-fermion states. Number of

n-fermion states is��Assume Ω available 1-boson states. Number of n-

boson states is��Example: 162Dy96 with 14 neutrons (Ω=44) and 16

protons (Ω=32) (132Sn82 inert core).�SM dimension: 7·1019 �IBM dimension: 15504�

€

Ω

n#

$ % &

' ( =

Ω!n! Ω− n( )!

€

Ω+ n −1n

$

% &

'

( ) =

Ω+ n −1( )!n! Ω−1( )!

Dynamical symmetries�Boson hamiltonian is of the form��In general not solvable analytically.�Three solvable cases with SO(3) symmetry: �

€

ˆ H IBM = εiˆ b i

+ ˆ b ii=1

6

∑ + υ i1i2i3i4ˆ b i1

+ ˆ b i2+ ˆ b i3

ˆ b i4i1i2i3i4 =1

6

∑

€

U 6( )⊃ U 5( )⊃ SO 5( )⊃ SO 3( )U 6( )⊃ SU 3( )⊃ SO 3( )U 6( )⊃ SO 6( )⊃ SO 5( )⊃ SO 3( )

U(5) vibrational limit: 110Cd62�

SU(3) rotational limit: 156Gd92�

SO(6) γ-unstable limit: 196Pt118�

Applications of IBM�

Classical limit of IBM�For large boson number N, a coherent (or intrinsic)

state is an approximate eigenstate, ��The real parameters αµ are related to the three

Euler angles and shape variables β and γ. �Any IBM hamiltonian yields energy surface: ��

€

ˆ H IBM N;αµ ≈ E N;αµ , N;αµ ∝ s+ + αµdµ+

µ∑( )

N

o

€

N;αµˆ H IBM N;αµ = N;βγ ˆ H IBM N;βγ ≡V β,γ( )

Phase diagram of IBM�

J. Jolie et al. , Phys. Rev. Lett. 87 (2001) 162501 �

The ratio R42 �

Bibliography �A. Bohr and B.R. Mottelson, Nuclear Structure. I

Single-Particle Motion (Benjamin, 1969).�A. Bohr and B.R. Mottelson, Nuclear Structure. II

Nuclear Deformations (Benjamin, 1975).�R.D. Lawson, Theory of the Nuclear Shell Model

(Oxford UP, 1980).�K.L.G. Heyde, The Nuclear Shell Model (Springer-

Verlag, 1990).�I. Talmi, Simple Models of Complex Nuclei

(Harwood, 1993).�

Bibliography (continued)�P. Ring and P. Schuck, The Nuclear Many-Body

Problem (Springer, 1980).�D.J. Rowe, Nuclear Collective Motion (Methuen,

1970).�D.J. Rowe and J.L. Wood, Fundamentals of

Nuclear Collective Models, to appear.�F. Iachello and A. Arima, The Interacting Boson

Model (Cambridge UP, 1987).�

Electric (quadrupole) properties�Partial γ-ray half-life: ��Electric quadrupole transitions: ��Electric quadrupole moments: �

€

B E2;Ii → If( ) =1

2Ii +1IfM f ekrk

2Y2µ θk,ϕk( )k=1

A

∑ IiM i

2

M f µ

∑M i

∑

€

eQ I( ) = IM = I 16π5

ekrk2

k=1

A

∑ Y20 θk,ϕk( ) IM = I

€

T1/ 2γ Eλ( ) = ln2 8π

λ +1

λ 2λ +1( )!![ ]2Eγ

c%

& '

(

) *

2λ+1

B Eλ( )+ , -

. -

/ 0 -

1 -

−1

Magnetic (dipole) properties�Partial γ-ray half-life: ��Magnetic dipole transitions: ��Magnetic dipole moments: �

€

B M1;Ii → If( ) =1

2Ii +1IfM f gk

l lk,µ + gkssk,µ( ) IiM i

k=1

A

∑2

M f µ

∑M i

∑

€

µ I( ) = IM = I gkl lk,z + gk

ssk,z( )k=1

A

∑ IM = I

€

T1/ 2γ Mλ( ) = ln2 8π

λ +1

λ 2λ +1( )!![ ]2Eγ

c%

& '

(

) *

2λ+1

B Mλ( )+ , -

. -

/ 0 -

1 -

−1

Extensions of IBM�Neutron and proton degrees freedom (IBM-2): �

F-spin multiplets (Nν+Nπ=constant) �Scissors excitations�

Fermion degrees of freedom (IBFM): �Odd-mass nuclei�Supersymmetry (doublets & quartets)�

Other boson degrees of freedom: �Isospin T=0 & T=1 pairs (IBM-3 & IBM-4)�Higher multipole (g,…) pairs�

Scissors mode �Collective displacement modes between neutrons and protons:�

Linear displacement (giant dipole resonance): Rν-Rπ ⇒ E1 excitation.�Angular displacement (scissors resonance): Lν-Lπ ⇒ M1 excitation.�

Supersymmetry�A simultaneous description of even- and odd-mass nuclei (doublets) or of even-even, even-odd, odd-even and odd-odd nuclei (quartets).�Example of 194Pt, 195Pt, 195Au & 196Au: �

Bosons + fermions�Odd-mass nuclei are fermions.�Describe an odd-mass nucleus as N bosons + 1

fermion mutually interacting. Hamiltonian: ��Algebra: ��Many-body problem is solved analytically for

certain energies ε and interactions υ. �€

ˆ H IBFM = ˆ H IBM + ε j ˆ a j+ ˆ a j

j=1

Ω

∑ + υ i1 j1i2 j2

ˆ b i1+ ˆ a j1

+ ˆ b i2 ˆ a j2j1 j2 =1

Ω

∑i1i2 =1

6

∑

€

U 6( )⊕U Ω( ) =ˆ b i1

+ ˆ b i2ˆ a j1

+ ˆ a j2

$ % &

' &

( ) &

* &

Example: 195Pt117�

Example: 195Pt117 (new data)�

Nuclear supersymmetry�Up to now: separate description of even-even and

odd-mass nuclei with the algebra��Simultaneous description of even-even and odd-

mass nuclei with the superalgebra�

€

U 6( )⊕U Ω( ) =ˆ b i1

+ ˆ b i2ˆ a j1

+ ˆ a j2

$ % &

' &

( ) &

* &

€

U 6/Ω( ) =ˆ b i1

+ ˆ b i2ˆ b i1

+ ˆ a j2

ˆ a j1+ ˆ b i2 ˆ a j1

+ ˆ a j2

# $ %

& %

' ( %

) %

U(6/12) supermultiplet �

Example: 194Pt116 &195Pt117�

Example: 196Au117�

nuclear structure collective models...

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