The algebraic approach
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Introduction The building blocks Dynamical symmetries Single nucleon description. Critical point symmetries Symmetry in n-p systems Symmetry near the drip lines. The Algebraic Approach. Lecture 1. Lecture 2. I. R. Shell Model. Geometrical Model. w. j. Single particle motion

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The Algebraic Approach

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The algebraic approach

Introduction

The building blocks

Dynamical symmetries

Single nucleon description

Critical point symmetries

Symmetry in n-p systems

Symmetry near the drip lines

The Algebraic Approach

Lecture 1

Lecture 2


The algebraic approach

I

R

Shell Model

Geometrical Model

w

j

Single particle motion

Describes properties in which a limited number of nucleons near the Fermi surface are involved.

Collective motion (in phase)

Vibrations, rotations, deformations

Describes bulk properties depending in a smooth way on nucleon number

Dynamical symmetry

Interacting Boson

Approximation

Truncation of configuration space

Algebraic

NUCLEAR MEAN FIELD

Three ways to simplify


The algebraic approach

Basic, attractive SD Interaction

(2J+1)+

6+

4+

2+

0+

0+ and 2+ lowest;

separated from the rest.


The algebraic approach

Pauli Principle

Consider f7/2 “shell”

with 6 neutrons

M

 7/2

 5/2

 3/2

 1/2

Maximum seniority = 2

Maximum (d)-boson number =1

Bosons counted from nearest closed

shell (i.e. particles or holes).

[Eg 130Ba Z = 56 N = 74 N= 3 N= 4 ; N=7]

WHY???


The algebraic approach

li

mi

ni

DYNAMICAL SYMMETRY

  • Describes basic states of motion available to a system - including relative motion of different constituents

  • Dynamical symmetry breaking splits but does not mix the eigenstates

G1 G2  G3  ……….

H= aC1 [G1] † bC2 [G2] † cC3 [G3] † ……


Some working definitions

Some ‘Working Definitions’

  • Have states s and d with  = -2,-1,0,1,2 - 6 -dim. vector space.

  • Unitary transformations involving the operators s, s†, d, d†

    => ‘rotations that form the group U(6).

  • Can form 36 bilinear combinations which close on commutation,

    s†s, s†d, d†s, (d†d)(L)

    (eg: [d†s,s†s] = d†s)

    - these are the generators

[Analogy: Angular momentum: Jx,Jy,Jz generate rotations and form group 0(3)]

[For 0(3), use Jz,J ; J = Jx ± i Jy

Then [J+,J-] = 2Jz ; [Jz,J] = ±J]


The algebraic approach

All C’s commute and H is diagonal

DYNAMICAL SYMMETRY

  • A Casimir operator commutes with all the generators of a group.

    • Eg:C1U(6) = N;C2U(6) = N(N+5)

  • Now look for subsets of generators which form a subgroup. Eg: (d†d)(L)- 25-U(5) (d†d)(1), (d†d)(3)-10-0(5) (d†d)(1)-3-0(3)

  • ie: U(6) U(5)  0(5)  0(3) - group chain decomposition

  • Now form a Hamiltonian from the Casimir operators of the groups.

H = C1U(6) + C2U(6) C2U(5) + C2O(5) +C2O(3)


Example of angular momentum

M= +Jz

M= -Jz

J2

J1

O(2)

O(3)

Example of angular momentum

  • For O(3), generators are Jz, J+ and J-

  • Then [J2, Jz] = [J2, J+] = [J2, J-] = 0

  • C2O(3) = J2

  • Subgroup O(2) simply Jz = C1O(2)

  • So H =  C2O(3) +  C1O(2)

  • E=  J(J+1) +  M


Only 3 chains from u 6

Only 3 chains from U(6)

I. “U(5)” - Anharmonic Vibrator

II. “SU(3)” - Axially symmetric rotor

III. “O(6)” - Gamma - unstable rotor


The algebraic approach

U(5)

R4/2= 2.0


The algebraic approach

SU(3)

R4/2= 3.33


The algebraic approach

O(6)

R4/2= 2.5


The algebraic approach

The first O(6) nucleus ………..

Cizewski et al, Phys Rev Lett. 40, 167 (1978)


The algebraic approach

and then many more….


Transition regions and realistic calculations

Transition Regions and Realistic Calculations

  • Most nuclei do not satisfy the strict criteria of any of the 3 Dyn. Symm.

  • Need numerical calculations by diagonalizing HIBA in s – d boson basis

  • Can use a very simple form of the most general H

Consistent Q Formalism


The algebraic approach

=0.03 MeV

Z=38-82

2.05 < R 4/2 < 3.15

N.V. Zamfir, R.F. Casten, Physics Letters B 341 (1994) 1-5


Summary

Summary

  • Algebraic approach contains aspects of both geometrical and single particle descriptions.

  • Dynamical symmetries describe states of motion of system

  • Analytic Hamiltonian is a sum of Casimir operators of the subgroups in the chain.

  • Casimir operators commute with generators of the group; conserve a quantum number

  • Each Casimir lifts the degeneracy of the states without mixing them.

  • Three and only three chains possible; O(6) was the surprise.

  • Very simple CQF Hamiltonian describes large ranges of low-lying structure


Evolution of nuclear shape

?

Previously, no analytic solution to describe nuclei at the “transitional point”

Evolution of nuclear shape

Vibrational

Transitional

Rotational

E = nħω

E = J(J+1)


Critical point symmetries

V(β)

Approximate potential at phase transition with infinite square well

β

Solve Bohr Hamiltonian with square well potential

Result is analytic solution in terms of zeros of special Bessel functions

Predictions for energies and electromagnetic transition probabilities

γ-soft

E(5)

Symmetric Rotor

Spherical Vibrator

X(5)

Critical Point Symmetries

Two solutions depending on γ degree of freedom

F. Iachello, Phys. Rev. Lett. 85, 3580 (2000); 87, 052502 (2001).


X 5 and e 5

τ = 1

Key Signatures

τ = 0

E(41)/E(21) = 2.91

ξ = 2

E(02)/E(21) = 5.67

R4/2 = 2.20

E(02)/E(21) = 3.03

E(03)/E(21) = 3.59

ξ = 1

X(5) and E(5)


Searching for x 5 like nuclei

P= NpNn

Np+Nn

Searching for X(5)-like Nuclei

β-decay studies at Yale

152Sm

R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001).

N.V. Zamfir et al., Phys. Rev. C 60, 054312 (1999)..

156Dy

M.A. Caprio et al., Phys. Rev. C 66, 054310 (2002).

162Yb

E.A.McCutchan et al., Phys. Rev. C 69, 024308 (2004).

166Hf

Good starting point: R4/2 or P factor

E.A.McCutchan. et al., Phys. Rev. C- submitted.

Other Yale studies: 150Nd - R.Krücken et al., Phys. Rev. Lett. 88, 232501 (2002).


Searching for e 5 like nuclei

Searching for E(5)-like Nuclei

Ce

58

3.06

2.93

2.80

2.69

2.56

2.38

2.32

Ba

134Ba

56

2.96

2.89

2.83

2.78

2.69

2.52

2.43

2.32

2.28

Xe

54

2.33

2.40

2.47

2.50

2.48

2.42

2.33

2.24

2.16

2.04

R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000).

Te

52

2.09

2.00

1.99

2.07

2.09

2.07

2.04

2.01

1.94

1.72

Sn

50

1.54

1.67

1.75

1.81

1.79

1.68

1.84

1.85

1.87

1.88

1.86

1.80

1.71

1.63

102Pd

Cd

48

2.33

1.79

2.11

2.27

2.36

2.38

2.33

2.29

2.30

2.38

2.39

2.38

46

Pd

1.79

2.12

2.29

2.38

2.40

2.42

2.46

2.53

2.56

2.58

N.V. Zamfir et al., Phys. Rev. C 65, 044325 (2002).

Ru

44

1.82

2.14

2.27

2.32

2.48

2.65

2.75

2.76

2.73

42

Mo

1.81

2.09

1.92

2.12

2.51

2.92

3.05

2.92

130Xe

P~2.5

Zr

40

1.60

1.60

1.63

1.51

2.65

3.15

3.23

Sr

38

1.99

2.05

3.01

3.23

Z/N

52

54

56

58

60

62

64

66

68

70

72

74

76

78

80

Good starting point: R4/2 or P factor


Symmetries and phases transitions in the ibm

Symmetries and phases transitions in the IBM

  • Challenges for neutron-rich:

    • New collective modes in three fluid systems (n-skin).

    • New regions of phase transition

    • New examples of critical point nuclei?

    • Rigid triaxiality?

D.D. Warner, Nature 420 (2002) 614


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