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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

Lecture 1

Lecture 2

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

Basic, attractive SD Interaction

(2J+1)+

6+

4+

2+

0+

0+ and 2+ lowest;

separated from the rest.

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???

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] † ……

- 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]

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)

M= +Jz

M= -Jz

J2

J1

O(2)

O(3)

- 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

I. “U(5)” - Anharmonic Vibrator

II. “SU(3)” - Axially symmetric rotor

III. “O(6)” - Gamma - unstable rotor

U(5)

R4/2= 2.0

SU(3)

R4/2= 3.33

O(6)

R4/2= 2.5

The first O(6) nucleus ………..

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

and then many more….

- 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

=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

- 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

?

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

Vibrational

Transitional

Rotational

E = nħω

E = J(J+1)

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)

Two solutions depending on γ degree of freedom

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

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

P= NpNn

Np+Nn

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).

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

- 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