ƒ The transition between order and chaos

Nuclear Structure at Finite Temperature

ƒ Statistical properties of CN and Shell model states

ƒ The transition between order and chaos

ƒ The damped rotation

Nuclear Structure at Finite Temperature

0 5 10 15

U [MeV]

10^-9 sec 10^-19sec

10^-15sec

np-nh states

| µ >

I

+ +

∑

=

µ

µα µ

α X

| α > ^{| µ} >

I

j

k l

vi^ijkl

Two-body interaction

compound states

[

^{2( )1/2}

]

exp ) 0 ( )

(E ρ aE

ρ =

τ=h/Γµ

n, p resonances τ≈10^-16-10^-18s

mean-field (shell model)

beyond mean-field

Bohr’s

compound states

Statistical Properties of CN states

Γ <d

Γ≥ d

NON overlapping resonances:

isolated resonances behaving stocastically (chaotic system)

E projecti le

(eV) p or n resonances

Level spacing distribution of p and n resonances

J

^π

=1/2

⁺

U ≈ 8-10MeV d ≈ 10 eV Γ ≈ 1 eV

s/d

The CN is CHAOTIC

strongly overlapping resonances:

Ericson fluctuations

(chaotic system)

Order and Chaos in classical & quantum mechanics

Order

^and

Chaos

are concepts well defined in classical mechanics, namely for systems governed by Hamilton’s equations

ORDERED Systems

ƒ

integrable

ƒ

regular, periodic orbits

ƒ

predictable

CHAOTIC Systems

ƒ

NON-integrable

ƒ

irregular, NON-periodic orbits

ƒ

NON-predictable

example: point-particle moving freely in 2-dimensions and reflected elastically from boudaries

Stadium or Sinai biliard rectangle circle

Quantum counterparts:

Schrödinger equations in 2-dimensions for a free particle

with wave-function becoming 0 at the surfaces of the boundaries

Wigner

(chaos)

se

^-s2

e

^-s

_Poisson

(order)

Level spacing

distributions

T=0 order:

shell model states (mean field)

T≠0 onset of chaos:

mixed states (residual interaction)

E_x= S_n≈ 8MeV

E_x ≤ 4.3MeV

E_x ≈ 0

Level spacing distributions

CHAOS ORDE R

CN n

γ

Compound Nucleus

CHAOS U ≈ 8 MeV Γ

_µ

τ=h/Γ

µ

|α

>=Σ

_µ

X

_α^{µ |µ}

>

Loss of Selection rules

Rotational Damping

strongly interacting

bands U= 1-5 MeV

Γ

_µ

^|α>

I

I-2

∆ω₀

Γ

_µ

Γ

_rot

I-2

E2 strength

The damping of Nuclear Rotation

Regular Bands mean field

ORDER U < 1 MeV

I I+2 I-2

np-nh states

| µ > I

Selection rules

(E,I,K,α, …)

Experimental Signature of Damped Rotation

Γ

rot ^I+2I

I-2

E

_γ

[keV]

Counts [arb.un.]

discrete transitions (regular bands)

E2 continuum spectrum (damped rotation)

statistical E1 γ -spectrum of

¹⁶⁴

Yb

2 0 3 0 4 0 5 0 6 0 7 0

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0

<U > = 2 M e V

<U > = 1 .4 M e V

U < 1 M e V

2 Γ

_µ

Γ

_{ro t}

d is c r e t e

Width [keV]

S p in [h ]

Γ

rot[keV]

Spin [h]

163

Er

Theory Exp

rotational

DAMPING

The damping width Γ

_rot

is a basic quantity for the understanding of nuclear structure at finite temperature

A. Bracco and S. Leoni, Rep, Prog. Phys. 65(2002)299