Nuclei as Mesoscopic Systems

Alexander VolyaFlorida State University

Mesoscopic system

• Quantum many-body system between microscopic (few-body) and macroscopic (thermodynamic limit)

• Quantum many-body system with identifiable individual quantum states, while sufficiently large to reveal regularities of statistical nature.

• Emergence of complexity.

A rich variety of mesoscopic systems

• Nano-wires• Quantum dots• Helium drops• Atomic clusters• Quantum computers

http://www.pa.msu.edu/~tomanek/

• Quantum Dot : 5 metallic gates fabricated on the surface of a GaAs; two dimensional electron gas inside.

• quantum dot can be seen as a cavity in which electrons bounce at the boundaries similar to a billiard table.

Nanotubes

http://pages.unibas.ch/phys-meso/

The nuclear world: the rich variety of natural mesoscopic phenomena

• Predicted: 6000 - 7000 particle-stable nuclides• Observed: 2932• even-even 737; odd-A 1469; odd-odd 726.• Lightest (deuteron), Heaviest • No gamma-rays known 785.• Largest number of levels known (578) • Largest number of transitions known 1319• Highest multipolarity of electromagnetic transition E6 in

• Resut of 100 years of reasearch 182000 citations in Brookhaven database, 4500 new entries per year.

Nuclear Chart

Single-Particle Motion

• Symmetry, surface and shells• Shells and supershells• Single-particle modes and magic numbers• Symmetry and chaos• Classical periodic orbits

T. P.Martin Physics Reports 273 (1966) 199-241

0 100000 200000

62

171

364

665

1098

Mass [amu]

Cou

nts/

chan

nel [

abun

danc

e in

the

beam

]

(NaI) Nan+

Salt Clusters, transition from small to bulk

• Symmetry • Surface• “Shells”

Shell Structure in atoms

From A. Bohr and B.R.Mottleson, Nuclear Structure, vol. 1, p. 191 Benjamin, 1969, New York

Nuclear Magic Numbers, nucleon packaging, stability, abundance of elements

From W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)

Mean field

Shell gaps N=2,8,20,

Nuclear Woods-Saxon solverhttp://www.volya.net/ws/

single-partic le levels

Nucleon in a box

oscillator square well Woods-saxon

Level density in the Woods-SaxonPotential: N=1000, 2000, and 3000

Nishioka et. al. Phys. Rev. B 42, (1990) 9377R.B. Balian, C. Block Ann. Phys. 69 (1971) 76

34382

28 20

4058

92138

198

Nuclear Fission limit

Supershellstructure

Supershells

Binding energy, deviation from average

Supershells and classical periodic orbits

171A 184A

Nucleon in the potential wellQuantum Billiard

• Shell ModelLevels in nuclei

single-partic le levels

Nucleon in a box

Chaotic motion• Non-symmetric

shape– Shape changes– Collective vibrations

Periodic orbitals and shell structureIn the realm of chaos

•Why some nuclei are more stable than others?•Why are there shell effects?

Single-nucleon motion in deformed potential

Spherical Deformed

Quantum chaosDistribution of energy spacing between

neighboring states

• Regular motion– Analog to integrable

systems– No level repulsion – Poisson distribution

P(s)=exp(-s)

• Chaotic motion– Classically chaotic– Level repulsion– GOE (Random Matrix)

P(s)=s exp(-p s2/4)

Circular billiard Irregular triangle billiard

Evolution of shells

• Melting of shell structure• Shells in deformed nuclei• Shells in weakly bound nuclei• Is the mean field concept valid

Shell structure in extreme limitsShells in nuclei far from stability

Melting of shell structure

T=0 and T=0.4 ev, Frauendorf S, Pahskevich VV. NATO ASI Ser. E: Appl. Sci., ed. TP Martin, 313:201. Kluwer (1996)

J. Dobaczewski et al., PRC53, 2809 (1996)

Deformation and shell gaps

Mesoscopic many-body complexity• Complexity and Chaos

– Typical level density– Chaotization process, geometric chaoticity– Random matrix theory– Enhancement of weak perturbations

• Collective Motion– Pairing and superconductivity– Phase transitions– Giant resonances– Fission

• Shapes– Shape change transitions– Rotations

• Thermodynamics and phase transitions– Features of small systems– Thermalization and level density– Yang-Lee theory, roots of partition functions

Many-nucleons, two-body scatterings

single-partic le levels

0 0.2 0.4 0.6 0.8 1interaction strength

0.0

2.0

4.0

6.0

ener

gy le

vels

Nucleons in the box collide (interact)•Jump from level to level•Many-body dynamics

‡Even more complicated motion

Quantum billiards and neutron resonances n + 232ThTransmission spectrum of a 3D-stadium billiard

T = 4.2 K

Spectrum of neutron resonances in 232Th + n

● Great similarities between the two spectra: universal behaviour

Chaotic motion in nuclei

0 1 2 30.0

0.5

1.0

s

PoissonP(s)GOE

NDE

0 1 2 3 40

2

4

6

8

10

s

PoissonP(s)

Wigner

“Cold” (low excitation) rare-earth nuclei

High-Energy region, Nuclear Data EnsembleSlow neutron resonant date Haq. et.al. PRL 48, 1086 (1982)

Pairing interaction in nucleine

utro

n pa

iring

gap

mass number A

∆=12Α−1/2

N=8

20

28

50 82 126

Rotation

Evidence of nuclear superfluidity

Pairing Hamiltonian

• Pairing on degenerate time-conjugate orbitals

• Pair operators P =(a1a1)J=0 (J=0, T=1)• Number of unpaired fermions is seniority s• Unpaired fermions are untouched by H

Approaching the solution of pairing problem

• Approximate– BCS theory

• HFB+correlations+RPA

– Iterative techniques• Exact solution

– Richardson solution– Algebraic methods– Direct diagonalization + quasispin symmetry1

1A. Volya, B. A. Brown, and V. Zelevinsky, Phys. Lett. B 509, 37 (2001).

BCS theory

Low-lying states in paired systems

• Exact treatment – No phase transition and Gcritical– Different seniorities do not mix– Diagonalize for pair vibrations

• BCS treatment

HFB+RPAHF+RPACollective excitations

quasiparticleexcitationEs=2=2 e

single-particle excitationsEs=2=2 ε

Elementary excitations

BCSHartree-FockGround state

G>GcriticalG<Gcritical

Cooper Instability in mesoscopic system

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 0.5 1 1.5 2

Ene

rgy

diffe

renc

e pe

r pa

rtic

le [

MeV

]

Pairing strength G

N=20 N=100

0 0.5 1 1.5 2Pairing strength G

0

0.5

1

1.5

2

Pair

ing

gap

∆

ExactBCS

BCS versus Exact solution

Statistical treatment of pairing

• Microcanonical• Canonical• Grand canonicalPartition functions

Statistical averages

Entropy

Is there thermalization?r

(MeV

)-1

E (MeV) E (MeV)

T (

MeV

)

Critical behavior-thermodynamic limit

0 0.5 1 1.5 2

Temperature [MeV]

0

20

40

60

Ene

rgy

[M

eV]

N = 10N = 30N = 50N = 100

0 0.5 1 1.5 20

40

80

120

160

Cv

N = 10N = 30N = 50N = 100

Pairing phase diagram

Microcanonical ensemble and thermodynamic limit

0 5 10 15

Excitation energy [MeV]

0

10

20

30

40

Ent

ropy

MicrocanonicalCanonical

0 10 20 300

10

20

30

40

50

60

70

0 20 40 600

40

80

120N = 30 N = 50 N = 100 G=1

Statistical approach to mesoscopic system

0 10 20 30 40 50

Excitation energy [MeV]

0

10

20

30

40

50

Tem

pera

ture

[M

eV]

MicrocanonicCanonicGrand canonic

N=12, half occupied ladder

Phase Transition in Mesoscopic System

ββ

ττ

http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html

Complex roots- similar to chargesAppear symmetrically, never exactly on real axis

Energy- similar to potentialRoots become polesMacroscopic accumulation of polescreates charged surface

Heat Capacity – E-filed

Classification of phase transitions zeros in the complex temperature plane

Main Characteristics• Angle of approach

• Congestion of roots

Classification

First order Second order Higher order

0

4

8

12τ

[M

eV-1

]

0

100

200

Cv

0

4

8

12

τ [

MeV

-1]

0 5 10

β [MeV-1

]

0

100

200

Cv

0 5 10 0 5 10 0 5 10

(a) G = 0.00 (b) G = 0.01 (c) G = 0.40 (d) G = 0.60

(e) G = 0.00 (f) G = 0.01 (g) G = 0.40 (h) G = 0.60

(i) G = 0.80 (j) G = 1.00 (k)G = 1.20 (l)G = 2.00

(m) G = 0.80 (n) G = 1.00 (o)G = 1.20 (p)G = 2.00

N=100 particles

End of lecturecontinue reading to learn more…

• Invariant correlational entropy• Phase diagrams• Open mesoscopic quantum systems• Superradiance, quasi-stationary states in

continuum

Invariant Correlational Entropy• Parameter-driven equilibration (pairing strength)• Averaged density matrix

• ICE

Advantages•Basis independent •Explore individual quantum states•Needs no heat bath•No equilibration, thermalization and particle number conservation issues. •Probe sensitivity of states to noise in external parameter(s)•Phase transitions -> peaks in ICE

Mg Phase diagram

strength of T= 0 pairing

stren

gth

ofT=

1 pa

iring

Normal

T= 1 pairing

T= 0 pairing

realistic nucleus

Exotic nuclei: Halo Nucleus 11Li11Li is halo, it is as big a lead

Two neutrons in 11Li are moving ondecaying orbitals!

p1/2

s1/2

p1/2

s1/2

filled

filled

Two “valence” states are possible

Nuclear reaction theoryQuantum billiards with particle-

leaksdecay

• Due to finite lifetime states acquire width (uncertainty in energy G=h/t)

• Internal complex motion ñ Radiation and decay ?

escape

Superradiance, collectivization by decay

Dicke coherent stateN identical two-level atomscoupled via common radiation

Analog in nucleiInteraction via continuumTrapped states ⇒ self-organization

Volume ¿ λ3

g ~ D and few channels•Nuclei far from stability•High level density (states of

same symmetry)•Far from thresholds

Shape vibration and GDR

PDR

GDR

(2+ x 3- )1-

p n

B(E1)

E (MeV)

15

3

Simple example: Two-spin system

• Two interacting spins – Spherical symmetry (triplet and singlet

states)• Magnetic field

– Preserves spherical symmetry• First spin in sz=1/2 state decays

– Reduces symmetry (Sz is preserved but not S2)

• Hamiltonian for Sz=0

• Complex energies

Features of open system• Incompatible symmetries• Many-body versus single-spin properties• Interaction of two resonances•Superradiance and separation of states• “Phase transition”

11Li an example of interacting resonances

11Li is stable it is held by interactionof resonances

11Li is borromean, if one nuclon isremoved it becomes unstable

Scattering and cross section near threshold

Scattering Matrix

Cross section

Solution in two-level model

Single-particle decay in many-body system

−100 −50 0 50 100E

−100

−80

−60

−40

−20

0

Γ

Total states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21

•Assume energy independent W•Assume one channel γ=A2

•System 8 s.p. levels, 3 particles•One s.p. level in continuum e=ε –iγ/2

Evolution of complex energies E=E-i Γ/2as a function of γ