nuclei as mesoscopic systemsmesoscopic system • quantum many-body system between microscopic...
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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