Download Emergent structures

Emergent Phenomena in
mesoscopic systems
S. Frauendorf
Department of Physics
University of Notre Dame
Emergent structures and
properties in nature
An emergent behaviour or emergent property can appear
when a number of simple entities (agents) operate in an environment,
forming more complex behaviours as a collective
Emergent structures are patterns not created by a single
event or rule. There is nothing that commands the system to
form a pattern, but instead the interactions of each part to its
immediate surroundings causes a complex process which
leads to order
The complex behaviour or properties are not a property
of any single such entity, nor can they easily be predicted
or deduced from behaviour in the lower-level entities:
they are irreducible.
1
Living systems-ant colony
A more detailed biological example is an ant
colony. The queen does not give direct orders
and does not tell the ants what to do. Instead,
each ant reacts to stimuli in the form of
chemical scent from larvae, other ants,
intruders, food and build up of waste, and
leaves behind a chemical trail, which, in turn,
provides a stimulus to other ants. Here each
ant is an autonomous unit that reacts
depending only on its local environment and
the genetically encoded rules for its variety of
ant. Despite the lack of centralized decision
making, ant colonies exhibit complex
behavior and have even been able to
demonstrate the ability to solve geometric
problems. For example, colonies routinely
find the maximum distance from all colony
entrances to dispose of dead bodies.
A termite "cathedral" mound produced by a termite colony:
a classic example of emergence in nature.
2
Physics
Emergence means complex organizational
structure growing out of simple rule. (p. 200)
Protection generates exactness and reliability,…
The universal properties of ordering of rigid bodies,
the flow of superfluids, and even the emptiness of
space are among the many concrete,
well documented examples of this effect. (p. 144)
Macroscopic emergence, like rigidity, becomes increasingly
exact in the limit of large sample size, hence the
idea of emerging. There is nothing preventing organizational
phenomena from developing at small scale,…. (p. 170)
3
Emergent phenomena
•
•
•
•
•
•
•
•
•
•
Liquid-Gas Phase boundary
Rigid Phase – Lattice
Superconductivity (Meissner effect, vortices)
Laws of Hydrodynamics
Laws of Thermodynamics
Quantum sound
e2
 25812.8
Quantum Hall resistance
h
Fermi and Bose Statistics of composite particles
…
…
4
Mesoscopic systems
N ~ 10  10
2
5
Emergence of a macroscopic phenomena with N.
Appearance of “finite size corrections” to familiar macroscopic
phenomena in very small probes (quantum dots, quantum wells,
quantum junctions, quantum wires).
Abundance in the cluster beam
Emergence of cubic crystal structure in salt clusters
T. P.Martin Physics Reports 273 (1966) 199-241
5
Ca clusters: the transition to the bulk is not smooth
fcc lattice: Close packing with
translational symmetry
Abundance in the cluster beam
Icosahedra: Close packing with small surface
bulk
T. P.Martin Physics Reports 273 (1966) 199-241
6
Water - dramatic example
7
8
Emergent phenomena - nuclei
• The nucleon liquid
• Superfluidity,
superconductivity
• Shell structure
• Spatial orientation
• Temperature
• Phases and phase
transitions


Extrapolation
to bulk
Finite nuclei
9
Neutron stars
Suprafluid, superconducting nuclear matter and more.
Studying the scaling of
clusters properties seems
instructive, because
these properties are well
known for the bulk.
B  108 Tesla
SGR 1806-20
7
Astrophysics:
What is the equation of state for nuclear matter?
Nuclei are only stable for A<300.
Clusters can be made for any N.
Liquid drop model:
Volume + Surface energy
EB  aV N  aS N
2/3
Transition to the bulk liquid
The liquid drop model scaling law seems reliable.
Binding energy of K clusters
Coulomb
energy
Neutral –one
component
EB
1 / 3
 aV  aS N
N
8
Ionization energy of Na clusters
IEbulk  Ecoulomb( N )
 ab  aC N
1 / 3
Other quantities
scale in the same way.
9
Nuclei: charged two-component liquid
EB
1 / 3
2 4 / 3
2 2
 aV  aS A  aC Z A  aS ( N  Z ) A
A
Strong correlation
How good is it? Symmetry energy ????
Is there a term
( N  Z ) 4 A4 ?
What is the bulk equation of state?
For example: compressibility
dE
d
Clusters may
provide examples
for scaling.
10
He droplets – getting really close to
nuclei
3
He clusters are most similar to nuclei.
Liquid at zero temperature
Electrical neutral: Limit N-> easily achieved.
4
He
produced for all N.
3
He
clusters probably exist only for N>50
Strong zero point motion.
Weakly bound nuclei
Very hard to experiment with, because of small energy scale.
11
Study of
4
He :
theory
Experiment?
P. Toennis, Lecture at the NATO Advanced Studies Series, Erice 1996
12
Superconductivity/Superfluidity
Described by the Landau – Ginzburg equations for
the order parameter
 (r )  (r ) / G | (r) |2  Density of Cooper pairs
Controlled by 
( inside the superconductor)
coherence length  0  vF /  (size of Cooper pair)
penetration depth of magnetic field
L  ( 0 mc2 / e 2 )1/ 2 G / 
G,  , Fermi energy  F  mvF2 / 2 , and
critical Temperature Tc related by BCS theory.
13
Phase diagram of a macroscopic
type-I superconductor
normal
H
Meissner effect
super
T
32
Type II superconductor
Solid state, liquid He:
Calculation of 
very problematic – well protected.
Take Tc from experiment.
0  vF /  ~ 15m  R
local
N  Tc
1K
~
~
~ 5
N
 F TF 10 K
BCS very good
Nuclei:
Calculation of  not possible so far.
Adjusted to even-odd mass differences.
0  vF /  ~ 40 fm  R ~ 5 fm
N
 Tc
1MeV
~
~
~
N
 F TF 40MeV
How to extrapolate to stars?
highly non-local
BCS poor
16
Vortices, pinning of magnetic field?
Mesoscopic regime
17
Superfluidity
1
2
Intermediate state of
Reduced viscosity
Atttractive interaction between 3 He Fermions generates
Cooper pairs -> Superfluid
Moments of inertia at low spin are well reproduced by
cranking calculations including pair correlations.
rigid
irrotational
Non-local superfluidity: size of the Cooper pairs larger
than size of the nucleus.
18
4
He is superfluid
at this T.
3
He is not superfluid
at this T.
19
free
SF6
4
Rotational spectrum of SF6 in a He droplet
Density of “normal” atoms
Moment of inertia larger
3
4
Rotational spectrum of OCS in a He- He droplet
60
4
He behave like a superfluid!
Title: SUPERFLUID HELIUM DROPLETS: AN ULTRACOLD NANOLABORATORY , By: Toennies, J. Peter, Vilesov, Andrej F., Whaley, K. Birgitta, Physics Today, 0031-9228,
February 1, 2001, Vol. 54, Issue 2
Shell
structure
21
Fermions in
spherical Potential
Esh  N 5 / 6 , E  N
Esh
 N 1/ 6
E
Nuclei:
magnitude OK,
damping with
N and T OK.
Clusters:
More washed out.
Dies out quicker.
Not quantitatively
understood.
22
T  400K
canonical
Frauendorf,
Pashkevich
Clusters allow us to
study shell structure
over a much larger range
than nuclei.
Explains the
gross shell structure
23
Supershell structure of Na clusters
N-dependent factor multiplied
for compensating the too
rapid damping with N!
24
Emergence of resistivity?
Imax>20
 rgid
Nuclear moments of inertia at high spin
Pair correlations are quenched.
M. Deleplanque, S.F. et al. Phys. Rev. C 69 044309 (2004)
Currents caused
by nucleons on
periodic orbits
25
Larmor: System in
Magnetic field behaves
like in rotating system
(in linear order).
eB
L 
2mc
susceptibi lity :
V    Rig
26
Emergence of thermodynamics
Region of high level density:
important for astrophysics, nuclear applications, …
Limits to predictability of quantal states:
uncertainties in the Hamiltonian
deterministic chaos
Give up individual quantal states:
# states
average level density  
energy intervall
dE
entropy S  ln  temper ature T 
dS
28
Crossover phenomena
N   exists
Phase transitions
solid-liquid
superfluid-normal
liquid-gas
N   does not exist
T=0 transitions between
different symmetries in
nuclei.
Spherical
deformed
IBA symmetries
Artificial limit by mean
field approximation
29
The Casten Triangle of IBM
M. Caprio, Quantum Phase Transitions, Minisymposium on Nuclei as
Mesoscopic Systems, Fall Meeting of DNP APS Nashville 2006
30
Super-normal phase transition
normal
H
super
T
32
Grand canonical
Canonical
Microcanonical
31
Grandcanonical ensemble
Canonical ensemble
33
Melting of Na clusters
34
Microcanonical
1 dS

T dE
q latent heat
35
M. Schmitd et al.
Transition from electronic to geometric shells
In Na clusters
T ~ 250K
T. P.Martin Physics Reports 273 (1966) 199-241
36
Similar emergent phenomena in nuclear and
non-nuclear mesoscopic systems.
37
Emerge with increasing particle number, while calculating
them microscopically becomes increasingly difficult.
New principles of organization (+ parameters) – to be found.
Region where micro and makro calculations are possible.
Comparing different types mesoscopic systems is instructive.
Studies are complementary: bulk limit accessible or not,
energy scale, external heat bath, ….
More contact between the communities!
More can be found in:
S. Frauendorf, C. Guet, Ann. Rev. Nucl. Part. Sci. 51, 219 (2001)
Quantization of magnetic flux in type II superconductors
Magneto-optical images of vortices in a NbSe2 superconducting
crystal at 4.3 K after cooling in magnetic field of 3 and 7 Oe.
15
Emergence of orientation
Example for spontaneous symmetry breaking:
Weinberg’s chair
Hamiltonian rotational invariant  eigenstate s of good
angular momentum : | IM 
density distributi on :
ρ(r, ,  ) | YIM ( ,  ) |2
Why do we see the chair shape?
Tiniest perturbation mixes |IM>
states to a stable-oriented wave
packet: the symmetry broken state.
Mesoscopic variant I: Molecules
3
2
Can be kicked and turned
like a chair.
NH3
1
18
Quantal states |IM>
can be measured:
Rotational bands
Classical
moments of inertia
of arrangement of
point masses.
16
E ( I )  BI ( I  1) I  J
  E ( I  1)  E ( I )  B( I  1)
HCl
Microwave absorption
spectrum
Mesoscopic variant II: Nuclei
Symmetry broken state described by the mean field.
How is orientation generated?
Riley
Deformed potential aligns the
partially filled orbitals
Partially filled orbitals are
highly tropic
Nucleus is oriented –
rotational band
1.0
overlap
0.8
0.6
0.4
0.2
Well deformed 174 Hf
0.0
-90
19
0
90

180
270
 - rays from the superdefor med
nucleus
152
Dy
E2 radiation
 - rays from the spherical
nucleus
199
Pb
M1 radiation
163
Er
deformed
200
Pb
spherical
Symmetry breaking
Spontaneous symmetry breaking
Emergence
Periodic crystal structure
rigidity,
transverse sound
Finite N:
Localization
Shell structure,
center of mass motion
Orientation
rotational alignment, …
rotational bands
d
dB
susceptibi lity :     Rig
B
27