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Prologue
1985
1993
1999
2007
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2012
XXIII
XXXI
XXXVII
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XLVI
XLVIII
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“Multi-Fragmentation Reactions Within the Nuclear Lattice Model”
“Pion Correlations in Proton-Induced Reactions …”
“Sub-Poissonian Fluctuations …”
“Fragment Size Rankings …”
“Double Beta Decay”
“The Physics and Astrophysics of FRIB”
… anything I want …
W. Bauer, Michigan State University
The nuclear fragmentation problem
and
Bormio's contributions to its solution
Wolfgang Bauer
W. Bauer, Michigan State University
Taking Things Apart – A Universal Problem
Initial state
“Yard Sale”
W. Bauer, Michigan State University
Wikipedia
U. Post
D. Dean
U. Mosel
WB
Bormio 1985
W. Bauer, Michigan State University
Nuclear Fragmentation Data
• Proton-induced multifragmentation
• Here: Tevatron data (blue circles)
• Minimum bias data
– Impact parameter integrated
– Mixing of different event classes
W. Bauer, Michigan State University
Equation of State / Phase Transition?
• State variables: pressure, temperature, density (internal
energy, chemical potential, strangeness, …)
• Equation of state: relationship between state variables,
.
f ( p,T , r ) = 0
– Thermodynamic equation describing state of matter under
given physical conditions
– Example: Ideal gas: pV = nRT
– Example: Ultra-relativistic fluid: p = cs2e
– More realistic equations of state need to contain phase
transitions, coexistence regions, critical points, …
W. Bauer, Michigan State University
EoS for C2H6 (Ethane)
(a)
(b)
(c)
(d)
ideal gas EoS
Redlich Kwong EoS, p(T,V)
Maxwell construction
First-order phase transition, terminating in critical point
W. Bauer, Michigan State University
Nuclear EoS
• Can be computed, if you know nuclear
force
– Short-range repulsive
– Medium-range attractive
– Long-range repulsive (Coulomb)
• Here: Skyrme
– Note:
Coexistence region,
critical point.
• Mean-field approximation
(Sauer, Chandra, Mosel,
NPA 264 (1976)
)
W. Bauer, Michigan State University
Nuclear Matter Phase Diagram
Two phase transitions in nuclear matter:
ç “Liquid Gas”
ç Hadron gas / QGP / chiral restoration
o Problems / Opportunities:
o
ç Finite size effects (finite size scaling!
ç Is there equilibrium? ( )
ç Measurement of state variables
NUCLEAR SCIENCE, A Teacher’s Guide to the Nuclear
Science Wall Chart, Figure 9-2
)
(r , T, S, p, … )
ç Migration of nuclear system through phase
diagram (non-equilibrium processes) ( )
ç Near critical point(s): Critical slowing down!
Not sufficient time for equilibrium phase
transition! ( )
W. Bauer, Michigan State University
Astrophysics Connection
o Exploration of the drip lines
below charge ~40 via
projectile fragmentation
reactions
o Determination of the isospin
degree of freedom in the
nuclear equation of state
o Astrophysical relevance
(origin of heavy elements!)
o Review: Li, Ko, WB, Int. J.
Mod. Phys. E 7
W. Bauer, Michigan State University
Percolation
o Short-range NN force: nucleons
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WB et al., PLB 150, 53 (1985)
WB et al., NPA 452, 699 (1986)
X.Campi, JPA 19, L917 (1986)
T. Biro et al., NPA 459, 692 (1986)
J. Nemeth et al., ZPA 325, 347 (1986)
…
in contact with nearest neighbors
Expansion (thermal, compression
driven, dynamical, …)
Bonds between nucleons rupture
Remaining bonds bind nucleons into fragments
One control parameter: bond breaking probability
Can be done on large lattices with “exact” infinite size limits
Not necessarily an equilibrium phase transition theory!
o Also applies to non-equilibrium situations
W. Bauer, Michigan State University
Breaking Probability
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Determined by the excitation energy deposited
Infinite simple cubic lattice:
– 3 bonds/nucleon
– It takes 5.25 MeV to break a bond
Proton-induced: Glauber theory
– pbreak proportional to path length through matter
General relation between pbreak and T:
G = generalized incomplete gamma function, B = binding
energy per nucleon T. Li et al., PRL 70
(generalization of Coniglio-Klein for Fermi systems)
•
Or … obtain E* or T from other model or directly from
experiment
W. Bauer, Michigan State University
Event-by-Event Analysis
• Near critical point, information on fluctuations is essential;
averaging destroys it
• Promising candidates: E-by-E moment analyses
M k (e) = å ne (i) i k
i
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e = event, ne(i) = # of times i is contained in e
E-by-E for different observables can generate N-dimensional scatter
plots
Big question: How to sort events into classes?
Natural choice: If you know control parameter, use it!
(easy for theory, impossible for experiment)
Closest choice: observable that is ~linear in control parameter.
Attempt: use charged particle multiplicity, m.
W. Bauer, Michigan State University
Continuous Phase Transition
• Near critical point, we expect scaling behavior: all physical
quantities have power-law dependencies on the control parameter
• No characteristic scales in observables
• Critical exponents of power-laws are main quantities of interest
ns (Tc ) ~ s -t
– t , Cluster size:
– b , Order parameter:
P ~ (T - Tc )b
g
– g , Divergence of s:
~ T - Tc
• Hyper-scaling assumption
t -1
2 -a =
= 2b + g
s
(Determine 2 critical exponents sufficient)
W. Bauer, Michigan State University
Finite Size Scaling
• Phase transitions strictly only defined
for (almost) infinite systems
• Lattice calculations work on finite
lattices and extrapolate to infinite
lattices (hardest part!)
• Finite size scaling exponent,
– Modify control parameter by L1/n
– Modify order parameter by Lb /n
• Opportunity for nuclear physics:
Learn about extreme finite size
scaling in real systems
M. Thorpe, MSU
W. Bauer, Michigan State University
Determining Critical Exponents
•
•
EOS-TPC: Gilkes et al., PRL 73
Complete reconstruction of events: all
charges recovered
• Assume charged-particle multiplicity is
proportional to control parameter
• Find critical value, mc; extract critical exponents
g = 1.4, b = 0.29
• Assuming validity of hyper-scaling:
universality class of transition is completely
determined
• Interesting data; incorrect interpretation
W. Bauer, Michigan State University
M2~|m-mc|- g
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1 A GeV Au + C
Data are integrated over all residue
sizes and excitation energies
Complete detection of all charges
Data: red circles
Assumption: total multiplicity is
proportional to control parameter
Percolation model: green histograms
Percolation model contains critical
events Strong evidence for
continuous phase transition
But: g = 1.8, b = 0.41
WB & A.Botvina, PRC 52
WB & A.Botvina, PRC 55
mc
W. Bauer, Michigan State University
ISiS BNL Experiment
• 10.8 GeV p or p + Au
• Indiana Silicon Strip
Array
• Experiment performed
at AGS accelerator of
Brookhaven National Laboratory
• Viola, Beauliau, et al.
W. Bauer, Michigan State University
Energy Deposition
• Reaction:
pi, p+Au @AGS
• Very good statistics
(~106 complete events)
• Philosophy: Don’t deal
with energy deposition
models, but take this
information from experiment!
• Detector acceptance effects crucial
– filtered calculations, instead of corrected data
• Parameter-free calculations
W. Bauer, Michigan State University
Experiment / Theory Comparison
• Filter very
important
– Sequential
decay
corrections
huge
• Good
agreement
W. Bauer, Michigan State University
Scaling
• Idea (Elliott et al.): If data follow
scaling function
with f(0) = 1 (think “exponential”),
then we can use scaling plot to see
if data cross the point [0,1] ->
critical events
W. Bauer, Michigan State University
Scaling of ISIS Data
• Most important: critical region
and explosive events probed in
experiment
• Possibility to narrow window of
critical parameters
– t
: vertical dispersion
– s : horizontal dispersion
–2 Tc: horizontal shift
• c Analysis to find
critical exponents
and temperature
M. Kleine
Berkenbusch et
al., PRL 88
t
s
Result:
= 0.5 +- 0.1
= 2.35 +- 0.05
Tc = 8.3 +- 0.2 MeV
Scaled control parameter
W. Bauer, Michigan State University
Freeze-Out Density
•
•
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pb = 1 -
2
p
G( 23 ,0, B / T )
Percolation model only depends on breaking probability, which can be mapped into a
temperature.
Q: How to map a 2-dimensional
phase diagram?
A: Density related to fragment
rc = (0.35 ± 0.1)r0
energy spectra;
Coulomb many-body expansion
of pre-fragments
WB, Nucl.Phys. A787, 595c (2007)
W. Bauer, Michigan State University
Buckyball Fragmentation
Binding energy
of C60:
420 eV
625 MeV
Xe35+
Cheng et al., PRA 54
W. Bauer, Michigan State University
Cross-Disciplinary Comparison
Left: Nuclear Multifragmentation
Right: Buckyball Fragmentation
Histograms: Percolation Models
o Similarities:
ç U - shape (b-integration)
ç Power-law for imf’s
(1.3 vs. 2.6)
ç Binding energy effects
provide fine structure
o
o
o
Data: Bujak et al., PRC 32
LeBrun et al., PRL 72
Calc.: WB, PRC 38
Cheng et al., PRA 54
W. Bauer, Michigan State University
W. Bauer, Michigan State University
W. Bauer, Michigan State University
Colleagues
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Students
Larry Phair
Holger Harreis
Marko Kleine Berkenbusch
Brandon Alleman
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Collaborators (Theory)
Scott Pratt
Kerstin Paech
Ulrich Mosel
Ulrich Post
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Collaborators (Experiment)
Vic Viola (and Indiana group)
Konrad Gelbke (and MSU group)
Don Gemmel (and ANL group)
Funding
US National Science Foundation
A.v.Humboldt Forschungspreis
W. Bauer, Michigan State University