Transcript Document

Nuclear Multifragmentation
and Zipf’s Law
Wolfgang Bauer
Michigan State University
Work in collaboration with:
Scott Pratt (MSU), Marko Kleine Berkenbusch (Chicago)
Brandon Alleman (Hope College)
Nuclear Matter Phase Diagram
 Two (at least) thermodynamic phase
transitions in nuclear matter:
– “Liquid Gas”
– Hadron gasQGP / chiral restoration
 Problems / Opportunities:
– Finite size effects
– Is there equilibrium?
– Measurement of state
variables (r, T, S, p, …)
– Migration of nuclear system through phase
diagram (expansion, collective flow)
 Structural Phase Transitions (deformation,
spin, pairing, …)
Source: NUCLEAR SCIENCE, A
Teacher’s Guide to the Nuclear
Science Wall Chart,
Figure 9-2
– have similar problems & questions
– lack macroscopic equivalent
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History
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Influence of Sequential Decays
Critical fluctuations
Blurring due to
sequential decays
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Width of Isotope Distribution,
Sequential Decays
 Predictions for width of
isotope distribution are
sensitive to isospin term in
nuclear EoS
 Complication:
Sequential decay almost
totally dominates
experimentally observable
fragment yields
Pratt, WB, Morling, Underhill,
PRC 63, 034608 (2001).
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Isospin: RIA Reaction Physics
 Exploration of the drip lines below
charge Z~40 via projectile
fragmentation reactions
 Determination of the isospin
degree of freedom in the
nuclear equation of state
 Astrophysical relevance
 Review:
r-process
rp-process
B.A. Li, C.M. Ko, WB,
Int. J. Mod. Phys. E 7(2),
147 (1998)
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Cross-Disciplinary Comparison
 Left: Nuclear
Fragmentation
 Right: Buckyball
Fragmentation
 Histograms:
Percolation Models
 Similarities:
– U - shape
(b-integration)
– Power-law for
imf’s
(1.3 vs. 2.6)
– Binding energy
effects provide
fine structure
Data: Bujak et al., PRC 32, 620 (1985)
LeBrun et al., PRL 72, 3965 (1994)
Calc.: W.B., PRC 38, 1297 (1988)
Cheng et al., PRA 54, 3182 (1996)
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Buckyball
Fragmentation
Cheng et al., PRA 54, 3182 (1996)
Binding energy
of C60:
420 eV
625 MeV
Xe35+
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ISiS BNL Experiment
 10.8 GeV p or p + Au
 Indiana Silicon Strip Array
 Experiment performed
at AGS accelerator of
Brookhaven
National
Laboratory
 Vic Viola
et al.
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ISIS Data Analysis
•Marko Kleine Berkenbusch
•Collaboration w. Viola group
 Reaction:
p, 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
Residue
Sizes
Residue
Excitation
Energies
– filtered calculations, instead of
corrected data
 Parameter-free calculations
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Comparison:
Data & Theory
2nd Moments
Charge Yield
Spectrum
 Very good agreement
between theory and data
– Filter very important
– Sequential decay corrections
huge
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Scaling Analysis
 Idea (Elliott et al.): If data
follow scaling function
    T  Tc 
N(Z,T )  Z f  Z
Tc 

with f(0) = 1 (think
“exponential”), then we can use
scaling plot to see if data cross
the point [0,1] -> critical events
 Idea works for theory
 Note:
– Critical events present, p>pc
– Critical value of pc was corrected for
finite size of system
M. Kleine Berkenbusch et al.,
PRL 88, 0022701 (2002)
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Effects of Detector Acceptance Filter
Unfiltered
Filtered
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Scaling of ISIS Data
 Most important: critical region and
explosive events probed in experiment
 Possibility to narrow window of critical
parameters
 : vertical dispersion
 : horizontal dispersion
– Tc: horizontal shift
 c2 Analysis to find
critical exponents
and temperature
Result:
  0.5  0.1
  2.35  0.05
Tc  (8.3  0.2) MeV
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Essential for Scaling of Data:
Correction for Sequential Decays
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The Competition …
Work based on Fisher liquid
drop model
nA  q0 A e
1
(A c0  A )
T
Same conclusion:
Critical point is reached
Result:
  0.54  0.01
  2.18  0.14


Tc  (6.7  0.2) MeV 
J.B. Elliott et al., PRL 88, 042701 (2002)
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IMF Probability Distributions
Moby Dick:
IMF: word with ≥ 10 characters
Nuclear Physics:
IMF: fragment with 20 ≥ Z ≥ 3
System Size is the
determining factor
in the P(n) distributions
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Zipf’s Law
 Back to Linguistics
 Count number of words in a book (in English) and
order the words by their frequency of appearance
 Find that the most frequent word appears twice as
often as next most popular word, three times as
often as 3rd most popular, and so on.
 Astonishing observation!
G. K. Zipf, Human Behavior and the Principle of Least Effort
(Addisson-Wesley, Cambridge, MA, 1949)
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English Word Frequency
201
181
161
1
fn 
n
141
f1
1.4
fn
121
101
81
61
41
21
1
1
21
41
61
81
101 121 141 161 181 201
n
W ord
the
of
and
a
in
to
it
is
was
to
i
for
you
he
be
with
on
that
by
at
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
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British language compound, >4000 texts
2
1
1
1
1
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DJIA-1st Digit
 1st digit of DJIA
is not uniformly
distributed from 1
through 9!
 Consequence of
exponential rise
(~6.9% annual
average
 Also psychological
effects visible
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Zipf’s Law in Percolation
 Sort clusters according
to size at critical point
 Largest cluster is n
times bigger than nth
largest cluster
M. Watanabe, PRE 53 (‘96)
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Zipf’s Law in Fragmentation
 Calculation with Lattice
Gas Model
 Fit largest fragments
to
An = c n-
 At critical T:
 crosses 1
 New way to detect
criticality (?)
Y.G. Ma, PRL 83 (‘99)
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Zipf’s Law: First Attempt
N (A,T )  aA  f [A (T  Tc )]
<A1>/<Ar>
at Tc : f (0)  1 
N (A,Tc )  aA 
rank, r
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Zipf’s Law: Probabilities (1)
 Probability that cluster of size A is the largest one =
probability that at least one cluster of size A is present
times probability that there are 0 clusters of size >A
P1st (A)  p1 (A)  p0 ( A)
 [1  p0 (A)] p0 ( A)
 N(A) = average yield of size A: N(A) = aA-
 N(>A) = average yield of size A: (V = event size)
N( A) 
V
V
i  A1
i  A1
 N(i)   aA  a ( ,1  A)  a ( ,1  V )

 Normalization constant a from condition:
V
 A  N(A)  V
A1
V
a  V /  A1  V / HV(1 )
A1
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Zipf’s Law: Probabilities (2)
 Use Poisson statistics for individual probabilities:
N(i) e N (i )
pn (i) 
n!
p0 (i)  e N (i ) ; p1 (i)  N(i) p0 (i); p2 (i) 
n
1
2
N(i) p1 (i)...
 Put it all together:
P1st (A)  [1  p0 (A)] p0 ( A)
 [1  e N (A) ] e[a ( ,1 A)a ( ,1V )]
 Average size of biggest cluster
V
A1st   A  P1st (A)
A1
(Exact expression!)
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Zipf’s Law: Probabilities (3)
 Probability for given A to be 2nd biggest cluster:
P2nd (A)  p2 (A)  p0 ( A)  p1 (A)  p1 ( A)
 [1  p0 (A)  p1 (A)] p0 ( A)  [1  p0 (A)] p1 ( A)
 Average size of 2nd
biggest cluster:
V
A2nd   A  P2nd (A)
A1
 And so on … (recursion
relations!)
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Zipf’s Law: -dependence
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A1 / An
Verdict: Zipf’s Law does not work
for multifragmentation, even at the
critical point! (but it’s close)
Series1
2.00
Series2
2.18
Series3
2.33
Series4
2.50
Series5
2.70
Series6
3.00
Series7
5.00
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Expectation
if Zipf’s Law
was exact
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8
6
4
2
0
1
2
3
4
5
6
7
n
Resulting distributions: Zipf Mandelbrot
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W.B., Pratt (2005)
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Human Genome
 1-d partitioning problem of gene length distribution
on DNA
 Human DNA consist of 3G base pairs on 46
chromosomes, grouped into codons of length 3 base
pairs
– Introns form genes
– Interspersed by exons; “junk DNA”
QuickT i me™ and a
T IFF (Uncompressed) decom pressor
are needed to see this picture.
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Computer Hard Drive
 Genome like a computer hard
drive.
 Memory is like chromosomes.
 A files analogous to genes.
 To delete a file, or gene,
delete beginning.
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Recursive Method
Number of ways a length A string can split into m pieces
with no piece larger than i.
i
mj
N  A, m, i    N  A  j, m  1, i 
A
j
Probability the lth longest piece has length i
A l 1 A
m!
N  A  l   k i, m  l , i  1N  A  Asmall  ik , k , imax 

Asmall k 0 l 1 k!l   k !m  l !
Pl , i  
N  A, m, imax 
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Simulation
 Random numbers are
generated to determine
where cuts are made.
 Here length is 300 and
number of pieces is 30.
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Assumption: Relaxed Total Size
 The number of pieces falls exponentially.
 i
ni   Ce
 From this assumption the average piece size is obtained.
1
i

 Also, the average size of the longest piece.
 2A 
P  i ln

 i 
1
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Power Law – Percolation Theory
 Assumes pieces fall according to a power law.
na   Ca

 Average length of piece N is:
 N 1 
1


  1   C  1

N 
P 


 N     1 
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Gene Data
Alleman, Pratt, WB 2005
Data from
Chromosomes 1, 2, 7,
10, 17, and Y.
Plotted against
Exponential and
Power Law models in
Green.
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Summary
 Scaling analysis (properly corrected for decays and
feeding) is useful to extract critical point
parameters.
 “Zipf’s Law” does not work as advertised, but
analysis along these lines can dig up useful
information on critical exponent , finite size
scaling, self-organized criticality
 Gene length distribution as a 1d partitioning
problem is interesting and not solved
Research funded by US National Science Foundation
Grant PHY-0245009
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