Transcript Document

New Observations on Fragment
Multiplicities
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
 Goal: Determine Order &Universality Class
 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, …)
– have similar problems & questions
– lack macroscopic equivalent
22nd WWND - Wolfgang Bauer
Source: NUCLEAR SCIENCE, A
Teacher’s Guide to the Nuclear
Science Wall Chart,
Figure 9-2
2
Width of Isotope Distribution,
Sequential Decays
 Predictions for width of
isotope distribution are quite
sensitive to isospin term in
nuclear EoS
 Complication:
Sequential decay almost
totally dominates
experimentally observable
fragment yields
Pratt, Bauer, 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, W. Bauer,
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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Compression
 Symmetric A+A collisions
 Bubble and toroid formation
 Imaginary sound velocity
vs2  0
 Could also be a problem/opportunity
for CBM @ FAIR!
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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, 022701 (2002)
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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: 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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Freeze-Out Density
 Percolation model only depends on breaking
probability, which can be mapped into a
2
p

1

( 23 ,0, B / T )
temperature. b
p
 Q: How to map a 2-dimensional phase diagram?
 A: Density related to fragment energy spectra
WB, Alleman, Pratt
nucl-th/0512101
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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
Bauer, Pratt, PRC 59, 2695 (1999)
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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
1
f1
fn    n
n
fn
181
161
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
14
15
16
17
18
19
20
2
1
1
1
1
British language compound, 4124 texts, >100 million words 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, 4187 (1996)
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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, 3617 (1999)
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Zipf’s Law: First Attempt
Change
System
Size
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)
V
N( A) 

i  A1
N(i) 
V

ai  a ( ,1  A)  a ( ,1  V )
i  A1
 Normalization constant a from condition:
V
a  V / A
1 
V /H
(1  )
V
V
 A  N(A)  V
A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!
Bauer, Pratt, Alleman, Heavy Ion Physics, in print (2006)
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Zipf’s Law: -dependence
A1 / An
Verdict: Zipf’s Law does not work
for multifragmentation, even at the
critical point! (but it’s close)
20
2.00
Series1
18
2.18
Series2
16
Series3
2.33
14
Series5
2.70
Series4
2.50
Series6
3.00
12
Series7
5.00
Expectation
if Zipf’s Law
was exact
10
8
6
4
2
0
1
2
3
4
5
n
6
7
8
9
10
Resulting distributions: Zipf Mandelbrot
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Zipf-Mandelbrot
 Limiting distributions for cluster size vs. rank
Arth 
 Exponent
c
r  k 

WB, Alleman, Pratt
nucl-th/0511007
1
~
 1
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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
Research funded by US National Science Foundation
Grant PHY-0245009
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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.
 Files analogous to genes.
 To delete a file, or gene,
delete entry point (= start
codon).
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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 
N 
P 
 N 
 C 

   1
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 1
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Gene Data
Data from Human
Chromosomes 1, 2, 7,
10, 17, and Y.
Plotted against
Exponential and
Power Law models
Alleman, Pratt, Bauer 2005
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Influence of Sequential Decays
Critical fluctuations
Blurring due to
sequential decays
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