New Observations on Fragment Multiplicities Wolfgang Bauer Michigan State University Work in collaboration with: Scott Pratt (MSU), Marko Kleine Berkenbusch

New Observations on Fragment New Observations on Fragment MultiplicitiesMultiplicities

New Observations on Fragment New Observations on Fragment MultiplicitiesMultiplicities

Wolfgang BauerMichigan State University

Work in collaboration with:Scott Pratt (MSU), Marko Kleine Berkenbusch (Chicago)Brandon Alleman (Hope College)

22nd WWND - Wolfgang Bauer 2

Two (at least) thermodynamic phase transitions in nuclear matter:– “Liquid Gas”– Hadron gasQGP / chiral restoration

Goal: Determine Order &Universality Class

Problems / Opportunities:– Finite size effects– Is there equilibrium?– Measurement of state

variables (, 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

Nuclear Matter Phase DiagramNuclear Matter Phase DiagramNuclear Matter Phase DiagramNuclear Matter Phase Diagram

Source: NUCLEAR SCIENCE, A Teacher’s Guide to the Nuclear Science Wall Chart,Figure 9-2


Width of Isotope Distribution,Width of Isotope Distribution,Sequential DecaysSequential Decays

Width of Isotope Distribution,Width of Isotope Distribution,Sequential DecaysSequential 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 yieldsPratt, Bauer, Morling, Underhill,PRC 63, 034608 (2001).


Isospin: RIA Reaction PhysicsIsospin: RIA Reaction PhysicsIsospin: RIA Reaction PhysicsIsospin: RIA Reaction Physics

rp-processrp-process

r-processr-process Exploration of the drip lines belowcharge Z~40 via projectilefragmentation reactions

Determination of the isospin degree of freedom in thenuclear equation of state

Astrophysical relevance

Review:B.A. Li, C.M. Ko, W. Bauer, Int. J. Mod. Phys. E 7(2), 147 (1998)


Cross-Disciplinary ComparisonCross-Disciplinary ComparisonCross-Disciplinary ComparisonCross-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 energyeffects providefine 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)


Buckyball Buckyball FragmentationFragmentation

Buckyball Buckyball FragmentationFragmentation

625 MeVXe35+

Cheng et al., PRA 54, 3182 (1996)

Binding energy of C60: 420 eV


Symmetric A+A collisions Bubble and toroid formation Imaginary sound velocity

Could also be a problem/opportunity for CBM @ FAIR!

CompressionCompressionCompressionCompression

vs2 < 0


ISiS BNL ISiS BNL ExperimentExperiment

ISiS BNL ISiS BNL ExperimentExperiment

10.8 GeV p or + Au Indiana Silicon Strip Array Experiment performedat AGS accelerator ofBrookhaven National Laboratory

Vic Viola et al.


ISIS Data ISIS Data AnalysisAnalysisISIS Data ISIS Data AnalysisAnalysis

Reaction: p, +Au @AGS

Very good statistics (~106 complete events)

Philosophy: Don’t deal with energydeposition models, but take thisinformation from experiment!

Detector acceptance effects crucial– filtered calculations, instead of corrected

data Parameter-free calculations

•Marko Kleine Berkenbusch•Collaboration w. Viola group

ResidueSizes

ResidueExcitationEnergies


Comparison: Comparison: Data & TheoryData & TheoryComparison: Comparison: Data & TheoryData & Theory

Very good agreement between theory and data– Filter very important– Sequential decay corrections huge

2nd Moments

Charge Yield Spectrum


Scaling AnalysisScaling AnalysisScaling AnalysisScaling Analysis 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

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)

N(Z,T )=Z−τ f Zσ T −Tc

Tc

⎛

⎝⎜⎞

⎠⎟


Detector Acceptance FilterDetector Acceptance FilterDetector Acceptance FilterDetector Acceptance FilterUnfilteredUnfiltered FilteredFiltered


Scaling of ISIS DataScaling of ISIS DataScaling of ISIS DataScaling of ISIS Data Most important: critical

region and explosive events probed in experiment

Possibility to narrow window of critical parameters τ: vertical dispersion σ: horizontal dispersion– Tc: horizontal shift

2 Analysis to findcritical exponentsand temperature

Result:

σ =0.5 ±0.1τ =2.35 ±0.05

Tc =(8.3±0.2) MeV

Result:

σ =0.5 ±0.1τ =2.35 ±0.05

Tc =(8.3±0.2) MeV


Essential: Sequential DecaysEssential: Sequential DecaysEssential: Sequential DecaysEssential: Sequential Decays


Result:

σ =0.54 ±0.01τ =2.18 ±0.14

Tc =(6.7 ±0.2) MeV

Result:

σ =0.54 ±0.01τ =2.18 ±0.14

Tc =(6.7 ±0.2) MeV

The Competition …The Competition …The Competition …The Competition …Work based on Fisher liquid drop model

Same conclusion: Critical point is reached

J.B. Elliott et al., PRL 88, 042701 (2002)

nA =q0A−τe

1T(AΔμ−c0εAσ )


Freeze-Out DensityFreeze-Out DensityFreeze-Out DensityFreeze-Out Density 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 energy spectra

pb =1−2Γ( 32 ,0,B /T )

WB, Alleman, Prattnucl-th/0512101


IMF Probability DistributionsIMF Probability DistributionsIMF Probability DistributionsIMF Probability Distributions

Moby DickMoby Dick:IMF: word with ≥ 10 characters

Nuclear PhysicsNuclear Physics:IMF: fragment with 20 ≥ Z ≥ 3

System SizeSystem Size is thedetermining factorin the P(n) distributions

Bauer, Pratt, PRC 59, 2695 (1999)


Zipf’s LawZipf’s LawZipf’s LawZipf’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)


1

21

41

61

81

101

121

141

161

181

201

1 21 41 61 81 101 121 141 161 181 201

Word Rankthe 1of 2

and 3a 4in 5to 6it 7is 8

was 9to 10i 11

for 12you 13he 14be 15

with 16on 17

that 18by 19at 20

1

21

41

61

81

101

121

141

161

181

201

1 21 41 61 81 101 121 141 161 181 201

English Word FrequencyEnglish Word FrequencyEnglish Word FrequencyEnglish Word Frequency

1.4f1fn

n

fn ∝1n⇒

f1fn=nfn ∝

1n⇒

f1fn=n

British language compound, 4124 texts, >100 million words


DJIA-1st DigitDJIA-1st DigitDJIA-1st DigitDJIA-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


Zipf’s Law in PercolationZipf’s Law in PercolationZipf’s Law in PercolationZipf’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)


Zipf’s Law in FragmentationZipf’s Law in FragmentationZipf’s Law in FragmentationZipf’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)


Zipf’s Law: First AttemptZipf’s Law: First AttemptZipf’s Law: First AttemptZipf’s Law: First Attempt

N (A,T )=aA−τ f[Aσ (T −Tc)]at Tc : f (0) =1⇒

N(A,Tc) =aA−τ

rank, r

<A

1>/<

Ar>

ChangeSystemSize


Zipf’s Law: Probabilities (1)Zipf’s Law: Probabilities (1)Zipf’s Law: Probabilities (1)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

N(A) = average yield of size A: N(A) = aA-τ

N(>A) = average yield of size >A: (V = event size)

Normalization constant a from condition:

P1st (A)=p≥1(A)⋅p0 (> A)=[1−p0 (A)] ⋅p0 (> A)

P1st (A)=p≥1(A)⋅p0 (> A)=[1−p0 (A)] ⋅p0 (> A)

N (> A) = N(i) =i=A+1

V

∑ ai−τ =i=A+1

V

∑ aζ(τ,1+ A)−aζ(τ,1+V)N (> A) = N(i) =i=A+1

V

∑ ai−τ =i=A+1

V

∑ aζ(τ,1+ A)−aζ(τ,1+V)

A⋅N(A)A=1

V

∑ =V

a =V / A1−τ

A=1

V

∑ =V / HV(1−τ )a =V / A1−τ

A=1

V

∑ =V / HV(1−τ )


Zipf’s Law: Probabilities (2)Zipf’s Law: Probabilities (2)Zipf’s Law: Probabilities (2)Zipf’s Law: Probabilities (2) Use Poisson statistics for individual probabilities:

Put it all together:

Average size of biggest cluster

(Exact expression!)

pn (i)=N(i) n e− N(i )

n!p0 (i) =e− N(i ) ; p1(i) = N(i) p0 (i); p2 (i) =

12 N(i) p1(i)...

pn (i)=N(i) n e− N(i )

n!p0 (i) =e− N(i ) ; p1(i) = N(i) p0 (i); p2 (i) =

12 N(i) p1(i)...

P1st (A)=[1−p0 (A)] ⋅p0 (> A)

=[1−e−N(A) ] ⋅e−[aζ (τ ,1+A)−aζ (τ ,1+V )]

P1st (A)=[1−p0 (A)] ⋅p0 (> A)

=[1−e−N(A) ] ⋅e−[aζ (τ ,1+A)−aζ (τ ,1+V )]

A1st = A⋅P1st(A)A=1

V

∑A1st = A⋅P1st(A)A=1

V

∑


Zipf’s Law: Probabilities (3)Zipf’s Law: Probabilities (3)Zipf’s Law: Probabilities (3)Zipf’s Law: Probabilities (3) Probability for given A to be 2nd biggest cluster:

Average size of 2nd biggest cluster:

And so on … Recursion relations!

P2nd (A)=p≥2 (A)⋅p0 (> A) + p≥1(A)⋅p1(> A)=[1−p0 (A)−p1(A)] ⋅p0 (> A) + [1−p0 (A)] ⋅p1(> A)

P2nd (A)=p≥2 (A)⋅p0 (> A) + p≥1(A)⋅p1(> A)=[1−p0 (A)−p1(A)] ⋅p0 (> A) + [1−p0 (A)] ⋅p1(> A)

A2nd = A⋅P2nd(A)A=1

V

∑A2nd = A⋅P2nd(A)A=1

V

∑

Bauer, Pratt, Alleman, Heavy Ion Physics, in print (2006)


0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Zipf’s Law: Zipf’s Law: ττ-dependence-dependenceZipf’s Law: Zipf’s Law: ττ-dependence-dependence

2.002.182.332.502.703.005.00 Expectation

if Zipf’s Lawwas exact

A1 / An

n

Verdict: Zipf’s Law does not workfor multifragmentation, even at thecritical point! (but it’s close)

Verdict: Zipf’s Law does not workfor multifragmentation, even at thecritical point! (but it’s close)

Resulting distributions: Zipf Mandelbrot


Zipf-MandelbrotZipf-MandelbrotZipf-MandelbrotZipf-Mandelbrot Limiting distributions for cluster size vs. rank

Exponent

Arth =c

r + k( )Arth =

cr + k( )

~1

τ −1 ~

1

τ −1

WB, Alleman, Prattnucl-th/0511007


SummarySummarySummarySummary

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 τ, finite size scaling, self-organized criticality

Research funded by US National Science FoundationGrant PHY-0245009


Human GenomeHuman GenomeHuman GenomeHuman 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”

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.


Computer Hard DriveComputer Hard DriveComputer Hard DriveComputer 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).


Recursive MethodRecursive MethodRecursive MethodRecursive Method

( ) ( )imjANA

mjimAN

i

j

,1,,, −−=∑

( ) ( )( ) ( )( ) ( )

( )max

1

0 1max

,,

,,1,,!!!

!

,imAN

ikikAANilmiklANlmklk

m

ilP

A

A

l

k

A

lsmall

small

∑∑∑−

= =′

−−−′−−′−′−−′

=

Number of ways a length A string can split into m pieces with no piece larger than i.

Probability the lth longest piece has length i


SimulationSimulationSimulationSimulation

Random numbers are generated to determine where cuts are made.

Here length is 300 and number of pieces is 30.


Assumption: Relaxed Total SizeAssumption: Relaxed Total SizeAssumption: Relaxed Total SizeAssumption: Relaxed Total Size The number of pieces falls exponentially.

From this assumption the average piece size is obtained.

Also, the average size of the longest piece.

μ1

=i

( ) iCein μ−=

( ) ⎟⎠

⎞⎜⎝

⎛=i

AiP

2ln1


Power Law – Percolation TheoryPower Law – Percolation TheoryPower Law – Percolation TheoryPower Law – Percolation Theory Assumes pieces fall according to a power law.

Average length of piece N is:

n a( ) =Ca−τ

P N( ) =Γ

N −1τ −1

⎛⎝⎜

⎞⎠⎟

Γ N( )⋅

Cτ −1

⎛⎝⎜

⎞⎠⎟

1τ −1


Data from Human Chromosomes 1, 2, 7, 10, 17, and Y.

Plotted against Exponential and Power Law models

Gene Data

Alleman, Pratt, Bauer 2005


Influence of Sequential DecaysInfluence of Sequential DecaysInfluence of Sequential DecaysInfluence of Sequential Decays

Critical fluctuations Blurring due to sequential decays

Documents

New Observations on Fragment Multiplicities Wolfgang Bauer Michigan State University Work in collaboration with: Scott Pratt (MSU), Marko Kleine Berkenbusch