INFORMATION ON THE NUCLEAR EQUATION OF STATE FROM MULTIFRAGMENTATION STUDIES

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1 INFORMATION ON THE NUCLEAR EQUATION OF STATE FROM MULTIFRAGMENTATION STUDIES Wolfgang Bauer Michigan State University Work in collaboration with: Scott Pratt (MSU), Marko Kleine Berkenbusch (Chicago) Brandon Alleman (Hope College)

2 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 (ρ, 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 Source: NUCLEAR SCIENCE, A Teacher s Guide to the Nuclear Science Wall Chart, Figure 9-2 W. Bauer 2

3 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, (2001). W. Bauer 3

4 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 r-process Review: B.A. Li, C.M. Ko, W. Bauer, Int. J. Mod. Phys. E 7(2), 147 (1998) rp-process W. Bauer 4

5 initial final Percolation Model Applies to systems with local finite-range interactions Nuclear fragmentation Many other exploding systems Combustion processes Movement of liquids through porous materials Traffic flow Biological systems Percolation model has critical point in infinite size limit, but can also be used for finite systems Finite-size scaling Can be realized on lattices, but also random percolation without a lattice Universality class (= set of critical exponents) is dependent on spatial dimensionality Location of critical point depends on coordination number Connection with minimum spanning tree and molecular dynamics algorithms W. Bauer 5

6 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) W. Bauer 6

7 Buckyball Fragmentation Cheng et al., PRA 54, 3182 (1996) Binding energy of C 60 : 420 ev 625 MeV Xe 35+ W. Bauer 7

8 Compression Symmetric A+A collisions Bubble and toroid formation Imaginary sound velocity v s 2 < 0 Could also be a problem/opportunity for FAIR! W. Bauer 8

9 ISiS BNL Experiment 10.8 GeV p or π + Au Indiana Silicon Strip Array Experiment performed at AGS accelerator of Brookhaven National Laboratory Vic Viola et al. W. Bauer 9

10 Reaction: p, Very good statistics (~10 6 complete events) ISIS Data Analysis Marko Kleine Berkenbusch Collaboration w. Viola group 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 Residue Sizes Residue Excitation Energies W. Bauer 10

11 Comparison: Data & Theory 2 nd Moments Charge Yield Spectrum Very good agreement between theory and data Filter very important Sequential decay corrections huge W. Bauer 11

12 Scaling Analysis Idea (Elliott et al.): If data follow scaling function $ T! T N(Z,T ) = Z!" f Z # c ' % & T c ( ) 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>p c Critical value of p c was corrected for finite size of system M. Kleine Berkenbusch et al., PRL 88, (2002) W. Bauer 12

13 Unfiltered Detector Acceptance Filter Filtered W. Bauer 13

14 Scaling 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 T c : horizontal shift χ 2 Analysis to find critical exponents and temperature Result:! = 0.5 ± 0.1 " = 2.35 ± 0.05 T c = (8.3 ± 0.2) MeV W. Bauer 14

15 Essential: Sequential Decays W. Bauer 15

16 Work based on Fisher liquid drop model n A = q 0 A!" e 1 T (A#µ!c 0$ A % ) Same conclusion: Critical point is reached The Competition Result:! = 0.54 ± 0.01 " = 2.18 ± 0.14 T c = (6.7 ± 0.2) MeV J.B. Elliott et al., PRL 88, (2002) W. Bauer 16

17 Freeze-Out Density Percolation model only depends on breaking probability, which can be mapped into a temperature. p b = 1! 2 " #( 3,0, B / T ) 2 Q: How to map a 2-dimensional phase diagram? A: Density related to fragment energy spectra WB, Alleman, Pratt nucl-th/ W. Bauer 17

18 Moby Dick: IMF: word with 10 characters Nuclear Physics: IMF: fragment with 20 Z 3 IMF Probability Distributions System Size is the determining factor in the P(n) distributions Bauer, Pratt, PRC 59, 2695 (1999) W. Bauer 18

19 Back to Linguistics Zipf s Law 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) W. Bauer 19

20 English Word Frequency f n! 1 n " f 1 f n = n 1.4 f 1 f n n British language compound, 4124 texts, >100 million words W. Bauer 20

21 Sort clusters according to size at critical point Largest cluster is n times bigger than n th largest cluster Zipf s Law in Percolation M. Watanabe, PRE 53, 4187 (1996) W. Bauer 21

22 Calculation with Lattice Gas Model Fit largest fragments to A n = c n -λ At critical T: λ crosses 1 New way to detect criticality (?) Y.G. Ma, PRL 83, 3617 (1999) Zipf s Law in Fragmentation W. Bauer 22

23 Zipf s Law: First Attempt Change System Size <A 1 >/<A r > N(A,T ) = aa!" f [A # (T! T c )] at T c : f (0) = 1 $ N(A,T c ) = aa!" rank, r W. Bauer 23

24 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 P 1st (A) = p!1 (A) " p 0 (> A) = [1# p 0 (A)]" p 0 (> A) N(A) = average yield of size A: N(A) = aa -τ N(>A) = average yield of size >A: (V = event size) V N(> A) =! N(i) =! ai "# = a$(#,1+ A) " a$(#,1 + V ) i= A+1 i= A+1 Normalization constant a from condition: V a = V / # A 1!" (1!" ) = V / H V A=1 V V " A! N(A) = V A=1 W. Bauer 24

25 Zipf s Law: Probabilities (2) Use Poisson statistics for individual probabilities: p n (i) = N(i) n e! N (i) Put it all together: Average size of biggest cluster n! p 0 (i) = e! N (i) ; p 1 (i) = N(i) p 0 (i); p 2 (i) = 1 2 P 1st (A) = [1! p 0 (A)]" p 0 (> A) (Exact expression!) N(i) p 1 (i)... = [1! e! N (A)![a# ($,1+ A)!a# ($,1+V )] ]" e V " A=1 A 1st = A! P 1st (A) W. Bauer 25

26 Zipf s Law: Probabilities (3) Probability for given A to be 2nd biggest cluster: P 2nd (A) = p!2 (A) " p 0 (> A) + p!1 (A) " p 1 (> A) Average size of 2nd biggest cluster: And so on = [1 # p 0 (A) # p 1 (A)]" p 0 (> A) + [1 # p 0 (A)]" p 1 (> A) V " A=1 A 2nd = A! P 2nd (A) Recursion relations! W.B., Pratt, Alleman, Heavy Ion Physics, in print (2006) W. Bauer 26

27 A 1 / A n Zipf s Law: τ-dependence Verdict: Zipf s Law does not work for multifragmentation, even at the critical point! (but it s close) 5.00 Expectation if Zipf s Law was exact n Resulting distributions: Zipf Mandelbrot W. Bauer 27

28 Zipf-Mandelbrot Limiting distributions for cluster size vs. rank Exponent A rth = c ( r + k)!! ~ 1 " # 1 WB, Alleman, Pratt nucl-th/ W. Bauer 28

29 Summary Scaling analysis (properly corrected for decays and feeding) is useful to extract critical point parameters. Result:! = 0.5 ± 0.1 " = 2.35 ± 0.05 T c = (8.3 ± 0.2) MeV 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 Foundation Grants PHY , PHY W. Bauer 29