Conclusive Evidence for a Continuous Phase Transition

1
Conclusive Evidence for a Continuous Phase
Transition

• Wolfgang Bauer
• National Superconducting Cyclotron Laboratory
• and Department of Physics and Astronomy
• Michigan State University

2
Nuclear Matter Phase Diagram

• Two (at least) 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

Source NUCLEAR SCIENCE, A TeachersGuide to the
Nuclear Science Wall Chart,Figure 9-2
3
Dynamics

• Transient formation of non-compact structures
• Sheet instabilitiesMoretto et al., PRL 69, 1884
  (1992)
• Bubble and ring formationWB, Schulz, Bertsch,
  PRL 69, 1888 (1992)

• Thermal equilibriumassumptions notvalid
• Need transporttheory
• Various event class averages
• Connections to underlying phase diagram poorly
  understood

Molecular dynamics approaches (Wilets,
Boal, Aichelin, Bonasera,
Feldmeier,
Horiuchi,
Ohnishi, )
4
Volume

• Information from interferometry
• Two-particle correlation are sensitive to
  space-time extension of emitting
  sourceC(P,q)?d3x Fp(r) f(r,q)2

C(q)
Relative Momentum, q
W.B., Gelbke, Pratt, Annu.Rev.Nuc.Part.Sci. 42,
77 (1992)
5
Temperature
J. Pochodzalla, CRIS 96

• Measure nuclear temperature indirectly via
• Slopes of charged particle spectra
• Bound-state populations
• Unbound states
• Fragment isotopic yieldsHe-Li
  thermometerAlbergo et al., Nuovo C. A89,1
  (1985)

Central question At which time do we measure the
temperature with each thermometer?
6
Temperature from Fragment Spectra

• Nucleon momentumdistribution at temp. T
• Fragment momentum sum of momenta of nucleons in
  it
• Problem equivalent to solving Pearson random walk
  in momentum space
• Limiting distribution (Boltzmann with
  )
• Fragment slope temperature, Teff, is notequal
  to T, but is a monotonous function of it?
  Nuclear Thermometer
• ApproximationWB, Phys. Rev. C 51, 803 (1995)

Teff /ef
Tin/ef
7
Isospin RIA Reaction Physics

• Exploration of the drip lines belowcharge Z40
  via projectilefragmentation reactions
• Determination of the isospin degree of freedom
  in thenuclear equation of state
• Astrophysical relevance
• ReviewLi, Ko, WB, Int.J. Mod. Phys. E 7(2),
  147 (1998)

r-process
rp-process
8
Width of Isotope Distribution,Sequential Decays

• Predictions for width of isotope distribution are
  sensitive to isospin term in nuclear EoS
• ComplicationSequential decay almost totally
  dominates experimentally observable fragment
  yieldsPratt, WB, Morling, Underhill,PRC 63,
  034608 (2001).

9
First-Order Phase Transition

• Coexistence of two phases (e.g.
  icewater,watersteam)
• Addition of heat does not change
  temperatureLatent heat(H2O Lf 80 kcal/kg,
  Lv 540 kcal/kg)
• Different specific heat capacities in the
  different phases -gt different slopes T vs. Q
• Pressure kept constant!

Verlag Harri Deutsch, Frankfurt 1999
10
Observation of First-Order Phase Transition?

• Low E Liquid-likeT E1/2
• High E Gas-like T E
• 1st order transition Liquid-gas coexistence
• Temperature does not change in phase mixture
  while liquid is converted to vapor.
• Analogy Boiling of water
• BUT
• Pressure not kept constant
• Finite system

J.Pochodzalla et al. (ALADIN), Phys. Rev. Lett.
75,1040 (1995)
11
Buckyball-Melting

• C60 Cluster
• Soccer Ball Geometry

• Molecular dynamics calculations
• Hoover-Nose heat bath

S.G. Kim D. Tomanek,PRL 72, 2418 (1994)
12
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
• t, Cluster size ns(pc) ? s-t
• b, Order parameter P ? (p-pc)b
• g, Divergence of s ? p-pc-g
• Hyper-scaling assumption
  2-a (t - 1) / s 2b g

13
Finite Size Scaling

• Phase transitions strictly only defined for
  (almost) infinite systems
• Lattice calculations work on finite lattices and
  extrapolate to infinite lattices
• Finite size scaling exponent, n
• Modify control parameter by L1/n
• Modify orderparameter by Lb /n

Thorpe, MSU
14
Multi-Component Systems

• What happens when physically different components
  are in the system undergoing phase transition?
• (protons neutrons, different flavor quarks
  gluons, )
• Possible
• Change of character of phase transitionMüllerSe
  rot, PRC 52, 2072 (1995)
• Shift in critical value of controlparameter,
  same criticalexponentsHarreisWB, PRB 62, 8719
  (2000)

15
E-by-E

• Near critical point, information on fluctuations
  is essential, averaging destroys it
• Promising candidates E-by-E moment analyses
  Mk(e) Si ne(i)
  ik.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 charged particle
  multiplicity, m.

16
Determining Critical Exponents?

• EOS-TPCGilkes et al., PRL 73, 1590 (1994)
• Complete reconstruction of events all charges
  recovered
• Assume charged-particle multiplicity is
  proportional to control parameter

• Find critical value, mc extract
  criticalexponents b and g g 1.4,
  b 0.29
• Assuming validity of hyper-scalinguniversality
  class of transition is completely determined

17
Percolation
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)

• Short-range NN force nucleons incontact w.
  nearest neighbors
• Expansion (thermal, compression driven,
  dynamical, )
• Bonds between nucleons rupture
• Remaining bonds bind nucleons into fragments
• One control parameter bond breaking probability

time
18
Breaking Probability

• Determined by the excitation energy deposited
• Infinite simple cubic lattice
• 3 bonds/nucleon
• It takes 5.25 MeV to break a bond

• p,p induced eikonal approximation
• pbreak proportional to path length through matter
• General relation between pbreak and TG
  generalized incomplete gamma function, B
  binding energy per nucleonT. Li et al., PRL 70,
  1924 (1993)
• Obtain E or T from other model or from experiment

19
AA Collisions Hybrid Model
1 A GeV Au C

• First stage Intra-nuclear cascade
• Produces distribution of residue sizes and E
• Convert E into temperature and percolation
  breaking probability
• Second stage Percolation model with lattice size
  charge of residue
• Produces fragments
• Total multiplicity INC pre-equil. percolation
  output

20
EoS Data / Percolation
all fragments
M2

• 1 A GeV Au C
• Data are integrated over all residue sizes and
  excitation energies
• Complete detection of all charges
• Data black circles
• Percolation model red histograms
• Percolation model contains critical events ?
  Strong indirect evidence for 2nd order phase
  transition
• But g 1.80, b 0.41
• WB A.Botvina, PRC 52, R1760 (1995)WB
  A.Botvina, PRC 55, 546 (1997)

Without largest fragment
Total multiplicity
21
ISiS BNL Experiment

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

22
Collision Cartoon
23
Influence of Sequential Decays
Critical fluctuations
Blurring due to sequential decays
24
ISIS Data Analysis

• Marko Kleine Berkenbusch
• Collaboration w. Viola group

• Reaction p, pAu _at_AGS
• Very good statistics (106 complete events)
• Philosophy Dont deal with energydeposition
  models, but take thisinformation from
  experiment!
• Detector acceptance effects crucial
• filtered calculations, instead of corrected data
• Parameter-free calculations

25
ComparisonData Theory

• Charge yield spectrum
• Second moments
• Very good agreement between theory and data
• Filter very important
• Sequential decay corrections huge

26
Scaling 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 -gt critical events
• Idea works for theory
• Note
• Critical events present, pgtpc
• Critical value of pc was corrected for finite
  size of system

27
Effects of Detector Acceptance Filter
Unfiltered
Filtered
28
Scaling of ISIS Data

• Most importantcritical region and explosive
  events probed in experiment
• Possibility to narrow window of critical
  parameters
• t vertical dispersion
• s horizontal dispersion
• Tc horizontal shift
• c2 Analysis to find critical exponents and
  temperature

Kleine Berkenbusch et al., PRL 88, 0022701 (2002)

• Result s 0.5 -0.1 t 2.35 - 0.05 Tc
  8.3 - 0.2 MeV

29
Conclusions

• Non-equilibrium effects make extraction of EoS
  information hard
• Sequential decays blur our view of the important
  physics,but can be dealt with effectively
• View of multi-fragmentation as a critical
  phenomenonis still alive and well
• Circumstantial evidence for the liquid-gas phase
  transitionis at least as substantial as that for
  the QGP transition
• Detailed questions on the values of the state
  variables and quantities like latent heats need
  to be answered
• Finite-size corrections can be calculated (and
  measured!) opportunity for us to contribute to
  larger science community