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Probing Reaction Dynamics with Two-Particle Correlations

This study investigates the dynamics of reactions by analyzing two-particle correlations. The experiment focuses on p-p correlations and uses transport theory to study neutron and proton emission times and symmetry energy. The study also examines the momentum and rapidity dependence of correlation functions and fits the data to the Koonin-Pratt equation. The results provide insights into the size of the p-p source and its sensitivity to initial conditions and symmetry energy. Additionally, the study compares data to theoretical models and explores the emission of neutrons and protons in different scenarios.

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Probing Reaction Dynamics with Two-Particle Correlations

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  1. Probing reaction dynamics with two-particle correlations Zbigniew Chajęcki National Superconducting Cyclotron Laboratory Michigan State University

  2. Outline • p-p correlations (work with M. Kilburn, B. Lynch and collaborators) • NSCL 03045 Experiment • transport theory (BUU) • neutron and proton emission times and symmetry energy(particle emission chronology) • transport theory • Summary Z. Ch. - NuSYM 2011, June 17-20, 2011

  3. Experimental correlation function (p,p) correlation function |q| = 0.5 |p1 - p2| p1 Experimental correlation function: x1 r x2 p2 few fm P(p1,p2) P(p1)P(p2) |q| = 0.5 |p1 - p2| Z. Ch. - NuSYM 2011, June 17-20, 2011

  4. Femtoscopy (p,p) correlation function (p,p) correlation function S-wave interraction S(r) S(r) S-wave interraction r1/2 Coulomb Coulomb r r uncorrelated uncorrelated 0 0 |q| = 0.5 |p1 - p2| |q| = 0.5 |p1 - p2| Theoretical CF: Koonin-Pratt equation p1 S.E. Koonin, PLB70 (1977) 43 S.Pratt et al., PRC42 (1990) 2646 x1 r … 2-particle wave function … source function x2 p2 few fm Z. Ch. - NuSYM 2011, June 17-20, 2011

  5. NSCL experiments 05045: HiRA + 4 detector = High Resolution Array beam • 4π detector => impact parameter + reaction plane • HiRA => light charge particle correlations (angular coverage 20-60º in LAB, • 63 cm from target (= ball center)) Reaction systems: 40Ca + 40Ca @ 80 MeV/u 48Ca + 48Ca @ 80 MeV/u Z. Ch. - NuSYM 2011, June 17-20, 2011

  6. Momentum and rapidity dependence C(q) Measured correlation functions depend on rapidity and the transverse momentum of the pair Next step:extract the sizes Z. Ch. - NuSYM 2011, June 17-20, 2011

  7. Fits to the data C(q) Koonin-Pratt Equation Brown, Danielewicz, PLB398 (1997) 252 Danielewicz, Pratt, PLB618 (2005) 60 Two ways of characterizing the size of the p-p source S(r) - Gaussian shape Imaged S(r) (Brown, Danielewicz) Z. Ch. - NuSYM 2011, June 17-20, 2011

  8. Fits to the data C(q) Koonin-Pratt Equation Brown, Danielewicz, PLB398 (1997) 252 Danielewicz, Pratt, PLB618 (2005) 60 Two ways of characterizing the size of the p-p source S(r) - Gaussian shape Imaged S(r) (Brown, Danielewicz) Both methods give consistent fits Z. Ch. - NuSYM 2011, June 17-20, 2011

  9. Fits to the data r1/2 Source distribution : S(r) x103 Correlation function C(Q) Z. Ch. - NuSYM 2011, June 17-20, 2011

  10. Fit results Small rapidity:reflect the participant zone of the reaction Large rapidity:reflect the expanding, fragmenting and evaporating projectile-like residues Higher velocity protons are more strongly correlated than their lower velocity counterparts, consistent with emission from expanding and cooling sources Sensitivity to the initial size Z. Ch. - NuSYM 2011, June 17-20, 2011

  11. Modeling heavy-ion collisions : transport models Danielewicz, Bertsch, NPA533 (1991) 712 B. A. Li et al., PRL 78 (1997) 1644 • BUU - Boltzmann-Uehling-Uhlenbeck • Simulates two nuclei colliding Micha KilburnNSCL/MSU • Parameter space • not only about the symmetry energy • also important to understand e.g. an effect of cross section (free x-section, in-medium x-section), reduced mass • Production of clusters: d,t, 3He (alphas) Z. Ch. - NuSYM 2011, June 17-20, 2011

  12. Comparing data to theory (pBUU) . BUU Pararameters • No dependence on symmetry energy • Rostock in-medium reduction • Producing clusters BUU does reasonably well Except at larger rapidities - Spectator source Where evaporation and secondary decays are important! Micha Kilburn, NSCL/MSU Z. Ch. - NuSYM 2011, June 17-20, 2011

  13. Averaged emission time of particles in transport theory Z. Ch. - NuSYM 2011, June 17-20, 2011

  14. Emission of p’s and n’s: Sensitivity to SymEn 52Ca 48Ca Stiff EoS L-W Chen et al., PRL90 (2003) 162701 Soft EoS Soft Stiff EoS (γ=2) Soft EoS (γ=0.5) p’s and n’s emitted at similar time fasteremission times p’s emittedafter n’s later emission times Stiff Z. Ch. - NuSYM 2011, June 17-20, 2011

  15. n-p correlation function (n,p) correlation function (n,p) correlation function S(x) S(x) x x 0 0 Theoretical CF: Koonin-Pratt equation p1 S.E. Koonin, PLB70 (1977) 43 S.Pratt et al., PRC42 (1990) 2646 x1 r … 2-particle wave function … source function x2 p2 few fm q = 0.5(p1 - p2) Z. Ch. - NuSYM 2011, June 17-20, 2011

  16. Emission of p’s and n’s: Sensitivity to SymEn 52Ca 48Ca Stiff EoS L-W Chen et al., PRL90 (2003) 162701 Soft EoS Soft EoS (γ=0.5) Stiff EoS (γ=2) p’s emitted after n’s later emission times p’s and n’s emitted at similar time fasteremission times Z. Ch. - NuSYM 2011, June 17-20, 2011

  17. Possible emission configurations (stiff sym. pot.) n n n n p p p p Catching up Catching up qx<0 qx>0 Moving away Moving away qx<0 qx>0 (n,p) correlation function q=pp -pn =(qx, qy=0,qz=0); r=(x, y=0,z=0) qx<0 qx>0 S(x) x 0 q = 0.5(pp - pn) Z. Ch. - NuSYM 2011, June 17-20, 2011

  18. Emission of p’s and n’s: Sensitivity to SymEn 52Ca 48Ca Stiff EoS L-W Chen et al., PRL90 (2003) 162701 Soft EoS Soft EoS (γ=0.5) Stiff EoS (γ=2) p’s emitted after n’s later emission times p’s and n’s emitted at similar time fasteremission times Z. Ch. - NuSYM 2011, June 17-20, 2011

  19. Sensitivity to particle emission (soft sym. pot.) n n p p Experimentally, we measure the CF, not the source distribution! Moving away Catching up qx<0 qx>0 (n,p) correlation function qx<0 qx>0 S(x) x 0 q=pp -pn =(qx, qy=0,qz=0); r=(x, y=0,z=0) qx = 0.5(px,p - px,n) Z. Ch. - NuSYM 2011, June 17-20, 2011

  20. Relating asymmetry in the CF to space-time asymmetry Stiff EoS Soft EoS (n,p) correlation function qx<0 qx>0 S(x) <x> x 0 qx = 0.5(px,p - px,n) Classically, average separation b/t protons and neutrons Not expected if n,p emitted from the same source (no n-p differential flow) =0 Protons emitted later Voloshin et al., PRL 79:4766-4769,1997Lednicky et al., PLB 373:30-34,1996 Z. Ch. - NuSYM 2011, June 17-20, 2011

  21. IBUU: more calculations Figure obtained from calculations with momentum-independent potential Calculations with momentum-dependent nuclear potential L-W Chen et al., PRL90 (2003) 162701 Soft AsyEoS Stiff AsyEoS L-W Chen et al., PRC69 (2004) 054606 Z. Ch. - NuSYM 2011, June 17-20, 2011

  22. IBUU: averaged emission time Momentum dependent (isoscalar) Momentum independent Momentum dependent (isoscalar & isovector) 52Ca+48Ca @ 80 MeVA Z. Ch. - NuSYM 2011, June 17-20, 2011

  23. IBUU vs pBUU: Averaged emission time pBUU IBUU 52Ca+48Ca @ 80 MeVA Z. Ch. - NuSYM 2011, June 17-20, 2011

  24. pBUU: Averaged emission time Danielewicz, Bertsch, NPA533 (1991) 712 momentum dependent WITHOUT CLUSTERS WITH CLUSTERS No effect of symmetry energy on averaged emission time of particles Clusters affect the space-time picture of the HIC (t-3He correlations could show possible sensitivity to the relative emission time analogously to n-p correlations) Z. Ch. - NuSYM 2011, June 17-20, 2011

  25. Summary • Two particle correlations provide a unique probe to study the space-time extend of the source • add constrains on the in-medium cross-section • importance of the clusters, symmetry energy • validate theoretical models • The average relative emission time of n’s and p’s potentially sensitive to the symmetry energy and can be “measured” with two particle correlations • Transport models • Predictions are model dependent • Collaboration between theorists and experimentalists beneficial for both sides Z. Ch. - NuSYM 2011, June 17-20, 2011

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