Study of spin dynamics in heavy-ion collisions at intermediate energies

Transcription

1 核学会 04 年会 Zhangjiang campus, Nov. 7, 04 Study of spin dynamics in heavy-ion collisions at intermediate energies Collaborators: Jun Xu ( 徐骏 ) SINAP, CAS, China: Yin Xia, Wen-Qing Shen Texas A&M-Commerce, USA: Bao-An Li Jun Xu and Bao-An Li, Phys. Lett. B 74, 46 (0) Yin Xia, Jun Xu*, Bao-An Li, and Wen-Qing Shen, Phys. Rev. C 89, (04) Yin Xia, Jun Xu*, Bao-An Li, and Wen-Qing Shen, arxiv: [nucl-th]

2 Spin-orbit potential and magic number Otto Haxel p hq Uq Wq ( p ),( q n, p) m Schrödinger equation: hq q eqq The spin-orbit (SO) potential W ( p q ) helps to explain the magic number and shell structure. From in-medium spin-orbit nuclear interaction

3 Skyrme-Hartree-Fock model Hartree-Fock method W 0 Wq Relativistic mean field model Dirac equation Non-relativistic expansion W q P. G. Reinhard and H. Flocard, Nucl. Phys. A, 995 C q, C ( m C) m m g g the isospin dependence of the SO potential kink of Pb isotopes charge radii M. M. Sharma et al., Phys. Rev. Lett., 995

4 the density dependence of the SO potential bubble nucleus J. M. Pearson and M. Farine, Phys. Rev. C 50, 85 (994). M. Grasso et al., Phys. Rev. C, 009 Generally W q W0 ( aq bq) ( q q) 0 density dependence W a b 0 T. Lesinski et al., Phys. Rev. C 76, 04 (007). M. Zalewski et al., Phys. Rev. C 77, 046 (008). M. Bender et al., Phys. Rev. C 80, 0640 (009). isospin dependence = 80 ~ 50 MeVfm 5,,, and still under debate

5 The spin-orbit interaction may affect ) Properties of drip-line nuclei ) Astrophysical r-process ) Location of SHE 4) G. A. Lalazissis et al., Phys. Rev. Lett., 998 B. Chen et al., Phys. Lett. B, 995 M. Morjean et al., Phys. Rev. Lett., 008 J.P. Schiffer et al., Phys. Rev. Lett., 004 A hot topic in the studies of nuclear structure! Shell structure of M. Bender et al., Phys. Rev. C, 999

6 Spin related experiments Spin-polarized beam at RIKEN, GSI, NSCL, GANIL can be produced with pick-up or removal reactions. The spin alignment of projectile fragment can be measured through the angular distribution of its γ or β decay.

7 Analyzing power measurement at AGS and RHIC A LU LU RD RD LD LD RU RU The analyzing power can be as large as 00% at certain angles and energies Providing a possible way of identifying nucleon spin Silicon detector Recoil Carbon, kev Polarized Proton Beam Ultra-thin Carbon Ribbon approx. 00 atomic layers thick Forward proton p+ C@00 MeV

8 Spin-orbit potential at low- and high-energy HIC low energies (TDHF): high energies: A. S. Umar et al., Phys. Rev. Lett., 986 b impact parameter Y n re pin b p b in perpendicular to the reaction plane J. A. Maruhn et al., Phys. Rev. C, 006 Z. T. Liang and X. N. Wang, Phys. Rev. Lett., 005 Phys. Lett. B, 005

9 Hall effect Lorentz force F qv B Spin Hall effect Spin-orbit potential U ~ L // L U so q p W p q ( p ) ( pwq ) ~ ( p) // p Spin up +y density distribution spin distribution y Spin down -y Spurious spin polarization! x z

10 Spurious spin twist! Frame dependent!

11 Wait We have time-odd terms Hartree-Fock Galilean invariance J : number density s time-even : spin density : spin-current density j : momentum density j time-odd so p s U // U // j ~ ( p) p Time-odd terms have opposite effect to time-even terms! (exact cancellation without Fermi motion)

12 Introduction to IBUU transport model Phys. Rev. C 9, 67(R) (984) Phys. Rep. 60, 89 (988) U pp Isospin degrees of freedom: n, p U0, np U sym S.K. Charagi and S.K. Gupta, PRC 4, 60 (990) Isospin-dependent Pauli Blocking Isospin-dependent momentum dependence, particle production (π, Δ, N*),

13 How is IBUU initialized? Density initialization Skyrme-Hartree-Fock Read, MC sample, and boost (x,y,z) to the C.M. frame Momentum initialization Local density => local Fermi momentum Boost (px,py,pz) to the C.M. frame L.W. Chen, C. M. Ko, B. A. Li, and JX PRC 8, 04 (00) Macroscopic quantities p ( 0, pf ) SHF parameter set

14 How does the system evolve? r( t dt) r( t) p( t) E dt p( t dt) p( t) U[ ( t)] Update spatial density distribution at each time step System size x (-0,0), y (-0,0), z(-4,4) 40*40*48 cells (ix,iy,iz) ( ix, iy, iz) ~ du( ( ix, iy, iz)) dx n test U[ ( ix ( ix, iy, iz ) ~, iy, iz)] U[ ( ix dx U, For neutrons and protons ) ( n p 9 n test, iy, iz)], dx Coulomb potential from each pair of protons----standard (More complicated for momentum-dependent mean-field potential)

15 How are the collisions treated? Bertsch s approach (Phys. Rep. 60, 89 (988)) Go into the C.M. frame of the two nucleons Collision can happen if and If collision happen, change the direction of P cm in the C.M. frame Boost the momenta of the two nucleons to lab frame Check phase space density; if Pauli blocked, return to the initial momenta

16 How is Pauli blocking treated? Update the isospin-dependent phase-space density at each time step f ( ix, iy, iz, ipx, ipy, ipz) (size of each cell (dx,dy,dz,dpx,dpy,dpz)) When a collision is attempted, check the occupation number of the phase space at (ix,iy,iz,ipx,ipy,ipz) (interpolation) n occup d h * dx * dy * dz * dpx * dpy * dpz f ( ix, iy, iz, ipx, ipy, ipz), d Normalization factor If ran()>n occup, the collision is Pauli blocked; otherwise not.

17 Introduce spin-orbit interaction to IBUU Additional spin-dependent mean-field potential q n, p J s j,,, and from test particle method (C. Y. Wong, PRC 5, 460 (98))

18 Single-particle Hamiltonian: dr phq dt dp hq dt d, h dt i q s Similar to B, j ijk k i i h ~ ds dt s ~ s B B Equations of motion: q n, p

19 a unit vector for each nucleon expectation value of the spin spin- and isospin-dependent Pauli blocking h n occup f d * dx * dy * dz * dpx * dpy * dpz Projection on y direction (total angular momentum) y ( y ) ( y ) f sin Nucleon spin may flip after nucleon-nucleon scattering (randomized?) sin probability probability s s y s y ( ix, iy, iz, ipx, ipy, ipz) Spin- and isospin-dependent phase space distribution function Spin-up Spin-down ( ix, iy, iz, ipx, ipy, ipz), d

20 Time-odd terms overwhelm time-even terms! Results and discussions b = 8 fm W 0 = 50 MeVfm 5 γ = 0 a = b = Time-even: // p Time-odd: // j <

21 Transverse flow p x ~ y Spin up-down differential transverse flow sensitive to nuclear interaction U U 0 U spin ( ) or ( ) reflects different transverse flows of spin-up and spin-down nucleons JX and B.A. Li PLB (0) F ud is sensitive to W 0, the strength of the spin-orbit interaction.

22 Energy and impact parameter dependence The transverse flow repulsive NN scatterings attractive mean-field potential Spin up-down differential transverse flow density gradient (surface) angular momentum (current) violent NN scatterings Y. Xia, JX*, B.A. Li, and W.Q. Shen, PRC (04)

23 Iisospin dependence of SO coupling W Globally a neutron-rich system a b... 0 q q 0 n p j By comparing the spin up-down differential transverse flow for neutrons and protons using different isospin dependence of SO coupling. n j p

24 h 0 h 0 h 0 Free nucleons: 8 / 0 Different densities are reached at different beam energies Nucleons from high transverse momentum (p T ) are emitted at early stages.

25 density dependence of SO coulping W a b... 0 q q 0 Low-p T nucleons: emitted at later stages carry information of lower densities high-p T nucleons: emitted at early stages carry information of higher densities The strength of the SO coupling at a certain density can be extracted from HIC at the corresponding collision energy. In this way the strength and density dependence of SO coupling can be disentangled.

26 Directed flow: System size dependence heavier system higher density, pressure Larger v Spin up-down differential directed flow: Surface effect NN scatterings wash out spin effect

27 Effects of spin-orbit interaction on v Elliptic flow: +: in-plane hydro -: squeeze out Dynamics more complicated than v Spin splitting at large centralities

28 Spin splitting at higher p T Spin up-down differential directed flow: Spin up-down differential elliptic flow: Both are sensitive probes of SO coupling!

29 The spin-dependent light cluster production the spin- and isospin-dependent BUU (SIBUU) model the improved coalescence model ( with spin and isospin DOF) light clusters with explicit spin The probability for producing a cluster is determined by the overlap of its Wigner phase-space density with the nucleon phase-space distributions at freeze out. Free nucleons: local densities are less than /8 0 Freeze out Propagate to the same time in the C.M. frame and coalesce Widely-used algorithm for lightly bound clusters

30 deuteron Wigner phase-space density Internal wave function Triton or Helium Internal wave function RMS radius

31 The multiplicity of a M-nucleon cluster is R. Mattiello et al., Phys. Rev. Lett 995 Phys. Rev. C 997. W is the Wigner phase-space density of the M-nucleon cluster, spin-isospin statistical factor (spin-averaged) A G (S ) Z! N!/ A! A /8 for deuteron / for triton G is the G : coalescence with a given isospin G: coalescence with a given spin and isospin G / for helium G He( S / ) G p& n p& n p& n p& n p & n & n p & n & n n & p & p n & p & p

32 H wave function H ~ S= T=0 spin isospin S S 0 S T pp pn np nn pn np T T 0 S z = + S z = 0 S z = - Assign all many-nucleon states which are allowed from the Pauli principle the same weight. 8 wave function(considering the spinisospin and exchange of antisymmetric), of 8 are feasible. G= /8 (no information about spin) ~ ( ) p n n p ~ ( ) p n p n n p n p ~ ( ) p n n p 4 ~ ( ) p n p n n p n p 5 ~ ( ) p n n p 6 ~ ( ) p n p n n p n p 7 ~ ( ) p n n p 8 ~ ( ) p n p n n p n p p& n p& n p& n p& n G / ( S ) G / 4( S 0) G / 4( S 0) G / ( S ) z z z z

33 H & He wave function H / He ~ spin isospin S=/ T=/ S T S T S / S / S / S 6 6 { S( H) / & S( He) / } T ppp ppn npp pnp T / 4 wave function(considering the spinisospin and exchange of antisymmetric), of 4 are feasible. G= / (no information of spin) nnp pnn npn nnn ppn pnp npp 6 pnn npn nnp 6 pnp npp pnn npn T / T / He ( SZ ) ~ ( p n p p n p n p p 6 n p p p p n p p n ) ~ ( p n p n p p p p n p p n ) ~ ( p n p p n p n p p n p p p p n p p n ) He n & p & p n & p & p Similar for G He /( / ) S z /( / ) S z

34 Transverse flow of light clusters H n & p H p & n & n He n & p & p F ( H ) n p n p ud F ( He) n p p n p p ud F ( H ) p n n p n n ud Additive relations for light cluster transverse flow

35 b= 8 fm Spin splitting of light clusters collective flows observed Easily experimentally measured/identified Useful probe of SO coupling

36 Spin dynamics from QMD model C.C. Guo, Y.J. Wang, Q.F. Li, and F.S. Zhang, Phys. Rev. C 90, (04) Our conclusions on spin dynamics are robust and model independent.

37 Tensor force Another interaction related to nucleon spin

38 Add tensor force to transport? k r k k r k t k k k r r k k k t v e T 0 } { Hartree-Fock framework: Skyrme-type tensor force: r d r H ij P P P v ij E T j i r T T, ) ( ) ( i T i i H T h ~ *

39 i i v i i i v v i i i i i v v i i i i i i F i J T s * * * * * * Spin density Spin kinetic density Spin current density Pseudovector tensor kinetic density Only consider vector component of v J q q q o e o e q q q q o e o e q q q q o e o e q q o e o e T s s t t s s t t J F s t t J F s t t J T s t t J T s t t s t t s t t H p n q, Potential energy density T h Equation of motion,, 0 J J J J J J J v v v v

40 Spin-orbit interaction+tensor interaction

41 Conclusion and outlook ) Introduce spin and spin-orbit interaction to a transport model for intermediate-energy heavy-ion collisions ) Local spin polarization observed ) Spin splittings of transverse flow, directed flow, elliptic flow 4) Different spin states of light clusters 5) Spin up-down differential flows are sensitive probes of in-medium spin-orbit interaction 6) Tensor force in heavy-ion collisions

42 Heavy-ion collisions has the advantages of adjusting the energy, density, and momentum current, and might be a more promising way to study the properties of the nuclear spin-orbit interaction! Thanks Wolfgang Trautmann for useful discussions on spin measurement and encouraging us to keep going! Thank you! xujun@sinap.ac.cn

43 Momentum dependence treatment MDI mean-field potential C.B. Das, S. Das Gupta, C. Gale, and B.A. Li, PRC (00) L.W. Chen, B.A. Li, and C.M. Ko PRL (005) f(r,p) calculation: time consuming! From effective interaction J. Xu and C.M. Ko, PRC (00) In-medium cross section B.A. Li and L.W. Chen, PRC (005)

44 Current status of IBUU With magnetic field L. Ou and B.A. Li, PRC (0) (time consuming!)

45 W a b... 0 q q 0 F ud is larger with a weaker density dependence of LS force F ud decreases with increasing spin-flip probability

46 W a b... A globally neutron-rich system 0 q q 0 n j n p j p Relative strength of like and unlike coupling affects: ) Relative F ud for n and p ) Total magnitude of F ud