Coulomb Correction to Density and Temperature of Fermions and Bosons from Quantum Fluctuations

Coulomb Correction to Density and Temperature of Fermions and Bosons from Quantum Fluctuations Hua Zheng a, Gianluca Giuliani a and Aldo Bonasera a,b! a)cyclotron Institute, Texas A&M University! b)lns-infn, Catania-Italy. * Trento 2014 1

Outline Motivations Methods to determine density Conventional thermometers New thermometer Applications to F&B Summary 2

Methods to determine the density SAHA s equation ρ = N V Coalescence model Two particles correlation Guggenheim approach Quantum fluctuation 6

Methods to determine the density SAHA s equation Justified for very low density region and high temperature S. Albergo et al., IL NUOVO CIMENTO, vol 89 A, N. 1 (1985) S.Shlomo, G. Ropke, J.B. Natowitz et al., PRC 79, 034604(2009) 7

Methods to determine the density Coalescence model! A. Mekjian, PRL Vol 38, No 12 (1977) L. Qin, K. Hagel, R. Wada, J.B. Natowitz et al., PRL 8, 172701(2012) K. Hagel, R. Wada, L. Qin, J.B. Natowitz et al., PRL 8, 062702(2012) 8

Conventional thermometers The slopes of kinetic energy spectra (Tkin) Discrete state population ratios of selected clusters (Tpop) Double isotopic yield ratios (Td) S. Albergo et al.,il Nuovo Cimento, Vol 89A, N. 1 (1985) M. B. Tsang et al., PRC volume 53, (1996), R57 J. Pochodzalla et al., CRIS, 96, world scientific, A.Bonasera et al., IL Nuovo Cimento, Vol 23, p1, 2000 A. Kelic, J.B. Natowitz, K.H. Schmidt, EPJA 30, 203 (2006) 11

Nucl. Phys. A 843 (20) 1 12

Nucl. Phys. A 843 (20) 1 A Quadrupole is defined in the direction transverse to the beam axis 12

Nucl. Phys. A 843 (20) 1 A Quadrupole is defined in the direction transverse to the beam axis 2 2 Q xy =px -p y 12

Nucl. Phys. A 843 (20) 1 A Quadrupole is defined in the direction transverse to the beam axis Its variance is 2 2 Q xy =px -p y 12

Nucl. Phys. A 843 (20) 1 A Quadrupole is defined in the direction transverse to the beam axis 2 2 Q xy =px -p y Its variance is σ = d p(p -p ) f(p) 2 3 2 2 2 xy x y When a classical Maxwell-Boltzmann distribution of particles at temperature is assumed T cl σ 2 2 xy =N (2 mt cl ) 12

Density and temperature of fermions from quantum fluctuations Quadrupole fluctuations: Fermi Dirac distribution σ 2 2 xy =N (2mT) F QC % 4 T 7 T T & 35 ε 6 ε ε -2 2 2 4 ( ) [1+ π ( ) +O( ) ] ( low T approx) 2 f f f = N (2mT) ' & T 1.71 + 0.2( ) 1 ( higher order) &( ε f Wolfgang Bauer, PRC, Volume 51, Number 2 (1995) H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 13

Density and temperature of fermions from quantum fluctuations Quadrupole fluctuations: Fermi Dirac distribution σ 2 2 xy =N (2mT) F QC % 4 T 7 T T & 35 ε 6 ε ε -2 2 2 4 ( ) [1+ π ( ) +O( ) ] ( low T approx) 2 f f f = N (2mT) ' & T 1.71 + High T 0.2( ) 1 1 ( higher order) &( ε f Wolfgang Bauer, PRC, Volume 51, Number 2 (1995) H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 13

Density and temperature of fermions from quantum fluctuations Quadrupole fluctuations: Fermi Dirac distribution σ 2 2 xy =N (2mT) F QC % 4 T 7 T T & 35 ε 6 ε ε -2 2 2 4 ( ) [1+ π ( ) +O( ) ] ( low T approx) 2 f f f = N (2mT) ' & T 1.71 + Multiplicity fluctuations: High T 0.2( ) 1 1 ( higher order) &( ε f 2 2 N ( Δ N) ( N) T ( ) T, V, x Δ = = µ N Wolfgang Bauer, PRC, Volume 51, Number 2 (1995) H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 13

Density and temperature of fermions from quantum fluctuations Quadrupole fluctuations: Fermi Dirac distribution σ 2 2 xy =N (2mT) F QC % 4 T 7 T T & 35 ε 6 ε ε -2 2 2 4 ( ) [1+ π ( ) +O( ) ] ( low T approx) 2 f f f = N (2mT) ' & T 1.71 + Multiplicity fluctuations: " 2 T x # 3 = $ ε 0.442 f # + + #& (1 x) High T 0.2( ) 1 1 ( higher order) &( ε f 2 2 N ( Δ N) ( N) T ( ) T, V, x Δ = = µ ( low T approx) 2 0.442 0.345x 0.12 x ( higher order) 0.656 N Wolfgang Bauer, PRC, Volume 51, Number 2 (1995) H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 13

Density and temperature of fermions from quantum fluctuations Quadrupole fluctuations: Fermi Dirac distribution σ 2 2 xy =N (2mT) F QC % 4 T 7 T T & 35 ε 6 ε ε -2 2 2 4 ( ) [1+ π ( ) +O( ) ] ( low T approx) 2 f f f = N (2mT) ' & T 1.71 + Multiplicity fluctuations: " 2 T x # 3 = $ ε 0.442 f # + + #& (1 x) High T 0.2( ) 1 1 ( higher order) &( ε f 2 2 N ( Δ N) ( N) T ( ) T, V, x Δ = = µ Low T ( low T approx) 2 0.442 0.345x 0.12 x ( higher order) 0.656 0.635x N Wolfgang Bauer, PRC, Volume 51, Number 2 (1995) H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 13

Density and temperature of fermions from quantum fluctuations H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 14

Density and temperature of fermions from quantum fluctuations Density: ε f ρ = 36( ) ρ 0 2/3 H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 14

Density and temperature of fermions from quantum fluctuations Density: ε f ρ = 36( ) ρ 0 2/3 Testing the method H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 14

Density and temperature of fermions from quantum fluctuations Density: ε f ρ = 36( ) ρ 0 2/3 Testing the method u u CoMD simulations:! Experimental data 40 40 Ca+ Ca, b = 1fm, t = 00fm/c H. Zheng, A. Bonasera, PLB, 696(2011) 178-181 H. Zheng, A. Bonasera, PRC 86, 027602 (2012) 14

GEMINI statistical model decay simulations. First decay step H.Zheng et al. submitted(2014).

GEMINI statistical model decay simulations. First decay step S=2aT;a=A/15=A/k_0(rho/rho_0)^(-2/3) 1/MeV; A=80 Z=40 H.Zheng et al. submitted(2014).

GEMINI statistical model decay simulations. First decay step S=2aT;a=A/15=A/k_0(rho/rho_0)^(-2/3) 1/MeV; A=80 Z=40 Double ratio: pnth &dtha H.Zheng et al. submitted(2014).

GEMINI statistical model decay simulations. First decay step S=2aT;a=A/15=A/k_0(rho/rho_0)^(-2/3) 1/MeV; A=80 Z=40 Double ratio: pnth &dtha 25 20 2 4 6 8 12 14 16 18 20 k 0 =k infinity =7.3 MeV 2 4 6 8 12 14 16 18 20 25 k 0 =k infinity =15 MeV 20 T(MeV) 15 15 5 5 0 0-2 -2 i -3-3 -4-4 -5 2 4 6 8 12 14 16 18 20 E*/A(MeV) 2 4 6 8 12 14 16 18 20 E*/A(MeV) -5 H.Zheng et al. submitted(2014).

GEMINI statistical model decay simulations. First decay step S=2aT;a=A/15=A/k_0(rho/rho_0)^(-2/3) 1/MeV; A=80 Z=40 Double ratio: pnth &dtha Classical&Quantum fluctuations p&n 25 20 2 4 6 8 12 14 16 18 20 k 0 =k infinity =7.3 MeV 2 4 6 8 12 14 16 18 20 25 k 0 =k infinity =15 MeV 20 25 20 2 4 6 8 12 14 16 18 20 k 0 =k infinity =7.3 MeV 2 4 6 8 12 14 16 18 20 2 k 0 =k infinity =15 MeV 2 T(MeV) 15 15 T(MeV) 15 1 1 5 5 5 5 0 0 0 1 0 1-2 -2-1 1 i -3-3 i -4-4 -2 1-5 2 4 6 8 12 14 16 18 20 E*/A(MeV) 2 4 6 8 12 14 16 18 20 E*/A(MeV) -5 H.Zheng et al. submitted(2014). 2 4 6 8 12 14 16 18 20 E*/A(MeV) 2 4 6 8 12 14 16 18 20 E*/A(MeV)

1 2 4 6 8 12 14 16 18 2 4 6 8 12 14 16 18 1-1 -1-2 -2 i -3-3 -4-4 -5 0 k 0 =k infinity =7.3 MeV k 0 =k infinity =15 MeV 0-5 ) -0.2-0.2 ( p + n )/( -0.4-0.4 p - n -0.6-0.6-0.8-0.8-1 2 4 6 8 12 14 16 18 T(MeV) 2 4 6 8 12 14 16 18-1 T(MeV) 11

Exp. data (G.Bonasera et al.) in preparation 12

Density and temperature of fermions from quantum fluctuations B. C. Stein et al, arxiv: 1111.2965v1 H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 15

Density and temperature of fermions from quantum fluctuations S32+Sn112 B. C. Stein et al, arxiv: 1111.2965v1 H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 15

Density and temperature of bosons from quantum fluctuations Quadrupole fluctuations: Bose-Einstein distribution Multiplicity fluctuations: N Δ = = µ 2 ( N) T ( ) T, V NT ρκt, H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 16

Density and temperature of bosons from quantum fluctuations Quadrupole fluctuations: Bose-Einstein distribution Multiplicity fluctuations: Density: N Δ = = µ 2 ( N) T ( ) T, V NT ρκt, H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 16

Density and temperature of bosons from quantum fluctuations H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 17

Density and temperature of bosons from quantum fluctuations 18

Density and temperature of bosons from quantum fluctuations N Δ = = µ 2 ( N) T ( ) T, V NT ρκ T 18

Multiplicity fluctuation using Landau s phase transition theory H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 19

The results of Fermions and bosons We introduce the Coulomb correction 20

Coulomb correction Similar to the density determination of the source in electron-nucleus scattering The distribution function is modified 21

Coulomb correction 22

Coulomb correction results for Fermions Add another figure rhop/rhon 25

T (MeV) 12 =0.065 m s m s =0.115 8 6 m s =0.165 m s =0.215 ms=(n-z)s/as 4 2 0-1 -0.5 0 0.5 1 ( - )/( + J.Mabiala priv. comm. n p n ) p 23

T (MeV) 12 =0.065 m s m s =0.115 8 6 m s =0.165 m s =0.215 ms=(n-z)s/as Two fluids p&n, different densities 4 2 0-1 -0.5 0 0.5 1 ( - )/( + J.Mabiala priv. comm. n p n ) p 23

T (MeV) 12 =0.065 m s m s =0.115 8 6 m s =0.165 m s =0.215 ms=(n-z)s/as Two fluids p&n, different densities Order parameter? 4 2 0-1 -0.5 0 0.5 1 ( - )/( + J.Mabiala priv. comm. n p n ) p 23

Coulomb correction for Bosons (T<Tc) 23

Coulomb correction for Bosons (T<Tc) 24

Coulomb correction results for Bosons 26

P.Marini et al. 27

P.Marini et al., Circumstantial Evidence of Boson/Fermion mixtures in Nuclei 28

Bose condensate in trapped fermions and bosons mixture 36

Summary ØWe reviewed the three conventional thermometers! ØA new thermometer to take into account the quantum effect of fermions and bosons is proposed! ØSome evidences of quantum nature of fermions and bosons are found in the model and experimental data! ØMore investigations need to be done 27

Thank you! 28

Experimental data J. Mabiala et. al., submitted to PRL 29 K. Huang, Statistical Mechanics (2Ed, Wiley, 1987)

Quantum nature phenomena 3

Quantum nature phenomena Cosmic microwave background radiation http://asd.gsfc.nasa.gov/arcade/cmb_spectrum.html 3

Quantum nature phenomena Cosmic microwave background radiation Specific heat of Au http://asd.gsfc.nasa.gov/arcade/cmb_spectrum.html Phys Rev 98, 1699 (1955) 3

Quantum nature phenomena Cosmic microwave background radiation Specific heat of Au http://asd.gsfc.nasa.gov/arcade/cmb_spectrum.html Phys Rev 98, 1699 (1955) C. Tournmanis s lecture 3

Methods to determine the density SAHA s equation ρ = N V Coalescence model Two particles correlation Guggenheim approach Quantum fluctuation 6

Trapped Fermions/Bosons systems 4

Trapped Fermions/Bosons systems Li6 T / T f = 0.6 T / T f = 0.21 PRL 5, 040402 (20) 4

Trapped Fermions/Bosons systems Li6 T / T f = 0.6 T / T f = 0.21 Rb87 PRL 5, 040402 (20) PRL 96, 130403 (2006) 4

Other works B. Borderie et al., arxiv: 1207.6085v1 30

Nuclear collision Measured in experiment event by event:! Mass (A) Charge (Z) Yield Velocity Angular distribution Time correlation The physical quantities in EoS:!! Pressure (P) Volume (V) or Density ( ) Temperature (T) 5

Density and temperature of fermions from quantum fluctuations H. Zheng, A. Bonasera, PRC 86, 027602 (2012) J. B. Natowitz et al., PRC Volume 65, 034618 31

Methods to determine the density Two particles correlation S.E. Koonin, Phys. Lett Vol 70B, No 1 (1977) S. Pratt, M.B. Tsang, PRC Vol 36, No 6 (1987) W.G. Gong, W. Bauer, C.K. Gelbke and S. Pratt, PRC Vol 43, No 2 (1991) 9

Methods to determine the density Guggenheim formulae E.A. Guggenheim, J. Chem. Phys Vol 13, No7 (1945) T. Kubo, M. Belkacem. V. Latora, A. Bonasera, Z. Phys. A. 352, 145 (1995) P. Finocchiaro et al., NPA 600, 236 (1996) J.B. Elliott et al., PRL Vol 88, No4 (2002), J.B. Elliott et al., PRC 87, 054622 (2013) L.G. Moretto et al., J. Phys. G: Nucl. Part. Phys. 38, 1131 (2011) J.B. Natowitz et al., Int. J. Mod. Phys. E Vol 13, No1, 269 (2004)

Why Boson condensate? ØLarge production of alpha particles in experiment! ØEvents with large multiplicity alpha-like or d-like fragments are found in experiment! ØThe light nuclei display the alpha-structure, (12C)! ØRecent experimental data and microscopic quantum statistical model suggests there is a boson condensate to reproduce the data (Joe)! ØThe configuration of nuclei can be alpha clusters 32

Some relevant references about boson condensate 33

CoMD α Pauli principle M. Papa et al., Journal of computational physics 208 (2005) 403-415 H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 34

CoMD α α1 α 2 Ξ = 1 e ij Vc 1 σπρvij dt E Π = (1 + f )(1 + f ) Π ' > Π k α1 α 2 Pauli principle Alpha-alpha collision M. Papa et al., Journal of computational physics 208 (2005) 403-415 H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 34

Density and temperature of bosons from quantum fluctuations H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57 35