Hua Zhenga, Gianluca Giuliania and Aldo Bonaseraa,b

a)Cyclotron Institute, Texas A&M University

b)LNS-INFN, Catania-Italy.

1

Coulomb Correction to the Density and the Temperature of Fermions and

Bosons from Quantum Fluctuations

23/4/21 IWNDT2013, College Station, Tx

Outline

MotivationMethods to determine densityConventional thermometersNew thermometer Application to F&BCoulomb correction to F&BSummary

2

http://asd.gsfc.nasa.gov/arcade/cmb_spectrum.html

Cosmic microwave background radiation

3

Phys Rev 98, 1699 (1955)

Specific heat of Au

Quantum nature phenomena

C. Tournmanis’s lecture

Trapped Fermions/Bosons systems

4

PRL 105, 040402 (2010)

/ 0.21fT T

/ 0.6fT T

Li6

PRL 96, 130403 (2006)

Rb87

Nuclear collision

5

Measured in experiment event by event:

Mass (A)Charge (Z)YieldVelocityAngular distributionTime correlation

The physical quantities in EoS:

Pressure (P)Volume (V) or Density ( )Temperature (T)

Methods to determine density

6

N

V

SAHA’s equation

Coalescence model

Two particles correlation

Guggenheim approach

Quantum fluctuation

Methods to determine the density

7

SAHA’s equation

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)

It is justified for very low density region and high temperature

3 / 2 3( 1), ( , ) ( , )

( , ) exp[ ](2 1) (2 1)

AT Z N

p nZ Np n

A A Z B A ZA Z

s s T

l wr r r

-

=+ +

¥

1 4

2 33

,3 / 21 4 1 4

2 3 2 3

1 4 2 3

(2 1)(2 1)( ) ( ) exp[ ]

2 (2 1)(2 1)

( ) ( )

Z Np n

T A

YY

Y Y

A A s s B

A A s s T

f f f f f

r rl

D D

D

=+ + D+ +

D = + - +

¥

Methods to determine the density

8

Coalescence model

33 1 30

13 1/ 2

4( , , ) (1,0)3{ } [ ]

! ! [2 ( )]N A AAnp

A C

Pd N Z N E A d NR

dE d N Z m E E dEd

0 /3 3 3 31

3 3

( , ) (2 1) (1,0)( ) [ ]

2

E TN A Anp A

d N Z N A s e h d NR

dp V dp

0

13 3/ 1

30

! ! 3[ (2 1) ]

2 4E T A

A

Z N A hV s e

P

A. Mekjian, PRL Vol 38, No 12 (1977), PRC Vol 17, No 3 (1978)T.C. Awes et al., PRC Vol 24, No 1 (1981)L. Qin, K. Hagel, R. Wada, J.B. Natowitz et al., PRL 108, 172701(2012)K. Hagel, R. Wada, L. Qin, J.B. Natowitz et al., PRL 108, 062702(2012)

Methods to determine the density

9

Two particles correlation

( , )( , )

( ) ( )

P p pC p p

P p P pa b

a ba b

=

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)

Methods to determine the density

10

Guggenheim approach

, 1/ 33 71 (1 ) (1 )

4 4l g

c c c

T T

T T

r

r= + - ± -

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, 113101 (2011)J.B. Natowitz et al., Int. J. Mod. Phys. E Vol 13, No1, 269 (2004)

Conventional thermometers The slopes of kinetic energy spectra (Tkin) Discrete state population ratios of selected clusters (Tpop) Double isotopic yield ratios (Td)

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S. Albergo et al.,IL Nuovo Cimento, Vol 89A, N. 1 (1985)M. B. Tsang et al., PRC volume 53, (1996), R1057J. Pochodzalla et al., CRIS, 96, world scientific, p1A. Bonasera et al., IL Nuovo Cimento, Vol 23, p1, 2000A. Kelic, J.B. Natowitz, K.H. Schmidt, EPJA 30, 203 (2006)

All of them are based on the Maxwell-Boltzmann distribution. No quantum effect has been considered so far.

New thermometer

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A new thermometer is proposed in S. Wuenschel, et al., Nucl. Phys. A 843 (2010) 1 based on momentum fluctuations

2 2xy x yQ =p -p

A Quadrupole is defined in the direction transverse to the beam axis

Its variance is 2 3 2 2 2

x y= d (p -p ) f(p)xy p LHS: analyze event by event in experiment

RHS: analytic calculation by assuming one distribution

When a classical Maxwell-Boltzmann distribution of particles at temperature was assumed

2 2xy =N (2 )clmT

clT

Density and temperature of fermions from quantum fluctuations

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Quadrupole fluctuations: Fermi Dirac distribution2 2xy

-2 2 2 4

f f f2

1.71

f

=N (2mT)

4 T 7 T T( ) [1+ ( ) +O( ) ] ( )

35 6N (2mT)

T0.2( ) 1 ( )

QCF

low T approx

higher order

2f0.656

2( )

T 30.442

0.442 0.345 0.12 ( )(1 )

x low T approx

x x higher orderx

Multiplicity fluctuations:2

2,

( )( ) ( ) ,T V

N NN T x

N

H. Zheng, A. Bonasera, PLB, 696(2011) 178-181H. Zheng, A. Bonasera, PRC 86, 027602 (2012)

High T1

Low T0.635x

Wolfgang Bauer, PRC, Volume 51, Number 2 (1995)

Density and temperature of fermions from quantum fluctuations

14

f

Density:

CoMD simulations:

Experimental data

40 40Ca+ Ca, b = 1fm, t = 1000fm/c

Testing the method

H. Zheng, A. Bonasera, PLB, 696(2011) 178-181H. Zheng, A. Bonasera, PRC 86, 027602 (2012)

Density and temperature of fermions from quantum fluctuations

15H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57

B. C. Stein et al, arXiv: 1111.2965v1

S32+Sn112

PRL 105, 040402 (2010)

/ 0.21fT T

/ 0.6fT T

Li6

Density and temperature of bosons from quantum fluctuations

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Multiplicity fluctuations:

2,( ) ( ) ,T V T

NN T NT

H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57

Quadrupole fluctuations: Bose-Einstein distribution

Density:

17H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57

Density and temperature of bosons from quantum fluctuations

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Density and temperature of bosons from quantum fluctuations

2,( ) ( )T V T

NN T NT

19H. Zheng, G. Giuliani and A. Bonasera, NPA 892 (2012) 43-57

Multiplicity fluctuation using Landau’s O(m6) phase transition theory

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The results of Fermions and bosons

We introduce the Coulomb correction

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Coulomb correction

Similar to the density determination of the source in electron-nucleus scattering

2

2

1.44 4( ) ( )p sZ Z

V q F qq V

p´=

h

2

2

1.44 4[ ] /

1( )

1p sZ Z

Tp V

f p

e

pe m

´+ -

=

±

h

The distribution function is modified

B. Povh et al., Particles and Nuclei, 6th ed. (Springer, Berlin, 2008)H. Zheng, G. Giuliani and A. Bonasera, arXiv: 1305.5494, PRC 88, 024607 (2013)

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Coulomb correction

5 / 2'

0

2 2

1/ 2'

0

1

4 1(2 )115

1

Ay

yVT

xy

Ay

yVT

dyy

emTdyy

e

n

n

s

¥

+ -

¥

+ -

±=

±

ò

ò

'

1/ 2'

02 2

1/ 2'

0

( ) ( 1)1

1

Ay

yVT

Ay

yVT

Ay

yVT

edyy

N eN dyy

e

n

n

n

+ -¥

+ -

¥

+ -

D ±=

±

ò

ò

3 / 21/ 2

'3 0

(2 ) 14

21

Ay

yVT

gV mN dyy

he

np

¥

+ -=

±ò

Need one more condition

H. Zheng, G. Giuliani and A. Bonasera, arXiv: 1305.5494, PRC 88, 024607 (2013)

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Coulomb correction for Bosons (T<Tc)

5 / 2'

0

2 2

1/ 2'

0

1

4 1(2 )115

1

Ay

yVT

xy

Ay

yVT

dyy

emTdyy

e

s

¥

+

¥

+

-=

-

ò

ò

3 / 21/ 2

'3 0

(2 ) 14

21

Ay

yVT

gV mN dyy

he

p¥

+=

-ò

H. Zheng, G. Giuliani and A. Bonasera, PRC 88, 024607 (2013)

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Coulomb correction for Bosons (T<Tc)

H. Zheng, G. Giuliani and A. Bonasera, PRC 88, 024607 (2013)R.P. Smith et al., PRL 106, 250403 (2011)

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Coulomb correction results for Fermions

H. Zheng, G. Giuliani and A. Bonasera, arXiv: 1305.5494

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Coulomb correction results for Bosons

H. Zheng, G. Giuliani and A. Bonasera, PRC 88, 024607 (2013)K. Hagel, R. Wada, L. Qin, J.B. Natowitz et al., PRL 108, 062702(2012

Deuteron is over bound in the model. The densities of deuteron may be over estimated.

Summary

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We reviewed the methods to determine density and three conventional thermometers

A new thermometer to take into account the quantum effects of fermions and bosons is proposed

Some evidences of quantum nature of fermions and bosons are found in the model and experimental data

Coulomb correction to the temperature and the density of fermions and bosons from quantum fluctuations is discussed

Thank you!

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