Hua Zheng a, Gianluca Giuliani a and Aldo Bonasera a,b a)Cyclotron Institute, Texas A&M University...
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Transcript of Hua Zheng a, Gianluca Giuliani a and Aldo Bonasera a,b a)Cyclotron Institute, Texas A&M University...
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)
11
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
12
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
13
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
16
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
18
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
20
The results of Fermions and bosons
We introduce the Coulomb correction
21
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)
22
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)
23
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)
24
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)
25
Coulomb correction results for Fermions
H. Zheng, G. Giuliani and A. Bonasera, arXiv: 1305.5494
26
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
27
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!
28