J. B. Natowitz Department of Chemistry and Cyclotron Institute , Texas A&M University, College Station

Experimental Investigations of The Equation of State of Low Density Nuclear Matter

Exploring The Nuclear Matter Phase Diagram With Collisional Heating

• Collisions of normal density nuclei create initially compressed and excited systems, which expand and cool.

• During this process, the properties of the expanding system is manifested in the matter flow, in the energy spectra, and in the yield patterns and nature of produced species which emerge from the collision zone.

THERMAL SHOCK COMPRESSION

FREEZEOUT

SEPARATION

SECONDARY EMISSION

EXPANSIONPRE-EQUILIBRIUM EMISSIONEQUILIBRIUM EMISSION ?

AMD Calculation

TI

ME

• Dynamic Evolution

• Excitation Energy ?

• Temperatures ?

• Degree of expansion

• Composition ?

• Chemical and Thermal Equilibrium ?

• Equation of State ?

• Liquid-gas phase transition?

Light Charged Particle Emission - High Total Multiplicity Collisions

NIMROD4 Pi Charged Particles4 Pi Neutrons

THERMAL SHOCK COMPRESSION

FREEZEOUT

SEPARATION

SECONDARY EMISSION

EXPANSIONPRE-EQUILIBRIUM EMISSIONEQUILIBRIUM EMISSION ?

Event Selection

Neutron + Charged Particle multiplicity distribution for 64Zn+124Sn. Bin4 corresponds to the most violent

collision events

Most Violent Collision Events

@ 30% Top Highest Multiplicity

Mn

MCP

Source Analysis of Emission ( Energy, Angle)

Source Fitting – 4He from 40Ar + 124Sn

PLF

1 2 3

10

4

7

5 6

98

1211

Angular Distribution

NN TLF

Elab, MeV

Reaction Tomography-Particles

TLF

NN

Experiment

From Fitting

Velocity Plot Protons 40Ar+124Sn PLF

V parallel

V p

erpe

ndic

ular

Evaporation-like

Evaporation-like

Coalescence

NN

Sum of Sources

12

14

16

18

20

22

0 50 100 150 200 250

A

Tcri

t, M

eV

TC =16.6 0.86 MeV

Critical Temperature of Symmetric Nuclear Matter

Phys.Rev.Lett. 89 (2002) 212701Phys.Rev. C65 (2002) 034618

employing Skyrme interactions with the = 1/6 density dependence,

this value of Tc leads to K = 232 22 MeV.

Using Gogny interactions with = 1/3 leads to K = 233 37 MeV.

These results for K lead to m* value = 0.674 A value of K = 231 5 MeV, was derived by D. H. Youngblood, H. L. Clark, and Y.-W. Lui, Phys. Rev. Lett. 82, 691 (1999) by comparison of data for the GMR breathing mode energy of five different nuclei.

K. Hagel et al. Phys. ReV. C 62 034607 (2000)

J.B. Natowitz et al., Phys.Rev. C 66 031601 (2002)

Derived Average Freeze-Out Densities

Coalescence Model Non-Dissipative

Analyses Expanding Fermi Gas 47A MeV

R ~ 10 km

SUPERNOVA

NEUTRON STAR

STARS Giant NucleiAnd Sites of Nucleosynthesis

Large Changes in Temperature, Density, Proton/Neutron content

C.J. Horowitz, A. Schwenk nucl-th/0507033

Alpha Particle Mass Fraction vs Density

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0001 0.001 0.01 0.1Density, fm-3

X a

lph

a

Shen-Toki T = 4 Yp = 0.45

Shen-Toki T = 5

Shen-Toki T = 8

Lattimer SKM* T = 5.03 Yp = 0.5

Lattimer T = 8.05 SKM*

ASTROPHYSICAL EQUATIONS OF STATE AT LOW DENSITY DOMINATED BY ALPHA CLUSTERING

Density

Cluster Formation and The Equation of State of Low-Density Nuclear Mattersymmetric nuclear matter, T=2, 4, 8 MeV C.J. Horowitz, A. Schwenk nucl-th/0507033

. nucl-th/0507064 The Virial Equation of State of Low-Density Neutron Matter C.J. Horowitz and A. Schwenk

Clustered Gas VEOS

SF

SE

T/2

T/2

Skyrme, Fermi gas etc.

SYMMETRY ENERGY(T,)

Many Nucear and Astrophysical Phenomena Strongly Affected by the Symmetry Energy

At Normal Density aa ~ 23 MeV for Finite Nuclei ~30 MeV for Symmetric Nuclear Matter

V perpendicu

lar

S c h e m a t ic V e lo c i t y P lo t -I n t e r m e d ia t e E n e r g y H e a v y I o n C o l l is io n A s y m m e t r i c E n t r a n c e C h a n n e lE a r l y E m is s io n N N -L i k e

E a r l y E m is s io nP r o je c t i l e -L ik e

V p

erp

en

dic

ula

r

Schematic Veloc ity P lot-Intermediate Energy Heavy Ion Collis ion A symmetric Entranc e Channel

Early Emiss ion N N -L ike

E arlyE miss ionP rojec tile-L ike

EARLY EMISSION TOTAL MASS in Z=1,2 Charged

Particles 47A MeV Projectiles on 124Sn

0

5

10

15

20

25

30

35

0 20 40 60 80

A projectile

Acp

(Z=

1-,2

) 124 EXPT ACPCentral NN + PLFChimera ACP R=3 300fm/cAMD ACP 2300 fm/c

NN SOURCE EMISSION- Experimental Data and Calculated Yields from AMD and Chimera QMD Codes

Average Freeze-out Density 64Zn + 124Sn ~ 0.06 fm-3

“Gas” density ~ ANN/(Atot-ANN) * 0.06 fm-3

~ 0.01 fm-3

COALESCENCE

Isoscaling Analyses and Symmetry Energy

A Comparison of the Yields of Emitted Species for Two Different Sources of Similar Excitation Energy and Temperature but Differing in Their Neutron to Proton Ratios

M.B. Tsang, W.A. Friedman, C.K. Gelbke, W.G. Lynch, G. Verde and H.S. Xu, Phys.Rev. C64 (2001) 041603

Fsym

T

α ═ (4F/T)[(Z/A)21 – (Z/A)2

2]

n = 0.62 x 1036 T3/2 exp[- 20.6/T] Y(4He)/ Y(3He) cm-3

p = 0.62 x 1036 T3/2 exp[ -19.8/T] Y(4He)/ Y(3H) cm-3

nuc tot = p + n + 2d + 3t + 33He + 4

Density

LOW DENSITY CHEMICAL EQUILIBRIUM MODEL(Albergo)

43

/surf

YdYHeYtYHeV R

Temperature

THHe = 14.3/ [ln (1.59R)]R = [ YR = [ Yd d ] [ Y] [ Y44He He ]]

[ Y[ Yt t ] [ Y] [ Y33He He ]]

Isoscaling Analyses and Symmetry Energy

Clusterization in Low Density Nuclear Matter

0

5

10

15

20

25

30

35

0.001 0.01 0.1 1

Rho , nuc/fm3

Esym

, MeV

Expt

Gogny

1̂.05

HS calc

Density corr

Poly. (HS calc)

Poly. (Densitycorr)

C.J. Horowitz, A. Schwenk nucl-th/0507033

Private Communication O’Connor, Schwenk, Horowitz Manuscript in Preparation August 2007

Neutron Rich

Proton Rich

• p + 112Sn and 124Sn • d + 112Sn and 124Sn • 3He + 112Sn and 124Sn • 4He + 112Sn and 124Sn • 10B + 112Sn and 124Sn • 20Ne + 112Sn and 124Sn • 40Ar + 112Sn and 124Sn • 64Zn+ 112Sn and 124Sn

Projectile Energy - 47A Mev

Reaction System ListThesis – L. Qin TAMU

||V

||V

Vpar cm/ns

Vp

e rp c

m/ n

s

Significant Temperature Evolution With Velocity Relatively Small Changes with Projectile Size

DOUBLE ISOTOPE RATIO THHe

CHEMICAL EQUILIBRIUM TEMPERATURES

THHe = 14.3/ [ln (1.59R)](albergo)

R = [ YR = [ Yd d ] [ Y] [ Y44He He ]] [ Y[ Yt t ] [ Y] [ Y33He He ]]

Reaction Tomography-Temperatures

“Gas” DensityTLF REMOVED

L. Qin – PhD Thesis, In Progress

CHEMICAL EQUILIBRIUM DENSITIES (Albergo) FROM ISOTOPE RATIOS

Fm-3

Reaction Tomography-Densities

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43

S1

S4

S7

S10

S13

S16

S19

25-30

20-25

15-20

10-15

5-10

0-5

1 5 9 13 17 21 25 29 33 37 41 45

S1

S4

S7

S10

S13

S16

S19

25-30

20-25

15-20

10-15

5-10

0-5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43

S1

S4

S7

S10

S13

S16

S19

25-30

20-25

15-20

10-15

5-10

0-5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43

S1

S4

S7

S10

S13

S16

S19

25-30

20-25

15-20

10-15

5-10

0-5

-6.2

5

-4.7

5

-3.2

5

-1.7

5

-0.2

5

1.2

5

2.7

5

4.2

5

5.7

5

7.2

5

8.7

5

10

.25

11

.75

13

.25

14

.75

10.25

8.75

7.25

5.75

4.25

2.75

1.25

4He Fsym

25-30

20-25

15-20

10-15

5-10

0-5

64Zn

40Ar20Ne

10B

4He DERIVED VALUES OF Fsym

as a FUNCTION of VELOCITY

47 MeV/u Projectiles on 112Sn, 124Sn

V parallel

NN

V p

erp

end

icu

lar

F sym Roepke Calculation 4 April 08

0

5

10

15

20

25

30

35

40

0.0001 0.001 0.01 0.1 1 10

density, fm-3

F s

ym,

MeV Roepke T = 4 Fsym

Roepke T = 6

Roepke T = 10

Gogny D1S T = 0

31.6(rho/rho0)^0.69

F sym Roepke Calculation 4 April 08

0

5

10

15

20

25

30

35

40

0.0001 0.001 0.01 0.1 1 10

density, fm-3

F s

ym,

MeV Roepke T = 4 Fsym

Roepke T = 6

Roepke T = 10

Qin NN

Gogny D1S T = 0

31.6(rho/rho0)^0.69

K. Hagel et al. PHYSICAL REVIEW C 62 034607 (2000)

J.B. Natowitz et al., Phys.Rev. C66 (2002) 031601

Derived Average Freeze-Out Densities

Coalescence Model Non-Dissipative

Analyses Expanding Fermi Gas

F sym Roepke Calculation 4 April 08

0

5

10

15

20

25

30

35

40

0.0001 0.001 0.01 0.1 1 10

density, fm-3

F s

ym,

MeV

Roepke T = 4 Fsym

Roepke T = 6

Roepke T = 10

Qin NN

Gogny D1S T = 0

31.6(rho/rho0)^0.69

He-Zn + Sn TLF at Liquid Densities

F sym Roepke Calculation 4 April 08

0

5

10

15

20

25

30

35

40

45

0.0001 0.001 0.01 0.1 1 10

density, fm-3

F s

ym,

MeV

Roepke T = 4 Fsym

Roepke T = 6

Roepke T = 10

Qin NN

Gogny D1S T = 0

31.6(rho/rho0)^0.69

He-Zn + Sn TLF at Liquid Densities

Liquid scaled to NMDanielewiczEsym(nuclides) = Esym(NM)

(1 + 2.7/A 1/3)

P. Danielewicz

Few Body Syst.Suppl. 14 (2003) 361-366 Eur.Phys.J. A22 (2004) 261-269

M. Beyer, G. Roepke et al., Phys.Lett. B488, 247-253 (2000)

IN MEDIUM BINDING ENERGIES and MOTT TRANSITION

Alpha Mass Fractions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0001 0.001 0.01 0.1

Shen-Toki T = 4 Yp = 0.45Shen-Toki T = 5 Shen-Toki T = 8Roepke T = 4 Yp = 0.42Roepke T = 5 Roepke T = 8 Series6Series8Series9

Multiplicities with Free Cluster Bindiing Energies (Albergo model-like)T= 10 MeV, A = 250

0.01

0.1

1

10

100

1000

0 0.05 0.1 0.15

Density nucleons/fm3

Mu

ltip

lici

ty

Nucleons

d

t + 3He

4He

Beyer Model Multiplicities( In Medium Binding Energies)

T= 10 MeV A = 250

0.01

0.1

1

10

100

1000

0 0.05 0.1 0.15

Density, nucleons/fm3

Mu

ltip

licit

y

Nucleons

d

T + 3He

4He

Note: Same at low densityRho LE ~.005 fm-3

M. Beyer et al. nucl-th/0310055

Light Clusters in Nuclear Matter of Finite

Temperature

Calculated Density vs Roepke DensityFree Binding Energies

0

0.005

0.01

0.015

0.02

0 0.005 0.01 0.015

Density fm-3

Calc

. Den

sity

fm

-3 T = 4T = 5T = 8T = 4T = 5T = 8Series4

Calculated.Temperature vs Roepke Density Free Binding Energies

0

2

4

6

8

10

12

0.0001 0.001 0.01 0.1

Roepke Density fm-3

Cal

c. T

M

eV

Series1

Series2

Series3

T = 4

T = 5

T = 8

Correlations Bose Condensates Superfluidity Efimov States

E. Bell1, M. Cinausero2, Y. El Masri 6,D. Fabris3, K. Hagel1, J. Iglio1, A. Keksis1, T. Keutgen6, M. Lunardon3, Z. Majka4, A. Martinez-Davalos,5 A. Menchaca-Rocha5, S. Kowalski1,T. Materna1, S. Moretto3, J. B. Natowitz1, G. Nebbia3, L. Qin1, G. Prete,2 R. Murthy1, S. Pesente3, V. Rizzi,3 D. V. Shetty1, S. Soisson1, B. Stein1, G. Souliotis1, P. M. Veselsky1,A. Wieloch1, G. Viesti3, R. Wada1, J. Wang1, S. Wuenshel1, and S. J. Yennello1

1Texas A&M University, College Station, Texas 2INFN Laboratori Nazionali di Legnaro, Legnaro, Italy 3INFN Dipartimento di Fisica, Padova, Italy 4Jagellonian University, Krakow, Poland 5UNAM, Mexico City, Mexico 6UCL, Louvain-la-Neuve, Belgium

Major Contributors

• M. Barbui, A. Bonasera, C. Bottosso, M. Cinausero, Z. Chen, D. Fabris, Y. El Masri, K. Hagel, T. Keutgen, S. Kowalski, M. Lunardon, Z. Majka, S. Moretto, G. Nebbia, J. Natowitz L. Qin, S. Pesente, G. Prete, V. Rizzi, P. Sahu, S. ShlomoJ. Wang, G. Viesti

• S. Shlomo, A. Ono, G. Roepke

• A. Schwenk, E. O’ConnorAND THE NIMROD COLLABORATION

•TAMU, PADOVA, LEGNARO, KRAKOW, LOUVAIN la NEUVE, CATANIA, LANZHOU