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International Nuclear Physics Conference INPC2007 Tokyo, Japan, June 3-8, 2007. In-medium properties of nuclear fragments at the liquid-gas phase coexistence. A.S. Botvina 1,2,3. ( In collaboration with W.Trautmann, I.Mishustin, N.Buyukcizmeci, R.Ogul ).

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in medium properties of nuclear fragments at the liquid gas phase coexistence

International Nuclear Physics Conference


Tokyo, Japan, June 3-8, 2007

In-medium properties of nuclear fragments at the liquid-gas phase coexistence

A.S. Botvina1,2,3

(In collaboration with W.Trautmann, I.Mishustin, N.Buyukcizmeci, R.Ogul)

1Institute for Nuclear Research, Russian Academy of Sciences,

Moscow, Russia

2Frankfurt Institute for Advanced Studies, J.W.Göthe University,

Frankfurt am Main, Germany

3Gesellschaft für Schwerionenforschung, Darmstadt, Germany


Multifragmentation of nuclei takes place in reactions initiated by all high energy

particles (hadrons, heavy-ions, photons), where high excitation energy of residual

nuclei is reached.

Experimentally established:1) few stages of reactions leading to multifragmentation,

2) short time ~100fm/c for primary fragment production, 3) freeze-out density is

around 0.1ρ0 , 4) high degree of equilibration at the freeze-out.


Thermal multifragmentation of nuclei: Production of hot fragments at temperature T ~ 3---8 MeV and density ρ ~ 0.1 ρ0 (ρ0≈0.15 fm-3) Interpretation: liquid-gas phase transition in finite nuclei. Investigation of properties of fragments surrounded by nuclear species.

statistical multifragmentation model smm
Statistical Multifragmentation Model(SMM)

J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin, K. Sneppen, Phys. Rep. 257 (1995) 133

Ensemble of nucleons and fragments

in thermal equilibrium characterized by




neutron number N0

proton number Z0 , N0+Z0=A0

excitation energy E*=E0-ECN

break-up volume V=(1+k)V0





All break-up channels are enumerated by the sets of

fragment multiplicities or partitions, f={NAZ}

Statistical distribution of probabilities:

Wf ~ exp {Sf (A0, Z0, E*,V)}

under conditions of baryon number (A), electric charge (Z) and energy (E*) conservation


Probability of channel:

mass and charge


Energy conservation

entropy of channel

Fragments obey Boltzmann statistics, liquid-drop description of individual fragments, Coulomb interaction in the Wigner-Seitz approximation

free energy of channel:

individual fragments:




multifragmentation of

relativistic projectiles

A.S.Botvina et al.,

Nucl.Phys. A584(1995)737

H.Xi et al.,

Z.Phys. A359(1997)397

comparison with

SMM (statistical



Statistical equilibrium

has been reached in

these reactions


The surface (B0) and symmetry (γ) energy coefficients

in the multifragmentation scenario

Fsym = γ·(N-Z)2/A

Fsuf = B0f(T)A2/3

isoscaling and the symmetry coefficient
Isoscaling and the symmetry coefficient γ

ALADIN: 12C+ 112,124Sn A.Le Fevre et al., Phys.Rev.Lett 94(2005)162701


α·T≈ -4γ (Z12/A12-Z22/A22)








The symmetry energy coefficient γ and isospin of fragments

A.S.Botvina et al., PRC72(2005)048801

G.Souliotis et al., PRC75(2007)011601


One can distinguish effects of the surface and symmetry energies since

the charge yield of fragments is very sensitive to the surface:

A.S.Botvina et al., PRC74(2006)044609

Fsuf = B0((Tc2-T2)/(Tc2+T2))5/4A2/3

Fsym = γ·(N-Z)2/A

properties of hot fragments the surface energy term b 0 z analysis of imf yields
Properties of hot fragments: the surface energy termB0Z-τ analysis of IMF yields

projectiles with different isospin



A.S.Botvina et al., PRC74(2006)044609


We analyze all previous observables: distributions of IMF , Zmax , T , ...

vs Zbound , and involve additionally new τ - observables for each

projectile (Xe, Au, U)

for single isolated nuclei:

C -- Cameron mass formula (1957)

MS -- Myers-Swiatecki mass formula


(include separate volume and surface

contributions to the symmetry energy)

We obtain an evolution of the surface energy of hot

fragments toward region of full multifragmentation



Multifragmentation reactions can be interpreted as a manifestation of the liquid-gas

type phase transition in finite nuclei, and allow for investigating the phase diagram of

nuclear matter. One can investigate properties of hot nuclei/fragments surrounded by

other nuclear species.

By analyzing experimental data it was found:

-- decreasing the symmetry energy of primary hot fragments by ~ 40% when the

systems evolve toward full multifragmentation (with increasing excitation energy

and decreasing the freeze-out density): ALADIN, FRS, MARS;

-- as a result of the same process the surface energy of these fragments becomes

independent on their isospin, this means that the difference between surface and

volume symmetry energies (as adopted in some mass formulas for isolated nuclei)

disappears also: ALADIN.

Important applications in astrophysics:

since mass distributions of fragments in

stellar matter, and electro-weak reactions

are very sensitive to the symmetry energy

A.Botvina and I.Mishustin, PRC72(2005)048801