cluster emission in intermediate-energy nuclear reactions and the

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M_2006 W. Udo Schröder

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Cluster emission in intermediate-energy nuclear reactions and the nuclear “liquid-gas phase transition”

Basic goal: Quantal A body system

Lack of knowledge & understanding

Beyond the mean field, correlationsNuclear stability at extremes (A, Z, E*, I) Nuclear thermodynamics/transportIsospin (asymmetry) effectsRole of the diffuse nuclear surface

Accessibility in (peripheral) heavy-ion collisions at intermediate energies, (perhaps also in fusion)

Illustration of the disintegration of 197Au after annihilation of an antiproton (impact from right). Snapshot of a hydro-dynamical calculation. [Nix80]

W. Udo Schröder, Departments of Chemistry and Physics, University of Rochester,

Rochester, NY 14627 USA

J. Tõke, W. Gawlikowicz, J. Lu, WUS (Rochester) R.G. Charity, L.G. Sobotka, (St. Louis) R.T. deSouza (Bloomington)

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Outline

• Introduction: Intermediate energy HI reaction domain How to map nuclear phase diagram (?)

• Experiment: Cluster emission in the dissipative reaction environment

• Models: ConventionalExpansion and surface phenomena, stability and decay of hot nuclei

• Conclusions

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Macroscopic clusters: Surface ablation with lasers

(Zhigelej et al., 2000)

Goals: Mechanism (thermodynamical, dynamical), dependence of cluster multiplicities & sizes on energy

surface physics, nano materials

Low energy

High energy

Low energy: few large clustersHigh energy: many small clusters vaporization

Equilibrium molecular dynamics models

Reaction Scenarios

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4Reaction Scenarios

Near the Barrier(E/A =5-20 MeV)

Projectile

Target

Projectile-like fragment

Target-like fragment

Dissipative Collisions leading to Focussing and Orbiting

2 possible emitters: PLF, TLF

Fermi/Intermediate Energies(E/A =20-100 MeV)Participant

Fireball

PeripheralParticipant-Spectator Scenario (Fireball)3 emitters: PLF, TLF, IVS

Fermi/Intermediate Energies: CentralMulti-Fragmentation (Fireball at high energies)1 emitter: CN

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Interaction and Relaxation Times Scales

Thermal relaxation times (BUU) decrease with T. Slower relaxation at higher density ρ.

tevap experimental CNsystematics.

trxn dynamical NEMsimulations Bi+Xe

Limits to heat generation in nucleiThermal stability: t< trxninteraction/acc time

ρ =0.5ρ0

ρ = 2ρ0

tevap

t-Reaction/Accel. Bi+Xe 40A MeV

trel

trxn

trel: Abu-Samreh&Köhler, NPA 552, 101(1993), trxn WUS

0.4 1 2

2 20 0 0

310 57( , ) 1 0.11 160 /

ρ ρ ρρρ ρ ρ

−⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

relTt T T

T T

24 13( ) 5.6 10−≈ ⋅ MeV Tevapt T e s

Evap too fast for fission !

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Outline




• Conclusions

Nuclear EOS

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(after Bertsch & Siemens (PL 126B,9): Skyrme interaction)

Spinodal (for uniform ρ):

( ) 1 22max

tkkUnstable modes k k c e ωκρ ρ

− +< = − → ∼

21 2p f

Compressibility κ ρ ρρ ρ

− ∂ ∂= ⋅ = ⋅

∂ ∂

k: wave number, ωk: frequency

This is one of several reasonable theoretical estimates. Needs experimental verification/falsification.

exponential growth

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Mechanical Nuclear Instabilities

Density multipoles, sharp sphere (bulk), high E*

(schematic treatment, surface, Coulomb barrier)

Doorway states: assumptionsisentropic expansion fragmentation

( )nLL rδρ

Has to overcome 3-body saddle

Isentropic? No entropy generation? No dissipation? Calls for experimental test

Cluster decay = signature of spinodal decomposition?

Nörenberg et al., Eur. Phys. J. A9, 327 (2000)

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How to Map the Nuclear Phase Diagram

Phases of infinite nuclear matter

Attempt to reach unstable (spinodal) region with nuclear reactions cluster decay ?

How to produce an equilibrium situation (heat bath?)

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Limits to Nuclear Heating

Natowitz et al. PRC65, 34618 (2002)

Limiting temperatures Tlimfrom sequential particles

Tlim cannot be exceeded.

T > 7 MeV: resolution problems (PE)

Negative heat capacities?

Potential reasons for limiting T:

• small tevap

• preequilibriumemission

• expansion cooling

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Decay via Intermediate-Mass Clusters (?)

8( )

25 30cluster

cluster

Z oxygen

E MeV

≈

≈ −Statistical independence of cluster emissionπ Power law(s)

mIMF = multiplicity of IMF clusters

Projectile dissipates 2 MeV/nucleon for each cluster.

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Outline




• Conclusions

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Some Experimental Data

Expt. collaboration:

J. Tõke, W. Gawlikowicz, J. Lu, WUS (Rochester)

R.G. Charity, L.G. Sobotka (St. Louis),

R.T. deSouza (Bloomington)

Examples:Examples:

Phenomenology of the HI reactions209Bi+136Xe (εLab= 7,……, 62 MeV)197Au+86Kr (εLab= ……35, 42, 54 MeV)

:

:

Survey experiments

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SuperBall/Dwarf Calorimeter

Experiments at NSCL/MSU

4π measurement of neutrons + LCPsexcitation energy, impact parameter

4π measurement of charged particles, sampling of massive PLF/TLF

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Dissipative Reaction Environment PLF Distributions

Projectile Trajectories

NEM: Stable <ZPLF> ≈ Zproj

PLF remnant distributions from evaporative decay

<ZPLF>, <σ2ZPLF>

Beam Direction

ΘPT

NEMNEM

NEM

NEM

Deca

y Deca

y

NEM

NEM

2 massive fragments survive PLF + TLF209Bi+136Xe

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Thermal Observables: Joint Multiplicity Distributions

E*209Bi+136Xe

( ) ( )

( ) ( )

* ,

( ) !!

' ' ,

n LCP

lab

n LCP

P E P P m m

f E

P b P P m m

⎡ ⎤= ⎣ ⎦≠

⎡ ⎤= ⎣ ⎦

209Bi+136Xe, ELab/A=62 MeV

P(mn, mLCP) (log contours)

Dotted: average ridge lines for E/A=28, 40, 62 MeV.

E*, bIndependent of ELab

Also (SB): P(TKEn), 5 ∆Θ

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Unusual Statistical Features of Cluster Emission

E*

E*

E*

E* E*

E* E*

inclusive

E* E* Regions in mn/mlcpplane correspond to regions in total excitation energy E*of reaction fragments

For mIMF = 3,4…,8 no change in illuminated E* region.

Challenge: statistical competition clusters and light particles?Valid for all cluster components???

spontaneous multi-fragmentation?

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Detector Response to Charged Particles

TLF

PLF

PLF+TLF

Average cluster emission patterns (Zcluster > 2, deduce A from A≈Aattractor)

Energy resolution, angular resolution (granularity), detection/ID thresholds all distort the ideal (circular) kinematical pattern

Beam Direction

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Light Charged Particles and Clusters

Sources are kinematically correlated with PLF (projectile-like fragment)TLF (target-like fragment)IVS (Intermediate velocity source = “CN”, pre-equilibrium)

Similar invariant velocity distributions for LCP & IMF clusters (3 ≤ Z≤10)

Concentrate first on sequential cluster emission from PLF (well prepared)

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Moving-Source Parameterization of Cluster Emission

Semi-peripheral Central

“b“ Selection: P(mn , mlcp)

Cluster multiplicities increase with decreasing b, increasing mIVS/(mPLF+mTLF). IVS clusters have much harder (“PE” type) spectra than PLF clusters.

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Outline




• Conclusions

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Challenges to Conventional Statistical Models

Fundamental problems for stat. cluster emission (mcluster > 1):

• high emission barriers: VB ≈ (50-60) MeV, T [ 5-6 MeV• nuclear interactions non negligible even at ρ < ρ0/3.

Shortcomings of conventional statistical models

No, or unrealistic, reaction scenarios

EESM: “scales” the barrier at the expense of compressional, and not thermal energy.

SMM, MMMC: “scale” the barrier one nucleon at time – rely on randomized gas flux (perfectly reflecting container walls).limited account of entropy (neglect of ρ < ρ0 , level density)

Fisher’s droplet model: does not treat emission. Questionable adaptations

Percolation: does not treat emission.

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Surface Matter/State Density

Number (i a ) of states/nucleon is larger in the diffuse surface than in the dense bulk.Tõke & Swiatecki NPA 372, 141(1981)

bulk

surface

Nuclear Matter Distribution

r (fm)

ρ C(r

) (1

01

9C

/cm

3)

Light nuclear clusters (e.g., 12C) are mostly surface

( ) 2 ( ) * ( , *) /, * a A E S A E kA E e e⋅Ω = =Fermi gas density of states

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o

ρρ

/ oρ ρ

Thermal Expansion: Equilibrium Density

( )

*

*

*

2

1 (1 9 8 )4

0

ρ

ρ

ρ

∂ ∂ =

= ⋅

→

= + −

therm

eq Total

o Bin

E

ding

S a E

E

S

E

0

2 323( ) ( )ρ ρα α −

= +

= +

SurVolume

V

ace

S o

f

a

Level Density Paramete

A

ra a

A

a

Density drops to ρ0/3 before disassembly nucleons

2bind

* 2 / 3 2therm 0

*c r 0omp

*0E E aE ( ) T1 )E ( ρ ρ ρρ −− ⋅= + = +

EOS-harmonic approximation

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Effects of Irreversible Nuclear Expansion

Caloric Curve

(Effects of m*) Sobotka PRL 2004

Weisskopf, ρ0

HIFG, ρequ

T non monotonic, plateau

Evaporation time

Softening/melting of surface reduction in energy of surface normal modes stability loss.

Reduction of T increased evaporation time increased stability/lifetime

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Multipole Surface Vibrations as MF Doorways ?

22 3

1 3

5 ( 1)( 1)( 2)

4 2 (2 1)s

cb Z

C AAλ

λλ λπ π λ

−= − + −

−b Bohr & Mottelson II, Ch. 6

actually an over-estimate !

5 42 2

2 2

24

&

( ) 17.5

( ) 0.7 1 7.6 10

cs

c

C

Shlomo Kolomietz

T Tb T MeV

T T

Tb T MeV

MeV−

⎛ ⎞−= ⋅ ⎜ ⎟⎜ ⎟+⎝ ⎠

⎛ ⎞⎛ ⎞= ⋅ − ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

With increasing E* decreasing γdecreasing energy of λ modeπdegeneracy of different λdevelop at same E*(mn, mlcp)

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Cluster Decay of Excited Nuclei

saddle config

mon

on

ucle

us

Entropy per nucleon for

• Mono-nucleus: 208Pb* parent• Dinucleus: 192W residue in nuclear contact with 16O cluster

True saddle-point

Vnucl

VCoul

VTotal

(fm)

dinucleus

Effective Emission BarriersClu

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BEffS TP e e

−∆∝ = EffB T S= − ∆ ,∆ = −Saddle diN monoNS S S

EffB(MeV)

T (MeV)

Without surface effects

With surface effects

Dramatic entropy gains at high T favor cluster emission

197Au

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Fragment Emission Probabilities

ClusterZ

Cluster

p

pp

For E*/A > 5 MeV 16O emission from 197Au more likely than proton emission !

mIMF>1 because Q<0 for heavy CN*

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Academic Considerations: Boxed Fermi Matter

If one could confine nuclear matter (A, Z) in a box with perfectly reflecting walls (V = Vliqu+Vgas = const., T= const.) equilibration to maximum entropy minimum of free energy F

ρ ρ ρ ρ −

= + −

= − = − −

= − ⋅

⋅

* * 2compr therm

* 2 2 2 / 3 2compr bind 0

*t

0 0

otal E E 2aT

E aT E (1 ) a ( )

F T S

T

E

2 2 3

20

0 0

1ερ ρ

−⎛ ⎞ ⎛ ⎞

= − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

gas gasgas gas Bind gas

gas gas

A AF A A a T

V V

2 2 3

20

0 0

1ερ ρ

−⎛ ⎞ ⎛ ⎞

= − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

liqu liquliqu liqu Bind liqu

liqu liqu

A AF A A a T

V V

Isotherms of Nuclear MatterIsotherms of Confined Nuclear Fermi MatterClu

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Real gas of interacting nucleons in confined volume V, T-const.

Spinodal

Compression leads to liquefaction. Results automatically from minimization of

F=Fliqu+Fgas

Maxwell construction not necessary.

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Liquid-Gas Coexistence Line of Fermi Matter

/ρ ρo

T( )MeV

Critical exponent α=0.5 for ρ(T-Tc)

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Conclusions: Dynamics & Thermodynamics of Cluster Decay

• There are different modes of cluster emission: non-equilibrium and equilibrium

• Found that expansion of hot nuclear matter changes high-T statistical decay pattern significantly ∆S, ∆Ssurf

• Entropy-driven decay favors clusters over simple particles, surface melting equivalent to phase transition

• Exact mechanism of cluster formation and decay not yet determined.

• If one can establish mechanism, then new access to (isospin) nuclear EOS.

• New important study of nuclear surface becomes possible

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cluster emission in intermediate-energy nuclear reactions and the

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