Collective nuclear motion at finite temperature investigated with fission reactions induced by 238 U...

Collective nuclear motion at finite temperature investigated with fission

reactions induced by 238U at 1 A GeV on deuterium

Jorge Pereira Conca Universidad de Santiago de Compostela (USC)

Collective phenomena in hot nuclei

VibrationsRotations Fission

Nucleus: N-body mesoscopic system

Presence and nature of collective nuclear phenomena

Deexcitation of hot compound nuclei (wide range of excitation energies)

Nuclear Physics: nature of the nuclei

•Individual excitations of nucleons

•Collective excitations of nucleons

Single-particle model

Collective model

Spectroscopic technique (Low Lying states)

Study of collective nuclear phenomena from the competition fission vs. evaporation

Experimental observable: fission and evaporation production cross section

Excited nuclei (pre-fragments):

•Well defined initial conditions (Small deformations and low angular momenta)

•Wide range of excitation energies

Spallation reaction 238U at 1AGeV on deuterium

Collective phenomena: competition between fission and evaporation

•High fissilities

•Experiment: measurement of fission and evaporation residue productions in 238U(1AGeV)+d

•Analysis of fission residues

•Investigation of collective phenomena from the competition fission vs. evaporation

Rotational and vibrational motion

Dynamics of fission

Study of the spallation reaction 238U(1AGeV)+d in inverse kinematics

at GSI, Germany

Separation and Identification of the reaction products with the FRS

•(A/Z) identification:βγBρ

ZA

1

4220 D

xMx1(BρBρ )

1/2

22

2

ToFcL

1ToFcL

βγ

•Z identification: ΔEZ

•Longitudinal velocities:

BρAZ

βγv

A/Z-resolution ~10-3

Separation of fission residues with the FRS

Measurement of fission residue productions in 238U(1AGeV)+d

Measurement of fission residue productions in 238U(1AGeV)+d

Production cross sections of 780 fission residues were measured (uncetainties ~ 10-20%)

Evaporation residue productions (E. Casarejos Ph.D. USC, 2001)

Velocity measurements of fission residues

Velocities of fission residues were determined with uncertainties 15%

Particularities of the reaction 238U (1AGeV)+d

•Competition between fission and evaporation extends over a wide excitation energy distribution

•High fissilities of decaying pre-fragments

•Strong correlation between the mass of evaporation residues with excitation energy

•Influence of shell effects and collective excitations at different deformations

(Calculation)

Spherical

Deformed

•Kinematical properties of fission residues provide valuable information of the fissioning nuclei





Dynamics of fission

Influence of rotational and vibrational motion on the competition fission vs. evaporation

•Influence of Shell Structure

•Level density enhancement due to the presence of rotational and vibrational bands: (E)=Kcoll.int(E)

Correction of the level density and binding energy due to structural effects

Fermi-gas model

At low excitation energies:

5/41/4FG *E12a

*aE2expπ(E)ρ

Level density

fissev ΓΓ

Collective enhancement (rotations and vibrations) at different deformations modifies

the competition fission VS. evaporation

•Large deformations: Rotations

•Small deformations: Vibrations

Description of spallation residue productions with model calculations

1. First step: pre-fragment formation (ISABEL intranuclear cascade)Y.Yariv and Z.Fraenkel, Phys. Rev. C20 (1979)

Pre-fragment (Z,N,A; J; E)

2. Second step: pre-fragment deexcitation (ABLA)

Level density: Fermi-gas model

Shell and pairing effects

Collective enhancementA.Junghans et al., Nucl. Phys. A629 (1998)

f(E))(βσ)β(E,K0.15)β(E,K 22

2rot2coll

f(E))(βσN)(Z,βS)β(E,K0.15)β(E,K 22

eff2

2vib2coll

Decay probability from statistical model with dynamical correction for fission according to Kramers

and Grangé, Jun-Quing and Weidenmüller

25S

(E)=Kcoll.int(E)

Influence of rotational and vibrational motion on the evaporation residue productions

Fermi-gas model

Shell effects

Coll. Enh. S=25

(A.Junghans)

Kvib underestimates survival probability against fission for small-deformed nuclei

Influence of rotational and vibrational motion on the evaporation residue productions

•Highly precise measurements enable to investigate the sensitivity of the evaporation residue productions to the presence of collective motion (rotations and vibrations)

•Vibrational enhancement factor Kvib must be increased (S=75) with respect to the formulation of A.Junghans et al. in order to properly describe the survival probability against fission

•Evaporation residue productions offer a complementary approach to the spectroscopic techniques for analyzing the presence of collective excitations





Dynamics of fission

Dynamics of fission

Statistical model: Available phase-space at the saddle point

Dynamical model: Time evolution of the probability flow across the saddle point

Coupling of collective deformation degree of freedom Q with internal degrees of freedom through dissipation

Fission probability

Langevin equation:

F(t)βMT

dtdQ

βdQ

(Q)VdM1

dtQd2

2

~

Drift force Friction forceDiffussion (stochastic) force

Fission: diffusion process governed by the Reduced Dissipation Coefficient

•Fission probability needs time to go up to the stationary value (transient effects)

Transient effects increase evaporation residue productions with respect to fission

(specially at high energies)

•During this time the compound nucleus can evaporate nucleons

•Dynamics of fission from the ground-state to the saddle-point: evaporation residue productions

•Dynamics of fission beyond the saddle point: kinematical properties and production cross sections of fission residues

Dynamical calculation of the fission probability

Solution of FPE:

•Stationary solution of FPE (Kramers): K Fission width is reduced compared to BW

BWK ΓΓ K

0

2

0 221

ωβ

ωβ

K

•Time-dependent solution of FPE (Grangé, Jun-Qing, Weidenmüller): f(t) Time evolution of fiss(t) before the stationary regime (K )(transient effects)

Fokker-Planck equation (FPE) ~ Langevin equation

)τΘ(tΓ(t)Γ fKstep

)exp(-t/τ-1Γ(t)Γ fKexp

(t)WΓ(t)Γ FPEKFPE

•Step function

•Exponential in-growth function

•Approximated solution of FPE for a parabolic potential (B.Jurado et al., Phys.Lett. B (2003))

)()( tft Kfiss

Influence of dissipation on the evaporation residue productions

•Transient effects due to dissipation are expected at high energies E*

•At lower energies, dissipation hinders the fission decay according to Kramers factor

Strong sensitivity to

Strong sensitivity to (t)

•Experimental data enable to investigate (t): validation of fpe(t)

•Measured residue productions are compatible with =2x1021s-1 (small deformations)

•Strong correlation between mass and excitation energy of evaporation residues enables to

separate dissipation effects due to Kramers factor and transient times

Effects of dissipation at deformations beyond the saddle point

H.Hofmann and J.R.Nix, Phys. Lett. B122 (1983): Dissipation enlarges the saddle-to-scission time ssc

γγ1ττ 20sscssc ΔV/TR

ω2

τ0

0ssc

02ωβ

γ

At large deformations, dissipation damps the fission motion due to the friction force

ssc can be determined from saddle-to-scission neutron multiplicities ssc

sscν

1i nssc E*)J,A,(Z,Γ

τ

•Dynamics of fission from the ground-state to the saddle-point (Fission probability)

•Dynamics of fission beyond the saddle-point

Analysis of saddle-to-scission neutron multiplicities in 238U (1AGeV)+d

•Post-saddle neutron multiplicities were calculated with a deexcitation code: sad is very sensitive to E*sad

Saddle-to-scission ssc = Post-saddle sad – Post-scission sc

Comparison of isotopic distributions for fission and evaporation residues

Evaporation

Fission

Analysis of saddle-to-scission neutron multiplicities in 238U (1AGeV)+d

•Post-saddle neutron multiplicities were calculated with a deexcitation code: sad is very sensitive to E*sad

Saddle-to-scission ssc = Post-saddle sad – Post-scission sc

•Post-scission neutron multiplicities:

Zfiss, A’fiss = “Reconstructed” Fissioning nucleus

Zfiss, Afiss = Real fissioning nucleus

Hypothesis: Real fissioning nuclei are located along the evaporation corridor

No neutrons are evaporated “post scission”

vfiss(Z,A)

fiss(Z,A)

Afiss = A’fiss + sc

Evaluation of dissipation at large deformation from saddle-to-scission neutron multiplicities

Hilsher’s systematics sc(Zfiss,E*) D.Hilsher

and H.Rossner, Ann. Phys. Fr. 17 (1992) 471

)(Zν)(Zν)(Zν fissscfisssadfissssc

γγ1ττ 20sscssc

sscν


τ

All ssc(Zfiss) are compatible with >3; ( 6 )

Agreement of sc with Hilsher’s systematics validates the hypothesis

=24

=10

=6

=3

=1

These values are larger than =2x1021s-1 at small

deformations

•Saddle-to-scission neutron multiplicites >6x1021s-1

(large deformations)

•Evaporation residues =2x1021s-1 (small deformations)

Values of at small and large deformations are compatible with the results obtained from GDR -ray spectra (Shaw et al.; Diószegi et al.) and from pre-

scission neutron multiplicities (Fröbrich et al.)

Indications of deformation-dependent dissipation

Summary and conclusions

•Production cross sections and kinematical properties of 780 fission residues have been measured with high accuracy at the FRS

•U+d system constitutes an optimum scenario to investigate collective phenomena

•The measured productions of evaporation residues in regions of small deformation revealed a greater contribution of Kvib to the survival probability against fission

•Measured evaporation residue productions provided valuable information to investigate the dynamics of fission at small deformations

• =2x1021s-1

• Validation of the formulation of B.Jurado fpe(t)

•Kinematical properties and production cross sections of fission residues enabled to investigate the dynamics of fission at large deformations

•Post-scission neutron multiplicities agreed with Hilscher’s systematics

•Saddle-to-scission multiplicities are compatible with >6x1021s-1

•Possible indications of deformation-dependent dissipations

Determination of production cross sections

(Z,A) =Y(Z,A).fsc2.feff.fchs.fTi.fmr.fT

•fsc2: Interactions of nuclei in the plastic SC2

•feff: Detection efficiencies

•fchs: Charge-state contamination

•fTi: Nuclear reactions in the Ti target container

•fmr: Multiple-reactions in the target

•fT:Angular transmission

Influence of dissipation on total fission cross section

Spallation reactions induced by 238U at 1GeV/u on different targets

Excitation energy of pre-fragments E* Target mass

Transient effects are important for heavier targets

(high E*)

•Total fission cross section for 238U(1AGeV)+d is not sensitive to dissipation

•Total fission cross sections at different E* are well described with FPE(t) and =2x1021s-1

Collective enhancement of the level density

1

cr

cr

dEE

exp1f(E)

f(E)3

β1TARm

5

2f(E)

T(E)σ 22

022rot2

(E)σβS(E)σ rot

22effvib

2

•Phenomenological description of Kcoll(E) (A.R.Junghans et al. Nucl. Phys. A629 (1998) 635):

•Transition from vibrational to rotational enhancement:

Damping of Kcoll with excitation

energy:

Z 0.005ΔN 0.003Δ0.022βeff

0.15βtrans

1)f(E)(σ1K 2coll (E)ρ(E)Kρ(E) intcoll

: spin cut-off parameter

Rotational VS. Vibrational excitations

•Rotational modes: large deformation

•Vibrational modes: small deformations

•Shell effects enhance survival probability against fission

•Vibrational modes enhance survival probability against fission

•Rotational modes enhance fission probability

Reconstruction of kinematics

Kinematics of fission and fragmentation

ABρZ

βγv

Limited -acceptanceLimited angular-acceptance

Separation of fission and fragmentation from their kinematics measured with the FRS

Limited angular acceptance of the FRS enables the separation of fission and fragmentation from their kinematics

•Longitudinal velocity spectra of each nucleus was measured

• Fission and fragmentation components were separated by fitting the spectra to specific functions

•Fit parameters: mean velocities and yields of fission and fragmentation residues

New approach to determine the Angular Transmission

Determination of the angular acceptance a with Monte Carlo calculations

New analytical approach to determine the Angular Transmission

•Calculation of angular transmission for fission and fragmentation according to their kinematics

Fragmentation Fission

)σ(θ)α

(T fr 2

2

2exp1

2

cos1 ffT

2

cos1 bbT

Ingredients of the Monte Carlo code

Level density (Fermi-gas state density)4/54/1~12)exp(π

ρEa

S

))()((~2 EhPEkδUEaS 13/21 MeV A095.0MeVA 073.0~ sBa

U

)exp(1)( EEk a~/A4.0/1 3/4

241 2gP /6~ag

;,

,1

11)(

2

.

crit

crit

critEE

EEEE

Eh

MeV10critE

Entropy

Asymptotic level-density parameter (Ignatyuk)

Shell correction calculated in the finite-range liquid-drop model

Damping of shell effects with E

Effective pairing energy shift MeVA/12

Washing out of the pairing correlations

fBFission barriers (Sierk)

ISABEL VS. INCL CASCADES

ISABEL and INCL

Fission cross sections

overestimated with INCL

INCL overestimates angular momentum

Large angular momenta J reduce the fission barriers

Approximated solution of the FPE for a parabolic potential

W(x,v,t)vm

βkTU(x)

xmβv

vtv

tW(x,v,t)

2

21

Time-dependent probability flux of trajectories across the saddle-point

,v,t)dvW(xvΦ bxb

Probability that the system is at x<xb

bx

b W(x,v,t)dvdx,t)Π(x,t)Π(x

Φ(t)λ(t)Γ

b

xfissfiss

b

,t)W(x,t)(xv,v,t)dvW(x,t)(xv(t)Φ bbbbxb

),(

),(),()(

txtxWtxv

tb

bbfiss

Kb

bbfiss tx

txWtxvt

),(

),(),()( K

bb

bbfiss txWtxv

txWtxvt

),(),(),(),(

)(

Approximated solution of the FPE for a parabolic potential

Kbb

bbfiss txWtxv

txWtxvt

),(),(),(),(

)( ),(),( txvtxv bb

),(),( txWtxW bparb

Kbpar

bparfiss txW

txWt

),(

),()(

2

2

2exp

2

1),,(

x

par

xtvxW

1)(

22

1 11

122

1

2

22 tSinh

tSinhe

M

T t

g

x

221 4 g

Dissipation from saddle point to scission point

Pre-saddle particle multiplicity: Excitation energy at saddle point

Damping of the fission motion from the ground-state configuration to the saddle point:

=2x1021s-1

•Monte Carlo calculation (ISABEL+ABLA): post-saddle neutron multiplicity

14(Z)ν sadpost

•Hilscher’s systematics: post-scission neutron multiplicity

5E*)(Z,ν sciss-post

Saddle-to-scission neutron multiplicity: 9E*)(Z,ν(Z)νν sciss-postsadpostssc

sscν


τ γγ1ττ 20

sscssc ΔV/TRω

2τ

0

0ssc

>10x1021s-1

Collective nuclear motion at finite temperature investigated with fission reactions induced by 238 U...

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