Fission rate and_time_of_higly_excited_nuclei

Fission rate and time of highly excited nuclei inmulti-dimensional stochastic calculations

Yu. A. Anischenko, A. E. Gegechkori and G. D. Adeev

Omsk State University, Russia

September 2, 2010

Zakopane Conference of Nuclear Physics 2010

Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 1 / 23

Motivation

The evolution of the nucleus should be described taking into account manycollective variables.

Investigate the influence of the inclusion of collective variablesresponsible for nuclear shape evolution in the dynamical model.

Study the impact of the orientation degree of freedom on the fission rateand time of the compound nuclei.


The stochastic approachMacroscopic description of fissiondynamics

Collective degrees of freedom thatdescribe the gross features of thefissioning nucleus. Similar to a massiveBrownian particle

Internal degrees of freedom thatconstitute «heat bath»

Langevin EquationLangevin equation describes time evolution ofthe collective variables like the evolution ofBrownian particle that interacts stochasticallywith a «heat bath».Suppose that equilibration time of thecollective variable is much greater than of theintrinsic degrees of freedom.

Random character


The decay of the compound nuclei


Collective coordinates

Fission is a multi-dimensional process

Bare minimum of collective coordinatesthe elongation of the nucleus

constriction coordinate that describes nuclear neck thickness

mass-asymmetry coordinate defined as the ratio of the masses ofnascent fragments

Additional coordinatesorientation degree of freedom(in case of high angular momenta)a

charge-asymmetry coordinate

aJ. P. Lestone, Phys. Rev. C 59, 1540 (1999).


Funny hills parametrization1

Equation of the nuclear surface(profile function)

ρ2s (z) =

{(c2− z2)(As + Bz2/c2 + αz/c), if B > 0;

(c2− z2)(As + αz/c)exp(Bcz2), if B < 0,(1)

As =

c−3−B/5, if B > 0;

− 43

B

expBc3+(

1+ 12Bc3

)√−πBc3erf(

√−Bc3)

, if B < 0. (2)

B = 2h +c−1

2. (3)

Evolution of nuclear shapes

c - the elongation of the nucleus

h - constriction coordinate

α - mass-asymmetry parameter relatedto the ratio of the masses of nascentfragments

1M. Brack et. al., // Rev. Mod. Phys., 44 320 (1972).Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 6 / 23

Multidimensional Langevin EquationSystem of coupled Langevin equations

p(n+1)i = p(n)

i − τ

[12

p(n)j p(n)

k

(∂ µjk (~q)

∂qi

)(n)

−K (n)i (~q)− γ

(n)ij (~q)µ

(n)jk (~q)p(n)

k

]+ θ

(n)ij ξ

(n)j

√τ, (4)

q(n+1)i = q(n)

i +12

µ(n)ij (~q)(p(n)

j + p(n+1)j )τ, (5)

Input parameters

Inertia Tensor: ‖µij(~q)‖ = ‖mij(~q)‖−1

Friction Tensor: γij(~q)

Conservative Force: Ki(~q) =− ∂F(~q)∂qi

, where F(~q) - free energy

Random Force: θijξj , where θij is the random force amplitude

〈ξ (n)i 〉= 0, (6)

〈ξ (n1)i ξ

(n2)j 〉= 2δijδn1n2 , (7)

Dij = θik θkj = T γij (8)


Potential

V (q, I,K ) = (Es(q)−E(0)s )+(Ec(q)−E(0)

s )+Erot(q, I,K )(9)Es(q) - surface energy of the deformed nucleus

Ec(q) - Coulomb energy of the deformed nucleus

E(0)s - surface energy at the ground state

E(0)c - Coulomb energy at the ground state

Erot - rotational energy of the deformed nucleusrelative to the non-rotating sphere

Free Energy

F(q, I,K ,T ) = V (q, I,K )−a(q)T 2 (10)

a(q) - level density parameter

T =√

Eint/a(q) - temperature of the nucleus

The potential energy of the nucleus was calculatedwith Sierk’s parametersa

aA. J. Sierk, Phys. Rev. C 33, 2039 (1986).

Potential surfaces for 254Fm

The potential energy surface for

the compound nucleus 254Fm in

coordinates {c,h} (a) and

{c,α}(b)Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 8 / 23

Inertia tensorWerner-Wheeler approximationInertia tensor is calculated according to Werner-Wheeler approximation forincompressible irrotational flow.a

υz(z) = ∑i

Ai(z;q)qi , (11)

υρ (z,ρ) =−ρ

2 ∑i

∂Ai(z;q)

∂zqi , (12)

υφ (z,ρ) = 0, (13)

aDavies K. T., Sierk A. J. and Nix J. R., // Phys. Rev. C 13 2385 (1976).

Expressions

mij =34

zmax∫zmin

ρ2s (Ai Aj +

ρ2s

8A′i A′j)dz · (M0R2

0) (14)

Ai(z,q) =− 1ρ2

s (z,q)

∂

∂qi

z∫zmin

ρ2s (z ′,q)dz ′ (15)


Friction

Friction tensor was calculated using the ”wall+window” model of the modifiedone-body dissipation mechanism with a reduction coefficient from the ”wall”formula2

One body dissipation

γij =

ksγwall

ij , neck doesn’t exist

ksγwallij f (RN) + γww

ij (1− f (RN)), neck exists

(16)

γwwij = γ

winij + ksγ

wallij (17)

f (RN) = sin2(πRN

2RM) (18)

ks - reduction coefficient from the wall formula 0.2≤ ks ≤ 1.0

2J. Blocki, Y. Boneh, J. R. Nix, et al., Ann. Phys. (N. Y.) 113, 330 (1978).Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 10 / 23

Initial and final conditionsInitial ConditionsThe initial values were chosen according to the vonNeumann method with the generating function

P(q0,p0, I,K , t = 0)∼ P(q0,p0)σ(I)P0(K ),

whereP(q0,p0)∼ exp

{− V(q0 ,l)+Ecoll(q0 ,p0)

T

}δ(q0−qgs(I,K )

)Scission configurationScission criterion is the criterion ofinstability of the nucleus with respect tovariations in the neck thickness:

∂ 2V∂h2 c,α=const

= 0

RN = 0.3R0, (19)

RN - neck thickness and R0 - radius at ground state.

Initial angular distribution

Spin distribution σ(I) ofcompound nuclei is parametrizedaccording to the triangulardistribution with Imax value takenfrom experemental dataa. Initialdistribution of K is uniform: [−I, I]

aB. B. Back et al., Phys. Rev. C 32,195 (1985).


Orientation degree of freedom

Rotational energy

Erot(q, I,K ) =h2I(I + 1)

2J⊥(q)+

h2K 2

2Jeff(q)

The effective moment of inertia:

J−1eff = J−1

‖ − J−1⊥

Rigid body moments of inertia:

J⊥(‖)(q) = J(sharp)⊥(‖) (q) + 4Ma2

M ,

where J(sharp)⊥(‖) - rigid body moments of inertia

calculated in liquid drop model with sharp boundary.aM - Sierk’s parametersa

aA. J. Sierk, Phys. Rev. C 33, 2039 (1986).

K-state

I - total angularmomentum

K - spin about thesymmetry axis

M = 0 - projection of thetotal angular momentumon the direction of thebeam.


Dynamical evolution of the orientation degree offreedom(K-state)

Orientaion degree of freedom is treated as thermodynamically fluctuatingoverdamped coordinate.3 Its evolution is defined by a reduced Langevinequation:

K (n+1) = K (n)− γ2K I2

2∂Erot

∂Kτ + Γ

(n)K γK I

√T τ, (20)

where γK = 0.077(MeV10−21s)−12 is a parameter that controls the coupling

between K and the thermal degrees of freedom;ΓK is a random number from a normal distribution with unit variance.

3J. P. Lestone and S. G. McCalla, Phys. Rev. C 79, 044611 (2009)Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 13 / 23

Fission Rate

General

Bohr-Wheeler expression: RBWf = T

2πexp(−Bf /T )

Kramers formula: RKf =

(√1 + γ2− γ

)hωgs

T RBWf , where γ = γ

2ωsd

where γ is nuclear friction coefficient

Time-dependent fission rateIn Langevin calculations the time-dependent fission rate is defined as follows:

Rf (t) =1

N−Nf (t)∆Nf (t)

∆t, (21)

where N - the total number of simulated particles (trajectories).Nf (t) the number of particles which reach the scission point during the time t .∆Nf (t) the number of particles which reach the scission point during the timeinterval t → t + ∆t .


Ensemble of trajectories

-2-1012345

0.0 50.0 100.0 150.0 200.0

c/r 0

t,10−21c

Figure: Sample Evolution of the coordinate responsible for nucleus elongation. Toproperly obtain information about fission rate the ensemble of the trajectories mustcontain about 1 million trajectories that reached scission configuration.


Time dependence of the fission rate

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.0 10.0 20.0 30.0 40.0 50.0

R(t

),10

21s−

1

t,10−21s

1D

Figure: Time dependence of the fission rate for the reaction 16O + 208Pb −→ 224Thwith Elab = 90MeV . Only one collective coordinate c is taken into account.


Time dependence of the fission rate: h and α added

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.0 10.0 20.0 30.0 40.0 50.0

R(t

),10

21s−

1

t,10−21s

1D3D

Figure: Time dependence of the fission rate for the reaction 16O + 208Pb −→ 224Thwith Elab = 90MeV . Three collective coordinates c,h,α are taken into account.


Time dependence of the fission rate: K-state added

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.0 10.0 20.0 30.0 40.0 50.0

R(t

),10

21s−

1

t,10−21s

1D3D

3D+K

Figure: Time dependence of the fission rate for the reaction 16O + 208Pb −→ 224Thwith Elab = 90MeV . Coordinates c,h,α and K-state are taken into account.


Time dependence of the fission rate

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

R(t

),10

21s−

1

t,10−21s

Figure: Time dependence of the fission rate for the reaction 16O + 208Pb −→ 224Thwith Elab = 130MeV . Particle evaporation is taken into account. Along the entirestochastic Langevin trajectory in the space of collective coordinates, we monitored,fulfillment of the energy-conservation law in the formE∗ = Eint + Ecoll(q,p) + V (q, I,K ) + Eevap(t)


The mean fission time

The concept of the mean first-passage time is applied in calculating meanfission time4:

〈tf 〉= τMFPT [q0→ ∂G] = limN→∞

1N

N

∑n=1

τFn ,

where τFn , n = 1, . . . ,N is time required to escape G area for the first time for

implementation of the Brownian process q(t). Suppose that q0 ∈ G andτMFPT [q0→ ∂G] are finite. q0 = qgs(I,K ). The region G includes the potentialwell and borders on the saddle- point configuration.

4D. Boilley, B. Jurado and C. Schmitt, Phys. Rev. E, 70, 056129 (2004)Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 20 / 23

The fission time distribution

0

50

100

150

200

250

300

350

0.0 20.0 40.0 60.0 80.0 100.0

Yie

ld

tf ,10−21s

tmp

〈tf 〉

Figure: The fission time distribution for the reaction 16O + 208Pb −→ 248Th withElab = 140MeV . tmp - most probable time


Fission lifetime

100

1000

80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0

〈t f〉,

10−

21s

Elab

224Th

1D

Figure: Calculations of the mean fission time 〈tf 〉 as a function of Elab for the reaction16O + 208Pb −→ 224Th. Only one collective coordinate c is taken into account.


Fission lifetime: h and α added

100

1000

80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0

〈t f〉,

10−

21s

Elab

224Th

1D3D

Figure: Calculations of the mean fission time 〈tf 〉 as a function of Elab for the reaction16O + 208Pb −→ 224Th. Three collective coordinates c,h,α are taken into account.


Fission lifetime: K-state added

100

1000

80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0

〈t f〉,

10−

21s

Elab

224Th

1D3D+K

3D

Figure: Calculations of the mean fission time 〈tf 〉 as a function of Elab for the reaction16O + 208Pb −→ 224Th. Coordinates c,h,α and K-state are taken into account.


Neutron multiplicity5

2

4

6

8

10

80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0

〈n〉

Elab

224Th

ks = 0.25ks = 1.0

experiment

Figure: The mean neutron multiplicity 〈tf 〉 calculated as a function of Elab for thereaction 16O + 208Pb −→ 224Th. The symbols in green colour are experimentaldata. The Langevin calculations carried out for different values of the reductioncoefficient: ks = 0.25(red) and ks = 1.0(blue)

5H. Rossner et. al., Phys. Rev. C 45, 719 (1992)Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 25 / 23

ConclusionsIn the present work the influence of the dimensionality of the deformationspace on the fission rate and mean fission time is investigated. {c,h,α}parametrization is used as the shape parametrization of the nuclear surface.

A considerable increase of the stationary fission rate can be obtainedwhen we take into account the constriction h and mass-asymmetrycoordinate α . The difference between 1D and 3D cases is about 30–70%for the heavy fissioning nuclei with A∼ 220.For the light nuclei with A∼ 170 these effects are more considerable andthe difference is up to 500–1000%.The orientation degree of freedom impact on the fission rate and timealmost fully canceled the effect produced by inclusion of nuclear neck andmass-asymmetry coordinates in the 1D Langevin calculations. Thedifference of 5-25% between 4D and 1D calculations was found as theresult of this research.To learn more about the role of the dissipation effects the calculations have also been performed forone-body viscosity with the reduction coefficient from the ”wall” formula ks = 1.0 and two-bodyviscosity. It was shown that the ratios of the stationary fission rates obtained in the calculations withthe different dimensionalities: remain almost the same for different dissipation mechanisms. Thus weconclude that the fission rate is mostly determined by the structure of the potential energy surface ofthe system.Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 26 / 23

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Fission rate and_time_of_higly_excited_nuclei