The document describes a multi-dimensional stochastic calculation of fission rates and times of highly excited nuclei. It presents a system of coupled Langevin equations to describe the time evolution of collective variables like nuclear shape. The model includes collective coordinates for nuclear elongation, constriction, and mass asymmetry. Additional coordinates for orientation and charge asymmetry are also considered. Ensemble calculations of over 1 million trajectories are used to obtain fission rate information as a function of time. Including additional collective variables beyond elongation is found to impact fission rates and times.

1. Fission rate and time of highly excited nuclei in
multi-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

2. Motivation
The evolution of the nucleus should be described taking into account many
collective variables.
Investigate the influence of the inclusion of collective variables
responsible for nuclear shape evolution in the dynamical model.
Study the impact of the orientation degree of freedom on the fission rate
and time of the compound nuclei.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 2 / 23

3. The stochastic approach
Macroscopic description of fission Random character
dynamics
Collective degrees of freedom that
describe the gross features of the
fissioning nucleus. Similar to a massive
Brownian particle
Internal degrees of freedom that
constitute «heat bath»
Langevin Equation
Langevin equation describes time evolution of
the collective variables like the evolution of
Brownian particle that interacts stochastically
with a «heat bath».
Suppose that equilibration time of the
collective variable is much greater than of the
intrinsic degrees of freedom.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 3 / 23

4. The decay of the compound nuclei
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 4 / 23

5. Collective coordinates
Fission is a multi-dimensional process
Bare minimum of collective coordinates
the elongation of the nucleus
constriction coordinate that describes nuclear neck thickness
mass-asymmetry coordinate defined as the ratio of the masses of
nascent fragments
Additional coordinates
orientation degree of freedom(in case of high angular momenta)a
charge-asymmetry coordinate
a
J. P. Lestone, Phys. Rev. C 59, 1540 (1999).
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 5 / 23

6. Funny hills parametrization1
Equation of the nuclear surface(profile function)
2 (c 2 − z 2 )(As + Bz 2 /c 2 + α z /c ), if B 0;
ρs ( z ) = (1)
(c 2 − z 2 )(As + α z /c ) exp (Bcz 2 ), if B < 0,

c −3 − B /5, if B 0;
As = 4 B (2)
− 3 exp Bc 3 + 1+ 1 3
√ √
−π Bc 3 erf( −Bc 3 )
, if B < 0.
2Bc
c −1
B = 2h + . (3)
2
Evolution of nuclear shapes
c - the elongation of the nucleus
h - constriction coordinate
α - mass-asymmetry parameter related
to the ratio of the masses of nascent
fragments
1
M. Brack et. al., // Rev. Mod. Phys., 44 320 (1972).
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 6 / 23

7. Multidimensional Langevin Equation
System of coupled Langevin equations
(n )
(n+1) (n ) 1 (n ) (n ) ∂ µjk (q ) (n ) (n ) (n ) (n ) (n ) (n ) √
pi = pi −τ p pk − Ki (q ) − γij (q )µjk (q )pk + θij ξj τ, (4)
2 j ∂ qi
(n+1) (n ) 1 (n ) (n ) (n+1)
qi = qi + µij (q )(pj + pj )τ, (5)
2
Input parameters
Inertia Tensor: µij (q ) = mij (q ) −1
Friction Tensor: γij (q )
∂ F (q )
Conservative Force: Ki (q ) = − ∂ q , where F (q ) - free energy
i
Random Force: θij ξj , where θij is the random force amplitude
(n)
ξi = 0, (6)
(n1 ) (n2 )
ξi ξj = 2δij δn1 n2 , (7)
Dij = θik θkj = T γij (8)
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 7 / 23

8. Potential Potential surfaces for 254 Fm
(0) (0)
V (q, I , K ) = (Es (q)− Es )+(Ec (q)− Es )+ Erot (q, I , K )
Es (q) - surface energy of the deformed nucleus (9)
Ec (q) - Coulomb energy of the deformed nucleus
(0)
Es - surface energy at the ground state
(0)
Ec - Coulomb energy at the ground state
Erot - rotational energy of the deformed nucleus
relative 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 surface for
The potential energy of the nucleus was calculated the compound nucleus 254 Fm in
with Sierk’s parametersa coordinates {c , h} (a) and
a
A. J. Sierk, Phys. Rev. C 33, 2039 (1986). {c , α}(b)
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 8 / 23

9. Inertia tensor
Werner-Wheeler approximation
Inertia tensor is calculated according to Werner-Wheeler approximation for
incompressible irrotational flow.a
˙
υz (z ) = ∑ Ai (z ; q )qi , (11)
i
ρ ∂ Ai (z ; q )
υρ (z , ρ) = − ∑ ˙
qi , (12)
2i ∂z
υφ (z , ρ) = 0, (13)
a
Davies K. T., Sierk A. J. and Nix J. R., // Phys. Rev. C 13 2385 (1976).
Expressions
˜max
z
3 ρ2 ˜ ˜
˜
mij = ρs (Ai Aj + s Ai Aj )d z · (M0 R0 )
˜2 ˜ ˜ ˜ 2
(14)
4 8
˜min
z
z
∂ 1 2
Ai (z , q ) = − 2 ρs (z , q )dz (15)
ρs (z , q ) ∂ qi
zmin
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 9 / 23

10. Friction
Friction tensor was calculated using the ”wall+window” model of the modified
one-body dissipation mechanism with a reduction coefficient from the ”wall”
formula2
One body dissipation
wall
 
 ks γij , neck doesn’t exist 
γij = (16)
wall ww
ks γij f (RN ) + γij (1 − f (RN )), neck exists
 
ww win wall
γij = γij + ks γij (17)
π RN
f (RN ) = sin2 ( ) (18)
2RM
ks - reduction coefficient from the wall formula 0.2 ≤ ks ≤ 1.0
2
J. 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

11. Initial and final conditions
Initial Conditions Initial angular distribution
The initial values were chosen according to the von
Neumann method with the generating function
P (q0 , p0 , I , K , t = 0) ∼ P (q0 , p0 )σ (I )P0 (K ),
where
V (q0 ,l )+Ecoll (q0 ,p0 )
P (q0 , p0 ) ∼ exp − T
δ q0 − qgs (I , K )
Scission configuration Spin distribution σ (I ) of
Scission criterion is the criterion of compound nuclei is parametrized
instability of the nucleus with respect to according to the triangular
variations in the neck thickness: distribution with Imax value taken
from experemental dataa . Initial
∂ 2V
=0 distribution of K is uniform: [−I , I ]
∂ h2 c ,α=const
a
B. B. Back et al., Phys. Rev. C 32,
RN = 0.3R0 , (19) 195 (1985).
RN - neck thickness and R0 - radius at ground state.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 11 / 23

12. Orientation degree of freedom
Rotational energy K-state
2 2 2
h I (I + 1)
¯ ¯
h K
Erot (q, I , K ) = +
2J⊥ (q) 2Jeff (q)
The effective moment of inertia:
Jeff1 = J −1 − J⊥ 1
− −
Rigid body moments of inertia:
(sharp)2
J⊥( ) (q) = J⊥( ) (q) + 4MaM , I - total angular
momentum
(sharp) K - spin about the
where J⊥( ) - rigid body moments of inertia symmetry axis
calculated in liquid drop model with sharp boundary. M = 0 - projection of the
total angular momentum
aM - Sierk’s parametersa
on the direction of the
a beam.
A. J. Sierk, Phys. Rev. C 33, 2039 (1986).
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 12 / 23

13. Dynamical evolution of the orientation degree of
freedom(K-state)
Orientaion degree of freedom is treated as thermodynamically fluctuating
overdamped coordinate.3 Its evolution is defined by a reduced Langevin
equation:
γ 2 I 2 ∂ Erot (n) √
K (n+1) = K (n) − K τ + ΓK γK I T τ, (20)
2 ∂K
1
where γK = 0.077(MeV 10−21 s)− 2 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.
3
J. 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

14. Fission Rate
General
T
Bohr-Wheeler expression: RfBW = 2π exp (−Bf /T )
¯
Kramers formula: RfK = ˜ ˜ hωgs RfBW , where γ = 2ω
1 + γ2 − γ T
˜ γ
sd
where γ is nuclear friction coefficient
Time-dependent fission rate
In Langevin calculations the time-dependent fission rate is defined as follows:
1 ∆Nf (t )
Rf ( t ) = , (21)
N − Nf (t ) ∆t
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 time
interval t → t + ∆t.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 14 / 23

15. Ensemble of trajectories
5
4
3
2
c /r0
1
0
-1
-2
0.0 50.0 100.0 150.0 200.0
t , 10−21 c
Figure: Sample Evolution of the coordinate responsible for nucleus elongation. To
properly obtain information about fission rate the ensemble of the trajectories must
contain about 1 million trajectories that reached scission configuration.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 15 / 23

16. Time dependence of the fission rate
0.009
0.008 1D
0.007
R (t ), 1021 s−1
0.006
0.005
0.004
0.003
0.002
0.001
0
0.0 10.0 20.0 30.0 40.0 50.0
t , 10−21 s
Figure: Time dependence of the fission rate for the reaction 16 O + 208 Pb −→ 224 Th
with Elab = 90MeV . Only one collective coordinate c is taken into account.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 16 / 23

17. Time dependence of the fission rate: h and α added
0.009
0.008 1D
3D
0.007
R (t ), 1021 s−1
0.006
0.005
0.004
0.003
0.002
0.001
0
0.0 10.0 20.0 30.0 40.0 50.0
t , 10−21 s
Figure: Time dependence of the fission rate for the reaction 16 O + 208 Pb −→ 224 Th
with Elab = 90MeV . Three collective coordinates c , h, α are taken into account.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 17 / 23

18. Time dependence of the fission rate: K-state added
0.009
0.008 1D
3D
0.007
3D+K
R (t ), 1021 s−1
0.006
0.005
0.004
0.003
0.002
0.001
0
0.0 10.0 20.0 30.0 40.0 50.0
t , 10−21 s
Figure: Time dependence of the fission rate for the reaction 16 O + 208 Pb −→ 224 Th
with Elab = 90MeV . Coordinates c , h, α and K-state are taken into account.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 18 / 23

19. Time dependence of the fission rate
0.018
0.016
0.014
R (t ), 1021 s−1
0.012
0.01
0.008
0.006
0.004
0.002
0
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0
t , 10−21 s
Figure: Time dependence of the fission rate for the reaction 16 O + 208 Pb −→ 224 Th
with Elab = 130MeV . Particle evaporation is taken into account. Along the entire
stochastic Langevin trajectory in the space of collective coordinates, we monitored,
fulfillment of the energy-conservation law in the form
E ∗ = Eint + Ecoll (q, p) + V (q, I , K ) + Eevap (t )
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 19 / 23

20. The mean fission time
The concept of the mean first-passage time is applied in calculating mean
fission time4 :
1 N F
tf = τMFPT [q0 → ∂ G] = lim τn , ∑
N →∞ N
n =1
F
where τn , 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 and
τMFPT [q0 → ∂ G] are finite. q0 = qgs (I , K ). The region G includes the potential
well and borders on the saddle- point configuration.
4
D. 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

21. The fission time distribution
350
300
250
tf
200
Yield
tmp
150
100
50
0
0.0 20.0 40.0 60.0 80.0 100.0
tf , 10−21 s
Figure: The fission time distribution for the reaction 16 O + 208 Pb −→ 248 Th with
Elab = 140MeV . tmp - most probable time
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 21 / 23

22. Fission lifetime
1D
1000
tf , 10−21 s
100 224 Th
80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0
Elab
Figure: Calculations of the mean fission time tf as a function of Elab for the reaction
16 O + 208 Pb −→ 224 Th. Only one collective coordinate c is taken into account.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 22 / 23

23. Fission lifetime: h and α added
1D
3D
1000
tf , 10−21 s
100 224 Th
80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0
Elab
Figure: Calculations of the mean fission time tf as a function of Elab for the reaction
16 O + 208 Pb −→ 224 Th. Three collective coordinates c , h, α are taken into account.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 23 / 23

24. Fission lifetime: K-state added
1D
3D+K
1000
3D
tf , 10−21 s
100 224 Th
80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0
Elab
Figure: Calculations of the mean fission time tf as a function of Elab for the reaction
16 O + 208 Pb −→ 224 Th. Coordinates c , h, α and K-state are taken into account.
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 24 / 23

25. Neutron multiplicity5
10 ks = 0.25
ks = 1.0
8 experiment
6
n
4
224
Th
2
80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0
Elab
Figure: The mean neutron multiplicity tf calculated as a function of Elab for the
reaction 16 O + 208 Pb −→ 224 Th. The symbols in green colour are experimental
data. The Langevin calculations carried out for different values of the reduction
coefficient: ks = 0.25(red ) and ks = 1.0(blue)
5
H. Rossner et. al., Phys. Rev. C 45, 719 (1992)
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26. Conclusions
In the present work the influence of the dimensionality of the deformation
space 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 obtained
when we take into account the constriction h and mass-asymmetry
coordinate 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 and
the difference is up to 500–1000%.
The orientation degree of freedom impact on the fission rate and time
almost fully canceled the effect produced by inclusion of nuclear neck and
mass-asymmetry coordinates in the 1D Langevin calculations. The
difference of 5-25% between 4D and 1D calculations was found as the
result of this research.
To learn more about the role of the dissipation effects the calculations have also been performed for
one-body viscosity with the reduction coefficient from the ”wall” formula ks = 1.0 and two-body
viscosity. It was shown that the ratios of the stationary fission rates obtained in the calculations with
the different dimensionalities: remain almost the same for different dissipation mechanisms. Thus we
conclude that the fission rate is mostly determined by the structure of the potential energy surface of
Anischenko, Gegechkori and Adeev (Omsk)
the system. Multi-dimensional stochastic calculations September 2010 26 / 23

27. Thank you!
Thank you!
Anischenko, Gegechkori and Adeev (Omsk) Multi-dimensional stochastic calculations September 2010 27 / 23