Fission rate and_time_of_higly_excited_nuclei

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

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.


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.

The decay of the compound nuclei


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).


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).

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)


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)

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

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).

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.


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).


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)

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.


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.


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.


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

with Elab = 90MeV . Three collective coordinates c , h, α are taken into account.


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

with Elab = 90MeV . Coordinates c , h, α and K-state are taken into account.


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

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 )


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)

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


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.


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
16 O + 208 Pb −→ 224 Th. Three collective coordinates c , h, α are taken into account.


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
16 O + 208 Pb −→ 224 Th. Coordinates c , h, α and K-state are taken into account.


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)

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

Thank you!

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Fission rate and_time_of_higly_excited_nuclei

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