Stochastic resonance in nuclear fission
Fission decay of highly excited periodically driven compound nuclei is considered in the framework of Langevin approach. We used residual-time distribution (RTD) as a tool for studying the dynamic features in the presence of periodic perturbation. The structure of RTD essentially depends on the rela...
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irk-123456789-798942015-04-07T03:01:54Z Stochastic resonance in nuclear fission Berezovoj, V.P. Bolotin, Yu.L. Dzyubak, A.P. Yanovsky, V.V. Zhiglo, A.V. Anomalous diffusion, fractals, and chaos Fission decay of highly excited periodically driven compound nuclei is considered in the framework of Langevin approach. We used residual-time distribution (RTD) as a tool for studying the dynamic features in the presence of periodic perturbation. The structure of RTD essentially depends on the relation between Kramers decay rate and the frequency w of periodic perturbation. In particular, the intensity of the first peak in RTD has a sharp maximum at certain nuclear temperature depending on w. This maximum should be considered as fist-hand manifestation of stochastic resonance in nuclear dynamics. 2001 Article Stochastic resonance in nuclear fission / S.V. Naydenov, A.V. Tur, A.V. Yanovsky, V.V. Yanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 226-229. — Бібліогр.: 16 назв. —англ. 1562-6016 PACS:05.40.+j;25.85-w http://dspace.nbuv.gov.ua/handle/123456789/79894 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Anomalous diffusion, fractals, and chaos Anomalous diffusion, fractals, and chaos |
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Anomalous diffusion, fractals, and chaos Anomalous diffusion, fractals, and chaos Berezovoj, V.P. Bolotin, Yu.L. Dzyubak, A.P. Yanovsky, V.V. Zhiglo, A.V. Stochastic resonance in nuclear fission Вопросы атомной науки и техники |
| description |
Fission decay of highly excited periodically driven compound nuclei is considered in the framework of Langevin approach. We used residual-time distribution (RTD) as a tool for studying the dynamic features in the presence of periodic perturbation. The structure of RTD essentially depends on the relation between Kramers decay rate and the frequency w of periodic perturbation. In particular, the intensity of the first peak in RTD has a sharp maximum at certain nuclear temperature depending on w. This maximum should be considered as fist-hand manifestation of stochastic resonance in nuclear dynamics. |
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Article |
| author |
Berezovoj, V.P. Bolotin, Yu.L. Dzyubak, A.P. Yanovsky, V.V. Zhiglo, A.V. |
| author_facet |
Berezovoj, V.P. Bolotin, Yu.L. Dzyubak, A.P. Yanovsky, V.V. Zhiglo, A.V. |
| author_sort |
Berezovoj, V.P. |
| title |
Stochastic resonance in nuclear fission |
| title_short |
Stochastic resonance in nuclear fission |
| title_full |
Stochastic resonance in nuclear fission |
| title_fullStr |
Stochastic resonance in nuclear fission |
| title_full_unstemmed |
Stochastic resonance in nuclear fission |
| title_sort |
stochastic resonance in nuclear fission |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Anomalous diffusion, fractals, and chaos |
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http://dspace.nbuv.gov.ua/handle/123456789/79894 |
| citation_txt |
Stochastic resonance in nuclear fission / S.V. Naydenov, A.V. Tur, A.V. Yanovsky, V.V. Yanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 226-229. — Бібліогр.: 16 назв. —англ. |
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Вопросы атомной науки и техники |
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2025-07-06T03:50:28Z |
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2025-07-06T03:50:28Z |
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STOCHASTIC RESONANCE IN NUCLEAR FISSION
V.P. Berezovoj1, Yu.L. Bolotin1, A.P. Dzyubak1, V.V. Yanovsky2, A.V. Zhiglo1
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
2Institute of Single Crystals, National Academy of Sciences of Ukraine, Kharkov, Ukraine
Fission decay of highly excited periodically driven compound nuclei is considered in the framework of Langevin
approach. We used residual-time distribution (RTD) as a tool for studying the dynamic features in the presence of
periodic perturbation. The structure of RTD essentially depends on the relation between Kramers decay rate and the
frequency ω of periodic perturbation. In particular, the intensity of the first peak in RTD has a sharp maximum at
certain nuclear temperature depending on ω. This maximum should be considered as fist-hand manifestation of
stochastic resonance in nuclear dynamics.
PACS:05.40.+j;25.85-w
INTRODUCTION
The process of nuclear fission, connected with cru-
cial reconstruction of a nucleus occupies an important
position in nuclear dynamics and is a most perspective
source of energy (together with nuclear fusion). Scien-
tists have been studying this phenomenon for more than
half of century. The book by A.I. Akhiezer and
I.Ya. Pomerantchuk "Nekotorye problemy teorii yadra”
[1], published in 1950, contained practically first pre-
sentation of the fission theory. The complicity of the
process became move apparent as study progressed. De-
spite much time has passed since discovery, we cannot
claim that we understand all aspects of the process. Be-
cause of that complexity of the phenomenon under con-
sideration a wide range of models and methods (some-
times mutually contradicting) were used for its descrip-
tion. Each piece of progress in comprehending of nucle-
ar structure (or, more generally, of the structure of arbi-
trary nonlinear system) caused immediate response in
fission physics. For instance, alteration of universally
recognized views on existence of shell structure of high-
ly excited nuclei [2] had resulted in radical reconsidera-
tion of fission barrier geometry as well as made it possi-
ble to explain a series of critical experiments. An intro-
duction of the concept of dynamical chaos into nuclear
dynamics enabled scientists to review the penetration
through fission barrier clue role in induced fission [3])
from new positions.
Kramers [4] was the first who considered nuclear
fission as a process of overcoming the potential barrier
by the brownian particle. A slow fission degree of free-
dom (with large collective mass) is considered as brow-
nian particle, and fast nucleon degrees of freedom - as a
heat bath. Adequacy of such description is based on the
assumption that the while of equilibrium achievement in
the system of nucleons degrees of freedom is much less
than the characteristic time scale of collective motion.
As was shown (see [5]) rate of the thermal activation is
essentially varied in the presence of the weak periodic
perturbation of the fixed frequency, whish depend on
the bath temperature. The effect has received a title of a
stochastic resonance (SR) and is increasingly becoming
a concept of universal validity.
Originally SR was introduced nearly 20 years ago to
explain the periodicity of the Earth’s ice ages [6,7] and
has found its numerous applications into such diverse
fields like physics, chemistry and biology (see [5]).
The mechanism of SR can be explained in terms the
motion of a particle in a symmetric double-well poten-
tial subjected to noise and time periodic forcing. The
noise causes incoherent transitions between the two
wells with a well-known Kramers rate [4] rk. If we apply
a weak periodic forcing noise-induced hopping between
the potential wells can become synchronized with peri-
odic signal. This statistical synchronization takes place
at the condition
1 /kr π ω− ≈ ,
(1)
where ω is a frequency of periodic forcing. Two
prominent feature of SR arise from synchronization
condition (1):(i) signal-to-noise ratio does not decrease
with increasing noise amplitude (as it happens in linear
system), but attains a maximum at a certain noise
strength (optimal noise amplitude can be found from
(1) as rk is simply connected with it);(ii) the residence-
time distribution (RTD) demonstrates a series of peaks,
centered at odd multiples of the half driving period Tn =
2(n – ½)π/ω, exponentially decreasing amplitude. No-
tice that if a single escape from a local potential well is
the event of interest – RTD reveals the dynamics of con-
sidering system more transparently than the signal-to
noise ratio. These signatures of SR are not confined to
the special models, but occur in general bi- and monos-
table systems and for different types of noise.
The aim of the present work is to demonstrate the
possibility of observation of SR in nuclear dynamics. As
a specific example we consider the process of induced
nuclear fission in the presence of weak periodic pertur-
bation.
226 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 226-229.
1. DISSIPATIVE DYNAMICS IN PRES-
ENCE PERIODIC PERTURBATION
The most general way of description of dissipative
nuclear dynamics is Fokker-Planck equation [8]. How-
ever for demonstration of qualitative effects it is conve-
nient to use Langevin equation [9] that is equivalent to
Fokker-Planck equation but is more transparent. As it
has been shown the description based on Langevin
equation adequately represents nuclear dissipative phe-
nomena such as heavy-ion reactions and fission decay
[10-12] and possesses a number of advantages over
Fokker-Planck description.
Because we intend to only qualitatively demonstrate
SR in nucleus let us consider the simplest type of
Langevin equation — one-dimensional problem with in-
ertial M and friction γ parameters independent on coor-
dinates. Fission coordinate R is considered as a coordi-
nate of Brownian particle. The rest degrees of freedom
play a role of heat bath being modeled by random force
ξ(t).
The particle motion in the presence of external peri-
odic force cosA tω is described by Langevin equation
for canonically conjugate variables { , }P R ,
/ /
/ / cos ( )
/
dR dt P M
dP dt P dV dR A t t
M
β ω ξ
β γ
=
= − − + +
=
(2)
ξ ( )t is stochastic force possessing statistical properties
of white noise
( ) 0, ( ) ( ) 2 ( ),t t t D t t D Tξ ξ ξ δ γў ў= = − =
(3)
The nuclear temperature ( )T MeV * /E a= where
*E is an excitation energy and the level density param-
eter /10a A= ( A being a mass number). The deforma-
tion potential V is given as [10]
2
2
37.46( 1) ( ), 0 1.27
( )
8.0 18.37( 1.8) ( ), 1.27
R MeV R
V R
R MeV R
м − < <
= н
− − >о
(4)
these are parameters of 205 At nucleus [10].
The discrete form of the Langevin equation is given
by [11,12]
1
1
/
(1 )
( ) 2cos ( )
n n n
n n
n n
n
R R P M
P P
dV R MTA t t
dR N
τ
β τ
β τω τ η
+
+
= +
= − −
й щж ц − +к ъз ч
и шл ы
(5)
Here nt nτ= and ( )ntη is a normalized Gaussian dis-
tributed random variable which satisfies
( ) 0, ( ) ( )
nn nnt t t Nη η η δ
ў ў= = (6)
Efficiency of numerical algorithm (5) was checked
for the following cases:
0, 0,V A= = where numerical and analytical results for
2P< > and 2R< > can be compared [10];
0, 0V A=№ , where numerical and analytical values for
Kramers decay rate kr can be compared. According to [4]
2min
max
[ 1 ]exp( / ),
2kr V Tω
β β
π
ββ
ω
= + − − ∆
=
(7)
Here minω is the angular frequency of the potential (4)
in the potential minimum and maxω – at the top of barri-
er, V∆ is the height of the potential barrier. Numerical
value of the Kramers decay rates i
kr for the time bin I is
calculated by sampling the number of fission events
(Nf)i in the ith time bin width ∆t normalized to the num-
ber of events Ntotal - Σi(Nf)j which have not fissioned
( )1
( )
f ii
k
total f j
N
r
N N t
=
− ∆е . (8)
Comparison of (7) with asymptotic value of (8)
was used for determination of the time interval τ , which
provides saturation for numerical integration (5). From
the result one could see, that 20 steps per nuclear time
/ MeVh provides a sufficient saturation.
3. MANIFESTATION SR IN NUCLEAR FIS-
SION
Now let us proceed to the description of expected ef-
fect- manifestation of SR in nuclear fission.. For usually
considered case of symmetric double well in the ab-
sence of periodic forcing, RTD N(t) has the exponential
form (see [5]) ( ) exp( )kN t r t−: . In the presence of the
periodic forcing, one observes a series of peaks, cen-
tered at odd multiples of the half driving period Tω =
2π/ω .The height of these peaks decrease exponentially
with their order number.
These peaks are simply explained [15]. The best
time for the particle to escape potential well is when the
potential barrier assumes a minimum. A phase of peri-
odic perturbation φ may be chosen in such a special
way that the potential barrier ( ) cos( )V r AR tω φ− + as-
sumes its first minimum at t = 1/2 Tω. Thus 1/2 Tω is a
preferred residence time interval. Following a “good op-
portunity” to escape occurs after a full period, when po-
tential barrier again achieves its minimum. The second
peak in the RTD is therefore located at 3/2 Tω. The loca-
tion of the other peaks is evident. The peak heights de-
cay exponentially because the probabilities of the parti-
cle to jump over a potential barrier are statistically inde-
pendent. As is shown for symmetric double-well poten-
tial [14], the strength P1 of the first peak at 1/2 Tω (the
area under peak) is a measure of the synchronization be-
tween the periodic forcing and the switching between
the wells. So, if the mean residence time (MRT) of the
particle in one potential well is much larger than the pe-
riod of the driving, the particle is not likely to jump the
first time the relevant potential barrier assumes its mini-
mum. The RTD exhibits in such a case a larger number
227
of peaks where P1 is small. If the MRT is much shorter
than the period of the driving RTD has already decayed
practically to zero before the time 1/2 Tω is reached and
the weight P1 is again small. Optimal synchronization,
i.e., maximum P1 , is reached when the MRT matches
half the period of the driving, i.e., condition (1). This
resonance condition can be achieved by varying the
noise intensity D (or ω ).
We will show that the analogous correlation between
periodic forcing and escape time takes place for a decay
of excited states (fission) with a single potential mini-
mum as well. For RTD constructing (and following 1P
calculation) we use the numerical solutions of Langevin
equation (5). Let us study evolution of 1P within the
temperature interval
2 21 6MeV T MeV a b+Ј Ј .
Let us fix a frequency of periodic perturbation
0.0267 /MeVω = h ( / 2 117 /T MeVω = h ) – a reso-
nance frequency at 3T MeV= (following Eq. (1)). The
result of numerical procedure for RTD under fixed pa-
rameters of periodic perturbation [ 3, 0.0267A ω= = ]
are presented in Fig. 1. Nuclear friction β in all numer-
ical calculations is chosen to be 1 /MeV h .
Fig. 1. RTD for T=1 and T=4
In accordance with the expected behavior at
1T MeV= (for low kr ), one can distinctly see three
peaks located near / 2(~ 117),3 / 2 (~ 353)t T Tω ω= and
5 / 2 (~ 588)Tω , and at T=4MeV almost all RTD is con-
centrated near 0t = (with width less than / 2Tω ). The
corresponding variations of 1P (which represent the
measure of SR) are depicted in Fig. 2 for two frequen-
cies of periodic perturbation, corresponding to tempera-
tures 2 and 3 MeV. Maximum of intensities 1P are close
to the predicted values of temperature.
The above calculated 1P can be estimated theoreti-
cally for single-well situation using a model similar to
two-state model [5]. Let us evaluated RTD for a single-
well case. Rate equation for a number of a fissile nuclei
should be written as
cos t
k
dn nr e
dt
ε ω−= − , (9)
where /A Tε = . At low temperature Eq. (9) properly
describes simulated process. The solution of Eq. (9) is
0 0
( )ln exp( cos )
t
k
n t r t dt
n
ε ω ў ў= − −т
0
1
2
( 1) ( ) sinn k
n
n
rr t I t
n
ε ω
ω
Ґ
=
= − + −е , (10)
where 0 0 ( ) ;k kr r I rε= > ( )nI ε is modified Bessel func-
tions. RTD in this model is given by ( ) dnN t
dt
= − , so
that
0( ) exp( )k
rN rπ ε
ω π ω
= − +
(11)
Fig. 2. P1(t) at ω=0.0267 (crossed) and ω=0/007
(solid); 1 corresponds to Eg.(12) and 2 is for the nu-
merical calculation
Using Eqs. (9)-(11) we obtain instead of Eq. (1) new
condition for resonant temperature ( )REST ω , which pro-
vides maximal value for ( / ) /N π ω ω , whose depen-
dence on ω/2 and T properly represents 1( , )P Tω cal-
culated above:
1 0 1( ) ( )
1k
I I
r
λ ε επ
ω λ
− −
=
−
` (12)
Here /V Aλ ∆є . Numerical solution to Eq. (12) for
( )REST ω is presented in Fig. 3 [together with a solution
to 1 2 /kr π ω− = , which much better than (1) approxi-
mate curve (12)].
Evaluated 1( )P T depicted in Fig. 2 is to be compared
with numerical results. The scale of 1( )P T is chosen in
such away that the height in its maximum for
0.0267ω = coincides with the numerical data. Higher
REST in the latter case is connected with non-equilibrium
distribution within a long interval near 0t = (which can
be easily seen in Fig. 1).
The first maximum in RTD is shifted from /π ω , so
it may seem more reasonable to evaluate the height in
true maximum. The calculation shows that this height
dependence on ω resembles that presented in Fig. 2,
excepting the region of high T , where the curve ( )N t
does not possess any maximum. Nevertheless ( / )N π ω
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
228
is easily defined observable and studying its dependence
on T allows one to determine the important nuclear
characteristics, as for example, nuclear friction.
Fig. 3. Resonant conditions (12) and (13) for TRES(ω)
Let us now briefly consider the possible sources of
periodic perturbation. One of possibilities to consider a
fissile nucleus as a component of a double nuclear sys-
tem formed, for example, in heavy-ion collisions [15].
In this case, the deformation potential will experience
periodic perturbation similar to tide waves on the Earth
caused by the Moon rotation. In the case of asymmetric
fission, alternating electric field may be the source of
periodic perturbation.
4. OPEN PROBLEMS
In conclusion, we shall stop on a number of open
questions connected to prospect of research of the nu-
clear stochastic resonance.
In Section 2 we used model of Brownian motion,
where the heat bath influences the motion of the Brown-
ian particle (collective motion), while the bath itself is
not affected by its coupling to the collective motion. In
particular, temperature of the bath remains constant. At
the description of the fission process we assume that the
bath represent the nuclear internal degrees of freedom of
nuclei. The statistical properties of this bath (including
temperature) are influenced by the coupling to the col-
lective motion. We must then permit that the tempera-
ture of the bath rises as its energy increases during the
course of the fission process. Consequence it is that the
coefficient D of Eq. 3, which determines the strength
of the fluctuation of the Langevin force ( )tξ is not con-
stant, but is continually re-adjusted. Such self-consistent
scheme usually used at the description of the fission
process [12], should essentially improve considering
model.
There is a natural question, whether it is possible to
observe a nuclear stochastic resonance in spontaneous
fission. The equivalent statement of a question consists
in a possibility of observation of stochastic resonance on
a quantum scale. Indeed, quantum mechanics provides
an additional channel to overcome a potential barrier.
This additional channel is provided by quantum tunnel-
ing. Because quantum noise persists even at absolute
zero temperature, the transport on a quantum scale
should naturally be aided by quantum fluctuations as
well. As show estimations [16] for strongly damped sys-
tems quantum contributions can enhance the classical
effect up to two orders of magnitude.
All these questions require special consideration.
This work was partial supported by National Fund
for Fundamental Research Grant F7/336-2001.
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229
V.P. Berezovoj1, Yu.L. Bolotin1, A.P. Dzyubak1, V.V. Yanovsky2, A.V. Zhiglo1
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
2Institute of Single Crystals, National Academy of Sciences of Ukraine, Kharkov, Ukraine
PACS:05.40.+j;25.85-w
INTRODUCTION
REFERENCES
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