Journal of Modern Physics, 2010, 1, 379-384
doi:10.4236/jmp.2010.16054 Published Online December 2010 (http://www.SciRP.org/journal/jmp)
Regulating the Nuclear Reactor through Changes of the
Fraction of Delayed Neutrons: Theoretical Probabilities
Dmitry V. Filippov1, Leonid I. Urutskoev1, Valery I. Rachkov2, Olga I. Gadzaova1,
Larion A. Lebedev3
1Moscow State University of Printing Arts, Russia
2State Atomic Energy Corporation “Rosatom”, Moscow, Russia
3State Research and Development Center for Expertise of Projects and Technologies, Moscow, Russia
Received August 1, 2010; revised October 19, 2010; accepted October 17, 2010
Abstract
In recent years а significant number of both theoretical and experimental works devoted to the influence of
external electromagnetic fields and ionization on the probability of beta decays have been published. The
present work investigates the feasibility of using this physical effect as the main mechanism for controlling
the reactor. In this paper a system of equations is written and studied that allows one to describe the work of
a nuclear reactor in the case where the probability of beta decay and, therefore, the fraction of delayed neu-
trons is a function of time. It is shown that in the case of a constant fraction of delayed neutrons, the pro-
posed system of equations is identical to the known system. As can be seen from analysis of a solution of the
new system of equations for the proposed method of reactor control, acceleration by instantaneous neutrons
is impossible even theoretically.
Keywords: Fraction of Delayed Neutrons, Bound-State Beta-Decay, Kinetic Equation
1. Introduction
During recent years, our understanding has been that the
impact of atomic electron shell disturbances on the peri-
ods of nuclear decay caused by weak electromagnetic
interactions could be quite significant. For example, the
163Dy, 193Ir, 205Tl nuclei, which are absolutely stable in
neutral atoms, become -active under full atom ioniza-
tion [1], while complete ionization of 187Re increases the
possibility of -decay by a factor of 109 [2]. The prob-
abilities for β-decay increase not only under ionization
but when an atom is exposed to a superstrong magnetic
field [3]. Since the physical mechanism of the production
of delayed neutrons (DN) from nuclear emitters is di-
rectly connected to β-decay processes, the question was
reasonably raised [4] of whether it is possible to change
the DN fraction. [5,6], it was proved quite convincingly
that the DN fraction definitely increased under ionization
of atoms.
The appearance of DN during uranium fission is a ba-
sic physical effect making it possible to create a nuclear
reactor and underlying the operation control for reactors
of any type. Particularly significant in this respect is the
influence of DN on the behavior of the reactor with cir-
culating fuel [7]. Although at present the possibility of
changing the DN fraction through external impacts
causes no doubts when describing the atomic reactor
kinetics, it is nevertheless considered that the DN frac-
tion of each particular nuclear emitter does not depend
on external conditions. This discrepancy can be ex-
plained by the fact that the theoretical foundations of
reactor operation had been developed long before they
received reliable experimental data witnessing the influ-
ence of external physical impacts on the probability of
nuclear processes that involve weak interactions. At pre-
sent, when describing the reactor kinetics, consideration
is taken only of the variation of the average DN fraction
during the reactor process due to a different chemical
composition of the active zone. This work is aimed at
qualitative analysis of whether the method based on
changing the DN fraction could form (at least theoreti-
cally) the basis for regulating the said nuclear reactor.
The classical equations of reactor kinetics [7] were ba-
sically written under invariable DN fraction conditions.
Hence, analyzing them in the framework of the variable
DN fraction would not be quite correct. This work for-

D. V. FILIPPOV ET AL. 380
mulates the reactor kinetics equations as based on the
whole quantity of the DN nuclear emitters (including
those whose decays do not result in forming neutrons).
These equations are analyzed in the case of a change in
the fraction of DN. It is shown that when using external
impacts (e.g. superpowerful magnetic field) to change
the fraction of DN, then theoretically, in a way it is pos-
sible to regulate the power of the reactor.
2. Effect of Bound-State Beta-Decay on the
Fraction of Delayed Neutrons
A theory of bound-state β–-decay, in which the beta elec-
tron does not leave the atom but occupies a free orbit,
was constructed in [8-11]. The ratios of the decay con-
stants (ratios of the β–-decay probabilities) for transitions
to bound and free states (λb and λc, respectively) were
calculated in [9-11]. For β−-decays of low energy in fully
ionized heavy atoms, the ratio λb/λc may be as large as
103-104. Thus, we see that, in the presence of free elec-
tron orbits, the probability of the β−-decay of nuclei may
increase by three orders of magnitude or more. The the-
ory of bound-state β−-decay was experimentally con-
firmed in [1,2].
However, the allowance for bound-state beta-decay
may prove to be of importance not only in the case where
the nucleus involved has an anomalously low boundary
energy for beta decay, 187Re, for example, possesses this
property [2]; but also in the case where the decay process
being considered proceeds via various channels, includ-
ing those of decay to highly excited levels of the daugh-
ter nucleus. In the former case, the decay half-life
changes, while, in the latter case, the change in the decay
half-life is small, but the ratio of the intensities of decays
through different channels may undergo a substantial
redistribution. This effect will lead to a change in the
relationship between the intensities of the lines of
gamma radiation from the daughter nucleus. If, in addi-
tion, an emitter of DN appears as the initial beta-decay-
ing nucleus, the DN fraction will change.
The ratio of the probabilities of bound-state β−-decay
to free states can be calculated by a method similar to the
classic method for calculating the ratio of the К-capture
to the β+-decay probability [12]. In the following, we will
use the system of units in which e. For
allowed beta decays, the decay constants for transitions
to a bound and a free state are proportional to the same
matrix elements and differ only in phase spaces of elec-
tron–neutrino final states. From [11,12], it is well known
that, for beta decay to a free state, the phase space is
proportional to the integral Fermi function:
cm 1 
 
2
2
1
,,εε 1εεε
E
f
ZEF ZEd
, (1)
(E is the beta-transition energy) and is the sum of all
possible energies and momentum directions for the elec-
tron (neutrino). The Fermi function in (1) grows with
energy faster than in proportion to E2.
For bound-state beta-decay the neutrino spectrum is
concentrated at a single energy value, since the energy of
an electron moving along an orbit is fixed, and the phase
space is determined by a possible arbitrary direction of
the neutrino momentum. The phase space is then propor-
tional to the product of the square of the neutrino mo-
mentum,

2
2
2
11εpE
c

(ε is the energy of the electron moving in the respective
orbit) and the probability of the intersection of the free
electron orbit and the nucleus involved. In turn, the in-
tersection probability is proportional to |Ψe(R)|2, where
Ψe(R) is the density of the electron wave function in the
region occupied by the nucleus. For hydrogen-like orbit:

3
21
0~
e
Z
m 




, (2)
where α = 1/137 is the fine structure constant, Z is the
charge number of the nucleus and m is the quantum
number of state.
It follows that the appearance of a free electron orbit
enhances the decay constant for the allowed beta transi-
tion of energy E (0) by the quantity λ,


22
2
0
1
2
λ,
eRE
fZE



 . (3)
Since, with increasing energy E, the Fermi function (1)
grows faster than in proportion to E2, the ratio λ/λ0 de-
creases as the energy increases,
0
1, 0
λE




. (4)
Thus, the lower the decay energy E, the more pro-
nounced the increase in the decay constant because of
decay to a bound state. It is noteworthy that the energy
dependence of the decay constant takes the same form,
irrespective of which electron orbit is free, since it is the
factor |Ψe(R)|2 that absorbs the effect of the distinction
between the orbits. In order to derive the estimate in (3),
we only employed the fact that the neutrinos accompa-
nying bound-state beta-decay are monoenergetic.
In the particular case where a hydrogen-like orbit of
the atom being considered is free (the case of a fully
ionized atom), we find from (2) and (3) that
 

32
0
1
λ~2
λ,
ZE
fZE



. (5)

D. V. FILIPPOV ET AL.381
We note that the ratio λ/λ0 is greater for forbidden
than for allowed transitions since, in expression (5), the
numerator features the beta-decay form factor at the
maximum neutrino energy, while the denominator is
equal to the same form factor averaged over all neutrino
energies according to (1). For uniquely forbidden transi-
tions, the ratio λ/λ0 was considered in [11].
The fission of 235U leads to the formation of a large
number of fragments whose atomic weights range be-
tween A = 72 and A = 160. The mass and charge distri-
butions of fission fragments have been well understood.
The majority of the fragments are unstable neutron-rich
nuclei [13]. Among these, about 50 nuclei are sources of
DN and decay according to the scheme in the Figure 1.
The beta decay of the initial nucleus (DN emitter)
through a channel characterized by lower beta-transition
energies leads to the formation of an intermediate nu-
cleus in an excited state. At an excitation energy above
the neutron binding energy (Qn), the intermediate nu-
cleus emits a neutron. Neutron emission from the inter-
mediate nucleus is virtually prompt, the delay time being
determined by the lifetime of the initial nucleus. We de-
fine:  as the constant of full decaying DN nuclear emit-
ters and n as the constant of decaying DN nuclear emit-
ters through the β-decay channel with the appearance of
neutrons.
As can be seen from Figure 1, the β-decays of DN
nuclear emitters resulting in the appearance of neutrons
(taking place on the excited energy levels of intermediate
nuclei) have significantly lesser decay energies than
β-decays without the appearance of neutrons [13]. Hence,
relative changing of n is much greater than relative
changing of  (4) [3-6]. Let the following changing of
β-decay take place:
00
, ,
nn n
 
then
00
λλ
λλ
n
n

. (6)
Let us define the fraction of DN η in the following
way:
λ
λ
n
R , (7)
then the said changing of the β-decay probabilities re-
sults in a greater fraction of DN  = 0 +  (0 is the
unexcited fraction of DN):
0
000
λλ λ0
λλ λ
n
n

  



 . (8)
Therefore, the appearance of a free electron orbit in an
atom that emits DN leads to an increase in the DN frac-
Q

Q
n

n
Initial nucleus
Z, N
Intermediate nucleus
Z+1, N–1
Final nucleus
Z+1, N–2

Figure 1. Scheme of the decay of a nucleus emitting delayed
neutrons (Qβ is the maximum beta-decay energy, while Qn is
the neutron binding energy in the intermediate nucleus).
tion. In [5,6] the relative increase in the DN fraction of
nuclei was calculated which originated from the fission
of uranium and plutonium from the first three groups.
3. Increase in the DN Fraction in a
Superstrong Magnetic Field
In [14-16] it is shown that upon placing of an atom in an
external homogeneous constant strong magnetic field
23 39
02.3510 e
H
Hcme Gs

 , the properties of
the atom qualitatively change. It follows from [14-16]
that in such a field the density of electronic states in the
nucleus increases and changes the ionization energy of
the atom. In a superstrong magnetic field, the motion of
atomic electrons in a plane perpendicular to the magnetic
field occurs on the Landau levels. In the direction along
the magnetic field the electron moves in one-dimensional
Coulomb potential, averaged over the cross movement.
In [3] the density of the electron orbit is calculated in the
nucleus in a superstrong magnetic field. In contrast to (2)
in the nucleus the density of electronic states with quan-
tum number m of longitudinal motion is

2
0~
2
He
eH Z
m 






. (9)
Consequently, first, in superstrong magnetic field, the
density of excited electronic states (m > 1) of the nucleus
increases so much that bound-state -decay of nuclei
becomes significant not only for a fully ionized but also
for a neutral atom (formally the sum m–1 diverges).
Secondly, in the superstrong field, the density of unoc-
cupied electronic states in the field of the nucleus (9)
becomes proportional to the magnetic field H. Conse-
quently, the probability of bound-state decay also be-
comes proportional to the strength of the magnetic field
H. The probability of bound-state decay with complete
ionization of an atom is limited to the size of the nuclear
charge Z (2), (3), and in a superstrong magnetic field, the

D. V. FILIPPOV ET AL. 382
probability of bound-state decay can indefinitely increase
with a sufficiently large magnetic field (9).
From (3) and (8) subject to (9) it follows that upon
placing of an atom with a nucleus of DN emitter into a
superstrong magnetic field with strength H such that

2
λ
λI
eH Z

, (10)
(where I <<  is an increase of the total permanent
-decay due to the complete ionization of the atom), the
increase in the fraction of DN is

2
~
HI
eH
Z

 . (11)
Consequently, for a sufficiently large magnetic field
eH > (Z)2 the fraction of DN upon placing of an atom
with a nuclear emitter of DN in a superstrong magnetic
field grows more strongly than in the case of the full
ionization of the atom.
The limiting increase in the fraction of DN in the
magnetic field, the strength of which satisfies the ine-
quality inverse to (10), is


2
β
β
max
,
λ
~~
λ,
n HI
I
n
f
ZE
E
EfZE

   
 

 
, (12)
and does not depend on the magnetic field but is deter-
mined only by the ratio of the energy En of -decay to
give a neutron and the energy E of decay giving no neu-
tron. For the first three groups of nuclear emitters of DN,
which are the products of uranium fission, (/)max >
25.
4. Kinetic Equations Taking into
Consideration Possible Changes
in the Delayed Neutron Fraction
The power of the reactor is proportional to the neutron
density n. It is well known [7] that DN impacts on the
reactor dynamics can be correctly described by kinetic
equations in the framework of a homogeneous isotropic
model. For thorough analysis of the reactor behavior
under changing  (β-decay constants) we will use the
well accepted approximation of one efficient DN group.
So let us first define the following variables: n is the
density of all neutrons in the active reactor zone; Y is the
density of all DN nuclear emitters in the active zone in-
cluding those whose decay does not result in neutron
formation. This value significantly differs from the den-
sity of nuclear emitters which is typically used in the
classic kinetic equations where only nuclei decaying
along with the appearance of the neutrons are taken into
account.
Let us define χ as the prompt neutron cascade multi-
plication coefficient which is the ratio of the rate of the
appearance of prompt neutrons to the rate of the absorp-
tion of all neutrons (the ratio of the number of instanta-
neous neutrons produced during a unit time in a unit
volume to the number of all neutrons absorbed during the
same time and in the same volume); R as the ratio of the
number of produced nuclear DN emitters to the number
of produced prompt neutrons; T as the effective life time
of the generation of prompt neutrons such that nT–1 is, by
definition, the rate of the appearance of prompt neutrons
(number of prompt neutrons produced during the unit
time in the unit volume); n as the constant of decaying
DN nuclear emitters through the β-decay channel with
the appearance of neutrons, that is nY is defined as the
rate of the DN appearance (the number of DN produced
during the unit time in the unit volume). It is well known
that a small number of DN nuclear emitters produce
more than one DN. We will take into account this prop-
erty in n. Finally we define λ as the decay constant of
DN nuclear emitters through all the β-decay channels
under which the decay results in a nucleus not being DN
emitter. It should be noted that β-decays resulting in the
presence of a new nuclear emitter are usually not taken
into account here: a nucleus that is a nuclei composition
is well described by the Y density factor. Taking into
account the above notations, the equations for neutrons
and nuclear emitters of DN are as follows:
1λn dn nnY
dt TT
 
, (13)
dY n
RY
dt T

 .
The first term in the right-hand part of the first equa-
tion describes the appearance of instantaneous neutrons,
the second term describes the absorption of neutrons and
the third one characterizes the DN appearance. The first
term in the right-hand part of the second equation de-
scribes the formation and the second term describes the
disappearance of DN nuclear emitters.
Now let us suppose that some impact is imposed on
the reactor active zone that slightly affects the process of
enforced fission and absorption of neutrons but increases
the probability of the β-decay (e.g. ionization or an in-
tensive magnetic field). In such a case, n and  ratios
are changed in Equations (13) while the other remain the
same. Relative changing of  and n (6) results in a
greater fraction of DN (8).
Further we will introduce the reactivity of nuclear re-
actor :
0
1


. (14)

D. V. FILIPPOV ET AL.383
System (13) can be represented as
0λn
dn n
dt Τ

Y
, (15)
dY n
RY
dt Τ

.
Now let us consider the behavior of the reactor which,
being unexcited, operated in the stationary regime, that is
 = 0. We are primarily interested in solutions satisfying
the following initial conditions:
  
0
00
00
0
n
Yn
n
RRT


T
. (16)
We will take into account the fact that changes  and
n occur instantly (during the time   ). Let us exam-
ine the behavior of the reactor with new time-independ-
ent  = 0 + , n = n0 + n, and  = 0. Then from
(15), taking into account (8), we obtain the following
equation:
2
0
2
λ
λ0
dn dnn
dt ΤΤ dt




 
, (17)
which describes the reactor behavior when it deviates
from equilibrium. Should such be reached under certain
impact on the active zone (with excited values of  and )
then the deviation may occur in the case of the impact
cut off (disturbance). Thus we can, generally speaking,
consider  and  in (17) both positive when putting
on the impact and negative when taking it off.
Equation (17) describes an unstable singular saddle-
shaped point. It is not difficult to find solutions of the
equation by solving its characteristic equation and de-
termining the characteristic number :

0
2
0
1λ
λ14 1
2λ
T
TT




 


 

. (18)
In the approximation T << 0 we get
0
0
λ, ,
T




 (19)
and such solutions (19) are applicable to both cases of
small  <<  and greater  >  excitations.
When including ( > 0) the said impact, the density
of the n neutrons will grow with the value + (19) but
then it will stop. Since the power in the reactor is propor-
tional to the density of the neutrons n [7] it becomes pos-
sible to change its power by exerting impacts on the ac-
tive reactor zone by means of external fields. Now let us
compare such a method of regulation with the classical
one.
5. Classical Kinetic Equations
To come to the classical equations of kinetics, let us first
define the text-book [7] value of C as the DN density of
nuclear emitters which have β-decay through the chan-
nels giving a neutron:
nY
CY
R


, (20)

.
In the classical approach to the problem, n, , and
hence, β are constants, but in this case the substitution of
(20) into (15) leads to the well known equations [7]
ρβdn nC
dt Τ


 , (21)
βλ dCn C
dt Τ
.
Obviously, if β is not a constant then the second equa-
tion in (21) should be different:
βln
λ
dC nd
CС
dt Τdt


Thus, the system of Equations (15) seems to be more
general than (21) because the latter holds only for con-
stant n, , and β. Using the constant reactivity ρ, from
(21) we get
2
20
dn dnn
dt TT
dt
  


 

 . (22)
It is not difficult to solve this equation by calculating
the characteristic values  from the following character-
istic equation:

2
1βρ λρ
14 1
2βρ
T
ΤT








 


   


. (23)
When  <<  and T <<  we find
λρ , ,
T


 
 (24)
and when  ~  >> T we have
λρρ β
, Τ


 
 . (25)
Equation (22) is similar to (17) and they become coin-
ciding in the first order with respect to the small distur-
bances  << 0 if we set

.
However, in the case of large disturbances  ~ 0,
these equations are qualitatively different. It is worth

D. V. FILIPPOV ET AL. 384
noting that Equations (17) and (22) are applicable when
 > 0 and  > , respectively, while in the superstrong
magnetic field the fraction of DN can become much lar-
ger than the value (12).
If in the classical case (22) the reactivity becomes
greater then the fraction of DN,  > , then the root signs
of  in (25) become opposite and the larger of them
which is ~T–1 becomes positive. That is, the reactor starts
to accelerate prompt neutrons, and thus gets out of con-
trol. In our case (17), we always have  > 0 and the root
sign of –~T–1 in (18) can never change (19). Hence, in
the new method of regulating a reactor, it shall never
accelerate prompt neutrons (with significant κ~T–1) but
its power will always increase along with the value of κ
proportional to , i.e. inversely proportional to the life-
time of DN nuclear emitters.
6. Conclusions
Thus, if external impacts (e.g. superpstrong magnetic
field) change the DN fraction, then the power of the re-
actor can be theoretically regulated. The reactor sets up
initially subcritical, but it switches on and works under
an external impact on the active zone. Such method will
be much safer than the traditional one just because, even
in the case of significant excitations, the reactor will nei-
ther be accelerated by instantaneous neutrons nor be-
come “uncontrollable”.
This work was partially supported by the #2.1.1/2840
grant of The RF Education and Science Agency.
7. References
[1] M. Jung, F. Bosch, K. Beckert, et al. “First Observation
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[2] F. Bosch, T. Faestermann, J. Friese, et al. “Observation
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[3] D. V. Filippov, “Increase in the Probability of Forbidden
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