Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave prop...
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nasplib_isofts_kiev_ua-123456789-1114752025-02-09T15:47:35Z Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons Аналiз нейтронно-температурних коливань у розмножувальних системах з запiзнiлими нейтронами Анализ нейтронно-температурних колебаний в размножающих системах с запаздывающими нейтронами Vodyanitskii, A.A. Rudakov, V.A. Ядерно-физические методы и обработка данных External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave propagation along the coolant flow with amplitude rising in the range of high flow velocity and with amplitude decreasing in the range of low velocity and high frequencies, as well as, the oscillation propagation in the opposite direction. The temperature oscillations undergo a weak decrease of amplitudes. The strongly coupled neutron-temperature oscillations are excited, as the microscopic fission cross-sections are dependent on the "effective" temperature of thermal neutrons.The influence of delayed neutrons is studied. Comparison between the analytical treatment data and the analysis of experimental measurements confirms these conclusions. Зовнішні джерела з заданою частотою в нейтронній розмножувальній системі збуджують нейтронні і температурні коливання. Проведено аналітичний розгляд і аналіз коливань в умовах як слабкjого, так і сильного зв'язків між гілками. Нейтронна гілка коливань описує їх розповсюдження уздовж течії теплоносія з наростанням амплітуди в області великих швидкостей його руху, а також, з затуханням амплітуди - в протилежному напрямку. Конвективна гілка температурних коливань зазнає слабкого затухання амплітуди. При залежності мікроскопічних перетинів поділу ядра від температури теплових нейтронів збуджуються сильно зв'язані нейтронно-температурні коливання. Враховано вплив запізнілих нейтронів. Аналітичний розгляд і аналіз експериментальних вимірювань підтверджують ці висновки. Внешние источники с заданной частотой в нейтронной размножающей системе возбуждают нейтронные и температурные колебания. Проведены аналитическое рассмотрение и анализ колебаний в условиях как слабой, так и сильной связей между ветвями. Нейтронная ветвь колебаний описывает их распространение вдоль течения теплоносителя с нарастанием их амплитуды в области больших скоростей его движения, а также с затуханием амплитуды - в противоположном направлении. Конвективная ветвь колебаний испытывает слабое затухание амплитуды. При зависимости микроскопических сечений деления ядер от температуры тепловых нейтронов возбуждаются сильно связанные нейтронно-температурные колебания. Учтено влияние запаздывающих нейтронов. Сравнение аналитического рассмотрения с анализом экспериментальных измерений подтверждает эти выводы. 2011 Article Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons / A.A. Vodyanitskii, V.A. Rudakov// Вопросы атомной науки и техники. — 2011. — № 5. — С. 30-38. — Бібліогр.: 22 назв. — англ. 1562-6016 PACS: 24.60.-k https://nasplib.isofts.kiev.ua/handle/123456789/111475 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерно-физические методы и обработка данных Ядерно-физические методы и обработка данных |
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Ядерно-физические методы и обработка данных Ядерно-физические методы и обработка данных Vodyanitskii, A.A. Rudakov, V.A. Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons Вопросы атомной науки и техники |
| description |
External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave propagation along the coolant flow with amplitude rising in the range of high flow velocity and with amplitude decreasing in the range of low velocity and high frequencies, as well as, the oscillation propagation in the opposite direction. The temperature oscillations undergo a weak decrease of amplitudes. The strongly coupled neutron-temperature oscillations are excited, as the microscopic fission cross-sections are dependent on the "effective" temperature of thermal neutrons.The influence of delayed neutrons is studied. Comparison between the analytical treatment data and the analysis of experimental measurements confirms these conclusions. |
| format |
Article |
| author |
Vodyanitskii, A.A. Rudakov, V.A. |
| author_facet |
Vodyanitskii, A.A. Rudakov, V.A. |
| author_sort |
Vodyanitskii, A.A. |
| title |
Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons |
| title_short |
Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons |
| title_full |
Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons |
| title_fullStr |
Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons |
| title_full_unstemmed |
Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons |
| title_sort |
analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2011 |
| topic_facet |
Ядерно-физические методы и обработка данных |
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https://nasplib.isofts.kiev.ua/handle/123456789/111475 |
| citation_txt |
Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons / A.A. Vodyanitskii, V.A. Rudakov// Вопросы атомной науки и техники. — 2011. — № 5. — С. 30-38. — Бібліогр.: 22 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-11-27T14:02:05Z |
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| fulltext |
NUCLEAR-PHYSICAL METHODS AND PROCESSING OF DATA
ANALYSIS OF NEUTRON-TEMPERATURE OSCILATIONS
IN NEUTRON MULTIPLYING SYSTEMS WITH DELAYED
NEUTRONS
A.A. Vodyanitskĭı∗, V.A. Rudakov
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received July 21, 2011)
External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and
analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between
the oscillation branches.The neutron oscillation branch describes the wave propagation along the coolant flow with
amplitude rising in the range of high flow velocity and with amplitude decreasing in the range of low velocity and
high frequencies, as well as, the oscillation propagation in the opposite direction. The temperature oscillations
undergo a weak decrease of amplitudes. The strongly coupled neutron-temperature oscillations are excited, as the
microscopic fission cross-sections are dependent on the ”effective” temperature of thermal neutrons.The influence of
delayed neutrons is studied. Comparison between the analytical treatment data and the analysis of experimental
measurements confirms these conclusions.
PACS: 24.60.-k
1. INTRODUCTION
The in-reactor wave processes can possess features
characteristic for phenomena of spatial damping and
amplification. Therefore, to develop in-reactor con-
trol systems it is necessary to investigate physical
processes, taking place in the neutron-multiplying
system, and their wave properties characteristic for
fission reactors [1] - [6].
The goal of the paper is to analyze the propa-
gation and spatial damping (amplification) of neu-
tron field and cool-ant oscillations excited by exter-
nal sources. Oscillations in the coolant-containing
neutron-multiplying systems (cooled reactors) may
be described by the set of nonequilibrium statisti-
cal physics equations. The paper presents the analy-
sis of wave processes of neutron-temperature oscil-
lation transport in the cool-ant-containing neutron-
multiplying system.
The coolant moving through the core makes a con-
vective heat and neutron transport and causes some
non-equilibrium that can lead to the oscillation am-
plification in the dissipative media. This effect is not
evident for our system because the neutron diffusion
and capture by nuclei act as stabilizing factors. Be-
sides, the choice of a steady state being stable in time
does not guarantee the spatial damping of oscillations
unlike the linear and nonlinear theory of waves in
weakly nonequilibrium media.
The previous papers have considered a problem of
excitation of neutron field oscillations by the acoustic
wave [7]. In addition, there investigated was a prob-
lem of neutron field modulation by the external lo-
calized source with a given frequency through the
acoustic, neutron and temperature channels [8]. The
paper [20] briefly considers the conditions for propa-
gation of neutron-temperature oscillations in the mul-
tiplying systems without taking into account the de-
layed neutrons and temperature dependence of dif-
fusion coefficients. In the present paper, within the
framework of the model applied, we comprehensively
studied and analyzed the conditions for propagation,
damping and amplification of oscillations. The ki-
netics of centers, emitting the delayed neutrons and
the dependence of linearized diffusion terms on the
temperature perturbations, are taken into account.
To solve the engineering tasks one uses an incon-
sistent approximation in definition and solving the
problem. The influence of heat release in the fission
reactions on the neutron diffusion is related with the
neutron density via some transfer function introduced
in the design formula as an unknown and should be
determined from the experimental analysis of quan-
tities being measured. To the equation of coolant
heat balance one adds the equation of a fuel heat
balance with an unknown heat-transfer coefficient.
Thus, there is no self-consistent description in the en-
gineering tasks. Probably, such an approximation is
quite acceptable in the engineering interpretation of
the models of noise diagnostics including some data
from the experiments of [3] - [5] (more complete list
of references see in [9]). Note that the first works on
the neutron physics of multiplying systems were per-
∗Corresponding author E-mail address: vodyanitskii@kipt.kharkov.ua
30 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5.
Series: Nuclear Physics Investigations (56), p.30-38.
formed in the understandable language of theoretical
physics [10]-[15].
2. SELF-CONSISTENT SYSTEM OF
EQUATIONS
The influence of hydrodynamic parameter variations
on the neutron density oscillations in the multiplying
system with coolant and delayed neutrons requires
the description by the methods of non-equilibrium
statistical physics. To analyze this influence let us
consider the set of equations for diffusion of thermal
neutrons with their multiplication and capture and
the set of hydrodynamic equations for the coolant
being a moderator of fast neutrons [15].
∂N
∂t
+∇(~vN)=∇(D∇N)+Da4N+νc(Kβ−1)N+S, (1)
ρcp(∂T/∂t + ~v · ∇T ) = ∇(κ∇T ) +
+Q(~r, t; N) + qT , (2)
ρ(∂~v/∂t + (~v · ∇)~v) = −∇P + η4~v, (3)
∂ρ/∂t + div(ρ~v), P = P (ρ, T ). (4)
In diffusion equation (1) the term ∇(~vN) =
~v∇N + N∇~v, obtained in [16], represents the con-
servation of neutron density during the neutron con-
vection with a velocity of the coolant taking into ac-
count its compressibility. Then, Kβ = K(1 − β),
where K = νϕθ is the neutron multiplication fac-
tor including three factors from which ν is the num-
ber of generated neutrons in the one fission act,
ψ is the probability to avoid the neutron capture
in the process of its slowing-down. The third fac-
tor, the coefficient of thermal neutron utilization
θ = νf/(νf + νa) , is equal to the relation of the
inverse time (”frequency”) of neutron capture by fis-
sion nuclei νf = σfNF v to the sum of frequencies
of neutron absorption by fission nuclei and in other
processes νa = Σiσ
i
aN i
av. Here i is the index of nu-
clear kind; σf is the nuclear fission reaction cross-
section; Nf is the nuclear concentration and v is the
average velocity of thermal neutrons. The effective
coefficient of neutron multiplication, Kβ , includes a
factor where β is the sum of probabilities for forma-
tion of all the sources of delayed neutron emission
source. The coefficient of diffusion D = (1/3)vl and
the diffusion-type transport coefficient Da = Kτνc,
related with the age of slowing-down neutrons, in the
linear approximation form an effective coefficient of
diffusion equal to Def = D + Kτνc. As is shown
by the authors of [14], the age parameter is equal
to τ = l2ln(E0/E), where l2 = 1/(3ξsΣsΣtr) or
to 1/l = ΣαNασα
s in terms of [12] for the scatter-
ing by different-type nuclei with concentrations Nα
and scattering cross-sections σα
s . In these formulae
ξs ≈ 1, and Σtr is the macroscopic transport cross-
section Σtr ≈ Σs, approximately equal to the scatter-
ing cross-section Σs = Nsσs, where σs is the micro-
scopic scattering cross-section, Ns is the density of
scattering nuclei. In diffusion equation (1) S is the
density of thermal neutron sources, either external
or delayer neutrons, for which S = ΣiνiNi, where Ni
is the concentration of i-type fission products radi-
ating delayed neutrons and 1/τi = νi is the inverse
”keeping” or delay time of neutron radiation. In the
last case the set of equations under consideration is
complemented by the following equation
∂Ni/∂t = βi
fN −Ni/τi, (5)
where βi
f = νβiνf and νβi is ”the multiplication of
probability of the i-type radiation source formation
in the process of fission on the probability of neu-
tron emission by these fission product” [13] and νf is
denoted above.
The hydrodynamic equations for the coolant are
with standard notations. Note that Q = Q(~r, t) is
the thermal-power density per time unit of fission
nuclear reactions. For the heat release we introduce
EN = Q/(ρcp), where N denotes the neutron den-
sity. The last relation in the line (4) is the coolant
state equation.
3. MODEL CHOICE
Propagation of neutron-temperature oscillations
occurs when the multiplying medium is in the station-
ary or quasi-stationary state. The simplest model of
the stationary state is a one-dimensional model with
a coolant having the constant density ρ0 and veloc-
ity U , neutron density N0, temperature T0 and con-
centration of centers radiating delayed neutrons Ni0.
These quantities satisfy the equations (here we ne-
glect the inhomogeneity in the velocity and pressure
supposing their changes being insignificant)
U
dN0
dz
=
d
dz
(
D
dN0
dz
)
+Da
d2N0
dz2
+νc(Kβ−1)N0 +S,
U
dT0
dz
= EN0 + χ
d2T0
dz2
, βi
fN0 =
Ni0
τi
.
The choice of the model is accompanied by the inves-
tigation of the stationary state stability. (Mathemat-
ical aspects of the theory on the existence, uniqueness
and stability of stationary states in nuclear reactors,
as well as, in a slab model are under consideration in
the monograph [17]).
Let us formulate the set of equations for perturba-
tions of stationary values of system parameters. As-
sume that N = N0 +n(z, t), Ni = N0i +ni(z, t), T =
T0 + T1(z, t), ρ = ρ0 + ρ1(z, t) and ~v = (0, 0, U + U1).
We will take the linear terms in each of equations (1)
- (5) considering the dependence of coefficients from
the system parameters. The linear dependence of the
capture frequency on the medium density is natural if
the relative concentrations of nuclei are not varying,
i.e. when ρj = Njmj/ρ = const, where Nj and mj
, are the density and mass of j-type nuclei. Under
these conditions all the macroscopic cross-sections of
capture and scattering and inverse times of neutron
capture are directly proportional to the medium den-
sity Σj = σjNj = rΣj0, 1/Tc = νc = νcor, where
r = ρ/ρ0. Here and below the index 0 denotes the
values of quantities in the stationary state.
31
The terms, describing the multiplication of neu-
trons and their capture in equation (1) gain the de-
pendence on the medium density and on the ”effec-
tive” neutron temperature Tn in the form νc(Kβ −
1) = rϑδ
nνc0(Kβ0), where ϑn = Tn/T0.
The contribution of temperature perturbations to
the multiplication and capture of neutrons in the
same equation (1) gives the term
T1d
(
νc(Kβ−1)
) /
dT = T1
(
νc(Kβ−1)+νa
)
δ/T0, (6)
where νc = νa + νf is the frequency of all the cap-
ture types. The frequency of thermal neutron absorp-
tion by the nuclei without fission νa, in view of the
dependence of absorption cross-sections on the aver-
age thermal velocity in the form of σα ∼ 1/v, does
not depend on the energy. In formula (6) assumed
was a dependence of the frequency of capture by fis-
sion nuclei on the ”effective” neutron temperature
in the form of νf ∼ T δ. There used was equation
(38) from Appendix 1 and the value of the deriva-
tive (∂νf/∂Tf ) ·∂Tn/∂T = νf (Tf )δ/T , containing, as
a multiplier, the inverse medium temperature 1/T .
(Details on the physical situation concerning the ”ef-
fective” temperature in nuclear reactors see in Ap-
pendix 1).
Here we give the results obtained in Appendix
2 on the temperature perturbation contribution into
the diffusion transport of neutrons. The depen-
dence of diffusion coefficients on density variations
ρ1 = ρ− ρ0 by the equation of motion (9) we express
in terms of temperature perturbations neglecting a
medium local acceleration. Then the linearized terms
of diffusion, being dependent on the temperature per-
turbations in the one-dimensional approximation, are
added giving the sum
L̂T
{
∂
∂z
(
D
∂N
∂z
)
+ Da
∂2N
∂z2
}
=
= D′
G,T
(
N ′′
0 T1 + N ′
0
∂T1
∂z
)
+ D′
Ga,T N ′′
0 T1 + Rext,
(7)
where:
Rext = {D0∂ (P extN ′
0) /∂z + N ′′
0 Da0P
ext} /(ρs2),
N ′
0 = dN0/dz, N ′′
0 = d2N0/dz2, D′
G,T = D0G/T0,
D′
Ga,T = Da0Ga/T0. Parameters G and Ga are given
in Appendix 2 after formulae (39) and (40) and con-
tain not only the temperature perturbation contribu-
tion but also the medium density perturbation con-
tribution. In the expression given the dependence of
stationary values of T0, D0 and G on the ”nonuni-
form” coordinate z is neglected. The neglect condi-
tion is fulfilled for perturbations having the scales of
characteristic lengths significantly less than the char-
acteristic dimensions of uniformities in the stationary
state.
The linearized equations of coolant motion and
continuity
ρ0∂U1/∂t=−s2∂ρ1/∂z−(
P ′T
)
ρ∂T1/∂z−∂P ext/∂z,
(8)
∂ρ1/∂t + ∂(ρ1U + ρ0U1)/∂z = 0, (9)
require explications. As is shown in [9], the pertur-
bations in coolant local acceleration and coolant den-
sity inertia cause the excitation of acoustic waves.
However, the sound branch of oscillations is sepa-
rated from the neutron-temperature oscillations and
the above-mentioned time derivatives are neglected.
Equation (8) includes the force action of the external
pressure gradient on the coolant.
The perturbations in the medium density ρ1 and
in the coolant velocity will be removed from the set
of perturbations in the linearized system of equa-
tions by equations (8) and (9) by neglecting the local
medium acceleration and the term in the continuity
equation ∂ρ/∂t. At the same time, the perturbations
of neutron density, temperature and concentration of
sources radiating delayed neutrons satisfy the set of
equations
−iω̂Dn(z, t)−ĤT1−
∑
i
Niνi = R̂P ext
/
E ≡ b1, (10)
En(z, t)−i
(
ω̂χ+iΓ
)
T1 = EN0P
ext
/
ρs2−qT ≡ b2, (11)
∂ni/∂t = βi
fn− νini. (12)
Here Rext is given in the line after formula (7) and the
operator quantities and some parameters are equal to
the following
−iω̂D = ∂/∂t + U∂/∂z −Def∂
2/∂z2 − νc(Kβ − 1),
−iω̂χ = ∂/∂t + U∂/∂z + E′
T N0,
(13)
ĤE =−ΓΩ̂ +ED′
G,T
(
N
′′
0 +N
′
0∂/∂z
)
+EN0(λ+νa)δ/T0,
(14)
Γ = γ
(
P ′T
)
ρ
EN0
ρs2
, λ = νc(Kβ − 1), Ω̂ = λ + ikU.
(15)
The set of equations (10)-(12) describes both the mul-
tiplying medium stationary state stability and the
propagation of coupled neutron-temperature oscilla-
tions excited by the external sources of force P ext and
thermal action qT .
For the conditions of oscillation propagation an
asymptotic solution is found in the form of quasi-
classical exponents or exponents with expanded in-
dexes exp
(
i
∫ z0 k(z′)dz′
) ≈ exp(ik(z1)z0), where
z1 = εz0 is a ”slow” coordinate for ε ¿ 1, and z0 = z
is the ”fast” one [18]. The coupling of neutron and
temperature perturbations is performed with the help
of two factors. The first takes place in the thermal
balance equation in the form of the heat releasing
from the nuclear fission reactions proportional to the
thermal neutron density. The second factor intro-
duces the temperature perturbation into the neutron
diffusion equation and is proportional to the sum of
four terms:
H =−Γ
Ω
E
+D′
G,T
(
N
′′
0 + ikN
′
0
)
+D′
Ga,T N ′′
0 +N0(λ+νa)
δ
T0
,
(16)
where the notations correspond to these used in for-
mulae (13)-(15). The first term in (16) on the right
part expresses the medium density perturbation con-
tribution into the macroscopic capture cross-section
32
Σc =
∑
i N i
cσ
i
c ∼ ρ. As is noted above, the medium
density-and-temperature perturbations are interre-
lated by equation (8) with a left part being equal
to zero. The second and the third terms in (16) de-
scribe the contribution of dependence of diffusion co-
efficients on the medium density and medium temper-
ature given in Appendix 2 (see the text with formulae
(39) and (40)). The last term in (16) is obtained by
taking into account the microscopic nuclear fission
cross-sections as a function of the thermal neutron
”effective” temperature.
4. CHARACTERISTIC EQUATION FOR
NEUTRON MULTIPLYING MEDIA WITH
DELAYED NEUTRONS
The propagation of waves with a given frequency
oscillations ω from the external sources is determined
by the values of complex wave numbers k from the
characteristic (or dispersion) equation obtained as a
solvability condition for a homogeneous system of
equations corresponding to equations (10)-(12) in the
k-representation. The characteristic equation has the
form
(
ωD−i
∑
j
νjβ
j
f
/
(νj−iω)
)(
ωχ+iΓ
)
= a1+a2+A, (17)
where the quantities
ωD = ω − kU + ik2Def − i(Kβ − 1)νc, (18)
ωχ = ω − kU + ik2χ− iE′
T N0,
are the same as in formulae (13)-(15) with the sub-
stitution ∂/∂t⇒−iω, ∂/∂z⇒ ik. In the right part
of (17), the sum is presented by the coupling factors
a1 =Γ(λ + ikU), a2 =−D′
G,T (N
′′
0 + ikN ′
0), (19)
A=−EN0(λ+νa)δ
/
T0, δ < 0. (20)
In the left part of equation (17) in the parenthesis the
term with an index of summation is a contribution
to the propagation of neutron oscillations of delayed
neutrons from all the sources of their radiation.
Equation (17) without delayed neutron contribu-
tion is obtained in [8]. We will start the analysis of
the solutions from the study of the effect equation
of delayed neutrons. For low oscillation frequencies
ω, which are much less than the inverse time of de-
layed neutron retention time ω ¿ 1/τi = νi, the sum∑
i νiβ
i
f
/
(νi − iω) ≈ ∑
i βi
f = Kνcβ is well simpli-
fied (here we used the notations K = νθ, θ = νf
/
νc,
β =
∑
i βi and βi
f = νβiνf ).
For high frequencies ω À 1/τi = νi, as compared
to the inverse times 1/τi, both the reactive contribu-
tion and the dissipative contribution are insignificant
due to the smallness of νi/ω ¿ 1: Σiνiβ
i
f
/
(νi−ω) ≈
Kνc
(
iΣiβiνi
/
ω+Σiβi(νi/ω)2
)
.
If the frequency ω is close to one of the frequen-
cies νi then the influence of delayed neutrons may be
significant. The frequency band f = ω/2π in this
case ranges to the tenth of hertz for the times of fis-
sion reaction half-life equal to 2 s and 0.45 s [11]. The
contribution of a j-type emitter into the quantity ωD
in equation (17) is estimated as (1 + i) βjKνc/(2ω)
and can be significant for the case of a great proba-
bility βj of formation in the process of j-type emitter
fission. The neutron multiplication factor includes
only prompt neutrons and their part from the to-
tal number of thermal neutrons is decreased by the
sum of probabilities for all-type emitter formation
β =
∑
j βj (a part of thermal neutrons arrives from
the delayed ones).
As is seen from the above analysis, the contri-
bution of delayed neutrons includes the reactive and
dissipative parts. In the case of general statement, it
can be combined in the dispersion equation with the
terms of the same type. Introduce the notations
ωβ = ω + ωKνc
∑
i
νiβi
/(
ω2 + ν2
i
)
, (21)
Kβω = Kβ + Kβω, βω =
∑
i
ν2
i βi
/(
ω2 + ν2
i
)
, (22)
with which the investigation of limiting cases is eas-
ier. The dispersion function of proper oscillations
together with delayed neutrons, taking into account
the introduced notations, is written in the following
form
DN (ω, k) = ωD − i
∑
i
νiβ
i
f
/
(νi − iω) =
= ωβ − kU + ik2Def − iνc(Kβω − 1). (23)
5. PROPAGATION OF OSCILLATIONS IN
NEUTRON MULTIPLYING MEDIUM
WITH DELAYED NEUTRONS
Let us present the results of dispersion equation
solution and their analysis. Of importance are not
only the conditions for perturbation theory applica-
tion in the range of values of the multiplying medium
but also its physical characteristics, e.g. dependence
on the temperature of medium of the nuclear fission
microscopic cross-sections and on the frequencies of
capture by these thermal neutron nuclei. It is possi-
ble to estimate the influence of these processes in the
noise diagnostics of the multiplying system by study-
ing the conditions of neutron-temperature oscillation
propagation.
In the disperse equation with taking into account
the introduced notations
DN (ω, k)
(
ωχ + iΓ
)
= a1 + a2 + A, (24)
the sum of partial coupling coefficients of neutron-
and-temperature perturbations is given in formula
(17) with notations on the line of (18), DN is in for-
mulae (23). The solution of this equation we search
by the perturbation theory in the approximation of
weak coupling between neutron and temperature os-
cillations. The wave number of neutron oscillations is
found in the form k = kN +δkN , where |kN | À |δkN |.
33
In the zeroth approximation k = k1,2 are the solutions
of quadratic equation
ωβ − kU + ik2Def − (Kβ − 1)νc = 0. (25)
The real and imaginary parts of the complex roots of
these equations are obtained in [8]
k1,2 =
(
1
/√
Def
)[−iU
/(
2
√
Def
)±
±
√
−U2/(4Def) + iωβ + νc(Kβ − 1)
]
, (26)
are written as
Rek1,2 =± 1√
2
[√
ωβ+
√
ω2
β +B2 +
√√
ω2
β +B2−ωβ
]
,
(27)
Imk1,2 =
1√
Def
{
−ib±
±1
2
[√
ωβ +
√
ω2
β + B2 −
√√
ω2
β + B2 − ωβ
]}
.
(28)
Here the notations b =U
/(
2
√
Def
)
and B = νc
(
Kβ −
1
)− b2 are introduced.
When an above-critical state can be realized and
when B > 0 (B À ωβ or B ¿ ωβ) it is easy to
find approximated expressions using the perturbation
theory. In particular, in the case of low values of the
squared frequency ω2
β ¿ B2 the real and imaginary
parts of wave numbers take the values
Rek1,2 = ±
√
B/Def
(
1 + ω2
/
(8B2)
)
, (29)
Imk1,2 =
(
1
/ √
Def
)(−b + ω
/
(2
√
B)
)
,
where the notations of formulae (27) and (28) are
taken. The frequency dependence of the real part of
the wave number about its low values looks like a
branch of a parabola with vertical axis and the imag-
inary part of the wave number is a linear function.
As the frequency increases with its high values
ω2
β À B2, the approximated values of the real and
imaginary parts of wave numbers are equal
Rek1,2 = ±
√
ωβ/Def
(
1 + B
/
(2ωβ
)
,
Imk1,2 =
(
1
/ √
Def
)(−b±√
2ωβ
(
1−B
/
(2ωβ
))
.
(30)
Besides, the fundamental approximation of mod-
ules Rek1,2 expresses their growth depending on the
frequency by the parabolic law with a horizontal
parabola axis. As the frequency Imk1 increases and
becomes positive, and Imk2 decreases and remains
negative.
The expressions for Rek1,2 and Imk1,2 determine
the complex wave numbers of proper neutron oscilla-
tions including the diffusion kinetics of thermal neu-
trons with capture and multiplication by fission nu-
clei, as well as, the processes of delayed neutron radi-
ation. The waves are propagating in opposite direc-
tions. The wave, propagating in the direction oppo-
site to the direction of the coolant flow rate, under-
goes the spatial damping at all the values of parame-
ters being in formula (28). The neutron wave, moving
along the flow having a low rate and a high oscillation
frequency, undergoes the amplitude decreasing too.
When the flow rate increases or the frequency de-
creases , this wave changes the propagation character
from the weakening to its amplification. Indeed, the
condition of amplification Imk1 < 0 for the wave with
Rek1 > 0 leads to the inequality ω2
β < 4b2
(
B + b2
)
or
in the reference variables
ω2
β < νc
(
Kβ − 1
)(
U2/Def
)
. (31)
If the frequency values exceed the inverse decay time
of all the fission products being the sources of delayed
neutron radiation ω À 1/τi, condition (31) simplifies
and takes the form
ω2 < νc
(
K(1− β)− 1
)(
U2/Def
)
, β =
∑
i
βi,
where is the sum of probabilities for formation of
these sources.
To find additives to the wave number of neutron
oscillations (being small because of a weak coupling
between the neutron field oscillations and coolant
hydrodynamic oscillations), in formula (20) used is
the expansion DN (ω, k) = DN (ω, kN ) + δDN , where
DN (ω, kN ) = 0 and δDN = (−U + 2ikNDef)δkN ,
ωχ(kN ) = ω − kNU + iξ (last term is small, ξ = Γ+
k2
N − E′
T N0 > 0 and |E′
T N0/U | ¿ |kN |, where
E′
T N0 = ET N0δ/T ). Substituting this expansion
into equation (24) we find the sought correction to
the wave number
δkN =
(
a(kN )
)/[
ωχ(kN )
(
U−2ikNDef
)]
, (32)
containing the dependence on the delayed neutrons
in the quantity kN , one of the complex roots in
formula (26). In formula (32) the terms a(kN ) =
a1+a2(kN )+A are determined in lines (19-20). Prop-
agation of temperature oscillations in the basic ap-
proximation occurs with a wave number of the con-
vective transport kT = ω/U . By analogy with the
above case, the correction is found from the equation
of neutron oscillations written in the form k−ω
/
U =
a
/
[UDN (ω, k)]+ iξ
/
U . Assuming that both terms in
the right part are small and k = kT +δT , |kT | À |δkT |
by the perturbation theory of we find
δkT = iξ
/
U − a(kT )
/ [
UDN (ω, kT )
]
, (33)
where DN (ω, kT ) = ωβ −ω + ik2
T Def − iνc(Kβω − 1).
The first term in (33) leads to the oscillation damping
unlike the second term, which under conditions of su-
percriticality νc(Kβ−1) > kT Def gives the correction
decreasing the oscillation damping.
Note here the value of the factor of coupling be-
tween the neutron and the temperature oscillations.
The dependence of the frequency of thermal neutron
capture by the fission nuclei νf ∼ T δ
n, even with low
values of the index |δ|, leads to the strong coupling
34
between the temperature perturbations and the neu-
tron density. In dispersion equation (24) the term A
performs a strong coupling, and is proportional not
to the supercriticality coefficient νc(Kβ−1), but sim-
ply νaK, to νc(Kβ − 1)δ/T that exceeds by several
orders of magnitude.
By virtue of a mentioned property, characteristic
for nuclear-physical kinetics of thermal neutron cap-
ture by the fission nuclei, it is interesting to carry
out the analysis of two limiting situations with the
strong coupling between the proper neutron wave os-
cillations and the coolant oscillations. For this pur-
pose in dispersion equation (24) we transpose into the
left part the terms independent on the wave number
k,
(ω + iξ)
(
ωβ − iνc(Kβ − 1)
)− α = kϕ(k, ω), (34)
where a = α + iγ; α = a1 + Rea2 + A is the real part
of the sum of coupling factors and the polynomial is
equal to the equation
ϕ(k, ω) = U
[
ωβ + iγ − iνc(Kβ − 1)− ω − iξ
]
+ (35)
+ikDef(ω + iξ)− kU2 + ik2UDef .
After the transformation, the dispersion equation
takes the form
ωωβ − α + 2ξλ + iωλ = kϕ(k, ω), (36)
where the parameter λ = νc(Kβ − 1) is proportional
to the supercriticality of the neutron multiplying sys-
tem. Of a particular interest are the strongly coupled
neutron-temperature oscillations with high values of
the frequency and coupling factors. Therefore, when
searching the solution of a dispersion equation one
supposes that the oscillation frequency contribution
is much higher than the contribution into the dis-
persion equation of diffusion, convection and balance
between the multiplication and capture of neutrons.
It means that the wave numbers of oscillations (at
least, one of the branches) will be low and at a given
frequency of the external oscillation source undergoes
weak spatial variations. Such a behavior is important
for the oscillation recording in the core of the multi-
plying medium.
The left part of equation (36) is small at low val-
ues of the parameters δα = ωωβ − α and ξλ. Then,
assuming that |k| is low, in the right part it is possi-
ble to restrict oneself by the term linear on k, namely:
kϕ(0), where ϕ(0) = U
[
ωβ+iγ− iνc(Kβ−1)−ω−iξ
]
.
Neglecting the low terms we have
k1 = q1 + κ1 =
(
δα + iξωβ − iωλ
) /
ϕ(0). (37)
Besides the roots determined there are another two
roots with the same assumptions about the parame-
ter smallness. These roots do not contain at all the
smallness of the above-mentioned parameters and are
close to zeros k2 and k3 of quadratic trinomial (35),
ϕ(k2,3) = 0, with minor corrections. However, here
we do not given them and restrict ourselves by the
following interpretation of conditions of calculation
validity. The equality to zero of the parameter δα =
ωωβ − α = 0 with high values of its parameters, as
well as the equality ξωβ − ωλ = 0 in root expres-
sion (37) does not mean ”automatically” the amplifi-
cation or nonpassage of oscillations with frequencies
from the vicinity of their values ω1 and ω2, q(ω1) = 0
and κ(ω2) = 0. There required is an adequate prob-
lem definition and choice of parameter values such
as, for example, in guide [19] about amplification or
nonpassage of waves in the stable and absolutely- or
convectively unstable medium. For example, for a
medium for the problem with initially boundary con-
ditions it is a choice of ”immediately including” ex-
ternal source in the form of g(x, t) ∼ δ(x)exp(−iω0t)
at t > 0 and g(x, t) = 0 at t < 0 corresponding to
the initially boundary problem with the point local-
ization and given frequency ω0. Far from the source,
the steady-state condition at t → ∞ with amplifi-
cation or nonpassage of oscillations at |x| → ∞ is
possible only in the problem ” on the convective un-
stable (or stable) system. In the case of absolute in-
stability the perturbation is increasing unrestrictedly
in all the points of space, so that it is impossible at
all to reach a steady-state condition” [19] p.332. Note
that the low wave number values violate the condition
of applicability of the quasi-classical approximation∣∣(d/dz)(1/k(z)
∣∣ =
∣∣(1/k(z)2(dk(z)/dz)
∣∣ ¿ 1, within
the framework of which the treatment accepted in
this paper is valid. Therefore, a special asymptotic
expansion [21] of the desired solution and numerical
computation are required.
Besides, it is of interest to consider the case with
high modulo wave numbers. We solve the problem
on a spatial behavior of strongly coupled neutron
and temperature oscillations. Their coupling occurs
through the density and temperature oscillations of
both the media and the neutron field. To be exact:
the neutron density variation in the term being the
released heat contribution causes temperature varia-
tions in the medium. Under conditions of neglecting
the inertia and weak convective transfer of a hydrody-
namic medium pulse to these temperature variations
”fitted” are the medium density variations together
with the density of fuel nuclei. In the neutron diffu-
sion equation among the terms of neutron self-action
we take into account only the diffusion of neutrons
and neglect their multiplication and nuclear capture
(the matter concerns the spatial behavior of excita-
tions over the stationary state of the active multiply-
ing medium not the stationary state itself). In the
right part of equation (36) the largest wave-number
cubic term remains. Neglecting the small terms we
obtain from (34) the equation
δα ≡ ωωβ − α = ik3UDef .
When δα > 0 the solution of this equation is kj =
ρ exp
(
i2πj/3− iπ/6
)
, where j is the integer and ρ =
3
√
δα/(UDef) or k0 = ρ exp
(−iπ/6
)
, k1 = iρ and k−1 =
ρ exp(−i5π/6).
35
It is possible to consider the definition of a bound-
ary problem with oscillations being stationary in time
at a given frequency. Then the position in the com-
plex plane of a single root k0 for oscillations in the
region where z > 0, on the right of their excitation
source site, expresses the amplification of these os-
cillations. Two other roots determine the damped
oscillations in the infinite medium. Similar consider-
ations are convenient if δα < 0. In the problems on
the excitation of oscillations in finite systems, for con-
struction of solutions with in-time stable behavior,
one uses all three roots of the dispersion equations.
The application of a method of variation of arbitrary
constants using the exponents makes it possible to
solve the problems with an arbitrary dependence on
the spatial variable (see also in [19] § 65 on the finite
system instability).
6. CONCLUSIONS
The conditions of excitation and propagation of
neutron-temperature oscillations in the multiplying
system with delayed neutrons are studied. It is ob-
tained and investigated the characteristic equation
that is equation of the third degree in relation to the
wave number of complex kind. The roots behavior of
equation depending on external source frequency and
other parameters of the system is studied, in particu-
lar, values of coefficient of reproduction of neutrons,
their ”effective” temperature, times of capture and of
coolant velocity.
The calculation analysis data show that in the
case of a weak coupling between the neutron field
and temperature the dependence of the wave number
real part on the frequency, near its low values, looks
like a branch of a parabola with vertical axis and the
imaginary part of the wave number is a linear func-
tion. As the frequency of external source oscillations
increases, the values of the wave number real part, as
well as, the neutron oscillation spatial damping are
increasing too (at high frequencies it is as square root
of ω ). The signal amplitudes are noticeably decreas-
ing when the frequency increases.
Numerical analysis shows [22], that for standard
parameter values of WWER-1000 type reactors, the
neutron wave frequencies can be within the range of
values from fractions and units of Hz to several tens
of Hz with taking into account the time of thermal
neutron capture by fission nuclei as a function of their
”effective” temperature.
If the frequency oscillations is close to one of the
inverse times of delayed neutron emission their influ-
ence may be significant. Found out the noticeable
increase of real values of wave numbers and coeffi-
cients of the spatial damping of neutron branch of
oscillations at diminishing of coefficient of diffusion
of neutrons. Under similar conditions, the neutron
fields can experience the transport by the convective
temperature mode undergoing a slight damping with
a weak connection of neutron and temperature per-
turbations.
Dependence of time of capture of thermal-neutron
from their ”effective” temperature by nuclei of divi-
sion plays a key role at the analysis of conditions of
propagation of neutron-temperature oscillations. As
a result of its account the strongly coupled neutron-
temperature oscillation are formed.
The mentioned dependence is a problem of nu-
clear kinetics of the neutron field evolution in time.
Namely, it is necessary to formulate the meth-
ods describing the different non-stationary physical
processes during slowing-down of fast neutrons from
the nuclear fission reactions to the thermal energies
simultaneously with their scattering and nuclear cap-
ture. The conditions for propagation of analyzed os-
cillations revealed in experiment [22] allows one to
solve the problem of the ”effective” temperature exis-
tence and dependence of the time of thermal neutron
capture by fission nuclei on this temperature.
The authors are grateful to S.V.Peletminsky and
Yu.V.Slusarenko for helpful discussions of the matter
and investigation results.
The subject of the investigation fulfilled is in-
cluded in the Government Program according to Con-
tract No X-2-204.
APPENDIX 1
”EFFECTIVE” TEMPERATURE OF
THERMAL NEUTRON
The question concerning the dependence of the fission
nuclear capture frequency νf on the ”effective” neu-
tron temperature Tn and its relation with the medium
temperature T is very complicated. One knows that
in the reactor core the neutron temperature is by
50-100 degree higher than the medium temperature.
In the region of thermal neutron energies, a steady-
state connection was established [13] from the condi-
tion of a balance between the flow of neutrons slowing
down in the scattering processes and the flow of ther-
mal neutrons captured by nuclei. The foregoing is
expressed by the equation
Tn =
(
1 + ∆ · Σa
/
Σs
)
T. (38)
Here represented are the medium temperature T , the
”effective” neutron temperature Tn and the macro-
scopic capture cross-sections
∑
a = σaNa and scat-
tering cross-sections
∑
s = σsNs, where σa and σs are
the corresponding microscopic cross-sections of nuclei
with concentrations Na and Ns. The parameter 4
depends on the moderator type (water, graphite etc.)
and equals to the tenth part of unity. Phase densi-
ties (their energy distributions ) of slowing-down neu-
trons are not Maxwellian ones. It has been found [13]
that the value of ”effective” temperature, at which
the neutron distribution is maximum, exceeds the
medium temperature .
The evolution processes of neutron slowing-down
flow during 10−5 s and their capture time is less than
10−4 s (the lifetimes of thermal neutrons). As the
periods of neutron slowing-down exceeds 2 · 10−2 s,
36
it can be assumed that, in the oscillation processes
too, the ”effective temperature of neutrons is related
with the medium temperature by the relationship of
(38)-type established for the stationary conditions.
APPENDIX 2
Linearization of diffusion terms Let us go on to
the problem of linearization in the neutron diffu-
sion equation. Classical diffusion of thermal neu-
trons and their transport, represented by the term
of a diffusion type and related with the age of
neutrons being slowing-down, are written in equa-
tion (1) in the form of two terms with the co-
efficients D and Da. Regard the explicit depen-
dence of the temperature and density in the diffu-
sion constants. By changing them for dimension-
less ones (relatively to the values in the stationary
state) ϑ = T/Ta and these introduced above ϑn and
r we obtain, D = (1/3)v/(σtrNtr) = ϑ1/2+µD0/r,
Da = νcKτ = ϑδ
nDa0/r. The values of derivatives
of diffusion constant are equal to ∂D/∂ρ = −D/ρ,
∂D/∂T = −(µ + 1/2)D/T and ∂Da/∂ρ = −Da/ρ,
∂Da/∂T = δ − 1/ ln(E0/T )Da/T . Here µ ≈ 0.1 rep-
resents the weak dependence of the scattering cross-
section on the medium temperature σs ≈ σtr ∼ ϑ−µ.
The above-mentioned formulae takes into account the
dependence of the inverse time of thermal neutron
capture by fission nuclei on the density and medium
temperature T and the dependence on the ”effective”
temperature Tn. The result of diffusion term L̂ lin-
earization takes the form
L̂∇(D∇N)=∇{
D0∇n +
[
T1G/T0 + P ext
/(
ρs2
)]
D0∇N0}.
(39)
Here introduced is the notation for the factor
G = 1/2 + µ+
(
γcT0
/
(%s2)
)
(P ′T )ρ, where the first
two terms show the dependence of the diffusion con-
stant on the temperature, and the last term is ob-
tained by changing its density perturbation for ρ1 =
−(
T1/s2
)
γc(P ′T )ρ− P ext
/
s2 via the perturbation of
temperature T1 in accordance with equation (9) with-
out an inertial term and with zero boundary condi-
tions ρ1(~r0) = T1(~r0) = P ext(~r0) = 0. Linearization
of the diffusion-type term, related with the slowing-
down neutron age, leads to the expression (a zero in-
dex denotes, as before, the stationary values of quan-
tities)
L̂Da4N)=D04n +
(
Da0Ga
T1
T0
+
D0
ρs2
P ext
)
4N0,
(40)
where the factor Ga = δ +
(
γcT0
/
ρs2
)
(P ′T )ρ−
1/ ln(E0/T ) contains the second terms obtained from
the parameter of neutron age τ = l2 ln(E0/T ). When
in the fission reaction the energy of neutron generated
is E0 ≈ 5 MeV and the temperature T ≈ 0.05 eV
this term is equal to ln(E0/T ' 1/20. The result
of diffusion term linearization in the one-dimensional
approximation is given in formulae (39), (40) and (7)
of the paper text.
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АНАЛИЗ НЕЙТРОННО-ТЕМПЕРАТУРНИХ КОЛЕБАНИЙ В РАЗМНОЖАЮЩИХ
СИСТЕМАХ С ЗАПАЗДЫВАЮЩИМИ НЕЙТРОНАМИ
А.А. Водяницкий, В.А. Рудаков
Внешние источники с заданной частотой в нейтронной размножающей системе возбуждают нейтрон-
ные и температурные колебания. Проведены аналитическое рассмотрение и анализ колебаний в усло-
виях как слабой, так и сильной связей между ветвями. Нейтронная ветвь колебаний описывает их
распространение вдоль течения теплоносителя с нарастанием их амплитуды в области больших скоро-
стей его движения, а также с затуханием амплитуды – в противоположном направлении. Конвективная
ветвь колебаний испытывает слабое затухание амплитуды. При зависимости микроскопических сече-
ний деления ядер от температуры тепловых нейтронов возбуждаются сильно связанные нейтронно-
температурные колебания. Учтено влияние запаздывающих нейтронов. Сравнение аналитического рас-
смотрения с анализом экспериментальных измерений подтверждает эти выводы.
АНАЛIЗ НЕЙТРОННО-ТЕМПЕРАТУРНИХ КОЛИВАНЬ У РОЗМНОЖУВАЛЬНИХ
СИСТЕМАХ З ЗАПIЗНIЛИМИ НЕЙТРОНАМИ
А.А. Водяницький, В.А. Рудаков
Зовнiшнi джерела з заданою частотою в нейтроннiй розмножувальнiй системi збуджують нейтроннi
i температурнi коливання. Проведено аналiтичний розгляд i аналiз коливань в умовах як слабкого,
так i сильного зв’язкiв мiж гiлками. Нейтронна гiлка коливань описує їх розповсюдження уздовж
течiї теплоносiя з наростанням амплiтуди в областi великих швидкостей його руху, а також, з зату-
ханням амплiтуди – в протилежному напрямку. Конвективна гiлка температурних коливань зазнає
слабкого затухання амплiтуди. При залежностi мiкроскопiчних перетинiв подiлу ядер вiд температури
теплових нейтронiв збуджуються сильно зв’язанi нейтронно-температурнi коливання. Враховано вплив
запiзнiлих нейтронiв. Аналiтичний розгляд i аналiз експериментальних вимiрювань пiдтверджують цi
висновки.
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