Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer
The technique of restoring fast neutron spectra on the basis of amplitude spectra registered by a scintillation detector has been developed. The typical instability of restoring, that is boundary ejection and periodic perturbations, has been found out. The quality criteria of restoring, such as valu...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2000 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/82158 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer / S.N. Olejnik, N.A. Shlyakhov, V.P. Bozhko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 18-21. — Бібліогр.: 9 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859901067196628992 |
|---|---|
| author | Olejnik, S.N. Shlyakhov, N.A. Bozhko, V.P. |
| author_facet | Olejnik, S.N. Shlyakhov, N.A. Bozhko, V.P. |
| citation_txt | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer / S.N. Olejnik, N.A. Shlyakhov, V.P. Bozhko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 18-21. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The technique of restoring fast neutron spectra on the basis of amplitude spectra registered by a scintillation detector has been developed. The typical instability of restoring, that is boundary ejection and periodic perturbations, has been found out. The quality criteria of restoring, such as values of empirical risk and conditional characteristics of a matrix in the set of linear equations have been proposed. The main reasons of solution instabilities and methods of their overcoming have been found.
|
| first_indexed | 2025-12-07T15:57:13Z |
| format | Article |
| fulltext |
RESTORING OF FAST NEUTRON ENERGETIC SPECTRA
ON THE BASIS OF AMPLITUDE DISTRIBUTION OF SIGNALS
MEASURED BY SINGLE CRYSTAL NEUTRON SPECTROMETER
S.N. Olejnik, N.A. Shlyakhov, V.P. Bozhko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: Olejniksn@kipt.kharkov.ua
The technique of restoring fast neutron spectra on the basis of amplitude spectra registered by a scintillation
detector has been developed. The typical instability of restoring, that is boundary ejection and periodic perturbations,
has been found out. The quality criteria of restoring, such as values of empirical risk and conditional characteristics
of a matrix in the set of linear equations have been proposed. The main reasons of solution instabilities and methods
of their overcoming have been found.
PACS: 29.30.Hs
1. INTRODUCTION
A wide application of nuclear installations generating
penetrating radiation in science, engineering and
medicine requires a control on quantitative
characteristics of such radiation. The integrated type
means of measurement (dosimeters, radiometers etc.), in
fact, are not sufficiently informative. The fullest
information on neutron and gamma fields can be
obtained with the help of a special equipment−
spectrometers of appropriate radiation [1,2].
The authors have developed a model of a single
crystal spectrometer of fast neutrons operating on the
basis of a method of neutron registration considering
"recoil protons" [3]. Such spectrometers are described in
papers [1,2] and are characterized by simplicity of the
experimental equipment, high counting efficiency,
satisfactory energy resolution and universality of use.
They consist of a scintillator (stilben, liquid or solid
organic scintillator etc.), a photoelectronic multiplier
and electronic blocks of selection, blocks for shaping
and operating signals, registered from the detector.
The spectrometers operate satisfactorily in the
neutron energy range 0,2-20 MeV. The initial neutron
spectra are restored by the special mathematical
processing of the measured amplitude distributions of
signals. The restoring is carried out with the help of
complicated and rather bulky software of the
spectrometer [4].
At present both various approaches to solving the
problem of restoring, and appropriate algorithms and
programs for applying this or that method are described
in [5-8]. The characteristic feature of restoring is a well-
known problem of solution instability [5-7]. Its appears
in deviation of a spectra restored from the initial one,
moreover, the character of deviations can be in a wide
range, according to a class of selected approximating
functions, algorithms of restoring and experimental data
obtained [5,6,8].
Therefore, to apply the published programs [5] for
restoring the certain spectra of fast neutrons it was
necessary to conduct a number of "numerical"
experiments imitating experimental factors, which affect
the solution, and to reveal an effect of parameters used
in computing procedures.
The results allowed working out a technique of
restoring fast neutrons on the basis of measured
distributions. This technique permitted to optimize
parameters of computing procedures, to develop criteria
of optimum calculation variants and to conduct a priori
evaluation of solution quality.
The main mathematical premises of an algorithm of
restoring spectral dependences, the technique of making
a stable solution and examples of typical instabilities
obtained in the process of restoring smooth neutron
spectra are indicated below.
2. MATHEMATICAL FUNDAMENTALS OF
RESTORING ALGORITHM
The problem of determining a fast neutron spectrum
on the basis of measured amplitude distributions is one
of the mathematical problems on interpreting results of
indirect experiments [5]. The solutions of such problems
are based on a theory of restoring dependences selected
from a limited volume of data. These dependences can
be determined with the same mathematical scheme −
minimization of average risk on empirical data [5,6].
In general the problem under consideration is
described by the Fredholm integral equation of kind I
[5]:
∫=
b
a
dttgtxRxf )(),()( , (1)
where g(t) is the required dependence − neutron
spectrum, R(x,t) is the core of integral equation, f(x) is
the amplitude spectrum of signals registered by the
detector. In this case, the function of two variables
R(x,t) is formed of the functions of a detector response
to monoenergetic neutrons.
The solution of Eq. (1) belongs to a class of reverse
problems of incorrect type characterized by instability of
a solution, which depends on both the components of
18 ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, № 2.
Серия: Ядерно-физические исследования (36), c. 18-21.
.
Eq. (1), and the algorithm of the solution with its
program realization.
Two programs were used to restore neutron spectra.
In these programs the approximation of spectral
dependence g(t) was presented either as a linear
combination of Chebyshev polynomials or as a normal
system of cubic spline-functions (programs POLIL and
SPLIL, respectively) [5].
The results of "numerical" experiments have shown,
that the best approximation of the solution g(t) can be
achieved with the help of the program SPLIL.
Therefore, the further presentation of research results
will be connected with this program.
In a spline-solution algorithm the section [a,b] is
divided into (N+1) parts by equidistant points (a1...,aN) −
knots of conjugation. The number N is determined in the
process of algorithm fulfilment. On each section [aj,aj+1]
(j=0...,N; a0=a; aN+1=b) the approximate solution is
represented by a cubic polynomial Pj(t). The coefficients
of these polynomials are agreed in such a way that the
solution of Eq. (1) was continuous and had the first and
the second continuous derivatives along the whole
section [a,b].
The solution (1) is found with a normal system of
cubic splines:
(t),P jα*jg(t)
4N
1j
∑=
+
=
(2)
where αj
* is the coefficients of Eq. (2), are determined
from the condition of minimizing the empirical risk
functional:
, σ2
i/
l
i
2
(t)dtP j
4N
1j
α j
b
a
t),xiR(y jl
1) (αIe
1
∑
=
∑
+
=
∫−= (3)
where σi is the dispersion of measurements yi=f(xi)+εi,
Mεi=0, Dεi<∞, i=1,...,l.
The minimum value of the empirical risk functional
(3) is achieved in the solution of a normal set of linear
algebraic equations:
,ySαSS TT = (4)
where α=(α1,...,αN+4)Ò is the vector of cubic spline
coefficients, y=(y1,...,yl) is the vector of experimental
values of dependence observed, S is the matrix of a size
lx(N+4) of image values of fundamental splines in data
points xi (i=1,...,l), and α*=(STS)-1STy.
The value of empirical risk magnitude (3) determines
the quality of the constructed approximation of solution
of Eq. (1). The evaluation of approximation quality is
determined by the expression:
,
l
lnη1)4)ln(N4)(lnl(N1
)α*(Iecrit(N)
−++−+−
= (5)
where 1−η is the probability, when the evaluation is
true.
The value of N, when evaluation (5) of the spline
quality is of the least value, is the optimum number of
cubic spline conjugations; and the spline itself is taken
for the approximate solution of the integral equation (1).
3. TECHNIQUE OF CALCULATINS AND
RESULTS
The technique of restoring neutron spectra includes a
number of special programs, developed by the authors.
These programs imitate neutron spectra as Gaussian
distributions, and a response function of the detector −
as linear associations F(Ee)=aE+b, where the constats
a<0 and b>0. F(En)=0, where En is the neutron energy
(corresponds in (1) to the variable t), Eeis the amplitude
of a signal from the detector in the scale of equivalent
electron energies [3] (corresponds in (1) to the variable
x).
For the a priori evaluation of restoring quality the
following dependences are analyzed: a) crit(N) is the
criterion of solution quality (1) defined by Eq. (5) and b)
cond(N) is the degree of conditioning the matrix of
linear equation system (4). To analyze the parameter of
the cond(N) in the used program SPLIL, a subprogram
for solution of the linear equation system GELS [5] was
replaced by the subprograms DECOMP and
SOLVE [9].
Besides, the upper El and lower Eh thresholds of
cutting off the amplitude spectrum corresponding to the
real conditions of measurements were introduced as an
element of a restoring technique.
Practically, all the typical peculiarities of solutions
are seen in the process of restoring three types of
imitative neutron spectra (Gaussian distributions): a
"broad" spectra with a dispersion σ1=3 MeV, an
"average" − with σ2=1 MeV and a "narrow" − with σ
3=0.25 MeV. The positions of maximum of these
Gaussian distributions (Em=7 MeV) coincided, and the
definition regions corresponded to the values Em±4σ,
respectively.
0 2 4 6 8 10 120
100
200
300
400
500
IN
TE
NS
IT
Y
Ee,MeV
Fig. 1. Amplitude spectra. Symbols: “•” - σ
1=3 MeV, “ � ” - σ2=1 MeV, “+” - σ3=0.25 MeV.
Fig. 1 shows the "registered" amplitude spectra,
corresponding to the Gaussian distributions, which
served as those for restoring initial neutron spectra. The
results of such restoring are presented in Fig. 2 as
symbols, and initial imitative neutron spectra − as
continuous curves.
It is seen, that restoring the "average" spectra with σ
2=1 MeV (see Fig. 2) is the most qualitative; the region
of its definition is exceeded completely with a shaping
19
range (En=1-14 MeV) of the core R(x, t) in integral
equation (1). In the case if the range of spectra
definition is wider than the range of shaping R(x, t),
there is a typical solution instability − boundary ejection
(curve with σ1=3 MeV in Fig. 2).
0 2 4 6 8 10 12 14
-500
0
500
1000
1500
3
2
1
IN
TE
NS
IT
Y
En,MeV
Fig. 2. Neutron spectra. Symbols: ”o” - σ1=3 MeV,
“+” - σ2=1 MeV, “✼” - σ3=0.25 MeV.
Another limiting case takes place in the process of
restoring the "narrow" spectra (curve with σ
3=0,25 MeV in Fig. 2); besides a noticeable deviation of
the dependence restored from the initial one, there is a
typical instability of the solution − periodic (wave−type)
perturbation outside the region of initial spectra
definition, and a boundary ejection too.
The degree of restored spectrum adequacy for three
cases under consideration, on the whole, correlates with
a behaviour of parameters crit(N) and cond(N) the
analysis of which can give a priori qualitative
information on the solution stability.
2 4 6 8 10 12 14 16 18 20
101
102
103
104
105
106
107
108
109
101 0
crit-3
crit-2crit-1
cond
N
cr
it
100
101
102
103
104
105
106
107
108
109
101 0
101 1
101 2
cond
Fig. 3. Parameters of restoring the neutron spectra.
Symbols: “o” - σ1=3 MeV, “+” - σ2=1 MeV, “✼” - σ
3=0.25 MeV
Fig. 3 presents the dependences crit(N) and cond(N)
expressing the process of determining the solutions of
Eq. (1) shown by the appropriate curves in Fig. 2. The
approximation to a stable solution is accompanied by
the fast changing in the parameter crit(N) before
reaching a minimum and by rather smooth changing in
the field of the minimum (curves crit−1 and crit−2 in
Fig. 3). The unstable solution is characterized by the
slow change of crit(N), and its minimum is usually on
the boundary of changing matrix grade (nmax) of the set
of linear Eq. (4) (curve crit−3 in Fig. 3). In this case
nmax=20.
The research of solution instabilities (1) show that the
main factor of their origin when restoring the "broad"
neutron spectra (σ1=3 MeV), is the disagreement of a
definition range of the "measured" peak spectra and the
sub-integral core R(x,t). The availability of such a
difference can lead to full loss of adequacy of the
solution. The result of such restoring is represented in
Fig. 4, when the definition range of the core R(x,t) by
the variable "x" is 1…14 MeV, and that of the
"measured" amplitude spectra is 3…14 MeV.
0 2 4 6 8 10 12 14-1500
-1000
-500
0
500
1000
IN
TE
NS
IT
Y
En,MeV
Fig. 4. Inadequate restoring of neutron spectra with
σ1=3 MeV.
For the "narrow" spectra (curve with σ3=0,25 MeV
in Fig. 3) it is possible to reach the suppression of
periodic instability to the right of the peak by
introduction of the upper threshold Eh, the value of
which is determined by the maximum value of the
registered signal amplitude (see Fig. 1). In this case, the
quality of restoring (the approximation to the initial
spectra) is improved, and perturbations to the left of the
peak decrease (see Fig. 5).
0 2 4 6 8 10 12 14
0
500
1000
1500
IN
TE
N
S
IT
Y
En,MeV
Fig. 5. Results of restoring the neutron spectra with
σ3 = 025 MeV. Symbols: “+” – Eh “switch off”
(corresponds to curve 3 in Fig. 2); “� ” - Eh=8 MeV.
Simultaneous introduction of upper and lower
thresholds can lead to unexpected effects. The results of
restoring the "narrow" spectra σ3=0,25 MeV for two
values of upper and lower thresholds are given in Fig. 5.
In the process of restoring the neutron spectra with
El=Em-4σ=6 MeV and Eh=Em+4σ=8 MeV (Fig. 5) the
20
obtained solution does not correspond to the spectra
form we are looking for (continuous curve in Fig. 6),
though the position of a peak maximum is given
correctly.
The increase in the width of the Eh-El window results
in a satisfactory approximation of the solution to the
initial dependence, however, outside the region of
dependence definition there are periodic and boundary
instabilities (see Fig. 6).
5 6 7 8 9
-3000
-2000
-1000
0
1000
2000
IN
TE
NS
IT
Y
En,MeV
Fig. 6. Results of restoring the neutron spectra with
σ3 = 0.25 MeV. Symbols: “� ” - El=5 MeV,
Eh=9 MeV; “+” - El=6 MeV, Eh=8 MeV.
While restoring the real spectra of a complicated
form the satisfactory solution within the framework of
the above mentioned technique can be achieved in case
if the spectrum has not any narrow peaks. In Fig. 7, one
can see an illustration of the results of restoring then
imitative spectrum of the Pu-Be-source simulated by a
set of Gaussian distributions. The defined solution
(symbols in Fig. 7), in main, corresponds to the initial
spectrum (continuous curve). However, small periodical
perturbations can arise in the right part of the restored
spectrum. The methods of avoiding these perturbations
are described above.
0 2 4 6 8 10 12 14
0
1000
2000
3000
4000
IN
TE
NS
IT
Y
En,MeV
Fig. 7. Neutron spectra of the Pu-Be-source.
CONCLUSIONS
The studies on quality of restoring the neutron spectra
from amplitude distributions based on response
functions of detecting neutron equipment have been
done within the framework of the most general
mathematical approach to the numerical solution of the I
kind Fredholm equation.
A number of special programs and approaches has
been developed. For the first time they served as a base
for simulation of calculations that permitted to find
numerical characteristics of computing parameters,
which depend on both the form of energetic neutron
spectra, and response functions. This circumstance has
allowed developing a criterion for determining the stable
solutions of the integral equation describing the real
experimental data.
REFERENCES
1. V.I. Kukhtevich, O.A. Trykov, L.A. Triyov. Single
crystalline scintillation spectrometer. Moscow,
«Atomizdat», 1971, 136 p. (in Russian).
2. U.I. Kolevatov, V.P. Semenov, L.A. Trykov.
Spectrometry of neutrons and gamma-radiation in
radiation physics. Moscow, «Energoatomizdat», 1990,
296 p. (in Russian).
3. B.V. Rybakov, V.A. Sidorov. Spectrometry of fast
neutrons. Moscow, 1958, p. 9-17 (in Russian).
4. S.N. Olejnik, N.A. Shlyakhov, V.P. Bozhko. A
program complex on restoring spectra of fast neutrons
from amplitude distribution of neutron detectors //
International conference on spectrometry and nuclear
structure. Moscow, May − June 1997. The theses, 317 p.
(in Russian).
5. V.N. Vapnik. Algorithms and programs of
distribution restoring. Moscow, «Nauka», 1984, 816 p.
(in Russian).
6. V.N. Vapnik. Restoring of dependences by empirical
data. Moscow, «Nauka», 1979, 514 p. (in Russian).
7. A.F. Verlan, V.S. Sizikov. Integral equations:
methods, algorithms, programs. The reference book.
Kiev,
«Naukova dumka», 1986, 543 p. (in Russian).
8. E.A. Kramer-Ageev, V.S. Troshin, E.G. Tikhonov.
Activation methods of neutron spectrometry. Moscow,
«Mir», 1976, 232 p. (in Russian).
9. G. Forsythe, M. Malcolm, C. Moler. Computer
methods for mathematical computations. Moscow,
«Mir», 1980, p. 61-70 (in Russian).
21
S.N. Olejnik, N.A. Shlyakhov, V.P. Bozhko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
1. INTRODUCTION
2. MATHEMATICAL FUNDAMENTALS OF RESTORING ALGORITHM
3. TECHNIQUE OF CALCULATINS AND RESULTS
CONCLUSIONS
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-82158 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:57:13Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Olejnik, S.N. Shlyakhov, N.A. Bozhko, V.P. 2015-05-26T08:04:56Z 2015-05-26T08:04:56Z 2000 Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer / S.N. Olejnik, N.A. Shlyakhov, V.P. Bozhko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 18-21. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 29.30.Hs https://nasplib.isofts.kiev.ua/handle/123456789/82158 The technique of restoring fast neutron spectra on the basis of amplitude spectra registered by a scintillation detector has been developed. The typical instability of restoring, that is boundary ejection and periodic perturbations, has been found out. The quality criteria of restoring, such as values of empirical risk and conditional characteristics of a matrix in the set of linear equations have been proposed. The main reasons of solution instabilities and methods of their overcoming have been found. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Nuclear reactions Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer Восстановление энергетических спектров быстрых нейтронов из измеренных однокристальным нейтронным спектрометром амплитудных распределений сигналов Article published earlier |
| spellingShingle | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer Olejnik, S.N. Shlyakhov, N.A. Bozhko, V.P. Nuclear reactions |
| title | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer |
| title_alt | Восстановление энергетических спектров быстрых нейтронов из измеренных однокристальным нейтронным спектрометром амплитудных распределений сигналов |
| title_full | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer |
| title_fullStr | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer |
| title_full_unstemmed | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer |
| title_short | Restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer |
| title_sort | restoring of fast neutron energetic spectra on the basis of amplitude distribution of signals measured by single crystal neutron spectrometer |
| topic | Nuclear reactions |
| topic_facet | Nuclear reactions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82158 |
| work_keys_str_mv | AT olejniksn restoringoffastneutronenergeticspectraonthebasisofamplitudedistributionofsignalsmeasuredbysinglecrystalneutronspectrometer AT shlyakhovna restoringoffastneutronenergeticspectraonthebasisofamplitudedistributionofsignalsmeasuredbysinglecrystalneutronspectrometer AT bozhkovp restoringoffastneutronenergeticspectraonthebasisofamplitudedistributionofsignalsmeasuredbysinglecrystalneutronspectrometer AT olejniksn vosstanovlenieénergetičeskihspektrovbystryhneitronovizizmerennyhodnokristalʹnymneitronnymspektrometromamplitudnyhraspredeleniisignalov AT shlyakhovna vosstanovlenieénergetičeskihspektrovbystryhneitronovizizmerennyhodnokristalʹnymneitronnymspektrometromamplitudnyhraspredeleniisignalov AT bozhkovp vosstanovlenieénergetičeskihspektrovbystryhneitronovizizmerennyhodnokristalʹnymneitronnymspektrometromamplitudnyhraspredeleniisignalov |