Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В
We demonstrated that the standard contour plot of the wavelet transform with wavelet “Mexican Hat” of the inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²C(γ,р)¹¹B does not give reliable visual indication of the maxima in experimental spectra. We used the scale disconvolutio...
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irk-123456789-805152015-04-19T03:02:48Z Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В Omelaenko, A.S. Ядерная физика и элементарные частицы We demonstrated that the standard contour plot of the wavelet transform with wavelet “Mexican Hat” of the inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²C(γ,р)¹¹B does not give reliable visual indication of the maxima in experimental spectra. We used the scale disconvolution of the initial spectra to construct more informative image in the energy-scale plane. Показано, що стандартні контурні графіки вейвлет-трансформа з вейвлетом “Мексиканська шляпа” інклюзивного ⁴He(e,e′)-спектра і спектра загубленої енергії в реакції ¹²C(γ,р)¹¹B не забезпечують надійної візуальної індикації максимумів експериментальних спектрів. Для побудови більш інформативного образу в масштабно-енергетичній площині використанано масштабну розгортку початкових спектрів. Показано, что стандартные контурные графики вейвлет-трансформа с вейвлетом “Мексиканская шляпа” инклюзивного ⁴He(e,e′)-спектра и спектра потерянной энергии в реакции ¹²C(γ,р)¹¹B не обеспечивают надежной визуальной индикации максимумов экспериментальных спектров. Для построения более информативного образа в масштабно-энергетической плоскости использована масштабная развертка исходных спектров. 2004 Article Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В / A.S. Omelaenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 58-61. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 13.75Gx http://dspace.nbuv.gov.ua/handle/123456789/80515 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Omelaenko, A.S. Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В Вопросы атомной науки и техники |
| description |
We demonstrated that the standard contour plot of the wavelet transform with wavelet “Mexican Hat” of the inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²C(γ,р)¹¹B does not give reliable visual indication of the maxima in experimental spectra. We used the scale disconvolution of the initial spectra to construct more informative image in the energy-scale plane. |
| format |
Article |
| author |
Omelaenko, A.S. |
| author_facet |
Omelaenko, A.S. |
| author_sort |
Omelaenko, A.S. |
| title |
Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В |
| title_short |
Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В |
| title_full |
Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В |
| title_fullStr |
Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В |
| title_full_unstemmed |
Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В |
| title_sort |
wavelet analysis of inclusive ⁴he(e,e′) spectrum and missing energy spectrum in reaction ¹²с(γ,p)¹¹в |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Ядерная физика и элементарные частицы |
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http://dspace.nbuv.gov.ua/handle/123456789/80515 |
| citation_txt |
Wavelet analysis of inclusive ⁴He(e,e′) spectrum and missing energy spectrum in reaction ¹²С(γ,p)¹¹В / A.S. Omelaenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 58-61. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-06T04:31:36Z |
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2025-07-06T04:31:36Z |
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1836870582810968064 |
| fulltext |
WAVELET ANALYSIS OF INCLUSIVE 4He(e,e′) SPECTRUM
AND MISSING ENERGY SPECTRUM IN REACTION 12С(γ,p)11В
A.S. Omelaenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
We demonstrated that the standard contour plot of the wavelet transform with wavelet “Mexican Hat” of the
inclusive 4He(e,e′) spectrum and missing energy spectrum in reaction 12C(γ,р)11B does not give reliable visual
indication of the maxima in experimental spectra. We used the scale disconvolution of the initial spectra to construct
more informative image in the energy-scale plane.
PACS: 13.75Gx
1. INTRODUCTION
It is just twenty years ago term “wavelet” was
introduced into the scientific usage [1]. For this time
wavelet analysis (WA) become a powerful common tool
for a many investigations in the fields of applied
sciences, physics and mathematics. A theoretical
treatment of the wavelet analysis is developed in several
monographs beginning from [2]. In Russian the
substantial review article of the basic theory of the
wavelet analysis appeared in 1996 [3]. A useful guide
for application of the discrete wavelet transform in
computational practice is presented in [4]. The recent
article [5] contains the concise digest of the continuous
version of the WT and its applications to the e+e-
annihilation into hadron states with quantum numbers of
ρ and ω mesons, and to the p-wave ππ scattering.
The objective of our investigation is to assess the use
wavelet analysis for nuclear reactions. For this
assessment we choose spectrum of inclusive scattering
of the high-energy electrons on 4He nuclei [6] and
missing energy spectrum in the typical low energy
nuclear reaction 12С(γ,р)11В taken from work [7]. We
used the so-called wavelet “Mexican Hat” considered as
the most suitable due to it’s good localization and small
number of oscillations. In particular we have
demonstrated that at least for two examples at
consideration the visual inspection of the standard
wavelet transformation (WT) of a one-dimension initial
spectrum (diffuse two-dimension energy-scale image)
practically couldn’t give a reasonable idea about the
position of the maxima in the original spectrum. Trying
to shed some light on this unexpected circumstance it is
relevant to remember that WT is defined as projection
of initial spectrum on the wavelet with scale a at
position b. Of course, such a value depends on the
characteristic details of a spectrum being at the same
time sensitive to the strong energy dependence of the
wavelet used. To relax influence of the latter factor we
have considered the contribution of all wavelets with
scale a (that is integrated of on their position) to the
point t of initial spectrum as a function to plot the
energy-scale image (scale unfolded spectrum). The
energy-scale image of such an averaged value exhibits
all characteristic features of a complicated nuclear
reaction spectrum.
58 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 58-61.
2. WAVELET TRANSFORMATION AND
RECONSTRUCTION
The underlying notion of the wavelet theory with
continuous variables is a biparametric family of self-
similar soliton-like functions generated by dilatation
(the scale parameter a) and translation (the parameter b)
of some analyzing function ψ(t) (by tradition the
variable t is called “time” even in the case of energy or
some other variable):
.
a
btatba,
−= − ψψ / 21)( (1)
The main points of the wavelet theory is formula for
WT being a convolution of initial function f(t) with
wavelet ψ:
tdtf
a
tt*aCt,aw // ′′
−′
= ∫
∞
∞−
−− )()( 2121 ψψ (2)
and the inverse transformation (reconstruction of the
function f(t)):
.
a
datdta,w
a
tt
Ctf /
/
25
21 )()(
′′
′−= ∫ ∫
∞
∞−
∞
∞−
− ψψ (3)
The normalizing constant Cψ can be presented as the
following integral on variable ω:
( ) ω ,ωω= −∞
∞−
∫ dC
12ψψ
(4)
involving the Fourier transformation of the wavelet
used:
.dtet tiω
∞
∞−
∫=ω )()( ψψ (5)
The graphic representation of the w(a,t) as contour
plots in scale-time plate is used in WA as significant
tool for visual resolving the structures in the initial
spectrum. Besides, implementation of some restrictions
at integration on time and scale variable give broad
lands for processing of data by separation of the scale
band contributions, noise filtering and so on.
-6 -4 -2 0 2 4 6
-0,5
0,0
0,5
1,0
ψ
M
H
t
Fig. 1. Wavelet “Mexican Hat”
200 400 600 800
0
2
4
6
d2 σ
/d
E
d
Ω
, m
kb
n.
G
eV
-1
.s
r-1
ω (MeV )
a)
200 400 600
0,0
0,2
0,4
0,6
0,8
1,0
ω , MeV
-0,4
0
0,4
0,8
1
b)
200 400 600
0
100
200
300
400
ω , MeV
a,
M
eV
-25,0
10,2
45,4
70,0
c)
Fig. 2. The spectrum of inclusive 1269 MeV
scattering from 4He nucleus at angle 30° ([5]), a), its
wavelet transform and the scale unfolded spectrum (c)
59
0 5 10 15 20 25
0
50
100
150
200
250
300
350
400
0 5 10 15 20 25
0
5
10
15
E
x
, MeV
a,
M
eV
-45,0
-20,0
5,00
30,0
55,060,0
0 5 10 15 20 25
0
5
10
15
20
25
E
x
, MeV
a,
M
eV
-150
50,0
250
450
650
700
co
un
ts
E
x
, MeV
Fig. 3. Missing mass spectrum for reaction 12С(γ
,р)11В from work [6] (a), its wavelet transform (b) and
scale unfolded spectrum (c)
0 10 20 30
0
100
200
300
400
-100
0
100
200
300
400
-100
0
100
200
300
400
co
un
ts
co
un
ts
co
un
ts
c)
Ex, MeV
AMIN =5.00
AMAX =35.0
b) AMIN =1.0
AMAX =5.0
a)
AMIN =1.0
AMAX =35.
Fig. 4. Missing mass spectrum for reaction 12С(γ
,р)11В from work [6] (black squares with line) and the
same spectrum after reconstruction within scale bands
amin=1, amax=35 (a); amin=1, amax=5 (b); amin=5, amax=35
(c); all shifted downwards to avoid confusion with
initial spectrum
3. SCALE UNFOLDED SPECTRUM (SUS)
To get some different approach to building of a
scale-energy image we have changed order of
integration in Eq. (3) and have written down the
spectrum reconstructed through the lens of the wavelet
analysis in form of a convolution on the scale a:
.data,ftf ∫
∞
∞−
= )()( (6)
In Eq. (6) f(a,t) is a contribution to the initial spectrum
per scale unit at current a (scale unfolded spectrum):
td
a
t-ttfta,f ′
′
δ′= ∫
∞
∞−
ψ)()( (7)
with the function
.td
a
tt
a
tt
aC
a
a
t-t ′′
′′−′
′′−=
′
δ ∫
∞
∞−
*sign
3 ψψ
ψ
ψ (8)
In Eqs. (7,8) the argument of δψ is written in fashion
characteristic for a wavelet function (Eq. (1)). Indeed,
such a specification is manifested by the next formula:
( ) ω .
′
ωω
π
=
′
δ ∫
∞
∞−
d
a
t-tiˆ
Caa
t-t exp
2
1
2
2
ψ
ψ ψ (9)
It is interesting to note that independently of the wavelet
used integrating of this function on scale parameter a
yields the Dirac δ function:
).( t-tad
a
t-t ′δ=
′
δ∫
∞
∞−
ψ (10)
So, Eq. (10) signifies that the function δψ((t – t′)/a) can
be treated as a result of the scale unfolding of the δ
function itself on the base of wavelet ψ. Further, from
the fact that the wavelets are introduced as functions an
average values of which are equal to zero it follows
.td
a
t-ttd
a
t-t 0=′
′
δ=
′
δ ∫∫
∞
∞−
∞
∞−
ψψ (11)
Similar to the standard WT the function defined by
Eq. (7) can be depictured as contour plot in scale-time
plane. But it should be stressed that the time arguments
for the both functions have the quite different meaning.
In the case of WT one has to do with the coordinate of
the projection of an initial signal on the wavelet with his
inherent strong time dependence. It is necessary
integration on time to get contribution to the initial
spectrum at parameter a (Eq. (3)). On the contrary, our
function f(a,t) represents direct contribution to the t
point of the spectrum due to all wavelets with fixed
scale value a irrespective of their localization. For f(a,t)
influence of the energy dependence of the wavelet used
60
is smoothed over with holding good of scale
dependence being the distinguishing feature of the
wavelet conception. So, one can anticipate that the
contour image of the f(a,t) would give more robust
reflection of the specific details of the initial spectrum.
4. WT AND SUS: NUMERICAL COMPARISON
In this communication we have restricted our self by
calculations with the wavelet “Mexican Hat” (MH,
Fig. 1):
.ttt )2)exp(1()( 22
ΜΗ −−=ψ (12)
For MH Cψ = CMH = 2π and Eq. (8 ) turns into
.
a
t-t
a
t-t
aa
t-t
′
−
−
−
′
×
×
π
=
′
δ Μ Η
222
2
4
1exp246
32
1
(13)
The wavelet transformation images the inclusive
electron scattering spectrum from 4He [5] (Fig. 2, a))
and the missing energy spectrum for 12С(γ,р)11В [6]
(Fig. 3, a)) are shown on panels b) of Figs. 3,4. In
calculations the initial spectra were presented by broken
lines coming through the neighboring experimental
points fk ≡ f(tk). That is f(t) = f(t;f1,…,fk,…,fn), n being
the number of points.
For inclusive spectrum the most characteristic
feature presented by the contour WT plot in the energy-
scale plate is the wide overall maxima at ω ∼ 350 MeV
and a∼ 400 MeV. From low scales corresponding “hill”
adjoins to the narrow gully. This gully ensures
appearance of two well known from the physical point
of view maxima at reconstruction of the initial
experimental spectrum: the large quasi-elastic peak and
the lower one corresponding to the excitation of the first
nucleon resonance.
On the other hand, the quasi-elastic peak and the
resonance peak are displayed immediately by the image
of the scale unfolded inclusive spectrum (Fig. 2,b)) as
clearly seen hills. Minimum between them corresponds
to so-called “dip” region of the spectrum. The SUS
image allows also distinguishing on the left hand side of
quasi-elastic peak some manifestation of an additional
dip region. The white spots of the dip-regions have a
specific elongated form and are symmetric orientation
relatively to the quasi-elastic peak. There are seen the
numerous small-scale structures due to the noise-like
high-frequency contributions. The structure at the right
side of the SUS image can be treated as excitation of the
Roper resonance being obscured by experimental
problems at the end of the measured spectrum.
For the missing energy spectrum from work [7]
comparison of the standard WT image (Fig. 3,b)) and
our SUS image (Fig. 3,c)) is not so contrasting as in the
case of the inclusive scattering. Nevertheless, one can
make certain that on the SUS image the location of the
on the missing mass spectrum for reaction 12С(γ,р)11В
(Fig. 4a), reconstruction within the scale bands
containing the characteristic resonance structures
(Fig. 4b) and contribution of large-scale background
(Fig. 4c). For this goal we have used coming from
Eqs. (6,7) formula with relevant limits amin, amax at
integration on scale and with integration on energy in
limits of the experimental spectra:
.daEdEE,aEfEf ′′δ′= ∫ ∫ ),()()( ΜΗ (14)
Errors were calculated according to
2
nk1
ki
);(
∑ ∫ ∫
′
′
δ
∂
′∂
δ=δ 2
k k
daEd
a
E-E
f
f,...,f,...,fEf
ψ
with summing on the experimental points.
5. DISCUSSION
The numerical calculations that we describe in this
paper show that for complicated nuclear spectra with
overlapping resonances the standard WT contour scale-
time images are not a robust tool of visual indication
with respect to the properties of the initial signal.
Indeed, the WT image reflects characteristic feathers of
both the initial signal and the wavelet used in analysis
and influence of the latter may surpass the role of the
small resonance structures. It should be kept in mind
that experimental spectra are of restricted length.
The main result of this paper is the suggestion to use
the SUS image that is free from this shortage. Using
SUS images demonstrates some additional potentialities
of the WA.
Reconstruction of the original function in the
framework of the wavelet analysis involves integration
on two time variables and the scale. Because in
applications one is dealing with finite-length time
intervals instead of (− ∞ … + ∞) used in the theory
there will appear some errors at integration on time (this
effect is known as “cone of influence” [8]). From
Eqs. (1,2) it follows that coming from WT standard
reconstruction involves two such approximate time
integrations. Some advantage of our approach is that
one of time integrations for given wavelet is taken
analytically in theoretical infinity limits at calculation of
the δψ(a,t,t′). It is interesting to note that the latter is the
scale-unfolding of the Dirac δ function.
REFERENCES
1. A. Grossman, J. Morlet. Decomposition of Hardy
functions into square integrable wavelets of constant
shape // SIAM J. Math. Anal. 1984, v. 15, p. 723-
736.
2. I. Daubechies. Ten lectures on wavelets. Society for
industrial and applied mathematics. 1992, 357 p.
3. N.M. Astaf’eva. Wavelet analysis: basic theory and
some applications // Uspehi Fiz. Nauk. 1996, v. 166,
p. 1145-1170 (in Russian).
4. I.M. Dremin, O.V. Ivanov, V.A. Nechitailo.
Wavelets and their applications // Uspekhi Fiz.
Nauk. 1996, v. 171, p. 465-501 (in Russian).
5. T.S. Belozerova, P.G. Frick, V.K. Henner. Wavelet
analysis of data in particle physics: vector mesons in
61
e+e- annihilation // Yad. Fiz. 2003, v. 66, p. 1269-
1281.
6. E.L. Kuplennikov et al. Examination of the cross
sections 4He(e,e′) reaction in the quasifree, dip and ∆
(1232)-resonance region // Yad. Fiz. 1994, v. 57,
p. 771-776.
7. P.D. Hartry et all. 12C(γ,p)11B cross section from 80
to 157 MeV // Phys. Rev. C, 1995, v. 51, p. 1982-1990.
8. Ch. Torrence and G.P. Compo. A practical guide to
wavelet analysis // Bull. Amer. Met. Society. 1998,
v. 79, p. 61-78.
ВЕЙВЛЕТ-АНАЛИЗ ИНКЛЮЗИВНОГО 4He(e,e′)-СПЕКТРА
И СПЕКТРА ПОТЕРЯННОЙ ЭНЕРГИИ В РЕАКЦИИ 12С(γ,p)11В
А.С. Омелаенко
Показано, что стандартные контурные графики вейвлет-трансформа с вейвлетом “Мексиканская шляпа”
инклюзивного 4He(e,e′)-спектра и спектра потерянной энергии в реакции 12C(γ,р)11B не обеспечивают
надежной визуальной индикации максимумов экспериментальных спектров. Для построения более
информативного образа в масштабно-энергетической плоскости использована масштабная развертка
исходных спектров.
ВЕЙВЛЕТ-АНАЛІЗ ІНКЛЮЗИВНОГО 4He(e,e′)-СПЕКТРА
ТА СПЕКТРА ЗАГУБЛЕНОЇ ЕНЕРГІЇ В РЕАКЦІЇ 12С(γ,p)11В
О.С. Oмeлaєнкo
Показано, що стандартні контурні графіки вейвлет-трансформа з вейвлетом “Мексиканська шляпа”
інклюзивного 4He(e,e′)-спектра і спектра загубленої енергії в реакції 12C(γ,р)11B не забезпечують надійної
візуальної індикації максимумів експериментальних спектрів. Для побудови більш інформативного образу в
масштабно-енергетичній площині використанано масштабну розгортку початкових спектрів.
62
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