Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn
The cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn is expected by different methods. Results of the got cross section it is well comported inter se the Penfold-Leiss and Tikhonov's methods. The calculation of cross section is conducted the Penfold-Leiss method with smoothing out by the method of...
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Zhaba, V.I. Parlag, A.M. 2017-01-17T17:23:26Z 2017-01-17T17:23:26Z 2015 Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn / V.I. Zhaba, A.M. Parlag // Вопросы атомной науки и техники. — 2015. — № 3. — С. 34-37. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 24.75+1, 25.85.-w, 25.85. Ec, 25.85. Ca https://nasplib.isofts.kiev.ua/handle/123456789/112124 The cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn is expected by different methods. Results of the got cross section it is well comported inter se the Penfold-Leiss and Tikhonov's methods. The calculation of cross section is conducted the Penfold-Leiss method with smoothing out by the method of iterations. Number of iterations n = 1; 3; 5. In the programmatic package of TALYS-1.4 got cross section for five models of closeness of levels. Theoretical and experimental results well coincide in a maximum. Переріз реакції ¹¹⁵In(γ, n) ¹¹⁴mIn розраховано різними методами. Результати отриманого перерізу методами Пенфольда-Лейсса і Тіхонова добре узгоджуються між собою. Розрахунок перерізу методом Пенфольда-Лейсса проведено зі згладжуванням методом ітерацій. Число ітерацій n = 1; 3; 5. У програмному пакеті TALYS-1.4 отримано переріз для п'яти моделей густини рівнів. Теоретичні та експериментальні результати добре співпадають у максимумі. Сечение реакции ¹¹⁵In(γ, n) ¹¹⁴mIn рассчитано разными методами. Результаты полученного сечения методами Пенфольда-Лейсса и Тихонова хорошо согласуются между собой. Расчет сечения методом Пенфольда-Лейсса проведен со сглаживанием методом итераций. Число итераций n = 1; 3; 5. В программном пакете TALYS-1.4 получены сечения для пяти моделей плотности уровней. Теоретические и экспериментальные результаты хорошо совпадают в максимуме. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядерная физика и элементарные частицы Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn Методи розрахунку перерiзу реакцiї ¹¹⁵In(γ, n) ¹¹⁴mIn Методы расчета сечения реакции ¹¹⁵In(γ, n) ¹¹⁴mIn Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn |
| spellingShingle |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn Zhaba, V.I. Parlag, A.M. Ядерная физика и элементарные частицы |
| title_short |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn |
| title_full |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn |
| title_fullStr |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn |
| title_full_unstemmed |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn |
| title_sort |
methods of calculation of cross section of reaction ¹¹⁵in(γ, n) ¹¹⁴min |
| author |
Zhaba, V.I. Parlag, A.M. |
| author_facet |
Zhaba, V.I. Parlag, A.M. |
| topic |
Ядерная физика и элементарные частицы |
| topic_facet |
Ядерная физика и элементарные частицы |
| publishDate |
2015 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Методи розрахунку перерiзу реакцiї ¹¹⁵In(γ, n) ¹¹⁴mIn Методы расчета сечения реакции ¹¹⁵In(γ, n) ¹¹⁴mIn |
| description |
The cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn is expected by different methods. Results of the got cross section it is well comported inter se the Penfold-Leiss and Tikhonov's methods. The calculation of cross section is conducted the Penfold-Leiss method with smoothing out by the method of iterations. Number of iterations n = 1; 3; 5. In the programmatic package of TALYS-1.4 got cross section for five models of closeness of levels. Theoretical and experimental results well coincide in a maximum.
Переріз реакції ¹¹⁵In(γ, n) ¹¹⁴mIn розраховано різними методами. Результати отриманого перерізу методами Пенфольда-Лейсса і Тіхонова добре узгоджуються між собою. Розрахунок перерізу методом Пенфольда-Лейсса проведено зі згладжуванням методом ітерацій. Число ітерацій n = 1; 3; 5. У програмному пакеті TALYS-1.4 отримано переріз для п'яти моделей густини рівнів. Теоретичні та експериментальні результати добре співпадають у максимумі.
Сечение реакции ¹¹⁵In(γ, n) ¹¹⁴mIn рассчитано разными методами. Результаты полученного сечения методами Пенфольда-Лейсса и Тихонова хорошо согласуются между собой. Расчет сечения методом Пенфольда-Лейсса проведен со сглаживанием методом итераций. Число итераций n = 1; 3; 5. В программном пакете TALYS-1.4 получены сечения для пяти моделей плотности уровней. Теоретические и экспериментальные результаты хорошо совпадают в максимуме.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/112124 |
| citation_txt |
Methods of calculation of cross section of reaction ¹¹⁵In(γ, n) ¹¹⁴mIn / V.I. Zhaba, A.M. Parlag // Вопросы атомной науки и техники. — 2015. — № 3. — С. 34-37. — Бібліогр.: 14 назв. — англ. |
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2025-11-27T08:31:15Z |
| last_indexed |
2025-11-27T08:31:15Z |
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| fulltext |
METHODS OF CALCULATION OF CROSS SECTION OF
REACTION 115In(γ, n)114mIn
V. I.Zhaba, A.M.Parlag ∗
Uzhhorod National University, 88000, Uzhhorod, Voloshin Str., 54
(Received January 27, 2015)
The cross section of reaction 115In(γ, n)114mIn is expected by different methods. Results of the got cross section it
is well comported inter se the Penfold-Leiss and Tikhonov’s methods. The calculation of cross section is conducted
the Penfold-Leiss method with smoothing out by the method of iterations. Number of iterations n = 1; 3; 5. In
the programmatic package of TALYS-1.4 got cross section for five models of closeness of levels. Theoretical and
experimental results well coincide in a maximum.
PACS: 24.75+1, 25.85.-w, 25.85. Ec, 25.85. Ca
1. INTRODUCTION
The purpose of the majority of physical researches
that are carried spent on brake beams electronic
accelerators,- studying of power dependence of cross
sections of different photonuclear processes. As the
spectrum received from the accelerator γ-quantum’s
has continuous character, therefore in experiment not
cross section is measure of reaction, it is so-called
output. The output is intensity photonuclear to the
process, the doze attributed to unit γ-quantums that
have passed through a target with researched sub-
stance at various values of the top border of a brake
spectrum [1].
The output is directly connected with effective
cross section of reaction by integrated equation Fred-
holm’s (or Voltera’s) the first sort:
Y (Eγmax) =
∫ Eγmax
Enop
σ(E)Φ(E, Eγmax)dE , (1)
where Em - a threshold of reaction, Eγmax - the max-
imal energy of a spectrum brake γ- quantum’s, σ(E)
- cross section of reaction, Φ(E, Eγmax) - a spec-
trum of brake radiation (Shiff’s spectrum [2]). So,
the cross section of reaction can be received from
experimental data about output as a result of the
decision of a return task (1). For the numerical de-
cision of this task mathematical methods have been
developed. The most widespread methods are ”a dif-
ference of photons”, ”the least structure” Cook’s [3],
”a return matrix” (Penfold-Leiss method) [4], ”regu-
larizations” (Tikhonov’s method) [5-8]. Penfold-Leiss
and Tikhonov’s methods differ the form of the effec-
tive spectrum of photons hardware function of the
method. It is the direct decision of a return task.
But other methods of definition of cross section are
also possible: a combination of outputs of reaction
and a method of a reduction. Conditions of correct
statement of Adamar’s task [6,7] is: 1) Existence of
the decision in space of possible values for any curve
of an output from space of her possible values; 2)
The decision should be the only thing; 3) continuous
dependence of the decision on the initial data. One
more condition is that the cross section should be
positive.
2. METHODS OF CALCULATION OF
SECTION OF REACTION
In a method of a return matrix [4] required cross sec-
tion σ(Ej−∆E)/2 is defined on experimental outputs
Y (Eγmax):
σ(Ej −∆Ej/2) =
Ej −∆Ej/2
f(Ej −∆Ej/2)∆Ej
j∑
i=1
BjiYi ,
(2)
where Bij – elements of a return matrix. The stan-
dard deviation (2) is defined as
δσ(Ej −∆Ej/2) =
Ej −∆Ej/2
f(Ej −∆Ej/2)∆Ej
×√√√√ j∑
i=1
(Bji)
2
(∆Yi)
2
, (3)
where ∆Yi - a standard deviation of experimental
value Yi.
One more method of the decision of the integrated
equation (1) is three dot method or measurement
of photonuclear cross sections with the help ”quasi-
monochromatic” γ-quantum’s of brake radiation [9].
For the fixed values j elements Bj,j−1, Bj,j−3, Bj,j−5
have negative values, and also at i < j− 2 values Bji
are decreased very quickly. Using these properties of
∗Corresponding author E-mail address: parlag.oleg@gmail.com
34 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.34-37.
elements Bj,i for the analysis of cross section σ(Eγ),
it is possible to write down
σj1 =
Ej −∆Ej/2
f(Ej −∆Ej/2)∆Ej
×
(Bj,j −Bj,j−2)
Yj − Yj−2
2∆E
, (4)
σj2 =
Ej −∆Ej/2
f(Ej −∆Ej/2)∆Ej
∆E
2
×
(Bj,j −Bj,j−2)
Yj − 2Yj−1 + Yj−2
∆E2
. (5)
Numerical experiment in the work [9] has shown, that
cross section (Ej − ∆E/2) can be approximated by
expression
σ̃ = σ(Ej −∆Ej/2) = σj,1 + σj,2 . (6)
According to this method the cross section of pho-
tonuclear reaction can be found on three experimen-
tal values of output Y (Eγmax), received on a brake
beam for three different values of maximal energy
Eγmax.
In Cook’s method [3] set of σj s will be considered
as acceptable solution to equation (1) if
χ2(σj) =
n∑
i=1
(
∑
Bi,jσj − Yi)
2
(∆Yi)
2 ≤ n . (7)
The ”most smooth” set of decisions is chosen from
physical decisions. For this purpose an auxiliary func-
tion S(σj) called ”the structure function” will be de-
fined. Within certain wide limits, the exact definition
of S(σj) is arbitrary. Several definitions of S(σj) have
been extensively explored, namely:
S1(σj) =
n−1∑
j=1
(σj+1 − σj)
2 , (8)
and
S2(σj) =
n∑
j=2
(σj+1 − 2σj + σj−1)
2 . (9)
In Tikhonov’s method a choice of the approached
cross section the principle of smoothness [7] of set
of ”formal” decisions σ(k), satisfying a condition is
used
m∑
j=1
[∫ Ej
Em
a(Ej , k)σ(k)dk − y(Ej)
]2
[dy(Ej)]
2 ≤ m (10)
is chosen such, for which special functional
Ω[σ] =
∫ Emax
Em
[
σ2(k) +
(
dσ(k)
dk
)2
]
dk (11)
if has the minimal value. That is the finding of func-
tion σ(k) is reduced to search of a minimum func-
tional
α
M
=
∥∥∥∥∥
∫ E
Em
a(E, k)σ(k)dk − y(E)
∥∥∥∥∥+ αΩ[σ] , (12)
where α – parameter regularizations.
In work [1] research of influence by miscellaneous
of methods of smoothing of an experimental curve of
an output on power dependence of cross section of re-
action (γ, γ′) is carried out. By the example of reac-
tion 115In(γ, n)115mIn it is shown, that smoothings
by a method of iterations and a method of approxi-
mation give three variants of power dependence of dif-
ferential cross section. In some power area the cross
section has negative values, and it testifies about ”not
physical” results. To remove this discrepancy in cal-
culations, it is necessary to impose a requirement: the
cross section should be the positive.
3. CALCULATIONS OF CROSS SECTION
OF REACTION 115In(γ, n)114mIn AND
CONCLUSIONS
Calculation of differential cross section of reaction
was spent on outputs from work [10]. Results of
calculation of cross section are resulted by Penfold-
Leiss method on Fig.1. Smoothing cross section by a
method of iterations was used. Number of iterations
n = 1; 3; 5. The received cross section σ(E) > 0
also is quite comparable to the data in [11,12]. On
Fig.1 comparison calculations with the data of op-
eration [11] is made. Nuclear performances of an
isomers 114mIn is specified in Table 1, where Em
– a threshold (γ, n) – responses that gives in for-
mation of an isomer; T1/2 – a half-life period; Eγ
– energy γ- quantum’s that radiates an isomer; Jm,
Jg – the complete moment isomeric and the basic
states; ”+”, ”−” – paired relationship of a state.
Fig.1. Cross section of reaction 115In(γ, n)114mIn
(Penfold-Leiss method)
Table 1. Nuclear characteristics of an isomers In
Isomer Em T1/2 Eγ Jg Jm
MeV keV
113mIn 0.39 1.658 h 392 9/2+ 1/2-
114mIn 9.23 43 ms 310 1+ 8-
115mIn 0.33 4.486 h 336 9/2+ 1/2-
For Tikhonov’s method a nucleus of the equa-
tion (1) will be Shiff’s spectrum Φ(E, Eγmax). Thus
35
it is necessary to take into account function of re-
sponse of the absolute chamber f(E). On Fig.2 the
cross section received by a method of Tikhonov’s
is resulted. The difference between calculations by
Penfold-Leiss and Tikhonov’s methods makes 5...7%.
Fig.2. Cross section of reaction 115In(γ, n)114mIn
(Tikhonov’s method)
For calculation of cross section of reaction
115In(γ, n)114mIn is possible to use programmatic
package TALYS-1.4 [13,14]. At calculations of cross
section in TALYS-1.4 is possible to choose a level
density model per nuclide considered in the reaction
(parameter ldmodel). There are 3 phenomenological
level density models and 2 options for microscopic
level densities: ldmodel 1 – constant temperature +
Fermi gas model; ldmodel 2 – back-shifted Fermi gas
model; ldmodel 3 – generalised superfluid model; ld-
model 4 – microscopic level densities from Goriely’s
table; ldmodel 5 – microscopic level densities from Hi-
laire’s table. In package TALYS-1.4 the cross section
of reaction 115In(γ, n)114mIn in the interval of en-
ergies 10...25MeV with step 0.1MeV is numerically
designed. In Fig.3 is resulted the received results for
five models of density of levels nuclide (ldmodel 1-
5). For ldmodel 1-3 value of cross section is relatives,
and a maximum ∼ 58...59mb. Also differ from ex-
perimental in 1.52 times. A maximum thus displaced
to the left on 0.1MeV . The results of treatment of
peaks of power dependences of section of reaction are
driven to the Table 2, where next denotations are
used: χ2 – value of function of selection; R2 – coeffi-
cient of determination. As an approximating function
the Gauss function was chosen:
y = y0 +
A
w
√
π/2
e−(x−xc)
2/w2
, (13)
where A, w, xc - parameters of Gauss function.
Fig.3. Cross section of reaction 115In(γ, n)114mIn
(TALYS-1.4)
Consequently, the cross section of reaction is
expected by the Penfold-Leiss method, by the
Tikhonov’s method and in the package of TALYS -
1.4. The got results well coincide in a maximum.
Table 2. Results of treatment of peaks of section of reaction
Model χ2 R2 Area Emax Width Offset σmax
MeV mb
ldmodel 1 2.21121 0.98574 168.40 15.632 3.8456 1.7953 34.941
ldmodel 2 2.14085 0.98681 175.97 15.672 3.9378 1.7112 35.656
ldmodel 3 2.09347 0.98714 177.93 15.698 3.9794 1.6275 35.674
ldmodel 4 2.19684 0.98902 199.53 15.742 4.0273 2.1202 39.530
ldmodel 5 2.67697 0.98810 206.03 15.633 3.9095 2.1692 42.048
[11] 2.37070 0.98893 232.66 15.715 5.0989 1.6392 36.407
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ÌÅÒÎÄÛ ÐÀÑ×ÅÒÀ ÑÅ×ÅÍÈß ÐÅÀÊÖÈÈ 115In(γ, n)114mIn
Â.È. Æàáà, À.Ì. Ïàðëàã
Ñå÷åíèå ðåàêöèè 115In(γ, n)114mIn ðàññ÷èòàíî ðàçíûìè ìåòîäàìè. Ðåçóëüòàòû ïîëó÷åííîãî ñå÷åíèÿ
ìåòîäàìè Ïåíôîëüäà-Ëåéññà è Òèõîíîâà õîðîøî ñîãëàñóþòñÿ ìåæäó ñîáîé. Ðàñ÷åò ñå÷åíèÿ ìåòîäîì
Ïåíôîëüäà-Ëåéññà ïðîâåäåí ñî ñãëàæèâàíèåì ìåòîäîì èòåðàöèé. ×èñëî èòåðàöèé n = 1; 3; 5.  ïðî-
ãðàììíîì ïàêåòå TALYS-1.4 ïîëó÷åíû ñå÷åíèÿ äëÿ ïÿòè ìîäåëåé ïëîòíîñòè óðîâíåé. Òåîðåòè÷åñêèå è
ýêñïåðèìåíòàëüíûå ðåçóëüòàòû õîðîøî ñîâïàäàþò â ìàêñèìóìå.
ÌÅÒÎÄÈ ÐÎÇÐÀÕÓÍÊÓ ÏÅÐÅÐIÇÓ ÐÅÀÊÖI� 115In(γ, n)114mIn
Â.I. Æàáà, Î.Ì. Ïàðëàã
Ïåðåðiç ðåàêöi¨ 115In(γ, n)114mIn ðîçðàõîâàíî ðiçíèìè ìåòîäàìè. Ðåçóëüòàòè îòðèìàíîãî ïåðåðiçó ìå-
òîäàìè Ïåíôîëüäà-Ëåéññà i Òiõîíîâà äîáðå óçãîäæóþòüñÿ ìiæ ñîáîþ. Ðîçðàõóíîê ïåðåðiçó ìåòîäîì
Ïåíôîëüäà-Ëåéññà ïðîâåäåíî çi çãëàäæóâàííÿì ìåòîäîì iòåðàöié. ×èñëî iòåðàöié n = 1; 3; 5. Ó ïðî-
ãðàìíîìó ïàêåòi TALYS-1.4 îòðèìàíî ïåðåðiç äëÿ ï'ÿòè ìîäåëåé ãóñòèíè ðiâíiâ. Òåîðåòè÷íi òà åêñïå-
ðèìåíòàëüíi ðåçóëüòàòè äîáðå ñïiâïàäàþòü ó ìàêñèìóìi.
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