Середні резонансні параметри ядер Ni і Zn
Отримано повнi набори середнiх резонансних параметрiв S0, S1, R'0, R'1, S1,3/2 для ядер нiкелю i цинку з природним складом iзотопiв. Їх визначено з аналiзу середнiх експериментальних диференцiальних перерiзiв пружного розсiяння нейтронiв низьких енергiй розробленим авторами методом. Провед...
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Правдивий, М.М. Корж, І.О. Скляр, М.Т. 2010-11-05T13:28:11Z 2010-11-05T13:28:11Z 2010 Середні резонансні параметри ядер Ni і Zn / М.М. Правдивий, І.О. Корж, М.Т. Скляр // Укр. фіз. журн. — 2010. — Т. 55, № 2. — С. 170-174. — Бібліогр.: 11 назв. — укр. 2071-0194 PACS 21.60.Ev https://nasplib.isofts.kiev.ua/handle/123456789/13378 539.171.4 Отримано повнi набори середнiх резонансних параметрiв S0, S1, R'0, R'1, S1,3/2 для ядер нiкелю i цинку з природним складом iзотопiв. Їх визначено з аналiзу середнiх експериментальних диференцiальних перерiзiв пружного розсiяння нейтронiв низьких енергiй розробленим авторами методом. Проведено аналiз отриманих результатiв, рекомендованих параметрiв та деяких лiтературних даних, на основi якого зроблено висновок, що рекомендованi для обох ядер параметри S1 є заниженими у два рази. Получены полные наборы средних резонансных параметров S1, R'0, R'1, S1,3/2 для ядер никеля и цинка с естественным составом изотопов. Они определены из анализа средних экспериментальных дифференциальных сечений упругого рассеяния нейтронов низких энергий разработанным авторами методом. Проведен анализ полученных результатов, рекомендованных параметров и некоторых литературных данных, в результате которого сделан вывод, что рекомендованные для обоих ядер параметры S1 занижены в два раза. The complete sets of average resonance parameters S1, R'0, R'1, and S1,3/2 for nickel and zinc nuclei with a natural isotope composition have been obtained. They were determined by analyzing the average experimental differential cross-sections of low-energy neutron elastic scattering with the help of a method developed by the authors. The analysis of obtained results, recommended parameters, and some literary data has been carried out. The conclusion has been made that the recommended values of the parameter S1 are underestimated by a factor of two for both nuclei concerned. uk Відділення фізики і астрономії НАН України Ядра та ядерні реакції Середні резонансні параметри ядер Ni і Zn Средние резонансные параметры ядер Ni и Zn Average Resonance Parameters of Ni and Zn Nuclei Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Середні резонансні параметри ядер Ni і Zn |
| spellingShingle |
Середні резонансні параметри ядер Ni і Zn Правдивий, М.М. Корж, І.О. Скляр, М.Т. Ядра та ядерні реакції |
| title_short |
Середні резонансні параметри ядер Ni і Zn |
| title_full |
Середні резонансні параметри ядер Ni і Zn |
| title_fullStr |
Середні резонансні параметри ядер Ni і Zn |
| title_full_unstemmed |
Середні резонансні параметри ядер Ni і Zn |
| title_sort |
середні резонансні параметри ядер ni і zn |
| author |
Правдивий, М.М. Корж, І.О. Скляр, М.Т. |
| author_facet |
Правдивий, М.М. Корж, І.О. Скляр, М.Т. |
| topic |
Ядра та ядерні реакції |
| topic_facet |
Ядра та ядерні реакції |
| publishDate |
2010 |
| language |
Ukrainian |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Средние резонансные параметры ядер Ni и Zn Average Resonance Parameters of Ni and Zn Nuclei |
| description |
Отримано повнi набори середнiх резонансних параметрiв S0, S1, R'0, R'1, S1,3/2 для ядер нiкелю i цинку з природним складом iзотопiв. Їх визначено з аналiзу середнiх експериментальних диференцiальних перерiзiв пружного розсiяння нейтронiв низьких енергiй розробленим авторами методом. Проведено аналiз отриманих результатiв, рекомендованих параметрiв та деяких лiтературних даних, на основi якого зроблено висновок, що рекомендованi для обох ядер параметри S1 є заниженими у два рази.
Получены полные наборы средних резонансных параметров S1, R'0, R'1, S1,3/2 для ядер никеля и цинка с естественным составом изотопов. Они определены из анализа средних экспериментальных дифференциальных сечений упругого рассеяния нейтронов низких энергий разработанным авторами методом. Проведен анализ полученных результатов, рекомендованных параметров и некоторых литературных данных, в результате которого сделан вывод, что рекомендованные для обоих ядер параметры S1 занижены в два раза.
The complete sets of average resonance parameters S1, R'0, R'1, and S1,3/2 for nickel and zinc nuclei with a natural isotope composition have been obtained. They were determined by analyzing the average experimental differential cross-sections of low-energy neutron elastic scattering with the help of a method developed by the authors. The analysis of obtained results, recommended parameters, and some literary data has been carried out. The conclusion has been made that the recommended values of the parameter S1 are underestimated by a factor of two for both nuclei concerned.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13378 |
| citation_txt |
Середні резонансні параметри ядер Ni і Zn / М.М. Правдивий, І.О. Корж, М.Т. Скляр // Укр. фіз. журн. — 2010. — Т. 55, № 2. — С. 170-174. — Бібліогр.: 11 назв. — укр. |
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2025-11-26T03:09:12Z |
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2025-11-26T03:09:12Z |
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| fulltext |
M.M. PRAVDIVY, I.O. KORZH, M.T. SKLYAR
AVERAGE RESONANCE PARAMETERS OF Ni AND Zn
NUCLEI
M.M. PRAVDIVY, I.O. KORZH, M.T. SKLYAR
Institute for Nuclear Research, Nat. Acad. of Sci. of Ukraine
(47, Nauky Ave., Kyiv 03680, Ukraine; e-mail: sklyar@ kinr. kiev. ua )
PACS 21.60.Ev
c©2010
The complete sets of average resonance parameters S0, S1, R′
0, R′
1,
and S1,3/2 for nickel and zinc nuclei with a natural isotope compo-
sition have been obtained. They were determined by analyzing the
average experimental differential cross-sections of low-energy neu-
tron elastic scattering with the help of a method developed by the
authors. The analysis of obtained results, recommended parame-
ters, and some literary data has been carried out. The conclusion
has been made that the recommended values of the parameter S1
are underestimated by a factor of two for both nuclei concerned.
1. Introduction
Within the period of researches of average nuclear res-
onance parameters, five editions of the Atlas of recom-
mended resonance parameters have already been pub-
lished [1]. In comparison with the previous edition [2],
there appeared a lot of new experimental data, which al-
lowed one to put in order the dependences of parameters
S0, S1, and R′0 on the atomic weight A and to consid-
erably reduce discrepancies between the parameters of
certain nuclei and the results of calculations obtained
in the framework of the optical model. However, there
remained the unresolved problems concerning the min-
ima of parameters S0 (at 100 < A < 140) and S1 (at
A ≤ 70). In those ranges, the parameters of neighbor
nuclei are 5 to 10 times different, which contradicts the
ideology of the optical model and constrains its progress.
Isotopes of nickel and zinc are located in that A-range,
where the magnitudes of the strength function S1 for
many nuclei reveal considerable discrepancies with the
results of calculations carried out in the framework of the
optical model. For those nuclei, recommended are the
parameters S0, S1 and R′0, R′1 we used to calculate the
weighted average values for nuclei with natural isotope
compositions. Their comparison with the data of work
[3] has demonstrated that the values of the parameters
S0 and S1 are different by a factor of 2 to 5. In this
connection, we tested the agreement of those parameters
with experimental data and determined new parameter
sets; it was done with the help of a method developed
by us for the analysis of the differential cross-sections of
elastic low-energy neutron scattering. This method has
been successfully used earlier for the determination of
resonance parameters of even isotopes of cadmium and
tin [4], as well as some other nuclei.
2. Determination Technique for Average
Resonance Parameters
Neutron scattering by nuclei at energies up to about
450 keV mainly occurs at the orbital moments l = 0
and 1. In this case, the differential cross-sections of elas-
tic scattering can be expanded in a series of Legendre
polynomials as follows:
σel(µ) =
σel
4π
{1 + ω1P1(µ) + ω2P2(µ)}, (1)
where µ = cos θ, θ is the scattering angle, σel is the inte-
gral cross-section of elastic scattering, Pl are the Legen-
dre polynomials, ω1 and ω2 are the expansion coefficients
of the differential cross-sections. These coefficients are
referred to as the angular moments of the scattering in-
dicatrix, and they are equal to ωl = (2l+ 1)P̄l, where P̄l
are the Legendre polynomials averaged over the angles
with the weight of a differential scattering cross-section.
For even-even nuclei and provided that σt ≈ σel, we
obtained the following expressions for the expansion co-
efficients [5]:
ω1 =
6πλ2
σel
(1− η0Re − η1Re + η0Reη1Re + η0Imη1Im), (2)
ω2 =
2
σel
(σs1 + πλ2T1,3/2), (3)
where ηl = ηlRe + iηlIm are the diagonal elements of the
average scattering matrix, σs1 are the cross-sections of
potential neutron scattering with l = 1, and T1,3/2 are
the penetrability coefficients for l = 1 and j = 3/2.
In the optical model, the cross-sections σel consist
of the corresponding partial cross-sections of compound
170 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2
AVERAGE RESONANCE PARAMETERS OF Ni AND Zn NUCLEI
and potential neutron scattering, σel = σc0 +σc1 +σs0 +
σs1 which are expressed in terms of matrix elements ηl.
In the resonance theory, the average cross-sections also
consist of the corresponding cross-sections of resonance
and potential scatterings which are expressed, in turn,
in terms of average resonance parameters. In the case
of narrow resonances (Γ� D), the partial cross-sections
given by the optical model coincide with the correspond-
ing cross-sections obtained in the resonance theory [6].
This allows the matrix elements ηl to be expressed in
terms of resonance parameters. Thus, should the quan-
tities σel, ω1, and ω2 in Eqs. (1)–(3) be expressed in
terms of average resonance parameters, the fitting of
those quantities to their experimental values can be used
to determine the average resonance parameters S0, S1,
R′0, R′1, and S1,3/2 (they are fitting parameters). From
the relation S1 = 1/3(S1,1/2 + 2S1,3/2), the parameter
S1,1/2 can be found.
For carrying out the calculations, we used the cor-
responding program for a fitting on the basis of the χ2-
minimization method. Three quantities – σel, ω1, and ω2
– were fitted simultaneously, and the χ2-criterion could
be monitored for each quantity separately. The tech-
nique for the determination of average resonance param-
eters is explained in work [5] in detail.
3. Determination of Resonance Parameters and
Their Analysis
The complete sets of average resonance parameters S0,
S1, R′0, R′1, S1,3/2 for nickel and zinc nuclei with the
natural composition of isotopes were determined by fit-
ting the quantities σel, ω1, and ω2 to the corresponding
experimental values taken from work [7] (we carried out
the additional averaging of data at the beginning of the
energy range). The same data were used to make all
the fittings described below and to estimate the quality
of their description by means of the resonance parame-
ters given by other authors. To check the reliability of
the data of work [7] and to make a general evaluation of
experimental data in the energy range under investiga-
tion, we present our results together with the available
experimental data of other authors in the figures given
below.
For nickel and zinc isotopes, recommended are the pa-
rameters S0, S1, and R′0 [1] which were used to calcu-
late the weighted average values for nuclei with natural
isotope compositions. The obtained values were fixed
to determine, by fitting, the other parameters from the
complete set. In addition, we calculated the quantities
σel, ω1, and ω2 using the parameter sets taken from work
Fig. 1. Energy dependences of the quantities σel, ω1, and ω2
for a 28Ni nucleus. Symbols correspond to experimental data,
curves are the results of calculations with various sets of resonance
parameters (see the text)
[3] which were obtained from the same experimental data
of work [7], but another method was applied. In every
case, the description quality of experimental data was
examined by the χ2-value and visually in the plots.
28Ni. In Fig. 1, the experimental energy dependences
of the quantities σel, ω1, and ω2 obtained in works [7–10]
for nickel nuclei are given. To improve the presentation
of the σel-dependence, the first and third points of work
[7] were reduced by a factor of two, and the second one
by a factor of four. It should be noted that the resonance
structures of the total cross-sections only start to reveal
themselves in the studied energy range [11], and the res-
onances in the cross-sections σel reported in work [7]
manifest themselves at energies of about 15 and 65 keV,
which testifies to their insufficient averaging. In addi-
tion, the authors noted that the cross-sections measured
below an energy of about 80 keV are underestimated
owing to a substantial resonance self-shielding. The fig-
ure demonstrates that there are the appreciable discrep-
ancies between experimental data obtained by various
authors; they are especially considerable for the quanti-
ties ω1 and ω2. The curves in the figure correspond to
the results of calculations with the use of different sets
of resonance parameters. Curves 2 exhibit the depen-
dences for σel, ω1, and ω2 calculated with the following
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2 171
M.M. PRAVDIVY, I.O. KORZH, M.T. SKLYAR
Fig. 2. Energy dependence of the quantities σel, ω1, and ω2 for a
nucleus 30Zn
set of parameters taken from work [3]: S0 = 1.40(30),
S1 = 2.49(52), R′0 = 5.90(49), R′1 = −0.11(50), and
S1,3/2 = 1.44(23) (hereafter, the parameters Sl and R′l
are given in units of 10−4 and fm, respectively, and the
numbers in parentheses indicate the corresponding er-
ror). From the figure, one can see that only the experi-
mental data for ω1 are described satisfactorily, whereas
the other data demonstrate considerable discrepancies.
In particular, the calculated cross-sections are less than
the experimental ones at the beginning of the energy
range, which can be explained by a small value of the
parameter S0.
The following average values are recommended for
nickel parameters: S0 = 3.59, S1 = 0.50, and R′0 = 6.49
[1]. Using the fitting procedure, we determined the other
parameters: R′1 = 3.79 and S1,3/2 = 1.52. The de-
pendences for the quantities σel, ω1, and ω2 calculated
with this set of parameters are presented in the figure
by curves 3. It is evident that the description of exper-
imental data is satisfactory on the whole, being better
than that for the first set. However, from the relation
S1 = (S1,1/2 +2S1,3/2)/3, it follows that S1,1/2 = −1.54.
According to formula (3), the description of the quan-
tity ω2 is mainly determined by the parameters R′1 and
S1,3/2. The contribution of the parameter R′1, owing
to its smallness, is appreciable only at energies higher
than 200 keV. Practically, the parameter S1,3/2 is deter-
mined unambiguously from the experimental ω2-values,
and the figure testifies that its value is optimal in this
case. Therefore, the reason for why the value of the pa-
rameter S1,1/2 is unphysical is a small magnitude of the
parameter S1. To be convinced of that, we carried out
additional calculations. Provided that S1,1/2 = 0 and
S1,3/2 = 1.52, the value of S1 cannot be less than 1.01.
For S1 = 0.50 and S1,1/2 = 0, we have S1,3/2 = 0.75.
When the parameters S0 = 3.59, S1 = 0.50, R′1 = 6.49,
and S1,3/2 = 0.75 were fixed, the fitting procedure gave
rise to R′1 = 4.03. The calculated values for ω2 are de-
picted in the figure by curve 4 (the dependences for σel
and ω1 are close to curves 3 ). One can see that they
describe neither the data of work [7] nor the data of
work [8, 9], the energy dependence of which are abnor-
mal and cannot be described at all using resonance pa-
rameters. Therefore, there are reasons to consider the
recommended value for the parameter S1 as underesti-
mated.
The insufficient averaging of the data of work [7] and
the underestimation of cross-sections at the beginning
of the energy range due to the resonance self-shielding
result in a mutual disagreement among the quantities
σel, ω1, and ω2, which revealed itself in considerable χ2-
values at their description, especially in those for the
cross-sections. Under such conditions, χ2 is not a reli-
able criterion for the quality of a description of experi-
mental data, so that the determination of resonance pa-
rameters by the automatic fitting and using those data
cannot produce the reliable results as well. This is ev-
idenced by the results of work [3], in which the values
of the parameters S0 and S1 are several times differ-
ent from those recommended for this A-region [1], and
the description of experimental data is rather doubtful.
Therefore, we determined the resonance parameters by
stage-by-stage calculations of σel, ω1, and ω2 with se-
quential variations of each parameter until the optimal
description of experimental data was achieved. The de-
scription was estimated both visually (in the plots) and
by the χ2-criterion. As a result, the following values were
obtained: S0 = 3.80, S1 = 0.95, R′0 = 6.25, R′1 = 3.36,
and S1,3/2 = 1.40. The calculated values for σel, ω1,
and ω2 are presented by curves 1 in the figure. Accord-
ing to the χ2-criterion, our description of experiments
was better than that with the parameters of work [3].
Our parameters agree well with their dependences on
the atomic weight A, except the parameter S1, the value
of which, however, agrees with new data for neighbor
nuclei [1, 2].
30Zn. In Fig. 2, the experimental energy dependences
of σel, ω1, and ω2 obtained in works [7, 8, 10] are de-
172 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2
AVERAGE RESONANCE PARAMETERS OF Ni AND Zn NUCLEI
Average resonance parameters for Ni and Zn nuclei
Nucleus S0×104 S1×104 R′
0, fm R′
1, fm S1,1/2×104 S1,3/2×104
28Ni 3.80(35) 0.95(26) 6.25(26) 3.36(75) 0.05(1.0) 1.40(32)
30Zn 1.80(25) 0.95(20) 6.50(25) 1.27(90) 0.05(0.9) 1.40(35)
picted. To improve the presentation of cross-sections
σel in the figure, the second and third points taken from
work [7] were reduced by a factor of two. The authors in-
dicated that the cross-sections measured below of about
50 keV were underestimated owing to the resonance self-
shielding. From the figure, one can see that the appre-
ciable discrepancies between the ω1- and ω2-values ob-
tained at the end of the energy range in various works
are observed. The curves in the figure correspond to the
results of calculations with different sets of resonance
parameters.
Curves 2 in the figure correspond to the results of cal-
culations with the parameters taken from work [3]: S0 =
1.15(9), S1 = 1.42(22), R′0 = 6.73(12), R′1 = 0.75(35),
and S1,3/2 = 1.23(15). Among all the data, only the
quantity ω1 is well described. The other data demon-
strate appreciable discrepancies with experiment.
For zinc nuclei, the recommended average parameters
are S0 = 2.00 and S1 = 0.58. By fitting, the other pa-
rameters were determined: R′0 = 6.35, R′1 = 2.11, and
S1,3/2 = 1.34. The calculated quantities σel, ω1, and ω2
are presented in the figure by curves 3. One can see that
the cross-sections σel at energies above 150 keV are ap-
preciably smaller than the experimental ones, whereas
the quantities ω1 and ω2 are described satisfactorily.
However, as it was for nickel nuclei, there appears an is-
sue concerning the value of the parameter S1. From the
obtained parameter set, it follows that S1,1/2 = −0.94.
If S1 = 0.58 and S1,1/2 = 0, the maximal possible value
is S1,3/2 = 0.87. Having fixed the parameters S0 = 2.00,
S1 = 0.58, and S1,3/2 = 0.87, we obtained, by fitting, the
following other parameters: R′0 = 6.35 and R′1 = 2.18.
The description of the quantities σel and ω1 remained
practically unchanged. The calculated values of ω2 are
shown by curve 4 in the figure. One can see that the
description of ω2 became considerably worse, both visu-
ally and according to the χ2-value. This means that the
value S1,3/2 = 0.87 is too small to provide a satisfac-
tory description of ω2-values, so that a conclusion can
be drawn that the recommended value of the parameter
S1 is underestimated.
We determined a new set of resonance parameters in
the same way as it was done for nickel nuclei. As a result,
the following parameters were obtained: S0 = 1.80, S1 =
0.95, R′0 = 6.50, R′1 = 1.27, and S1,3/2 = 1.40. The
results of calculations are shown in the figure by curves
1. One can see that the description of experimental data
became appreciably better in comparison with that for
the previous sets, both visually and according to the χ2-
value. The obtained parameters S0 and R′0 agree well
with their dependences on A [1], but the parameter S1,
as it was for nickel, is almost twice as large. The results
obtained for both nuclei are listed in Table.
Hence, the values of the parameter S0 obtained by us
for both nuclei confirmed the recommended values, but
not those given in work [3]. The values obtained for the
parameter S1 confirmed neither the recommended val-
ues nor those reported in work [3]. The recommended
value of the parameter R′0 for nickel nuclei was con-
firmed. From the recommended values of the parameter
R′0 for isotopes 64Zn and 66Zn (the latter amounts to
77% of a natural mixture), the weighted average value
R′0 = 5.58 was obtained, which is much less that the rec-
ommended values for this A-region [1]. By fixing this R′0-
value and the recommended values for the parameters S0
and S1, we determined, by fitting, the other parameters:
R′1 = 2.41 and S1,3/2 = 1.31. The difficulties associated
with the magnitude of the parameter S1 remained in the
obtained parameter set; moreover, it does not describe
experimental cross-sections at all, because the calculated
cross-sections are much less than the experimental ones.
The values of R′1 and S1,3/2 taken from three parame-
ter sets are mutually agreed, in general, within the error
limits.
4. Conclusions
In this work, by analyzing the experimental differential
cross-sections for low-energy neutron elastic scattering,
new complete sets of average resonance parameters S0,
S1, R′0, R′1, and S1,3/2 for nickel and zinc nuclei with the
natural isotope composition have been determined. In
general, the resonance parameters characterize specific
nuclei, being determined for nuclear isotopes. However,
data for the natural isotope mixture are also very im-
portant, in particular, for the specification of parameter
dependences on the atomic weight A and for the im-
provement of theoretical calculations in the framework
of the optical model. As a result of the analysis made,
the recommended values for the resonance parameter S1
were found to be underestimated by a factor of about two
for both nuclei. At last, it was proved that the available
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2 173
M.M. PRAVDIVY, I.O. KORZH, M.T. SKLYAR
experimental data can be described better without dras-
tic and unjustified variations of the parameters S0 and
S1, as it was done in work [3].
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nance Parameters and Thermal Cross Sections. Z = 1–
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and G.S. Samosvat, preprint JINR P3-85-133 (JINR,
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Received 24.06.09.
Translated from Ukrainian by O.I. Voitenko
СЕРЕДНI РЕЗОНАНСНI ПАРАМЕТРИ ЯДЕР Ni I Zn
М.М. Правдивий, I.О. Корж, М.Т. Скляр
Р е з ю м е
Отримано повнi набори середнiх резонансних параметрiв
S0, S1, R′
0, R
′
1, S1,3/2 для ядер нiкелю i цинку з природним
складом iзотопiв. Їх визначено з аналiзу середнiх експеримен-
тальних диференцiальних перерiзiв пружного розсiяння ней-
тронiв низьких енергiй розробленим авторами методом. Про-
ведено аналiз отриманих результатiв, рекомендованих параме-
трiв та деяких лiтературних даних, на основi якого зроблено
висновок, що рекомендованi для обох ядер параметри S1 є за-
ниженими у два рази.
174 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2
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