About absoluteness of data on elastic electron scattering with ¹²C nucleus
The results obtained in Mainz in 1982 year were check up. The analysis of data from this work was made at momentum transfer range q = 0.25 - 0.75 fm⁻¹ using the model independent form factor (the expansion of form factor in a power series of q²) and the form factor corresponding to the distribution...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2007 |
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| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | About absoluteness of data on elastic electron scattering with ¹²C nucleus / A.Yu. Buki, I.S. Timchenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 45-47. — Бібліогр.: 8 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859846518879551488 |
|---|---|
| author | Buki, A.Y. Timchenko, I.S. |
| author_facet | Buki, A.Y. Timchenko, I.S. |
| citation_txt | About absoluteness of data on elastic electron scattering with ¹²C nucleus / A.Yu. Buki, I.S. Timchenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 45-47. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The results obtained in Mainz in 1982 year were check up. The analysis of data from this work was made at momentum transfer range q = 0.25 - 0.75 fm⁻¹ using the model independent form factor (the expansion of form factor in a power series of q²) and the form factor corresponding to the distribution of charge density in the shell model framework. We found a 3% systematical overestimation in Mainz data
Переглядаються результати вимірювань пружного розсіяння електронів на ядрі ¹²C, виконані в Майнці (1982 р). Аналіз даних цієї роботи проводиться в діапазоні переданих імпульсів q=0.25 - 0.75 фм⁻¹ з розкладанням форм-фактора по степенях q², а також з використанням оболонкової моделі розподілу щільності заряду в ядрі. Знайдено 3% систематичне завищення результатів вимірювань даної роботи
Пересматриваются результаты измерений упругого рассеяния электронов на ядре ¹²C, выполненные в Майнце (1982 г). Анализ данных этой работы проводится в диапазоне переданных импульсов q = 0.25 - 0.75 фм⁻¹ с разложением форм-фактора по степеням q², а также с использованием оболочечной модели распределения плотности заряда в ядре. Найдено 3% систематическое завышение результатов измерений рассматриваемой работы.
|
| first_indexed | 2025-12-07T15:39:08Z |
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| fulltext |
ABOUT ABSOLUTENESS OF DATA ON ELASTIC
ELECTRON SCATTERING WITH 12C NUCLEUS
A. Yu. Buki∗, I.S. Timchenko
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received April 16, 2007)
The results obtained in Mainz in 1982 year were check up. The analysis of data from this work was made at
momentum transfer range q = 0.25− 0.75 fm−1 using the model independent form factor (the expansion of form
factor in a power series of q2) and the form factor corresponding to the distribution of charge density in the shell
model framework. We found a 3% systematical overestimation in Mainz data.
PACS: 13.85.Dz
1. INTRODUCTION
The results of electronuclear experiments are usu-
ally brought to absolute values by means of their
normalization using especially precise (master) data
from elastic electron scattering. These data are ob-
tained from elastic electron scattering with 12C or 1H
nuclei, and, sometimes, with 4He nucleus. During the
experiment for the purpose of normalization in addi-
tion to measurements with the nucleus under study
we also measure elastic electron scattering cross sec-
tions of one of the nuclei, for which we possess refer-
ence data. The obtained cross sections are reduced to
the nucleus ground state form factor values Fel(qi).
Using the found values Fel(qi) we calculate the nor-
malization factor
Ki =
F 2
el,0(qi)
F 2
el(qi)
, (1)
where Fel,0(qi) is the reference form factor; qi is the
momentum transferred to the nucleus.
The importance of the reference form factor preci-
sion in the processing of experimental data was shown
in work [1] performed in Darmstadt. Earlier the root
mean square radii for 4He to 209Bi nuclei (24 nuclei
in all), normalized to measurements with 12C nucleus
from [2, 3], were obtained in this laboratory. In view
of uncertainties about the precision of data from [2, 3]
new measurements of elastic electron scattering cross
section of 12C nucleus were carried out in Darmstadt.
Using this result the renormalization for all available
data was performed and the revised values of charge
radii were obtained.
The latest and, obviously, the most precise work
on elastic electron scattering with 12C nucleus was
carried out in Mainz lab [4]. These data were used in
the processing of our measurements results. However,
a question about the probability of a systematical er-
ror in 12C nucleus data from ref.[4] has arisen. The
present paper is dedicated to the study of the above
mentioned problem.
2. DATA ANALYSIS
In Mainz lab the elastic electron scattering cross
section measurements of 12C nucleus were carried
out at q = 0.25− 2.75 fm−1. However, the measure-
ments, which the authors of this work consider abso-
lute, were made at q = 0.25− 0.75 fm−1 (the rest of
the measurement results were relative and standard-
ized to these data). Below we shall only analyze data
from the momentum transfer range q ≤ 0.75fm−1.
The data table of electron initial energies E0,
scattering angles θ and elastic electron scattering
cross sections dσ/dΩ measured on 12C nucleus can
be found in ref.[4]. To use the results of this work
for the normalization procedure, it is necessary to
find the squared form factor of nucleus ground state
F 2
el,0(qi) at different momenta transfer qi. For this
purpose:
- (a) let us transform E0, θ and dσ/dΩ values to
the corresponding values of F 2
el,0(qi) and qi;
-(b) let us select the analytical function F 2
th(q),
which will approximate the obtained F 2
el,0(qi) in the
momentum transfer range we are interested in. This
is necessary to avoid measuring the form factors Fel
at the same qi value as reference form factors Fel,0(qi)
during normalization using eq.(1).
Let us transform the E0, θ and dσ/dΩ values to
the values of F 2
el,0(qi) and qi, using well-known for-
mulas
F 2
el,0 =
dσ/dΩ
σMott
, (2)
q =
2E0
h̄c
· sin(θ/2)√
η
· ξ, (3)
where
σMott =
(
Ze2
2E0
)2
· cos2(θ/2)
η · sin4(θ/2)
∗Corresponding author. E-mail address: abuki@ukr.net.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2007, N5.
Series: Nuclear Physics Investigations (48), p.45-47.
45
is the scattering cross section on the nucleus with the
charge Z;
η = 1 +
2E0sin
2(θ/2)
M
is the kinematical correction, M is the nucleus mass;
ξ = 1 +
3
2
· Ze2
√
5/3 ·Rrms · E0
is the correction, which takes into account the in-
fluence of the nucleus coulomb field on the incoming
electron. Note that the formulas shown here are from
ref.[4].
For approximation of the obtained values F 2
el,0(qi)
we use simple presentations of nucleus ground state
form factor F 2
th(q). As known, some of these presen-
tations describe the data at small momenta transfer
[5] well enough and allow to obtain the values of the
root mean square radius Rrms with fairly good preci-
sion. Such is the expansion of form factor in a power
series of q2, which is
F 2
th(q) = 1− 1
3
· a · q2 +
1
60
· b · q4, (4)
as well as the form factor of the nucleus ground state
corresponding to the distribution of charge density in
the shell model framework. For 12C nucleus this form
factor can be expressed as follows [6]
F 2
th(q) =
(
1− c2 · q2
9
)2
· exp
(
−d2 · q2
2
)
. (5)
Here a, b, c and d are parameters of fitting related
to the root mean square radius: in the case of the
form factor expansion in a power series of q2 (eq.(4))
Rrms =
√
a, and for the form factor with the distri-
bution of charge density in the shell model framework
(eq.(5)) Rrms =
√
2
3c2 + 3
2d2.
By definition
lim
q→0
F 2
el(q) = 1.
This approach was used in some of the first ee′-
scattering works and in works with especially dif-
ficult conditions of measurements (for instance, the
measurements of electron scattering on 3H nuclei im-
planted in titanium base [7]). Thus, a variable mul-
tiplier k was introduced in analytic presentation of
form factor which is fit to elastic electron scattering
data. The k value which was obtained as a result of
the fitting is precisely the normalization factor for ab-
solutization of measured data. Using this experience,
we shall write the expression for the fitting function
as
F 2(q) = k · F 2
th(q). (6)
If there is no systematic deviation in the data un-
der study, it is possible to assume the variable factor
k = 1.0. Also, it is possible to leave the k factor as a
variable parameter, however in this case we have to
obtain its value close to 1.0 within the limits of the
parameter errors.
The example of fitting eq.(4) to Mainz data with
and without eq.(6) is shown in Fig.1. The statistical
precision of the data is 0.45% − 0.49% therefore the
errors boundaries aren’t visible in the figure. The re-
sults of fitting the equations (4,5,6) to these data are
shown in table.
Since the value of the parameter k appeared
to be different from 1.0 approximately by 10 stan-
dard deviations, it is necessary to check whether
the obtained result is dependent on the analy-
sis conditions chosen. There are 16 experimental
points in the examined momentum transfer range,
and among them there are two points for each of
q = 0.25; 0.35; 0.45; 0.55; 0.74 fm−1.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0
0.2
0.4
0.6
0.8
1.0
F2
el,0
q, fm-1
Fig.1. The squared form factor of 12C nucleus
ground state. The closed circles are the values
obtained from the data of ref. [4]; solid line is the
fitting of eq.(4) with variable parameter k to these
data; dashed is the same fitting with the fixed k = 1.0
The result of fittings
k∗) Rrms χ2
i k Rrms χ2
i
power series in q2 1.0 3.07± 0.05 962.0 1.026± 0.003 2.42± 0.01 0.71
shell model 1.0 2.33± 0.03 5.0 1.029± 0.003 2.45± 0.05 0.75
∗) The analysis with fixed value k = 1.0 is shown in the left part of the table.
To verify whether the dependence of the ob-
tained result on the selection of fitting range is
possible, we made a number of fittings: 1 –
all 16 points at q = 0.25− 0.75 fm−1; 2 – 13
points at q = 0.35− 0.75 fm−1; 3 – 8 points
at q = 0.25− 0.45 fm−1 and 4 – 8 points at
q = 0.50− 0.75 fm−1. The results of this analysis
are shown in Fig.2.
3. DISCUSSION AND CONCLUSIONS
First of all, it is necessary to note that in the case
of the fitting with the fixed value k = 1.0 we ob-
46
tained the improper χ2
i (χ2 per degree of freedom),
while in the case of the fitting with variable param-
eter k, χ2
i ≈ 0.7 (see table 1). As to the obtained
values Rrms, within the limits of errors the identical
values of this magnitude were found for two different
presentations of form factor (eq.(4) and eq.(5)) and
variable k. The values Rrms obtained in this case are
close to 2.456 – the value of the root mean square ra-
dius of 12C nucleus (this value is the weighted mean
of the results from a series of works [4, 8]). In case
k being fixed, there is considerable discrepancy in
the values of Rrms. Figure 2 shows that the val-
ues of variable multiplier k and Rrms which is ob-
tained in this case within the limits of its errors
does not depend on the selection of the fitting range.
1 2 3 4
2.3
2.4
2.5
2.6
2.7
Rrms
number of fitting
a)
b)
1 2 3 4
1.00
1.01
1.02
1.03
1.04
1.05
1.06
k
Fig.2. The results of the fittings of eq.(6) with
using eq.(4) (open circles) and with using eq.(5)
(close circles) to the different ranges of data. The
horizontal scale the represents numbers of the fitting
variants (see text). a) k is the normalization factor;
b) Rrms is the root mean square radius
Thus, the application of traditional methods of
the data processing to measured results of ref. [4]
gives the acceptable values of χ2
i and Rrms. How-
ever, this results in the discrepancy in (2.6 − 2.9)%
with the normalization of Mainz data (the absolute
precision of the data is 0.4%, according to ref. [4]).
We consider that this discrepancy in the normaliza-
tion should be taken into account using the data of
work [4] as master data.
REFERENCES
1. G. Fey, H. Frank, W. Schutz and H. Theissen.
Nuclear Rms Charge Radii from Relative Elec-
tron Scattering Measurements at Low Energies
// Z. Phys. 1973, v.265, p.401-403.
2. H.A. Bentz. Kernradien von 12C, 13C, 14N und
16O aus Electronenstreuung zwischen 30 und
60 MeV // Z. Phys. 1971, v.243, p.138-153.
3. R. Endfer, D. Turck. Measurement of the rms
Radius of 12C by Elastic Electron Scattering at
53 MeV // Z. Phys. 1967, v.205, p.90-95.
4. W. Reuter, G. Fricke, K. Merle and H. Miska.
Nuclear charge distribution and rms radii of 12C
from absolute electron scattering measurements
// Phys. Rev. 1982, C26, p.806-818.
5. H. Uberal. Electron scattering from complex nu-
clei. New York: ”Academic Press”, 1971, Part A
- 467p.; Part B - 867p.
6. I.S. Gulkarov. Nucleus research by electrons.
Moscow: ”Atomizdat”, 1977, 208p. (in Russian).
7. D.H. Beck, S.B. Kovalski et al. Tritium form fac-
tors at low q // Phys. Rev. 1984, C30, p.1403-
1408.
8. R. Barret, D. Jackson. Nuclear sizes and struc-
ture. Kiev: ”Naukova dumka”, 1981, 420p. (in
Russian).
ОБ АБСОЛЮТНОСТИ ДАННЫХ УПРУГОГО РАССЕЯНИЯ
ЭЛЕКТРОНОВ НА ЯДРЕ 12С
А.Ю. Буки, И.С. Тимченко
Пересматриваются результаты измерений упругого рассеяния электронов на ядре 12С, выполнен-
ные в Майнце (1982 г.). Анализ данных этой работы проводится в диапазоне переданных импульсов
q = 0.25− 0.75 фм−1 с разложением форм-фактора по степеням q2, а также с использованием обо-
лочечной модели распределения плотности заряда в ядре. Найдено 3% систематическое завышение
результатов измерений рассматриваемой работы.
ПРО АБСОЛЮТНIСТЬ ДАНИХ ПРУЖНОГО РОЗСIЯННЯ
ЕЛЕКТРОНIВ НА ЯДРI 12С
О.Ю. Буки, I.С. Тiмченко
Переглядаються результати вимiрювань пружного розсiяння електронiв на ядрi 12С, що були ви-
конанi в Майнцi (1982 р.). Аналiз даних цiєї роботи проводиться в дiапазонi переданих iмпульсiв
q = 0.25− 0.75 фм−1 з розкладанням форм-фактора по степенях q2, а також з використанням обо-
лонкової моделi розподiлу щiльностi заряду в ядрi. Знайдено 3% систематичне завищення результатiв
вимiрювань даної роботи.
47
|
| id | nasplib_isofts_kiev_ua-123456789-110164 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:39:08Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buki, A.Y. Timchenko, I.S. 2016-12-31T06:50:59Z 2016-12-31T06:50:59Z 2007 About absoluteness of data on elastic electron scattering with ¹²C nucleus / A.Yu. Buki, I.S. Timchenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 45-47. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 13.85.Dz https://nasplib.isofts.kiev.ua/handle/123456789/110164 The results obtained in Mainz in 1982 year were check up. The analysis of data from this work was made at momentum transfer range q = 0.25 - 0.75 fm⁻¹ using the model independent form factor (the expansion of form factor in a power series of q²) and the form factor corresponding to the distribution of charge density in the shell model framework. We found a 3% systematical overestimation in Mainz data Переглядаються результати вимірювань пружного розсіяння електронів на ядрі ¹²C, виконані в Майнці (1982 р). Аналіз даних цієї роботи проводиться в діапазоні переданих імпульсів q=0.25 - 0.75 фм⁻¹ з розкладанням форм-фактора по степенях q², а також з використанням оболонкової моделі розподілу щільності заряду в ядрі. Знайдено 3% систематичне завищення результатів вимірювань даної роботи Пересматриваются результаты измерений упругого рассеяния электронов на ядре ¹²C, выполненные в Майнце (1982 г). Анализ данных этой работы проводится в диапазоне переданных импульсов q = 0.25 - 0.75 фм⁻¹ с разложением форм-фактора по степеням q², а также с использованием оболочечной модели распределения плотности заряда в ядре. Найдено 3% систематическое завышение результатов измерений рассматриваемой работы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядерная физика и элементарные частицы About absoluteness of data on elastic electron scattering with ¹²C nucleus Про абсолютність даних пружного розсіяння електронів на ядрі ¹²C Об абсолютности данных упругого рассеяния электронов на ядре ¹²C Article published earlier |
| spellingShingle | About absoluteness of data on elastic electron scattering with ¹²C nucleus Buki, A.Y. Timchenko, I.S. Ядерная физика и элементарные частицы |
| title | About absoluteness of data on elastic electron scattering with ¹²C nucleus |
| title_alt | Про абсолютність даних пружного розсіяння електронів на ядрі ¹²C Об абсолютности данных упругого рассеяния электронов на ядре ¹²C |
| title_full | About absoluteness of data on elastic electron scattering with ¹²C nucleus |
| title_fullStr | About absoluteness of data on elastic electron scattering with ¹²C nucleus |
| title_full_unstemmed | About absoluteness of data on elastic electron scattering with ¹²C nucleus |
| title_short | About absoluteness of data on elastic electron scattering with ¹²C nucleus |
| title_sort | about absoluteness of data on elastic electron scattering with ¹²c nucleus |
| topic | Ядерная физика и элементарные частицы |
| topic_facet | Ядерная физика и элементарные частицы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110164 |
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