Response function extrapolation of ²H nucleus in the region of high transfer energy
The experimental response functions of the ²H nucleus are extrapolated along the w-region by the power function for q=1.05 fm⁻¹.
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| Date: | 2000 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Response function extrapolation of ²H nucleus in the region of high transfer energy / A.Yu. Buki, I.A. Nenko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 13-15. — Бібліогр.: 9 назв. — англ. |
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| author | Buki, A.Yu. Nenko, I.A. |
| author_facet | Buki, A.Yu. Nenko, I.A. |
| citation_txt | Response function extrapolation of ²H nucleus in the region of high transfer energy / A.Yu. Buki, I.A. Nenko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 13-15. — Бібліогр.: 9 назв. — англ. |
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| description | The experimental response functions of the ²H nucleus are extrapolated along the w-region by the power function for q=1.05 fm⁻¹.
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| first_indexed | 2025-12-07T18:52:13Z |
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RESPONSE FUNCTION EXTRAPOLATION OF 2H NUCLEUS IN THE
REGION OF HIGH TRANSFER ENERGY
A.Yu. Buki, I.A. Nenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The experimental response functions of the 2H nucleus are extrapolated along the ω-region by the power
function for q=1.05 fm−1.
PACS: 25.30.Fj, 27.10.+h
1. In the modern e,e'-experiment measured are the
transversal (RT(q,ω)) and longitudinal (RL(q,ω)) response
functions of the atomic nuclei (e.g., see [1]). The mo-
ments of these functions are calculated by sum rules,
being one of the most model-independent theoretical
methods.
By definition the moment of a response function is
∫
∞
=
0
k)k( ωω)ω,()( dqRqS , (1)
where k is the moment number, q and ω are the 3-
momentum and the energy transfer to the nucleus, re-
spectively. But values of the RT-function (Re
T) and the RL-
function (Re
L) practically are measured as high as ω≈
1/2q. Therefore, for determination of the moments value,
it is necessary to extrapolate the experimental data along
ω by some function Rt(q,ω).
The problem of this extrapolation is very important.
This can be seen from the fact that the behaviour of the
Rt-function determines the maximum number, kmax
corresponding to the finite moment value. In this
connection it should be noted that G. Orlandini and
M. Traini (paper [1], p.275) classified the kmax problem as
one of main crucial points until clarifying of which “the
sum rules risk remaining of only academic interest”.
2. In this paper the Rt-function of 2H nucleus is
studied. We consider as such function the expression in
the form
Rq
t (ω)=Cq ω−α, (2)
suggested for the RL-function in the theoretical papers
[1], [3], [4] and for the RT-function in [3]. Here α is the
parameter and Cq for q = const is the parameter too. For
determination their values we used the Re
L- and Re
T-data
of the 2H nucleus for q = 1.05 fm−1 (Fig. 1) produced at
the linear accelerator LUE-300 KhFTI [2]. Since it is not
clear from what value of the energy transfer the expres-
sion (2) starts to correspond to the behaviour of the
experimental data, then at first for the fitting we used a
more general expression proposed by V.D. Efros in [5]:
∑
=
−−−=
N
n
n
nq
t CqR
1
)1(
,
α ωω)ω,( . (3)
The fittings of this expression to different groups of the
experimental data demonstrated that the magnitude N = 1
is sufficient both for the Re
L- and the Re
T-data in the
region of ω ≥ 34 MeV. That is the experimental data of
the 2H nucleus reduce the multiparameter expression (3)
to the form (2) beginning from the ω value
corresponding to the quasi-elastic peak half-altitude.
Note that such extrapolation function simplification is
not rule. So, in the case of studying of the 4He nucleus
Rt
T-function the similar analysis required N = 3 [6].
Fig. 1. Experimental value of the 2H nucleus
response functions: ( ° ) is RL(q,ω), ( • ) is RT(q,ω). The
curve demonstrates the fitting result of the expression
(2) with the Re
L-data for ω ≥ 34 MeV .
We examine a dependence of the parameter αL on the
choice of lower (ωmin) and upper (ωmax) limits of the ω
range, where the expression (2) is fitted with the Re
L-data.
The variations of ωmax have shown that αL and ∆αL
values are unchangeable for different ωmax > 70 MeV. It
is explained by a low relative accuracy of the
experimental data in this region of the energy transfer.
Fig. 2 shows αL, ∆αL and χ i
2 values as functions of
ωmin for ωmax = 100 MeV. One can see that χ i
2 quickly
increases with decreasing ωmin, beginning from 34 MeV,
where the expression (2) is not able to approximate the
experimental data (see Fig. 1). The function αL(ωmin) = a
+ bωmin was fitted with αL ± ∆αL value defined on
different ωmin from the interval ωmin = 34 − 56 MeV. The
fitting result is b = −0,0006 ± 0,0060 MeV-1 (Fig.2). The
inequality |∆b| > b shows that the dependence of αL
on ωmin is not observable (at least the linear dependence)
in the mentioned interval. From here we can conclude
that the expression (2) with the parameter αL = const
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, № 2.
Серия: Ядерно-физические исследования (36), с. 13-15.
13
describes the response function adequately, and
evidences the independence (in limits of the statistical
errors ∆sαL) of αL value obtained on the choice of ωmin.
Fig. 2. The parameter αL ( ° ) and χ i
2 (its values ( • )
are given in arbitrary units) are shown as functions of ω
min, being is the lower limit of the fitting region of the
expression (2) with the Re
L-data. The solid line
demonstrates the fitting result of the linear function with
αL value.
Analysis of influence on αL value of the errors of the
cross-section normalization, the background-correction
and the radiation-correction shows that the statistical
error ∆sαL makes a main contribution to the ∆αL value
and specifies it almost completely.
The solution of the problem of the approximating the
Re
T-data by the expression (2) is slightly differed
(methodologically as well as by results) from the above-
mentioned case of the longitudinal response function and
therefore it is not presented. Also, we have considered
the RT-function in the form proposed in paper [4]:
Rt
T,q (ω) = Cq ω−β eν ω. (4)
However, the fitting of expression (4) with Re
T-data for
the different ωmin gives a wide spread in values of the
parameters: β = 1.5÷2.6; ν=−0.02 ÷ −0.10 MeV−1. The
more detail examination of the expression (4) apparently
requires the using of the additional experimental data.
With regard to the expression (2), one can conclude
that it describes adequately the used Re
L- and Re
T-data
with the values of parameters:
αL = 2.82±0.07; αT = 2.93±0.15.
3. Let us extract from the equation (1) the part where
R(q,ω) = Rt(q,ω) and write down it for the case of q =
const as
St
q
(k)(ωx) = ∫∞→
f
x
kt
q
f
ω
ω
ωω)ω(
ω
lim dR , (5)
where ωx is the lower limit of the interval in which the Rt-
function is determined. The substitution of Eq.(2) into
integral (5) transforms one to
St
q
(k)(ωx) =
++
→ ∞
−++
−−
k1α-
fω
ω
k1α-
xω
1α
q
f
lim
k
C
. (6)
We can see that St
q
(k)(ωx), and therefore the total integral
Sq
(k), are converged if k is less then ent(α) i.e. integer
part of α value:
kmax =
=−
≠−
)α(αif2α
)α(αif1)α(
ent
entent
(7)
According to Eq. (7) the derived α values
correspond to kmax = 1.
4. The extrapolation functions in the light of sum
rules give support to experimental defining of the Sq
(k)
moments. These moments usually are calculated either
in the laboratory coordinate system (l.c.s.) or nuclear
coordinate system (n.c.s.) connected with the nuclear
centre-of-mass after electron interaction. The
nonrelativistic coordinate systems (c.s.), in particular
l.c.s., are defined by a ωel observable:
ω = ω′+ωel, (8)
where ω′ is the energy transfer in the n.c.s. For the l.c.s.
ωel = q2/(2AM), where A is the atomic weight of nucleus,
M is the nucleon mass. The moment values and in
particular their errors are not always simply to convert
from one c.s. to another, thus we suggest better to obtain
the experimental Sq
(k) values both in the n.c.s. and in the
l.c.s. to compare them with the calculated ones. To
obtain Sq
(k) it is necessary to have the Rt-function in the
corresponding c.s.. The Rt-function was considered in
the n.c.s. as in ref. [1], [3,4] and in the present paper.
Let us find this function in other c.s.
The properties of the Rt-function follow from the
invariance of the response function zero moment Sq
(0)
(here and below it means the invariance respectively to
the choice of some c.s. of those which are correlated by
Eq. (8)). Write down this moment as
Sq
(0) = Se
q
(0)(ωx) + St
q
(0)(ωx), (9)
where the integral St
q
(0)(ωx) is defined by Eq. (5) and the
integral Se
q
(0)(ωx) covers the range of the experimental
values of the response function and therefore it can be
represented as a histogram area. Since the area of every
histogram pole is the product of Se
q(ωi) and ωi − ωi−1 and
these quantities are not depend on the choice of c.s. then
Se
q
(0)(ωx) is not connected with any c.s.. From the
invariance of Sq
(0) and Se
q
(0)(ωx) follows the St
q
(0)(ωx)
invariance.
The value of the Rt-function at the point ωx which is
the lower limit of the integral St
q
(0)(ωx) can be represented
as
R x
d S
d
x
q
t q
t,(0)
( )
( )
ω
ω
ω
ω ω
=
=
, (10)
If it is right that St
q
(0)(ωx) and dω are invariant then the
Rt-function is invariant too. The quantity ωx can assume
arbitrary values therefore Rt
q(ωx) is invariant for the all
domain of the function.
We denote that Rt
q
el(ω) the invariant Rt-function for
some fixed q. This function can be derived from
expression (2) if in a power base of this expression to
make independent relatively to c.s. The ωel value defines
the belonging of the ω to some c.s. Thus ω − ωel does not
depend on c.s. and the desired function can be written as
Rt
q
el (ω) = Cq (ω − ωel)−α. (11)
14
Now we consider this problem by another point of view.
The R-function describes a nuclear response on the
energy transfer due to the electron scattering on it and in
the common case this function depends on c.s. The
energy gained by nucleus is distributed on the kinetic
energy of the nuclear of centre-of-mass motion, ωel and
on the energy of the changing inner nucleus state (the
excitation energy) which coincidences with the total
energy transfer in the n.c.s., ω′. The excitation energy of
nuclear states is the structure characteristic of a nucleus
and this energy does not depend on c.s. So, if there is R(
ω′) and ω′ is the invariant parameter then this function
itself is the invariant one. The response function in some
c.s., R(ω) = R(ω′+ωel) is invariant function R(ω′) shifted
along the transfer energy on the ωel magnitude. The
power base in the expression (11) is ω-ωel that is equal
to ω′. From this fact follows: R(ω′) ≡ Rt,
q
el (ω). We note,
that R(ω′+const) is the invariant too, but there is no
special meaning at such presenting when const ≠ 0.
The total moment Sq
(k) is not invariant at k ≠ 0.
Therefore there is a question: how the kmax value is
connected with choice of c.s.? To answer this question
we substitute the Rt-function, in the form (11), into the
integral (5). As the result we derive:
++−−−
∞
−++−−−
×
−−+=
=
→
∑
1α)elωfω(
ω
1α)elωx(ω
1α
n
elωq
0
)xω(el(k),t,
q
lim
f
knkn
kn
Ck
n k
n
S
. (12)
The term where n = 0 is the most slowly decreasing with
the increase of ωf, thus it defines a condition when the
total sum has a finite value. This summand itself is
identically equal to expression (2). So, Eq.(7) is valid for
the form (11). Therefore from the invariance of α
parameter follows the invariance of kmax.
5. We compare the obtained data with the theoretic
calculations. In [3] αL = αµ = 2.5 was defined basing on
the relatively simple model. The parameter αµ is related
with the contribution into the RT-function of electron
scattering on nucleon magnetic moments. This contri-
bution at the q = 1 fm-1 according to the calculation of
[4] is about 90%, thus αµ ≈ αT and αL ≈ αT. The
experimental data give αT − αL = 0.11 ±0.17. The more
exact calculation with the using the nucleon Reid soft-
core potential gives αL = 3 - 4 [1]. Taking into account
the spread of calculation values this result does not
contradict to the experimental one: αL = 2,82.
The moment S(k) is calculated for k = 2, 3 in a
number of some theoretic papers (see [1], [7], [8], [9]).
We have derived that the moment S(k) is diverged at the
k > 1, basing on the response function extrapolation by
form (2) and experimental values of α. In view of the
complexity and importance of the kmax problem we
suppose that it is expedient to restrict ourselves in this
paper by the fact of so low experimental kmax value 1/ and
to point out that it is a good idea to make research, using
the greater experimental data body.
1/ The fact that kmax = 1 follows from Eq.(7) for the
magnitudes of α = 2.5, 3.0 of [3], [1] too.
In terms of the experimental determination of S(k)
values we note that since, in accordance with [1], the
extrapolation function does not depend on the transfer
momentum and faintly depends on the atomic number of
nucleus, the defined parameters αL, αT and also
expression (11) for the Rt-function in the l.c.s. can be
used for studying not only 2H nuclei but nuclei of other
elements too. The revision of the Rt-function should
have the most effect while determining the moments
with k = 1 when the extrapolation part is of about 15-
40% of the measured value.
We would like to thank A.V. Zatserklyany and L.G.
Levchuk for their critic notes.
REFERENCES
1.G. Orlandini and M. Traini. Sum rules for
electron-nucleus scattering. // Rep. Prog. Phys.
1991, v. 54, p. 257-338.
2.A.Yu. Buki, N.G. Shevchenko, I.A. Nenko et.al.
The moments of response functions for 2H nucleus at
q=1.05 fm-1 // Yad. Fiz, to be published (in Russian).
3.V. Tornow et.al. A study of electronuclear sum
rules and medium-weight nuclei // Nucl. Phys. 1980,
v. A348, p. 157-178.
4.W. Leidemann and H. Arenhövel. Electronuclear
sum rules at constant momentum transfer for the
deuteron // Nucl. Phys. 1982, v. A.381, p. 365-380.
5.A.Yu. Buki, V.D. Efros, N.G. Shevchenko et al.
Electronuclear transversal sum rule in 4He at
moderate transfer momenta and high-ω
extrapolation of response functions. Preprint IAE-
5397/2. M., 1991, 19 p.
6.A.Yu. Buki, N.G. Shevchenko, V.N. Polyschuk,
A.A. Khomich. The transversal momenta of 4He
response functions in the range of momentum
transfer 0.75 - 1.5 fm-1 // Yad.Fiz. 1995, v. 58,
p. 1353-1361 (in Russian).
7.V.D. Efros. Sum rules in the electron scattering
by nuclei // Yad. Fiz. 1973, v. 18, p. 1184-1202 (in
Russian).
8.E.V. Inopin, S.N. Roshchupkin. Sum rule for the
second moment of the response function in electron-
nuclei scattering // Yad. Fiz. 1974, v. 19, p. 1543-
1548 (in Russian).
9.R. Shiavilla et al. Coulomb sum rule of A = 2, 3,
and 4 nuclei // Ph. R. 1989, v. C40, p. 1484-1490.
15
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The experimental response functions of the 2H nucleus are extrapolated along the -region by the power function for q=1.05 fm1.
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-82162 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:52:13Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buki, A.Yu. Nenko, I.A. 2015-05-26T08:18:50Z 2015-05-26T08:18:50Z 2000 Response function extrapolation of ²H nucleus in the region of high transfer energy / A.Yu. Buki, I.A. Nenko // Вопросы атомной науки и техники. — 2000. — № 2. — С. 13-15. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 25.30.Fj, 27.10.+h https://nasplib.isofts.kiev.ua/handle/123456789/82162 The experimental response functions of the ²H nucleus are extrapolated along the w-region by the power function for q=1.05 fm⁻¹. We would like to thank A.V. Zatserklyany and L.G.
 Levchuk for their critic notes. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Nuclear reactions Response function extrapolation of ²H nucleus in the region of high transfer energy Экстраполяция функций отклика ядра ²H в области больших передаваемых энергий Article published earlier |
| spellingShingle | Response function extrapolation of ²H nucleus in the region of high transfer energy Buki, A.Yu. Nenko, I.A. Nuclear reactions |
| title | Response function extrapolation of ²H nucleus in the region of high transfer energy |
| title_alt | Экстраполяция функций отклика ядра ²H в области больших передаваемых энергий |
| title_full | Response function extrapolation of ²H nucleus in the region of high transfer energy |
| title_fullStr | Response function extrapolation of ²H nucleus in the region of high transfer energy |
| title_full_unstemmed | Response function extrapolation of ²H nucleus in the region of high transfer energy |
| title_short | Response function extrapolation of ²H nucleus in the region of high transfer energy |
| title_sort | response function extrapolation of ²h nucleus in the region of high transfer energy |
| topic | Nuclear reactions |
| topic_facet | Nuclear reactions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82162 |
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