The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target
We have developed the fitting program used for deuteron polarization calculation. The simulation has been done for the deuterated 1,2-propanediol-D8 as a target material. Analyses of the uncertainties sources have been performed. The polarization calculated was P=+34.0% and P = -36.6% for above-ment...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target / A.A. Belyaev, A.P. Dzyubak, A.A. Luchanin // Вопросы атомной науки и техники. — 2000. — № 2. — С. 38-40. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-822672015-05-28T03:01:52Z The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target Belyaev, A.A. Dzyubak, A.P. Luchanin, A.A. Еxperimental methods We have developed the fitting program used for deuteron polarization calculation. The simulation has been done for the deuterated 1,2-propanediol-D8 as a target material. Analyses of the uncertainties sources have been performed. The polarization calculated was P=+34.0% and P = -36.6% for above-mentioned target material. 2000 Article The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target / A.A. Belyaev, A.P. Dzyubak, A.A. Luchanin // Вопросы атомной науки и техники. — 2000. — № 2. — С. 38-40. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 13.75Gx http://dspace.nbuv.gov.ua/handle/123456789/82267 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Еxperimental methods Еxperimental methods |
| spellingShingle |
Еxperimental methods Еxperimental methods Belyaev, A.A. Dzyubak, A.P. Luchanin, A.A. The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target Вопросы атомной науки и техники |
| description |
We have developed the fitting program used for deuteron polarization calculation. The simulation has been done for the deuterated 1,2-propanediol-D8 as a target material. Analyses of the uncertainties sources have been performed. The polarization calculated was P=+34.0% and P = -36.6% for above-mentioned target material. |
| format |
Article |
| author |
Belyaev, A.A. Dzyubak, A.P. Luchanin, A.A. |
| author_facet |
Belyaev, A.A. Dzyubak, A.P. Luchanin, A.A. |
| author_sort |
Belyaev, A.A. |
| title |
The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target |
| title_short |
The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target |
| title_full |
The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target |
| title_fullStr |
The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target |
| title_full_unstemmed |
The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target |
| title_sort |
employment of fitting programs for deuteron polarization calculation in a nuclear polarized target |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Еxperimental methods |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/82267 |
| citation_txt |
The employment of fitting programs for deuteron polarization calculation in a nuclear polarized target / A.A. Belyaev, A.P. Dzyubak, A.A. Luchanin // Вопросы атомной науки и техники. — 2000. — № 2. — С. 38-40. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-06T08:46:16Z |
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2025-07-06T08:46:16Z |
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| fulltext |
THE EMPLOYMENT OF FITTING PROGRAMS FOR DEUTERON
POLARIZATION CALCULATION IN A NUCLEAR POLARIZED
TARGET
A.A. Belyaev, A.P. Dzyubak, A.A. Luchanin,
National Science Center «Kharkov Institute of Physics and Technology», Kharkov, Ukraine
We have developed the fitting program used for deuteron polarization calculation. The simulation has been
done for the deuterated 1,2-propanediol-D8 as a target material. Analyses of the uncertainties sources have been
performed. The polarization calculated was P=+34.0% and P= −36.6% for above-mentioned target material.
PACS: 13.75Gx
The accuracy of target polarization measurement for
the polarized deuteron in situ is important for realization
of physics experiments. In the best way the polarization
of a target is determined by calibration of the enhanced
signal against the signal of the spin system in thermal
equilibrium (TE) with the lattice. But for deuterons the
precision calibration is very difficult under conditions of
thermal equilibrium since the TE signal has very low
intensity. Therefore, the development of the nuclear
magnetic resonance (NMR) line-shape analysis for a
calculation of the deuteron polarization is the problem
of current interest. The measurements procedure
consists in recording the absorption spectrum of the
nuclear magnetic resonance from the Q-meter for
computing with subsequent determining the polarization
value [1]. The multiple accumulation of these signals is
executed for the purpose of raising the accuracy.
The typical materials for the polarized target are the
deuterated alcohols. The C-D and O-D bonds leads to
broadening Zeeman levels. The energy levels of such a
spin system can be written as [2,3]
),223(
)}2cos()(2sin1)(2cos3{
−×
×+−+=
m
qmdmE φθηθνν
where θ is the polar angle between the C-D and O-D
bonds and the magnetic field direction;
m= −1, 0, 1 is the spin magnetic quantum number;
ν q =eqeQ/8 is the energy of the quadrupole interaction;
eq is the magnitude of the electric field gradient along
the bound direction;
eQ is the electric quadrupole moment of deuteron,
φ is the azimuth angle;
η is the parameter of asymmetry of a quadrupole
deuteron moment.
For ease let us use the normalized frequencies
,)(2 ddx ννν −=
which can be written as
,),()12( QQmx ννφθλ−−=
where 2)2cos2sin12cos3(),( φθηφθλ +−= .
In this variables the full spectrum has the F(x) and F(
−x) components, which are symmetrical relatively x=0.
The F(x) and F(−x) components are related to two
Zeeman transitions m↔m−1. In normalized frequencies
F(x) is written as [4]:
,)]([)]([
1
)]([)]([
1
)(
22
11
∫ −+
+∫ −=
r
p
dttkKtxf
q
dtr
q tkKtxf
p
xF
ψ
π
ψ
π
where
,2η=p
η+= 3r ,
,221)(1 tt −=ψ
2)12()(2 ηψ −−= tt ,
,)()( 31 pqttk ψ=
,)()( 32 qpttk ψ=
.1)22()(3 −= trtψ
f(x) is the Gauss or Lorentz function, where σ is the
dispersion. K(x) is the elliptic integral of first kind. The
contribution of the second type of the quadrupole bond
is taken into account similarly, replacing
)()(* tt iψρψ ⋅= , (i=1,2)
and ,12 QQ ννρ =
where 2Qν and 1Qν are given above.
In assumption of the equal spin temperature in the
deuteron system, the relative population of the magnetic
levels R = exp(-hνd/kTs), where νd and Ts are the central
NMR frequency and the spin temperature, respectively.
Therefore, the full spectrum with taking into account the
C-D and O-D bonds is
,)1()}]()({12[2)( RxiFxiRFi iCxS +−+∑ ==
where Ci is the relative deuteron number in the bond.
This spectrum is defined by 6 parameters:
R, η1, η2, ρ, σ and C (as C1+C2=1),
which are defined by comparison of the model with the
experimental NMR−absorption signal and characterize
the substance under study. Thus, the η1 and η2 define the
asymmetry of the quadrupole moment of C-D and O-D
38 ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, № 2.
Серия: Ядерно-физические исследования (36), c. 38-40.
.
bonds, R−parameter is used to calculate deuteron
polarization:
P=(1− R2)/(1+R+R2).
We have developed the program complex to
calculate deuteron polarization by analyzing the
deuteron NMR line-shape in frozen alcohols, using
above algorithm.
The program consists of two parts. In the first part
we have to process the data file (DMR−line) received
from Data Acquisition System to remove the apparatus
distortions. These are the following.
1) Frequency sweeping has no linear dependence on
digital step size. And we have to recalculate this
frequency sweep to make dependence as linear one.
2) The measured DMR−line is real line plus some
additional baseline. And the baseline usually is a
polynomial-like line, which has to be simulated and
removed.
3) The DMR−line size usually is narrower (and
therefore has additional not needed parts) and not in the
window center of Analog-to-Digital Converter. Then we
have to recalculate DMR−line to make it of right size
and put at right place.
This part of program has been designated as having
the interactive and graphical windows. In the interactive
window one is prompted the questions what one wants
to do with DMR−line and the graphical window shows
what is happening. If the DMR−line transforms are
accepted then the new DMR−line is written into a new
file and the interactive session is closed.
The frequency is extrapolated for each channel by
the polynomial of the third degree. With the help of
splines and with the regard for the frequency
dependence the signal is formed from the primary
DMR−signal. It has the linear frequency dependence of
the sweep. The results are written down in the separate
massive.
It is not in general possible to choose the optimal
values of the resonance frequency and the deviation
width for the Q−meter system. As result the measured
signal is not found in the center of the sweep and has a
different frequency width in the channel. It makes the
fitting process difficult and increases the error of
parameter definition. It has become necessary to
correct these effects. After the frequency linearization
program displaces the signal in the sweep center,
remove the channels, not having signal information and
do further signal extension over the whole channel
range. The area of the obtained signal normalized per
unity and result is recorded into the certain file. The
result of the correction after these operations is shown
in Fig. 1 and in Fig. 2.
On the second stage the model spectrum is fitted to the
corrected spectrum. The program minimizes the
functional describing the deviation of the theory
function from the experimental data. In our case the
functional
2
1 exp
)),((
)(2 ∑
=
−
=
n
i
ieaixf
а
σ
χ
Fig. 1. Deuteron magnetic resonance signal.
Correction of the frequency dependence. The solid line
is the real Q-meter signal. The circles are the signal
after correction
Fig. 2. Deuteron magnetic resonance signal.
Correction of the boundaries, the base line and the
extension over the whole channel range. The circles are
signal before correction. The solid line is signal
obtained after full correction.
was minimized, where ei is the measurement result, a is
the vector of parameters, σexp is the uncertainties in the
individual ei measurements.
The calculations were performed using the MINUIT
package from CERN program library. This package
enables to find the best parameter values for multi-
parameter functions. It is possible to perform a fitting
for parameters with user’s restricted limits. Two files are
prepared for this task. Two files are prepared for this
task. The first file consists of the DMR−signal to be
corrected. The second one contains command script to
be executed by MINUIT at run−time. The script
includes the work title, the parameter namelist, the initial
values, the boundaries, the list of command describing
the calling sequence of MINUIT fitters.
To find the correct full set initial parameters defining
the DMR-line shape it is needed a long run time and we
use SCAN algorithm because of it. After it is done the
parameters of the measured DMR−signal are only
39
slightly changing. And the parameter, related to the
polarization has to be fitted only. To do this the
MIGRAD algorithm is used, which is more carefully
and makes fitting more correctly.
We have made a number of program tests to estimate
the errors introduced by the MINUIT program into the
polarization value.
The source of errors is determined by the fact that the
χ2–functional has a lot of the local extremes, which lead
to the multivalued fitting parameters. This gives the
dispersion deuteron polarization value. We have
determined the area of parameter values, obtained after
fitting, depending on the change of their initial values.
For this we have generated a DMR-signal with the
known parameters. Further these parameters were taken
as initial ones and the fitting program determined the
parameters of DMR-signal. During the next stage the
changes (20–40%) were put into the values of initial
parameters and DMR–signal parameters were
determined again. Besides, we added Gaussian noise of
different intensity into the DMR−signal and calculate
the parameters again. We have got the following result
of such simulation with the absence of changing in the
initial parameter values.
The yielded parameters were absolutely aligned with
initial ones. With 40% changing of the R- parameter the
value of polarization being determined was changing not
more than 0.3%. With simultaneous variation of all
initial parameters up to 40% the changing of
polarization value did not exceed 4%. In the absence of
changing in initial parameter values the addition of
Gaussian noise with intensity of 10% leads to the
change of the polarization value not more than by 1%.
Under conditions of a real physical experiment with
the calculation of a polarization value the initial
parameter values of DMR−signal are unknown. So it is
necessary to make some calculation to know the
parameters of DMR−spectrum and every time it is
necessary to take the parameters of the previous
calculation as an initial but with insignificant changes.
When stability of results is reached one can take the
average value as a polarization value.
The result is shown in Figs. 3, 4. The fitting was
made using the DMR−signal for 1,2 propanediol−D8
with the positive and negative deuteron polarization.
Fig. 3. The result of the fitting procedure by
MINUIT program, applied to PD = −36.6 ± 1.6%. The
circles are the DMR signal after full correction .The
solid line is the fitted DMR signal.
Fig. 4. The same as Fig. 3 for positive deuteron
polarization PD = 34.0 ± 1.4%.
The experimental signal from Q−meter consisted of
1000 enhanced polarization sweeps. All frequency range
was sampled on 1024 channels. Only 256 channels were
used to fit the theoretical absorption function to the
experimental DMR−signal. The following value of the
parameters for 1,2 propanediol−D8 were obtained: η1=
0.18; η2=0.017; ρ0.78; σ=0.019.
From this figures one can see that the program gives
a good fitting and can be used for finding the deuteron
polarization. At the conditions the equilibrium (thermal)
the DMR−signal cannot be measured correctly enough
and the method of calculation of deuteron polarization
by analyzing DMR–line shape is an excellence
alternative possibility.
REFERENCES
1. A.A. Belyaev et al. The measurement of the
deuteron polarization in the polarized target. Preprint
KIPT 87-27. M.: CNIIatominform, 1987. 9 p. (in
Russian).
2. A. Abragam. Nuclear magnetism. M. IL, 1963.
551 p. (in Russian).
3. M.H. Cohen. and F. Reif. Nuclear quadrupole
effects in solids // Solid State Physics. 1957, v. 5,
p. 321.
4. O. Hamada. at all. Analysis of deuteron NMR
spectrum in propanediol for polarization measurement //
Nucl. Instr. Meth. 1981, v. 189, p. 561-568.
40
REFERENCES
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