In this paper I access the degree of approximation of known symbolic approach to solving of Ginzburg-Landau (GL) equations using variational method and a concept of vortex lattice with circular unit cells, refine it in a clear and concise way, identify and eliminate the errors. Also, I will improve its accuracy by providing for the first time precise dependencies of the variational parameters; correct and calculate magnetisation, compare it with the one calculated numerically and conclude they agree within 98.5% or better for any value of the GL parameter
k and at magnetic field
Much of basic superconductor behavior in magnetic field can be understood from the phenomenological model expressed by two Ginzburg-Landau (GL) equations [
However, a simple check shows that magnetisation calculated with this method ( [
In this paper I omit the time-dependent terms in the GL equations written in SI units [
the lower, the upper and thermodynamic critical magnetic fields: Bc1, Bc2 and
independent of k or B, defined externally (through ξ and BCS theory [
Form 1 [ | Form 2 [ |
---|---|
magnetic induction
The variational method [
where the notation
Averaged over the cell area Helmholtz dimensionless free energy density F of a circular unit cell (with radius R) has two contributions [
where
circular (Wigner-Seitz) unit cell carrying one magnetic flux quantum, thus
The GL equations are obtained by minimising the free energy of superconductor F with respect to e.g., fr and to qr (form 1,
Complementing the Equations (1) (2) and (3) (4) Maxwell equations are:
From Equations (4) and (5) one gets:
since
Since
where
where
The boundary conditions can be found e.g., in [
In order to stay focused and limit the size of the paper, below I simply accept Equations (14), (15) and will study namely this case in more detail. With the constants c1 and c2 defined, the solution allows calculating the local quantities as well as the mean quantities: equilibrium magnetic field, magnetisation, etc. In this paper I focus on magnetisation.
In the variational method Equation (5) in fact replaces the unknown
From Equation (12) at
with
Minimising the free energy density F of superconductor, Equations (6)-(8), (16) (17), with respect to the variational parameters
At any
(
At
Minimisation of F with respect to
The obtained (by minimizing F) dependence
The agreement of the obtained data with [ [
On the other hand at
Based on this study, I conclude that none of the Equations (13)-(15) in [
The applied (thermodynamic, equilibrium) magnetic field
from Equations (14)-(17) according to [
with [
and
Finally, after re-normalisation of Equation (25), one obtains:
Namely the magnetization
The error (present in Equations (10)-(26) at
for any k and
cells [
In this paper the error is eliminated in the following way. The Equations (21)-(26) are only valid at
the 1st derivative through this point of the true magnetisation curve should be constant set by Equation (28) and that only depends on k and on the Abrikosov parameter
From [ [
From Equations (30) and (16) (17) it is easy to check that for
A smooth transition from Equation (26) to Equation (27) can be achieved in several ways. In this paper I use the following approach. So far calculated from Equation (26) magnetisation
In this paper a compliance with this condition (Equation (28)) is achieved by introducing the correction:
In conclusion, the correction makes the obtained solution compliant with [
In Figures 5-8 the magnetisation m1 is compared to that calculated numerically [
As clear from Figures 5-8, for any value of the GL parameter
tween m1 and m2 is achieved: the relative difference
everywhere (except in the narrow range:
Representative for the range of
The relative difference
tally fails to describe the data (m2) quantitatively, since k < 3 and thus
is too high. The data represented by magnetisation m1 (and m2) are in good
agreement with these represented by m4 at
this case). It is clear from the figure that m5 also fails to describe quantitatively the data represented by m1 (and m2): e.g., at
reading [
Representative for the range of
In
1.1% and 1.3% respectively in the entire range
scribe the data (m2) quantitatively, since k < 3 and thus
data represented by m1 (and m2) are in good agreement with these represented by m4 at
are below 0.4% (
ference
competitive and not what one expects after reading [
Representative for the range of
(except in the range:
data (m2) quantitatively, since k > 3 and thus
represented by almost the same line in the figure. The data represented by m1 (and m2) are in good agreement with these represented by m4 at
that both
ing [
Representative for the range of
tively, since k > 3 and thus
almost the same line in the figure. The data represented by m1 (and m2) are in good agreement with these represented by m4 at
is clear from
m1 (and m2): with larger relative difference
ative (on
Representative for the range of
and
below 0.6% in the entire range
(m3) describes the data (m2) quantitatively, since k > 3 and thus
small [
at
low 0.5%, with
magnetisation first derivative (on
Representative for the range of
ure 8(a). The relative difference
in
represented by m4 at
at
0.3%, with
the data represented by m1 (and m3) with larger relative difference
tion 1st derivative (on
difference
error in the magnetisation’s first derivative (on
The fragment of the magnetisation curve at k = 200 exemplifies that the error in the 1st derivative of the magnetization m5 is present at highest values of k and
results in the noticeable difference
caused by the digitalisation of the data [ [
To summarise, over the entire ranges of the GL parameter
achieved: the relative difference
the narrow range:
Presented in Annex data for the dependencies of
In this paper I calculate values of the field bc1 from Equations (25) (26) typically at
with
The red dashed line is calculated from the symbolic expression for isolated flux line [
In this range of k the bc1 changes by 5 orders of magnitude, so I find the agreement between
As noted [
This correction is dealt with in section 4.2. Moreover,
Finally, in order to illustrate the common feature of the ideal superconducting
materials (in this case homogeneous, bulk, elliptically shaped, edge- and pin-free, placed in uniform magnetic field), in
Known symbolic method [
and on magnetic flux density
If you are interested to support this study, contact the author at:
o.a.chevtchenko@gmail.com. If you are interested in the results, contact us at: http://www.hts-powercables.nl. This research is funded privately and all rights belong to the author. Many thanks to A. A. Shevchenko for helping with difficult parts of this study, to R. Bakker for the inspirational atmosphere, to prof.-em. J. J. Smit for his support of this approach. A word of disgrace goes personally to Mr Jose Labastida, head of scientific department and to prof.-em. Helga Nowotny, former president of ERC, who chose to deny a funding for this study and thus created obstacles on the way to this result that could otherwise be published years earlier.
Chevtchenko, O.A. (2017) Accurate Symbolic Solution of Ginzburg-Landau Equations in the Circular Cell Approximation by Variational Method: Magnetization of Ideal Type II Superconductor. Journal of Modern Physics, 8, 982-1011. https://doi.org/10.4236/jmp.2017.86062
Accurate dependencies of the variational parameters
Note that in order to simplify comparisons with [
k = 0.75 | k = 0.85 | k = 1.2 | k = 2 | 5 ≤ k ≤ 200 | |||||
---|---|---|---|---|---|---|---|---|---|
7.26E−6 | 1.000E+0 | 5.95E−6 | 1.000E+0 | 2.96E−6 | 1.000E+0 | 1.05E−6 | 1.000E+0 | 5.80E−10 | 1.000E+0 |
3.46E−4 | 1.001E+0 | 3.46E−4 | 1.001E+0 | 3.46E−4 | 1.001E+0 | 3.46E−4 | 1.001E+0 | 3.456E−4 | 1.001E+0 |
1.12E−3 | 1.001E+0 | 1.12E−3 | 1.001E+0 | 1.12E−3 | 1.002E+0 | 1.12E−3 | 1.002E+0 | 1.121E−3 | 1.002E+0 |
4.00E−3 | 1.004E+0 | 3.68E−3 | 1.004E+0 | 3.68E−3 | 1.004E+0 | 2.00E−3 | 1.003E+0 | 3.682E−3 | 1.005E+0 |
1.00E−2 | 1.008E+0 | 1.00E−2 | 1.009E+0 | 1.00E−2 | 1.010E+0 | 3.68E−3 | 1.005E+0 | 1.287E−2 | 1.014E+0 |
2.50E−2 | 1.017E+0 | 2.50E−2 | 1.018E+0 | 2.50E−2 | 1.020E+0 | 7.00E−3 | 1.008E+0 | 2.500E−2 | 1.024E+0 |
5.00E−2 | 1.029E+0 | 5.00E−2 | 1.030E+0 | 5.00E−2 | 1.034E+0 | 1.00E−2 | 1.011E+0 | 5.000E−2 | 1.042E+0 |
7.00E−2 | 1.036E+0 | 7.00E−2 | 1.038E+0 | 7.00E−2 | 1.043E+0 | 2.00E−2 | 1.019E+0 | 7.000E−2 | 1.053E+0 |
1.00E−1 | 1.046E+0 | 1.00E−1 | 1.048E+0 | 1.00E−1 | 1.054E+0 | 3.50E−2 | 1.029E+0 | 1.000E−1 | 1.065E+0 |
1.50E−1 | 1.056E+0 | 1.50E−1 | 1.058E+0 | 1.50E−1 | 1.064E+0 | 5.00E−2 | 1.038E+0 | 1.500E−1 | 1.073E+0 |
2.00E−1 | 1.058E+0 | 2.00E−1 | 1.060E+0 | 2.00E−1 | 1.065E+0 | 7.00E−2 | 1.048E+0 | 2.000E−1 | 1.071E+0 |
2.25E−1 | 1.056E+0 | 2.25E−1 | 1.058E+0 | 2.25E−1 | 1.062E+0 | 1.00E−1 | 1.060E+0 | 2.250E−1 | 1.066E+0 |
2.50E−1 | 1.053E+0 | 2.50E−1 | 1.055E+0 | 2.50E−1 | 1.057E+0 | 1.50E−1 | 1.070E+0 | 2.500E−1 | 1.059E+0 |
3.00E−1 | 1.042E+0 | 3.00E−1 | 1.042E+0 | 3.00E−1 | 1.042E+0 | 2.00E−1 | 1.068E+0 | 3.000E−1 | 1.040E+0 |
3.50E−1 | 1.025E+0 | 3.50E−1 | 1.024E+0 | 3.50E−1 | 1.021E+0 | 2.25E−1 | 1.064E+0 | 3.500E−1 | 1.016E+0 |
4.00E−1 | 1.002E+0 | 4.00E−1 | 9.996E−1 | 4.00E−1 | 9.939E−1 | 2.50E−1 | 1.058E+0 | 4.000E−1 | 9.860E−1 |
4.50E−1 | 9.738E−1 | 4.50E−1 | 9.701E−1 | 4.50E−1 | 9.621E−1 | 3.00E−1 | 1.041E+0 | 4.500E−1 | 9.519E−1 |
5.00E−1 | 9.404E−1 | 5.00E−1 | 9.355E−1 | 5.00E−1 | 9.255E−1 | 3.50E−1 | 1.018E+0 | 5.000E−1 | 9.136E−1 |
5.50E−1 | 9.018E−1 | 5.50E−1 | 8.959E−1 | 5.50E−1 | 8.843E−1 | 4.00E−1 | 9.891E−1 | 5.500E−1 | 8.712E−1 |
6.00E−1 | 8.579E−1 | 6.00E−1 | 8.511E−1 | 6.00E−1 | 8.384E−1 | 4.50E−1 | 9.558E−1 | 6.000E−1 | 8.245E−1 |
6.50E−1 | 8.082E−1 | 6.50E−1 | 8.009E−1 | 6.50E−1 | 7.873E−1 | 5.00E−1 | 9.181E−1 | 6.500E−1 | 7.730E−1 |
7.00E−1 | 7.521E−1 | 7.00E−1 | 7.444E−1 | 7.00E−1 | 7.305E−1 | 5.50E−1 | 8.761E−1 | 7.000E−1 | 7.162E−1 |
7.50E−1 | 6.884E−1 | 7.50E−1 | 6.806E−1 | 7.50E−1 | 6.668E−1 | 6.00E−1 | 8.296E−1 | 7.500E−1 | 6.529E−1 |
8.00E−1 | 6.152E−1 | 8.00E−1 | 6.075E−1 | 8.00E−1 | 5.943E−1 | 6.50E−1 | 7.782E−1 | 8.000E−1 | 5.812E−1 |
8.50E−1 | 5.288E−1 | 8.50E−1 | 5.217E−1 | 8.50E−1 | 5.095E−1 | 7.00E−1 | 7.214E−1 | 8.500E−1 | 4.978E−1 |
9.00E−1 | 4.217E−1 | 9.00E−1 | 4.156E−1 | 9.00E−1 | 4.054E−1 | 7.50E−1 | 6.579E−1 | 9.000E−1 | 3.957E−1 |
9.50E−1 | 2.708E−1 | 9.50E−1 | 2.667E−1 | 9.50E−1 | 2.598E−1 | 8.00E−1 | 5.858E−1 | 9.500E−1 | 2.534E−1 |
9.75E−1 | 1.421E−1 | 9.75E−1 | 1.398E−1 | 9.75E−1 | 1.362E−1 | 8.50E−1 | 5.020E−1 | 9.750E−1 | 1.327E−1 |
9.80E−1 | 9.740E−2 | 9.80E−1 | 9.586E−2 | 9.80E−1 | 9.333E−2 | 9.00E−1 | 3.991E−1 | 9.800E−1 | 9.099E−2 |
9.85E−1 | 1.000E−4 | 9.85E−1 | 1.000E−4 | 9.85E−1 | 1.000E−4 | 9.50E−1 | 2.556E−1 | 9.825E−1 | 5.995E−2 |
9.60E−1 | 2.154E−1 | 9.830E−1 | 5.162E−2 | ||||||
9.70E−1 | 1.656E−1 | 9.850E−1 | 1.000E−4 | ||||||
9.80E−1 | 9.180E−2 | ||||||||
9.85E−1 | 1.000E−4 |
k = 50 | k = 100 | k = 200 | |||
---|---|---|---|---|---|
7.025E−9 | 9.989E−1 | 9.090E−6 | 1.000E+0 | 1.318E−10 | 1.000E+0 |
9.000E−6 | 1.001E+0 | 3.287E−5 | 1.002E+0 | 1.000E−6 | 1.000E+0 |
1.500E−5 | 1.001E+0 | 1.073E−4 | 1.003E+0 | 9.090E−6 | 1.001E+0 |
3.287E−5 | 1.002E+0 | 3.456E−4 | 1.007E+0 | 3.287E−5 | 1.001E+0 |
1.073E−4 | 1.004E+0 | 1.121E−3 | 1.014E+0 | 1.073E−4 | 1.003E+0 |
3.456E−4 | 1.008E+0 | 3.682E−3 | 1.029E+0 | 3.456E−4 | 1.006E+0 |
1.121E−3 | 1.016E+0 | 1.287E−2 | 1.056E+0 | 1.121E−3 | 1.014E+0 |
3.682E−3 | 1.031E+0 | 2.500E−2 | 1.075E+0 | 5.000E−3 | 1.033E+0 |
1.000E−2 | 1.052E+0 | 5.000E−2 | 1.095E+0 | 2.000E−2 | 1.067E+0 |
2.500E−2 | 1.079E+0 | 7.000E−2 | 1.099E+0 | 5.000E−2 | 1.093E+0 |
5.000E−2 | 1.098E+0 | 1.000E−1 | 1.095E+0 | 1.000E−1 | 1.094E+0 |
7.000E−2 | 1.103E+0 | 1.500E−1 | 1.072E+0 | 1.500E−1 | 1.071E+0 |
1.000E−1 | 1.099E+0 | 2.000E−1 | 1.041E+0 | 2.000E−1 | 1.040E+0 |
1.500E−1 | 1.076E+0 | 2.250E−1 | 1.024E+0 | 2.250E−1 | 1.023E+0 |
2.000E−1 | 1.044E+0 | 2.500E−1 | 1.006E+0 | 2.500E−1 | 1.006E+0 |
2.250E−1 | 1.027E+0 | 3.000E−1 | 9.725E−1 | 3.000E−1 | 9.718E−1 |
2.500E−1 | 1.010E+0 | 3.500E−1 | 9.402E−1 | 3.500E−1 | 9.395E−1 |
3.000E−1 | 9.759E−1 | 4.000E−1 | 9.098E−1 | 4.000E−1 | 9.092E−1 |
3.500E−1 | 9.435E−1 | 4.500E−1 | 8.816E−1 | 4.500E−1 | 8.810E−1 |
4.000E−1 | 9.130E−1 | 5.000E−1 | 8.553E−1 | 5.000E−1 | 8.548E−1 |
4.500E−1 | 8.847E−1 | 5.500E−1 | 8.310E−1 | 5.500E−1 | 8.305E−1 |
5.000E−1 | 8.583E−1 | 6.000E−1 | 8.084E−1 | 6.000E−1 | 8.079E−1 |
5.500E−1 | 8.339E−1 | 6.500E−1 | 7.875E−1 | 6.500E−1 | 7.869E−1 |
6.000E−1 | 8.113E−1 | 7.000E−1 | 7.679E−1 | 7.000E−1 | 7.674E−1 |
6.500E−1 | 7.902E−1 | 7.500E−1 | 7.497E−1 | 7.500E−1 | 7.492E−1 |
7.000E−1 | 7.706E−1 | 8.000E−1 | 7.326E−1 | 8.000E−1 | 7.321E−1 |
7.500E−1 | 7.523E−1 | 8.500E−1 | 7.166E−1 | 8.500E−1 | 7.161E−1 |
8.000E−1 | 7.352E−1 | 9.000E−1 | 7.015E−1 | 9.000E−1 | 7.010E−1 |
8.500E−1 | 7.191E−1 | 9.500E−1 | 6.874E−1 | 9.400E−1 | 6.893E−1 |
9.000E−1 | 7.040E−1 | 9.750E−1 | 6.805E−1 | 9.500E−1 | 6.869E−1 |
9.500E−1 | 6.898E−1 | 9.800E−1 | 6.792E−1 | 9.600E−1 | 6.837E−1 |
9.750E−1 | 6.829E−1 | 9.825E−1 | 6.785E−1 | 9.700E−1 | 6.815E−1 |
9.800E−1 | 6.817E−1 | 9.830E−1 | 6.784E−1 | 9.790E−1 | 6.786E−1 |
9.850E−1 | 6.817E−1 | 9.850E−1 | 6.784E−1 | 9.840E−1 | 6.774E−1 |
k = 0.75 | k = 0.85 | k = 1.2 | k = 2 | ||||
---|---|---|---|---|---|---|---|
7.258E−6 | 1.001E+0 | 5.954E−6 | 9.946E−1 | 2.962E−6 | 1.000E+0 | 3.287E−5 | 1.001E+0 |
9.000E−6 | 1.001E+0 | 7.000E−6 | 9.946E−1 | 9.000E−6 | 1.000E+0 | 1.073E−4 | 1.001E+0 |
1.500E−5 | 1.001E+0 | 9.000E−6 | 9.946E−1 | 1.000E−5 | 1.000E+0 | 3.456E−4 | 1.008E+0 |
3.287E−5 | 1.001E+0 | 3.287E−5 | 9.946E−1 | 3.287E−5 | 1.000E+0 | 1.121E−3 | 1.019E+0 |
1.073E−4 | 1.001E+0 | 1.073E−4 | 9.946E−1 | 1.073E−4 | 1.000E+0 | 2.000E−3 | 1.027E+0 |
3.456E−4 | 1.006E+0 | 3.456E−4 | 1.006E+0 | 3.456E−4 | 1.007E+0 | 3.682E−3 | 1.041E+0 |
1.121E−3 | 1.013E+0 | 1.121E−3 | 1.014E+0 | 1.121E−3 | 1.016E+0 | 7.000E−3 | 1.061E+0 |
4.000E−3 | 1.031E+0 | 3.682E−3 | 1.031E+0 | 3.682E−3 | 1.035E+0 | 1.000E−2 | 1.076E+0 |
1.000E−2 | 1.056E+0 | 1.000E−2 | 1.059E+0 | 1.000E−2 | 1.066E+0 | 2.000E−2 | 1.116E+0 |
2.500E−2 | 1.097E+0 | 2.500E−2 | 1.102E+0 | 2.500E−2 | 1.115E+0 | 3.500E−2 | 1.157E+0 |
5.000E−2 | 1.142E+0 | 5.000E−2 | 1.149E+0 | 5.000E−2 | 1.168E+0 | 5.000E−2 | 1.188E+0 |
7.000E−2 | 1.168E+0 | 7.000E−2 | 1.177E+0 | 7.000E−2 | 1.198E+0 | 7.000E−2 | 1.215E+0 |
1.000E−1 | 1.197E+0 | 1.000E−1 | 1.206E+0 | 1.000E−1 | 1.229E+0 | 1.000E−1 | 1.237E+0 |
1.500E−1 | 1.225E+0 | 1.500E−1 | 1.234E+0 | 1.337E−1 | 1.247E+0 | 1.500E−1 | 1.242E+0 |
2.000E−1 | 1.236E+0 | 2.000E−1 | 1.244E+0 | 1.500E−1 | 1.252E+0 | 2.000E−1 | 1.224E+0 |
2.250E−1 | 1.237E+0 | 2.250E−1 | 1.244E+0 | 2.000E−1 | 1.252E+0 | 2.250E−1 | 1.211E+0 |
2.500E−1 | 1.236E+0 | 2.500E−1 | 1.242E+0 | 2.250E−1 | 1.247E+0 | 2.500E−1 | 1.196E+0 |
3.000E−1 | 1.230E+0 | 3.000E−1 | 1.231E+0 | 2.500E−1 | 1.239E+0 | 3.000E−1 | 1.165E+0 |
3.500E−1 | 1.218E+0 | 3.500E−1 | 1.216E+0 | 3.000E−1 | 1.218E+0 | 3.500E−1 | 1.133E+0 |
4.000E−1 | 1.203E+0 | 4.000E−1 | 1.198E+0 | 3.500E−1 | 1.194E+0 | 4.000E−1 | 1.101E+0 |
4.500E−1 | 1.187E+0 | 4.500E−1 | 1.179E+0 | 4.000E−1 | 1.168E+0 | 4.500E−1 | 1.070E+0 |
5.000E−1 | 1.170E+0 | 5.000E−1 | 1.159E+0 | 4.500E−1 | 1.141E+0 | 5.000E−1 | 1.041E+0 |
5.500E−1 | 1.152E+0 | 5.500E−1 | 1.138E+0 | 5.000E−1 | 1.115E+0 | 5.500E−1 | 1.014E+0 |
6.000E−1 | 1.134E+0 | 6.000E−1 | 1.118E+0 | 5.500E−1 | 1.090E+0 | 6.000E−1 | 9.885E−1 |
6.500E−1 | 1.117E+0 | 6.500E−1 | 1.098E+0 | 6.000E−1 | 1.066E+0 | 6.500E−1 | 9.643E−1 |
7.000E−1 | 1.099E+0 | 7.000E−1 | 1.079E+0 | 6.500E−1 | 1.043E+0 | 7.000E−1 | 9.417E−1 |
7.500E−1 | 1.082E+0 | 7.500E−1 | 1.061E+0 | 7.000E−1 | 1.021E+0 | 7.500E−1 | 9.204E−1 |
8.000E−1 | 1.066E+0 | 8.000E−1 | 1.043E+0 | 7.500E−1 | 1.000E+0 | 8.000E−1 | 9.004E−1 |
8.500E−1 | 1.050E+0 | 8.500E−1 | 1.025E+0 | 8.000E−1 | 9.803E−1 | 8.500E−1 | 8.815E−1 |
9.000E−1 | 1.035E+0 | 9.000E−1 | 1.009E+0 | 8.500E−1 | 9.614E−1 | 9.000E−1 | 8.637E−1 |
9.500E−1 | 1.020E+0 | 9.500E−1 | 9.928E−1 | 9.000E−1 | 9.434E−1 | 9.500E−1 | 8.469E−1 |
9.750E−1 | 1.012E+0 | 9.750E−1 | 9.850E−1 | 9.500E−1 | 9.263E−1 | 9.600E−1 | 8.437E−1 |
9.800E−1 | 1.011E+0 | 9.800E−1 | 9.835E−1 | 9.750E−1 | 9.181E−1 | 9.700E−1 | 8.405E−1 |
9.850E−1 | 1.011E+0 | 9.850E−1 | 9.835E−1 | 9.800E−1 | 9.165E−1 | 9.800E−1 | 8.373E−1 |
9.850E−1 | 9.165E−1 | 9.850E−1 | 8.373E−1 |
k = 5 | k = 10 | k = 20 | k = 50* | ||||
---|---|---|---|---|---|---|---|
2.000E−7 | 1.001E+0 | 4.000E−6 | 1.000E+0 | 1.000E−8 | 1.001E+0 | 7.025E−9 | 9.989E−1 |
1.000E−6 | 1.001E+0 | 9.090E−6 | 1.000E+0 | 1.000E−6 | 1.001E+0 | 9.000E−6 | 1.001E+0 |
9.090E−6 | 1.001E+0 | 3.287E−5 | 1.000E+0 | 9.090E−6 | 1.001E+0 | 1.500E−5 | 1.001E+0 |
3.287E−5 | 1.001E+0 | 1.073E−4 | 1.004E+0 | 3.287E−5 | 1.002E+0 | 3.287E−5 | 1.002E+0 |
1.073E−4 | 1.004E+0 | 3.456E−4 | 1.010E+0 | 1.073E−4 | 1.004E+0 | 1.073E−4 | 1.004E+0 |
3.456E−4 | 1.010E+0 | 1.121E−3 | 1.022E+0 | 3.456E−4 | 1.010E+0 | 3.456E−4 | 1.008E+0 |
1.121E−3 | 1.022E+0 | 3.682E−3 | 1.046E+0 | 1.121E−3 | 1.020E+0 | 1.121E−3 | 1.016E+0 |
3.682E−3 | 1.047E+0 | 1.000E−2 | 1.076E+0 | 3.682E−3 | 1.039E+0 | 3.682E−3 | 1.031E+0 |
1.000E−2 | 1.085E+0 | 2.500E−2 | 1.110E+0 | 1.000E−2 | 1.063E+0 | 1.000E−2 | 1.052E+0 |
2.500E−2 | 1.132E+0 | 5.000E−2 | 1.135E+0 | 2.500E−2 | 1.091E+0 | 2.500E−2 | 1.079E+0 |
5.000E−2 | 1.169E+0 | 7.000E−2 | 1.142E+0 | 5.000E−2 | 1.112E+0 | 5.000E−2 | 1.098E+0 |
7.000E−2 | 1.181E+0 | 1.000E−1 | 1.139E+0 | 7.000E−2 | 1.117E+0 | 7.000E−2 | 1.103E+0 |
1.000E−1 | 1.183E+0 | 1.500E−1 | 1.117E+0 | 1.000E−1 | 1.113E+0 | 1.000E−1 | 1.099E+0 |
1.500E−1 | 1.165E+0 | 2.000E−1 | 1.085E+0 | 1.500E−1 | 1.090E+0 | 1.500E−1 | 1.076E+0 |
2.000E−1 | 1.134E+0 | 2.250E−1 | 1.067E+0 | 2.000E−1 | 1.058E+0 | 2.000E−1 | 1.044E+0 |
2.250E−1 | 1.116E+0 | 2.500E−1 | 1.050E+0 | 2.250E−1 | 1.041E+0 | 2.250E−1 | 1.027E+0 |
2.500E−1 | 1.099E+0 | 3.000E−1 | 1.015E+0 | 2.500E−1 | 1.024E+0 | 2.500E−1 | 1.010E+0 |
3.000E−1 | 1.063E+0 | 3.500E−1 | 9.813E−1 | 3.000E−1 | 9.894E−1 | 3.000E−1 | 9.759E−1 |
3.500E−1 | 1.029E+0 | 4.000E−1 | 9.498E−1 | 3.500E−1 | 9.565E−1 | 3.500E−1 | 9.435E−1 |
4.000E−1 | 9.961E−1 | 4.500E−1 | 9.204E−1 | 4.000E−1 | 9.257E−1 | 4.000E−1 | 9.130E−1 |
4.500E−1 | 9.657E−1 | 5.000E−1 | 8.931E−1 | 4.500E−1 | 8.970E−1 | 4.500E−1 | 8.847E−1 |
5.000E−1 | 9.374E−1 | 5.500E−1 | 8.678E−1 | 5.000E−1 | 8.703E−1 | 5.000E−1 | 8.583E−1 |
5.500E−1 | 9.111E−1 | 6.000E−1 | 8.443E−1 | 5.500E−1 | 8.456E−1 | 5.500E−1 | 8.339E−1 |
6.000E−1 | 8.866E−1 | 6.500E−1 | 8.224E−1 | 6.000E−1 | 8.226E−1 | 6.000E−1 | 8.113E−1 |
6.500E−1 | 8.639E−1 | 7.000E−1 | 8.020E−1 | 6.500E−1 | 8.013E−1 | 6.500E−1 | 7.902E−1 |
7.000E−1 | 8.426E−1 | 7.500E−1 | 7.830E−1 | 7.000E−1 | 7.814E−1 | 7.000E−1 | 7.706E−1 |
7.500E−1 | 8.227E−1 | 8.000E−1 | 7.652E−1 | 7.500E−1 | 7.628E−1 | 7.500E−1 | 7.523E−1 |
8.000E−1 | 8.041E−1 | 8.500E−1 | 7.485E−1 | 8.000E−1 | 7.455E−1 | 8.000E−1 | 7.352E−1 |
8.500E−1 | 7.867E−1 | 9.000E−1 | 7.328E−1 | 8.500E−1 | 7.292E−1 | 8.500E−1 | 7.191E−1 |
9.000E−1 | 7.702E−1 | 9.500E−1 | 7.180E−1 | 9.000E−1 | 7.139E−1 | 9.000E−1 | 7.040E−1 |
9.500E−1 | 7.548E−1 | 9.750E−1 | 7.110E−1 | 9.500E−1 | 6.995E−1 | 9.500E−1 | 6.898E−1 |
9.700E−1 | 7.488E−1 | 9.800E−1 | 7.096E−1 | 9.750E−1 | 6.926E−1 | 9.750E−1 | 6.829E−1 |
9.800E−1 | 7.459E−1 | 9.825E−1 | 7.089E−1 | 9.800E−1 | 6.912E−1 | 9.800E−1 | 6.817E−1 |
9.850E−1 | 7.459E−1 | 9.830E−1 | 7.088E−1 | 9.830E−1 | 6.904E−1 | 9.850E−1 | 6.817E−1 |
9.850E−1 | 7.088E−1 | 9.850E−1 | 6.904E−1 |
*same values as in
Dimensional quantity | Dimensionless quantity | Remark |
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