_{1}

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 [_{c}_{1} can be accurately calculated symbolically, see Section 6.1. In strong magnetic fields the symbolic approximation [

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 [_{c}_{2} and _{a}, magnetisation M (from the induced currents), the resulting (total) field:

_{r} defined by GL equations in

the lower, the upper and thermodynamic critical magnetic fields: B_{c}_{1}, B_{c}_{2} and

independent of k or B, defined externally (through ξ and BCS theory [

Form 1 [ | Form 2 [ |
---|---|

magnetic induction _{0} is magnetic permeability of vacuum) and by the GL coefficients

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_{em}―is related to super-current and to magnetic field; the first term of F_{core}-to the local density of the Cooper pairs, and the second term:

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., f_{r} and to q_{r} (form 1, _{r} and b_{kr} (form 2). In the circular cell approximation both the order parameter and magnetic flux density have axial symmetry and for this case two stationary GL equations in two forms are listed in

Complementing the Equations (1) (2) and (3) (4) Maxwell equations are:

From Equations (4) and (5) one gets:

since

Since

where _{1} and c_{2} are the integration constants set by the boundary conditions. So far the solution is the same as obtained in [

where

The boundary conditions can be found e.g., in [_{1} and c_{2}, such as based upon [

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 c_{1} and c_{2} 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 [

_{4} and moreover it crosses the horizontal axis at 0.985 (instead of at 1 as Equation (27) implies). The error in

In this paper the error is eliminated in the following way. The Equations (21)-(26) are only valid at_{c}_{2} on the true magnetisation curve

the 1^{st} 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:_{1} becomes equal to that set by Equation (28) and the transition from m_{1} to m_{4} is smooth since the higher derivatives are preserved. Furthermore, used for the comparison (in Section 5) magnetisation m_{2} from [ [

In conclusion, the correction makes the obtained solution compliant with [_{c}_{2} and the same direction of the magnetisation curve at this point). It should be noted that this correction of the magnetisation uses the symbolic form of the theoretical result [_{2} also agree with the conditions of Equations (27), (28) only means that these conditions are just). Corrected this way magnetisation m_{1} is in excellent agreement with the conditions of Equations (27), (28) and is further compared to that calculated numerically (and with other symbolic methods) in Figures 5-8.

In Figures 5-8 the magnetisation m_{1} is compared to that calculated numerically [

_{1} to magnetisation calculated using other methods. Namely, in Figures 5-8 I compare m_{1} to m_{2} - m_{5} being respectively magnetisation calculated from: this work (m_{1}); [_{2}); [ [_{2} with limited validity (m_{3}); [_{4}) and [_{5}).

As clear from Figures 5-8, for any value of the GL parameter

tween m_{1} and m_{2} is achieved: the relative difference

everywhere (except in the narrow range:

Representative for the range of

The relative difference

_{3}) to-

tally fails to describe the data (m_{2}) quantitatively, since k < 3 and thus

is too high. The data represented by magnetisation m_{1} (and m_{2}) are in good

agreement with these represented by m_{4} at

this case). It is clear from the figure that m_{5} also fails to describe quantitatively the data represented by m_{1} (and m_{2}): e.g., at

reading [

Representative for the range of

In

1.1% and 1.3% respectively in the entire range _{3}) fails to de-

scribe the data (m_{2}) quantitatively, since k < 3 and thus

data represented by m_{1} (and m_{2}) are in good agreement with these represented by m_{4} at

are below 0.4% (_{5} fails to describe quantitatively the data represented by m_{1} (and m_{2}): e.g., at b = 0.25 the relative dif-

ference

competitive and not what one expects after reading [

Representative for the range of

(except in the range: _{3}) describes the

data (m_{2}) quantitatively, since k > 3 and thus

represented by almost the same line in the figure. The data represented by m_{1} (and m_{2}) are in good agreement with these represented by m_{4} at

that both

_{5} still fails to describe quantitatively the data represented by m_{1} (and m_{2}): e.g., at

ing [

Representative for the range of

_{3}) describes the data (m_{2}) quantita-

tively, since k > 3 and thus

almost the same line in the figure. The data represented by m_{1} (and m_{2}) are in good agreement with these represented by m_{4} at

is clear from _{5} describes quantitatively the data represented by

m_{1} (and m_{2}): with larger relative difference

ative (on

Representative for the range of

and

below 0.6% in the entire range

(m_{3}) describes the data (m_{2}) quantitatively, since k > 3 and thus

small [_{1} (and m_{2}) are in good agreement with these represented by m_{4}

at

low 0.5%, with_{5} describes quantitatively the data represented by m_{1} (and m_{2}): with larger relative difference

magnetisation first derivative (on

Representative for the range of

_{3} instead of m_{2} in Fig-

ure 8(a). The relative difference

in _{1} (and m_{3}) are in good agreement with these

represented by m_{4} at

at _{1} (and m_{2}) are in good agreement with these represented by m_{4} at

0.3%, with_{5} describes quantitatively

the data represented by m_{1} (and m_{3}) with larger relative difference

tion 1^{st} derivative (on_{5} describes quantitatively the data represented by m_{1} (and m_{2}) with larger relative

difference

error in the magnetisation’s first derivative (on

The fragment of the magnetisation curve at k = 200 exemplifies that the error in the 1^{st} derivative of the magnetization m_{5} is present at highest values of k and

results in the noticeable difference_{2}

caused by the digitalisation of the data [ [

To summarise, over the entire ranges of the GL parameter _{1} and m_{2} is

achieved: the relative difference

the narrow range:

Presented in Annex data for the dependencies of

In this paper I calculate values of the field b_{c}_{1} from Equations (25) (26) typically at_{c}_{1}. The black solid line is calculated from the fit to the numerical results [

with

The red dashed line is calculated from the symbolic expression for isolated flux line [

In this range of k the b_{c}_{1} changes by 5 orders of magnitude, so I find the agreement between

As noted [

This correction is dealt with in section 4.2. Moreover, _{min}, (derived from Equations (16) (17) through the minimisation procedure) corresponding to the data in Annex. The box on each curve corresponds to the value of magnetic field _{min} can be used as a reference when comparing different methods of solving GL equations.

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 _{3}Sn; NbTi and REBCO respectively) and assuming vortex lattice with hexagonal unit cells. The solid lines represent m_{1} as proposed here; the dashed line ? m_{3} (interpolated from the fit [ [_{40} for the magnetisation in all cases is derived from Equation (28) (assuming_{3} gives errors as expected [

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 |
---|---|---|