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A163973 Decimal expansion of Van der Pauw's constant = Pi/log(2) 2
4, 5, 3, 2, 3, 6, 0, 1, 4, 1, 8, 2, 7, 1, 9, 3, 8, 0, 9, 6, 2, 7, 6, 8, 2, 9, 4, 5, 7, 1, 6, 6, 6, 6, 8, 1, 0, 1, 7, 1, 8, 6, 1, 4, 6, 7, 7, 2, 3, 7, 9, 5, 5, 8, 4, 1, 8, 6, 0, 1, 6, 5, 4, 7, 9, 4, 0, 6, 0, 0, 9, 5, 3, 7, 2, 1, 3, 0, 5, 1, 0, 2, 2, 5, 9, 0, 8, 3, 8, 7, 9, 6, 0, 4, 0, 1, 6, 0, 8, 9, 6, 5, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Van der Pauw developed a method for measuring the sheet resistance of a four-terminal conducting sheet of arbitrary shape. Assuming the terminals to be point contacts at the periphery of the structure, he proved a general theorem which yields an analytical expression for the sheet resistance Rs. In the special case that the structure is invariant for a rotation of ninety degrees, the formula of Van der Pauw is Rs = (Pi/log(2))*(V/I).

A general theorem for the sheet resistance Rs of a Van der Pauw structure with finite contacts which is invariant for a rotation of ninety degrees was proved by Versnel. His theorem states that Rs = [K(k1)/K'(k1) - K(k2)/(2*K'(k2))]^(-1)*(V/I) with K(k) and K'(k) complete elliptic integrals with modulus k (Abramowitz and Stegun use parameter m = k^2).

Versnel found, with a little help from the author, expressions for Rs = C(d)*(V/I) for several Van der Pauw structures if d, the ratio of the sum of the lengths of the contacts and the length of the boundary of the sheet, tends to zero, see the formulae (first two terms are given). For point contacts, i.e. d = 0, Van der Pauw's constant appears.

REFERENCES

L.J. van der Pauw, A method of measuring specific resistivity and Hall effect of disc of arbitrary shape. Philips Research Reports, Vol. 13. no. 1, pp 1-9, February 1958.

W. Versnel, Analysis of symmetrical Van der Pauw structures with finite contacts. Solid State Electronics, Vol. 21, pp. 1261-1268, Pergamon Press Ltd, 1978.

W. Versnel, Analysis of theGreek cross, a Van der Pauw structure with finite contacts. Solid State Electronics, Vol. 22, pp. 911-914, Pergamon Press Ltd, 1979.

LINKS

Table of n, a(n) for n=1..103.

Wikipedia.org, Van der Pauw method

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 17, pp. 589-626.

Eric. W. Weisstein, Elliptic Integral, from Wolfram MathWorld.

FORMULA

1) Circle with contacts in the middle of each side:

C(d) = Pi/log(2) + (Pi^3/(64*(log(2))^2))*d^2

2) Square with contacts in the middle of each side:

C(d) = Pi/log(2) + (Pi*K^2/(8*(log(2))^2))*d^2

3) Square with complementary contacts:

C(d) = Pi/log(2) + (Pi*K^4/(64*(log(2))^2))*d^4

with K = K(sqrt(2)/2) = 1.8540746773.

4) Greek cross with contacts at the cross ends:

C(d) = Pi/log(2) + 2*Pi/(log(2))^2*exp(Pi/2-Pi/d)

5) Greek cross with contacts between the cross ends:

C(d) = Pi/log(2) + ((Pi/(2^12*log(2)^2)*((-3/4)!/(-1/4)!)^8))*d^4

EXAMPLE

4.5323601418271938

MATHEMATICA

RealDigits[N[Pi/Log[2], 103]][[1]] (* Mats Granvik, Apr 04 2012 *)

CROSSREFS

Cf. A000796 (Pi), A002162 (log(2)), A093341 (K), A131223 (2*Pi/log(2)).

Sequence in context: A201337 A114263 A094850 * A124118 A016716 A004485

Adjacent sequences:  A163970 A163971 A163972 * A163974 A163975 A163976

KEYWORD

cons,easy,nonn

AUTHOR

Johannes W. Meijer, Aug 13 2009

STATUS

approved

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Last modified April 19 05:22 EDT 2014. Contains 240738 sequences.