Lonal Bu ' dards

i« aB Reference Book riot to be

JUL 8 1964

taken from the |i br3ry>

^.ecknical vtete 241

CALCULATIONS FOR COMPARING

TWO-POINT AND FOUR-POINT

PROBE RESISTIVITY MEASUREMENTS

ON RECTANGULAR BAR-SHAPED

SEMICONDUCTOR SAMPLES

LYDON J. SWARTZENDRUBER

^

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

Technical Note 241

ISSUED JUNE 1, 1964

CALCULATIONS FOR COMPARING

TWO-POINT AND FOUR-POINT

PROBE RESISTIVITY MEASUREMENTS

ON RECTANGULAR BAR-SHAPED

SEMICONDUCTOR SAMPLES

Lydon J. Swartzendruber

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C., 20402 - Price 25 cents
Contents

1. Introduction , 1

2. Expressions for the Correction Factors.... 2 2.1 General potential expression for a bar with plated ends 2 2.2 General potential expression for a bar with unplated ends 4

2.3 Correction factor for the case of unplated ends 4

2.4- Correction factor for the case of plated ends 5

3. Computation of the Correction Factors , 5 3.1 Series convergence. 5 3.2 Fortran program for use in the case of plated ends 7 3.3 Fortran program for use in the case of unplated ends 7

4. Errors Due to Geometrical Factors 7 4.1 Errors due to orobe displacement 7 4.2 Errors due to dimensional inaccuracy in the bar size and shaoe'8

4. 3 Effect of non-uniform end plating 8 4.4 Effect of unequal probe spacing 9

5. Conclusion 11

Appendix I - Program for Plated Ends Correction Factor ........13

Appendix II- Program for Unplated Ends Correction Factor 18

Appendix Ill-Program for Non-Uniform End Plating Error 23

II
CALCULATIONS FOR COMPARING TWO-POINT AND FOUR-POINT PROBE RESISTIVITY

MEASUREMENTS ON RECTANGULAR BAR-SHAPED SEMICONDUCTOR SAMPLES

Lydon J. Swartzendruber

Fortran codes are given which enable the calculation

of four-point probe correction factors for use with barshaped samples. Samples with either plated or unplated

ends are considered. The errors that arise due to probe

misplacement, inaccurate sample size and shape, and non- uniform end plating are also considered. Use of the re- sults permits accurate comparison of two-point and fourpoint probe resistivity measurements. The codes are in

Fortran II language and were written for an IBM 7090

comouter.

1. INTRODUCTION

When measuring the resistivity of a semiconductor, such as silicon

or germanium, the four-point probe as described by Valdes^- is very con- venient and is in common use. However, the four-point probe may give inaccurate results for several reasons, such as charge carrier injection

by the current probes and surface leakages. The accuracy attained by a four-point probe measurement is hence dependent on such factors as the

semiconductor sample material, the probe material, the sample surface

finish, and the probe pressure. Since the two-point probe is generally

a more accurate method when a bar-shaped sample can be formed, a comparison of two-point and four-point probe measurements on the same bar sample will indicate the accuracy being obtained by use of the four-point

probe. Thus the effect of different experimental conditions can be as- sessed, and the range of validity of four-point probe measurements can be

determined. Also, such a comparison can be used to calibrate four-point

probe measuring apparatus.

When using the two-probe method on a rectangular bar-shaped sample,

as illustrated in Figure 1, the sample resistivity is given by

V ah ,_. P--T— (1)

where p is the sample resistivity, V is the potential difference measured between the two voltage probes, I is the uniform current being

passed through the sample, a is the sample width, h is the sample height,

and s is the separation between the two probes.

L.B. Valdes , "Resistivity Measurements on Germanium for Transistors",

Proc. IRE, 42, 420 (1954).
When using the four-point probe method, as illustrated in Figure 2,

the sample resistivity is given by

p = j

(2tts) I (2)

where p is the resistivity, V is the potential difference measured between the two voltage probes, I is the current being passed through the

two outer probes, s is the probe spacing, and F is a correction factor

which is a function of the sample width, height, and length, the probe

spacing, and the position of the probe on the sample. In calculating

the correction factor, F, ideal conditions are assumed, i.e., no carrier

injection, no surface leakage, etc.

It is of interest to note that, as can be seen by the reciprocity

theorem, the same correction factor will apply when the roles of the

current and voltage probes are reversed.

The correction factor, F, has been derived by Hansen^for a fourpoint probe on a rectangular bar sample having all surfaces in contact

with an insulating medium, and has been extended by Reber^to the case where the bar ends are plated with a highly conducting material and for

a more general type of probe placement than shown in Figure 2. General

ootential formulas are presented below, along with the correction factor

formulas for the most interesting cases. To facilitate calculation of

the correction factors, Fortran programs suitable for use on the IBK

7090 and similar machines are given. Several representative cases are worked out and a discussion of the errors introduced by geometrical

factors and poorly plated end contacts is given.

2. EXPRESSION FOR THE CORRECTION FACTORS

2.1 General Potential Expression for a Bar with Plated Ends

When the ends of the bar are plated with a highly conductive

material, with the negative current probe located at r = (x , h, z ),

(see Figure 2 for definition of coordinate system), and with the positive

current probe located at r+ = (x+ , h, z+), the potential, <J>(x,y,z), at

any arbitrary point between the two current probes (z+<z<z ), as shown

by Reber3, is given by

CO CO d>(x,y,z) = V V cos (^^) cos (-^Tr^-) \c sinh(az)+D cosh(az)} - — z T * J * l l a 2h l a a ' ah m=o n=o

(m,n)^o,o

2 . . E. B. Hansen, "On the Influence of Shape and Variations in Conductivity

of the Sample on Four-Point Measurements," Appl. Sci. Res., 8_, 93 (1960)

3 J. M. Reber, "Potential Distribution in a Rectangular Bar for Use with

Four-Point Probe Measurements ," Solid-State Electronics, (to be published)
Figure 1. Two-point probe on a rectangular barshaped sample.

Figure 2. Four-point probe on a rectangular barshaped sample. The probe is centered

along the length of the bar and displaced a distance A from the center of

the bar.
where A nnrx mux

C

a

= sinh(ab) {sinh[a(z__-b)]cos(——-)-sinh[ a (z++b)]cos( )}

" mux mux

Da

= cosh dab) f sinh ta(z__-b)]cos( )+sinh[a(z++b)]cos( ±0}

21 p cos(-r-) a~ aha(lt6o,m)(l+6o,n)

a = [(UH)

2 + (£1)

2

] 1/2

a 2n

and

r 1 if s=o

5o,s = . _ , , . * o if sj<o . (3)

Above, and in what follows, the letter b is used to denote one-half the

length of the bar.

2.2 General Potential Expression for a 3ar with Unplated Ends

If, instead of having the ends of the bar plated, they are unplated,

then we have the same expression for <j>(x,y,z) with (z+<z<z ) as given by

equation (3), except Ca and D a are replaced by

A mux TT1TTX c =_ , / , \ {cosh[a(z -b)]cos( )+cosh[a(z++b)]cos( )} a cosh (, ab.) l — a

x a and

A rnirx mux

n = • ; ) r \ {cosh[a(z -b)]cos( )-cosh[a(z++b)]cos( )}. (4) a smn(ab) l — a

+ a

'

2.3 Correction Factor for the Case of Unplated Ends

The correction factor, F]_, for an in-line four-point probe array

when placed on a bar sample as shown in Figure 2, with all surfaces of

the bar in contact with an insulating medium, was first worked out by

Hansen^. It can also be obtained by using the general solution of equation (4) and is given by

Fl

= 27Ts( ah

+ A

l + B

l

) v/here

_ 8_ r r cosh[g(b-3s/2)]sinh(8s/2)

1 ah * * (l+6o,m)(l+6o,n)Bcosh(3b)

m=o n=o * * (m,n)^(o,o)
» ao (-l) m" 1 sin

2 (^-)coshC Y (b-3s/2)]sinh(Ys/2)

~ _ o__ r y a ^___

1 ah '•_ ^ (l+6o,n)Ycosh(Yb) m=l n=o '

,2-n. r 2 , ... v2-,l/2

6 = ( —) Cm + (na/2h) ] ,

3.

Y

= (Tr/a)[m

2 + (na/h)

2

] 1/2

. (5)

The resistivity of the sample is found by using this correction factor in

equation (2). This solution has the nice feature that if the sidewise

displacement of the probe, A, is zero, the series for Bj_ is identically

zero. This solution only applies, however, when the probe is centered

lengthwise on the sample.

2.4 Correction Factor for the Case of Plated Ends

The correction factor, F2 , for an in-line four-point probe array

placed on a bar sample as shown in Figure 2 with the bar having its ends

plated with conductive material, is shown by Reber^to be

where

and

F

2 = 2irs(s/ah + A

2 + B )

(L.) Y Y sinh[3(b-3s/2)]sinh(3s/2)

2

~ ah ;

*• ^ (l+5o,m)(l+6o,n)B sinh(Sb)

m=o n=o (m,n)^o,o

00 00 (-l)

rn" 1sin2 (^.)sinh[Y(b-3s/2)]sinh( Y s/2)

3

2

= ( ih ) ^ < (l+6o,n) Y sinh(Yb) (6)

m=l n=o

and where 3, Y, and <5 are as defined in equations (3) and (5). This

equation can be obtained by using the general solution of equation (3).

Note that it is the same as equation (5) except that all hyperbolic co- sine terms have been replaced by hyperbolic sine terms. The resistivity

of the sample can then be found by using this correction factor in

equation (2).

3. COMPUTATION OF THE CORRECTION FACTORS

3.1 Series Convergence

In calculating the correction factors, Fj_ and F2» we need to know

the number of terms in each series which must be summed in order to

achieve a specified accuracy. In equations (5) and (6), let the summations indices m and n in the series Aj_, Bi, A2 or B2 become very large,

5
then all the hyperbolic sine terms and all the hyperbolic cosine terms

can be replaced by their limiting exponential forms. If the summation

has been carried out for every combination of m and n up to m=M and n=ll

,

then good apnroximations for the remainders, R^ for series Aj_ or A2 , and

Rg for series Bj_ or 32, are given by

A ah

t exp i27TS a

r 2 [m + ,na,2 1/2

1

K 2\\ J ;

m=N n=N

2tt r — 2 ,na*2-,l/2 [m + ( w) ] dm dn (7)

and

R.. = 8_

ah

exo I 1TS r 2 m + ( na )2] l/2

;

7T r 2 — [m

m=N n=iJ

f na,2n l/2

k h; J dm dn, (3)

Now in (7) make the substitution x=m and y=na/2h and let M« equal either

(a/2h)N or N, whichever is smaller. In (8) make the substitution x=m

and y=na/h and let Mjj equal (a/h)N or N , whichever is smaller. Then we have for the remainders

2tts 2 2,1/2

A ira

. exp{- (x +y ) j

f 2 ' "27172"

(x + y ) dx dy

x=M

A y=MA

and

B ira

r tts , 2 2,l/2i

f expi- —- (x +y ) \

x=M

3 y=M

fi

, 2 2~T72"

(x + y ) dx dy.

(9)

(10)

These integrals can be estimated by converting them to polar form and

integrating from a circle of radius MA or Mg to infinity. This increases

the value of the integrals, but still gives a reasonable upper bound for

the remainder. One obtains, then,

and

8 t 2tts „ v R. < — exp( M.)

A tts a A

R

3

< - eX? ( " TV

(11)

(12)

Since the factor F is always one or greater, the values of HA and MB re- quired to make 2ttsR^ and 2irsRg small in comparison to one will yield the

required number of terms that should be summed to insure the accuracy of

the computation.
3.2 Fortran Program for Use in the Case of Plated Ends

The program for computing F2 , the correction factor for a fourooint probe on a bar sample with plated ends, is given in Appendix I. The formats for the required data cards are also illustrated, as is the

data printout. The program corresponds to the probe being centered on

the bar lengthwise and displaced sidewise a distance DELTA. The values

of the sample width W, height H, overall length of the bar EL, probe

spacing S, sidewise displacement DELTA, and the desired accuracy ACC,

are read in on data cards as specified in statements 146 and 147. The

number of different conditions to be calculated is read in as specified

in statements 143 and 144, and hence any number of various conditions

can be computed on a single run. The units on W, H, EL, S and DELTA are arbitrary, but must be consistent. ACC should be given in decimal units,

i.e., if 0.1% accuracy is desired ACC = 0.001. The program consists of

a main program, which appears last, four function subprograms, and two

subroutines. The printout is as specified in statements 151 and 152.

The number of terms computed for any single value of F2 is automatically

limited to 10^, which should be adequate for most practical purposes.

If more than 10^ terms would have been necessary to achieve the required

accuracy, the accuracy actually achieved is computed and printed out

with the answer. The code is as written in FORTRAN II language to be run

on National Bureau of Standards' modified BELL SYSTEM. Some slight

changes in input and output statements would be required to run the program on the II B MONITOR SYSTEM. A short table of values for F2 com- puted by use of the program is given in Table I.

3.3 Fortran Program for Use in the Case of Unplated Ends

This program computes Fj_, the correction factor for a four-point

probe on a bar sample with unplated ends and is given in Appendix II.

It is identical to the program for plated ends, Appendix I, except for

statements 29, 31, 40, 43, 59, 61, 70 and 73, which are changed respectively to the statements given in Appendix II. The same data cards

illustrated in Appendix I are used. The code is as written in FORTRAN II

language to be run on National Bureau of Standards' modified BELL SYSTEM.

Some slight changes in input and output statements would be required to run the program on the II B MONITOR SYSTEM. A short table of values com- puted by the program for Fj_ is given in Table II.

4. ERRORS DUE TO GEOMETRICAL FACTORS

4.1 Errors Due to Probe Displacement

When a four-point probe measurement is made, the correction factor

will usually be calculated for the probe symmetrically placed in the

center of the har. Since experimentally the probe cannot be exactly centered, it will be displaced from the center along the length of the bar

a distance e and sidewise a distance A. The effect of a sidewise displacement, A, is included in the program for the correction factor and
can be directly calculated. For example, as can be seen from Table I,

for a probe spacing of 0.0625 in. and a bar 0.1 in. wide- by 0.1 in. high

by 0.5 in. long, an inadvertent sidewise displacement of 0.001 in. would

introduce an error of about 0.01% in the correction factor.

The effect of a lengthwise displacement can also be estimated by

use of the program for the correction factor. If the probe is displaced

lengthwise an amount e on a sample with unplated ends, the correction

factor will be increased. However, this increase will be less than the

increase for a symmetrically placed probe caused by a decrease in sample

length of 2e. For example, from Table II, for a probe spacing of 0.0625

in. and a sample 0.1 in. wide by 0.1 in. high by 0.5 in. long, a de- crease in length of 0.1 in. causes the correction factor to increase

about ,0.02%. Thus the probe could be displaced lengthwise a distance

0.05 in. off center without introducing errors larger than about 0.02%.

The same argument applies for a sample with plated ends except that the

correction factor decreases with decreased length.

4.2 Errors Due to Dimensional Inaccuracy in the Bar Size and Shape

We wish to know the dimensional tolerances required on the bar

length, width and height in order to achieve a specified accuracy in the

measurement. For a two-point probe measurement the accuracy will depend

on how well the cross sectional area of the sample is known. Thus if

the width and height are both known within, say, 1%, then the error due

to bar size tolerance can be as high as 2%.

The required bar size tolerances when using the four-point probe

are easily calculated by use of the correction factor programs. From its

known dimensions, the irregular bar can be bounded by two perfect bars,

one which completely encloses the irregular bar, and one which is com- pletely enclosed by the irregular bar. Then the correction factor lies

between that obtained for the two perfect bars. An example of the

effect of the tolerance due to a single dimension can be obtained from

Table I. There it is seen that, for a bar 0.1 in. high by 0.1 in. wide

by 0.5 in. long, with a probe spacing of 0.0625 in., a height or width

change of 0.1% causes a change in correction factor of 0.1%, while a length change of 20% causes a correction factor change of only about

0.01%.

4.3 Effect of Non-Uniform End Plating

In general, a perfect plating is hard to achieve. When making twopoint probe measurements, a non-uniform plating on the bar ends will disturb the uniformity of current through the sample, thereby introducing

an error into measurement. In order to see how necessary a uniform

plating is, consider the worst possible case of non-uniform end plating

as shown in Figure 3. Here the end plating is reduced to a point current

source, and the measurement is being made with the probe along the edge,
as shown. The potential, V, measured by the probe can be found with the

aid of equation (4) to be

V.plSpdti)

where

00 CO

m=o n=o

(m,n^o,o)

.m (-1)' (-D

(l+Som)(l+6on)

n rsinh(yd) -sinh y(d-s)

Y sinh (yb)

y = (ir/a) Cm

2 + (na/h)2

] 1/2

b = 1/2

A Fortran program to calculate 6 is given in Appendix III. It is seen that 6 is a measure of the error introduced into a measurement of p if

uniform current were to be assumed. A graph of 6 vs. the ratio of probe

displacement, d, to half bar length, b, is shown in Figure 4 for several

different cases of width, length and height. This figure shows the importance of uniform end plating. If the bar length is 20 times the probe

spacing and the width and height are each equal to 4 probe spacings, an error greater than 1% occurs over most of the bar length, and the error

can be as high as 26% near the bar ends. However, if the experimental

conditions are chosen as in curve (1) of Figure 4, then any error due to

non-uniform end plating can be neglected over a large portion of the bar.

4.4 Effect of Unequal Probe Spacing

One wishes to know the effect of probe spacing tolerance on the

value of the correction factor. In the limiting case of s>>a and s>>h,

the resistivity is given by equation (1). In this case only the spacing

of the two inner probes is of importance and

Ap/p = -As/s

where As/s is the fractional change in spacing of the two inner probes

and Ap/p is the fractional change in the measured resistivity.

In the limiting case where s<<a, s<<h, and s<< the bar length, the

factor F of equation (2), as given by Valdes , is

P = s (I + I . _i L_>

s l

s 3 sl +s2 s 2 +s 3

where S^ is the spacing between probes (1) and (2) (see Figure 2), S2

is the spacing between the two inner probes (2) and (3), S3 is the

spacing between probes (3) and (4), and S can be interpreted as the mean probe spacing. If S2 = S3 = S and the spacing Sj_ varies slightly, the

fractional deviation in measured resistivity is
~v^

Figure 3. Geometry to illustrate the worst possible case of non- uniform end plating. The current passes in and out of

the bar ends only at the corners as shown. The largest

effect is for a two-point probe located along the corner

as shown.

-0.4

-0.3-

O

h- -0.2 -

0.2 0.4 0.6 0.8 PROBE DISPLACEMENT, d/b

1.0

Figure 4. Graph of the deviation, &

t caused by the non-uniform

plating as shown in Figure 3, vs. the probe displacement d/b, where d is as shown in Figure 3 and b is the

half bar length, 1/2. Curve (1) is for a=0.1, h=0.1,

£=1.0 and s=0.0625. Curve (2) is for a=4.0, h=4.0,

£=20.0 and s=1.0. Curve (.3) is for a=6.0, h=6.0, £=20.0

and s=1.0. Curve (4) is for a=10.0, h=10.0, £=20.0 and

s=1.0.

10
to .. 3 fl

p 4 S

If S3=S and Sj_+S2=2S but S2 varies slightly from S, then the fractional

deviation in measured resistivity is

p 4 s Thus, in the limiting cases, the percentage error in measured resistivity

is roughly the same as the percentage error in probe SDacing, and one ex- pects this also to be true for the intermediate cases. The effect of a sidewise displacement of a single needle is of second order and will

normally be negligible if ordinary tolerances are held.

5. CONCLUSION

Using the Fortran programs given, the correction factors necessary

to make four-point probe measurements on rectangular bar-shaped samnles

can be obtained. If the four-point probe measurement is valid, it should

compare with a two-point probe measurement to within the accuracy limits

set by the geometrical tolerances, as discussed in Section 4. By making

such a comparison, the range of validity of a four-point probe measurement can be established for a particular semiconductor and the effect of

such factors as surface treatments and probe pressures can be determined.

This work has been supported by the Advanced Research Projects

Agency under order No. 373-62.

11
TABLE I

Correction Factor F2 for a Sample with Plated Ends

Width

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.10001

0.1001

0.100

0.100

Overall Probe

Height Length

0.500

Spacing

0.0625

Delta

0.0

F2

0.100 2.8703

0.100 0.500 0.0625 0.0001 2.8703

0.100 0.500 0.0625 0.001 2.8706

0.100 0.500 0.0625 0.01 • 2.9050

0.100 0.500 0.0625 0.02 3.0085

0.100 0.500 0.0625 0.05 3.4163

0.100 0.400 0.0625 0. 2.8699

0.100 0.300 0.0625 0. 2.8614

0.100 0.200 0.0625 0. 2.6181

0.100 0.500 0.0625 0. 2.3700

0.100 0.500 0.0625 0. 2.8677

0.10001 0.500 0.0625 0. 2.8701

0.1001 0.500

TABLE

0.0625

II

0. 2.8635

Correction Factor F]_ for a Sample with Unplated Ends

Width

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.10001

0.1001

0.100

0.100

Overall Probe

Height Length

0.500

Spacing

0.0625

Delta

• 0.0

Fl

0.100 2.8703

0.100 0.500 0.0625 0. 0001 2.8703

0.100 0.500 0.0625 0. 001 2.8706

0.100 0.500 0.0625 0. 01 2.9050

0.100 0.500 0.0625 0. 02 3.0085

0.100 0.500 0.0625 0. 05 3.4163

0.100 0.400 0.0625 0. 2.8707

0.100 0.300 0.0625 0. 2.8791

0.100 0.200 0.0625 0. 3.1217

0.100 0.500 0.0625 0. 2.8700

0.100 0.500 0.0625 0. 2.8677

0.10001 0.500 0.0625 0. 2.8701

0.1001 0.500 0.0625 0. 2.8686

12
APPENDIX I

Fortran Program for Calculation of ^j the Four-Point Probe Correction

Factor 'Jhen the Bar Ends are Plated.

A. Format of Data Cards

First Card : Variable

NDATA

Second Card: Variable

Ends in Column

W

H

EL

S

DELTA

ACC

Ends in Column

10

20

30

40

50

60

Meaning of

Example Variable

5 The number of

data cards that

follow.

Meaning of

Example Variable

0.100 the samole width

0.101 the sample height

0.500 the overall sample length

0.0625 the probe spacing

0.001 the sidewise

probe spacing

0.0001 the desired com- putation accuracy

The third and subsequent cards have the same format as the second card.

B. Example of Data Printout

Parts 1 and 2 below will be printed side by side in the printout.

SERIES A ACCURACY and SERIES B ACCURACY refer to the accuracy actually

achieved by the program in the computation of these two series. SERIES

A TERMS and SERIES 3 TERMS show how many terms were actually calculated

by the program for each series.

1.

PROBE

WIDTH HEIGHT LENGTH SPACING DELTA

0. 10 00 0. 10000 0.50000 0.06250 0.

0. 10000 0.10000 0.5C0GQ 0.06250 0.01000

0. 10 000 0. 10000 0.50000 0.06250 0.00100

o.ioooo 0. 10000 0.50000 0.06250 0.00010

0. 10000 C. 10000 0.5C000 0.06250 0.00001

13
2.

CORRECTION SERIES A SERIES B SERIES A SERIES 8

FACTOR ACCURACY ACCURACY TERMS TERMS

2.87025926 0.00010 0.00010 17

2.90499508 0.00010 0.00010 17 17

2.87060627 0.00010 0.00010 17 17

2.87026271 0.00010 0.00010 17 17

2.87025928 0.00010 0.00010 17 17

C. Program

1 FUNCTION FUN0 1IUN 8ETA(M,N) BklA(MfiN)

2 COMMON W,H,EL,S,ACC,ACCA, A0,ACCC,A00,MA,MB

3 EM=M= V

4 FN = N

BETA=(6.23 318i>3/W)*SURTF(EM»#2+{EN»W/(2.»H) )**?.)

ft F TURN

j

6 RETURN

END

7 FUNCTION GAMMA (M f N) ' 8 COMMON W,H,EL,S,ACC, ACCA, AO, ACCB , A00,MA,MB

9 FM = M LC EN=N

11 GAMWA=(3.14159265/W)»SQRTF(EM**2 + (EN*Wi /h)»*2)

12 RETURN

END

13 FUNCTION TERMA(M,N)

14 CCMMOM WfHtELfStACCf ACCAf AOtACCBfAOOtMAffMB

15 IF (M+N-l) 16,18,20

16 TtRNA=0.0

17 RETURN

18 DEM=2.

19 GC TO 21

20 0EN=1.

II AA=S/2.

22 Ai3 = EL/2.

23 AC=AB-3.*AA

2 4 Q=BETA(V,NJ

25 IF (Q*A0W5.) 26,26,34

?6 SH=EXPF<Q*AA)

27 SHA=0.5*(SH-1./SH)

28 SH=EXPF(Q*AB)

29 SH8=0.3*(SH-1./SH)

14
30 SH=EXP

31 SHC=0.

32 TCRVA=

33 R-ETURN

14 IF {Q*

3 b TERMA=

36 RETURN

3/ IF (Q*

38 SH=EXP

39 SHA=SH

40 TER.VA=

41 RETURN

42 SH=EXP

43 TERMA=

44 RETURN

END

F{(j*AC)

5*(SH-i./SH)

SHC*SHA/(CEN*Q*SHB)

S-75.) ;>7,35,35

CO

(2.*AB-S)-75.) 38,42,42

F(-0*S)

*( l.-SH)

( -SH4 + E X PF ( -Q* ( 2 .* AB-S )

) -t XPF ( -Q *2 . » i AB-S ) ) ) / < 2 . *Q* DEN

)

F{->>*S)

(SH*( L.-SH) )/(2.*Q*DEN)

45 FUNCTl

46 CGMVUM

47 IF(N-1

4>3 DEfi = 2-

49 GC TO

50 0EN=1.

5L AA=S/2

52 AB=EL/

53 AC=AB54 Q=GAMM

55 IF (Q*

36 SH=EXP

57 SHA=0.

53 SH=EXP

i 9 ShB=0.

60 SH=EXP

61 SHC=0.

62 TERMB=

63 RETURN

64 IF (Q»

65 TERyB=

66 RETURN

t>7 IF (Q»

68 SH=EXP

69 SHA=SH

70 TER.VB = 71 RETURN

72 SH=EXP

73 TERMB=

74 RETURN

END

JN TEKMB(M,N)

Vt , H , EL , S , ACC , ACC A , AfJ , ACCB , AflO , MA , MB

) 4 3,50,50

L

2.

3.**A

A I M , N ) AB-75.) 56,56,64

F(Q*AA)

5MSH-1./SH)

F(Q*AB)

5MSH-1./SH)

F(Q*AC)

5MSH-1./SH)

SHC*SHA/(DEN*G*SH3)

S-75.) 6 7,65,65

0.0

(2.*AB-S)-75. ) o0,72,72

F(-Q»S)

*( l.-SH)

ISHA+EXPF(-Q*(2.*AB-S) ) -EXPF ( -Q*2 . * ( AB-S )

)

)/<2.*Q*DEN)

F(-Q*S)

(SH*(1.-SH) )/<2.*G*DEN)

15
SUBROUTINE SKIES A

COMMON W,H,EL,S,ACC,ACCA,AC,ACCe,AOC,MA,MB

P=3. 14159265

EM=(W/(2.*P»S) )«LOGF(8./ACC)

IF (2.»H/W-1.) 80,80,82

M = l-W+1 -

7 5

76

77

78

79

8C N=fcM+l.

81 GO TO 83

62 N=(2.*H/W)*EM+1.

23 IF (N-17) 84,^1,86

84 N=17

8 3 GO TO 91

86 IF (330-N) 87,91,91

87 N=330

88 EN = N

89 ACCA=S.*EXPF(-2.*P*S»EN/W)

90 GC TO 92

91 ACCA=ACC

iZ SUM=0.0

93 DC 98 1=1,

N

94 11 = 1-1

')b DC 97 J=l,N

96 JJ=J-1

97 SUM=SUM+TERMA(II,JJ)

98 CONTINUE

99 AC=(8./(W»H) )*SUM

100 MA = N

LCI RETURN

END

SUBROUTINE SRI ESB ( DELTA

)

COMMON w,H,EL,S,ACC,ACCA,AO,ACCB,AOO,MA,M.b

IF (DELTA) 109,105,109

ACCB=ACC

102

103

104

105

106 MB=0

107 ACO=0.0

108 RETURN

109 P=3. 14159265

110 EM=(W/(P»S))*LOGF( 32./ACC)

111 IF (H/W-l.) 112,112,114

112 N=EM+1.

113 GC TO 115

114 N=(H/W)*EM+1.

115 IF (N-17) 116,123,118

116 N=17

117 GC TO 123

118 IF (330-N) 119,123,123

119 I l 9 to=nn N=330

16
120 EN=N

121 ACCB=32.»EXPF(-P*S*EN/W)

122 GO TO 124

123 ACCB=ACC

124 SUM=0.0

125 ARG=P*UELTA/W

126 SIGN=-1.0

127 DC 134 1=1,

N

128 SIGN=-SIGN

129 AI=I

130 T=SIGN*(SINF( AI*ARG) )**2

131 DO 133 J=1,N

132 JJ=J-1

133 SUM=SUM+T*TERMB{ I, J J

)

134 CONTINUE

135 AC0=(8./(W*H) )*SUM

136 MB=N

137 RETURN

END

IJ8 COMMON W,H,EL,S,ACC,ACCA, AO, ACCB , AGO,MA,MB

139 PRINT 140

140 OFORMAT ( 1H 1 , 33X5HPRGBE2 IX 1QHC0RRECT I0NSX8HSERI ES A4X8HSERIES B2X8H

141 ISERIES A2X8HSERIES 3/1H , 4X5HW IDTH4X6HHE IGHT4X6HLENGTH3X7HSPACI NC5

L42 2X5HDELTA12X6HFACTQK8X8HACCURACY4X8HACCURACY4X5HTERMS5X5HVERMS/1H ) 143 READ 144.NDATA

144 FORMAT /( I 3 ) 145 DO 153 I=1,NDATA

146 READ 147, W,H, EL, S, DELTA, ACC

147 FCRMA1 (6F10.0)

148 CALL SRIESA

149 CALL SRIESB(DELTA)

150 F=6.283185 3*S*(S/(W»H)+A0+A00)

151 PRINT 152, K,H, EL, S, DELTA, F , ACCA, ACCB, MA, MB

152 FORMAT ( 1H , 5F10.5,F20. 8 , 2F12. 5, 2 1 10)

153 CONTINUE

CALL SYSTEM

END

17
APPENDIX II

Fortran Program for Calculation of Fj_, the Four-Point Probe Correction

Factor When the Bar Ends are Unplated.

A. Format of Data Cards

The format for the data cards is identical with that given in

Appendix I.

B. Example of Data Printout

Parts 1 and 2 below will be printed side by side in the printout.

SERIES A ACCURACY and SERIES B ACCURACY refer to the accuracy actually

achieved by the program in the computation of these two series. SERIES

A TERMS and SERIES B TERMS show how many terms were actually calculated

by the program for each series.

1.

PROBE

VsICTH HEIGHT LENGTH SPACING 1 • ELTA

0. LCOCC . 1 C C 0.5C0C0 0.062 50 C.t 0. IC0C0 0. 1CCC0 0.5C000 0.06250 0.,01.000

0. ICCCO 0. LCCC0 C.5C0 00 0.06250 0..00100

0. ICCCO 0. 1CCC0 0.5C0CO 0.06250 0..000 10

0. 1C0C0 0. 1CCC0 C.5C0C0 0.0625C c.•COCCI

2.

CRRECTICN SERIES A SERIES P SERIES A SERIES B

FACTOR ACCURACY ACCURACY TERMS TERMS

2.87029204 O.OOCIO 0.00010 17

2.90503 103 C.00CI0 0.C0010 17 17

2.P7C63912 C.C0CL0 0. C00 10 17 17

2.37029549 O.OOCIO O.OOCIO 17 17

2.870292C7 o.cociu 0.00010 17 17

C. Program

1 FUNCTION RETA{M,i\)

2 COMMON W,H, CL;S,ACC,ACCA, AO,ACCH f ACG.f MA.K8

3 EM = M

A EN=N

5 DETA=(6. 2831853/W)*S0P.TF( E.v * *2+ ( EN*'a/ < 2.*U) )**2)

RET C P- N

E N D

13
( FUNCTION GAMMA(N,N)

B COMMON fc,HfEL,S t ACC,ACCA t Aa,ACCBf ACCtMA.MB

EM = V

IC EN=N

11 GAMMA=( :!.14159265/V.')»!>QRTF(E ,***2 + ( EN*W/H)*«2)

12 RETURN

END

13

14

15

16

17

ie

19

2C

21

22

23

24

2 5 26

27

28

29

JC

31

32

33

34

35

36

J7

*e

39

4C

41

42

4 3 44

FUNC

CCNN

IF ( TERN

RETL

DEN*

GC 7 OEN=

AA=S

AE = E AC = £ C = DE

IF { SF = E SFA = SF = E SHB = SH = E SFC = TERM

RETL

IF ( TERN

RETL

IF ( SF = E SFA = TERN

RETL

SF = E TERN

RETL

END

TI

GN

N +

A — M — RN

2.

C

1.

/2

L/

BTA

G*

XP

0.

XP

c.XP

0.

A=

RN

C*

A^

RN

G*

XP

Sh

A = RN

XP

A = RN

CN TERWA(f,N)

H f H, CLf S f ACCACCA, AO f ACCEt AOOf^AfMB

N-l) 16,18,20

0.0

21

2.

3 . * A A

( f , \ ) AR-75.J 26,26,34

F<Q»AA)

5«(.SH-1./SHJ

F ( G * A 6 ) 5MSH+1./SH)

F(Q*AC)

5MSH+1./SH)

SHC*SHA/ ( CEN*C»SHE )

S-75.) 37,35,35

0.0

[2.*AE-S)-75. ) 33,42,42

F(-Q*S)

*( L.+SH)

(SHA + EXPF(-G»(2.**B-S) )+FXPF(-Q*2.*( AB-S) ) )'/(2.*Q«Cfc~N)

F(-Q«S1

(SHM1. + SF) )/(2.*C*DEN)

4 5 FUNCTION TEKMB(FtN)

46 CCNNCN tt,H,EL,S,ACC,ACCA,AO,ACCE ,ACO,.w A,MB

47 lF(N-l) 4B,50,5C

4d DEN=2.

49 GC TO 51

5 C C F N = I

.

19
51

52

53

54

55

56

57

58

59

6C

61

62

63

64

65

66

67

68

69

7C

71

72

73

74

A* = S AE = E AC = A Q = GA

IF < Sh = E SKA = SH = E SHB = SH = E SFC = TERN

RETL

IF { TERN

RETL

IF ( SH = E SHA = TERN

RETL

Sh = E TERN

RETL

END

/2

L/

BNM

G*

XP

0.

XP

0.

XP

0.

B = RN

C*

B=

RN

C*

XP

SH

B = RN

XP

B=

RN

2.

3.*AA

A(Ni,N)

AE-75.) 56,56,64

F(0»AA)

5*(SH-1./SH)

F(Q*AB)

5«<SH*1./SH)

F(0*AC)

5*(SH+i./SH)

SHC«SHA/(CEN*C*SHB)

S-75.) 67,65,65

0.0

(2.»AB-S)-75.) 68,72,72

F(-0*S)

*( l.+Sh)

(SHA*FXPF(-Q*12.*A3-S) ) +EX PF ( -Q*2.* ( AB-S ) ) )/(2.»G*CEN)

F(-G*S)

(SH*( l.+Sh) )/(2.*G*DEN)

75 SLBPCUTINE SRIESA

76 CCNNGN U ,H , FL , S , ACC , ACC A , AO, ACCE , AGO, NA,N6

77 P = 3. 14159265

78 EN=(W/(2.*P*5) )*LCGF(8./ACC)

79 IF (2.«F/W-1.) 80,30,82

8C N=EN+1.

81 GG TC 83

82 N=(2.*H/W)»EM+1.

83 IF (N-17) 84,91,86

84 N=17

85 GC TC 91

86 IF (330-N) 87,91,91

87 N=330

88 EN=N

89 ACCA=8.*EXPF(-2.*P*S*EN/W)

9C CC TC 92

91 ACC*=ACC

92 SLN=0.0

93 DC 9 8 1=1,

N

94 11=1-1

95 CC 97 J=1,N

96 JJ=J-l

97 SLM=SUM+TFRNA( II, JJ)

98 CCNTINUE

20
99 AQ=te./(W*H) )*SL«

ICC MA=N

IC1 RETURN

END

1C2 SUBROUTINE SK IES 8 ( DELTA ) 1C3 COMMON WiHiELtSt ACC , ACC A , AO , ACCfi , AGG,PA,f*'B

LC4 IF (DELTA) 109,105,109

IC5 ACCB=ACC

1C6 M6=0

1C7 AOO^O.O

1C8 RETURN

1C9 P=3. 14159265

110 EM=(W/(P»S) )*L0GF(32./ACC)

111 IF (H/W-l.) 117,112,114

112 N=EM+1.

113 GO TO 115

114 N=(H/W)*EM+i.

115 IT (N-17) 116,123,118

116 N=17

117 GO TO 123

118 IF (330-N) 119,123,123

119 N=330

120 EN=N

121 ACCB=32.«EXPF(-P*S*EN/W)

122 GO TO 124

123 ACCB=ACC

124 SUM=C.O

125 ARG=P»DELTA/W

126 SIGN=-1.0

127 DO 134 1=1,

N

128 SIGN=-SIGN

129 AI=I

130 T=SIGN*(SINF(AI*ARG) )**2

131 DO 133 J=1,N

132 JJ=J-1

133 SUM=SUM+T*TERMB( I, JJ)

134 CONTINUE

135 A00=(8./(W*H) )*SUM

136 MB=N

137 RETURN

END

21
138 COMMON W,H,EL,S,ACC,ACCA, AO, ACC8» AOO»MA,MB

139 PRINT l^tO 140 OFORMAT ( 1H 1 , 33X5HPRGBE2 IX 10HCORRECT I0N6X8HSER I ES A4X8HSEHIES B2X8H

141 ISERIES A2X8HSERIES E/1H , 4X5HW ICTH4X6HHE IGHT4X6HLENCTH3X7HSPAC ING5

14 2 2X5HDELTA12X6HF4CT0R8X8HACCURACY4X8HACCURACY4X5HTERMS5X5HrERMS/lH ) 143 READ 144,NCATA

144 FORMAT (13)

145 DO 153 I=1,NDATA

146 READ 147,W,H,EL,S f DELTA,ACC

147 FORMAT (6F10.0)

148 CALL SRIESA

149 CALL SRIESB(DELTA)

150 F=6.2831853*S»(S/(W*H)+AO+AGO)

151 PRINT 152,h,H,EL f S t CELT'A,F, ACCAfACCBffAtFJi

152 FCRNAT ( 1H , 5F 1C . 5, F20. 8 , 2F 12 . 5 , 2 1 1 C

)

153 CONTINUE

CALL SYSTEM

END

22
APPENDIX III

Fortran Program for the Calculation of 6.

A. Format of Data Cards

First Card : Variable Ends in Column Example

NDATA 3 2

Meaning of

Variable

The number of data

cards that follow

Meaning of

Variable Ends in Column Example Variable

W 10 0.100 the sample width

H 20 0.100 the sample height

EL 30 0.500 the overall sample

length

S 40 0.050 the probe spacing

D 50 0.005 the position of the

probe (as shown in

Figure 3)

The third and subsequent cards have the same format as the second card.

B. Example of Data Printout

In the printout A0 is the desired value for 6.

W = O.lCOCOf- = 0.1OCC0L = 0.5uOCOS = 0.0625 0C = 0. AG^

W=0. 1C 0001- = 0. 10 000 L = 0.b00C0S= 0.062 50l>0. 5 0C0AQ;

W=0. lCCG0h=0. 10C00 L = 0. 5 0000 S= 0.06 2500=0. lOOOOAG^

W=0« lCCC0h = 0. lOOCOL = 0.5f)OCOS=0.06250D=C. 15000AO = W=0. lCCCCI- = 0. LOCOOL = 0.5bOCOS=0.06250D=C.20COOACh

-0.CC5165C3

-0. (10404362

-0.C1410C08

-0.C57662C0

-0. 18221583

C. Program

FLMCTICN SAypA(N,N)

CCPPCK W,H,EL,S,C,PI t AO

Ef = N Ef\=f\

G/WA=( FI/W)*SGPTF(EM»*2+(EN*W/I- )**2)

KETIPN

END

23
FUNCTION TERM(M,N)

CCKPCN V«,H,EU,S,D,PI,AO

DENCM=1.

IF (M+N-1) 1,2,3

TERN=O.C

RETURN

DENCN-2.

C=GAMA(M,N)

A = Q*C

B=C*(C-S)

C=Q*EU/2.

IF (C.-75.) A, A,

5

A=EXPF(A)

A=0.5*(A-l./A)

B=EXPF(6)

B=0.5«(B-l./e)

C=EXPF(C)

C=0.5*(C+l./C)

TfcRN=(A-B}/(C»DENGM«C)

RETURN

IF (C-7 5.) 6,6,7

AA=EXPF(C-E)

EE=EXPF(C-A)

TERV=(1./RB-1./AA)/(Q«DENCM)

RETURN

IF (C-A-75.) 8,8,9

AA=EXPF(C-A)

TERN=1./ (AA*G*DENCM)

RETURN

TERN=G.C

RETURN

ENC

CCNNCN WtHtELtSfDfPI, AQ

PI=3. 14159265

REAC 1,NDATA

FCRNAT (in

DC 2 I-1 V NCATA

REAC 3,W,H,EU,S,D

FCRNAT (5F10.C)

CAUU SUN

PRINT 4,W,H,EU,S,C,A0

FCRVAT ( 1H ,2FW=F1C. J ,2HH=F 10 .5, 2HU=F 10.5

13FAC=F12.3 ) CCNTINUE

CAUU SYSTPy

END

,2HS=F10.5,2HD=F10.5

2H
SUP ROUTINE SUM

COMMON W t Hf EL,S,0,PItAO

5 EGNA=-l

.

AO=0.0

K =I=-l

6 1=1+1

SIGNA=-SIGNA

SIGNE=-1.

6 = 0-

J = -l

4 J = J + 1 SIGNB=-SIGNB

e=(SIGNA)*(SIC\'E)*l4.0/S)*TERM(I,J)

A = A+B

K = K+1

IF { J-5) 4,4,10

1C IF (K-LCOQC) 1,2,2

1 IF (ABSF(B)-l.OE-d) 3,3,4

3 AO=AO+A

IF (AESF ( 'O-l.OE-5) 5,5,6

5 RETURN

2 A0=1.

RETURN

END

25

* U. S. GOVERNMENT PRINTING OFFICE : 1964 O - 732-560