calibration of a scintillation beta ray …

CALIBRATION OF A SCINTILLATION BETA RAY SPECTROMETER

by

JERRY WRIGHT MOULDER, B.S.

A THESIS

IN

PHYSICS

Submitted to the Graduate Faculty of Texas Technological College

in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

Approved

Accepted

TEXAS rrCHNOLOGlCAL

LUSttlOCK, TEXAS ; > < d i i t^\>t k I

COLLET

'^G'WZ'i

9 (0(0

ACKNOWLEDGMENT

I am deeply indebted to Dr. D. A. Howe for his

direction of this thesis.

11

TABLE OF CONTENTS

ACKNOWLEDGMENT ii

LIST OF TABLES iv

LIST OF FIGURES v

I. INTRODUCTION 1

II. THEORY 9

III. DESCRIPTION OF COMPUTER PROGRAM 1?

IV. RESULTS AND CONCLUSIONS 25

LIST OF REFERENCES 35

APPENDIX 36

I. Standard Deviation of a Computed (i uantity . . 37

II. Computer Program 39

iii

LIST OF TABLES

Table Page

1. Correlation of Computer Quantities with Quantities in this Thesis • 19

IV

LIST OF FIGURES

Figure Page

1. Beta Ray Spectrum of Sn " ^ 3

2. Beta Ray Spectrum of Bi * 4

3. Beta Ray Spectrum of Cs^^ 5

4. Graphical Determination of the Line Center , 6

5. Computer Program Flow Chart » 18

6. Shifted Curve Fit to the Cs" * Internal Conversion Line 23

11-5 7. Curve Fit to the Sn ^ Internal Conversion

Line 26 8. Curve Fit to the Cs^^ Internal Conversion

Line 21 9. Curve Fit to the Bi ^ Internal Conversion

Line 28

10. Energy Versus Channel Number Calibration Curve 29

11. Weighted Curve Fit to the Bi " Internal Conversion Line 30

12. Unweighted Curve Fit to the Bi * Internal Conversion Line 31

15. Curve Fit to the Cs^'^ Line and Additional Points 32

14. Curve Fit to the Cs" * Line 33

CHAPTER I

INTRODUCTION

A typical scintillation beta ray spectrometer con

sists of a scintillator, photomultiplier tube, amplifier,

and multichannel analyzer. Beta particles from the

radioactive source produce photons of light or scintil

lations in the scintillator. These scintillations strike

the photocathode of the photomultiplier tube, producing

electrons which are attracted to the first dynode of the

photomultiplier tube. Secondary electrons are produced

as a result of the impact of these electrons on the

dynode. The secondary electrons are accelerated to the

next dynode where the same process occurs. This happens

on every dynode, resulting in a significant current pulse

at the last one. This current pulse produces a voltage

pulse across a resistance. The voltage pulses are ampli

fied further and are stored in different channels of the

analyzer according to the heights of the pulses. The

pulse heights are related to the energy of the beta

particles emitted by the source.

The important question to be answered is; What is

the relationship of channel number and energy? Any method

of calibration of the spectrometer answers this question.

Without an answer, the results of an experiment could not

be expressed in terms of energy and consequently no

information concerning energy levels of nuclei could be

2

obtained.

Three beta-active sources, Sn" " , Bi * , and Cs *

are used to calibrate the spectrometer. All of these

sources have a beta ray energy spectrum with one feature

in common. (See Figures 1,2, and 3.) They decay by

electron capture or beta particle emission to excited

states of the daughter nuclei. The daughter nuclei

usually go to their ground states by gamma ray emission.

But part of the time they decay by a process called in

ternal conversion. A nucleus, in an excited state,

ejects a K, L, or M shell electron instead of a gamma ray.

This mode of decay occurs often enough in these sources

so that pronounced peaks appear in the beta spectrum.

These internal conversion lines appear at an energy of

370 KeV for Sn" " , 631 KeV for Cs" * , and 988 KeV for 207 1 Bi . The calibration procedure is to determine what

channel numbers correspond to these three energies.

Previously it was necessary to graph the portion of

each spectrum containing the internal conversion line, to

draw the "best" line through the points on the slopes of

the line, and to read from the intersection of these two

lines, the channel number. See Figure 4. This method

was subject to significant error. The subjective nature

of the procedure alone would warrant investigation of a

better way of determining the channel number. A shift of

up to one channel number from one person to another was

COUNTS 5000

4000

3000 -

2000 -

1000 -

Internal Conversion Line

10 20 30 40 30 50 70 80 ChA.\:;£L NUMBER

Fig. 1—Beta Any ^ pectrum of Sn ^

COUNTS

5000 r

4000


3000 —

2000 -

1000

80 90 100 120 CHA:«NEL NUMBER

Fig* 2—Beta Hay Spectrum of Bi 207

COUNTS 6000

5000 -

4000 -

Internal Conversion Line 3000 —

2000 —

1000 -

10 20 30 40 60 70 80 90

CHANNEL NUMBER

Fig. 3—Beta Ray Spectrum of Cs 137

COUNTS

1800 r-

Line Center - 1 1 9 . 5

1600

1400

1200

1000

800

600

400 1 1 1 1 1 \ I I \ I 100 104 108 112 11

•^ z '

120 124 128 132 136 140 CHANNEL NU71BER

Fig. 4—Graphical Determination of the Line Center

noticed. A person's judgment as to what was the "best"

line to draw and his past experience definitely affected

the value. Another cause of concern was the amount of

time involved.

After the three channel numbers were found, a plot

of energy versus channel number was made. For acceptable

data, this plot should be linear. If not, the photo

multiplier tubes, preamplifiers, or amplifiers used in

the experiment were not operating properly. In the

checkout of the spectrometer before a run was to be made,

the components of the spectrometer just mentioned were

checked and this usually involved the multichannel ana

lyzer and the use of the three calibration sources. So

the calibration procedure described was performed often.

For these reasons, a more accurate and easier method of

calibration seemed mandatory.

The purpose of this^thesis is to investigate a new

calibration procedure. The internal conversion lines have 2

a Gaussian shape. So it should be possible to fit a

Gaussian curve to the internal conversion line and knowing

the constants of the curve it should be possible to cal

culate the channel number at the peak value of the curve.

Also, from tne manner of the curve fit, it should be pos

sible to find the standard deviation of the computed

channel number. A curve fit of the data would require a

computer and thus the development of a computer program.

8

The remainder of this thesis will concern the procedure

just outlined. The procedure was applied to a 4/7" scin

tillation spectrometer using two Nuclear Enterprise NE102

detectors. The pulses from the two photomultiplier tubes

were amplified and added by a Tennelec amplifier. The

output from the amplifier was analyzed by a RIDL 400

channel analyzer. The application of this method to

other systems will be mentioned later.

CHAPTER II

THEORY

The general form of a Gaussian curve is,

Z - AexpC-(X-Xo)^/cJ (1)

where A is the maximum value of Z, X© is the abscissa for

which Z = A, /c"is the half-width at Ve of A. (Z is the

number of counts as a function of X, the channel number.)

Taking the logarithm of both sides of Eq. 1 yields,

Y = ln(Z) = ln(A) - X?/C + 2XoX/C - X^/C. (2)

This quadratic expression will be fitted to the logarithm

of the input data.^ Now a parabola has the form,

Y - C^ + C2X + C,X^. (3)

Comparing with Eq. 2,

C^ » -1/C, C^ = 2Xo/C, C^ = ln(A) - xf/C . (4)

Eq. 3 can be written,

3 Y = ;^ C.f.(X), Where f. (X) - X""*" .

i-1

Let N be the number of data points to be used in the

curve fit. The different data points are identified by a

subscript, j. So,

3 Y. . ± C,f,(X.) . (3)

i«l

10

There are N equations like Eq. 5 with three unknowns.

These equations are never realized in curve fitting since

in general X. and Y, are subject to uncertainties. In the

case here, X. is exact, while Y. *ln(Z.) is subject to J %} 0

counting statistics. The standard deviation of Z. is 0

Z..^ Since Y. differs from ^C.f.(X.), 0 0 "^-^ 1 1 0

Y - £ C,f,(X.) = 5 . (6) 0

i=l

where Q^ is called a residual. The set of N equations

like Eq. 6 may be solved for the C's by various methods

The most widely used method is the method of least

squares. That is, the C's are chosen such that the

sum of the squares of the residuals is a minimum,

N M = U- ^ minimum.

J=l

This is true when, - ^ = 0 for k = 1, 2, and 3.

So,

N

But from Eq. 6,

ac, ' :^ oj 3 0 o. ^ j=l

^ O.i , _f ex :)

(7)

Then Eq. 7 becomes, cancelling the 2, and substituting

11

from Eq. 6,

Since f^(X) « X" "" ,

N / 3 \

- £ [ £ Vi( j)- j V^j) -0 d-i\ i=l /

N 3 N

£ £ Vi(^j)V^j) « £ j kf j). j=l i'l i=l

i-1

N 3 N

£vr"'=£ vr- ^ j»l i=l j=l

This system of 3 linear equations will be solved by the

elimination method by the computer. The computer program

will be discussed later.

A close examination of the above procedure will show

that minimizing the sum of the squares of the residuals,

Q . is actually a "weighted" least squares fit to the

internal conversion line. This weighting was introduced

because of the taking of the logarithm of Eq. 1. Con

sider the following,

^d = Zj-^^"<-^^0-^°^'/°)- (9)

Now minimizing Vj . is not the same as minimizing

£ /\ . If Z. is uncertain by an amount, Z ., Y .=ln(Z .) 0 j J • J 0 J

is uncertain by an amount AZ./Z.. See Appendix I.

Since Q . is the "error" in Y. and j . is the "error" in

12

Z., then^

0 "0 J

So minimizing ^ 0j ^^ actually minimizing ^ Jf/Z^.

Each ]f. is weighted by VZ.. An unweighted least squares

N

fit would require minimizing, N « ^ Z^ O^ • ^— J J

Then setting ^ = 0 yields,

J = l ^

which reduces to,

N 3 N

-4 ^ Z^CX^""^"^ = ^ Z^Y.X^"^. (10) ^ : z i j i j ^ j j j ^ ^ j=l i=l j=l

This is an alternate expression to Eq. 8.

A widely used method for determining the goodness of

a curve fit is the chi square test. Chi square is de

fined by the equation,-^ 2 P / ((observed value). - (expected value)j

/C " y^ (expected value).

or N (^Z^-AexpC-(X^-Xo)^/C)J

2 X - 1 - : 1 1^ Q^) ^ ' - ^ AexpC-(X^-Xo)^/C)

using the previous notation.

13 o p

Comparison of the '^ for the curve fit with y^ found in tables yields a probability, P that a repetition

of the experiment would show greater deviations from the

assumed curve. The usual interpretation of the tables

is that if P lies between .1 and .9 the assumed curve very

probably corresponds to the observed data. But if P is

less than .02 or more than .98, it is very unlikely that

the assumed curve corresponds to the observed data.

Assume for the moment that C^, Cp, and C^ have been

found. The standard deviations of the C's are desired.

From Eq. 6, again since f.(X) = X ~ "i

-2c,Y.X?-2C-,Y.X.-2C,y. .

N 2 So M - £ S j = ^Y^c2 £x%2C5C2 £x5+(2C^C5+c|) £ X^

0=1

*2C,C3 £x2.Nc2-2C3 £y.x2-2C2 £XjX.-2C, £ Y . .

Setting ^ . . i ^ . 0 yields, 3C^ SCg 3C,

s£^d^ °2iô-* "î^^* 'î^^r ^^^d = °- ''

Cjix^* S^'^d" ^^d*"î " °- ^ ""

14

Equations 12, 13* and 14 will be solved for C, and Cp

since these two coefficients will later be used, (See

Eq. 4) to find X© the abscissa for which Z = A in Eq. 1.

Xo is the line center. From Eq. 13,

C2- (^Vd-°3^^d^/^^! '5 and from Eq. 14,

Ci=i[£Y.-C3(£x^*£x2)j. (16)

After substitution of these two expressions into Eq. 12

and after a lengthy manipulation,

Cj .[NC £ X 2 Y ^ ) ( £X2)-( £ Y J ) ( £ X J ) ( £ X 2 )

-N(£X.Y.)(£X3)-(£Y.)(£X2)2

.N( xJ)2-( £x^)(£xj)2-(£x^)2(£x2)

-(£xj)2( £x^)-( £xj)5 .

N(£xJ)( £xj)

(17)

The square of the standard deviation of C^, denoted by

.2 C. Si; is,

N

1^14 (18)

i=l

where S^ is the square of the standard deviation of Y^. i

See Appendix I. Let the denominator of Eq. 17 be denoted

by VH. Then,

15 Bc

Hence,

Be 3 ^ 1 '

H

H

^£^j-^£^j)( £x5)-NX^(£x^).(£x2)

Nxf £x2-( £x .)( £x^)-Nx^( £x^)-( £x2)'

Since the Y's are functions of the Z's, the standard

deviation of Y. depends upon the standard deviation of

Z.. The relationship is,

2

So

2 n^i

Q2 1

1

(20)

Further algebra is unnecessary since the computer program

uses Eq. 19 and 20, together with Eq. 18 to compute SQ .

Substitution of Eq. 17 into Eq. 15 yields, after some

algebra,

C2 - H N( f X't)( £x,Y,)-2( £x,Y,)( £x.)( £x2) 0 0 J 3

-(£x,Y,)(£x2)2.(£x,Y,)(£x.) J 0 0 0

-N( £X2Y,)( £x^)^( £ Y )( £x )( £x^) J J

* ( £ Y , ) ( £ X ' ) C £ ^ ^

Again, S^

N

i-l "aY" ' i

'd

(21)

16

and,

- 1 ^ - H|X,[N( £XJ)-2( :£XJ)( £x2)-( £x2)2.( £^^^

-NX2( £x5)+( £'x^)( £x5)+( £x2)( £x5) I (22)

Eqs. 20 and 22 are substituted into Eq. 21 to find

2 Sp .

^2

The l i n e c e n t e r , from Eq. 4 i s

*» " " 2C^ '

The standard deviation of X© can be found from.

(23)

2 2 Substitution of S^ and S^ from previous equations

^2 ^3

yields the standard deviation of the line center.

The next step is the development of a computer

program for the available IBM 1620 computer, that will

curve fit the internal conversion line, calculate the

line center, and the standard deviation of the line

center.

CHAPTER III

DESCRIPTION OF THE COMPUTER PROGRAM

Figure 5 shows the flow diagram of the computer

program. The program is given in Appendix II. Table I

correlates quantities used in thesis to those used in the

computer program. A few of the steps in the flow chart

need some clarification.

After the coefficients of the parabola have been

found from the curve fit, these are used to find the

Gaussian curve using Z=exp(Y), where Y is the quadratic

expression, Eq. 3. Then this curve, the input data, and

the fractional error are punched. After the standard

deviation of the line center has been computed, the

fractional errors, divided by the standard deviation of

the input data (called DIF(L) in the program) are com

pared with each other and the largest one is used as a

criterion for performing another curve fit without the

data point whose DIF(L) is the largest. This allows data

points that are not a part of the internal conversion

line to be included in the input data. As the earlier

beta spectra show, it is possible to include data that are

not part of the internal conversion line. This provision

means that within reason, data around the lower portion

of the line may be included and that after discarding

these points, a good curve fit is possible.

17

18

RiiAD AND TAKE

LOGARITHM OF

INi'UT DATA

SYS SOLVE

;n OF Ei UATIONS

FOR OEFFICIENTS

PUNCH

COEFFICIENT^

COMPUTE HALF-WIDTH RESOLUTION, CHI SQUARE, MAXIMUM AMPLITUDE

AND LINE CENTZR

COMPUTE GAUoSIAN CURVE

FROM THE QUADRATIC EXPRESSION

^ ^

PUNCH HALF-WIDTH MAXIMUM AMPLITUDE,

RESOLUTION, LINE CENTER,

CHI SviUARE

COiXPUTE STANDARD DEVIATION OF THE CGEFEICIENTS USED TO FIND TdZ

STANDARD' DEVIATION OF THE

L

FIND LARGEST FRACTIONAL ERROR/

STANDARD DEVIATION

OF LINE CENTER (DIFCD)

COMPUTE

DEVIATION

OF LINE CENTER L

PUNCH

STANDARD

DEVIATION

LARGEST

CIF(L)

REARRANGE

INTUT DATA ARRAY,

LOGARITHM OF

INPUT DATA ARRAY,

A <D CHANTS Aj NJM3r*R

ARRAY

Yif-, 5—Con-.put( r Program Flow Chart

19

TABLE I

P(I) - Input data

Q(I) - Logarithm of input data

CH(I) - Channel number

NXF - First channel number

NXL - Last channel number

MOLY - Order of the polynomial (MOLY = 2)

ALN - Fitted curve

ARR - Absolute value of the difference of the input

data and the curve

ERR - Fractional error

DIF(L) - Fractional error divided by the standard devia

tion of the input data

PEAK - Line center (Xo)

ZLN - Maximum amplitude (A)

AMP - Maximum amplitude (A)

CO - Number of channel numbers (N)

N

CNl . ^ X ^

i=l

N

CN2 - £ i i-l

N 3 CN3 - £ 4

i-l

CN4

DX

HWX

SDCl

SDC2

SDXO

RES

20

N

• i.A i»l

- C

- Standard deviation of C,

- Standard deviation of Cp

- Standard deviation of the line center

- Resolution

21

After the largest DIF(L) has been found it is com

pared with a quantity found through experience to be of

the correct magnitude such that the largest percent error

(fractional error times 100%) is roughly 4 or 5%. This

is considered a good curve fit. If DIF(L) is larger than

this quantity, another curve fit is performed without the

point whose DIF(L) is the largest. The complete program

is repeated each time without the point whose DIF(L) is

too large until the largest DIF(L) is smaller than the

quantity mentioned. This quantity is 1.0 x 10 ^. When

this occxirs the program is completed. Another method to

stop the computer prograjn would be to use the chi square

test. Whenever the probability, P, mentioned earlier,

showed that the observed data was Gaussian, the program

would be stopped. Analysis of several sets of data shows

that the chi square test would be a better means of

stopping the program.

The flow chart also shows that additional quantities

are computed. They include the half-width of the curve,

maximum amplitude, chi square, resolution, standard

deviation of the two coefficients, length of the data run,

DIFCL), and the channel number corresponding to DlFCi-),

that is, L.

The computer program just described is the one deemed

most successful. But other computer programs were tried

before this one was settled on as the "best", r arly in

22 1*57 the work, the curve fit to the Cs ' line was not good.

Consideration of Figure 3 resulted in an attempt to

modify the data before a curve fit was made. The con

tinuous beta spectrum of Cs^'^ in the region of the

internal conversion line was described by an exponential

function. This curve was subtracted from the original

data, and then a curve fit was made. This had little ef

fect on improving the fit. The problem was resolved by

using only the top portion of the line. The problem

would not have appeared if the technique of repeated

curve fits had been developed at the stage of the work

that the problem arose. The problem occurred only for

the Cs- * line.

There exist other means of determining which data

point to discard. One workable method is to neglect that

data point whose fractional error is the largest. Another

usable method is to discard that data point which is

greater than, say, three standard deviations from the

curve. This is the most logical, since it is very un

likely that a data point will be three or even two stand

ard deviations away from the curve. For typical data,

this latter method tended to discard data in the middle

and upper portion of the line. Figure 6 shows that this

results in a shifted curve fit. Using DIF(L) to discard

data, a closer fit was made and a line center of 80.673

was found. It is possible that source thickness or

23

COUNTS 2000 r

1800

1600

1400

1200

1000 -

800

1 I L 1 68 70 12 74 7o ^ "57 ii HH ttto 55 90 b4 OD

CHANN NbTiBER

F i g . 6 . ^ h i f t e d Curve F i t to t h e Cs^57 E t e r n a l Conversion i j i n e

24

multiple internal conversion lines would give rise to a

slightly non-Gaussian line which might explain why this

method of discarding data points does not work. For low

counting rates, that is a maximum of less than a thousand

counts at the peak of the line, little difference was

found between this method and method used in the dis

cussed computer program. This problem deserves some

further study.

CHAPTER IV

RESULTS AND CONCLUSIONS

Figures 7, 8, and 9 compare three curve fits with

the input data of tin, cesium, and bismuth respectively.

The figures also list pertinent quantities computed. Chi

square analysis gives a value of .47, .71, and .55

respectively for the probability that a set of data taken

under the same conditions would show greater deviations

from the Gaussian curve. This implies that the observed

data is very probably Gaussian. These graphs are typical.

Of major importance is the standard deviation of the line

center. The high accuracy of the line center means that

for purposes of calibration, the line center is exactly

determined by the program. This is in sharp contrast to

the previous graphing procedure. Figure 10 shows a plot

of energy versus channel number using the line centers

shown of the previous figures. This graph is the impor

tant graph used to determine if the spectrometer is

operating properly. As mentioned in the beginning, a

linear plot is desired.

Figures 11 and 12 show the effect of using a

weighted versus an unweighted curve fit. It is clear

that little difference is evident. Figure 15 shows a

curve fit when points that are not a part of the internal 157 conversion line are included for the Cs "" line. After

25

26

COUNTS 3500

3000

Line Center - 30.459

Standard Deviation - ±.004

Half Width at Ve of Max- 4.24

Resolution - 23.2%

2500

2000 —

1500

1000 —

1 1 J L 1

26 21 28

Fig. 7—Curve Fit to the Sn

3C

113

31 32 33 34 35

CHANNEL NUMBER


21

COUNTS 4000f—

Line Center - 33«614

Standard Deviation - 1.003

Half Width at Ve of Max - 5.65

Resolution - 17%

3500

3000

2500 -

2000 —

150()-

1000-

Fig. 8-

51 52 53 34 53 36 57 58 59 60 61 CHANNEL rrUMBER

-Curve Fit to tho Cs' ' ' Internal Conversion Line

23

COUNTS 2500


Standard Deviation - ±.005

Resolution - 15.2%

Half Width at Ve of Max

- 8.10

2000 —

1500 —

1000 —

500

1 1 1 i i 1 i J 1 82 84 86 QS 90 92 94

CHANNS 96 98 NUTIB ER

Fig. 9—Curve Fit to the Bi ' Internal Conversion Line

W.-'l

29

•

Energy, KeV

1000 -

900

100 -

30 50 60 70 80 90 100 110

CHANNEL NUMBER

Fig* 10**£nergy Versus Channel Number Calibration Curve

30

COUNTS 2200r-

2000 —

1800 —

1600-

1400-

1200-

1000 —

800 —


90 92 94 96 98 100 102 104 106 108 110 CHANNEL NUMBER

Fig. 11—Weighted Curve Fit to the Bi Internal Donversion Line

207

31

COUNTS

2200r-

2000

1800

1600 —


1400 —

1200

1000

800

1 1 1 1 i 1 1 i 1 1 88 90 92 94 96 98 100 102 104 106 108

CHANNEL N'UMBER

Fig.' 12—Unweighted Curve Fit to the Bi Internal Conversion Line

207

32

COUNTS 2100

1900

1700

1500

1300 -

1100 -

900 —

700 -

Line Center •79.887 Standard Deviation '±.008

600 68 70 72 7^ 76 73 80 Q2 84 86 83 90

CHANNEL NUMBER

Fig. 13—Curve Fit to the Cs '' Line and Additional Points

COUNTS

2100

1900

1700

1500

1300

1100

900

700

J \ L

Line Center -80.676

Standard Deviation - 1.006

33

1 I I I 1 12 74 76 76 80 82 84 86 88 90

CHANNEL NUTIBER

Fig. 14—Curve Fit to the Cs * Line

3^

neglecting these points, the resultant curve fit is

depicted in Figure 14.

From these results, it should be clear that the

calibration method discussed here is an accurate one as

well as an easy one. The procedure has recently been ap

plied to the calibration of the spectrometer before and

after the beta spectrum of Y' and Re was measured.

The computer program has been used to determine areas

under Gaussian curves by Mr. Horton Struve, who is working

in gamma ray spectroscopy. Soon this program will be in

corporated into a beta spectrxim analysis program. This

combined program will allow fast data analysis. The com

puter program has possible uses outside the realm of

scintillation beta ray spectroscopy. Any experimental

data that has a Gaussian form can be fitted by the program

and properties of the data can be found.

LIST OF REFERENCES

1. Waak, Thomas, "Beta Ray Scintillation Spectrometer," Unpublished Master's thesis. Department of Physics, Texas Technological College, 1966, p. 38.

2. Siegbahn, Kai, Editor, Alpha-, Beta-, and Gamma-Ray Spectroscopy, (North Holland Publishing Company, Amsterdam, 1962) p. 535.

3. Pennington, R. H., Introductory Computer Methods and Numerical Analysis, CMacmillan Company. New York. 1965; p. 377.

4. Yuan, L. C. L. and Wu, Chien-Shiung, Editors, Methods of Experimental Physics Volume 3 Nuclear Physics Part B (Academic Press, New York, 1963) P. 774.

5. Yuan and Wu, dp. cit. p. 786.

6. Hoel, P. G., Introduction to Mathematical Statistics (Wiley and Sons, New York, 1962) p. 401.

7. Baird, D. C. , Experimentation: An Introduction to Measurement Theory (Prentice Hall Inc., 1962) p. 62.

35

APPENDIX

I. Standard Deviation of a Computed Quantity

II. Computer Program

36

37

APPENDIX I: STANDARD DEVIATION

OF A COMPUTED QUANTITY

The purpose of this appendix is to derive the gen

eral equation for the standard deviation of a computed

quantity in terms of the standard deviations of the

measured quantities. The formula was used often in the

theory section of this thesis.

The relation will be derived for the case of two

measured quantities but will hold true for any number of 7

quantities.' The problem is this: Suppose that a large

number of measurements of two quantities has been made

xinder the same conditions. From these two quantities,

say X and Y, a function Z is computed. The large number

of measurements allows the calculation of a standard

deviation for X and Y, denoted respectively by S^ and Sy.

S™, the standard deviation of Z is desired.

Now, Z is some function of X and Y,

Z = f(X,Y) So, <i2 = -|| dX + -|| dY.

This means that an uncertainty, SX in X and SY in Y is

propagated to Z by an amount SZ given by

By the usual definition,

c2 ^i^^)^ o2 f C6Y)^ ^ . o2 . ^ iSZ)^ . Sx - ^ N ' ' N — ^ ^ ^Z N

38

where N measurements were made. ( SX is simply X-X

where X is the mean value of X.) Then,

,2 S'z ^ £ [df] S X .(If )SY]

The third term will be zero if the errors SX and SY

are independent. Eq. 25 becomes,

4''(-lif4^(-Mj4' C26) In the general case of M measured quantities, with

independent errors, Eq. 26 is extended to

4-^(-iij4- (27) i=l ^ ^

Eq. 21 is used to find the standard deviations of the

two coefficients of the parabola, the standard deviation

of the line center, and the error in Y ^ » In(Z^) when

the error in Z. is YzT .

£-^c:jyf: IlilllllHH^HHI

39 APPENDIX I I : COMPUTER PROGRAM

40

C fll) IS THE INPUT DATA C QUI IS THE LOGARITHM OF THE INPUT DATA

OINENSION MPC6).NL(5) eiKENSlON CH(200),0IF(20O| • OIMENSION QI200l,P(200l,C(5t5|

83 M0tY«2 NP-I READ 3 5 « M F a ) > N L I l l r j NXF«NF(1) NXL«NL(1| READ 2«TIME,(PCn,I«NXF,NXL) 00 51 I'NXFtNXL

51 CN(n«I 00 50 I *NXFtNXt

50 Q ( I I > L 0 6 F I P i n ) M0tYl«M0I .V4l M0LV2«M0tY42 N«l

C SET UP LINEAR EQUATIONS 8 00 3 I«1«M0LY1 00 3 J«l9N0LY2

3 CilfJ)»0« 00 4 L>NXF«NXL X«CHfLI 00 4 I«itNOLYl ll«I-l IFfI1}13«13«1I

13 XP«1«0 60 TO 12

11 XP«X««I1 12 C U f M 0 L Y 2 ) « C ( I » M 0 L Y 2 ) 4 Q I D ' X P

00 4 J » I » I C ( I t J ) > C ( I » J ) 4 X P

4 XP»X»XP 00 5 J=1«M0LYI 00 5 I ' l t J

5 C ( I f J ) » C ( J » I ) N«MOLYl

C SUBROUTINE SIHPLE C USEO TO SOLVE SYSTE S OF LINEAR EQUATIONS 6 C(ltJ) ARE THE COEFFICIENTS OF THE UNKNOWN VARIABLES. C FOR N EQUATIONS C(NtN*l) IS THE INHOMOGENEGUS TERM

N1«N^1 NF«N^1 NB«N-1 00 10 K^l^NB N1«K*1

41

00 I0 I«N1#I I ' 00 18 J»NlfNF

10 C t i ^ J ) « C f l t J t - € I I f K | « C f K « J | / C ( K , K ) CtNtllF»«CfN»NFI/CfN«N) 00 20 I«1«NB

N1«J«I Cf J fNFHCf J»NF}/Cf J«J) 00 20 K«Nl0N

20 C I J f N P I * C(J»NF| - CU«K)«C(K,NF) /CfJ«J) C C I J t N ^ l l J>I«N CONTAINS THE SOLUTION

PUNCH 6 t f C ( I t N 0 L Y 2 ) » I » l « M 0 L Y l ) XCHI 'O. 00 30 L«NXF«NXL X«CHfL) YLN«CClfN4l)«C(2fN4l)«X^C(3ffN4ll*X«X ALN«EXPF(YLNI ARR«ABSFfPfL)*ALNl ERR«ARR/PfL) DIF(L)«ERR/SQRTF(P(L)) XCHI«XCHI«ARR«ARR/ALN PUNCH 9«PCL)«ALN,ERR

30 CONTINUE 0X»-1.0/Cf3tN4l) HMX«SQRTF(OX) PEAKsOX«C(2*N4l)/2«C XsPEAK YLN«CClfN^l)4C(2tN+l)«X*C(3,N*l)»X«X ZLN»EXPF(YLNI RES«HMX«SQRTF(LOGF(2.0))•200./PEAK PUNCH91«RES PUNCH14,HWXtPEAK PUNCH15fTINEtZLN PUNCH93«XCHI C0«NXL-NXF*1 CN1«0 CN2«0 CN3-0 CN4«0 DO 31 L»NXF,NXL Y«CHCL) CN1«CN1*Y CN2«CN2^Y«Y CN3«CN34Y«Y»Y CN4«CN44Y»Y«Y«Y

?1 CONTINUE H»l.0/(C0«CN4«CN2-CC«CN3»CN3-CN2«CN2»{2.0»CNUCN2)-CNl»CNl»CN2) TUNsO. SUMsO.

46

00 32 L*NXF«NXL Z>CH(L) CY«C0«Z»2«CN2-CN1«CN2-C0«2«CN3-CN2«CN2 CY1«CY»CY S Y « l . 0 / P a i SUM«SUM«CY1«SY 0Y1»Z«IC0«CN4-2.0«CN1«CN2-CN2«CN2-CN1«CN1) 0Y2»CN1«CN34CN2«CN3-CG«Z«Z«CN3 0Y3«0Y1*0Y2 0Y«0Y3«0Y3 TUN«TUM40Y«SY

32 CONTINUE S0C1«H«SQRTFISUN} S0C2«H«SQRTF(TUM} PUNCH 90*SDCl PUNCH16«S0C2 SXl«1.0/l4.0«CI3«N^l}«Cf3fN^l))«S0C2«S0C2 SX2«fC(2fN4l)«C(2tN4in/C4.0«Cf3fN«l)«»4)«S0CI«S0Cl SX«SX1>SX2 SDXO*SQRTF(SX) PUNCH19»SDX0 NN«NXF DO 7 L«NXF,NXL IF(OIF(MN)-OIF(L)l 1,7,7

1 NM«L 7 CONTINUE PUNCH17,0IF(NN) PUNCH17,NM IFfOIF<MM)-.0010) 46,45,45

45 NXL«NXL-l DO 44 I«NM,NXL Q(n«Q(l4l) P<I)«P(I41)

44 CH(I)»CH(l4l) GC TO 8

46 CONTINUE 19 F0RMAT(35HSTANDARD DEVIATION OF LINE CENTER= ,E11.4) 17 F0RHAT(E11.4) 16 F0RMATI26HSTANDARD DEVIATION CF €2= ,E11.4) 90 F0RHAT(26HSTAN0ARC DEVIATION CF Cl= ,E1I.A) 18 F0RMAT(4(2X,E11.4)) 9 F0RMATI2(2X,F8.C)2X,F6.3) 14 F0RMAT(12HHALF klCTH* ,F7.2,13HLINE CENTER= ,F9.A) 15 F0RMAT(5HTIME ,F8.0,5l-Af<P= ,E11.4) 2 F0RHAT(9F8.0) 6 F0RMAT(23HP0LYN0MIAL C0EFFICIENTS/6(2X,EII.4)) 35 F0RMAT(2I4) 91 FCRMAT(11HRESCLUTICN=,F6.2) 93 FCRMAT(11HCHISQUARE»,F7.2)

C GAUSSIAN CURVE FITTING PROGRAM FOR A SCINTILLATION C BETA RAY SPECTROMETER C Pin IS THE INPUT DATA C QUI IS THE LOGARITHM OF THE INPUT DATA

OlMENSliN MFf6l,NLf5) OIMENSION CHt200),OIF(200) OIMENSION QC200 ) ,P (200 ) ,C (5 ,5 )

83 M0LY«2 •lP«i READ 35#MFCl )«NLai NXF«MFfll . NXL«NLfll READ 2 , T I M E , f P ( I ) , I ' N X F , N X L ) 00 51 I«NXF,NXL

51 C H i n » I 00 50 I»NXF,NXL

50 Qin«LOGFCPnn M0LY1«M0LY41 M0LY2-M0LY42 M»l

C SET UP LINEAR EQUATIONS 8 00 3 I*1,M0LY1 t 00 3 J«1,M0LY2

3 cn,j)*o. 00 4 L«NXF,NXL X»CH(L) 00 4 I«1,M0LY1 I1«I-1 IFII1I13«13,11

13 XP«1.0 GO TO 12

11 XP«X««I1 12 C(I,M0LY2)«C(I,M0LY2U Q(L)«XP

00 4 J«l,I C(I,J)»C(I,J)4XP

4 XP«X«XP DO 5 J=1,M0LY1 DO 5 1 1,J

5 Cn,J)»C(J,I) N«M0LY1

C SUBROUTINE SIMPLE C USEO TO SOLVE SYSTEMS OF LINEAR EQUATIONS C C(I,J) ARE THE COEFFICIENTS OF THE UNKNOWN VARIABLES. C FOR N EQUATIONS C(N,N*1) IS THE INHOMOGENEOUS TERM

N1«N*1 NF-N41 NB«N-1 00 10 K»1,NB Nl-K^l

41

00 IOI«NI#N 00 to J«N1,NF

10 CH,J|*CII,Jl-Cn,KMCU,J|/CCK,K| C(NtNP)«CIN,NF)/C(N,NI 06 20 1*1,NB J«iN«I N1«J41 CrJ tNF)»CCJ ,NF | /C fJ ,J | 00 20 K«N1,N

20 CfJ ,NPI« CCJ.NFI- CCJ,KMCCK,NFl /C(J,J) C C U f N ^ l l J«1,N CONTAINS THE SOLUTION

PUNCH 6,(€iIfM0LY2),I«l,M0LYl) XCHI«0. 00 30 L»NXFtNXL X«CHIL> YLN»C(l,N4l}4C(2,N4l)«X4C(3,N4l)«X«X * ALN>EXPF(YLNI ARR«ABSFfP(L)-ALNI ERRsARR/PiL) OIF(L)«ERR/SQRTF(P(L)) XCHI-XCHI4ARR»ARR/ALN PUNCH 9,P(LI,ALN,ERR

30 CONTINUE 0X»*l«0/CI3,N4l) HNX«SQRTF(OX) PEAK'0X«C(2,N4l)/2«0 X«PEAK YLN«Cfl,N4l)4C(2,N4^1)«X4C(3,N4l)«X«X ZLN«EXPF(YLNI RES»HWX»SQRTF<L0GF(2.C))»2C0./PEAK PUNCH91,RES PUNCH14,HMX,PEAK PUNCH15,TIME,ZLN PUNCH93,XCHI C0«NXL-NXF4l CN1«0 CN2«0 CN3«0 CN4s0 DO 31 L«NXF,NXL Y«CH(L) CN1«CN1*Y CN2«CN2*Y«Y CN3«CN3*Y«Y«Y CN4«CN4*Y»Y»Y«Y

31 CONTINUE H«1.0/«C0«CN4«CN2-C0*CN3*CN3-CN2»CN2»(2.0»CNI*CN2)-CNl«CNl»CN2) TUM«0. SUM«0.

42

00 32 L«NXF,NXL Z»CHfL) CY«C0«Z»Z«CN2-CN1«CN2-C0«Z«CN3-CN2«CN2 CYlaCY«CY SY-l.O/PILI SUM«SUM>CY1«SY OY1«Z«CCO«CN4-2.0«CN1«CN2-CN2«CN2-CN1«CN1) 0Y2«CN1*CN3>CN2«CN3-C0«Z«Z«CN3 0Y3«0Y140Y2 0Y«0Y3«0Y3 TUM«TUM>OY«SY

32 CONTINUE SOCl«H«SQRTFfSUM| S0C2«H*SQRTFfTUM) PUNCH 90,S0C1 PUNCH16,S0C2 SXl«1.0/(4.0*Cf3,N4l)«CI3,N4in«S0C2*S0C2 SX2»(C(2,N4l)«C(2,N4l))/(4.0«C(3,N«l)««4)«S0Cl«SDCl SX*SXl'fSX2 SDXO>SQRTF(SX) PUNCH19,SDX0 MM«NXF DO 7 L«NXF,NXL IF(OIF(MM)-OIFCLn 1,7,7

1 MM«L 7 CONTINUE PUNCH17,0IF(MM) PUNCH17,MM IF(0IF(MM)-.0010) 46,45,45

45 NXL»NXL-1 DO 44 I>MM,NXL

P<n = P{I*l) 44 CHn)«CH(I*l)

GO TO 8 46 CONTINUE 19 F0RMATC35HSTANDAR0 DEVIATION CF LINE CENTER= ,E11.4) 17 F0RMAT(E11.4) 16 FCRMAT(26HSTANDARC DEVIATION CF C2= ,E11.4) 90 F0RMATI26HSTANDAR0 DEVIATION OF Cl= ,EI1.4) 18 F0RMAT(4(2X,E11.4)) 9 F0RMAT(2(2X,F8.0)2X,Fe.3) 14 F0RMAT(12HHALF WIDTH* ,F7.2,13HLINE CENTER= ,F9.4) 15 F0RMAT(5HTIME ,F8.0,5FAMP= tEll.4) 2 F0RMAT(9F8.0) 6 F0RMAT(23HP0LYN0MIAL COEFF IC IENTS/6(2X,ElI.A)) 35 FCRMAT(2I4) 91 F0RMATaiHRES0LUTI0N=,F6.2) 93 FCRMAT(11HCHISCUARE=,F7.2)

calibration of a scintillation beta ray …

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