calibration of a scintillation beta ray …
TRANSCRIPT
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
Internal Conversion Line
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^ °2i^o-* "^i^^* '^i^^r ^^^d = °- ''
Cjix^* S^'^d" ^^d*"^i " °- ^ ""
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
Internal Conversion Line
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
Line Center - 88.357
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 —
Line Center - 99.032
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 —
Line Center - 98.944
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)