This page needs some serious editing because some of the headings from Corel Ventura's Desktop Publishing program were erroneously transferred into the HTML version. I can fix this, but not tonight (February 27, 2004 at 9:00 p.m.) |
The operating principle of many of the Helgeson internal, external, or waste monitors is firmly based on the unchanging laws of mathematics. Specifically, it is based on the behavior of the Poisson Distribution which describes the frequency with which certain random events will occur. The Helgeson counters are designed so that the user will have complete control of the counter's performance.
Radioactivity is always present in our environment. We call this the "background radioactivity." Therefore, any radiation detector measures this radioactivity. As detector sensitivity increases, the frequency of observing natural background radiation also increases. The background contains both beta and gamma radiations. The beta radiations originate primarily from the daughters of radon and thoron gases. These are naturally occurring radionuclides in the uranium and thorium decay chains. The gamma radiations originate from several naturally occurring sources: radioactive materials such as uranium, thorium and their daughter products and from potassium which is a vital part of the human body. Gamma radiations also come from cosmic rays. All of these are called background. The purpose of almost all of the Helgeson monitors is to measure very low levels of radioactivity, such as those found inside the body, those found on the external surfaces of a person or those found in supposedly "clean" wastes These low levels of radioactivity must be measured in the presence of background radiation. Therefore, we need to know how the measurement of radiation counts will vary with time. We must also state the "action point," or the level of radioactivity requiring notification.
To obtain the best sensitivity, we must work very close to the background. If we are measuring the internal deposition of radioactive materials in a person, we risk telling this individual that he has internal (or external) contamination when, in reality, he is not contaminated. This is called a "false positive" result. Likewise, we risk telling a person that he is free of contamination when, in reality, he is contaminated. This is called a "false negative" result. We call the point at which the counter makes this determination the "Calculated Decision Value" (CDV). If we measure a person or a barrel of supposedly "clean" waste, the exact same methods may be used.
This chapter discusses:
@MAJOR HEADING = Poisson and <$IGaussian Distribution;plot of>Gaussian Distributions
The <$IPoisson Distribution;demonstration program>Poisson Distribution shows
how random events may occur. It is a
@LETTER BULLET = 640 - 2*SQRT(640) = 589 deaths
to
@LETTER BULLET = 640 + 2*SQRT(640) = 691 deaths.
(Here is an excellent example of a
The shape of the <$IPoisson Distribution;demonstration program>Poisson
Distribution curve depends to a great extent on the long-term mean. If the
long-term mean is close to or less than 1.0, then the distribution will be
highly skewed to the left. The graph of frequency (Y-axis) versus the number of
events (X-axis) will have high values for 0, 1, and maybe 2, but will fall off
towards zero very rapidly as shown in Table
<$R[C#,TABLE301.TXT]3>-<$R[T#,TABLE301.TXT]1>, on the opposite page, where the
mean is 1.0 events. Note that when the long-term mean is 1.000, the probability
of observing 0 or 1 are the same, namely, 36.8%.
Columns 1 and 2 are graphed in Figure
<$R[C#,POISAT1P.IMG]3>-<$R[F#,POISat1P.IMG]1>, on the opposite page. The X
value is displayed on the X-axis and the Probability of Observing X only is
displayed on the Y-axis.
If the long-term mean is large, such as 20 or more, then the shape of the
distribution is symmetrical around the long-term mean. Look at the figure
again. Note that the distribution of data points represented by the "+"
look quite symmetrical around a mean of 11. This shows that with values as low
as 11, the <$IPoisson Distribution;demonstration program>Poisson Distribution
is reasonably well represented by the <$IGaussian Distribution;plot of>Gaussian
Distribution.
@BODYTEXTPGBRK = The Normal <$IGaussian Distribution;plot of>Gaussian
Distribution represents most of the variation in physical measurements, such as
the height of all of the male people living in San Francisco who are between
the ages of 30 and 90, the length of a piece of wood or the percent of sugar in
1000 cans of Coca Cola. All of these will be fractional numbers and the graph
of these frequencies normally will be a smooth, unbroken symmetrical curve
around the average value.
<$&TABLE301.TXT>Because the Poisson and <$IGaussian Distribution;plot
of>Gaussian Distributions are so similar for larger numbers, such as 20 or
greater, the arithmetic for the <$IGaussian Distribution;plot of>Gaussian
Distribution may be used for estimating values of the <$IPoisson
Distribution;demonstration program>Poisson Distribution.
For a practical d<$Ifiles;demonstration>emonstration of the shape of the
Poisson and <$IGaussian Distribution;plot of>Gaussian Distributions, use the
"Poisson" program and choose different values for the
<$Ibackground>bac<$Iunits;kilograms>kground and <$Iaction point>action point to
see how the shapes are plotted.<$&POISAT1P.IMG>
@MAJOR HEADING = Bac<$Iunits;kilograms>kground and Its Statistical
Distribution<$M[Statistical Dist]>
The <$Ibackground>bac<$Iunits;kilograms>kground is a function of the
environment around the counter. If the counter is in a well shielded and
ventilated room, the <$Ibackground>bac<$Iunits;kilograms>kground should be
quite constant. If these conditions do not prevail, variations may occur in the
<$Ibackground>bac<$Iunits;kilograms>kground as a function of the time of day,
the season, and the temperature. It is important to study the variation of
<$Ibackground>bac<$Iunits;kilograms>kground as a function of time so you will
know how long a <$Ibackground>bac<$Iunits;kilograms>kground to take and how
frequently it should be taken. The longer the bac<$Iunits;kilograms>kground,
the lower is the minimum <$Isensitivity;analytical>sensitivity. The following
discussion illustrates the influence of <$Itime;counting>counting time on the
ability to determine the true bac<$Iunits;kilograms>kground.
Assume that the true <$Ibackground>bac<$Iunits;kilograms>kground
@ONE EQUATION = <$Eroman B~=~roman C sub roman b over roman T>
The total number of counts, C
We also know that if we make another
<$Ibackground>bac<$Iunits;kilograms>kground count of T seconds, we will not
obtain exactly the same <$Ianswer>answer due to the random variability of
radioactive counting. However, we can make a statement about the average
counting rate and our estimate of the range of counting rates which should be
observed in successive T-second <$Ibackground>bac<$Iunits;kilograms>kground
counts:
@ONE EQUATION = <$Eroman B~=~ { roman C sub roman b ~+-~roman t sub roman p ~*~
sqrt{roman C sub roman b } }over roman T>
Expressing this in terms of the counting rate, B, instead of the total count
C
@ONE EQUATION = <$Eroman B~=~ roman B ~+-~roman t sub roman p ~*~ sqrt{roman B
/ roman T }>
where t
@MINOR HEADING = Two-Sided Limits
If we make a table, it should look something like this:
@PROBABILITY =
@PROBABILITY = t
@PROBABILITY = 1.0 68.269%
@PROBABILITY = 1.645 90%
@PROBABILITY = 1.96 95% (exactly)
@PROBABILITY = 2.0 95% (approximately)
@PROBABILITY = 2.0 95.45% (exactly)
@PROBABILITY = 2.57586 99%
@PROBABILITY = 3.2906 99.9%
@BODYTEXTPGBRK =
<$&TwoSide.IMG>Figure <$R[C#,TwoSide.IMG]3>-<$R[F#,TwoSide.IMG]2>, above, shows
uneven two-sided limits. These are called the two-sided values. If the average
counting rate is 100 counts per second, then we can state that for a 1-second
count, about 95% of the time the true counting rate lies between:
For 1-second,
@FIRST EQUAT'N = <$E{100~*~1~+-~2~*~sqrt{100~*~1}} over
1~=~{100~+-~2~*~sqrt{100}} over 1~=~{100~+-~2~*~10} over 1 ~=>
@EQUATION = 100. + 20. = 120. counts per second.
@LAST EQUATION = 100. - 20. = 80. counts per second.
The following are ranges for 10-second and 100-second counts:
For 10-seconds,
@FIRST EQUAT'N = <$E{100~*~10~+-~2~*~sqrt{100~*~10}} over
10~=~{1000~+-~2~*~sqrt{1000}} over 10~=~{1000~+-~2~*~31.6} over 10 ~=>
@EQUATION = 100. + 6.32 = 106.32 counts per second.
@LAST EQUATION = 100. - 6.32 = 93.68 counts per second.
For 100-seconds,
@FIRST EQUAT'N = <$E{100~*~100~+-~2~*~sqrt{100~*~100}} over
100~=~{10000~+-~2~*~sqrt{10000}} over 100~=~{10000~+-~2~*~100} over 100 ~=>
@EQUATION = 100. + 2.0 = 102.0 counts per second.
@LAST EQUATION = 100. - 2.0 = 98.0 counts per second.
@BODYTEXTPGBRK = These same equations are re-written using the
For 1-second,
@FIRST EQUAT'N = <$E100~+-~2~*~sqrt{100~/~1}~=~100~+-~2~*~sqrt{100}
=~100~+-~2~*~10 ~=>
@EQUATION = 100. + 20. = 120. counts per second.
@LAST EQUATION = 100. - 20. = 80. counts per second.
For 10-seconds,
@FIRST EQUAT'N =
<$E100~+-~2~*~sqrt{100~/~10}~=~100~+-~2~*~sqrt{10}~=~100~+-~2~*~3.16 ~=>
@EQUATION = 100. + 6.32 = 106.32 counts per second.
@LAST EQUATION = 100. - 6.32 = 93.68 counts per second.
For 100-seconds,
@FIRST EQUAT'N =
<$E100~+-~2~*~sqrt{100~/~100}~=~100~*~+-~2~*~sqrt{1}~=~100~+-~2~*~1 ~=>
@EQUATION = 100. + 2.0 = 102.0 counts per second.
@LAST EQUATION = 100. - 2.0 = 98.0 counts per second.
Summarizing the meaning of the rate equations, we can say that the standard
deviation is equal to the square root of the (Counting Rate divided by the
Counting Time):
@ONE EQUATION = <$Eroman StdDev~=~sqrt{ roman Counting roman Rate~/~ roman
Counting roman Time}>
Remember, these are the
@MINOR HEADING = One-Sided Limits<$&One-Sided Limits>
In our work we are not really interested in two-sided limits. However, we want
to know the probability that the <$Ibackground>bac<$Iunits;kilograms>kground
will exceed a certain value just by chance. These are called
@PROBABILITY =
@PROBABILITY = t
@PROBABILITY = 1.0 84.1345%
@PROBABILITY = 1.281 90%
@PROBABILITY = 1.645 95%
@PROBABILITY = 2.326 99%
@PROBABILITY = 3.0933 99.9%
@PROBABILITY =
Consider these conditions:
@BULLETSPACETX = The <$Ibackground>bac<$Iunits;kilograms>kground is 10 counts
per second. A 5,000 Bq 137-cesium <$Isources>source measuring 10 cm by 10 cm is
placed three inches above one of the corners of a large p<$Iproportional
counters>roportional counter. Two sides of the <$Isources>source are directly
above the outer edges of the sensitive area of the p<$Iproportional
counters>roportional counter. A <$Isources>source in this position increases
the counting rate by 5 counts per second versus 8 counts per second if the
<$Isources>source is placed directly above the center of the p<$Iproportional
counters>roportional counter. We want to be able to detect 5,000 Bq of
137-cesium contamination in a person during a time of 7 seconds.
@BODYTEXTPGBRK =
Let us see what a one-sided distribution looks like graphically. First consider
just the bac<$Iunits;kilograms>kground. This example shows that if we make a 7
second count using a p<$Iproportional counters>roportional counter with an
average <$Ibackground>bac<$Iunits;kilograms>kground of 10 counts per second and
an <$Iaction point>action point of 5 counts per second above the
bac<$Iunits;kilograms>kground, then 90% of the counts would be below 11.71
counts per second and 10% would be above that value. Stated differently, if the
average <$Ibackground>b<$Itime;background counting>ac<$Iunits;kilograms>kground
counting rate is 10 counts per second and you make a series of 7 second counts,
then 90% of the time you will obtain the following:
@ONE EQUATION = <$&UPPERONE.IMG><$E10~+~1.281~*~sqrt{10~/~7}~=~11.53 ~roman
counts ~roman per ~roman second.>
Compare this value to the Poisson program value of 11.71 counts per second.
Additionally, 95% of the time you will observe counting rates which lie between
the values of:
@ONE EQUATION = <$E10~+-~1.96~*~sqrt{10~/~7}~=~10.0~+-~1.20 ~roman counts
~roman per ~roman second.>
@ONE EQUATION = <$E~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=~8.80 ~roman to ~11.20~
roman counts~ roman per ~roman second.>
@BODYTEXTPGBRK = If counting rates as low as 8.8 counts per second in a 7
second count are observed, such values are considered to be appropriate.
@BODYTEXTPGBRK = <$&LOWERONE.IMG>
@MAJOR HEADING = Bac<$Iunits;kilograms>kground <$Iunits;kilograms><$&Bkg
Collection>Collection Methods
There are several methods for collecting bac<$Iunits;kilograms>kground. There
is also a statistical approach to allocating a limited amount of time so errors
are minimized. We shall start our discussion with this last topic. We are
quoting generously from Dr. Price's book "Nuclear Radiation Detection"
but the same material may be found in many textbooks on statistics. (Reference
<$FPrice, William J., "Nuclear Radiation Detection," p 60, McGraw-Hill
Book Company, Inc., 1958>)
@MINOR HEADING = Allocation of Time for Sample Versus
Bac<$Iunits;kilograms>kground
If a limited amount of time is available for making both a sample (person,
barrel, etc.) count as well as a <$Ibackground>bac<$Iunits;kilograms>kground
count, the simple application of statistical principles and differential
calculus gives the ratio of <$Itime;counting>counting times. Let us define the
<$Ibackground>b<$Itime;background counting>ac<$Iunits;kilograms>kground
counting rate, counts per second, as <$Er sub b>, the total counting rate
(sample plus bac<$Iunits;kilograms>kground) as <$Er sub T>, the
<$Ibackground>b<$Itime;background counting>ac<$Iunits;kilograms>kground
<$Itime;counting>counting time as <$Et sub b>, and the total
<$Itime;counting>counting time as <$Et sub T>. The standard deviation of the
net sample counting rate is
<$Esigma sub s~=~ left ( r sub b over t sub b~+ ~r sub T over t sub T right )
sup 0.5>
By differentiation,
<$E2 sigma sub s ~ d~ sigma sub s~=~-~r sub b over {t sub b ~sup 2}~d t sub
b~-~r sub T over {t sub T ~sup 2}~d~t sub T>
Setting <$Ed~sigma sub s~=~0>, the condition for minimum error, and <$Ed~t sub
b~+~d ~t sub T~=~0>, the condition for constant time, the result
<$Et sub b over t sub T~=~left (~r sub b over r sub T ~right ) sup 0.5>
is obtained for the optimum use of the <$Itime;counting>counting time. To
determine this ratio at the start of the experiment us approximation. Adequate
values of the two rates may be determined by short counts.
@MINOR HEADING = Measure the Bac<$Iunits;kilograms>kground for a Fixed Period
of Time
When most people think of <$Ibackground>bac<$Iunits;kilograms>kground
collection, they think of counting the
<$Ibackground>bac<$Iunits;kilograms>kground for a <$Imarkers>fixed period of
time. The <$Ibackground>bac<$Iunits;kilograms>kground count usually is made
just before or just after the count of the person or sample. This is a
completely acceptable method and may be chosen from Chapter <$R[C#,Parameters
Chapt]8>, "Examine and/or Change Parameters."
@MINOR HEADING = Measure the Bac<$Iunits;kilograms>kground for a Minimum
and
<$Ibackground;mode 2>Choosing a minimum and maximum
<$Ibackground>b<$Itime;background counting>ac<$Iunits;kilograms>kground
<$Itime;counting>counting time gives somewhat better results, especially if the
<$Ibackground>bac<$Iunits;kilograms>kground is <$Ivariable>variable. Since 1974
the older versions of Helgeson software for the "Do-It-Yourself" Lay
Down Diagnostic Counter have collected bac<$Iunits;kilograms>kgrounds for 16
minutes. If the operator wanted to <$Icount;starting a count of a person>start
a count and the current <$Ibackground>bac<$Iunits;kilograms>kground had been
running for less than 8 minutes, the current
<$Ibackground>bac<$Iunits;kilograms>kground was abandoned and the previous
<$Ibackground>bac<$Iunits;kilograms>kground was used. If the
<$Ibackground>bac<$Iunits;kilograms>kground had been collected for 8 minutes or
more, the current <$Ibackground>bac<$Iunits;kilograms>kground was saved. The
rationale was that for an 8 minute
@BODYTEXTPGBRK = This same rationale is applicable to other counting systems.
When used with any of the "<$IQuicky Counter>Quicky" models, where the
<$Itime;subject counting time>subject <$Itime;counting>counting time is two
minutes or less, then we recommend that the
@MINOR HEADING = Measure the Bac<$Iunits;kilograms>kground <$Ibackground;mode
3>Over a Sliding Window
<$&BKGTIME1.IMG>Another good method for ensuring a statistically valid
<$Ibackground>bac<$Iunits;kilograms>kground is the practice of a "sliding
window." The total <$Ibackground>bac<$Iunits;kilograms>kground time is
<$Isliding window background mode;number of sampling intervals>divided into
"n" short intervals. If the <$Ibackground>bac<$Iunits;kilograms>kground
<$Isliding window background mode;principle>collected within the
"n
In order to demonstrate properly the "sliding window" concept, we
deliberately changed the values in the 25
@BODYTEXTPGBRK = Figure <$R[C#,BKGTIME3.IMG]3>-<$R[F#,BKGTIME3.IMG]8> shows the
range of 40 to 80 intervals. Note that the upper 4-sigma limit on the
60
@BODYTEXTPGBRK = The Site Health Physicist may elect
@MAJOR HEADING = <$Iaction point>Action Point
The <$Iaction point>Action Point, A, is defined as the amount of radioactive
contamination for which an alarm will sound if a person is truly contaminated.
Its value, A, in terms of counts per second is determined by placing a
calibration <$Isources>source of the energy of concern at a distance from the
detector to simulate actual counting.
In the following example an <$Iaction point>action point, A, of 5 counts per
second above the <$Ibackground>bac<$Iunits;kilograms>kground of 10 counts per
second (for a total of 15 counts per second), a Type 1 error of 10%, and a Type
2 of 1% where chosen. Less than 1% of the time, a result of less than 11.7
counts per second would be obtained
<$E( ~roman B~+~ roman A~)~+-~2~*~sqrt{(~roman B~+~ roman A)~/~ ~roman T }>
<$E( ~10~+~5~)~-~2~*~sqrt{(~10~+~5)~/~ ~7 }~=~12.07 ~roman counts~/~roman sec
roman~~and>
<$E( ~10~+~5~)~+~2~*~sqrt{(~10~+~5)~/~ ~7 }~=~17.93 ~roman counts~/~roman sec .>
will be obtained 95% of the time.
Note that there is a wide distribution of possible counting rates when the true
counting rate is 15 counts per second.
Now the questions are:
@BULLET SPACE = Where do we set the limit for calling one result positive and
another negative?
@BULLET SPACE = What are our chances of being wrong in either case?
@BODYTEXTPGBRK = The <$Ianswer>answers lead directly to the discussion of the
Type 1 and Type 2 errors, the CDV, and the <$Itime;counting>counting time.
@MAJOR HEADING = Calculated Decision Value, Type 1 & 2 <$&Type 1 & 2
Error>Errors, and Counting Time
If the true counting rate is 10 counts per second and the
<$Itime;counting>counting time is 7 seconds, 95% of the time we will obtain
counting rates as low as:
10 - 2*SQRT(10/7) = 7.6 counts per second
and as high as:
10 + 2*SQRT(10/7) = 12.4 counts per second,
This is just the bac<$Iunits;kilograms>kground. So how do we make a
determination as to when a result should be called positive?
This question introduces the terms Type 1 error and Type 2 error. The terms
come from the teachings of statistics and are probably more easily understood
if we call them the False Positive and False Negative points. To illustrate,
let us set the Type 1 error equal to 10%. In simpler terms, there is a 10% risk
of calling a result positive when it is truly negative. This means that by
random chance one of every ten results will be above our decision value.
Let us also set the Type 2 error, or the risk of
Remember that we said that we can use the arithmetic of the <$IGaussian
Distribution;plot of>Gaussian Distribution to estimate (very closely, most of
the time) the values of the <$IPoisson Distribution;demonstration
program>Poisson Distribution when the total number of events is 20 or more.
Therefore, using our one-sided t(p) values we can calculate the Calculated
Decision Value, CDV.
The equation which gives us the CDV based on the
<$Ibackground>bac<$Iunits;kilograms>kground only is:
CDV = (B) + t(p,Type 1)*SQRT(B/T)
We know, however, that we can calculate the CDV based on the <$Iaction
point>action point and the Type 2 error. The equation is:
CDV = (B + A) - t(p,Type 2)*SQRT((B+A)/T)
Since both of these equations represent the same number, we may set them equal
to each other and eliminate the CDV. From the resulting equation we may
calculate the <$Itime;counting>counting time, T.
B + t(p,Type 1)*SQRT(B/T) = B + A - t(p,Type 2)*SQRT((B+A)/T)
Subtracting B from both sides of the equation:
t(p,Type 1)*SQRT(B/T) = A - t(p,Type 2)*SQRT((B+A)/T)
Re-arranging,
t(p,Type 1)*SQRT(B/T) + t(p,Type 2)*SQRT((B+A)/T) = A
t(p,Type 1)*SQRT(B) + t(p,Type 2)*SQRT(B+A) = A * SQRT(T)
and finally,
T = {[t(p,Type 1)*SQRT(B) + t(p,Type 2)*SQRT(B+A)]/A}
As a test, substitute the numbers from the example to see how closely the
<$Itime;counting>counting times compare:
@EQUATION =
@EQUATION = Type 1 error = 10%, therefore t(10%) = 1.281
@EQUATION = Type 2 error = 1%, therefore t( 1%) = 2.326
@EQUATION = T = {[1.281 * SQRT(10) + 2.326 * SQRT(10 + 5)]/5}
@EQUATION = T = {[1.281 * 3.16228 + 2.326 * 3.87298]/5}
@EQUATION = T = {[4.05087 + 9.00856]/5}
@EQUATION = T = {[13.0594]/5}
@EQUATION = T = 6.86 seconds, which rounds up to 7 seconds.
@EQUATION =
@BODYTEXTPGBRK = This agrees with the Poisson program results.
@MAJOR HEADING = Calculation of the Minimum Counting Time
The <$Ibackground>bac<$Iunits;kilograms>kground is different for each detector.
The <$Iaction point>Action Points, Type 1 and Type 2 errors may be different,
too. As a result, we must calculate the <$Itime;counting>counting time for each
detector and select the longest as the system <$Itime;counting>counting time.
@BODYTEXTPGBRK = <$&COUNTIME.TXT>Table
<$R[C#,COUNTIME.TXT]3>-<$R[T#,COUNTIME.TXT]2>, below, shows system
<$Itime;counting>counting time calculations with variations to each of the
previously-cited parameters are varied. Note that the longest
<$Itime;counting>counting time occurs for Detector Number 1. Also note that the
lower and upper Calculated Decision Values (CDV) are the same. Normally, there
is only one pair where this is true. All of the others are different because of
the differences in the various parameters. Figures
<$R[C#,CLOSECDV.IMG]3>-<$R[F#,CLOSECDV.IMG]9> and
<$R[C#,WIDECDV.IMG]3>-<$R[F#,WIDECDV.IMG]10> show two extremes in the data
shown in Table <$R[C#,COUNTIME.TXT]3>-<$R[T#,COUNTIME.TXT]2>, below.
<$&CLOSECDV.IMG>
@BODYTEXTPGBRK = <+><$&WIDECDV.IMG>
@MAJOR HEADING = Rationale for "Gross Counting"
Gross counting is defined as accepting all of the counts between wide energy
ranges, such as from 100 keV through 3.0 MeV. No attempt is made to do any
energy discrimination within the chosen range.
In order to obtain meaningful information by the gross counting method there
are certain requirements.
@MINOR HEADING = Uniformity of Radionuclides
You should be able to document that there is uniformity in the types and
relative amounts of the various radionuclides in the samples of concern. For
example, if you can show from historical sampling information that there is
uniformity, then gross counting without spectroscopy is acceptable.
@MINOR HEADING = Isotopic Composition
@BODYTEXTPGBRK = You must have a good knowledge of the average number of gammas
per disintegration from the mixture of radionuclides. Figure
<$R[C#,WSTMXTR1.IMG]3>-<$R[F#,WSTMXTR1.IMG]11>, below, shows the nuclides and
the concentrations found in some typical "green" wastes at a nuclear
power electric generating facility. These same data are shown numerically in
Table <$R[C#,WSTMIXT1.TXT]3>-<$R[T#,WSTMIXT1.TXT]3>. Note that
c<$I60-cobalt>obalt-60 and cobalt-58 make up almost 75% of the radioactivity
found. The next few pages show how to use isotopic composition to calculate the
effective number of gamma rays emitted per disintegration of a
mixture.<$&WSTMXTR1.IMG>
@BODYTEXTPGBRK = <$&WSTMIXT1.TXT>
@MINOR HEADING = Determining the Average Gammas per Disintegration
Using the data from Table <$R[C#,WSTMIXT1.TXT]3>-<$R[T#,WSTMIXT1.TXT]3>, we
shall calculate the average number of gammas emitted per disintegration. We
should choose the upper and lower energy levels, rather than just take the data
from the table for all energies because the detectors may not be able to
measure all of the energies emitted by the radionuclide(s). For example,
although the radiochemical analysis of the "green" waste showed
cesium-137, barium-137m to be present, it is unlikely that the 31.82, 32.19,
and 36.49 keV gamma rays could penetrate the walls of the drum or detector.
Therefore, these gamma rays should be excluded when calculating the average
number of gammas per disintegration.
The analytical results of the representative sample are given in columns A and
B. Column A identifies the radionuclide and Column B gives the concentration in
Columns D, E, and F contain standard disintegration information. The energy in
keV, the branching ratio and the gammas per disintegration at the stated energy
are taken from "RADDECAY" data. The "RADDECAY" program is in
the public domain and is periodically updated by its author. The
"RADDECAY" program is a normal part of the "HELGE" software.
Column G may now be calculated by multiplying Column B by Columns E, F and
37,000. The results are the values of gammas per second per gram, abbreviated
We are ready now to sum the data in column G (refer to Table
<$R[C#,WSTMIXT2.TXT]3>-<$R[T#,WSTMIXT2.TXT]4> on page
<$R[C#,WSTMIXT2.TXT]3>-<$R[P#,WSTMIXT2.TXT]23>). This table is actually an
<$Iextensions>extension of Table <$R[C#,WSTMIXT1.TXT]3>-<$R[T#,WSTMIXT1.TXT]3>.
Look at the first line below the repeated c<$I60-cobalt>obalt-60 data. This
line is labeled "Total gammas per second per gram." Column G contains
the corresponding value of 0.19356. This number, divided by the total
disintegrations per second per gram (0.145197 disintegrations per second per
gram), gives us the <$Ianswer>answer we are looking for, the average gammas per
disintegration per gram, 1.333117, found on the next line: <$Eroman { 0.19356
over 0.145197~=~1.33117 }>. (That degree of precision is not justifiable but
is used to show the differences between using all of the gammas and using a
portion, as is explained in the next two para<$Igraphs>graphs
There is a problem with this value, however. It covers all energy ranges, even
the 31.82, 32.19, and 36.49 keV gammas from the Cs-137 - Ba-137 decay chain.
Depending on how the calibration was performed, these energies could or could
not have been included in the determination of the calibration factor, counts
per second per Becquerel.
@BODYTEXTPGBRK = Column I contains only those gamma rays which have energies of
100-keV or greater. The sum of column I is given in Table
<$R[C#,WSTMIXT2.TXT]3>-<$R[T#,WSTMIXT2.TXT]4> on the row labeled "Total
gammas per second per gram" and has a value of 0.19289. This number,
divided by the total disintegrations per second per gram, 0.125, gives us the
<$Ianswer>answer we are looking for, 1.328, the average gammas per
disintegration per gram. Let us compare the two numbers:
@3.7.3 HEADER = gammas/sec-gram gammas/dis-gram
@GAMMAS/SECGR = All gammas 0.19356 1.333
@GAMMAS/SECGR = Gammas within the energy window: 0.19289 1.328
<$&WSTMIXT2.TXT>
We now have all the information necessary to perform gross counting:
Total Activity per gram = 0.145197 Becquerels per gram, or
<$Eroman { Total~nCi~per~gram ~=~ {0.145197~ Bq ~ per ~gram)} over {37~Bq ~ per
~ nCi} ~=~ 0.0039243 ~nCi ~per ~ gram }>
Composite Calibration Factor for a 6-detector system (see Figure
<$R[C#,WMCALIB1.IMG]2>-<$R[F#,WMCALIB1.IMG]1>) and Chapter 11:
@3.7.3 HEADER = Factor = 0.0172979 cps per Bq or 0.640022 c/s per nCi.
@BODYTEXTPGBRK =
Using Becquerels the calibration factor is:
<$Eroman {Counting ~rate ~from ~1~ gram ~of ~mixture ~=~ 0.0172979 ~*~
0.145197~ =~0.0025116~cps~per~gram }>,or using nanoCuries it is:
<$Eroman {Counting ~rate ~from ~1~ gram ~of ~mixture ~=~ 0.0039243 ~*~
0.640022~ =~0.0025116~cps~per~gram }>.
Assume that the net weight of the barrel of waste is 20 pounds. Therefore, the
weight in grams is:
<$Eroman { 20 ~pounds~*~ 453.592~ grams ~per ~pound ~=~ 9,071.54~ grams}>
Therefore, the counting rate above <$Ibackground>bac<$Iunits;kilograms>kground
from the total radioactivity in the 20 pound barrel is:
<$Eroman {0.0025116 ~counts~ per~ sec~ per~ gram ~*~ 9,071.85~ grams ~=~22.78~
counts ~per~ second}>.
The <$Icalibration;factors>calibration factors are determined by measuring
standard radioactive <$Isources>sources in a uniform distribution within a
phantom barrel. The results are expressed in counts per second per Becquerel as
their basic units but may be converted to counts per second per nanoCurie for
those persons not routinely using the new nomenclature of Becquerels, Seiverts,
etc.
@WARNING = Attention:
@MINOR HEADING = Total Spectrum Versus Photo<$Ipeak>peak
There is one more justification for using gross counting. When you use the
total <$Ispectrum>spectrum you will be using many more counts than just using
the counts under a <$Ipeak;photopeak>photo<$Ipeak>peak. This increases the
<$Ibackground>b<$Itime;background counting>ac<$Iunits;kilograms>kground
counting rate, but it also increases the calibration factor, thereby improving
the minimum <$Isensitivity;analytical>sensitivity of the system.
@BODYTEXTPGBRK =
@MAJOR HEADING = Sequential Analysis
@MINOR HEADING = <$Isequential analysis;introduction to>Introduction to
Sequential Analysis
While performing Whole Body Counts, following the Three Mile Island incident,
Helgeson Scientific Services recognized the need for a Whole Body Counter which
was capable of counting large numbers of people in a short period of time - a
capability which existing equipment could not accommodate. In the event of
future accidents, such a high speed counter could measure a large population
segment quickly and frequently. Therefore, Helgeson Scientific Services
developed the "<$IQuicky Counter>Quicky" In Vivo Counter which utilizes
a statistical technique called Sequential Analysis, as well as other analytical
techniques. Since then, other suppliers of whole body counting equipment have
recognized the potential of such a machine and have built similar counters.
Several methods for determining the amount of radioactivity in an
"When a series of observations are obtained one after another, as is common
in chemical and physical research, it is generally possible to adopt an
alternative procedure in which, after each observation is made, a simple
statistical test is applied to determine whether the results obtained so far
indicate a definite conclusion from the experiment, or whether more
observations are needed to make the experiment decisive. The <$Isequential
analysis;time savings realized by>experiment thus terminates as soon as a
definite conclusion can be drawn, and the average number of observations
required in experiments carried out in this manner tends to be definitely less
than when the number has to be decided before hand. Consequently, this
sequential method of performing comparative experiments has definite
advantages, particularly when the observations are expensive or time consuming.
This basic principle has been applied to Whole Body Counting where the
"number of samples" taken is actually the number of seconds or minutes
of <$Itime;counting>counting time, which is a <$Ivariable>variable and depends
upon the bac<$Iunits;kilograms>kground, the amount of radioactivity in the
person being counted, the level of radioactivity at which an alarm should be
given, and upon the risks of failing to find true incorporation of
radioactivity or of failing to state that a person is below this level of
concern.<$Igraphs><$FThis paragraph and parts of several following paragraphs
are quoted directly or paraphrased from "The Design and Analysis of
Industrial Experiments," by Owen L. Davies, Chapter 3, pp. 57-98, Hafner
Publishing Company, N.Y., 1956.>"
Most whole body counting work (and by <$Iextensions>extension, counting for
external contamination or counting of wastes) is done for a pre-determined
time. If a person has little or no internally deposited radioactivity due to
his work, there really is no reason to spend any more <$Itime;counting>counting
time than is necessary to prove that the level of radioactivity is definitely
less than some pre-determined decision value. "An alternative and more
economical procedure may be adopted when a series of short counts are made
serially, that is to say, when the result of each separate trial is known
before the next is carried out. In these tests the <$Itime;counting>counting
time is not <$Imarkers>fixed in advance, but the test is applied to the
accumulating data after each observation, the experiment being terminated as
soon as a decision between the alternative hypotheses can be made with the
desired degree of certainty."
data after each observation, the experiment being terminated as soon as a
decision between the alternative hypotheses can be made with the desired degree
of certainty."
"The intuitive sequential procedure discussed above can be put on an
objective basis by a sequential significance test. The quantities required to
plan such a test are precisely those needed for a non-sequential test. The test
decides as each new observation comes to hand, in the light of all the
information up to that time, whether:
@PARAGRAPHNUMB = 1.
@FOLLOWINGPARA = to accept the Null Hypothesis that no change of importance has
occurred;<$&Zone1.TXT>
@PARAGRAPHNUMB = 2.
@FOLLOWINGPARA = to accept the Alternative Hypothesis that a real change has
occurred; or<$&ZONE2.TXT>
@PARAGRAPHNUMB = 3.
@FOLLOWINGPARA = to continue taking observations.<$&Zone3.TXT>
The advantage of testing after each new observation, instead of reserving the
decision until the completion of a <$Imarkers>fixed number, lies, as one might
expect, in the smaller number of observations needed on the average to detect a
given difference. Frequently this is only one-half of that required by the
non-sequential test, and if an unexpectedly large effect occurs, the sequential
test may yield a decision after only one or two observations."
@MINOR HEADING = The Sequential Test<$&FRSTGRAF.IMG>
"<$Isequential analysis;principle of>Sequential tests are best explained
graphically. As each new observation is obtained, the value of the function of
all observations recorded up to that time is calculated and plotted against the
number of observations on a chart such as that shown in Figure
<$R[C#,FRSTGRAF.IMG]3>-<$R[F#,FRSTGRAF.IMG]12>. On the chart are two boundary
lines, the positions of which depend upon the risks
@PARAGRAPHNUMB = 1.
@FOLLOWINGPARA = in which the Null Hypothesis is accepted, the lower right
triangle of the graph;
@PARAGRAPHNUMB = 2.
@FOLLOWINGPARA = in which the Alternative Hypothesis is accepted, the upper
left triangle of the graph; and
@PARAGRAPHNUMB = 3.
@FOLLOWINGPARA = in which there is no decision, the area between the two
lines."
"The sequential test then consists in plotting the function of these
observations
"In practice the chart can often be dispensed with, the boundary values
being calculated in advance for each value of
@MINOR HEADING = The <$Isequential analysis;single-sided alternative
hypotheses>Single-Sided Alternative Hypotheses
In
"The function
@EQUATION = <$Eh sub 0~=~{-b~sigma sup 2} over delta>, <$Eh sub 1~=~{a~sigma
sup 2} over delta>, <$Es~=~mu sub 0~+~delta over 2>
@EQUATION = <$Ea~=~roman {ln}^left (~{~(1~-~ beta~)} over alpha~right )> and
<$Eb~=~roman {ln}^left (~{~(1~-~ alpha~)} over beta~ right )> ."
@BODYTEXTPGBRK =
@MINOR HEADING = An Example<$&ORIGDATA.TXT>
<$Isequential analysis;an example>Let us use data from Davies, (op.cit.). Let
us quote from his paragraph 3.211, page 60:
@BODYTEXTPGBRK = "It is instructive to compare the sequential and
non-sequential types of test upon the same data. An inspection scheme for
primers for explosives was required. It was assumed that
Table <$R[C#,ORIGDATA.TXT]3>-<$R[T#,ORIGDATA.TXT]5> shows the calculations of
T
"Since the test is to detect a difference, it will not be affected if a
constant amount is subtracted from each observation. For purposes of
convenience, therefore, instead of considering the actual density we consider
the amount by which the density exceeds 1.40, that is to say, 1.40 is
subtracted from each observation.
@BODYTEXTPGBRK = Then
Now we can see how the cumulative sums of the coded values proceed to make the
decision to "Accept the Batch" after the fifth sample. Remember, that
when Davies performed a non-sequential test of the same data, eleven samples
were required to make the same determination.<$IESC key><$&DAVIESCD.IMG>
The reader can readily observe that if