Radiation Counting Statistics

Created by Barry Schaeffer of Oregon State University
 

Radiation Counting Statistics

by Barry Schaeffer (Oregon State University)

(Some of the equations given below are quoted from Chapter 9 of  Introduction to Health Physics by Herman Cember.  Much of this discussion is my attempt to present a clear subset of the ideas which Cember presents).

Radiation: A Probabilistic Phenomenon

One of the first equations which every nuclear engineer or health physicist encounters in their studies is this radioactive decay formula:

`A = A_0e^-lambdat`

We can derive this relationship by statistical reasoning, if we assume that radioactive decay is a random event in which the average rate of decay over time is characteristic of a specific radionuclide. 

As Cember puts it, the probability of transformation p during a time interval Dt is directly proportional to the length of the time interval:

    `p  alpha Deltat`

We arbitrarily assign a constant of proportionality (l) , and this expression becomes:

    `p =   lambdaDeltat`

It seems obvious that the probability that any of the radionuclide atoms will survive the time interval is 1-p.  The probability that any of the atoms survives n successive periods of length  Dt   each is:

`(1-p)^n = (1-lambdat/n)^n`

We can determine the probability that a single atom survives for a time equal to t if we evaluate this expression as n approaches infinity.

p(t) = `e^(-lambdat)`

If A0 is the initial number of radioactive atoms, then we arrive at our conclusion that the number that survive for a time t is

A = `A_0e^(-lambdat)`

Key Concepts: If we think hard about what this means, we'll see some important concepts about the nature of radioactive substances and the quantity of the radiation they produce:

  • radioactive transformation is a random process that obeys the laws of probability
  • the average rate of this transformation over time is a characteristic of a specific radionuclide that we call the rate constant (l)
  • Because the process is one of random probability, we should expect that individual measurements (in a series of consecutive radiation counts) will be randomly distributed around an average value
  • Because the count rate fluctuates, it doesn't make sense to talk about a true rate of transformation, since the random nature of the process means that there is no way to consistently measure a single rate).  It only makes sense to talk about a true average rate.
  • We estimate the true average rate from the observed count rate.
  • The error is defined as the difference between the true average rate and the measured rate.
  • We can determine the frequency of occurrence of an error of any given size by applying the laws of probability.

Distributions of Radiation Measurements

In Chapter 9 of Cember's book, he briefly presents a derivation of the sampling distribution of a population of randomly occurring events.  From this, he demonstrates that the derivation produces the bell-shaped normal distribution.

For our purposes right now, we are most interested in the range of radiation count measurements that we could observe when we take readings from a single sample.

To summarize some of the key attributes of a normal distribution of such radiation measurements:

  • The normal distribution curve shows how the observed values are distributed throughout their range, such that the y-axis value of the curve is the probability of observing any given radiation count. The total area under the normal curve, therefore, is always equal to one.
  • The x-axis value of the curve is the range of observed radiation counts for the sample.
  • The mean the most important parameter that describes the distribution is the true average of the observed values.
  • The standard deviation tells us how widely distributed are the range of observed values
  • The fraction of the area between any two observed radiation count values corresponds to the cumulative probability of recording any radiation count in the range between those two values.
  • We can measure distances above and below the mean value in units of standard deviations.  If we do this, we discover that the fraction of the area between any two ordinates is fixed.  For example, we find that 34% of the area under the curve lies between the mean and 1s, while 14% of the area is between 1`sigma` and 2`sigma`

    Confidence Intervals

    As Cember puts it, "the standard deviation is a measure of the disperson of randomly occurring events around a mean value." In the normal curve plot above, you can see that the dark blue region (i.e. `+-1sigma`) comprises 68% of the area under the probability curve.

    If we make a large number of the same rate count measurements, we would see that 68% of the measurements would fall between 1`sigma` above and below the mean.

    We therefore say that `+-1sigma` is the 68% confidence interval, because any one of our measurement will fall into this range 68% of the time. We can also say that we are 68% certain that the true average transformation rate of our sample lies within this range.

    Radiation Counts and Poisson Statistics

    Each radioactive transformation (in most cases) is a highly improbable event. For example, any individual atom of 32P (which has a half-life of 14.3 days) has a chance of decaying in any given second of .00000056.

    Because radioactive transformations are highly improbable, the distribution of the rate counts closely approaches a kind of normal distribution called the Poisson distribution. This kind of distribution has wide application for phenomena that are measured over an interval of time, like radiation counting or rainfall effects simulation.

    The standard deviation for a Poisson distribution is equal to the square root of the mean number of counts that are recorded during a given measurement time interval. The sample "size" in this situation is the measurement time interval for which we are taking a radiation count.

    s = `sqrt(n)`

    Because the counting rate r = n/t, we can simplify the expression for the standard deviation to:

    r = `sqrt(r/t)`

    Cember gives a useful example to explain this:

    • We take a radiation count, and observe 10,000 counts during a 10-minute counting interval. The mean counting rate is 10,000 counts/10 minutes = 1,000 cpm.
    • The standard deviation for this ten-minute measurement is the square root of 10,000 or 100 counts per 10 minutes. The standard deviation is therefore 100 counts / 10 minutes = 10 cpm.
    • If the same sample had been measured over a one minute inteval and had given 1000 counts, we would derive 1000 cpm with a standard deviation of sqrt(1000)/1 minute = 32 cpm.< /li >

    Precision of Measurement

    Precision is a gauge of the reproducibility of a measurement. In the example we just reviewed, we are more likely to observe 1000 cpm during any one 10-minute counting inteval than a 1-minute counting interval.

    We can calculate a measure of precision called the Coefficient of Variation, defined as the ratio of the standard deviation to the mean:

    CV = `sigma` / mean

    Generally, the CV is multiplied by 100, so that we can think of the CV as the standard deviation percentage of the mean. The larger the CV, the greater is the relative scatter of our data, hence the less reproducible is our true average measurement. In the example we just reviewed:

    • where the mean count rate of 1000 cpm was determined by a 10-minute counting interval, the precision (i.e. the CV) was (10 cpm / 1000 cpm) x 100 = 1%
    • where the mean count rate of 1000 cpm was determined by a 1-minute counting interval, the precision was (32 cpm / 1000 cpm) x 100 = 3.2%< /li >

    Error of Sums/Differences of Radiation Counts

    We have a special term for the square of the standard deviation (i.e. the variance). The variance has a special significance whenever we need to compute the sum or difference of two different radiation measurements.

    When we add or subtract two quantities (e.g. A and B), each of which has a characteristic variance, the following relationship applies:

    `sigma_(A+-B)` = `sqrt(sigma_A^2 + sigma_B^2)`

    Perhaps the most common example of this occurs when we must approximate the true average count rate from a sample after we subtract the background count rate in our lab environment.

    As Cember puts it, we are "interested in the net counting rate; that is, the difference between the Gross counting rate of the sample (which includes background) and the Background counting rate. The standard deviation of the net counting rate is given as:

    `sigma_n = sqrt(sigma_g^2 + sigma_b^2) = sqrt((r_g/t_g) + (r_b/t_b))`

    where:

    • `sigma_g` = standard deviation of gross counting rate
    • `sigma_b` = standard deviation of background counting rate
    • `r_g` = gross counting rate
    • `r_b` = background counting rate
    • `t_g` = time during which gross count was made
    • `t_b` = time during which background count was made

    • Example: Computing a Net Count Rate

    A 5-minute sample count measurement gave 625 counts, while a 1-hour background measurement yielded YY counts. What is the net counting rate of the sample? What is the standard deviation of the net counting rate?

    rn = (625 counts)/(5 minutes) = XX cpm

    sn = sqrt (125/5 + YY/60) = ZZ

    Our net counting rate is therefore XX plus/minus ZZ cpm.

    Do I want to include the example given about the efficiency and its sigma for a counting system, using a counting standard. Pp. 402-403 of Cember.

    This example raises a question about the geometry of the counting system (can you really get 2-pi or 4-pi efficiency)? Is that the whole point of the example, and should it all be explained in a separate section?

    Do I want to cover Example 9.8 from Cember (P.403), where he discusses the statistical error for a count ratemeter being a function of its time constant, with no clarification or backup.