7.3 Radiation Statistics
Radioactive decay is a binary process, any given atom may decay or it may
not. For large quantities of atoms a Poisson distribution very accurately
describes the process. This distribution has a standard deviation
s
, which
is equal to the square root of the mean rate of decay. The predicted
precession of the count rate is defined as ± one standard deviation.
The mean of a sample is:
∑
n
m =
N
Where N number of samples:
The predicted precision of the sample is
s
(n) =
√
m
The one standard deviation spread of a single sample is:
n = n ±
√
n
From these equations it is evident that the predicted precision of the Gauge
is directly related to the square root of the number of detector counts
accumulated during a measurement. Further, the precision can be improved
either by counting a longer time period or by averaging the count rate for
a number of measurements and this improvement is the square root of the
number of measurements made.
While the precision of the Gauge in count rate shows the trend, what is
of interest is the precession of the density and moisture measurement. In
order to obtain this information, it is necessary to know the change in the
measured parameter in terms of a change in count rate. This is the slope of
the calibration equation.
Or
Then the differential is:
57
𝑪𝑪𝑪𝑪 = 𝑨𝑨 ∗ 𝒆𝒆
!𝑩𝑩𝑩𝑩
− 𝑪𝑪
𝒏𝒏 = 𝑫𝑫𝑫𝑫 ∗ 𝑨𝑨 ∗ 𝒆𝒆
!𝑩𝑩𝑩𝑩
− 𝑪𝑪
𝑺𝑺 =
𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅 ∗ 𝑨𝑨 ∗ 𝒆𝒆
!𝑩𝑩𝑩𝑩
− 𝑪𝑪
