Net Count Rates
The background count rate, , is given by
. (1)
where
background counts accumulated in time . The uncertainty in the background count rate, , is given by
. (2)
The respective net alpha count rate, , and net beta count rate, , are given by
(3)
. (4)
The associated uncertainties are
(5)
(6)
where
gross alpha count rate
gross beta count rate
sample acquisition time
background acquisition time.
Equations (3) and (4) are written to reflect the fact that the observed gross count rate is a paired observation of the sample’s inherent count rate and the background count rate. Writing the expression in this manner assures that the uncertainties are calculated correctly. For further reference see [Currie ‘68] or [Evans ‘55].
Spillover and Net Count Rate Corrections
The count rates can be corrected for spillover. Note that spillover is a function of mass. This dependency is explained later. Let
(7)
represent the ratio of the counts in the alpha channel, , to the counts which appear in the beta channel, , and
(8)
represent the ratio of the counts in the beta channel, , to the counts which appear in the alpha channel, . The uncertainties are
(9)
(10)
The corrected count rates are then
(11)
(12)
and their uncertainties
(13)
. (14)
Efficiency and Activity
Writing the efficiency as a function of mass accounts for any self-attenuation in the sample. The efficiency, , is then written as
(15)
where decay-corrected activity at a given mass for a standard, observed count rate. can be represented by one of the following models.
(1) Linear
(16)
(2) Exponential
(17)
(3) Inverse Linear
(18)
(4) Inverse Quadratic
. (19)
For the simple case of one standard with “zero mass” . If it is desired to correct for self-attenuation data can be collected using standards of various masses and stated activities. A fit of the data using one of the models is performed and the coefficients are obtained. The fit is accomplished using the algorithms in [Press, Teukolsky et al ‘92].
The inverse of equation (15) gives the activity:
(20)
and the uncertainty in the activity is given by
(21)
Spillover
Spillover is modeled in the same way as efficiency. The only difference is that equation (15) is written as
. (22)
The coefficients for are arrived at using the same fitting algorithms.
Decay Corrections
In performing efficiency calibrations, the activity of the standard being used is decay corrected according to
(23)
where is the decay corrected activity and is the stated activity of the standard. is the factor that corrects for decay prior to counting and is given by
(24)
where
,
and is the half life of the standard’s nuclide. The uncertainty is given by
. (25)
See [ANSI ‘91] and [Debertin ‘88].
Critical Level, Detection Limit and Less-Than Level
The critical level, , the detection limit, , and less-than level, , are defined in terms of count rate. These equations are defined most rigorously in order to accommodate a wide variety of simplified cases. With a little algebra, the more common expressions can be derived.
The critical level is given by
. (26)
The detection limit is given by
. (27)
The less-than level is given by
. (28)
See [Currie ‘68], [Currie ‘84], and [Lochamy ‘81].
MDA
The MDA as viewed is Ld divided by the efficiency where
.
and k = 1.645
Sources
[ANSI ‘91] - “Calibration and Use of Germanium-Spectrometers for the Measurement of Gamma-Ray Emission Rates of Radionuclides”, ANSI42.14-1991, 1991.
[Currie ‘68] - L.A. Currie, “Limits for Qualitative Detection and Quantitative Determination,” Anal. Chem. 40, No. 3, 586-693
[Currie ‘84] - L.A. Currie, “Lower Limit of Detection: Definition and Elaboration of a proposed Position for Radiological Effluent and Environmental Measurements”, NUREG/CR-4007, U.S. Nuclear Regulatory Commission, Washington, D.C.
[Debertin ‘88] - K. Debertin and R.G. Helmer, “Gamma- and X-ray Spectrometry with Semiconductor Devices”, North-Holland, Amsterdam, 1988.
[Evans ‘55] - R. Evans, “The Atomic Nucleus”, McGraw-Hill, New York, N.Y., 1955.
[Lochamy ‘81] - J.C. Lochamy, “The Minimum Detectable Activity Concept”, proceedings of the National Bureau Standards 75th Anniversary Symposium, and EG&G Ortec System Applications Studies, PSD No. 17, September 1981.
[Press, Teukolsky et al ‘92] - W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, “Numerical Recipes in C”, Cambridge University Press, Cambridge, 1992.