Answer to Question #10110 Submitted to "Ask the Experts"

How can I determine the minimum detectable activity (MDA) for a high purity germanium detector system for various radionuclides? How can I draw the efficiency curve from a given spectrum?

Determination of the minimum detectable activity (MDA) of a radionuclide being assessed through gamma ray spectrometry is more complicated than is the case for other common counting techniques. The complications arise from a couple of sources.

The first has to do with the fact that the background in gamma spectrometry for fixed counting interval is represented by the number of background counts in the photopeak region of interest. This background consists of the usual contribution from ambient gamma radiation but may also be affected by the presence of radionuclides other than the one of interest in the sample—e.g., high-energy gamma rays from other radionuclides may add Compton counts to the region of interest. If possible, the background should be measured using a sample that is as similar as possible to the samples of interest to be measured, but without any measurable amount of the radionuclide of interest present. The second notable consideration is that in gamma ray spectrometry the background counts are generally determined by taking a few channels on each side of the photopeak region of interest and fitting a curve (usually a straight line but sometimes a polynomial) connecting the points to form a continuous curve under the photopeak. The points on either side are used in this way to estimate the background that will be subtracted in the photopeak region. This manipulation affects the calculation of the MDA.

The MDA depends on the value of the critical level, and the critical level, for routine counting (not gamma spectrometry), is generally as defined originally by Currie (Currie LA. Limits for qualitative detection and quantification determination. Analytical Chemistry 40(3):587–593; 1968), and is given by

L_C = k(R_b/T_s + R_b/T_b)^1/2,        (1)

where k is the number of standard deviations above the mean expected net count rate of zero when no activity is present at which the critical level is specified to lie;

R_b is the background count rate;

T_b is the background counting time, and

T_s is the sample counting time.

The critical level is the minimum net count rate that confirms the presence of radioactivity. When equal probabilities of false positive and false negative identifications of activity are assumed, and sample and background counting times are the same (t), the lower limit of detection, L_D, is related to the critical level by:

L_D = k²/t + 2L_C,           (2)

and the MDA is  obtained by dividing this detection limit by the counting efficiency for the radionuclide of interest.

In equation 2 the value of k is most often taken as 1.645; when the counting time is long compared to the value of k², equation 2 reduces to the simple form

L_D = 2L_C,         (3)

and for equal values of background and sample counting times, t, this reduces to the common expression

L_D = 2k(2R_B/t)^0.5 = 4.65(R_B/t)^0.5       (4).

Various groups and individuals have attempted to apply the above logic to gamma spectrometry measurements. In perhaps the simplest approach, for cases in which the background spectral region of interest does not contain a peak, the background count rate is sometimes simply interpreted as the gross counts in the region of interest divided by the counting time, and this is used for R_B in equation 4. This simple value may be acceptable for some cases, but it is not a technically correct representation for the actual situation in gamma spectrometry because it does not properly account for the uncertainty that is propagated when the background is subtracted from the gross counts.

In gamma spectrometry, the background associated with a given photopeak is associated with the continuum of counts that lie under that peak. Most commonly that continuum is determined by assuming that it is represented by a straight line that can be drawn under the peak by considering a few channels on each side of the peak as representative of the background for that region. Usually equal numbers of channels are selected on each side of the photopeak, and the counts are used to extend a line between these points that represents the background line for the photopeak region of interest. Thus, if N is the number of channels on each side of the photopeak, and C is the number of counts contained in the 2N channels on both sides of the peak, then, if there are P channels in the photopeak region of interest, the number of background counts expected in the region of interest is estimated as (P/2N)C. This can then be used in conjunction with definitions of the critical level and propagation of uncertainties to determine an appropriate estimate of the critical level and the detection limit.

The International Organization for Standardization has dealt with this issue. In ISO-11929-3, Determination of the detection limit and decision threshold for ionizing radiation measurements -- Part 3: Fundamentals and application to counting measurements by high resolution gamma spectrometry, without the influence of sample treatment, 2000, a recommended expression is generated. The expression has been discussed by others, and you can find a brief presentation on the internet by Aurelian Luca to a pdf document. This cited document refers to the year 2000 ISO publication. The publication was updated in 2010 and is available for purchase through ISO. The expression given by ISO for the critical level (also called decision level or decision threshold) when no peak exists in the background region of interest and when the counting time is much greater than k², is

L_C = k[(R_o/t)(1+b/2l)]^0.5,        (5)

where R_o is the background counting rate (for the channels selected on either side of the photopeak), b is the number of channels in the photopeak region of interest, and 2l is the combined number of background-defining channels on both sides of the photopeak. The detection limit, again for counting times much greater than the value of k², is equal to two times the critical level—i.e., L_D = 2L_C. These values of L_C and L_D represent net count rates associated with equal values of false positive and false negative results. In most cases the accepted values of false positive and false negative results are set at 0.05, and k = 1.645. The MDA would be obtained by dividing L_D by the photopeak counting efficiency (see later discussion on efficiency).

If the background photopeak region of interest contains a peak, the above expression does not apply, since it assumes a fixed line defines the photopeak background; a background peak in the photopeak region of interest is not consistent with this assumption and must be corrected for through additional steps. The more recent ISO-11929-3 document of 2010 considers this and presents an alternative expression to account for the presence of a peak in the background. You can see a proposed version of the expression in the above cited paper by Luca.

I am not quite sure what you mean by your second question. In particular I am not sure what spectrum you are referring to when you say "a given spectrum." If you mean that you generated the spectral data by counting a known standard that emits multiple gamma ray energies, the usual procedure for determining the efficiency vs. gamma energy is rather straightforward.

For each energy of interest you must determine the net counts in the respective photopeak. The activity of the responsible radionuclide in the standard, usually in disintegrations per second or disintegrations per minute at the time of use, is then multiplied by the fractional disintegration yield of the gamma ray being considered to obtain the number of gamma rays of the respective energy emitted per unit time by the standard. The net counts in the respective photopeak are divided by the counting time to obtain counting rate, and this is divided by the gamma emission rate to obtain the gamma counting efficiency, net photopeak counts per emitted gamma ray of the energy under consideration. With a multienergy gamma source, the same procedure is carried out for several gamma ray energies that cover the range of energies that you will be dealing with for the samples you anticipate.

It is important that when you do these efficiency determinations, the sample mass, geometry, and other physical characteristics simulate the actual samples to be measured to the extent practical. The efficiencies are determined, as noted, in dimensions of net counts per emitted gamma ray of a specified energy. The efficiency results are often plotted on a log-log plot of efficiency vs. gamma ray energy. Usually such curves will have depressed efficiencies at low energies when photon attenuation in the sample matrix and detector components is significant. The curve will rise through a maximum and then decline in a rather smooth fashion, often approaching a straight line on the log-log plot.

The obvious advantage of having efficiencies expressed as net photopeak counts per gamma ray of a specific energy emitted is that the efficiency at a specified energy may be applied to any radionuclide that emits that energy gamma ray. The activity of the radionuclide responsible may then be readily obtained by dividing the net photopeak count rate by the photopeak gamma efficiency and then dividing the result by the fractional gamma ray yield for the radionuclide. The efficiency curve can be used graphically, or is frequently fitted to a mathematical function over the energy range of likely concern, so that efficiencies for gamma ray energies that lie between points used to develop the curve may be readily interpolated.

I hope this satisfactorily addresses your questions.

George Chabot, PhD, CHP