Behavior of Several Germanium Detector

Full-Energy-Peak Efficiency Curve-FittingFunctions

Robert C. McFarlandAnalytics, Inc.

sult in interpolated efficiencies differing by as muchas 6% when using different curve-fitting functions inthe 700-keV to 900-keV region? Second, since the co-incidence- free standard only covered the energyrange of 59.5 keY to 1115.5 keV,how much uncer-tainty would be introduced by extrapolating thecurve fits to higher energy where there were no ex-perimental points? Third, what effect would one in-accurate data point have on an efficiency curve?Fourth, if it is suspected that a particular point is lessaccurate than the Qthers in a data set, is it better touse it or delete it? Since most commercial softwarepackages do not allow weighted fits, weighted fitswill not be considered here.

The coincidence-free, Marinelli-beaker standardin reference 2 emitted gamma rays at 59.5, 88, 122,165.9,279.2,391.7,661.6,810,834.8,1099.3, and1115.5 keY. Using a least-squares procedure, the effi-ciency data from one count was fitted to a fifth-or-der polynomial of the form:

In(E) = co + Clln(E) + C2 (In(E))2 +

article presents in qualitative terms someof the behaviors of three frequently used ger-

~~i~rl1 detector full-energy-peak efficiencycurve-fitting functions. It is written from the

viewpoint of doing routine gamma-ray counting atthe accuracy of several percent using commerciallyavailable software and standards with only 8 to 11major gamma rays. Some of the limitations of thiscalibration procedure are discussed.

The germanium detector full-energy-peak effi-ciency curve-fitting functions investigated are: thepolynomial function in log-log coordinates, the ex-ponential (or four factor) function, and the spline.Most commercial gamma-ray-spectroscopy softwarepackages provide for the utilization of one or moreof these curve-fitting functions without providingany discussion of their behavior in the documenta-tion. The effects of variations in the data points oninterpolated and extrapolated efficiencies (in the en-ergy range 60-2000 keY) using these curve fits arecompared.

An excellent discussion of the physical basis fora number of germanium detector full-energy-peak,efficiency fitting functions as well as a general discus-sion of the mathematics of curve fitting can befound in the book by Debertin and Helmer.l Whileusing some of those efficiency curve-fitting proce-dures in the analysis of the efficiency data from thecoincidence- free, Marinelli-beaker standard pre-sented previously,2 several questions arose. The firstquestion was why did one particular set of data fromthe coincidence-free, Marinelli-beaker standard re-

C3 (In(E))3 + C4 (In(E))4 + C5 (In(E))5

whereE = counting efficiencyE = energy.

The uncertainty due to counting statistics was:1:1 %(1 a) for all measured points. The maximum differ-ence between the measured efficiency and calculated

35Reprint of "From the Counting Room", Vol. 2, No.4, 1991

Radioactivity & Radiochemistry

The Counting Room: Special Edition

10 200 300 400 500 600 700 800900 100011001200

Energy (keV)

Figure 1 Efficiency data and polynomial fit. Figure 2 Comparison of polynomial and exponential curve fits.

efficiency was +3.39% at the 661.6-keV point. The810-keV point fell below the curve 3.21 %. Figure 1shows the experimental points and the polynomialcurve fit.

After noticing that these differences were higherthan generally encountered in this energy region,two additional fitting procedures (the exponential orfour-factor fit and a spline fit) were used in an at-tempt to improve the agreement between calculatedand measured efficiencies. The exponential or four-factor function has the form:

graphs are presented to facilitate a qualitative discus-sion of efficiency curve fits. Although no quantita-tive significance should be attached to this data, as itapplies only to one detector, one standard and onecounting run, it is useful in demonstrating some ofthe problems with efficiency curve fitting.

Inspection of Figure 1 leads to the hypothesisthat the differences are caused by a 5% low-effi-ciency point at BID keY. The polynomial fit was recal-culated omitting the BI0-keV point and compared tothe original polynomial fit. Figure 5 compares theoriginal polynomial function to the function ob-tained by omitting the BI0-keV point. Notice thatomitting the BI0-keV point made only about 2% dif-ference in the energy region bounded by the experi-mental points, but the two functions diverged rap-idly as the energy increased beyond 1200 keY. Sincethere were three higher energy points in the data set,it was not expected that the polynomial fit would bethat sensitive to the BI0-keV point even when ex-trapolating beyond the data points. Next, the expo-nential fitting procedure was repeated omitting theBI0-keV point and compared to the original fit (thedashed line in Figure 5). The exponential fit is not assensitive as the polynomial fit to the BI0-keV pointas the maximum difference is only about 1 % even at2000 keY. This is not really surprising since the expo-nential function does not allow as many degrees offreedom as the fifth-order polynomial. In all casesthe fifth-order-polynomial functions gave better fitsto the data than did the exponential fits as measured

E = 1 / (Cl * EC2 + C3 * p).

The simple spline fit without smoothing, as de-scribed by Debertin and Helmer,! uses a set of cubicpolynomials joined at and passing through each ex-perimental point.

Figure 2 shows the ratio of calculated efficienciesfrom the polynomial function to the exponentialfunction in the energy region bounded by the experi-mental points. The maximum difference betweenthe two functions is 2.5%. Use of the fifth-orderpolynomial allows the polynomial curve to oscillateas shown in the graph. The ratio of the polynomialto the spline fit is shown in Figure 3. Use of the cubicspline and the fifth-order polynomial allows moreoscillation in the difference function. The maximumdifference is approximately 4.4%. Figure 4 comparesthe spline and the exponential function showing amaximum difference of approximately 6%. The

36 Reprint of "From the Counting Room", Vol. 2, No.4, 1991

"Behavior of Several Germanium Detector FuIl-Energy-Peak Efficiency Curve-Fitting Functions"

Robert C. McFarland

Figure 3 Comparison of spline and polynomial curve fits. Figure 5 Effect of deleting the 81 Q-keV point.

Figure 6 Exponential, spline, and polynomial fits omitting the

81Q-keV point.

Figure 4 Comparison of spline and exponential curve fits.

spline. Omission of the 810-keV point brought allthree fits into better agreement. As shown in Figure6, the maximum difference was approximately 2%.The calculated efficiencies at 1836 keY from the poly-nomial and exponential fits (omitting the 810-keVpoint) were surprisingly close to the measured valuefor this particular data.

The data in Figure 5 indicates that extrapolationof an efficiency curve to higher energies could intro-duce a large uncertainty in high-energy measure-ments. The data in Figure 6 shows that an extrapola-tion might be performed if there is no other choiceand if the data points give a very close fit. As long asthere are an adequate number of points to define the

by the sum of the absolute magnitudes of the per-cent differences. The extra degree of freedom of thepolynomial function allowed it to fit the experimen-tal data more closely. However, when extrapolatingbeyond the region of the data points that degree offreedom may allow too much variation in somecases.

Figure 6 shows a comparison of the three fittingfunctions after omitting the 810-keVpoint. The ex-ponential and spline functions were divided by thepolynomial function only for ease in graphing. Bydefinition, the spline fit does not exist outside the en-ergy region of the experimental points, therefore, nocomparison was possible past 1120 keY for the

Reprint of "From the Counting Room", Vol. 2, No.4, 1991 37

Radioactivity 6- Radiochemistry

The Counting Room: Special Edition

Figure 7 Effect of omitting the 59.5-keV point.

Figure 8 Effect of variation in the 88-keV point using polynomial fit

curve, the exponential fit with fewer degrees of free-dom appears to give better results when extrapolat-ing to higher energies or when using data sets con-taining possibly one inaccurate point.

On the other end of the spectrum, if the stand-ard did not have the 59.5-keV gamma ray, could thecurve fits be reliably extrapolated from 88 keY to59.5 keY? Figure 7 shows the comparison betweenthe original polynomial fit and the fit obtained byomitting the 59.5-keV point. Omitting the 59.5-keVpoint caused a 15% difference between the curves at59.5 keY. More surprising is the difference at highenergy: 20% at 2000 keY due solely to the deletion ofa point at 59.5 keY. Also, the deletion of the 59.5-keVpoint caused some differences of about 3% in the en-ergy range 88 keY to 1115.5 keY. Is the exponentialfitting as sensitive to the 59.5-keV point? The dashedline in Figure 7 shows the ratio of the exponential fitdeleting the 59.5-keV point to the original fit includ-ing the 59.5-keV point. At 59.5 keY the difference be-tween the exponential functions was approximately9%. The exponential fit was not as sensitive above100 keVas the polynomial to the 59.5-keV point.

In order to test the effects of an inaccurate pointon the low-energy extrapolation of the polynomialand exponential functions, curve fits were recalcu-lated using the more evenly spaced efficiency data setfrom the 88-,122-,165.9-,279.2-,391.7-,661.64-,834.8-, and 1115.5-keV gamma rays. The efficiencyat 88 keY was varied from 0.95 to 1.05 times themeasured value and the curves extrapolated to60 keY. For comparison, the values at each energywere divided by the values obtained from the curvefit which included the 59.5-keV point. The results forthe polynomial curves are shown in Figure 8 and theexponential curves in Figure 9. Variations in the 88-keY efficiency caused great differences in the extrapo-lated 60- keY efficiency with both functions. Largedifferences were also observed at the high-energyend of the polynomial function due solely to vari-ations in the 88-keV efficiency.

To investigate the effect of variations in an inte-rior point, curve fits were recalculated with the834.8-keV efficiency varied between 0.95 and 1.05times the measured value. Figure 10 shows the re-sults of the polynomial fits and Figure 11 the expo-nential fits. Inside the energy region bounded by the

Figure 9 Effect of variation in the 88-keV point using exponential fit

38 Reprint of "From the Counting Room", Vol. 2, No.4, 1991

"Behavior of Several Germanium Detector Full- Energy-Peak Efficiency Curve-Fitting Functions"

Robert C. McFarland

Figure 10 Effect of variation in the 834.8-keV point using polyno-

mialfit

Figure 12 Effect of variation in the 1115.5-keV point using polyno-

mial fit.

.,~"'1oti::1..11~1c.,

~1.c

~1i.:o""6~o~ (:0-x

WCc

~OD~

Figure 11 Effect of variation in the 834.8-keV point using expo-

nential fit.

Figure 13 Effect of variation in the 1115.5-keV point using expo-

nential fit.

experimental points, both functions exhibited someresistance to the variation: that is, deviations wereless than the 5% introduced into the 834.8-keVpoint. The exponential fit was slightly less sensitiveto the variation in the data than the polynomial fit.At higher energies the variation at 834.8 keY causedlarge differences with the polynomial function.

For completeness, the effect of variation in thehighest energy point (1115.5 keY) was investigated.The efficiency at 1115.5 keY was varied between0.95 and 1.05 times the measured value. The resultsfor the polynomial functions are shown in Figure 12and the exponential functions in Figure 13. Thecurves are very similar to the curves obtained by

varying the 834.8-keV point especially for the expo-nential functions.

Besides computer operator exhaustion, what hasbeen derived from these 23 tedious curve fits? First,consider the case: the efficiency data set to be fit con-tains an inaccurate boundary point. Apparently, lessuncertainty is introduced by using the point in thecurve fit rather than reducing the energy range cov-ered by measured points. If the polynomial functionis used to fit the data, extrapolation outside thebounds of the measured points will probably not bevery accurate. Low-energy extrapolation with eitherfunction is very sensitive to inaccuracies in the low-est energy point. If absolutely forced to extrapolate

Reprint of "From the Counting Room", Vol. 2, No.4, 1991 39

Radioactivity & Radiochemistry

The Counting Room: Special Edition

to higher energy, it is probably best to use the expo-nential function. These functions are best describedas interpolative functions and extrapolation is not re-ally justified. Last case: the data contains an inaccu-rate interior point. It is probably best to delete thepoint (assuming it can be positively identified). How-ever, the effects of deleting a point should be care-fully checked. The spline fit and the polynomial fitare most sensitive to inaccurate interior points.Points that are close in energy but vary in efficiencycan introduce large variations when using the splinefit.

from multiple gamma-ray-emitting radionuclidesproduce less accurate points. Coincidence summingand inaccuracies in the nuclear decay data forgamma-ray transitions can also affect the efficiencydata? Spectral interferences (such as overlappingpeaks or Compton edges) are frequent causes of in-accurate data points.

Where should one look for possibly inaccurateefficiency data points? Data points with the lowestsignal-to-background ratio should be investigated.In a mixed standard, gamma rays from decayedshort-lived components can give such points. Also,weak (low probability of emission) gamma rays

References1. K. Debertin and R.G. Helmer; Gamma- and X-Ray Spectrometry

With Semiconductor Detectors, Physical Science Publishers B.V.;

North Holland: The Netherlands, (1988).

2. R.C. McFarland, "Coincidence Summing Considerations When Us-

ing Marinelli-Beaker Geometries in Germanium Gamma-Ray

Spectroscopy," Radioact Radiochem., 2 (3), 4-7, (1991).

R&R

40 Reprint of "Prom the Counting Room", Vol. 2, No.4, 1991