Experimental data on photopeak integration methods in activation analysis

NUCLEAR INSTRUMENTS AND METHODS

115 (I974) I97-212; ©

NORTH-HOLLAND

PUBLISHING

CO.

EXPERIMENTAL DATA ON PHOTOPEAK INTEGRATION METHODS IN ACTIVATION ANALYSIS J. HERTOGEN*, J. D E D O N D E R t

and R. GIJBELS +

Instztute for Nuclear Sciences, RijksuMLerstte*t Gent, Gent, Belgzum Received 16 July 1973 Experimental data on the performance of various peak integration methods in Ge(Li) gamma-ray spectrometry are presented. Included are the s u m m a n o n methods proposed by Covell, Sterhnskl, Quittner, Op de Beeck and Ralston and Wilcox. Another summation techmque is based on quadratic background

esnmation. Special attention is paid to systematic errors due to changing peak-to-background ratios and to instrumental shift. A slmphfied modlfiCatlon of the Ralston-Wilcox method is proposed and evaluated.

1. Introduction In recent activation analysis much work is spent in extending the scope of Ge(Li) gamma-ray spectrometry. A critical step in the counting procedure is the precise and accurate extraction of the data from the numerous and often complex spectra. For this purpose digital computers are commonly used. The important features of computer analysis and various programs have been reviewed in several papers1-5). In quantitative analysis the ultimate task remains the evaluation of the photopeak areas, or of some related quantity. Unfortunately, many successful techniques to perform this are too time-consuming or are beyond the possibilities of smaU and medium-size digital computers. Moreover, an immediate access to a computer is not always available. Hence, many analysts have to rely on less sophisticated methods of photopeak area integration. A number of well-known procedures of this kind were proposed in the past years6-~3). Most of them are modifications of the original Covell method6), all being characterized by the fact that the peak data are not fitted to a function prior to area determination. In most of the original papers emphasis was laid on the reproducibility of the proposed method. In two papers a'14) more attention was paid to the sensitivity of the various methods to systematic errors. In particular the errors due to a changing spectrum shape and peak-to-background ratio were mvestigatedo As an introduction to the present work it is worthwhile to give a brief account of Baedecker's papert4). In the first part the reproducibihty of various methods for nine photopeaks was evaluated by measuring a neutron

activated sample twelve times under the same conditions. In the second part the same sample was counted seven t~mes together with increasing amounts of 6°C0 background activity. A positive point is the author's approach: while taking real spectra as test cases rather than a few favourable and well defined peaks, he lowered the risks of underestimating the problems occurring in common laboratory practice. Nevertheless, some remarks have to be made concerning the experimental approach of the second part. When operated at relatively high counting rates, as was the case, measuring systems including an MCA are sensitive to three types of effects, degradation of the resolution characteristics, loss of counts due to pulse pile-uptS'16), and possibly erroneous performance of the live-timer circuitry (even in the case of constant activity level)IT-t9). The question arises whether the overall negative errors for decreasing peak-to-background ratios as observed in ref. 14, reflect inherent shortcomings of the integration method, or reflect failures of the counting equipment. As the magnitude of instrumental errors is more or less typmal for the apparatus used, they have to be minimized as much as possible in a general assessment of peak area integration methods. The impact of resolution changes, pulse pile-up, etc. should be better evaluated in separately designed experiments. In the approach of Turkstra et al. a) those instrumental errors are avoided by processing synthetic spectra, obtained by adding separately counted data. This summing technique has also been applied successfully by other authors6'2°'21). Turkstra et al. a) tested some total peak area methods on rather simple Ge(Li) spectra, measured with a 400 and 4000 channel analyzer. Special attention was paid to the RalstonWilcox method22).

* Aspirant of the N F W O . ti, Technician at the Umverslty. + Research Associate of the I.I.K.W.

197

198

J. HElZTOGEN et al.

The m a i n p u r p o s e o f the present study ts to g a t h e r m o r e e x p e r i m e n t a l d a t a on the p e r f o r m a n c e o f various integration procedures. It is questionable indeed whether the limited a m o u n t o f d a t a available at the m o m e n t allow a s o u n d evaluation o f the different methods. 2. Outline of the e x a m i n e d m e t h o d s S o m e p r e l i m i n a r y r e m a r k s m a y be m a d e c o n c e r n i n g the m e a n i n g o f totalpeak area (TPA) a n d partial peak area (PPA). As there is no s t a n d a r d definition o f the t o t a l p e a k area, this t e r m is used in a general sense when the integration is carried o u t over all channels t h a t are higher t h a n the estimated underlying backg r o u n d c o n t i n u u m . F o l l o w i n g the t e r m i n o l o g y o f Lederer23), the investigated T P A m e t h o d s are all o f the pseudo-bacl
400

350

300 i

- -

4) A q u a d r a t i c base-line u n d e r the p e a k can be constructed by the usual least squares fitting o f an equal n u m b e r o f channels, m, at b o t h sides o f the p e a k (see fig. 1A). The selection o f the p e a k b o u n d a r y channels a t and b 1 wtll be discussed later. In one run m was set equal to 3, in a second equal to 5. The mimma m this structure provide a first estimate of the boundaries ai and hi. Multlplets are revealed by dewatmns from this mmlmum-maxlmum-mimmum pattern.

1i

I,

aI

11

1 12

bl

~o

4.10 3 -1

convo[utlon

3.10 3

function (3,6,3)

2.10 3

1,10 3

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-1.10 3

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3) Quittner's method 9'1°): 23 channels were fitted to a q u a d r a t i c on each side o f the p h o t o p e a k to evaluate the p a r a m e t e r s o f the cubic base-line. - N o t a t i o n : Q T for T P A ; Q a , b, c.

.....?.....il,.rJ.l.E~:1.r:~oF~

3.103

i

i

:

i

I

I

)I

*4" ) i i

r

I

+ t i i

1

1

a 1

b 1

Fig. 1. A) Explanation of the symbols used for methods (4), (5) and (6). The imposed peak boundaries are denoted as at and bl. To evaluate the quadranc base-hne (open circles), m channels at both peak sides are fitted to a second order polynomial. Channels li and le are the last peak channels having a higher content than the estimated background. Total peak areas are obtained by summing the net counts from channels (al+ 1) to (b~-I) [method (4)1, or from ll to 12 [method (6)]. B) Main features of Op de Beeck's peak search routine [method (10)] The 12-points (3, 6, 3) square wave convolunon function - used for this parucular case - and the value of the convolution integers over this interval are displayed m the inset. The open circles represent the resulting convoluted spectrum. Because of the even number of channels of the square wave function, the convoluted spectrum was shifted half a channel downward.

PHOTOPEAK

INTEGRATION

- Notation: P3, P5 for TPA integration from (a t + 1) to ( h i - 1); P3b, c and P5b, c. 5) In some experiments the Wasson-Sterlinskt method ~) was examined The base-line was determined in the same manner as in (4) with m = 5. - Notatmn: WSTb, c. 6) The choice of m in (4) is rather arbitrary. In many cases the number of points useful for the background evaluation is larger than five. To determine m, the following procedure was adopted: channels a~ (i = 1 to 4) and b~ (i = 1 to 4) are fitted to a quadratic. If then

[c(a~)-c*(a,)l

< 3

c*(a~)

,

\i=5

~=5

and

E [c(b,)-c*(bj]

< 3

c*(b~) ,

the fittln~ is repeated for channels a,(i = 1 to 7) and b~(i= 1 to 7). c(a,) and c(b,) are the counts in the background channels around the peak, while c*(a~) and c* (bL) represent the value of the quadratic polynomial m these channels The same test is performed for a,(t = 8 to 10) and b,(i= 8 to 10). In this way m can take the values 4, 7 and 10. While taking the points in groups of three, one can lower the risk that a slightly anomalous point blocks the procedure. When this modificatmn was used for TPA integration, the summation was stopped when a point in the peak wings was smaller than the calculated baseline. - Notation: P M T for TPA integration from l 1 to 12 (see fig. 1A): PMb, c. 7) Ralston and Wilcox 22) proposed to generate a

base-line spectrum by replacing an original data point n, by the average over an interval ( 2 l + l) about ~, whenever n , - ~ > c ~ (~)~, where ~z --

1

J=l+I

21+1

;=~-z

~_,

~j •

The resulting secondary spectrum is treated m the same way with a smaller value for ~. Usually the procedure is carried out 10 times, :~ taking the values 9.2, 8.2, ..., 0.2. The success of the technique depends upon the choice of l, as pointed out in refs. 3, 22 and 24. Although this paper mainly deals with rather simple techniques, the results of the original method (with l = 30) are included to complement the data of Turkstra et alP). It has to be emphasized that only the essential part of

MEIHODS

199

the procedure, namely the base-line generation, was retained. For example, the cross points of the measured and base-line spectrum were not taken as the peak boundaries, because their spacing IS too wide in several cases. It was preferred to determine the integration width as a function of the fwhm. - Notation: R W T for TPA; RWb, c. 8) In view of the relatively long execution time and the necessary memory space, the method is not so suitable for small computers. For this reason a simplified modification of the Ralston-Wdcox technique is used m the author's laboratory2S). Here the base-line is not generated as a whole, but only over 30 channels at both sides of the peak maximum. In most cases the results of this modification and of the original method are quite identical. However, problems arise when at is applied to spectrum regions showing a high peak density, and also when the peak separation is ca 30 channels. This will be illustrated an a separate section. - Notation: M R W T for TPA; M R W b , c. 9) This TPA method is a part of a general computer programZ6'27). Important steps are data smoothing and peak location by scanning the smoothed first derivative11'12). StatlsUcal tests on adjacent channels at the peak wings allow to establish the peak boundaries. A straght-hne background is estimated from the average of 1 to 5 channels at the boundaries Characteristic for this method is the variable integration width for the determination of the peak area. - Notation. OB. 10) This TPA method makes use of the weighted square wave convoluted spectrum (WSWCS) which is calculated from the original data with the aid of a square wave convolution funcnon, and by weighting the results by their estimated variance2S). The WSWCS is similar to an inverted smoothed second derwatwel). The WSWCS is used to locate single peaks, to detect multiplets and to estimate the boundaries of both single peaks and multiplet groups. The maln features of this procedure are illustrated in fig. lB. Final adjustment of the boundaries is done in the original spectrum in order to place them far enough from the peak centrolds and thus get an unbiased estimate of the total peak area. This freedom of bias for different peak-to-background ratios is gained at the expense of precision, since a relatively large background as subtracted. A straight-line background is estimated in the original data from the average of three channels at the boundaries. Notation: NOB. -

200

J. H E R . T O G E N et al.

3. Influence of background level and spectrum shape

instrumental errors mentioned in the introduction, this procedure has the additional advantage to eliminate spread in the peak intensities owing to statistical fluctuations in the decay and detection processes. This is obviously important for smaller peaks. The reference spectrum is a 4000 channel 169yb175Yb spectrum (up to 400 keV), measured with a small Ge(Li) low energy photon detector29). 177Lupeaks and Ba-K X-rays were respectively due to impurities in the

3.1. EXPERIMENTAL

To reveal possible systematic errors due to a decreasing peak-to-background ratio and to changes of the spectrum shape, the same experimental approach as used by Turkstra et al. 3) was followed. A reference spectrum was successively added in the analyzer memory to six background spectra, collected during increasing counting times. Besides the avoidance of the

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10 `I 2

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1

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1500

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11

12

2500

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...... i ........ i ..... 3000

i....... I ..... -~ ......... 3500

Channel. number Fig. 2. T h e two end m e m b e r s o f the series o f G e ( L 0 spectra used to evaluate the p e r f o r m a n c e o f the various peaks are m t e g r a t m n m e t h o d s (section-3.1). T h e reference s p e c t r u m A is a 169yb-17ayb s p e c t r u m m e a s u r e d ~ l t h a small planar detector (gain: ca 0.1 keV/ channel). Spectrum B is a c o m p o s i t e spectrum, obtained by a d d i n g s p e c t r u m A to the longest background counting. T h e peaks m s p e c t r u m B m a r k e d with a n asterisk are due to the b a c k g r o u n d activity.

PHOTOPEAK

INTEGRATION

Yb-source and to fluorescence effects in the glass counting vial containing the Yb-source. The background spectra were composed of STCo, 22Na, 54Mn and 6°Co actiwty accumulated at fairly high counting rates. The latter fact caused the tailing of the 57Co

201

METHODS

peaks, but it had evidently no effect on the Yb-peaks. The reference and the last composite spectrum are shown in fig. 2. The spectra were processed with a DEC PDP-9 16k computer. After the determination of the maxi-

[12 O W

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16

-4

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-3

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-8

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PEAK 3 TPA = 5,500

PEAK 4 TPA = 28,000 1

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PEAK 11 -70 r PA = 860 a

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Fig. 3 Deviations of the measured area from the expected area for the six composite spectra, some selected results for a few typical peaks. The area or peak f r a c n o n calculated with a particular procedure in the reference spectrum, was considered as being the true one for thin method. F o r the sake of clarity and for reasons explained in the text, the same peak-to-background ratio was taken as abscissa for all the methods The two divergent lines visuahze the magnitude of the expected errors ( 4 - l a ; counting statmtics only). A: CVc, A" STc, D: OB, mR: NOB, O: QT, 0 : PMT, + : RWT.

202

J. H E R T O G E N et al

The spectra were not smoothed prior to area integratton, except for method (9) where smoothmg was a built-in part of the program

m u m peak channel, the peak areas were evaluated with the methods listed above The distances between the peak m a x i m u m and the boundaries aa and bi were supplied as mput data for each peak when applying methods (4), (5) and (6). The boundaries were determined from a graphTcal plot of the reference spectrum and were kept at these values for the composite spectra. For a single peak the boundaries enclosed an interval approximately equal to 3 fwhm. In a preliminary experiment the repercussion of this particular choice was examined. A number of peaks was processed four times through the series, each time having different mitial boundary values. In the last run the enclosed inter~.al was ca 3 f w h m + 6 channels. There were obviously small differences among the vartous sets of data, but they were essentially of a statistical nature. Where systematic deviations were observed, they turned out to be nearly the same m each run. TPA integration for methods (7) and (8) was also carried out from (a~ + l) to ( b 1 - 1). Because none of the described techmques is able to treat mult~plets and closely spaced peaks to an acceptable degree of accurac3., the bunch of peaks between channels 450 and 650 is not considered further

3.2. RESULTS AND DISCUSSION First the results for only eight isolated and not too small peaks are considered (3, 4, 5, 7, 8, 9, 10, 14). Although peak 5 is a doublet, it was retained as representative for cases where two peaks of the same radioisotope or of two long-lived species of the same element coincide. Then there is no need for elaborate multiplet resolving Ftg. 3 presents the behavlour of some methods for three typical peaks among them. The percent deviations from the corresponding area in the reference spectrum are plotted as a function of the decreasing peak-to-background ratio for TPA. For convemence the same abscissa value was used for ali methods, namely the ratio of P5 m the reference spectrum to the background counts between a~ and b 1 . The two divergent hnes indicate the magmtude of the expected standard devlattons for TPA from countmg statistics; more details are given below, These types of figure showed for some peaks a qualitatively slmtlar behavtour for all the methods, the

l

CVa

k=-045 s=220

STa

J~

s : l 88

,J-]JI r]q CVb

E=-077 s=188

,%,#

k=O08

J.

r-q

r] STb [--[h

. ~=-008

P5

-~8 -5 -4 -3 -2 -1

,:2o,

,q

P3

s=l 67

P

_lq CVc

I

k=-123 s= I 65

,q 123~56~

I E=o03 P3b s=065

STc s=115

-4-3-2-I 0

q -4

-3 -i -1

0

~

k'=-O09 s:084

' '4 !I

-5-/,

- -1 0 1 2 3 4

Fig. 4. H]stograms of the normahzed de~latlons for eJght peaks in the six composite spectra (peaks 3, 4, 5, 7, 8, 9, 10, 14). The normallzed, wetghted errors - denoted m the text as the k-values - were obtained by dividing the actually observed percent devlauons for each redly]dual result by the expected percent standard deviation (secUon 3.2); ]c is the average normalized error, and s the spread around ]c (standard deviation).

PHOTOPEAK

INTEGRATION

deviations being either dominantly positive (8), or negative (4, 5, 10, 14). For other peaks (3, 7, 9) the errors seem to be scattered randomly, but detailed investigation learns that actually two groups of methods with opposite trends are present. A prominent aspect is the absence of an overall negatwe or positive trend as a function of the increasing background. The same conclusion may be drawn from the data of Turkstra et alP). Furthermore, in most cases the spread of the results is far from random; so, one may tentatively conclude that Poisson statistics alone cannot account for all the errors in the peak area estimahon, even when the necessary precautions against pulse pile-up and shift are taken. It is, however, difficult to extract from those graphs a clear picture of the average performance of the involved methods. Some kind of statistical treatment of the results ~s, therefore, desirable. As the precision of the various peaks is considerably different, their results can hardly be treated together without prehmmary we~ghtlng. For this reason a so-called k-value for

203

METHODS

each individual result was determined as follows:

kuz = Au/SDjI, where '3uz= 100 [(A,jt/A,13-1]; A stands for area, and i,j and l respectively for the method, the spectrum number (from 2 to 7) and the peak number. The expected percent standard deLiation SDjt, needs some further comments. To a certain extent the standard deviation - if estimated from counting statistics depends upon the integration method. For a given peak it is evidently smaller for the PPA than for the TPA. Normalisation of the observed errors of the two methods to their specific standard deviation, perhaps results in nearly the same k-value; but the PPA is then the better in absolute terms To avoid such ambiguous sttuations, the observed deviations for equivalent photopeaks (i.e. with the same j and l) are divided by the same SDjz, so that smaller k-values reflect smaller absolute errors. Moreover, the specific standard deviations are only approximate values, because it is not so clear how to calculate them exactly for methods (3) - (8). In the present experiment the additional I

RWT

I

~=-002 MRWT s=l 30

k=012 s= 107

__F

PM b

j-

R=-oo7 s:070

PMc

l

E'=O02

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k=-023

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,

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s=069

,

k=-021 s=063

-- -

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/

k= 0,05 s =0.72

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1

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--

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i

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1_ ..

i i

I

~=o16

s=088

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k=029 s:112

s=073

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k-=0.27 s=0.77

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I !

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F

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0 1

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-%--

i

Fig. 5. C o n t i n u e d f r o m fig. 4. T h e last row ( m e t h o d s m a r k e d with an asterisk) deals only with the results o f six peaks (peaks 3, 7, 8, 9, 10, 14). T h e results for P M b * a n d P M T * for these six peaks are given for c o m p a r i s o n with the results o f Q m t t n e r ' s m e t h o d .

204

J. H E R T O G E N

background counts ( = AB) are the only a priori known source of variance with respect to the reference spectrum. Hence, SDjz was calculated following the equation SDjt = 100(2ABjI)~/TPAI,t. The data AB and T P A for method P5 were arbitrarily chosen as a common basis for comparison. In Figs. 4 and 5 (except the last row) the distribution of the k-values of some methods for the eight peaks are presented. The total number of values equals 48 ( = 8 x 6). The histograms are numerically characterized by the average k-value (k) and the spread around k(s). For completeness it has to be stated that these parameters were calculated after grouping the data in intervals of a quarter of the abscissa unit. The main advantage of these histograms is their summarizing feature, but they cannot be interpreted in a strictly statistical manner. The average k-value and the spread are merely a measure for the range of the observed errors. For example, a k near to zero obviously does not mean that the deviations for all peaks are random. The figures are quite self-explanatory and further discussions are limited. One can see at a glance that starting from the same spectral information final results of rather different quahty are obtained. The results for CVa and STa (W = 7 channels, which covers only a small portion of the peaks at the end of the spectrum) are presented merely to illustrate the effect of inadequate application of a method. A distractive feature of these two techniques, as compared to PPA methods, is the smaller spread for W = 2 f w h m than for W = l . 5 f w h m . Remarkable, however, is the simultaneous shift of to negative values for the Covell method. A notable result is the significantly smaller spread for P5 than for P3 (fig. 4), showing the effect of taking four channels more for the background estimation. Increasing thls number when feasible, such as proposed for method (6) did not give appreciable improvement (fig. 5). On the other hand, the limitation of the integration width to ll and 12 as shown in fig. 1 A turned out to be very effective in reducing the spread, while it retained at the same time all the aspects of TPA integration. F r o m fig. 5 it follows further that the improvement in precision was negligible when applying the Wasson-Sterlinski method instead of the usual PPA method. The Ralston-Wilcox method and its modification gave almost identical results, but the negative deviations for peaks 4 and 5 were somewhat larger for the original method; this is probably due to the influence

et al.

of the nearby intense 57Co peak. Contrary to the conclusions of Turkstra et al.3), it was found that this method competes well with others. It is not so clear why methods CVc and OB are the sole to show dominantly negative errors for most of the peaks. Possible reasons in the case of the second method could be the effects of spectrum smoothinga°'3~), or difficulties in establishing appropriate boundaries. Indeed, the application of statistical tests to determine the boundaries often leads to narrower peak widths in the case of a high background, and this may cause some negative bias*). But these two arguments cannot hold for CVc. The less negative deviations for STc, on the other hand, become plausible if one considers it as a summation of successive Covell areas; hence, the errors cancel out a bit. A common characteristic of the three methods is the straight-hne background assumption. This, however, is also the case for method (10), for which an overall negative bias was not observed here. The Qulttner method was not statable to process peaks 4 and 5 due to mutual interference and to the presence of small peaks at the left side of 4. The results for the other s~x peaks were treated as explained above (fig. 5, last row). For comparison the results of method (6) for the same six peaks were calculated too. Quittner's methods certainly stand among the better ones. TABLE 1 Results a for T P A of m u l t i p l e t 2. PMT

QT

P3 b

0.986 0.952 0.960 0.922 0.907 0.863

0.953 0.931 0.902 0.891 0 907 0.880

0.981 0 948 1.019 1.020 1.013 0.930

a G i v e n as the ratio between the measured a n d expected area for the six c o m p o s i t e spectra. b A p p r o x i m a t e l y a s t ra l ght -hne base-hne.

As this method was initially introduced to account for non-linearities in the base-line, one should expect better results for the base-line estimation under the multiplet 2. In this case the underlying continuum is definitely non-linear and both the curvature and the slope change in the course of the series. From table 1 it can be seen, however, that the expected gain in accu-

PHOTOPEAK

INTEGRATION

205

METHODS

TABLE 2

TABLE 3

Results a for peak 6.

Results for smaller peaks.

PMT

Qb

QT

0.989 0 942 0.992 0.902 0.864 0.738

1.29 1.53 1.87 2.20 2.49 3.00

1.48 1.99 2.70 3.23 3.90 4.88

a Measured area/expected area.

racy and precision is absent. In the first part of his experiments, Baedeckerl~), on the other hand, found a better performance of the Qulttner method for a peak superimposed on the Compton edge of the 1333 keV peak of 6°Co. The same improvement as compared to equivalent straight-line methods, however, was not observed in the second part although one should expect a f o r t i o r i better results, because 6°Co activity was added to the spectra. In view of the good precision of the Quittner method for single peaks on an almost linear continuum portion, the question arises whether this is finally the result of the small corrections for non-linearity, or of a statistically better estimate of the underlying background. I n d ~ d , one has to keep in mind that the latter was evaluated here indirectly from forty-six channels. The main advantage of the common quadratic fitting [methods (4)-(6)] over Quittner's approach is that in cases of strong curvature, the latter usually fails, while the first procedure gives an almost straight-line background. The accuracy is then not so good, but the errors remain within an acceptable range. The results for peak 6 which is situated on the tail of the 122 keV STCo peaks, serve as an example (table 2). A lot of problems in applied gamma-ray spectrometry arise when dealing with weak spectral components. An extreme case was studied by Yule2°), who investigated some typical computer methods. In the following, special attention is paid to the smaller peaks 11 and 15. The peaks 12 and 13 could only be detected in the first two or three composite spectra and are not considered any more. The results for 11 and 15 were treated as explained above, but the average k-value and the associated spread were calculated separately for each method and peak. The results are presented in fig. 3 (peak 11) and table 3. A criterion for peak rejection was implied only for

ll original T P A ca 860 /~ s

15 original T P A ca 1600 ]~ s

CVb CVc CVb a CVc a

+0.21 - 0.20 + 1.48 +0.052

2.19 1.92 1.25 0.91

+0.92 + 1.25 + 1.53 +0.03

1.18 1.29 0.68 1.61

STb STc STb a STc a

+ 1.03 +090 +1.53 +0.72

1.39 0.89 1 01 0.66

+ 2.44 +1.15 +2.11 +0.99

1.09 1.12 0.45 1.07

P5 P5b PMb PMT

-0.52 -0.24 -0.58 -0.72

0.83 0 61 0.35 0.56

-0.35 -0.18 +0.06 -0.58

0.77 1.05 0.47 0 35

RWT RWb MRWT

-0.34 -0.31 -0.37

0.21 0.34 0.18

-0.35 -0.42 -0.54

0 27 0.58 0.35

Qb QT

0.03 -0.37

0.28 0 15

-

a After smoothing of the spectral data. b Failed due to presence of background peak at right wing

methods (9) and (10). The other methods were in fact forced to find a peak at the predetermined position. Although methods (9) and (10) discriminated quite soon against peak 11, the results for other methods were retained simply because some of them continued to give reasonable data. The spread of the results for methods (1) and (2) is serious and the errors may amount to high values. Preliminary smoothing aS) of the spectral data caused a noticeable Improvement for W = 2 fwhm This is not so surprising, as the two rather broad peaks are favourable cases to demonstrate the effect of smoothing. A common characteristic of the other methods are the dominantly negative errors. The spread of the results for methods (3), (7) and (8) is quite small. The systematic negative errors for the Ralston-Wdcox method can be understood from the calculation procedure. One can indeed expect some negative errors, which are a function of the peak-to-background ratio. For smaller and broader peaks the situation worsens of course. The results of Quittner's method for peak 11 seem to confirm its good performance when dealing with smaller peaks, as was already stated in refs 9 and 10.

206

J. H E R T O G E N et al.

J

.........

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B

% Error /"~

20

!

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Fig. 6. A) Expanded view (semi-log plot) of the complex X-ray part of the reference spectrum (fig. 2, spectrum A from channel 480 to 650) and the base-line calculatect by means of the original and modified Ralston-Wdcox technxques. In the latter case the height of the estimated base-line m a particular channel depends upon the choice of the peak, about whxch the calculauon is centered. B) Systematic errors observed for the modified Ralston-Wilcox technique (section 4). The experimental procedure is sketched m the reset. Composite spectra were generated an such a way that the relauve posiUon of the shaded peak varied m the course of the experiment. The parameter p is numerically equal to the distance between the two peaks The presented errors are those observed for the shaded peak The five sets of data correspond to different Intensity ratios between the unshaded peak plus the Compton edge and the shaded peak. The errors observed for 1 5 0 > p > 6 0 are due to the superposltion of the shaded peak on the Compton edge. Some dlustratwe data: TPA of the shaded peak is 23 740 counts; m the last set (filled triangles) the ratio of the height of the shaded peak to the height of the Compton edge was equal to 1.7.

PHOTOPEAK

INTEGRATION

TABLE 4

Although the results for the two peaks are quite consistent, the rather scarce amount of data gathered here for smaller peaks do not allow to draw firm conclusions.

Percent error for peak on C o m p t o n edge. D a t a from fig. 6 B.

4. A further investigation of the modified Ralston-Wilcox method

1 2 3 4 5

For most of the integration methods only a narrow part of the spectrum around the photopeak is of direct interest. This ts obviously not the case for the RalstonWilcox type methods, where one has to be aware of possible long range influences. This will most hkely occur when the peaks are superimposed on Compton edges or when they are situated in a very crowded portion of the spectrum In fig. 6A a section of the reference spectrum (from channel 480 to 560) is presented together with the base-hne calculated following the original and the modified technique. The total base-hne is a good approximation but systematic errors are evident. The modified technique, on the other hand, apparently fails. The erroneous behaviour of this technique is estabhshed more systematically in the following experiment. 54Mn was counted five Umes together with S9Fe; the Compton edge of the 1099 keV peak of 59Fe was situated about 100 channels hagher than the S~Mn peak. The timing in this series was p r o p o m o n a l to 0.1, 0.5, 1, 5 and 10. The third one is further denoted as the reference spectrum. Composite spectra were generated in a computer memory such that

MRW

CV

+ 0.2 + 1.0 + 1.9 + 11.6 +20.4

+0.1 --0.5 -0.3 -0.2 +4 9

and p is the number of channels a spectrum was shifted with respect to the reference spectrum. In this way 150 synthetic spectra were obtained, composed of five groups in which p took 30 different values between + 160 and - 5 2 . The Compton edge and a S4Mn peak were thus mowng from left to right under the S4Mn peak of the reference spectrum, which was present in all the composite spectra. The total peak area of this peak was calculated each time and compared to the original value As the background is evaluated from 121 channels around the peak, the SgFe peaks can be neglected. The results and some relevant data are given in fig. 6B. A striking feature is the considerable amount of errors when the peak separation is about 30 channels. This can be easily explained with the aid of fig. 7. The contents of channels in the sections AB and DE are never replaced by averaged values, and the high points at the left side of B prevent that the points at the right are averaged down to the background level The latter points have in turn the same effect on the channels under the processed peak. It must be emphasized that

Cf . . . . p = Ct, 3 -Jr"Ct+p,j, where C,. s are the counts m channel i of spectrum j,

/

207

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. . . . . --X---__...we[g X

20

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30

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30

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30

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=E

Fig. 7 Modified R a l s t o n - W d c o x m e t h o d e x p l a n a u o n o f the errors w he n the s e p a r a u o n between two pe a ks ~s ca 30 channels The crosses are the obviously erroneous base-hne. P r e l i m m a r y ~ e l g h t m g of the spectral d a t a as shown p r o v i d e d a means to reduce the errors.

208

J. HERTOGEN et al

this kind of error is typical for the modified and not for the original Ralston-Wilcox method. The dewations when the peak is superimposed on a Compton edge, on the other hand, are approximately the same for the original method, the positive errors (fig. 6B) are even expected to be h~gher. In table 4 some results from fig. 6B (for a peak separatmn equal to 98) are compared with the corresponding results of the Covellmethod, the integration width being the same for both methods.

observed for TPA methods (6) and (10), but the results clearly show the weak point of PPA techniques. Application of Covell's and Sterlinsk~'s method in an inadequate way by taking too small a part of the peak, leads to flagrant errors even for minor shift. The procedures based on Sterlinski's principle, are very sensitive to shift, due to distortions of the statistical weights of the peak channels.

After some prehminary experiments it was found that weighting of the spectral data with the trapezoidal functmn in&cated in fig. 7 was able to reduce cons~derably the above mentmned errors of the modified technique. A better performance of the RalstonWilcox method may also be achieved by adjusting the value of l, as was pointed out in refs 3, 22 and 24. However, adjusting the/-value and weighting the data do not always provide better results, and they make the Ralston-Wilcox technique even more complex.

When Ge(Li) g a m m a spectrometry is apphed to routine analysis of a series of nearly identical samples, a prehminary qualitative treatment of the spectra ~s often omitted. A complete quantitative reduction is restricted to a number of preselected analytical peaks, whose surroundings and interferences are assumed to be known 24'~4). In this type of programming one has to chose whether the peak boundaries - if required for the background evaluation - are determined in situ, or are supplied as input information under the form of a given peak m a x i m u m to boundary distance. The

6. Application to routine analysis

5. Instrumental shift The foregoing experiments are arranged in such a way that systematic errors due to instrumental shift are excluded. To arrive at realistic overall conclusions this effect must be taken into account. As shift is a rather indefimte term, its possible impacts on peak area integration are usually treated indefinitely as well. Inadequate pole-zero cancellatmn or base-line restorattan m the amplifier cause tailing or broadening of the peak, while in longer countmgs continuous or discontinuous zero point or gain shifts are hkely to occur. It was tried here to simulate discontinuous shift experimentally, as it can easily be kept under control. The 122keV and 136keV peaks of 57Co were counted first for 600 s with the planar Ge (Li) detector. In a second run the source was counted for 540 s in the same conditions, and for 60 s after the peak was shifted down l to 2 channels by changing the amplifier gain. The two spectra were then added in the analyzer memory. In the same manner other measurements such as 4 0 + 8 0 + 4 8 0 s , 2 4 0 + 3 6 0 s , . . (downward shift), 3 6 0 + 1 5 0 + 9 0 s, . . (upward shift) were combined. A narrow pulser peak was inserted between the 57Co peaks to monitor the introduced shifts. F~g. 8 presents the percent devlatmns for the 122 keV peak: the 136 keV peak gave similar results. The absolute errors are peculiar to this experiment. More relevant quantitative information is the considerable difference in sens~tivaty to shift for the various types of methods. No appreciable deviations were

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PHOTOPEAK

INTEGRATION

TABLE 5 Influence of a small interfering peak at the left wing of peak 1. Some selected results a. P5

OB

PM b

0.975 0.971 0.955 0.933 0.898 0.853

0.988 0.986 0.964 0.957 0.942 0.923

1.003 1 011 1.017 1.014 1.008 1.019

a Measured area/expected area. b With fixed mtegratmn interval, set to cover the total peak wldth.

second choice guarantees a reproducible boundary selection, but some obvious disadvantages are the problems arising from the presence of unexpected interfering peaks and the integration over too many insignificant channels when deahng with smaller peaks. The latter fact, however, can be avoided by limiting the integration width to ca 2 fwhm, or by lnsertmg an additional check of the channel content such as proposed for method PMT (6). Fixing the boundaries on the other hand allows to build in a fair amount of a priori knowledge about the spectra. This can be illustrated in the following case. While measuring the background spectra in the experiment described in section 3.1, a spurious peak appeared at the left wing of peak 1 in the composite spectra. Neglecting this interference in the boundary selection for method (4) resulted m increasing negative errors (table 5, P5). Negative, though smaller errors were also observed for method (9), which searches for minima in the peak structure (table 5, OB). When the left boundary was moved downward and the integration interval adjusted so as to account for the expected interference, its influence could be largely eliminated in quite a simple fashion (table 5, PM). This kind of situation is not so uncommon in practice. For example, m the complex Ge(L1)-spectra of neutron irradiated granitic or trachytic rock samples a lot of the analytical peaks, even above 150 keV, suffer from such small interfering species at the peak wings. In a more versatile approach to routine analysis, it is not even necessary to treat all the peaks in the same way: one can guide the program to calculate for some peaks the total area, and for others a partial area, if this choice proved to gwe more reliable results. In a further step one can include two or three different inte-

METHODS

209

gration methods as options33'34). Spectrum smoothing over an interval around the peak of interest can also readily be supplied as an option. As a consequence of the improving resolution of the present counting equipment, smoothing by means of a Sawtzky-Golay filter 35) may cause peak distortions of the often narrow low-energy lines, while it may occasionally be useful for the relatwely broader high-energy peaks. This type of programming requires, of course, a lot of human interference before the quantitative reduction. Fortunately this has to be effected only once for slmdar measurements in routine analysis. On the other hand, it facilitates the necessary corrections of the data and the immediate calculations of the elemental abundances This holds especially when lack of memory space and access to fast peripheric storage units limits the design of so-called complete activation analysis programs. The integration width, W, is a supplementary parameter for PPA methods, and the only one for methods (1) and (2). The most obvious way is a calculation as a function of the resolution for each energy. Starting from a statistical treatment Reber and Major a6) and Heydorn and Lada 37) evaluated the optimum value for W for PPA methods and for the Covell method respectively. Apart from the dependence on the resolution, the optimum value turned out to be also a function of the peak-to-background ratio an both cases. As a consequence a particular choice of W is often a compromise in routine analysis, because of differences in background level and peak intensities among various related spectra. 7. Remarks on the spectral shape at lower energies and its implication for base-line evaluation A well-known aspect of intense high-energy lines is the higher count level at the left side of the peak, due to multiple Compton interactions, scattering under small angles in surrounding materials, etc. One has to account for this level when chosing a mathematical expression for the peak shape2a). Some similar peculiarities of the spectrum shape for lower-energy photons, however, are not always realized. To explain this, one has to bear in mind the general formula for the Compton interaction: E' = ElI1 + e ( 1 - - c o s 0)1 ,

where c~= E/mo c 2. The maximum energy of a Compton electron, T,(max), and the minimum energy of a backscattered photon, E'(min), are, respectively, equal to

210

J. HERTOGEN et al.

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Fig. 9 A) SchemaUc representanon o f the detector response for E~ < 200 keV. B) Synthenc multlplet C) and D ) E x p a n d e d view o f the lo~er part o f fig 9 B The dots are the background obtained by' adding the hnear base-hnes of the mdl~qdual components o f the s:rnthetlc muluplet. The hnear base-hne calculated from the multlplet as a whole clearly underestimates the total background actlvtty. Fig. 9 D shows the results o f the smoothed staircase base-hne techmque, proposed by G u n n m k and N1day a8) The agreement between the expected (open circles] and the calculated base-hne (crosses) is qmte good.

E[2~./(l+2a)] and E/(l+2~) (for 0 = 180:). If E_<205.5 keV, then T¢(max) < E'(min) and the resulting detector response is sketched in fig. 9A. It can be checked that the width of the interval BP decreases for smaller E. Hence, at lower energies all the backscattered photons are compressed in a narrow energy interval. These effects are clearly illustrated m the reference spectrum (fig. 2), e.g. the section between channels 250 and 1000. A considerable number of interactions may occur in the source itself As a consequence, the relatl~.e height of the section BP depends also on the atomic composition and the dimensions

of the source. In spectra measured ~lth a conventional detector, this level has often a pronounced positive slope, due to preferential detection of photons that are scattered under small angles in the front side of the detector housing and occasionally m the insensitive upper layer of the G e ( L 0 diode. This may seriously harm the selection of the left peak boundary. The situation worsens when evaluating the base-hne of mulUplet structures m semiconductor X-ray analysm and m photon spectrometry up to ca 120 keV. In fig. 9C, the dotted line represents the real background of the displayed synthetic multiplet (fig. 9B). It was

PHOTOPEAK

INTEGRATION

obtained by adding the straight-line background of the four separate components. It is clear that the straight-line base-line for the entire multiplet is a poor estimate and will introduce appreciable errors at the position of the smaller components. One may doubt that the real background under the multiplet can be described by a simple low-order polynomial and subtracted before passing to a non-linear curve fitting32), at least, if this low-energy step is not included m the response function. In a more modest way of treating doublets or triplets, it is often tried to remove the contribution of species which have another free photopeak. The minimum requirement is a knowledge of the true net area of the multiplet. This is obviously not always the case when a simple straight-line background is assumed. In this connection it is worthwhile to mention the quite successful heuristic procedure proposed by Gunnink and Niday3S). After the qualitative analysis, a s t a i r c a s e base-line is constructed, the various steps being proportional to the roughly estimated intensity of the components. Repeated smoothing (moving average) of this step function gives a fluent base-line. Apphcation of this technique to the multiplet resulted in the crossed base-line (fig. 9 D). It clearly fits well to the real background, obtained after processing the single components an the same way (open circles). 8. Conclusions

l) The errors in the calculated peak area as a function of decreasing peak-to-background ratio turned out to be not so negatively biased as expected from the results of Baedecker14). On the contrary, positive as well as negative trends were observed for a number of peaks. Therefore, a statistically weighted average deviation for intense and smaller single peaks was assumed to be a realistic criterion for the performance of the examined peak integration methods. The decrease of the peak-tobackground ratios caused dominantly negative errors for only two of the examined techniques, when they were judged by this criterion. After all, this is a rather positive conclusion. Moreover, another positive point is apparent from the obtained results: even when the peak-to-background ratio changed over one or two orders of magnitude, the errors were seldom larger than twice the standard deviation estimated from counttng statistics. This emphasizes the inherent reliaNlity of Ge (Li) gammaray spectrometry, when dealing with well resolved photopeaks. 2) It is generally known that TPA integration is the

METHODS

211

safest approach when degradation of the system resolution occurs. However, in view of the considerable gain in precision it should be a bad philosophy to discard the PPA option, also when peak distortion is absent. 3) One can doubt if a direct comparison of the examined procedures is justified. As a matter of fact, the computer-oriented methods among them [(7), (9) and (10)] are not only designed for the quantitative but also for the quahtatlve analysis of gamma-ray spectra. As a consequence, they have to be judged by their performance on both levels. If for the sake of clarity, only the quantitative aspect is considered, none of the techmques here evaluated clearly comes out as the best one. The Ralston-Wilcox method has the advantage over most of the others, that interfering species at the peak wings have a negligible mftuence on the calculated base-line. The latter method and Qulttner's method were able to provide precise and quite accurate results down to very low peak-to-background ratios. Nevertheless, both techniques may occasionally lead to rather senseless results, so that actually all the results have to be checked carefully. Covell's method showed a significant negative bias for decreasing peak-to-background ratios. This phenomenon could not be explained so far. When properly apphed, Sterlinski's method, on the other hand, competed very well with the more complex techniques. The latter two methods, however, are very sensitive to resolution losses and shift. When dealing with small peaks on a high background one can better switch to another technique. Method (6), proposed in this paper, proved to be a quite precise and accurate TPA and PPA method, suitable for routine analysis with smaller computers. Although the precision of Op de Beeck's second method (10) was not so excellent, it turned out to be a valuable - and with respect to the quahtatlve analysis, more powerful - alternative for the well-known Yule-method 11,12). A drawback of all the examined methods is that they can only process single peaks. Unfortunately, multlplets in Ge(Li) spectra are not so uncommon as claimed by several authors. At the present time, a relatively simple technique for quantitative multiplet resolving should advance Ge(Li) spectrometry far more than any new procedure to treat single peaks. We are grateful to Prof. J. Hoste for his Interest in this work. The financial support of the "Belgisch Nationaal Fonds voor Wetenschappehjk Onderzoek"

212

J. H E R T O G E N et al.

and the "Interuniversitair Inst~tuut voor Kernwetens c h a p p e n " is h i g h l y a p p r e c i a t e d .

References 1) H. P. Yule, 3[odern trends in actiLatzon anal3sis, J. R. De Voe fed.) (N.B.S. Special Pubhcauon 312, Washington, 1969) pp. 1155-1204. z) H. P. Yule, Activatton attalysls tn geochemistry and cosmochemistry, A. O. Brunfelt and E. Stemnes (Eds.) (Umversitetsforlaget, Oslo, 1971) pp. 145-166. a) j. Turkstra, M. C. J. van Rensburg and W. J. de Wet, Report PEL-204 (South African Atomic Energy Board, Pehndaba, August 1970). 4) F. Adams, J. Op de Beeck, P. van den Wlnkel, R. GIjbels, I3 De Soete and J. Hoste, Critic. Re~. Anal. Chem. 1 (1971) 455. 5) p. Qmttner, Gamma-ray spectroscopy (Adam Hflger Ltd., London, 1972). 6) D. F. Covell, Anal. Chem. 31 (1959) 1785. 7) S. Sterlinski, 1bid. 40 (1968) 1995. s) S. Sterlinskl, 1bid. 42 (1970) 151. 9) p. Qmttner, lbld. 41 (1969) 1504. 10) p. Qutttner, Nucl. Instr. and Meth. 76 (1969) 115. 11) H. P. Yule, Anal. Chem. 38 (1966) 103. 12) H. P. Yule, 1bid. 40 (1968) 1480. 13) G. Guzzi, J. Pauly, F. Gwardl and B. Dorpema, Report E U R 3469 e (1967). 14) p. A. Baedecker, Anal. Chem. 43 (1971) 405. 15) j. E. Chne, IEEE Trans. Nucl. Sc. 15 (1968) 198. 16) O. U. Anders, Nucl. Instr. and Meth. 68 (1969) 205. 17) D. E. Crouch and R. L. Heath, Report IDO-16923 (1963). is) j. Barto~ek, G. Wlndels and J. Hoste, Nucl. Instr. and Meth. 103 (1972) 43.

19) K. Huysmans, R. Gijbels and J. Hoste, J. Radioanalyt. Chem., in press 20) H. P. Yule, Conf. Int. Les tendances modernes de l'analyse par activation, Saclay (October 1972) Paper M 29. 21) W. W. Bowman, Nucl. Instr. and Meth. 96 (1971) 135. 22) H. R. Ralston and G. E. Wdcox, Modern trends in actzvation analysis, J. R. DeVoe (Ed.) (N.B.S. Special Pubhcatlon 312, Washington, 1969) pp. 1238-1243. 28) C. M. Lederer, Report UCRL-18948 (1969). 24) R. Dams, J. A. Robbins and J. W. Winchester, Report COO-1705-6 (Univ. of Michigan, 1970). 25) j. De Donder, program NABAS (Ghent Umv., 1971). 26) j. p. Op de Beeck, P. A. Baedecker and J. Siegel, Report MIT-905-108 (Mass. Inst. Technol., 1967). 27) j. p. Op de Beeck, J. Radioanalyt. Chem. 11 (1972) 283. 2s) j. p. Op de Beeck, communication presented at the 3rd Symp. Recent developments in neutron activatton analysis Cambridge, U.K. (July 1973). 29) j. Hertogen and R. Gi3bels, Anal. Chim. Acta 56 (1971) 61. 8o) H. Tomlngana, M. Dojyo and M. Tanaka, Nucl. Instr. and Meth. 98 (1972) 69. m) H. P. Yule, Anal. Chem. 44 (1972) 1245. 82) E. Junod, Nucl. Instr. and Meth. 105 (1972) 13. 33) G. A. Borchardt, G. W. Hoagland and R. A Schmltt, J. Radloanalyt. Chem. 6 (1970) 241. a4) j. Hertogen and J. De Donder, program LESDEP (Ghent Umv., 1971). 85) A. Savitzky and M. J. E. Golay, Anal. Chem. 36 (1964) 1627. a6) j. D. Reber and J. K. Major, Nucl. Instr. and Meth. 23 (1963) 162. 37) K. Heydorn and W. Lada, Anal. Chem. 44 (1972) 2313. zs) R. G u n n m k and J. B. Niday, Report UCRL-51061, voh 1 (1972).