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Numerical correction of multichannel analyzer data for system nonlinearity
NUCLEAR
INSTRUMENTS
AND
METHODS
87 ( 1 9 7 o ) 4 5 - 5 7 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
N U M E R I C A L C O R R E C T I O N OF M U L T I C H A N N E L A N A L Y Z E R DATA F O R S Y S T E M NONLINEARITY* C. E. T U R N E R , Jr.t
Institute for Atomic Research and Department o f Physics, Iowa State University, Ames, Iowa 50010, U.S.A. Received 12 June 1970 considerations o f importance for achieving a rapid, accurate
A correction technique using a c o m p u t e r for elimination o f differential a n d integral nonlinearity in a multichannel pulseheight-analyzer ( M C P H A ) system is presented. N o n l i n e a r h i s t o g r a m data are redistributed into a n e w set o f channels having equal width by use o f an accurate differential nonlinearity m e a s u r e m e n t and a suitable a p p r o x i m a t i o n . Experimental
differential nonlinearity measurement are treated carefully for a time-to-amplitude converter (TAC) system. Tests of the procedure with a system comprised of a TAC, biased amplifier, and MCPHA are described.
1. Introduction
calibration-nonlinearity data to a suitable function provides the parameters needed to determine accurately unknown energies or time delays. The various other attempts at linearity correction encountered in the literature involve a microscopic D N L measurement of the system. Some authors ~-3"5) have made use of the fact that the integral with respect to counts over a suitable D N L curve is a linear function of the measured variable. The calibration is determined by integration of the D N L data between the channel locations of two accurately known reference peaks. The value of the variable associated with any other channel is found by integration of the D N L counts between it and a reference channel. Others have fit a straight line 4) o r r/th order polynomial a) to the D N L data by the method of least squares. In conjunction with a suitable calibration this method gives the variable of interest as a polynomial function of channel number. However, a reasonably low order polynomial cannot account for the real kinks often observed in D N L curves of timeto-amplitude converters (TAC) [subsection 4.2.3]. The above correction methods are useful when one simply requires accurate energy or time differences between distinct spectrum features. However, the data remain nonlinear. The channels do not have constant width. For more sophisticated data analysis requirements this can be a problem. For example, in the extraction of nuclear lifetimes from delayed coincidence distributions, numerical calculation of a line slope, of the various moments of the time distribution, of convolution integrals, etc., is much more tractable with the assumption of constant channel width. I f significant variation in channel width cannot be avoided instrumentally, then some form of correction technique is necessary, preferably one which results in data distributed in channels of constant width.
A linear response is a most desirable feature of the analogue-to-digital converter (ADC) and other electronic components of a multichannel pulse-heightanalyzer ( M C P H A ) system. Such a response simplifies data analysis and eliminates sources of systematic error. Linearity has been achieved to varying degrees of approximation which have often proved to be inadequate for the accuracy desired of the measurements. The demands of greater accuracy have led to the development of both correction techniques t-5) and more sophisticated instrumentation6-S). This work presents a novel approach to the problem of linearity correction with M C P H A systems. The differential nonlinearity ( D N L ) of a M C P H A system can be examined either microscopically or macroscopically9). In the former case one is concerned with the channel-to-channel deviations from linearity while in the latter case only the deviation of the severalchannel-average width from constancy is considered. In the following, D N L will be considered to indicate the microscopic viewpoint unless otherwise indicated. A common method of correcting the macroscopic D N L of a M C P H A system is to obtain several accurate calibration points over the range of the variable of interest which may be typically either energy 1°) or time H, 12). For energy determinations the macroscopic D N L can also be measured by use of a precision pulser 13-16) or other techniques9'17). When electronic methods of nonlinearity determination are used, fewer calibration points (in principle, only two) are required for scale definition. A least-squares fit of the combined * W o r k p e r f o r m e d in part in the A m e s L a b o r a t o r y of the U.S. A t o m i c Energy C o m m i s s i o n , C o n t r i b u t i o n no. 2692. t Present address: Lawrence Radiation Laboratory, Livermore, California.
45
46
C. E. TURNER, JR.
Over the last several years integral nonlinearity specifications of A D C ' s have improved from +_0.25%* to 0.005% t and D N L specs have dropped from +_2%* to 0.15 % t. But the best A D C specs presently obtainable are often not matched by those of other system components (table 1). Also an older M C P H A , still functioning within its original but unsatisfactory specifications, may be the only one available to the experimenter for economic reasons. In such cases nonlinearity must be corrected in order to obtain accurate results. For the sake of accuracy, programming simplicity, and economy, the development was undertaken of a generally applicable procedure which could bring the effective linearity of an entire M C P H A system close to that of the newer A D C ' s , which would result in linear data having essentially constant channel width, and which would entail a minimum of extra equipment and experimental time. These objectives have been accomplished for the case of a system utilizing a TAC. This paper discusses a computer correction which is based on a D N L measurement of the system. Experimental considerations of importance for achieving a rapid, accurate D N L measurement are treated carefully for a T A C system. The results of various performance tests of the numerical correction process with a T A C system demonstrate the validity and effectiveness of the method. The correction procedure can, in principle, also be used with energy spectroscopy systems if the detector itself is linear and the D N L measurement is carefully obtained by feeding pulses of the proper shape from a sliding pulse generator to the preamplifier inputS). However, the method presented in this paper depends
for statistical accuracy on having a large number of counts in each channel of the D N L measurement. For energy measurements with spectra typically involving a few thousand channels such a D N L measurement could become lengthy depending on the number of channels involved, the number of counts per channel desired, and the sliding pulser rate used. Since energy spectra do not normally require complicated data analysis where constant channel width is especially desirable, the "integral with respect to counts" correction procedure used by Strauss et al. 5) has an experimental advantage. In this method the correction accuracy is determined by the total number of counts in the D N L measurement between the reference channel and the channel location of the spectral peak of interest. Hence, a considerably lower counts/ channel in the D N L measurement can give the accuracy desired.
* Radiation Instrument Development Laboratory, Inc., Model 34-12B. 1" Tennelec, Inc., Model TC 500.
2. The correction method
2.1. BAsIc APPROACH The data accumulated with an A D C is essentially a histogram which has nonconstant channel width due to instrumental nonlinearity. Determination of the relative channel widths of a M C P H A system simply requires a measurement of the system's D N L by common techniques (subsection 3.2). With this information it is possible to adjust numerically histogram data to have constant channel width, at least approximately. 2.2. NUMERICAL CORRECTION PROCEDURE Let D(t) be some physically measurable distribution of the parameter t which can be converted to a pulse amplitude distribution for analysis by an ADC. Assume that the channel boundaries are given by t,,
TABLE 1 Linearity specifications of the equipment used.
Instrument MCPHA TAC
Biased amplifier Sliding pulser
Integral nonlinearity 4-0.25% of full scale (f.s.) < 4-0.2% 10%-i00% Ls. <4-1.0%
5%-100% f.s.
<0.2%
1%-100% f.s.
< ±0.1%
Differential nonlinearity 4-2% for top 98% <0.5% 5%-100% Ls. except <0.5% 25%-100% f.s. on 100-nsec range <~4-1% (limit of measuring capability) < 4-0.25%
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
i = 0, 1, ..., M. The channel widths are
Wi = h - t i - ~ ,
i = 1,2 .... , M .
(1)
=
(2)
WdWN.
[t is assumed that the w~ are known from a microscopic D N L measurement of sufficient accuracy. In a measurement of the unknown distribution D(t) the observed data will be
N~ =
f:
D(t')df,
i=
1,2 ..... M ,
(3)
47
DATA
ignoring for the moment statistical fluctuations. It would be preferable to have linear data
Let Wrq be some suitable normalization width, e.g., an average over several channels in the most linear region of the ADC. Then the relative channel widths are
w,
ANALYZER
~tR
"P j d t
D(t')dt',
Nj = tR+(j--
j--
1,2,...,
(4)
l)At
where At is a suitable interval, e.g. WN, and tR is a reference point. I f the functional form for the variation of D(t) across the channels were known, then linear data could be obtained by use of eq. (4). In principle, an approximate variation of D(t) across a channel could be obtained by fitting a polynomial with parameters ct over a region of a few channels. Let this approximation be indicated by Di(t, ~). The approxi-
WM_5
(a)
Nk'l -,IWM_,I.Nb._w/,'4
°
~
z
tM_4 tM-3 tM-2 IM_I tM
(b) D =M-5 ~ Dj=M-2
IN THIS EXAMPLE:
_WN Dj=M--W-M-M' Di=M z
Dj=M-I=(~)'Di=M
z
oj= .2 :(
+(2WN-WM )'Di,M-I WM.I
~',.-, / / / DJ=M-I ,,N,\ / , ~X.,'XX~ \ ~ \ ¢, ',-., OJ=M ,,\, //, \\NI//I
WM_I+WM-2wN~
, / t \\NI/// \ \ N
+
( ~WN "WMI' I WM i)"
WM-2
O~ = M-- ~
/
/
/
~ \ \ /.F.~ ~, \ X r / j
\ \ N I / / / ~ . \ N V / / ~. \ N I / / J \ \ ~ V I /
,.x...
.
.
~ \ ~ 1 / / i
ETC. t
tM-4WN tM-2WN tM tM-SWN
I'M-WN
Fig. 1. Illustration o f the numerical nonlinearity correction procedure for M C P H A data Ni with the particular set o f channel widths W~ s h o w n in (a) u n d e r the a p p r o x i m a t i o n that the distribution D(t) is constant, De = Ni/Wi, across channel i. Correction begins at tM a n d proceeds to the left. T h e c o r r e s p o n d i n g similarly shaded areas in (a) and (b) are equal.
48
C. E. T U R N E R ,
mately linear data are then I tR +jdt N j ,~ Di(t',o~i)dt', j = 1,2 . . . . . (5) ,J t R + ( j - - 1)At The subscript i is associated with the experimental channels of varying width, while the subscript j indicates adjusted data having constant channel width. I f the variation of D(t) across At is not large, it can he ignored. In this simplest approximation
Di(t' , cq) ~ Ni/W~.
sonably high or low values of W N. Fig. lb shows the results of the correction procedure applied to the data of fig. la. The areas within the solid lines in fig. 1 represent the channel count content. The corresponding shaded areas in figs. l a and b are equal and represent the simplest approximation to the expected count content for channels of equal width WN. The bookkeeping exemplified in fig. lb gets more complex for arbitrary Wi and is best understood by a careful study of figs. 7 and 8 of the appendix.
(6)
This investigation is restricted to application of the approximation (6) to correct for system nonlinearity. The approximation gives good results even when there is considerable variation over the channels, as in the case of calibration peaks with fwhm ~ three to four channels (subsection 4.2). Fig. 1 illustrates the numerical adjustment involved for a particular set of W i. The integration (5) with the approximation (6) begins with channel M (tR = t~t) and proceeds to lower channels since A D C ' s are usually most linear at the top. WN is determined from the average of the top several channels, say 10. of
2.3.
LIMITATIONS ON ACCURACY
The essential element for the utilization of the numerical correction described in section 2.2 is knowledge of the channel widths 14/,.. Thus a measurement of the D N L of the system is required under conditions which reproduce as nearly as possible those used for obtaining the data to be corrected, if identical conditions are achieved, the accuracy of Wi is determined by statistical effects and the fractional standard deviation of Wi will be A Wi/Wi = ,j'Ni/Ni, where N i is the number of counts in the ith channel of the D N L measurement. With Ni ~ 40000 counts the best possible D N L achievable would be ~ 0.5%. In practice the lower limit AWJW~ may not be reached
M
the D N L
JR.
data, i.e.,
WN = ~ I4//. This averaging i=M--9 prevents statistical fluctuations from producing unrea-
LUCITELIGHTPIPE ~ "-] / L~
NATON
REFRIGERATION ,/[H.V~. UNIT (-48°C)
SOURCE
VARIABLE RATE RANDOM NOISE PULSER
TIME r----ITRIGGER I. . . . MARK ' ~ J IGENERATOR ~4
START•
T AC
1
. STOP
BIASED AMR MCPHA Fig. 2. A simplified block d i a g r a m o f the T A C system omitting electronic calibration, digital stabilization, a n d pileup elimination electronics. T h e solid-line inputs to the S T A R T a n d S T O P triggers represent the n o r m a l m e a s u r e m e n t configuration. T h e dashed-line inputs represent the differential nonlinearity m e a s u r e m e n t configuration.
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
because of the statistical inaccuracy of the data to be corrected, error introduced by using the approximation (6) and systematic differences between the D N L measurement and the measurement to be corrected. 3. Application to TAC systems 3.1. GENERAL CONSIDERATIONS The numerical correction technique described above is particularly suited to T A C systems. Here the number of channels used is usually low ( < 400 channels) and this reduces the time required to make a D N L measurement. In addition the accidental coincidence distribution obtained under certain conditions (subsection 3.3) provides the D N L not only of the M C P H A but also of the TAC, biased amplifier, etc., under conditions almost exactly the same as used for the physical measurement. Systematic differences are expected to be associated mainly with rate effects which can be minimized. 3.2.
DIFFERENTIAL NONLINEARITY OF
TAC
SYSTEMS
Fig. 2 presents schematically the essential features of the TAC system used in this investigation. The complete system which contained circuitry for pileup rejection, digital stabilization and electronic calibration is described elsewhere18). With a T A C system one measures the distribution in time of coincidences between two events such as nuclear radiations signaling the birth and decay of an excited nuclear state or an accelerator beam trigger pulse and a subsequent nuclear radiation, etc. The T A C produces an output pulse which has an amplitude approximately proportional to the time difference between S T A R T and STOP signals received from triggers associated with the two detectors. The biased amplifier presents a certain range of the T A C output amplitude distribution with appropriate amplification and pulse shaping to the M C P H A . All three of these T A C system instruments have nonlinearities associated with them. The D N L of the M C P H A alone or possibly the M C P H A plus biased amplifier can be obtained by the sliding pulser technique for certain special pulse shapes and rates (section 4). However, such measurements omit the D N L of the T A C and there is uncertainty about the effect of pulse shape differences between the sliding pulser output and the T A C or biased amplifier outputs. It is better to obtain the D N L of the entire system from the random coincidence spectrum. In this case one relies on the fact that the time distribution of accidental coincidences is uniform if certain conditions are met (sub-
ANALYZER
49
DATA
section 3.3) in addition to the obvious requirement of uncorrelated S T A R T and STOP pulses. Several alternatives are possible for obtaining uncorrelated pulses in the S T A R T and STOP channels. One can use the basic detection system with one radioactive source as long as the energy discrimination can be set to provide non-coincident events at a suitable count rate. Alternatively, various combinations of one of the basic detectors, an independent source-detector system, a random noise pulser, and a constant frequency pulser can be used. The arrangement shown in fig. 2 consisting of a variable-rate random-noise pulser in the S T A R T channel and a variable-rate periodic pulser in the STOP channel is preferred (subsection 3.3). 3.3.
RATE EFFECTS
In order to reduce the time required to obtain the desired accuracy in the D N L measurement one normally uses singles rates as high as are possible without distortion of the observed distribution. Such distortion can arise from statistical or electronic sources. For the case of a START-STOP T A C the use of high rate random STOP pulses can cause a nonuniform accidental coincidence distribution. I f the STOP pulses occur randomly and have a constant mean rate a, then Poisson statistics apply. At t = 0 a S T A R T pulse activates the TAC. The probability for observing a START-STOP interval of duration between t and t + dt is the combined probability that there will be no pulse in the STOP channel during the time interval t, but one pulse between t and t + d t . The result from application of the Poisson distribution is 19) dPt = a
e-"tdt.
(7)
Eq. (7) clearly shows that small time intervals tare more probable. In order to maintain a uniform accidentals distribution for any T A C measurement using random STOP inputs, the STOP rate should be such that d P t does not change significantly over the range T of the TAC, i.e., dPo/dP T = e -"r ~
1,
or equivalently, aT
,~ 1.
(8)
For T = 100 nsec a random stop rate less than l0 s Hz is desirable. Note that eq. (7) also applies to the intervals between random pulses in the STOP channel. Small STOP intervals are more probable than large. Thus it is more probable for a S T A R T pulse to occur during a short
50
C.E.
TURNER, JR.
STOP interval and short S T A R T - S T O P intervals are more probable. The accidentals nonuniformity arises basically because STOP intervals shorter than the T A C range are possible and because STOP intervals of different length do not have the same probability. I f the STOP input is not random but has a constant period greater than the T A C range T, Poisson statistics no longer apply. The S T A R T pulses occur at random during STOP intervals of constant length and a uniform accidentals distribution results. Therefore a D N L measurement undistorted by statistical effects can be obtained more rapidly with a high-rate constantfrequency STOP input.* The above statistical considerations were based on a START-STOP TAC. For other TAC types similar considerations should be made before using high rate inputs. Electronic distortions can arise for a variety of reasons which will depend somewhat on the particular equipment used. Analyzer pileup distortion can be
eliminated or considerably reduced by inhibiting the T A C during the analyzer deadtime. The use of a constant frequency STOP signal with a period shorter than the T A C range will obviously result in severe distortion. The deadtimes of the system components can cause difficulties. Also the use of high singles rates may cause distortions due to overdriving the TAC input circuitry in some cases. In general, each system must be evaluated separately and care exercised in the choice of the singles rates used. The deadtime asymmetry of the START-STOP TAC also influences the choice of input rates. Such a T A C normally has a fixed deadtime associated with each S T A R T input even when no coincidence occurs. There is usually no deadtime associated with the STOP input pulse unless a coincidence occurs. Thus it is generally preferred to route the higher singles rate channel to the STOP input provided that condition (8) is met for random inputs. However, this may not be practical on the longer time scales if the higher rate is associated with the event which occurs first physically. This is due to the large delays which would have to be introduced into the STOP channel in order to get an inverted time distribution, i.e., one where small amplitude TAC output pulses correspond to longer physical time delays. The setup indicated in fig. 2 for D N L measurements optimizes all of the above considerations by use of a
* This conclusion was reached independently in a paper by Cova and Bertolacciniz°) which appeared just after the completion of this work. The analysis above gives a qualitative physical explanation of effects which were established in 2o) by a detailed theoretical examination of the statistical characteristics of a START-STOP TAC. The situation considered in this work is the most important special case of the more general treatment in 2o).
(a) + SLIDING PULSER DATA 18000
+
÷
u9 IZ
. . ~ 1
i~:''"
÷
•
-I-'~" ' +.,~
16000
14000
1.05
bl
X NORMALIZED AVERAGE WIDTH
"1tr'-,
x x
x
x
x
x
x
x
x
x
x
_> .J I.u r'r
x
x
I.O0
0,95
2O
,~o
6'o
8'0
,oo
,~o
40
~o
,;o
2~0
CHANNEL NUMBER
Fig. 3. (a) The microscopic differential nonlinearity of the MCPHA measured with a sliding pulser. (b) The relative ten-channelaverage widths obtained from the data in (a) normalized to the average channel width.
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
very-high-rate constant-frequency pulser* in the STOP channel and a variable-rate random-noise pulser ~ in the START channel. The complete system ~8) used in this investigation also incorporated digital stabilization and MCPHA busy inhibit of the TAC, both of which help prevent electronic rate distortion. 4. Performance with a TAC system 4.1. LINEARITY CHARACTERISTICS OF T A C SYSTEM COMPONENTS
The linearity specifications of the MCPHA +, TAC X, * Tektronix, Inc., Model 184. t Canberra Industries, Inc., Model 1407R. + Radiation Instruments Development Laboratories, Model 34-12B. × E G & G , Inc., Model TH200A.
ANALYZER
DATA
51
biased amplifier*, and sliding pulser t used in this study are given in table 1. Fig. 3 presents the optimum linearity obtained from the MCPHA during tests of the various conversion gain settings with the sliding pulser. Reasonable variations of the pulse shape used did not affect the results appreciably. Fig. 3a shows the microscopic DNL with some indication of odd-even effects which are not uncommon in older MCPHA's. Fig. 3b shows a macroscopic DNL of __+3.5% which is outside specifications. This deviation may be due to non-optimum pulse shape for the MCPHA. In figs. 4a and b are shown two DNL measurements for the combined TAC, biased amplifier, and MCPHA system obtained by means of accidental coincidences
Inc., * E G & G , Inc., Model AN109/N. t Berkeley Nucleonics Corp., Model GL-3.
90000
(a)
A WIDTH DATA
Z A
o 75ooo
~
~
" ~"
"2---J
M. ~ ' -
-~''- .......
~'2
"~-
~;.~,,;.%'L" ---='~-. . . . . .
60000
12000"
(b)
+ UNCORRECTED DATA +
hZ 0 (,,.)
t0000 +
800 0~O3 I-Z
12000 <
(c)
© CORRECTED DATA
800C NORMALIZED
I.OE i
(d)
1" Fa 1.00
A V E R A G E WIDTH
x
BEFORE
CORRECTION
O
AFTER
CORRECTION
-
x
w >
x
x x
x
x
x~
x
*
~
x
x x
-J
hi r,"
0.95
x
i
0
20
;-"
i
40
60
i
80
i
i
I00
120
CHANNEL
i
140
i
160
i
180
i
200
i
220
i
240
i
260
NUMBER
Fig. 4. Illustration o f the numerical nonlinearity correction as applied to two independent accidental coincidence spectra (a) and and (b), i.e., independent differential nonlinearity measurements o f the T A C system. Use o f the data (a) for width definition in order to correct the data (b) for nonlinearity yields the corrected data (c). The macroscopic differential nonlinearity o f the data before and after correction is presented in (d).
52
c.E.
TURNER,
as described in section 3. The lower points (× 's) in fig. 4d, associated with the data of fig. 4b, show a macroscopic D N L of +2.5% which is closer to the MCPHA specifications. The pulse presented to the MCPHA by the biased amplifier is closer to the
optimum shape specified for the MCPHA (0.8/~sec rise, 2-3 psec wide, 2/~sec decay). A comparison of the "best" shapes obtainable from the biased amplifier and sliding pulser is given in fig. 5. The differences seen in figs. 3, 4, and 5 point out the necessity of making the DNL measurement under conditions as nearly identical as possible to the physical measurement to be corrected.
(I) BEST SLIDINGPULSER S H A P E (2) BEST BIASEDAMP PULSE SHAPE
ibJ l
0
2
4
6 8 I0 MICROSECONDS
JR.
4.2. TESTS OF THE CORRECTIONMETHOD Tests were made to ascertain that the computer program functioned properly and to check the effectiveness of the correction technique for various relative statistical accuracies of the width and correctable data. Additional studies were carried out to check for ratedependent and configuration-dependent systematic variations in the D N L measurements and to determine the stability of the D N L measurements over extended periods of time. Finally the criticality of the approximation (6) was determined by application of the
12
Fig. 5. A comparison of the pulse shapes obtainable from the sliding pulser (1) and the biased amplifier (2) which are closest to that preferred by the M C P H A . The preferred shape is similar
to (2).
.k
10800
IO)
++
+ UNCORRECTEDDATA
+ ÷ +
+
+
10600 + (/) I.-
Z =1 0 ¢.,)
10400
+AVERAGE -- --+-+
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+
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"t~" + 4.+ .a.l.+W + ÷ - + - - ÷ - +- - ~ + - , % -~7- - + ~ ÷÷
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++
I0000
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+
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t~ 0
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+ ++ +++
c~
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9800
÷
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ItoZ
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(b)
(9 (9 (9 (9 (9--~-AVERAGE~_~( ~ ( ~ _ . ~ ( ~
10400
(9~-(9~---(9-(9
LIJ I-(.~ M.I
~ CORRECTEDDATA
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m e
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± ~ r :I:zlG - -
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"n.- 1 0 2 0 0 0 0
2' 0
40
60
8'0
Io 0
1. 2 0
.
CHANNEL
1 .4 0
.
1 6. 0
180
200
2
0
2 4' 0
60
NUMBER
Fig 6.(a) a n d (b). Expanded views of figs. 4(b) and (c), respectively, showing the nonlinearity and statistical scatter about the average with respect to the one- and two-standard deviation limits.
NUMERICAL
CORRECTION
OF
MULTICHANNEL
correction method to TAC calibration data where the observed distribution varies rapidly across the channels. 4.2.1. Statistical effects testing To test for proper operation of the correction program the same D N L data was used as both width and correctable data. A perfectly constant line resulted as expected. Independent D N L measurements were then used as width and correctable data. In this case the variation of the correctable data across a channel is negligible and the approximation (6) should be very good. Thus, statistical and systematic effects can be tested independently of the approximation validity. Figs. 4 and 6 show the results of correcting a D N L measurement of ~ 1% accuracy by width data of 0.4% accuracy. A sharp peak a few channels wide at the low end of the spectrum is off scale. Figs. 4a and 4b exhibit, respectively, width and correctable data corresponding to data sets No. 2H and No. 1 of table 2. Fig. 4c shows the data of fig. 4b after linearity correction. Fig. 4d presents the macroscopic D N L of the uncorrected and the corrected data. After correction the deviations of the macroscopic D N L from perfect linearity are all less than 0.5%. The average of the
ANALYZER
53
DATA
magnitudes of the deviations before and after correction are 1.4% and 0.26%, respectively. For perfectly linear data obtained from linear instruments one would expect one third or 79 of the 237 channels to have a count content (N~) beyond one standard deviation (a = x/N~) from the average (Ni) simply for statistical reasons. For the data of fig. 6a there are 130 channels beyond Ni_+ a before correction as indicated in table 2. After correction fig. 6b shows only 63 channels outside
Ni+a. Table 2 indicates the influence on the correction effectiveness of the relative statistical accuracies possessed by the width and correctable data. The improvement in macroscopic D N L is shown by the average of the magnitudes of the deviations of the normalized ten-channel-average widths from unity expressed as a percent before and after correction. The improvement in the microscopic D N L is displayed by listing the number of channels outside N~___a before and after correction. Data set No. I is totally independent of data set No. 2. The data sets No. 2A through No. 2H were obtained by nondestructive printout at various times during the data accumulation period and are thus not completely independent.
TABLE 2 Comparison o f the macroscopic and microscopic differential nonlinearity improvement for width and correctable data sets o f various relative statistical accuracies.
Width label
Data label
Average counts N~
a/Nt(%)
Average % deviation before
Average % deviation after
1 I I I I 1 1 1
2A 2B 2C 2D 2E 2F 2G 2H
258 662 1025 2054 4119 8260 12395 76500
6.2 3.9 3.1 2.2 1.6 1.1 0.90 0.36
2.0 1.6 1.4 1.4 1.3 1.3 1.3 1.3
1.7 0.99 0.82 0.46 0.42 0.35 0.29 0.22
67(33 + 34) 74(38 + 36) 69(33 + 36) 82(36 + 46) 94(42 + 52) 115(51 +64) 134(55 + 79) 192(74 + 118)
48(21 + 27) 52(26 + 26) 52(24 + 28) 56(25 + 31) 76(39 + 37) 97(48+49) 105(53 + 52) 153(78 + 75)
2H 2H 2H 2H 2H 2H 2H 2H 2H
1 2A 2B 2C 2D 2E 2F 2G 2H
10475 258 662 1025 2054 4119 8260 12395 76500
0.98 6.2 3.9 3. I 2.2 1.6 1.1 0.90 0.36
1.4 1.9 1.5 1.3 1.4 1.3 1.3 I. 3 1.3
0.26 1.5 0.91 0.62 0.44 0.33 0.26 0.21 0.00
130(56 + 74) 67(33 + 34) 74(38 + 36) 69(33 + 36) 82(36 + 46) 94(42 + 52) 115(5 i + 64) 134(55 + 79) 192(74+ 118)
63(28 + 35) 46(23 + 23) 47(23 + 24) 47(21 + 26) 49(23 + 26) 52(25 + 27) 51 (25 + 26) 52(27 + 25) 0
Number o f channels outside N t ± a * beforet afterf
* For perfectly linear uncorrected data approximately one third or 79 of the 237 channels would be beyond ~'i ± a . ~f The numbers are given in the order: total (below + above).
54
C. E. T U R N E R ,
However, for the purposes of table 2, No. 2H is essentially independent of No. 2A through No. 2G. In no case does the correction process worsen the linearity. This result is expected to be true as long as the width accuracy is better than the system nonlinearity. Thus noticeable improvement in both macrosopic and microscopic D N L is seen even for correctable data such as No. 2A where the statistics are worse than the nonlinearity. As expected the corrected macroscopic D N L improves with better accuracy of either width or correctable data. The improvement in microscopic D N L lessens somewhat as the width and correctable data get closer in statistical accuracy, probably because of compounding statistical errors. For example, data sets No. 2F, 2G, and 2H corrected by No. 1 show an increase (above the simple one-third expectation) in the number of points beyond N i___~r. On the whole, however, the accuracy is better so the average percent deviation continues to improve. Generally it appears that the microscopic D N L can be corrected to better than the accuracy of the correctable data with sufficiently accurate width data. However, the average percent deviations after correction are somewhat higher than would be expected for normal linear data. For example, with N~ ~ 10 000, one would expect a standard deviation of ~ 0.1% in ten-channel averages whereas ~ 0.2% is seen in table 2. However, for practical purposes the difference between corrected data and perfectly linear data is probably negligible provided the approximation (6) is good. The listing of the linearity test program L I N E used in conjunction with the correction subroutine L I N T Y to obtain the results shown in table 2 and figs. 4 and 6 is available elsewhere 21). L I N E was also used for the systematic effects testing described in subsections 4.2.2 through 4.2.4.
JR.
4.2.3. Systematic confiyuration dependence The waves seen at the lower end of the D N L measurements in fig. 4 are not uncommon 22-24) for T A C ' s used with ranges of ~ 100 nsec. These oscillations indicate correlations between S T A R T and STOP signals. In the present work it was found that they could be minimized by using as far as possible a common powerline and power supply for all the electronics associated with the S T A R T and STOP channels. Also when modular fast electronics are used, it is unwise to select the two sections of a dual discriminator, dual OR circuit, etc. for signal processing in the S T A R T and STOP channels, respectively. Note that the correction procedure described here can readily take small oscillations into account. A linearity correction obtained by fitting a polynomial to the accidental coincidence spectrum can not correct for these oscillations. It would be meaningless to try to correct the sharp peaks at the extreme ends of the spectrum. In other tests no significant difference in D N L was observed when the constant frequency pulser and/or the random noise pulser were replaced with independent radioactive source plus photomultiplier assemblies. 4.2.4. Stability of the differential nonlinearity The D N L of the system was found to be quite stable. Two D N L measurements of comparable statistical accuracy 0~i ~ 103) taken 16 days apart showed no significant differences when one was used as width data and the other as correctable data. Even longer term stability can be inferred from the fact that the TAC calibration determined with the use of the same D N L measurement remained constant (___0.25%) over a period of 45 days. Calibration runs were made every other day. 4.2.5. Correction of TAC calibration;
4.2.2. Systematic rate dependence In rate-effect tests, where the condition (8) was fulfilled, electronic distortion was found to be mainly a function of the coincidence rate. For a T A C range of 100 nsec various combinations of random S T A R T rates and a constant frequency STOP rate of up to 2 M H z showed no electronic distortion for coincidence rates of at least 10 kHz. At such coincidence rates accurate D N L measurements can be made in a few to several hours*. * The analysis in 2o) was able to give estimates of the counting time required to achieve a D N L measurement of specified accuracy. It should be noted that M C P H A rate dependence may be the true limiting factor, however.
Approximation validity The effect of the linearity correction on TAC calibration data is shown in table 3. The electronic calibration method in the case considered here gave symmetric peaks (fwhm ~ 3-4 channels) whose separation was known very accurately to be 5 nseclS). A linear leastsquares fit to the centroid versus delay curve was performed with and without linearity correction. The differential nonlinearity width data had N~ 125 000 counts. This implies a corrected D N L no better than 0.3%. The deviations of the experimental centroids from the calculated least-squares lines are shown in table 3. The very significant reduction in the deviations after linearity correction provides good evidence that approximation (6) is adequate even for
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
TABLE 3 C o m p a r i s o n of the deviations of T A C calibration peak centroids from a least-squares line before and after linearity correction; also the channel count variation across a typical calibration peak.
Peak number
Delay (nsec)
I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000
(
Centroid deviation before (channel)
Centroid deviation after (channel)
--0.71742 -0.23491 -0.03667 0.11681 0.13322 0.23836 0.18126 0.25261 0.21426 0.25867 0.21005 0.20091 0.10255 0.02483 -0.14935 -0.29303 -0.50009
-0.00333 0.01303 0.01314 0.01143 - 0.00378 0.02153 - 0.02367 -0.00421 - 0.02046 - 0.03506 -0.00818 - 0.00119 0.00926 0.01773 -0.00226 0.00177 0.01808
Channel number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
N~
0 0 1
11
201
1052 3265
6055 7873 6808 3661 966 48 l 0 0 o
DATA
55
rather rapid across-channel variation of the type seen in column 6 of table 3 for a typical calibration peak. The standard deviations before and after correction were 0.30 channel and 0.016channel, respectively. Before correction the integral nonlinearity is estimated as ~ (2×0.717)/220 = 0.65% from the maximum deviation of 0.717 channel out of a 220 channel ( ~ 85 nsec) range. After correction the integral nonlinearity is expected to be < (0.016/220)= 0.074%. The linearity correction raised the calibration constant 1.3% and reduced the standard calculated statistical error from 0.1% to 0.006%. 5. Conclusions
The linearity correction procedure developed here is capable of improving the integral nonlinearity o f a T A C system using an older M C P H A to within a factor of two of the best A D C specifications encountered. The microscopic D N L is corrected to better than the statistical accuracy of the data subjected to correction with an ultimate accuracy determined by the statistics of the D N L measurement. The deviation of the macroscopic D N L from constancy after correction is only about a factor of two times what would occur for linear data with statistical fluctuations.
SUBR~UTINELINTY~ (MI,LI, L2,JF, KF ) ~ /
8
R N ~ ,RNWIO+ NWID(K)-1.0 CTS(J),RCTS+(Y(K}/NWID(K}W NWID(K)-RNWID) RCT$-(RNW~O/NWIO {K))W-Y( K)
REALNWID C~MM~N X(400}, Y(400), C H(400), CTS(4OO),WX(400), WlO ( 4 0 0 ) , NWID (400)
t
WN~RM • 0 . 0 RNWID • 0 . 0 RCTS " 0.0
5 CTS( J ) ,RCTS/RNW]D RNWID • RNWID-I.O RCTS • RCTS-CTS(J) J-J-I
t NL2 sL2-9
l
ANALYZER
I
O~ I 0 0 I , N L 2 , L 2 IO0 WN~IRM=WNORM÷WZD { r ) i
l I WN¢B.-WN~R./tLZ-N' ~+,)J f
7 RNW~ • RNWm+N WII) (K) RCTS • RCTS + Y(K) G~ TI~ 3
9 JF-J KF, K D~lO Z'JF, L2 IOCH(I)- X(I)
(-,°-.o 3
DO I Z = I o M I CH ( I ) " 0 . 0 CTS (Z) - O 0 I NW~D{I) • W I O { I ) / W N ~ R M
Fig. 7. Statement by statement flow chart of the nonlinearity correction subroutine L I N T Y .
56
c.E.
C C C C C C C C C C C
C C C C C C
C C C C C C C
C C C C C
C C
TURNER, JR.
SUBROUTINE LIKTY ADJUSTS THE DATA ( X , Y I BY USE CF A DIFFERENTIAL NONLINEARITY MEASUREMENT (WX,WIO) TO PROVIDE APPROXIMATELY lINEAR DATA (CH,CTS) NOTE: X ( I ) , Y ( 1 ) SHOULD CDRRESPCKD TO W X ( 1 ) , W I C ( I I , N W I D { I ) MI : NUMBER CF CHANNELS IN THE DIFFERENTIAL NONLINEARITY MEASUREMENT L I , L 2 : STORAGE INDICES OF THE LOWER AND UPPER CHANNEL LIMITS DF THE LINEARITY CORRECTION REGION JF,KF : FINAL VALUES OF J,K NWID : ARRAY FOR THE NORMALIZED CHANNEL WIDTHS COMPUTED IN LINTY IX,Y) : ARRAYS CONTAINING THE (CHANNEL,COUNT) DATA TO BE CORRECTED (CH,CTS) : ARRAYS FCR THE CORRECTE~ {CHANNEL,COUNT| DATA (WX,WID) : ARRAYS CONTAINING THE (CHANNEL,CCUNT) DATA OF THE OIFFERENTIAL NONLINEARITY MEASUREMENT NL2 : INDEX SPECIFYING THE LOWER LIMIT CF THE REGION FOR CALCULATING WNORM WNCRM : NORMALIZATION WIDTH OBTAINED BY ~VERAGING OVER A FAIRLY LINEAR REGION OF THE WID DATA ; IN THIS WORK THE REGION WAS CHOSEN TO BE THE UPPERMOST IC CHANNELS OF THE LIKEARITY CCRRECTION REGION J : INDEX FOLLOWING THE FORMATION OF C H ( J ) , C T S ( J ) K : INDEX OF THE UNCORRECTED DATA X ( K ) , Y ( K ) BEING USED TO FORM CH(J) ,CTS(J) RNWID : NORMALIZED WIDTH REGISTER; KEEPS TRACK CF WIDTH AVAILABLE FCR FORMATICN OF AN ACJUSTEC CHANNEL AND/OR THE WIDTH REMAINING UNUSED AFTER FORMATION GF AN ADJUSTED CHANNEL RCTS: COUNTS REGISTER; KEEPS TRACK OF THE NUMBER OF COUNTS ASSOCIATED WITH RNWID COMPILATION AND EXECUTION TIME: L E S S THAN 5 SEC FOR SEVERAL DATASETS ON IBM 360/65 SUBROUTINE L INTY( MI, L I , L 2 , J F , KF ) REAL NWID COMMON X(400},Y(4CO),CH(40C),CTS(40CI,WX(40O),WID(4OO),NWID(400) WNORM=O,
100
I
2 3 4 5
6 7
8
g
10
RNWID=C.O RCTS=O .0 NL2 = L2 - g DO i 0 0 I=NL2,L2 WNORM=WNCRM÷WID(I) WNORM=WNCRM/( L 2 - N L 2 * I ) 00 1 I = I , M I CH(II=O.O CTSqI)=O.O NWID(I)=WID( I)/WNORM J=L2÷I K=L 2÷ 1 J=J-1 K=K-I IF(RNWID-I.0| 6 , 5 , 5 CTS ( J)=RCTS/RNW ID RNWI D=RNWI D-I • RCTS=RCTS-CTS(J ) J=J - I I F ( K - L I ) g,%,4 IF{RNWID÷NWID(K)-I.0) 7 , 8 , ~ RNWID=RNWID÷NW ID(K) RC TS=RCTS+Y(K) GO TO 3 RNWID=RNW ID÷NW ID ( K ) - i ,0 CTS( J )=RCTS÷( Y(K)/NWID(K) ) * ( NWID (K) -RNW I D) RCTS=(RNWICINW I D ( K ) ) ~ Y ( K ) IF(K-LI) 9,9,2 JF=J KF=K DO i 0 I = J F , L 2 CH(I)=X(I) RETURN END
Fig. 8. Computer subroutine LINTY for nonlinearity correction of MCPHA system data.
NUMERICAL CORRECTION
OF M U L T I C H A N N E L A N A L Y Z E R DATA
After linearity correction the adjusted data have essentially constant channel width and further analysis techniques can validly utilize this simplifying assumption. This feature is quite useful if sophisticated unfolding procedures are to be used to extract accurate nuclear lifetimes from delayed-coincidence distributions25). The time and equipment necessary to accomplish the correction are not excessive. The correction takes only a few seconds on a fast computer. The D N L measurement takes only a few to several hours after preliminary setup-testing for system optimization. Random-noise sources, high-rate pulsers, and even sliding pulsers are generally required for standard measurements or equipment testing and, therefore, are already available at most laboratories. In any case they are inexpensive compared to a very linear MCPHA. Since the correction depends intimately on the channel width measurement, care must be exercised to avoid systematic differences between the D N L measurement and the physical measurement of interest. Rate-dependence deserves special consideration when attempting to obtain a rapid D N L measurement. Systematic differences are most easily avoided with a TAC system but the method is generally applicable where a suitable D N L measurement can be obtained. The correction can significantly reduce errors which would otherwise be of the order of the average macroscopic D N L . The continued interest and encouragement of Professor E. N. Hatch as well as his helpful comments on the manuscript are gratefully acknowledged,
Appendix A statement by statement flow chart is given in fig. 7
57
of the commented F O R T R A N nonlinearity correction subroutine L I N T Y presented in fig. 8.
References 1) W. Schweimer, Nucl. Instr. and Meth. 32 (1965) 190. 2) W. F. Mruk, Nucl. Instr. and Meth. 34 (1965) 293. 3) C . A . Baker, G . J . Batty and L.E. Williams, Nucl. Instr. and Meth. 59 (1968) 125. 4) C. Camhy-Val, M. Dreux, A. M. Dumont and J. Marchal, Nucl. Instr. and Meth. 70 (1969) 25. 5) M. G. Strauss, F. R. Lenkszus and J. J. Eicholtz, Nucl. Instr. and Meth. 76 (1969) 285. ~) N. Abbattista, M. Coli and V. L. Plantamura, Nucl. Instr. and Meth. 60 (1968) 337. 7) H. J. Schuster, Nucl Instr. and Meth. 63 (1968) 182. 8) I. De Lotto, P . F . Manfredi, P. Maranesi, F. Vaghi and R. Vecchio, Nucl. Instr. and Meth. 65 (1968) 228. 9) H. Lycklama and T. J. Kennett, Nucl. Instr. and Meth. 59 (1968) 56. lo) H. Lycklama, L. B. Hughes and T. J. Kennett, Can. J. Phys. 45 (1967) 1871. 11) j. L. lrigarag, J. Roturier and G. Y. Petit, Nucl. Instr. and Meth. 40 (1966) 221. 12) C. Dardini, G. laci, M. Lo Savio and R. Visentin, Nucl. Instr. and Meth. 47 (1967) 233. t3) W. W. Black and R. L. Heath, Nucl. Phys. A90 (1967) 650. ta) j. Greenblatt, K. S. Kuchela and N. K. Sherman, Nucl. Instr. and Meth. 49 (1967) 86. 15) A.J. Levy and R. C. Ritter, Nucl. Instr. and Meth. 49 (1967) 359. t6) W. W. Black, Nucl. Instr. and Meth. 53 (1967) 249. 17) R . E . Berg and E. Kashy, Nucl. Instr. and Meth. 39 (1966) 169. 18) C. E. Turner, Jr., U.S.A.E.C. Report no. IS-T-350 (1970). 19) R . D . Evans, The atomic nucleus (McGraw-Hill Book Co., Inc., New York, 1955) ch. 26, p. 754. 20) S. Cova and M. Bertolaccini, Nucl. Instr. and Meth. 77 (1970) 269. 21) C. E. Turner, Jr., U.S.A.E.C. Report no. IS-2253 (1970). 22) H. Weisberg, Nucl. Instr. and Meth. 32 (1965) 133. 23) A. Tamminen and P. Jauho, Nucl. Instr. and Meth. 65 (1968) 132. 24) F. Du Chaffaut, P. Charmet and R. Trabaud, Nucl. Instr. and Meth. 65 (1968) 285. 25) C. E. Turner, Jr. and R. A. Anderl, to be published.
INSTRUMENTS
AND
METHODS
87 ( 1 9 7 o ) 4 5 - 5 7 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
N U M E R I C A L C O R R E C T I O N OF M U L T I C H A N N E L A N A L Y Z E R DATA F O R S Y S T E M NONLINEARITY* C. E. T U R N E R , Jr.t
Institute for Atomic Research and Department o f Physics, Iowa State University, Ames, Iowa 50010, U.S.A. Received 12 June 1970 considerations o f importance for achieving a rapid, accurate
A correction technique using a c o m p u t e r for elimination o f differential a n d integral nonlinearity in a multichannel pulseheight-analyzer ( M C P H A ) system is presented. N o n l i n e a r h i s t o g r a m data are redistributed into a n e w set o f channels having equal width by use o f an accurate differential nonlinearity m e a s u r e m e n t and a suitable a p p r o x i m a t i o n . Experimental
differential nonlinearity measurement are treated carefully for a time-to-amplitude converter (TAC) system. Tests of the procedure with a system comprised of a TAC, biased amplifier, and MCPHA are described.
1. Introduction
calibration-nonlinearity data to a suitable function provides the parameters needed to determine accurately unknown energies or time delays. The various other attempts at linearity correction encountered in the literature involve a microscopic D N L measurement of the system. Some authors ~-3"5) have made use of the fact that the integral with respect to counts over a suitable D N L curve is a linear function of the measured variable. The calibration is determined by integration of the D N L data between the channel locations of two accurately known reference peaks. The value of the variable associated with any other channel is found by integration of the D N L counts between it and a reference channel. Others have fit a straight line 4) o r r/th order polynomial a) to the D N L data by the method of least squares. In conjunction with a suitable calibration this method gives the variable of interest as a polynomial function of channel number. However, a reasonably low order polynomial cannot account for the real kinks often observed in D N L curves of timeto-amplitude converters (TAC) [subsection 4.2.3]. The above correction methods are useful when one simply requires accurate energy or time differences between distinct spectrum features. However, the data remain nonlinear. The channels do not have constant width. For more sophisticated data analysis requirements this can be a problem. For example, in the extraction of nuclear lifetimes from delayed coincidence distributions, numerical calculation of a line slope, of the various moments of the time distribution, of convolution integrals, etc., is much more tractable with the assumption of constant channel width. I f significant variation in channel width cannot be avoided instrumentally, then some form of correction technique is necessary, preferably one which results in data distributed in channels of constant width.
A linear response is a most desirable feature of the analogue-to-digital converter (ADC) and other electronic components of a multichannel pulse-heightanalyzer ( M C P H A ) system. Such a response simplifies data analysis and eliminates sources of systematic error. Linearity has been achieved to varying degrees of approximation which have often proved to be inadequate for the accuracy desired of the measurements. The demands of greater accuracy have led to the development of both correction techniques t-5) and more sophisticated instrumentation6-S). This work presents a novel approach to the problem of linearity correction with M C P H A systems. The differential nonlinearity ( D N L ) of a M C P H A system can be examined either microscopically or macroscopically9). In the former case one is concerned with the channel-to-channel deviations from linearity while in the latter case only the deviation of the severalchannel-average width from constancy is considered. In the following, D N L will be considered to indicate the microscopic viewpoint unless otherwise indicated. A common method of correcting the macroscopic D N L of a M C P H A system is to obtain several accurate calibration points over the range of the variable of interest which may be typically either energy 1°) or time H, 12). For energy determinations the macroscopic D N L can also be measured by use of a precision pulser 13-16) or other techniques9'17). When electronic methods of nonlinearity determination are used, fewer calibration points (in principle, only two) are required for scale definition. A least-squares fit of the combined * W o r k p e r f o r m e d in part in the A m e s L a b o r a t o r y of the U.S. A t o m i c Energy C o m m i s s i o n , C o n t r i b u t i o n no. 2692. t Present address: Lawrence Radiation Laboratory, Livermore, California.
45
46
C. E. TURNER, JR.
Over the last several years integral nonlinearity specifications of A D C ' s have improved from +_0.25%* to 0.005% t and D N L specs have dropped from +_2%* to 0.15 % t. But the best A D C specs presently obtainable are often not matched by those of other system components (table 1). Also an older M C P H A , still functioning within its original but unsatisfactory specifications, may be the only one available to the experimenter for economic reasons. In such cases nonlinearity must be corrected in order to obtain accurate results. For the sake of accuracy, programming simplicity, and economy, the development was undertaken of a generally applicable procedure which could bring the effective linearity of an entire M C P H A system close to that of the newer A D C ' s , which would result in linear data having essentially constant channel width, and which would entail a minimum of extra equipment and experimental time. These objectives have been accomplished for the case of a system utilizing a TAC. This paper discusses a computer correction which is based on a D N L measurement of the system. Experimental considerations of importance for achieving a rapid, accurate D N L measurement are treated carefully for a T A C system. The results of various performance tests of the numerical correction process with a T A C system demonstrate the validity and effectiveness of the method. The correction procedure can, in principle, also be used with energy spectroscopy systems if the detector itself is linear and the D N L measurement is carefully obtained by feeding pulses of the proper shape from a sliding pulse generator to the preamplifier inputS). However, the method presented in this paper depends
for statistical accuracy on having a large number of counts in each channel of the D N L measurement. For energy measurements with spectra typically involving a few thousand channels such a D N L measurement could become lengthy depending on the number of channels involved, the number of counts per channel desired, and the sliding pulser rate used. Since energy spectra do not normally require complicated data analysis where constant channel width is especially desirable, the "integral with respect to counts" correction procedure used by Strauss et al. 5) has an experimental advantage. In this method the correction accuracy is determined by the total number of counts in the D N L measurement between the reference channel and the channel location of the spectral peak of interest. Hence, a considerably lower counts/ channel in the D N L measurement can give the accuracy desired.
* Radiation Instrument Development Laboratory, Inc., Model 34-12B. 1" Tennelec, Inc., Model TC 500.
2. The correction method
2.1. BAsIc APPROACH The data accumulated with an A D C is essentially a histogram which has nonconstant channel width due to instrumental nonlinearity. Determination of the relative channel widths of a M C P H A system simply requires a measurement of the system's D N L by common techniques (subsection 3.2). With this information it is possible to adjust numerically histogram data to have constant channel width, at least approximately. 2.2. NUMERICAL CORRECTION PROCEDURE Let D(t) be some physically measurable distribution of the parameter t which can be converted to a pulse amplitude distribution for analysis by an ADC. Assume that the channel boundaries are given by t,,
TABLE 1 Linearity specifications of the equipment used.
Instrument MCPHA TAC
Biased amplifier Sliding pulser
Integral nonlinearity 4-0.25% of full scale (f.s.) < 4-0.2% 10%-i00% Ls. <4-1.0%
5%-100% f.s.
<0.2%
1%-100% f.s.
< ±0.1%
Differential nonlinearity 4-2% for top 98% <0.5% 5%-100% Ls. except <0.5% 25%-100% f.s. on 100-nsec range <~4-1% (limit of measuring capability) < 4-0.25%
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
i = 0, 1, ..., M. The channel widths are
Wi = h - t i - ~ ,
i = 1,2 .... , M .
(1)
=
(2)
WdWN.
[t is assumed that the w~ are known from a microscopic D N L measurement of sufficient accuracy. In a measurement of the unknown distribution D(t) the observed data will be
N~ =
f:
D(t')df,
i=
1,2 ..... M ,
(3)
47
DATA
ignoring for the moment statistical fluctuations. It would be preferable to have linear data
Let Wrq be some suitable normalization width, e.g., an average over several channels in the most linear region of the ADC. Then the relative channel widths are
w,
ANALYZER
~tR
"P j d t
D(t')dt',
Nj = tR+(j--
j--
1,2,...,
(4)
l)At
where At is a suitable interval, e.g. WN, and tR is a reference point. I f the functional form for the variation of D(t) across the channels were known, then linear data could be obtained by use of eq. (4). In principle, an approximate variation of D(t) across a channel could be obtained by fitting a polynomial with parameters ct over a region of a few channels. Let this approximation be indicated by Di(t, ~). The approxi-
WM_5
(a)
Nk'l -,IWM_,I.Nb._w/,'4
°
~
z
tM_4 tM-3 tM-2 IM_I tM
(b) D =M-5 ~ Dj=M-2
IN THIS EXAMPLE:
_WN Dj=M--W-M-M' Di=M z
Dj=M-I=(~)'Di=M
z
oj= .2 :(
+(2WN-WM )'Di,M-I WM.I
~',.-, / / / DJ=M-I ,,N,\ / , ~X.,'XX~ \ ~ \ ¢, ',-., OJ=M ,,\, //, \\NI//I
WM_I+WM-2wN~
, / t \\NI/// \ \ N
+
( ~WN "WMI' I WM i)"
WM-2
O~ = M-- ~
/
/
/
~ \ \ /.F.~ ~, \ X r / j
\ \ N I / / / ~ . \ N V / / ~. \ N I / / J \ \ ~ V I /
,.x...
.
.
~ \ ~ 1 / / i
ETC. t
tM-4WN tM-2WN tM tM-SWN
I'M-WN
Fig. 1. Illustration o f the numerical nonlinearity correction procedure for M C P H A data Ni with the particular set o f channel widths W~ s h o w n in (a) u n d e r the a p p r o x i m a t i o n that the distribution D(t) is constant, De = Ni/Wi, across channel i. Correction begins at tM a n d proceeds to the left. T h e c o r r e s p o n d i n g similarly shaded areas in (a) and (b) are equal.
48
C. E. T U R N E R ,
mately linear data are then I tR +jdt N j ,~ Di(t',o~i)dt', j = 1,2 . . . . . (5) ,J t R + ( j - - 1)At The subscript i is associated with the experimental channels of varying width, while the subscript j indicates adjusted data having constant channel width. I f the variation of D(t) across At is not large, it can he ignored. In this simplest approximation
Di(t' , cq) ~ Ni/W~.
sonably high or low values of W N. Fig. lb shows the results of the correction procedure applied to the data of fig. la. The areas within the solid lines in fig. 1 represent the channel count content. The corresponding shaded areas in figs. l a and b are equal and represent the simplest approximation to the expected count content for channels of equal width WN. The bookkeeping exemplified in fig. lb gets more complex for arbitrary Wi and is best understood by a careful study of figs. 7 and 8 of the appendix.
(6)
This investigation is restricted to application of the approximation (6) to correct for system nonlinearity. The approximation gives good results even when there is considerable variation over the channels, as in the case of calibration peaks with fwhm ~ three to four channels (subsection 4.2). Fig. 1 illustrates the numerical adjustment involved for a particular set of W i. The integration (5) with the approximation (6) begins with channel M (tR = t~t) and proceeds to lower channels since A D C ' s are usually most linear at the top. WN is determined from the average of the top several channels, say 10. of
2.3.
LIMITATIONS ON ACCURACY
The essential element for the utilization of the numerical correction described in section 2.2 is knowledge of the channel widths 14/,.. Thus a measurement of the D N L of the system is required under conditions which reproduce as nearly as possible those used for obtaining the data to be corrected, if identical conditions are achieved, the accuracy of Wi is determined by statistical effects and the fractional standard deviation of Wi will be A Wi/Wi = ,j'Ni/Ni, where N i is the number of counts in the ith channel of the D N L measurement. With Ni ~ 40000 counts the best possible D N L achievable would be ~ 0.5%. In practice the lower limit AWJW~ may not be reached
M
the D N L
JR.
data, i.e.,
WN = ~ I4//. This averaging i=M--9 prevents statistical fluctuations from producing unrea-
LUCITELIGHTPIPE ~ "-] / L~
NATON
REFRIGERATION ,/[H.V~. UNIT (-48°C)
SOURCE
VARIABLE RATE RANDOM NOISE PULSER
TIME r----ITRIGGER I. . . . MARK ' ~ J IGENERATOR ~4
START•
T AC
1
. STOP
BIASED AMR MCPHA Fig. 2. A simplified block d i a g r a m o f the T A C system omitting electronic calibration, digital stabilization, a n d pileup elimination electronics. T h e solid-line inputs to the S T A R T a n d S T O P triggers represent the n o r m a l m e a s u r e m e n t configuration. T h e dashed-line inputs represent the differential nonlinearity m e a s u r e m e n t configuration.
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
because of the statistical inaccuracy of the data to be corrected, error introduced by using the approximation (6) and systematic differences between the D N L measurement and the measurement to be corrected. 3. Application to TAC systems 3.1. GENERAL CONSIDERATIONS The numerical correction technique described above is particularly suited to T A C systems. Here the number of channels used is usually low ( < 400 channels) and this reduces the time required to make a D N L measurement. In addition the accidental coincidence distribution obtained under certain conditions (subsection 3.3) provides the D N L not only of the M C P H A but also of the TAC, biased amplifier, etc., under conditions almost exactly the same as used for the physical measurement. Systematic differences are expected to be associated mainly with rate effects which can be minimized. 3.2.
DIFFERENTIAL NONLINEARITY OF
TAC
SYSTEMS
Fig. 2 presents schematically the essential features of the TAC system used in this investigation. The complete system which contained circuitry for pileup rejection, digital stabilization and electronic calibration is described elsewhere18). With a T A C system one measures the distribution in time of coincidences between two events such as nuclear radiations signaling the birth and decay of an excited nuclear state or an accelerator beam trigger pulse and a subsequent nuclear radiation, etc. The T A C produces an output pulse which has an amplitude approximately proportional to the time difference between S T A R T and STOP signals received from triggers associated with the two detectors. The biased amplifier presents a certain range of the T A C output amplitude distribution with appropriate amplification and pulse shaping to the M C P H A . All three of these T A C system instruments have nonlinearities associated with them. The D N L of the M C P H A alone or possibly the M C P H A plus biased amplifier can be obtained by the sliding pulser technique for certain special pulse shapes and rates (section 4). However, such measurements omit the D N L of the T A C and there is uncertainty about the effect of pulse shape differences between the sliding pulser output and the T A C or biased amplifier outputs. It is better to obtain the D N L of the entire system from the random coincidence spectrum. In this case one relies on the fact that the time distribution of accidental coincidences is uniform if certain conditions are met (sub-
ANALYZER
49
DATA
section 3.3) in addition to the obvious requirement of uncorrelated S T A R T and STOP pulses. Several alternatives are possible for obtaining uncorrelated pulses in the S T A R T and STOP channels. One can use the basic detection system with one radioactive source as long as the energy discrimination can be set to provide non-coincident events at a suitable count rate. Alternatively, various combinations of one of the basic detectors, an independent source-detector system, a random noise pulser, and a constant frequency pulser can be used. The arrangement shown in fig. 2 consisting of a variable-rate random-noise pulser in the S T A R T channel and a variable-rate periodic pulser in the STOP channel is preferred (subsection 3.3). 3.3.
RATE EFFECTS
In order to reduce the time required to obtain the desired accuracy in the D N L measurement one normally uses singles rates as high as are possible without distortion of the observed distribution. Such distortion can arise from statistical or electronic sources. For the case of a START-STOP T A C the use of high rate random STOP pulses can cause a nonuniform accidental coincidence distribution. I f the STOP pulses occur randomly and have a constant mean rate a, then Poisson statistics apply. At t = 0 a S T A R T pulse activates the TAC. The probability for observing a START-STOP interval of duration between t and t + dt is the combined probability that there will be no pulse in the STOP channel during the time interval t, but one pulse between t and t + d t . The result from application of the Poisson distribution is 19) dPt = a
e-"tdt.
(7)
Eq. (7) clearly shows that small time intervals tare more probable. In order to maintain a uniform accidentals distribution for any T A C measurement using random STOP inputs, the STOP rate should be such that d P t does not change significantly over the range T of the TAC, i.e., dPo/dP T = e -"r ~
1,
or equivalently, aT
,~ 1.
(8)
For T = 100 nsec a random stop rate less than l0 s Hz is desirable. Note that eq. (7) also applies to the intervals between random pulses in the STOP channel. Small STOP intervals are more probable than large. Thus it is more probable for a S T A R T pulse to occur during a short
50
C.E.
TURNER, JR.
STOP interval and short S T A R T - S T O P intervals are more probable. The accidentals nonuniformity arises basically because STOP intervals shorter than the T A C range are possible and because STOP intervals of different length do not have the same probability. I f the STOP input is not random but has a constant period greater than the T A C range T, Poisson statistics no longer apply. The S T A R T pulses occur at random during STOP intervals of constant length and a uniform accidentals distribution results. Therefore a D N L measurement undistorted by statistical effects can be obtained more rapidly with a high-rate constantfrequency STOP input.* The above statistical considerations were based on a START-STOP TAC. For other TAC types similar considerations should be made before using high rate inputs. Electronic distortions can arise for a variety of reasons which will depend somewhat on the particular equipment used. Analyzer pileup distortion can be
eliminated or considerably reduced by inhibiting the T A C during the analyzer deadtime. The use of a constant frequency STOP signal with a period shorter than the T A C range will obviously result in severe distortion. The deadtimes of the system components can cause difficulties. Also the use of high singles rates may cause distortions due to overdriving the TAC input circuitry in some cases. In general, each system must be evaluated separately and care exercised in the choice of the singles rates used. The deadtime asymmetry of the START-STOP TAC also influences the choice of input rates. Such a T A C normally has a fixed deadtime associated with each S T A R T input even when no coincidence occurs. There is usually no deadtime associated with the STOP input pulse unless a coincidence occurs. Thus it is generally preferred to route the higher singles rate channel to the STOP input provided that condition (8) is met for random inputs. However, this may not be practical on the longer time scales if the higher rate is associated with the event which occurs first physically. This is due to the large delays which would have to be introduced into the STOP channel in order to get an inverted time distribution, i.e., one where small amplitude TAC output pulses correspond to longer physical time delays. The setup indicated in fig. 2 for D N L measurements optimizes all of the above considerations by use of a
* This conclusion was reached independently in a paper by Cova and Bertolacciniz°) which appeared just after the completion of this work. The analysis above gives a qualitative physical explanation of effects which were established in 2o) by a detailed theoretical examination of the statistical characteristics of a START-STOP TAC. The situation considered in this work is the most important special case of the more general treatment in 2o).
(a) + SLIDING PULSER DATA 18000
+
÷
u9 IZ
. . ~ 1
i~:''"
÷
•
-I-'~" ' +.,~
16000
14000
1.05
bl
X NORMALIZED AVERAGE WIDTH
"1tr'-,
x x
x
x
x
x
x
x
x
x
x
_> .J I.u r'r
x
x
I.O0
0,95
2O
,~o
6'o
8'0
,oo
,~o
40
~o
,;o
2~0
CHANNEL NUMBER
Fig. 3. (a) The microscopic differential nonlinearity of the MCPHA measured with a sliding pulser. (b) The relative ten-channelaverage widths obtained from the data in (a) normalized to the average channel width.
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
very-high-rate constant-frequency pulser* in the STOP channel and a variable-rate random-noise pulser ~ in the START channel. The complete system ~8) used in this investigation also incorporated digital stabilization and MCPHA busy inhibit of the TAC, both of which help prevent electronic rate distortion. 4. Performance with a TAC system 4.1. LINEARITY CHARACTERISTICS OF T A C SYSTEM COMPONENTS
The linearity specifications of the MCPHA +, TAC X, * Tektronix, Inc., Model 184. t Canberra Industries, Inc., Model 1407R. + Radiation Instruments Development Laboratories, Model 34-12B. × E G & G , Inc., Model TH200A.
ANALYZER
DATA
51
biased amplifier*, and sliding pulser t used in this study are given in table 1. Fig. 3 presents the optimum linearity obtained from the MCPHA during tests of the various conversion gain settings with the sliding pulser. Reasonable variations of the pulse shape used did not affect the results appreciably. Fig. 3a shows the microscopic DNL with some indication of odd-even effects which are not uncommon in older MCPHA's. Fig. 3b shows a macroscopic DNL of __+3.5% which is outside specifications. This deviation may be due to non-optimum pulse shape for the MCPHA. In figs. 4a and b are shown two DNL measurements for the combined TAC, biased amplifier, and MCPHA system obtained by means of accidental coincidences
Inc., * E G & G , Inc., Model AN109/N. t Berkeley Nucleonics Corp., Model GL-3.
90000
(a)
A WIDTH DATA
Z A
o 75ooo
~
~
" ~"
"2---J
M. ~ ' -
-~''- .......
~'2
"~-
~;.~,,;.%'L" ---='~-. . . . . .
60000
12000"
(b)
+ UNCORRECTED DATA +
hZ 0 (,,.)
t0000 +
800 0~O3 I-Z
12000 <
(c)
© CORRECTED DATA
800C NORMALIZED
I.OE i
(d)
1" Fa 1.00
A V E R A G E WIDTH
x
BEFORE
CORRECTION
O
AFTER
CORRECTION
-
x
w >
x
x x
x
x
x~
x
*
~
x
x x
-J
hi r,"
0.95
x
i
0
20
;-"
i
40
60
i
80
i
i
I00
120
CHANNEL
i
140
i
160
i
180
i
200
i
220
i
240
i
260
NUMBER
Fig. 4. Illustration o f the numerical nonlinearity correction as applied to two independent accidental coincidence spectra (a) and and (b), i.e., independent differential nonlinearity measurements o f the T A C system. Use o f the data (a) for width definition in order to correct the data (b) for nonlinearity yields the corrected data (c). The macroscopic differential nonlinearity o f the data before and after correction is presented in (d).
52
c.E.
TURNER,
as described in section 3. The lower points (× 's) in fig. 4d, associated with the data of fig. 4b, show a macroscopic D N L of +2.5% which is closer to the MCPHA specifications. The pulse presented to the MCPHA by the biased amplifier is closer to the
optimum shape specified for the MCPHA (0.8/~sec rise, 2-3 psec wide, 2/~sec decay). A comparison of the "best" shapes obtainable from the biased amplifier and sliding pulser is given in fig. 5. The differences seen in figs. 3, 4, and 5 point out the necessity of making the DNL measurement under conditions as nearly identical as possible to the physical measurement to be corrected.
(I) BEST SLIDINGPULSER S H A P E (2) BEST BIASEDAMP PULSE SHAPE
ibJ l
0
2
4
6 8 I0 MICROSECONDS
JR.
4.2. TESTS OF THE CORRECTIONMETHOD Tests were made to ascertain that the computer program functioned properly and to check the effectiveness of the correction technique for various relative statistical accuracies of the width and correctable data. Additional studies were carried out to check for ratedependent and configuration-dependent systematic variations in the D N L measurements and to determine the stability of the D N L measurements over extended periods of time. Finally the criticality of the approximation (6) was determined by application of the
12
Fig. 5. A comparison of the pulse shapes obtainable from the sliding pulser (1) and the biased amplifier (2) which are closest to that preferred by the M C P H A . The preferred shape is similar
to (2).
.k
10800
IO)
++
+ UNCORRECTEDDATA
+ ÷ +
+
+
10600 + (/) I.-
Z =1 0 ¢.,)
10400
+AVERAGE -- --+-+
--
--
-H" + +
I,,IJ I,,-
÷ +
%
+
++
lad
,~-+ +'~+
__+-~ +++
+
+
+4- + + + + + +
++
+
÷
+ +
+
_
+
'W-_
+++++
+
+ +
+
*-+~ -- ---+÷++ +
+
+~
+-+ +
~- + ++
-w-
__
+
•
+
+
+
L
+ +
-+
+:++ +
+
+++ +e +
++
+++ 4- 4-
"t~" + 4.+ .a.l.+W + ÷ - + - - ÷ - +- - ~ + - , % -~7- - + ~ ÷÷
._+÷
%
++
I0000
~
+
+
+
+---+
++ +++
+
,-" 10200
t~ 0
+ +
+
+
+
+
+
+ ++ +++
c~
Z
9800
÷
÷÷
++
ItoZ
10600
0 C)
(b)
(9 (9 (9 (9 (9--~-AVERAGE~_~( ~ ( ~ _ . ~ ( ~
10400
(9~-(9~---(9-(9
LIJ I-(.~ M.I
~ CORRECTEDDATA
(9
(9
(9
(9 (9
(9
(9-%-@-(9--~
~
.
(9
(9(9
(9
(9_@,,~_ej
- ~ =~ ~ ~ ~
(9~(9(9~
(9
(9 - -
~
(9
(
~ . . . .
~(9 ( 9
9 (9 ~
(9 ( 9 ( 9
(9 (9
(9_(9_ ( 9 ~ _
m e
__(9ev-(9~9-- - -,---~
% (9(9
(9
(9
/
t~ (9 e~ (9~(9 . . _ o ~ ~
(9 (9 (9
~
o
--
/
± ~ r :I:zlG - -
(9 (9
(9
"n.- 1 0 2 0 0 0 0
2' 0
40
60
8'0
Io 0
1. 2 0
.
CHANNEL
1 .4 0
.
1 6. 0
180
200
2
0
2 4' 0
60
NUMBER
Fig 6.(a) a n d (b). Expanded views of figs. 4(b) and (c), respectively, showing the nonlinearity and statistical scatter about the average with respect to the one- and two-standard deviation limits.
NUMERICAL
CORRECTION
OF
MULTICHANNEL
correction method to TAC calibration data where the observed distribution varies rapidly across the channels. 4.2.1. Statistical effects testing To test for proper operation of the correction program the same D N L data was used as both width and correctable data. A perfectly constant line resulted as expected. Independent D N L measurements were then used as width and correctable data. In this case the variation of the correctable data across a channel is negligible and the approximation (6) should be very good. Thus, statistical and systematic effects can be tested independently of the approximation validity. Figs. 4 and 6 show the results of correcting a D N L measurement of ~ 1% accuracy by width data of 0.4% accuracy. A sharp peak a few channels wide at the low end of the spectrum is off scale. Figs. 4a and 4b exhibit, respectively, width and correctable data corresponding to data sets No. 2H and No. 1 of table 2. Fig. 4c shows the data of fig. 4b after linearity correction. Fig. 4d presents the macroscopic D N L of the uncorrected and the corrected data. After correction the deviations of the macroscopic D N L from perfect linearity are all less than 0.5%. The average of the
ANALYZER
53
DATA
magnitudes of the deviations before and after correction are 1.4% and 0.26%, respectively. For perfectly linear data obtained from linear instruments one would expect one third or 79 of the 237 channels to have a count content (N~) beyond one standard deviation (a = x/N~) from the average (Ni) simply for statistical reasons. For the data of fig. 6a there are 130 channels beyond Ni_+ a before correction as indicated in table 2. After correction fig. 6b shows only 63 channels outside
Ni+a. Table 2 indicates the influence on the correction effectiveness of the relative statistical accuracies possessed by the width and correctable data. The improvement in macroscopic D N L is shown by the average of the magnitudes of the deviations of the normalized ten-channel-average widths from unity expressed as a percent before and after correction. The improvement in the microscopic D N L is displayed by listing the number of channels outside N~___a before and after correction. Data set No. I is totally independent of data set No. 2. The data sets No. 2A through No. 2H were obtained by nondestructive printout at various times during the data accumulation period and are thus not completely independent.
TABLE 2 Comparison o f the macroscopic and microscopic differential nonlinearity improvement for width and correctable data sets o f various relative statistical accuracies.
Width label
Data label
Average counts N~
a/Nt(%)
Average % deviation before
Average % deviation after
1 I I I I 1 1 1
2A 2B 2C 2D 2E 2F 2G 2H
258 662 1025 2054 4119 8260 12395 76500
6.2 3.9 3.1 2.2 1.6 1.1 0.90 0.36
2.0 1.6 1.4 1.4 1.3 1.3 1.3 1.3
1.7 0.99 0.82 0.46 0.42 0.35 0.29 0.22
67(33 + 34) 74(38 + 36) 69(33 + 36) 82(36 + 46) 94(42 + 52) 115(51 +64) 134(55 + 79) 192(74 + 118)
48(21 + 27) 52(26 + 26) 52(24 + 28) 56(25 + 31) 76(39 + 37) 97(48+49) 105(53 + 52) 153(78 + 75)
2H 2H 2H 2H 2H 2H 2H 2H 2H
1 2A 2B 2C 2D 2E 2F 2G 2H
10475 258 662 1025 2054 4119 8260 12395 76500
0.98 6.2 3.9 3. I 2.2 1.6 1.1 0.90 0.36
1.4 1.9 1.5 1.3 1.4 1.3 1.3 I. 3 1.3
0.26 1.5 0.91 0.62 0.44 0.33 0.26 0.21 0.00
130(56 + 74) 67(33 + 34) 74(38 + 36) 69(33 + 36) 82(36 + 46) 94(42 + 52) 115(5 i + 64) 134(55 + 79) 192(74+ 118)
63(28 + 35) 46(23 + 23) 47(23 + 24) 47(21 + 26) 49(23 + 26) 52(25 + 27) 51 (25 + 26) 52(27 + 25) 0
Number o f channels outside N t ± a * beforet afterf
* For perfectly linear uncorrected data approximately one third or 79 of the 237 channels would be beyond ~'i ± a . ~f The numbers are given in the order: total (below + above).
54
C. E. T U R N E R ,
However, for the purposes of table 2, No. 2H is essentially independent of No. 2A through No. 2G. In no case does the correction process worsen the linearity. This result is expected to be true as long as the width accuracy is better than the system nonlinearity. Thus noticeable improvement in both macrosopic and microscopic D N L is seen even for correctable data such as No. 2A where the statistics are worse than the nonlinearity. As expected the corrected macroscopic D N L improves with better accuracy of either width or correctable data. The improvement in microscopic D N L lessens somewhat as the width and correctable data get closer in statistical accuracy, probably because of compounding statistical errors. For example, data sets No. 2F, 2G, and 2H corrected by No. 1 show an increase (above the simple one-third expectation) in the number of points beyond N i___~r. On the whole, however, the accuracy is better so the average percent deviation continues to improve. Generally it appears that the microscopic D N L can be corrected to better than the accuracy of the correctable data with sufficiently accurate width data. However, the average percent deviations after correction are somewhat higher than would be expected for normal linear data. For example, with N~ ~ 10 000, one would expect a standard deviation of ~ 0.1% in ten-channel averages whereas ~ 0.2% is seen in table 2. However, for practical purposes the difference between corrected data and perfectly linear data is probably negligible provided the approximation (6) is good. The listing of the linearity test program L I N E used in conjunction with the correction subroutine L I N T Y to obtain the results shown in table 2 and figs. 4 and 6 is available elsewhere 21). L I N E was also used for the systematic effects testing described in subsections 4.2.2 through 4.2.4.
JR.
4.2.3. Systematic confiyuration dependence The waves seen at the lower end of the D N L measurements in fig. 4 are not uncommon 22-24) for T A C ' s used with ranges of ~ 100 nsec. These oscillations indicate correlations between S T A R T and STOP signals. In the present work it was found that they could be minimized by using as far as possible a common powerline and power supply for all the electronics associated with the S T A R T and STOP channels. Also when modular fast electronics are used, it is unwise to select the two sections of a dual discriminator, dual OR circuit, etc. for signal processing in the S T A R T and STOP channels, respectively. Note that the correction procedure described here can readily take small oscillations into account. A linearity correction obtained by fitting a polynomial to the accidental coincidence spectrum can not correct for these oscillations. It would be meaningless to try to correct the sharp peaks at the extreme ends of the spectrum. In other tests no significant difference in D N L was observed when the constant frequency pulser and/or the random noise pulser were replaced with independent radioactive source plus photomultiplier assemblies. 4.2.4. Stability of the differential nonlinearity The D N L of the system was found to be quite stable. Two D N L measurements of comparable statistical accuracy 0~i ~ 103) taken 16 days apart showed no significant differences when one was used as width data and the other as correctable data. Even longer term stability can be inferred from the fact that the TAC calibration determined with the use of the same D N L measurement remained constant (___0.25%) over a period of 45 days. Calibration runs were made every other day. 4.2.5. Correction of TAC calibration;
4.2.2. Systematic rate dependence In rate-effect tests, where the condition (8) was fulfilled, electronic distortion was found to be mainly a function of the coincidence rate. For a T A C range of 100 nsec various combinations of random S T A R T rates and a constant frequency STOP rate of up to 2 M H z showed no electronic distortion for coincidence rates of at least 10 kHz. At such coincidence rates accurate D N L measurements can be made in a few to several hours*. * The analysis in 2o) was able to give estimates of the counting time required to achieve a D N L measurement of specified accuracy. It should be noted that M C P H A rate dependence may be the true limiting factor, however.
Approximation validity The effect of the linearity correction on TAC calibration data is shown in table 3. The electronic calibration method in the case considered here gave symmetric peaks (fwhm ~ 3-4 channels) whose separation was known very accurately to be 5 nseclS). A linear leastsquares fit to the centroid versus delay curve was performed with and without linearity correction. The differential nonlinearity width data had N~ 125 000 counts. This implies a corrected D N L no better than 0.3%. The deviations of the experimental centroids from the calculated least-squares lines are shown in table 3. The very significant reduction in the deviations after linearity correction provides good evidence that approximation (6) is adequate even for
NUMERICAL
CORRECTION
OF M U L T I C H A N N E L
TABLE 3 C o m p a r i s o n of the deviations of T A C calibration peak centroids from a least-squares line before and after linearity correction; also the channel count variation across a typical calibration peak.
Peak number
Delay (nsec)
I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000
(
Centroid deviation before (channel)
Centroid deviation after (channel)
--0.71742 -0.23491 -0.03667 0.11681 0.13322 0.23836 0.18126 0.25261 0.21426 0.25867 0.21005 0.20091 0.10255 0.02483 -0.14935 -0.29303 -0.50009
-0.00333 0.01303 0.01314 0.01143 - 0.00378 0.02153 - 0.02367 -0.00421 - 0.02046 - 0.03506 -0.00818 - 0.00119 0.00926 0.01773 -0.00226 0.00177 0.01808
Channel number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
N~
0 0 1
11
201
1052 3265
6055 7873 6808 3661 966 48 l 0 0 o
DATA
55
rather rapid across-channel variation of the type seen in column 6 of table 3 for a typical calibration peak. The standard deviations before and after correction were 0.30 channel and 0.016channel, respectively. Before correction the integral nonlinearity is estimated as ~ (2×0.717)/220 = 0.65% from the maximum deviation of 0.717 channel out of a 220 channel ( ~ 85 nsec) range. After correction the integral nonlinearity is expected to be < (0.016/220)= 0.074%. The linearity correction raised the calibration constant 1.3% and reduced the standard calculated statistical error from 0.1% to 0.006%. 5. Conclusions
The linearity correction procedure developed here is capable of improving the integral nonlinearity o f a T A C system using an older M C P H A to within a factor of two of the best A D C specifications encountered. The microscopic D N L is corrected to better than the statistical accuracy of the data subjected to correction with an ultimate accuracy determined by the statistics of the D N L measurement. The deviation of the macroscopic D N L from constancy after correction is only about a factor of two times what would occur for linear data with statistical fluctuations.
SUBR~UTINELINTY~ (MI,LI, L2,JF, KF ) ~ /
8
R N ~ ,RNWIO+ NWID(K)-1.0 CTS(J),RCTS+(Y(K}/NWID(K}W NWID(K)-RNWID) RCT$-(RNW~O/NWIO {K))W-Y( K)
REALNWID C~MM~N X(400}, Y(400), C H(400), CTS(4OO),WX(400), WlO ( 4 0 0 ) , NWID (400)
t
WN~RM • 0 . 0 RNWID • 0 . 0 RCTS " 0.0
5 CTS( J ) ,RCTS/RNW]D RNWID • RNWID-I.O RCTS • RCTS-CTS(J) J-J-I
t NL2 sL2-9
l
ANALYZER
I
O~ I 0 0 I , N L 2 , L 2 IO0 WN~IRM=WNORM÷WZD { r ) i
l I WN¢B.-WN~R./tLZ-N' ~+,)J f
7 RNW~ • RNWm+N WII) (K) RCTS • RCTS + Y(K) G~ TI~ 3
9 JF-J KF, K D~lO Z'JF, L2 IOCH(I)- X(I)
(-,°-.o 3
DO I Z = I o M I CH ( I ) " 0 . 0 CTS (Z) - O 0 I NW~D{I) • W I O { I ) / W N ~ R M
Fig. 7. Statement by statement flow chart of the nonlinearity correction subroutine L I N T Y .
56
c.E.
C C C C C C C C C C C
C C C C C C
C C C C C C C
C C C C C
C C
TURNER, JR.
SUBROUTINE LIKTY ADJUSTS THE DATA ( X , Y I BY USE CF A DIFFERENTIAL NONLINEARITY MEASUREMENT (WX,WIO) TO PROVIDE APPROXIMATELY lINEAR DATA (CH,CTS) NOTE: X ( I ) , Y ( 1 ) SHOULD CDRRESPCKD TO W X ( 1 ) , W I C ( I I , N W I D { I ) MI : NUMBER CF CHANNELS IN THE DIFFERENTIAL NONLINEARITY MEASUREMENT L I , L 2 : STORAGE INDICES OF THE LOWER AND UPPER CHANNEL LIMITS DF THE LINEARITY CORRECTION REGION JF,KF : FINAL VALUES OF J,K NWID : ARRAY FOR THE NORMALIZED CHANNEL WIDTHS COMPUTED IN LINTY IX,Y) : ARRAYS CONTAINING THE (CHANNEL,COUNT) DATA TO BE CORRECTED (CH,CTS) : ARRAYS FCR THE CORRECTE~ {CHANNEL,COUNT| DATA (WX,WID) : ARRAYS CONTAINING THE (CHANNEL,CCUNT) DATA OF THE OIFFERENTIAL NONLINEARITY MEASUREMENT NL2 : INDEX SPECIFYING THE LOWER LIMIT CF THE REGION FOR CALCULATING WNORM WNCRM : NORMALIZATION WIDTH OBTAINED BY ~VERAGING OVER A FAIRLY LINEAR REGION OF THE WID DATA ; IN THIS WORK THE REGION WAS CHOSEN TO BE THE UPPERMOST IC CHANNELS OF THE LIKEARITY CCRRECTION REGION J : INDEX FOLLOWING THE FORMATION OF C H ( J ) , C T S ( J ) K : INDEX OF THE UNCORRECTED DATA X ( K ) , Y ( K ) BEING USED TO FORM CH(J) ,CTS(J) RNWID : NORMALIZED WIDTH REGISTER; KEEPS TRACK CF WIDTH AVAILABLE FCR FORMATICN OF AN ACJUSTEC CHANNEL AND/OR THE WIDTH REMAINING UNUSED AFTER FORMATION GF AN ADJUSTED CHANNEL RCTS: COUNTS REGISTER; KEEPS TRACK OF THE NUMBER OF COUNTS ASSOCIATED WITH RNWID COMPILATION AND EXECUTION TIME: L E S S THAN 5 SEC FOR SEVERAL DATASETS ON IBM 360/65 SUBROUTINE L INTY( MI, L I , L 2 , J F , KF ) REAL NWID COMMON X(400},Y(4CO),CH(40C),CTS(40CI,WX(40O),WID(4OO),NWID(400) WNORM=O,
100
I
2 3 4 5
6 7
8
g
10
RNWID=C.O RCTS=O .0 NL2 = L2 - g DO i 0 0 I=NL2,L2 WNORM=WNCRM÷WID(I) WNORM=WNCRM/( L 2 - N L 2 * I ) 00 1 I = I , M I CH(II=O.O CTSqI)=O.O NWID(I)=WID( I)/WNORM J=L2÷I K=L 2÷ 1 J=J-1 K=K-I IF(RNWID-I.0| 6 , 5 , 5 CTS ( J)=RCTS/RNW ID RNWI D=RNWI D-I • RCTS=RCTS-CTS(J ) J=J - I I F ( K - L I ) g,%,4 IF{RNWID÷NWID(K)-I.0) 7 , 8 , ~ RNWID=RNWID÷NW ID(K) RC TS=RCTS+Y(K) GO TO 3 RNWID=RNW ID÷NW ID ( K ) - i ,0 CTS( J )=RCTS÷( Y(K)/NWID(K) ) * ( NWID (K) -RNW I D) RCTS=(RNWICINW I D ( K ) ) ~ Y ( K ) IF(K-LI) 9,9,2 JF=J KF=K DO i 0 I = J F , L 2 CH(I)=X(I) RETURN END
Fig. 8. Computer subroutine LINTY for nonlinearity correction of MCPHA system data.
NUMERICAL CORRECTION
OF M U L T I C H A N N E L A N A L Y Z E R DATA
After linearity correction the adjusted data have essentially constant channel width and further analysis techniques can validly utilize this simplifying assumption. This feature is quite useful if sophisticated unfolding procedures are to be used to extract accurate nuclear lifetimes from delayed-coincidence distributions25). The time and equipment necessary to accomplish the correction are not excessive. The correction takes only a few seconds on a fast computer. The D N L measurement takes only a few to several hours after preliminary setup-testing for system optimization. Random-noise sources, high-rate pulsers, and even sliding pulsers are generally required for standard measurements or equipment testing and, therefore, are already available at most laboratories. In any case they are inexpensive compared to a very linear MCPHA. Since the correction depends intimately on the channel width measurement, care must be exercised to avoid systematic differences between the D N L measurement and the physical measurement of interest. Rate-dependence deserves special consideration when attempting to obtain a rapid D N L measurement. Systematic differences are most easily avoided with a TAC system but the method is generally applicable where a suitable D N L measurement can be obtained. The correction can significantly reduce errors which would otherwise be of the order of the average macroscopic D N L . The continued interest and encouragement of Professor E. N. Hatch as well as his helpful comments on the manuscript are gratefully acknowledged,
Appendix A statement by statement flow chart is given in fig. 7
57
of the commented F O R T R A N nonlinearity correction subroutine L I N T Y presented in fig. 8.
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