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Precise calibration of a Ge(Li)-spectrometer using a digital to analog converter
NUCLEAR
INSTRUMENTS
AND
92 (1971) 421-432;
METHODS
PRECISE USING
CALIBRATION A DIGITAL
R. J. MCKEE, National
H.
Carleton
PUBLISHING
CO.
and
CONVERTER*
and
Council of Canada,
MES
NORTH-HOLLAND
OF A Ge(Li)-SPECTROMETER TO ANALOG
C. K. HARGROVE
Research
0
A. G. SMITH
Ottawa,
Canada
A. THOMPSON
University,
Ottawa,
Canada
M. DIXIT Enrico Fermi Institute,
Uniuersity Received
of Chicago, 11 December
Chicago,
Illinois 60637,
U.S.A.
1970
A new method to precisely calibrate the linear electronics used with Ge(Li)-spectrometers has been developed. The calibration is done by measuring the system’s differential response with a mercury relay pulse generator whose amplitude is distributed over the range of the spectrum by a computer controlled 15 bit digital to analog converter. It is shown that the integral of the differential response, collected simultaneously with the data, linearizes the electronics throughout the whole spectrum range in a model independent way, except for small errors introduced
by electronic noise and accidental coincidence with detector events. A test of the calibration system using a small planar detector is presented in which a linear fit of peak position to adopted energy was done for seven y-ray lines between 100 and 500 keV which are known to very high precision. The rms deviation from the linear fit for two separate runs is less than 3 eV. In addition, a new value of 136.473&0.015 keV is reported for one of the principal transitions of 57Co.
1. Introduction
of high precision one can bracket each photon peak with two pulser peaks set as closely as possible to reduce interpolation error and convert, by linear interpolation, the position of the photon peak to volts. However, this is a laborious operation and not readily automated. Strauss et a14) have developed a calibration method which dispenses completely with the gain correction curve. Instead of using the conventional pulser, they use a sliding pulse generator; that is, a pulser whose amplitude increases linearly with time. The resulting spectrum measures the differential response of the amplifier-analyzer system and can be used in place of the gain correction curve for the following reason: the number of counts in a channel is proportional to the channel width and hence is also proportional to the energy per channel. Thus, the integral of the differential spectrum is linear with energy provided that the voltage ramp of the sliding pulser is also linear. Not only are interpolation problems eliminated by the differential response spectrum, but it is simple to obtain and easy to use in analysis. Strauss et al. have tried out their method by measuring y-rays from several sources in the energy region of 0.27 to 2.75 MeV and their results are within 40 to 110 eV of the generally accepted values. The calibration system that we have developed is similar to the one described by Strauss. Instead of a sliding pulser we use a conventional pulse generator
Since their invention over ten years ago, the lithiumdrift germanium detector has improved to such an extent that it is now possible to buy small planar detectors with a resolution of better than one keV for photons in the energy range of 100 to 500 keV. With such a spectrometer it should be possible, in principle, to measure photon energies to a precision of better than 10 eV in this low energy region. The best that has been achieved so far are precisions on the order of 30 to 60 eV1-‘). One of the problems in high precision Ge(Li) spectrometer work is the calibration of the amplifier and multichannel analyzer system. In the course of our work on muonic X-rays we have developed and used a novel scheme for solving the calibration problem. With it we have calibrated our peak positions in a model independent way to a precision of better than 10 eV. A conventional method of calibration is obtained by a precision pulse generator and voltmeter and a series of accurately known y-ray lines5.6). In this method several points are measured along the spectrum and the resulting gain correction curve is used to correct the data for non-linearities in the amplifier-analyzer system. Errors are introduced by the necessity of interpolation between the measured points. In work * Research Canada
supported by the National and the U.S. National Science
Research Council Foundation.
of
421
R. J.
422
MCKEE
whose amplitude is controlled by a 15 bit (32768 step) digital to analog converter (DAC). Since the number of DAC settings per channel can be made fairly large, a differential response curve - called the DAC spectrum in this paper - can be generated if a continuous range of DAC settings is used and the number of counts for each DAC setting is kept constant. Since we also use a 13 bit (8192 channel) analog to digital converter (ADC) for our multi-channel analyzer, the DAC calibration scheme is versatile and fully computer controlled. The use of the DAC calibration method has the following advantages: 1. It linearizes the electronics in a model independent way even removing local non-linearities. 2. The DAC calibration spectrum is concurrent with the data. 3. The computer controlled DAC allows for the automatic accumulation of the DAC spectrum using versatile pulse patterns. 4. The calibration method is easily applied to the analysis of the data. 5. It may be easily set to cover any range of the ADC. A test of the DAC calibration scheme using seven very accurately known (within 9 to 14 eV) y-ray lines in the energy region of 100 to 500 keV showed striking results: the deviation from linearity, after DAC correction, was less than 4 eV. The details of these results are presented in sect. 5. In sect. 2 the systematic errors of the DAC calibration method are explored. Sect. 3 presents a study of statistical errors. The apparatus is described in sect. 4 and conclusions are given in sect. 6. 2. Systematic
errors
In this section it is shown very generally that under ideal conditions the integral of the DAC spectrum is linear with the DAC setting and thus linear with energy, provided that the DAC itself is perfectly linear. It is also shown that a small departure from linearity is introduced by electronic noise. We then introduce a background in the DAC spectrum and show that it increases the departure from linearity. In order to reduce the background effect, a DAC window is introduced; that is, pulser events are required to fall into a channel range specified by the DAC setting. Finally it is shown that the DAC window also introduces a small error. With several DAC settings per channel, it is reasonable and convenient to treat the DAC setting as a continuous quantity. Let F(x) be the ADC channel
et al.
corresponding to DAC setting x; that is, assuming no noise or background, the pulser event appears in channel F(x). Let g(C) be the inverse function: the DAC setting corresponding to channel C. The range of DAC settings is given by a < x < b. Let C, = F(a) and C, = F(b). Since the edges of the DAC spectrum are not useful, the calibration is restricted to a somewhat smaller channel range from C, to C, where c, < c, < c, < c,. The electronic noise distributes the pulser events to some extent. Let p(C-F(x)) be the pulser probability distribution along the ADC for DAC setting x, where it is assumed that the distribution is the same for all DAC settings. It is also assumed that the noise is small: p(C-F(x)) = 0 for IC-F(x)1 larger than a few channels. For the time being background is ignored. Specifically, C, is far enough from C’, and C, from C, so that p(C, -C,) = p(C, - C,) = 0. Assume now a single DAC cycle: one count per DAC setting. With dx DAC settings between x and x+dx, there are dN(C) = p(C- F(x))dx counts contributed to channel C in the DAC spectrum. The total DAC spectrum for one cycle is h
N(C)
=
p(C-F(x))dx.
(1)
i’0 The replacement C’ = F(x) x = g(C’) results in N(C) =
s a
-cc
dg
-p dC’
which
(C-C’)dC’,
is equivalent
to
C, i C I CZ,
(2)
where C, and C, are replaced by + cc since p is zero outside this range. A Taylor series expansion of g(C’) about C’ = C gives the DAC spectrum in terms of the moments of the distribution. Let C”p(C)dC
(3)
be the normalized n’th moment where c2 is the second moment: r~is defined such that m2 = 1 and is a measure of the half width of the distribution. Since the distribution is normalized, m, = 1 also. An easy calculation shows that -I < m, < 1; m, is a measure of the asymmetry of the distribution. The final result for the DAC spectrum is N(C)
=
f (-O);mng’n+l)(C), n=O n.
C, I Cc
c,,
(4)
where g@) (C) symbolizes the n’th derivative. The next step is to integrate N(C) and show that it is
PRECISE
CALIBRATION
linear in the DAC setting except for small corrections. But the integral of the DAC spectrum is unwieldly to use; we find it more convenient to adjust the ADC channel itself. For this purpose, the standard channel S(C), linear in the integral, is introduced: S(C) = C1 +(C,-C,)
/I1 1(C’)dC’/jI; c,
SC
N(C’)dC’ SC,.
, (5)
Note that S(C,) = C, and S(C,) = C,. The quantity S(C)- C is the small correction made to the ADC channel in order to linearize it. Replacing g(C) by x and defining J(x)
=
f
(-a>“mn2,
n=O
n!
it is easily seen that S(C) = c, +(C7-C,)
f’(x)-f’(xl) J’(xz)-f(x,)’
(7)
where x1 =g(C,) and x2 =g(C,). In the limit of no noise, 0 = 0 and ,f(x) =x which proves that the standard channel is linear in the DAC setting. Noise, however, introduces corrections. It is not the size of the corrections that is important, but their variation along the ADC. Introducing l/p as the average number of DAC steps per channel, the variation in the first order term off(x) is Sf’= - am,(dx/dC-p-l), where dx/dC corresponding
detector events due to the fact that the DAC spectrum is collected simultaneously with the data. Since c reflects the half width of the total distribution, it can become many times larger than the half width of the pulser peak alone. The first order correction may still be insignificant if the total distribution is very symmetric (nll z 0); but the second order correction not only has no symmetry damping term, but is also proportional to 0’. In a region of local non-linearity, where the most accuracy is needed in calculating S(C), the second order term may become larger than the precision aimed for. The background effect is reduced by the introduction of a DAC window. For DAC setting x it is required that the pulser count fall in the channel range SI+flx+ A before the event is accepted and the DAC stepped to a new setting. The constants c( and fl are chosen to keep the peak part of the pulser distribution as much as possible in the center of the window throughout the entire DAC spectrum and A is chosen large enough to always encompass the peak proper. However: the movement of the peak in the window as the DAC is stepped introduces a small error of its own. Let C,(C) be the center of the window: Co(C) = m+/%(C).
linearity. The channel is
&Y(C) = - cm, z,
(9)
where (10)
and is the fractional change in the differential linearity. The first order correction expressed by eq. (9) is negligible. Assuming a non-linearity of 1% (z = 0.0 l), an asymmetry of 20% (Im, 1= 0.2) in the pulser peak distribution, and CJless than 5 channels then 16SI < 0.01 channels. Higher order corrections are smaller yet. Now consider what happens when a background is introduced. The pulser distribution, which must include the background, is no longer restricted to a few channels. In fact, the data indicates that the background is distributed over a large range of the ADC. The origin of the background is primarily accidental coincidences between the pulser events and
(111
Let R be the pulser rejection rate; it is the fraction of pulser counts that fall outside the window. It is here defined as if the peak were always in the center of the window:
(8)
is the local differential variation in the standard
z = /3dx/dC-1,
423
OF A Ge(Li)-SPECTROMETER
4
R=l-
p(C)dC.
s
(12)
-A
Under computer control the DAC window is easily manipulated and the rejection rate easily measured, although the experimental R differs somewhat from the definition above, since the peak moves around in the window. The definition given in eq. (12) is made to ease the mathematics. In order to further simplify matters it is assumed that CYand p are chosen such that C,(C,) = C, and C,(C,) = CZ. The actual values used in the experiment keep the peak, on the average, better centered; the simplification made here guarantees the center position only at C, and C,. But with CIand /3 defined this way, C,(C) assumes the desired properties of the standard channel: linear in the DAC setting and equal to the ADC channel at C, and C,. The inclusion of the DAC window modifies the probability distribution used in the calculations; since a pulser count must occur in the window before the
424
R.
DAC is stepped, the new distribution P(C-F(x))
J.
MCKEE
is:
standard
C inC,(F(x))fA
,
where sZ(C) is the normalization
factor
p(C’-C)dC’. f
(14)
G-A
The standard channel, eq. (7), is evaluated by procedures similar to the ones already used. Three approximations are used in the calculation. At the edges of the window the pulser distribution is purely background. A good approximation is to treat p(C- F(x)), as well as g”‘(C) and Q(C), constant over the small channel range C,,(C) - C at the edges of the window. For example,
= 1 -R+(C,-C)&,,
(15)
with e0 = p(d) -p( - A) and the distribution is treated as constant in the last two integrals. Secondly, if the background is fairly uniform, (C, - Cl) p (+ A) is on the order of the rejection rate. Thus, E= (C, - C)p( f A) e 1 and terms of order s2 are dropped. Thirdly, for good electronics the differential non-linearity expressed byeq. (10) is small. Hence, terms of order z2 are also ignored. The final evaluation of f(x) that appears in the standard channel is f
m,
(-y;“n !$
: ‘no n=l
x
(
(-1>” n!
>
A”E,_~n~Om,
$ [(Co-C) g],
where the cxom,/mo term has been dropped since it is small compared to Acl and the factor bdx/dC has been replaced by 1. It is difficult to evaluate eq. (17) exactly since the background is not known very well. But assuming (C, - C,)p() A) = R it is easily seen that 2R &s(C) = - 1-R
m”
(16)
where E, = p(A)-(l)“p(d). The moments of the distribution are now modified by the DAC window; for example m, = 1 -R. This means that G essentially still reflects the half width of the peak alone; therefore, a discussion of the first series in eq. (16) reduces to the earlier discussion before the introduction of the background. The second series in eq. (16) is a correction due to the fact that the distribution doesn’t stay centered in the window. The corresponding correction in the
Co-C ~ C,-C,
A ’
(18)
For example, with a pulser rejection rate of lo%, Co-C = 1.5 channels, and C,- C’i = 3000 channels 6S(C) is 10m4A, entirely negligible. Even if E, is an order of magnitude larger than what is estimated here, 6S(C) is still negligible if A is kept small enough. In summary it was shown that the deviation from linearity of the integral of the DAC spectrum is due to electronic noise and is negligible if the noise is kept reasonably small. It was also shown that, if background is present, a DAC window must be used and the error introduced by the use of the window is negligible if the background is not too big or the window width too long. Since the origin of the DAC window error is due to the movement of the pulser peak in the window, it may be better to use a higher order polynomial to determine the window center instead of the linear relationship used here. In either case, great care must be taken to insure that the pulser peak stays in the window for all DAC settings, otherwise the calibration may be severely biased. 3. Statistical
n=o
+ i
(17)
(13)
C,,+A
l/L?(C) =
%(Co-C), m.
outside,
i
due to the first term in this series is
&s(C) = -
rlo
=
channel
=
W(x))p(C-F(x)),
f(x)
et al.
errors
More important are the statistical errors of the DAC spectrum. Like the systematic errors discussed above, the origin of the statistical errors is the finite width of the pulser peak distribution. Two types of statistics are investigated here: the statistical uncertainty in the integral of the DAC spectrum and the statistical uncertainty in the counts of a single channel. The first is used to determine the minimum number of DAC cycles needed to achieve a specified precision in the calibration. The second is used as a check on the DAC spectrum since the statistics of a single channel are related to the pulser peak width, both of which are measurable. Qualitatively it is easy to understand what is expected. The statistical uncertainty on the integral of the DAC spectrum up to channel C comes from
PRECISE CALIBRATION
pulser counts for DAC settings near g(C) or, more exactly, for DAC settings in the range g(C++r) where r is the full width at half maximum (fwhm) of the pulser peak distribution. The reason for this is if the DAC setting is considerably less than (or greater than) g(C), the pulser count falls in the integral (or outside of it) and thus does not enter into the statistical uncertainty of the integral. With p-’ DAC settings per channel, there are on the average r/P pulser counts in the critical region Cf 3 r for each DAC cycle. With n DAC cycles, the statistical uncertainty on the integral is on the order of ,,/(nr/j?). Since the channel may be found by dividing the integral by n/P (the counts per channel) the resulting statistical uncertainty on the standard channel is J(fj?/n). If a precision of 6s of a channel is desired, the minimum number of DAC cycles required is about fp/6S2. For example, with 10 DAC steps per channel, f = 4 channels, and a desired precision of 0.1 channel, a minimum of 40 DAC cycles are necessary. This is an important consideration in estimating how long to collect the data. This result also shows another important role of the DAC window. Without the window, the effective width of the pulser peak may be much larger, which necessitates more running time to achieve the desired precision. In order to make the calculations, some simplifications are assumed which are not very essential to the final results. The first is to assume the ADC channel is linear in the DAC setting: F(x) = cc+ j?x.The next is to choose the pulser distribution as Gaussian:
=-J-
p( C-F(x))
aJ(2
- (C-2;,))z
exp [
n)
1
) (19)
with r related to 0 by r = 2ojln4.
(20)
The effect of the background in the window is merely to increase r by a small amount; so for simplicity the background is ignored. Let p(x) be the probability that, for DAC setting x, a pulser count falls between channels Ci and Cj where C, < Ci < Cj < C,. cj P(C-F(x))dC.
P(X) =
(21)
I Ci
If there are n pulser counts at DAC setting x, what is the probability that k of these counts fall in the channel range Ci to Cj? The answer is given by the binomial distribution: n.I
Pk(n) =
PWk(l-P(X)) “-k.
k!(n-k)!
The average
number
ck) =
of counts
for DAC
setting
k$,kPk(n) =np(x>.
G-4 x is (23)
The statistical fluctuation in (k) is the rms deviation whose square is symbolized by (6k2): (Sk’)
= ((k-(k))2)=
i
(k-(k))‘Pk(n)
k=l
=
(24)
Thus, the expectation value of the integral of the DAC spectrum from Ci to Cj is (k) summed over all DAC settings. (A(A))
IO,
425
OF A Ge(Li)-SPECTROMETER
=
1
b
(25)
where 1= Cj - Ci. Since the DAC settings are independent, the statistical fluctuation in (A(a)) is found by summing the rms deviations in quadrature over all DAC settings.
0.8 1
r\ =063.
2 : “a
0.4
&
-
(&4’(;1))
0.2
=
b (6k2) ca
dx
.
(26)
-
Finally, 000
2
3 Pu~sm
4
5
6
7
8
the Gaussian
distribution
gives the results
9
(27)
FWH~I (C~~ANNEL)
Fig. 1. A plot of the statistical error in a single channel of the DAC spectrum as a function of the fwhm of the pulser peak distribution under the assumption that the latter is Gaussian. The plot was calculated using eqs. (27) and (28).
<6A2(A>> = (n/B> CAerfc(u) +(2 a/Jn) (1 -e-“‘)] where u = l/20 and erfc(u) = 1 - erf(u). Eq. (27) shows that, on the average,
, (28)
there are n/j3
426
R.
J.
MCKEE
counts per channel in the DAC spectrum. Consider two extreme cases in the statistical fluctuation given by eq. (28). For g 4 I., erfc(u) z e-“‘/(u Jrc) for large arguments and therefore
(S/p(Q)
l
z
V’(711n4)
“!I = 0.48 /3
c. B
’
(29)
Except for the factor 0.48 this result agrees with the earlier qualitative argument. For n DAC cycles the statistical uncertainty in the standard channel is 6.S = ,:1(0.48 rfl/n).
(30)
For the other extreme case, r~ $ i, erf(u) z u for small arguments. There results
(&12(n)) = HA//l = (A(1)).
(31)
In the limit of no noise (f = 0) there are no statistical fluctuations. In the other limit of the noise evenly distributing the pulser counts, the statistical fluctuation approaches the conventional result that is encountered in counting random events. The statistical fluctuations of counts in a single channel is found by setting J. = 1 in eq. (28). Fig. 1 presents a plot of (6A’(l))/(A (1)) as a function of F. The plot can be used as a check on the DAC spectrum since both quantities may be measured experimentally:
Preamplifier
et al.
the abscissa is the x2 per channel of the DAC spectrum and I- is measured by accumulating a DAC spectrum for one DAC setting and noting the width of the distribution. 4. Apparatus and experimental procedure The electronic equipment is outlined in fig. 2. It is a relatively standard lithium-drift germanium spectrometer system except for the addition of a precision digital to analog converter as the voltage source for the pulser. The detector and pulser pulses are fed through a preamplifier, to an amplifier, and into an analog to digital converter. The digital output of the ADC is read into a PDP-9 computer along with a tag which identifies the word as being from the diode or the pulser. The DAC is controlled by the computer. In addition, pre- and post-pileup circuits prevent the analysis of events which were preceded by pulses within 50 psec and followed by pulses within 4 ,usec, the time for the ADC to recognize a pulse. These two circuits are not shown in fig. 2. All the apparatus shown in the diagram, except the computer, was located inside a copper hut whose air temperature was regulated to within + 1 “C. The Ge(Li) detector, manufactured by Ortec, was a 0.5 cm3 planar diode with an area of I cm2 and a depletion depth of 5 mm; it was dc coupled to a cooled FET preamplifier and linear amplifier’). An 8192 channel
Ampllfler ADC
Ge(Li)
Fig.
2. A schematic
drawing
of the electronic
equipment
used in the DAC
c
Computer
calibration
method.
PRECISE CALIBRATION
427
OF A Ge(Li)-SPECTROMETER
analog to digital converter was used with both gain and zero digital stabilizers’). The stabilizers were operated on y-ray peaks outside the DAC range. A mercury relay, driven by a 50 cps oscillator which could be turned on or off by the computer, was used to generate the pulses. The zero point and range of the pulser in the DAC spectrum were adjusted by potentiometers Pl and P2 which offset and attenuated the DAC output voltage, respectively. The heart of the calibration system was the 15 bit DAC manufactured by Analog Devices’). It is unipolar with the output voltage range from 0 to + 10 V full scale (DAC setting of 32767). Its output stage is a difference operational amplifier with an offset input. Its linearity is rated at +i of the least significant bit or 15 ppm (parts per million). In addition its temperature coefficient is rated at 17 ppm/“C and its long term (one month) full scale stability at + 10 ppm. The settling time is 100 psec for the full scale step. The linearity of the DAC was checked against a six digit digital voltmeter”). The DAC was stepped from 0 to 32000 in 1000 step intervals and the voltage measured for each setting. The deviations from a straight line fit revealed a uniform linearity with an rms deviation of slightly less than & of a DAC setting or 7 ppm. An additional check was made by measuring each bit setting with a potentiometer and standard cell; the results agreed with the DVM test. The system was operated in the following manner. After the ADC finished analyzing an event, the computer read and reset the ADC and the tag register. If the event was identified by the tag register as a pulser event the computer then checked to see if the ADC channel was inside the DAC window of + 10 channels defined by the DAC setting. If it did, the pulser count was added onto the DAC spectrum and the computer stepped the DAC to a new setting. At the beginning of a DAC cycle, the settings were started at the minimum setting (a) and incremented in steps of 48 after each accepted pulser event until the settings were about to exceed the maximum setting (b); the next setting was started in the second division (a+ 1) and again incremented by 48. The procedure was continued until all settings between a and b inclusive were covered and then the next DAC cycle started. This method of stepping the DAC was done to minimize the effect of short term drifts on the DAC spectrum. The new DAC voltage was present at the pulser about 0.2 msec after the last accepted pulser event. This gave the capacitor Cl time to fully charge before the mercury relay switched again. The pulser decay constant was made long compared to the preamplifier time constant.
Data collection in the DAC spectrum was always ended at the end of a completed DAC cycle. If the pulser event fell outside the window, the event was scaled as a pulser reject and the DAC was not stepped. With a total preamplifier rate of about 2000 per second, the rejected pulser events were about 3% of the total pulser events. 5. Results The data used in testing the DAC calibration system was obtained during a muonic X-ray experiment at the University of Chicago synchrocyclotron. The test data consists of a series of y-ray lines of well known energy from several y-ray sources which were used in the experiment to calibrate muonic X-rays. The usefulness of the calibration system is demonstrated by fitting a straight line to the energies and DAC corrected channel positions of the y-ray peaks and noting the deviations from linearity that result. The uncorrected data is also analyzed in the same manner for comparison. Estimations of the systematic and statistical errors are also made. TABLE 1 Parameters
of the two sets of data.
Constant
Running time (hours) Number of DAC cycles (n) Rejection rate (R) Channel corresponding to DAC setting of zero (a) Channel per DAC setting @) DAC half window width in channels (d) Initial DAC setting (a) Final DAC setting (6) Channel corresponding to DAC setting a (C,) Channel corresponding to DAC setting b (Cb) Initial channel used in calibration (Cl) Final channel used in calibration (Cs) fwhm of pulser peak (I-)
Run 1 Run 2
68 172 0.033
73 173 0.034 728 0.1112 10 1370 30680 880 4140 900 3900 4
Several runs from the X-ray experiment, obtained under the same operating conditions, were chosen for analysis. The runs were grouped into two nearly equal sets and summed. Pertinent quantities for both summed runs are presented in table 1. The data used here were collected over a period of about two weeks. Drifts were looked for by systematically analyzing certain of the y-ray lines in each run, but no peak shifts outside statistics were observed. The y-ray spectrum contained eight lines inside the
428
R.
I.
MCKEE
et al.
TABLE 2
Gamma-ray
Gamma-ray
TABLE 3
lines used in the DAC calibration.
source
Energy
The ADC and DAC corrected positions of the eight photon peaks for both runs. The errors are purely statistical. The number in parantheses is the DAC corrected peak positions.
(keV)
136.478+0.030” 165.85210.010” 238.624&0.009c 295.938&0.009c 308.429&0.010c 316.486&0.010c 411.795f0.009e 468.053*0.014c
5 J. Rapaport, Nuclear Data B3 (1970) 103. The number quoted here is the sum of the adopted values for the cascade y-rays 14.408&0.005 keV and 122.07&0.03 keV. b Refs. 11, 16 and 17. e Ref. 12.
range of the DAC spectrum. The identity and energies of the lines are summarized in table 2. Except for the 57Co line, the y-ray energies listed in the table are primary standards as defined in the review article on y-ray sources by Marion”). Murray et al.“) have measured the four 1921r energies and the 228Th energy by a direct comparison with the 412 keV line of 19*Au using an iron free beta-ray spectrometerL3). The relative precision of these six energies is therefore better than the errors quoted in table 2. In an earlier paper14) the same group report the precise determination of the 412 line of 19*Au itself by direct comparison with the annihilation radiation15) using the magnetic spectrometer. The ‘39Ce line listed in the table is an average”) of a magnetic spectrometer measurementr6) and a crystal diffraction spectrometer measurement”). The energies of all seven of the primary standards as listed in table 2 are directly based upon the 1963 value of 511.006 + 0.005 keV for the rest mass of the electron”). While the fundamental physical constants have been readjusted several times since then”) no attempt is made in the present work to rescale the seven energies, since the primary purpose here is to demonstrate the linearity of the DAC calibration scheme. In addition to the eight lines listed in the table, two more y-ray lines appeared in the spectrum outside the limits of the DAC range; these were the 122 keV line of 57Co and the 511 keV annihaliation line from 22Na. They were used to stabilize the electronics. The fwhm of the eight y-ray peaks used in the analysis ranged from 0.59 to 1.OOkeV. The channel position of each line was found by
Gamma-ray line
57Co 13sCe z2sTh 19’LIr tszIr lg21r ts8Au ls21r
(136)” (166) (239) (296) (308) (316) (412) (468)
985.50+0.10(985.37) 1237.94+0.01(1237.51) 1863.00+0.02(1861.82) 2355.01+0.04(2353.57) 2462.15*0.04(2460.70) 2531.29&0.02(2529.88) 3348.44&0.01 (3347.55) 3830.40&0.04(3830.20)
985.49+0.10(985.37) 1237.92_rtO.Ol(l237.36) 1862.99~0.02(1861.76) 2354.93&0.04(2353.48) 2462.12~0.04(2460.69) 2531.27+0.02(2529.80) 3348.42+0.01 (3347.56) 3830.35*0.04(3830.22)
a The peak positions of this line were adjusted upward by 0.07 of a channel to compensate for the 5% contamination of the 136 keV line of ls21r. See text for details. The uncertainty of the peak position has also been increased to 0.1 to allow for systematic errors in fitting a single line shape to a doublet.
fitting the peak region with a mathematical peak shape plus background terms. Details of the line shape and fitting procedure is given in the appendix. The position of each peak was corrected using the DAC spectrum by applying the standard channel definition given in eq. (5). Linear interpolation between adjacent channels was used to determine fractions of a channel. The y-ray peak positions, uncorrected and corrected. are given in table 3. For both sets of data, a weighted least squares fit to a straight line was made using the adopted energies of table 2. The 136 keV line of “Co was not used in the straight line fit for two reasons. Its known precision is two to three times worse than the seven primary standards and there is a weak 136.314f0.019 keV line2’) of lg21r under it. From the known intensity of the ratios21) of the ig21r lines and a measurement counter efficiency the contamination from the 1921r line amounted to about 5%. With the 1921r line 164 + 36 eV of lower than the 57Co line and with a dispersion 117 eV/channel, the centroid of the two peaks is lower by 0.07 channels than the 57Co peak alone. Hence it is possible to correct for the ig21r contamination. This has been done in the results from the peak fitting for the 57Co line as shown in table 3, but with an increased peak position uncertainty. When the linear fit to the seven energies of the primary standards is made using the uncorrected peak positions as shown in table 4, the rms deviation from linearity is several times the precision to which the seven energies are known. But when the DAC corrected
PRECISE
CALIBRATION
01
TABLE 4 Results of a weighted least squares fit to E = cro+al C for the seven primary standard y-ray lines where E is the energy and C the uncorrected peak positions.
Parameters
ao(keV) ar(keV/channel) x2 of fit
Gamma-ray
line
Weight
s7Co (I 36)c lsgCe (166) =*Th (239) ls21r (296) lgslr (308) rgelr (316) lg”Au (4 12) rg%Ir (468) (AEz)*
(x 104)”
Run 1
Run 2
2 1.484 0. I16560 158
21.486 O.I16561 154
A E(eV)u
AE(eV)b
0 0.99 1.16 0.97 0.82 0.95 1.21 0.46
- 124 - 73 11 46 44 45 - 17 - 97 49
- 122 - 73 14 40 44 41 - 15 - 99 49*
a The energy equivalent of the statistical errors on the peak positions (table 3) are folded into the quoted errors from the literature (table 2). The weights are the reciprocal of the squares of the augmented errors. b AE = E(calculated using an and nt) - E(table 2). C Not used in determining the fit as explained in the text. * Weighted rms deviation. 57Co not included.
TABLE 5 Results of a weighted least squares fit to E = ao + al C for the DAC corrected positions. The weights are the same as in table 4.
Parameters
ao(keV) al(keV/channel) xs of fit
Gamma-ray
57Co 13gCe s=Th lg21r lg21r rg21r
line
Run 1
Run 2
21.6116 0.116558 0.28
21.6361 0.116551 0.16
AE(eV)
AE(eV)
(136) (166) (239) (296) (308) (316)
- 14 1.3 - 2.4 1.0 -3.1 3.3
4 -0.8 1.3 -2.4 2.0 -0.2
lg8Au (412) rg21r (468)
0.3 - 0.9
1.2 -2.5
* a Weighted
rms deviation.
2.1” “7Co not included.
1.6=
429
A Ge(Li)-SPECTROMETER
peak positions are used instead, as shown in table 5, the improvement is very striking: the rms deviation drops by more than an order of magnitude for both sets of data. The excellent results using the DAC suggests that the sources of error contributing to the deviations from linearity are small. These points can be explored independently. The systematic errors in the DAC spectrum introduced by the electronic noise and DAC window, discussed in sect. 2, are negligible. Fig. 3 shows a plot of S(C)C, the DAC correction, as a function of C for the first set of data. By recalling that S(C) z C,,(C), the derivative of this curve, after allowing for statistical fluctuations, is the fractional change in the differential linearity, z, defined in eq. (10). Inspection of fig. 3 shows that jz/ < 0.005. Since 0 % 2 channels, the effect of noise, according to eq. (9), is less than 1 eV. Since the largest DAC correction is 1.5 channels, the DAC window effect, by eq. (18), is much less than 1 eV. The uncertainty introduced by the statistics in the DAC spectrum is considerably larger. According to eq. (30) it is 4 eV for both sets of data. The statistical error in a single channel of the DAC spectrum may be checked for consistency by using fig. 1. Several 1000 channel regions of the DAC spectra for both sets of data were fitted with horizontal straight lines and the X2/channel noted. The average of all the fits gives 0.81 which agrees very well with the value of 0.83 obtained from the plot using I- = 4 channels. The DAC calibration system is, in principle, limited only by the degree of linearity of the DAC and Ge(Li) detector. The DAC linearity measurement described in sect. 4 showed an rms deviation of + of a DAC step
-2/ 1000
1500
2000 A DC
2500
3000
3500
48
00
CHANNEL
Fig. 3. The DAC calibration curve between 3900 for the first set of data. S(C) is defined is the ADC channel.
channels 900 and in eq. (5) where C
430
R.
J.
MCKEE
which is equivalent to 3 eV for the data analyzed. While no similar independent check of the counter exists, the very good linear fit of the seven y-ray lines after DAC correction indicates that the linearity of the Ge(Li) diode is roughly as good as or better than the linearity of the DAC. The statistical uncertainties in the peak position, defined as the variation in the peak position parameter which increases x2 by 1, average to 3 eV for the seven lines for both sets of data. Systematic errors introduced by the peak fitting method are related to the question of detector linearity. Processes occurring in the detector that might contribute to non-linearity, such as multiple Compton scattering or charge carrier trapping, will also influence the peak shape and hence the peak fitting. But the clean and very symmetric peak shapes generated by the small detector are well fitted by a consistent set of shape parameters which not only minimizes systematic uncertainties in finding the relative peak positions but also increases our confidence in the high degree of linearity of the detector. The internal consistency of the energies of the seven primary standards must also be considered. An idea of their relative precision may be obtained by folding out the 9 eV error of the ig8Au line from the other six listed in table 2. When this is done, the relative errors of all seven lines range up to 10 eV with an average of 4 eV. Another source of error is pointed out in a paper by Gunnink et a1.5). If a planar Ge(Li) diode is orientated so that the electric field in the detector is parallel (or anti-parallel) to the incoming y-ray, the photopeak will be shifted due to the energy gain or loss of the photo-electron in moving through the field. Extrapolation of their measurement of this effect to account for the slightly higher electric field in our detector gives a shift of about 15 eV for a 500 keV y-ray. Since all the y-ray sources except the “Co source were located in front of our detector and since the shift is nearly linear with y-ray energy, the effect of source orientation on the linear fit is negligible. The 136 keV line of 57Co is shifted relative to the others by about 3 eV. Combining all the known errors gives 7 eV for the expected rms deviation from a straight line fit to the DAC corrected data. The fact that the linear fit to both sets of data gives the smaller result of 2 eV may be partly due to an overestimation of some of the errors discussed above and partly due to accident since only seven y-ray lines were used. When the energy of the “Co line is re-evaluated using the calibration coefficients of table 5, a discre-
et al.
pancy of 18 eV results between the two sets of data. The discrepancy is apparently due to a slight downward shift in the peak positions in the second set, both before and after DAC correction, although these changes differ very little from the statistical uncertainties that result from peak fitting and DAC calibration. The average of the two results gives 136.473 f0.015 keV for the 57Co line. Since the deviation from linearity of the electronics appears quadratic according to fig. 3, is it really necessary to employ a DAC to calibrate the system? Instead, why not make a quadratic fit to the energies of the seven primary standards using the uncorrected peak positions? When this is done the quadratic fit is as good as the linear fit to the corrected data. However, the data analyzed here is merely a single example of the use of the DAC calibration method. It cannot be overemphasized that this method is independent of the nature of the non-linearities - in this case, quadratic. The DAC will linearize even local regions of nonlinearity, a feat which no quadratic or other mathematical model will do with comparable confidence. 6. Conclusions It has been shown that the DAC calibration scheme described in this paper gives a model independent method of linearizing the electronics associated with lithium-drift germanium detectors. With the linearized system it has been possible to fit the energies of y-ray lines between 100 and 500 keV to a straight line with an rms deviation of less than 3 eV. Using this procedure a “Co line was measured to be 136.473 kO.015 keV. The system automatically corrects for integral and differential non-linearities with an accuracy limited in principle only by the precision of the DAC. The ultimate precision is probably set by the linearity of the detector itself. For large detectors this linearity is not well understood. It should be possible to achieve absolute accuracies of a few electron volts in the measurement of y-ray energies with this method.
We would especially like to thank A. Carter for originally suggesting to us the sliding pulser as a calibration technique. We would also like to thank J. P. Legault for building the computer interfaces, R. Gabriel for his measurement of the DAC using a standard cell, and D. Switzer and M. Wenger for supervising some of the runs used for analysis in this paper.
PRECISE
CALIBRATION
Appendix PEAK FITTING FUNCTION The mathematical line shape that is used by the computer program to determine centroids is the ratio of two fourth degree polynomials: N(C) = NCIP (C - C,), where C, is the centroid,
Cin P(C)
(Al)
IV, the amplitude,
and p(C) is
-6+fA,
(A2)
=
Lo
C outside.
The quantity A is the peak base width and 6 is the base offset; see fig. 4 for clarification. Two more length parameters, r the fwhm and y its offset, are also defined in fig. 4. The asymmetry parameters y and 6 are defined positive for offsets on the low channel side and negative for offsets on the high channel side. Also by definition A, = B, = 1 and A 1 = B1 = 0 which implies that dN(C)/dC = 0 at C = Co; therefore, for any asymmetry N(C) is maximum at Co and N(C,) = IV,,. The remaining six coefficients of the polynomials are determined in terms of the four length parameters, r, A, y, and 6, and two dimensionless parameters a and b which are essentially the quartic coefficients. Here, for completeness, are the relationships between Ak and B,(k = 2,3,4) and the six parameters: A,
1
B,
= (2/A)4 a, (A31 = (2/r)4 b ;
X,=-SIfi+A, Y* =-y++r, P,
= - (l+A,X&
D,
=
A,
= (P+ X3 -P_
X:)/D,,
= (X: P--X:
P+)/D,
x:x3
-x2_
431
OF A Ge(Li)-SPECTROMETER
There are several reasons why this particular line shape was chosen. In fitting a data peak, the parameters, as well as Co and N,, are varied many times before a good fit is obtained. This necessitates the calculation of the line shape function each time a parameter is varied. With no transcendental functions involved such as in a Gaussian line shape the iterations go much faster. The iterations also converge rapidly. The polynomial line shape also has the nice feature, that it is a maximum at the defined centroid position Co, no matter what the asymmetry is. Finally, it is versatile; it can assume a large range of peak shapes. For example, the Lorentzian line shape is contained in it; the parameters can also be adjusted to approximate a Gaussian. A seventh parameter f is introduced to allow for a step in the background under the peak. The step function is defined as:
s C
B.,(C)= (f/lW
N(C')dC'
-‘*I
.
(A6)
Forf < 0 the step is downward. Two more background parameters, u and v, are used to fit the general trend of the background: B(C)=u+v(C-C,),
(A7)
where C, is the initial channel of the channel interval being fitted. Background regions on both sides of the peak, each usually as extensive as the peak region, are chosen for the fit. The total mathematical function that is fitted to a data peak with background is N(C)+B,(C)+B(C). But with so many parameters, how does one achieve consistency in fitting a series of peaks whose shapes
x: ; (A4)
A, C
;
Qk = 1+2(A2Y:+A3Y3+(2A4-B4)Y;, D,
= Y: Y_3- Y_2 r: ;
B,
=
(Q+ J"-Q- Y:W,,
CB,
=
V:Q--Y~Q+W,.
CA3
Eq. (A3) is a defined relationship while eqs. (A4) and (A5) follow by the definitions of the four length parameters as shown in the figure.
Fig. 4. A schematic drawing of an idealized data peak without background in which the four length parameters r, d, y and 6 are defined.
432
R. TABLE
J.
MCKEE
6
References
The consistent set of shape parameters used to fit the eight y-ray source peaks for both sets of data.
r= I-/A y/r 6/A a b f
3.79+1.250x 0.29 = = 0 = 0.10 = -0.1 = 0.24 = -0.02
et al.
IO-3C
-
change with channel number? If just the width is changing, consistency may be obtained by keeping the other length parameters A, y, and 6 in constant ratio to r for every peak with a, b, and f the same for all peaks in the spectrum. The consistency may be further enhanced if the width is kept linear with channel number. The data from the small detector was analyzed in this fashion. Initially each of the eight y-ray peaks were analyzed by letting all the parameters vary until a good fit was obtained. Weighted averages of parameters or ratios of parameters were made to get a consistent set. The counts under each peak were chosen as the weights. In addition f as a function of peak position was fitted to a straight line. The final averaged parameters for the eight y-rays are shown in table 6. Using this consistent set of parameters, the peaks were refitted with the shape parameters held fixed; only CO, NO, u, and v were allowed to vary. The results shown in table 3 are from the second set of fits. The x2 values for these fits were only slightly worse than for the first set in which no constraints were used.
1) W. W. Black and R. L. Heath,
Nucl. Phys. A90 (1967) 650. J. Legrand, J. P. Boulanger and J. P. Brethon, Nucl. Phys. A107 (1968) 177. 3) J. D. King, N. Neff and H. W. Taylor, Nucl. Instr. and Meth. 52 (1967) 349. “) M. G. Strauss, F. R. Lenkszus and J. J. Eichholr, Nucl. Instr. and Meth. 76 (1969) 285. 5) R. Gunnink, R. A. Meyer, J. B. Niday and R. P. Anderson, Nucl. lnstr. and Meth. 65 (1968) 26. C. K. Hargrove, E. P. Hincks, J. D. 6) H. L. Anderson, McAndrew, R. J. McKee, R. D. Barton and D. Kessler, Phys. Rev. 187 (1969) 1565. No. 117 and Tennelec No. TC203BLR. 7) Ortec preamplifier Scientific ADC No. NS624 and two Northern 9 Northern Scientific stabilizers No. NS409. 9) Analog Devices DAC No. 15RBClOPS. lo) Hewlett Packard No. 3460A. While the absolute accuracy of the DVM is rated at 40ppm its linearity is considerably better than this. 11) J. B. Marion, Nucl. Data A4 (1968) 301. 9 G. Murray, R. L. Graham and J. S. Geiger, Nucl. Phys. 63 (1965) 353. G. T. Ewan and J. S. Geiger, Nucl. Instr. 13) R. L. Graham, and Meth. 9 (1960) 245. 19 G. Murray, R. L. Graham and J. J. Geiger, Nucl. Phys. 45 (1963) 177. 9 But see ref. 12 for a discussion of the adjustment of the r9sAu energy in order to conform with the 1963 value of the electron mass. I. Bergstrom and F. Brown, 16) J. S. Geiger, R. L. Graham, Nucl. Phys. 68 (1965) 352. is also made 17) J. J. Reidy, to be published. This measurement by direct comparison with the la8Au line. Rev. Mod. Phys. 37 9 E. R. Cohen and J. W. M. DuMond, (1965) 537. of the physical constants (as of Aug. 9 The latest compilation 1970) is contained in: Review of particle properties, Particle Data Group, Phys. Letters 33B (1970). 20) The energy of the 136 keV line of lszIr was calculated using the other precisely known transitions found in ref. 12 and the energy level diagram of rsLPt also found in ref. 12. 21) L. L. Baggerly, P. Marimier, F. Boehm and J. W. M. DuMond, Phys. Rev. 100 (1955) 1364.
“)
INSTRUMENTS
AND
92 (1971) 421-432;
METHODS
PRECISE USING
CALIBRATION A DIGITAL
R. J. MCKEE, National
H.
Carleton
PUBLISHING
CO.
and
CONVERTER*
and
Council of Canada,
MES
NORTH-HOLLAND
OF A Ge(Li)-SPECTROMETER TO ANALOG
C. K. HARGROVE
Research
0
A. G. SMITH
Ottawa,
Canada
A. THOMPSON
University,
Ottawa,
Canada
M. DIXIT Enrico Fermi Institute,
Uniuersity Received
of Chicago, 11 December
Chicago,
Illinois 60637,
U.S.A.
1970
A new method to precisely calibrate the linear electronics used with Ge(Li)-spectrometers has been developed. The calibration is done by measuring the system’s differential response with a mercury relay pulse generator whose amplitude is distributed over the range of the spectrum by a computer controlled 15 bit digital to analog converter. It is shown that the integral of the differential response, collected simultaneously with the data, linearizes the electronics throughout the whole spectrum range in a model independent way, except for small errors introduced
by electronic noise and accidental coincidence with detector events. A test of the calibration system using a small planar detector is presented in which a linear fit of peak position to adopted energy was done for seven y-ray lines between 100 and 500 keV which are known to very high precision. The rms deviation from the linear fit for two separate runs is less than 3 eV. In addition, a new value of 136.473&0.015 keV is reported for one of the principal transitions of 57Co.
1. Introduction
of high precision one can bracket each photon peak with two pulser peaks set as closely as possible to reduce interpolation error and convert, by linear interpolation, the position of the photon peak to volts. However, this is a laborious operation and not readily automated. Strauss et a14) have developed a calibration method which dispenses completely with the gain correction curve. Instead of using the conventional pulser, they use a sliding pulse generator; that is, a pulser whose amplitude increases linearly with time. The resulting spectrum measures the differential response of the amplifier-analyzer system and can be used in place of the gain correction curve for the following reason: the number of counts in a channel is proportional to the channel width and hence is also proportional to the energy per channel. Thus, the integral of the differential spectrum is linear with energy provided that the voltage ramp of the sliding pulser is also linear. Not only are interpolation problems eliminated by the differential response spectrum, but it is simple to obtain and easy to use in analysis. Strauss et al. have tried out their method by measuring y-rays from several sources in the energy region of 0.27 to 2.75 MeV and their results are within 40 to 110 eV of the generally accepted values. The calibration system that we have developed is similar to the one described by Strauss. Instead of a sliding pulser we use a conventional pulse generator
Since their invention over ten years ago, the lithiumdrift germanium detector has improved to such an extent that it is now possible to buy small planar detectors with a resolution of better than one keV for photons in the energy range of 100 to 500 keV. With such a spectrometer it should be possible, in principle, to measure photon energies to a precision of better than 10 eV in this low energy region. The best that has been achieved so far are precisions on the order of 30 to 60 eV1-‘). One of the problems in high precision Ge(Li) spectrometer work is the calibration of the amplifier and multichannel analyzer system. In the course of our work on muonic X-rays we have developed and used a novel scheme for solving the calibration problem. With it we have calibrated our peak positions in a model independent way to a precision of better than 10 eV. A conventional method of calibration is obtained by a precision pulse generator and voltmeter and a series of accurately known y-ray lines5.6). In this method several points are measured along the spectrum and the resulting gain correction curve is used to correct the data for non-linearities in the amplifier-analyzer system. Errors are introduced by the necessity of interpolation between the measured points. In work * Research Canada
supported by the National and the U.S. National Science
Research Council Foundation.
of
421
R. J.
422
MCKEE
whose amplitude is controlled by a 15 bit (32768 step) digital to analog converter (DAC). Since the number of DAC settings per channel can be made fairly large, a differential response curve - called the DAC spectrum in this paper - can be generated if a continuous range of DAC settings is used and the number of counts for each DAC setting is kept constant. Since we also use a 13 bit (8192 channel) analog to digital converter (ADC) for our multi-channel analyzer, the DAC calibration scheme is versatile and fully computer controlled. The use of the DAC calibration method has the following advantages: 1. It linearizes the electronics in a model independent way even removing local non-linearities. 2. The DAC calibration spectrum is concurrent with the data. 3. The computer controlled DAC allows for the automatic accumulation of the DAC spectrum using versatile pulse patterns. 4. The calibration method is easily applied to the analysis of the data. 5. It may be easily set to cover any range of the ADC. A test of the DAC calibration scheme using seven very accurately known (within 9 to 14 eV) y-ray lines in the energy region of 100 to 500 keV showed striking results: the deviation from linearity, after DAC correction, was less than 4 eV. The details of these results are presented in sect. 5. In sect. 2 the systematic errors of the DAC calibration method are explored. Sect. 3 presents a study of statistical errors. The apparatus is described in sect. 4 and conclusions are given in sect. 6. 2. Systematic
errors
In this section it is shown very generally that under ideal conditions the integral of the DAC spectrum is linear with the DAC setting and thus linear with energy, provided that the DAC itself is perfectly linear. It is also shown that a small departure from linearity is introduced by electronic noise. We then introduce a background in the DAC spectrum and show that it increases the departure from linearity. In order to reduce the background effect, a DAC window is introduced; that is, pulser events are required to fall into a channel range specified by the DAC setting. Finally it is shown that the DAC window also introduces a small error. With several DAC settings per channel, it is reasonable and convenient to treat the DAC setting as a continuous quantity. Let F(x) be the ADC channel
et al.
corresponding to DAC setting x; that is, assuming no noise or background, the pulser event appears in channel F(x). Let g(C) be the inverse function: the DAC setting corresponding to channel C. The range of DAC settings is given by a < x < b. Let C, = F(a) and C, = F(b). Since the edges of the DAC spectrum are not useful, the calibration is restricted to a somewhat smaller channel range from C, to C, where c, < c, < c, < c,. The electronic noise distributes the pulser events to some extent. Let p(C-F(x)) be the pulser probability distribution along the ADC for DAC setting x, where it is assumed that the distribution is the same for all DAC settings. It is also assumed that the noise is small: p(C-F(x)) = 0 for IC-F(x)1 larger than a few channels. For the time being background is ignored. Specifically, C, is far enough from C’, and C, from C, so that p(C, -C,) = p(C, - C,) = 0. Assume now a single DAC cycle: one count per DAC setting. With dx DAC settings between x and x+dx, there are dN(C) = p(C- F(x))dx counts contributed to channel C in the DAC spectrum. The total DAC spectrum for one cycle is h
N(C)
=
p(C-F(x))dx.
(1)
i’0 The replacement C’ = F(x) x = g(C’) results in N(C) =
s a
-cc
dg
-p dC’
which
(C-C’)dC’,
is equivalent
to
C, i C I CZ,
(2)
where C, and C, are replaced by + cc since p is zero outside this range. A Taylor series expansion of g(C’) about C’ = C gives the DAC spectrum in terms of the moments of the distribution. Let C”p(C)dC
(3)
be the normalized n’th moment where c2 is the second moment: r~is defined such that m2 = 1 and is a measure of the half width of the distribution. Since the distribution is normalized, m, = 1 also. An easy calculation shows that -I < m, < 1; m, is a measure of the asymmetry of the distribution. The final result for the DAC spectrum is N(C)
=
f (-O);mng’n+l)(C), n=O n.
C, I Cc
c,,
(4)
where g@) (C) symbolizes the n’th derivative. The next step is to integrate N(C) and show that it is
PRECISE
CALIBRATION
linear in the DAC setting except for small corrections. But the integral of the DAC spectrum is unwieldly to use; we find it more convenient to adjust the ADC channel itself. For this purpose, the standard channel S(C), linear in the integral, is introduced: S(C) = C1 +(C,-C,)
/I1 1(C’)dC’/jI; c,
SC
N(C’)dC’ SC,.
, (5)
Note that S(C,) = C, and S(C,) = C,. The quantity S(C)- C is the small correction made to the ADC channel in order to linearize it. Replacing g(C) by x and defining J(x)
=
f
(-a>“mn2,
n=O
n!
it is easily seen that S(C) = c, +(C7-C,)
f’(x)-f’(xl) J’(xz)-f(x,)’
(7)
where x1 =g(C,) and x2 =g(C,). In the limit of no noise, 0 = 0 and ,f(x) =x which proves that the standard channel is linear in the DAC setting. Noise, however, introduces corrections. It is not the size of the corrections that is important, but their variation along the ADC. Introducing l/p as the average number of DAC steps per channel, the variation in the first order term off(x) is Sf’= - am,(dx/dC-p-l), where dx/dC corresponding
detector events due to the fact that the DAC spectrum is collected simultaneously with the data. Since c reflects the half width of the total distribution, it can become many times larger than the half width of the pulser peak alone. The first order correction may still be insignificant if the total distribution is very symmetric (nll z 0); but the second order correction not only has no symmetry damping term, but is also proportional to 0’. In a region of local non-linearity, where the most accuracy is needed in calculating S(C), the second order term may become larger than the precision aimed for. The background effect is reduced by the introduction of a DAC window. For DAC setting x it is required that the pulser count fall in the channel range SI+flx+ A before the event is accepted and the DAC stepped to a new setting. The constants c( and fl are chosen to keep the peak part of the pulser distribution as much as possible in the center of the window throughout the entire DAC spectrum and A is chosen large enough to always encompass the peak proper. However: the movement of the peak in the window as the DAC is stepped introduces a small error of its own. Let C,(C) be the center of the window: Co(C) = m+/%(C).
linearity. The channel is
&Y(C) = - cm, z,
(9)
where (10)
and is the fractional change in the differential linearity. The first order correction expressed by eq. (9) is negligible. Assuming a non-linearity of 1% (z = 0.0 l), an asymmetry of 20% (Im, 1= 0.2) in the pulser peak distribution, and CJless than 5 channels then 16SI < 0.01 channels. Higher order corrections are smaller yet. Now consider what happens when a background is introduced. The pulser distribution, which must include the background, is no longer restricted to a few channels. In fact, the data indicates that the background is distributed over a large range of the ADC. The origin of the background is primarily accidental coincidences between the pulser events and
(111
Let R be the pulser rejection rate; it is the fraction of pulser counts that fall outside the window. It is here defined as if the peak were always in the center of the window:
(8)
is the local differential variation in the standard
z = /3dx/dC-1,
423
OF A Ge(Li)-SPECTROMETER
4
R=l-
p(C)dC.
s
(12)
-A
Under computer control the DAC window is easily manipulated and the rejection rate easily measured, although the experimental R differs somewhat from the definition above, since the peak moves around in the window. The definition given in eq. (12) is made to ease the mathematics. In order to further simplify matters it is assumed that CYand p are chosen such that C,(C,) = C, and C,(C,) = CZ. The actual values used in the experiment keep the peak, on the average, better centered; the simplification made here guarantees the center position only at C, and C,. But with CIand /3 defined this way, C,(C) assumes the desired properties of the standard channel: linear in the DAC setting and equal to the ADC channel at C, and C,. The inclusion of the DAC window modifies the probability distribution used in the calculations; since a pulser count must occur in the window before the
424
R.
DAC is stepped, the new distribution P(C-F(x))
J.
MCKEE
is:
standard
C inC,(F(x))fA
,
where sZ(C) is the normalization
factor
p(C’-C)dC’. f
(14)
G-A
The standard channel, eq. (7), is evaluated by procedures similar to the ones already used. Three approximations are used in the calculation. At the edges of the window the pulser distribution is purely background. A good approximation is to treat p(C- F(x)), as well as g”‘(C) and Q(C), constant over the small channel range C,,(C) - C at the edges of the window. For example,
= 1 -R+(C,-C)&,,
(15)
with e0 = p(d) -p( - A) and the distribution is treated as constant in the last two integrals. Secondly, if the background is fairly uniform, (C, - Cl) p (+ A) is on the order of the rejection rate. Thus, E= (C, - C)p( f A) e 1 and terms of order s2 are dropped. Thirdly, for good electronics the differential non-linearity expressed byeq. (10) is small. Hence, terms of order z2 are also ignored. The final evaluation of f(x) that appears in the standard channel is f
m,
(-y;“n !$
: ‘no n=l
x
(
(-1>” n!
>
A”E,_~n~Om,
$ [(Co-C) g],
where the cxom,/mo term has been dropped since it is small compared to Acl and the factor bdx/dC has been replaced by 1. It is difficult to evaluate eq. (17) exactly since the background is not known very well. But assuming (C, - C,)p() A) = R it is easily seen that 2R &s(C) = - 1-R
m”
(16)
where E, = p(A)-(l)“p(d). The moments of the distribution are now modified by the DAC window; for example m, = 1 -R. This means that G essentially still reflects the half width of the peak alone; therefore, a discussion of the first series in eq. (16) reduces to the earlier discussion before the introduction of the background. The second series in eq. (16) is a correction due to the fact that the distribution doesn’t stay centered in the window. The corresponding correction in the
Co-C ~ C,-C,
A ’
(18)
For example, with a pulser rejection rate of lo%, Co-C = 1.5 channels, and C,- C’i = 3000 channels 6S(C) is 10m4A, entirely negligible. Even if E, is an order of magnitude larger than what is estimated here, 6S(C) is still negligible if A is kept small enough. In summary it was shown that the deviation from linearity of the integral of the DAC spectrum is due to electronic noise and is negligible if the noise is kept reasonably small. It was also shown that, if background is present, a DAC window must be used and the error introduced by the use of the window is negligible if the background is not too big or the window width too long. Since the origin of the DAC window error is due to the movement of the pulser peak in the window, it may be better to use a higher order polynomial to determine the window center instead of the linear relationship used here. In either case, great care must be taken to insure that the pulser peak stays in the window for all DAC settings, otherwise the calibration may be severely biased. 3. Statistical
n=o
+ i
(17)
(13)
C,,+A
l/L?(C) =
%(Co-C), m.
outside,
i
due to the first term in this series is
&s(C) = -
rlo
=
channel
=
W(x))p(C-F(x)),
f(x)
et al.
errors
More important are the statistical errors of the DAC spectrum. Like the systematic errors discussed above, the origin of the statistical errors is the finite width of the pulser peak distribution. Two types of statistics are investigated here: the statistical uncertainty in the integral of the DAC spectrum and the statistical uncertainty in the counts of a single channel. The first is used to determine the minimum number of DAC cycles needed to achieve a specified precision in the calibration. The second is used as a check on the DAC spectrum since the statistics of a single channel are related to the pulser peak width, both of which are measurable. Qualitatively it is easy to understand what is expected. The statistical uncertainty on the integral of the DAC spectrum up to channel C comes from
PRECISE CALIBRATION
pulser counts for DAC settings near g(C) or, more exactly, for DAC settings in the range g(C++r) where r is the full width at half maximum (fwhm) of the pulser peak distribution. The reason for this is if the DAC setting is considerably less than (or greater than) g(C), the pulser count falls in the integral (or outside of it) and thus does not enter into the statistical uncertainty of the integral. With p-’ DAC settings per channel, there are on the average r/P pulser counts in the critical region Cf 3 r for each DAC cycle. With n DAC cycles, the statistical uncertainty on the integral is on the order of ,,/(nr/j?). Since the channel may be found by dividing the integral by n/P (the counts per channel) the resulting statistical uncertainty on the standard channel is J(fj?/n). If a precision of 6s of a channel is desired, the minimum number of DAC cycles required is about fp/6S2. For example, with 10 DAC steps per channel, f = 4 channels, and a desired precision of 0.1 channel, a minimum of 40 DAC cycles are necessary. This is an important consideration in estimating how long to collect the data. This result also shows another important role of the DAC window. Without the window, the effective width of the pulser peak may be much larger, which necessitates more running time to achieve the desired precision. In order to make the calculations, some simplifications are assumed which are not very essential to the final results. The first is to assume the ADC channel is linear in the DAC setting: F(x) = cc+ j?x.The next is to choose the pulser distribution as Gaussian:
=-J-
p( C-F(x))
aJ(2
- (C-2;,))z
exp [
n)
1
) (19)
with r related to 0 by r = 2ojln4.
(20)
The effect of the background in the window is merely to increase r by a small amount; so for simplicity the background is ignored. Let p(x) be the probability that, for DAC setting x, a pulser count falls between channels Ci and Cj where C, < Ci < Cj < C,. cj P(C-F(x))dC.
P(X) =
(21)
I Ci
If there are n pulser counts at DAC setting x, what is the probability that k of these counts fall in the channel range Ci to Cj? The answer is given by the binomial distribution: n.I
Pk(n) =
PWk(l-P(X)) “-k.
k!(n-k)!
The average
number
ck) =
of counts
for DAC
setting
k$,kPk(n) =np(x>.
G-4 x is (23)
The statistical fluctuation in (k) is the rms deviation whose square is symbolized by (6k2): (Sk’)
= ((k-(k))2)=
i
(k-(k))‘Pk(n)
k=l
=
(24)
Thus, the expectation value of the integral of the DAC spectrum from Ci to Cj is (k) summed over all DAC settings. (A(A))
IO,
425
OF A Ge(Li)-SPECTROMETER
=
1
b
(25)
where 1= Cj - Ci. Since the DAC settings are independent, the statistical fluctuation in (A(a)) is found by summing the rms deviations in quadrature over all DAC settings.
0.8 1
r\ =063.
2 : “a
0.4
&
-
(&4’(;1))
0.2
=
b (6k2) ca
dx
.
(26)
-
Finally, 000
2
3 Pu~sm
4
5
6
7
8
the Gaussian
distribution
gives the results
9
(27)
FWH~I (C~~ANNEL)
Fig. 1. A plot of the statistical error in a single channel of the DAC spectrum as a function of the fwhm of the pulser peak distribution under the assumption that the latter is Gaussian. The plot was calculated using eqs. (27) and (28).
<6A2(A>> = (n/B> CAerfc(u) +(2 a/Jn) (1 -e-“‘)] where u = l/20 and erfc(u) = 1 - erf(u). Eq. (27) shows that, on the average,
, (28)
there are n/j3
426
R.
J.
MCKEE
counts per channel in the DAC spectrum. Consider two extreme cases in the statistical fluctuation given by eq. (28). For g 4 I., erfc(u) z e-“‘/(u Jrc) for large arguments and therefore
(S/p(Q)
l
z
V’(711n4)
“!I = 0.48 /3
c. B
’
(29)
Except for the factor 0.48 this result agrees with the earlier qualitative argument. For n DAC cycles the statistical uncertainty in the standard channel is 6.S = ,:1(0.48 rfl/n).
(30)
For the other extreme case, r~ $ i, erf(u) z u for small arguments. There results
(&12(n)) = HA//l = (A(1)).
(31)
In the limit of no noise (f = 0) there are no statistical fluctuations. In the other limit of the noise evenly distributing the pulser counts, the statistical fluctuation approaches the conventional result that is encountered in counting random events. The statistical fluctuations of counts in a single channel is found by setting J. = 1 in eq. (28). Fig. 1 presents a plot of (6A’(l))/(A (1)) as a function of F. The plot can be used as a check on the DAC spectrum since both quantities may be measured experimentally:
Preamplifier
et al.
the abscissa is the x2 per channel of the DAC spectrum and I- is measured by accumulating a DAC spectrum for one DAC setting and noting the width of the distribution. 4. Apparatus and experimental procedure The electronic equipment is outlined in fig. 2. It is a relatively standard lithium-drift germanium spectrometer system except for the addition of a precision digital to analog converter as the voltage source for the pulser. The detector and pulser pulses are fed through a preamplifier, to an amplifier, and into an analog to digital converter. The digital output of the ADC is read into a PDP-9 computer along with a tag which identifies the word as being from the diode or the pulser. The DAC is controlled by the computer. In addition, pre- and post-pileup circuits prevent the analysis of events which were preceded by pulses within 50 psec and followed by pulses within 4 ,usec, the time for the ADC to recognize a pulse. These two circuits are not shown in fig. 2. All the apparatus shown in the diagram, except the computer, was located inside a copper hut whose air temperature was regulated to within + 1 “C. The Ge(Li) detector, manufactured by Ortec, was a 0.5 cm3 planar diode with an area of I cm2 and a depletion depth of 5 mm; it was dc coupled to a cooled FET preamplifier and linear amplifier’). An 8192 channel
Ampllfler ADC
Ge(Li)
Fig.
2. A schematic
drawing
of the electronic
equipment
used in the DAC
c
Computer
calibration
method.
PRECISE CALIBRATION
427
OF A Ge(Li)-SPECTROMETER
analog to digital converter was used with both gain and zero digital stabilizers’). The stabilizers were operated on y-ray peaks outside the DAC range. A mercury relay, driven by a 50 cps oscillator which could be turned on or off by the computer, was used to generate the pulses. The zero point and range of the pulser in the DAC spectrum were adjusted by potentiometers Pl and P2 which offset and attenuated the DAC output voltage, respectively. The heart of the calibration system was the 15 bit DAC manufactured by Analog Devices’). It is unipolar with the output voltage range from 0 to + 10 V full scale (DAC setting of 32767). Its output stage is a difference operational amplifier with an offset input. Its linearity is rated at +i of the least significant bit or 15 ppm (parts per million). In addition its temperature coefficient is rated at 17 ppm/“C and its long term (one month) full scale stability at + 10 ppm. The settling time is 100 psec for the full scale step. The linearity of the DAC was checked against a six digit digital voltmeter”). The DAC was stepped from 0 to 32000 in 1000 step intervals and the voltage measured for each setting. The deviations from a straight line fit revealed a uniform linearity with an rms deviation of slightly less than & of a DAC setting or 7 ppm. An additional check was made by measuring each bit setting with a potentiometer and standard cell; the results agreed with the DVM test. The system was operated in the following manner. After the ADC finished analyzing an event, the computer read and reset the ADC and the tag register. If the event was identified by the tag register as a pulser event the computer then checked to see if the ADC channel was inside the DAC window of + 10 channels defined by the DAC setting. If it did, the pulser count was added onto the DAC spectrum and the computer stepped the DAC to a new setting. At the beginning of a DAC cycle, the settings were started at the minimum setting (a) and incremented in steps of 48 after each accepted pulser event until the settings were about to exceed the maximum setting (b); the next setting was started in the second division (a+ 1) and again incremented by 48. The procedure was continued until all settings between a and b inclusive were covered and then the next DAC cycle started. This method of stepping the DAC was done to minimize the effect of short term drifts on the DAC spectrum. The new DAC voltage was present at the pulser about 0.2 msec after the last accepted pulser event. This gave the capacitor Cl time to fully charge before the mercury relay switched again. The pulser decay constant was made long compared to the preamplifier time constant.
Data collection in the DAC spectrum was always ended at the end of a completed DAC cycle. If the pulser event fell outside the window, the event was scaled as a pulser reject and the DAC was not stepped. With a total preamplifier rate of about 2000 per second, the rejected pulser events were about 3% of the total pulser events. 5. Results The data used in testing the DAC calibration system was obtained during a muonic X-ray experiment at the University of Chicago synchrocyclotron. The test data consists of a series of y-ray lines of well known energy from several y-ray sources which were used in the experiment to calibrate muonic X-rays. The usefulness of the calibration system is demonstrated by fitting a straight line to the energies and DAC corrected channel positions of the y-ray peaks and noting the deviations from linearity that result. The uncorrected data is also analyzed in the same manner for comparison. Estimations of the systematic and statistical errors are also made. TABLE 1 Parameters
of the two sets of data.
Constant
Running time (hours) Number of DAC cycles (n) Rejection rate (R) Channel corresponding to DAC setting of zero (a) Channel per DAC setting @) DAC half window width in channels (d) Initial DAC setting (a) Final DAC setting (6) Channel corresponding to DAC setting a (C,) Channel corresponding to DAC setting b (Cb) Initial channel used in calibration (Cl) Final channel used in calibration (Cs) fwhm of pulser peak (I-)
Run 1 Run 2
68 172 0.033
73 173 0.034 728 0.1112 10 1370 30680 880 4140 900 3900 4
Several runs from the X-ray experiment, obtained under the same operating conditions, were chosen for analysis. The runs were grouped into two nearly equal sets and summed. Pertinent quantities for both summed runs are presented in table 1. The data used here were collected over a period of about two weeks. Drifts were looked for by systematically analyzing certain of the y-ray lines in each run, but no peak shifts outside statistics were observed. The y-ray spectrum contained eight lines inside the
428
R.
I.
MCKEE
et al.
TABLE 2
Gamma-ray
Gamma-ray
TABLE 3
lines used in the DAC calibration.
source
Energy
The ADC and DAC corrected positions of the eight photon peaks for both runs. The errors are purely statistical. The number in parantheses is the DAC corrected peak positions.
(keV)
136.478+0.030” 165.85210.010” 238.624&0.009c 295.938&0.009c 308.429&0.010c 316.486&0.010c 411.795f0.009e 468.053*0.014c
5 J. Rapaport, Nuclear Data B3 (1970) 103. The number quoted here is the sum of the adopted values for the cascade y-rays 14.408&0.005 keV and 122.07&0.03 keV. b Refs. 11, 16 and 17. e Ref. 12.
range of the DAC spectrum. The identity and energies of the lines are summarized in table 2. Except for the 57Co line, the y-ray energies listed in the table are primary standards as defined in the review article on y-ray sources by Marion”). Murray et al.“) have measured the four 1921r energies and the 228Th energy by a direct comparison with the 412 keV line of 19*Au using an iron free beta-ray spectrometerL3). The relative precision of these six energies is therefore better than the errors quoted in table 2. In an earlier paper14) the same group report the precise determination of the 412 line of 19*Au itself by direct comparison with the annihilation radiation15) using the magnetic spectrometer. The ‘39Ce line listed in the table is an average”) of a magnetic spectrometer measurementr6) and a crystal diffraction spectrometer measurement”). The energies of all seven of the primary standards as listed in table 2 are directly based upon the 1963 value of 511.006 + 0.005 keV for the rest mass of the electron”). While the fundamental physical constants have been readjusted several times since then”) no attempt is made in the present work to rescale the seven energies, since the primary purpose here is to demonstrate the linearity of the DAC calibration scheme. In addition to the eight lines listed in the table, two more y-ray lines appeared in the spectrum outside the limits of the DAC range; these were the 122 keV line of 57Co and the 511 keV annihaliation line from 22Na. They were used to stabilize the electronics. The fwhm of the eight y-ray peaks used in the analysis ranged from 0.59 to 1.OOkeV. The channel position of each line was found by
Gamma-ray line
57Co 13sCe z2sTh 19’LIr tszIr lg21r ts8Au ls21r
(136)” (166) (239) (296) (308) (316) (412) (468)
985.50+0.10(985.37) 1237.94+0.01(1237.51) 1863.00+0.02(1861.82) 2355.01+0.04(2353.57) 2462.15*0.04(2460.70) 2531.29&0.02(2529.88) 3348.44&0.01 (3347.55) 3830.40&0.04(3830.20)
985.49+0.10(985.37) 1237.92_rtO.Ol(l237.36) 1862.99~0.02(1861.76) 2354.93&0.04(2353.48) 2462.12~0.04(2460.69) 2531.27+0.02(2529.80) 3348.42+0.01 (3347.56) 3830.35*0.04(3830.22)
a The peak positions of this line were adjusted upward by 0.07 of a channel to compensate for the 5% contamination of the 136 keV line of ls21r. See text for details. The uncertainty of the peak position has also been increased to 0.1 to allow for systematic errors in fitting a single line shape to a doublet.
fitting the peak region with a mathematical peak shape plus background terms. Details of the line shape and fitting procedure is given in the appendix. The position of each peak was corrected using the DAC spectrum by applying the standard channel definition given in eq. (5). Linear interpolation between adjacent channels was used to determine fractions of a channel. The y-ray peak positions, uncorrected and corrected. are given in table 3. For both sets of data, a weighted least squares fit to a straight line was made using the adopted energies of table 2. The 136 keV line of “Co was not used in the straight line fit for two reasons. Its known precision is two to three times worse than the seven primary standards and there is a weak 136.314f0.019 keV line2’) of lg21r under it. From the known intensity of the ratios21) of the ig21r lines and a measurement counter efficiency the contamination from the 1921r line amounted to about 5%. With the 1921r line 164 + 36 eV of lower than the 57Co line and with a dispersion 117 eV/channel, the centroid of the two peaks is lower by 0.07 channels than the 57Co peak alone. Hence it is possible to correct for the ig21r contamination. This has been done in the results from the peak fitting for the 57Co line as shown in table 3, but with an increased peak position uncertainty. When the linear fit to the seven energies of the primary standards is made using the uncorrected peak positions as shown in table 4, the rms deviation from linearity is several times the precision to which the seven energies are known. But when the DAC corrected
PRECISE
CALIBRATION
01
TABLE 4 Results of a weighted least squares fit to E = cro+al C for the seven primary standard y-ray lines where E is the energy and C the uncorrected peak positions.
Parameters
ao(keV) ar(keV/channel) x2 of fit
Gamma-ray
line
Weight
s7Co (I 36)c lsgCe (166) =*Th (239) ls21r (296) lgslr (308) rgelr (316) lg”Au (4 12) rg%Ir (468) (AEz)*
(x 104)”
Run 1
Run 2
2 1.484 0. I16560 158
21.486 O.I16561 154
A E(eV)u
AE(eV)b
0 0.99 1.16 0.97 0.82 0.95 1.21 0.46
- 124 - 73 11 46 44 45 - 17 - 97 49
- 122 - 73 14 40 44 41 - 15 - 99 49*
a The energy equivalent of the statistical errors on the peak positions (table 3) are folded into the quoted errors from the literature (table 2). The weights are the reciprocal of the squares of the augmented errors. b AE = E(calculated using an and nt) - E(table 2). C Not used in determining the fit as explained in the text. * Weighted rms deviation. 57Co not included.
TABLE 5 Results of a weighted least squares fit to E = ao + al C for the DAC corrected positions. The weights are the same as in table 4.
Parameters
ao(keV) al(keV/channel) xs of fit
Gamma-ray
57Co 13gCe s=Th lg21r lg21r rg21r
line
Run 1
Run 2
21.6116 0.116558 0.28
21.6361 0.116551 0.16
AE(eV)
AE(eV)
(136) (166) (239) (296) (308) (316)
- 14 1.3 - 2.4 1.0 -3.1 3.3
4 -0.8 1.3 -2.4 2.0 -0.2
lg8Au (412) rg21r (468)
0.3 - 0.9
1.2 -2.5
rms deviation.
2.1” “7Co not included.
1.6=
429
A Ge(Li)-SPECTROMETER
peak positions are used instead, as shown in table 5, the improvement is very striking: the rms deviation drops by more than an order of magnitude for both sets of data. The excellent results using the DAC suggests that the sources of error contributing to the deviations from linearity are small. These points can be explored independently. The systematic errors in the DAC spectrum introduced by the electronic noise and DAC window, discussed in sect. 2, are negligible. Fig. 3 shows a plot of S(C)C, the DAC correction, as a function of C for the first set of data. By recalling that S(C) z C,,(C), the derivative of this curve, after allowing for statistical fluctuations, is the fractional change in the differential linearity, z, defined in eq. (10). Inspection of fig. 3 shows that jz/ < 0.005. Since 0 % 2 channels, the effect of noise, according to eq. (9), is less than 1 eV. Since the largest DAC correction is 1.5 channels, the DAC window effect, by eq. (18), is much less than 1 eV. The uncertainty introduced by the statistics in the DAC spectrum is considerably larger. According to eq. (30) it is 4 eV for both sets of data. The statistical error in a single channel of the DAC spectrum may be checked for consistency by using fig. 1. Several 1000 channel regions of the DAC spectra for both sets of data were fitted with horizontal straight lines and the X2/channel noted. The average of all the fits gives 0.81 which agrees very well with the value of 0.83 obtained from the plot using I- = 4 channels. The DAC calibration system is, in principle, limited only by the degree of linearity of the DAC and Ge(Li) detector. The DAC linearity measurement described in sect. 4 showed an rms deviation of + of a DAC step
-2/ 1000
1500
2000 A DC
2500
3000
3500
48
00
CHANNEL
Fig. 3. The DAC calibration curve between 3900 for the first set of data. S(C) is defined is the ADC channel.
channels 900 and in eq. (5) where C
430
R.
J.
MCKEE
which is equivalent to 3 eV for the data analyzed. While no similar independent check of the counter exists, the very good linear fit of the seven y-ray lines after DAC correction indicates that the linearity of the Ge(Li) diode is roughly as good as or better than the linearity of the DAC. The statistical uncertainties in the peak position, defined as the variation in the peak position parameter which increases x2 by 1, average to 3 eV for the seven lines for both sets of data. Systematic errors introduced by the peak fitting method are related to the question of detector linearity. Processes occurring in the detector that might contribute to non-linearity, such as multiple Compton scattering or charge carrier trapping, will also influence the peak shape and hence the peak fitting. But the clean and very symmetric peak shapes generated by the small detector are well fitted by a consistent set of shape parameters which not only minimizes systematic uncertainties in finding the relative peak positions but also increases our confidence in the high degree of linearity of the detector. The internal consistency of the energies of the seven primary standards must also be considered. An idea of their relative precision may be obtained by folding out the 9 eV error of the ig8Au line from the other six listed in table 2. When this is done, the relative errors of all seven lines range up to 10 eV with an average of 4 eV. Another source of error is pointed out in a paper by Gunnink et a1.5). If a planar Ge(Li) diode is orientated so that the electric field in the detector is parallel (or anti-parallel) to the incoming y-ray, the photopeak will be shifted due to the energy gain or loss of the photo-electron in moving through the field. Extrapolation of their measurement of this effect to account for the slightly higher electric field in our detector gives a shift of about 15 eV for a 500 keV y-ray. Since all the y-ray sources except the “Co source were located in front of our detector and since the shift is nearly linear with y-ray energy, the effect of source orientation on the linear fit is negligible. The 136 keV line of 57Co is shifted relative to the others by about 3 eV. Combining all the known errors gives 7 eV for the expected rms deviation from a straight line fit to the DAC corrected data. The fact that the linear fit to both sets of data gives the smaller result of 2 eV may be partly due to an overestimation of some of the errors discussed above and partly due to accident since only seven y-ray lines were used. When the energy of the “Co line is re-evaluated using the calibration coefficients of table 5, a discre-
et al.
pancy of 18 eV results between the two sets of data. The discrepancy is apparently due to a slight downward shift in the peak positions in the second set, both before and after DAC correction, although these changes differ very little from the statistical uncertainties that result from peak fitting and DAC calibration. The average of the two results gives 136.473 f0.015 keV for the 57Co line. Since the deviation from linearity of the electronics appears quadratic according to fig. 3, is it really necessary to employ a DAC to calibrate the system? Instead, why not make a quadratic fit to the energies of the seven primary standards using the uncorrected peak positions? When this is done the quadratic fit is as good as the linear fit to the corrected data. However, the data analyzed here is merely a single example of the use of the DAC calibration method. It cannot be overemphasized that this method is independent of the nature of the non-linearities - in this case, quadratic. The DAC will linearize even local regions of nonlinearity, a feat which no quadratic or other mathematical model will do with comparable confidence. 6. Conclusions It has been shown that the DAC calibration scheme described in this paper gives a model independent method of linearizing the electronics associated with lithium-drift germanium detectors. With the linearized system it has been possible to fit the energies of y-ray lines between 100 and 500 keV to a straight line with an rms deviation of less than 3 eV. Using this procedure a “Co line was measured to be 136.473 kO.015 keV. The system automatically corrects for integral and differential non-linearities with an accuracy limited in principle only by the precision of the DAC. The ultimate precision is probably set by the linearity of the detector itself. For large detectors this linearity is not well understood. It should be possible to achieve absolute accuracies of a few electron volts in the measurement of y-ray energies with this method.
We would especially like to thank A. Carter for originally suggesting to us the sliding pulser as a calibration technique. We would also like to thank J. P. Legault for building the computer interfaces, R. Gabriel for his measurement of the DAC using a standard cell, and D. Switzer and M. Wenger for supervising some of the runs used for analysis in this paper.
PRECISE
CALIBRATION
Appendix PEAK FITTING FUNCTION The mathematical line shape that is used by the computer program to determine centroids is the ratio of two fourth degree polynomials: N(C) = NCIP (C - C,), where C, is the centroid,
Cin P(C)
(Al)
IV, the amplitude,
and p(C) is
-6+fA,
(A2)
=
Lo
C outside.
The quantity A is the peak base width and 6 is the base offset; see fig. 4 for clarification. Two more length parameters, r the fwhm and y its offset, are also defined in fig. 4. The asymmetry parameters y and 6 are defined positive for offsets on the low channel side and negative for offsets on the high channel side. Also by definition A, = B, = 1 and A 1 = B1 = 0 which implies that dN(C)/dC = 0 at C = Co; therefore, for any asymmetry N(C) is maximum at Co and N(C,) = IV,,. The remaining six coefficients of the polynomials are determined in terms of the four length parameters, r, A, y, and 6, and two dimensionless parameters a and b which are essentially the quartic coefficients. Here, for completeness, are the relationships between Ak and B,(k = 2,3,4) and the six parameters: A,
1
B,
= (2/A)4 a, (A31 = (2/r)4 b ;
X,=-SIfi+A, Y* =-y++r, P,
= - (l+A,X&
D,
=
A,
= (P+ X3 -P_
X:)/D,,
= (X: P--X:
P+)/D,
x:x3
-x2_
431
OF A Ge(Li)-SPECTROMETER
There are several reasons why this particular line shape was chosen. In fitting a data peak, the parameters, as well as Co and N,, are varied many times before a good fit is obtained. This necessitates the calculation of the line shape function each time a parameter is varied. With no transcendental functions involved such as in a Gaussian line shape the iterations go much faster. The iterations also converge rapidly. The polynomial line shape also has the nice feature, that it is a maximum at the defined centroid position Co, no matter what the asymmetry is. Finally, it is versatile; it can assume a large range of peak shapes. For example, the Lorentzian line shape is contained in it; the parameters can also be adjusted to approximate a Gaussian. A seventh parameter f is introduced to allow for a step in the background under the peak. The step function is defined as:
s C
B.,(C)= (f/lW
N(C')dC'
-‘*I
.
(A6)
Forf < 0 the step is downward. Two more background parameters, u and v, are used to fit the general trend of the background: B(C)=u+v(C-C,),
(A7)
where C, is the initial channel of the channel interval being fitted. Background regions on both sides of the peak, each usually as extensive as the peak region, are chosen for the fit. The total mathematical function that is fitted to a data peak with background is N(C)+B,(C)+B(C). But with so many parameters, how does one achieve consistency in fitting a series of peaks whose shapes
x: ; (A4)
A, C
;
Qk = 1+2(A2Y:+A3Y3+(2A4-B4)Y;, D,
= Y: Y_3- Y_2 r: ;
B,
=
(Q+ J"-Q- Y:W,,
CB,
=
V:Q--Y~Q+W,.
CA3
Eq. (A3) is a defined relationship while eqs. (A4) and (A5) follow by the definitions of the four length parameters as shown in the figure.
Fig. 4. A schematic drawing of an idealized data peak without background in which the four length parameters r, d, y and 6 are defined.
432
R. TABLE
J.
MCKEE
6
References
The consistent set of shape parameters used to fit the eight y-ray source peaks for both sets of data.
r= I-/A y/r 6/A a b f
3.79+1.250x 0.29 = = 0 = 0.10 = -0.1 = 0.24 = -0.02
et al.
IO-3C
-
change with channel number? If just the width is changing, consistency may be obtained by keeping the other length parameters A, y, and 6 in constant ratio to r for every peak with a, b, and f the same for all peaks in the spectrum. The consistency may be further enhanced if the width is kept linear with channel number. The data from the small detector was analyzed in this fashion. Initially each of the eight y-ray peaks were analyzed by letting all the parameters vary until a good fit was obtained. Weighted averages of parameters or ratios of parameters were made to get a consistent set. The counts under each peak were chosen as the weights. In addition f as a function of peak position was fitted to a straight line. The final averaged parameters for the eight y-rays are shown in table 6. Using this consistent set of parameters, the peaks were refitted with the shape parameters held fixed; only CO, NO, u, and v were allowed to vary. The results shown in table 3 are from the second set of fits. The x2 values for these fits were only slightly worse than for the first set in which no constraints were used.
1) W. W. Black and R. L. Heath,
Nucl. Phys. A90 (1967) 650. J. Legrand, J. P. Boulanger and J. P. Brethon, Nucl. Phys. A107 (1968) 177. 3) J. D. King, N. Neff and H. W. Taylor, Nucl. Instr. and Meth. 52 (1967) 349. “) M. G. Strauss, F. R. Lenkszus and J. J. Eichholr, Nucl. Instr. and Meth. 76 (1969) 285. 5) R. Gunnink, R. A. Meyer, J. B. Niday and R. P. Anderson, Nucl. lnstr. and Meth. 65 (1968) 26. C. K. Hargrove, E. P. Hincks, J. D. 6) H. L. Anderson, McAndrew, R. J. McKee, R. D. Barton and D. Kessler, Phys. Rev. 187 (1969) 1565. No. 117 and Tennelec No. TC203BLR. 7) Ortec preamplifier Scientific ADC No. NS624 and two Northern 9 Northern Scientific stabilizers No. NS409. 9) Analog Devices DAC No. 15RBClOPS. lo) Hewlett Packard No. 3460A. While the absolute accuracy of the DVM is rated at 40ppm its linearity is considerably better than this. 11) J. B. Marion, Nucl. Data A4 (1968) 301. 9 G. Murray, R. L. Graham and J. S. Geiger, Nucl. Phys. 63 (1965) 353. G. T. Ewan and J. S. Geiger, Nucl. Instr. 13) R. L. Graham, and Meth. 9 (1960) 245. 19 G. Murray, R. L. Graham and J. J. Geiger, Nucl. Phys. 45 (1963) 177. 9 But see ref. 12 for a discussion of the adjustment of the r9sAu energy in order to conform with the 1963 value of the electron mass. I. Bergstrom and F. Brown, 16) J. S. Geiger, R. L. Graham, Nucl. Phys. 68 (1965) 352. is also made 17) J. J. Reidy, to be published. This measurement by direct comparison with the la8Au line. Rev. Mod. Phys. 37 9 E. R. Cohen and J. W. M. DuMond, (1965) 537. of the physical constants (as of Aug. 9 The latest compilation 1970) is contained in: Review of particle properties, Particle Data Group, Phys. Letters 33B (1970). 20) The energy of the 136 keV line of lszIr was calculated using the other precisely known transitions found in ref. 12 and the energy level diagram of rsLPt also found in ref. 12. 21) L. L. Baggerly, P. Marimier, F. Boehm and J. W. M. DuMond, Phys. Rev. 100 (1955) 1364.
“)