Precision measurement of X-ray line spectra by energy dispersion in a gas microstrip detector

Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

Precision measurement of X-ray line spectra by energy dispersion in a gas microstrip detector J.E. Bateman* CLRC, Rutherford Appleton Laboratory, Instrumentation Division, Chilton, Didcot, Oxon OX11 0QX, UK Received 2 October 2001; received in revised form 18 January 2002; accepted 12 February 2002

Abstract It is shown that when a gas microstrip detector (GMSD) is operated in such a way as to gain freedom from detector wall effects, the response of the detector to an X-ray line is stable and can be fitted reliably with a lognormal (LN) distribution function. The LN function permits the fitting of adjacent, overlapping X-ray lines with an accuracy in position of a few eV with an attendant penalty in the statistics required for a given precision in the amplitude measurement. Experimental data and Monte Carlo simulation data are presented to indicate the possible range of usefulness of this technique for energy-dependent X-ray line spectroscopy in applications (such as X-ray fluorescence analysis). The usefulness of the LN fits for detector studies is also noted. r 2002 Elsevier Science B.V. All rights reserved. PACS: 07.85F; 29.40C Keywords: Gas microstrip detector; Energy dispersive X-ray spectroscopy; Data analysis

1. Introduction In analytical techniques such as X-ray Fluorescence Analysis (XFA) and X-ray Absorption Fine Structure (XAFS) the detected X-rays occur as distinct lines (emitted by the atoms under study). The analytical techniques depend on the identification of the line position (in wavelength or energy) and the quantification of the X-ray flux in each line. Operation in the wavelength domain offers very high potential resolution in wavelength at the expense of a very low sensitivity to the X-ray flux because of the small solid angle of a typical monochromator. In the energy domain much *Tel.: +44-01865-863759; fax: +44-01235-445321. E-mail address: [email protected] (J.E. Bateman).

greater solid angle is possible but the line resolution is generally much poorer. The detector of choice is the cryogenic semiconductor (Ge or Si) detector which can give a FWHM energy resolution down to the order of 150 eV and is widely used in materials analysis systems. The main disadvantages of this device are the complexity and cost of the cryostat necessary for operation. Technical limitations arise from the finite front contact thickness (which raises the useful threshold of operation to E1 keV) and the slow amplifier time constants necessary to achieve the low noise (which limit the data capture rates to E100 kHz). Gas counters have been largely superseded in this field due to their poor energy resolution. However, at low X-ray energies (o10 keV) the operational flexibility inherent in the gas counter (e.g. the gas

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 5 6 5 - X

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622 800 Normal PH Spectrum PH Spectrum with Anticoincidence

600

Counts / Channel

filling can be tailored to a given application, internal counting can give ‘‘windowless’’ operation for very soft X-rays, etc.) has kept them alive in certain applications (e.g. the simpler XFA systems). The recent development of the Gas Microstrip Detector (GMSD) [1] has opened up new technical possibilities for gas detectors which are explored in this presentation.

611

Argon + 25% IB 400

Vc=-549V, Vd = -1300V

200

2. The gas microstrip detector 0

The GMSD consists of an array of fine metallic lines produced on a semiconducting glass substrate by standard micro-lithographic processes. Connected alternately to a high electrical potential (anodes and cathodes) the metal strips function as amplifiers of free electrons in the gas surrounding them. A drift cathode at a suitable distance from the plate (typically 10 mm) defines the gas volume from which X-ray-induced electron clouds are collected onto the anodes for amplification. Gas gains of up to 5000 enable individual X-rays to be detected as a pulse in an amplifier connected to either a single strip or groups of strips. The practical importance of the GMSD technology is that all the high-resolution gain-defining elements are located on the glass plate which is now a simple component that can be cleaned, handled and connected up very simply. Further, the cylindrical geometry of the conventional, proportional counter is replaced by planar geometry which is yet further simplified by the weak dependence of the gas gain on the drift electrode position [2]. A final advantage of the GMSD is its high rate capability. Using the divided electrode structure it is possible to achieve counting rate densities of E10 MHz/ cm2, giving several tens of MHz for a typical detector area of several cm2 [3].

3. X-ray line detection in the GMSD The energy resolution achieved in a GMSD is determined by various factors but chiefly by the photoelectron and avalanche statistics [4] and varies with the X-ray energy approximately as 1=2 Ex : Fig. 1 (upper line) shows the pulse height

0

100

200 Channel No.

300

400

Fig. 1. A comparison of the pulse height spectra obtained from a GMSD with and without the anti-coincidence circuits of Fig. 2 in operation.

spectrum given by Mn K X-rays in a GMSD. The FWHM of the Ka peak (5.9 keV) is 14.5%. This is a typical value for any well-constructed GMSD and it is difficult to make any significant improvement on this value in a practical detector [4]. As well as a central portion which is very approximately Gaussian, the line response function (LRF) (i.e. the response of the detector to an X-ray line) of the GMSD has a long tail going down to zero energy. This tail is characteristic of all gas counters and is due to wall effects in which part of the electron cloud of an event is lost by collision with the wall or loss from the scavenged volume of the counter. By constructing a GMSD in the form shown in Fig. 2, one can eliminate the low energy tail of the LRF [4]. In Fig. 2 the X-rays enter the drift space of the GMSD parallel to the plate and are collimated sufficiently well to prevent any X-ray converting close to either the plate or the drift electrode (E1 mm). A central part of the plate is used as the main signal source and a section of plate on either side is used as a guard structure to veto events in which charge has been shared between the central section and a guard section. If the threshold on the guard sections is sufficiently low (o20% of the peak energy) this eliminates the ‘‘tail’’ events almost completely as the lower curve in Fig. 1 (heavy line) illustrates. The resulting PSF is little narrower in its core shape but it has now

612

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

Fig. 2. A schematic diagram of the mode of operation of a GMSD which permits pulse height spectra from an effectively ‘‘wall-less’’ counter to be obtained.

become a relatively symmetric function, which is fitted rather well by a lognormal (LN) curve. The LN distribution is such that the logarithms of the pulse heights are normally distributed. This mode of operation of the GMSD effectively makes a ‘‘wall-less’’ counter. The shape of the core LRF in a gas counter is generally assumed to be a normal distribution. This is (in general) a very good approximation but the effects of avalanche statistics, of electron loss to negative ion formation in the drift and nonhomogeneity in the electric field all contribute to give the LRF a small positive skew [4]. A particular property of the LN distribution is that it generates a positive skew which is dependent only on the single parameter sR the standard deviation (SD) of the logarithimic transform of the pulse height (PH) distribution. This parameter (also found as b in Eq. (1) below) is notated as sR because it represents the relative resolution (SD/ peak pulse height) of the PH spectrum. Thus 14.5% FWHM for Mn Ka transforms into sR ¼ 0:061:

The application of pulse height spectra to quantitative analysis (e.g. XFA) requires the determination of the centroid (for X-ray line identification) and the amplitude (for the signal intensity). When fitting overlapping lines the first parameter (the centroid) is not sensitive to the presence of spurious backgrounds due to the strongly symmetric core function which can be adequately modelled with a normal form. However, the determination of the amplitude is a different matter. In order to evaluate the true amplitude of the line an ad hoc pedestal must be assumed. This pedestal is not a simple background but rather the accumulation of the ‘‘wall-tails’’ of all lines to the right of the line under consideration. Thus fitting of the amplitudes of overlapping lines must be essentially an iterative process. (The presence of the pedestal also injects stochastic noise into the measurement and this can become a serious limitation for weak lines to the left of strong lines.) ‘‘Wall-less’’ operation, by removing the tails in the LRF, makes a simple, one-step fit mathematically rigorous without the need for any ad hoc manipulations of the data. The tests described below show that the LN distribution applied to a ‘‘wall-less’’ spectrum appears to give a stable representation of the LRF over a practical range of bias conditions for a typical GMSD and permits the successful fitting of the energy and amplitude of overlapping X-ray lines.

4. Lognormal fits to Mn X-ray lines The radioactive (internal conversion) X-ray source 55Fe yields the Mn Ka and Kb lines at 5.9 and 6.49 keV. The relative intensity of the two lines can vary depending on the effect of differential absorption in the source, detector window and detector volume, but the Kb is generally found to be E20% of the total rate. In a detector filled with argon, a reflection of the two peaks appears at 2.9 and 3.49 keV due to the escape of the argon K X-rays from the counter volume without conversion. This provides a useful set of four lines (we neglect the presence of the Ar Kb in the escape spectrum) for calibrating the energy response of

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

Counts per Channel

10000

Data: PH Spectrum Fit: Mn K Line (Ar escapes) Fit: Mn K Lines

10µm anodes Vc=-619.0 V, Vd=-1.30 kV Ar(75%) - IB(25%) 6 keV X-rays Anticoincidence LLD=10%

7500

613

2

2

y=a/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(c)) /2/b ) + 2 2 2 d/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(f)) /2/b ), r =0.99972 a=3.630E5, b=0.06056, c=257.00, d=9.944E4, f=284.4 σ =2183, σ =2.10E-4, σ =0.09793, σ =1972, σ =0.2830 a b c d f

5000 2

2

y=a/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(c)) /2/b ) + 2 2 2 d/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(f)) /2/b ), r =0.99766 a=4.1683E4, b=0.090958, c=126.37, d=6779.7, f=153.04 2500 σ =448.7, σ =8.0361E-4, σ =0.1374, σ =377.1, σ =1.002 a b c d f

0 0

100

200

300

400

Channel Number Fig. 3. A plot of a ‘‘wall-less’’ pulse height spectrum of a 55Fe X-ray spectrum with the LN fits to the Mn lines and the Ar escape lines superimposed.

the detector. Fig. 3 shows a PH spectrum of the Mn X-rays in a GMSD filled with argon +25% isobutane. The anti-coincidence circuits are in operation with a threshold of 590 eV giving a very clean spectrum containing E5
105 events. Clearly, the energy resolution of the GMSD (855 eV FWHM) is inadequate to resolve the Mn Kb from the Mn Ka peak (separation 590 eV); however, as Fig. 3 shows the sum of two LN distributions fits very well to the data and a further pair of LN distributions fit the Ar escape peaks equally well. Fig. 4 plots the channel positions obtained from the fits against the known energies of the Mn lines. The Ka peaks have the best statistics, so they are used to generate the straight line fit. All four points lie close to the straight line fit. A spectral line has only two innate properties, its amplitude and its position (we assume that any broadening due to atomic processes is negligible). These are represented by the parameters a and c in the formula for the LN distribution (Eq. (1)). The parameter b is the relative SD of the detector LRF (actually the SD of the normal curve in lnðxÞ space) which determines both the width and

skewness of the distribution.   dn a ðlnðxÞ  lnðcÞÞ2 ¼ pffiffiffiffiffiffi exp : dx 2b2 2pbx

ð1Þ

The errors dictated by the statistical noise are evaluated by the Marquart–Levenburg fitting routine and displayed in Fig. 3. (The Marquardt– Levenberg algorithm is used in the implementation in the commercial EasyPlot package [5].) The position errors are 0.097channels (2.2 eV) in the Ka and 0.28ch (6.4 eV) in the Kb. The amplitude errors are 2183 counts (in 3.63
105) for the Ka and 1972 counts (in 99 439) for the Kb. The relative SD of the LRF is assumed constant over the energy span of the Ka and Kb lines (0.59 keV). One can allow the parameter b to fit separately to the Kb line but no significant improvement is seen. For the Ka line b ¼ 0:0606 giving a FWHM (2:36  b  100) of 14.3%. At the escape peaks b ¼ 0:091 giving a FWHM of 21.5% showing the typical increase of the relative LRF width as the X-ray energy decreases.

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

Peak Channel No. (from lognormal fit)

614

Mn Kβ

300

Mn K α

200 Mn Kβ+ Ar esc

1

2

y = +43.557x +11.096, r =1.0000

Mn Kα+ Ar esc

100

0 0

2

4

6

X-ray Line Energy (keV) Fig. 4. A plot of the centroid channel identified by the LN fits (Fig. 3) against the known energies of the X-ray lines.

5. Lognormal fits to high-energy X-ray lines ‘‘Wall-less’’ operation of the GMSD greatly enhances the pulse height spectra obtainable when high-energy X-rays are detected. Fig. 5 shows the spectrum obtained from an 241Am fluorescence X-ray source with the silver target exposed. Fitting and subtracting the background scatter from the housing leaves two very clear groups of X-ray lines—the Ag K lines with their escapes and the Ni/Fe/Cr lines from the stainless steel housing with their escapes. In spite of the very poor statistics, satisfactory fits can be obtained with the LN function as the plot of the peak channel versus the X-ray line energy shows (Fig. 6). The energy resolution (b) is held constant within each of the two groups of lines in the fits of Fig. 5. Measurements of the relative FWHM using the range of X-ray energies available from the fluorescence source (Cu, Rb, Mo and Ag K) lines [4] show that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1061:5 FWHM ¼ þ 5:142 %: ð2Þ Ex This expression shows the typical statistical behaviour with an additional 5.14% (2.3% RMS) which represents (mostly) the fluctuation of the GMSD gain over the active area. The

calibration of the LRF width can be used either to give the starting value of a three parameter fit to each peak or to remove one parameter and fit only the position and amplitude.

6. Application of lognormal fits to detector studies The Mn K lines from the 55Fe radioactive source are routinely used as a convenient test stimulus for X-ray detectors. The ability of the LN fits to resolve the Ka and Kb lines now gives a new level of precision in studying the energy resolution of the GMSD. Fig. 7 compares the plots of the FWHM measured in the conventional way and by means of a LN fit (236sR ) to the two lines as a function of the drift potential (Vd ) in the set-up of Fig. 2 filled with argon +25% isobutane. The characteristic feature of an optimum FWHM [4] (at around Vd ¼ 1200 V) is reproduced in both cases with the LN value approximately 1% lower than the conventional one. This improvement is entirely due to the separation of the Ka and Kb lines which the LN fitting effects. Simulating the two line spectra with two LN distributions of the appropriate relative intensity and calculating the FWHM (conventionally defined) as the resolution of the LN is

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

300

Original PH Spectrum PH spectrum with background subtracted

10µm anodes Vc=-630.1 V, Vd=-2.0 kV

Counts / channel

615

Fit: Argon Escapes of Ni/Fe/Cr Group

Ar(75%) - IB(25%) Ag Fluorecent X-ray Source Anticoincidence LLD at 3.7% of Ag peak

200

2

2

Fit: y=a/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(c)) /2/b ) + 2 2 2 sqrt(d )/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(f)) /2/b ) + 2 g/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(h)) /2/b, 2 r =0.98541 a=553.21, b=0.037071, c=175.77, d=4416.5, f=203.19, g=726.70, h=227.46 2 2 Fit: y=a/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(c)) /2/b ) + 2 2 2 sqrt(d )/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(f)) /2/b ) + 2 g/sqrt(2*pi)/b/x*exp(-(ln(x)-ln(h)) /2/b, 2 r =0.96610 a=1719.8, b=0.069182, c=58.357, d=570.60, f=48.392, g= 432.51, h=69.290

100

0 0

100

200

300

Channel No. Fig. 5. The pulse height spectra obtained with ‘‘wall-less’’ operation from a silver fluorescence source of X-rays. The LN fits to the various groups of X-ray lines are shown after background subtraction. (The X-ray line spectra are not rendered perfectly clean by ‘‘wall-less’’ operation because the strong 241Am X-ray stimulating source produces a continuum of X-rays in addition to the fluorescent lines—hence the need for background subtraction.)

increased, yields the plot seen in Fig. 8. Here the conventional measure of the FWHM is seen to rise to a fairly constant E2% greater than 236sR as the resolution deteriorates above 10% (the approximate separation of the two lines). In the case of an isolated line the conventional definition of the FWHM always exactly equals 236sR : Thus, with the typical energy resolution available from gas counters, the presence of the Kb line routinely causes a significant overestimate of the counter resolution. Being skewed, the peak position (mode) of the LN distribution does not coincide with the centroid, so that even for an isolated line there is a small systematic error in using the peak value. In the case of two lines close together, such as the Mn K lines, the mode shows a small systematic movement as the detector resolution changes (Fig. 8). The degradation of the FWHM seen in Fig. 7 as the drift field increases above the optimum has been analysed elsewhere [4] and shown to be due to an increased skewing of the PH distribution due to

inhomogeneity in the paths followed by the X-raygenerated photoelectrons to the anode. The increased skewing reveals itself in a failure of the LN curve to match the experimental LRF and a systematic error in the position of the Kb line as evaluated by the LN fits. Fig. 9 shows the fitted line positions of the four lines identifiable in the 55 Fe spectrum in an argon gas mixture with Vd ¼ 1200 V and Vd ¼ 3000 V. Apart from the extra gain caused by the higher Vd the fitted Kb line positions are (unlike the Vd ¼ 1200 V case) significantly in error with the line fitted to the Ka positions. Plotting this error as a function of Vd shows (Fig. 10) that beyond a Vd of E1200 V the error increases rapidly as the LRF distorts from the LN shape. A further diagnostic available from the fitting procedure is the relative intensity of the Kb and Ka lines. While the relative intensity is basically defined by the transition probabilities in the Mn atom, absorption in the source and detector window and differential efficiency in the detector

616

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

Peak Channel No. (from lognormal fits)

250 Ag Kβ 200

Ag Kα Ag Kα (esc)

150 2

y = a*x, std-err:0.753, r =0.99994 a=9.1667

100 Ni Kα Fe Kα Cr Kα Fe Kα (esc)

50

0

0

10

20

30

X-ray Line Energy (keV)

Fig. 6. A plot of the centroids of the LN fits to the various peaks in the pulse height spectrum (Fig. 5) plotted against the X-ray line energies emitted from the fluorescence source.

do modulate the value. However, in a given experimental configuration the fraction of the Kb counts should remain a constant fraction of the total. Fig. 11 shows this fraction plotted as a function of Vd : Between 1100 and 1500 V this fraction remains constant (within the statistical errors) at E0.22. It is clear from the data presented in Figs. 7, 10 and 11 that for an argon+25% isobutane gas mixture there is an ideal operating region of 1100 Vo  Vd o1300 V in which the detector is capable of resolving the Mn Ka and Kb lines with excellent energy resolution and proportionality.

7. Quantitation In the fitted parameters of Fig. 3 two points are immediately noticeable: the position errors are very low, and the amplitude errors are high compared with the Poisson errors expected from the amplitudes i.e. 602 for Ka and 100 for Kb (ON). The factor of 3.63 increase in the SD of the fitted Ka means that 13.1 (3.632) times the number of counts must be collected to achieve the same amplitude resolution as would be achieved by a

given number of events in the peak as detected by an ideal detector. The corresponding factor in the Kb case is 39.1. The ratio of the observed variance in the fitted number of counts to the Poisson value (i.e. s2N =N) is known as the excess noise factor (F ) and it represents the factor by which the counting time must be extended to achieve the same resolution in the counts as would be achieved by an ideal detector counting N events. For a pair of closely spaced X-ray lines the excess noise factor (F ) of each line fit is a function of a ¼ sE =dE; (sE ¼ sR E is the energy resolution of the LRF in keV) at the mean line position and dE is the energy spacing between the lines (in keV) and the partition fraction between the two lines (fp ). Since setting up a range of X-ray lines with close (and variable) spacing is a large undertaking it was decided to explore the dependence of F on a and fp by means of a Monte Carlo model. With the energy of the lower line set to 5 keV at which sR ¼ 0:08 (from Eq. (2)) and a variable separation between the lines, LN pulse height distributions were simulated with 105 (N) events in the combined lines divided according to fp (the fraction of the counts in the lower line). The Marquardt–Levenberg fitting procedure was used to evaluate the error in the number of counts in

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

617

FWHM of 5.9 keV X-ray Line (%)

16.0

15.5

15.0

14.5

14.0

13.5

0

1000

2000

3000

-VD (V) No Anticoincidence - Conventional FWHM Definition With Anticoincidence, Lognormal Fits - FWHM = 236σR

Fig. 7. The relative FWHM (%) of the Mn Ka line as measured by the conventional method in a GMSD and also by means of a LN fit to a ‘‘wall-less’’ pulse height spectrum.

2

30 FWHM of simulated pulse height spectrum FW HM = 236 σR Relative error in Kα energy as measured by the mode of the simulated pulse height spectrum

20 FWHM (%)

Relative Error (%)

1

0 10 -1

-2

0

5

10

15

20

0 25

236σ R (%)

Fig. 8. A plot of the FWHM measured for the Mn Ka line in the presence of the Kb line by the conventional method against the LN fit value. The presence of the Kb line causes the conventional method to overestimate the FWHM by a significant amount. The discrepancy between the mode of the pulse height distribution and the correct Ka position is also shown. sR is the relative SD of the LRF (b in Eq. (1)).

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

618

400 X-ray Line Fits - VD = -3000V y = +54.010x1 + 8.6310 X-ray Line Fits - VD = -1200V

Fit Centroid (chs)

300

y = + 42.400x1 + 10.670

Mn KΕ Mn K∆

Straight Lines fitted to K∆ (Ar esc) and K∆ only

200

Mn KE (Ar esc)

100

0

Mn K∆ (Ar esc)

0

2

4

6

Xray Line Energy (keV) Fig. 9. The LN fit centroids of the four lines in the 55Fe spectrum are plotted against the X-ray line energies for two different drift potentials (1200 and –3000 V). At high drift field the distortion of the LRF becomes significant.

0.3

Ratio of Counts in Mn K Lines

Error in Fit to Mn Kβ Line (keV)

0.6

0.4

0.2

0

500

1500

2500

-VD (V)

Fig. 10. A plot of the error induced in the fitted Kb line energy by high drift potentials (from plots such as Fig. 9).

each line (sN ) and F was evaluated using the relation, F ¼ 1 þ s2N =fp N: Fig. 12 shows the prediction of the model for the excess noise factors in two closely spaced lines as a function of a in the case of fp ¼ 0:8; the approximate value for the Mn Ka, Kb. The values of F measured experimentally (Fig. 3) for the Mn lines are superimposed for comparison. The fact that the experimental values are approximately twice as bad as the model values probably reflects the fact that the model can only include the strictly

mean: 0.21583 sdev: 0.00469

0.2

Kβ/(Kα+Kβ)

0.1

0 500

1500

2500 -VD (V)

Fig. 11. The fraction of the fitted counts in the Kb line of the 55 Fe pulse height spectrum as a function of the drift potential in the GMSD.

stochastic effects and not any contributions from such factors as an imperfect match between the experimental LRF and the LN distribution. Fig. 13 illustrates the general behaviour of F for the lower line under the influence of the Poissonian noise as predicted by the model. Generally, F rises steeply with a at all fp and F minimises for all the values at fp ¼ 0:5: As expected, the largest values of F are obtained at low fp (few counts in the line)

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

and high a (poor resolution). The curves for the upper peak are essentially similar if (1  fp ) is plotted on the abscissa, although in some corners of parameter space the asymmetry of the LN distribution produces small differences. Clearly, F tends to infinity as fp tends to zero and F ¼ 1 at fp ¼ 1: The errors in the fitted line energies are very small: for example at a ¼ 0:6; fp ¼ 0:8 (corre-

Excess Noise Factor (F)

1000

Exp. Data: Mn Kβ Exp. Data: Mn Kα Model Kβ Model Kα

100

10

1

0

0.2

0.4

0.6

0.8

619

sponding to the Mn lines) errors of 3.6 and 9.6 eV are predicted by the model which compare with the measured values of 2.2 and 6.4 eV (Fig. 3). Thus one can view the fitting process as one which transfers the position noise of the initial spectrum into amplitude noise in the fitted parameters. The implication of the above analysis is that, over a useful range of situations, the overlapping PH distributions from adjacent X-ray lines can be separated by fitting LN LRFs to the experimental spectra from a GMSD adapted for ‘‘wall-less’’ operation. The extra energy resolution is obtained at the cost of a greatly increased statistical requirement. Taking the a value typical of the Mn lines (0.6) shows that the statistics must be increased by a factor between 5 and 100 to achieve the same amplitude resolution as an ideal detector for 0:1ofp o0:9: Alternatively, for a fixed number of counts the amplitude resolution is degraded by a factor between 2.2 and 10. The line energy resolution is o10 eV in this range of parameters.

1.0

α = σE/δE

Fig. 12. The excess noise factor induced in the fitted amplitude values (counts) of the Mn Ka and Kb lines as a function of the parameter a ¼ sE =dE; as predicted by the Monte Carlo model. The experimental values derived from Fig. 3 are shown for comparison.

8. X-ray fluorescent analysis An example of the possible application of this technique is XFA. In XFA the ultimate requirement is (usually) to resolve the K lines of adjacent

1000 α = 0.8 α = 0.6 α = 0.4 α = σ /δ E = 0.2 E

Excess Noise Factor (F)

Lower Line

100

10

1

0

0.2

0.4

0.6

0.8

1.0

Fraction of Counts in Lower Line (fp) Fig. 13. The predictions of the Monte Carlo model for the behaviour of the excess noise factor (F ) induced in a fitted line intensity as a function of a and the fraction of counts in the line (fp ).

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J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

elements. Examining the separation between the Ka energies of adjacent elements (neglecting the presence of the Kb lines) one finds that it increases 1=2 approximately as EK : As Eq. (2) shows, the energy width of the LRF (sE ) has a similar dependence on Ex so that in the case of adjacent K lines the parameter a ¼ sE =DE which determines the ‘‘separability’’ of lines is approximately constant. Fig. 14 plots a for the case of separating adjacent Ka lines as a function of EK (for the lower line) throughout the periodic table. As can be seen, a is relatively constant at E0.65 for adjacent elements and E0.33 for next-but-one elements. Thus for adjacent K lines the values of F predicted for the fitting process lie just above the a ¼ 0:6 curve in Fig. 13. In other words for situations in which the weaker line contains at least 10% of the counts, one requires to accumulate between 10 and 200 times as many counts as would be required with an ideal detector in order to effect the separation to the same statistical precision.

permits the simple fitting process to be mathematically exact. The benefits can be exemplified by considering the two PH spectra of Fig. 1. Fitting first the two escape lines in Fig. 1 we obtain the following fitting errors from the Marquardt– Levenberg procedure. For the Ka (esc) line in the normal spectrum the error in the centroid (sc ) is 0.77ch in 148.8ch and the amplitude error (sa ) 284.2 counts in 2999 counts. The corresponding errors for the ‘‘wall-less’’ spectrum are sc ¼ 0:40ch (in 148.9) and sa ¼ 79:4 counts (in 2728). The Kb (esc) line shows a similar behaviour with sc ¼ 8:06ch (in 175.4), sa ¼ 123:1 counts (in 409.4) in the normal spectrum compared with sc ¼ 3:05ch (in 179.2) and 64.5 counts (in 461.1). Thus, (approximately) the errors are halved (or better) in the ‘‘wall-less’’ case. In the case of escape peaks the ad hoc assumption made was simply that of a uniform pedestal and the improvement in fitting precision obtained with the ‘‘wall-less’’ data is simply due to the removal of the stochastic noise in this pedestal. However, in the case of the Ka and Kb lines we have a more complex guess to make. The pseudo-background that has to be fitted under the LRFs is now declining from a finite value to zero across the peak. The simplest assumption that can

9. Discussion As noted above, the important property of ‘‘wall-less’’ operation is that the tail-free LRF now 1.0

α for Kα lines of Elements Z and Z+1 α for Kα lines of Elements Z and Z+2

α = σE/δE

0.8

0.6

0.4

0.2

0 0

2

4

6

8

10

Lower X-ray (Kα) Line Energy (keV) Fig. 14. The parameter a for successive pairs of Ka fluorescence lines detected in a GMSD, plotted as a function of the energy of the lower line up to Cu in the periodic table. The values for second neighbours are also plotted.

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

be made is a linear ramp and this is used to give the following fitting errors. For the Ka in the normal spectrum we get sc ¼ 0:58ch (in 291.3) and sa ¼ 880 counts (in 21 751) as compared with sc ¼ 0:44ch (in 292.8) and sa ¼ 643 counts (in 24 816) in the ‘‘wall-less’’ case. Similarly, for the Kb we get (normal) sc ¼ 1:2ch (in 316.4) and sa ¼ 671 counts (in 8170) compared with (‘‘wall-less’’) sc ¼ 1:4ch (in 321.2) and sa ¼ 589 counts (in 5999). Clearly in this case the stochastic errors are little different (since most of the signal is in the lines); however, we can observe significant systematic errors in both the line positions and (more important) the amplitudes. Thus the Ka normal and ‘‘wall-less’’ amplitudes are different by 3065 counts which is nearly four times the stochastic error. Similarly, the Kb amplitudes differ by 2171 counts, or greater than three times the stochastic noise. The centroids of the Ka line similarly differ by nearly three times the stochastic error and those of the Kb lines by over three times the stochastic error. In the case of the normal pulse height spectrum the systematic shifts depend on the ad hoc function assumed for the pseudobackground. In the ‘‘wall-less’’ case this problem does not arise. Thus one sees that for low-energy lines, wallless operation cleans up the stochastic noise introduced by the tails of the LRF while for the highest energy lines, it removes the systematic errors induced by an (otherwise necessary) ad hoc background model. The magnitude of the ‘‘walltails’’ is very much a function of the detector dimensions. A narrow beam (a few millimetres diameter) incident on a detector with large dimensions (say a 50 mm diameter proportional counter) will have only about 1% of the signal in the tails. For narrow counter volumes of a few millimetres (for which the GMSD is particularly useful) the fraction in the tail can approach 50% and ‘‘wall-less’’ operation is essential if useful energy resolution is to be attained. A typical example of such an application is the Wide Angle X-ray Scattering GMSD which can achieve sub-millimetre position resolution simultaneously with good energy resolution [6].

621

10. Conclusions The results described in this report may be summarised as follows. *

*

*

*

*

*

‘‘Wall-less’’ operation of a GMSD with an argon/isobutane gas filling yields (under appropriate bias conditions) a stable point spread function (LRF) in response to an X-ray line. This LRF permits mathematically rigorous, single-step fitting of overlapping X-ray lines without the need for the arbitrary ad hoc manipulations of the data typical of the usual ‘‘spectrum stripping’’ approach to fitting. It also extends the possibility of precision fitting of X-ray lines to small active volumes of gas counters in which wall effects can be excessive. This LRF can be modelled by a LN distribution to a high degree of accuracy which permits fitting of the amplitude and position of the overlapping pulse height distributions generated by closely spaced X-ray lines. This process permits the separation of the Mn Ka and Kb lines emitted by an 55Fe X-ray source and enhances the use of this common test source in the characterisation of GMSDs. Experimental measurements and Monte Carlo modelling show that line energies can be established within a precision of 10 eV for a wide range of practical cases. The signal-tonoise ratio in the amplitudes of the fitted lines degrades as a function of the parameters a (¼ sE =dE—the energy resolution over the line separation) and fp (the fraction of the counts in a pair of lines which is in the line under consideration). The excess noise factor (F ) measures the factor by which the statistics in the data set need to be increased to recover the Poissonian errors. For adjacent K lines in the periodic table the value of a is approximately constant at E0.65. With this value the adjacent lines can be separated by fitting with a penalty of F E10 until the fraction of events in the line (fp ) drops below 0.3 when it rises steeply reaching F E200 at fp ¼ 0:1: Since the GMSD is a high-rate device and is contemplated for use in high-intensity SR

J.E. Bateman / Nuclear Instruments and Methods in Physics Research A 488 (2002) 610–622

622

*

beams, it is anticipated that the demand for extra statistics will not prove a barrier to the application of this technique in many experiments. The separation of X-ray lines by the technique described operates perfectly well in the sub-keV energy region. In this domain the window layer on semiconductor detectors poses a real limitation to their usefulness. A GMSD with a thin plastic window can operate down to energies below the carbon edge at E300 eV.

References [1] A. Oed, Nucl. Instr. and Meth. A 263 (1988) 351. [2] J.E. Bateman, J.F. Connolly, G.E. Derbyshire, D.M. Duxbury, J. Lipp, J.A. Mir, R. Stephenson, J.E. Simmons, E.J. Spill, Rutherford Laboratory Report, RAL-TR-

[3]

[4]

[5] [6]

1999-057 (http://www-dienst.rl.ac.uk/library/1999/tr/raltr1999057.pdf). J.E. Bateman, J.F. Connolly, G.E. Derbyshire, D.M. Duxbury, J. Lipp, J.A. Mir, R. Stephenson, J.E. Simmons, E.J. Spill, R.C. Farrow, B.R. Dobson, A.D. Smith, Gas Microstrip X-ray Detectors for Applications in Synchrotron Radiation Experiments, Rutherford Appleton Laboratory Report, RAL-TR-1999-056 (http://www-dienst.rl.ac.uk/library/1999/tr/raltr-1999056.pdf), Presented at the International Workshop on Micro-Pattern Gas Detectors, Orsay, France, 28–30 June 1999. J.E. Bateman, J.F. Connolly, D.M. Duxbury, G.E. Derbyshire, J.A. Mir, E.J. Spill, R. Stephenson, Energy Resolution in X-ray Detecting Gas Microstrip Detectors, Nucl. Instr. and Meth. A 484 (2002) 384. EasyPlot for Microsoft Windows V 4.0 by Spiral Software. J.E. Bateman, J.F. Connolly, G.E. Derbyshire, D.M. Duxbury, J. Lipp, J.A. Mir, J.E. Simmons, E.J. Spill, R. Stephenson, B.R. Dobson, R.C. Farrow, W.I. Helsby, R. Mutikainen, I. Suni, Nucl. Instr. and Meth. A 477 (2002) 340.