Rapid methods for the calibration of solid-state detectors

A complementary pair of rapid methods for the energy calibration of solid-state detectors have been developed. Each method requires only a single measurement from either (i) a glass sample containing elements chosen to produce fluorescence lines over a suitable energy range when exposed to a polychromatic beam of X-rays, or (ii) a powder diffraction standard in the presence of diffraction slits. The fluorescent glass method has the advantage of allowing simultaneous energy calibration of a number of detectors without requiring diffraction slits. There is the potential for the glass material to be incorporated into virtually any sample holder to allow a continuous in situ calibration. Complementary observations of a powder diffraction standard allow simultaneous calibration of diffraction 2θ and monitoring of the detector for drift.

1. Introduction

Energy-sensitive solid-state detectors are in routine use both in the laboratory and at synchrotron sources for a variety of applications including energy-dispersive powder diffraction (Clark, 1996) and X-ray fluorescence measurements (Derbyshire et al., 1999). These detectors produce an output voltage pulse proportional to a detected X-ray energy which is then usually amplified and fed to a multichannel analyser where it is typically mapped onto 2¹⁰–2¹² channels. Before analysing the data it is usually necessary to convert from channels to energy [and then possibly on to d-spacing (Cole, 1970)]. To do this a calibration curve relating channels to energy is required. An energy calibration curve of the form

where E is the energy, c is the channel number and a₀, a₁ and a₂ are appropriate coefficients, is usually sufficient, although in most cases the quadratic term is found to be unnecessary. There are traditionally two methods used to determine these coefficients, as follows.

(i) X-rays or gamma-rays of known energy are shone into the detector (Buras et al., 1989). A series of different X-ray or gamma-ray energies is used to cover the range of energies in use. The position of the resulting peaks (in channels) is determined and a list of channel numbers and related energies is produced. Least-squares estimates of the energy-calibration-curve coefficients are then determined from these values. Gamma-rays are usually produced by a suitable radioactive source while X-rays are usually produced as fluorescence from suitable elements excited by gamma-radiation.

(ii) Multiple orders of a reflection from the same set of lattice planes of an oriented single crystal are recorded (Skelton, 1977). Knowing the appropriate crystallographic data and the diffraction angle, a list of energies and channel numbers can be produced and least-squares estimates of the calibration-curve coefficients can be determined as above.

The main problem with the gamma-stimulated X-ray method is that each element usually has to be selected separately and the detector exposed to each set of fluorescence lines in series. This is generally time-consuming and it can become extremely tedious when multi-element detectors are in use because for many detector geometries each element of the detector has to be calibrated separately. For example, on station 16.4 of the SRS Daresbury, calibration of a single energy for a single detector takes 10–15 min. For six energies and three detectors a full calibration can take up to a whole day. Also, the range of X-ray energies available from commercially available sources is limited to below 60 keV. Gamma sources giving only a few lines suffer from the same problem as the above X-ray method, while sources containing multiple lines pose the problem of identifying the correct lines and accurately measuring their positions when they may well be overlapped with other lines. The Skelton method relies on having a suitably oriented single crystal and a suitable post-sample slit system to allow accurate diffraction data to be collected, and having the detector and post-sample slits at a suitable diffraction angle for making the measurement. Quite often these conditions cannot be met, either due to a lack of slits, e.g. when making spectroscopic measurements, or due to geometric restrictions that limit the placement of the crystal, the slits and the detector, e.g. when using multiple-element detectors.

2. Experimental

We have developed a method of calibrating the whole energy range of a number of detectors simultaneously. It is based upon a method recently developed for calibrating area X-ray detectors (Moy et al., 1996). A glass containing selected elements is held in a polychromatic beam of X-rays and the fluorescence lines from the selected elements are used for the energy calibration. The energy range of interest for our experiments was 30–120 keV so elements with suitable fluorescence were selected to cover this range (Table 1). The health and safety implications of including radioactive materials led us to produce two glasses, one containing uranium, the other not. The glasses were made by mixing appropriate quantities of the relevant oxides, carbonates or, in the case of uranium, acetate, together with a glass-making flux (Table 1). This mixture was heated to 1423 K over one hour and then held at that temperature for another hour before the liquid was quenched to glass by pouring the molten mixture into a suitable mold. This procedure was found to give a clear glass with no diffraction lines. Fig. 1 shows a spectrum collected from the uranium-containing glass on station 16.4 (Clark, 1996) of the Daresbury synchrotron radiation source. The data-acquisition time of 15 min for the complete calibration of three detectors compares very favorably with the gamma-source procedure described above. The spectra are seen to be composed entirely of the expected fluorescence lines and escape peaks with no contribution from diffraction lines caused by precipitation of crystalline compounds during the glass-making process. Fig. 2 shows the linearity of the fit obtained between channels and energy. This gave an energy calibration curve with the following parameters: a₀ = 6.22 (6), a₁ = 0.02703 (8) and a₂ = 5 (2) × 10⁻⁸.

Table 1
A list of the characteristic fluorescence lines, their positions and the proportions of materials used to produce both the uranium-containing and non-uranium-containing glasses

	Peak position			Glass with U	Glass with no U
Fluorescence line	(channel)	(keV)	Added as	(wt%)	(wt%)
Ge Kα1,2	139.2	9.877	GeO2	13.6	13.6
Ge Kβ1,3	179.9	10.981
Mo Kα1,2	418.4	17.444	MoO2	3.6	4.9
Mo Kβ2	495.5	19.960
Sn Kα2	696.1	25.044	SnO2	3.1	3.9
Sn Kα1	704.6	25.272
Sn Kβ1,3	822.3	28.474
Sn Kβ2	844.8	29.122
Ba Kα2	945.2	31.817	BaCO3	3.0	3.9
Ba Kα1	959.0	32.194
Ba Kβ1,3	1112.0	36.353
Ba Kβ2	1143.1	37.270
W Kα2	1906.6	57.982	WO3	4.6	5.8
W Kα1	1955.6	59.312
W Kβ1,3	2243.8	67.146
W Kβ2	2311.2	69.100
Pb Kα2	2451.0	72.804	PbO2	4.4	5.8
Pb Kα1	2530.6	74.969
Pb Kβ1,3	2894.9	84.774
Pb Kβ2	2986.6	87.367
U Kα2	3252.6	94.659	UO2(CH3.COO)2	19.9
U Kα1	3392.5	98.440
U Kβ3	3825.1	110.424
U Kβ1	3849.3	111.303
			Li2CO3	17.8	23.3
			B2O3	30.0	38.8

Figure 1 Energy-dispersive pattern collected from the uranium-containing glass using a polychromatic beam of X-rays to stimulate the fluorescence. The Pulser peak was added electronically as an aid to checking the stability of the detector electronics. The unassigned peaks at low energy are due to escape events (Stuan Olsen, 1974).

Figure 2 Result of the calculation of polynomial coefficients that best describe the fit between channel number and energy for the fluorescence lines present in Fig. 1.

A powder diffraction pattern was collected shortly after this calibration from a sample of NBS 640b silicon, which serves as a detector angle calibration standard. Table 2 contains a list of the observed and calculated peak positions obtained from the refinement of this particular detector angle (7.631° 2θ) and unit cell [a = 5.4325 (5)]. The excellent agreement between these values over the entire energy range observed indicates the quality of the detector energy calibration. This is in contrast to energy calibrations limited to <60 keV based on commercial fluorescence sources. These limited calibrations introduce errors up to 2 keV at 100 keV. When powder diffraction standards are routinely used to determine the detector angle, as at the SRS Daresbury, then a complementary method of detector energy calibration monitoring can be used. If one assumes the cell parameters of the diffraction standard are well enough known, in principle any collection of indexable reflections can be used to solve for the detector energy response polynomial over the energy range of observed reflections. One needs at least as many reflections as the number of parameters a₀, a₁, a₂. For example, the ten Si reflections in Table 2 were used to calculate the following values: a₀ = 6.13 (6), a₁ = 0.02719 (5) and a₂ = 1 (1) × 10⁻⁸.

Table 2
Values of the calculated and observed peak positions for a sample of NBS 640b silicon used to determine the diffraction angle

The refined value of the diffraction angle was 7.634° 2θ.

H	K	L	Observed peak positions (keV)	Calculated peak positions (keV)	Difference
1	1	1	29.71	29.70	0.0087
2	2	0	48.49	48.50	−0.014
3	1	1	56.89	56.88	0.0113
4	0	0	68.60	68.59	0.008
3	3	1	74.76	74.75	0.0132
4	2	2	84.02	84.01	0.0045
5	1	1	89.11	89.11	0.0055
4	4	0	97.02	97.01	0.0151
5	3	1	101.48	101.45	0.0292
6	2	0	108.39	108.46	−0.0649

The a₂ parameter almost vanishes in the powder-diffraction-based calibration as it did in the fluorescent-glass-based calibration. Both calibrations of the same detector appear close to linear with agreement between the coefficient values within twice the standard error. The above energy calibration based on the Si reflections used the fluorescent glass calibration to establish trial Si peak energies from which the detector angle, 2θ, was calculated. Alternately, if the number of reflections exceeds the number of detector energy response parameters, it is possible to also solve for 2θ. Naturally the recovered value of 2θ would be based upon less information in a refinement of four parameters from ten observations than in the case of Table 2 when all ten silicon reflections are used to refine only 2θ. Likewise the fluorescent glass provides 24 lines over the energy range of interest. Refining these lines for detector response gives values based on more information than found in ten Si lines and is independent of any detector 2θ considerations. Thus the powder diffraction method to find both 2θ and detector response serves best as a backup procedure to the combined use of the fluorescent glass for energy response calibration and the diffraction standard for 2θ calibration.

Even though Si diffraction produces only ten observable lines in this energy range, refining them can provide both 2θ and a quick calibration of the detector energy response. The 2θ values recovered by this method are less stable than if all lines are used to refine only 2θ, and therefore the use of the silicon or other diffraction standard as the only calibrant for both energy and 2θ is recommended only for situations where no other method is available. However, if 2θ is known from an energy calibration over a limited range then the energy parameters recovered over an extended range from a diffraction standard are very stable. We have found the powder-diffraction-based method to be particularly suited to recovering extended energy calibrations (beyond the 60 keV provided by our radioactive dial source) for data sets collected before the advent of the fluorescent glass method. In practice the radioactive dial source calibration at <60 keV allows calibration of 2θ from reflections observed at <60 keV. Using this 2θ, one may then refine all the lines to retrieve an energy calibration of the detector over the whole range for which silicon reflections are observed. Another potential reason for use of the lower-precision powder diffraction standard method is that detector 2θ often needs adjustment during an experiment. Each time a precise new 2θ is determined in the normal way from all the reflections, a quick check that the detector energy response remains unchanged can also be provided. This check can be performed either by separate refinement for 2θ with the detector parameters or else by examining the residuals on the refinement of 2θ alone as a function of energy. Changes in the detector parameters generally cause anomalous systematic patterns in the residuals as a function of energy. Any suggestion of detector drift can be investigated further with recourse to the fluorescent glass method. Spreadsheets for performing both sorts of calibrations may be retrieved from the CCP14 web site (http://www.ccp14.ac.uk/projects/glass-calib/ ).

3. Conclusions

We have presented a fluorescent glass method of calibrating multiple energy-sensitive detectors simultaneously. It allows the calibration lines to be selected to match the energy range of the experiment and eliminates the problems and expense associated with the use of radioactive sources. As a cross check on this method, the range of reflections observed from powder diffraction standards in excess of those needed for 2θ refinement can be routinely used as a complementary energy calibration monitor.