Systematic Errors

Systematic error is a series of errors in accuracy that are consistent in a certain direction, or over time.  In general, systematic errors cause a bias in measurements that result in an ‘offset’ of measured data points from the ‘true’ value of the quantity being measured.  Systematic errors are commonly harder to detect than random errors, and fall under several sub-categories, as follows:

Instrumental errors

• Thermal instability of instrument – Linear variation in temperatures in the laboratory can have systematic effects on the d-spacing of the diffractor crystals in the WDS spectrometers, potentially affecting the accuracy of the analysis.  Thermal changes ca also affect the gas pressure of the Ar-CH₄ gas flowing through the flow-proportional counters, thereby changing the efficiency of the detector, and changing the accuracy and precision of the data.  Temperature variations can also cause significant drift in Z-dimensional control of the stage, which in turn can lead to the optical defocussing of the sample and subsequent decreasing of the measured X-ray intensities.
  
  
• Less than ideal matrix correction – If an inappropriate standard has been selected for a given element, the resultant matrix correction may not be optimal, leading to an erroneous calculation of concentration from a measured X-ray intensity.
  
  
• Charging of sample or standard – If the carbon-coat is insufficient to permit proper grounding of the sample, the measured X-ray intensities will be systematically reduced for all elements being analyzed, and will, therefore, affect the accuracy of the analyses.
  
  
• Beam damage – As discussed above, the electron beam can damage samples, depending on the composition of the samples, for example, the reduction of carbonates, and alkali migration in Na- and K-rich samples.  In such cases, reduction of the X-ray signal with increasing time results in lower average intensities and calculated concentrations.  Such phenomena can be corrected using time-dependent-intensity (TDI) corrections, whereby theoretical ‘temporal titration’ curves are calculated to extrapolate back to what an X-ray intensity would have been at measurement-time = 0.
  
  
• Peak shift – Over long analysis times, the peak positions of X-rays can drift or shift slightly; in order to systematize the errors induced by such drift, users collect X-ray data by analysing standards as though they are unknowns.  If the concentrations yielded for an element(s) in the standard vary significantly over time, one may deduce that the peak-position(s) has drifted and can correct for the loss in signal intensities.
  
  
• Beam current stability – The stability of the beam current can change over time for a variety of reasons, some of which are out of the user’s control, for example, the changing of a heating/cooling cycle by a building’s climate-control system.  Commonly, this problem is fixed (or at least tracked) by the software used to control the microprobe, as the X-ray intensities are expressed in units of counts/second/unit area (beam spot-size)/nano-amp, i.e., the counts are normalized based on a live measurement of the beam current.
  
  
• Spectrometer reproducibility – As the moving parts of a spectrometer age, they will experience mechanical damage owing to the repetition of duty-cycles.  As wear-and-tear takes its toll, the goniometers may drive to slightly different positions from the ‘ideal’ positions, potentially resulting in loss of signal in the long-term, thereby affecting both the accuracy and precision unfavourably.
  
  
• Pulse height adjustment – Periodically, the high-voltage and gains of the detectors must be adjusted owing to the aging of electronic components – this results in a change of measured intensities as signals are electronically amplified and ‘reshaped’, hopefully for the better, but this process still can introduce errors from time to time.
  
  
• Dead time – “Dead time is the time interval after a photon enters the detector during which the system cannot respond to another pulse…In order to obtain accurate values of the peak intensities, corrections must be applied for the dead time associated with the measurement of an X-ray….In the WDS spectrometer, the dead-time correction is applied after the intensity is measured.  For the proportional counter used in WDS, [the following equation is used to correct for dead time, and this correction must be used]…before that intensity is used to form a K-ratio:  N = N’ / (1 – τN’), where N’ is the measured count rate, N is the true count rate which we wish to calculate, and τ is the dead time in seconds….” (Goldstein et al. 1992).
  
  
• Spectrometer focus -  We have already discussed this subject above with respect to baseplate alignment, as the baseplate alignment does affect the focus of the spectrometer itself.  If one of the steel belts inside the spectrometer stretches too much, or the tensioner wire than holds the weight of the detector stretches, the X-rays from the diffractor crystal will not focus properly through the slit in front of the detector, thus affecting both accuracy and precision of the intensity data.
  
  
• Diffractor crystal deterioration – Commonly, the diffractor crystals become scratched, cracked, contaminated and suffer beam damage over the long-term.  These types of issues reduce the crystal’s ability to diffract incident X-rays properly, which can lead to reduced intensities, and reproducibility of measurements (usually associated with a few elements, systematically, that have very similar Bragg angles/L-values).
  
  
• Filament change -  Given that every gun-filament is different in physical shape, responds differently to being heated by the electricity applied to it, ages differently and has slightly different emission characteristics and therefore need to be individually aligned, errors in both accuracy and precision can occur as the filament ages.  If a filament is changed in the middle of an analytical run, it is wise to recalibrate the primary standards, as intensity counts commonly improve with a new, well-aligned filament.
  
  
• Window/crystal/detector/column contamination – As the microprobe is used and samples are changed regularly, there are opportunities for contamination to enter the main chamber, and to contaminate Be-windows, diffractor crystals, detector windows, and the column itself.  One of the most common contaminants is finger oils from users who handle their samples or sample-holders without wearing rubber gloves.  These oils evaporate in the vacuum of the microprobe, and re-condense on the inside of the probe parts.  To minimize these effects, the column needs to be cleaned about twice a year.  Contamination within the column can affect the aim of the beam and focussing qualities of the beam, leading to errors in both accuracy and precision with time.