Lineshape of X-ray Detectors Used in PIXE
In the last few years, considerable progress has been made in understanding the processes that give rise to the response function, or lineshape, of Si(Li) and Ge X-ray detectors.
The lineshape is approximately Gaussian. The peak due to a photon energy E is centred at channel
x = a1 + a2E
The Gaussian shape arises from the combination of electronic noise and the Fano statistics of hole-electron pair creation in the semiconductor. The standard deviation of the Gaussian therefore takes the form,
s = (a3 + a4E)0.5
where the constants a3 and a4 represent these two processes
Every such peak corresponding to X-ray energy above 1.836 keV is accompanied by a corresponding peak displaced leftward by a well-defined amount; this arises from the escape of the K X-ray of silicon, and so it is called the escape peak, and is expected to appear at the channel corresponding to energy (E - 1.74) keV. However, many authors have observed that it seems to be displaced by 1.75-1.76 keV. This was for a long time a puzzling observation, because the silicon Ka X-ray energy is very accurately known as 1.740 keV. However we have shown that the apparent displacement is caused by strong low-energy tailing on the escape peak, over and above the normal tailing that is discussed below. This extra "tailing" is due to the high-energy shake-off satellites of the silicon K X-ray, which appear as low-energy satellites on the escape peak.
Non-Gaussian components of the peak
In practice there are significant deviations from the Gaussian form of the main peak, due both to basic electron transport processes and also to imperfections in the fabrication of the detector. It is important that these be described within the overall lineshape function; if they are not so described then there is the possibility that a spectrum fitting code may report these effects as spurious small concentrations of elements that are not in fact present. Most of these effects augment the low energy flank of the Gaussian, and are collectively referred to as "low-energy tailing". The figure above shows a widely used empirical representation of this tailing that assists in least-squares fitting of the spectrum. In addition to the Gaussian G, the features are: an exponential D; a long shelf S; a truncated shelf ST; some authors combine the exponential and the truncated shelf into a single truncated exponential function.
The physical processes responsible for the non-Gaussian components include:
- the escape of Auger electrons and photo-electrons that are created near the front surface through that surface, are basic electron transport processes that cannot be removed by detector design. The intensity of the overall effect increases with decreasing X-ray energy; this effect was suggested by various authors as being the cause of the long, approximately flat shelf-like feature that extends to near zero energy. More detail on this electron transport feature is given in a separate section below. In the same manner the entry of Auger electrons and photo-electrons created within the metal contact introduces artifacts in the spectrum; for a gold contact, this results in a broad hump in the 2 - 3.5 keV region.
- escape of thermalized ionization electrons due to diffusion out of the front surface; these electrons have diffusion lengths of about 0.1 mm, and so again, there is the possibility of charge loss especially for lower energy X-rays which interact near the surface. The contact-silicon interface can be designed to reflect diffusing electrons back into the detector, thus minimizing this effect. Several authors take the view that this effect causes the short flat shelf or truncated exponential tail feature that has been observed in some detectors, but not in all. We will not discuss this feature further because our detectors do not show it.
- especially near the periphery, some of the charge carriers created in an X-ray interaction may follow electric field lines that terminate on the curved surface of the detector instead of to the back face;
- polarization effects, i.e. build-up of charge, have been observed to cause "ghost" or "satellite" peaks; as detector technology has improved, the reports of these effects have diminished;
- trapping of ionization electrons by defects in the region near the surface; the subsequent de-trapping may be sufficiently late relative to the processing time of the amplifier that this trapped charge is excluded from the recorded event;
Electron Transport contribution to lineshape
The presence of a small step effect in the low energy shelf at about 1.8 keV was first demonstrated by our group both in the spectrum of 55Fe and also in monoenergetic photon spectra recorded at the LURE synchrotron in collaboration with the group of M.-C. Lepy.
This step is predicted by our Monte Carlo simulation, despite the relative crudeness of the approach taken there to modeling electron interactions. The shelf feature that runs from about 1.8 keV to the peak energy E represents those events where the K photo-electron silicon escaped wholly or partly from the detector. Similarly, the shelf that runs from about (E - 1.8 keV) to the peak represents full or partial loss of the silicon K Auger electrons. In addition there is a feature where partial loss of both electrons is described.
Dr. Tibor Papp demonstrated the origin of these features by performing experiments on the energy spectra of photo-electrons emitted from silicon slabs bombarded by X-rays. His PE spectra correspond to the electron transport shelves in Si(Li) detector spectra.
Effects of natural X-ray line width
The brief discussion above has assumed so far that the lineshape arises entirely from the detector's properties. This is a fairly good approximation, but certainly not a perfect one. Any characteristic X-ray line emitted by an excited atom has a natural distribution in energy that is Lorentzian, and this is characterized by a half-width that is inversely related to the lifetime of the excited state. As a result, the lineshape of a detector is the convolution of the lineshape described above with a Lorentzian function. The main feature, which we have approximated as Gaussian, is thus actually Voigtian. The long shallow high- and low-energy tails of a Voigtian contribute to what is often thought (wrongly) to be pure continuum background between peaks, and so neglect of natural lineshape affects not only peaks but also background when spectra are fitted.
The GUPIX software package contains a table of natural widths and allows the user to adopt a Voigtian description in the overall description of the lineshapes in the spectrum.