Spectral Redistribution: Processes, Models and Parameters

A spectral redistribution function describes the probability of an instrument response in each pulse-height channel to photons of of any particular energy. In this section we discuss the spectral redistribution function and its models.

To guide this discussion, a typical spectral redistribution function for a front-illuminated CCD (the response of a flight-like detector to a radioactive ⁵⁵Fe source) is shown in Figure  4.3. The main components of the redistribution function are indicated: in addition to the two primary photopeaks at 5.9 and 6.4 keV (from the K$\alpha$ and K$\beta$ lines of the Manganese daughter of the radioactive decay), one sees Si-K escape and flourescence peaks, an assymetrical shoulder onn the low-energy side of the photopeaks, a low-energy continuum, and an upturn (tail) at very low energies. (The Manganses L lines from the source are also visible). The off-nominal features (everything except the primary photopeaks) together contribute about 2% of the integrated area under the response function at this energy. The task of modelling the spectral response function is to predict this curve in sufficient detail to meet AXAF calibration requirements.

  

Figure 4.3: Principal components of the CCD spectral response function. The (asymmetric) primary photopeaks, silicon K-escape and fluorescence features, and the extended low-energy tail are indicated. Note that the upturn in the tail at very low energies (E< 400 eV) is not due to electronic noise, but is a characteristic of the detector itself. Note that the ordinate is logarithmic.

We divide our discussion into three parts: the energy scale (or ``gain''); the spectral resolution (essentially the width of the photopeak); and the off-nominal features (everything else).

Consider first the first moment of the spectral redistribution function, that is, the variation of the mean response (essentially the centroid of the primary photopeak) with incident energy. To high accuracy, the output of an ACIS front-illuminated detector (and its associated detector electronics) is highly linear. This linear relationship is sometimes referred to as the energy scale or ``gain'' relation. While the linearity of the energy scale rests on the well-understood physics discussed briefly below, (not to mention near-ideal detectors and meticulous electrical engineering!), the program of the energy scale calibration is the empirical one of determining the slope and intercept of this relationship for each of the forty CCD output nodes as a function of focal plane and detector electronics temperatures. Results of this determination are presented in sections  4.3.1 and  4.8.2. It is worth noting, that, as is shown in section  4.3.1, the back-illuminated detectors exhibit significant non-linearity that we have not yet modelled in physical terms.

The second moment of the redistribution function (the detector's ``spectral resolution'', as measured by the width of the photopeak) is, in the case of front-illuminated detectors, equally amenable to simple modelling. In the mean, the quantity of charge liberated by an X-ray interacting in the CCD is proportional to the energy of the incident X-ray:  

$$N_{e} = \frac{E}{w}$$ (13)

where N_e is the number electrons liberated, E is the photon energy and $w \approx 3.7 eV/e^{-}$, the mean ionization energy per electron-hole pair, is a function of the temperature of the silicon. This proportionality, of course, is a sine qua non of the energy scale linearity discussed above.

The two most important factors affecting the spectral resolution of the CCD are readout noise and the stochastic nature of the ionization process. The readout noise characterizes the accuracy with which one can measure the quantity of charge deposited in any given pixel. For ACIS detectors, the readout noise is found to be a normally distributed, zero mean random deviate (see, e.g.,  [Pivovaroff et al.1996]) of standard deviation $\sigma_{r} \sim$ 2-3 electrons, RMS, per read. The variance $\sigma_{N}^{2}$on the charge liberated N_e, is  

$$\sigma_{N}^{2} = F \times N_{e} = F \times \frac{E}{w}$$ (14)

where F, the Fano factor, has the value F = 0.115, and is characteristic of crystalline silicon. Using the room-temperature measurement of Scholze et al (1996)  [Scholze et al.1996], viz, $w = 3.64 \pm 0.03 eV/e^{-}$ and the temperature dependence reported by Canali et al (1972)  [Canali et al.1972], who find that w increases linearly with temperature between 77K to 300 K, we expect w=3.71 eV/e^- at the CCD operating temperature of T=153K.

The effect of these two processes alone, then, is that the primary photopeak produced in response to a beam of monochromatic incident photons of energy E will be a Gaussian distribution with mean proportional to E and standard deviation, expressed on the input energy scale, given by  

$$\sigma_{E} = \sqrt{(w\sigma_{r})^{2} + (F \times w \times E)}$$ (15)

The form of equation  4.3 is an accurate model of the spectral resolution of front-illumianted ACIS detectors, as is shown in section  4.3.1. This accuracy is the more remarkable in light of the fact that equation  4.3 neglects the summation of multiple pixels to compute the event amplitude (see below). The insensitivity of the spectral resolution to multi-pixel summation arises because the readout noise $\sigma_{r}$ is low enough, and the depletion depth large enough, that $\sigma_{N}^{2} \gg \sigma_{r}^{2}$ at all energies where a significant fraction of events must be summed. Note also that equation  4.3 implies that the the spectral resolution can be computed from i) a measurement of the readout noise and ii) knowledge of w and F. As discussed in section  4.3.1, the best fit model parameters differ significantly from the values expected on this basis. Finally, the resolution of the back-illuminated devices deviates considerably from the predictions of equation  4.3. This deviation has yet to be modelled in detail.

As figure  4.3 shows, a number of processes complicate the CCD spectral redistribution function. In addition to the K-escape and fluorescence phenomena common to all photoelectric X-ray detectors, the CCD response function is influenced by incomplete charge collection and, to a more limited extent, by photoelectric interactions in the gates and channel stops. We describe these briefly here.

The amplitude of the K-escape peak may be modelled reasonably accurately through straightforward simulation of photon transport in the detector. With somewhat less accuracy, the (lower ) amplitude of the silicon flourescence line can also be modelled; see section  4.3.2, below. The locations and widths of these peaks are also modelled in the obvious way.

In practice, then our model of the portion of the spectral redistribution function discussed to this point consists of a sum, for each incident energy, of a set of Gaussian distributions of fixed relative centroid locations, with absolute locations, relative amplitudes and widths constrained by fits to measurements. Together, these components account for all but about 1% of the spectral redistribution function for the front-illuminated devices.

Finally, we come to the remaining off-nominal features: the shoulder, low-energy continuum and low-energy tail. The physical origin of these features is believed to be well-understood, but their amplitude is not easily desribed analytically. Although the analysis presented in this report neglects these features, we review the the important physical processes here.

In spite of the relatively small volume occupied by initial the photoelectric charge cloud, there is a non-zero probability that charge can be shared among multiple pixels, because events can occur near pixel boundaries. Although the event amplitude computation algorithm sums together pixels in which signficant charge is detected, a pixel is only included in the sum if it exceeds a so-called split-event threshold, typically set at 15 electrons (equivalent to about 55 eV). The split-event threshold excludes empty (but noisy) pixels from the event amplitude summation, but has the undesirable consequence of excluding some bonafide photo-ionization from the sum. As a result, the spectral redistribution function can contain a low-energy shoulder.

Moreover, some charge from events which interact below the depletion region (in front-illuminated devices) will diffuse to the back of the device and will never be collected in the charge transfer channel. Events of this sort can generally be identified because the charge that is collected must diffuse large distances, and hence can occupy several pixels (more than 3). Charge can also, in, principle, be lost to the channel stops and to the gate structure. These processes produce a ``low-energy continuum'' tail on the spectral response function.

Part of this continuum is produced by photons which interact in other regions of the detector outside of the depletion region. In particular, interactions occuring in the channel stops and gate structure can be detected in the depeletion region. These phenomena are especially important at the low-energy end of the spectral redistribution function,