Spectral Resolution and Energy Efficiency for FOXSI III Silicon Detectors

Lance Davis and Connor O'Brien

Spring 2018

Introduction

FOXSI-3 is the third flight of the Focusing Optics X-ray Solar Imager (FOXSI) sounding rocket, which will take place on August 21st, 2018. The goal of the sounding rocket is to image the sun in the soft x-ray regime of 4 to 15 keV in order to probe the source of coronal heating and the mechanics of solar microflares. FOXSI flights 1 and 2 used exclusively silicon x-ray detectors, whereas this latest flight has added new, fine-pitch Cadmium-Tellurium (CdTe) detectors as well. While previous flights have operated at -20 °C, these new detectors have a higher operating temperature of -10°C, which will affect the performance of the older silicon detectors. The goal of our project is to measure the spectral resolution and energy efficiency of FOXSI silicon detectors four and five at -10 °C in order to assess how this new operating temperature affects the performance of the detectors.

Theory

The first half of our experiment involved calculating the spectral resolution of the detectors. Spectral resolution is the broadening of a monoenergetic source into a gaussian due to instrumental effects, and is also known as energy resolution. This resolution was probed using spectral lines from radioactive sources. When the flux of such a source is measured by the detector, statistical noise causes the measured peak to widen into a Poisson distribution about the energy peak, which can be approximated as a Gaussian for large N (such as the number of incident photons observed by one of the detectors), as shown below.

An illustration of the incident monoenergetic peak and the resultant Gaussian distribution at energy H. The Full Width Half Maximum is the width of the peak at half of its maximum value, and is the standard deviation of the distribution.

The equation of a Gaussian distribution is given by the function

(1)

where P is the height of the peak at energy H, E is the energy of the incident photon, and is the standard deviation.

The width of this Gaussian is the energy resolution of the detector at the peak energy. We numerically define the energy resolution of our detectors using the relation

(2)

where FWHM is the full-width half-maximum; at half-maximum, the full width of a Gaussian distribution is equal to 2.35[4]. Usually, the variance of this distribution σE2 would be proportional to the variance in the number σn2 of charge carrier pairs produced at a given energy; σE is related to σn by E=wσn where w, which is material specific, is the average energy required to create a charge carrier pair. In Poisson statistics, n = sqrt(N) where N is the number of charge pairs produced; N can be calculated by N=E/w [4]. For silicon, w = 0.00366 keV [8].

However, the processes that give rise to each individual charge carrier pair in the detector are not independent; therefore we cannot describe the total number of charge carrier pairs with Poisson statistics. The departure of observed statistical noise from pure Poisson statistics is described by the Fano Factor, which is given by the relation

(5)

where Nfano is the fano noise given by Equation 4, r is the energy-independent electronic noise, and f1 is a factor that describes the energy-dependent noise f1E [11]. We assume the energy dependence of the energy-dependent noise is linear because the amount of charge carriers liberated by a photon increases approximately linearly at the 4-15 keV energy scale. For our setup, r mainly takes into account thermal noise and power line noise, while f1 mainly describes the shot noise observable in the system.

After identifying the energy resolution of the detectors, we then wished to measure their energy efficiencies. The energy efficiency is defined as the ratio of the number of photons that the detector actually measures to the number of photons incident on the detector, and can be energy-dependent. This is given by the relation

(6).

This relation assumes that each recorded event was a single photon. The X-ray source for this experiment was beamline 3.3.2 at the ALS. This beam was formed by redirecting continuum radiation created by synchrotron radiation [9]. Synchrotron radiation is emitted when particles are accelerated in a curved path.

Because our silicon detector counts single photons and has a thickness of 500 μm, literature suggests that the energy efficiency between 6 to 10 keV will be near unity [6]. Our detectors use a threshold energy of 4 keV, which is used to prevent the detectors from being flooded with low energy photons and only allowing photons within the energy range of FOXSI 3’s science goals to be detected. Because of the spectral resolution of the fast pulse shaper that precedes the discriminator which form the threshold circuit, the resulting efficiency drop-off resembles an error function centered at 4 keV. This experiment sought to characterize the behavior of this threshold above 4 keV.

Experimental Setup

The silicon detectors used are made of an n-type silicon wafers onto which acceptor doped (p+) and donor doped (n-) strips are placed orthogonally onto opposite sides. By placing the strips orthogonally, each crossing forms a pixel in an image. On both the p+ or the n- side, there are 128 strips with a pitch of 75 μm. Each side of the detector is divided into two regions, called ASICs; each ASIC has 64 strips. For this project we focused on the p+ side, which has better resolution.

Diagram of FOXSI silicon detector.

We used detectors 4 and 5 (numbering used internally to track detectors; number based on position in electronics board) for gauging the spectral resolution. These detectors were housed in the same electronics board that will be flown. This setup was encased by an aluminized mylar, Faraday cage material, which insulated the detectors to help maintain the desired -10℃ operating temperature. The detectors were cooled using cold nitrogen gas. The flow of nitrogen gas into the enclosure was regulated by the temperature control unit which maintained the specified temperature of -10±1℃.

Our silicon detectors measure individual photons. When a single energetic photon is incident upon the detector, electron-hole pairs are produced. A bias voltage of 200 V is applied across the silicon detector to separate the electron-hole pairs. This process creates a current proportional to the amount of electron-hole pairs. This current is measured by the electronic board. The data is then digitized into discrete ADC bins. The ADC value is then sent to an FPGA which sends the data to a computer interface where it is written to file to be used in later analysis.

Sealed radioactive sources, Am-241 and Fe-55, and fluorescence from metal foils of Cu and Ni, were used to measure the spectral resolution. The metal foils, secured on top of the radioactive Am-241 source, gave out the characteristic lines of the metal foil through X-ray fluorescence. The spectral lines produced by these four sources which are used in gain calibration are given in the table below. Data for each source were collected at -10℃ until a statistically significant number of counts were obtained for each strip.

Table of X-ray sources used and the spectral lines used for gain calibration from each source.

The next part of this project was the energy efficiency measurement. A silicon drift detector with an assumed energy efficiency [6], as well as the flux observed in the continuum of the beamline X-ray source, were used as references for calculating the incoming flux. Separately, this detector and our detector were illuminated by a monoenergetic X-ray source, which was beamline 3.3.2 at the ALS; for more details, see [9]. The continuum spectra in the range of 4 to 20 keV was on the order of 109 photons/s/mm2. This amount of flux would saturate our detector, meaning the detector no longer counts single photon events. The flux was reduced by passing the beamline through a 2x2 μm slit and when needed, a 60 or 100 μm aluminum foil to attenuate the flux to the order of 103 to 104 photons/s/mm2, which our detector could measure reliably. This reduced flux was measured by both detectors. The air gap between the detector and the beamline was roughly 15 cm.

Flux measurements were made at 4.75, 5, 5.5, 6, 7, 8, 9, and 10 keV. Flux below 4.75 keV began to enter the falling tail of the continuum spectra, as well as the air gap absorbing a large percentage of incoming photons [10]; statistics were becoming poor below 4.75 keV. Above 10 keV, a smaller fraction of higher energy photons were being absorbed by the aluminum than lower energy photons, causing a higher incoming flux; the detector started to saturate above 10 keV. Only detector 4 was brought to the ALS for testing.

Incoming flux generated by beamline 3.3.2 at the ALS. Plot shows energy (x-axis) vs intensity (y-axis).

Spectral Resolution

The data for the spectral resolution were analyzed using IDL analysis code made for the FOXSI mission, parts of which were modified for our purposes. Histograms for data observed by each of the 64 strips from each of the four ASICs were made. These histograms plotted ADC bins vs counts.

Histogram of Fe-55 for Detector 4 ASIC 2 showing counts (y-axis) vs ADC bin (x-axis).

Next, an algorithm was used to fit a Gaussian curve in order to find the peaks of the spectral lines used in this experiment. The ADC bin where the peaks occurred was assigned the energy of that peak. A plot of ADC peaks vs energy is then made, where a quadratic function was fit to the data points. This function is the gain calibration curve. Using this curve, each ADC bin was calibrated to a corresponding energy bin. At this point, each strip was analyzed; if the ratio of the observed conversion from ADC bin to predicted conversion given by the gain calibration curve was greater than 5% away from unity, the strip was discarded from further analysis; in total, 5/128 strips from detector 4 and 11/128 strips from detector 5 were discarded.

Plots of spectral line peaks (x-axis) vs ADC bins (y-axis) for the first four strips. The blue curve is the fit gain calibration curve.

Ratios of data points to the fit. The top right plot, Channel 1, was discarded as it had ratios exceeding 5% away from unity.

The FWHMs of the peaks, now in units of energy, were then calculated by fitting a Gaussian curve to each peak. For the Ni and Cu lines, the Am-241 spectrum was scaled to match the intensity of high energy lines, 20.6 keV, 26.3 keV, and 59.6 keV, which were much less affected by the absorption in Ni and Cu, seen in the Ni and Cu spectra. The background Am-241 continuum seen in the spectra collected for the Ni and Cu foils was accounted for by subtracting the scaled Am-241 spectrum. The FWHM values for Ni and Cu were then calculated. Note that this is a crude approximation, as we have not accounted for how the foils would affect the energy emission of Am-241 in the range of 6 to 10 keV.

Gaussian fit (curved line) of the Am-241, 13.9 keV spectral line. The histogram shows the number of counts in each energy bin.

The fit minimized χ2; the errors for the FWHM were found by subtracting the FWHM value found at χ2min+1 by the FWHM value found at χ2min. The resolutions were plotted as a function of energy. The red line shows the best fit to Equation 5; the blue dotted lines show the error in this fit. The parameters for the fit are given in the table below.

Spectral resolution as a function of energy for ASICs 2 and 3 on Detectors 4 and 5. The red line shows the best fit to Equation 5; the blue dotted lines show the error in this fit.

In comparison to the spectral resolution operated at -20℃ [2], the low energies were more affected. For the spectral lines at 59.6 keV, 26.3 keV, and partially the line at 17.6 keV, the energy resolution remained constant. For the other spectral lines, the energy resolution increased anywhere from 0.03 to 0.25 keV. Each plot has several data points outside of the errors of the model fit. This does not imply immediately that this is a poor model however, as Equation 5 has been used to model the spectral resolution for these detectors at lower operating temperatures [2,11]. It is believed that with better data quality, the data points will better fit the model. Improved quality could be achieved by increasing the number of counts collected. More statistics would lead to a better gain calibration and FWHM calculations. Also, a more precise removal of the Am-241 continuum background from the Ni and Cu lines would lead to a more precise FWHM calculation for those lines.

Table of the parameters used to fit Equation 5 in the plot of spectral resolution vs energy.

The energy dependent noise term, f1, was calculated to be on the order of 104 times smaller for the fit in detector 4, ASIC 2 and detector 5, ASIC3 when compared to the other plots, as well as the f1 term found when operating the detectors at -20℃ [2]. One possible explanation for this is that the resolution for the 59.6 keV Am-241 line was lower for these two plots. This issue may be resolved if a larger integration time was used in order to gain a higher number of counts for this emission line.

It is important to note that the flight goal resolution for the silicon detectors was 1 keV. As Figure 4 shows, for the energy range of 4 to 15 keV, the resolution is roughly 0.55 keV, well below the 1 keV goal. While operating at -10℃, the contribution from electronic noise has increase by about 0.03 to 0.09 keV in comparison to when the detectors were operated at -20℃ [2].

Energy Efficiency

In order to measure the energy efficiency of the FOXSI detectors, the first step is to calculate the incident flux measured by the reference Silicon drift detector (SDD). First, we perform gain calibration on the reference detector using a process very similar to gain calibration for the FOXSI detector. Since the SDD is a single silicon bulk crystal with a very small resolution, there are no individual strips to calibrate and we are therefore able to manually identify the peak bin when the SDD is illuminated with 4.75 keV, 5 keV, 5.5 keV, 6 keV, 7 keV, 8 keV, and 9 keV monochromatic beams. We then perform a linear fit of peak bin to peak energy to obtain a gain calibration curve, shown below.

Gain calibration curve for reference SDD. We found that each bin corresponds to 0.0389 keV and that bin zero corresponds to -0.0536 keV.

To find the rate of incident counts, the total time the SDD was actively recording incident photons, also known as the live time, needs to be calculated. In the SDD provided by the ALS, there is a fast triggering channel and a slow triggering channel. The fast channel has a very fast shaping time during which it can’t accept new counts, therefore we assume that the number of counts recorded by the fast channel is the number of photons incident on the SDD, whether or not they were actually recorded in the data. The slow channel has a slow shaping time, which results in it being able to discern the energy of the photons it records. The data recorded by the SDD is data recorded by the slow channel. While the slow channel is reading the energy of the incident photon, it cannot accept any new counts. The time it spends shaping the pulse where it cannot accept new counts is known as the dead time of the SDD’s measurement. The provided SDD calculates dead time by taking the ratio of the total slow channel counts to the total fast channel counts, giving the ratio of live time to total time the SDD was operating. Taking this ratio and multiplying by the time the SDD was active yields the live time of the SDD [6]. Dividing the counts within a given energy peak by the live time gives the actual incident countrate supplied by the x-ray source. Similarly, we sum the total number of counts in a given peak measured by the FOXSI detector, and divide by the live time, which is a quantity measured by onboard electronics and included in the data packet. We then calculate the energy efficiency as a function of energy, shown below.

Energy efficiency as a function of energy for detector 4, ASICs 2 and 3.

From the calculated energy efficiency of detector 4, ASIC 2 and 3, we immediately note that the efficiency for the region well above the threshold is well below unity efficiency. Furthermore, it is observed to drop as energy increases, the opposite dependence as expected. The only places in our analysis where such a discrepancy could have occurred is in calculating the number of counts registered by the FOXSI detector, and the live time calculation for the SDD.

The live time calculation outlined in preceding paragraphs is that detailed by the manufacturer, but it differs in key ways from how FOXSI dead time is calculated. Primarily, it assumes that the ratio between the slow channel and the fast channel counts gives the live time percentage, something not supported in other literature [6]. If the live time is actually larger than the calculated value for high count rates, it would reduce the incident count rate more in the high energy regime and result in the expected higher efficiency values. The counts measured by the FOXSI detector during our tests may also be suspect. Between each strip, there is a 25 micron section of the detector that cannot register data. After further analysis of other data sets taken at the ALS, the count rates observed by the FOXSI detector during our energy efficiency tests was on the order with count rates of data taken while the non-data-taking parts of the detector were illuminated. While the position of our detector during the energy efficiency tests would indicate that the x-ray source was illuminating a part of the detector that could take data, it is possible the system that positions the detector in front of the source exhibited unforeseen hysteresis behavior that resulted in the wrong section being illuminated. This would result in drastically reduced observed count rates. If accounted for, this would explain the low observed count rates but not the odd energy dependence of the calculated efficiency. It is possible that one or both of these factors are affecting our data.

Conclusion

The measurements found that the spectral resolution in the energy range of interest to the FOXSI mission, 4 to 15 keV, was on average 0.55 keV, roughly 0.05 keV higher than if the detectors were operated at -20℃ [2]. This energy resolution is below the 1 keV resolution required by the science goal of the FOXSI mission. If this calibration were to be repeated, it would be useful to gain further statistics for the spectral lines. This would provide a tighter and more accurate fit to the FWHM measurements.

The energy efficiency calculations were believe to be incorrect, as the efficiency does not match expected [6] or previous results [9]. The reason for this is thought to be due to not correctly calculating the live time, and thus the count rate, in the reference detector. To improve upon this work, more information would be needed in how to understand the data collected by the reference detector, namely how to calculate the live time. It would also be useful to extend the range of measurements to 4 to 20 keV, the full range of the beamline at ALS, as this would provide efficiency information for the entire energy range of the FOXSI detectors.

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