An Improved Paralyzable Detector Model

Abstract

Certain radiation detectors are 'paralyzed' with high input count rates. When applied to count rates close to the event discriminator working rate the one-parameter dead time model fails. Here we present a corrected paralyzable detector model accounting for the event discriminator's finite response time. This two-parameter analytical model, when compared to the experimental data from a commercial x-ray detector, gives an improved description of the input and output count rate relations. Furthermore, it can independently determine the discriminator response time and the pulse shaper dead time, critical parameters for understanding a detector's performance. Finally, this model also provides a post-acquisition pile-up correction that greatly reduces artifacts in high-throughput spectra. In some situations, applying this model to optimize the acquisition and post-acquisition correction allows a user to acquire data an order of magnitude faster without compromising accuracy.

Figures (9)

• Figure 1: Fundamental models illustration and radiation detection system schematic. a) The same event sequence arrives at different types of detectors. The red and white triangles represent detected events and missed coincidence events, respectively. The rectangular box represents the recovery time (the dead time $\tau$). b) Output count rate $C_{\text{out}}$ as a function of input count rate $C_{\text{in}}$ for different types of detectors. The green dashed line marks the $C_{\text{out}}$ asymptote of the non-paralyazble detector, and the red dashed lines mark the position of maximal $C_{\text{out}}$ of the paralyzable detector. c) A schematic of a typical EDS system, where the progression of signal and approximate time constant is illustrated as an inset after each stage goldstein_x-ray_2003ballabriga_design_2009. This schematic can be generalized to other radiation detection systems by substituting the SDD with other types of detectors, such as neutron and electron detectors.
• Figure 2: Paralyzable detector model fits. The experimental data (black dots) and corresponding fits (red curves) using the paralyable detector model (Eq. \ref{['eq:ideal_paralyzable']}). The data is obtained on an Oxford X-MaxN 100TLE EDS detector for its six dead time options, and the measured dead time is labeled at the bottom right corner of each plot. The classic paralyzable detector model cannot accurately represent the data of shorter dead times.
• Figure 3: EDS system with event discriminator. a) Simplified schematic of an EDS system with event discriminator and hardware pile-up rejection goldstein_x-ray_2003. The output of the pre-amplifier (pre-amp) goes into two parallel pathways, one with a pulse shaping linear amplifier used for energy measurement, and another one with a fast amplifier used for input event discrimination. The fast amplifier has a shorter response time $t_{\text{dis}}$, which is better for discriminating adjacent events in the expense of pulse height-energy linearity. b,c) Probability density function (PDF) and cumulative distribution function (CDF) of having a sequential event arriving within a time $t$. The time axis is in the unit of $1/C_{\text{in}}$. Sequential events arriving within the response time of event discriminator $t_{\text{dis}}$ are missed by the system and can become pile-up events that the system fails to reject. Sequential events arriving between $t_{\text{dis}}$ and the pulse shaping linear amplifier dead time $\tau$ are successfully rejected. If no sequential event arrives within $\tau$, the leading event becomes a clean output signal with no pile-up. For a complete event category diagram, refer to Fig. \ref{['fig:EventDiagram']} in the appendix.
• Figure 4: Fits using model corrected for a real-world event discriminator. Experimental data in Fig. \ref{['fig:ParalyzableModelFit']} is fitted again using the corrected model in Eq. \ref{['eq:corrected_model_simplified']} which accounts for an event discriminator and pile-up rejection system. a) A simultaneous fit to all six dead time options to determine the common parameter $t_{\text{dis}}$. It is assumed that all six dead time options use the same event discriminator. b-g) $C_{\text{out}}$ vs $C_{\text{hit}}$ fit (red) and predicted pile-up rate (blue) for each dead time option. Compared to the classic paralyzable detector model in Fig. \ref{['fig:ParalyzableModelFit']}, the corrected model accurately represents the data even to the shortest dead time.
• Figure 5: Maximizing clean events throughput. Maximizing the count rate of clean output events as a function of the ratio of dead time $\tau$ and the response time of the event discriminator $t_{\text{dis}}$. The dots along the curves represent the ratios measured on our Oxford X-MaxN 100TLE EDS detector for different dead time options. These plots guide users to maximize the data throughput and estimate the pile-up effect in the acquired spectrum. The $D_\%$ corresponding to the max output rate deviates more from the prediction of the classic paralyzable detector model (dashed line) as the ratio $n$ gets smaller. Note that the measured $D_\%$ given in Eq. \ref{['eq:measured_deadtimepct']}, and the dimensionless formulas can be derived with the probabilities shown in Appendix A.