# Analysis and Mitigation of Pulse-Pile-Up Artifacts in Plasma Pulse-Height X-ray Spectra

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## Abstract

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## 1. Introduction

## 2. Sample Spectra with PPU

## 3. Pulse Pile-Up Reduction Techniques

## 4. Determination of Photon Energy

## 5. Modeling Pile up for Uncorrelated Trapezoidal Pulses in the Two-Photon Approximation

## 6. Ppu Examples Using the Two-Photon Uncorrelated Trapezoidal-Pulse Model

## 7. Experimental Validation of Pulse-Pile-Up Model in Two Photon Approximation

## 8. Using the Two-Photon Model to Reduce Pulse Pile-Up Tails

Algorithm 1: Pile-up Reduction Algorithm |

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**X-ray spectra at two count rates, 14 kcps (blue) and 65 kcps (red), for a 5-keV electron beam impacting a carbon target. Noise was reduced using a 50-eV-wide weighted moving-average filter. The slow-channel dead time was 212 ns. For comparison, the predicted spectrum (yellow) was derived using the two-photon PPU model, to be described in Section 7.

**Figure 2.**X-ray spectra from seed plasma in the PFRC-2 (H${}_{2}$ fill-gas) formed by a continuous radiofrequency (rf) 550 W source. ${t}_{r}=1.0$ $\mathsf{\mu}$s (blue) and 5.6 $\mathsf{\mu}$s (red). From the least piled-up spectrum, ${t}_{r}=1.0$ $\mathsf{\mu}$s, the true count rate $\mu =9.54$ kcps is obtained. The X-ray spectrum predicted with a two-photon model for ${t}_{r}=5.6$ $\mathsf{\mu}$s is shown in yellow and described in Section 7. The inset shows a magnified view of the low energy region of the spectra to make clear the peak shift at high $\mu {t}_{d}$.

**Figure 3.**Voltage vs. time plot for two rectangular, triangular, trapezoidal, and Gaussian pulses (blue and orange) arriving within ${t}_{d}$ and them added together (green).

**Figure 4.**Sum (shown in green) of two trapezoidal voltage pulses (blue and orange) for different values of $\Delta t$. The sum follows different patterns and formulae for the three cases: $\Delta t<{t}_{f}$, ${t}_{f}<\Delta t<{t}_{d}$, and ${t}_{d}<\Delta t$. For ${t}_{d}<\Delta t$, the pulses are resolved and no PPU occurs.

**Figure 6.**Probability density function for apparent energy, ${p}_{2\gamma}\left(E\right|{E}_{1},{E}_{2})$ vs. E for several values of the triangularity a with ${E}_{1}=0.5$, ${E}_{2}=1$. An upward arrow indicates a Dirac delta function at that value.

**Figure 7.**Probability distribution vs. energy of piled-up output of a narrow Gaussian for 5 different values of triangularity.

**Figure 8.**X-ray spectrum, Zn target illuminated by X-ray tube, 30 kV: 25 mm${}^{2}$ Amptek (PUR disabled) DppMCA data [17]. Horizontal axis—energy channel number (264/keV); vertical axis—counts, log scale. Zn K-$\alpha $ at 8.6 keV, ch 2270.

**Figure 9.**PPU spectra for truncated exponential ($\propto {e}^{-E/{E}_{0}}$, ${E}_{0}=1$ keV) spectrum with cutoff at 1 keV.

**Figure 10.**Apparent 65 kcps (blue) graphite-target X-ray-tube spectra and pile-up-reduced spectra (red) extracted using the two-photon pile-up-reduction algorithm. Noise was reduced using 50-eV-wide weighted moving-average filter. Negative values that could not be shown in logarithmic plot are seen in the linear scale inset.

**Figure 11.**Prediction of piled up spectra using tail cut off (> 5 keV) spectrum as the original spectrum.

**Figure 12.**True pile-up-less X-ray spectra (with differing peaking times: 1.0 $\mathsf{\mu}$s (blue) and 5.6 $\mathsf{\mu}$s (red) from the seed plasma in PFRC filled with H${}_{2}$ gas, extracted using the two-photon pile-up reduction algorithm.

**Table 1.**Comparison of both models with measured data from graphite target X-ray tube. The tail is assumed to be the region beyond 5 keV. See Figure 1.

Spectra | 14 kcps Spectrum | Apparent 65 kcps Spectrum | Predicted 65 kcps Spectrum | |
---|---|---|---|---|

Results | ||||

Percentage of area under Tail | $0.0222\%$ | $0.0515\%$ | $0.0536\%$ |

**Table 2.**Comparison of both models with measured data from seed plasma in PFRC filled with ${H}_{2}$ with RMF turned off. Peaking time was 5.6 $\mathsf{\mu}$s. The tail is assumed to be the region beyond 650 eV.

Spectra | 1 $\mathsf{\mu}$s Spectrum | Apparent 5.6 $\mathsf{\mu}$s Spectrum | Predicted 5.6 $\mathsf{\mu}$s Spectrum | |
---|---|---|---|---|

Results | ||||

Percentage of area under Tail | $11.7\%$ | $15.7\%$ | $15.4\%$ |

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**MDPI and ACS Style**

Ahsan, T.; Swanson, C.P.S.; Galea, C.; Vinoth, S.P.; Qian, T.; Rubin, T.; Cohen, S.A.
Analysis and Mitigation of Pulse-Pile-Up Artifacts in Plasma Pulse-Height X-ray Spectra. *Plasma* **2023**, *6*, 58-71.
https://doi.org/10.3390/plasma6010006

**AMA Style**

Ahsan T, Swanson CPS, Galea C, Vinoth SP, Qian T, Rubin T, Cohen SA.
Analysis and Mitigation of Pulse-Pile-Up Artifacts in Plasma Pulse-Height X-ray Spectra. *Plasma*. 2023; 6(1):58-71.
https://doi.org/10.3390/plasma6010006

**Chicago/Turabian Style**

Ahsan, Taosif., Charles P. S. Swanson, Chris Galea, Sangeeta P. Vinoth, Tony Qian, Tal Rubin, and Samuel A. Cohen.
2023. "Analysis and Mitigation of Pulse-Pile-Up Artifacts in Plasma Pulse-Height X-ray Spectra" *Plasma* 6, no. 1: 58-71.
https://doi.org/10.3390/plasma6010006