Analytical model for the photomultiplier single photoelectron response including the electron back-scattering contribution

This paper derives and validates a comprehensive analytical model for photomultiplier single photoelectron responses that replaces ad hoc noise descriptions with a physically grounded function accounting for electron back-scattering at the first dynode, alongside analytical descriptions for fully amplified peaks and low-charge signals.

The Gist

Imagine you are trying to listen to a single, tiny whisper in a very noisy room. To do this, you have a super-sensitive microphone (a Photomultiplier Tube, or PMT) that can turn that one whisper into a loud shout so you can hear it.

However, the microphone isn't perfect. Sometimes, when the whisper hits the first part of the machine, it bounces off instead of being caught immediately. This paper is about figuring out exactly how those "bounced" whispers sound, so we can tell the difference between a real signal and a glitch.

Here is the breakdown of the paper using simple analogies:

1. The Problem: The "Ghost" Signals

When a PMT detects a single particle of light (a photon), it turns it into an electron and then multiplies it millions of times to create a measurable electrical signal.

• The Ideal Scenario: The electron hits the first metal plate (the first dynode) and creates a perfect, loud "boom." This is the main peak in your data.
• The Messy Reality: Sometimes, the electron hits the metal plate and bounces off (back-scatters) like a rubber ball hitting a wall. It doesn't give all its energy to the plate. It only gives a little bit, creating a "half-boom" or a "whisper" instead of a shout.
• The Confusion: In the past, scientists looked at these "half-booms" and the tiny noises in between the silence and the main shout, and they just called it "noise" or "garbage." They used a generic math formula (like a smoothed-out curve) to ignore it.

The Paper's Goal: The authors say, "Wait a minute! That 'garbage' isn't random. It's actually a specific physical process." They wanted to write a new math recipe that explains exactly why those half-booms happen, based on how electrons bounce.

2. The Solution: The "Bouncing Ball" Model

The authors created a new analytical model (a set of equations) that treats the electron like a ball bouncing off a wall.

• The Full Amplification (The Loud Shout): Most electrons hit the first plate and stay, creating a full cascade of electrons. This creates a nice, tall bell curve in the data.
• The Partial Amplification (The Half-Whisper): About 30% of the time, the electron bounces off the first plate. It loses some energy. Because it lost energy, it creates fewer electrons downstream. This creates a "hump" or a "box" shape in the data between the silence and the main shout.
  • Analogy: Imagine throwing a ball at a pinball machine. Sometimes it hits the bumper and bounces back immediately (lost signal). Sometimes it hits the bumper, loses some speed, and keeps going but hits fewer targets (partial signal). The authors figured out the exact math for how many targets it hits when it loses speed.
• The "Pre-Pulses" (The Early Birds): Sometimes, a photon misses the first plate entirely and hits the second one. This creates a signal that arrives a tiny fraction of a second earlier than the others. The model accounts for these "early birds" too.

3. The Experiment: Testing the Theory

To prove their model works, the team built a test setup:

• They used two different types of high-tech microphones (Hamamatsu PMTs).
• They shone a laser so dim that it only sent one photon at a time (like a single drop of water falling into a bucket).
• They used a very quiet amplifier (to hear the whispers clearly) and a super-fast camera to record the electrical signals.

They compared the data they collected against their new "Bouncing Ball" math model.

4. The Results: It Works!

The results were impressive.

• Old Way: Scientists used a "one-size-fits-all" curve that didn't really explain why the data looked the way it did.
• New Way: Their new model, which accounts for the bouncing electrons, fit the data perfectly. It could explain the shape of the "half-booms" and the "early birds" without needing to make up fake numbers.

Why does this matter?
Think of a photomultiplier like a scale used to weigh gold dust. If your scale has a weird "ghost weight" in the middle that you don't understand, you might miscount your gold.

• In medical imaging (like PET scans), this helps doctors see tumors more clearly.
• In physics experiments (like searching for dark matter), it helps scientists distinguish between a real particle from the universe and a random electronic glitch.

Summary

This paper is like writing a new instruction manual for a microphone. Instead of saying, "Ignore the weird static in the middle," the authors said, "That static is actually the sound of the microphone's internal parts bouncing." By understanding the bounce, they created a better tool for scientists to measure the universe's tiniest whispers.

Technical Summary

1. Problem Statement

Photomultiplier tubes (PMTs) remain the standard for detecting very low light levels over large areas. A critical characteristic for their calibration and simulation is the Single Photoelectron (SPE) response.

• Current Limitations: Existing models generally describe the fully amplified photoelectron peak well (using scaled Poisson or Gamma distributions) but fail to accurately model the spectral region between the electronics pedestal and the main SPE peak.
• The "Noise" Gap: This intermediate region is often attributed to generic "noise" and modeled using ad-hoc functions (e.g., exponentially modified Gaussians) without a rigorous physical basis.
• Physical Mechanism: A significant fraction of primary photoelectrons striking the first dynode undergo inelastic back-scattering. Instead of depositing all their kinetic energy to generate secondary electrons, they scatter and lose only a fraction of their energy. This results in "partially amplified" signals that create a distinct, non-Gaussian distribution in the low-charge region of the spectrum.

2. Methodology

The authors developed a comprehensive analytical model based on the physical description of electron back-scattering found in The Photomultiplier Handbook (A. G. Wright). The model separates the amplification process into two stages: the first dynode and the subsequent $N-1$ dynodes.

A. Theoretical Framework

The model constructs the total SPE response function, $\mathcal{P}_{SPE}(x)$, as a weighted sum of four distinct physical components:

1. Fully Amplified (FA) Electrons:

   • Primary electrons that deposit all energy at the first dynode.
   • Modeled as a scaled continuous Poisson distribution convolved with the Gaussian resolution of the remaining dynode chain.
   • Formula: $\mathcal{P}_{FA}(x) = \rho e^{-\rho \mu} (\rho \mu)^{\rho x} / \Gamma(1 + \rho x)$.

2. Partially Amplified (PA) Electrons (Back-scattering):

   • Primary electrons that inelastically back-scatter, depositing only a fraction $(1-\kappa)$ of their energy.
   • The discrete probability of generating $n$ secondary electrons is derived by integrating a Poisson distribution over a uniform energy loss distribution.
   • After convolution with the subsequent dynode chain, this forms a rounded "box" shape described by error functions ($\text{erf}$).
   • This component is the core innovation, replacing ad-hoc noise functions with a physics-based derivation.

3. Pre-pulses ($\gamma_{1Dy}$):

   • Photons that bypass the photocathode and hit the first dynode directly.
   • These generate signals with a gain roughly $1/G_1$ smaller than the main peak.
   • Modeled as a scaled Poisson distribution with modified gain and resolution parameters.

4. Exponential Low-Charge Events ($\mathcal{P}_{Exp}$):

   • Electrons with poor trajectories missing dynodes or other low-charge anomalies.
   • Modeled as an exponentially modified Gaussian centered at the origin.

B. Experimental Validation

The model was tested using two distinct PMT models:

• Hamamatsu R5912-100 (8-inch, used in XENONnT and LZ).
• Hamamatsu 6233 (3-inch).

Setup:

• Light Source: Pico-second lasers (405 nm and 507 nm) with sub-nanosecond pulses.
• Readout: Low-noise amplifier (RFBAY LNA-150) and a high-speed digitizer (2 GS/s).
• Conditions: Measurements were taken at various High Voltage (HV) settings, illumination positions, and wavelengths to test parameter stability.
• Data Processing: Dark counts were subtracted, and waveforms were integrated to generate charge spectra.

3. Key Contributions

• First Analytical Derivation of PA Signals: The paper provides the first rigorous analytical function for the partially amplified electron spectrum derived directly from back-scattering physics, rather than empirical fitting.
• Unified Framework: It integrates fully amplified peaks, back-scattered tails, pre-pulses, and low-charge tails into a single, consistent mathematical framework with a limited number of intrinsic parameters.
• Parameter Reduction: The model reduces the description of the complex SPE spectrum to a few physically meaningful parameters:
  • $G_1$: Gain of the first dynode.
  • $\eta$: Fraction of back-scattered electrons.
  • $R$: Resolution parameter of the dynode chain.
  • $\zeta$: Ratio of full vs. average gain at the second dynode (for pre-pulses).
  • $\alpha$: Decay constant for exponential low-charge events.

4. Results

The model was fitted to experimental data with high precision:

• Fit Quality: The model achieved excellent agreement with data across all tested conditions, with reduced chi-square values ($\chi^2/\text{ndf}$) ranging from 1.01 to 1.36.
• Hamamatsu R5912-100:
  • The back-scattering fraction ($\eta$) was found to be approximately 27%, consistent with theoretical expectations for high-Z metals.
  • The exponential low-charge component ($A_{Exp}$) was significant (~4%), while pre-pulses were negligible.
  • Parameters $G_1$ and the total gain showed expected linear dependence on HV.
  • The fraction $\eta$ was found to be independent of HV but dependent on the illumination position on the photocathode.
• Hamamatsu 6233:
  • Showed a lower back-scattering fraction ($\eta \approx 17\%$).
  • A significant pre-pulse contribution ($A_{\gamma 1Dy} \approx 7\%$) was observed, which was successfully resolved by the model.
  • The resolution parameter $R$ was successfully extracted as a free parameter due to the clear separation of the pre-pulse peak.

5. Significance and Impact

• Physics-Based Calibration: By replacing empirical "noise" functions with a physically derived model, this work allows for more accurate determination of PMT gain, quantum efficiency, and effective acceptance in experiments.
• Simulation Improvement: The analytical functions can be directly implemented in Monte Carlo simulations (e.g., for Dark Matter or Neutrino experiments like XENONnT and LZ) to better model the low-energy background and signal reconstruction.
• Diagnostic Tool: The ability to extract specific fractions ($\eta$, $A_{Exp}$, $A_{\gamma 1Dy}$) allows researchers to diagnose specific PMT behaviors, such as dynode surface properties or optical alignment issues, without relying on black-box fitting.
• Robustness: The model's validity across different PMT sizes, materials, and operating conditions demonstrates its universality for single-photon counting applications.

In conclusion, this paper successfully bridges the gap between theoretical electron physics and experimental PMT characterization, providing a robust, parameter-efficient tool for the next generation of low-light detection experiments.