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A four parameters modified Gaussian function describing shape of asymmetric alpha lines

This paper presents a new four-parameter analytical function for accurately modeling asymmetric alpha peaks, which has been integrated into the established ALF software to enable the deconvolution of up to ten overlapping peaks.

Original authors: Jerzy Wojciech Mietelski

Published 2026-07-09

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CC BY 4.0

Original authors: Jerzy Wojciech Mietelski

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are listening to a choir of alpha particles singing their way into a detector. In a perfect world, every singer hits the exact same note, creating a beautiful, symmetrical bell curve—a classic Gaussian shape. But in the real world, things get messy. As these tiny particles travel from their source to the detector, they bump into things: a "dead layer" on the detector's entrance window and the source material itself. These collisions steal a bit of their energy, but not always the same amount.

Think of it like runners in a race where some trip over a loose stone and some don't. The runners who don't trip arrive exactly on time, keeping the high-energy side of the peak sharp and symmetrical. But the runners who trip arrive late, dragging the low-energy side of the peak out into a long, messy tail. This creates an "asymmetric" shape that looks like a bell that's been dragged through the mud on one side.

For decades, scientists struggled to describe this messy shape with math. They usually had to use a complicated toolkit of many different formulas to patch together the different parts of the curve. But Jerzy Wojciech Mietelski, a researcher from the Institute of Nuclear Physics in Poland, decided to try a different approach. He introduced a single, clever mathematical function that acts like a "shape-shifter."

This new function is a modified Gaussian, but with a twist. It has only four adjustable knobs (parameters) to turn:

Amplitude: How tall the peak is.
Position: Where the peak sits on the chart.
Shape Parameter 1: Controls how the curve behaves on the right side (the clean, Gaussian side).
Shape Parameter 2: Controls the messy, long tail on the left side.

Here is the magic trick: On the right side of the peak, the function behaves like a standard, perfect bell curve. But on the left side, it morphs into a more complex formula that naturally creates that long, exponential tail caused by the energy loss. It's like having a single piece of clay that can be smooth and round on one side but stretched and tapered on the other, all without needing to glue two different shapes together.

Mietelski didn't just dream this up; he put it to work in a computer code called ALF, which has been running in his lab since 1993. The process works in two steps, like tuning a radio. First, the computer looks at a single, clear peak and figures out the exact settings for the shape parameters (the "mud" and the "clean" sides). Once it knows what the "mud" looks like for that specific experiment, it locks those settings in. Then, it uses that fixed shape to untangle a mess of overlapping peaks—up to 10 of them at once—by just adjusting their height and position.

The paper reports that this method has been tested over 30 years and has successfully analyzed more than 15,000 spectra. It even passed rigorous "proficiency tests" where the lab had to prove they could measure things correctly against other experts. The authors note that the results are so reliable that the area under the curve (which tells you how much radioactive material is there) matches the total count of particles almost perfectly.

Surprisingly, this "alpha" shape-shifter wasn't just for alpha particles. The paper mentions that the team tried using the same math to separate beta radiation signals from a liquid scintillation spectrometer. In one test involving 90Sr (Strontium-90) in equilibrium with 90Y (Yttrium-90), the method managed to separate the signals with an error margin of just 3%.

So, while the paper doesn't claim this is a magic cure for every problem in physics, it presents a robust, four-parameter tool that has been proven to work for decades. It turns a messy, asymmetric problem into a solvable puzzle, allowing scientists to see clearly through the noise of overlapping signals, whether they are looking at alpha particles from Antarctic lichen or beta particles from animal bones.

Technical Summary: A Four-Parameter Modified Gaussian Function for Asymmetric Alpha Lines

Problem Statement
In alpha spectrometry, spectral lines frequently exhibit asymmetric shapes rather than ideal Gaussian distributions. This asymmetry arises primarily from energy degradation as alpha particles traverse the dead layer (entrance window) of the semiconductor detector and the source material itself. The extent of this degradation depends on the random emission angle of the particle relative to the detector. Consequently, the low-energy side of the peak becomes wider, approaching an exponential fall, while the high-energy side remains Gaussian, modified only by statistical fluctuations.

Historically, there has been no single, generally accepted mathematical formula to describe this shape. Practical solutions typically rely on multi-parameter sets of functions applied to specific sections of the spectral line. While various software packages (e.g., ALFUN-A, MicroSAMPO, ALFA) exist to address these challenges, the specific analytical function used in the widely adopted ALF computer code had not been detailed in the literature prior to this work.

Methodology
The paper presents an original analytical function designed to describe the asymmetric alpha peak shape using only four free parameters. This function was implemented in the ALF computer code, a Fortran-based tool developed around 1993 (modified from an earlier gamma-ray code, P.I.M.P.) for alpha spectrum evaluation.

The core of the methodology is the following modified Gaussian function, $N(x)$ , where $x$ represents the channel number:

$N(x) = \alpha \cdot \exp\left( \frac{-(x-M)^2}{\beta \{|x-M| - (x-M)\} + \gamma} \right)$

Parameters:

$\alpha$ : Amplitude (counts at the maximum).
$M$ : Position of the peak maximum.
$\beta$ and $\gamma$ : Shape parameters (both $>0$ ) specific to the source and detector geometry.

Behavior of the Function:

High-energy side ( $x > M$ ): The function converges to a standard Gaussian distribution.
Low-energy side ( $x < M$ ): The function becomes more complex, transitioning toward an ordinary exponential function at significant distances from the peak maximum ( $x \ll M$ ).
Parameter Roles: $\gamma$ governs the shape of the right side of the peak, while $\beta$ and $\gamma$ jointly govern the left side.

Fitting Procedure (Deconvolution):
The ALF code utilizes a non-linear function fitting procedure to deconvolve up to 10 overlapping alpha peaks. The process is executed in two distinct steps:

Initial Shape Determination: A single, well-defined spectral line (or a fragment containing the maximum) is fitted to determine the four parameters ( $\alpha, M, \beta, \gamma$ ). This uses a standard non-linear least-squares fit via 10 iterations of a linear approximation (Taylor series expansion).
Peak Deconvolution: Once $\beta$ and $\gamma$ are determined, they are treated as fixed constants for the entire spectrum, as they depend on the source quality and source-detector geometry rather than individual peaks. The second step involves fitting the amplitudes ( $\alpha_i$ ) and positions ( $M_i$ ) for all overlapping peaks simultaneously. This is achieved by expanding the sum of functions into a series and solving for the increments of the vector variables using a Cracovian algorithm.

The Cracovian algorithm is highlighted as a specific advantage, as it calculates the uncertainties of the fitted parameters within the same procedure used to determine the parameter values.

Key Contributions and Results

Analytical Simplicity: The paper introduces a compact, four-parameter function that effectively models the complex asymmetry of alpha lines without requiring complex multi-segment fitting.
Validation and Longevity: The function has been in routine use since 1993. Over the last 30 years, it has been applied to more than 15,000 spectra. Validation was performed by comparing calculated peak areas against the sum of counts, with all tests passing positively.
Deconvolution Capability: The method successfully separates partially overlapping peaks, as demonstrated with Th-fraction samples from animal bones.
Unexpected Utility: The authors note that the function proved surprisingly useful for analyzing beta radiation spectra (specifically $^{90}\text{Sr}$ / $^{90}\text{Y}$ ) obtained with a liquid scintillation spectrometer using a logarithmic channel scale, achieving results within a 3% margin of error.

Significance
The paper claims significance primarily through the successful, long-term validation of this specific four-parameter function within the ALF software. By formalizing the previously undocumented analytical basis of ALF, the work provides a robust, mathematically defined tool for the deconvolution of overlapping alpha peaks. The method offers a balance between analytical rigor and computational efficiency, allowing for the precise determination of peak areas in complex spectra where traditional Gaussian fitting fails. The use of the Cracovian algorithm further ensures that parameter uncertainties are rigorously calculated alongside the fit.

Technical Summary: A Four-Parameter Modified Gaussian Function for Asymmetric Alpha Lines

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