A systematic approach to Covariance matrix formulation in charged particle activation experiments

This paper presents a systematic framework for constructing covariance and correlation matrices in charged particle activation experiments by explicitly calculating both statistical and systematic uncertainties through sensitivity coefficients and parameter propagation, thereby demonstrating the critical importance of accounting for correlated uncertainties in the interpretation and comparison of experimental cross-section data.

The Gist

Imagine you are a detective trying to solve a mystery: How likely is a specific nuclear reaction to happen?

In the world of nuclear physics, scientists don't just guess; they measure. They shoot tiny particles (like protons) at a target and count how many "hits" occur. This number is called the cross-section. It's like measuring how big a target you need to hit to get a bullseye.

For decades, scientists have been good at counting the hits. But there was a problem with how they reported their "confidence" in those numbers. They would say, "We are 95% sure the answer is X, with a margin of error of Y."

The Problem: The "Silent Correlation"
The old way of calculating error was like adding up the mistakes in a recipe. If you used a slightly inaccurate scale for the flour, a slightly inaccurate timer for the baking, and a slightly inaccurate oven temperature, you'd add all those small errors together to get a "total error."

But here's the catch: Some errors are shared.
Imagine you are baking 10 different cakes at the same time.

• Independent Error: You might accidentally drop a sprinkle of sugar on Cake #1 but not Cake #2. That's a random, one-off mistake.
• Shared Error: But what if the entire batch of flour you bought was slightly heavier than the label said? Then every single cake you baked is slightly too heavy. The error isn't random; it's correlated. If Cake #1 is heavy, Cake #2 is definitely heavy too.

In nuclear experiments, things like the detector's efficiency (how good the camera is at seeing the hits) or the beam intensity (how strong the particle stream is) are like that bad batch of flour. They affect every measurement you take. If you ignore this "shared error," you might think your data is more precise than it really is, or you might draw the wrong conclusions when comparing your results to computer models.

The Solution: The Covariance Matrix
This paper by Tanmoy Bar is essentially a user manual for building a "Shared Error Map."

Instead of just listing the total error, the author proposes a systematic way to create a Covariance Matrix. Think of this matrix as a giant spreadsheet that doesn't just tell you how wrong each number might be, but also how much each number's mistake is linked to the others.

Here is how the paper breaks it down, using simple analogies:

1. The Ingredients (The Parameters)

To calculate the cross-section, you need many ingredients:

• Counting Stats: How many particles you saw (Random noise, like static on a radio).
• Detector Efficiency: How well your "camera" sees the particles.
• Beam Flux: How strong the particle stream is.
• Target Thickness: How thick the wall of atoms is you are shooting at.
• Time: How long you shot, how long you waited, and how long you counted.

2. The Sensitivity Coefficients (The "How Much It Matters" Factor)

The paper introduces a concept called Sensitivity Coefficients.
Imagine you are driving a car.

• If you turn the steering wheel a tiny bit, the car turns a tiny bit. (Low sensitivity).
• If you slam on the brakes, the car stops instantly. (High sensitivity).

The author calculates exactly how much the final answer (the cross-section) changes if one of your ingredients (like the detector efficiency) is slightly off. This helps figure out which "ingredients" are the most dangerous sources of error.

3. The Jacobian Matrix (The Recipe Calculator)

This is the mathematical engine. It takes all those "Sensitivity Coefficients" and the known uncertainties of your ingredients and crunches the numbers to see how the errors propagate.

• Statistical Errors: These are the "sprinkles of sugar" (random). They only affect the specific measurement they happened to. In the matrix, these only show up on the diagonal (the main line).
• Systematic Errors: These are the "bad batch of flour" (shared). They create a web of connections. If your detector was 2% off, it was 2% off for all your measurements. The matrix captures this web, showing that if Measurement A is high, Measurement B is likely high too.

4. The Result: A Complete Picture

By the end of the paper, the author shows how to turn this complex math into a Correlation Matrix.

• 0.0 means: "My mistake in this measurement has nothing to do with that one."
• 1.0 means: "My mistake here is exactly the same as the mistake there."
• 0.5 means: "They are somewhat related."

Why Does This Matter?

The paper argues that without this "Shared Error Map," scientists are flying blind.

• For Nuclear Astrophysics: We need to know exactly how stars burn. If we ignore the shared errors, we might think a star's recipe is different than it actually is.
• For Medical Isotopes: If we are making medicine for cancer treatment, we need to know the reaction rates are accurate.
• For Safety: If we are designing nuclear reactors, we need to know the uncertainties are real, not just hidden correlations.

The Takeaway

Tanmoy Bar's paper is a guide to honesty in science. It says: "Don't just give us a number and a total error bar. Show us the structure of the error. Tell us which errors are random and which ones are shared across the board."

By using this systematic approach, scientists can stop guessing and start making reliable comparisons, ensuring that the data used to build models of the universe or design life-saving medical treatments is as solid as possible. It's about moving from "We think we are right" to "We know exactly how right (or wrong) we are, and how our mistakes are connected."

Technical Summary

1. Problem Statement

In nuclear physics, particularly in Charged Particle Activation Method (CPAM) experiments, cross-section measurements are crucial for nuclear astrophysics, isotope production, and reactor design. Historically, experimental uncertainties have been reported as a single "total uncertainty" derived from the quadratic summation of statistical and systematic errors.

The core problem identified by the author is that this traditional approach ignores correlations between measurements taken at different energy points. In activation experiments, many uncertainty sources (e.g., detector efficiency, beam flux, target thickness, decay data) are common to all energy points. Treating these as independent leads to:

• Biased model estimations.
• Inconsistent nuclear data evaluations.
• Incorrect statistical weighting in global model fits.
• Inability to properly propagate uncertainties for applications like reactor design or astrophysical reaction rate calculations.

The paper addresses the lack of a standardized, reproducible workflow for constructing covariance and correlation matrices that explicitly account for these interdependencies.

2. Methodology

The author proposes a systematic, Jacobian-based formalism to construct covariance matrices. The methodology involves the following steps:

A. Theoretical Framework

The cross-section ($\sigma$) is defined as a function of multiple parameters ($x_k$), including counting statistics ($C_\gamma$), detector efficiency ($\varepsilon$), beam flux ($\phi_b$), target thickness ($N_t$), decay constants ($\lambda$), and time intervals (irradiation, cooling, counting).
$$\sigma_i = \xi(x_k)$$

B. Sensitivity Coefficients (Jacobian Matrix)

The propagation of uncertainty is calculated using partial derivatives (sensitivity coefficients) of the cross-section with respect to each parameter. These form the Jacobian matrix ($J$):
$$J_{ik} = \frac{\partial \sigma_i}{\partial x_k}$$

• Multiplicative parameters (e.g., flux, efficiency, counts) have constant sensitivities of $\pm 1$ (relative).
• Decay and time parameters have complex, non-linear sensitivities due to exponential decay corrections ($e^{-\lambda t}$). The paper derives specific expressions for $\frac{\partial \sigma}{\partial \lambda}$ and time derivatives, noting that while their contribution is often small, they must be formally included.

C. Detector Efficiency Uncertainty

A specific sub-procedure is detailed for determining the uncertainty in detector efficiency ($\varepsilon$).

1. Standard source data is fitted to an exponential function: $\varepsilon(E_\gamma) = \varepsilon_0 e^{-E_i/E_0} + \varepsilon_c$.
2. The covariance matrix of the fitting parameters ($\varepsilon_0, E_0, \varepsilon_c$) is extracted.
3. This matrix is propagated to the specific $\gamma$-ray energy of interest using the chain rule to determine $u_\varepsilon$.

D. Covariance Matrix Construction

The total covariance matrix ($V_\sigma$) is the sum of statistical ($V^{stat}$) and systematic ($V^{sys}$) components:
$$V_\sigma = V^{stat} + V^{sys} = J \eta J^T$$

• Statistical Component: Only affects diagonal elements (uncorrelated between energy points), arising from counting statistics ($\sqrt{C}$).
• Systematic Component: Affects both diagonal and off-diagonal elements. It is calculated by summing the contributions of correlated parameters (like flux and efficiency) across all energy points.
• Correlation Matrix: Derived by normalizing the covariance matrix elements by the standard deviations ($\rho_{ij} = V_{ij} / \sqrt{V_{ii}V_{jj}}$).

3. Key Contributions

• Systematic Workflow: Provides a step-by-step guide for researchers to move from raw experimental data to a full covariance matrix, explicitly separating correlated and uncorrelated uncertainties.
• Explicit Efficiency Propagation: Demonstrates how to rigorously propagate the covariance of efficiency fitting parameters into the final cross-section uncertainty, a step often overlooked or simplified.
• Handling of Time/Decay Parameters: Derives the specific sensitivity coefficients for decay constants and time intervals, showing how they differ from simple multiplicative factors.
• Reproducibility: Offers a transparent mathematical framework (using standard Jacobian propagation) that can be implemented in any data analysis software (e.g., ROOT, Python, Origin).

4. Results and Application

The methodology was applied to experimental data from a previous study (Ref. [11]) involving $^{144}\text{Sm}(p, \gamma)^{145}\text{Eu}$ cross-section measurements in the 2.6–4.1 MeV energy range.

• Efficiency Fitting: The detector efficiency curve was re-fitted, yielding a covariance matrix for the fit parameters ($\varepsilon_c, \varepsilon_0, E_0$).
• Uncertainty Quantification: At 893.73 keV, the detector efficiency was determined to be $0.03127$ with a relative uncertainty of 3.59%.
• Matrix Generation:
  • The Statistical Covariance Matrix was diagonal (uncorrelated).
  • The Systematic Covariance Matrix showed significant off-diagonal elements, indicating strong correlations between energy points due to shared parameters (flux, efficiency, target thickness).
  • The resulting Correlation Matrix showed values close to +1 for systematic effects, confirming that errors in normalization parameters affect all data points coherently.

5. Significance

• Data Reliability: By including covariance information, nuclear data libraries (such as those from IAEA/OECD) can perform more accurate uncertainty propagation in reactor simulations and astrophysical models.
• Model Validation: The correlation matrix allows for a more rigorous comparison between experimental datasets and theoretical models (e.g., statistical model calculations), preventing the over-weighting of correlated data points.
• Standardization: The paper advocates for a shift from reporting "total error bars" to reporting full covariance matrices, which is essential for the next generation of nuclear data evaluation.
• Low-Energy Physics: The approach is particularly vital for low-energy nuclear astrophysics where cross-sections are extremely small ($\sim$pb-nb) and systematic normalization errors dominate over statistical ones.

In conclusion, this work establishes that ignoring correlations leads to an incomplete and potentially misleading representation of experimental data. The proposed formalism ensures that the "statistical weight" of data is assigned correctly, enhancing the reliability of nuclear physics applications.