New Procedure for the Evaluation of Fission Product Yields: Application to the Spontaneous Fission of $^{252}$Cf

This paper presents a new evaluation procedure for independent and cumulative fission product yields in the spontaneous fission of $^{252}$Cf, utilizing a Bayesian Kalman filter to integrate experimental data with the Hauser-Feshbach Fission Fragment Decay model to produce mean values, full covariances, and consistent predictions for prompt and delayed neutron and $\gamma$-ray multiplicities.

The Gist

Imagine a nuclear reactor or a nuclear battery as a giant, chaotic kitchen. When a heavy atom like Californium-252 splits (fissions), it's like a massive cake suddenly shattering into two main chunks. But the story doesn't end there. These chunks are hot, unstable, and immediately start throwing off sparks (neutrons) and heat (gamma rays) to cool down. Eventually, they settle into stable, final forms.

The scientists in this paper are trying to create the ultimate "recipe book" for what those final forms look like. They want to know exactly how much of every possible ingredient (fission products) ends up in the final dish.

Here is how they did it, explained simply:

1. The Problem: Guessing vs. Knowing

For a long time, scientists had two ways to guess the recipe:

• The Experimental Way: Measuring the actual ingredients in a lab. This is accurate but messy. Different experiments sometimes give different numbers, and some ingredients are so rare they are hard to find.
• The Theoretical Way: Using a computer model (called BeoH) to simulate the physics of the explosion. This is clean and consistent, but it's just a simulation. It might miss a tiny detail that real life has.

The old way was to pick one or the other, or to try to force them to agree by hand, which is slow and prone to error.

2. The Solution: The "Smart Adjuster" (The Kalman Filter)

The authors invented a new method to combine the lab measurements and the computer model. They used a mathematical tool called a Bayesian Kalman Filter.

Think of the computer model (BeoH) as a blindfolded chef trying to bake a cake. The chef has a recipe (physics laws), but they don't know the exact temperature of the oven or the humidity in the room.

• The Experimental Data are like taste testers who tell the chef, "This cake is too sweet," or "It's a bit dry."
• The Kalman Filter is the smart assistant standing between the chef and the taste testers.

The assistant listens to the taste testers, looks at the chef's current recipe, and says, "Okay, Chef, turn the heat down 2% and add a pinch more flour." The chef tries again. The assistant does this over and over, not just once, but in steps, getting the cake closer and closer to the perfect taste every time.

3. What They Actually Did

The team applied this "Smart Adjuster" to the spontaneous fission of Californium-252 (a heavy atom that splits on its own without needing a neutron to hit it).

• The Inputs: They fed the assistant two things:
  1. The computer model's predictions.
  2. A massive collection of real-world data from a database called EXFOR (which contains thousands of past experiments).
• The Process: They let the assistant tweak the computer model's settings (like how much energy the pieces have, how they spin, and how they share heat) until the model's output matched the real-world data as closely as possible.
• The Result: They didn't just get a new list of numbers. They got a list of numbers plus a map of how uncertain those numbers are and how they relate to each other.

4. The "Map of Relationships" (Covariances)

This is a key part of their breakthrough. Usually, data books just give you a number (e.g., "We have 5% of this ingredient"). They don't tell you if that number is reliable or if it's connected to another number.

The authors created a correlation map.

• Analogy: Imagine you are baking a cake. If you change the amount of sugar, the sweetness changes. But if you change the sugar, the texture might change too, and the baking time might need to adjust.
• In their new data, if you know the amount of one fission product, the map tells you how that changes your confidence in the amount of other products. It connects the dots, showing that if one number goes up, another is likely to go down, or that they are both likely to be wrong in the same way.

5. Did It Work?

They tested their new "adjusted recipe" in two ways:

1. Against Real Lab Data: They compared their new numbers to the actual experiments they used to train the model. The results matched very well.
2. Against the "Official" Library (ENDF/B-VIII.0): They compared their new numbers to the current standard library used by nuclear engineers.
   • The Surprise: Even though they didn't use the official library to train their model, their new numbers agreed surprisingly well with the official library.
   • The Difference: Where they disagreed (mostly on very rare ingredients where there was no lab data), their new method provided a more consistent picture because it was forced to obey the laws of physics and the available data simultaneously.

6. The Bonus: Predicting Other Things

Because their model is based on real physics, they could predict things they didn't even try to fit.

• They calculated how many neutrons are emitted immediately (prompt) and later (delayed).
• They calculated how many gamma rays are emitted.
• The Result: Even though they didn't tell the computer to match these specific numbers, the "Smart Adjuster" tuned the model so well that these predictions also matched real-world data very closely. It's like tuning a radio to get the perfect song, and suddenly the volume and clarity of the voice on the radio also become perfect.

Summary

The authors built a new, smarter way to create nuclear data. Instead of just listing numbers, they created a system that blends computer simulations with real experiments to produce a "recipe" that is consistent, physically realistic, and comes with a detailed map of how all the ingredients relate to one another. They proved this works for Californium-252, and they say the same method can be used for other nuclear reactions in the future.

Technical Summary

1. Problem Statement

Accurate knowledge of Independent Fission Yields (IFYs) and Cumulative Fission Yields (CFYs) is critical for nuclear fuel cycle applications, reactor design, and fundamental studies of the fission process. Current evaluated nuclear data libraries (e.g., ENDF/B-VIII.1) rely heavily on experimental data systematics (such as the England-Rider evaluations) with limited consistency between different fission observables.

• Limitations of Current Methods: Existing evaluations often treat fission yields in isolation, lacking full covariance matrices that describe correlations between different fission products. Furthermore, they do not consistently constrain yields using other simultaneous observables like prompt neutron multiplicity ($\nu_p$) or $\gamma$-ray multiplicities.
• Goal: To develop a unified evaluation procedure that combines nuclear reaction modeling with experimental data to produce consistent IFYs, CFYs, and their full covariance matrices, while ensuring consistency with other prompt and delayed fission observables.

2. Methodology

The authors propose a new evaluation framework centered on the BeoH code and a Bayesian Kalman Filter optimization technique.

A. Theoretical Model: BeoH

BeoH is a deterministic fission fragment decay code based on the Hauser-Feshbach statistical theory.

• Initial Conditions: It models the scission of the compound nucleus ($^{252}\text{Cf}$) by defining initial distributions for mass ($A$), charge ($Z$), total kinetic energy (TKE), spin ($J$), and parity ($\pi$).
  • Mass distribution is modeled as a sum of Gaussian modes (symmetric and asymmetric).
  • Charge distribution follows Wahl systematics with adjustable scaling factors ($f_Z, f_N$) to account for odd-even staggering.
  • Excitation energy sharing between light and heavy fragments is governed by a temperature ratio $R_T(A)$.
• Decay Process:
  • Prompt Emission: Fragments de-excite via prompt neutron and $\gamma$-ray emission using Hauser-Feshbach theory.
  • Delayed Emission: The code calculates cumulative yields by simulating $\beta$-decay chains and $\beta$-delayed neutron/$\gamma$ emission using evaluated decay data (ENDF/B-VIII.0 decay library).
• Outputs: The model generates IFYs, CFYs, prompt neutron multiplicity ($\nu_p$), delayed neutron multiplicity ($\nu_d$), and prompt $\gamma$-ray multiplicity ($N_\gamma$).

B. Optimization: Bayesian Kalman Filter

To optimize the BeoH model parameters against data, the authors employ a Kalman filter, assuming multivariate Gaussian distributions for likelihood and priors.

• Linear Approximation: The relationship between model parameters ($x$) and outputs ($f(x)$) is linearized: $f(x_1) = f(x_0) + C\delta x$, where $C$ is the sensitivity matrix.
• Iterative Process:
  1. Sensitivity Calculation: Sensitivities of CFYs and $\nu_p$ to model parameters are calculated by varying parameters by $\pm 1\%$.
  2. Update Step: The filter updates parameters ($\delta x$) to minimize the difference between model predictions and target data (experimental or library values), accounting for experimental covariances ($V$) and prior parameter uncertainties ($X$).
  3. Non-linearity Check: Because the model is non-linear, the authors perform full BeoH calculations with updated parameters after each filter step to verify the linear approximation holds (checking $\delta_{KB}$, the percent difference between Kalman and full BeoH results).
• Multi-Step Fitting: To prevent small-yield nuclei from biasing the fit, a step-wise optimization is used, starting with high-yield nuclei (>5%) and progressively adding lower-yield nuclei down to 0.2%.

C. Data Sources

Two distinct optimization targets were tested:

1. Experimental Data: Selected CFYs from the EXFOR database (with outlier removal and uncertainty screening).
2. Evaluated Library: CFYs from the ENDF/B-VIII.0 library.

3. Key Contributions

• Unified Evaluation Framework: Demonstrates a method to evaluate fission yields that is intrinsically consistent with other fission observables (prompt/delayed neutrons and gammas) through a single physical model.
• Full Covariance Matrices: Generates complete covariance matrices for both IFYs and CFYs, capturing correlations between different fission products and across different energy points (though this paper focuses on spontaneous fission).
• High-Resolution Energy Grid: While applied here to spontaneous fission (no incident energy dependence), the methodology is designed to provide yields on a fine energy grid for neutron-induced fission up to 20 MeV.
• Validation of Consistency: Shows that optimizing for CFYs alone (or CFYs + $\nu_p$) results in predictions for other observables (like $\nu_d$ and $N_\gamma$) that remain consistent with experimental data, even though those specific observables were not explicitly included in the cost function.

4. Results

The study applied the procedure to the spontaneous fission of $^{252}\text{Cf}$.

• Parameter Optimization:
  • The optimization successfully adjusted BeoH parameters (e.g., Gaussian widths, TKE, and temperature ratios $R_T$) to match target data.
  • Changes to parameters were generally small (<10%), except for specific $R_T$ values in the mass region where data sensitivity was highest.
  • The linear approximation remained valid ($\delta_{KB} < 1\%$) for the majority of the fit, confirming the robustness of the Kalman filter approach.
• Comparison of Targets:
  • Fit to EXFOR: The model reproduced experimental data well, with $\chi^2/N$ values generally low. The fitted BeoH calculations often fell within the uncertainties of the experimental data.
  • Fit to ENDF/B-VIII.0: The model successfully adjusted to reproduce the library values. Discrepancies were noted for specific isotopes (e.g., $^{115}\text{Sn}$) where the BeoH baseline was significantly lower than the library value, highlighting potential issues in the library or the model physics for those specific cases.
• Consistency with Other Observables:
  • Prompt Neutrons ($\nu_p$): The fitted values were consistent with experimental data and the ENDF/B-VIII.0 evaluation.
  • Delayed Neutrons ($\nu_d$): Despite not being included in the optimization, the predicted $\nu_d$ showed good agreement with experimental data (specifically Cox et al.), validating the model's ability to capture odd-even staggering effects via the $f_N$ and $f_Z$ parameters.
  • Prompt Gammas ($N_\gamma$): Some discrepancies were observed, attributed to differences in experimental timing/energy cuts and the model's sensitivity to fragment spin distributions.
• Covariances:
  • The resulting covariance matrices showed strong positive correlations between neighboring mass chains and structured correlations within isotopic chains.
  • Uncertainty Reduction: The inclusion of model constraints significantly reduced the relative uncertainties of the CFYs compared to the current ENDF/B-VIII.0 evaluation, particularly in high-mass regions where experimental data is sparse.

5. Significance and Future Work

• Impact: This work establishes a pathway for next-generation nuclear data evaluations that are physically consistent and provide rigorous uncertainty quantification (covariances). This is essential for advanced reactor simulations and safety analyses where correlations between fission products matter.
• Scalability: The authors note that the method is already being extended to neutron-induced fission ($^{235}\text{U}$, $^{238}\text{U}$, $^{239}\text{Pu}$) up to 20 MeV, which introduces energy dependence and multi-chance fission channels.
• Future Directions:
  • Refining experimental uncertainty templates to better handle outliers and systematic errors.
  • Propagating the new covariance matrices through application codes to quantify their impact on reactor physics parameters.
  • Including delayed neutron multiplicity ($\nu_d$) directly in the optimization loop to further constrain charge distribution systematics.

In conclusion, the paper presents a robust, physics-based methodology for evaluating fission product yields that bridges the gap between theoretical modeling and experimental data, offering a significant improvement in the consistency and reliability of nuclear data libraries.