Help & Methodology Reference
The Space RHA SEE Cross-Section Fitting Tool is a web-based analysis platform for single event effects (SEE) cross-section data. It provides end-to-end workflow from raw test data through Weibull curve fitting, proton/neutron cross-section estimation, in-orbit rate prediction, and mission-level probability analysis — all within a single browser-based interface requiring no software installation.
The tool is organized into four tabs: Cross-Section Fitting (data entry, Weibull fit, and visualization), Proton Estimation (Edmonds Upper Bound model), Rate Prediction (spectral integration against preset or custom radiation environments), and Mission Analysis (event probability over time).
The standard model for fitting SEE cross-section versus LET data is the 4-parameter Weibull cumulative distribution function. This functional form was adopted by the radiation effects community because it naturally captures the threshold behavior, saturation, and shape of measured cross-section curves.
The four parameters are:
| Parameter | Symbol | Description |
|---|---|---|
| LET Threshold | LETth | Onset LET below which no events are observed (MeV-cm²/mg) |
| Saturation Cross-Section | σsat | Limiting cross-section at high LET (cm²) |
| Width | W | Scale parameter controlling the transition width from onset to saturation |
| Shape (Exponent) | s | Shape parameter controlling the steepness of the curve |
The function returns zero for LET ≤ LETth. This model is used throughout the radiation effects community for characterizing SEE susceptibility and is a required input format for rate prediction codes such as CREME96.
This tool provides three optimization approaches for fitting the Weibull model to measured data. All use the Nelder-Mead simplex algorithm with multi-start initialization to improve convergence to the global minimum.
Minimizes the sum of squared residuals between the measured cross-section values and the Weibull model predictions. This is the most common fitting approach and works well when data points have roughly uniform uncertainty. The objective function is:
Uses the Poisson likelihood function directly, which is statistically more appropriate for SEE data since event counts follow Poisson statistics. The MLE method maximizes the log-likelihood:
Here, ni is the observed event count and Φi is the particle fluence for each data point. MLE naturally handles zero-event data points and accounts for the discrete nature of the count data.
Minimizes the chi-squared statistic weighted by the Poisson variance of each measurement point. This gives less weight to points with higher statistical uncertainty (low event counts):
where ν is the number of degrees of freedom (data points minus parameters) and vari is the Poisson-derived variance for each point.
After any automated fit, you can fine-tune all four Weibull parameters using the interactive sliders. This is useful when engineering judgment or knowledge of device physics suggests a slightly different fit than the purely mathematical optimum. The plot updates in real time as sliders are adjusted.
Accurate uncertainty quantification is essential for SEE cross-section measurements. Because SEE event counts are inherently Poisson-distributed — discrete, rare events occurring at a constant average rate per unit fluence — standard Gaussian error bars are inappropriate, especially at low counts. This tool implements exact Poisson confidence intervals using the chi-squared distribution relationship.
For an observed count of n events, the exact 95% Poisson confidence interval on the true mean λ is given by:
The cross-section confidence interval is then obtained by dividing by the particle fluence Φ:
The common approximation σ ± σ/√n (based on the Gaussian assumption that variance equals the mean for Poisson data) breaks down badly for low count experiments. For example, with n = 0 events the Gaussian interval is [0, 0] — implying absolute certainty that the device is immune — whereas the correct 95% Poisson upper bound is λ = 3.69, meaning there is still a 5% chance the true cross-section exceeds 3.69/Φ. Similarly, for n = 1 or n = 2, the Gaussian interval is asymmetric and undersized. The exact Poisson intervals used in this tool are valid for any count, including zero.
When zero events are observed at a given LET, the tool reports a 95% Poisson upper bound cross-section of 3.69/Φ. This value comes from the two-sided 95% confidence interval: λupper = −ln(0.025) ≈ 3.689 (equivalently, χ²0.975, 2/2). This is critical for demonstrating that a device is below a specification limit: observing zero events in 1×107 ions/cm² establishes an upper bound of 3.69×10-7 cm² at 95% confidence. The fluence required to demonstrate immunity below a given cross-section level can be planned accordingly.
In many real-world scenarios, full heavy-ion cross-section versus LET curves are not available. A device may have been tested at only a single LET, or only a saturated cross-section is known from a go/no-go qualification test. The single-point estimation feature uses the empirical correlation between σs and LET0 developed by Ladbury et al. to estimate a conservative LET threshold and produce a bounding Weibull fit suitable for rate prediction.
Analysis of a large dataset of SEL susceptibility data (compiled from the CERN database of over 80 parts and the JPL qualification database of hundreds of parts) revealed a significant inverse correlation between the saturated cross-section σs and the onset LET (LET0). This correlation follows a power law trend that persists over more than 100× in LET0 and six orders of magnitude in σs, with R² ≈ 0.44:
The variation about this trend is well modeled by a lognormal distribution with constant standard deviation:
The inverse relationship also holds — given a known LET0, the distribution of σs follows:
The physical intuition is straightforward: parts with large σs (large sensitive areas) tend to have low LET thresholds, and vice versa. Given only σs, we can estimate LET0 at a desired confidence level using the lognormal distribution. A conservative (rate-bounding) estimate uses the lower tail of the LET0 distribution, because a lower LET0 produces a higher SEE rate.
The tool offers two fit types for constructing the Weibull curve from the estimated LET0 and σs:
Setting the width parameter W ≈ 0 produces a step function — the cross-section jumps immediately from zero to σs at LET0. This yields the absolute maximum rate for the given σs and LET0, following the Figure of Merit (FOM) rate bound:
where CE is the environment rate constant. The actual rate RF for any finite-width Weibull is always less than RB:
Use this option when you need an absolute worst-case bound, for example during early design trades where maximum conservatism is appropriate.
In reality, SEL cross-section curves are never step functions — they rise gradually over some range of LET. Ladbury et al. showed that the 40 empirical w–s pairs from the CERN database can serve as a coarse-grained representation of the likely σ vs. LET behavior for SEL-susceptible parts generally. The ratio of the actual rate to the step-function bound:
follows a β distribution whose parameters are power laws in LET0 (fitted from the 40 w–s pairs for each LET0 value, with R² > 0.97):
For low LET0, the β distribution is concentrated near zero (r ≪ 1), meaning the step function greatly overestimates the rate. As LET0 increases, the distribution broadens and shifts right — the Weibull width becomes less important relative to LET0.
The tool selects the rate ratio r at the chosen confidence level from the β distribution, then backs out a representative Weibull width and shape by matching to the closest w–s pair in the CERN database. This produces a realistic curve shape that is still conservative (bounded at the desired CL) but typically yields rates several times lower than the step function — a more useful estimate for practical risk assessment.
In the Cross-Section Fitting tab, enter the known σsat value, select a confidence level (50% for median, 90% recommended for most applications, 95% or 99% for high-criticality parts), and choose a fit type (Step Function or Database-Informed). Click "Estimate Conservative Fit" to see the estimated LET0 and bounding Weibull parameters. Click "Apply as Current Fit" to load these parameters into the sliders and plot, enabling downstream analysis (Edmonds proton estimation, rate prediction, mission analysis) using the conservative fit.
Ladbury et al. validated the LET0–σs model by comparing predicted LET0 values against 11 independently measured parts from the JPL database. All but one fell within the 90% confidence interval, and seven of the eleven fell close to the median prediction — confirming the model's predictive utility across different vendors, processes, and part functions.
The coarse-grained w–s representation was validated by showing that the 40 database pairs provide dense coverage of the r-value space: for LET0 ≥ 10 MeV-cm²/mg, removing any single part changes the bounding rate by less than 10%. Even for the worst case (LET0 = 1 MeV-cm²/mg), coverage is achieved for 20% allowed error on r.
A critical capability of this tool is estimating proton and neutron SEE cross-sections directly from heavy-ion test data, without requiring separate proton beam testing. This is accomplished using the Edmonds Upper Bound (UB) model.
Proton beam testing is expensive and facility access is limited. For many programs, heavy-ion test data is available but proton data is not. Several methods have been developed to estimate proton cross-sections from heavy-ion data, including the PROFIT code, SIMPA, sensitive volume approaches, and more recently, machine learning techniques. Hansen et al. (2022) provide a comprehensive review of these approaches.
Among the available methods, the Edmonds Upper Bound model is widely regarded as the most practical and conservative approach. Hansen's review found that while more complex methods can achieve tighter estimates in specific cases, the Edmonds model provides a reliable upper bound that requires no assumptions about device geometry, charge collection mechanisms, or sensitive volume shape. This makes it uniquely suited to rapid assessment and screening applications where conservatism is appropriate.
Edmonds derived an upper bound on the proton SEE cross-section by starting from the fundamental relationship between proton nuclear reactions and the resulting secondary ion LET spectra. The key insight is that the proton cross-section can be bounded using only the heavy-ion cross-section curve σHI(L) and tabulated nuclear reaction parameters.
The model takes the form:
where:
| Term | Description |
|---|---|
| σHI(L) | Heavy-ion cross-section as a function of LET, given by the Weibull fit |
| β(E) / a | Energy-dependent coefficient tabulated from nuclear reaction data (combines the nuclear reaction cross-section and secondary particle LET distribution) |
| L | LET of secondary ions (MeV-cm²/mg) |
| a | Unit conversion factor between LET and deposited charge |
A key feature of the Edmonds approach is that the integral ∫(1/L²)σHI(L)dL is computed once from the heavy-ion Weibull fit and is then multiplied by different β(E)/a coefficients to obtain estimates at different energies. The integral itself is independent of particle energy — it depends only on the heavy-ion cross-section curve.
The β/a coefficients used in this tool are taken directly from Table I of Edmonds & Irom (2008):
| Particle / Energy | β/a (MeV-cm²/mg) | Application |
|---|---|---|
| 14 MeV neutron | 1.06 × 10-5 | Terrestrial / atmospheric SEE |
| 50 MeV proton | 4.60 × 10-6 | Trapped proton belts |
| 100 MeV proton | 5.39 × 10-6 | Solar proton events |
| 200 MeV proton | 7.13 × 10-6 | GCR / high-energy protons |
Hansen et al. (2022) reviewed and compared the major proton estimation methods in detail. Their analysis highlights several advantages of the Edmonds approach:
No device geometry assumptions. Unlike sensitive volume (SV) based methods such as PROFIT and SIMPA, the Edmonds model does not require knowledge of device geometry, charge collection depth, or funnel length. Edmonds showed through simulations that for most devices (particularly those not using physical boundaries for isolation), there is no well-defined sensitive volume, making SV-based assumptions inherently questionable.
Guaranteed upper bound. The model produces a rigorous upper bound rather than a point estimate. For radiation hardness assurance purposes, a conservative bound is often more valuable than a potentially tighter but uncertain point estimate.
Simplicity and reproducibility. The calculation requires only the Weibull fit parameters and the tabulated β/a coefficients. There are no tunable parameters, fitting choices, or subjective inputs beyond the heavy-ion cross-section data itself.
Validated against experimental data. Edmonds validated the model against measured proton cross-sections for multiple device types and consistently found that the calculated values serve as upper bounds to the measured proton data.
The tool provides a quick susceptibility assessment based on the heavy-ion LET threshold:
| LET Threshold | Assessment | Rationale |
|---|---|---|
| LETth > 15 | Likely Immune | Proton nuclear reaction secondaries rarely exceed 15 MeV-cm²/mg |
| 5 < LETth ≤ 15 | Low Susceptibility | Some sensitivity possible but cross-section will be small |
| LETth ≤ 5 | Proton Susceptible | Device is likely sensitive to proton-induced SEE |
The tool computes in-orbit SEE rates using spectral integration of the device cross-section against the particle environment spectrum. This follows the standard Integral Rectangular Parallelepiped (IRPP) methodology used in CREME96 and similar rate prediction codes.
The heavy-ion SEE rate is computed by integrating the Weibull cross-section curve against the directional differential LET flux spectrum for each RPP chord-length distribution. The rate table contains fluxes pre-computed at multiple Z/X ratios (R-values) from the environment integral flux spectrum, accounting for the angular dependence of effective LET through the device sensitive volume:
The integration accounts for particles arriving from all angles, with effective LET modified by the chord-length through the RPP geometry. The upper bound rate (computed at R = 1.0, a cube) is also reported as a worst-case estimate.
The proton rate is computed by integrating the estimated proton cross-section (from the Edmonds model) against the integral proton flux spectrum above a user-specified energy threshold (default: 15 MeV). The proton cross-section is treated as a step function from zero at the threshold energy to the Edmonds estimate at the saturation energy:
The total SEE rate is the sum of the heavy-ion and proton contributions. The Mean Time Between Events (MTBE) is the reciprocal of the combined rate, expressed in the most convenient unit (hours, days, or years).
The RPP (Rectangular Parallelepiped) model approximates the device sensitive volume as a box with dimensions X (length), Y (width), and Z (depth). The Z/X ratio is the aspect ratio of the sensitive volume depth to its lateral dimension. CREME96 uses this geometry to compute the chord-length distribution for ions arriving from all directions, which determines how effective LET varies with incidence angle.
A thin, wide sensitive volume (small Z/X, e.g. 0.05–0.20) represents a planar geometry where off-normal particles traverse shorter chord lengths, reducing their effective LET and thus the computed rate. A cube (Z/X = 1.0) gives the highest rate for a given cross-section curve because all chord lengths are maximized — this is the upper bound.
Edmonds' recommendations for the use of CREME96 provide detailed guidance on selecting RPP dimensions. The fundamental requirement is that the RPP dimensions be chosen so that the angular dependence implied by the geometry matches the actual angular dependence of the device. Edmonds emphasizes that the RPP dimensions need not correspond to any physically recognizable device dimensions — they are fitting parameters chosen to reproduce the correct directional response.
The key diagnostic is the cosine law: if SEE test data collected at various tilt angles show that the device satisfies the effective LET cosine law (i.e., the cross-section at a given effective LET is the same regardless of whether that LET was achieved by changing ion species or by tilting), then each RPP lateral dimension should be at least several times the RPP thickness. In other words, the RPP should be a thin slab with a small Z/X ratio. This is the case for most modern planar CMOS technologies.
Conversely, if the data show that the minimum LET for upsetting the device is smaller at normal incidence than at larger tilt angles (regardless of azimuth), the RPP thickness is larger than the lateral dimensions, and Z/X > 1 may be appropriate. This can occur in some SOI or 3D device structures.
| Z/X Range | Typical Scenario |
|---|---|
| 0.01 – 0.10 | Very thin sensitive volumes (e.g., SOI with thin body, some advanced FinFET nodes) |
| 0.10 – 0.30 | Most modern bulk planar CMOS devices; data follows cosine law |
| 0.30 – 0.60 | Thicker collection depths, some power devices or analog circuits |
| 0.60 – 1.00 | Nearly isotropic response; cube gives upper bound rate |
Given the computed SEE rate, the tool calculates the probability of experiencing at least one event over a specified mission duration. This uses the Poisson probability:
where Rate is the combined SEE rate (events/device/day) and T is the mission duration in days. The mission analysis tab plots this probability curve and marks milestones at 1, 5, 10, and 15 years, plus the user-specified mission duration.
This calculation assumes a constant rate environment (the "steady-state" assumption). For missions with time-varying environments (e.g., different orbit phases, solar cycle variations), the rate should be understood as a time-averaged value or the analysis should be performed for each environment segment separately.
| Name | Description |
|---|---|
| GCR Solar Minimum | Galactic cosmic ray environment at solar minimum (worst case GCR), geosynchronous orbit, 100 mil Al shielding |
| GCR Solar Maximum | GCR environment at solar maximum (reduced GCR flux), geosynchronous orbit, 100 mil Al shielding |
| GCR Worst Week | Anomalously large cosmic ray event (extreme GCR), geosynchronous orbit, 100 mil Al shielding |
| ISS Orbit | International Space Station orbit (51.6° inclination, ~400 km), includes trapped and GCR components |
| 1470 km 53° | Medium Earth orbit at 1470 km altitude, 53° inclination — passes through inner radiation belt |
| Name | Description |
|---|---|
| None | No proton contribution — use when device is proton-immune or for HI-only analysis |
| Proton Flare GEO | Solar proton event environment, geosynchronous orbit, 100 mil Al shielding |
| Proton Worst Day GEO | Worst-day solar proton flare, geosynchronous orbit, 100 mil Al shielding |
| Peak 5-min Flare | Peak 5-minute solar proton flux (unshielded) — short-duration extreme case |
| Proton ISS | Trapped proton environment for ISS orbit |
| 1470 km 53° | Trapped proton environment at 1470 km, 53° inclination |
Click "Import Custom HI .LET File" to load a CREME96 LET spectrum output file. The parser accepts the native CREME96 format (with LETMIN, LETMAX, LBINS metadata in comment lines or the second data line) as well as simple two-column text files with LET and differential flux values.
Click "Import Custom Proton Flux File" to load a proton integral flux spectrum. Accepted formats include CREME96 proton flux output (with EMIN, EMAX, EBINS metadata) and simple two-column text files with energy (MeV) and integral flux (#/cm²/s). Supported file extensions are .FLX, .flx, .txt, and .dat.
After importing, a "Custom Proton" card appears in the environment selector. The imported spectrum is used directly in the proton rate integration.
Data can be entered manually row-by-row using the input fields, or pasted directly from Excel or other spreadsheet applications.
Select a range of cells in your spreadsheet containing LET, Events, and Fluence data (3 columns), copy them (Ctrl+C), then click on a cell in the data table and paste (Ctrl+V). The tool will parse tab-delimited rows and populate multiple rows at once. You can also use Tab and Enter keys to navigate between cells in the data table.
| Column | Units | Description |
|---|---|---|
| LET | MeV-cm²/mg | Effective LET at the device surface (accounts for angle of incidence if applicable) |
| Events | counts | Number of SEE events observed (integer, 0 or greater) |
| Fluence | ions/cm² | Total particle fluence delivered at this LET |
The tool provides multiple export options:
Export PNG — Available on each plot. Saves a publication-quality PNG image of the current plot with all data, fits, error bars, and annotations.
Export Text Report — Generates a plain-text summary of all analysis results, including Weibull parameters, Edmonds estimates, rate predictions, and mission probabilities.
Export PDF Report — Generates a formatted, multi-page PDF report with Space RHA branding, embedded plot images, data tables, and all analysis results. Requires an internet connection (for the jsPDF library) when running locally; works automatically when accessed through the deployed website.