SEE Cross-Section Fitting Tool

Preliminary analysis aid. Check the result against your own analysis and, where needed, a test on the intended part. Full disclaimer.

1. What the tool calculates

From heavy ion counts and fluences at several LETs: a four parameter Weibull cross section curve with exact Poisson error bars, or a bounding Weibull from a single cross section point when that is all you have; a proton and 14 MeV neutron cross section estimated from the heavy ion curve with the Edmonds upper bound; an on orbit rate against preset or imported CREME96 environments; and the probability of at least one event over a mission. Four tabs: Cross-Section Fitting, Proton Estimation, Rate Prediction, Mission Analysis. Exports as PNG, text and PDF.

2. Inputs

Enter rows by hand or paste three columns from a spreadsheet (tab delimited; Tab and Enter move between cells). Include the runs with no events: they enter the likelihood as observations and limit the curve at their LET. In direct cross-section mode the columns are LET, cross section and, optionally, its lower and upper 95 percent bounds; no count or fluence is assumed. Other inputs: the fitting method, the confidence level for the single point estimator, the RPP Z/X ratio (default 0.20, slider locked until you unlock it), the heavy ion and proton environments (preset, or imported: CREME96 .LET, .flx and .tfx files, or two column text), the proton threshold energy (default 15 MeV) and the mission duration.

3. Calculation: Weibull fit and Poisson intervals

The default fit is Poisson maximum likelihood on the counts. Two earlier objective functions remain for comparison with earlier analyses. All three are minimized with multi start Nelder-Mead:

Maximum likelihood is the default and includes the runs with no events as observations. Least squares and reduced chi squared are the tool's earlier methods and are kept for comparison. Sliders let you move any parameter after a fit; the plot follows, and the tool reports how far the moved curve is from the maximum likelihood fit (2ΔlnL, with 9.49 as the edge of the 95 percent region for four parameters).

How much the data determine. The tool counts constraints: the distinct LETs with events, plus one if a run with no events lies below the lowest of them. Let ν be the constraints less the free parameters. With fewer than two LETs with events no curve is fitted, and the single point estimator of section 4 applies. With ν below 1 the fit is descriptive, one of many curves that fit, and the rate range of section 6 is the result to report. With ν of 1 or more and at least as many LETs with events as free parameters the fit is predictive. The count is a necessary condition, not a sufficient one: runs crowded at nearly the same LET, or all on the plateau, can satisfy it and still leave the curve undetermined. For that reason, once the intervals are known, a fit that leaves two or more parameters unbounded within the slider ranges is classed descriptive regardless of its count. The earlier methods, and direct cross sections without bounds, give no intervals; a fit by one of them is classed by its count alone and is not called predictive. A parameter can be held at its current value, which takes it out of the count; state a held value with the result. A fixed minimum number of points is not used, because what the data determine depends on where the points and the runs with no events fall.

Goodness of fit is the Poisson deviance, D = 2 Σ [n_i ln(n_i/λ_i) − (n_i − λ_i)], with a p value from a parametric bootstrap: 200 data sets are drawn from the fitted curve at the fluences of the test, each is refitted, and p = (1 + k)/201, with k the number of replicates whose deviance is at least the observed one. The p value is not reported when ν is below 1. With 200 replicates a p value near 0.1 has a standard error of about 0.02. R² on log₁₀ σ is shown as a description of the curve and is not a test; Pearson chi squared is not used, because it is generally unreliable at low counts. Each parameter gets a 95 percent profile likelihood interval inside the slider ranges; an interval that reaches the end of a range is marked open, which means the data do not bound the parameter on that side within the slider range; the upper end of the L_th interval is at most the lowest LET with events. The 95 percent level of these intervals, and of the rate range of section 6, rests on the chi squared approximation to the likelihood ratio; it is generally approximate at low counts and where a parameter lies at the edge of its range. A run with no events limits the curve at its LET; it does not by itself bound L_th from below.

Direct cross sections. When only σ is known, the curve is fitted by least squares on log₁₀ σ and no count or fluence is assumed. With a lower and an upper 95 percent bound on every point the fit is weighted by them (each point has the standard deviation log₁₀(upper/lower)/3.92 decades, taken as symmetric about the entered value) and gives intervals and a rate range; without bounds it gives neither, since a cross section alone does not carry its own uncertainty. Bounds on some points only are refused, and so are bounds that do not bracket their cross section. A row with an upper bound and no cross section is an upper limit: it adds a penalty only where the curve exceeds it, and in a weighted fit it is given the root mean square uncertainty of the points with bounds. The chi squared of a weighted fit is reported with its degrees of freedom, the number of points with bounds less the number of free parameters.

Correction, October 2026. Until then least squares was the default although this page named maximum likelihood; direct cross sections were given a synthetic count of σ × 10⁷ so that they could enter a counting fit, which assigned them a statistical precision that the entered data did not contain; three points were enough for a four parameter fit; and the goodness of fit was a Pearson chi squared divided by the points less four, at least one. Fits made with the earlier default can be reproduced by selecting least squares.

Error bars are exact two sided 95 percent Poisson intervals from the chi squared relation, divided by fluence:

For zero events the upper bound is λ = −ln(0.025) = 3.69, so zero events in 1e7 ions/cm2 bounds the cross section at 3.69e-7 cm2 at 95 percent. The Gaussian σ ± σ/√n approximation gives [0, 0] for n = 0, which is wrong, and is undersized for n of 1 or 2; the exact intervals are valid at any count [5].

4. Calculation: single point bounding fit

When only a saturated cross section is known, from a go or no go test or a single LET run, the tool builds a bounding Weibull from the empirical correlation between σ_s and onset LET in historical latchup data [9], [10]. Parts with large sensitive areas tend to have low thresholds. Given σ_s, the onset is drawn at the chosen confidence from the lower tail of a lognormal, because a lower onset means a higher rate.

The EHSP trend is a global errors in variables fit to 303 part entries from IEEE REDW, RADECS, TNS, NASA GSFC and proprietary reports, with extrapolated thresholds treated as interval censored and scatter that grows with cross section. Class trends (bulk CMOS, ADC and DAC, MCU and processor, power, logic, FPGA, amplifier, flash, DRAM, BiCMOS, BCD) can be selected when the part class is known. The exponents agree with the CERN trend within their uncertainties (minus 0.161 [minus 0.182, minus 0.139] against minus 0.192); the larger EHSP prefactor comes from censoring the extrapolated thresholds, which removes a downward bias. Derivation, census, model ladder, bootstrap and validation are in the SEL threshold predictor methodology.

Two curve shapes are offered for the bounding Weibull.

Step function. W near zero: the cross section jumps to σ_s at L₀. This is the figure of merit bound and the largest rate any Weibull with that σ_s and L₀ can give:

Database informed (default). Real curves are not steps. The ratio r = R_F / R_B = L₀² / (L₀ + w · 0.288^1/s)² over the 40 empirical (w, s) pairs of the CERN database follows a beta distribution whose parameters are power laws in L₀ (R² above 0.97):

The tool takes r at the chosen confidence, then picks the closest (w, s) pair from the database to give a realistic shape. At low L₀ the distribution sits near zero and the step function overestimates the rate badly; as L₀ rises the width matters less. The database informed fit is typically several times below the step bound and still conservative at the stated confidence. Use the step function only when you want an absolute worst case or deliberately stacked conservatism.

Choose 50 percent for a median estimate, 90 percent for most work, 95 or 99 for high criticality parts. Apply as Current Fit loads the result into the sliders so the proton estimate, rate and mission tabs run from it. Both shapes are bounds, not best estimates. If you have only one non zero point, the output is a bound and should be reported as one; a curve fitted to data is generally preferable when the data can be obtained.

Validation in [9]: against 11 independently measured JPL parts, all but one fell inside the 90 percent interval and seven of eleven near the median. The 40 pair coverage of r space is dense: for L₀ at or above 10 MeV cm2/mg, dropping any one part changes the bounding rate by under 10 percent; at L₀ = 1 coverage holds to 20 percent.

5. Calculation: proton and neutron cross section from heavy ion data

Proton beam time is scarce and heavy ion data often exist without it. The Edmonds upper bound [1], [2] bounds the proton cross section from the heavy ion curve and tabulated nuclear reaction parameters, with no assumption about device geometry, collection depth or sensitive volume:

The integral depends only on the heavy ion curve and is computed once; the energy dependence sits in β/a, taken from Table I of Edmonds and Irom [2]:

Hansen's review of the methods [3] finds that sensitive volume approaches (PROFIT, SIMPA) and machine learning [4] can be tighter for specific devices but need geometry that most devices do not have in any well defined sense, while the Edmonds model is a rigorous bound with no tunable inputs and has held as an upper bound against measured proton data. For screening, use a bound when the available data do not support a well-constrained point estimate. A quick classification is also shown: L_th above 15 MeV cm2/mg, likely proton immune (recoils rarely exceed 15); 5 to 15, low susceptibility; at or below 5, proton susceptible.

6. Calculation: rate and mission probability

The heavy ion rate is Edmonds' effective flux formulation [8]: the integral of an effective flux, which carries the environment and the sensitive volume aspect ratio, with respect to the measured cross section, so the fitted curve is used as measured and the geometry is one explicit parameter. Heavy ion flux tables are precomputed for several Z/X ratios from each environment's integral spectrum with the full chord length distribution of the box (the same integral as CREME96's integral RPP; in September 2026 the folding and six of the eleven tables were corrected from a mean chord approximation that had run 25 to 80 percent low at Z/X = 0.2, the GCR solar minimum table was regenerated from a 2024 CREME96 run because the old one was 5 to 25 percent low against the exact folding of its own spectrum, and the flux in the rate sum moved from the lower edge to the log midpoint of each LET bin, which had overstated rates by 10 to 15 percent; the tool now matches the closed form I-HICUP rate of the alpha law page to within a few percent), so effective LET varies with incidence angle through the box. The upper bound reported with each rate takes, at each LET, the largest effective flux over the tabulated Z/X ratios (1.0 down to 0.01). In the 14 preset environment tables the thinnest tabulated volume, Z/X = 0.01, carries that flux at every LET above about 1.5 MeV cm2/mg. In a check of those tables over the slider ranges of W and s, the rate at Z/X = 0.01 stayed within 0.5 percent of the upper bound for onsets of 1 MeV cm2/mg and above, and a cube (Z/X = 1.0) gave the smallest rate of the tabulated ratios, so the cube is not a bound. For onsets below 1 the bound draws on more than one ratio, and the rate at Z/X = 0.01 fell as much as 19 percent below it. The tool also reports the range of the combined rate over the likelihood region of the fit: the largest and smallest rate at the selected Z/X found over the Weibull parameters within 2ΔlnL of 2.71 of the fit (one-sided 95 percent each way). For each curve shape the search finds the allowed interval of σ_sat by a one-dimensional search, and it searches the shape numerically over the slider ranges, their edges included; a held parameter stays at its value. Where an extreme lies on an edge of the slider ranges, the range is conditional on those ranges. With a proton environment and the Edmonds source, the proton term is computed anew for each parameter set; its integral is summed in steps of 0.5 MeV cm2/mg and is generally not converged for onsets near zero with s below 1. The range measures the uncertainty of the fit only; geometry, environment and the proton model are outside it. Correction, October 2026: for an onset of exactly zero, which is the lowest slider value and a value a maximum likelihood fit can return, the rate sum started its LET grid at zero and returned a rate of zero. The grid now starts at the onset or at 1e-6 MeV cm2/mg, whichever is higher, and the cross section the curve has reached at that LET is the first term of the sum; rates for onsets above 1e-6 are unchanged. Total rate is heavy ion plus proton; MTBE is its reciprocal in the most convenient unit. The mission tab plots P(≥1) against time with markers at 1, 5, 10 and 15 years and at your duration, for a constant environment; for phased missions run each phase and add the expected counts.

Choosing Z/X

The RPP dimensions are fitting parameters chosen so that the box reproduces the device's directional response; they need not match any physical dimension [8]. The diagnostic is the cosine law. If tilted data give the same cross section at the same effective LET as a heavier ion at normal incidence, the volume is thin and Z/X should be small; that is most planar CMOS. If the minimum LET to upset is lower at normal incidence than at tilt regardless of azimuth, the volume is deeper than it is wide and Z/X above 1 may be right; that happens in some SOI and 3D structures, and for those the alpha law tool offers an alternative, one possible approach that has not yet been validated.

The default 0.20 represents a planar geometry. Changes should be supported by angular data, a process cross section, or TCAD; unlock the slider to adjust it. When in doubt report the upper bound over Z/X alongside.

Environments

All presets except the legacy 1470 km proton table are one set of CREME96 runs made on 11 and 12 September 2026: solar minimum (the CREME96 solar quiet mode), AP8MIN, quiet geomagnetic transmission unless the label says stormy, 100 mil aluminum. The environment plot shows the integral spectrum, the flux above each LET, which is what CREME96 writes and what the rate integral uses. Import your own: heavy ion .LET files (native format with LETMIN, LETMAX, LBINS metadata, or two columns of LET and differential flux); proton files as either a CREME96 FLUX or TRANS output (.flx or .tfx; use the transported .tfx so the shielding is included), read natively: the proton block of the differential spectrum, in particles per m2 s sr MeV, is integrated above each energy and converted to per cm2 per day over 4 pi steradian, the way the presets were made; or two columns, energy in MeV and the integral flux above that energy, in the unit picked next to the import button (per cm2 per s, per cm2 per day, or per m2 per s per sr). The proton rate works in integral flux per cm2 per day throughout. The import message gives the flux above 10, 30 and 100 MeV so the conversion can be checked, and the spectrum appears as a Custom Proton card.

Correction, September 2026: earlier versions used an imported proton file as it came, without the unit conversion or, for a CREME96 file, the integration. Proton rates from imported files were understated by four to five orders of magnitude (a factor of 86,400 for a two-column file in per cm2 per s). The presets were not affected. Recompute any proton rate made with an imported file.

7. Assumptions

• Counts are Poisson and independent; the fluence is what reached the die.
• The LET column is effective LET for a thin volume. If the part is not planar, tilted points do not belong on the same curve as normal incidence points; the alpha law tool is one possible way to fit them, not yet validated.
• One Weibull, one sensitive volume. Stepped cross sections from multiple volumes need more than one curve.
• The single point estimator's correlation is for latchup in unhardened CMOS. Using it for another effect is a guess, not an estimate.
• The Edmonds estimate is an upper bound on the proton cross section under the model's assumptions, not a measurement.
• Environment rate tables are behind 100 mil of aluminum unless you imported your own.

8. Implementation details

• Nelder-Mead with several starting simplices; the best converged optimum is kept.
• Poisson intervals use the exact chi squared quantiles, not a normal approximation.
• The Edmonds integral is evaluated numerically on the fitted Weibull from L_th upward; the four β/a values give four estimates from one integral.
• The proton rate integrates a step cross section from the threshold energy (default 15 MeV) to saturation against the integral proton spectrum.
• Exports: PNG per plot with data, fit, error bars and annotations; a plain text report of every parameter and result; a multi page PDF with plots and tables (jsPDF, loaded from the network when run locally).

9. Limitations

• A Weibull fit is not evidence that the device response is Weibull. Three points fitted with four parameters can look perfect while the data determine little; the tool classes such a fit as descriptive, and the rate range, not the curve through the points, is the result to report.
• The rate is most sensitive to the threshold region, where data are usually thinnest and where the environment flux is steepest.
• The single point method bounds; it does not characterize. Its correlation carries a factor of two to three in L₀ at one sigma.
• The Edmonds bound can sit an order of magnitude above a measured proton cross section for some devices; when the proton term drives the answer, test with protons.
• The mission probability assumes a constant rate; solar particle events are not constant.

10. References

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