How far off is the cosine law for your part, and what rate does the alpha law give?

Heavy ion upset rates have been computed for forty years by tilting the beam, converting the angle to an effective LET with the cosine law, and integrating the resulting curve, usually with an integral RPP (IRPP) code. That conversion assumes the sensitive volume is a thin sheet. A fin is not a thin sheet, and for FinFETs the IRPP model is known to be insufficient. This tool implements the alpha law of Larry Edmonds of the Jet Propulsion Laboratory, published in 2002, which replaced the cosine law with an exact result for a more general charge collection picture and included the rate integration code to go with it. It fits that law to cross section data taken at several tilts and rotations, recovers the normal incidence curve the law implies, shows how far the data depart from the cosine law, and computes on orbit rates with confidence intervals. For FinFETs it is one possible approach, not a validated one; until a better model exists, it is what we have. It also tells you which additional beam runs would most sharply test the model, because the way to earn trust in it is to predict a measurement and then take it.

What the alpha law is, and whose it is. The physics, the plotting format and the rate integration on this page are the work of L. D. Edmonds at JPL (IEEE Transactions on Nuclear Science 49(3), 1522, 2002), with the automated parameter search due to J. D. Patterson and Edmonds. This tool is an implementation, not a new model. A and B above one, the regime of a volume taller than it is wide, is theirs as well: Patterson and Edmonds scanned A and B from 0 to 10, found a 512 Mbit SDRAM at A = 1.38 and B = 6.56, and noted that the effective LET then falls with tilt. What this page adds is the application to fins, and three things that are not in the papers and should not be laid at their door: the parametric bootstrap with its confidence intervals on the parameters and the rate (Efron's bootstrap, used on SEE data in the spirit of the Monte Carlo studies of Ladbury and of Hansen), the four parameter Weibull option (the DIAL_W variant Edmonds names but did not publish; the Weibull itself is the field's convention, see Petersen), and the interpolated LET grid. Those were added for this site, written with Claude, Anthropic's AI model, under the direction of G. Allen (Space RHA, 2026; see how this site is made). Edmonds showed that if a device's charge collection efficiency falls off on concentric ellipsoids around the sensitive node, the directional cross section obeys σ(L, θ, φ) = α · σ_N(L / α),   α² = (A² cos²φ + B² sin²φ) sin²θ + cos²θ where σ_N is the normal incidence curve, θ is tilt, φ is rotation, and A and B are the vertical to lateral aspect ratios of the ellipsoid in the two azimuthal planes of the die (Z / X = A and Z / Y = B in the RPP that Edmonds' equation 14 assigns to it). A = B = 0 is the cosine law exactly. A = B = 1 is isotropic. Most of the planar literature keeps A and B below 1, but nothing in the derivation requires it, and Patterson and Edmonds had already found a part above it: A or B above 1 describes a volume taller than it is wide, which is a fin, and it predicts an effective LET that falls with tilt, so that near threshold the cross section falls with tilt, which is what bulk FinFETs measurably do (at saturation the law has it rising with the projected area).

About the name. "Alpha law" is a name, not a description, chosen for its resemblance to "cosine law": take the cosine law, replace the cosine with a different function of the angles, and it becomes the alpha law. Neither is a law of nature; both are models. Some in the field prefer "alpha model" for that reason, and the two names refer to the same thing. This page uses "alpha law" because that is how the community compares it to the cosine law, and the phrase will last as long as "cosine law" does.

What the tool does. It first scans A and B with no assumption about the shape of the curve, looking for the pair that collapses every measurement onto one line. It then fits a ladder of models, cosine law through full alpha law, with a chosen normal incidence form, reports which the data prefer and by how much, bootstraps the parameters, integrates the directionally averaged cross section against a CREME96 environment with the same algorithm as Edmonds' DIAL_E code, and hands back the normal incidence curve and a Weibull fitted to the isotropic average, so the result can be used elsewhere. The rate can be computed from the fitted function or, as Edmonds did it, from a table drawn through the collapsed points with A and B from the model free scan, so that no functional form enters at all. The Weibull is a fitting function here, not a claim about the physics.

The other angular model. The same data are also fitted with HICUP, the rectangular parallelepiped model of Connell, Sexton and Prinja (Sandia and the University of New Mexico, 1995), in which the ion crosses a box along an exact distribution of chord lengths and the critical energy is Weibull distributed. It is the model behind CREME96's integral RPP rate, made directional, and its angle integrated rate, I-HICUP, is the closed form of that calculation. The two models make different physical pictures of the same measurements; the page shows both fits, both rates, and a hand off of the RPP parameters to CREME96 style codes.

The two node case. A DICE latch, or any cell hardened so that no single node can flip it, breaks a rule that every single volume model obeys, and neither the alpha law nor HICUP can fit its data at any parameter values. For those parts the page also fits Edmonds' dual-node model (JPL Publication 11-6, 2011): two alpha law nodes whose upset regions become circles in a transformed plane, with the cross section the lens they share, which opens when the beam runs down the node line at grazing incidence. The report's four data sets are preloaded so that its tables can be reproduced before your own data go in.

This section exists because the method asks for something most heavy ion campaigns do not collect, and the tool is only as good as the campaign behind it.

Why the cosine law fails for a fin

The cosine law says that tilting the beam by θ is the same as raising the LET by 1 / cosθ, because the ion's path through a thin sheet grows by that factor. That is a statement about geometry, and it is right for a sheet. A fin is a wall a few tens of nanometers wide and taller than it is wide. Tilt the beam across the wall and the track exits through the sidewall into the isolation oxide; the chord through silicon gets shorter, not longer, and the deposited charge falls. Tilt it along the wall and the chord grows more or less as the sheet picture says. So the same part obeys something close to the cosine law in one rotation and something close to its inverse in the other. Vanderbilt measurements at the 14 and 16 nm node established this experimentally, with normal incidence the worst case for across fin tilting at low LET. The alpha law carries both behaviors in two numbers, A for tilts in one azimuthal plane of the die and B for tilts in the perpendicular plane, the two rotation settings 90 degrees apart. On each axis the parameter says which way the effective LET moves with tilt: below one it rises, as in the cosine law (A = 0 is the cosine law exactly); above one it falls, which is the fin; at one nothing changes with tilt. The two axes need not be in the same regime, and on the worked example they are not, A = 2.3 and B = 0.26. One more thing the law says, which the phrase "falls with tilt" hides: on an axis with A above one the cross section falls with tilt near threshold, because the effective LET falls, but at saturation it rises by the factor alpha, the projected area; on the example at 60 degrees along the A axis the fitted curve gives a factor 60 fall at LET 3 and a factor 2 rise at LET 200. The rate comes from the threshold region, so the fall is what matters, but both are in the model and both are in the plots.

The campaign

Normal incidence first. At least three ions spanning from below threshold to as high an LET as the facility can deliver with adequate range, because the normal incidence curve is what everything else is referred to. Five or more is much better. Saturation must be reached or the Weibull saturation is an extrapolation, and the tool will say so.

Then tilt, at two rotations. Tilt at 45 and 60 degrees, and ideally 30, at two rotations 90 degrees apart. If you know where the fins run, align the rotations with them, since those are the axes on which A and B are defined. If you do not, use the die edges and add a 45 degree rotation if beam time allows; the ellipsoidal form predicts a specific interpolation between the two principal axes, and the third rotation tests it. With only one rotation the data cannot determine B at all, and the tool will tell you to treat B as zero, the conservative choice, or to set it equal to A.

Use true LET and beam normal fluence. Facilities habitually report an effective LET and a device plane fluence for tilted runs. Both bake the cosine law into the raw data, which is the assumption under test. Enter the LET of the ion at the die and the fluence in the plane perpendicular to the beam. If you only have device plane numbers, select that convention and the tool converts, but check that the facility did not already convert once. The tool fits both conventions internally; if the one you selected fits much worse, it says so.

Per bit counts, not per event. The alpha law is a statement about a single cell. It applies to bit cross sections, what Edmonds calls U type, and it need not apply to event cross sections where multiple cell upsets have been clustered, because clustering has an angular dependence of its own. Fit bit data, handle multiple cell statistics separately. Fit all ones and all zeros patterns separately too, since they can have different sensitive nodes with different aspect ratios.

Mind the range at grazing angles. The ion must cross the overlayers and the full sensitive depth after projection by cosθ. A short range ion at 60 or 70 degrees can produce an apparent angular effect that is a range artifact. Nsengiyumva and co-workers flag overlayer range effects above about 80 degrees; stay below that, and choose the longest range ion available at each LET.

Count. Aim for a hundred or more upsets per exposure. The tool's next measurement recommender only proposes runs that can reach a hundred counts inside 1e8 ions per square centimeter for exactly this reason. And put a realistic systematic floor on the errors. With thousands of counts the Poisson error is a percent, and no two beam runs agree to a percent; the default floor of ten percent reflects dosimetry, LET and run to run scatter, and without it the fit is dominated by whichever exposure happened to be longest.

The campaign for a two node cell

A DICE cell or any part hardened so that no single node flips it needs a different campaign, because its response lives where a single node campaign never looks. The report's data show it: cross sections three or four decades below the grazing peak at tilts under 60 degrees, and at a fixed tilt of 65 degrees a factor of more than a thousand between one rotation and another (its Table 1). The measurements that pin the dual-node model are rotation sweeps at fixed tilt and LET, in steps of ten degrees or finer through the node line, at tilts from about 55 degrees to as close to grazing as the beam and the overlayers allow (the report's later data sets go to 87 degrees); tilt sweeps at fixed rotation along the node line; and an LET sweep along the node line at the tilt where the window is widest, because the low LETs carry the rate and the node thresholds and the exponent P are set by them. If the layout is unknown, a coarse rotation sweep at 75 degrees and one LET finds the node line first. Record the directions with no response as zero rows rather than dropping them: under the report's weighting they are what pins the edges of the alignment window. The node separation T should come from the layout; when it cannot, sweep it, since the fit cannot determine it on its own. Nine parameters need a few hundred directions, which is why the report's sets are the size they are; a campaign of a dozen tilted runs will fit, but the intervals will say what it did not measure. All ones and all zeros patterns are separate node pairs and separate fits. The range caution above matters more here than anywhere: at 85 degrees the path through the overlayers is eleven times its normal incidence length, and the longest range ion at each LET is the only safe choice.

Reading the results

Look at the collapse plot first. If the alpha law holds for your part, every exposure, at every tilt and rotation, lies on one line. Scatter about the line is what the model cannot explain, and it should look like counting noise. Systematic structure, for example fine tilt runs that wobble above and below the line, is the signature of multiple fin crossings that a single ellipsoid cannot follow, and the reduced chi square will say so. The scan heat map shows whether A and B are pinned: a sharp minimum is good; a valley along the B axis means you have one rotation and B is unconstrained.

Then the model ladder. The cosine law is the null hypothesis, isotropic and axisymmetric are the intermediate cases, and the full alpha law is the alternative. A delta AIC above ten in favor of the alpha law is decisive. If it is not decisive, your part may not need the correction, which is a perfectly good result. The sum rule row is a consistency check: if the fitted exponents p and q differ materially, no single sensitive volume model of any kind fits the data, and neither this tool nor an RPP calculation will give a reliable rate.

Finally the rate ratio. If the alpha law holds for the part, a ratio below one means the cosine law analysis was conservative, which is common for fins tilted across their width, and a ratio above one means it was optimistic. The bootstrap interval on the rate is the statistical uncertainty of the fit only. It does not include the environment model, part to part variation, or the same LET, different energy effects that no LET based model can represent.

What to hand to other tools

The normal incidence Weibull is the part's curve with the angular distortion removed. An integral RPP code such as CREME96 applies the angular dependence of whatever box it is given (a thin box is the cosine law, a cube is nearly isotropic), so the curve and the box have to go together. Edmonds' own recipe for that is the first hand off: the curve as measured, per bit, with the box of his equation 15, whose dimension ratios are the ellipsoid's (Z / X = A, Z / Y = B) and whose top area is the saturation cross section; the rate panel prints the box and the rate such a code returns with it, which on his examples was within about 15 percent of the direct integration. The directional average Weibull is the second: the isotropic average of the directional cross section, for a tool that computes rate as the integral of a cross section against the omnidirectional differential flux with no angular integration of its own (an RPP code can come close to that with a cube, which adds the least angular dependence an RPP can). The exact table is available as a download and is preferable to the Weibull approximation where a tool can accept it. The HICUP panel gives a third: the fitted box, funnel and normal incidence Weibull in the form CREME96 style codes take, which reproduces the I-HICUP rate in such a code because the two integrate the same thing. The dual-node panel gives a fourth: the report's Table 4 recipe, a Weibull through the dual-node directional average and a cube whose side is the square root of its limiting cross section, which is as close to isotropic as an RPP code gets; the report found it twenty percent below the direct sum, so quote the direct rate and use the hand off only where a code must be fed.

Earning trust in the method

The honest position is that the alpha law is one possible approach for FinFETs, not an established one. What is known is that the integral RPP model is insufficient for them. The alpha law with A or B above one has been shown to fit FinFET angular data well, and has not yet been shown to predict a flight rate; until it has, or a better model appears, its rate is the best available estimate rather than a settled answer. Two things would change that. One is a flight experiment in a cosmic ray dominated orbit with enough bits to count. The other is cheaper: fit the model on the runs you have, let the tool tell you which unmeasured direction would most sharply distinguish it from the cosine law, go and measure that direction, and see whether the point lands on the prediction. Do that a few times on a few parts and the case makes itself. The hold out feature is there so that the prediction is written down before the measurement is made.

What this tool does not do

It assumes LET and direction describe the ion completely. That excludes low energy proton direct ionization, nuclear reactions at angle, and the same LET, different energy effects that appear in highly scaled parts; for those, Edmonds' energy based extension or a Monte Carlo code is needed. The single node models fit one ellipsoid or one box, so a device whose fin and sub fin volumes feed one node with different aspect ratios is beyond them; the collapse scatter and the sum rule diagnostic will show the strain but will not resolve it. The dual-node model covers the case of two nodes that must both collect, one pair at a time, through the geometric overlap of their collection regions; a cell with several vulnerable pairs, or one whose nodes couple through the well potential rather than through shared charge, is beyond it too. None of them handles multiple cell upset statistics. And every fit is to one part in one lot at one bias, with everything that implies.

Model

The directional cross section is counts divided by beam normal fluence, per bit. The alpha law is applied with the ellipsoidal shape function of Edmonds' equation 5, with A and B free on the interval 0 to 10, the range Patterson and Edmonds scanned. Any positive shape function with α(0, φ) = 1 satisfies Edmonds' direction independence sum rule, his equation 7, because substituting u = L / α in the integral of (1 / L) dσ / dL removes the direction dependence; the ellipsoid is the two parameter member of that family. The normal incidence curve is either a four parameter Weibull, σ_sat (1 − exp(−((L − L₀) / w)^s)), or the two parameter exponential used in the original paper and by Kobayashi and Ikuta, σ_sat exp(−L_1/e / L). Both are fitting functions with no physical content (Edmonds, 2005, on the Weibull). In the paper the curve is chosen only after A and B have been found by the model free scatter method and the shape of the collapsed points is visible, so that a poor choice of function cannot corrupt A and B; here the parametric fit finds A, B and the curve together, and the model free collapse scan is run first to seed it and to show whether the two agree. When the data stop short of saturation, the Weibull's saturation and width trade off against each other and the curve above the last point is an extrapolation of the function, not a measurement.

Collapse scan

Following Patterson and Edmonds, for each (A, B) on a grid the exposures are transformed to (L / α, log(σ / α)), sorted by the first coordinate, and scored by the sum of absolute steps in the second divided by its total rise. A perfectly monotone collapse scores one. This is model free in the sense that it commits to no form for the curve, and it is guaranteed to find the global minimum on the grid, which is why it is used to seed and to sanity check the parametric fit.

Fit

Weighted least squares on the directional cross section, with the error on each point the Poisson error from its counts combined in quadrature with a fractional systematic floor, minimized by a bounded Nelder and Mead simplex on logistic transformed parameters with a dozen jittered restarts and a polishing pass. The same code runs in Python in the development harness and in the browser, and the two agree to all displayed digits on the example. Model comparison uses AIC. The sum rule diagnostic refits with independent exponents on α inside and outside the normal incidence function.

Whether the minimum is the minimum. Two things are shown with every fit. The multistart count is the number of the thirteen simplex starts that ended within one unit of chi square of the best, which is the direct test for local minima; on the worked example ten of thirteen do, one stops at B = 0 on the flat side of the B profile described next, three units of chi square above, and two stall far above (chi square 250 and 1360) where the Weibull's saturation and width trade off and at a bound on its exponent, which is why the count is reported and not assumed. The profile chi square holds A, and separately B, at 37 values in eighth octave steps from about a sixth to four times the fitted value and refits everything else at each, starting both from the base solution and from the neighboring profile point and keeping the lower, so the curve is continuous and passes through the fitted chi square at the fitted value. Where it rises by 2.30 and 5.99 above the minimum, the crossings interpolated between profile points, are the joint 68 and 95 percent intervals for the pair (A, B), scaled by chi square per degree of freedom when that exceeds one, the usual allowance for errors the error model does not carry; on the worked example, with chi square per degree of freedom 9.2, they are A from 2.1 to 2.5 and 1.9 to 2.6. A profile that is flat on one side means the data do not pin that parameter on that side, and the interval then reports the edge of the scanned range rather than a number. The model free scan is an exhaustive grid, so its minimum is the global minimum on the grid by construction, and its contours at 5, 10 and 25 percent above the minimum are drawn so that a shallow valley is visible for what it is. On the worked example the A profile is sharp (chi square doubles within a factor 1.5 of the fitted A) while the B profile rises by only three units between B = 0.26 and B = 0.02, and the scan is within ten percent of its minimum for any B from 0 to 0.95. Neither is a numerical artifact. Along the B axis the exposures are close to the cosine law, and the shape function moves by less than fifteen percent for any B between 0 and 0.3 at the tilts measured, so no objective built on these points can separate those values; what matters is that the rate does not depend on them much either, moving by eleven percent between B = 0.26 and B = 0.02 when the rest of the fit is allowed to readjust. The valley is a property of the data and of the physics, and the profile and the scan agree on it.

Uncertainty

A parametric bootstrap: for each replicate the expected counts at every exposure are taken from the fitted model, Poisson resampled, multiplied by a lognormal factor with the systematic floor as its width, and refitted from the base solution. Parameter intervals and the rate interval are percentiles over replicates. The interval is statistical only. The bootstrap is Efron's; resampling Poisson counts about a fitted cross section curve to see what the data can and cannot pin down is the approach of Ladbury's Monte Carlo data quality studies and of Hansen's Monte Carlo rate calculations, and the OMERE uncertainty estimate of Sukhaseum and others propagates cross section fit uncertainty into the rate the same way.

Rate

The rate calculation is Edmonds' direct numerical integration, the algorithm of his DIAL_E program in Appendix C of the 2002 paper, rewritten in JavaScript. The directional average cross section is his equation 12, evaluated by the same 45 by 45 midpoint rule over one quadrant of solid angle using the symmetry of the ellipsoid. The rate is his equation 13, the summation by parts of the integral flux against differences of the average cross section, with the CREME96 integral omnidirectional flux converted from per square meter per second per steradian to per square centimeter per day by the same factor 108.57. His cutoff LET is provided as an input with his defaults (the fitted threshold for the Weibull, L_1/e over 30 for the exponential). Two things differ. DIAL_E sums on the LET points of the CREME96 file, and Edmonds notes that the sum is conservative when those points are coarse; this page interpolates the flux in log log onto a two thousand point grid, which brings the sum within half a percent of the converged integral, still on the conservative side (on the raw 210 point tables used here the sum is about six percent high, on a 400 point grid three). And DIAL_E drops the boundary term H(L₁) σ_avg(L₁) because a CREME96 table starts near 0.001 MeV cm2/mg where every cross section is zero; the tables on this site start at 0.1, so the term is kept. The integrator was checked three ways: against the analytic limits (an isotropic device returns the normal incidence curve exactly, a saturated cosine law device returns half its saturation cross section), and against a line for line port of the DIAL_E listing run on the same LET table with the exponential form, which it reproduces to five digits. The printed listing itself is arithmetically sound; its only defects are typesetting, a comma moved inside the closing quote in the OPEN and WRITE statements and typographic quotes, which a compiler rejects and a reader fixes in a minute.

The table curve

Edmonds names a DIAL_T variant that takes the normal incidence curve as a table and interpolates it, and in the paper the curve is drawn through the collapsed points by hand once A and B are known. The rate panel offers that path. A and B are the minimum of the Patterson and Edmonds scatter score, the grid scan refined by a simplex, so no functional form enters. The collapsed points (L / α, σ / α) are then joined by a weighted isotonic regression of log σ along L / α (the pool adjacent violators algorithm of Ayer and others, weights (σ / error)²), which is the monotone curve closest to the points, interpolated linearly in log log. Above the last point the table holds its last value, so a curve that is still rising there gives a rate that is low by whatever lies beyond the data; below the first point it is extended as a power law with the slope of the first segment. The panel reports the rate from the other curve alongside, and the share of the rate that comes from below the data. The interval for the table curve is a nonparametric bootstrap: counts resampled about the observed values with the systematic factor, A and B refined and the table rebuilt for every replicate. The cosine law counterpart is the same table drawn at A = B = 0, which is the curve through the effective LET points that a conventional analysis produces.

Inputs for an integral RPP code

Edmonds wrote his 2002 paper with a section for people who prefer CREME96 for the rate calculation (Section VI-C). An integral RPP code takes a normal incidence curve and a box, and the box's chord length distribution supplies the angular dependence, so the way to carry the fitted A and B into such a code is to give the box the ellipsoid's dimension ratios, Z / X = A and Z / Y = B (his equation 14), with the top area equal to the per bit saturation cross section so that the code's own normalization is respected: X = ((B / A) σ_sat)^1/2, Y = ((A / B) σ_sat)^1/2, Z = (A B σ_sat)^1/2 (his equation 15), a zero A or B replaced by a small number such as 0.01, and the curve entered per bit. He calls the method not self consistent, because the box is chosen to imitate the alpha law rather than fitted, and found it within about 15 percent of the direct integration for a range of hardness levels, 13 percent low on his Toshiba DRAM example. The rate panel prints these inputs for the fitted curve and, when the curve is a Weibull, the rate an integral RPP code returns with them, evaluated with the closed form of that integration (the I-HICUP calculation with this box and no funnel, which the SEE rate tool's exact effective flux tables reproduce to a percent). On the worked example the box is 0.040 by 0.344 by 0.090 micrometers and the code's rate is 0.89 of the direct integration, the same shortfall Edmonds reported; and the box HICUP fits on its own, with no knowledge of A and B, is 0.036 by 0.366 by 0.083 micrometers, the same box within ten percent on every side. That agreement between a box derived from the alpha law fit and a box fitted independently is the clearest evidence on this page that the two models are describing the same geometry.

HICUP (Heavy Ion Cross section for single event UPset) is the directional RPP model of Connell, Sexton and Prinja at Sandia and the University of New Mexico, first published in 1995 and revised the same year in the form used here. The sensitive volume is a box l by w by h with a funnel length S_f added to every chord, and the energy an ion deposits is its LET times the chord plus the funnel. For a direction Ω the projected area A_p is exact, and the area over which the chord is at least s is the projected area of the box shrunk by s along the beam, a quadratic in s; the paper writes it as A_p minus a removal cross section g₁ h s minus g₂ s², and the point value cross section for a fixed critical energy E is that area at the critical chord E / L minus S_f. The critical energy is then taken to be Weibull distributed across the cells, F(E) = 1 minus exp(minus ((E minus E₀) / W)^b), which at normal incidence is the usual Weibull in LET with L₀ = E₀ / (h + S_f) and width W / (h + S_f). The average over that distribution has a closed form in the incomplete gamma function (their Appendix B), so the directional cross section at any tilt, rotation and LET is a formula in seven parameters. The scaling laws of their equations 14 to 16 invert it: each tilted measurement maps to an effective LET set by the longest chord and the funnel and an effective cross section set by the projected area and the removal term, and the corrected points fall on the normal incidence curve, where the inverse cosine law leaves them scattered. The angle integrated cross section, I-HICUP, saturates at half the surface area of the box and starts at L₀ (h + S_f) / (diagonal + S_f), and the rate is its integral against the omnidirectional flux, evaluated here with the same summation by parts as the alpha law rate so the two are comparable. That rate is the closed form of what CREME96's integral RPP calculation performs numerically (an RPP with a funnel and the normal incidence Weibull as the distribution of thresholds), which is why the fitted parameters can be handed to such a code directly.

The transcription was checked before it was used: the removal cross section against the shrunken box identity and against a Monte Carlo of 400,000 chords through the box, the closed form Weibull average against direct quadrature, the reductions to the Weibull at normal incidence, to a point value for a sharp threshold and to the cosine law for a thin cell without a funnel, the scaling laws as an exact round trip, the paper's Table I device against its Figure 3 (4 percent rms), the angle average's saturation and threshold, the rate against the classic sum over thresholds and against this page's cosine law integrator in the thin limit, and parameter recovery from synthetic counts with bootstrap coverage. The checks and their numbers are on the implementation checks page. The fit is the paper's chi square (their equation 12, the fitting routine there is Letaw's) by the same bounded simplex as the alpha law, in two stages: the normal incidence shape first with the geometry held, then everything free with random restarts, because the seven parameters trade off (A against W and b when the data stop short of saturation, h against S_f when there are few tilts). With one rotation the aspect ratio cannot be determined and is held at one; with fewer than three tilts the funnel is held at zero unless freed. The bootstrap is the alpha law's, with fewer replicates because each refit is slower.

How the two models relate. Both reduce to the cosine law for a thin planar volume and both go beyond it by giving the volume a height: the alpha law through the ellipsoidal shape function with A and B, HICUP through a box with l, w, h and the chord distribution inside it. The alpha law is an effective LET scaling and makes no claim about the box; HICUP is a geometric construction and makes no claim about charge collection efficiency. A tall box (h larger than w) in HICUP produces the same fall of the cross section with tilt that A or B above one produces in the alpha law, and a large aspect ratio l / w the same azimuthal asymmetry that A different from B produces, so the two fits can be read against each other. Two cautions from the literature apply to both. Edmonds showed that the RPP depth and the funnel are fitting parameters, not measurements of the device, and that the same data can be fitted with different depths and correspondingly different curves; the box HICUP returns should be read that way, as a parameterization of the angular data that integrates correctly, not as a drawing. And the same authors extended HICUP a year later to HIC-UP-TS, which replaces the line source of charge by a track with a 1 / r² radial profile to describe same LET, different energy effects; that extension is not implemented here, so like the alpha law this fit assumes LET and direction describe the ion completely.

Which to use. The practical point is worth stating plainly. HICUP assumes a Weibull for the normal incidence curve; that is what makes it a conventional least squares problem, and it is also its weakness, because the Weibull is a fitting function with no claim to the shape of a real curve. Give the alpha law the same assumption, the Weibull option above, and it becomes the same kind of least squares problem with the same weakness, and on the worked example it lands within a few percent of the same rate. What the alpha law keeps that HICUP does not have is the option of assuming no function at all: A and B from the collapse scan, which uses only the ordering of the points, and the rate from the table curve drawn through them. That is the path to take when the normal incidence points do not look like a Weibull, as they do not on the example, where the table curve rate is twelve percent above the Weibull's. HICUP is on this page for the comparison and for the hand off to codes that take an RPP, not because it fits better; when the two agree, the agreement says the angular treatment is not the uncertainty, and the shape of the curve below the data is.

Every model above has one sensitive volume, and every such model obeys a rule that Edmonds states as equation 1 of JPL Publication 11-6: the integral of σ / L² over LET is the same in every direction. For the alpha law the proof is one substitution, u = L / α in the integral of α σ_N(L / α) / L², and it holds for the cosine law, for HICUP through its scaling laws, and for any model in which direction enters as an effective LET and an efficiency factor. Cells hardened so that no single node can flip them, the DICE latch and the Xilinx dual-node cells of the report, need two particular nodes to collect their critical charge from the same ion, and their data break the rule badly: little or nothing at normal incidence, a spike when the beam runs down the line joining the nodes at grazing incidence, and on the report's Table 2 device an integral at 75 degrees tilt along the node line that is four times its normal incidence value while across the node line it is unchanged. On this page the sum rule row is the same fingerprint: fitted exponents p and q that differ by more than about half say that no single node model was ever going to fit, and the reading panel says so when the dual-node fit runs.

The model is Section VII of the report, transcribed equation by equation. Each node keeps the alpha law, with the same A and B for both because they sit in the same silicon, and Edmonds' exponential normal incidence form S_0i exp(−(L_0i / L)^P) with one exponent P shared (equations 40 and 43). A linear change of coordinates in the hit plane turns each node's upset region into a circle of radius √(σ_i / π), and the center distance between the two circles, equation 45, carries the node separation T along the node line, a vertical offset h between the two centroids and the beam direction: at normal incidence with h = 0 it is √(A / B) T, and it collapses toward zero as the beam approaches the node line at grazing incidence. The dual-node cross section is the area the two circles share, the lens of equation 46 and Appendix C, with a rotation offset that places the laboratory rotation zero on the node line (equation 41). The node labels are interchangeable (swapping them flips the sign of h and turns the node line half a turn), so the fit is reported with node 1 the softer node, as the report's tables have it. Nothing in the derivation constrains A and B, so the FinFET regime with A or B above one passes through the same equations, although the report's devices were planar 90 and 65 nm parts and the model has not been tested on a FinFET cell.

The fit. The report used a commercial regression package with hand supplied starts and a recorded norm; here it is the same bounded simplex as the rest of the page, on logistic transformed parameters with A, B, the node sizes and the floor in the log, in two stages: the symmetric geometry first (A, B, one node curve, P, h = 0), then every free parameter from that solution with random restarts, ten in the harness and five to eight on the page depending on the size of the data set. T is held throughout, because the report found that floating it produces a field of local minima in which the regression almost never finds the global one; a list of trial values can be swept and the smallest objective kept, which is the procedure the report recommends. Three objectives are offered. The counting error chi square with the systematic floor is the alpha law's, and it skips rows with no response because they carry no error. The report's weighted least squares in square micrometers, with weight L⁻³ in Section VIII and L⁻³ (1 + 10⁶ σ)^1/2 in Section IX, counts the zero rows, which is how the report's data sets carry the directions where the window is shut; the L⁻³ factor favors the low LETs because the raw omnidirectional flux, not an effective flux, is what the average cross section is integrated against, and the low LETs dominate the rate. An additive floor can be freed for data with a residual response that the lens cannot make. The parametric bootstrap is the alpha law's recipe, with a zero row resampled as a lognormal draw about the model and returned to zero when the draw falls below three tenths of the smallest measured value; forty replicates, twenty above four hundred rows, each refitted from the base solution.

The average and the rate. The directional average is evaluated as the report's Appendix F listing evaluates it: tilt at the odd degrees from 1 to 89, rotation at the odd degrees from 1 to 359, the rotation mean times sinθ dθ summed over the upper hemisphere and the lower hemisphere taken as its mirror, which is exact for h = 0 and the report's approximation otherwise. A 15 by 60 grid serves the bootstrap replicates and differs from the full grid by less than a tenth of a percent. The rate is the same summation by parts as the alpha law rate on the same two thousand point LET grid, so the three rates on the page are like for like. One remark is owed on the report's own numbers. Its Fig. 21 spreadsheet evaluates the average cross section at the upper edge of each LET bin, which is conservative by about twelve percent against the converged integral on the report's 61 point grid: the printed 1.49e-9 per bit per day for the Table 2 device in solar minimum cosmic rays is 1.33e-9 when the integral is converged, and the page reports the converged value. The hand off panel follows the report's Table 4 recipe, a Weibull fitted to the directional average over LET 1 to 30 and an isotropic cube of side equal to the square root of the limiting cross section, so that a code with its own RPP machinery adds no angular dependence of its own; the report found the cube about twenty percent below its direct sum, and the direct rate is the primary number.

The transcription was checked before use; the checks and their numbers are on the implementation checks page. The lens area against a 200,000 point Monte Carlo over twelve random circle pairs and its four analytic limits; the reductions (T = 0 and h = 0 return the smaller single node cross section, the normal incidence center distance is √(A / B) T); the invariant integral obeyed by the alpha law to 0.04 percent and violated by the Table 2 device; the directional average against the 61 values the report prints for Fig. 21, worst difference 0.34 percent on the three printed figures, and its spreadsheet recipe reproduced to 1.49e-9; the site's solar minimum cosmic ray spectrum against the report's flux column to 1.2 percent; the Table 4 and Table 6 rates by the report's bin edge rule to within six percent (2.04e-10 against 2.05e-10, 5.31e-11 against 5.33e-11); the Table 1 single LET data against the report's brute force average; a refit of the 106 rows of Fig. 10 from random starts with the Section VIII weighting, which reaches the report's minimum (objective within two percent) with A, B and P within ten percent of Table 2, the remaining difference lying along the weakly determined direction the report describes; parameter recovery from synthetic dual-node counts, A and B within fifteen percent, the rotation offset within three degrees, chi square per degree of freedom 0.65; and the 45 by 180 quadrature against a 90 by 360 grid.

What the model does not do. It describes one pair of nodes, so a cell whose all-1's and all-0's patterns expose different pairs is two fits, as in the report; a cell with several vulnerable pairs, or a pair whose coupling is through the well potential rather than through the geometric overlap of two collection regions, is outside it. The normal incidence response need not be zero: compact layouts whose regions overlap at normal incidence have a normal incidence cross section as well as the grazing spike, and the reading panel reports which case the fit is in. The node sizes are weakly determined when only the lens they share is seen, and the report set conservative values by hand in that situation; the bootstrap spread of the size ratio is reported so that the same judgment can be made here. Nine parameters need dense angular data, and the campaign section says what to take. And like every model on this page it assumes LET and direction describe the ion completely.

The worked example is a per bit SEU data set on an AMD Versal device taken at the Berkeley 88-Inch Cyclotron in November 2023: seven ions at normal incidence from LET 1.7 to 63, then tilts of 45 and 60 degrees at two rotations 90 degrees apart, about 4.8e8 bits under test. At LET 3.3 and 60 degrees the two rotations differ by a factor of nearly seventy. The fit puts A near 2.3 on one rotation axis, which is the inverse cosine regime of a tall volume, and B near 0.26 on the other, which is close to the cosine law, and the ratio of the alpha law rate to the cosine law rate in a solar minimum cosmic ray environment is about one third. A word on the normal incidence curve, because the rate depends on it. The seven normal incidence points climb four decades between LET 1.7 and 63 and do not look like a Weibull: the rise is steep at the bottom (about the fourth power of LET between 1.7 and 2.7) and still a factor 2.5 between the last two ions, with no sign of saturation. The four parameter Weibull forced through them puts its saturation just past the last point because it has to saturate somewhere, and misses the three lowest points by 35 to 45 percent, which is where three quarters of the cosmic ray rate comes from (96 percent from LET below 20, all of it from below 34). With the table curve, Edmonds' own procedure, the rate comes out 12 percent higher than with the Weibull, and the cosine law comparison moves more, because the effective LET points a cosine analysis produces scatter widely and the Weibull was averaging through them. Both numbers are shown so the choice of curve is visible rather than buried. The fitted Weibull parameters agree with an independent hand analysis of the same spreadsheet to within a few percent; that agreement says the fit is reproducible, not that the Weibull is the right function.

HICUP on the same points is a dead heat with the alpha law: chi square 173 against 175 with six free parameters each, and the two fits pull the same way on 24 of the 25 points, so the residual structure is the Weibull toe at normal incidence and not the angular treatment. The box it wants is about 370 by 36 by 83 nm, an aspect ratio of ten and a height more than twice its width, which is HICUP's way of saying what A = 2.3 and B = 0.26 say in the alpha law; forced to a square cell it fails exactly as the axisymmetric alpha law does, so the azimuthal asymmetry is in the data. The I-HICUP rate in solar minimum cosmic rays is 4 percent below the alpha law rate. That agreement between two different physical pictures is the best evidence this page has that the angular correction for this part is right; the remaining uncertainty is the shape of the curve below the lowest LET measured, which no angular model can supply.

The four dual-node data sets in the example list are the report's own, transcribed from its tables and figures with the tilt, rotation and LET of every row and the cross sections it prints (reproduced there with the manufacturer's permission; the report is a public JPL publication). The early SRAM6 all-1's set of Fig. 10, 106 rows, is the fitting example of the report's Section VIII and loads with its settings, T = 2 micrometers, two node curves and the 1 / L³ weighting, so that the fit can be compared with Table 2: A 0.237 against 0.223, B 0.691 against 0.695, P 0.684 against 0.708, the rest along the weakly determined direction the report describes, and a solar minimum cosmic ray rate of 1.2e-9 per bit per day against the report's 1.49e-9 with its bin edge rule. The later SRAM6 sets, 976 rows for all-1's (Fig. 22) and 576 for all-0's (Fig. 27), are the report's Section IX with its second weighting and T held at the values it used, 1.079 and 4.010 micrometers; they are large, and the page fits them with fewer restarts and replicates, which takes a minute or two. The SRAM11 set of Table 1, 43 directions at a single LET of 58.5, is the brute force example of Section V; at one LET the node thresholds cannot be determined, so it loads with both held at zero and a sweep of T, and what it shows is the alignment structure itself: the 80 degree spike along the node line and nothing across it. None of the four is a FinFET, and none carries counts or errors, so a fractional error is assumed (ten percent, thirty for Table 1) and the bootstrap intervals are indicative.

Related tools: SEE Rate Assessment, SEL Threshold Predictor, Analog SET Guideline, Standards and Data Sources. Corrections and additional angular data sets are welcome through the community forum.