A heavy ion test produces counts. A mission needs a rate. The chain between them has six links, and every one of them is a place where a factor of two or ten can hide. This guide walks the chain in order, names the assumption at each link, and finishes with a worked example from a published NASA case study that you can reproduce in the site's tools to the last digit.
1. Cross section from counts
At each test condition you have a number of events N observed in a fluence F of ions per square centimetre at a known LET. The cross section is sigma = N / F, in square centimetres per device (or per bit, if you divide by the bit count). Two things to watch. First, the fluence is what reached the die, after any degrader or window; the beam calculator handles that. Second, N is a Poisson count, so its uncertainty is roughly its square root: ten events is a 30 percent measurement, a hundred is 10 percent. Zero events in a fluence F does not give a cross section of zero; it gives an upper bound of about 2.3 / F at 90 percent one sided confidence. That bound, not zero, is what goes into a rate.
2. Effective LET and angle
Tilting the device by an angle theta from normal incidence makes the ion cross a thin sensitive volume along a longer path, so the deposited charge scales as LET / cos(theta). Test campaigns use this to reach higher effective LET than the beam alone provides, and cross section curves are plotted against effective LET. The assumption is that the sensitive volume is thin compared with its width; it holds for most planar CMOS memories and logic, it fails for deep structures such as latchup paths and power devices, and it is wrong in a particular way for FinFET and other non planar devices. When it fails, tilted data disagree with normal incidence data, and the alpha law tool is the right instrument.
3. Fit a Weibull
The cross section curve is almost always fitted with the four parameter Weibull
where sigma_sat is the saturated cross section, L0 the threshold LET, W the width and s the shape. The fit should be weighted by the Poisson uncertainty of each point, and null results should enter as bounds rather than as zeros. Two cautions. The threshold is the least well determined parameter, because there is rarely data near it, and it is the parameter the rate is most sensitive to in a benign environment. And a fit to three or four points can look excellent while being unconstrained; the cross section tool reports the confidence region so that this is visible.
4. Choose the environment and read the integral flux
The environment is a table of integral flux versus LET: how many particles per square centimetre per second, from all directions, have LET above each value. It depends on the orbit, the phase of the solar cycle, the shielding depth and whether you are asking about the mean environment or a solar particle event. The standard source is a cosmic ray model such as CREME96 run for the mission orbit behind a stated shielding depth, conventionally 100 mil (2.54 mm) of aluminium. Always compute at least two cases: the mean environment at solar minimum, when the galactic cosmic ray flux is highest, and a worst case solar particle event, the worst day or the worst week. The mean case gives the rate the mission lives with; the event case gives the rate it has to survive for a day. They differ by orders of magnitude. Behind 100 mil of aluminium at solar minimum, the integral heavy ion flux above LET 10 is about one particle per square centimetre per day; above LET 37 about 5e-5; above LET 75 about 3e-6.
5. Integrate
The rate is the integral over LET of the cross section times the differential flux, equivalently the integral of the directional averaged cross section against the integral flux. For a thin sensitive volume the directional average is done with the cosine law, the same LET / cos(theta) assumption as in step 2, and the tools do this for you. For a step shaped bound or a deep structure, use no angular enhancement: sigma_sat times the integral flux above the threshold. The difference between the two is not academic; for a step at LET 85 the cosine law overestimates the rate by about five orders of magnitude, because it credits grazing ions at low LET with reaching a threshold they never reach in a deep volume.
Units: if sigma is per device in cm2 and the flux is per cm2 per day, the rate is events per device per day. Multiply by the number of devices and by the mission duration for expected events. For per bit cross sections multiply by the bit count first.
6. From rate to mission numbers
For recoverable effects the expected number of events over a mission of T days is lambda T, and the mission's tolerance is a rate (interruptions per day the system can absorb, given the recovery time). For destructive effects the right quantity is the probability that at least one occurs, P = 1 - exp(-lambda T), summed over every copy of the part, compared against a budget. The SEECA workbench holds that comparison for every function in a design.
7. Worked example: a commercial analog switch
A published NASA case study reports heavy ion transients on a commercial single pole analog switch with a Weibull fit of saturated cross section 2e-3 cm2, threshold 32.1 MeV cm2/mg, width 71 and shape 2.5, and computes a rate for a near Earth interplanetary environment at solar minimum behind 100 mil of aluminium of 1.55e-4 transients per device per day. Entering the same four parameters into this site's SEECA workbench (the COTS camera preset carries them) with the galactic cosmic ray solar minimum environment gives 1.56e-4 per device per day: the cosine law integration reproduces the published figure to better than one percent. Over a five year mission that is 0.28 expected transients, which the paper's application could tolerate because its requirement was one interruption per 30 minutes.
The same paper quotes 0.48 per day for an October 1989 worst day environment. The site's worst day set gives 0.12 and its peak five minute set gives 0.45. That factor of four is the honest spread among solar particle event models and it is why the worst case column is a scale, not a forecast.
8. Where the factors of ten hide
- The threshold. In a benign environment most of the rate comes from just above threshold, where the flux is steep. Moving L0 from 32 to 25 MeV cm2/mg raises the example above by a quarter to a half depending on the width. Moving a threshold below about 15 brings trapped and solar protons into play, which in a low orbit can change the rate by orders of magnitude.
- Counts. A curve built from a handful of events per point has 30 to 50 percent scatter before any fitting.
- The environment model. Solar particle event models differ by factors of several. Trapped proton models differ by a factor of two. The mean galactic cosmic ray flux is the best known input of all.
- The angular assumption. Cosine law for a deep structure, or no enhancement for a thin one, is a large systematic.
- Application conditions. Bias, temperature, clock frequency and operating mode change cross sections, sometimes by an order of magnitude. Test in the flight condition or bound the difference.
- Sensitive time. A part that is only sensitive while it is doing something, or a bit that is rewritten before it is read, sees a fraction of the rate. Only the design knows the fraction.
9. Proton rates
Proton induced events come from nuclear reactions, so the cross section is measured against proton energy, usually rising from a threshold of tens of MeV to saturation by 200 MeV, and fitted with a Bendel or a Weibull form. The rate is the integral of that curve against the trapped and solar proton spectra for the orbit. It matters for parts with heavy ion thresholds below about 15 MeV cm2/mg and for low orbits that cross the South Atlantic Anomaly. If you only have heavy ion data, the Edmonds method in the cross section tool estimates the proton cross section from the heavy ion curve.
Frequently asked questions
What units should a single event rate be in?
Events per device per day is the working unit for a system engineer; per bit per day for memories, converted to per device by the bit count. Events per mission and probability per mission are what the requirement is written in.
Do I need a Weibull, or can I use the measured points?
You can integrate a tabulated cross section directly, but the environment flux between and below your measured LETs matters, so you still need a model for the threshold region. The Weibull is that model; it is not sacred, but it is what everybody else uses, which makes results comparable.
Why do different tools give different rates from the same data?
Usually the environment: a different orbit, solar cycle phase, shielding depth or event model. Then the angular treatment, cosine law versus rectangular parallelepiped with a stated depth versus none. Then the treatment of null points. Compare the integral flux tables before comparing the rates.
Is a null result a rate of zero?
No. It is an upper bound on the cross section, about 2.3 divided by the fluence at 90 percent confidence, and it says nothing about LETs above the highest one tested. A rate computed from a null result is a bound, and it should be labelled as one.
Related tools and pages
- SEE cross section fitting tool: counts and fluence to Weibull with confidence regions.
- SEE rate tool: spectral integration against CREME96 and OMERE style environments, heavy ion and proton.
- Heavy ion beam calculator: LET and fluence at the die after degraders and overlayers.
- Alpha law tool: when tilted and normal incidence data disagree.
- SEECA workbench: rate against what the mission can tolerate.
References
More guides: COTS Parts in Space · Heavy Ion SEE Test Planning · Radiation Hardness Assurance for Commercial Space · Single Event Effects Explained · Total Dose, Displacement Damage and Single Event Effects Compared · Glossary · All guides
Questions, corrections or a case you would like written up: the community forum or contact.