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Single event upsets |
Particles having low magnetic rigidity (i.e. momentum per unit charge) are preferentially turned back by the field, so they are unable to penetrate beyond some depth in the magnetosphere. For each point in the magnetosphere and for each direction of approach to that point, there exists a threshold value of magnetic rigidity, called the geomagnetic cutoff. Below this value, no charged particle can reach the specified point from the specified direction. Above this cutoff value, particles arrive at the specified point from the specified direction as though the magnetic field were not present at all (Lemaître and Vallarta, 1933). Regions in the outer magnetosphere and near the poles can be reached at much lower magnetic rigidities than are required to reach points near the Earth's equator.
The magnetic rigidity is related to the particle's energy by:
As the geomagnetic cutoff varies with the particle's arrival direction, the geomagnetic cutoff transmission has to be averaged over all arrival directions. In the SPENVIS implementation of CREME, this is achieved by a numerical integration over the arrival direction. This is done for each point on the spacecraft orbit, and for each particle energy and charge. SPENVIS also allow to set a vertical cutoff as an effective average over arrival directions, which is described by the CREME formula P_{c} = 16/L^{2} or its update (see Tylka et al., 1997) P_{c} = 14.5/L^{2}. Above L=5 no shielding is assumed.
On the Earth's surface the effect of the Earth's shadow is simple: particles
can arrive from above, not below. For high energies, the portion of the
geometry factor that is occulted falls off with altitude h as:
The effect of geomagnetic storms can be modelled as follows:
The differential energy spectrum f(E) inside the spacecraft
and behind a thickness t (in g cm^{-2} of Al or equivalent) of
shielding, is given by:
Range-energy and stopping power data are provided by Adams et al. (1987). The differential energy spectra inside spacecraft estimates by the equations above are satisfactory for estimating SEU rates provided the shielding thickness does not exceed 50 g cm^{-2}. For greater shielding thicknesses, this method seriously underestimates the SEU rate (Adams, 1983). Precise estimates can be made by an exact transport calculation, following the methods of Tsao et al. (1984).
The transformation from a differential energy spectrum to a differential LET
spectrum is
The final step is to repeat the calculation of f(S) to the differential energy spectra for all the elements in cosmic rays (i.e. protons to uranium) and sum the resulting integral LET spectra to form one composite integral LET spectrum. This spectrum can then be used to estimate SEU rates that result from the direct ionisation of charged particles.
The simple model described above predicts that when the critical charge has been collected, an SEU will occur. The amount of charge collected depends linearly on the LET of the ionising particle and the length of its path withing the sensitive volume. There is, however, another effect that extends the size of the sensitive volume. The intense trail of ionisation left by the charged particle alters the electric field pattern in the neighbourhood of the feature. The field forms a funnel along the particle track and this enhances the efficiency with which charge is collected. This funnel effect can be partly accounted for in the simple model discussed above if the dimensions of the sensitive volume are experimentally determined.
For estimating the soft upset rates due to the direct ionisation by particles originating outside the spacecraft two methods are implemented in SPENVIS: the rectangular parallellopiped (RPP) and the integral rectangular parallellopiped (IRPP) method. Which one is applied, depends on the cross-section method i.e. RPP when the critical charge is given and IRPP when the cross sections are provided by a Weibull function or a table. We describe both methods briefly.
In the RPP method it is assumed that above a unique critical charge Q_{c} all bits of equal size, are upset. In this idealized situation the upset cross section curve is presented as a step function, equal to zero below
some threshold value of the LET and equal to a constant value above it (see figure 1 dashed line).
The upset rate U (in bit^{-1} s^{-1}) has been described by Adams (1983) and is given by:
The equation for U contains the implicit assumption that the LET of each ion is essentially constant over the dimensions of the critical volume. Of course, this is not true for stopping ions very near the end of their range. The equation assumes that the maximum LET of the stopping ion applies over its entire path length in the sensitive volume. This assumption can result in calculated energy depositions that exceed the residual energy of the ion. The problem is especially acute for large sensitive volume dimensions and threshold LET values just below the maximum LET of an ion that is much more abundant than all heavier ions. Fortunately, this circumstance rarely arises. The equation for Uis accurate if the flux of stopping ions is small compared to fast ions having the same LET. Care should be taken in the use of this formula when the threshold LET is just below the edge of a "cliff" in the integral LET spectrum.
Furthermore, the equation for U assumes one continuously sensitive critical node per bit. In general, there may be several critical nodes per bit, each with its own
sensitive volume dimensions and critical charge. In addition, these nodes may only be sensitive part of the time, making it necessary to calculate partial
upset rates for each node and then combine the results, weighted by the fractional lifetime of each node.
When using the RPP method to user has to enter the critical charge Q_{c} in pC.
More accurate estimations of the upset rate can be made using the experimental cross section curve which shows a more
gradually increase to a saturated cross section, σ_{lim}, instead of the sharp cut-on adopted in
the RPP method (see figure 1 full line). In this case the upset rate is given by the sum of a step-wise set of differential upset-rate calculations and is called the IRPP method (Petersen, 1997):
Default devices are the silicon 93L422AM device (Petersen, 1998) with a Weibull fit as cross section method and the gallium-arsenide MESFET 1K SRAM device (Weatherford et al., 1991) with a critical charge as input.
Figure 1. Idealised (dashed line) and representative cross section curves (full line). |
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1) The maximum LET of the stopping ion is applied over its entire path length in the sensitive volume which may result in calculated energy depositions that exceed the residual energy of the ion and thus overestimates the upset rate.
2) Ions with an energy below the minimum energy for an upset may still produce an upset as long as the deposit energy (E_{dep}=L⋅p) is larger than the minimum energy.
To meet the first shortcoming an improved algorithm has been implemented that accounts for the variation of the LET inside the volume. In the approach the required minimum path length p_{min} for depositing the minimum energy E_{min} for an upset is obtained from the condition:
To account for the fact that ions with the same LET value have different energies, we implemented the "slowing and stopping ion" algorithm that uses the ion energy spectra instead of the LET spectrum to calculate the upset rates by direct ionisation. In this way only ions with an energy above the minimum energy can cause an upset. Using the differential ion energy spectra as input, the upset rate is given by,
If experimental values for σ(E) are available they can be entered in a table of cross section (cm^{2}/bit vs. energy (MeV) values which are then either fitted to a 2-parameter Bendel or a 4-parameter Weibull function or are linearly interpolated. In case no fit is required the user can just enter the Bendel or Weibull function parameter values.
The 2-parameter Bendel function is given by (e.g. Stapor et al., 1990):
The 4-parameter Weibull function is given by (e.g. Petersen, 1992):
For the default silicon device 93L422AM (Tylka et al., 1996) a Bendel fit is used while for the gallium-arsenide MESFET
1K SRAM device (Weatherford et al., 1991). experimental data are implemented and linearly interpolated
When no experimental data are avalaible, the PROFIT or SIMPA method can be used to calculate the proton-induced upset cross-sections
on basis of heavy ion-induced cross-section data.
PROFIT method (Calvel et al., 1996):
SIMPA method (Doucin et al., 1994): (not yet available)
The proton differential energy spectra can be obtained with the trapped proton models implemented in SPENVIS and with the exomagnetospheric proton models (both for cosmic rays and solar flares) that are implemented in SPENVIS (the exomagnetospheric components have to be modulated to the spacecraft orbit). The combined proton differential energy spectrum incident on the skin of the spacecraft can then be propagated to the electronics inside using a simplified version of the equation given above for the differential energy spectrum inside the spacecraft .
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Petersen, E.L., Single-event analysis and prediction, Short Course Notes, ch. III, IEEE Nuclear and Space Radiation Effects Conference, Snowmass, 21 July 1997.
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Tsao, C. H., Silberberg, R., Adams, J. H., Jr. and Letaw, J. R., Cosmic Ray Effects on MicroElectronics, Part III: Propagation of Cosmic Rays in the Atmosphere, NRL Memorandum Report 5402, 1984.
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Last update: Mon, 12 Mar 2018