The current case study regards the impact on the expected number of single event effects (SEEs) occurring in a device, when both the physical mechanism dealt by means of the so-called restricted energy-loss treatment and time-dependent solar modulation models are combined together.

For the current case study, the expected number (S) for SEEs is computed for a Rad-hard 14-bit 400MSPS D/A converter (i.e., the part type DAC5675A discussed in [HIREX (2010)]) and it is based on the SEE cross section as a function of the mass electronic stopping power from HIREX (2010). In [HIREX (2010)], the SEE cross section for the part type DAC5675A (shown in Figure 5 at full scale current of 5 mA) is expressed by means of a four parameters Weibull function (for the formal equation see, for instance, Sect. 7 of [HIREX (2010)] and the discussion in Sect. 11.4.8.1 of [Leroy and Rancoita (2016)]) as a function of "LET" in MeV cm2 mg-1. The four parameters of the Weibull function at full scale current of 5 mA are reported in Table 2 of [HIREX (2010)]. The SEE cross section measurements were carried with N-, Ne-, Ar-, Fe-, Kr and Xe-ions at about (9.3-9.4) MeV/nucleon. It has to be remarked that, in this webpage, the unrestricted linear energy transfer - i.e., the electronic stopping power (as discussed in this webpage, see also [ICRU-85 (2011)]) - is indicated with the term "LET" in units of [MeV cm2 g-1] (e.g., see Section 2 of [ESCC-25100 (2014)]).

As discussed previously, since a device is usually not thick enough to fully absorb secondary δ-rays, the energy loss suffered by the incoming particle differs from the one deposited inside the traversed medium. As a consequence, the escaped δ-rays result in an effective detectable maximum transferred energy (indicated as W0 in the following) and, in turn, the energy deposition will reach a constant value, called the Fermi plateau at very high energy. Thus, when the energy deposited into an absorber or a device has to be considered, one has to deal with the so-called restricted energy loss - for instance, see the treatment by Fano (1963) - in which the effective detectable maximum transferred energy is accounted for (e.g., see Sect. 2.1.1.4 in [Leroy and Rancoita (2016)]). Finally, it has to be remarked that the electronic stopping power accounts for the energy lost by the incoming particle. In this webpage, it was already treated how, in general, a) the usage of restricted energy-loss is appropriate to account for the deposited energy and b) may result in lower number of expected SEEs with respect to that estimated when the energy loss by a particle is taken into account.

A further effect (discussed in this webpage) was investigated, i.e. that due to the time-dependent solar modulation which affects the omni-directional intensities of galactic cosmic rays (GCRs) species. In particular, five solar modulation model were considered:

HelMod - used within SR-NIEL framework - is a well-known Monte Carlo code developed to describe the transport of GCRs through the heliosphere from the interstellar space down to the Earth. Furthermore, the join GalProp-HelMod (2016) effort allowed one to determine local interstellar spectra (LIS) of protons, antiprotons, ions and electrons (e.g., [HelMod (2019a)]).

For the current case study, Fe-ions spectral fluences in units of [cm2 MeV]-1 were obtained, at 1 AU, for Carrington rotations (CR) 2148 (March 11, 2014 - April 7, 2014) and 2190 (April 29, 2017 - May 26, 2017) with solar conditions close to those of AMS-02 proton observations regarding the Bartels rotations 2464 (March 6, 2014 - April 1, 2014) and 2506 (April 13, 2017 - May 9, 2017) and discussed in this webpage. As illustrated here, CR2148 (CR2190) occurs during high solar activity (low solar activity). The CR duration is 2 356 992 s. In Figure 1 and 2, the spectral fluences of the investigated modulation models at 1 AU are shown from about 2.49 GeV, i.e., from 0.11 GeV above the kinetic energy threshold for allowing an Fe-ion to emerge, when the incoming particle strikes perpendicularly an Al absorber with a thickness of 50 mils (see the discussion in this webpage).

Figure 1. Fe-ion spectral fluences at 1AU as a function of the ion energy in MeV from 2.49 GeV for CR2148 for the presently investigated modulation models: CREME96 (blue curve), CREME2009 (orange curve), ISO-15390 (green curve), ISO-DLR (red curve) and HelMod (black curve).

Figure 2. Fe-ion spectral fluences at 1AU as a function of the ion energy in MeV from 2.49 GeV for CR2190 (upper panel) for the presently investigated modulation models: CREME96 (blue curve), CREME2009 (orange curve), ISO-15390 (green curve), ISO-DLR (red curve) and HelMod (black curve).

Furthermore, in the following we will consider the particles as impinging perpendicularly on the front side of the absorbing medium for estimating both a) the residual Fe-ion fluences (shown in Figures 3 and 4 for CR2148 and CR2190, respectively) after traversing an Al absorber with a thickness of 50 mils and b) the number of expected SEEs.

Figure 3. Fe-ion residual spectral fluences at 1AU as a function of the ion energy in MeV for CR2148 for the presently investigated modulation models: CREME96 (blue curve), CREME2009 (orange curve), ISO-15390 (green curve), ISO-DLR (red curve) and HelMod (black curve); they were obtained using an sr-niel calculator.

Figure 4. Fe-ion residual spectral fluences at 1AU as a function of the ion energy in MeV for CR2148 for the presently investigated modulation models: CREME96 (blue curve), CREME2009 (orange curve), ISO-15390 (green curve), ISO-DLR (red curve) and HelMod (black curve); they were obtained using an sr-niel calculator.

The expected number S of SEEs during Carrington rotations 2148 and 2190 was computed as:

(1)

where Φ(E) is the spectral fluence (e.g., see Figures 1- 4) for the investigated solar modulation models, and σi(E) is the SEE cross section for the part type DAC5675A at full scale current of 5 mA [HIREX (2010)] as function of ion-energy with i= SR-NIEL Framework mass electronic stopping power, mass restricted energy loss for W0=30, 300 keV; σi(E) was obtained using the SEE cross section, σ(LET), as a function of the mass electronic stopping power from HIREX (2010) using the methodology discussed in this webpage. Furthermore, Eq. (1) was computed for energies above 2.49 GeV for particle in free space and above 1 keV for particle emerging from an an Al absorber with a thickness of 50 mils.

In Tables 1 and 2, the value of the parameter S computed using Eq. (1) are reported as a function of solar modulation model (during CR2148 and CR2190) or the Fe-ion spectral fluences in free space and after traversing an Al absorber with a thickness of 50 mils, respectively.

Model CREME96 CREME2009 ISO-15390 ISO-DLR Helmod
CR2148 CR2190 CR2148 CR2190 CR2148 CR2190 CR2148 CR2190 CR2148 CR2190
Stopping Power
SR-NIEL framework
0.151 0.278 0.200 0.293 0.218 0.244 0.223 0.329 0.162 0.292
Restricted energy
loss W0=300keV
0.129 0.248 0.176 0.264 0.193 0.217 0.198 0.298 0.140 0.263
Restricted energy
loss W0=30keV
0.105 0.203 0.143 0.216 0.158 0.178 0.161 0.244 0.113 0.215

Table 1. As a function of solar modulation model, number of SEEs (S) using the SR-NIEL Framework mass electronic stopping power and the mass restricted energy loss for W0=30, 300 keV, for Fe-ions spectral fluences (see Figure 1 and 2) in free space during CR2148 and CR2190.

Model CREME96 CREME2009 ISO-15390 ISO-DLR Helmod
CR2148 CR2190 CR2148 CR2190 CR2148 CR2190 CR2148 CR2190 CR2148 CR2190
Stopping Power
SR-NIEL framework
0.157 0.299 0.211 0.318 0.232 0.262 0.237 0.358 0.168 0.314
Restricted energy
loss W0=300keV
0.135 0.270 0.188 0.290 0.207 0.236 0.212 0.327 0.146 0.286
Restricted energy
loss W0=30keV
0.110 0.224 0.155 0.242 0.172 0.196 0.175 0.274 0.120 0.238

Table 2. As a function of solar modulation model, number of SEEs (S) using the SR-NIEL Framework mass electronic stopping power and the mass restricted energy loss for W0=30, 300 keV, for Fe-ions spectral fluences (see Figure 3 and 4) after traversing an Al absorber with a thickness of 50 mils during CR2148 and CR2190.

By inspection of the Tables 1 and 2, one can remark that i) in particular, with respect to HelMod model, S (i.e., the expected number of SEEs) is slightly underestimated by CREME96, while the ISO-15390 model overestimates (underestimates) the expected number of SEEs during CR2148 (CR(2190) and ii) systematically the values of S increases by (3.7-12.1)% after the Fe-ions traverse an Al absorber with a thickness of 50 mils. As illustrated here, CR2148 (CR2190) occurs during high solar activity (low solar activity). Furthermore, by inspecting Table 2, the usage of the mass electronic stopping power results in larger expected numbers of SEEs with respect to those found when the restricted energy-loss is employed.

Bibliography

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[HelMod (2019b)] HelMod (2019b). The Fe-ion differential intensities were obtained from HelMod code [Ver. 4.0.0 - 01.May.2019] on Oct. 8 (2019). The Fe LIS (local interstellar spectrum) is not final and is based on current GalProp code normalization. At present, the Fe-LIS spectrum is under investigation by GalProp-HelMod (2016) join team and, possibly, an absolute scale decreasing of a few percents might be found.

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