Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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330 Principles of Radiation Interaction in Matter and Detection

Panasyuk and Suslov (1996)] or, equivalently, Equation (2) in [ISO-15390 (2004)]}:

s,N

, t) = 0.37 + 3 × 10

−4

(t − ∆t)

1.45

[GV], (4.61)

where R

is the particle rigidity (in GV) and

(t−∆t) is a smoothed mean-monthly

average of the Wolf numbers

∗

referred to the time t−∆t (e.g., see [Nymmik (2000)]);

∆t depends on the particle rigidity, on wether a solar cycle is even or odd, and on the

solar cycle phase. ∆t expresses the time lag of GCR-ﬂux variation relative to solar

activity (see discussion at page 315) and can be calculated using Equations (4–6)

of [ISO-15390 (2004)] {or, equivalently, Equations (10-11) in [Nymmik, Panasyuk

and Suslov (1996)]}. Outside the magnetosphere, for electrons and nuclei at 1 AU,

the modulated diﬀerential ﬂuxes [J(1 AU, R

, i, t)], i.e., the diﬀerential intensities

per unit rigidity and solid angle, are obtained from:

J(1 AU, R

, i, t) =

)

− φ

s,N

, t)

∆

,t)

[s m

sr GV]

−1

, (4.62)

where β [Eq. (4.70)] is the particle velocity divided by the speed of light; C

, α

and γ

are the parameters

of the non-modulated rigidity spectrum of the ith parti-

cle; ∆

, t) {e.g., see Equation (8) of [ISO-15390 (2004)]} depends on φ

s,N

, t)

and on the charge-number (i.e., the atomic number of the nuclide), β and rigidity

of the particle. For instance [Grandi (2008)], the measured proton-ﬂuxes could be

compared with those expected [using Eq. (4.62)] at 1 AU as function of the pro-

ton kinetic-energy expressed in GeV for i) CAPRICE, hydrogen data [Boezio et

al. (1999)] at the top of the atmosphere (August 1994) collected when the Sun

magnetic-ﬁeld lines in the northern hemisphere are directed outwards (A > 0), ii)

AMS-01 data [AMS Collab. (2000a)] at 1 AU (June 1998, A > 0) and iii) BESS,

proton data [Haino et al. (2004)] at the top of the atmosphere (August 2002) with re-

versed magnetic-ﬁeld (A < 0). A few systematic departures of the computed curves

from the data were observed at low and high energies, but, in general, they do not

exceed the uncertainties of the model (e.g., see [ISO-15390 (2004)]). In addition, it

has to be remarked that, since the balloon experiments

‡

provided their data at the

top of the atmosphere, the ﬂuxes may diﬀer from those at 1 AU, due to the propa-

gation of particles through the magnetosphere. Note that a detailed description of

this model is beyond the purpose of the present book.

Nowadays, the modulation of CRs is often described by means of the so-called

omnidirectional distribution function

††

f(~r, ~p

, t) of CR particles with momentum

, at the position

and time

(see page 324 and, for instance, see [Fisk (1976);

∗

The Wolf number was discussed at page 306.

These parameters are listed, for instance, in Tables 1-2 of [ISO-15390 (2004)].

‡

The CAPRICE spectrometer was ﬂown from 56.5

N and 101.0

W to 56.15

N and

117.2

W [Boezio et al. (1999)]; BESS-TeV spectrometer was launched from 56.5

N and

101.0

W [Haino et al. (2004)].

††

Sometimes, the omnidirectional distribution is introduced as a function of the particle rigidity

[Eq. (4.71)].

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Radiation Environments and Damage in Silicon Semiconductors 331

Potgieter, Le Roux, Burlaga and McDonald (1993); Potgieter (1998); Boella, Ger-

vasi, Mariani, Rancoita and Usoskin (2001); Burger and Sello (2005)] and references

therein). Furthermore, it has to be added that diﬀerent authors attempted to include

a latitudinal-dependent SW speed (e.g., see [Burger and Sello (2005)]) and/or Fisk-

type heliospheric magnetic-ﬁeld (see page 313) on estimating the modulation eﬀect

(e.g., see [Burger and Hattingh (2001)]). However, even though since then more

work has been done, Parker original theory is still important because he introduced

the fundamental concepts for dealing with the propagation and modulation of the

GCRs in heliosphere (e.g., see [Jokipii (1971); Jokipii, Levy and Hubbard (1977);

Jokipii and Thomas (1981)], Chapter V of [Toptygin (1985)], Sections 6.5.3–6.5.4

of [Cravens (1997)], Chapter 13 of [Gombosi (1998)], [Burger and Hattingh (2001)],

Chapter 14 of [Schlickeiser (2002)], Section 7.7 of [Kallenrode (2004)]).

4.1.2.4 Solar, Heliospheric and Galactic Cosmic Rays in the

Interplanetary Space

The cosmic rays

are particles traveling through the interstellar and interplane-

tary space. Inside the solar cavity and, mainly from the corona, the Sun is a major

source of particles and blows oﬀ a plasma [i.e., the solar wind (SW), see Sect. 4.1.2.1,

[Lang (2001); Gosling (2006)] and references therein], consisting primarily of elec-

trons and protons, but also of alpha particles (approximately 2.8–4 % [Alurkar

(1997)]) and other ions (approximately 1%), of which carbon, nitrogen, oxygen,

neon, magnesium, silicon and iron are the most abundant (e.g., Chapter 6 in [Grieder

(2001)]). The SW transports an embedded magnetic-ﬁeld and is estimated to extend

up to ≈ 100 AU (Sect. 4.1.2.2) inside the galactic disk. This interplanetary magnetic-

ﬁeld (e.g., see Sect. 4.1.2.1) aﬀects the passage of charged particles [the so-called

Galactic Cosmic Rays (GCRs)] coming into the solar cavity from our galaxy

‡‡

(see,

for instance, Sect. 4.1.2.3 and [Toptygin (1985); Boella, Gervasi, Potenza, Rancoita

and Usoskin (1998); Boella, Gervasi, Mariani, Rancoita and Usoskin (2001)] and

references therein). The eﬀect is referred to as solar modulation and depends on

the solar activity. At 1 AU in the ecliptic plane [White (1970); Encrenaz, Bibring

and Blanc (1991); Alurkar (1997)], the mean wind velocity is about (400–500) km/s

with mean proton (and electron) density of about 5 particles/cm

−3

which depends

on the solar activity (e.g., ﬂares and sunspots).

Furthermore, some evidence has been found for particle acceleration processes,

in many cases in association with shock waves, throughout the heliosphere (e.g., see

Chapter 3 in [Schlickeiser (2002)]). Solar ﬂares, during which a sudden release of

The reader can see, for instance, Section 24 in [PDB (2008)] and, also, [Ginzburg and Syrovatskii

(1964); Gaisser (1990)], Chapter 2 of [Schlickeiser (2002)].

‡‡

The cosmic rays are considered to be originated in our galaxy, although a few of them, in parti-

cular at ultra high-energy, can have an extragalactic origin [Allard, Parizot and Olinto (2007)]. A

discussion about the local interstellar spectra, i.e., the particle spectra beyond the solar modula-

tion region can be found in [Beliaev, Nymmik, Panasyuk, Peravaya and Suslov (1996)] (see also

references therein).

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332 Principles of Radiation Interaction in Matter and Detection

very large amounts of energy occurs in the solar chromosphere, can also proceed

by ejecting accelerated particles into space. This variety of processes results in

generating particles like i) the Solar Energetic Particles (SEPs) with energies

∗∗

to > 10 MeV/nucleon, accelerated in the solar corona and lasting a few days (see

Chapter 3 in [Toptygin (1985)] and [Smart and Shea (1989); Gabriel (2000); Nymmik

(2006)]), ii) the Energetic Storm Particles (ESPs) with energies up to > 500 MeV,

accelerated in propagating interplanetary shock and lasting hours [Lario and Decker

(2001)], and iii) the Anomalous Cosmic Rays (ACRs). The solar wind and solar

energetic particles have typically low energies and are mostly prevented to reach

the Earth by the magnetosphere or are absorbed in the upper atmosphere. In [ISO-

15391 (2004)], a probabilistic model is provided for the 4–10

MeV SEP-event proton

ﬂuences and p eak ﬂuxes in the near-Earth space beyond the Earth magnetosphere

under varying solar activity

The ACRs (mostly He, N, O, Ne ions) are believed to come from interstellar neu-

tral particles, which have entered into the heliosphere, subsequently ionized by the

SW or UV radiation and ﬁnally accelerated to energies larger than 10 MeV/nucleon,

probably, at the solar wind termination shock (see [Mewaldt et al. (1993); Mewaldt

(1996); Tylka et al. (1996)] and references therein). However, collisions in the ter-

mination shock region cause some ions to reach multiply charge states [Mewaldt

(1996)] and be accelerated to even larger energies. For instance, SAMPEX has ob-

served anomalous cosmic ray oxygen ions at Earth level with energies of about

100 MeV/nucleon ([Mewaldt (1996)] and Chapter 6 in [Grieder (2001)]).

The GCRs mainly consist of protons and alpha particles (about 98%) [Simp-

son (1983); M¨uller (1989); Mewaldt (1994); ISO-15390 (2004); Stanev (2004)] (see

also Sect. 9.11.1), in addition there is a lesser amount of low-Z and medium-Z nu-

clei up to nickel, and an even lesser amount of heavier-Z nuclei

††

, where Z is the

atomic number (Sect. 3.1). At energies & 10 GeV/nucleon (e.g., see Sect. 4.1.2.3),

the solar modulation marginally aﬀects the energy spectra which exhibit an ap-

proximate power law behavior as a function of the kinetic energy per nucleon, E

e.g., their intensities b eing ∝ E

−γ

, where γ is the so-called diﬀerential spectral in-

dex

∗

{see Fig. 9.37, [M¨uller (1989)], and Chapters 5 and 6 in [Grieder (2001)]}

. γ

is ≈ 2.74 (2.60–2.63) for hydrogen (helium) {see Table 5.1 (5.3) in Section 5.4.1

of [Stanev (2004)]}. The spectral index depends on the element, thus relative nu-

clear abundances may depend on energy: for instance, observed and calculated

abundance-ratios are shown in Figures 7 and 8 of [Engelmann et al. (1990)] for

0.8 < E

< 35 GeV/nucleon. Above 50 GeV/nucleon, spectral indices of elements

∗∗

The reader can ﬁnd the deﬁnition of kinetic energies per nucleon in Sect. 1.4.1.

††

Parameters for the description of GCR ﬂuxes and time dependent modulation eﬀects using a

semi-empirical model are presented in [Nymmik, Panasyuk, Pervaja and Suslov (1992); Nymmik

and Suslov (1996); ISO-15390 (2004)] (see also the discussion at page 329).

∗

It is also referred to as spectral index.

In addition, the reader can refer to ﬂuxes and energy distributions of protons, helium, carbon

and iron in [Mewaldt (1994)] (see also [Engelmann et al. (1985, 1990)]), Chapter 5 of [Stanev

(2004)] and Section 24 of [PDB (2008)] (see also [ISO-15390 (2004)]).

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Radiation Environments and Damage in Silicon Semiconductors 333

Fig. 4.11 From hydrogen (Z = 1) to nickel (Z = 28), relative abundances (¥) and relative abundances multiplied by Z

(◦) of GCRs at

16.2 GeV/nucleon normalized to oxygen (i.e., the relative abundance of oxygen is equal to 1) as a function of the atomic number Z (from [Leroy and

Rancoita (2007)]). The oxygen ﬂux at this energy is 0.0106 ±0.0004 m

−2

−1

(GeV/nucleon)

−1

[Engelmann et al. (1990)]. The H and He relatives

abundances are averaged from [AMS Collab. (2000a,b); Sanuki et al. (2000)] (see also [AMS Collab. (2002)]). The Li (Z = 3) relative abundance is

at 5 GeV/nucleon [Mewaldt (1994)].

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334 Principles of Radiation Interaction in Matter and Detection

from hydrogen to iron range from 2.55 to 3.2 (e.g., see Table 3.2 in Chapter 3

of [Schlickeiser (2002)]). For energies from above several GeV to somewhat beyond

100 TeV, the overall intensity of primary nucleons [I

nucl

tot,isot

)] is approximately

given by {see Equation (24.2) in [PDB (2008)]}:

nucl

tot,isot

) ≈ 1.8 × 10

isot

tot,isot

1 GeV

−γ

s sr GeV/nucleon]

−1

, (4.63)

where E

tot,isot

= E

tot

isot

in GeV/nucleon is the energy-per-nucleon

[including

the rest-mass energy, see Eq. (4.66)] of an isotope with I

isot

nucleons and, ﬁnally,

≈ 2.7 is the diﬀerential spectral index of the CR ﬂux. In Fig. 4.11, the rela-

tive abundances of GCRs up to nickel (Z = 28) are shown at 16.2 GeV/nucleon

‡

normalized to oxygen (i.e., the oxygen relative abundance is equal to 1), as a func-

tion of the atomic number Z. The oxygen ﬂux, at this energy, is 0.0106 ± 0.0004

−2

−1

(GeV/nucleon)

−1

[Engelmann et al. (1990)]. The H and He relatives

abundances are averaged from [AMS Collab. (2000a,b); Sanuki et al. (2000)] (see

also [AMS Collab. (2002)]). Since in matter, at suﬃciently high-energies (see

Sect. 2.1.4) the collision energy-loss depends on Z

[Eq. (2.1)], in the same ﬁgure

the relative abundances are also shown multiplied by Z

(e.g., see [Mewaldt (1994);

Codegoni et al. (2004b); Consolandi, D’Angelo, Fallica, Mangoni, Modica, Pensotti

and Rancoita (2006)]). In the cosmic radiation, electrons and positrons are less than

about 1 % of the proton ﬂux at about 10 GeV (e.g., see [M¨uller (1989); Barwick et al.

(1998); AMS Collab. (2000c)] and references therein). In the 5.0 to 50 GeV energy

range, the positron and electron diﬀerential spectral indexes are about 3.31 and

3.09, respectively [Barwick et al. (1998)]. Above 1 GeV, the positrons are less than

10 % of the combined electron–positron ﬂux [M¨uller (1989); Barwick et al. (1998)],

but above 5–10 GeV their fraction has been observed to increase ([M¨uller (1989);

Aharonian and Atoyan (1991); Barwick et al. (1998)] and Chapter 2 of [Schlickeiser

(2002)]).

The GCRs are a major radiation concern for missions with long duration outside

the Earth magnetosphere. In addition, since they have energies large enough, most

of them pass through the magnetosphere and are observed at LEOs.

The ﬂux of GCR’s is a possible hazard to spacecraft electronics, because GCR’s

can penetrate shielding materials [Barth (1997)]. In fact after a standard aluminum

shield of 100 mils, fast energetic charged particles

∗∗

will be able to inﬂict radiation

damages. Each isotopic element of these penetrating charged particles releases a

dose and loses an amount of energy by the NIEL process

‡‡

, which are non-negligible

fractions of those due to the proton component. As a consequence, even for not

heavy-shielded VLSI components, the proton component is expected to contribute

At high energies per nucleon, we have E

≈ E

tot,isot

‡

The Li, Z = 3, relative abundance is at 5 GeV/nucleon [Mewaldt (1994)].

∗∗

These are, for instance, electrons, protons, alpha-particles and iron ions above about 1.2 MeV,

22.5 MeV, 22.5 MeV/nucleon and 90 MeV/nucleon, respectively.

‡‡

The reader can see Sect. 4.2.1, [Colder et al. (2002); Codegoni et al. (2004b); Consolandi,

D’Angelo, Fallica, Mangoni, Modica, Pensotti and Rancoita (2006)] and references therein.

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Radiation Environments and Damage in Silicon Semiconductors 335

Table 4.1 Dependence (from [Leroy and Rancoita

(2007)]) of the dose rate (Sv/day), due to protons

from the inner radiation belt, on the altitude and

inclination of the orbit {e.g., see [Miroshnichenko

(2003)] and references therein}.

Orbit altitude

km 0

◦

300 0.05 0.05 0.05

500 0.16 0.16 0.16

2500–3000 110.00 26.00 18.00

7500 2.60 3.40 2.40

with an important, but not dominant, fraction of both the dose and the NIEL

deposition.

Furthermore, after intro ducing the radiation weighting factor to account for

the biological eﬀectiveness, heavier elements like Fe, Si, Mg and O make an even

larger contribution to the dose equivalent. Moreover, it has been estimated that,

for ≈ 3 g/cm

Al shielding, about 75 % of the dose equivalent is due to GCRs

with kinetic energies lower than 1 GeV/nucleon ([Mewaldt (1994)] and reference

[13] therein). Lockwood and Hapgood (2007) pointed out that, i) while in Earth’s

biosphere we typically receive 2 mSv per year from cosmic radiation, ii) the eﬀective

GCR doses in interplanetary space are greater than in the biosphere by factors of

about 90 and 250 at sunspot maximum and minimum, respectively, and that iii)

even in the most favorable time, a manned trip to Mars would let the crew to receive

up to the lifetime radiation allowance for men and more than the double for women.

4.1.2.5 Trapped Particles and Earth Magnetosphere

The magnetosphere (see Sect. 4.1.2.2) is the volume of space where the Earth’s

magnetic-ﬁeld extends its eﬀects [Knecht and Shuman (1985); Kivelson and Bagenal

(2006); Schulz (2007)]. Close to the Earth, the magnetic ﬁeld is approximately

represented by a dipole, slightly oﬀset from the center of the Earth and inclined

by about 11.3

◦

from the axis of rotation. Its magnetic dipole axis intersects the

Earth’s surface in two points, the boreal (northern) pole at 78.3

◦

N 69

◦

W and

austral (southern) pole at 78.3

◦

S 111

◦

E. The SW compresses the magnetosphere

side toward the Sun to a distance of (6–10) Earth radii (one Earth radius, Re, is

about 6371 km) as a function of the solar activity and stretches the opposite side

into a tail (the so-called magnetotail) to perhaps 1000 Re (Fig. 4.8). The Earth’s

radiation belts in the inner magnetosphere

∗

, one of the ﬁrst major discoveries of

the satellite age [Van Allen, Ludwig and McIlwain (1958); Hess (1968)], are due

∗

For a general description, one can see, for instance, [Spjeldvik and Rothwell (1985)].

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336 Principles of Radiation Interaction in Matter and Detection

to the structure of the magnetic-ﬁeld lines near the Earth and consist of trapped

particles that gyrate around geomagnetic-ﬁeld lines, travel back and forth along

geomagnetic-ﬁeld lines between conjugate points in opposite hemispheres, and drift

around the Earth.

The overall particle ﬂux is due to a combination of Primary Cosmic Rays

‡‡

(PCRs), trapped ACRs, trapped solar-wind particles, soft and energetic Secon-

dary Cosmic Rays (SCRs), mainly consisting of electrons and protons [Buenerd et

al. (2000)] generated by the interactions of PCRs with the atmosphere (e.g., see

Section 12 in [Leroy and Rancoita (2000)], Chapter 1 in [Grieder (2001)], [Stanev

(2004)], Chapters 6 and 7 in [Grupen (2005)] and references therein). Nevertheless,

the Earth magnetosphere prevents the arrival of less energetic cosmic and solar

rays. The value of the rigidity, below which the PCRs cannot reach an observation

position, is called local rigidity of geomagnetic cut-oﬀ

‡

, R

cut

(see, for instance, [Sto-

ermer (1930); Fermi (1950); Shea, Smart and McCracken (1965)] and Section 6.2.3.1

in [Knecht and Shuman (1985)]). The transport of the PCRs, from the Earth mag-

netosphere boundary at 1 AU through the magnetosphere to the atmosphere, is

described by the so-called transmission function (TF), which depends on the PCR

rigidity, location of the observation point, solid angle and pointing of the detecting

instrument (e.g., see [Bobik et al. (2006)] and references therein). The TF becomes

0, when the incoming PCRs cannot reach the observation location any more, thus

the particle rigidity is lower than R

cut

Inside the Earth magnetosphere there are two Van Allen belt regions (e.g., see

[Hess (1968)]) of trapped fast particles (mostly electrons and protons). In between

the two regions, there is the so-called slot region. The inner radiation belt, discovered

by Van Allen [Van Allen, Ludwig and McIlwain (1958)], is relatively compact. In

the region above the equator, it extends perhaps up to about 2.4 Re and is centered

at about 1.5 Re. It is populated by protons with energies in the (10–100) MeV range

and a density of about 15 protons per m

[Lang (2001)]. There are also electrons

of lower energies, but exceeding 0.5 MeV. The population is fairly stable, but is

subject to occasional perturbations due to geomagnetic storms, and varies along a

11 year solar cycle (e.g., see Sect. 4.1.2.1). The main sources of particles are so-

lar energetic particles and interactions of cosmic ray in the atmosphere (e.g., see

[Hess (1968)]). In the inner radiation belt many satellites, designed to provide tele-

phony (whether for mobile or ﬁxed phones) and internet services for a broad range

of users, move on LEOs where they encounter trapped energetic charged parti-

cles. These ones represent an hazard primarily for astronauts and also for VLSI

electronics. The doses change considerably at small altitudes and depend on the

inclination of the orbit plane. For instance, the dose rate for protons of the inner

radiation belt is shown in Table 4.1 (e.g., see Sect. 1.1 of [Miroshnichenko (2003)]

‡‡

Above about 100 MeV/nucleon, the PCRs are mostly GCRs.

‡

For a practical introduction to the cut-oﬀ rigidity of cosmic rays, the reader can see Chap-

ter 1 in [Grieder (2001)]. For practical calculations, the reader can refer to the web sites:

http://hpamsmi2.mib.infn.it/∼wwwams/rig.html and/or http://pfunc.selﬁp.biz/riho/index.html.

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Radiation Environments and Damage in Silicon Semiconductors 337

and references therein). These dose rates take into account the radiation weighting

factor for protons

†

, but can be adapted to provide rate estimates in other absorbers

(e.g., in silicon).

Further out, centered at about (4.5–5.0) Re, the outer radiation belt mainly

contains electrons with energies up to 10 MeV. This belt is mostly generated by

the injection of particles following geomagnetic storms, which make it much more

dynamic than the inner belt (it is also subject to day-night variations). It has an

equatorial distance of about (2.8–12) Re, with a maximum for electrons above 1 MeV

occurring at about 4 Re. Unlike the inner belt, this population ﬂuctuates widely,

rising when magnetic storms inject fresh particles from the tail of the magneto-

sphere, then gradually falling oﬀ again. The outer radiation belt coincides with the

geostationary orbit of many communications satellites.

A feature of the Van Allen belts is the so-called South Atlantic Anomaly (SAA):

the oﬀset of the geomagnetic dipole axis towards the northwest Paciﬁc causes an

anomalously weak geomagnetic-ﬁeld strength in a region which, for an altitude of

about 500 km (i.e. at LEO), extends at latitudes between about -55

◦

–0

◦

and longi-

tudes between about -80

◦

–20

◦

[Barth (1997)]. In this region, trapped radiation par-

ticles can reach lower altitude before bouncing back to the northern hemisphere. The

orbit parameters of a spacecraft (inclination and altitude) determine the number of

passes made per day through this region. At an orbit below an altitude of about

(500–550) km a considerable part of absorbed radiation dose is caused by passing

the SAA.

A few years ago, the analysis of SAMPEX data

has conﬁrmed the existence of

belts containing trapped ACRs, i.e., heavier nuclei (like N, O and Ne) with energies

of the order of 10 MeV/nucleon, already observed by COSMOS satellites [Grigorov

et al. (1991)]. These trapped particles have steep energy spectra and are unlikely

to be a signiﬁcant radiation hazard for (lightly-) shielded systems, i.e., with about

50 mils Al equivalent (see for instance [Tylka et al. (1996)]).

PCRs are particles reaching the Earth from the outer space. However the ﬂux

of PCRs is only a tiny fraction of the total particle ﬂux observed at the Earth’s

surface or at LEOs, since a large amount of SCRs can be generated by the intera-

ctions of the PCRs with the upper and middle atmosphere. Part of the SCRs can

remain trapped by the magnetic ﬁeld near the Earth. Their ﬂux depends on the ﬂux

and energy spectrum of the PCRs and, consequently, on the solar modulation (see

Sect. 4.1.2.4). Thus, the observed ﬂux of SCRs is also related to phenomena like the

solar activity and magnetic storms. Trapped energetic SCRs are always present and

populate a radiation belt at low altitude. The existence of trapped and quasi-trapped

high-energy (up to few GeV/c’s) particles

‡

, generating the so-called AMS Radiation

†

For instance, the radiation weighting factor for protons is 2 (Table 29.1 of [PDB (2008)]). The

biological eﬀects of radiation are treated, for instance, in [Stather and Smith (2005)].

One can see, e.g., [Cook et al. (1993); Cummings et al. (1993); Selesnick et al. (1995); Mewaldt,

Selesnick and Cummings (1997)].

‡

These particles are mostly protons, electrons and positrons.

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338 Principles of Radiation Interaction in Matter and Detection

belt located at LEO, was observed by the AMS mission (Space Shuttle ﬂight STS-91,

June 1998, see [AMS Collab. (2000a,b,d, 2002)] and references therein). As already

mentioned, while the PCRs come from outside the magnetosphere, the SCRs are

originated in the Earth atmosphere. They could be distinguished by reconstructing

their trajectories

∗

from the observation point, i.e., the AMS spectrometer, back to

the border of the magnetosphere (e.g., PCRs) or to the border of the atmosphere

(e.g., SCRs).

Let us indicate by E

tot

the total kinetic-energy of an isotope with I

isot

nucleons

[i.e., with mass number I

isot

, Eq. (3.1)] and rest mass m

isot

∼ I

isot

nucl

where m

nucl

is the nucleon mass (see page 219). The kinetic energy per nucleon,

, is

tot

isot

. (4.64)

From Eq (1.15), the isotope momentum, p

isot

, is

isot

tot

+ m

isot

)

− m

isot

. (4.65)

In addition [see Eq. (1.12)], the isotope total-energy is

tot

isot

= E

tot

+ m

isot

. (4.66)

Moreover under the assumption that the direction of momentum is perpendicular

to the direction of the magnetic ﬁeld, the momentum p

isot

and magnetic rigidity

isot

of the isotope of nuclear charge Ze (e is the electron charge) can be expressed

as a function of the kinetic energy per nucleon by

isot

tot

+ m

isot

)

− m

isot

Zec

, (4.67)

from which we get:

isot

+ I

isot

nucl

)

− (I

isot

nucl

)

Zec

isot

+ m

nucl

)

− m

nucl

isot

+ 2m

nucl

)

. (4.68)

∗

On this subject, the reader can see, for instance, [Bobik et al. (2000, 2001, 2005)] and references

therein.

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

Radiation Environments and Damage in Silicon Semiconductors 339

Conversely, by squaring the two sides of Eq. (4.67), we obtain

isot

+ m

isot

tot

+ m

isot

and, ﬁnally,

isot

+ m

isot

− m

isot

= c





−m

nucl

isot

+ m

nucl





. (4.69)

It has to be noted that, using Eqs. (1.6–1.10), from Eqs. (4.65–4.67) we can derive

isot

tot

isot

Zec

isot

Zec)

+ m

isot

+ [(m

isot

c)/(Ze)]

. (4.70)

Nowadays, it is common to use the quantity (referred to as particle rigidity)

isot

≡

isot

(4.71)

in units of GV, where p

isot

is the isotope momentum in GeV/c. Equa-

tions (4.68, 4.69) can be rewritten, without any assumption on the momentum

and magnetic-ﬁeld directions, as:

isot

+ 2m

nucl

)

, (4.72)

and

= c





−m

nucl

isot

+ m

nucl





, (4.73)

respectively. Equation (4.70) can be rewritten as

isot

)

+ [(m

isot

)/(Ze)]

. (4.74)

From Eqs. (1.15, 4.71), the rigidity of a proton with kinetic energy E

k,p

+ 2m

)

, (4.75)

where m

is the rest mass of the proton. Since m

≈ m

nucl

(Sect. 3.1), from an

inspection of Eqs. (4.72, 4.75) the rigidity of an isotope heavier than proton with

kinetic energy per nucleon E

isot

≈

isot

, (4.76)