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

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350 Principles of Radiation Interaction in Matter and Detection

from which we get

κ =

min

D(E)

D(1 MeV)

φ(E) dE

¸µ

min

φ(E) dE

−1

(4.84)

and, conversely,

1 MeV

min

D(E)

D(1 MeV)

φ(E) dE. (4.85)

As already mentioned, in some calculations the value used for E

min

is about

10 keV. Values of the hardness parameter for some neutron reactor spectra can

be found in [Angelescu et al. (1994)] and, for E

min

= 10 keV, in [Kelly, Luera,

Posey and Williams (1988)].

By replacing the spectral ﬂuence with the neutron kinetic-energy distribution

F (E) obtained by integrating the spectral ﬂuence (assumed to be independent of

the position) over the exposed area F (E) =

φ(E) dS, we can rewrite Eq. (4.81) and

obtain the damage energy lost per unit length, i.e., the non-ionization energy-loss

in units of MeV per cm:

dis

= n

D(E) F (E) dE [MeV/cm]. (4.86)

For a neutron with kinetic energy E, the NIEL amounts to

= n

D(E). (4.87)

Equations (4.86, 4.87) can be used to express the relationship between the da-

mage function and the non-ionizing energy-loss per unit path for any kind of beam

particles with a kinetic-energy distribution F (E) or at ﬁxed kinetic-energy E. Equa-

tion (4.87) can be rewritten in terms of the diﬀerential cross section dσ

(E)/dE

for any incoming particle with kinetic energy E resulting in a nuclear recoil with

kinetic energy E

= n

max

dσ

(

)

) dE

, (4.88)

where E

max

is the maximum energy transferred to the recoil nucleus and the sum-

mation is extended over the possible k reaction channels which allow the generation

of a recoil silicon (for example in the elastic scattering) or a residual nucleus.

4.2.1.1 Knock-on Atoms and Displacement Cascade

When a semiconductor (as any solid material) is subjected to incident radiation

(i.e. neutrons, electrons, protons, ions. . . ), the resulting degree of damage depends

on a number of factors. For instance, the type of interaction (strong or electromag-

netic), the nature and energy of the incoming particle determine the initial-damage,

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Radiation Environments and Damage in Silicon Semiconductors 351

i.e., the number of atoms directly displaced (PKAs) by the incoming particles and

their energy spectrum. These PKAs, in turn, can generate a cascade of displa-

cements, i.e., the cascading damage. That is, the primary displaced atoms, upon

leaving their lattice positions with large enough energies, can interact and displace

other atoms. In this way, a displacement cascade is generated upon full dissipation

of the primary recoil-energy. In the cascade, the energy dissipation is a combination

of elastic interactions (referred to as nuclear or elastic energy-loss

‡‡

) mostly result-

ing in displacements and inelastic processes (referred to as electronic or inelastic

energy-loss), in which moving displaced-atoms excite or ionize atomic electrons on

their passage

. The point defects (see Sect. 4.2.2), created in the silicon lattice at

this initial stage, are often referred to as primary. Secondary defects are created

during the diﬀusion of these primary defects.

The primary recoil energy is deposited by the ionization energy, E

loss

, (including

that deposited by recoiling atoms in the cascade when it occurs) and damage energy

(also referred to as partition or defect producing energy), E

, accounting for di-

splacements and sub-threshold collisions, which transfer energies lower than E

. In

these latter interactions, the knock-on atom cannot escape from its lattice location

and the energy is dissipated in lattice vibrations, as mentioned in Sect. 4.2. The

non-ionizing energy-loss (NIEL) is the damage-energy deposited by an incoming

particle per unit length (e.g., in units of MeV/cm in this section, if not speciﬁed

otherwise). As discussed in Sect. 4.2.1, permanent radiation-damage is induced by

atoms displaced from their lattice locations and, as a consequence, is related to the

amount of damage energy.

As already discussed, the energy and diﬀerential cross section of the incoming

particle determine the energy spectrum of the primary recoil. However, the se-

condary interactions can be treated in the framework of atom–atom interactions

∗

which, for recoil atoms at low energy, are described by largely screened Coulomb

potentials. Kinchin and Pease (1955) estimated the average number of displaced

atoms generated by a PKA with recoiling energy above E

, assuming (i) two-body

hard-sphere atomic collisions, (ii) a sharp displacement threshold E

and (iii) no in-

elastic energy-losses (i.e. no ionization energy-loss in secondary collisions). Torrens

and Robinson (see [Torrens and Robinson (1972); Robinson and Torrens (1974);

‡‡

For a further discussion see, for instance, Sects. 2.1.4 and 2.1.4.1; see, also, Chapter 2 of [Ziegler,

Biersack and Littmark (1985a); Ziegler, J.F. and M.D. and Biersack (2008a)].

For incoming high-energy particles, this type of energy-loss process is commonly referred to

as collision energy-loss process (e.g., see Sects. 2.1, 2.1.1, 2.1.4 and discussion in the chapter on

Electromagnetic Interaction of Radiation in Matter ; in addition see Chapter 3 of [Ziegler, Biersack

and Littmark (1985a); Ziegler, J.F. and M.D. and Biersack (2008a)]).

∗

The reader can see, for instance, Sections 2.2–2.4 of [Dienes and Vineyard (1957)], Section 2.2.8

of [van Lint, Flanahan, Leadon, Naber and Rogers (1980)], [Ziegler, Biersack and Littmark (1985a);

Ziegler, J.F. and M.D. and Biersack (2008a)] and references therein.

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352 Principles of Radiation Interaction in Matter and Detection

Fig. 4.15 Fractions of kinetic energy of recoil silicon deposited in processes producing de-

fects (dashed line) and ionization-losses (dashed and dotted line) [calculated by means of

Eqs. (4.94, 4.95)] (from [Leroy and Rancoita (2007)]). The experimental data (¥) are from [Sattler

(1965)].

Norgett, Robinson and Torrens (1975)] and references therein) have estimated

the average number of displacements < N

F P

>, i.e., of Frenkel-pairs, by means of

extensive computer simulations of collision cascades accounting for inelastic losses

and the damage energy left available. The number of Frenkel-pairs generated in a

cascade is given by the modiﬁed Kinchin–Pease formula

< N

F P

> =







1 for E

< E

≤ 2 E

/ζ,

ζE

/(2E

) for E

> 2 E

/ζ,

(4.89)

where ζ ≈ 0.8 is a constant independent of energy (except for recoil energy close to

[Norgett, Robinson and Torrens (1975)]). The factor ζ is called displacement

eﬃciency (e.g., see discussion at pages 425 and 431–435 in [Torrens and Robin-

son (1972)]). For silicon, as discussed at page 347, the threshold energy to create

a displacement is about 25 eV. In Eq. (4.89), the factor 2 was determined in the

framework of the above mentioned Kinchin–Pease analytical treatment, in which

a sharp displacement threshold energy, E

, is assumed. When a more realistic in-

teratomic potential is used instead of the hard-sphere model, low-energy transfers

are favored, thus resulting in a smaller number of displaced atoms per cascade. The

displacement eﬃciency is the parameter accounting for such a decrease and for the

eﬀects of thermal vibrations in the lattice.

The energy depositions E

loss

and E

can be computed by means of analytical

For Cu, a comparison between < N

F P

> computed using the modiﬁed Kinchin–Pease formula

and that created by electrons and ions can be found in [Merkle, King, Baily, Haga and Meshii

(1983)].

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Radiation Environments and Damage in Silicon Semiconductors 353

approximation of Robinson [Robinson (1970)] used, for instance, in the SRIM

‡‡

code

(see pages 115–166 in [Ziegler, Biersack and Littmark (1985a)] and also [Ziegler, J.F.

and M.D. and Biersack (2008b)]). In a silicon medium, for a recoil silicon nucleus

with kinetic energy E

in eV and atomic number Z

= 14 we have

1 + k

g(²

)

, (4.90)

where:

= 1.014 × 10

−2

−7/3

= 2.147 × 10

−5

, (4.91)

g(²

) = ²

+ 0.40244 ²

3/4

+ 3.4008 ²

1/6

, (4.92)

= 0.1462. (4.93)

In the latter expressions ²

, g(²

), k

are dimensionless. Using Eq. (4.90), the

fractions of energy deposited by non-ionization (f

niel

) and by ionization (f

ion

) can

be computed from:

niel

1 + k

g(²

)

, (4.94)

ion

loss

= 1 −

g(²

)

1 + k

g(²

)

. (4.95)

In Fig. 4.15, f

ion

and f

niel

are shown and compared with the experimental data

of Sattler (1965) for neutrons energies between 20 keV and 3.2 MeV (e.g., see Sec-

tion 2.4.1 in [van Lint, Flanahan, Leadon, Naber and Rogers (1980)] and, also,

[Smith, Binder, Compton and Wilbur (1966)]). At these energies, the neutron–

silicon total cross section is almost determined by the elastic cross section [Smith,

Binder, Compton and Wilbur (1966)], i.e., the interactions result in a recoil sili-

con. In general, the agreement between f

ion

, computed by means of Eq. (4.95), and

the experimental data is quite satisfactory. As one can see in Fig. 4.15, for recoiling

energy up to about 10 keV the fraction f

niel

is larger than about 70 %, i.e., most of

the energy is deposited by NIEL. Above about 100 keV, f

niel

is lower than 50 %.

Furthermore, since recoil atoms collide with equal mass atoms in the lattice,

a large fraction of the incoming recoil energy can be transferred in a collision,

‡‡

Nowadays, an algorithm has been developed for the GEANT4 Monte Carlo package for the

computation of screened Coulomb interatomic scattering [Mendenhall and Weller (2005)] for non-

relativistic ion–ion interactions. The algorithm allows one to use an arbitrary screening function,

for instance the well-established ZBL screening function (see page 48 of [Ziegler, Biersack and

Littmark (1985a)]).

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354 Principles of Radiation Interaction in Matter and Detection

but soft collisions can also occur depending on the recoil energy and the eﬀect of

interpenetration of orbital electron clouds. Monte-Carlo simulations of displacement

cascade generated by a 50 keV recoil silicon (e.g., see Fig. 4.14) show that there

are a few collisions with large energy transfers interspersed with many low-energy

transfers. These latter are grouped in terminal clusters of displacements.

As mentioned before, Eq. (4.90) expresses the partition energy (Sect. 4.2.1) for

reactions resulting in a recoil silicon as PKA. In general {see Equations (5–9) in

[Norgett, Robinson and Torrens (1975)]}, the partition energy for a recoil nucleus of

atomic number Z

rec

and atomic rest-mass m

rec

(in atomic mass units) in a medium

with atomic number Z

and atomic rest-mass m

can be evaluated by means of

the Norgett–Robinson–Torrens expression:

rec

1 + k

g(²

)

, (4.96)

wherein E

rec

is the kinetic energy of the recoil nucleus and

rec

+ m

rec

, (4.97)

0.88534 a

2/3

rec

+ Z

2/3

[see Eq. (2.72)], (4.98)

g(²

) = ²

+ 0.40244 ²

3/4

+ 3.4008 ²

1/6

, (4.99)

= 0.1337 Z

2/3

rec

−1/2

rec

, (4.100)

where ²

is the reduced energy (see, for instance, page 80) accounting for the intera-

ction between the atom of the medium and the recoil atomic particle and is calcu-

lated using the Lindhard and Sharﬀ screening length a

, E

rec

is the kinetic energy of

the recoil nucleus, e is the electronic charge in esu, a

= 0.529177208×10

−8

cm is the

Bohr radius (see Appendix A.2); ²

, g(²

) and k

are dimensionless. Equation (4.97)

can be rewritten as:

32.536 × 10

−3

rec

2/3

rec

+ Z

2/3

rec

+ m

rec

(4.101)

with E

rec

in eV. The damage energy calculated by means of Eq. (4.96) in case of

a recoil silicon as PKA diﬀers from that of Eq. (4.90) by less than 4 % for recoil

silicon with energies lower than 150 keV (i.e., when more than 45 % of E

rec

≡ E

deposited as damage energy). The number of Frenkel-pairs generated in a cascade

is given by the modiﬁed Kinchin–Pease formula, thus, can be calculated by means

of Eqs. (4.89, 4.96). Equations (4.96–4.100) were derived according to the method

of Lindhard, Nielsen, Scharﬀ and Thomsen (1963) to evaluate the inelastic energy-

loss. As discussed by Norgett, Robinson and Torrens (1975), this latter applies to

interactions in which Z

rec

does not diﬀer too much from unity and for energies

lower than ≈ 25 Z

4/3

rec

keV.

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Radiation Environments and Damage in Silicon Semiconductors 355

4.2.1.2 Neutron Interactions

In neutron interactions, the heavy recoil nuclei are of primary importance in de-

termining the amount of damage energy and permanent damages in silicon (as in

any other semiconductor). For fast neutrons in the energy range between 50 keV

and 14 MeV, relevant reactions are the (n,n) or elastic scattering, (n,n’) or inelastic

neutron scattering, (n,p) and (n,α) (for a discussion see [Smith, Binder, Compton

and Wilbur (1966)]; see, also, Sect. 4.1.3.1).

Since in a nuclear reactor spectrum the high-energy part strongly decreases

above (3–5) MeV, the elastic scattering [about (1–10) b] is the most likely process

to produce displacements (as already mentioned). The kinetic energy of the recoil

silicon depends on the angular diﬀerential cross section. For an isotropic angular

distribution in the center-of-mass system, all recoil energies in the lattice (i.e. the

laboratory-system) from 0 to the maximum transferred energy T

max

are equally

probable and the mean energy transferred is 0.5 T

max

. At the energy of 0.762 MeV,

the angular distribution is almost isotropic, but neutrons with kinetic energies of

several MeV tend to scatter in a forward direction (e.g., see Figure 2.21 in [van Lint,

Flanahan, Leadon, Naber and Rogers (1980)]) and, therefore, the mean transferred

energy is much lower than 0.5 T

max

In a two body elastic scattering the maximum transferred energy is [see

Eq. (1.28) and discussion in Sect. 1.3.1]

max

= 2M

(γ

− 1)

1 +

+ 2γ

−1

, (4.102)

where M

is the mass of the target (recoil) atom assumed to be at rest, m is the rest

mass of the incoming particle and γ is the Lorentz factor of the incoming particle

with kinetic energy E

= (γ − 1)mc

. For γ ≈ 1, Eq. (4.102) becomes:

max

≈ 4M

(γ − 1)

1 +

−2

4 mM

(m + M

)

. (4.103)

In a neutron–silicon elastic scattering, since for fast neutrons γ ≈ 1, from Eq. (4.103)

max

is given by

max

4 m

+ M

)

≈ 0.133 E

, (4.104)

where E

is the kinetic energy of the fast neutron, M

and m

are the silicon and

neutron rest-masses, respectively. For instance as discussed above, on average a neu-

tron with kinetic energy close to 0.762 MeV imparts ≈ 0.5 ×0.133 E

= 0.051 MeV

to the recoil silicon, thus, as shown in Fig. 4.15, about 58 % of its recoil energy

is dissipated as damage energy. In an elastic scattering of a neutron with a silicon

nucleus at rest, the minimum kinetic energy of the neutron to impart 25 eV is about

190 eV [Eq. (4.104)].

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356 Principles of Radiation Interaction in Matter and Detection

4.2.1.3 Interactions of Protons, α-particles and Heavy-Isotopes

When protons traverse a material, they can interact through atomic Coulomb scat-

tering (b eing charged particles) and strong interaction. Furthermore, since the pro-

ton rest-mass is m

≈ m

, from Eq. (4.104) the maximum transferred energy in

elastic interactions becomes

max

≈ 0.133 E

, (4.105)

where E

is the kinetic energy of the incoming proton. As for neutrons, the minimum

kinetic energy of the protons to impart an energy of 25 eV to a recoil silicon at rest

is about 190 eV [Eq. (4.105)].

In strong p–silicon interactions, the elastic scattering is a relevant fraction of

the total cross section also at high-energies (see for instance [Srour and McGarrity

(1988)]), while the non-elastic cross section becomes relevant for protons with kinetic

energies above about (5–10) MeV. As shown in [Jun et al. (2003)], at about 10 MeV,

the nuclear interactions account for about 15% of the deposited damage-energy,

while above 20 MeV the strong (elastic and non-elastic) interactions become the

dominant mechanisms (i.e. at 20 MeV the strong interactions account for about 50%

of the deposited damage-energy). Below 10 MeV, the atomic Coulomb scattering is

the dominant mechanism for producing displacements.

Charged atomic particles passing in a medium undergo energy-loss processes by

Coulomb scattering (e.g., see [Northcliﬀe (1963)], chapter on Electromagnetic Inte-

raction of Radiation in Matter and references therein). It is common practice to

distinguish between light and heavy ions. However, it must be considered that this

distinction is arbitrary so far the fundamental mechanism of the Coulomb scatte-

ring for the energy-loss process is considered. Nonetheless, there is a real practical

diﬀerence. Light ions (i.e., protons and α-particles) can be regarded as fully ionized

and charge invariant over most of the energy region involved in the nuclear and

particle physics (e.g., & 1 MeV see Sect. 2.1.4, [Fano (1963)] and also references

therein). Heavier atoms can be fully ionized (by the stripping of their electrons)

and proceed as bare nuclei only for velocities largely exceeding the orbital veloc-

ities of their electrons. At high energies, light and heavy ions lose energy predo-

minantly by electronic (i.e., collision) energy-loss processes (see [Northcliﬀe (1963)]

and Sect. 2.1.4).

At low energy, the dominant mechanism of energy-loss is by nuclear energy-

loss, i.e., the energy of the incoming atomic particle is released in matter by

Coulomb interactions occurring with the nuclei of the medium. These interactions

are described (as the interactions of the PKA inside the semiconductor medium)

by largely screened Coulomb potentials (e.g., see [Ziegler, Biersack and Littmark

(1985a); Ziegler, J.F. and M.D. and Biersack (2008a)] and references therein, see also

Sect. 2.1.4.1). In the Ziegler–Biersack–Littmark treatment (Chapter 2 in [Ziegler,

Biersack and Littmark (1985a); Ziegler, J.F. and M.D. and Biersack (2008a)]) it

is shown that the nuclear stopping power is properly accounted by the Rutherford

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Radiation Environments and Damage in Silicon Semiconductors 357

scattering with unscreened Coulomb potential for universal reduced energies, ²

r,U

above 30; whereas, the universal reduced energy is given by [Eq. (2.76)]

r,U

+ m

tot

, (4.106)

in which E

tot

is the kinetic energy of the incoming atomic particle i with rest mass

(in atomic mass units) and atomic number Z

, the stationary PKA has rest mass

(in atomic mass units) and atomic number Z

, e is the electronic charge in esu,

the universal screening length a

0.88534 a

0.23

+ Z

0.23

(4.107)

[Eq. (2.74)] and a

is the Bohr radius. Equation (4.106) can be rewritten as

r,U

32.536 × 10

−3

0.23

+ Z

0.23

)

+ m

tot

, (4.108)

where E

tot

is in units of eV. In a silicon medium, the condition ²

r,U

> 30 is satisﬁed

above

Rut

' (75–80) keV/nucleon for nickel isotopes and at lower energies for

isotopes

∗∗

with Z < 28. To a ﬁrst approximation, since in the Rutherford scattering

the diﬀerential cross section is inversely proportional to the square of the transferred

energy T [see Eq. (1.56) and discussion in Sect. 1.5], the mean transferred energy

hT i, i.e., the mean kinetic energy of the recoil silicon hE

i, above the displacement

threshold energy E

i ≡ hT i

max

− E

max

for

tot

isot

> E

Rut

, (4.109)

where I

isot

is the number of nucleons of the isotope and the maximum transferred

energy T

max

is given for γ ≈ 1 by Eq. (4.103). An approximate expression

∗

for the

mean number of displaced atoms, induced by the recoil, (i.e. for ≈ hN

F P

i) is

F P

i ≈

max

2 (T

max

− E

)

1 + ln

max

2 E

¶¸

for

tot

isot

> E

Rut

(4.110)

(see Equations (29, 30, 35) of [Vavilov and Ukhin (1977)] and Equation (2-31)

of [Dienes and Vineyard (1957)]). Furthermore, for γ ≈ 1 and T

max

À E

, we can

rewrite Eq. 4.109 as

i ≈ E

4 m

tot

+ m

)

. (4.111)

Equation (4.111) shows that (for γ ≈ 1) the mean kinetic energy of the recoil silicon

increases with the logarithm of the kinetic energy of the incoming isotope. The initial

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

∗∗

These isotopes (including nickel) are the most abundant in GCRs; for isotopic abundances, the

reader can see Sect. 4.1.2.4 and Fig. 4.11.

∗

This expression was derived assuming ζ ' 2 (in the framework of the Kinchin–Pease model) for

Eq. (4.89) and T

max

> 2E

(see Section 2.4 of [Dienes and Vineyard (1957)]).

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358 Principles of Radiation Interaction in Matter and Detection

displacement is followed by a cascading-process, in which additional displacements

are induced by the recoil silicon. From Eqs. (4.89, 4.96, 4.109) and since the fraction

of the damage energy is larger than 50% for recoil silicon with energies lower than

≈ 100 keV [e.g., see Eq. (4.90) and Fig. 4.15], we do not expect more than 4–8

Frenkel-pairs created per PKA

over most of the range of energies wherein the

Coulomb interaction is the dominant scattering mechanism. This energy range is

of interest for semiconductor applications. Thus, the developed cascade of Frenkel-

pair is typically smaller compared to that induced by strong interacting particles

[e.g., see the above discussion about fast neutron and protons with energies larger

than (10–20) MeV]. Nonetheless, the dominant mechanism for generating atomic

displacements is via silicon-to-silicon cascading-displacement.

As previously said, for protons with kinetic energy larger than about 20 MeV

[Jun et al. (2003)], the NIEL deposition results mainly from the elastic and non-

elastic strong interactions on silicon. Since isotopes undergo (short-range) strong-

and (long-range) electromagnetic-interactions passing through matter (see for in-

stance Section 17.8 in [Eisberg and Resnick (1985)]), let us estimate the energies

above which about 50% of the NIEL deposition is expected to be determined by

strong isotope–silicon interactions. To a ﬁrst approximation, the total cross section

for strong interactions between two isotopes i and t is proportional to (R

+ R

)

(e.g., see Section 11.11 in [Segre (1977)] and Sect. 3.2.1), where R

and R

are the

nuclear radii of the isotopes in units of r

[Eq. (3.12)] computed as:

i,t

' M

1/3

i,t

(4.112)

in which M

and M

are the nucleon numbers of the isotopes. The total cross section

for electromagnetic interactions is proportional to Z

, where Z

and Z

are the

atomic numbers of the two isotopes. However, because at relativistic energies the

region of space at the maximum electric ﬁeld is relativistically contracted (e.g., see

page 37), the collision time

∗∗

is proportional to 1/(v

), where v

and γ

are the

velocity and Lorentz factor of the incoming isotope, respectively, while the target

isotope is supposed to be at rest. Thus, the cross section ratio C

C,st

of the Coulomb

to strong interaction behaves as

C,st

∝

i,t

or, equivalently,

C,st

∝

i,t

− 1

, (4.113)

where β

= v

/c, c is the speed of light and

i,t

+ R

This range of values for hN

F P

i is in agreement with that computed using Eq. (4.110).

∗∗

This treatment follows that applied in Sect. 2.1.1 to the scattering of a charged particle with

atomic electrons resulting in electronic energy-losses.

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Radiation Environments and Damage in Silicon Semiconductors 359

Fig. 4.16 Isotop e–silicon interactions: kinetic energies (indicated as “20 MeV proton equivalent”)

of isotopes in units of GeV/amu corresponding to the values of the Lorentz factor γ

i,st

[Eq.( 4.114)]

versus the atomic mass-number [Eq. (3.1)] up to

Kr. The data indicated as “10 MeV proton

equivalent” correspond to the kinetic energies of isotopes calculated replacing the Lorentz factor

with that corresponding to 10 MeV proton in Eq. (4.114). The curves are the spline ﬁt to the

calculated kinetic energies for p,

He,

Li,

Be,

Ne,

Mg,

Si,

Ar,

Fe,

and

Kr on silicon.

C,st

decreases as the energy of the incoming isotope increases, i.e., the strong inte-

raction cross section becomes dominant at high-energies. By means of Eq. (4.113),

for an isotope–silicon interaction, we can estimate the value of the Lorentz factor

i,st

for which the dominant mechanism of the NIEL deposition results from strong

interactions, as it occurs for protons at 20 MeV [Jun et al. (2003)], i.e,

i,Si

i,st

− 1

≈

H,Si

p,20MeV

− 1

where γ

p,20MeV

is the Lorentz factor of a proton with 20 MeV of kinetic energy. From

this latter expression, we obtain

i,st

p,20MeV

− 1

i,Si

H,Si

+ 1. (4.114)

In Fig. 4.16, the kinetic energies in units of GeV/amu

‡

, corresponding to the

values of the Lorentz factor γ

i,st

of the isotopes [Eq.( 4.114)], are shown as function

of the mass number [Eq. (3.1)] for isotope–silicon interactions. For instance, the

dominant mechanism of the NIEL deposition is expected to result from strong

interactions i) for α-particles,

Li and

Be isotopes with kinetic energies & 0.12, 0.42

‡

The values of the masses in units of amu used in the calculation are those from [Ziegler, J.F.

and M.D. and Biersack (2008b)]. The reader can ﬁnd a deﬁnition of kinetic energies per amu in

Sect. 1.4.1.