Nuclear and Non-Ionizing Energy-loss of Electrons with Low and Relativistic Energies in Materials and Space Environment

Abstract

The treatment of the electron-nucleus interaction based on the Mott
differential cross section was extended to account for effects due to screened
Coulomb potentials, finite sizes and finite rest masses of nuclei for electrons
above 200 keV and up to ultra high energies. This treatment allows one to
determine both the total and differential cross sections, thus, subsequently to
calculate the resulting nuclear and non-ionizing stopping powers. Above a few
hundreds of MeV, neglecting the effect due to finite rest masses of recoil
nuclei the stopping power and NIEL result to be largely underestimated; while,
above a few tens of MeV the finite size of the nuclear target prevents a
further large increase of stopping powers which approach almost constant
values.

Full text

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arXiv:1111.4042v4 [physics.space-ph] 6 Dec 2011
December 7, 20111:55WSPC - Proceedings Trim Size: 9in x 6inConsolandi2011
1
To appear on the Proceedings of the 13th ICATPP Conference on
Astroparticle, Particle, Space Physics and Detectors
for Physics Applications,
Villa Olmo (Como, Italy), 3–7 October, 2011,
to be published by World Scientific (Singapore).
NUCLEAR AND NON-IONIZING ENERGY-LOSS OF
ELECTRONS WITH LOW AND RELATIVISTIC ENERGIES
IN MATERIALS AND SPACE ENVIRONMENT
M.J. Boschini1,2, C. Consolandi∗,1, M. Gervasi1,3, S. Giani4, D. Grandi1,
V. Ivanchenko4, P. Nieminem5, S. Pensotti3, P.G. Rancoita1and M. Tacconi1
1INFN-Milano Bicocca, P.zza Scienza,3 Milano, Italy
2CILEA Via R. Sanzio, 4 Segrate, MI-Italy
3Milano Bicocca University, P.zza della Scienza, 3 Milano, Italy
4CERN, Geneva, 23, CH-1211, Switzerland
5ESA, ESTEC, AG Noordwijk (Netherlands)
∗E-mail: cristina.consolandi@mib.infn.it
The treatment of the electron–nucleus interaction based on the Mott dif-
ferential cross section was extended to account for effects due to screened
Coulomb potentials, finite sizes and finite rest masses of nuclei for electrons
above 200keV and up to ultra high energies. This treatment allows one to
determine both the total and differential cross sections, thus, subsequently to
calculate the resulting nuclear and non-ionizing stopping powers. Above a few
hundreds of MeV, neglecting the effect due to finite rest masses of recoil nu-
clei the stopping power and NIEL result to be largely underestimated. While,
above a few tens of MeV, the finite size of the nuclear target prevents a further
large increase of stopping powers which approach almost constant values.
1. Introduction
Nuclei and electrons populate the heliosphere. Most of the nuclei are galac-
tic cosmic rays (GCR), while electrons can additionally be originated by
the Sun and Jupiter’s magnetosphere, which is a major source of relativistic
electrons in the heliosphere (e.g., see Ref.1,2and references therein). Pro-
tons and electrons are also major constituents of the Earth’s radiation
belts. These particles can interact with materials and onboard electro-
nics in spacecrafts, inducing displacements of atomic nuclei, thus inflicting
permanent damages. As the particle energy increases, for instance above
≈ 20MeV for protons and ≈ 130 MeV/nucleon for α-particles (e.g., see

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Section 4.2.1.4 and Figure 4.26 at page 418 of Ref.3), the dominant me-
chanism for displacement damage is determined by hadronic interactions;
for electrons and low-energy nuclei the elastic Coulomb scattering is the
relevant physical process to induce permanent damage.
The non-ionizing energy-loss (NIEL) is the energy lost from particles
traversing a unit length of a medium through physical processes resulting
in permanent atomic displacements. The displacement damage is mostly
responsible for the degradation of semiconductor devices - like those using
silicon - where, for instance, depleted layers are required for normal ope-
ration conditions (e.g. see Ref.4). The nuclear stopping power and NIEL
deposition - due to elastic Coulomb scatterings - from protons, light- and
heavy-ions traversing an absorber were previously dealt5,6with (see also
Sections 1.6, 1.6.1, 2.1.4–2.1.4.2, 4.2.1.6 of Ref.3). In the present work, the
nuclear stopping power and NIEL deposition due to elastic Coulomb scat-
terings of electrons are treated up to ultra relativistic energies.
The developed model (i.e., see Sects. 2–2.4) for screened Coulomb elastic
scattering up to relativistic energies is included into Geant4 distribution7
and is available with Geant4 version 9.5 (December 2011). In Sects. 3, 4,
the nuclear and non-ionizing stopping powers for electrons in materials are
treated, while a final discussion is found in Sect. 5.
2. Scattering Cross Section of Electrons on Nuclei
The scattering of electrons by unscreened atomic nuclei was treated by
Mott8(see also Sections 4–4.5 in Chapter IX of Ref.9) extending a method
of Wentzel10(see also Born11) and including effects related to the spin of
electrons8. Wentzel’s method was dealing with incident and scattered waves
on point-like nuclei. The differential cross section (DCS) - the so-called
Mott differential cross section (MDCS) - was expressed by Mott8as two
conditionally convergent infinite series in terms of Legendre expansions. In
Mott–Wentzel treatment, the scattering occurs on a field of force generating
a radially dependent Coulomb - unscreened (screened) in Mott8(Wentzel10)
- potential. It has to be remarked that Mott’s treatment of collisions of fast
electrons with atoms (e.g., see Chapter XVI of Ref.9) involves the knowledge
of the wave function of the atom, thus, in most cases the computation
of cross sections depends on the application of numerical methods (see a
further discussion in Sect. 2.2). Furthermore, the MDCS was derived in the
laboratory reference system for infinitely heavy nuclei initially at rest with
negligible spin effects and must be numerically evaluated for any specific
nuclear target. Effects related to the recoil and finite rest mass of the target

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nucleus (M) were neglected. Thus, in this framework the total energy of
electrons has to be smaller or much smaller than Mc2.
As discussed by Idoeta and Legarda12(e.g., see also Refs.13,14), Mott
provided an “exact” differential cross section because no Born approxi-
mationaof any order is employed in its derivation. Various authors have
approximated the MDCS for special situations, usually expressing their re-
sults in terms of ratios, R, of the so-obtained approximated differential cross
sections with respect to that one for a Rutherford scattering (RDCS) - the
so-called Rutherford’s formula, see Section 1.6.1 of Ref.3- for an incoming
particle with z = 1 given by:
dσRut
dΩ
=
?Ze2
?
pβc
?2
Ze2
1
(1 − cosθ)2=
?2
?Ze2
2pβc
?2
1
sin4(θ/2)
(1)
=
2mc2β2γ
1
sin4(θ/2),
where m is the electron rest mass, Z is the atomic number of the target
nucleus, β = v/c with v the electron velocity and c the speed of light; γ is
the corresponding Lorentz factor; p and θ are the momentum and scattering
angle of the electron, respectively; finally, since the interaction is isotropic
with respect to the azimuthal angle, it is worth noting that dΩ can be given
as
dΩ = 2πsinθdθ.(2)
The MDCS is usually expressed as:
dσMott(θ)
dΩ
=dσRut
dΩ
RMott,(3)
where RMott(as above mentioned) is the ratio between the MDCS and
RDCS. In particular, Bartlett–Watson15determined cross sections for nu-
clei with atomic number Z = 80 and energies from 0.024 up to 1.7MeV (see
also Ref.16). McKinley and Feshbach17expanded Mott’s series in terms of
power series in αZ (with α the fine-structure constant) and (αZ)/β; these
expansions, which give results accurate to 1% up to atomic numbers Z ≈ 40
(e.g., see discussions in Refs.18,19), were further simplified to obtain an ap-
proximate analytical formula with that accuracy for αZ ≤ 0.2. Feshbach20
tabulated values of the differential cross section as a function of scattering
aIn quantum mechanical potential scattering, the scattered wave may be obtained from
the so-called Born expansion. The Born approximation is the first term of the Born
expansion (see, for instance, references indicated in Section 1.6.1 Ref.3).

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angle for nuclei with atomic number up to 80 and electrons with kinetic
energies larger than 4MeV. Curr18reported values of the differential cross
section as a function of scattering angle accurate at 1% for (αZ)/β ? 0.6;
while Doggett and Spencer21tabulated the MDCS for energies from 10
down to 0.05MeV. Recently, Idoeta and Legarda12provided a further se-
ries transformations and made a systematic comparison with those from
McKinley and Feshbach17, Curr18, Doggett and Spencer21. For electrons
with kinetic energies from several keV up to 900MeV and target nuclei with
1 ? Z ? 90, Lijian, Quing and Zhengming22provided a practical interpo-
lated expression [Eq. (16)] for RMottwith an average error less than 1%; in
the present treatment, that expression - discussed in Sect. 2.1 - is the one
assumed for RMottin Eq. (3) hereafter.
The analytical expression derived by McKinley and Feshbach17- men-
tioned above - for the ratio with respect to Rutherford’s formula [Equa-
tion (7) of Ref.17] is given by:
RMcF= 1 − β2sin2(θ/2) + Z αβπ sin(θ/2)[1 − sin(θ/2)](4)
with the corresponding differential cross section (McFDCS)
dσMcF
dΩ
=dσRut
dΩ
RMcF, (5)
where dσRut/dΩ is from Eq. (1). It has to be remarked that for positrons,
the ratio RMcF
pos becomes
RMcF
pos = 1 − β2sin2(θ/2) − Z αβπ sin(θ/2)[1 − sin(θ/2)](6)
(e.g., see Equation (6) of Ref.23). Furthermore, for Mc2much larger than
the total energy of incoming electron energies the distinction between la-
boratory (i.e., the system in which the target particle is initially at rest)
and center-of-mass (CoM) systems disappears (e.g., see discussion in Sec-
tion 1.6.1 of Ref.3). Furthermore, in the CoM of the reaction the energy
transferred from an electron to a nucleus initially at rest in the laboratory
system (i.e., its recoil kinetic energy T) is related to the maximum energy
transferable Tmaxas
T = Tmaxsin2(θ′/2) (7)
[e.g., see Equations (1.27, 1.95) at page 11 and 31, respectively, of Ref.3],
where θ′is the scattering angle in the CoM system. From Eqs. (2, 7) one
obtains
dT =Tmax
4π
dΩ′. (8)

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Since θ is ≈ θ′for Mc2much larger than the electron energy, one finds that
Eq. (7) can be approximated as
T ≃ Tmaxsin2(θ/2),
T
Tmax
(9)
=⇒ sin2(θ/2) =
(10)
and
dT ≃Tmax
4π
dΩ. (11)
Using Eqs. (4, 10, 11), Eqs. (1, 5) can be respectively rewritten as:
dσRut
dΩ
=Tmax
4π
dσRut
dT
?2πTmax
?2πTmax
=⇒dσRut
dT
=
?Ze2
?Ze2
?
pβcT2
,(12)
dσMcF
T
=
pβcT2
×1 − β2
T
Tmax
+ Z αβπ
?
T
Tmax
?
1 −
?
T
Tmax
??
=⇒dσMcF
T
=
?Ze2
?Ze2
pβc
?2πTmax
?2πTmax
T2
?
1−β
T
Tmax(β+Zαπ)+Zαβπ
?
T
Tmax
?
(13)
=
pβcT2
RMcF(T)
with
RMcF(T) =
?
1−β
T
Tmax(β+Zαπ)+Zαβπ
?
T
Tmax
?
(14)
[e.g., see Equation (11.4) of Ref.,24see also Ref.19and references
therein]. Similarly, for positrons one finds
dσMcF
pos
T
=
?Ze2
pβc
?2πTmax
T2
?
1−β
T
Tmax(β−Zαπ)−Zαβπ
?
T
Tmax
?
[e.g., see Refs.19,23and references therein]. Finally, in a similar way the
MDCS [Eq. (3)] is
dσMott(T)
dT
=dσRut
dT
RMott(T)
=
?Ze2
pβc
?2πTmax
T2
RMott(T)(15)

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with RMott(T) from Eq. (18).
2.1. Interpolated Expression for RMott
As mentioned in Sect. 2, Curr18derived RMottas a function the atomic
number Z of the target nucleus and velocity βc of the incoming electron at
several scattering angles from θ = 30
and Zhengming22provided a practical interpolated expression [Eq. (16)]
which is a function of both θ and β for electron energies from several keV
up to 900MeV, i.e.,
◦up to 180
◦. Recently, Lijian, Quing
RMott=
4
?
j=0
aj(Z,β)(1 − cosθ)j/2, (16)
where
aj(Z,β) =
6
?
k=1
bk,j(Z)(β − β)k−1, (17)
and β c = 0.7181287c is the mean velocity of electrons within the above
mentioned energy range. The coefficients bk,j(Z) are listed in Table 1
of Ref.22for 1 ? Z ? 90.
At 10, 100 and 1000MeV for Li, Si, Fe and Pb, values of RMottwere
calculated using both Curr18and Lijian, Quing and Zhengming22meth-
ods and found to be in a very good agreement. It has to be remarked
that with respect to the values of RMcFobtained from Eq. (4) at 100MeV
one finds an average variation of about 0.2%, 3.2% and 8.8% for Li, Si
and Fe nuclei, respectively. However, the stopping power determined using
Eq. (52) (i.e., with RMott) differs by less than 0.5% with that calculated
using Eq. (53) (i.e., with RMcF). RMottobtained from Eq. (16) at 100MeV
is shown in Fig. 1 for Li, Si, Fe and Pb nuclei as a function of the scattering
angle. Furthermore, it has to be pointed out that the energy dependence
of RMottfrom Eq. (16) was studied and observed to be negligible above
≈ 10MeV [as expected from Eq. (17)].
Finally, from Eqs.(7, 16) [e.g., see also Equation (1.93) at page 31
of Ref.3], one finds that RMottcan be expressed in terms of the transferred
energy T as
RMott(T) =
4
?
j=0
aj(Z,β)
?
2T
Tmax
?j/2
. (18)

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2.2. Screened Coulomb Potentials
As already mentioned in Sect. 2, a complete treatment of electron intera-
ctions with atoms (e.g., see Chapter XVI of Ref.9) involves the knowledge of
the wave function of the target atom and, thus - as remarked by Fernandez-
Vera, Mayol and Salvat14-, a relevant amount of numerical work when the
kinetic energies of electrons exceed a few hundreds of keV.
The simple scattering model due to Wentzel10- with a single expo-
nential screening function [e.g., see Equation (2.71) at page 95 of Ref.3,
Equation (21) in Ref.25and Ref.10] - was repeatedly employed in treat-
ing single and multiple Coulomb scattering with screened potentials
(e.g, see Ref.25- and references therein - for a survey of such a topic
and also Refs.5,6,26–28). Neglecting effects like those related to spin and
finite size of nuclei, for proton and nucleus interactions with nuclei it was
shown that the resulting elastic differential cross section of a projectile
with bare nuclear-charge ez on a target with bare nuclear-charge eZ dif-
fers from the Rutherford differential cross section (RDCS) by an additional
term - the so-called screening parameter - which prevents the divergence
of the cross section when the angle θ of scattered particles approaches
0◦[e.g., see Refs.5,6,26–28(see also references therein) and Section 1.6.1
of Ref.3]. It has to be remarked that the RDCS for z = 1 particles can also
be employed to describe the scattering of non-relativistic electrons with
unscreened nuclei (e.g, see Refs.8,12and references therein). As derived by
Moli` ere26for the single Coulomb scattering using a Thomas–Fermi poten-
tial, for z = 1 particles the screening parameter As,M[e.g., see Equation (21)
of Bethe27] is expressed as
As,M=
?
?
2p aTF
?2?
1.13 + 3.76 ×
?αZ
β
?2?
(19)
where α, c and ? are the fine-structure constant, speed of light and reduced
Planck constant, respectively; p (βc) is the momentum (velocity) of the
incoming particle undergoing the scattering onto a target supposed to be
initially at rest - i.e., in the laboratory system -; aTFis the screening length
suggested by Thomas–Fermi (e.g., see Refs.29,30)
aTF=CTFa0
Z1/3
(20)
with
a0=
?2
me2

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the Bohr radius, m the electron rest mass and
CTF=1
2
?3π
4
?2/3
≃ 0.88534
a constant introduced in the Thomas–Fermi model [e.g., see Equa-
tions (2.73, 2,82) - at page 95 and 99, respectively - of Ref.3and
Ref.6, see also references therein]. The modified Rutherford’s formula
[dσWM(θ)/dΩ], i.e., the differential cross section - obtained from the
Wentzel–Moli` ere treatment of the single scattering on screened nuclear po-
tentials - is given by [e.g., see Equation (2.84) of Ref.3, Section 2.3 in Ref.25
and Ref.6(see also references therein)]:
dσWM(θ)
dΩ
=
?zZe2
=dσRut
dΩ
2pβc
?2
1
?As,M+ sin2(θ/2)?2
sin4(θ/2)
?As,M+ sin2(θ/2)?2
F2(θ).
(21)
=dσRut
dΩ
(22)
with
F(θ) =
sin2(θ/2)
As,M+ sin2(θ/2).(23)
F(θ) - the so-called screening factor - depends on the scattering angle θ
and screening parameter As,M. As discussed in Sect. 2.4, in the DCS the
term As,Mcannot be neglected [Eq. (22)] for scattering angles (θ) within a
forward (with respect to the electron direction) angular region narrowing
with increasing energy from several degrees (for high-Z material) at 200keV
down to less than or much less than a mrad above 200MeV.
An approximated description of elastic interactions of electrons with
screened Coulomb fields of nuclei can be obtained factorizing the
MDCS, i.e., involving Rutherford’s formula [dσRut/dΩ] for particles with
z = 1, the screening factor F(θ) and the ratio RMottbetween RDCS and
MDCS:
dσMott
sc
(θ)
dΩ
≃dσRut
dΩ
F2(θ) RMott
(24)
[e.g., see Equation (1) of Ref.12, Equation (A34) at page 208 of Ref.13, see
also Ref.14and citations from these references]. Thus, the corresponding

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?
?
? ???
Fig. 1.
function of scattering angle.
RMottobtained from Eq. (16) at 100MeV for Li, Si, Fe and Pb nuclei as a
screened differential cross section derived using the analytical expression
from McKinley and Feshbach17can be approximated with
dσMcF
sc
(θ)
dΩ
≃dσRut
dΩ
F2(θ) RMcF.(25)
It has to be remarked - as derived by Zeitler and Olsen31- that spin and
screening effects can be separately treated for small scattering angles; while
at large angles (i.e., at large momentum transfer), the factorization is well
suited under the condition that
2Z4/3α21
β2γ≪ 1
(e.g., see Refs.12,31). Zeitler and Olsen31suggested that for electron ener-
gies above 200keV the overlap of spin and screening effects is small for
all elements and for all energies; for lower energies the overlapping of the
spin and screening effects may be appreciable for heavy elements and large
angles.

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2.3. Finite Nuclear Size
As suggested by Fernandez-Vera, Mayol and Salvat14, above 10MeV the
effect of the finite nuclear size has to be taken into account in the treat-
ment of the electron–nucleus elastic scattering. With increasing energies,
deviations from a point-like behavior (see, for instance, Figure 4 of Ref.,14
Ref.32,33and references therein) were observed at large angles where the
screening factor [Eq. (23)] is ≈ 1.
The ratio between the actual measured and that expected from the
point-like differential cross section (e.g., the MDCS) expresses the square
of the nuclear form factor (|F|) which, in turn, depends on the momentum
transfer q, i.e., that acquired by the target initially at rest:
q =
?T(T + 2Mc2)
c
,(26)
with T from Eq. (7) or, for Mc2larger or much larger than the elec-
tron energy, from its approximate expression Eq. (9) [e.g., see Equa-
tions (31, 57, 58) of Ref.33, Section 3.1.2 of Ref.3, Refs.14,28,32,34].
The factorized differential cross section for elastic interactions of elec-
trons with screened Coulomb fields of nuclei [Eq. (24)] accounting for the
effects due to the finite nuclear size is given by:
dσMott
sc,F(θ)
dΩ
=dσMott
sc
(θ)
dΩ
|F(q)|2
≃dσRut
dΩ
F2(θ) RMott|F(q)|2
(27)
[e.g., see Equation (18) of Ref.14, Ref.28and also references therein]. Thus,
using the analytical expression derived by McKinley and Feshbach17
[Eq. (4)] one obtains the corresponding screened differential cross section
[Eq. (25)] accounting for the finite nuclear size effects, i.e.,
dσMcF
sc,F(θ)
dΩ
=dσMcF
sc
dΩ
(θ)
|F(q)|2
≃dσRut
dΩ
F2(θ) RMcF|F(q)|2
(28)
=dσRut
dΩ
F2(θ)|F(q)|2
×?1−β2sin2(θ/2) + Z αβπ sin(θ/2)[1 − sin(θ/2)]?. (29)

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In terms of kinetic energy, one can respectively rewrite Eqs. (27, 28) as
dσMott
sc,F(T)
dT
dσMcF
sc,F(T)
dT
=dσRut
dT
F2(T) RMott(T) |F(q)|2
(30)
≃dσRut(T)
dT
F2(T) RMcF(T) |F(q)|2
(31)
with dσRut/dT from Eq. (12), RMott(T) from Eq. (18), RMcF(T) from
Eq. (14) and, using Eqs. (7, 9, 23),
F(T) =
T
TmaxAs,M+ T.
The nuclear form factor accounts for the spatial distribution of charge
density probed in the electron–nucleus scattering [e.g., see Equation (58)
of Ref.33, Section 3.1.2 of Ref.3, Refs.14,28,32,34and references therein]. For
instance, among those spherically symmetric treated in literature, one finds
that for i) an exponential charge distribution (Fexp) [e.g., see Equation (6)
of Ref.28, Equation (93) at page 252 of Ref.33and references therein], ii)
a Gaussian charge distribution (Fgau) [e.g., see Equation (6) of Ref.28and
references therein] and iii) an uniform–uniform folded charge distribution
over spheres with different radii (Fu) [e.g., see Equation (22) of Ref.14,
Ref.32and references therein]. For instance, the form factor Fexpis
Fexp(q) =
?
1 +1
12
?qrn
?
?2?−2
,(32)
where rnis the nuclear radius [e.g., see Equation (6) of Ref.28]. To a first
approximation, rncan be parameterized by
rn= 1.27A0.27fm (33)
with A the atomic weight [e.g., see Equation (7) of Ref.28]. Equation (33)
provides values of rn in agreement up to heavy nuclei (like Pb and U)
with those available, for instance, in Table 1 of Ref.34. The nuclear form
factor is 1 for q = 0 and rapidly decreases with increasing q [e.g., see
Eq. (32), Equation (6) of Ref.28and Equation (22) of Ref.14for Fexp, Fgau
and Fu, respectively]. Furthermore, from inspection of Eqs. (7, 9, 26) small
q are those corresponding to scattering angles within the forward (with
respect to the electron direction) angular region which, in turn, narrows
with increasing electron energy. For instance, in lithium the square values
(|F(q)|2) of these form factors are in agreement within 1% up to θ′? 124.1◦
(2.4◦) at 20MeV (1GeV); in silicon up to θ′? 138.4◦(2.4◦) at 20MeV
(1GeV); in iron up to θ′? 108.0◦(2.1◦) at 20MeV (1GeV). However, as

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?
?
?
?
?
?
? ?????
Fig. 2.
of nuclear stopping power of electrons in silicon calculated neglecting i) nuclear size effects
(i.e., for |Fexp|2= 1) (dashed curve) and ii) effects due to the finite rest mass of the
target nucleus (dashed and dotted curve) [i.e., in Eq. (53) replacing dσMcF
with dσMcF
sc,F(T)/dT from Eq. (31)] both divided by that one obtained using Eq. (53).
As a function of the kinetic energy of electrons from 200keV up to 1TeV, ratios
sc,F,CoM(T)/dT
discussed in Sect. 2.4, these upper angles are larger or much larger with
respect to those required to obtain 99% of the total cross section. Thus,
the usage of any of the above mentioned nuclear form factors - e.g., Fexpas
in the present treatment - is expected to be appropriate in the treatment
of the transport of electrons in matter, when single scattering mechanisms
are relevant, for instance in dealing with the nuclear stopping power and
non-ionization energy-loss deposition.
2.4. Finite Rest Mass of Target Nucleus
The DCS treated in Sects. 2–2.3 is based on the extension of the MDCS
to include effects due to interactions on screened Coulomb potentials of
nuclei and their finite size. However, in the treatment, the electron energies
were assumed to be small (or much smaller) with respect to that (Mc2)
corresponding to the rest mass (M) of target nuclei.
The Rutherford scattering on screened Coulomb fields - i.e., under the
action of a central force - by massive charged particles at energies larger or
much larger than Mc2was treated by Boschini et al.5,6in the CoM system

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(e.g., see also Sections 1.6, 1.6.1, 2.1.4.2 of Ref.3and references therein). It
was shown that the differential cross section [dσWM(θ′)/dΩ′with θ′the
scattering angle in the CoM system] is that one derived for describing the
interaction on a fixed scattering center of a particle with i) momentum
p′
requal to the momentum of the incoming particle (i.e., the electron in
the present treatment) in the CoM system and ii) rest mass equal to the
relativistic reduced mass µrel [e.g., see Equations (1.80, 1.81) at page 28
of Ref.3]. µrelis given by
µrel=mM
M1,2
(34)
=
mMc
?
m2c2+ M2c2+ 2M?m2c4+ p2c2
, (35)
where p is the momentum of the incoming particle (the electron in the
present treatment) in the laboratory system: m is the rest mass of the
incoming particle (i.e., the electron rest mass); finally, M1,2is the invariant
mass - e.g., Section 1.3.2 of Ref.3- of the two-particle system. Thus, the
velocity of the interacting particle is
β′
rc = c
?
?
?
?
?
1 +
?µrelc
p′r
?2?−1
(36)
[e.g., see Equation (1.82) at page 29 of Ref.3]. For an incoming particle with
z = 1, dσWM(θ′)/dΩ′is given by
dσWM′(θ′)
dΩ′
=
?
Ze2
2p′rβ′rc
?2
1
?As+ sin2(θ′/2)?2,(37)
with
As=
?
?
2p′raTF
?2?
1.13 + 3.76 ×
?αZ
β′r
?2?
(38)
the screening factor [e.g., see Equations (2.87, 2.88) at page 103
of Ref.3]. Equation (37) can be rewritten as
dσWM′(θ′)
dΩ′
=dσRut′(θ′)
dΩ′
F2
CoM(θ′)(39)
with
dσRut′(θ′)
dΩ′
=
?Ze2
2p′rβ′rc
?2
1
sin4(θ′/2)
(40)

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the corresponding RDCS for the reaction in the CoM system [e.g., see
Equation (1.79) at page 28 of Ref.3] and
FCoM(θ′) =
sin2(θ′/2)
As+ sin2(θ′/2)
(41)
the screening factor. Using, Eqs. (7, 8), one can respectively rewrite
Eqs. (40, 41, 39, 37) as
dσRut′
dT
= π
?Ze2
p′rβ′rc
T
?2Tmax
T2
(42)
FCoM(T) =
TmaxAs+ T
=dσRut′
dT
?Ze2
(43)
dσWM′(T)
dT
dσWM′(T)
dT
FCoM(T) (44)
= π
p′rβ′rc
?2
Tmax
(TmaxAs+ T)2
(45)
[e.g., see Equation (2.90) at page 103 of Ref.3or Equation (13) of Ref.6].
As already mentioned (Sect. 2.2), the screening parameter Asprevents
the DCS to diverge - see last term in Eq. (37) -, i.e., for θ′of the order of
or smaller than
θ′
sc= arcsin
?
2
?
As
?
effects due to screening of the nuclear Coulomb field have to be accounted
for. θ′
scrapidly decreases with increasing the kinetic energies of elec-
trons. For instance, in iron θ′
scis ≈ 1.7◦(0.03rad) at 200keV and ≈ 0.004◦
(7.0×10−2mrad) at 200MeV; in silicon, it is ≈ 1.3◦(0.022rad) at 200keV
and ≈ 0.003◦(5.5×10−2mrad) at 200MeV; while, in lithium, it is ≈ 0.75◦
(13mrad) at 200keV and ≈ 0.002◦(3.3×10−2mrad) at 200MeV. Therefore,
in Eq. (39) the term As(i.e., the screening parameter [Eq. (38)]) cannot be
neglected for scattering angles within a forward angular region narrowing
with increasing energies from a few degrees (for low-Z material) at about
200keV down to less than or much less than a mrad above 200MeV. It is
worthwhile to remark that in silicon, for instance, θ′can be approximated
with θ up to a few hundred MeV.
To account for the finite rest mass of target nuclei, the factorized MDCS

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[Eq. (27)] has to be re-expressed in the CoM system as:
dσMott
sc,F,CoM(θ′)
dΩ′
=dσMott
sc
(θ′,β′
dΩ′
r,p′
r)
|F(q)|2
≃dσWM′(θ′)
dΩ′
≃dσRut′(θ′)
dΩ′
RMott
CoM(θ′) |F(q)|2
F2
CoM(θ′) RMott
CoM(θ′) |F(q)|2, (46)
where F(q) is the nuclear form factor (Sect. 2.3) with q the momentum
transfer to the recoil nucleus [Eq. (26)]; finally, as discussed in Sect. 2.1,
RMottexhibits almost no dependence on electron energy above ≈ 10MeV,
thus, since at low energies θ ⋍ θ′and β ⋍ β′
θ and β′
rwith θ′and β′
Using the analytical expression derived by McKinley and Feshbach17,
one finds that the corresponding screened differential cross section account-
ing for the finite nuclear size effects [Eqs. (28, 29)] can be re-expressed as
r, RMott
CoM(θ′) is obtained replacing
r, respectively, in Eq. (16).
dσMcF
sc,F,CoM(θ′)
dΩ′
≃dσRut′(θ′)
dΩ′
F2
CoM(θ′) RMcF
CoM(θ′) |F(q)|2
(47)
with
RMcF
CoM(θ′) =?1−β2
It has to be remarked that scattered electrons are mostly found in the
forward or very forward direction. For instance, using Eq. (48) one can
derive that in lithium ≈ 99% of electrons are scattered with θ′? 0.27◦
(0.007◦) at 20MeV (1GeV); in silicon with θ′? 0.46◦(0.009◦) at 20MeV
(1GeV); in iron with θ′? 0.6◦(0.013◦) at 20MeV (1GeV).
In terms of kinetic energy T, from Eqs. (7, 8) one can respectively rewrite
Eqs. (46, 47) as
rsin2(θ′/2)+Z αβ′
rπ sin(θ′/2)[1−sin(θ′/2)]?. (48)
dσMott
sc,F,CoM(T)
dT
dσMcF
sc,F,CoM(T)
dT
=dσRut′
dT
F2
CoM(T) RMott
CoM(T) |F(q)|2
(49)
≃dσRut′(T)
dT
F2
CoM(T) RMcF
CoM(T) |F(q)|2
(50)
with dσRut′/dT from Eq. (42), FCoM(T) from Eq. (43) and RMcF
placing β with β′
rin Eq. (14), i.e.,
CoM(T) re-
RMcF
CoM(T) =
?
1−β′
r
T
Tmax(β′
r+Zαπ)+Zαβ′
rπ
?
T
Tmax
?
.(51)