Nuclear and Non-Ionizing Energy-Loss for Coulomb Scattered Particles from Low Energy up to Relativistic Regime in Space Radiation Environment

1.1 MB·English

by  M. J. Boschini

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1 Accepted for publication in the Proceedings of the ICATPP Conference
1 on Cosmic Rays for Particle and Astroparticle Physics,
0
Villa Olmo (Como, Italy), 7–8 October, 2010,
2
to be published by World Scientific (Singapore).
n
a
J

0 Nuclear and Non-Ionizing Energy-Loss for Coulomb Scattered
1
Particles from Low Energy up to Relativistic Regime in Space

] Radiation Environment
h
p M.J.Boschini1,2,C.Consolandi∗,1,M.Gervasi1,3,S.Giani4,
-
e D.Grandi1,V.Ivanchenko4,S.Pensotti3,P.G.Rancoita∗∗,1,M.Tacconi1
c 1INFN-Milano Bicocca, P.zza Scienza,3 Milano, Italy
a
2CILEA Via R. Sanzio, 4 Segrate, MI-Italy
p
3Milano Bicocca University,Piazza della Scienza, 3 Milano, Italy
s
. 4CERN, Geneva, 23, CH-1211, Switzerland
s ∗E-mail: [email protected]
c
∗∗E-mail: [email protected]
i
s
y
h Inthespaceenvironment,instrumentsonboardofspacecraftscanbeaffectedby
p displacementdamageduetoradiation.Thedifferentialscatteringcrosssection
[ for screened nucleus–nucleus interactions - i.e., including the effects due to

screenedCoulombnuclearfields-,nuclearstoppingpowersandnon-ionization
6
energylossesaretreatedfromabout50keV/nucleonuptorelativisticenergies.
v
2
2 1. Introduction
8
4 InthespaceenvironmentnearEarth,lowenergyparticlesare,forinstance,
.
1 found trapped within the radiation belts and are partially coming from
1 the Sun. On the other hand, the energies of Galactic Cosmic Rays (GCR)
0
extend up to relativistic range. Protons are the most abundant, but alpha
1
: particlesandheaviernucleiarealsopresent(e.g.,see Sections4.1.2–4.1.2.5
v
of Ref. [1]). Abundances and energy spectra of GCRs depend on the po-
i
X sition inside the solar cavity and are affected by the solar activity. Above
r (30–50)MeV/nucleon, the dominant radiation consists of GCRs. At lower
a
energies, from 1MeV/nucleon up to about 30MeV/nucleon, one also finds
the so-called Anomalous Cosmic Rays (ACRs). GCRs can reach Earth’s
magnetosphere and interact with upper layers of the atmosphere. These
interactionsproducesecondaryparticles,likeforexampleprotonswith(10–
100)MeV energies, which may - in turn - become trapped within the ra-
diation belts. In addition, during transientphenomena like solar flares and
coronalmassejections,SolarEnergeticParticles(SEP)areproducedinthe
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energy range from few keV’s to GeV’s.
All these energetic particles can inflict permanent damages to onboard
electronic devices employedin space missions.While passing through mat-
ter,theycanloseenergybyCoulombinteractionswithelectrons(electronic
energy-loss) and nuclei (nuclear energy-loss) of the material.In particular,
the nuclear energy-loss - due to screened Coulomb scattering on nuclei of
the medium - is relevant for the creation of permanent defects inside the
lattice of the material; thus, for instance, it is mostly responsible for the
displacement damage which is a cause of degradation of silicon devices.
The developed model - presented in this article - for screened Coulomb
elastic scattering up to relativistic energies is included into Geant4 distri-
bution [2] and is available with Geant4 version 9.4 (December 2010).
2. Nucleus–Nucleus Interactions and Screened Coulomb
Potentials
At small distances from the nucleus, the potential energy is a Coulomb
potential, while - at distances larger than the Bohr radius - the nuclear
field is screened by the fields of atomic electrons. The interaction between
twonucleiisusuallydescribedintermsofaninteratomicCoulombpotential
(e.g., see Section 2.1.4.1 of Ref. [1] and Section 4.1 of Ref. [3]), which is a
function of the radial distance r between the two nuclei
zZe2
V(r)= ΨI(rr), (1)
r
whereez (projectile)andeZ (target)arethe chargesofthe barenucleiand
ΨI is the interatomic screening function. This latter function depends on
the reduced radius rr given by
r
rr = , (2)
aI
whereaI isthe so-calledscreening length (alsotermedscreening radius).In
theframeworkoftheThomas–Fermimodeloftheatom(e.g.,seeChapters1
and 2 of Ref. [4]) - thus, following the approachof ICRU Report 49 (1993)
-, a commonly used screening length for z = 1 incoming particles is that
from Thomas–Fermi (e.g., see Refs. [5, 6])
CTFa0
aTF = Z1/3 , (3)
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and-forincomingparticleswithz ≥2-thatintroducedbyZiegler,Biersack
and Littmark (1985) (and termed universal screening lengtha)
CTFa0
aU = z0.23+Z0.23, (4)
where
~2
a0 = me2
is the Bohr radius, m is the electron rest mass and
2/3
1 3π
CTF = ≃0.88534
2 4
(cid:18) (cid:19)
is a constant introduced in the Thomas–Fermi model.
The simple scattering model due to Wentzel [10] - with a single ex-
ponential screening-function ΨI(rr) {e.g., see Ref. [10] and Equation (21)
in Ref. [11]} - was repeatedly employed in treating single and multiple
Coulomb-scattering with screened potentials (e.g., see Ref. [11] - and refe-
rences therein - for a survey of such a topic and also Refs. [12–15]). The
resulting elastic differential cross section differs from the Rutherford diffe-
rentialcrosssectionbyanadditionalterm-theso-calledscreeningparame-
ter - whichpreventsthe divergenceofthe crosssectionwhenthe angleθ of
◦
scattered particles approaches 0 . The screening parameter As,M [e.g., see
Equation(21) ofBethe (1953)]- as derivedby Moli`ere(1947,1948)for the
single Coulomb scattering using a Thomas–Fermi potential - is expressedb
as
~ 2 αzZ 2
As,M = 1.13+3.76× (5)
(cid:18)2p aI(cid:19) " (cid:18) β (cid:19) #
where aI is the screening length - from Eqs. (3, 4) for particles with z =1
and z ≥ 2, respectively; α is the fine-structure constant; p (βc) is the
momentum (velocity) of the incoming particle undergoing the scattering
onto a targetsupposed to be initially atrest; c and~ are the speed oflight
andthereducedPlanckconstant,respectively.Whenthe(relativistic)mass
aAnother screening length commonly used is that from Lindhard and Sharff (1961)
(e.g.,seeRef.[8];seealsoRef.[9]andreferencestherein):
CTFa0
aL= (cid:0)z2/3+Z2/3(cid:1)1/2.
bIt has to be remarked that the screening radius originally used in Refs. [12, 13] was
thatfromEq.(3).
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- with corresponding rest mass m - of the incoming particle is much lower
than the rest mass (M) of the target nucleus, the differential cross section
-obtainedfromtheWentzel–Moli`eretreatmentofthe singlescattering-is:
dσWM(θ) zZe2 2 1
= (6)
dΩ (cid:18) pβc (cid:19) (2As,M+1−cosθ)2
zZe2 2 1
= (7)
(cid:18)2pβc(cid:19) As,M+sin2(θ/2) 2
(cid:2) (cid:3)
(e.g., see Section 2.3 in Ref. [11] and references therein). Equation (7) dif-
fers from Rutherford’s formula - as already mentioned - for the additional
2
term As,M to sin (θ/2). The corresponding total cross section {e.g., see
Equation (25) in Ref. [11]} per nucleus is
zZe2 2 π
WM
σ = . (8)
(cid:18) pβc (cid:19) As,M(1+As,M)
Thus, for β ≃1 (i.e., at very large p) and with As,M ≪1, from Eqs. (5, 8)
one finds that the cross section approaches a constant:
WM 2zZe2aI 2 π
σ ≃ . (9)
c ~c 1.13+3.76×(αzZ)2
(cid:18) (cid:19)
In case of a scattering under the action of a central potential (for in-
stance that due to a screened Coulomb field), when the rest mass of the
targetparticleisnolongermuchlargerthantherelativisticmassoftheinco-
ming particle,the expressionofthe differential crosssectionmust properly
be re-written- in the center of mass system - in terms of an “effective par-
′ ′
ticle”withmomentum(p )equaltothatofthe incomingparticle(p )and
r in
rest mass equal to the relativistic reduced mass
mM
µrel = ,
M1,2
whereM1,2 isthe invariantmass;mandM arethe restmassesofthe inco-
mingandtargetparticles,respectively(e.g.,seeRefs.[15–17]andreferences
therein). The “effective particle” velocity is given by:
2 −1
µrelc
βrc=cv 1+ p′ .
u" (cid:18) in (cid:19) #
u
t
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f
Thus, the differential cross section per unit solid angle of the incoming
particle results to be given by
dσWM(θ′) zZe2 2 1
= , (10)
dΩ′ (cid:18)2p′inβrc(cid:19) As+sin2(θ′/2) 2
with (cid:2) (cid:3)
~ 2 αzZ 2
As =(cid:18)2p′in aI(cid:19) "1.13+3.76×(cid:18) βr (cid:19) # (11)
′
and θ the scattering angle in the center of mass system.
TheenergycT transferredtotherecoiltargetisrelatedtothescattering
2 ′
angle as T = T sin (θ /2) - where T is the maximum energy which
max max
canbe transferredinthescattering(e.g.,seeSection1.5ofRef.[1])-,thus,
assuming an isotropic azimuthal distribution one can re-write Eq. (10) in
terms of the kinetic energy transferred from the projectile - , i.e., [−T],
where the negative sign indicates that energy is lost by the projectile - to
the recoil target as
zZe2 2 1 2π
WM ′ ′
dσ = sin(θ )dθ dφ
(cid:18)2p′inβrc(cid:19) As+sin2(θ′/2) 2 Z0
zZe2 2(cid:2) Tmax (cid:3)
=π d[−T]. (12)
(cid:18)p′inβrc(cid:19) [TmaxAs+T]2
Finally, from Eq. (12), the differential cross section with respect to the
kinetic recoil energy (T) of the target is given by:
dσWM(T) zZe2 2 T
max
=π . (13)
dT (cid:18)p′inβrc(cid:19) [TmaxAs+T]2
fByinspectionofEqs.(5, 7,10,11), one findsthat forβr≅1the crosssectionisgiven
byEq.(9).
cOne can show - e.g., see Section 1.5 of Ref. [1] - that the four momentum transfer is
givenby
t=−2MT.
Since t is invariant, then the kinetic energy transferred is also invariant. Furthermore,
sinceT =Tmaxsin2(θ′/2),thenonefindsthat
d[−T]=Tmaxd[−sin2(θ′/2)]= Tmax sin(θ′)dθ′
2
(e.g.,seeSection1.5ofRef.[1]).
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Furthermore, since
pc2
βrc=
E
pM
′
p = (14)
in M1,2
2p2M
T =
max M2
1,2
with p and E the momentum and total energy of the incoming particle in
the laboratory, then one finds
T 2E2
max
= .
(p′inβrc)2 p2Mc4
Therefore, Eq. (13) can be re-written as
dσWM(T) 2 2 E2 1
=2π zZe . (15)
dT p2Mc4 [TmaxAs+T]2
(cid:0) (cid:1)
Equation (15) expresses - as already mentioned - the differential cross sec-
tion as a function of the (kinetic) energy T achieved by the recoil target.

4
10
3
-1 g]10 in silicon
2 m102
c
V
Me101 208Pb
wer [100 15165FIne
ping po10-1 2812SCi
Stop10-2 11B
-3 alpha
10
proton
-4
10
-1 0 1 2 3 4 5 6 7 8
10 10 10 10 10 10 10 10 10 10
Kinetic Energy [MeV/nucleon]
Fig. 1. Nuclear stopping power - in MeVcm2g−1 - calculated using Eq. (17) in sili-
con is shown as a function of the kinetic energy per nucleon - from 50keV/nucleon up
100TeV/nucleon - for protons, α-particle and 11B-, 12C-, 28Si-, 56Fe-, 115In-, 208Pb-
nuclei.
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3
10
2 -1 mg]102 in lead
c
V 1
Me10 208
er [100 15165PInb
w Fe
o 28
g p10-1 Si
pin 12C
p -2
o10 11
St B
10-3 alpha
proton
-4
10
-1 0 1 2 3 4 5 6 7 8
10 10 10 10 10 10 10 10 10 10
Kinetic Energy [MeV/nucleon]
Fig. 2. Nuclear stopping power - in MeVcm2g−1 - calculated using Eq. (17) in lead
is shown as a function of the kinetic energy per nucleon - from 50keV/nucleon up
100TeV/nucleon - for protons, α-particles and 11B-, 12C-, 28Si-, 56Fe-, 115In-, 208Pb-
nuclei.
3. Nuclear Stopping Power
Using Eq. (15) the nuclear stopping power - in MeVcm−1 - is obtained as
dE Tmax dσWM(T)
− = n T dT (16)
A
(cid:18)dx(cid:19)nucl Z0 dT
2 2 E2 Tmax T
= 2n π zZe dT
A p2Mc4Z0 [AsTmax+T]2
(cid:0) 2(cid:1)2 E2 As As+1
= 2n π zZe −1+ln (17)
A p2Mc4 As+1 As
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)
with n the number of nuclei (atoms) per unit of volume and, finally, the
A
negative sign indicates that the energy is lost by the incoming particle
(thus, achieved by recoil targets). For energies higher than a few tens of
keVs, because As ≪1, Eq. (17) can be re-written as
dE 2 2 E2 1
− =2πn zZe ln −1 . (18)
(cid:18)dx(cid:19)nucl A p2Mc4 (cid:20) (cid:18)As(cid:19) (cid:21)
(cid:0) (cid:1)
It has to be noted that, as the incoming momentum increases to a value
for which p ≃E, the set of terms - in front of those included in brackets -
decreases and approachesa constant; while the term ln(1/As) increases as
ln(p) for E ≫ mc2,Mc2 [e.g., see Eqs. (11, 14)]. Thus, a slight increase of
thenuclearstoppingpowerwithenergyisexpectedbecauseofthedecrease
of the screening parameter with energy.
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1
10
er
pow 100
g
n
oppi 10-1
st
ar
e -2
cl 10
u
d n
uce 10-3 proton in2 0l8ePabd in lead
d 56
sal re 10-4 2085P6b in silicon Fe in lead
ver F28e in silicon alpha in lead
ni Si in silicon
U -5 universal stopping power
10
proton in silicon 28
alpha in silicon Si in lead
-6
10
-1 0 1 2 3 4 5 6 7 8 9 10
10 10 10 10 10 10 10 10 10 10 10 10
Universal reduced energy
Fig. 3. Universal stopping power (dashed line) as a function of the universal reduced
energy [Eq. (19)]. The other curves correspond to the dimensionless nuclear stopping
power obtained from Eq. (17) - for protons, α-particles, 28Si-, 56Fe-, 208Pb-nuclei in
silicon(Fig.1)andlead(Fig.2)absorbers-anddividedbytheparameterK[Eq.(21)].
For instance, in Fig. 1 (Fig. 2) the nuclear stopping power in silicon
(in lead) - in MeVcm2g−1 - is shown as a function of the kinetic energy
per nucleon - from 50keV/nucleon up 100TeV/nucleon - for protons, α-
particles and 11B-, 12C-, 28Si-, 56Fe-, 115In-, 208Pb-nuclei.
It has to be remarkedthat - atvery low energies - the Wentzel–Moli`ere
nuclear stopping power [Eq. (17)] differs from that obtained by Ziegler,
Biersack and Littmark (1985) using the so-called universal screening po-
tential (see alsoRef. [18]).However,they haveshown(e.g., seeFigure 2-18
inRef. [7]or,equivalently,Figure2-18inRef.[18])thatdifferentscreening
potentials - including the Bohr potential in which ΨI(rr) is assumed to be
an exponential function similarly to Wentzel’s assumption - result in nu-
clear stopping powers which exhibit marginal differences for ǫr,U above 10
(see alsoFig.3).ǫr,U isthe so-calleduniversal reduced energy expressedas:
R M
ǫr,U = zZ(z0.23+Z0.23) m+M Ek, (19)
(cid:18) (cid:19)
where E is in MeV andthe numericalconstantis R=32.536×103MeV−1
k
{e.g.,seeEquation(2-73)ofRef.[7]orEquation(2-88)ofRef.[18],seealso
Section 2.1.4.1 of Ref. [1]}. For instance, in silicon ǫr,U ≃ 10 corresponds
to E ≃13keV [67keV/nucleon]for protons [leadnuclei]. Ziegler,Biersack
k
andLittmark(1985)providedageneralexpressionforthenuclearstopping
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15
protons
10
Percent variation 05 GePSbi C
-5
-10
-15
0.1 1 10 100
Kinetic energy [MeV]
15
alpha-particles
10
Percent variation 05 GeSi C Pb
-5
-10
-15
0.1 1 10 100
Fig.4. Variation(inpercentage) oKifnenticu ecnleergayr [MsetVo/npupcleionng] powers-calculatedwithEq.(17)
forenergies from50keV/nucleon upto 100MeV/nucleon -withrespect toICRU tabu-
latedvalues[3]asafunctionofthekineticenergyinMeV/nucleon,forprotons(top)and
α-particles(bottom) traversingamorphouscarbon,silicon,germaniumandleadmedia.
power (e.g., see Section 2.1.4.1 of Ref. [1]), i.e.,
U
dE
− =KR(ǫr,U) [MeV/cm], (20)
dx
(cid:18) (cid:19)nucl

15
alpha-particles
10
Percent variation 05 Ge CSi
-5 Pb
-10
-15
0.1 1 10 100
Fig.5. Variation(inpercentage) oKifnenticu ecnleergayr [MsetVo/npupcleionng] powers-calculatedwithEq.(17)
with the expressions (22, 23) for the screening parameter - with respect to ICRU tab-
ulated values [3] as a function of the kinetic energy in MeV/nucleon, for α-particles
traversing amorphous carbon, silicon, germanium and lead media for energies from
50keV/nucleon upto100MeV/nucleon.
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with
ρzZ
K≃5.1053×103 [MeV/cm] (21)
A(1+M/m)(z0.23+Z0.23)
where ρ and A are the density and atomic weight of the target medium,
respectively; R(ǫr,U) - the so-called (universal) reduced nuclear stopping
power [termed, also, (universal) scaled nuclear stopping power] given in
Equations (2-89)–(2-90) in Ref. [18] (see also page 80 in Ref. [1]) - is di-
mensionless. Additionally, the present calculations can be compared with
values ofstopping powers- obtainedusing the universalscreenedpotential
-availableinSRIM(2008)[19].Usuallyanagreement-tobetterthanafew
percents - is achieveddownto about 150keV/nucleon,where - for instance
- onefinds ≈5.5(9.9)%forα-particles(leadions)insilicon.Atlargeener-
gies, the non-relativistic approach due to Ziegler, Biersack and Littmark
(1985) becomes less appropriate and deviations from stopping powers cal-
culated by means of the universal screening potential are expected and
observed for ǫr,U &(1.5–2.5)×104 (e.g., see Fig. 3).
The non-relativistic approach - based on the universal screening po-
tential - of Ziegler, Biersack and Littmark (1985) was also used by
ICRU (1993) to calculate nuclear stopping powers - currently available
on the web (e.g., see Ref. [20]) - due to protons and α-particles in ma-
terials. ICRU (1993) used as screening lengths those from Eqs. (3, 4) for
protons and α-particles, respectively. In Fig. 4, the variation (in percent-
age) of nuclear stopping powers - calculated with Eq. (17) - with respect
to ICRU tabulated values [3] is shown as a function of the kinetic energy
per nucleon (in MeV/nucleon) - for energies from 50keV/nucleon up to
100MeV/nucleon - for protons and α-particles traversing amorphous car-
bon, silicon, germanium and lead media. The stopping powers for protons
(α-particles) from Eq. (17) are less than ≈ 5% larger than those reported
by ICRU (1993) from 50keV/nucleon up to ≈ 8MeV (19MeV/nucleon) -
the upper energy corresponds to ǫr,U ≈1.6×104 (2.6×104) for protons in
an amorphous carbon (α-particles in a silicon) medium -. At larger ener-
giesthe stoppingpowersfromEq.(17)largelydifferfromthosefromICRU
- as expected - due to the complete relativistic treatment of the present
approach.
The simple screening parameter used so far [Eq. (11)] - derived by
Moli`ere (1947) - can be modified by means of a practical correction, i.e.,
~ 2 αzZ 2
A′ = 1.13+3.76×C , (22)
s (cid:18)2p′in aI(cid:19) " (cid:18) βr (cid:19) #