Download Ionization

'
onization
$
1
Ionization
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
2
Ionization
The basic mechanism is an inelastic collision of the moving charged
particle with the atomic electrons of the material, ejecting off an
electron from the atom :
p + atom → p + atom+ + e−
In each individual collision, the energy transfered to the electron is
small. But the total number of collisions is large, and we can well
define the average energy loss per (macroscopic) unit path length.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
3
Mean energy loss and energetic δ-rays
dσ(Z, E, T )
dT
is the differential cross-section per atom for the ejection of an
electron with kinetic energy T by an incident charged particle of
total energy E moving in a material of density ρ.
One may wish to take into account separately the high-energy
knock-on electrons produced above a given threshold Tcut (miss
detection, explicit simulation ...).
Tcut I (mean excitation energy in the material).
Tcut > 1 keV in Geant4
Below this threshold, the soft knock-on electrons are counted only
as continuous energy lost by the incident particle.
Above it, they are explicitly generated. Those electrons must be
excluded from the mean continuous energy loss count.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
4
The mean rate of the energy lost by the incident particle due to the
soft δ-rays is :
Z Tcut
dσ(Z, E, T )
dEsof t (E, Tcut )
= nat ·
T dT
(1)
dx
dT
0
nat : nb of atoms per volume in the matter.
The total cross-section per atom for the ejection of an electron of
energy T > Tcut is :
Z Tmax
dσ(Z, E, T )
dT
(2)
σ(Z, E, Tcut ) =
dT
Tcut
where Tmax is the maximum energy transferable to the free
electron.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
5
Mean rate of energy loss by heavy particles
The integration of equation 1 leads to the well known Bethe-Bloch
truncated energy loss formula [pdg] :
(zp )2
dE
2
2
= 2πre mc nel 2 ×
dx T <Tcut
β
2 2 2
Tup
2Ce
2mc β γ Tup
2
−
β
1
+
−
δ
−
ln
I2
Tmax
Z
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
6
where
classical electron radius:
mc2
energy-mass of electron
nel
electrons density in the material
zp
charge of the incident particle
Tup
min(Tcut , Tmax )
I
mean excitation energy in the material
δ
density effect function
Ce
shell correction function
nel = Z nat
&
Geant4 Tutorial
e2 /(4π0 mc2 )
re
Nav ρ
=Z
A
Tmax
2mc2 (γ 2 − 1)
=
1 + 2γm/M + (m/M )2
%
May 23, 2007
$
'
Stopping power [MeV cm2/g]
onization
7
µ+ on Cu
µ−
100
Bethe-Bloch
10
LindhardScharff
AndersonZiegler
Eµc
Nuclear
losses
Radiative
losses
Radiative
Minimum effects
ionization reach 1%
Without δ
1
0.001
0.01
0.1
1
10
0.1
1
10
100
1
&
Geant4 Tutorial
[MeV/c]
βγ
100
1000
10 4
10 5
10 6
10
100
1
10
100
[GeV/c]
Muon momentum
[TeV/c]
%
May 23, 2007
$
'
onization
8
mean excitation energy
There exists a variety of phenomenological approximations for I,
the simplest being I = 10eV × Z
In Geant4 we have tabulated the recommended values in
[icru84].
22
20
18
16
I/Z (eV)
ICRU 37 (1984)
14
Barkas & Berger 1964
12
10
8
&
Geant4 Tutorial
0
10
20
30
40
50
Z
60
70
80
90
100
%
May 23, 2007
$
'
onization
9
the density effect
δ is a correction term which takes into account of the reduction in
energy loss due to the so-called density effect. This becomes
important at high energy because media have a tendency to
become polarised as the incident particle velocity increases. As a
consequence, the atoms in a medium can no longer be considered as
isolated. To correct for this effect the formulation of Sternheimer
[Ster71] is generally used.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
10
the shell correction
2Ce /Z is a so-called shell correction term which accounts for the
fact that, under certain conditions, the probability of collision with
the electrons of the inner atomic shells (K, L, etc.) is negligible.
The semi-empirical formula used in Geant4, applicable to all
materials, is due to Barkas [Bark62]:
Ce (I, βγ) =
&
Geant4 Tutorial
a(I)
b(I)
c(I)
+
+
(βγ)2
(βγ)4
(βγ)6
%
May 23, 2007
$
'
onization
11
low energies
The mean energy loss can be described by the Bethe-Bloch formula
only if the projectile velocity is larger than that of orbital electrons.
In the low-energy region where this is not verified, a different kind
of parameterisation must be used.
For instance:
• Andersen and Ziegler [Ziegl77] for 0.01 < β < 0.05
• Lindhard [Lind63] for β < 0.01
See ICRU Report 49 [icru93] for a detailed discussion of
low-energy corrections.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
12
low energies
in G4hIonisation, a simple parametrisation gives the energy loss
as a function of τ = (T /Mproton c2 ) :
√
dE/dx = nel · (A τ + Bτ )
√
dE/dx = nel · C/ τ
for
for
τ ≤ τ0
τ ∈ [τ0 , τ1 ]
The parameters A, B, C are such that dE/dx is a continuous
function of the kinetic energy T at τ0 and τ1 .
Above T1 the truncated Bethe-Bloch formula is used.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
13
PhysicsTables
At initialization stage, the function BuildPhysicsTables()
computes and tabulates the dE/dx due to the soft δ-rays, as a
function on energy, and for all materials.
The dE/dx of charged hadrons is obtained from that of proton (or
antiproton) by calculating the kinetic energy of a proton with the
same β, and using this value to interpolate the proton tables.
Tproton
&
Geant4 Tutorial
Mproton
T
=
M
%
May 23, 2007
$
'
onization
14
minimum ionization
2.5
10
6
5
H2 liquid
4
He gas
3
2
1
0.1
Sn
Pb
1.0
0.1
0.1
0.1
&
Geant4 Tutorial
1.0
10
100
βγ = p/Mc
Fe
Al
C
1000
10 000
1.0
10
100
Muon momentum (GeV/c)
1000
1.0
10
100
Pion momentum (GeV/c)
10
100
1000
Proton momentum (GeV/c)
1000
10 000
〈– dE/dx〉min (MeV g –1cm2)
− dE/dx (MeV g−1cm2)
8
H2 gas: 4.10
H2 liquid: 3.97
2.0
2.35 – 0.64 log10(Z)
1.5
Solids
Gases
1.0
0.5
H
He Li Be B C NO Ne
1
2
5
10
Z
Fe
20
Sn
50
100
%
May 23, 2007
$
'
onization
15
Fluctuations in energy loss
h∆Ei = (dE/dx).∆x gives only the average energy loss by
ionization. There are fluctuations. Depending of the amount of
matter in ∆x the distribution of ∆E can be strongly asymmetric
(→ the Landau tail).
The large fluctuations are due to a small number of collisions with
large energy transfers.
&
Geant4 Tutorial
%
May 23, 2007
'
onization
$
16
The figure shows the energy loss distribution of 3 GeV electrons in 5 mm
of an Ar/CH4 gas mixture [Affh98].
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
17
Energy loss fluctuations : the models in Geant
The straggling is partially taken into account by the simulation of
energy loss by the production of δ-electrons with energy T > Tcut .
However continuous energy loss also has fluctuations.
Two different models of fluctuations are applied depending on the
value of the parameter κ which is the lower limit of the number of
potential interactions of the particle in the step:
κ=
∆E
Tcut
where ∆E is the mean continuous energy loss in a track segment of
length ∆x.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
18
Energy loss fluctuations : Gaussian regime
For κ ≥ 10 and if Tmax is not too big ( Tmax ≤ 2Tcut ) the
straggling function approaches a Gaussian distribution with Bohr’s
variance [icru93]:
2
2
β
Z
Ω2 = 2πre2 me c2 nel h2 ∆x Tcut 1 −
β
2
(Zh is the charge of the incident particle in units of positron charge.)
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
19
Energy loss fluctuations : Urban model
For κ < 10.
It is based on a rather simple model of the particle-atom
interaction.
The atoms are assumed to have only two energy levels E1 and E2 .
The particle-atom interaction can be :
• an excitation with energy loss E1 or E2
• an ionization with energy loss distribution g(E) ∼ 1/E 2 ,
with E ∈ [E0 , Tup ].
Z Tup
E0 Tup 1
g(E) dE = 1 =⇒ g(E) =
Tup − E0 E 2
E0
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
20
The macroscopic cross section for excitation (i = 1, 2) is :
fi ln[2mc2 (βγ)2 /Ei ] − β 2
dE
(1 − r)
Σi =
2
2
2
dx
Ei ln[2mc (βγ) /I] − β
I = mean excitation energy
Ei , fi = energy levels and oscillator strengths of the atom
r is a model parameter.
fi and Ei must satisfy the sum rules [Bichs88] :
f1 + f 2 = 1
f1 · ln E1 + f2 · ln E2 = ln I
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
21
For ionization :
Σ3 =
dE
dx
Tup − E0
T
E0 Tup ln( Eup
)
0
r
E0 = ionization energy of the atom
Tup = min(Tmax , Tcut )
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
22
In a step of length ∆x the nb of collisions ni , (i = 1, 2 for excitation,
3 for ionization) follows a Poisson distribution with :
hni i = ∆x Σi
The mean energy loss in a step is the sum of the excitation and
ionization contributions :
(
)
Z Tup
dE
· ∆x = Σ1 E1 + Σ2 E2 +
Eg(E)dE ∆x
dx
E0
E2 is approximately the K-shell energy of atoms : E2 = 10 eV Z 2
E0 = 10 eV
Zf2 = 2 is the number of K-shell electrons.
f1 and E1 can be obtained from the sum rules.
&
Geant4 Tutorial
%
May 23, 2007
'
onization
$
23
The parameter r is the only variable in the model which can be
tuned. This parameter determines the relative contribution of
ionization and excitation to the energy loss.
Its value has been parametrized from comparison of simulated
energy loss distributions to experimental data :
Tup
r = 0.03 + 0.23 ln ln
I
&
Geant4 Tutorial
%
May 23, 2007
'
$
onization
24
Sample the energy loss :
the loss due to the excitation is
∆Eexc = n1 E1 + n2 E2
where n1 and n2 are sampled from Poisson distribution.
The energy loss due to the ionization can be generated from the
distribution g(E) by the inverse transformation method. Then, the
contribution from the ionization will be :
∆Eion =
n3
X
j=1
E0
1 − uj
Tup −E0
Tup
n3 is the nb of ionizations sampled from Poisson distribution,
uj is uniform ∈ [0, 1].
The total energy loss in a step ∆x is ∆E = ∆Eexc + ∆Eion
The energy loss fluctuation comes from the fluctuations of the
collision numbers ni
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
25
This simple model of the energy loss fluctuations is rather fast and
it can be used for any thickness of the materials, and for any Tcut .
This has been proved performing many simulations and comparing
the results with experimental data, see e.g [Urban95].
Approaching the limit of the validity of Landau’s theory, the loss
distribution approaches smoothly the Landau form.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
26
Fluctuations on ∆E lead to fluctuations on the actual range
(straggling).
penetration of e− (16 MeV) and proton (105 MeV) in 10 cm of water.
WORL
RUN
EVENT
NR
NR
1
20
10
11
12
13
14
15
16
17
18
19
1
2
3
4
5
6
7
8
9
18/ 9/ 0
WORL
RUN
EVENT
NR
NR
−
e
e
p
−
pp
−
e
e
p
pp
p
p
−
p
−
e−
−
e
18/ 9/ 0
p
p
p
−
e
−
e
e−
1
20
10
11
12
13
14
15
16
17
18
19
1
2
3
4
5
6
7
8
9
e
p
p
e−
p
p
e−
−
e e−
−
e −
−
e e
e
p
−
p
p
e− −
e
1 cm
&
Geant4 Tutorial
1 cm
%
May 23, 2007
'
onization
$
27
Bragg curve. More energy per unit length are deposit towards
the end of trajectory rather at its beginning.
MeV/mm
proton 105 MeV in water: Bragg peak
5
p 105 MeV
e- 16 MeV
4
3
2
1
0
0
10
20
30
40
50
60
70
80
90
100
X(mm)
Edep along X(MeV/bin)
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
28
Energetic δ rays
The differential cross-section per atom for producing an electron of
kinetic energy T, with I Tcut ≤ T ≤ Tmax , can be written :
2
2
z
T
T
dσ
p 1
= 2πre2 mc2 Z 2 2 1 − β 2
+
dT
β T
Tmax
2E 2
(the last term for spin 1/2 only).
The integration of equation(2) gives :
2πre2 Zzp2
1
Tmax
β2
1
−
ln
−
σ(Z, E, Tcut ) =
β2
Tcut
Tmax
Tmax
Tcut
Tmax − Tcut
+
2E 2
(the last term for spin 1/2 only).
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
29
Mean free path
λ(E, Tcut ) =
X
i
nati · σ(Zi , E, Tcut )
!−1
nati : nb of atoms per volume of the ith element in the material.
At initialization stage, the function BuildPhysicsTables()
computes and tabulates :
• meanFreePath for all materials
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
30
sample the energy of the δ-ray
Apart from the normalisation, the cross-section can be factorised :
dσ
= f (T ) g(T )
dT
with T ∈ [Tcut , Tmax ]
with :
f (T )
g(T )
=
1
Tcut
= 1−β
2
−
1
Tmax
T
Tmax
1
T2
T2
+
2E 2
(the last term in g(T ) for spin 1/2 only).
The energy T is obtained by :
1. sample T from f (T )
2. accept/reject the sampled T with a probability g(T ).
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
31
compute the final kinematic
The function AlongStepDoIt() computes the energy lost by the
ionizing particle, along its step.
The function PostStepDoIt() samples the energy of the knock-on
electron.
Then, the direction of the scattered electron is generated with
respect to the direction of the incident particle :
θ is calculated from the energy momentum conservation.
φ is generated isotropically.
This information is used to calculate the energy and momentum of
both scattered particles and to transform them into the global
coordinate system.
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
32
delta rays
200 MeV electrons, protons, alphas in 1 cm of Aluminium
WORL
&
Geant4 Tutorial
RUN
EVENT
NR
NR
0.1cm
1
30
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
1
2
3
4
5
6
7
8
9
6/ 9/ 0
%
May 23, 2007
'
onization
$
33
muon : number of δ-rays per cm in Aluminium
Tables for MUON + in Aluminium
DRAY X-sec (1/cm)
&
Geant4 Tutorial
muon kinetic energy (GeV)
%
May 23, 2007
'
onization
$
34
Incident electrons and positrons
For incident e−/+ the Bethe Bloch formula must be modified
because of the mass and identity of particles (for e− ).
One use the Moller or Bhabha cross sections [Mess70] and the
Berger-Seltzer dE/dx formula [icru84, Selt84].
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
35
truncated Berger-Seltzer dE/dx formula
dE
dx
1
×
2
β
2(γ + 1)
±
(γ − 1, τup ) − δ
+
F
ln
(I/mc2 )2
= 2πre2 mc2 nel
T <Tcut
where
Tcut
τc
τmax
τup
energy cut for δ − ray
Tcut /mc2
maximum energy transfer: γ − 1 for e+ , (γ − 1)/2 for e−
min(τc , τmax )
The functions F ± are given in [Selt84].
&
Geant4 Tutorial
%
May 23, 2007
'
onization
$
36
de/dx due to ionization (Berger-Seltzer formula)
Tables for ELECTRON in Aluminium
LOSS (MeV/cm)
&
Geant4 Tutorial
electron kinetic energy (GeV)
%
May 23, 2007
$
'
onization
37
differential cross section per atom (for T I)
For the electron-electron (Möller) scattering we have:
dσ
d
=
2πre2 Z
×
2
β (γ − 1)
2
1
2γ − 1
1 1 2γ − 1
1
(γ − 1)
−
−
+
+
γ2
γ2
1− 1−
γ2
and for the positron-electron (Bhabha) scattering:
2πre2 Z
B1
1
dσ
2
=
+
B
−
−
B
+
B
2
3
4
d
(γ − 1) β 2 2
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
38
where
E
=
energy of the incident particle
γ
=
E/mc2
y
=
1/(γ + 1)
B1
=
B2
=
B3
=
2 − y2
B4
=
(1 − 2y)(3 + y 2 )
=
(1 − 2y)2 + (1 − 2y)3
T /(E − mc2 )
(1 − 2y)3
with T the kinematic energy of the scattered electron.
The kinematical limits for the variable are:
0 =
&
Geant4 Tutorial
Tcut
1
≤
≤
E − mc2
2
for e− e−
0 ≤ ≤ 1 for e+ e−
%
May 23, 2007
$
'
onization
39
Total cross-sections per atom
The integration of formula 2 gives the total cross-section per atom,
for Möller scattering (e− e− ) :
1
1
2πre2 Z
(γ − 1)2 1
−x + −
σ(Z, E, Tcut ) =
β 2 (γ − 1)
γ2
2
x 1−x
2γ − 1 1 − x
ln
−
γ2
x
and for Bhabha scattering (e+ e− ) :
2
2πre Z 1 1
− 1 + B1 ln x + B2 (1 − x)
σ(Z, E, Tcut ) =
2
(γ − 1) β
x
B
B3
4
(1 − x3 )
− (1 − x2 ) +
2
3
&
Geant4 Tutorial
%
May 23, 2007
$
'
onization
40
where
γ
=
E/mc2
B1
β2
=
B2
x
=
1 − (1/γ 2 )
y
=
&
Geant4 Tutorial
Tcut /(E − mc2 )
1/(γ + 1)
B3
B4
= 2 − y2
= (1 − 2y)(3 + y 2 )
= (1 − 2y)2 + (1 − 2y)3
= (1 − 2y)3
%
May 23, 2007
$
'
onization
41
sample the energy of the δ-ray
Apart from the normalisation, the cross-section are factorised as :
dσ
= f () g()
d
for e− e− scattering :
f ()
g()
=
=
1
0
2 1 − 20
2
4
γ
2 2
2
(γ
−
1)
−
(2γ
+
2γ
−
1)
+
9γ 2 − 10γ + 5
1−
(1 − )2
and for e+ e− scattering :
f ()
=
g()
=
1 0
2 1 − 0
B 0 − B 1 + B 2 2 − B 3 3 + B 4 4
B0 − B1 0 + B2 20 − B3 30 + B4 40
Here B0 = γ 2 /(γ 2 − 1) and all the other quantities have been defined
&
above.
Geant4 Tutorial
%
May 23, 2007
Ionization
41-1
References
[Bark62] W. H. Barkas. Technical Report 10292,UCRL, August 1962.
[Lind63] J. Lindhad and al., Mat.-Fys. Medd 33, No 14 (1963)
[Mess70] H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.
[Ster71] R.M.Sternheimer and al. Phys.Rev. B3 (1971) 3681.
[Ziegl77] H.H. Andersen andJ.F. Ziegler, The stopping and ranges of
ions in matter (Pergamon Press 1977)
[Selt84] S.M. Seltzer and M.J. Berger, Int J. of Applied Rad. 35, 665
(1984)
[icru84] ICRU Report No. 37 (1984)
[icru93] ICRU Report No. 49 (1993)
[Affh98] K. Affholderbach at al. NIM A410, 166 (1998)
Geant4 Tutorial
May 23, 2007
Ionization
41-2
[pdg] D.E. Groom et al. Particle Data Group . Rev. of Particle Properties. Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/
[Bichs88] H.Bichsel Rev.Mod.Phys. 60 (1988) 663
[Urban95] K.Lassila-Perini, L.Urbán Nucl.Inst.Meth. A362(1995) 416
[geant3] geant3 manual Cern Program Library Long Writeup W5013
(1994)
Geant4 Tutorial
May 23, 2007