II.2 Energy Loss of Electrons & Positrons
r  Electrons & positrons suffer energy losses by radiation in addition
to the energy losses by collisions (ionization)
" dE %
" dE %
" dE %
+$ '
$ ' =$ '
# dx & tot # dx & rad # dx & coll
(2.49)
r  The basic mechanism of energy loss via collisions is also valid for
e±, but Bethe-Bloch
must be modified for 2 reasons:
€
i) their small mass ⇒ incident particle may be deflected
ii) For e- we have collisions between identical particles
⇒ we must take into account indistinguishable particles
⇒ Obtain some modifications, e.g. Tmax=Te/2
II.2 Energy Loss of Electrons & Positrons
r  The modified Bethe-Bloch formula is:
0
Z 1 / % τ 2 * (τ + 2) (
2C 2
−
= ρ
ln '
* + F(τ ) − δ ( βγ ) −
2 /
2
2
' 2(I / m c ) *)
dx
2 Aβ
Z2
e
. &
1
where τ is the e± kinetic energy τ=Te/mec2
dE
and
K
F(τ ) = 1 − β +
2
τ2
8
− (2τ + 1) ln 2
(τ + 1)
2
for e-
'
β2 $
14
10
4
F(τ ) = 2 ln 2 −
+
+
for e+
& 23 +
2
3)
12 %
τ + 2 (τ + 2)
(τ + 2) (
(2.50)
(2.51)
(2.52)
r  So, in F(τ) a term ∝-β2 is left & 1.term bears some resemblance
with Bethe-Bloch for heavy particles, other 2 terms remain identical
r  e± are the only particles in which radiation contributes
substantially to energy loss (at least at current energies)
since emission probability varies as σ∝r2e=(e2/mec2)2
⇒ For µ± radiation loss is 40000 times smaller than for e±
II.2 Energy Loss of Electrons & Positrons
r  The density effect can be parameterized by an empirical
formula in terms of X=log10(βγ)
# 0
for X<X 0
%%
δ = $ 4.6052X + C + a(X1 − X) k for X 0 < X < X1
%
for X>X1
%& 4.6052X+C
(2.20)
r  The explicit values of X0, X1, C, a and k depend upon the
absorbing material, with C defined in terms of plasma frequency
#
&
I
C = − % 2 ln
+ 1(
%
(
ω p
$
'
with
ωp =
N e e2
(2.53)
π me
r  An empirical formula for the shell correction is
(
+ ( 3.850190η
)
) × 10
C(I, η ) = 0.422377η −2 + 0.0304043η −4 − 0.00038106η −6 × 10 −6 I2 [eV2 ]
with
−2
η = βγ
+ 0.1667989η
−4
− 0.00157955η
−6
−9
I [eV ]
3
3
(2.54)
II.2 Energy Loss of Electrons & Positrons
Material
I{eV]
-C
a
k
X1
X0
Graphite
78
2.99
0.2024
3.00
2.486
-0.0351
Mg
156
4.53
0.0816
3.62
3.07
0.1499
Cu
322
4.42
0.1434
2.90
3.28
-0.0254
Al
166
4.24
0.0802
3.63
3.01
0.1708
Fe
286
4.29
0.1468
2.96
3.15
-0.0012
Au
790
5.57
0.0976
3.11
3.7
0.2021
Pb
823
6.2
0.0936
3.16
3.81
0.3776
Si
173
4.44
0.1492
3.25
2.87
0.2014
NaI
452
6.06
0.1252
3.04
3.59
0.1203
N2
82
10.5
0.1534
3.21
4.13
1.738
O2
95
10.7
0.1178
3.29
4.32
1.754
H2O
75
3.5
0.0911
3.48
2.80
0.2400
Lucite
74
3.3
0.1143
3.38
2.67
0.1824
Air
85.7
10.6
0.1091
4.28
4.28
1.742
BGO
534
5.74
0.0957
3.08
3.78
0.0456
Plastic Scint
64.7
3.2
0.1610
3.24
2.49
0.1464
II.2 Energy Loss of Electrons & Positrons
<dE/dx> [MeV/cm]
r  Compare energy loss of electrons and positrons with Bethe-Bloch
formula for Pb
δ [MeV/cm]
Pb
e-
e+
Bethe-Bloch
βγ
βγ
r  Particularly, for larger βγ Bethe-Bloch underestimates the energy
loss for both e- and e+
r  For low values of βγ Bethe-Bloch yields too large dE/dx losses
II.2 Energy Loss of Electrons & Positrons
II.2 Energy Loss of Electrons & Positrons
r  Bremsstrahlung emission depends on strength of E-field felt by eè amount of screening from atomic e-s surrounding nucleus plays
important role
Ø Screening can be parameterized by:
ξ =
E0: initial total e- energy
100me c2 ω
EE0 Z
(2.55)
E: final total e- energy
hω: photon energy
Z: atomic number
1
3
r  ξ is related to radius of Thomas Fermi atom
Ü it is small for complete screening (ξ ≅ 0,E/E0 ≅1)
Ü while it is large for no screening (ξ » 1, E/E0 ≅0)
r  The bremsstrahlung cross section for relativistic e± (E>few MeV) is
dσ
dω
= 4Z2re2α
•  ε=E/E0
•  α=1/137
1
ω
{(
)
1 + ε 2 ( 14 φ1 (ξ ) −
)
1
3
ln Z − f(Z) *
+
- 23 ε ( 14 φ2 (ξ ) −
)
1
3
ln Z − f(Z) *
+
}
(2.56)
•  φ1(ξ), φ2(ξ) screening functions
•  F(Z) function for Coulomb correction
II.2 Energy Loss of Electrons & Positrons
r  Since equation (2.43) is the result of a Born approximation,
it does not hold for low energies
r  For heavy elements (Z≥5) the screening functions φ1(ξ) & φ2(ξ) are
usually calculated with the atomic Thomas-Fermi model & their
values are given numerically
r  A useful approximation is given by the following empirical formulae:
φ1 (ξ ) = 20.863 − 2 ln $1 + (0.55846ξ )2 & − 4 $1 − 0.6 exp(−0.9ξ ) − 0.4 exp(−1.5ξ ) &
%
'
%
'
φ2 (ξ ) = φ1 (ξ ) −
2
3
$1 + 6.5ξ + 6ξ 2 &
%
'
−1
0.5% precision
(2.57)
(2.58)
r  It is interesting to look at asymptotic values for ξ=0 & ξ=∞:
φ1 (0) = φ2 (0) +
2
3
= 20.863
for ξ → 0 (complete screening) (2.59)
φ1 (∞) = φ2 (∞) → 19.19 − 4 ln ξ for ξ → ∞ (no screening)
(2.60)
II.2 Energy Loss of Electrons & Positrons
ξ
ε→1
φ1(ε)
φ2(ε)
ε→0
φi(ξ)
r  Screening functions and
approximations for no screening
and complete screening in Xe
for 10 GeV e±
Screening functions
φi(ε)
r  Screening variable for 10 GeV e±
versus e± energy in Xe
ε=E/E0
φ2(ξ)
φ1(ξ)
ξ→∞
ξ→0
ε
ξ(ε)
II.2 Energy Loss of Electrons & Positrons
r  The function f(Z) is a small correction to the Born approximation,
accounting for Coulomb interactions of the emitting e- in E field
of nucleus, that can be approximated by
f(Z) = a2
{
1
2
1+ a
+ 0.20206 − 0.0369a2 + 0.0083a 4 − 0.002a6
}
Z
(2.61)
with a= 137
r  The cross sections for complete screening and no screening are:
dσ
α,
= 4Z re - 1 + ε 2 − 23 ε
dω
ω.
(
2 2
)
−1
&
) 1 /
3
ln(183Z
)
−
f(Z)
('
+* + 9 ε 01
& 2E E
)
1
0
= 4Z re
1 + ε − 23 ε (ln
− − f(Z) +
2
( m c ω 2
+
dω
ω
e
'
*
dσ
2 2
α
(
2
)
(ξ→0)
(2.62)
(ξ→∞)
(2.63)
r  Bremstrahlung cross section for
10 GeV e± in Xe and Pb
r  Coulomb correction yields a small
correction in heavy elements only
dσ/dω
II.2 Energy Loss of Electrons & Positrons
dσ(ξ→0)
Pb
dσ(ξ→∞)
dσ/dω
Coulomb interaction f(Z):
E/E0
Xe
dσ(ξ→0)
F(Z)
dσ(ξ→∞)
Z/137
E/E0
II.2 Energy Loss of Electrons & Positrons
r  Now we can obtain the energy loss due to radiation by just
integrating the differential cross section over allowable Eγ range:
" dE %
N0 ρ
−$
=
'
A
# dx & rad
ω0
∫
0
Φrad⋅E0
dσ
ω
E0 , ω dω
dω
(
)
E0=!ω0
(2.64)
r  Since dσ/dω ∝ ω-1, the integral is practically independent of ω & is
a function of the material only.
r  For mec2«E0«137 mec2Z1/3, ξ»1 we have no screening &
integral yields:
$ 2E
'
1
0
Φ rad = 4Z2re2α & ln
− − f(Z) )
2
&% m c
)(
3
e
r  For E0»137 mec2Z1/3, ξ≅0 we have complete screening &
$
'
integral yields:
−
1
2 2
1
3
Φ rad = 4Z re α & ln(183Z ) −
− f(Z) )
18
%
(
(2.65)
(2.66)
r  At intermediate values of ξ, the integral needs to be calculated
numerically
II.2 Energy Loss of Electrons & Positrons
r  Lets compare: dE/dx via bremsstrahlung ⇔ dE/dx via ionization:
Ø (dE/dx)ion: ∝ ln E, ∝ Z
Ø (dE/dx)rad: ∝ E, ∝ Z2
r  This explains rapid rise in radiation loss
r  Another difference is that in radiation
loss all energy can be radiated via 1 or
2 photons
p
e
r  The above discussion represents case of
mean energy loss from radiation in field
of nucleus
r  There is also a contribution from
bremsstrahlung of field of atomic er  Exact calculation shows that one obtains same formula here,
except Z2 is replaced by Z,
Ö So, to account for both fields, Z2 ⇒Z(Z+1) in above equations
II.2 Energy Loss of Electrons & Positrons
r  Critical energy: For each material we define the energy, where
(dE/dx)rad=(dE/dx)coll as critical energy Ec
r  Approximate formula for
critical energy:
610
for solids
Z + 1.24
710
Ec =
for gases
Z + 0.92
Ec =
E.g. Pb: Ec=7.3 MeV
Air: Ec=102 MeV
(2.67)
II.2 Energy Loss of Electrons & Positrons
r  Another independent quantity is the radiation length X0, defined as
the distance over which the e- energy is reduced by 1/e due to
radiation loss only
−dE = NΦ rad dx
r  We obtain differential equation:
r  In the high-energy limit, where collisions can be ignored wrt
radiation, the rhs is independent of E:
"
%
where x is distance traveled & X0=1/(NΦ)
x
E = E0 exp $ −
'
$# X '&
0
(2.68)
(2.69)
r  Plugging in the expression for Φrad yields:
1
≅ # 4Z(Z + 1) ρ
%$
X0
N
0
A
{
& r2α ln(183Z − 13 ) − f(Z)
(' e
}
(2.70)
II.2 Energy Loss of Electrons & Positrons
r  Expressing x in units of X0 we obtain for the integral Φrad
−
dE
dt
(2.71)
≅ E0
r  Thus radiation energy loss is roughly independent of material type
when expressed in terms of X0
2
eg, H2:
Al:
Pb:
r  For compounds we have:
1
X0
=
∑ wi
i
1
Xi0
(2.72)
X0=63 [g/cm ]
24 “
6.3 “
X0
where wi are the fractions by weight
of each element in mixture
Z
II.2 Energy Loss of Electrons & Positrons
r  Range of e- is rather different due to multiple scattering in nuclei
20-400% depending on energy and material
r  Fluctuations are larger due to larger possible energy transfers per
collision and bremsstrahlung ⇒ range straggling
r  Range curves are calculated on assumption of continuous slowingdown.
II.2 Energy Loss of Electrons & Positrons
r  Because of continuous energy spectrum, absorption is well
approximated by
I = I exp(− µ x)
0
(2.73)
r  ⇒ exp decay is most general characteristic of β decay, deviating
in more complicated forbidden decays
II.3 Multiple Coulomb Scattering
r  In addition to inelastic collisions with atomic e-, charged particles
undergo repeated elastic Coulomb scatterings off nuclei, but with
smaller probability
r  These collisions are described by Rutherford formula:
dσ
dΩ
= z21 z22re2
mc
1
β p 4 sin4 θ2
(2.74)
r  Vast majority of particles experience small angular deflection
because of sin4 θ/2 term ⇒ leading to zig-zag path
r  We distinguish 3 cases:
i) single scattering ⇒ Rutherford formula
ii) plural scattering (N<20) ⇒ most difficult case (not yet statistical)
iii) multiple scattering (N≥20) ⇒ if ΔE is small we can use statistical
treatment
r  General treatment of multiple scattering is complicated
II.3 Multiple Coulomb Scattering
r  Statistical treatment is valid if we neglect a small probability for
large-angle scattering
r  For small-angle (< 100) single
scattering the MS probability
distribution is approximately
Gaussian:
P(θ )dΩ = 2
θ
θ2
$
(
θ &
&
exp % −
) dθ
2
& θ &
'
*
(2.75)
<θ2>: mean-squared scattering angle, √<θ2>: RMS scattering angle
r  A reasonable estimate is obtained by empirical formula that is
valid to 5% for Z>20 & target thickness 10-3 X0 < x < 10 X0:
θ
2
= z
20 [MeV/c]
pβ
x #%
$1 +
X 0 %&
1
9
log10
r  For low velocities & heavy particles
errors are larger (10-20%)
x '%
(
X 0 %)
[radian]
(2.76)
II.3 Multiple Coulomb Scattering
r  There are other formulations that are generally valid for all
particles up to angles ≅ 300 with exception of slow electrons
(β<0.05), e.g. by Molière, Snyder & Scott
r  Electrons because of their small mass are particularly susceptible
to large-angle scatterings ⇒ probability is rather high such that
we get back scattering
r  Effect of backscattering is rather strong for low-energy e(<1MeV) & high-Z materials
r  One calls ratio of backscattered e- to incident e- albedo or
backscattering coefficient
r  Backscattering is important consideration for e- detectors
e.g. for non-collimated e- on high-Z material (NaI) up to 80%
of e- may be reflected back!
II.4 Detection of Photons
r  A photon traversing a medium can experience 3 different processes
i) Photoelectric absorption
ii) Compton scattering
iii) Pair creation
r  All 3 processes lead to a reduction of the initial intensity in the
original beam direction, e.g. the z-axis
I(z) = I0 ⋅ exp(− µ z)
(2.77)
where µ is linear absorption coefficient that is related to photon
absorption cross section σ by µ=σN0ρ/A
r  Frequently parameter µ/ρ is quoted
II.4 Detection of Photons
r  Lets look at µ/ρ for Pb
r
Absorption increases sharply at
energies corresponding to binding
energies of a particular shell (K,L)
r
Photoelectric absorption decreases
as ~1/Eγ3.5 & increases as Z5
(e.g. for energies between K&L)
r
Compton scattering decreases
as 1/Eγ & increases as Z
r
Pair creation requires minimum
energy of E≥2mec2
II.4 Detection of Photons
r  Measured photon cross section in C & Pb
II.4 Detection of Photons
II.4.1 Photoelectric Absorption
r  The cross section for photoelectric absorption is given by:
σ ph = 4π re2 Z5α 4
1
for η»1
η
(2.78)
where η=Eγ/mec2
r  For energies between K absorption edge & η<1 following
approximation hold
σ ph =
32
3
π re2 2Z5α 4
1
η
7
2
for ηK<η<1
(2.79)
II.4.2 Compton Effect
r  Photons may scatter off an e- in atomic shell
Ø Initially, e- is at rest ⇒ 4-momentum: pe=(mec2,0)
Ø For initial photon we have pγ=(|k|,k)
Denoting 4-momenta after the collision by p’e, & p’γ energymomentum conservation yields:
p e + pγ = p 'e + p 'γ
(2.80)
r  From this we can evaluate the energy of the scattered photon,
E’γ=|k’γ|, yielding using η= Eγ/mec2:
E'γ =
Eγ
1 + η (1 − cos θ )
(2.81)
where θ is scattering angle of the photon. The kinetic energy of eis
1 − cos θ
(2.82)
T 'e = Eγ − E'γ = Eγ ⋅ η
1 + η (1 − cos θ )
II.4.2 Compton Effect
r  There are 2 extreme cases:
i) Small angle scattering: θ~0 ⇒ Eγ≅E’γ & T’e~0
ii) Back scattering at θ=π
Here
E'γ =
Eγ
→
1 + 2η
me c2
2
for η»1 (2.83)
and the maximum electron energy is
(Compton edge):
T 'e = Eγ
2η
1+2η
→
Eγ (1 −
1
2η
)
for η»1 (2.84)
r  The electron recoil spectrum in the detector is continuous between
0 & Compton edge
r
The gap between Compton edge & energy of incident γ is constant
me c2
2
= 0.256 MeV
for η»1
II.4.2 Compton Effect
r  The total scattering cross section per e- at very low energies is
given by the classical Thompson scattering formula
σ Th =
8π
3
re2 Z = 0.665 × Z barn
re=2.818×10-15m (2.85)
r  For relativistic photon energies QED calculation leads to
Klein-Nishina formula
2$ 1 + η '
σ C = 2π re Z 4 & 2 )
43 % η (
2
+- 2(1 + η ) 1
/1
1 + 3η 5
− ln(1 + 2η ) 0 +
ln(1 + 2η ) −
7
,
2 (2.86)
η
(1 + 2η ) 76
-. 1 + 2η
-1 2η
r  Lets look at 2 extreme cases:
i) η«1 ⇒
σc
ii) η»1 ⇒
η
σ C = σ Th (1 − 2η )
σC =
3
8
σ Th
(2.87)
1 1
( + ln 2η ) (2.88)
η 2
II.4.2 Compton Effect
r  For Thompson scattering the distribution of the scattered γ-ray
is forward-backward symmetric:
η=1
=
Z
re2 (1 + cos2 θ )
(2.89)
dΩ
2
r  For relativistic Compton scattering the
angular distribution becomes
asymmetric peaking at the
forward direction (Klein-Nishina):
dσ C
Z 2 $ k# '
=
re & )
dΩ
4 % k(
2
Compton
η=100
Thompson
-k
0
  2
k#
+ 4(ε ⋅ ε # ) − 2 2 (2.90)
/ +
k
/. k #
21
r  For unpolarized photons this leads to:
dσ C
dσ/dΩ
dσ Th
Thompson
)
Z 2&
1
=
re (
+
dΩ
2 ' 1 + η (1 − cos θ ) *
2
Compton
,
η 2 (1 − cos θ ) 2 /
2
.1 + cos θ +
1
1 + η (1 − cos θ ) 10
.-
r  For η→0 we obtain Thompson’s formula!
cosθ
(2.91)
II.4.3 Pair Production
r  For photon energies η≥2 an e+e-pair is created with cross section
i) for low energies: 2«η«137/Z1/3
(
σ p = 4α re2 Z2 %' 79 ln 2η − f(Z) −
&
109
54
)(*)
(2.92)
ii) for high energies: η»137/Z1/3
* $ 183
σ p = 4α re Z , 79 & ln 1 − f(Z) −
,+ % Z 3
2
2
1
54
')/
( /.
(2.93)
r  f(Z) accounts for Coulomb correction of e± in E field of nucleus
r  For arbitrary screening at extreme relativistic energies we obtain
using the Born cross section approximation
dσ p
dE+
=
4α re2 Z2
(ω )
3
{(
φ (ξ )
E2+ + E2− ' 14 −
)(
)
ln Z
3
− f(Z) * +
,+
2
3
φ (ξ )
E + E − ' 24 −
)(
ln Z
3
− f(Z) *
,+
}
(2.94)
II.4.3 Pair Production
r  The mass absorption coefficient for pair creation µp/ρ=σN0/A
approaches an asymptotic value, µ0p/ρ, at high energies which can
be obtained from eq (2.93) by omitting the last two terms
µp0
ρ
= 4α re Z
2
2
N0 7
A 9
ln
183
Z
1
3
≅
7 1
9 X0
r  The probability that pair creation occurs at high energies
within 1X0 is P=1-e-7/9≈54%
(2.95)
II.4.3 Pair Production
II.5 Electron-Photon Showers
r  At high energy a photon is likely to convert into e+er  The e± particles loose energy via bremsstrahlung producing new γ’s
that are likely to convert into e+er  The result is a cascade or shower of e+, e-, & γ’s
r  The process stops once the energies of e+, e-, & γ’s become so
small that the energy loss of γ’s occurs preferentially via
photoelectric absorption & that of e+ & e- occurs preferentially via
ionization
r  A similar shower is obtained if we start with a high-energy e- or e+
r  The development of a shower is a statistical process;
lets consider a simple model
II.5 Electron-Photon Showers
r  Simple model of shower development:
Call initial photon energy E0
After 1 X0: energy of e+e- is then E~E0/2
After 2 X0: energy of e+e-γγ is then ~E0/4
After 3 X0: energy of e+e- e+e- e+e- γγ is then ~E0/8
..
Ø After t X0: we have N≅2t particles with energy of ~E0/2t
Ø
Ø
Ø
Ø
Ø Assuming that shower stops at Ec (critical energy) then
tmax=ln[E0/Ec]/ln2 for Ec=E(tmax) & Nmax=E0/Ec
r  E0≥100 MeV a reasonable parameterization is given by:
N = N0 t a exp(−bt)
where
N0 = 5.51E0 [GeV] Zba +1 / Γ(a + 1)
(2.96)
b = 0.634 − 0.0021Z
#%1.77 − 0.52 ln E
0
a= $
%&2.0 − Z / 340 + (0.664 − Z / 340) ln E0
for Z=13
'%
(
for Z ≥ 26 %)