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Interaction of charged particles
in matter
Charged particles in matter: (1) energy loss (2) scattering
Caused by: - inelastic collisions with electrons
- elastic scattering with nuclei and atoms
Other process: - emission of Cherenkov radiation
- nuclear reactions
- Bremsstrahlung
Divide the description (i) heavy charged particles (m, p, p, a, nuclei)
(ii) electrons and positrons
Electrons: - larger energy transfer and bending due to low mass
- scattering of identical particles
- Bremsstrahlung
requires separate description of e- and e+
2222)/( mcere
Bohrs calculation – classical assumption and Ansatz
Consider moving particle of charge ze passing by with v the stationary charge Zeze
vb
Ze
r q
y
Assumptions:
• momentum approximation: short inter-
action, only transversal moment. transfer
• target remains non-relativistic
vb
Zzed
v
b
b
Zze
bdybyvdydtdtFp
b
Zze
brr
ZzeF
x
x
2
2
3
2/
2/
2
2
2
3
2
2
2
2
2
cos cos
d cos
1 tan / : targetonto momentum
cos
cos/ cos: trans.force
q
q
q
p
p
Interaction of heavy charged
particles in matter
Bohrs calculation
p
e
e
p
TT
T
m
m
mZ
AmZ
vbm
ezZ
m
pE
2/
/ :negligible is nuclei nsfer toEnergy tra
2
2
)( :atom target of increaseEnergy
2
2
22
4222
bdbdxdVdxb
dbN
vm
ezdVNbEbdE
N
dxdbb
bvm
ezE
e
e
e
e
e
pp
2 and 4
)()(
density Electronen
thicknessand ,b and distancein electrons all tolossEnergy
2 :electron one of increaseEnergy
2
42
22
42
Interaction of heavy charged
particles in matter
and :yuncertaint and momentum angular :Q.M.
and
:collision central in transfer-E maximum:limit classical
and for arguments sBohr'
nonsense! - transfer E infinite distance small (ii)
ion.approximat momentum tocontrary in (i)
: worknot does to from nIntegratio
hLbpL
bvm
ezvm
bb
b
bN
vm
ez
dx
dE
b
cms
e
e
e
e
min
2
2
min
2
4222
maxmin
min
max
2
42
v/cβ)β-(1
1γ
22.1
ln4
0
p
Interaction of heavy charged
particles in matter
p2
32
2
42
ln4
ze
vmN
vm
ez
dx
dE ee
e
Bohrs formula
oo IvhbhI
vb
vbtvbt
/,
/1)/(
)/(/
.2
max
:atoms of potential ionisation average
atom. in e all offrequency averaged with :condition
tic)(relativis :time collision typical
electrons. offrequency revolution
to respect withlong is time collision :'invarianceAdiabatic '
.distortionadiabatic anonly is field-E
,collisions peripheralvery in transfer -E no
-
Interaction of heavy charged
particles in matter
Q.M. approximation
o
ee
e I
mvN
vm
ez
dx
dE 22
2
42
ln4
p
Quantum mechanical results from Bethe-Bloch
energies)(low correction shell :
energies)ic relativist (at correctiondensity : terms correction two
collision single in transferenergy max. : Materials abs. of massatomic :
potential ionisation averaged :I matter abs. of number charge :
v/c : mass electron :
particle incoming of charge : cm10 x 2.81 radius electron class. :
matter absorbing ofdensity : mol10 x 6.022 constant Avogadro
13-
1- 23
C
WA
Z
m
zr
N
Z
C
I
Wvmz
A
ZcmrN
dx
dE
e
e
a
eeea
p
max
2
2
2
max
22
2
222
1/1,
:
222
ln2
Interaction of heavy charged
particles in matter
average ionisation potential: - difficult to calculate exactly for all shells
- semi-empirical approximation I ~ Z
Density correction: - Electric field of moving charge is polarizing atoms and
shields charges at larger distances.
- Polarisation depends on density of matter.
- reduced dE/dx at higher energies
- empirical approx.
Shell correction: -correction at small energies of incoming particles,
velocity is comparable to Bohr- or e- -velocity.
- at low energies strong decrease of dE/dx.
Interaction of heavy charged
particles in matter
Energy dependence of energy loss
- for non-relativistic energy: ~1/ 2
- minimum at v~0.96c
- independent of mass
- slow relativistic increase
- reduced by density effect
Interaction of heavy charged
particles in matter
Interaction of heavy charged
particles in matter
Interaction of heavy charged
particles in matter
Electronic stopping: slowing down by inelastic
collisions between electrons in matter and
moving ion at Eion >100 keV.
Nuclear stopping: elastic collisions between
Ion and the full atoms. Eion<100 keV
e.g.: Si-ion, Eion=1 MeV, range in Si r=1-2mm
Interaction of heavy charged
particles in matter
Energy loss, stopping power and range of charged particles
- Largest energy deposition at end of track: Bragg peak
),(1 2 If
A
Zz
dx
dE
d
dE
- dE/d is nearly independent of matter for equal particles.
- average range for particle with kin. energy T :
),(1 2 If
A
Zz
dx
dE
d
dE
dEdx
dETS
T
0
1
)(
- Range is not a precise quantity but smeared, called range straggling. Number
of interactions is statistically distributed. At low energies empirical constants
are included in S(T).
- Remark: In matter with spatial symmetry (e.g. crystals) Bethe-Bloch formula
is not valid! Correlated scattering at certain geometrical conditions (critical angle
with respect of crystal axis) reduces energy loss and enlarge range of particle.
Channeling effect
Interaction of heavy charged
particles in matter
Interaction of heavy charged
particles in matterRange and straggling
Range von 12C ions in water unit: MeV/n e.g. 90 MeV, n=12
Elab=1,08 GeV
Bragg
peak
Range of heavy ions in matter
Examples, applications for stopping power and range:
Energy loss of projectil and reaction products in target
e.g. 48Ca+ 208Pb @ Ebeam=200MeV target thickness d=0.5 mg/cm2 (s= 44 mm)
energy loss of 48Ca in target 7.2 MeV
-> equals width of excitation function 208Pb(48Ca,2n) - reaction
beam
target
reaction product
e.g. p, d, a
detector: E-E telescope
remark: thickness are given in d=mass/area
Units (e.g.): mg/cm2 d = m/A = s
density, s thickness: s=d/
• beam and reaction products are slowed down.
• cross section and kinematics are changed.
• targets limit count rate and energy resolution.
Interaction of charged particles in matter
Example and application for stopping power and range:
Particle identification with E – E telescope
Wechselwirkung geladener
Teilchen in MaterieInteraction of charged particles in matter
