Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
PHYS 5012Radiation Physics and Dosimetry
Lecture 4
Tuesday 19 March 2013
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Interactions of Charged Particles with Matter
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
General Aspects of two-particle Collisions
Collisions between two particles involve a projectile and atarget.
Types of targets: whole atoms, atomic nuclei, atomicorbital electrons, free electrons.
Types of projectiles:I heavy charged particles (protons, α-particles, heavy
ions)I light charged particles (electrons, positrons)I photons (considered previously)I neutrons (not considered here)
Henceforth, we will consider only charged particleprojectiles. Two-particle collisions are then Coulombcollisions.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Two-Particle Collisions
3 categories:1. Nuclear reactions – final reaction products differ from
initial particles; charge, momentum and mass-energyconserved; e.g. deuteron bombarding nitrogen-14:147 N(d, p)15
7 N
2. Elastic collisions – final products identical to initialparticles; kinetic energy and momentum conserved;e.g. Rutherford scattering of α particle on goldnucleus: 197
79 Au(α, α)19779 Au
3. Inelastic collisions – final products identical to initialparticles; kinetic energy not conserved
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
In inelastic collisions, some kinetic energy is converted toexcitation energy in the form of:
I nuclear excitation of target resulting from heavycharged particle striking target nucleus; e.g.AZ X(α, α)A
Z X∗
I atomic excitation or ionisation of target resulting fromheavy or light charged particle colliding with targetorbital electron
I bremsstrahlung emission by light charged particleprojectile resulting from Coulomb interaction withtarget nucleus
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Nuclear Reactions
Schematic illustration of a general nuclear reaction. (Fig. 5.1 in Podgoršak.)
I intermediate compound produced temporarily;spontaneously decays into reaction products
I conservation of atomic number:∑
Zbefore =∑
Zafter
I conservation of atomic mass:∑
Abefore =∑
Aafter
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Conservation of momentum
p1 = p3 + p4 (1)
=⇒ p1 = p3 cos θ + p4 cosφ ‖ to p10 = p3 sin θ + p4 sinφ ⊥ to p1
Conservation of mass-energy(m1c2 + EK,1
)+ m2c2 =
(m3c2 + EK,3
)+(m4c2 + EK,4
)(2)
where EK = particle kinetic energy = (γ − 1)mc2
Q =(m1c2 + m2c2)− (m3c2 + m4c2) Q value (3)
Also, Q = EK,final − EK,initial
I Q > 0⇒ exothermic collisionI Q = 0⇒ elastic collisionI Q < 0⇒ endothermic collision
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Threshold Energy
A minimum projectile energy Ethr is required for anendothermic reaction to proceed.Conservation of 4-momentum, p = (E/c,p):
p1 + p2 = p3 + p4 ⇒ (p1 + p2)2 = (p3 + p4)2
and using p21 = (E1/c)2 − |p1|2 = m2
1c2 andp2
2 = (E2/c)2 = m22c2, gives
2E1E2 = (p3 + p4)2c2 − (m21c4 + m2
2c4)
Note that p3 + p4 is the centre-of-mass 4-momentum, pcm,and so (p3 + p4)2 = p2
cm = (Ecm/c)2 = (m3c2 + m4c2)2/c2
since the modulus of a 4-vector is invariant and has thesame value in any frame of reference. So the thresholdenergy E1 for the projectile is:
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Ethr =(m3c2 + m4c2)2 − (m2
1c4 + m22c4)
2m2c2 (4)
corresponding to a threshold kinetic energy:
EK,thr =(m3c2 + m4c2)2 − (m1c2 + m2c2)2
2m2c2 (5)
in terms of the Q value:
EK,thr = −Q[
m1c2 + m2c2
m2c2 − Q2m2c2
](6)
If Q m2c2 (as is often the case), then
EK,thr ≈ −Q(
1 +m1
m2
)(7)
The Q value is defined for general two-particle collisions.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Example: pair production and triplet production.For pair production, m1 = 0, m2 = m3 me and Q =−2mec2, so (
Eppγ
)thr = 2mec2
For triplet production, Q = −2mec2 but m2 = me, so(Etpγ
)thr = 4mec2
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Elastic Scattering
I initial and final particles remain the same (i.e.m3 = m1 and m4 = m2), so Q = 0
I kinetic energy transfer ∆EK from m1 to m2
I total kinetic energy conserved
Schematic illustration of elastic scattering. θ is the scattering angle, φ is therecoil angle and b is the impact parameter. (Fig. 5.2 in Podgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Classical kinematics
Kinetic energy transfer determined from conservation ofmomentum and energy:
∆EK =12
m2u22 = EK1
4m1m2
(m1 + m2)2 cos2 φ (8)
Head-on collisions:I b = 0 and φ = 0I maximum energy and momentum transferI θ = 0 (forward scattering) when m1 > m2
I θ = π (back-scattering) when m1 < m2
I projectile stops when m1 = m2
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Example: proton colliding with orbital electron.Maximum energy transfer (for a head-on collision), notingthat mp me:
∆Emax ≈ 4EKpme
mp≈ 2× 10−3EKp
Collisions between particles of the same mass (m1 = m2):
I distinguishable particles (e.g. electron colliding withpositron): ∆Emax = EK1 ⇒head-on collision transfersall projectile’s kinetic energy to target
I indistinguishable particles (e.g. free electron collidingwith bound electron): ∆Emax = 1
2 EK1
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Relativistic formula for energy transfer in a head-oncollision: conservation of mass–energy⇒
∆Emax = (γ2 − 1)m2c2 =2(γ + 1)m1m2
m21 + m2
2 + 2γm1m2EK1 (9)
where γm1c2 = initial energy of incident projectile,γ1m1c2 = final energy of incident projectile andγ2m2c2 = final energy of target particle.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Classical Rutherford scatteringRutherford scattering is the elastic scattering of a pointcharge by a stationary fixed point charge. The originalGeiger-Marsden experiment, which led Rutherford topropose the currently accepted Rutherford–Bohr atomicmodel, was conducted with α particles scattering off goldfoil. The classical derivation of the differentialcross-section is cumbersome and requires knowledge ofthe trajectory of the α as it is deflected by the Coulombfield of the gold nuclei. This depends on knowing theposition and momentum of the charge at all times, whichis forbidden in quantum mechanics. The classicalderivation gives
dσdΩ
=
(zZ~c
4
)2( α
EK
)2
sin−4(θ
2
)(10)
where z is the charge of the incident projectile (2 for analpha particle) and α is the fine structure constant.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Quantum derivation
Recall from Lec. 2 that general problem of scattering is tofind, for a given initial state i, the probabilities of variousfinal states f .
Fermi’s Golden Rule is equivalent to the first order Bornapproximation:
dσdΩ
=m2
(2π)2~4 |Mfi|2 (11)
for elastic scattering.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
The transition matrix element amplitudeMfi is calculatedusing spherical plane waves of the form
Ψ(r) ∝ exp(ip · r/~)
and a screened Coulomb potential known as the Yukawapotential:
V(r) =zZe2
4πε0rexp(−ηr)
This gives the same result for dσ/dΩ as the classicalderivation, eqn. (10), but is much less lengthy to derive.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Inelastic collisions
I hard collisions: Coulomb interactions with orbitalelectron for b ≈ a
I soft collisions: Coulomb interaction with orbitalelectron for b a
I radiative collisions: Coulomb interactions withnuclear field for b a
The three different types of collisions depend on the classical impact paramaterb and atomic radius a. (Fig. 6.1 in Podgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Stopping PowerI Stopping power measures how readily charged
particles come to rest in matterI incident charged particle loses all kinetic energy via
multiple Coulomb interactions (mostly elastic, butsometimes inelastic)
I gradual loss of kinetic energy called continuousslowing down approximation (CSDA)
I e.g. 1 MeV charged particle typically undergoes∼ 105 interactions before losing all its kinetic energy
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Linear stopping power, dE/dx = rate of energy loss perunit path length of charged particle
Mass stopping power, S = −ρ−1dE/dx, is the commonlyused measure of stopping power (in units MeV m2 kg−1)
2 types of stopping powers:
1. Collision stopping power, Scol – for hard and softcollisions involving both light and heavy chargedparticles; can result in atomic excitation andionisation
2. Radiative stopping power, Srad – for radiativecollisions; only light charged particles (i.e. electronsand positrons) experience appreciable energylosses; can result in bremsstrahlung emission
Stot = Scol + Srad total stopping power
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Collision Stopping Power for Heavy ChargedParticles
I for Ei <∼ 10 MeV, heavy charged particles undergo softand hard collisions
I small angle scattering (θ ' 0)
Schematic diagram of a heavy charged particle collision with an orbital electron.The scattering angle θ is exaggerated for clarity. (Fig. 6.3 in Podgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Classical DerivationMomentum transfer:
∆p =
∫F∆pdt =
∫ ∞−∞
Fcoul cosφ dt
where Fcoul = (ze2/4πε0)r−2, giving
∆p =ze2
4πε0
∫ +(π−θ)/2
−(π−θ)/2
cosφr2
dtdφ
dφ
Hyperbolic particle trajectory⇒ angular displacementvaries with time⇒ dφ/dt = ω and conservation of angularmomentum requires L = Mv∞b = Mωr2 ⇒
∆p =ze2
4πε0
1v∞b
∫ +(π−θ)/2
−(π−θ)/2cosφ dφ
= 2ze2
4πε0
1v∞b
cosθ
2
≈ 2ze2
4πε0
1v∞b
(12)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Energy transferred to electron in a single collision withimpact parameter b:
∆E(b) =(∆p)2
2me= 2
(e2
4πε0
)2 z2
mev2∞b2 (13)
Total energy loss obtained by integrating ∆E(b) over allpossible b and accounting for all electrons available forinteractions.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
no. electrons in volume annulus between b and b + db
= no. electrons per unit mass×mass in annulus
⇒ ∆n =
(ZNA
A
)dm
where
dm = ρ dV = ρ[π(b + db)2∆x− πb2∆x] ≈ 2π ρ b db ∆x
⇒ ∆n ≈ 2π ρ (ZNA/A) b db ∆x
Multiply ∆E(b) by this and integrate over b to get the totalenergy transfer to electrons.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Mass collision stopping power
Scol = −1ρ
dEdx
= 4πZNA
A
(e2
4πε0
)2 z2
mev2∞
∫ bmax
bmin
dbb
= 4πZNA
Are
2mec4z2
v2∞
lnbmax
bmin(14)
I Scol ∝ z2, where z = atomic number of heavycharged particle (e.g. z = 2 for an α particle)
I Scol ∝ v−2∞ , where v∞ = initial velocity of heavy
charged particle
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
I bmax ⇔ ∆Emin = minimum energy transfercorresponding to minimum excitation or ionisationpotential of orbital electron from (13)
∆Emin = 2re
2mec4z2
v2∞b2
max= I (15)
I = mean ionisation-excitation potential of medium
I ≈ 9.1Z(1 + 1.9Z−2/3) eV (16)
e.g. I ≈ 78 eV for carbon. But (16) is poor approximationfor compounds because chemical bonds are neglected(e.g. I ≈ 75 eV for water).
Typical I values for various compounds of interest arelisted in Table 6.4 in the textbook.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
I bmin ⇔ ∆Emax = maximum energy transfercorresponding to head-on collisions:∆Emax ≈ 4 me
M EK,i = 2mev2∞ (for M me), so
∆Emax = 2(
e2
4πε0
)2 z2
mev2∞b2
min= 2mev2
∞ (17)
Putting together (15) and (17) gives
bmax
bmin=
(∆Emax
∆Emin
)1/2
=
(2mev2
∞I
)1/2
(18)
=⇒ classical collision stopping power for heavy chargedparticles:
Scol = 4πZNA
Are
2mec2z2
v2∞/c2
12
ln2mev2
∞I
(19)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Generalised solution for the collision stopping power forheavy charged particles:
Scol = 4πNA
Are
2mec2z2
(v∞/c)2 Bcol (20)
≈ 3.070× 10−5 z2
Aβ2 Bcol MeV m2 kg−1
with A in units of kg and where β = v∞/c andBcol = atomic stopping number includes relativistic andquantum-mechanical corrections and is ∝ Z
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Bcol
classical Z ln(
2mev2
I
)1/2
(Bohr)
non-rel, qm Z ln(
2mev2
I
)(Bethe-Bloch)
rel, qm Z[ln(
2mec2
I
)+ ln
(β2
1−β2
)− β2
](Bethe)rel, qm, shell
polarisation, Z[ln(
2mec2
I
)+ ln
(β2
1−β2
)− β2 − CK
Z − δ]
(Fano)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Corrections to Bethe formula:
I CK/Z = shell correction accounting fornon-participation of K-shell electrons at lowenergies; negligible energy transfer when velocity ofincident particle is comparable to that of orbitalelectrons (K-shell electrons are fastest).
I δ = polarisation (density effect) correction incondensed media; accounts for reduced participationby distant atoms resulting from effective Coulombfield being reduced by dipole of nearby atoms;important for heavy charged particles at relativisticenergies (but important for light charged particles atall energies).
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Example: The stopping power of water for protons.Using the Bethe formula (relativistic and quantum-mechanical derivation, but without shell and polarisationcorrections), with z = 1 for protons and for H2O, A =18.0 g = 0.0180 kg, Z = 10, and I = 75 eV giving
Scol = 1.71× 10−2β−2[
9.520 + ln(
β2
1− β2
)− β2
]in units of MeV m2 kg−1. For 1 MeV protons, for instance,β2 = 0.00213, giving
Scol = 26.97 MeV m2 kg−1
which compares well with the exact value obtained fromthe NIST/pstar database: Scol = 26.06 MeV m2 kg−1.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
I Scol ∝ Z/A, but Z/A does not vary appreciablybetween different materials (Z/A ≈ 0.4− 0.5 typically)
I Scol ∝ z2 ⇒ an α-particle of a given β has 4 times thecollision stopping power of a proton
I Z dependence of Scol mostly through I, whichincreases with Z; Bcol has term − ln I, so stoppingpower gradually decreases with higher Z
Stopping powers of protons in aluminium (Z=13) and lead (Z=82) (data fromNIST/pstar).
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
I dependence of Scol on particle kinetic energy EK
varies strongly from non-relativistic to relativisticregimes
I peak in Scol occurs at non-relativistic energies and isresponsible for the Bragg peak in depth dose curvesfor heavy charged particles
Schematic plot of the mass collision stopping power for a heavy charged particleas a function of kinetic energy; M0 is rest mass of the charged particle, I is meanexcitation/ionisation energy of the target medium (Fig. 6.7 in Podgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Collision Stopping Power for Light ChargedParticles
3 differences from heavy particle collisions:1. relativistic effects important at lower energies2. larger fractional energy losses3. radiative losses can occur
Hard and soft collisions combined using Møller andBhabba cross sections for electrons and positrons,respectively.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Scol = 2πr2e
ZNA
Amec2
β2
[ln(
EK(1 + τ/2)
I
)+ F±(τ)− δ
](21)
where
F−(τ) = (1− β2)[1 + τ 2/8− (2τ + 1) ln 2] for electrons
and
F+(τ) = 2 ln 2− β2
12
[23 +
14τ + 2
+10
(τ + 2)2 +4
(τ + 2)3
]for positrons and where
τ =EK
mec2
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
For light charged particles, Scol dependence on Z issimilar to that for heavy charged particles, butdependence on EK differs:
Mass collision stopping power (solid curves) and radiative stopping power(dashed curves) for electrons. (Fig. 6.10 in Podgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Radiative Stopping PowerElectrons and positrons can undergo radiative losses asa result of Coulomb interactions with atomic nuclei.Larmor formula predicts radiative power P ∝ a2 ∝ Z2/m2.Bethe and Heitler derived the cross-section forbremsstrahlung radiation:
σrad ∝ αr2e Z2
which contributes to mass stopping power:
Srad =NA
AσradEi (22)
Ei = EK,i + mec2 = initial total energyEK,i = initial kinetic energy
Srad can be written in terms of a weakly varying functionBrad of Z and Ei (see Table 6.1 in in Podgoršak):
Srad = αr2e Z2 NA
ABradEi (23)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Radiative stopping powers for electrons in different material (solid curves) andcollision stopping powers (dashed curves) for the same material. (Fig. 6.2 inPodgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Total Mass Stopping Power
Stot = Scol + Srad (24)
I for heavy charged particles, Srad ≈ 0I for light charged particles, Scol > Srad for EK
<∼ 10 MeVtypically
I critical kinetic energy, (EK)crit, where Ecol = Erad
(EK)crit ≈800 MeV
Z(25)
Total mass stopping power (solid curves) and radiative and collision stoppingpower (dashed curves) for electrons. (Fig. 6.11 in Podgoršak.)
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
RangeI heavy charged particles experience small fractional
energy losses and small angle deflections in elasticcollisions
I light charged particles experience larger fractionalenergy losses and large angle deflections per elasticor inelastic collision
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
I range, R, of a particular charged particle in aparticular medium measures the expected lineardistance the particle will reach in that medium beforecoming to rest (i.e. cannot penetrate beyond R)
I depends on particle charge and kinetic energy, aswell as absorber composition
I CSDA range, RCSDA, measures average geometricpath length traversed by charged particles of aspecific type in a given medium (in units kg m−2) inthe continuous slowing down approximation
I RCSDA > R always
RCSDA =
∫ EK,i
0
dEK
Stot(EK)= −ρ
∫ EK,i
0
dEK
dEK/dx(26)
I RCSDA difficult to solve using analytic Stot(EK)solutions, (20) and (21) for heavy and light chargedparticles, respectively
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
CSDA range for heavy charged particles
I for heavy particles, Stot(EK) = Scol(EK) ∝ z2Bcol(β)/β2,where β is related to EK via EK = (γ − 1)Mc2, whereγ = (1− β2)−1/2, so EK = EK(β) and
RCSDA ∝∫β2 dEK(β)
z2Bcol(β)
I use dEK = Mg(β)dβ and let G(β) = Bcol(β)/β2:
RCSDA ∝Mz2
∫ β
0
g(β)
G(β)dβ =
Mz2 f (β)
I f (β) independent of heavy particle type (onlydepends on β)⇒ can calculate values of RCSDA forheavy particles relative to protons
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
RCSDA(β) =M
mpz2 RpCSDA(β) (27)
RpCSDA(β) = proton range (obtain from NIST),
M/mp = heavy charged particle mass / proton mass,z = heavy particle charge
CSDA Range of protons in water (ρ = 1 g cm−3 so depth in cm has same valueas R). From the NIST/pstar database.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Example 1: Range of an 80 MeV 3He2+ ion in soft tissue.We have z = 2 and M = 3mp, so R(β) = 3
4 Rp(β). Nowwe need to find the energy of a proton having the same βas the 3He2+ ion. For a fixed β, EK/M = const, so Ep
K =(mp/M)E = 80/3 MeV = 26.7 MeV. Using the NIST/pstardatabase, and using water as a soft tissue equivalent,
RpCSDA = 0.7173 g cm−2 = 7.173 kg m−2
=⇒ RCSDA = 0.5380 g cm−2 = 5.380 kg m−2
Since water has ρ = 1 g cm−3, the average distance a3He2+ ion can penetrate into soft tissue is ≈ 0.5 cm. Note:this exceeds the minimum thickness of outer layer of deadskin cells (epidermis, ∼ 0.007 cm), so 3He2+ ions can reachliving cells from outside the human body.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
Example 2: Range of a 7.69 MeVα particle in soft tissue.Using z = 2 and M = 4mp gives Rα(β) = Rp(β). Forthe same β, the proton energy is Ep
K = (7.69/4) MeV ≈1.923 MeV. For this proton energy, the NIST/pstardatabase gives Rp = 7.077 × 10−3 g cm−2. So the aver-age depth to which 7.69 MeVα particles can penetrate intosoft tissue is close to the thickness of the epidermis. Thismeans that external sources of these particles are lessof a health hazard than 3He2+ ions. However, 7.69 MeVαparticles are emitted by the radon daughter 214
84 Po, which ispresent in the atmosphere of uranium mines. These α’spose a serious radiological hazard when ingested throughthe lungs. This has been linked to the higher incidence oflung cancer among uranium miners.
Radiation PhysicsLecture 4
Interactions ofCharged Particleswith MatterGeneral Aspects
Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Stopping Power
Collision Stopping Power(Heavy Particles)
Collision Stopping Power(Light Particles)
Radiative Stopping Power
Total Mass StoppingPower
Range
CSDA range for light charged particles
I for light particles, need to also take into accountradiative losses
CSDA Range of electrons in water (ρ = 1 g cm−3 so depth in cm has same valueas R). From the NIST/pstar database.
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