phys 5012 radiation physics and dosimetry - lecture 4kuncic/lectures/rp4_slides.pdf · radiation...

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



Nuclear Reactions

Elastic Scattering


Stopping Power





Range

Interactions of Charged Particles with Matter



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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:



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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)



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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)



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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)



Nuclear Reactions

Elastic Scattering


Stopping Power





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).



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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).



Nuclear Reactions

Elastic Scattering


Stopping Power





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.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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)



Nuclear Reactions

Elastic Scattering


Stopping Power





Range

Radiative stopping powers for electrons in different material (solid curves) andcollision stopping powers (dashed curves) for the same material. (Fig. 6.2 inPodgoršak.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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.)



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.



Nuclear Reactions

Elastic Scattering


Stopping Power





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.

phys 5012 radiation physics and dosimetry - lecture 4kuncic/lectures/rp4_slides.pdf · radiation...

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