FYS-KJM5920 - Nuclear Measurement Methods and Instrumentation 1

Passage of radiation through matter

- Important for detectors, shielding, nuclear medicine - Quantities: energy loss, scattering, straggling, absorption etc.

Cross section •  Probability that a reaction takes place •  dσ = ”area” of target •  Ns = number scattered / sec

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dσdΩ(E,Ω) =

1FdNs

dΩdΩ = sinθdθdϕ

dΩ = 4π∫∫θ = 0 −πϕ = 0 − 2π

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θ

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ϕ

Point of interaction

Total cross section

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σ(E) = dσ∫ = dΩ∫ dσdΩ

= sinθdθdϕ0

2π

∫0

π

∫ dσdΩ(E,θ,ϕ)

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σ(E) = sinθdθdϕ0

2π

∫0

π

∫ dσdΩ(E) = 4π dσ

dΩ(E)

Isotropic scattering:

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Real life

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Target

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Ns(Ω) = FA⋅ NδxdσdΩ

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(Ntot = FA⋅ Nδxσ)

Number of centres seen

Number of scattered Particles into dΩ

Incident number of particles

Prob. of interaction

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Mean free path

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Probability of not having interaction between x - > x +dxP(x + dx) = P(x)(1− wdx)

P(x) +dPdx

dx = P − Pwdx

dP = −wPdxP = exp(−wx) Probability of survival from 0 - > x

Probability of suffering an interaction anywhere in the distance x :Pint (x) =1− exp(−wx)

Mean free path :

λ =xPdx∫Pdx∫

=1w

=1Nσ

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P(x) : Probability of not having interaction between 0 - > xwdx : Probability of having interaction between x - > x + dx

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Target thickness

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Physical thickness :

m = ρV = ρdA⇒ d =mAρ

[um]

Mass thickness :

t =mA

[mg/cm2]

A

d

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Main types

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•  Direct ionizing radiation E(x), I=const – α-particles – β – Protons – Heavy ions (20Ne, 40Ar, etc.)

•  Indirect ionizing radiation E=const, I (x) – Neutrons –  ν, π, Κ etc –  γ, X-rays

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γ–ray attenuation coeff. (µ/t)

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Heavy charged particles

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-dE/dx Bragg- toppen

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Bohr’s calculations

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•  Electron initially free and at rest •  Trajectory not disturbed by the electron •  Passing time so short that electron do

not move

ze b

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Momentum transfer (I)

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Electric field:

Perpendicular momentum transfer:

Gauss’ law:

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E(r) =ze

4πε 0r2

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pper = eEperdt−∞

+∞

∫ =eEper

vdx =

ev−∞

+∞

∫ Eperdx−∞

∞

∫

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! E ⋅ d! S = q

ε 0S∫

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Momentum transfer (II)

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Integral over cylinder:

Perpendicular momentum transfer:

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Eper−∞

+∞

∫ 2πbdx =zeε 0

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pper =ze2

2πε 0bv

b db

x

Eper E

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Energy transfer

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ε =pper2

2me

=z2e4

8π 2ε 02b2mev

2

But a lot of electrons within db and dx:

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N ⋅ dV = N ⋅ 2πb⋅ db⋅ dx

Energy transferred to projectile:

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−dE = εNdV =z2e4N

8π 2ε02mev

22πbdbb2

dx

dx

db b

Stopping power

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Integrating over impact parameter b:

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−dEdx

=z2e4N

4πε02mev

2dbb

=bmin

bmax

∫ z2e4N4πε0

2mev2 ln

bmaxbmin

Naively:

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bmin = 0bmax = ∞

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Discussion bmin and bmax

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electron

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Δpmax = 2mev

εmax =Δp2max2me

= 2mev2

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ε =z2e4

8π 2ε02b2mev

2 < εmax = 2mev2 ⇒ bmin =

14πε0

ze2

mev2

Short passage time t compared to electron period T:

v b

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vt = b⇒ t =bv

< T =1ν ⇒ bmax =

vν

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Bohr’s classical formula

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−dEdx

=z2e4N

4πε 02mev

2 ln

vν

14πε 0

ze2

mev2

=z2e4N

4πε 02mev

2 ln4πε 0mev

3

ν ze2

All essential features Relativistic by including γ in last term:

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−dEdx

=z2e4N

4πε 02mev

2 ln 4πε 0γ2mev

3

ν ze2 , γ 2 =1− vc'

( ) *

+ ,

2

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eCGS2 →

eSI2

4πε 0To transfer formulas in Leo from CGS ->SI unit system: 16 FYS-KJM5920 - Nuclear Measurement Methods and

Instrumentation

Bethe-Bloch’s formula

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−dEdx

= 2πNare2mec

2ρZAz2

β2ln2meγ

2v 2Wmax

I2'

( )

*

+ , − 2β2 −δ − 2

CZ

.

/ 0

1

2 3

The stopping power of aluminium for protons

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Thin target or absorber

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E0 E1

θ

N(E)

N(θ)

0

E1 E0

ΔE Straggling

Scattering

E

θ

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Range

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R(E0) =dEdx

"

# $

%

& '

0

E 0

∫−1

dE

Roughly

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−dEdx

∝ β−2 ∝ E −1

R(E0)∝ EdE0

E 0

∫ ∝ E02

2200 um

80 MeV FYS-KJM5920 - Nuclear Measurement Methods and

Instrumentation 19

Bragg-profiles

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12C in water 4He in air

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Dagsavisen 27.1.2008

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0 5 10 15 20 Dybde [cm]

Relativ dose

γ fra 60Co

0

1

2

3

4

12C-ioner 250 MeV/u

18 MeV røntgen

Dosedistribution 12C versus X-rays

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Raster scan

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Particle telescope

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ΔE E

Etot = E + ΔE ΔE

E

Low Etot High Etot

Punch through

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Bananas

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Bananas (60Ni-exp)

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Thickness of ΔE detector

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Rα(E) (um)

E (MeV)

ΔE E

40MeV

5 MeV 35 MeV

35 40

700 550

d D - d

D = R(40) = 700um D-d = R(35) = 550um d = 700-550 = 150um

α

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Thickness spectrum

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If we used range curve Rα(E), other particles than α’s will “lie” about the thickness of the front detector ΔE p d t

3He α

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31 106 129

N(d)

d (um)

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ΔEd

≈dEdx

∝z2

v 2 ∝z2ME

Energy deposited in d for a given E is :ΔE ∝ z2M

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