Lesson 15

Interaction of Radiation with Matter

Charged Particles

Basic Ideas

• For charged particles and photons, the interaction is between the radiation and the atomic electrons. Nine orders of magnitude more probable than interaction with nucleus.

• For neutrons, the story is different and interactions are with nuclei.

Basic Ideas

• Divide subject (and lectures) into interactions of charged particles (alpha, electrons, heavy ions) and uncharged particles (photons, neutrons).. Another way of saying this is short range particles and long range particles

Areal Density

• One does not usually speak of thickness in terms of a linear dimensions (nm, microns, cm, meters, etc) but in terms of areal density (mg/cm2, g/cm2, etc.)

• The conversion is trivial, i.e., Linear thickness = areal density/density• Original motivation for this unit is

operational.

Heavy Charged Particles(particles like protons or heavier)

• Straight line tracks in matter• Concept of range

Stopping power

• Stopping power =

€

−dE

dx

€

Formally

€

−dE

dx= Selectronic + Snuclear ≈ Selectronic

Bethe-Bloch equation

€

Energyloss = ΔE(b) =2q2e4

mev2b2

Converting to a differential

€

−dE(b) = ΔE(b)NedV = ΔE(b)Ne2πdbdx

Substituting

€

−dE

dx=

4πq2e4

mev2

Ne lnbmax

bmin

Evaluating bmin and bmax

When b=min, get max Etransfer

€

bmin =qe2

γmev2

Note the expressions are relativistic

€

bmax = γvf (Z)

Similarly

Back to the Bethe-Bloch equation

€

−dE

dx=

4πq2e4

mev2

Ne lnbmax

bmin

Substituting

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−dE

dx=

4πq2e4

mev2

Ne lnγ 2mev

2 f (Z)

qe2

Making this qm correct

€

−dE

dx= 4πNAre

2mec2ρ

Zq2

Aβ 2ln

Wmax

I

⎛

⎝ ⎜

⎞

⎠ ⎟−β 2

⎡

⎣ ⎢

⎤

⎦ ⎥

How to use this

€

−dE

dx= 4πNAre

2mec2ρ

Zq2

Aβ 2ln

Wmax

I

⎛

⎝ ⎜

⎞

⎠ ⎟−β 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

Wmax = 2mec2(γβ )2

€

I = Z(12 + 7Z−1)eV

I = Z(9.76 + 58.8Z−1.19)eV

Z<13

Z 13

See p. 503 of LSM for example

General conclusions

€

−dE

dx∝

Aq2

Efor E/A < 10 MeV/A

Bragg peak

dE/dx for compounds, mixtures

€

1

ρ

dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟total

=w1

ρ1

dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟

1

+w2

ρ 2

dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟2

+w3

ρ 3

dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟3

+L

Note this relation is an approximate relation

Straggling

Straggling

€

N(E)dE

N=

1

απ 1/ 2exp −

E− E( )2

α 2

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

α2 = 4πq2e4 N ex0 1+kI

mev2

ln2mev

2

I

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

Ranges

€

R(T ) = −dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟−1

dE0

T

∫

Practical approaches to ranges and dE/dx

Practical approaches to ranges and dE/dx

• SRIM (http://www.srim.org)

• OSU variant (Range)-See class website

• ORNL--STOPX

Ranges in air

• Ranges of α-particles in air

€

Rair(cm) = (0.005Eα (MeV )+ 0.285)Eα3 / 2(MeV )

€

Rair(mg/ cm2 ) = 0.40Eα3 / 2(MeV )

€

RZ

Rair

= 0.90+ 0.0275Z +(0.06− 0.0086Z )log10

Eα

4

⎛

⎝ ⎜

⎞

⎠ ⎟

Interaction of electrons with matter

• In addition to electron-electron interactions, have the possibility of radiative processes such as bremsstrahlung

Formalism

€

−dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟electron

= Selectronic + Sradiative

€

Selectronic = −dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟electronic

=2πZe4ρN

mev2

lnmev

2E

2I 2 1− β 2( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟− ln 2 2 1− β 2 −1+ β 2

( ) + 1− β 2( ) +

1

81− 1− β 2( )

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

Sradiative = −dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟radiative

=Z +1( )Ze4ρNE

137me2c4

4 ln2E

mec2

⎛

⎝ ⎜

⎞

⎠ ⎟−

4

3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

Sradiative

Selectronic

≈ZE

800MeV

Sradiative important only for Z=80-90 and E=10-100 MeV

Practical aspects

• Concept of “range” is problematic

“Range” of -particles

• When monoenergetic electrons interact with matter, there is a distribution of stopping distances

-particles emitted in -decay have a range of energies, from 0 to Emax

• Get exponential attenuation

Nt = N0e-t

€

m2 kg( ) = 1.7Emax−1.14

Practical aspects of beta-counting

• Backscattering• Bremsstrahlung• Cerenkov radiation

Backscattering

Self Absorption