phys586-lec07-photons1
Download
Report
Transcript phys586-lec07-photons1
Photon Interactions
When a photon beam enters matter, it
undergoes an interaction at random and is
removed from the beam
I I 0e x I 0e x /
is thelinear attenuat ion coefficent1/cm
is theattenuat ion length or mean free pat hcm
is themass attenuat ion coefficient cm2 / g
1
2
is themass attenuat ion length g / cm
/
1
Photon Interactions
2
Photon Interactions
Notes
is the average distance a photon travels before
interacting
is also the distance where the intensity drops by a
factor of 1/e = 37%
For medical applications, HVL is frequently used
Half Value Layer
Thickness needed to reduce the intensity by ½
I
0.693
1
ln ln x HVL
2
I0
Gives an indirect measure of the photon energies
of a beam (under the conditions of a narrowbeam geometry)
In shielding calculations, you will see TVL used a 3lot
Beam Hardening
4
Photon Interactions
5
Photon Interactions
What is a cross section?
What is the relation of to the cross section s
for the physical process?
s has units cm and has units1 / cm
Ns where N is thedensity of atoms
N Av
s is thelinear attenuation coefficient
2
A
cm2
as mentioned, in
is morecommon
g
6
Cross Section
Consider scattering from a hard sphere
What would you expect the cross
section to be?
α
θ
α
b
α
R
7
Cross Section
The units of cross section are barns
1 barn (b) = 10-28m2 = 10-24cm2
The units are area. One can think of the
cross section as the effective target area
for collisions. We sometimes take σ=πr2
8
Cross Section
One can find the scattering rate by
R I 0 NTs
N / s N / s / cm cm
N Av
NT
A
3
g / cm cm
2
/ cm
g
2
2
9
Cross Section
For students working at collider accelerators
R Ls
L is t heluminosit yin / cm2 / s
N s Ldt
2
Ldt
is
t
he
int
egrat
ed
luminosit
y
in
/
cm
In 2010 t he LHC delivered 35 pb-1 of int egrat edluminosit y
In 2011t he LHC expect st o deliver 1 fb-1
A t ypicalcross sect ionat 7 T eV might be s tt 160pb
10
Photon Interactions
In increasing order of energy the
relevant photon interaction processes
are
Photoelectric effect
Rayleigh scattering
Compton scattering
Photonuclear absorption
Pair production
11
Photon Interactions
Relative importance of the photoelectric effect,
Compton scattering, and pair production
versus energy and atomic number Z
12
Photoelectric Effect
An approximate expression for the
photoelectric effect cross section is
7/2
me c
s pe 4 2 0 Z
hv
8re2
25
2
0
6.65 10 cm
3
2
4
5
What’s important is that the photoelectric
effect is important
For high Z materials
At low energies (say < 0.1 MeV)
13
Photoelectric Effect
More detailed calculations show
n
Z
s pe ~ m
hv
n variesfrom 4 - 5
m variesfrom 3.5 - 1
we' ll just taken 5 and m 3.5
14
Photon Interactions
Typical
photon
cross
sections
15
Photoelectric Effect
The energy of the (photo)electron is
Ee E Eb
Binding energies for some of the heavier
elements are shown on the next page
Recall from the Bohr model, the binding
energies go as
me Z 2e4 1
13.6Z 2
En
eV
2
2
2
2
n
40 2 n
E1 13.6eV for H
16
Photoelectric Effect
17
Photoelectric Effect
The energy spectrum looks like
This is because at these
photon/electron energies the electron is
almost always absorbed in a short
distance
As are any x-rays emitted from the ionized
atom
18
Photoelectric Effect and X-rays
PE proportionality to Z5 makes diagnostic x-
ray imaging possible
Photon attenuation in
Air – negligible
Bone – significant (Ca)
Soft tissue (muscle e.g.) – similar to water
Fat – less than water
Lungs – weak (density)
Organs (soft tissue) can be differentiated by
the use of barium (abdomen) and iodine
(urography, angiography)
19
Photoelectric Effect and X-rays
Typical diagnostic x-ray spectrum
1 anode, 2 window, 3 additional filters
20
Photon Interactions
Sometimes easy to loose sight of real thickness
of material involved
Photon Interactions
X-ray contrast depends on differing
attenuation lengths
Photoelectric Effect
Related to kerma (Kinetic Energy Released in
Mass Absorption) and absorbed dose is the
fraction of energy transferred to the
photoelectron
T hv Eb
hv
hv
As we learned in a previous lecture, removal of
an inner atomic electron is followed by x-ray
fluorescence and/or the ejection of Auger
electrons
The latter will contribute to kerma and absorbed
dose
23
Photoelectric Effect
Thus a better approximation of the energy
transferred to the photoelectron is
T hv PKYK hvK
YK is theprobability of an x - ray when K shell is vacant
PK is thefractionof PE interactions occurringin the K shell
hvK is themean energy carried away by theK x - rays
We can then define e.g.
s
tr
pe
hv PKYK hvK
s pe
hv
trpe hv PKYK hvK pe
hv
24
Photoelectric Effect
Fluorescence yield Y for K shell
25
Cross Section
dΩ=dA/r2=sinθdθdφ
26
Cross Section
27
Cross Section
If a particle arrives with an impact
parameter between b and b+db, it will
emerge with a scattering angle between
θ and θ+dθ
If a particle arrives within an area of dσ,
it will emerge into a solid angle dΩ
28
Cross Section
From the figure on slide 7 we see
b R sin and 2
then sin sin( / 2 / 2) cos( / 2)
and b R cos( / 2)
or 2 cos1 (b / R )
This is the relation between b and θ for hard
sphere scattering
29
Cross Section
We have
ds
ds
d
d
where ds bdbd
and d sin dd
And the proportionality constant dσ/dΩ
is called the differential cross section
30
Cross Section
Then we have
ds
b db
d sin d
And for the hard sphere example
db
R
sin
d
2
2
2
ds Rb sin 2 R cos 2 sin 2 R 2
d
2 sin
2 sin
4
31
Cross Section
Finally
ds
R
s ds d d R 2
d
4
2
This is just as we expect
The cross section formalism developed here is
the same for any type of scattering (Coulomb,
nuclear, …)
Except in QM, the scattering is not deterministic
32
Cross Section
We have
ds
ds
d
d
where ds bdbd
and d sin dd
And the proportionality constant dσ/dΩ is
called the differential cross section
The total cross section σ is just
ds
s ds d
d
33