gamma ray imaging lab tour
DESCRIPTION
Gamma Ray Imaging Lab Tour. Monday, March 7 @ 1100-1200 Please be prompt The lab can be hard to find so allow enough time to get there. Landau Distribution. What is the distribution (probability density function) of energy loss in a given detector? - PowerPoint PPT PresentationTRANSCRIPT
1
Gamma Ray Imaging Lab Tour
Monday, March 7 @ 1100-1200 Please be prompt The lab can be hard
to find so allow enough time to get there
2
Landau DistributionWhat is the distribution (probability density
function) of energy loss in a given detector? So far we have just calculated the mean energy
loss The mean energy loss may be fine for dosimetry
(bulk deposition) but it is inadequate in describing the energy loss of single particles
There are large statistical fluctuations in the distribution of dE/dx due to a small number of collisions involving very energetic electrons
A real particle detector cannot really measure the mean energy loss – it measures E deposited in x
4
Landau DistributionLet’s define thick and thin detectors
If a detector is thick
For thick detectors, the energy loss distribution is a Gaussian distribution with mean given by Bethe-Bloch and sigma given by Bohr (non-relativistic)
Basically the central limit theorem the sum of N random variables as N→∞ is a Gaussian
For thin detectors, the energy loss distribution is given by the Landau or Vavilov distribution
maxWtdx
dE
7
Landau Distribution The Landau probability density function is given
by
In practice one uses a numerical approximation found in most math libraries
If
xz
A
ZcmrN
uduuuu
xf
eeAv
,,,
2
sinlnexp1
where),(
2
22
0
8
Landau DistributionNotes
Usually is used to represent the energy loss and p the most probable energy loss
The probability functions describing the distribution are frequently called straggling functions but in EPP they are called Landau functions
The long tail is called the Landau tail It comes from a few scatters having large
energy transfers (up to Wmax) There are also expressions for the most
probable energy loss p
9
Landau Distribution
Given the skewed distribution, one can see why either the most probable energy loss or restricted energy loss are preferred to describe the energy loss distribution for heavy charged particles
10
Restricted Energy LossBecause the mean energy loss is
unreliable, one improvement is to restrict the energy loss below some value Tcut (sometimes called )
Since Tcut instead of Tmax appears in the ln term, the mean energy loss will approach the Fermi plateau at high energies
13
Linear Energy TransferAs mentioned, in radiation physics, often
linear energy transfer (LET) is used for dE/dx
LET is defined as
LET is used in radiobiology and radiation protection dosimetry
cut
cut
TTT dx
dELL
14
Range Since we know the energy loss we can calculate
the range (pathlength) a heavy charged particle travels before stopping
This is called the CSDA (Continuously Slowing Down Approximation) range
It is a very good approximation to the real range The range is defined as a straight-line thickness
The projected range is the average value to which a charged particle will penetrate measured along the initial direction
Detour factor is the ratio of the projected range to range and is always < 1
dEdx
dER
T 1
0
0
15
RangeA useful formula is the Bragg-Kleeman
rule Can be used to determine the range in one
material if one knows the range in another material
Alpha from 214Po R in air ~ 6 cm R in tissue ~ 0.007 cm
2
1
1
2
2
1
A
A
R
R
16
RangeAnother useful relationship can be used
to find the range for different particles (ions) with the same velocity in different materials
z1, z2 are the charges of particles 1 and 2 M1, M2 are the masses of particles 1 and 2
Comparing protons and 12C in water R(12C) = 12/36 = ~1/3 (see slide 26)
2
121
22
2
1
M
M
z
z
R
R
17
Range In our discussion of dE/dx loss we included
only the contribution from electrons Electronic stopping power
We ignored the contribution from collisions with nuclei
Nuclear stopping power At very low energies, nuclear recoil energy loss
becomes more important and in fact dominates for heavier ions
Both the electronic and nuclear stopping power at low energies (<500 keV protons) is a quite complicated subject and software (SRIM) or fitting formulas based on experimental data are used
Very important for ion implantation
25
RangeAs we saw, energy loss is a statistical
process This means that the range is not the same for
every particle An approximation is to use a Gaussian
distribution about the mean range (point of 50% transmission)
It’s difficult to calculate so a parameterization or simulation (GEANT or MCNP) must be used
26
Bragg Curve
The 1/2 dependence of dE/dx means that most of the energy loss will be deposited towards the end of the trajectory rather than uniformly along it
A plot of the energy loss versus distance is called a Bragg curve
29
Application of RangeThe localized energy deposition of
heavy charged particles can be useful therapeutically = proton radiation therapy
31
Proton Therapy
Another particle physics connection – original idea from Robert Wilson, particle physicist
33
Proton TherapyNuclear reactions are important in this
energy range as well About 20% of 160 MeV protons stopping in
water have a non-elastic nuclear reaction where the primary proton is seriously degraded and secondary protons, neutrons and nuclear fragments appear
40
Proton TherapyEspecially useful for chordomas (tumors
in the skull base), ocular tumors, and prostate cancer
But “Proton and other particle therapies need to
be explored as potentially more effective and less toxic RT techniques. A passionate belief in the superiority of particle therapy and commercially driven acquisition and running of proton centers provide little confidence that appropriate information will become available…An uncontrolled expansion of clinical units offering as yet unproven and expensive proton therapy is unlikely to advance the field of radiation oncology or be of benefit to cancer patients.” from Brada et al. in J.Clin.Oncol. (2007)
42
Multiple Scattering
A charged particle traversing matter will undergo multiple (small angle) Coulomb scattering from nuclei Small angle scattering – Gaussian Larger angle scattering – Rutherford
scattering
43
Multiple Scattering The trajectory looks like
At low momentum, position and momentum resolution is usually dominated by multiple Coulomb scattering
000
20
2
0
ln038.016.13
2exp
2
1
X
x
X
xz
cp
MeV
P planeplane