1. Gamma Ray Imaging Lab Tour
Monday, March
Please be prompt
The lab can be hard to find so allow enough time to get there

2. Landau Distribution
What 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 DE deposited in Dx

3. Laudau Distribution

4. Landau Distribution Let’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

5. Landau Distribution
Is a 1 cm scintillator thick or thin?

6. Landau Distribution
The Landau distribution looks like

7. Landau Distribution
The Landau probability density function is given by
In practice one uses a numerical approximation found in most math libraries

8. Landau Distribution Notes
Usually D is used to represent the energy loss and Dp the most probable energy loss
The probability functions describing the D 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 Dp

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 Loss
Because the mean energy loss is unreliable, one improvement is to restrict the energy loss below some value Tcut (sometimes called D)
Since Tcut instead of Tmax appears in the ln term, the mean energy loss will approach the Fermi plateau at high energies

11. Landau Distribution
Restricted dE/dx and most probable energy loss

12. Landau Distribution
Theory and experiment

13. Linear Energy Transfer
As 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

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

15. Range A useful formula is the Bragg-Kleeman rule Alpha from 214Po
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 ~ cm

16. Range
Another 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)

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

18. dE/dx (Stopping Power)
For protons

19. dE/dx (Stopping Power)
For alphas

20. dE/dx (Stopping Power)
Argon on Copper

21. Range
protons

22. Range
protons

23. Range

24. Detour
Detour is the projected range / range <= 1
Protons

25. Range As 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/b2 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

27. Bragg Curve
Protons and Carbon

28. Bragg Curve
Alpha particles in air

29. Application of Range
The localized energy deposition of heavy charged particles can be useful therapeutically = proton radiation therapy

30. Proton Therapy

31. Proton Therapy
Another particle physics connection – original idea from Robert Wilson, particle physicist

32. Proton Therapy
Energy range of interest from 50 (eye) – 250 (prostate) MeV

33. Proton Therapy
Nuclear 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

34. Proton Therapy
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35. Proton Therapy
Modulator, aperture, and compensator
Modulator

36. Proton Therapy

37. Proton Therapy

38. Proton Therapy Lung cancer treatment
Intensity modulated radiation therapy vs proton therapy

39. Proton Therapy

40. Proton Therapy
Especially 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)

41. Proton Therapy
Existing and new proton centers in the US

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

44. Landau Distribution
For very thin detectors, the Landau distribution may not be appropriate