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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?

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Gamma ray imaging lab tour
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
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

Landau distribution1
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

Landau distribution2
Landau Distribution

  • Is a 1 cm scintillator thick or thin?

Landau distribution3
Landau Distribution

  • The Landau distribution looks like

Landau distribution4
Landau Distribution

  • The Landau probability density function is given by

  • In practice one uses a numerical approximation found in most math libraries

Landau distribution5
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

Landau distribution6
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

Restricted energy loss
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

Landau distribution7
Landau Distribution

  • Restricted dE/dx and most probable energy loss

Landau distribution8
Landau Distribution

  • Theory and experiment

Linear energy transfer
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


  • 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


  • A 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


  • 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)


  • 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

De dx stopping power2
dE/dx (Stopping Power)

  • Argon on Copper


  • protons


  • protons


  • Detour is the projected range / range <= 1

  • Protons


  • 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

Bragg curve
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

Bragg curve1
Bragg Curve

  • Protons and Carbon

Bragg curve2
Bragg Curve

  • Alpha particles in air

Application of range
Application of Range

  • The localized energy deposition of heavy charged particles can be useful therapeutically = proton radiation therapy

Proton therapy1
Proton Therapy

  • Another particle physics connection – original idea from Robert Wilson, particle physicist

Proton therapy2
Proton Therapy

  • Energy range of interest from 50 (eye) – 250 (prostate) MeV

Proton therapy3
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

Proton therapy5
Proton Therapy

  • Modulator, aperture, and compensator


Proton therapy8
Proton Therapy

  • Lung cancer treatment

    • Intensity modulated radiation therapy vs proton therapy

Proton therapy10
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)

Proton therapy11
Proton Therapy

  • Existing and new proton centers in the US

Multiple scattering
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

Multiple scattering1
Multiple Scattering

  • The trajectory looks like

  • At low momentum, position and momentum resolution is usually dominated by multiple Coulomb scattering

Landau distribution9
Landau Distribution

  • For very thin detectors, the Landau distribution may not be appropriate