Gamma ray imaging lab tour
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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

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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


Laudau distribution

Laudau Distribution


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


Range

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


Range1

Range

  • 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


Range2

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)


Range3

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


De dx stopping power

dE/dx (Stopping Power)

  • For protons


De dx stopping power1

dE/dx (Stopping Power)

  • For alphas


De dx stopping power2

dE/dx (Stopping Power)

  • Argon on Copper


Range4

Range

  • protons


Range5

Range

  • protons


Range6

Range


Detour

Detour

  • Detour is the projected range / range <= 1

  • Protons


Range7

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


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 therapy

Proton 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 therapy4

Proton Therapy

  • asdf


Proton therapy5

Proton Therapy

  • Modulator, aperture, and compensator

Modulator


Proton therapy6

Proton Therapy


Proton therapy7

Proton Therapy


Proton therapy8

Proton Therapy

  • Lung cancer treatment

    • Intensity modulated radiation therapy vs proton therapy


Proton therapy9

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


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