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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

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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

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Laudau Distribution

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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

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Landau DistributionIs a 1 cm scintillator thick or thin?

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Landau Distribution

The Landau distribution looks like

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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

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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

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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

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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

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Landau DistributionRestricted dE/dx and most

probable energy loss

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Landau Distribution

Theory and experiment

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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

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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

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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

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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

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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

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dE/dx (Stopping Power)For protons

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dE/dx (Stopping Power)For alphas

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dE/dx (Stopping Power)Argon on Copper

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Rangeprotons

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Rangeprotons

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Range

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DetourDetour is the projected range / range <=

1Protons

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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

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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

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Bragg CurveProtons and

Carbon

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Bragg Curve

Alpha particles in air

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Application of RangeThe localized energy deposition of

heavy charged particles can be useful therapeutically = proton radiation therapy

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Proton Therapy

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Proton Therapy

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

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Proton Therapy

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

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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

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Proton Therapy

asdf

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Proton Therapy

Modulator, aperture, and compensator

Modulator

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Proton Therapy

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Proton Therapy

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Proton Therapy

Lung cancer treatment Intensity modulated radiation therapy

vs proton therapy

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Proton Therapy

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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)

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Proton Therapy

Existing and new proton centers in the US

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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

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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

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Landau Distribution

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