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Interaction of Particles with Matter

Dr. Pietro GiampaTRIUMF

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General Basic Review:

Mediating Forces: 1. Gravitational2. Electromagnetic3. Weak4. Strong

Particles:1. Photons2. Leptons

a. Neutrinosb. Electron, Muon, Tau

3. Hadronsa. Light Mesons u,d-quarks: 𝛑∓, 𝛑0

b. Light Mesons s- and u,d-quarks: k∓, kl0

c. Light Mesons c- and u,d,s-quarks: D∓, D0, Ds∓, J

d. Light Mesons b- and u,d,s,c-quarks: B∓, B0, Bc∓

4. Carriersa. Gluonsb. W+, W-, Z Bosons

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General Basic Review:

Mediating Forces: 1. Gravitational2. Electromagnetic3. Weak4. Strong

Particles:1. Photons2. Leptons

a. Neutrinosb. Electron, Muon, Tau

3. Hadronsa. Light Mesons u,d-quarks: 𝛑∓, 𝛑0

b. Light Mesons s- and u,d-quarks: k∓, kl0

c. Light Mesons c- and u,d,s-quarks: D∓, D0, Ds∓, J

d. Light Mesons b- and u,d,s,c-quarks: B∓, B0, Bc∓

4. Carriersa. Gluonsb. W+, W-, Z Bosons

Weak E&M Strong Gravity

Relative Strength

10-7 10-2 1 x

Neutrinos Yes No No No

Leptons Yes Yes No No

Hadrons Charged

Yes Yes Yes No

Hadrons Neutral

Yes No Yes No

Photons No Yes No No

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General Basic Review:

Cross Section per Atom/Nucleus 𝜎 (microscopic): For a given particle of energy E on an atom with atomic/mass number Z/A

Mean Free Path λ:Average distance traveled by a particle between two successive collisions.

Macroscopic Cross Section 𝛴:Estimated as the inverse of the Mean Free Path.

Cummulative Interaction Probability P(l):Assume that all particles are normally incident on a homogeneous material.P(l) is the probability that a particles interacts before reaching a path length l.

Particle

Homogeneous Material

l

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General Basic Review:

Simple Example: Beam Loss via Graphite Collimator Start with a proton beam with 288 bunches and a beam density of 1.3 x 1011 protons/bunch with energy of 520 MeV. The proton beam is passed through a 1.2 m long Graphite (density 1.84 g/cm3) collimator.How many protons will have a nuclear collision with the collimator?

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General Basic Review:

Simple Example: Beam Loss via Graphite Collimator Start with a proton beam with 288 bunches and a beam density of 1.3 x 1011 protons/bunch with energy of 520 MeV. The proton beam is passed through a 1.2 m long Graphite (density 1.84 g/cm3) collimator.How many protons will have a nuclear collision with the collimator?

* If you look it up on the database the inelastic p-C cross-section @ 520 MeV is ~ 245 mb.* Calculate the mean free path using previous equation:

* Total number of proton 288 x 1.3x1011 = 3.74x1013, therefore, the number of interacting protons is:

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Interactions with Matter

PhotonsCharged Particles Hadrons Neutrinos

1. Coulomb Interactions: Ionization and Excitation

2. Stopping Power3. Multiple Scattering4. Radiative

Processes

1. Coherent (Rayleigh) Scattering

2. Incoherent (Compton) Scattering

3. Photoelectric Effect

4. Pair Production

1. Hadron/Nucleon Interaction hN

2. Hadron/Nucleus Interaction hA

1. Charge Current2. Neutral Current

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Charged Particles

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Charged Particles: Coulomb InteractionsAll energy transfer to the target medium are ultimately dominated by Coulomb interactions of charged particles that interact with atomic or nuclear interactions. (True only up to critical particle energies, above that we are dominated by radiative processes). [for e- e+ Coulomb interactions dominate up to 10 MeV, above that Bremsstrahlung]

Charged Particle = Ze (Mc2 >> mec2)

Elastic Scattering off Nuclei Process leads to Excitation / De-Excitation

Atom + X ↦ Atom* + X ↦ Atom + ɣ + X

Inelastic Scattering off Atomic ElectronsProcess leads to Ionization

Atom + X ↦ Atom+ + e- + X

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Charged Particles: “Average” Stopping Power

K = 0.307 MeV g-1 cm2

Tmax = max energy transferred in collision

z = Charge of incident particle Z = Charge number of mediumA = Atomic mass of mediumv = Velocity of incident particle

Charged Particle = Ze (Mc2 >> mec2)

I = Mean excitation energy of the mediumme = Electron mass = 511 keVC = Speed of light

β = v/cɣ = (1-β2)-2

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Charged Particles: “Average” Stopping Power

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Charged Particles: Bragg Peak

Energy Loss at Small Momenta:At small momenta the energy deposition increases (-dE/dx ~ 1/v2). The more a particle interacts, the more it slows down and the more energy it will deposit. (βɣ < 3.5)

The maximum dE/dx as a function of path length in the material is called the Bragg Peak.

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Bragg Peak Application: Proton Therapy

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Charged Particles: Multiple InteractionsMultiple (Coulomb) Interactions:Incident particles can scatter off the Coulomb field of the nucleus. This leads to multiple scattering, the deflection of trajectory is significant because of the Z dependence.

At low energy, position and momentum are dominated by Coulomb scattering.

For a thin layer, single scatter can be approximated to Rutherford scattering.

Multiple scattering (>20), the struggling can be simplified and interpreted via Moliere Theory

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Charged Particles: Electron & Positron Interactions

Elastic Scattering of Electrons off Atomic Electrons Process leads to Ionization

Atom + e- ↦ Atom+ + e- + e-

The Bethe-Bloch equation needs to be modified for electrons and positrons in order to account for the smaller mass, meaning deflections become more important. Note that the incident and target electron have the same mass, therefore Tmax = T/2. Throw some Quantum mechanics in there, after the collision the incoming electron and the ejected electron are indistinguishable.

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Radiative ProcessesBremsstrahlungWhen a relativistic accelerated charged particle passes through the Coulomb field of the nucleus, it can radiate photons (“breaking radiation” in German).

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Radiative ProcessesCherenkov RadiationAs a charged particle traverse matter, it produces a wake of polarized molecules along its path. If the particle is moving with a speed exceeding the speed of light in the medium c/n, in a certain direction the radiation becomes coherent and results in a significant amount of light produced.

Cherenkov Critical Angle

Cherenkov Photo Yield

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Photons

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Photons

Coherent (Rayleigh) Scattering: Photoelectric Effect:

Incoherent (Compton) Scattering: Electron-Positron Pair Production:

E0

E0

𝜎 ~ Z2

E0

E’0

Ee 𝜎 ~ Z

e-

E0 > Ui

e-

Ee = E0 - Ui

𝜎 ~ Z4-5

E0 > 1.022 MeV

e-

e+

Ee

Ee

𝜎 ~ Z2

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Photons: Cross Section

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Photons: Absorption LengthAbsorption LengthSimilar to what discussed at the beginning, but now we have to sum over all possible photon processes.

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Hadrons

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Hadrons: General OverviewElastic vs Non-elastic ProcessesThese two processes effectively dominate both the Hadron-Nucleon (hN) and Hadron-Nucleus (hA).

Elastic Interactions● These interactions do not change the internal structure of the projectile/target and do not produce new

particles. ● These interactions deflect in opposite direction the projectile and the target in the center of mass system (CMS).● There is no threshold for elastic interactions.

Non-elastic Interactions● Are those where new particles are produced or the internal structure of the projectile/target is changed.● The new product could be unstable.● Each different type of Non-elastic interaction has a energy threshold below which the reaction does not occur.

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Hadrons: Non-elastic InteractionsFast Stage (10-22 s)This is how hadronic showers start, the incoming hadron interacts with the nucleus to produce new particles. Pions are the bare standard because of the low production energy (290 MeV). Energetic produced particles can leave the nucleus (forward direction)Other stays within the nucleus and can modify its state (excimer)

Slow Stage (10-16 s) These are the processes that are normally follow the fast stage. Striving for equilibrium.Evaporation: Light elements (n, photons)Fission: Heavy Elements (alpha, ecc)

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Neutrinos

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Neutrinos3 Flavours (that we know of) and they are pretty much massless and have no charge. Therefore, the only portal for interactions is via the weak forces. Although, very very rare, neutrinos can interact with atomic nuclei.

Charged Current InteractionNeutrino combines with a neutron in the nucleus to generate a corresponding lepton and a proton (W boson).

Neutral Current InteractionNeutrino communicates with a protonin the nucleus via the Z boson.

Elastic Scattering InteractionNeutrino communicates with an electron via the Z boson.

Coherent Elastic ScatteringNeutrino interacts with the nucleus as a whole.observed for the first time in 2017.

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Particles Showers

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Particle ShowersElectromagnetic ShowersHigh energy electrons, protons and photons can generate Showers effects when enter a material. Bremsstrahlung dominates e+/e-

Pair production dominates photons.Energy decreases after each production. The shower stopes once the energy drops below the critical energy.E < Ec Shower Stops.

Hadronic ShowersHadronic particles at high energy can create showers of other hadronic particles. Non negligible portion of the initial energy can go to the binding energy + recoil.Roughly continues until the energy falls below the production energy of the Pion. Longer than EM showers.

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Thank youMerci

Follow us @TRIUMFLabwww.triumf.ca

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Backup Slides

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Charged Particles: “Average” Stopping Power

v a da

dx

Derivation of the Bethe-Bloch Formula for Energy LossMomentum transfer from an incoming particle of charge ze toan atomic electron is:

Where a is the shortest distance between the particle’s trajectory and an atomic electron, and v is the velocity of the incoming particle. Integrating from - to + infinity in x we get a total momentum:

The energy transferred to the electron and, consequently, lost by the incoming particle is then:

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Derivation of the Bethe-Bloch Formula for Energy LossSumming up over all electrons in the medium, the average energy lossesexperienced by the incoming particle after passing dx is:

Note the cutoffs in the integration at amin and amax. At large distances, for which the calculated T becomes smaller than the ionization/excitation potential I, no energy transfer becomes possible, this impose the amax cutoff:

The maximum energy an incoming particle can transfer to an electron is limited by the energy-momentum conservation laws. For an incoming particle of mass M and velocity v:

Charged Particles: “Average” Stopping Power

v a da

dx

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Derivation of the Bethe-Bloch Formula for Energy LossTherefore, the formula energy losses can be written as: (note the negative sign is included to reflect than the energy is being lost)

The number of electrons in a medium of density p and made of elements Z,A is given by:

Therefore, we can rewrite the equation for dE/dx with the correct expression of:

Charged Particles: “Average” Stopping Power

v a da

dx

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Charged Particles: “Average” Stopping Power

K = 0.307 MeV g-1 cm2

Tmax = max energy transferred in collision

z = Charge of incident particle Z = Charge number of mediumA = Atomic mass of mediumv = Velocity of incident particle

Charged Particle = Ze (Mc2 >> mec2)

I = Mean excitation energy of the mediumme = Electron mass = 511 keVC = Speed of light

β = v/cɣ = (1-β2)-2

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Charged Particles: “Average” Stopping Power

Quick fall of dE/dx follows approximately a β-2, dominated by strong force (kinematic form factor).

Precisely it follows a β-5/3

behaviour, the slower the particle the longer the experience the “nuclear” field.

Shell corrections must be applied, particle velocity can be of the order of the electron orbital velocity (βc ~ ve).

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Charged Particles: “Average” Stopping Power

Presence of a minimum ionization saddle point. Minimum Ionizing Particle.

Relativistic rise ~ln(β2 ɣ2), the transverse electric field increases due to Lorentz transformation.

Critical Energy Bound, Commonly described by Rossi’s definition: Solid (gas)

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Charged Particles: “Average” Stopping PowerFermi plateau, therefore density correction must be considered. Dominated by radiative processes.

Dominated by the polarization nature of materials follows a -δ/2 behaviour, effectively reduces the long range contribution to the relativistic rise.

High energy approximation (Plasma Physics):