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Electromagnetic interactionsPowerPoint Presentation

Electromagnetic interactions

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Electromagnetic interactions

- Energy loss due to collisions
- An important fact: electron mass = 511 keV /c2, proton mass = 940 MeV/c2, so it is much easier to give an electron a "kick" than a nucleus, i.e. will be dominated by interactions with the electrons.

- Other types of e.m. interaction,
- bremsstrahlung and creation of electron-positron pairs by high-energy photons are sensitive to the electric field strength, so the interaction with the nucleus dominates.

- Cerenkov/Transition radiation
- A third category of interactions is sensitive to bulk properties of the matter, like dielectric constant. These interactions give rise to Cherenkov and transition radiation

Taking into account quantum-mechanical effects and using first-order

perturbation theory the Bethe-Bloch equation is obtained:

Tmax is the maximum energy transfer to a single electron:

,

Tmax is often approximated by 2me22. re is the classical

electron radius (re = e2 / mec2 = 2.82 x10-13 cm)

(radius of a classical distribution of the electron charge

with electrostatic self-energy equal to the electron mass).

I is the mean ionization energy.

NB: for high momentum particles

Substituting this and also e2 / mec2 for re gives eq. (2.19) of Fernow

Hans Albrecht Bethe

Felix Bloch

- is the "density correction“:
It arises from the screening of remote electrons by close electrons, which results in a reduction of energy loss for higher energies (transverse electric field grows with !). The effect is largest in dense matter, i.e. in solids and liquids.

- C is the "shell correction" :
- Only important for low energies where the particle velocity has the same order of magnitude as the "velocity" of the atomic electrons.
- For improved accuracy more correction factors need to be added, but the
- particle data group claims that the accuracy in the form shown above for
- energy loss of pions in copper for energies between 6 MeV and 6 GeV
- about 1 %, with C set to 0.
- Note that the Bethe-Bloch equation provides only the mean of the
- "stopping power", but no information on fluctuations in it

dE/dx for pions as computed with Bethe-Bloch equation

dE/dx divided

by density

(approximately

material

independent)

slope due to 1/v2

high :

dE/dx

independent

of

due to

density

effect,

"Fermi

plateau"

relativistic rise

due to ln

- about proportional to ne,
as ne = na Z = NA Z / A, -> ne ≈ NA / 2

From PDG, Summer 2002

Some phenomena not taken into account in the formula are :

- Bremsstrahlung: photons produced predominantly in the electric field of the
nucleus. This is an important effect for light projectiles, i.e. in particular for

electrons and positrons

- Generation of Cherenkov or transition radiation. Cherenkov radiation occurs when
charged particles move through a medium with a velocity larger than the velocity

of light in that medium. Transition radiation is generated when a highly relativistic

particle passes the boundary of two media with different dielectric constants. The

energy loss is small compared to the energy loss due to exciation and ionization

- For electrons and positrons the Moller resp. Babha cross sections should be
used in the calculation of dE/dx, this leads to small corrections. Fernow

quotes, for -> 1, Tmax set to 2me22 and without density and shell corrections:

Electrons:

Heavy particles:

Range of stopping particles

For thick enough material particles will be stopped, the range can

be calculated from (M = mass projectile, Z1 = charge projectile):

The Bethe-Bloch equation with Tmax approximated by 2me22 can be

written as:

f(v) can be replaced by g(E/M), as :

-> The dependency of R Z12/M on E is

approximately material and projectile

independent( (dE/dx)/ is ~ material

independent)

Two different projectiles

with same energy:

Most of the energy

deposited at end of track

Fraction of particles

surviving

100 %

Sir William

Henry Bragg

Sir William

Lawrence Bragg

dE/dx

Bragg

curve

Averange

range R

Depth x in material

Fluctuations in energy loss

- The energy transfer for each collision is determined by a probability distribution.
- The collision process itself is also a process determined by a probability distribution.
- The number of collisions per unit length of material is determined by a Gaussian distribution
- the energy loss distribution usually is referred to as a "Landau" distribution. This is a distribution with a long tail for high values of the energy loss. The tail is caused by collisions with a high energy transfer.

Lev Davidovich Landau

From PDG, Summer 2002

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