Energy deposition by light charged particles

Light charged particles

 electrons

 posit rons

All forms of ionizing radiation eventually result in a distribution of low - energy electrons .

The interactions of light charged particles ar e of central importance in radiation biology .

The large difference in mass between electrons and h eavy charged particles has im portant consequences for interactions.

Light charged particles deposit energy through two mechanism s

 Collisional losses

 Radiative losses

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Collisional losses

 Electrons lose energy via interactions w i th orbital electrons in the mediu m .

 This leads to excitation of the atom or ioniz a tion.

 Energy loss via these m e chanism s is called “ collisional loss ”.

 Maximum energy transfer occurs in a “head-on” collision between two particles of m a sses m and M : and can be expressed as

Q  4 mME

max ( M  m ) 2

where E is the kinetic en ergy of the incident particle.

With light charged particles, m = M and so Q max = E .

 The electron collides with a p a rtic le of identical mass and thus large scattering angles are possible .

 This results in a track that is very tort uous instead of the straight path of a heavy charged particle.

Radiative Losses: bremsstrahlung

A second mechanism of energy loss is possi ble because of the sm all mass of the light charged particle (negligible with HCPs).

A charged particle undergoi ng a ch ange i n accel erat ion always emits “radiative” electromagnetic radiation called bremsstrahlung .

The larger the change in acceleration, the m o re energetic the brem sstrahlung phot on.

For electrons, the brem sstrahlung photons have a continuous energy distribution that ranges downward from a maxim u m e qual to the kinetic energy of t h e incom i ng electron.

The efficiency of bremsstrahlung in elements of different atomi c nu mb er Z

varies nearly as Z 2 .

Thus, for beta particles of a given ener gy, brem sstrahlung losses are considerably greater in high-Z m a terials, su ch as lead, t h an in low-Z ma terials, such as water.

Brem sstrahlung phot on

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 Brem sstrahlung increases with electron kinetic en ergy .

 Brem sstrahlung increases with atomic number, Z.

Stopping Power

As with heavy charged particles the stoppi ng power is the linear rate of energy loss due to excitations and ionizatio ns (“collisional energy loss”).

The collisional stopping-power for elect rons and positrons can be written

 dE  

 

  col

5 . 08 x 10  31 n

=  2

[ G  (  )  ln I ]

MeV cm -1

This equation looks somewhat sim i lar to the stopping power equation for heavy particles but now the particle char ge is 1, so -dE/dx is proportional to n/v 2 .

The stopping power of a medium for an electron or positron is:

 proportional to the density of electrons in the mat e rial and

 inversely proportional to the particle energy

Tthe total stopping pow e r for light charged particles is equal to the sum of both collisional and bremsstrahl ung stopping powers.

 dE  

 dE  

  dE  

  

  tot

  



  

 coll 



dx  rad

There is n o sim p le analytic expr ession for calculat i ng energy loss via brem sstrahlung (but it is easier to m e asure).

The collisional energy-loss rate in an elem ent is proportional to n and hence to Z .

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The radiative energy-loss rate increases l ogarithmically and at high energies beco mes the predom inant mechanism of energy loss for beta particles.

  dE 

 dx  ZE

  ra d 

  dE 

800

 dx



 coll

Radiation Yield

Radiation yield, Y, is defined as the average fraction of its energy that a beta particle radiates as bremsstrahlung in sl owing down completely .

 Radiation yield increases with electron kinetic en ergy, T .

 Radiation yield increases with atomic number, Z .

6 x 10  4 ZT

Y  1  6 x 10  4 ZT

An estimate of radiation yield can give an indication of the potential brem sstrahlung hazard of a beta-particle source.

To lim it the brem sstrahlung yie l d, low Z ma terials (e.g., alum inum ) should be used as shielding for betas and electrons since Y increases with Z.

At very high energies, the do minance of radiative over collisional energy losses gives rise to electron-photon cascade show ers .

 High-energy beta particles em it high-energy photons.

 These, in turn, produce Com p ton electrons and electron-posit r on pairs.

 These, in turn, produce additiona l brem sstrahlung photons, etc.

Range

Two factors co mplicate r a nge as applied to electrons.

 If Y is high, then the ‘range’ of the ener g y associated with the initial incident particles is large indeed due to the ve ry large mean free path of phot ons.

 Second, since electrons do not have a straight path, it is necess a ry to distinguish between distance traveled an d penetration distance .

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Empirical equations for electr on range in low-Z materials.

R = range in g cm -1

T = kinetic energy of the electron For: 0.01 ≤ T ≤ 2.5 MeV

R  0 . 412 T 1 . 27  0 . 0954 ln T

ln T

 6 . 63 

3 . 24 ( 3 . 29  ln R ) 2

For: T > 2.5 MeV

R  0 . 530 T  0 . 106

T  1 . 89 R  0 . 2

Examples Of Electron Tracks In Water

Electron tracks can b e sim u lated using known proba bility distribution functions and Monte Carlo techniques to calculate

 energy loss per interaction,

 scatter angle, and

 distance to next interaction,

Individual particles are tracked until thei r energies are below th e threshold for electronic excitation (7.4 eV for water).

Note the “tortuous” path of an electron due largely to the large angle s c attering possible with each co llision.

At higher energies, scattering tends to be m o re forward directed , at low energies (e.g. near t h e end of the track) scattering is alm o st isotropic.

This is in contrast to the tr acks of heavy charged particles.

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Figure 3. Calculated trac ks (projected into the plane of the figure) of 800 keV electrons in water. [Turner, 1995]

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Three cal culated tracks of 5-ke V electrons in water. Each electron starts from the origin and initially travels along the horizontal axis toward the right. Each dot gives the position of an energy deposit ion event.

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Delta rays are secondary electrons with enough energy to leave the local vicinity of the track and m a ke a track of its own.

Both light and heavy charged particles traveling through matter sometim e s produce delta rays.

A delta ray can also creat e another delta r a y.

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Calculated track segmen t s (0.7  m ) of protons an d alpha p a rticles having the sam e velocities in water.

Positron s

 The coulomb forces that result in the maj o r mech anism of energy loss (“collisional”) for light charged particles exist regardless of the charg e on the particle ( i.e. electron or posi tron).

 Whether the interaction involves a repul sive (electron-orbital electron) or attractive (positron-orbita l elect ron) force, the en erg y transfer is about the same.

 Thus, the track s of positrons in a material are sim i lar to those of electrons, and the stopping pow e rs and range are also roughly the same , for the sam e initial energies.

At or near the end of the positron ra nge, the positron co m b ines with a neg a tive electron in the material and the two particles annihilate.

The com b ined rest mass energy is convert ed into two 0.511 MeV photons traveling in opposit e directions.

These photons are very penetrating and lead to deposition of energy far from the original positron track.

Electron-positron pair annihilation and the creation of two photons.

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