7.1.2 Stopping Power

Created October 19, 1995

7.1.2.1 Stopping Power

The stopping of charged particles in matter is by collisional and radiative processes which occur in frequencies dictated by their interaction cross sections (or probabilities). What we observe then is a statistical average of the 2 processes occurring as the particle slows down.

Linear Stopping Power is given by

S= - dT/dx,

where T is the charged particle kinetic energy, -dT is the energy increment lost in infinitesimal material thickness of dx. The units of stopping power are keV/micron. The higher the stopping power, the shorter the range into the material the particle can penetrate. The quantity S is also referred to as specific energy loss .

The stopping power S increases as the particle velocity is decreased. The classical depiction of the charged particle interaction with an electron is depicted in the following drawing:

Classically, a passing charged particle will impart kinetic energy to the electron given by the following formula:

The faster the particle is, the less energy is given to the electron (the less time it has to spend imparting energy to the electron).

The classical (i.e, non-Quantum Mechanical) expression that describes the specific energy loss is known as the Bethe-Bloch Formula and is written :

The first term in the STOPPING NUMBER B is sufficient if v is much less than c, that is, for non-relativistic charged particles. The Bethe-Bloch formula is valid for all types of heavy charged particles provided the particle velocity is large relative to the orbital electron velocity. The stopping number varies slowly with particle energy and is proportional to the atomic number Z of the absorber material. Thus the general behavior of dT/dx can be inferred from the residual multiplicative factor. It can be seen that dT/dx varies as 1/v**2, or inversely with particle energy. The Bethe-Bloch formula validates the intuitive assumption that the higher Z and the more dense the absorber, the greater the stopping power.

The ionization/excitation parameter I is treated as an experimentally determined quantity. The ratio of I/Z is approximately constant for absorbers with Z greater than 13. For smaller atoms the electrons play a more important role in influencing the value of I.

The Bethe-Bloch formula for electrons must take into account two facts:

  1. the electron has a much smaller mass than the heavy particles; and

  2. the electron is identical to the particles with which it is interacting, thus giving the possibility of the exchange of identity, i.e., the incoming particle has a probability of becoming the atomic electron with the atomic electron becoming the outgoing particle. Taking this into account, as well as relativity, gives us the Bethe-Bloch formula for electrons. .

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    Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu