# 7.2.1 Charged Particle Range

Created October 19, 1995

### 7.2.1.1 Range of Charged Particles

The range of a charged particle can be derived from the stopping power formula and gives:

Examine the thought experiment involving alpha particle transmission depicted in the following drawing.

A source of monoenergetic alpha particles is directed at a particle detector through an absorber. We know the initially monoenergetic spectrum will demonstrate energy straggling, but the direction of the particles should not be altered and they should ALL make it to the detector, provided the thickness of the absorber is less than the thickness required to stop the slowest of the distribution. This is shown as the constant horizontal line in the graph at the bottom of the drawing. Once the absorber thickness is such that some of the slower particles are stopped, the width of the energy straggling curve presents itself in the slope of the fall-off from constant particle number detected down to zero.

The mean range is the range at which the number of particles detected is one-half the original value.

The same experiment can be performed with a beam of electrons (or a beta emitter). In this case, however, the electrons are more vulnerable to wide-angle scattering even in the thinnest of foils - so wide an angle that the scattered electrons will not be detected. The result is an immediate falloff in the number of electrons detected as a function of absorber thickness as demonstrated is this drawing:

### Factors which affect range R:

**Energy:**

- Range is approximately linear with energy since the Bethe-Bloch equation for stopping power is inversely proportional to E.

**Mass:** - for the same kinetic energy, the electron is much faster than the alpha due to its smaller mass, and therefore the electron has less time to spend near orbital electrons. This reduces the effect of Coulomb interactions (hence stopping power) and increases range.

**Charge:** - the more charge, the more stopping power and the lower range. Range is inversely proportional to the square of the charge of the charged particle.
For example, a tritium particle with z=1 will have 1/4 the stopping power of a He-3 particle with z=2.

**Density:** - The stopping power increases with increasing density. The range is inversely proportional to the density of the absorbing medium.

### RANGE STRAGGLING

The same statistical factors which account for energy straggling result in a straggling of range - the total path of energy depletion is different for each initially monoenergetic alpha. For protons or alphas, straggling results in a few percent variation about the mean range. The sharper the cut-off in the transmission, the less pronounced the range straggling. Differentiating the cut-off curve often is used to demonstrate a range staggling curve analogous to the energy straggling curve.

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