7.2.2 Scaling Laws

Reference: Knoll; Lapp and Andrews

Created October 19, 1995


7.2.2.1 Scaling Laws

It is not possible to fund range experiments for all possible incident particles, all possible absorbing materials at all possible energies. In order to estimate the range or energy loss characteristics of a particular example we must assume the validity of the Bethe-Bloch formula and assume that the stopping power per atom of compounds or mixtures is additive.

The latter assumption is known as the Bragg-Kleeman rule, and can be found in Knoll (1989, p. 43).

A formula from the Bragg-Kleeman rule can tell us the range of particles in a material for which we have no energy loss or range information if we are given range information in another material. The approximation becomes less valid as the separation in atomic weights increases. This rule is as follows:

This rule follows from the fact that the range in cm (linear range) is inversely proportional to the density and we are taking a ratio of ranges - one known, one unknown. The atomic number part is the approximation and the guts of the Bragg-Kleeman rule.

The estimation of a particle's range in a given material is a two-step process:

  1. determine the range in air; then

  2. determine the range in the material given the Bragg-Kleeman rule .

The range of alpha particles in cm depends upon the material and the energy of the alphas. The first part takes care of the energy in air; the second part takes care of the difference in materials between air and the material of choice.

7.2.2.2 Determination of the range in air:

In order to compute the range of alpha particles in any material, we must use an empirical equation for the range of alpha particles in air. The following equation for range in cm is valid for alphas in air in the energy range 4 < E < 15 MeV (Eqn 11):

R = (0.005E + 0.285) E**3/2 ; R in cm

(the above equation is taken from page 202 of Lapp and Andrews textbook.

Sorenson and Phelps (p. 173) use a simpler equation for the range 4 < E < 8 :

R = 0.325 E**3/2 ; R also in cm

It is helpful to note that the density of air is 0.001293 g/cm**3 and that the effective atomic number of air is about 14.6.

EXAMPLE(1):

The alpha particles of Americium-241 have energies of 5.49 MeV and 5.44 MeV. What is the range of these particles in air?

Using the above equation, we find R(cm) = 4.18 and 4.12 S &P; 4.00 and 3.96 L & A.

If the material is changed, use the scaling law as follows:

If medium o is air (sqrt A = 3.82),

Knowing more about the material in question will tell us its density and effective atomic mass number.

EXAMPLE(2):

For Americium-241 what is the range in tissue if the sqrt(A) = 3 for tissue and rho = 1.0 g/cm**3?

7.2.2.3 Range of Beta Particles

The range of beta particles, when expressed in terms of g/cm**2, is independent of the absorbing material and only depends upon energy. The square root factor of the Bragg-Kleeman rule is not applicable to the electron. This indicates that screening or the extent of the electron orbitals is less important for electrons than for heavy particles.

Since the mass electron range depends only on energy, we can define it in terms of semi-empirical equations. Two semi-empirical equations are:

for the range 0.01 < E < 3.0 MeV ,

and

for the range 1 < E < 20 MeV ,

In order to demonstrate that these two empirical equations overlap and represent approximately the same electron range, we show the following graph:

EXAMPLE (1):

What is the maximum range of the principal Beta particle of I-131?

The maximum energy of the principal Beta particle for I-131 is 0.606 MeV. The range is then:

R = 412 (0.606)**n mg/cm**2;

where n = 1.265 - 0.0954 ln (0.606)

plugging in and calculating, we find that R = 0.345 g/cm**2. If the material was water or tissue the range would be 0.345 cm.

EXAMPLE(2):

The following table gives an indication of the order of magnitude of the ranges of monoenergetic alphas and electrons in both soft tissue as well as air:

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