Question

3. Using the Beer-Lambert Law, calculate the penetration depth at 1 absorption length (1/e) of a photon at normal incidence entering a dense homogeneous body in the following cases:

a. 60 keV photon, beryllium

b. 8 keV photon, lead

c. If the angle of incidence is changed to 1° in parts (a) and (b) what is the depth of penetration normal to the surface?

Please show all of your work and references.

Answer 1

The intensity decreases exponentially with the distance traveled,

--- (1)  

where I ₀ is the initial X-ray beam intensity,  mass absorption coefficient μ = A/ρ and ρ is the density of the material. Here the exponential decay of photon intensity applies in the optical region of the electromagnetic spectrum as well. In this region, it is known as the Beer–Lambert law.

Given

(I/I ₀) = exp (–μρx)
ln (1/e) = (–μρx)
–1 = (–μρx)
x = 1/(μρ) --- (2)

a) From the X-Ray Mass Attenuation Coefficients chart at 60 keV photon for  beryllium μ = 1.493 * 10^1 cm²/g,,

ρ = 1.85 g/cm³

Using above quantities in eqn (2), we get

x = 1/(μρ) = 0.036 cm

(b)From the X-Ray Mass Attenuation Coefficients chart at 8 keV photon for lead μ = 2.287 * 10^2 cm²/g, The density of lead ρ = 11.34 g/cm³.

Using above quantities in eqn (2), we get

x = 1/(μρ) = 3.85 cm