Attenuation of X-rays in Matter (Cambridge (CIE) A Level Physics): Revision Note

A student investigates the absorption of X-ray radiation in a model arm. A cross-section of the model arm is shown in the diagram.

Parallel X-ray beams are directed along the line MM and along the line BB. The linear absorption coefficients of the muscle and the bone are 0.20 cm^-1 and 12 cm^-1 respectively.

Calculate the ratio:

$\frac{\mathrm{intensity}\mathrm{of}\mathrm{emergent}\mathrm{x}-\mathrm{ray}\mathrm{beam}\mathrm{from}\mathrm{model}}{\mathrm{intensity}\mathrm{of}\mathrm{incident}\mathrm{x}-\mathrm{ray}\mathrm{beam}\mathrm{on}\mathrm{model}}$

for a parallel X-ray beam directed along the line

(a) MM

(b) BB

and state whether the X-ray images are sharp, or have good contrast.

Answer:

Part (a)

Step 1: Write out the known quantities

• Linear absorption coefficient for muscle, μ = 0.20 cm^-1

• Distance travelled through the muscle, x = 8.0 cm

Step 2: Write out the equation for attenuation and rearrange

$I=I_0{e}^{-\mu x}$

$\frac{\mathrm{intensity}\mathrm{of}\mathrm{emergent}\mathrm{x}-\mathrm{ray}\mathrm{beam}\mathrm{from}\mathrm{model}}{\mathrm{intensity}\mathrm{of}\mathrm{incident}\mathrm{x}-\mathrm{ray}\mathrm{beam}\mathrm{on}\mathrm{model}}=\frac{I}{I_0}=e^{-\mu x}$

Step 3: Substitute in values and calculate the ratio

$\frac{I}{I_0}=e^{-\left(0.20\times 8\right)}=0.2$

Part (b)

Step 1: Write out the known quantities

• Linear absorption coefficient for muscle, μ_m = 0.20 cm^-1

• Linear absorption coefficient for bone, μ_b = 12 cm^-1

• Distance travelled through the muscle, x_m = 4.0 cm

• Distance travelled through the bone, x_b = 4.0 cm

Step 2: Write out the equation for attenuation for two media and rearrange

$\frac{I}{I_0}=e^{-{\mu }_m{x}_m}\times e^{-{\mu }_b{x}_b}$

Step 3: Substitute in values and calculate the ratio

$\frac{I}{I_0}=e^{-\left(0.20\times 4\right)}\times e^{-\left(12\times 4\right)}=6.4\times {10}^{-22}\approx 0$

Step 4: Write a concluding statement

• Each ratio gives a measure of the amount of transmission of the beam

• A good contrast is when:

• there is a large difference between the intensities

  • the ratio is much less than 1.0

• Therefore, both images have a good contrast