Attenuation of X-rays in Matter

Bones absorb X-ray radiation
- This is why they appear white on the X-ray photograph
When the collimated beam of X-rays passes through the patient’s body, they are absorbed and scattered
The attenuation of X-rays can be calculated using the equation:

$I = I_{0} e^{- μ x}$

Where:
- I₀ = the intensity of the incident beam (W m^-2)
- I = the intensity of the reflected beam (W m^-2)
- μ = the linear absorption coefficient (m^-1)
- x = distance travelled through the material (m)

The attenuation coefficient also depends on the energy of the X-ray photons
The intensity of the X-ray decays exponentially
The thickness of the material that will reduce the X-ray beam or a particular frequency to half its original value is known as the half thickness

Intensity-distance graph of X-rays for air and body

Attenuation of X-rays, downloadable AS & A Level Physics revision notes

Absorption of X-rays by different materials

Worked example

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:

$\frac{i n t e n s i t y o f e m e r g e n t x - r a y b e a m f r o m m o d e l}{i n t e n s i t y o f i n c i d e n t x - r a y b e a m o n m o d e l}$

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

$I = I_{0} e^{- μ x}$

$\frac{i n t e n s i t y o f e m e r g e n t x - r a y b e a m f r o m m o d e l}{i n t e n s i t y o f i n c i d e n t x - r a y b e a m o n m o d e l} = \frac{I}{I_{0}} = e^{- μ x}$

Step 3: Substitute in values and calculate the ratio

$\frac{I}{I_{0}} = e^{- (0.20 \times 8)} = 0.2$

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

$\frac{I}{I_{0}} = e^{- μ_{m} x_{m}} \times e^{- μ_{b} x_{b}}$

Step 3: Substitute in values and calculate the ratio

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

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

Attenuation of X-rays in Matter (CIE A Level Physics)

Revision Note