A-Level · Physics · AQA · Mark scheme decoded
AQA A-Level Physics: Exponential Attenuation of X-rays — mark scheme explained
The short answer
Exponential attenuation is a fundamental concept in medical physics, particularly when dealing with the interaction of X-rays with matter. This topic covers the linear coefficient ( μ ), mass attenuation coefficient ( μ m ), and half-value thickness (HVT). It also touches on the differential tissue absorption of X-rays, excluding detailed processes of absorption.
The question
An X-ray beam with an initial intensity of 100 units passes through a material with a linear attenuation coefficient of 0.2 m -1 . Calculate the intensity after passing through 5 meters of the material. [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Use the exponential attenuation formula: I = I 0 e -μx
- S2
Substitute the given values: I = 100e -0.2 × 5
- S3
Calculate the exponent: -0.2 × 5 = -1
- S4
Evaluate the exponential term: e -1 ≈ 0.368
- S5
Multiply by the initial intensity: I = 100 × 0.368 = 36.8 units
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Use the exponential attenuation formula: I = I 0 e -μx
- S2
Substitute the given values: I = 100e -0.2 × 5
- S3
Calculate the exponent: -0.2 × 5 = -1
- S4
Evaluate the exponential term: e -1 ≈ 0.368
- S5
Multiply by the initial intensity: I = 100 × 0.368 = 36.8 units
Final answer: 36.8 units
Common mistakes
- Confusing the linear attenuation coefficient (μ) with the mass attenuation coefficient (μ m ). — Always check the units when using attenuation coefficients. Use the relationship μ = μ m × ρ to convert between them if necessary.
- Incorrectly applying the exponential formula, especially with units. — Double-check your formula and ensure that all units are consistent. If necessary, convert units to match the required form of the equation.
- Forgetting to use natural logarithms when solving for thickness or attenuation coefficient. — Always remember to use ln when dealing with exponential equations. For example, if you have I = I 0 e -μx , take the natural logarithm of both sides to solve for x or μ.
- Misinterpreting the half-value thickness (HVT) as a fixed value for all materials. — Understand that HVT varies with different materials. Use the formula HVT = ln(2) / μ ≈ 0.693 / μ to calculate it for each specific material.
- Failing to consider the density of the material when comparing attenuation coefficients. — Use the mass attenuation coefficient (μ m ) when comparing different materials, as it accounts for differences in density. Calculate μ m using μ m = μ / ρ .
- Not understanding the significance of differential tissue absorption in medical imaging. — Always consider the real-world application of exponential attenuation in medical imaging. Understand how different tissues absorb X-rays differently and why this is important for diagnostic purposes.
Where the marks go
- Full worked solution (all marking points)5 marks