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
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For calculation questions, marks are typically awarded for correct substitution into the formula, showing intermediate steps, and providing the final answer with appropriate units. For conceptual questions, marks are given for clear and accurate explanations.
What the command words demand
- Calculate
- Perform a numerical calculation using given data and formulas.
- Determine
- Find the value of a quantity or parameter using provided information.
- Compare
- Identify similarities and differences between two or more items, concepts, or processes.
- Explain
- Provide a detailed account of how something works or why it happens.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes for each question involving exponential attenuation to ensure you have enough time to carefully apply formulas and check your work.
- Use the exponential attenuation formula: I = I 0 e -μx1 mark
- Substitute the given values: I = 100e -0.2 × 51 mark
- Calculate the exponent: -0.2 × 5 = -11 mark
- Evaluate the exponential term: e -1 ≈ 0.3681 mark
- Multiply by the initial intensity: I = 100 × 0.368 = 36.8 units1 mark
Final answer: 36.8 units
Work through every step correctly and you earn all 5 marks.
Another worked example
A material has a linear attenuation coefficient of 0.1 m -1 . Calculate the half-value thickness (HVT) for this material.
- Use the formula for HVT: HVT = ln(2) / μ ≈ 0.693 / μ1 mark
- Substitute the given value: HVT = 0.693 / 0.11 mark
- Calculate the HVT: HVT = 6.93 meters2 marks
Final answer: 6.93 meters
Work through every step correctly and you earn all 4 marks.
Common mistakes
Confusing the linear attenuation coefficient (μ) with the mass attenuation coefficient (μ m ).
Why it happens: Students often mix up these two coefficients because they are related and have similar names. It's important to remember that μ is in units of m -1 and μ m is in units of m 2 /kg.
Fix: 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.
Why it happens: Students may forget to include all parts of the formula or make errors in unit conversions. For example, they might use I = I 0 e -μx without ensuring that μ and x have consistent units.
Fix: 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.
Why it happens: When solving exponential equations, students sometimes forget that they need to take the natural logarithm (ln) of both sides to isolate the variable. This can lead to incorrect solutions.
Fix: 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.
Why it happens: Students might think that HVT is a universal constant, but it actually depends on the material and its linear attenuation coefficient. This can lead to incorrect calculations or comparisons.
Fix: 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.
Why it happens: Students might compare linear attenuation coefficients (μ) directly without considering the density of the materials. This can lead to incorrect conclusions about which material is more effective at attenuating X-rays.
Fix: 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.
Why it happens: Students might focus solely on the mathematical aspects and overlook the practical implications. This can lead to a lack of context when answering exam questions.
Fix: 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
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| X-ray Attenuation Calculation | Use the exponential attenuation equation to find the transmitted X-ray intensity through a material. | 5 |
| Calculate Half-Value Thickness | Use the attenuation coefficient to work out the half-value thickness of a material. | 4 |
| Calculate Attenuation Coefficient | Multiply the mass attenuation coefficient by density to find the linear attenuation coefficient with units. | 3 |
| X-ray Attenuation | Rearrange the exponential attenuation equation to find the material thickness for a given intensity reduction. | 6 |
| Attenuation Coefficient Calculation | Calculate the linear attenuation coefficient from the given half-value thickness using the correct formula. | 4 |
| Total across these question types | 22 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.