University Physics V · topic UPV-02.04
Compton scattering and photon momentum
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Compton scattering and photon momentum: predict a finite change
Compton scattering is the elastic two-body collision of a single photon of four-momentum ħk with a quasi-free electron, whose four-momentum balance lengthens the photon's wavelength by (h/mₑc)(1 − cos θ). Use the supplied topic model Y_out = en/(1+(en/511)*(1-cosd(theta))), where Y_out is SCATTERED E′ (keV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Scattering angle θ, measured in deg. Keep en (Incident photon energy E) = 662 keV fixed. For this exercise use only 19.125 deg ≤ theta ≤ 160.875 deg on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at theta = 50.31 deg and 70.155 deg. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δtheta across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
SCATTERED E′ against theta
Predictions from the supplied Compton scattering and photon momentum model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): SCATTERED E′ / keV
Horizontal axis (x): theta: Scattering angle θ / deg
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| theta: Scattering angle θ / deg | SCATTERED E′ / keV |
|---|---|
| 19.125 | 617.8233375154246 |
| 54.5625 | 428.6593504259467 |
| 90 | 288.3904518329071 |
| 125.4375 | 217.28796902755607 |
| 160.875 | 188.09520285596054 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of theta separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At theta = 50.31 deg, substitute into Y_out = en/(1+(en/511)*(1-cosd(theta))): Y₀ = 450.908 keV.
- At theta = 70.155 deg, the same fixed-parameter model gives Y₁ = 356.737 keV.
- Signed change: ΔY_out = Y₁ − Y₀ = -94.1703 keV; the input change is Δtheta = 19.845 deg.
- Average sensitivity is ΔY_out/Δtheta = -4.74529 keV per deg. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 450.908 keV, Y₁ = 356.737 keV; ΔY_out = -94.1703 keV; average sensitivity = -4.74529 keV per deg.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Compton scattering and photon momentum: infer an input with uncertainty
Compton scattering is the elastic two-body collision of a single photon of four-momentum ħk with a quasi-free electron, whose four-momentum balance lengthens the photon's wavelength by (h/mₑc)(1 − cos θ). Use the supplied topic model Y_out = en/(1+(en/511)*(1-cosd(theta))), where Y_out is SCATTERED E′ (keV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Scattering angle θ, measured in deg. Keep en (Incident photon energy E) = 662 keV fixed. For this exercise use only 19.125 deg ≤ theta ≤ 160.875 deg on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 250.40463 keV, with a bounded reading uncertainty ±9.3464 keV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.014175 deg, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
SCATTERED E′ against theta
Predictions from the supplied Compton scattering and photon momentum model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): SCATTERED E′ / keV
Horizontal axis (x): theta: Scattering angle θ / deg
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| theta: Scattering angle θ / deg | SCATTERED E′ / keV |
|---|---|
| 19.125 | 617.8233375154246 |
| 54.5625 | 428.6593504259467 |
| 90 | 288.3904518329071 |
| 125.4375 | 217.28796902755607 |
| 160.875 | 188.09520285596054 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(theta) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(19.125) = 617.823 keV and Y_out(160.875) = 188.095 keV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives theta_line = 19.125 + (250.405 − 617.823)(160.875 − 19.125)/(188.095 − 617.823) = 140.322 deg. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives theta = 105.593 deg; substituting back gives Y_out = 250.40463 keV.
- The interpolation error in the input is |theta_line − theta| = 34.7291 deg. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 241.05823 keV and 259.75103 keV. Sort the roots even if the response decreases: 101.266 deg ≤ theta ≤ 110.36 deg.
- The lower and upper input deviations are 4.32645 deg and 4.76743 deg. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer theta ≈ 105.593 deg; compatible interval [101.266, 110.36] deg. Endpoint interpolation gives 140.322 deg.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.