University Physics IV · topic UPIV-04.04
Recoil kinematics, the unshifted line, and scattering regimes
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.
Recoil kinematics and the Compton edge: predict a finite change
The recoil kinetic energy T(θ) = Eα(1 − cosθ)/[1 + α(1 − cosθ)] is what the Compton ledger hands the electron at each photon scattering angle, and its ceiling at θ = 180° is the Compton edge. Use the supplied topic model Y_out = 511*en/(511+en*(1-cosd(th))), where Y_out is SCATTERED E′(θ) (keV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. en is Photon energy E, measured in keV. Keep th (Scattering angle θ) = 120 ° fixed. For this exercise use only 230.375 keV ≤ en ≤ 1789.62 keV 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 en = 573.41 keV and 791.705 keV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δen across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
SCATTERED E′(θ) against en
Predictions from the supplied Recoil kinematics and the Compton edge 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′(θ) / keV
Horizontal axis (x): en: Photon energy E / keV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Photon energy E / keV | SCATTERED E′(θ) / keV |
|---|---|
| 230.375 | 137.43495074790223 |
| 620.1875 | 219.88478133605082 |
| 1009.9999999999999 | 254.74333662388946 |
| 1399.8124999999998 | 273.9874555618065 |
| 1789.6249999999998 | 286.1887847908151 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of en separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At en = 573.41 keV, substitute into Y_out = 511*en/(511+en*(1-cosd(th))): Y₀ = 213.704 keV.
- At en = 791.705 keV, the same fixed-parameter model gives Y₁ = 238.179 keV.
- Signed change: ΔY_out = Y₁ − Y₀ = 24.4755 keV; the input change is Δen = 218.295 keV.
- Average sensitivity is ΔY_out/Δen = 0.112121 keV per keV. Its positive sign means that this response increases 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 = 213.704 keV, Y₁ = 238.179 keV; ΔY_out = 24.4755 keV; average sensitivity = 0.112121 keV per keV.
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.
Recoil kinematics and the Compton edge: infer an input with uncertainty
The recoil kinetic energy T(θ) = Eα(1 − cosθ)/[1 + α(1 − cosθ)] is what the Compton ledger hands the electron at each photon scattering angle, and its ceiling at θ = 180° is the Compton edge. Use the supplied topic model Y_out = 511*en/(511+en*(1-cosd(th))), where Y_out is SCATTERED E′(θ) (keV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. en is Photon energy E, measured in keV. Keep th (Scattering angle θ) = 120 ° fixed. For this exercise use only 230.375 keV ≤ en ≤ 1789.62 keV 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 = 264.42505 keV, with a bounded reading uncertainty ±3.2646 keV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown en by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.155925 keV, 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′(θ) against en
Predictions from the supplied Recoil kinematics and the Compton edge 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′(θ) / keV
Horizontal axis (x): en: Photon energy E / keV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Photon energy E / keV | SCATTERED E′(θ) / keV |
|---|---|
| 230.375 | 137.43495074790223 |
| 620.1875 | 219.88478133605082 |
| 1009.9999999999999 | 254.74333662388946 |
| 1399.8124999999998 | 273.9874555618065 |
| 1789.6249999999998 | 286.1887847908151 |
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(en) = 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(230.375) = 137.435 keV and Y_out(1789.62) = 286.189 keV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives en_line = 230.375 + (264.425 − 137.435)(1789.62 − 230.375)/(286.189 − 137.435) = 1561.5 keV. 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 en = 1181.52 keV; substituting back gives Y_out = 264.42505 keV.
- The interpolation error in the input is |en_line − en| = 379.978 keV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 261.16045 keV and 267.68965 keV. Sort the roots even if the response decreases: 1119.02 keV ≤ en ≤ 1249.61 keV.
- The lower and upper input deviations are 62.5023 keV and 68.0943 keV. 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 en ≈ 1181.52 keV; compatible interval [1119.02, 1249.61] keV. Endpoint interpolation gives 1561.5 keV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.