University Physics V · topic UPV-16.02
Induced fission, resonances, and neutron moderation
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.
Induced fission and neutron moderation: predict a finite change
Induced fission is the decay of a compound nucleus formed by neutron capture, and moderation is the elastic slowing-down that delivers the neutron to the energy where that capture is most likely to end in fission. Use the supplied topic model Y_out = ln(e0*1000000/0.0253)/(1+(((a-1)/(a+1))^2+0.0000001)*ln(((a-1)/(a+1))^2+0.0000001)/(1-((a-1)/(a+1))^2+0.0000001)), where Y_out is COLLISIONS TO 0.025 eV (collisions). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; ln(x) means the natural logarithm. e0 is Fission neutron energy E₀, measured in MeV. Keep a (Moderator mass number A) = 12; n (Collisions elapsed n) = 20 fixed. For this exercise use only 0.871875 MeV ≤ e0 ≤ 3.62812 MeV 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 e0 = 1.47825 MeV and 1.86412 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δe0 across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
COLLISIONS TO 0.025 eV against e0
Predictions from the supplied Induced fission and neutron moderation 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): COLLISIONS TO 0.025 eV / collisions
Horizontal axis (x): e0: Fission neutron energy E₀ / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| e0: Fission neutron energy E₀ / MeV | COLLISIONS TO 0.025 eV / collisions |
|---|---|
| 0.871875 | 110.004468153047 |
| 1.5609374999999999 | 113.69590242555233 |
| 2.25 | 116.01348323614863 |
| 2.9390624999999995 | 117.70684222819486 |
| 3.628125 | 119.04186138701697 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of e0 separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At e0 = 1.47825 MeV, substitute into Y_out = ln(e0*1000000/0.0253)/(1+(((a-1)/(a+1))^2+0.0000001)*ln(((a-1)/(a+1))^2+0.0000001)/(1-((a-1)/(a+1))^2+0.0000001)): Y₀ = 113.351 collisions.
- At e0 = 1.86412 MeV, the same fixed-parameter model gives Y₁ = 114.821 collisions.
- Signed change: ΔY_out = Y₁ − Y₀ = 1.47007 collisions; the input change is Δe0 = 0.385875 MeV.
- Average sensitivity is ΔY_out/Δe0 = 3.80971 collisions per MeV. 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 = 113.351 collisions, Y₁ = 114.821 collisions; ΔY_out = 1.47007 collisions; average sensitivity = 3.80971 collisions per MeV.
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.
Induced fission and neutron moderation: infer an input with uncertainty
Induced fission is the decay of a compound nucleus formed by neutron capture, and moderation is the elastic slowing-down that delivers the neutron to the energy where that capture is most likely to end in fission. Use the supplied topic model Y_out = ln(e0*1000000/0.0253)/(1+(((a-1)/(a+1))^2+0.0000001)*ln(((a-1)/(a+1))^2+0.0000001)/(1-((a-1)/(a+1))^2+0.0000001)), where Y_out is COLLISIONS TO 0.025 eV (collisions). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; ln(x) means the natural logarithm. e0 is Fission neutron energy E₀, measured in MeV. Keep a (Moderator mass number A) = 12; n (Collisions elapsed n) = 20 fixed. For this exercise use only 0.871875 MeV ≤ e0 ≤ 3.62812 MeV 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 = 116.81473 collisions, with a bounded reading uncertainty ±0.33407 collisions. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown e0 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000275625 MeV, 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.
COLLISIONS TO 0.025 eV against e0
Predictions from the supplied Induced fission and neutron moderation 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): COLLISIONS TO 0.025 eV / collisions
Horizontal axis (x): e0: Fission neutron energy E₀ / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| e0: Fission neutron energy E₀ / MeV | COLLISIONS TO 0.025 eV / collisions |
|---|---|
| 0.871875 | 110.004468153047 |
| 1.5609374999999999 | 113.69590242555233 |
| 2.25 | 116.01348323614863 |
| 2.9390624999999995 | 117.70684222819486 |
| 3.628125 | 119.04186138701697 |
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(e0) = 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(0.871875) = 110.004 collisions and Y_out(3.62812) = 119.042 collisions. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives e0_line = 0.871875 + (116.815 − 110.004)(3.62812 − 0.871875)/(119.042 − 110.004) = 2.94889 MeV. 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 e0 = 2.55319 MeV; substituting back gives Y_out = 116.81473 collisions.
- The interpolation error in the input is |e0_line − e0| = 0.395701 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 116.48066 collisions and 117.1488 collisions. Sort the roots even if the response decreases: 2.4221 MeV ≤ e0 ≤ 2.69137 MeV.
- The lower and upper input deviations are 0.131084 MeV and 0.138178 MeV. 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 e0 ≈ 2.55319 MeV; compatible interval [2.4221, 2.69137] MeV. Endpoint interpolation gives 2.94889 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.