University Physics V · topic UPV-14.07
Nuclear stability, drip lines, and decay-mode maps
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.
Stability, drip lines and decay maps: predict a finite change
A nuclide is particle-bound when every one- and two-nucleon separation energy computed from the mass table is positive, and beta-stable when no isobaric neighbour lies lower in mass; the drip lines are the loci where those separation energies pass through zero. Use the supplied topic model Y_out = 9 - s*(n-8) + d*cos(pi*n), where Y_out is S_n AT CURSOR (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. s is Trend slope s, measured in MeV/neutron. Keep d (Pairing gap Δ) = 1.4 MeV; n (Cursor neutron number N) = 20 fixed. For this exercise use only 0.685 MeV/neutron ≤ s ≤ 1.315 MeV/neutron 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 s = 0.8236 MeV/neutron and 0.9118 MeV/neutron. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δs across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
S_n AT CURSOR against s
Predictions from the supplied Stability, drip lines and decay maps 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): S_n AT CURSOR / MeV
Horizontal axis (x): s: Trend slope s / MeV/neutron
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Trend slope s / MeV/neutron | S_n AT CURSOR / MeV |
|---|---|
| 0.6849999999999999 | 2.180000000000001 |
| 0.8424999999999999 | 0.2900000000000005 |
| 1 | -1.6 |
| 1.1575 | -3.4900000000000007 |
| 1.315 | -5.379999999999999 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of s separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At s = 0.8236 MeV/neutron, substitute into Y_out = 9 - s*(n-8) + d*cos(pi*n): Y₀ = 0.5168 MeV.
- At s = 0.9118 MeV/neutron, the same fixed-parameter model gives Y₁ = -0.5416 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = -1.0584 MeV; the input change is Δs = 0.0882 MeV/neutron.
- Average sensitivity is ΔY_out/Δs = -12 MeV per MeV/neutron. 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 = 0.5168 MeV, Y₁ = -0.5416 MeV; ΔY_out = -1.0584 MeV; average sensitivity = -12 MeV per MeV/neutron.
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.
Stability, drip lines and decay maps: infer an input with uncertainty
A nuclide is particle-bound when every one- and two-nucleon separation energy computed from the mass table is positive, and beta-stable when no isobaric neighbour lies lower in mass; the drip lines are the loci where those separation energies pass through zero. Use the supplied topic model Y_out = 9 - s*(n-8) + d*cos(pi*n), where Y_out is S_n AT CURSOR (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. s is Trend slope s, measured in MeV/neutron. Keep d (Pairing gap Δ) = 1.4 MeV; n (Cursor neutron number N) = 20 fixed. For this exercise use only 0.685 MeV/neutron ≤ s ≤ 1.315 MeV/neutron 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 = -2.4316 MeV, with a bounded reading uncertainty ±0.44226 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000063 MeV/neutron, 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.
S_n AT CURSOR against s
Predictions from the supplied Stability, drip lines and decay maps 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): S_n AT CURSOR / MeV
Horizontal axis (x): s: Trend slope s / MeV/neutron
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Trend slope s / MeV/neutron | S_n AT CURSOR / MeV |
|---|---|
| 0.6849999999999999 | 2.180000000000001 |
| 0.8424999999999999 | 0.2900000000000005 |
| 1 | -1.6 |
| 1.1575 | -3.4900000000000007 |
| 1.315 | -5.379999999999999 |
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(s) = 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(0.685) = 2.18 MeV and Y_out(1.315) = -5.38 MeV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 0.685 + (-2.4316 − 2.18)(1.315 − 0.685)/(-5.38 − 2.18) = 1.0693 MeV/neutron. 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 s = 1.0693 MeV/neutron; substituting back gives Y_out = -2.4316 MeV.
- The interpolation error in the input is |s_line − s| = 0 MeV/neutron. This is within the requested numerical tolerance.
- Invert both limits Y_out = -2.87386 MeV and -1.98934 MeV. Sort the roots even if the response decreases: 1.03245 MeV/neutron ≤ s ≤ 1.10616 MeV/neutron.
- The lower and upper input deviations are 0.036855 MeV/neutron and 0.036855 MeV/neutron. They agree on this branch; this does not establish linearity for changes in other inputs or outside the interval. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer s ≈ 1.0693 MeV/neutron; compatible interval [1.03245, 1.10616] MeV/neutron. Endpoint interpolation gives 1.0693 MeV/neutron.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.