University Physics V · topic UPV-14.04
Mass defect, binding energy, and separation energies
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.
Mass defect and separation energies: predict a finite change
The binding energy of a nucleus is the energy equivalent of its mass defect — how much less the bound nucleus weighs than the free protons and neutrons it contains — and a separation energy is the change in that binding when one nucleon is removed. Use the supplied topic model Y_out = 9-4*s+d, where Y_out is S_n AT EVEN N (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Pairing gap Δ, measured in MeV. Keep s (Smooth fall of S_n per neutron) = 0.5 MeV fixed. For this exercise use only 0.265625 MeV ≤ d ≤ 2.23438 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 d = 0.69875 MeV and 0.974375 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δd across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
S_n AT EVEN N against d
Predictions from the supplied Mass defect and separation energies 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 EVEN N / MeV
Horizontal axis (x): d: Pairing gap Δ / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Pairing gap Δ / MeV | S_n AT EVEN N / MeV |
|---|---|
| 0.265625 | 7.265625 |
| 0.7578125 | 7.7578125 |
| 1.25 | 8.25 |
| 1.7421875 | 8.7421875 |
| 2.234375 | 9.234375 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of d separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At d = 0.69875 MeV, substitute into Y_out = 9-4*s+d: Y₀ = 7.69875 MeV.
- At d = 0.974375 MeV, the same fixed-parameter model gives Y₁ = 7.97438 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.275625 MeV; the input change is Δd = 0.275625 MeV.
- Average sensitivity is ΔY_out/Δd = 1 MeV 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 = 7.69875 MeV, Y₁ = 7.97438 MeV; ΔY_out = 0.275625 MeV; average sensitivity = 1 MeV 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.
Mass defect and separation energies: infer an input with uncertainty
The binding energy of a nucleus is the energy equivalent of its mass defect — how much less the bound nucleus weighs than the free protons and neutrons it contains — and a separation energy is the change in that binding when one nucleon is removed. Use the supplied topic model Y_out = 9-4*s+d, where Y_out is S_n AT EVEN N (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Pairing gap Δ, measured in MeV. Keep s (Smooth fall of S_n per neutron) = 0.5 MeV fixed. For this exercise use only 0.265625 MeV ≤ d ≤ 2.23438 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 = 8.4665625 MeV, with a bounded reading uncertainty ±0.11517 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000196875 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.
S_n AT EVEN N against d
Predictions from the supplied Mass defect and separation energies 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 EVEN N / MeV
Horizontal axis (x): d: Pairing gap Δ / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Pairing gap Δ / MeV | S_n AT EVEN N / MeV |
|---|---|
| 0.265625 | 7.265625 |
| 0.7578125 | 7.7578125 |
| 1.25 | 8.25 |
| 1.7421875 | 8.7421875 |
| 2.234375 | 9.234375 |
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(d) = 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(0.265625) = 7.26563 MeV and Y_out(2.23438) = 9.23438 MeV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives d_line = 0.265625 + (8.46656 − 7.26563)(2.23438 − 0.265625)/(9.23438 − 7.26563) = 1.46656 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 d = 1.46656 MeV; substituting back gives Y_out = 8.4665625 MeV.
- The interpolation error in the input is |d_line − d| = 8.88178e-16 MeV. This is within the requested numerical tolerance.
- Invert both limits Y_out = 8.3513925 MeV and 8.5817325 MeV. Sort the roots even if the response decreases: 1.35139 MeV ≤ d ≤ 1.58173 MeV.
- The lower and upper input deviations are 0.11517 MeV and 0.11517 MeV. 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 d ≈ 1.46656 MeV; compatible interval [1.35139, 1.58173] MeV. Endpoint interpolation gives 1.46656 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.