A bound nucleus has less mass than its separated constituent nucleons. That difference records an energy change, not missing bookkeeping. Reaction calculations work when every mass refers to a consistent set of particles.
We develop size scaling, binding and separation energies, then apply complete charge, mass and momentum bookkeeping to reaction and decay models.
What you will learn to do
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy Learning goals: - Read nuclide labels and balance charge and nucleon number. - Use the approximate nuclear-radius law and explain its density implication. - Calculate binding with consistent atomic or nuclear masses. - Distinguish average binding from separation energy. - Evaluate Q and interpret its sign without inferring a reaction rate. - Partition two-body decay energy using momentum conservation. - Track alpha, beta and gamma changes including atomic beta-mass conventions. - Explain why a fixed-target endothermic threshold exceeds the energy deficit. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Why must a nuclear equation conserve charge as well as nucleon count in the usual nuclear reactions studied here?
Compare your reasoning
Matching only mass numbers can conceal the wrong elements or missing charged particles. Track both atomic number and mass number, while also accounting for emitted particles and energy.
Revisit the explanation →01 / Nuclear size, binding and reaction energy
Nuclear bookkeeping starts with charge and nucleon number
A nuclide is labelled by mass number A and proton number Z. It contains Z protons and N = A − Z neutrons. The mass number counts nucleons; it is not the same as the measured atomic mass in atomic mass units.
Isotopes share Z but differ in neutron number. Isobars share A but have different Z, while isotones share neutron number. These labels describe different comparisons and should not be substituted for one another.
In an ordinary nuclear reaction, balance total electric charge and nucleon number across the equation. This identifies missing nuclear species, but an equation that balances these counts is not automatically energetically allowed or dynamically likely.
Electrons, positrons, photons and neutrinos have zero nucleon number. Their charge, energy, momentum and other conserved properties still matter. Omitting them can preserve A while violating another conservation law.
Nucleon-number bookkeeping here concerns ordinary low-energy nuclear transformations, with all emitted particles included as appropriate.
WORKED EXAMPLE
Find neutron number without treating atomic mass as a count
A nuclide has A = 37 and Z = 17. How many protons and neutrons does it contain? What changes if one neutron is added without changing Z?
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Topic: Nuclear size, binding and reaction energy
Focus: Nuclear bookkeeping starts with charge and nucleon number
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
N=A-Z,\qquad\sum A_{\mathrm{initial}}=\sum A_{\mathrm{final}},\qquad\sum q_{\mathrm{initial}}=\sum q_{\mathrm{final}}
Meaning: Nucleon-number bookkeeping here concerns ordinary low-energy nuclear transformations, with all emitted particles included as appropriate.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Nuclear size, binding and reaction energy
Nuclear radius grows approximately with the cube root of nucleon count
For many nuclei, an effective size is approximated by R = r0A^(1/3), with r0 around 1.2 fm and 1 fm = 10⁻¹⁵ m. The radius is a model of a diffuse nuclear distribution, not a perfectly sharp rigid surface.
Cubing the relation gives a volume proportional to A. If nuclear mass is approximately A times a nucleon mass, the average density is therefore approximately independent of A. Larger nuclei contain more nucleons in proportionally larger volumes.
This is an empirical bulk scaling, with deviations for light nuclei, deformed nuclei and unusual distributions such as halo nuclei. It should not be used to claim that every isotope is an exact uniform sphere.
Atomic size is much larger, commonly on the scale of tenths of a nanometre, and is controlled by electron states. The A^(1/3) nuclear law does not describe the radius of the whole atom.
Approximate spherical bulk-density model; r0 ≈ 1.2 fm and mu ≈ 1.66 × 10⁻²⁷ kg for estimates.
WORKED EXAMPLE
Compare radii and volumes using different powers
Use r0 = 1.2 fm to estimate radii for A = 64 and A = 216. Compare their radius and volume ratios.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy
Focus: Nuclear radius grows approximately with the cube root of nucleon count
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
R\approx r_0A^{1/3},\qquad\rho\approx\frac{Am_u}{(4\pi/3)r_0^3A}=\frac{3m_u}{4\pi r_0^3}
Meaning: Approximate spherical bulk-density model; r0 ≈ 1.2 fm and mu ≈ 1.66 × 10⁻²⁷ kg for estimates.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Nuclear size, binding and reaction energy
A size estimate can be checked by reconstructing density
Insert the approximate nucleon mass and radius coefficient into the bulk-density expression. The result is of order 10¹⁷ kg/m³, reflecting how much mass is concentrated in a very small nuclear volume.
The A factors cancel because both mass and volume scale with nucleon count. Forgetting to cube the radius or mixing femtometres with metres can change the estimate by many orders of magnitude.
A radius inferred from scattering depends on the probed distribution and convention. Charge radius, matter radius and a sharp-sphere equivalent radius need not be numerically identical. The simple coefficient r0 packages that detail into an exam-level approximation.
This density is a property of the nuclear model, not the mass density of ordinary bulk matter, whose atoms are spaced much farther apart than their nuclear radii.
WORKED EXAMPLE
Check a nuclear-density order of magnitude
Estimate the uniform-sphere nuclear density using mu = 1.66 × 10⁻²⁷ kg and r0 = 1.2 × 10⁻¹⁵ m.
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Topic: Nuclear size, binding and reaction energy Focus: A size estimate can be checked by reconstructing density Learning goals: - Read nuclide labels and balance charge and nucleon number. - Use the approximate nuclear-radius law and explain its density implication. - Calculate binding with consistent atomic or nuclear masses. - Distinguish average binding from separation energy. - Evaluate Q and interpret its sign without inferring a reaction rate. - Partition two-body decay energy using momentum conservation. - Track alpha, beta and gamma changes including atomic beta-mass conventions. - Explain why a fixed-target endothermic threshold exceeds the energy deficit. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Nuclear size, binding and reaction energy
Binding energy measures a mass-energy deficit relative to free nucleons
A bound nucleus has less rest mass than its separated constituent protons and neutrons. The difference Δm = Zmp + Nmn − Mnucleus corresponds to binding energy B = Δmc², the energy required to separate it into free nucleons in the chosen reference state.
A positive binding energy means the bound composite lies lower in energy than the separated constituents. It is not an extra positive energy sitting inside the nucleus waiting to be released without changing the state.
To calculate with masses in atomic mass units, use 1 u c² ≈ 931.5 MeV. Subtract the masses first with sufficient precision, then convert the relatively small mass defect to energy.
Mass number A is not a replacement for an accurately supplied nuclear or atomic mass. Using A u for the measured mass would erase the very mass defect the calculation is meant to determine.
This expression uses a bare nuclear mass and free proton/neutron masses. Keep the mass convention consistent.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy
Focus: Binding energy measures a mass-energy deficit relative to free nucleons
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
B=[Zm_p+(A-Z)m_n-M_{\mathrm{nucleus}}]c^2
Meaning: This expression uses a bare nuclear mass and free proton/neutron masses. Keep the mass convention consistent.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.05 / Nuclear size, binding and reaction energy
Neutral atomic masses can be used when electron bookkeeping is consistent
Mass tables often provide neutral-atom masses rather than bare nuclear masses. To use those directly for nuclear binding estimates, replace each free proton by a neutral hydrogen atom and compare with the neutral atom: B ≈ [ZmH + Nmn − Matom]c².
The electron rest masses then cancel because the Z hydrogen atoms and the neutral atom contain the same number of electrons. Differences in electronic binding energies are neglected in this common nuclear-energy approximation; they matter if the required precision reaches that scale.
Do not combine an atomic mass with Z bare proton masses and call the difference the nuclear mass defect. That leaves an unmatched electron-mass contribution. Conversely, if a bare nuclear mass is supplied, the direct proton formula is appropriate.
A mass-excess table can also be used, but its definition and units must be checked. If total A is conserved, the integer A u contributions cancel in a reaction mass difference, leaving differences of the mass excesses.
Neutral atomic masses with electronic binding differences neglected. mH includes one electron; mp does not.
WORKED EXAMPLE
Calculate a mass defect with neutral atomic masses
Estimate the binding energy and binding energy per nucleon of carbon-12 using neutral atomic masses mH = 1.007825 u, mn = 1.008665 u and M(¹²C) = 12.000000 u. Use 1 u c² = 931.5 MeV and neglect electronic binding corrections.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy
Focus: Neutral atomic masses can be used when electron bookkeeping is consistent
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
B\approx[Zm_H+(A-Z)m_n-M_{\mathrm{atom}}]c^2
Meaning: Neutral atomic masses with electronic binding differences neglected. mH includes one electron; mp does not.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Nuclear size, binding and reaction energy
Average binding and the energy to remove one particle are different
Binding energy per nucleon B/A is an average measure of how strongly the nucleus is bound relative to free nucleons. It rises rapidly among light nuclei, reaches a broad maximum around the iron–nickel region and then decreases more slowly for heavy nuclei.
This trend helps explain why some combinations of light nuclei and some splits of heavy nuclei can lower total rest energy. It does not establish the rate or accessibility of a particular reaction; charge barriers, nuclear structure and conservation laws also matter.
The energy required to remove one neutron is a separation energy, Sn = [M(A − 1,Z) + mn − M(A,Z)]c². In terms of total bindings, Sn = B(A,Z) − B(A − 1,Z), not generally B/A.
The average and incremental energies differ because the remaining nucleus reorganizes and has its own binding. The same warning applies to proton or alpha separation energies, with the appropriate masses included.
WORKED EXAMPLE
Do not replace a separation energy by an average
In a supplied hypothetical mass model, a nucleus with A = 20 has total binding 160 MeV, while the nucleus left after removing one neutron has total binding 151 MeV. Find neutron separation energy and compare it with the original average binding.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy Focus: Average binding and the energy to remove one particle are different Learning goals: - Read nuclide labels and balance charge and nucleon number. - Use the approximate nuclear-radius law and explain its density implication. - Calculate binding with consistent atomic or nuclear masses. - Distinguish average binding from separation energy. - Evaluate Q and interpret its sign without inferring a reaction rate. - Partition two-body decay energy using momentum conservation. - Track alpha, beta and gamma changes including atomic beta-mass conventions. - Explain why a fixed-target endothermic threshold exceeds the energy deficit. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Nuclear size, binding and reaction energy
A reaction’s Q value is a rest-mass difference
For a fully specified reaction, Q = (sum of initial rest masses − sum of final rest masses)c². Positive Q means rest energy is released into other forms; negative Q means the products require additional energy.
With all emitted particles and photons accounted for, conservation of total energy gives final kinetic plus emitted radiation energy minus initial kinetic energy equal to Q. If an excited daughter is produced, use its mass or reserve its excitation energy in the ledger.
When nucleon species are conserved through a rearrangement and no additional rest-mass particles are created, Q can be expressed as final total nuclear binding minus initial total binding. Greater final binding corresponds to lower final mass.
A positive Q is an energetic condition, not a guarantee of an immediate reaction. A process can face a barrier or be suppressed by its dynamics. Conversely, a negative-Q reaction can occur if sufficient initial kinetic energy is supplied.
All relevant particles must be included. If radiation is included in the final energy explicitly, do not also count it as a separate final rest mass.
WORKED EXAMPLE
Use an electron-balanced atomic-mass reaction
For ²H + ³H → ⁴He + n, use neutral atomic masses 2.014102 u, 3.016049 u, 4.002603 u and neutron mass 1.008665 u. Estimate Q with 931.5 MeV/u, neglecting electronic binding differences.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy
Focus: A reaction’s Q value is a rest-mass difference
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
Q=\left(\sum m_i-\sum m_f\right)c^2,\qquad K_f+E_{\gamma,\mathrm{out}}-K_i=Q
Meaning: All relevant particles must be included. If radiation is included in the final energy explicitly, do not also count it as a separate final rest mass.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.08 / Nuclear size, binding and reaction energy
Momentum conservation decides how released energy is shared
Suppose a parent at rest decays into two nonrelativistic massive products with no other emitted particle. Their momenta are equal in magnitude and opposite in direction, because initial total momentum is zero.
With the same momentum magnitude p, each kinetic energy is p²/(2m). The lighter product therefore receives more kinetic energy. If their masses are m and M and the available kinetic release is Q, the light product receives QM/(m + M) and the heavy product Qm/(m + M).
Giving all Q to the light product is only an approximation when the other product is much heavier. Recoil is required even if it is small. A photon emission also gives the residual nucleus recoil, so the photon energy is slightly below the full internal transition energy.
For beta decay there are more than two final particles, including a neutrino or antineutrino. Energy sharing then allows a continuous electron or positron spectrum rather than the unique two-body kinetic partition.
No additional emitted particles; Q is the available kinetic release for the specified daughter states.
WORKED EXAMPLE
Share a hypothetical two-body decay energy
A parent at rest releases 6.0 MeV into two nonrelativistic products whose masses are in the ratio 1:5. Find their kinetic energies.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy
Focus: Momentum conservation decides how released energy is shared
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
K_m=Q\frac{M}{m+M},\qquad K_M=Q\frac{m}{m+M}\quad\text{(two-body nonrelativistic decay at rest)}
Meaning: No additional emitted particles; Q is the available kinetic release for the specified daughter states.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.09 / Nuclear size, binding and reaction energy
Alpha, beta and gamma changes obey different bookkeeping
Alpha emission removes two protons and two neutrons, so the daughter has A − 4 and Z − 2. Beta-minus decay converts a neutron into a proton while emitting an electron and an electron antineutrino, leaving A unchanged and increasing Z by one.
Beta-plus decay converts a proton into a neutron while emitting a positron and an electron neutrino, leaving A unchanged and decreasing Z by one. Electron capture also decreases Z by one by absorbing an atomic electron. Gamma emission lowers an excited nuclear state without changing A or Z.
With neutral atomic masses and small electronic-binding corrections neglected, beta-minus Q is approximately [Mparent − Mdaughter]c². Beta-plus Q is approximately [Mparent − Mdaughter − 2me]c². The extra two-electron mass term in the atomic-mass beta-plus formula is a frequent bookkeeping trap.
The neutrino is necessary for the full conservation account and energy distribution in beta decay. A nuclear electron is created in the decay process; the nucleus is not modelled as storing ordinary bound electrons among its protons and neutrons.
These are neutral-atomic-mass formulas with electronic binding and tiny neutrino-mass corrections neglected. Bare nuclear masses require explicitly including the emitted electron or positron mass instead.
WORKED EXAMPLE
Check whether atomic mass difference is enough for positron emission
A hypothetical neutral-parent minus neutral-daughter mass difference corresponds to 0.80 MeV. Can ground-state beta-plus emission occur under the simple atomic-mass approximation? Use mec² = 0.511 MeV.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy
Focus: Alpha, beta and gamma changes obey different bookkeeping
Learning goals:
- Read nuclide labels and balance charge and nucleon number.
- Use the approximate nuclear-radius law and explain its density implication.
- Calculate binding with consistent atomic or nuclear masses.
- Distinguish average binding from separation energy.
- Evaluate Q and interpret its sign without inferring a reaction rate.
- Partition two-body decay energy using momentum conservation.
- Track alpha, beta and gamma changes including atomic beta-mass conventions.
- Explain why a fixed-target endothermic threshold exceeds the energy deficit.
Equations in this section (LaTeX):
Q_{\beta^-}\approx(M_P-M_D)c^2,\qquad Q_{\beta^+}\approx(M_P-M_D-2m_e)c^2
Meaning: These are neutral-atomic-mass formulas with electronic binding and tiny neutrino-mass corrections neglected. Bare nuclear masses require explicitly including the emitted electron or positron mass instead.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.10 / Nuclear size, binding and reaction energy
Both splitting and joining require the same complete energy ledger
Fission splits a heavy nucleus into lighter products; fusion joins light nuclei into a more tightly bound configuration, sometimes with other emitted particles. Neither word determines Q by itself. Write the whole reaction and compare complete initial and final rest energies, including incident and emitted neutrons or other particles.
For a reaction that preserves the total numbers of protons and neutrons and uses consistent nuclear masses, the free-nucleon rest energies cancel. Its Q then equals final total binding minus initial total binding. Count total binding, not an unweighted average of binding energy per nucleon.
This explains how both processes can release energy in suitable cases. Some heavy nuclei can form fragments with larger total binding; some light nuclei can combine into products with larger total binding. A more tightly bound final configuration has lower rest energy, and the difference can appear as kinetic energy and radiation.
A positive Q does not remove a reaction barrier or guarantee that a reaction happens at an appreciable rate. Charged reactants must approach despite Coulomb repulsion, and nuclear structure affects available channels. Energy balance answers whether the stated endpoints release energy; it does not by itself provide the reaction dynamics.
WORKED EXAMPLE
Count bound and free final nucleons separately
A model nucleus has A = 240 and binding energy per nucleon 7.6 MeV. A model fission channel produces two A = 119 fragments, each with binding energy per nucleon 8.4 MeV, plus two free neutrons; total proton count is conserved. Find Q.
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WORKED EXAMPLE
Subtract the binding already in the fuel
For D + T → helium-4 + neutron, use nuclear binding energies 2.2 MeV, 8.5 MeV and 28.3 MeV for D, T and helium-4 respectively. Find Q with these rounded inputs.
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STOP & THINK
A proposed fusion channel has positive Q. Does that alone establish a high reaction rate at room temperature?
- Yes
- No
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy Focus: Both splitting and joining require the same complete energy ledger Learning goals: - Read nuclide labels and balance charge and nucleon number. - Use the approximate nuclear-radius law and explain its density implication. - Calculate binding with consistent atomic or nuclear masses. - Distinguish average binding from separation energy. - Evaluate Q and interpret its sign without inferring a reaction rate. - Partition two-body decay energy using momentum conservation. - Track alpha, beta and gamma changes including atomic beta-mass conventions. - Explain why a fixed-target endothermic threshold exceeds the energy deficit. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
11 / Nuclear size, binding and reaction energy
An endothermic fixed-target reaction needs energy for recoil as well
For an endothermic reaction, merely supplying −Q does not generally suffice when the target is initially at rest. The incoming projectile carries momentum, and the final system must retain that total momentum. Some kinetic energy is therefore unavoidable even at threshold.
For a nonrelativistic projectile of mass ma hitting a stationary target of mass mA, with a small mass-energy change compared with rest energies, the threshold is approximately Kthreshold ≈ −Q(1 + ma/mA). This is a stated approximation, not an exact formula for arbitrary relativistic reactions.
The extra factor approaches one for a very heavy target because it can carry the required momentum with relatively little kinetic energy. For comparable projectile and target masses, the recoil penalty is more important.
When high precision or relativistic energies matter, use total energy–momentum conservation with the exact initial and final masses. The threshold also presumes a specified final channel and no additional excitation beyond what was included in Q.
WORKED EXAMPLE
Estimate the recoil penalty in a fixed-target reaction
A hypothetical endothermic channel has Q = −2.0 MeV. A projectile of mass m strikes a stationary target of mass 4m, and the nonrelativistic small-mass-change approximation is valid. Estimate the projectile threshold energy.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Nuclear size, binding and reaction energy Focus: An endothermic fixed-target reaction needs energy for recoil as well Learning goals: - Read nuclide labels and balance charge and nucleon number. - Use the approximate nuclear-radius law and explain its density implication. - Calculate binding with consistent atomic or nuclear masses. - Distinguish average binding from separation energy. - Evaluate Q and interpret its sign without inferring a reaction rate. - Partition two-body decay energy using momentum conservation. - Track alpha, beta and gamma changes including atomic beta-mass conventions. - Explain why a fixed-target endothermic threshold exceeds the energy deficit. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Mass number counts nucleons; measured mass includes binding effects.
- The approximate R ∝ A^(1/3) law gives nearly constant bulk nuclear density.
- Binding energy is a deficit relative to separated free constituents.
- Neutral-atomic mass calculations need electron counts that cancel consistently.
- A one-particle separation energy is a difference of total bindings, not simply B/A.
- Positive Q releases rest energy but does not guarantee a rapid reaction.
- Recoil shares energy even when one product is much heavier.
- Beta-plus atomic-mass Q includes a two-electron rest-mass subtraction.
- Endothermic fixed-target thresholds must satisfy momentum as well as energy conservation.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A nuclide has A = 56 and Z = 26. How many neutrons does it contain?
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SELF CHECK 2
Under R = r0A^(1/3), by what factor does radius change if A becomes eight times larger?
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SELF CHECK 3
A nuclear mass defect is 0.020 u. Using 931.5 MeV per u, what binding energy does it represent?
Answer in MeV.
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SELF CHECK 4
A nucleus has total binding 120 MeV and its one-neutron-removed daughter has binding 112 MeV. What is the neutron separation energy?
Answer in MeV.
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SELF CHECK 5
Two products of a decay at rest have mass ratio 1:3 and share Q = 8 MeV nonrelativistically. What energy goes to the lighter product?
Answer in MeV.
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SELF CHECK 6
Which combination is consistent for an approximate binding calculation using a neutral atomic mass?
- Z bare proton masses plus neutrons minus neutral atomic mass
- Z neutral hydrogen masses plus neutrons minus neutral atomic mass
- Mass number A alone as the exact atom mass
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SELF CHECK 7
In beta-minus decay, the daughter’s A and Z change how?
- A unchanged, Z increases by one
- A decreases by four, Z decreases by two
- A increases by one, Z unchanged
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SELF CHECK 8
A channel has positive Q. Which conclusion follows directly?
- It must occur immediately
- Its final rest mass is lower, leaving energy available to other forms
- Every product receives all of Q
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Compare complete rest masses in a fission channelIntermediate · Hints and guided reasoningAccount for the binding already present in the fuelIntermediate · Hints and guided reasoningCompare complete mass totalsIntermediate · Hints and guided reasoningMass missing from a bound nucleusIntermediate · Hints and guided reasoningSize grows more slowly than nucleon countIntermediate · Hints and guided reasoningCompare binding per constituentIntermediate · Hints and guided reasoningA radius law used for an areaIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Why can mixing tabulated atomic masses with bare proton masses create an error in a binding-energy calculation?
Compare your reasoning
Atomic masses include electron contributions, while bare nuclear particles do not. Use a consistent convention or explicitly account for electron masses and the relevant binding corrections.
Revisit the explanation →A few days later: change the situation
Why is final total binding not itself the released energy in fusion?
Compare your reasoning
The initial nuclei are already bound. With the same nucleons and consistent nuclear masses, Q is final binding minus initial binding. A positive Q establishes an energy release, not a fast reaction rate.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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