Topic 15.7: Fission, Fusion, and Nuclear Decay

What Topic 15.7 requires

Topic 15.7 belongs to Unit 15, Modern Physics, which the CED weights at 12-15% of the multiple-choice section and estimates at about 14 to 22 class periods. It is the only topic in Unit 15 with two learning objectives, and one of only two with five suggested skills, so it carries more required content than anything else in the unit.

15.7.A, describe the physical properties that constrain the behavior of interacting nuclei, subatomic particles, and nucleons.

• 15.7.A.1 The strong force is exerted at nuclear scales and dominates the interactions of nucleons, meaning protons or neutrons.
• 15.7.A.2 Possible nuclear reactions are constrained by the law of conservation of nucleon number.
• 15.7.A.3 The behaviour of the constituent particles of a nuclear reaction is constrained by conservation of energy, energy-mass equivalence, and conservation of momentum.
• 15.7.A.4 For all nuclear reactions, mass and energy may be exchanged due to mass-energy equivalence. Relevant equation: $E = mc^2$.
• 15.7.A.5 Energy may be released in nuclear processes in the form of kinetic energy of the products or as photons.
• 15.7.A.6 Nuclear fusion is the process by which two or more smaller nuclei combine to form a larger nucleus, as well as subatomic particles.
• 15.7.A.7 Nuclear fission is the process by which the nucleus of an atom splits into two or more smaller nuclei, as well as subatomic particles.
• 15.7.A.8 Nuclear fission may occur spontaneously or may require an energy input, depending on the binding energy of the nucleus.

15.7.B, describe the radioactive decay of a given sample of material consisting of a finite number of nuclei.

• 15.7.B.1 Radioactive decay is the spontaneous transformation of a nucleus into one or more different nuclei, or to a lower energy level of the same nucleus.
• 15.7.B.1.i The time at which an individual nucleus undergoes radioactive decay is indeterminable, but decay rates can be described using probability.
• 15.7.B.1.ii The half-life $t_{1/2}$ is the time it takes for half of the initial number of radioactive nuclei to have spontaneously decayed.
• 15.7.B.1.iii The decay constant relates to the half-life by $\lambda = \dfrac{\ln 2}{t_{1/2}}$.
• 15.7.B.2 A material's decay constant may be used to predict the number of nuclei remaining after a period of time, or the age of a material if the initial amount is known. Relevant equation: $N = N_0 e^{-\lambda t}$. Derived equation: $\ln\left(\dfrac{N}{N_0}\right) = -\lambda t$.
• 15.7.B.3 Different unstable elements and isotopes may have vastly different half-lives, ranging from fractions of a second to billions of years.

Topic 15.7 carries no boundary statement. Five of the unit's eight topics carry one; 15.1, 15.4 and 15.7 do not. What limits this one instead is the wording of 15.7.B.2, which frames every decay calculation as either "how many are left" or "how old is it", and the neighbouring topic's boundary statement, which says AP Physics 2 does not expect students to memorise the half-lives of specific isotopes.

The five suggested skills are 1.B create quantitative graphs with appropriate scales and units, including plotting data; 2.B calculate or estimate an unknown quantity with units from known quantities; 2.C compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.A create experimental procedures that are appropriate for a given scientific question; and 3.B apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Two of the CED's four Unit 15 sample instructional activities belong to this topic: a desktop experiment in which students roll 200 dice, remove those landing on 1 each turn, graph dice against turns and find a "half-life" of about 3.8 turns; and a subtask exercise estimating how many tons of uranium must be mined to meet the electricity needs of the United States from nuclear power alone. The dice activity is worked through below.

The strong force, and what constrains a nuclear reaction

Essential knowledge 15.7.A.1 gives the strong force one sentence and asks nothing else of it: it acts at nuclear scales and dominates the interactions of nucleons. That is the whole of what AP Physics 2 requires. No range formula, no exchange particles, no coupling strengths. What it explains is why a nucleus holds together when it is a box of protons that repel each other electrically. At nuclear separations the strong force wins.

The rest of 15.7.A is a list of things that must balance. Three constraints, and it is worth noticing what is and is not on the list.

• Nucleon number (15.7.A.2). Protons plus neutrons in, protons plus neutrons out. A reaction that does not balance nucleon number is not a possible reaction, and this is the first check to run on any equation you are handed.
• Energy, including energy-mass equivalence (15.7.A.3, 15.7.A.4). Energy is conserved provided you count mass as a form of it. This is the clause that makes nuclear physics different from everything earlier in the course.
• Momentum (15.7.A.3). The same conservation law as Topic 4.3, applied to fragments flying apart. It is why a stationary nucleus that splits in two sends the pieces in opposite directions, and why the lighter fragment gets the greater speed.

Notice what 15.7.A does not list: charge and lepton number. Those appear in the next topic, at 15.8.A.2.i, which states that in all nuclear decays nucleon number, lepton number and charge are conserved. So the full checking toolkit for a decay equation lives in Topic 15.8, and 15.7 supplies the energy half of the story.

Essential knowledge 15.7.A.5 then says where released energy actually goes: into the kinetic energy of the products, or into photons. Both, in practice. That matters for the framing of a question, because "energy released" in a nuclear reaction is not an abstract quantity, it is the kinetic energy of fragments plus the energy of any gamma rays emitted, and those are things a detector measures.

Mass-energy equivalence, and where nuclear energy comes from

Essential knowledge 15.7.A.4 is the load-bearing statement of the topic: for all nuclear reactions, mass and energy may be exchanged due to mass-energy equivalence, with relevant equation

Read "exchanged" literally. It is not that mass is destroyed and energy appears from nowhere. It is that the same conserved thing has two accounting columns, and a nuclear reaction moves an amount from one to the other. Add up the masses of everything going in, add up the masses of everything coming out, and if the total fell, the difference left as energy.

The scale is what makes this worth a topic. In a chemical reaction the mass change is far too small to weigh. In a nuclear reaction the released energy per event is measured in millions of electron volts, so the mass change is a measurable fraction of the mass involved.

Two lines from the sheet's constants block do all the unit work here, and both must be read off the sheet rather than recalled.

• $1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg} = 931 \, \text{MeV}/c^2$. The unified atomic mass unit, with its energy equivalent printed alongside it.
• $1 \, \text{eV} = 1.60 \times 10^{-19} \, \text{J}$, and $c = 3.00 \times 10^8 \, \text{m/s}$.

The second half of that first line is the shortcut. A mass difference expressed in unified atomic mass units, multiplied by $931 \, \text{MeV}$, is the energy released in MeV. No powers of ten, no squaring the speed of light. It is worth reaching for that route by default.

A rounding warning, because this is exactly where a checked answer looks wrong. The sheet's two expressions for $1 \, \text{u}$ are rounded independently, so they do not agree to three digits. Take the printed $1.66 \times 10^{-27} \, \text{kg}$, multiply by $c^2 = 9.00 \times 10^{16} \, \text{m}^2/\text{s}^2$ and convert with the printed $1 \, \text{eV} = 1.60 \times 10^{-19} \, \text{J}$, and you get $934 \, \text{MeV}$, not the $931 \, \text{MeV}$ printed on the same line. A 0.3% gap. Use the $931 \, \text{MeV}/c^2$ figure when you want an answer in MeV, and expect a SI-route answer to differ in the third digit. That is rounding in the reference table, not an error in your working.

Binding energy: what the CED defines, and what it leaves you to build

This section exists because the CED's treatment of binding energy is easy to get wrong by assuming it says more than it does.

The term appears twice in AP Physics 2, and the two uses are not the same thing. Essential knowledge 15.3.A.5, over in Topic 15.3, gives the only definition the CED prints: "Binding energy is the energy required to remove an electron from an atom, causing the atom to become ionized. An atom in the lowest energy level (ground state) will require the greatest amount of energy to remove the electron from the atom." That is an atomic, electron-level definition. Then 15.7.A.8 uses the term of the nucleus, saying fission may occur spontaneously or may require an energy input depending on the binding energy of the nucleus, and never defines that use.

So the CED does not print a definition of nuclear binding energy. What it does hand you is everything needed to build one, and the structure is identical to the atomic case. Binding energy is the energy required to take the thing apart.

• Atomic binding energy (15.3.A.5): the energy to remove an electron from an atom.
• Nuclear binding energy, the same idea one level down: the energy required to separate a nucleus into its individual nucleons.

And 15.7.A.4 supplies the mechanism that makes it measurable. A bound nucleus has less mass than the sum of the separate nucleons that make it up, because energy was released when they came together. Weigh the difference, multiply by $c^2$, and that is the binding energy. It is the same subtraction used for a reaction, applied to a single nucleus and its parts.

Two consequences carry the reasoning in 15.7.A.8 and in fission and fusion generally.

A more tightly bound nucleus has less mass per nucleon. So a reaction whose products are more tightly bound than its reactants loses mass overall, and releases energy. This is the single sentence that explains why both fusion of light nuclei and fission of heavy nuclei can release energy: both move their material toward the tightly bound middle.

Binding energy per nucleon is the quantity that decides. Total binding energy grows with size simply because there are more nucleons; what tells you whether a reaction releases energy is binding energy per nucleon before and after.

State plainly what you are doing when you write this in an answer. The CED supplies $E = mc^2$, the mass-energy exchange statement, and the $931 \, \text{MeV}/c^2$ conversion. The nuclear binding-energy definition above is built from those pieces rather than quoted from the framework, and no binding-energy curve or table of nuclear masses appears anywhere in the AP Physics 2 reference material. If a question needs a nuclear mass, it will give it to you.

Fission and fusion, side by side

The CED's two definitions are deliberately parallel, and reading them together is the fastest way to keep them straight.

• 15.7.A.6, fusion: two or more smaller nuclei combine to form a larger nucleus, as well as subatomic particles.
• 15.7.A.7, fission: the nucleus of an atom splits into two or more smaller nuclei, as well as subatomic particles.

Both definitions end the same way. Products are not only nuclei: subatomic particles come out too, and 15.7.A.5 adds that energy may leave as kinetic energy of the products or as photons. So a complete reaction equation has more on the right-hand side than two nuclei, and a question that asks you to balance one will usually be counting those extra particles.

That last row is worth dwelling on, because 15.7.A.8 is the only essential knowledge statement in this topic about whether a process starts at all. It says fission may occur spontaneously or may require an energy input, depending on the binding energy of the nucleus. Two things follow.

Not all fission is spontaneous. A nucleus can be energetically able to split and still sit there indefinitely until something disturbs it, which is why induced fission is a thing you can switch on.

"Releases energy" and "happens by itself" are different questions. Skill 3.B questions in this topic often turn on exactly that distinction, and answering the first when you were asked the second is a way to lose a mark while writing true physics.

For fusion the CED says nothing about spontaneity, so do not assert anything about it as required content.

Radioactive decay is probability, not a schedule

Essential knowledge 15.7.B.1 defines radioactive decay as the spontaneous transformation of a nucleus into one or more different nuclei, or to a lower energy level of the same nucleus. That second clause is easy to skip and it is what gamma decay is, covered in Topic 15.8: the nucleus does not become a different nucleus, it just drops in energy.

Then 15.7.B.1.i makes the statement that the whole of 15.7.B rests on: the time at which an individual nucleus decays is indeterminable, but decay rates can be described using probability.

Take that seriously and two conclusions follow that students routinely get wrong.

A nucleus does not age. There is no countdown inside it, no accumulated wear, no "due" nucleus. A nucleus that has survived ten half-lives has exactly the same chance of decaying in the next second as a freshly made one.

Half-life is a property of the sample's statistics, not a lifetime. Essential knowledge 15.7.B.1.ii defines $t_{1/2}$ as the time for half of the initial number of radioactive nuclei to have spontaneously decayed. Nothing in that sentence says any particular nucleus lasts that long.

The exponential is what a constant per-nucleus probability looks like when you have many nuclei. If every surviving nucleus has the same chance of going in the next interval, the number decaying is proportional to the number left, and the population falls by the same factor in each equal interval rather than by the same amount. Halving after $t_{1/2}$, halving again after another $t_{1/2}$, and so on forever, never reaching zero in the model.

The CED's own desktop activity for this topic makes the point physically. Roll 200 dice, remove every die showing a 1, record how many are left, repeat. No die knows how many turns have passed and no die is ever "due", yet the group as a whole halves on a predictable schedule, which the CED gives as about 3.8 turns. The third worked example below derives that 3.8 from $\lambda = \ln 2 / t_{1/2}$, and the agreement is exact rather than approximate.

Essential knowledge 15.7.B.3 closes the objective by noting that half-lives range from fractions of a second to billions of years. That range is why the graph section that follows insists on the logarithmic form: a quantity spanning twenty orders of magnitude in time is not something you read off a linear axis.

The two decay equations, and the graph that linearises them

Two equations, both printed in the Modern Physics group of the AP Physics 2 sheet, and one that is not.

On the sheet:

Not on the sheet: $\ln\left(\dfrac{N}{N_0}\right) = -\lambda t$. The CED labels this one a derived equation in 15.7.B.2, in contrast with the "relevant equation" label it puts on $N = N_0 e^{-\lambda t}$ immediately above it. Counting the Modern Physics group confirms it: ten entries, and the logarithmic form is not among them. Deriving it is one line, taking the natural log of both sides of the exponential, and doing that line explicitly is what suggested skill 2.B is paying for.

Watch the symbol $\lambda$. On this sheet it means two different things in two different groups, and the sheet's own symbol table for the Modern Physics column says so, defining $\lambda$ as "wavelength or decay constant". In $N = N_0 e^{-\lambda t}$ it is a decay constant with units of inverse time. In $\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$ two entries earlier it is a wavelength in metres. Nothing distinguishes them except context.

Reading $\lambda = \ln 2 / t_{1/2}$. The decay constant is the probability per unit time that a given nucleus decays, so a large $\lambda$ means a short half-life. The two are inversely proportional, with $\ln 2 = 0.693$ as the constant of proportionality, and that number appears because "half" is what the definition in 15.7.B.1.ii picks out. Nothing about one half is special to the physics; it is a convention that makes the number easy to talk about.

The graph, which is where skill 1.B lives. $N$ against $t$ is an exponential curve, and reading a rate constant off a curve by eye is hopeless. The derived equation fixes that.

Plot $\ln(N/N_0)$ on the vertical axis against $t$ on the horizontal axis and you get a straight line with

• slope $= -\lambda$, negative because the population falls, and
• vertical intercept $= 0$, because at $t = 0$, $N = N_0$ and $\ln 1 = 0$.

The line passing through the origin is a genuine check on the data rather than a fitted parameter. If your best-fit line misses the origin, either $N_0$ is wrong or the counting is.

A variant worth recognising: plotting $\ln N$ rather than $\ln(N/N_0)$ gives the same slope $-\lambda$ with a vertical intercept of $\ln N_0$, which is useful when the initial count is what you are trying to find. Once you have $\lambda$ from the slope, the half-life is $\ln 2 / \lambda$, and skill 3.A questions about designing the measurement come down to saying you would count decays at several times, plot the logarithm, and take the slope.

Traps, and how 15.7 gets tested

Pair each of the five suggested skills with what it catches.

1.B, graph. Plotting $N$ against $t$ and trying to read $\lambda$ off the curve. Take logs first.

2.B, calculate. Unit collisions. A decay constant in inverse days used with a time in hours, or a mass difference in kilograms multiplied by $931 \, \text{MeV}$. Fix the units before substituting.

2.C, compare. Two samples, two half-lives, and the question is which has more nuclei left after a fixed time. Compare $t/t_{1/2}$, the number of halvings, not the raw times.

3.A, design. An experiment question wants the plot named, not just the measurement. Say you would count remaining nuclei at several times and plot the logarithm against time.

3.B, claim. Confusing "releases energy" with "happens spontaneously", which 15.7.A.8 explicitly separates.

Six errors worth naming.

• Treating half-life as a lifetime. After two half-lives a quarter remains, not none. After ten half-lives about a thousandth remains. The model never reaches zero.
• Subtracting instead of dividing. Three half-lives leaves $(1/2)^3 = 1/8$, not $1 - 3(1/2)$.
• Using $\lambda = 1/t_{1/2}$. The factor of $\ln 2$ is on the sheet. Leaving it out makes every answer about 44% wrong.
• Believing a nucleus can be overdue. 15.7.B.1.i rules it out. Past survival changes nothing.
• Reaching for a half-life from memory. The next topic's boundary statement says AP Physics 2 does not expect students to know the half-lives of specific isotopes, so a question that needs one will supply it.
• Reporting an MeV answer to four figures. The sheet's own $1 \, \text{u}$ conversions disagree in the third digit, so three significant figures is the honest ceiling.

On exam shape: the AP Physics 2 exam runs 3 hours, with 42 multiple-choice questions and 4 free-response questions, split 50/50 by weight. The Unit 15 overview notes that the first free-response question is the Mathematical Routines question, which asks students to calculate or derive an expression, use a representation, and make and justify claims, and it says that question can pull content from any of the seven units. A decay-curve question with a linearisation and a claim attached is that shape exactly.

For what each decay type does to a nucleus, and the conservation checks that go with it, see Topic 15.8. For nuclear structure and notation, see Topic 15.2. For the conservation-of-energy habits this topic extends, see Topic 3.4 and the conservation of energy guide. The Unit 15 overview puts the eight topics in order, and the AP Physics 2 equation sheet is where the ten Modern Physics entries live.