IB DP Physics · Theme E · Nuclear and quantum physics

Fission.

E.4 · Splitting heavy nuclei: the energy released, chain reactions, reactors and waste
Standard level and higher level

SL + HL · 4 hours E.3 Radioactive decay ↗

The journey

Three parts.

Part 01

Fission and its energy

What happens, why energy is released, and how much

Part 02

Chain reactions and reactors

Moderator, control rods, heat exchanger and shielding

Part 03

The products of fission

What is left behind, and how it is managed

E.3 gave the tools: mass defect, binding energy per nucleon and \(E = mc^{2}\).

E.4 FISSION · SL + HL

Part 01

Fission and its energy.

A heavy nucleus splits in two, and a little mass disappears

E.4 FISSION · SL + HL

Splitting a heavy nucleus

What fission is.

NEUTRON-INDUCED FISSIONPRESS → TO PLAY

A heavy nucleus splits into two lighter nuclei and a few neutrons
Neutron-induced: it first absorbs a neutron
Spontaneous: it splits by itself, a rare kind of radioactive decay
Either way, energy is released

E.4 FISSION · SL + HL

One reaction of many

The slide shows \({}^{1}_{0}\mathrm{n} + {}^{235}_{92}\mathrm{U} \rightarrow {}^{141}_{56}\mathrm{Ba} + {}^{92}_{36}\mathrm{Kr} + 3\,{}^{1}_{0}\mathrm{n}\). Check it as in E.3: nucleon numbers, \(1 + 235 = 141 + 92 + 3\); proton numbers, \(0 + 92 = 56 + 36 + 0\). A uranium nucleus can split in dozens of different ways, into different pairs of fragments with two or three neutrons. The two fragments are usually unequal in size.

Why a neutron works so well

A neutron has no charge, so the nucleus does not repel it. Even a very slow one can drift in. When it is absorbed, the nucleus gains its binding energy and is left wobbling violently, like a drop of liquid that has been struck. It stretches, narrows in the middle and, pushed apart by the repulsion between its two positively charged halves, breaks in two.

Spontaneous fission needs no neutron. It happens in the heaviest nuclei, but for uranium it is extremely rare compared with alpha decay.

In any fission equation, find a missing number by balancing nucleon number and proton number, exactly as for a decay equation.

The story · Berlin and Sweden, December 1938

Background story · not assessed

The nucleus that split.

Black and white photograph of Lise Meitner and Otto Hahn standing at a laboratory bench in front of tall windows — **Lise Meitner** and **Otto Hahn** in their laboratory, 1913

In Berlin, Otto Hahn and Fritz Strassmann fired neutrons at uranium and found barium: an element half its size
Their colleague of thirty years, Lise Meitner, had fled Germany that July
At Christmas in Sweden, she and her nephew Otto Frisch worked it out: the nucleus had split, releasing about 200 MeV
Frisch named it fission, after the division of a living cell

The Nobel Prize for the discovery went to Hahn alone.

Photograph: unknown photographer, 1913, public domain.

E.4 FISSION · SL + HL

Binding energy per nucleon

Why energy is released.

Uranium: about 7.6 MeV per nucleon
The fragments: about 8.4 MeV per nucleon
More tightly bound means less mass

The difference in binding energy is released.

MOVING UP THE CURVE

E.4 FISSION · SL + HL

Reading the curve

Binding energy per nucleon measures how tightly a nucleus is held together: the higher on the graph, the more energy each nucleon gave up when the nucleus was formed, and the more stable it is. The peak is near iron and nickel. Uranium sits on the downward slope to the right. Split it into two middle-sized nuclei and each of its 235 nucleons ends up about 0.8 MeV more tightly bound. That is roughly \(235 \times 0.8 \approx 190\) MeV released in all.

Where the slope comes from

Every proton in a nucleus repels every other, over the whole width of the nucleus, while the strong nuclear force only acts between neighbors. In a very large nucleus the repulsion adds up and the nucleons are less tightly bound on average. Two smaller nuclei have less of that repulsion to fight.

The other side of the peak

Very light nuclei can also move up the curve, by joining together. That is fusion, in E.5.

Saying that the fragments have less binding energy. They have more. Binding energy is energy already given out: more of it means a lower, more stable state.

Worked example · first method

Energy released, from binding energies.

Binding energy per nucleon / MeV \({}^{235}\mathrm{U}\): 7.59 · \({}^{141}\mathrm{Ba}\): 8.33 · \({}^{92}\mathrm{Kr}\): 8.51 Before: \(235 \times 7.59 = 1784 \,\mathrm{MeV}\) After: \(141 \times 8.33 + 92 \times 8.51\) \(= 1175 + 783 = 1958 \,\mathrm{MeV}\)

Energy released = binding energy after − before \(= 1958 - 1784\) \(= \class{res}{174 \,\mathrm{MeV}}\)

Free neutrons have no binding energy: leave them out.

E.4 FISSION · SL + HL

Total, not per nucleon

The numbers in the table are per nucleon. Multiply each by its nucleon number to get the total binding energy of that nucleus before adding or subtracting anything. Subtracting the per-nucleon values directly, \(8.4 - 7.6\), gives the gain for one nucleon, not for the reaction.

After minus before

The products are more tightly bound, so their total binding energy is larger. The increase is the energy that has been given out.

How big is 174 MeV?

Burning one atom of carbon releases about 4 eV. One fission releases over forty million times as much. That ratio is why a nuclear power station uses a few tonnes of fuel a year where a coal station burns millions of tonnes.

The incoming neutron and the three released are free particles with zero binding energy. They appear in the mass calculation on the next slide, but not in this one.

Worked example · second method

Energy released, from masses.

Masses / u \({}^{235}\mathrm{U}\) 235.043930 · n 1.008665 \({}^{141}\mathrm{Ba}\) 140.914411 · \({}^{92}\mathrm{Kr}\) 91.926156 Before: \(235.043930 + 1.008665 = 236.052595\) After: \(140.914411 + 91.926156 + 3 \times 1.008665\) \(= 235.866562\)

\(\Delta m = 235.866562 - 236.052595\) \(\Delta m = -0.186033 \,\mathrm{u}\) \(E_\mathrm{released} = -\Delta m\,c^{2}\) \(E_\mathrm{released} = -(-0.186033) \times 931.5 = \class{res}{173 \,\mathrm{MeV}}\)

E.4 FISSION · SL + HL

The same answer, a different route

Use final minus initial: \(\Delta m = m_\mathrm{products} - m_\mathrm{reactants} < 0\). The rest-energy change \(\Delta m c^{2}\) is negative and the released energy \(-\Delta m c^{2}\) is positive. Including the products’ kinetic energy and radiation, \(\Delta E_\mathrm{total} = 0\). The products have less mass than what went in. The missing mass, 0.186 u, about a fifth of the mass of one nucleon, has been converted to energy: \(E = mc^{2}\). The two methods agree, as they must: binding energy is mass defect expressed as energy.

Keep every figure

The answer comes from the difference of two numbers that agree to four significant figures. Round the masses to 236.05 and 235.87 and the difference becomes 0.18, an error of several per cent. Subtract first and round at the end.

In joules

\(173 \,\mathrm{MeV} = 173 \times 10^{6} \times 1.60 \times 10^{-19} = 2.8 \times 10^{-11}\) J for one fission.

Data booklet: \(1 \,\mathrm{u} = 931.5 \,\mathrm{MeV} \,c^{-2}\). Multiplying a mass defect in u by 931.5 gives the energy in MeV directly.

Forgetting that one neutron goes in and three come out. Count all the particles on both sides.

Worked example

A kilogram of uranium-235.

173 MeV per fission · molar mass 235 g mol⁻¹ \(N = \dfrac{m}{M} N_\mathrm{A} = \dfrac{1000}{235} \times 6.02 \times 10^{23}\) \(N = 2.56 \times 10^{24}\) nuclei \(E = N \times 173 \times 1.60 \times 10^{-13}\) \(E = \class{res}{7.1 \times 10^{13} \,\mathrm{J}}\)

Coal gives about \(3 \times 10^{7}\) J per kilogram
So 1 kg of uranium-235 matches over two thousand tonnes of coal
Most of the energy is the kinetic energy of the fragments

E.4 FISSION · SL + HL

Where the 173 MeV goes

The two fragments are both positively charged and start almost touching, so they repel each other fiercely and fly apart. They carry about 165 MeV as kinetic energy between them. The neutrons take a few MeV, and the rest leaves as gamma rays. The fragments travel only a few thousandths of a millimeter through the fuel before collisions bring them to rest, and their kinetic energy becomes thermal energy of the fuel rod. A reactor is a way of heating metal rods.

Specific energy

\(7.1 \times 10^{13}\) J kg⁻¹ is the figure in the fuel table of A.3: about a million times that of any chemical fuel.

\(1 \,\mathrm{MeV} = 1.60 \times 10^{-13}\) J. Converting once, at the end, avoids carrying powers of ten through the working.

Quick check

Fission and its energy.

01What is the difference between spontaneous and induced fission?

Induced fission first absorbs a neutron

02n + \({}^{235}\mathrm{U}\) → \({}^{144}\mathrm{Ba}\) + \({}^{89}\mathrm{Kr}\) + ? neutrons

3 neutrons

03Which is more tightly bound per nucleon: uranium or its fragments?

The fragments

04A mass defect of 0.20 u. Energy released?

\(0.20 \times 931.5 = 186 \,\mathrm{MeV}\)

05In what form is most of the energy released?

Kinetic energy of the fragments

06Why can a slow neutron enter a nucleus easily?

It has no charge: no repulsion

PRESS → TO REVEAL EACH ANSWER

Part 02

Chain reactions and reactors.

Keeping the reaction going, and keeping it steady

E.4 FISSION · SL + HL

Neutrons making more neutrons

Chain reactions.

EACH FISSION CAUSES TWO MORENOT TO SCALE

0 Neutron Uranium-235 Pair: has just split

Each fission needs one neutron and releases two or three
Those neutrons can cause further fissions: a chain reaction
If more than one goes on to cause a fission, the rate grows

E.4 FISSION · SL + HL

How fast it can grow

In pure uranium-235 the time from one generation of fissions to the next is about a hundred-millionth of a second. If every fission caused two more, eighty generations, less than a millionth of a second, would be enough to split every nucleus in a kilogram. That is a bomb. A reactor is built so that this cannot happen.

Not every neutron causes a fission

Some escape through the surface of the fuel. Some are absorbed by other nuclei without causing fission, especially by uranium-238, which makes up over 99% of natural uranium. What matters is how many, on average, go on to cause another fission.

Critical mass

A small lump of fuel has a large surface for its volume, and too many neutrons leak out. Above a certain mass, enough stay inside to keep the chain going. That least mass is the critical mass.

A chain reaction: the neutrons released by one fission go on to induce further fissions.

One neutron, on average

Controlling a chain reaction.

Fewer than one goes on

The reaction dies away

Sub-critical: shutting down

Exactly one goes on

A steady rate

Critical: a reactor running at constant power

More than one goes on

The rate grows

Supercritical: raising the power, or, unchecked, an explosion

A reactor needs two things: enough slow neutrons to keep the chain going, and a way to absorb the surplus.

E.4 FISSION · SL + HL

A fine balance

“Critical” sounds like an emergency, but in a reactor it is the normal working state: every fission leads, on average, to exactly one more, so the number of fissions each second, and the power, stay constant. To raise the power the operators let the number creep slightly above one for a while, then bring it back to one at the new level.

The two jobs

The neutrons released by fission are fast, and fast neutrons are poor at causing fission in uranium-235. They must be slowed: that is the moderator’s job. And since each fission releases two or three neutrons where only one is wanted, the extra ones must be removed: that is the job of the control rods. The next slides take these in turn.

A power reactor cannot explode like a bomb. Its fuel is only a few per cent uranium-235, far too dilute for a fast chain reaction.

Slowing the neutrons

The moderator.

Fission neutrons are fast: about 1 MeV
Slow, thermal neutrons are far more likely to cause fission in \({}^{235}\mathrm{U}\)
The moderator slows them by collisions with light nuclei
Water or graphite

A NEUTRON IN THE MODERATOR

E.4 FISSION · SL + HL

Why light nuclei

This is a collision problem from A.2. A ball that hits a much heavier ball bounces off with almost all its speed. A ball that hits one of its own mass can hand over most of its kinetic energy in a single collision. A neutron slows fastest by colliding with nuclei close to its own mass: the hydrogen in water, or carbon in graphite. Bouncing off uranium nuclei would hardly slow it at all.

Why slow is better

A slow neutron spends longer near a nucleus, so it is more likely to be captured. For uranium-235 a thermal neutron is hundreds of times more likely to cause fission than a fast one. “Thermal” means in thermal equilibrium with the moderator: a kinetic energy of about \(\frac{3}{2} k_\mathrm{B} T\), a fraction of an electronvolt.

A good moderator slows neutrons without absorbing them, which rules out many materials.

Saying that the moderator slows the reaction. It slows the neutrons, which makes the reaction go. Without a moderator the chain reaction would stop.

Absorbing the surplus

Control rods.

Rods of a material that absorbs neutrons strongly, such as boron or cadmium
Lowered further in: more neutrons absorbed, fewer fissions, less power
Raised: more fissions, more power

Moderator

Slows neutrons

So that they can cause fission

Control rods

Absorb neutrons

So that only one per fission goes on

E.4 FISSION · SL + HL

How the power is held steady

Instruments count the neutrons in the core. If the count starts to rise, the rods are moved a little further in and absorb more; if it falls, they are eased out. The adjustment is continuous and automatic.

Shutting down

In an emergency the rods are dropped fully into the core, in many designs simply by releasing them to fall under gravity. Within seconds the chain reaction stops. The core does not go cold, though: the fission products already in the fuel go on decaying and giving out thermal energy, at first about 7% of full power, so cooling must continue for a long time afterwards. Loss of that cooling was what destroyed the reactors at Fukushima in 2011.

Mixing up the two. Control rods absorb; the moderator slows without absorbing.

From fission to electricity

A nuclear power plant.

Fuel rods: fission heats them
Heat exchanger: hot coolant from the core boils water in a separate loop
Steam turns a turbine and generator
Shielding: steel and concrete absorb neutrons and gamma rays

A PRESSURIZED-WATER REACTORSCHEMATIC

E.4 FISSION · SL + HL

Why two loops

The water that flows through the core becomes radioactive, and it could carry traces of fission products if a fuel rod leaked. The heat exchanger passes its thermal energy through the walls of metal tubes to a second, separate supply of water, which never touches the core. Only that clean water, as steam, goes out to the turbine hall.

Why loop 1 does not boil

Its water leaves the core at about 300 °C, but it is kept at about 150 times atmospheric pressure, which raises its boiling point above that. It stays liquid, carries energy well, and keeps moderating the neutrons. Loop 2 is at a much lower pressure, so the same hot coil boils it.

From here on it is an ordinary power station

Steam drives a turbine, which turns a generator (D.4); the steam is condensed and pumped back. The reactor replaces the furnace of a coal or gas station and nothing else. Like them it is a heat engine, with an efficiency of about a third (B.4).

What the shielding stops

Neutrons and gamma rays are both very penetrating. A steel pressure vessel and several meters of concrete surround the core, to protect the people working there and to contain the radioactive material if something goes wrong.

Know the job of each of the four named parts: moderator, control rods, heat exchanger and shielding.

In this design the same water is both coolant and moderator.

The story · Chicago, 2 December 1942

Background story · not assessed

A reactor in a squash court.

Drawing of a large stack of dark blocks, as tall as a room, built inside a timber frame with a ladder leaning against it — **Chicago Pile-1**, drawn by Melvin A. Miller

Under the stands of a university sports field, Enrico Fermi’s team stacked 45 000 blocks of graphite with uranium between them
Graphite was the moderator. Strips of wood covered in cadmium were the control rods
At 15:25 the chain reaction became self-sustaining, at about half a watt
No shielding and no cooling: it was never meant to run at power

The message sent in code: “The Italian navigator has landed in the New World.”

Drawing: Melvin A. Miller, Argonne National Laboratory, public domain.

E.4 FISSION · SL + HL

Worked example

The fuel used in a day.

1000 MW of electrical power · efficiency 33% 173 MeV = \(2.77 \times 10^{-11}\) J per fission \(P_\mathrm{in} = \dfrac{P_\mathrm{out}}{\eta} = \dfrac{1.0 \times 10^{9}}{0.33} = 3.0 \times 10^{9} \,\mathrm{W}\) Fissions per second \(= \dfrac{3.0 \times 10^{9}}{2.77 \times 10^{-11}}\) \(= 1.1 \times 10^{20} \,\mathrm{s}^{-1}\)

In one day: 86 400 s \(N = 1.1 \times 10^{20} \times 86\,400 = 9.5 \times 10^{24}\) \(m = N \times 235 \,\mathrm{u}\) \(m = 9.5 \times 10^{24} \times 235 \times 1.661 \times 10^{-27}\) \(m = \class{res}{3.7 \,\mathrm{kg}}\) of \({}^{235}\mathrm{U}\)

E.4 FISSION · SL + HL

A sense of scale

A large power station, enough for a city of a million people, consumes less than four kilograms of uranium-235 a day: a lump the size of a tennis ball. A coal station of the same output burns about ten thousand tonnes of coal a day, several trainloads, and sends some thirty thousand tonnes of carbon dioxide into the air.

Fuel rods hold more than that

Reactor fuel is only 3 to 5% uranium-235; the rest is uranium-238, which does not sustain the chain reaction. So the fuel actually loaded is some thirty times the mass calculated here, and it stays in the core for several years.

The method: electrical power → thermal power (divide by the efficiency) → fissions per second (divide by the energy of one) → mass (multiply by the mass of one nucleus).

Quick check

Chain reactions and reactors.

01What carries a chain reaction from one fission to the next?

Neutrons

02How many neutrons per fission must go on, for steady power?

Exactly one, on average

03What does the moderator do?

Slows the neutrons by collisions

04What do the control rods do?

Absorb neutrons

05Why is there a heat exchanger?

To keep radioactive coolant out of the turbine

06The control rods are lowered further. What happens to the power?

It falls

PRESS → TO REVEAL EACH ANSWER

Part 03

The products of fission.

What comes out of a reactor, and what has to be done with it

E.4 FISSION · SL + HL

What is in spent fuel

The products of fission.

The fragments have too many neutrons for their size
So they are radioactive: beta-minus emitters, with gamma rays
Half-lives from seconds to millions of years
Neutrons also turn \({}^{238}\mathrm{U}\) into plutonium and other heavy nuclei

In spent fuel	Half-life
Iodine-131	8 days
Strontium-90	29 years
Cesium-137	30 years
Plutonium-239	24 000 years

E.4 FISSION · SL + HL

Why the fragments are neutron-rich

Stable heavy nuclei need many more neutrons than protons: uranium-235 has 143 neutrons to 92 protons, a ratio of 1.55. Stable nuclei of middle size need fewer: about 1.3 neutrons per proton. When uranium splits, the fragments inherit its high ratio. Barium-141 has 85 neutrons where the stable isotopes of barium have at most 82. Such nuclei correct the imbalance by beta-minus decay, turning neutrons into protons, usually several times in succession.

Why spent fuel is so much more dangerous than new fuel

Fresh uranium fuel is only weakly radioactive and can be handled with gloves. After years in a reactor the same rods are filled with intensely radioactive fission products and would deliver a fatal dose in minutes. They also go on producing thermal energy from their decay.

Middle half-lives are the worst

A very short half-life means fierce activity that is soon over. A very long one means feeble activity. Strontium-90 and cesium-137, at about thirty years, are active enough to be dangerous and last long enough to be a problem for centuries.

Strontium behaves chemically like calcium and is taken up by bone; iodine collects in the thyroid gland. Where a nuclide goes in the body matters as much as its activity.

What is done with it

Managing nuclear waste.

First years: under water in cooling ponds, which absorb radiation and remove heat
Then: sealed in steel and concrete casks, or reprocessed and set in glass
Finally: burial deep in stable rock

ACTIVITY OF SPENT FUELA SKETCH

E.4 FISSION · SL + HL

Two timescales

The graph has a logarithmic time axis: each step is ten times longer than the last. For the first few hundred years the activity is dominated by fission products such as strontium-90 and cesium-137. After ten half-lives, about 300 years, only a thousandth of them is left. What remains is a much lower activity from plutonium and other heavy nuclei, which lasts for tens of thousands of years. A store has to be safe on both timescales.

The impact of long-term storage

The waste must be kept from groundwater and from people for longer than any human institution has lasted. That means choosing rock that has been stable for millions of years, containers that will not corrode, and some way of warning generations who may not share our languages. It also means cost, transport of dangerous material, and the consent of whoever lives nearby. Against that, the volume is small: all the high-level waste a person’s lifetime of electricity would produce fits in a drinks can.

The guide expects you to consider the impact of long-term storage. Be ready with points on both sides: the long timescale and its risks, and the small volume and the possibility of deep, stable burial.

The story · Olkiluoto, Finland

Background story · not assessed

A cave built to be forgotten.

A wide tunnel cut through bare rock, lit by lamps, with a side chamber opening on the right and a wet gravel floor — A tunnel in the **Onkalo** repository. The name means “small cave”

Excavation into granite bedrock began in 2004; the tunnels reach about 450 m down
Spent fuel goes into cast-iron inserts inside copper canisters, each in its own hole, packed round with clay
The first permanent store for spent fuel in the world. Its capsules are meant to last 100 000 years

Photograph: kallerna, CC BY-SA 4.0.

E.4 FISSION · SL + HL

Worked example

How long it must be stored.

Cesium-137: half-life 30 years What fraction remains after 300 years? Number of half-lives \(= \dfrac{300}{30} = 10\) Fraction remaining \(= \left(\dfrac{1}{2}\right)^{10}\) \(= \class{res}{\dfrac{1}{1024}}\)

About a thousandth after three centuries
Plutonium-239 over the same time: \(\dfrac{300}{24\,000}\) of a half-life. Hardly any has gone

Ten half-lives: a factor of a thousand.

E.4 FISSION · SL + HL

Quick check

The products of fission.

01Why are fission fragments radioactive?

Too many neutrons for their size

02Which decay do they mostly undergo?

Beta-minus

03Why is spent fuel first stored under water?

To absorb radiation and remove heat

04Strontium-90 has a half-life of 29 years. Fraction left after 87 years?

\(\frac{1}{8}\)

05Why is plutonium a long-term problem?

Its half-life is 24 000 years

06Where is high-level waste finally stored?

Deep underground, in stable rock

PRESS → TO REVEAL EACH ANSWER

Summary

Fission on one slide.

Fission

A heavy nucleus splits

Spontaneous, or induced by a neutron
Two fragments and a few neutrons

Energy

\(E_\mathrm{released} = -\Delta m \, c^{2}\)

Or the rise in total binding energy
About 200 MeV per fission

Chain reaction

Neutrons cause more fissions

Steady when one per fission goes on

Moderator · control rods

Slow · absorb

Slow neutrons cause fission; surplus neutrons are removed

Heat exchanger · shielding

Transfer · protect

Steam for the turbine from a separate loop

Products

Radioactive, some for millennia

Cooling ponds, casks, deep burial

E.4 FISSION · SL + HL

Summary

Key words.

FissionThe splitting of a heavy nucleus

Induced fissionTriggered by absorbing a neutron

Chain reactionNeutrons from one fission cause others

Critical massThe least mass that sustains a chain reaction

Thermal neutronA slow neutron, in equilibrium with the moderator

ModeratorSlows neutrons by collisions

Control rodAbsorbs neutrons

Heat exchangerPasses thermal energy to a separate loop

Fission productsThe radioactive fragments

E.4 FISSION · SL + HL