doPhysics · Modern Physics
The Nucleus, Binding Energy and Nuclear Reactions
What a nucleus is made of, how big it is and why it holds together. The mass defect and the binding energy, and the curve of binding energy per nucleon. Nuclear reactions with their Q value and threshold, then fission, the chain reaction and the reactor, and fusion in the stars and on the Earth.
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Why this chapter
Why this chapter exists
One kilogram of uranium-235 gives as much heat as nearly three thousand tonnes of coal. The Sun has burned for billions of years without running out. Both facts come from the nucleus, a speck tens of thousands of times smaller than its atom. This chapter opens it: what it is made of, what holds it together, and why a nucleus weighs less than its parts. One curve of binding energy then explains why splitting uranium and joining hydrogen both release energy. The chapter ends with the reactor at Kudankulam and the fusion fire of the Sun.
You will learn
By the end of this chapter you will
- Describe a nucleus by \(Z\), \(N\) and \(A\), and sort nuclei into isotopes, isobars and isotones.
- Find the radius of a nucleus from \(R = R_0A^{1/3}\), and show that every nucleus has the same density.
- State the properties of the nuclear force, and find the mass defect and binding energy of a nucleus from atomic masses.
- Read the binding energy per nucleon curve, explain its shape, and use it to estimate the energy of fission and fusion.
- Balance a nuclear reaction, find its Q value from masses or kinetic energies, and find the threshold of an endothermic reaction.
- Explain fission, the chain reaction and the multiplication factor, and the job of each part of a reactor.
- Explain the proton-proton chain, the Coulomb barrier and why fusion needs such high temperatures.
Before you begin
Before you begin
From the chapter on Matter Waves and Special Relativity, you should know that energy has mass, \(E = mc^2\). From the chapter on the Bohr atom, you should know Rutherford's nucleus and the potential energy \(ke^2/r\) of two charges, with \(ke^2 = 1.44\,\text{MeV fm}\). From the chapters on momentum and collisions, you should know conservation of momentum and the least energy lost when two bodies stick together.
Lessons
1
What a Nucleus Is
A nucleus is built of protons and neutrons, named by two numbers, Z and A. Nuclei that share one of Z, N or A form families. The radius grows as the cube root of A, so every nucleus has the same enormous density.
Nucleons and nuclide notationIsotopes isobars and isotonesNuclear radiusConstant nuclear density
2
Mass Defect and Binding Energy
A short-range, charge-blind force holds the protons of a nucleus together against their repulsion. Because the nucleons are bound, the nucleus weighs less than its separate parts. The missing mass, times \(c^2\), is the binding energy.
The strong nuclear forceMass defectBinding energy
3
The Binding Energy Curve
The binding energy per nucleon, plotted against the mass number, rises steeply, peaks near iron and falls slowly towards uranium. Its shape explains which nuclei are stable, why heavy nuclei need extra neutrons, and why both splitting and joining nuclei can release energy.
Binding energy per nucleonThe stability beltEnergy from moving toward the peak
4
Nuclear Reactions and the Q Value
A nuclear reaction keeps the charge, the number of nucleons, the energy and the momentum. The Q value, from the masses before and after, says whether energy is released or needed. An endothermic reaction needs more than |Q|, because the products must keep moving.
Conservation in nuclear reactionsThe q valueThreshold energy of an endothermic reaction
5
Fission and the Reactor
A slow neutron splits uranium-235 into two unequal fragments and two or three fresh neutrons, releasing about 200 MeV. Those neutrons can start a chain reaction. A reactor holds the chain steady with a moderator, control rods and a coolant.
Nuclear fissionChain reaction and the multiplication factorThe nuclear reactor
6
Fusion: the Energy of the Stars
The Sun joins four protons into one helium nucleus, releasing 26.7 MeV. Two nuclei must first come close against their electric repulsion, which classically needs billions of kelvin; the Sun manages at fifteen million. On the Earth, ITER aims to hold such a fire in a magnetic bottle.
Nuclear fusionCoulomb barrier and fusion temperatureFusion on earth
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- atomic mass unit
- One twelfth of the mass of a carbon-12 atom: \(1\,\text{u} = 1.66 \times 10^{-27}\,\text{kg}\), equivalent to \(931.5\,\text{MeV}\).
- atomic number
- The number of protons in a nucleus, \(Z\). It fixes the charge of the nucleus and the element.
- binding energy
- The energy needed to pull a nucleus apart into free nucleons, \(B = \Delta m\,c^2\). The same energy is given out when the nucleus forms.
- binding energy curve
- The graph of binding energy per nucleon \(B/A\) against mass number \(A\). It peaks at about \(8.8\,\text{MeV}\) near iron-56.
- carbon-nitrogen cycle
- The route by which hotter stars turn four protons into helium-4, with carbon-12 as a catalyst that is returned at the end.
- chain reaction
- A sequence of fissions in which the neutrons from one fission cause the next ones.
- control rod
- A rod of cadmium or boron that absorbs neutrons. Moving it in or out sets the multiplication factor of a reactor.
- coulomb barrier
- The hill of electric potential energy, \(kq_1q_2/r\), that two positive nuclei must climb to come within reach of the nuclear force.
- critical mass
- The least mass of a fissile material, in a given shape, for which the chain reaction just sustains itself, \(k = 1\).
- endothermic
- Said of a reaction with \(Q < 0\): the products are heavier, and kinetic energy must be supplied. It needs a threshold energy greater than \(|Q|\).
- exothermic
- Said of a reaction with \(Q > 0\): the products are lighter, and energy is released as kinetic energy.
- femtometre
- \(1\,\text{fm} = 10^{-15}\,\text{m}\), the natural unit of nuclear sizes. It is also called a fermi.
- heavy water
- Water whose hydrogen atoms are deuterium, \(\text{D}_2\text{O}\). It is about 11 % denser than ordinary water, and it is the moderator of India's pressurised heavy water reactors.
- isobar
- One of two or more nuclides with the same mass number \(A\) and different atomic numbers, such as carbon-14 and nitrogen-14.
- isotone
- One of two or more nuclides with the same neutron number \(N\), such as carbon-13, nitrogen-14 and oxygen-15.
- isotope
- One of two or more nuclides with the same atomic number \(Z\) and different neutron numbers. Isotopes are the same element.
- line of stability
- The narrow band on a graph of \(N\) against \(Z\) where the stable nuclei lie. It follows \(N = Z\) for light nuclei and bends to \(N/Z \approx 1.5\) for heavy ones.
- mass defect
- The mass of the separate nucleons minus the mass of the nucleus, \(\Delta m = Zm_{\text{H}} + Nm_n - M\) with atomic masses.
- mass number
- The total number of nucleons in a nucleus, \(A = Z + N\). It is a count, not a mass.
- moderator
- A material of light nuclei, such as water, heavy water or graphite, that slows fission neutrons by elastic collisions.
- multiplication factor
- The ratio \(k\) of the number of fissions in one generation of a chain reaction to the number in the generation before.
- neutron
- A particle with no charge and a mass of \(1.008665\,\text{u}\), slightly more than the proton. Chadwick found it in 1932.
- neutron number
- The number of neutrons in a nucleus, \(N = A - Z\).
- nuclear density
- The density of nuclear matter, \(3m/4\pi R_0^3 \approx 2.3 \times 10^{17}\,\text{kg/m}^3\), the same for every nucleus.
- nuclear fission
- The splitting of a heavy nucleus, such as uranium-235 after it catches a neutron, into two middle-sized nuclei, with about 200 MeV released.
- nuclear fusion
- The joining of light nuclei into a heavier one, with energy released, as when hydrogen becomes helium in the Sun.
- nuclear reaction
- A process in which a particle strikes a nucleus and changes it into another, written \(a + X \to Y + b\) or \(X(a, b)Y\).
- nucleon
- A proton or a neutron, the two kinds of particle that make up a nucleus.
- nuclide
- One kind of nucleus, fixed by its \(Z\) and \(A\), written \({}^{A}_{Z}\text{X}\).
- plasma
- A gas so hot that its atoms are torn into bare nuclei and free electrons.
- proton-proton chain
- The three-step route by which the Sun turns four protons into one helium-4 nucleus, releasing \(26.7\,\text{MeV}\).
- q value
- The energy released in a nuclear reaction, \(Q = (m_{\text{initial}} - m_{\text{final}})c^2 = K_{\text{final}} - K_{\text{initial}}\).
- quantum tunnelling
- The passage of a particle through an energy barrier higher than its energy, possible because the particle is also a wave.
- strong nuclear force
- The attraction between nucleons that holds a nucleus together. It is strong at about 1 fm and almost zero beyond a few fm; it is the same for protons and neutrons, and it saturates.
- thermal neutron
- A neutron slowed until its energy is about that of the surrounding atoms, about \(0.025\,\text{eV}\) at room temperature.
- thermonuclear fusion
- Fusion driven by heat: the nuclei of a very hot gas move fast enough to come within reach of the nuclear force.
- threshold energy
- The least kinetic energy of a projectile that can start an endothermic reaction on a target at rest, \(K_{th} = |Q|(1 + m/M)\).
- tokamak
- A ring-shaped machine that holds a fusion plasma away from its walls with magnetic fields. ITER is the largest.