Nuclear Physics

Mass and energy are equivalent: E = m c2

Einstein's mass-energy relation states that mass and energy are equivalent, and it is written E = m c2, where E is energy in joules, m is mass in kilograms and c is the speed of light in a vacuum, 3.00 x 10^8 m s-1. The relation works in both directions: a system that gains energy gains mass, and a system that loses energy loses mass. The factor c2 is about 9.00 x 10^16 m2 s-2, so losing one kilogram of mass would release 9.00 x 10^16 J, and losing the tiny mass of a single nucleus releases millions of electronvolts. That is why mass changes are never noticed in mechanics, where the energy changes are far too small, but are central to nuclear physics.

The clearest large-scale example is electron-positron annihilation. A positron and an electron each have a mass of 9.11 x 10-31 kg. When they meet, both particles disappear and two gamma-ray photons are produced. The total mass destroyed is 2 x 9.11 x 10-31 = 1.82 x 10-30 kg, so the energy carried away by the photons is E = m c2 = 1.82 x 10-30 x 9.00 x 10^16 = 1.64 x 10-13 J, which is 1.02 MeV. The two photons share this energy equally, so each carries 0.51 MeV. This is exactly the process that produces the two 0.51 MeV gamma rays detected in a PET scanner.

At nuclear scale, masses are quoted in unified atomic mass units: 1 u = 1.66 x 10-27 kg, which is equivalent to about 930 MeV of energy. A proton has a mass of 1.00728 u, a neutron 1.00867 u and an electron 0.00055 u. Learn these values, because every calculation in this topic depends on comparing the mass of a nucleus with the total mass of the nucleons inside it.

Mass defect and binding energy

The mass defect delta m of a nucleus is the difference between the total mass of its separate protons and neutrons and the mass of the nucleus itself: delta m = (mass of the protons + mass of the neutrons) - (mass of the nucleus). The mass of a nucleus is always less than the total mass of its separate nucleons, so the mass defect is always positive and the nucleus is always the lighter of the two. It is worth realising how small the difference is: for most nuclei the mass defect is less than one per cent of the total mass, which is why four-figure masses are needed in the arithmetic.

The binding energy of a nucleus is the minimum energy needed to separate the nucleus into its individual protons and neutrons. Since energy must be supplied to pull the nucleus apart, the separate nucleons have more mass than the nucleus, and the energy supplied equals c2 multiplied by the mass defect: binding energy = c2 x delta m. Equivalently, it is the energy released when the separate nucleons come together to form the nucleus. A nucleus with a larger binding energy for the same number of nucleons is more stable, because more energy would be needed to break it up.

Take helium-4. Two protons have a total mass of 2 x 1.00728 = 2.01456 u, and two neutrons have 2 x 1.00867 = 2.01734 u, giving 4.03190 u altogether. The mass of a helium-4 nucleus is 4.00151 u, so the mass defect is 4.03190 - 4.00151 = 0.03039 u. In kilograms this is 0.03039 x 1.66 x 10-27 = 5.04 x 10-29 kg, and the binding energy is E = c2 x delta m = 9.00 x 10^16 x 5.04 x 10-29 = 4.54 x 10-12 J, which is 4.54 x 10-12 / 1.60 x 10-13 = 28.4 MeV.

Divide the binding energy by the number of nucleons to compare different nuclei fairly. Helium-4 has 4 nucleons, so its binding energy per nucleon is 28.4 / 4 = 7.1 MeV. The same calculation for iron-56 gives about 8.8 MeV per nucleon and for uranium-238 about 7.6 MeV per nucleon, so a nucleon in an iron nucleus is held more tightly than a nucleon in either a helium nucleus or a uranium nucleus.

• Mass defect delta m = (mass of separate protons and neutrons) - (mass of the nucleus), always positive.
• Binding energy = c2 x delta m, in joules, or converted to MeV using 1 MeV = 1.60 x 10-13 J.
• 1 u = 1.66 x 10-27 kg, equivalent to 931 MeV.
• Binding energy per nucleon = binding energy / nucleon number, in MeV per nucleon.

Binding energy per nucleon against nucleon number

Binding energy per nucleon is the binding energy of a nucleus divided by its nucleon number, measured in MeV per nucleon. It is the fair way to compare nuclei of different sizes, because a large nucleus has a large total binding energy simply because it contains more nucleons, without being any more stable per nucleon. Sketch the graph with nucleon number on the x-axis from 0 to about 240 and binding energy per nucleon on the y-axis from 0 to 10 MeV.

The shape is the same for every isotope. Hydrogen-1 has no binding energy at all, since a single proton has nothing to be bound to. The curve then rises extremely steeply, and there is a sharp spike at helium-4 (7.1 MeV per nucleon), which is unusually stable for such a small nucleus. From there it climbs more gently and with small peaks at carbon-12 (7.7) and oxygen-16 (8.0), reaching a broad maximum of about 8.8 MeV per nucleon near iron-56. Beyond iron the curve falls slowly but steadily, down to about 7.6 MeV per nucleon at uranium-238. The fall is gentle; the rise at the light end could hardly be steeper.

Read the graph as a map of stability. The higher the binding energy per nucleon, the more tightly each nucleon is held in, and the more stable the nucleus. Any nuclear process that increases the total binding energy of the nucleons releases the difference as kinetic energy. Iron-56 sits at the top of the curve, so it cannot release energy by fusing or by splitting: nucleons can only be pushed towards the maximum, which means fusing light nuclei together or splitting heavy nuclei apart. That single sentence is the whole reason stars shine and reactors run.

Nuclear equations and what must balance

A nuclear reaction is written as an equation with the nuclei on each side. The mass number is written as a superscript and the proton number as a subscript on the left of the symbol, so the nucleus of carbon-14, with 14 nucleons and 6 protons, is written 14 over 6 to the left of C. In plain text the same nucleus is written C-14. Two quantities are conserved in every nuclear reaction: the total nucleon number and the total charge (the total proton number). Charge is conserved because the proton number counts the charge: an electron carries one unit of negative charge, which is why beta emission raises the proton number of the nucleus by one while leaving the nucleon number unchanged.

The reaction that made Rutherford famous in 1919 is a good test of the rules. He allowed alpha particles from a radium source into air and found protons being produced, which could only have come from the nitrogen nuclei. In words the equation is: nitrogen-14 + helium-4 -> oxygen-17 + hydrogen-1. Checking: the nucleon numbers are 14 + 4 = 18 on the left and 17 + 1 = 18 on the right, and the proton numbers are 7 + 2 = 9 on the left and 8 + 1 = 9 on the right. Both balance, so the equation is correct. This was the first artificial nuclear reaction.

The two decay equations you must be able to write work the same way. In alpha emission the nucleus loses a helium nucleus, so radium-226 -> radon-222 + helium-4: nucleon numbers 226 = 222 + 4, and proton numbers 88 = 86 + 2. In beta emission a neutron turns into a proton and an electron, so carbon-14 -> nitrogen-14 + electron: nucleon numbers 14 = 14 + 0, and proton numbers 6 = 7 - 1, the electron carrying the charge of one proton. Gamma emission changes neither number, because a gamma photon carries no nucleons and no charge.

In a reaction where energy is released, the missing mass is shared out as kinetic energy and as the energy of any gamma photons. Momentum is conserved as well, which is why in fission the two fragments fly apart in opposite directions and rarely share the energy equally: the lighter fragment usually carries more kinetic energy, because it must have the same magnitude of momentum as the heavier one.

• Total nucleon number before = total nucleon number after.
• Total proton number before = total proton number after (charge is conserved).
• Alpha emission: nucleon number -4, proton number -2. Beta emission: nucleon number unchanged, proton number +1. Gamma emission: neither changes.
• A beta particle is written as an electron with nucleon number 0 and charge -1.

Radioactive decay is spontaneous and random

Radioactive decay is spontaneous and random. Spontaneous means that nothing has to be done to a nucleus to make it decay: heating it, cooling it, squeezing it or combining the atom chemically makes no measurable difference to the rate, and the decay of one nucleus does not trigger the decay of its neighbours. Random means that you cannot predict which nucleus in a sample will decay next, when any particular nucleus will decay, or in which direction the emitted particle will travel. The only thing that can be stated is a probability, and that probability per unit time is the decay constant.

The evidence for randomness is in the counting statistics. Set a Geiger-Muller tube to count for 10 s at a time and take 30 readings with the source in a fixed position: the readings scatter about a mean value instead of being identical. If decays happened at regular intervals, every 10 s count would be the same. Instead, each count is a random sample of a random process, and the scatter grows the longer you look. Take the mean of all the readings and a long counting time and the value settles down, which is why a single measurement is unreliable but a long one is reproducible.

The size of the fluctuations is itself predictable. For a mean count of about 100, individual readings typically lie within about ten counts either side of the mean, a spread of roughly 10 per cent, because the spread is about the square root of the mean. For a mean count of about 10 000 the spread is about 100 counts, only 1 per cent. So a longer count gives a smaller percentage uncertainty and a more reliable mean, exactly as the statistical treatment of random events predicts.

The spontaneity of the process is shown by control experiments. A source of carbon-14 in a living plant and the same isotope in a lump of charcoal decay at the same rate, so the chemical environment is irrelevant; a source cooled in liquid nitrogen decays at the same rate as one at room temperature; and the rate is unaffected by the pressure or by the physical state of the material. Together, random and spontaneous decay mean that a sample can only be described statistically, through the decay constant and the half-life.

Activity, the decay constant and A = lambda N

The activity A of a source is the number of nuclei that decay per unit time, so it is simply a rate of decay. It is measured in the becquerel, symbol Bq, where 1 Bq = 1 decay per second = 1 s-1. Activities cover an enormous range: a school demonstration source might have an activity of a few thousand Bq, the radon in the air of a house gives a few tens of Bq per cubic metre of air, and one gram of radium-226 has an activity of 3.7 x 10^10 Bq, a value so familiar that it was once used to define the curie.

The decay constant lambda of an isotope is the probability that a particular nucleus decays per unit time. Its unit is s-1 (or per second, or day-1 if all times are in days, but then the activity must use the same time unit). Because it is a probability for a single nucleus, it is a fixed property of the isotope: it does not depend on how many nuclei are present, on the temperature, or on the chemical form of the material. A large decay constant means an unstable nucleus and a rapidly decaying source.

If a sample contains N undecayed nuclei, then in a short time interval delta t the number of nuclei that decay is lambda N delta t, because each of the N nuclei has a probability lambda delta t of decaying. Dividing by delta t gives the activity: A = lambda N. Check the units: lambda has unit s-1 and N is a number, so A comes out in s-1, that is Bq. The equation rearranges to N = A / lambda, which is how you find the number of undecayed nuclei in a source from its activity. Because a detector intercepts a fixed fraction of the emitted particles, the count rate corrected for background is proportional to the activity, so measurements from a Geiger-Muller tube can be used in the same equations.

Worked example. A sample contains 5.0 x 10^20 undecayed nuclei of an isotope whose decay constant is 2.0 x 10-6 s-1. The activity is A = lambda N = 2.0 x 10-6 x 5.0 x 10^20 = 1.0 x 10^15 Bq. Turned around, a source of activity 6.0 x 10^8 Bq with a decay constant of 1.0 x 10-6 s-1 contains N = A / lambda = 6.0 x 10^8 / 1.0 x 10-6 = 6.0 x 10^14 undecayed nuclei.

• Activity A is the number of decays per second, measured in becquerel (Bq); 1 Bq = 1 s-1.
• Decay constant lambda is the probability per unit time that a particular nucleus decays, measured in s-1.
• A = lambda N, so N = A / lambda.
• Both A and lambda are properties of the isotope and the sample size; lambda does not depend on how much material there is.

Half-life and the exponential decay law

The half-life t(1/2) of an isotope is the average time taken for half the nuclei in any sample of that isotope to decay. Because the activity is proportional to the number of undecayed nuclei, the half-life is also the time taken for the activity, or for the background-corrected count rate, to fall to half its value. Half-life is a fixed property of the isotope: it is the same for a gram or a tonne of the material, and it cannot be changed by temperature, pressure or chemical combination.

The decay law is exponential: x = x0 e raised to the power (-lambda t), where x0 is the value at time zero and x is the value at time t. The quantity x can be the number of undecayed nuclei N, the activity A, or the count rate once the background has been subtracted; it cannot be the raw count rate, because the background does not decay and pulls the curve up at long times. Putting x = x0 / 2 at t = t(1/2) gives x0 / 2 = x0 e raised to the power (-lambda t(1/2)), so e raised to the power (lambda t(1/2)) = 2, and therefore lambda t(1/2) = 0.693. Rearranged, lambda = 0.693 / t(1/2), which lets you jump between the two constants without doing any exponential arithmetic.

Sketch the graph of activity against time and you get a curve that falls ever more slowly and flattens towards the time axis without ever touching it. In each equal interval of one half-life the value is multiplied by 1/2, so after one, two, three and four half-lives the fractions remaining are 1/2, 1/4, 1/8 and 1/16. This constant fractional decrease is what makes half-life so useful: any two readings that differ by a factor of two are one half-life apart, whatever the absolute values, and this is exactly how a half-life is read from a graph or from a table of counts.

Two warnings about the arithmetic. First, convert the half-life into the same unit of time as the activity before you combine them: a decay constant in s-1 must be paired with an activity in Bq, so a half-life given in days must be changed into seconds, using 1 day = 8.64 x 10^4 s. Second, the exponential equation needs a difference in time, not an absolute time: use t as the time elapsed between the two measurements, and make sure the exponent lambda t is a pure number with no unit.

Fusion, fission and the energy released

Nuclear fusion is the joining together of two light nuclei to form a heavier nucleus. Energy is released because the product has a larger binding energy per nucleon than the nuclei that formed it, so the nucleons end up more tightly bound and the difference in total binding energy appears as kinetic energy of the products. In the Sun, four hydrogen-1 nuclei combine over a sequence of steps to give one helium-4 nucleus, two positrons and two neutrinos, releasing 26.7 MeV. Fusion requires extremely high temperature and pressure, because the nuclei must approach closely enough to overcome the electrostatic repulsion between their positive charges; the Sun's gravity holds its core at about 1.5 x 10^7 K, which is why fusion is so hard to reproduce in a laboratory on Earth.

Nuclear fission is the splitting of a heavy nucleus into two lighter nuclei, and it releases energy for exactly the same reason: the fragments are nearer the top of the binding-energy-per-nucleon curve than the original nucleus. Fission of uranium-235 is started by the nucleus absorbing a slow neutron, which makes the nucleus unstable enough to split. One possible outcome is: uranium-235 + neutron -> barium-141 + krypton-92 + 3 neutrons. The nucleon numbers balance, 1 + 235 = 141 + 92 + 3, and the proton numbers balance too, 0 + 92 = 56 + 36 + 0. The mass defect for this reaction is 0.186 u, so the energy released is E = c2 x delta m = 9.00 x 10^16 x 0.186 x 1.66 x 10-27 = 2.8 x 10-11 J, which is about 173 MeV.

The energy released is the increase in total binding energy, and the two routes must agree. Multiplying the mass defect in kilograms by c2 gives the energy in joules; converting each of those joules to MeV using 1 MeV = 1.60 x 10-13 J gives the same 173 MeV, and adding up the binding energies of the fragments and subtracting that of the uranium-235 nucleus gives the same figure. A candidate who is asked for the energy in MeV may use the shortcut that a mass defect of 1 u is equivalent to about 930 MeV, and 0.186 x 930 = 173 MeV, which is a good check on the longer calculation.

The scale of the energy is worth remembering. About 200 MeV is released on average per fission of uranium-235, which is roughly fifty million times the energy released by burning one molecule of methane in oxygen. One kilogram of uranium-235 therefore releases about 8 x 10^13 J, comparable to burning about 3000 tonnes of coal. Because each fission also releases two or three neutrons, these neutrons can trigger further fissions and produce a chain reaction; in a reactor the rate is kept steady by control rods that absorb neutrons and by a moderator that slows the neutrons down so that they are more likely to be captured.

• Fusion: light nuclei join, releasing energy; needs very high temperature and pressure.
• Fission: a heavy nucleus splits after absorbing a slow neutron, releasing about 200 MeV per fission.
• Both release energy because the nucleons end up with a larger binding energy per nucleon.
• A chain reaction occurs when the neutrons released by one fission cause further fissions.