Unit 19: Nuclear and Particle Physics — Long Questions

Mass defect (Δm) experiments show that the mass of a nucleus is always less than the total mass of its separate constituent protons and neutrons. This missing mass is called the mass defect, Δm = (Zmₚ + Nmₙ) − M_nucleus.
Binding energy (B.E.): the missing mass is converted, during the formation of the nucleus, into energy via Einstein's relation B.E. = Δmc². Binding energy is defined as the work that must be done on the nucleus to break it apart into its free, separate protons and neutrons; a more useful measure is the binding energy per nucleon (B.E./A), the average energy needed to remove one nucleon from the nucleus.
Graphical curve (B.E./A vs A): The curve rises very steeply from hydrogen (B.E. ≈ 0) to helium, showing the large energy release when hydrogen fuses to helium. It then rises more gradually and reaches a maximum of about 8.7 MeV at iron (A ≈ 56), the region between A = 50 and 80 containing the most stable nuclides. Beyond this maximum, the curve falls slowly for heavier nuclei. Thus the curve shows that B.E./A is lower for both very light and very heavy nuclei than for medium-mass nuclei. This tells us that energy can be released either by splitting (fission) a heavy nucleus into two medium nuclei, or by fusing two very light nuclei into a heavier one — in both cases the products lie higher on the curve (more tightly bound) than the reactants.

Nuclear fission a reaction in which a heavy nucleus (such as uranium) splits into two nuclei of roughly equal size, along with the emission of two or three neutrons and a large amount of energy (about 200 MeV per fission of uranium), e.g. ²³⁵₉₂U + n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3n + Q.
Fission chain reaction: the two or three neutrons released in one fission event can go on to cause fission in further uranium nuclei, each of which releases more neutrons, and so the process can continue to multiply itself — this self-propagating sequence of fissions is called a fission chain reaction. For the chain to be self-sustaining, on average exactly one neutron from each fission must cause another fission; the rest escape the sample or are absorbed elsewhere.
Critical mass: the minimum mass of fissile material (e.g. uranium-235) in which, on average, one neutron out of all those produced in a fission event goes on to produce a further fission, so that the chain reaction proceeds at a steady rate. If the mass is less than critical, the chain reaction dies out; if much greater than critical (supercritical), the reaction accelerates very rapidly, producing a huge, uncontrolled explosion (as in an atomic bomb).

A nuclear power reactor has four main parts:
1. Core and fuel: The core holds the fuel in cylindrical rods. Enriched uranium (U-235 raised from the natural 0.7% to about 2.4–3.0%), or sometimes plutonium-239/uranium-233, is used as fuel; fission in the fuel produces about 200 MeV per nucleus, appearing as kinetic energy of the fission fragments.
2. Moderator: Fuel rods are surrounded by a low atomic weight substance such as ordinary water or heavy water, called the moderator. Its function is to slow down the fast neutrons produced in fission to thermal (slow) speeds, since slow neutrons are much more effective at causing further fission.
3. Control rods: Made of cadmium or boron, these absorb neutrons strongly. They are moved in or out of the core to regulate how many neutrons remain available to sustain the chain reaction, keeping it steady rather than runaway; in an emergency they can be dropped fully into the core (as safety rods) to stop the reaction and shut the reactor down.
4. Heat exchanger and shielding: Heat generated in the core (raising its temperature to about 500 °C) is carried by water, heavy water, or another coolant under high pressure to a heat exchanger, where it converts ordinary water into high-pressure steam (around 300 °C). This steam drives a turbine, which rotates a generator to produce electricity; the steam is then cooled (often using river or sea water) back into water. A thick concrete and steel shield surrounds the whole reactor to absorb radiation and protect personnel.

The binding energy per nucleon curve shows that both very light nuclei (e.g. hydrogen, with B.E./A ≈ 0) and very heavy nuclei (beyond iron, A > 80) have a lower binding energy per nucleon than medium-mass nuclei — they are therefore less tightly bound and relatively less stable. For light nuclei, there are simply too few nucleons for the short-range strong nuclear force to bind them as tightly as in a medium nucleus. For heavy nuclei, the large number of protons causes a growing electrostatic (Coulomb) repulsion that works against the strong nuclear force, reducing the net binding per nucleon.
Usefulness as an energy source: because the products of splitting a heavy nucleus (fission) or fusing two light nuclei (fusion) lie closer to the peak of the binding energy curve — i.e. they are more tightly bound than the starting nuclei — the mass difference between reactants and products is converted into a large amount of energy via E = Δmc². This is why heavy nuclei like uranium are used for energy production by fission in nuclear reactors, and why light nuclei (hydrogen isotopes) fusing is the source of energy in the Sun and stars, and is being pursued as a future fusion energy source.

A radioactive tracer is a radioactive isotope that behaves chemically/biologically just like the normal (stable) isotope of the same element, so it can be introduced into a system and its location and concentration followed by detecting the radiation it emits. By substituting radioactive atoms for ordinary atoms in a substance and tracking the 'tagged' atoms with a radiation detector, the path of a chemical or biological process can be followed.
Medical applications: Radio-iodine (iodine-131) is selectively absorbed by the thyroid gland and is used to check whether the thyroid is working normally — a diseased or overactive gland absorbs more than twice the normal amount — and the same isotope is used to treat thyroid cancer. Phosphorus-32 is taken up by bone and cobalt by the liver, so these serve as tracers for those organs. Sodium-24 is used to study the circulation of blood. Technetium-99m is used for thyroid uptake and kidney function tests. In some cancer treatments, radioisotope 'seed' capsules (e.g. phosphorus-32, strontium-90) are implanted directly in malignant tissue for localized, short-range radiation treatment, since rapidly multiplying cancer cells absorb more radiation and are destroyed more easily than normal cells.

Half-life (T½) is the time taken for the activity (or number of undecayed nuclei) of a radioactive sample to fall to half of any starting value.
Derivation: If a sample starts with N₀ undecayed nuclei, after one half-life N₀/2 nuclei remain. After a second half-life, half of these remain: (1/2) × (N₀/2) = (1/2)²N₀. After a third half-life: (1/2) × (1/2)²N₀ = (1/2)³N₀. In general, after n half-lives have elapsed, the number of undecayed nuclei remaining is N = (1/2)ⁿN₀, where the number of half-lives n = t/T½, t being the total elapsed time. So N = N₀(1/2)^(t/T½). This is consistent with the exponential decay law N = N₀e^(−λt), with T½ = 0.693/λ.
Remaining and decayed quantity: Once n = t/T½ is found, the remaining (undecayed) quantity is N = N₀(1/2)ⁿ, and the decayed quantity is simply N₀ − N = N₀[1 − (1/2)ⁿ].