OCR A-Level Physics/Fields, Particles and Frontiers of Physics/Nuclear Physics

Mass-Energy conservation

Einstein's mass-energy equation

Einstein showed that a particle's mass increases as its speed increases, although this is only noticeable at relative speeds. It links that change in energy, $\Delta E$, in joules equals the change in mass in kg, $\Delta m$ multiplied by the square of the speed of light in a vacuum, $c$.

$\Delta E=\Delta mc^{2}$

When energy is absorbed there is an increase in mass, and when there is a decrease in the mass of a system, and amount of energy equivalent to the change in mass is released.

For smaller quantities of energy we use the electron volt, eV, which is equivalent to 1eV = 1.6022 X $10^{-19}$

Annihilation

If a particle collides with its equivalent antiparticle (e.g. an electron and a positron), they destroy each other, giving a large amount of energy released as two photons of electromagnetic radiation. This is called annihilation. Since a particle has a mass equal to that of its equivalent antiparticle, the total energy before annihilation is equal to twice the mass of one of the particles multiplied by $c^{2}$;

${\textstyle \Sigma E=2mc^{2}}$

Since two photos are produced, the total energy is equal to twice the energy of each photon;

${\textstyle 2E_{\gamma }=2mc^{2}}$

${\textstyle E_{\gamma }=mc^{2}}$

So the energy of a photon resulting from annihilation is equal to the mass of one of the original particles multiplied by $c^{2}$.

The wavelength of the photons can also be determined using an equation from the quantum physics module;

${\textstyle E_{\gamma }={\frac {hc}{\lambda }}=mc^{2}}$

${\textstyle \therefore \lambda ={\frac {h}{mc}}}$

Where $h$ is the Planck constant which is equal to $6.63\times 10^{-34}$ Js.

Pair production

Diagram showing the pair production of an electron and a positron from a gamma photon

Whereas annihilation involves the conversion of mass into energy, pair production involves the conversion of energy into mass. If a photon has an energy equal to or greater than ${\textstyle 2mc^{2}}$, then it can produce a particle and an antiparticle, each of mass ${\textstyle m}$.

Mass defect

When the mass of a given nucleus is compared with the mass of its constituent nucleons, the total mass of the separated nucleons is always greater than the mass of the nucleus. This difference in mass is called the mass defect.

The mass defect is equal to the mass of the constituent nucleons minus the mass of the nucleus;

$m_{d}=m_{c}-m_{n}$

Binding energy

The difference between the mass of separate nucleons and the mass of the nucleus arises from the fact that all the nucleons are bound together by the strong nuclear force. This means that work has to be done to separate the nucleons, so the separated nucleons gain potential energy and by $\Delta E=\Delta mc^{2}$, they also have more mass.

The minimum energy needed to break up the nucleus into its constituent nucleons is called the binding energy of the nucleus. The binding energy is calculated using $\Delta E=\Delta mc^{2}$. The binding energy $E_{b}$ and the mass defect $m_{d}$ are therefore defined by the relation;

${\textstyle E_{b}=m_{d}c^{2}}$

Binding energy per nucleon (BEPN)

The binding energy will be greater if there are more nucleons to separate. If we divide the total binding energy of the nucleus by the number of nucleons (also known as the mass number $A$) we get the binding energy per nucleon, the greater this value the more stable the nucleus will be. The BEPN can be determined using this formula;

${\textstyle \textstyle {BEPN}={\frac {E_{b}}{A}}}$

The most stable isotope is Iron-56, which has a BEPN of 8.8 MeV.

Fusion

Nuclear fusion and binding energy

Nuclear fusion occurs when smaller nuclei join or fuse to produce heavier nuclei. As a result of the fusion reaction, the resulting nuclei have a binding energy per nucleon that is greater than the combined binding energy of the individual nuclei which has fused together. The difference in binding energy is released as kinetic energy of the resulting particles and as photons. Another way of looking at this is that the nucleus lose mass, which results in energy released.

Fusion occurs when nuclei of lighter elements fusion to increase their stability. Conversely, fission occurs when the nuclei of heavier elements split into small fragments to release energy and increase their nuclear stability.

Fusion reactions in stars

Fusion powers all stars. The reaction between two deuterons, which are the nuclei of an isotope of hydrogen called deuterium and comprise one proton and one neutron, looks quite simple: ${}_{1}^{2}\!\ H+{}_{1}^{2}\!\ H+3.6$MeV = ${}_{2}^{3}\!\ He+{}_{0}^{1}\!\ n+6.9$MeV

The problem with this reaction is that it requires an energy input of 3.6MeV per reaction. The two deuterons repel one another very strongly at close range due to electrostatic repulsion, so they will not normally fuse. Temperatures as high as 10 000 000 K are needed, so that the nuclei have a great deal of kinetic and approach close enough to experience the attractive strong nuclear force. Even then, many millions of close encounters of nuclei do not cause fusion, but there are a few that do.

The sun is a source of vast numbers of neutrinos and gamma-ray photons. Positrons formed will be quickly annihilated by electrons to produce gamma-ray photons in the Sun's plasma.

Fusion power on Earth

Fusion of two isotopes of hydrogen (deuterium and tritium) can be caused experimentally, but at present there is no commercial power station using fusion. Two big advantages of using fusion for power production would be that:

• There is no radioactive waste products are formed by the fusion process
• There is a virtually unlimited supply of the raw materials. About 1% of seawater molecules have a deuterium atom in them.

The energy carried by neutrons produced in fusion reactions will be used to generate electricity. The problem is to maintain a high enough temperature for long enough for sufficient fusion to take place. At present, any apparatus that can induce fusion requires much more electrical energy than it could produce.

In tokamak devices, a huge discharge through a deuterium-tritium gas mixture from a bank of capacitors is compressed by magnetic fields into a doughnut-shape ring, within which temperatures of perhaps a high as a hundred million degrees can be maintained for a few microseconds. This provides the temperatures needed for fusion.

Fission

Induced fission

In induced nuclear fission, the absorption of a slow-moving neutron causes a large nucleus to split into two smaller nuclei, more neutrons and an enormous amount of released energy. In a nuclear power station, the energy released by nuclear fission heats water to change it into steam, which is then blown into turbines at high pressure. The rotation of the turbines turns a generator to produce electricity.

Chain reaction

When uranium undergoes induced fission, it may split into a number of different isotopes, releasing a varying number of neutrons. If a neutron hits another uranium-235 nucleus it can induce further fission. if more than one neutron causes further fission, then the process can repeat itself, and the number of fissions can rapidly escalate in a chain reaction, this is what happens in a nuclear bomb.

Components of a fission reactor

In a nuclear reactor, the chain reaction is controlled by ensuring that on average only one of the neutrons produced by the fission of uranium-235 causes subsequent fission. In practice, some neutrons are absorbed by some uranium-238, which doesn't undergo fission, and some are absorbed by material in the reactor, leaving a small excess.

Control rods made of boron are used to absorb these neutrons and, by moving the rods in and out of the reactor, to control the reactors rate of operation.

Slowing down the neutrons

A neutron will only cause fission of a uranium-235 nucleus to occur if it is travelling at the correct speed. If it is going too fast, it is far less likely to cause fission than if it is going slowly. The neutrons, when they emerge from a fission reaction, are always moving fast so they have to be slowed down before they can cause a uranium-235 nucleus to undergo induced fission. This is done by atoms of the moderator material.

We assume the collisions are elastic collisions and so both momentum and kinetic energy. Some nuclear reactors use heavy water to moderate the speed of neutrons, but most reactors in the uk use carbon in the form of graphite.

Environmental impact of nuclear waste

Nuclear waste, or radioactive waste is radioactive material that is no longer useful. Soucres include military weapon production and testing, nuclear power stations and hospitals. Waste can be classified as high-level, intermediate-level or low-level waste depending on its activity. Waste can remain radioactive for a fraction of second to millions of years depending on the isotopes.

Everything on earth is exposed to ionising radiation as there is naturally occurring background radiation everywhere. However exposure to ionising radiation above the background level can cause harm to plant and animal life. So nuclear waste must be stored securely in the shielded containers until it's no longer radioactive.

High-level waste is material that produces large amounts of ionising radiation. It includes the fuel rods removed from the core of nuclear power station and waste resulting from the reprocessing of this fuel. High-level waste produces heat as a result of the rapid decay of some of the short-lived isotopes, so it needs coolin for a few years, as well as shielding to block radioactive emission over many of thousands of years while the longer half-life isotopes decay.

Intermediate-level waste includes material which has become radioactive because it has been in a nuclear reactor, such as the reactors metal cladding.

Low-level waste includes items which are only slightly radioactive due to becoming contaminated with small amounts of radioactivity, such as used cleaning materials and protective clothing.