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Nuclear Physics
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  1. The structure of matter
  2. Nuclides
  3. Isotopes
  4. Nuclear forces
  5. Chart of the nuclides
  6. Enriched and depleted uranium
  7. Conservation of mass and energy
  8. References

The structure of matter

Before we are able to understand the process of nuclear fission and the operation of nuclear reactors which harness this power, it is necessary to be able to understand and describe the nature of matter itself. All matter is composed of atoms, the smallest amount of matter that retains the properties of an element.

First, a run-down on the atomic timeline:

Figure 1: Bohr’s model of an atom

So what are these protons, neutrons and electrons? These are known as subatomic particles which form the basis of the atom. The properties of each are shown below:

ParticleLocationChargeMass
NeutronNucleusnone1.008665 amu
ProtonNucleus+11.007277 amu
ElectronShells around nucleus-10.0005486 amu

Table 1: Properties of subatomic particles

The unit measure for mass is the atomic mass unit (amu), one atomic mass unit is equal to \(1.66 \times 10^{-24}\) grams. Note that the mass of a neutron and a proton are both about 1 amu.

Nuclides

Each type of atom that contains a unique combination of protons and neutrons is called a nuclide. Each nuclide is denoted by the chemical symbol of the element with the atomic number written as a subscript and the mass number written as a superscript, as shown in Figure 2:

Figure 2: Nomenclature for identifying nuclides

Each element has a unique name, chemical symbol and atomic number. As such, only one is necessary to identify the element, so nuclides are commonly identified using the chemical name or symbol followed by the mass number i.e. \(U-235 \) or \(uranium-235 \) or even \(^{235}U \). Typically, this essay series will use the notation as shown in Figure 2.

Isotopes

Isotopes are nuclides that have the same atomic number and are therefore the same element but differ in the number of neutrons. Most elements have a few stable isotopes and several unstable, radioactive isotopes. For example, hydrogen has two stable isotopes (hydrogen-1 and hydrogen-2) and a single radioactive isotope (hydrogen-3).

Keep this idea of element stability in mind. It is central to the understanding of nuclear physics. Spend a bit of time thinking about what it might mean for an element to be unstable—you are not yet expected to know but this will prove a highly useful exercise regardless.

Nuclear forces

From Table 1 we saw that the Bohr model of the atom the nucleus consists of positively-charged protons and electrically neutral neutrons. As both exist in the nucleus, these are both referred to as nucleons.

You may remember from school that like charges repel and unlike charges attract. Now consider the repelling forces that must be acting upon the positively-charged protons within the nucleus—we have not yet accounted for what forces overcome such repelling forces which result from this scenario.

Two attracting forces are:

  1. Gravitational forces between any two objects that have mass; and
  2. Electrostatic forces between charged particles.

We can calculate the magnitude of both using the following principles from classical physics:

Gravitational force

Newton stated that the gravitational force between two bodies is directly proportional to the masses of the two bodies and inversely proportional to the square of the distance between the bodies:

$$ F_{g}=\frac{G \times m_{1} \times m_{2}}{r^{2}} $$

Where:

\(F_{g} \) = gravitational force (newtons)
\(m_{1} \) = mass of first body (kilograms)
\(m_{2} \) = mass of second body (kilograms)
\(G \) = gravitational constant (\(6.67 \times 10^{-11} Nm^{2}/kg^{2}) \)
\(r \) = distance between particles (metres)

Thus, the larger the masses of the objects or the smaller the distance between the objects, the greater the gravitational force. Despite extremely small nucleon masses, the distance between each may result in a significant gravitational force. As we see below, however, protons separated by 1 fm \((10^{-15} m) \) generate only a tiny gravitational force:

$$ \eqalign{ F_{g}&=\frac{(6.67 \times 10^{-11})(1.66 \times 10^{-27})^{2}}{(10^{-15})^{2}} \cr &\approx 1.84 \times 10^{-34}N} $$

Electrostatic force

Coulomb's Law can be used to calculate the force between two protons. The electrostatic force is directly proportional to the electrical charges of the two particles and inversely proportional to the square of the distance between the particles:

$$ F_{e}=\frac{K \times Q_{1} \times Q_{2}}{r^{2}} $$

Where:

\(F_{e} =\) electrostatic force (newtons)
\(K =\) electrostatic constant \((9.0 \times 10^{9} Nm^{2}/C^{2}) \)
\(Q_{1} =\) charge of first particle (coulombs)
\(Q_{2} =\) charge of second particle (coulombs)
\(r =\) distance between particles (metres)

$$ F_{e}=\frac{8.99 \times 10^{9} \times (1.6 \times 10^{-19})^{2}}{(1 \times 10^{-15})^{2}} \approx 230N $$

The gravitational force is negligible in magnitude compared to the electromagnetic force and as such can be neglected.

With the (relatively) large electrostatic force repelling the nucleons and only a minuscule gravitational force attempting to hold the nucleons together, we would expect to see unstable nuclei. Since stable atoms of neutrons and protons do exist, there must be another attractive force. We call this the nuclear force.

The nuclear force is a strong attractive force, independent of charge, which acts equally only between pairs of neutrons, pairs of protons, or a neutron and a proton. The nuclear force has a very short range—acting only over distances approximately equal to the diameter of the nucleus \((10^{-15} m) \). Table 2 below summarises the forces acting within the nucleus:

ForceInteractionRange
GravitationalVery weak attractive force between all nucleonsRelatively long
ElectrostaticStrong repulsive force between like charged particles (protons)Relatively long
Nuclear ForceStrong attractive force between all nucleonsExtremely short

Table 2: Summary of forces

Stable atoms will balance the attractive and repulsive forces within the nucleus; without this balance the atom is considered unstable and the nucleus will emit radiation in an attempt to reach a stable configuration.

Chart of the nuclides

This is a tabulated chart which lists the stable and unstable nuclides in addition to pertinent information about each one. Figure 3 below shows a portion of this chart, from atomic numbers 1 to 6:

Figure 3: Nuclide chart for atomic numbers 1 to 6

As you can see, protons (Z) are on the horizontal axis with neutrons (A-Z = N) on the vertical. The grey squares indicate stable isotopes while the white squares are considered artificially radioactive, meaning they do not occur naturally.

Figures 4 and 5 below provide further clarification on the nuclide detail shown in the chart:

Figure 4: Stable nuclide data

The Chart of the Nuclides presents the relative abundance of the naturally occurring isotopes of an element in units of atom percent. Atom percent is the percentage of the atoms of an element that are of a particular isotope.

Further, the atomic weight for an element is defined as the average atomic weight of the isotopes of the element. The atomic weight for an element can be calculated by summing the products of the isotopic abundance of the isotope with the atomic mass of the isotope. For example, the element lithium:

$$ \eqalign{ &Atomic \ Weight \ Lithium \cr &= (0.075) \times (6.015122 \ amu) \cr &+ \ (0.925) \times (7.016003 \ amu) \cr &= 6.9409 \ amu} $$

Figure 5: Unstable nuclide data

Enriched and depleted uranium

Natural uranium, mined from the earth, contains the following isotypes and their respective proportion of atoms:

Although each isotope has similar chemical properties, their nuclear properties are significantly different; uranium-235 is usually the desired material for reactors, as we will see later in this series.

A vast amount of equipment and energy is expended to perform a process known as enrichment, whereby feed material with the naturally-occurring proportions seen above is used to produce uranium where the isotope uranium-235 has a concentration greater than its natural value. This process will also result in a by-product of depleted uranium; where uranium-235 is less concentrated than in naturally-occurring uranium.

Conservation of mass and energy

Before moving onto the topic of radioactivity, we will round out with another piece of the atomic puzzle—a single nuclear conservation law stating that the sum of mass and energy is conserved. Mass does not appear and disappear at random, an increase in mass results in an accompanying decrease in energy and vice-versa.

Mass defect

The mass defect is the difference between the mass of the atom and the sum of the masses of its constituent parts. This can be calculated using the equation below:

$$ \Delta m = [ Z(m_{p}+m_{e})+(A-Z)m_{n}]-m_{atom} $$

Where:

\(\delta m =\) mass defect (amu)
\(m_{p} =\) mass of a proton (1.007277 amu)
\(m_{n} =\) mass of a neutron (1.008665 amu)
\(m_{e} =\) mass of an electron (0.000548597 amu)
\(m_{atom} =\) mass of nuclide \(_{Z}^{A}\textrm{X} \) (amu)
\(Z =\) atomic number (number of protons)
\(A =\) mass number (number of nucleons)

Binding energy

Binding energy is directly tied to the mass defect, as in the formation of a nucleus the mass defect is converted to binding energy (BE). Binding energy, then, is the amount of energy that must be supplied to a nucleus to completely separate its nuclear particles (nucleons); the amount of energy that would be released if the nucleus was formed from the separate particles. Note: \(1 \ MeV = 1.6022 \times 10^{-13} \ joules \).

$$ B.E. = \Delta m \left [ \frac{931.5 \ MeV}{1 \ amu} \right ] $$

Put simply, the binding energy is the energy equivalent of the mass defect.

Nucleus energy levels

The nucleons in the nucleus of an atom, like the electrons that circle the nucleus, exist in shells that correspond to energy states. There is a state of lowest energy (the ground state) and discrete possible excited states for a nucleus. These states may be illustrated in an energy level diagram as seen in Figure 6 below, for the isotope nickel-60:

Figure 6: Energy level diagram for Nickel-60

Each horizontal bar represents an excited energy state of the nucleus; the vertical distance between the bars and the ground state is proportional to the energy level between the two states, also known as the excitation energy. A nucleus in an excited state will not remain at that level indefinitely—nucleons in an excited nucleus will transition towards their lowest-energy configuration and, in doing so, emit a discrete bundle of electromagnetic radiation called a gamma-ray \((\gamma -ray)\).

References