HSC Physics - Quanta to Quarks notes - course summary

HSC Physics – Quanta to Quarks notes

This is a set of HSC Physics dot-point summary notes for Quanta to Quarks. HSC Physics tutoring at Dux College provides students with the right support to achieve a band 6 result in HSC Physics.

Contents

1 HSC Physics – Quanta to Quarks notes
2 The Rutherfod-Bohr Model
3 Birth of Modern Quantum Physics
4 Nuclear Physics

Option Module: From Quanta to Quarks

The Rutherfod-Bohr Model

Rutherford’s atom

In 1907, Rutherford set up a simple experiment where he bombarded a thin gold foil with alpha particles. He noticed that a very small portion of alpha particles (around 1 in 8,000) were reflected back from the gold foil. The vast majority of alpha particles sent at the gold foil were scattered or simply passed through the foil.

These results led to Rutherford’s conclusion that an atom must consist of a nucleus of densely packed positive charge.

He calculated that at the centre of each gold atom was a region about 10,000 times smaller than the radius of the atom, containing all of the positive charge of the atom. Today, we now know this is the nucleus of the atoms, containing densely packed protons and neutrons.

Rutherford, however, could not account for the behaviour of electrons. He correctly identified that the electrons orbited around the nucleus of each atom, but according to classical mechanics, accelerating charges would emit EM radiation, causing the electrons to lose energy, and eventually collide with the nucleus. This is clearly not the case, so Rutherford’s model was incomplete.

Bohr’s atom

Bohr’s main contribution to Rutherford’s model of the atom is his success at explaining the behaviour of electrons. The main limitation of Rutherford’s model of the atom is his inability to explain why electrons did not lose energy in the form of EM radiation emission as electrons were constantly accelerating in their circular orbit around the nucleus.

Bohr solved this problem by applying quantum theory (pioneered by Planck, Einstein and Boltzmann) to explain the nature of electrons around atoms.

Significance of the hydrogen spectrum

Bohr was able to derive the Balmer series (an empirical equation to predict the wavelengths of light emitted by excited hydrogen atoms) by applying quantum theory to the model of the atom. The Balmer series is described as:

$\frac{1}{\lambda }=R_{H}(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}})$

Where:

$\lambda$ is wavelength of emitted radiation
$n_{f}$ and $n_{i}$ are integers corresponding to the energy levels
of the electron in a hydrogen atom
$R_{H}$ is Rydberg’s constant ( $R_{H}=1.097\times 10^{7}m^{-1}$ ,
this is given in your formula sheet)

Note: this equation is in your HSC formula sheet.

Bohr’s model and its success at deriving the Balmer series gave it credibility as it was able to account for the behaviour of electrons in a hydrogen atom. However, Bohr still did not quite understand why electrons did not fall to the nucleus.

Up to this point, Bohr’s model is also known as the Rutherford-Bohr model of the atom, due to the incorporation of Rutherford’s initial ideas.

Bohr’s postulates

While Bohr believed that he knew the arrangement of electrons, he could not explain why the electrons were arranged in this way. He started with the problem of electrons in the Rutherford model and pointed out that the accelerating electrons must lose energy by radiation and collapse into the nucleus.

In his Nobel lecture stated in reference to Rutherford’s discovery of the nucleus, Bohr made close reference to the quantum theory and proposed two postulates:

Electrons in an atom exist in ‘stationary states’ in which they possess an unexplainable stability. Any permanent change in their motion must consist of a complete transition from one stationary state to another.
In contradiction to the classical electromagnetic theory, no radiation is emitted from an atom in a stationary state. A transition between two stationary states will be accompanied by emission or absorption of electromagnetic radiation (a photon): $hf=E_{1}-E_{2}$

Where E₁ and E₂ are the energy values of the stationary states

An electron in a stationary state has an angular momentum that is an integral multiple of $\frac{h}{2\pi}$ .

The significance of these postulates is that Bohr proposed that the energy of electrons in atoms are quantised (his third postulate), and that electrons orbiting an atom do not radiate energy, except in set packets emitted when an electron moves from one stationary state to another (postulates 1 and 2).

Bohr’s model of the hydrogen atom is almost correct, in that it implies electron orbitals can only take on certain discrete levels.

The diagram below explains how the various emission spectra of a hydrogen atom is explained in terms of Bohr’s model.

Photons of specific energies are released when an electron moves from a higher stationary state to a lower one. These photons have energies as described by $hf=E_{1}-E_{2}$ . For the Balmer series, the Balmer equation can give the exact wavelength of the photons emitted.

Difficulties with the Rutherford-Bohr model

Under the Rutherford-Bohr model of the atom, there was no explanation for there being no energy emission from accelerating electrons as Maxwell predicted. Instead it was simply assumed in terms of Bohr’s first postulate, as “inexplicable stationary states” which electrons can occupy without losing energy.

Further difficulties with the model include its inability to explain the following phenomena:

The spectra of larger atoms

The Bohr model could not explain the spectra of larger atoms with more than one electron. The Bohr-Rutherford model could only be applied to hydrogen atoms with 1 electron, and the mathematics behind predicting wavelengths of spectra could not applied to atoms with more than one electron.

The relative intensity of spectral lines

When observing spectra, some lines were much brighter than others. The Rutherford-Bohr model could not explain why some lines were more intense than others (why some electron transitions were preferred to others).

The existence of hyperfine spectral lines

When the spectral lines were examined closely, it was observed that each line actually consisted of many small lines, the existence of which the Bohr model could not explain as it only predicted on clear line for each transition.

The Zeeman Effect

The Zeeman Effect occurs when a magnetic field is passed through the discharge tube. The magnetic field increased the hyperfine splitting of spectral lines, further breaking them up. Again, the Bohr model was unable to explain the experimental evidence.

Although the Rutherford-Bohr model lay down the framework for the quantum model of the atom, which ended in a scientific revolution out of which quantum mechanics, a vital part of modern physics emerged; it was left to future scientists such as Pauli and Heisenberg to fully explain these phenomena.

Birth of Modern Quantum Physics

De Broglie and wave-particle duality

De Broglie suggested that light quanta behaved as particles and as waves at the same time. He proposed the idea that both types of behaviours (as a wave and a particle) were inextricably linked and one and the same in many respects. By thinking in this way, De Broglie came up with an expression for the momentum of a photon.

$E=hf=mc^{2}$

And since $p=mv$ for normal particles, a photon would have a ‘momentum’ of $p=mc$ .

$p=\frac{E}{c}=\frac{hf}{c}$

This behaviour of light is called the wave-particle duality, because by thinking of light as a particle in some situations, and as a wave in others, many physical phenomena could be explained.

Wave particle duality of electrons

De Broglie extended this idea of wave-particle duality to electrons. He thought of them as behaving like a wave, as well as a particle. He applied the above equation to electrons:

$p=\frac{hf}{c}$
$p=\frac{h}{\lambda }$

Rearranging and substituting $p=mv$ for particles, we get:

$\lambda =\frac{h}{mv}$

According to this framework of thinking of electrons as waves, we can calculate the wavelength (and frequency) of electrons (in fact, any other moving particle, such as a tennis ball). Thinking about particles as behaving as waves was called ‘matter waves’. Furthering this line of thought, De Broglie theorised that it should be possible to observe diffraction patterns of an electron beam that is shined onto the surface of a crystal.

In 1927, Davisson and Germer accidentally discovered that an electron beam was diffracted when shined onto the surface of an annealed block of nickel. This provided experimental support for De Broglie’s matter waves. The interference patterns they noticed in their experiments corresponded to the wavelengths predicted by $\lambda =\frac{h}{mv}$ . Observing a diffraction pattern was experimental evidence in support of the fact that electrons behaved as waves, because diffraction was a behaviour of waves (like reflection and refraction).

The significance of De Broglie’s treatment of electrons as waves successfully accounted for why electrons did not seem to lose energy as they orbited atomic nuclei. This would be discussed in the next section.

Diffraction

Diffraction refers to the behaviour of light as it passes through a thin grating, or reflected off a surface with fine lines. Diffraction is the cause of why we see a rainbow spectrum when we look at CDs.

$De Broglie wavelength diffraction$

When a wavefront passes through a small opening, it tends to spread out around the edges of the opening. As it does this, interference patterns form. By looking at the interference patterns, and analysing the distances between the regions of constructive and destructive interference, we can calculate the wavelength of the light.

$De Broglie noticed particles exhibit wave properties such as diffraction$

Bohr’s electron orbits explained

The cornerstone of De Broglie’s idea was that the electron orbiting the atom must have a standing-wave pattern of vibration so that its orbit does not destructively interfere with itself. Since the orbital level represents an energy level, only electron energy levels where the electron orbits the nucleus with a standing wave pattern consisting of a ‘whole’ number of wavelengths is possible. Intermediate electron energy levels cannot be stable as they would produce a destructive interfering wave character.

If the circumference is taken as 2pr, then there are n wavelengths in the circumference:

$n\lambda =2r$

And since the De Broglie wavelength is:

$\lambda =\frac{h}{mv}$
We have:

$\frac{nh}{mv}=2r$
$mvr=n\times \frac{h}{2\pi }$

The LHS is the angular momentum of the electron. This is Bohr’s quantisation condition that angular momentum can exist only in integer multiples of $\frac{h}{2}$ . The \textbf{quantised electron orbits of Bohr can be explained by De Broglie’s proposal that particles have a wave nature} and their wavelength is $\lambda =\frac{h}{mv}$ .

Contributions of Heisenberg and Pauli

Heisenberg and Pauli both thought Bohr’s atomic model was unsatisfactory as it combined aspects of classical physics with the non-classical concept of quantisation of energy. So they developed mathematical descriptions of the atom.

Heisenberg and the Uncertainty Principle

Heisenberg’s treatment of quantum mechanics was based on observable things, e.g. spectra and ignored unobservable things, e.g. atomic electron orbits. He used wave-particle duality to predict allowable energy levels for electrons.

Heisenberg also formulated his Uncertainty Principle that says that the position and momentum of a particle cannot be measured simultaneously. A consequence of this is that the electron orbits postulated by Bohr could not actually exist. It led to the idea of ‘electron clouds’ rather than orbits where an electron’s position could not be determined but became a statistical probability. This is represented in Heisenberg’s uncertainty principle:

$\Delta x\times \Delta p\ge \frac{h}{2}$

$\Delta x$ represents the uncertainty of a particle’s position, and $\Delta p$ represented the uncertainty of a particle’s momentum. On the RHS is a constant. This simple equation implies that as $\Delta x$ or $\Delta p$ approaches 0, the other must approach infinite. That is, the more precisely we measure position, the less precise momentum becomes, and vice versa.

This proposal by Heisenberg represented a huge contribution to atomic theory, as it made physicists rethink the way they designed their experiments, and caused them to have more realistic expectations regarding the precision they could hope to achieve with their measurement.

Pauli and the Exclusion Principle

Pauli took the quantum mechanical model that was entirely theoretical and applied it to the hydrogen atom to derive the Rydberg constant and to develop Balmer’s equation. He used Bohr’s idea of shells of electrons and developed the Exclusion Principle in which he introduced a new quantum number known as ‘spin’ to explain the maximum number of electrons in the energy levels.

This principle states that no two electrons can have the same set of four quantum numbers. Pauli’s Exclusion Principle accounted for the maximum number of electrons in each shell around a nucleus, and derived it to be $2n^2$ where is the shell number. Pauli’s Exclusion Principle was later discovered to apply to all Fermions (a class of subatomic particles, of which electron is one).

The consequences of this principle are far-reaching. For example, it explains why the heavier atoms are so much larger than the lighter ones. The extra electrons have to find space in new orbitals further and further out. It also explains why, even at very low temperatures, most conduction electrons in a metallic crystal are still in high energy states, and very mobile.

Nuclear Physics

Nucleons and Transmutation

Nucleons are particles that normally reside in the nucleus.

Properties	Proton	Neutron
Charge	$+1.6\times 10^{-19}C$	0
Mass	$1.673\times 10^{-27}Kg$	(slightly heavier than a proton)

Transmutation is the process by which the constituents of a nucleus in an atom are altered as a result of either radioactive decay (alpha or beta), nuclear fusion or nuclear fission.

Natural transmutation

Artificial transmutation

Natural radioactive transmutations are those that have not been artificially induced. Natural transmutations include radioactive decays of a lone element into another element.
Natural nuclear transmutations usually occur either through alpha or beta decays. The release of gamma rays does not bring about a transmutation.
The key cause to radioactive decay is an unstable neutron to proton ratio. Generally, atoms with a high $n/p$ ratio are subject to beta-minus decay, and atoms with a low $n/p$ ratio are subject to alpha decay and beta-plus decay.

Artificially induced nuclear transmutations would involve either nuclear fusion or nuclear fission.
Nuclear fusion involves slamming two light nuclei into each other at high speeds, causing them to fuse into a heavier single atom.
Nuclear fission involves firing neutrons or other particles at a heavy atom, causing it to split into two or more daughter nuclei. Both processes also occur naturally, (e.g. at the centre of the Sun) but in most contexts, they are artificial.

Alpha Decay

Alpha decay involves the emission of an alpha particle from the nucleus, which is essentially a helium nucleus without any electrons: ${}_{2}^{4}He^{2+}$

It is important to be able to write nuclear reactions that involve these transmutations, and the key rules to writing these reactions are to remember that the mass numbers and atomic numbers on both sides of the reaction should add up.

An example of alpha decay:

${}_{92}{}^{238}U\to {}_{90}{}^{234}Th+{}_{2}{}^{4}He$

No additional particles are emitted when an alpha emission occurs. Alpha emissions usually occur to atoms with an excess of protons. An equal number of protons and neutrons are lost (i.e. 2 of each)

Beta Decay

Beta decay is slightly more complex than alpha decay, and there are in fact two types of beta decay, beta-plus and beta-minus decay:

Beta-minus Decay

Beta minus decay involves when a neutron is converted to a proton, emitting a Beta particle (i.e. an electron ${}_{-1}^{0}e$ ) and an antineutrino ( ${}_{0}^{0}\bar{v}$ ), which is the antiparticle of the neutrino: a particle that has a tiny mass and no charge. The proton remains in the nucleus. Thus, the process may be summarised by:

${}_{0}^{1}n\to {}_{1}^{1}p+{}_{-1}^{0}e+{}_{0}^{0}\bar{v}$

A memory device to remember what beta minus decay entails is that it is “minus” because it involves the emission of a negatively charged electron.

Beta-plus Decay

Beta plus decay is really the opposite to beta minus decay, in which a proton is converted into a neutron, emitting a positron ( ${}_{1}^{0}\bar{e}$ ) and a neutrino ( ${}_{0}^{0}v$ ), which has a tiny mass and no charge. Again, the neutron remains in the nucleus. The process may be summarised by:

${}_{1}^{1}p\to {}_{0}^{1}n+{}_{1}^{0}\bar{e}+{}_{0}^{0}v$

Beta plus decay is “plus” because it involves the emission of a positively charged positron (antielectron).

Chadwick’s discovery of the neutron

In 1930, Bothe and Becker bombarded a beryllium sheet with alpha particles. What they found was that the beryllium emitted an unknown radiation that was not electrically charged. At the time, this radiation was thought to be gamma radiation. 2 years later, Curie and Joliot discovered that this radiation could knock out fast protons from a block of paraffin wax.

The energy of these protons was measured to be 5 MeV, implying that if the unknown radiation was gamma radiation, it would have had an energy of 50 MeV per photon (you don’t need to know the mechanics of this calculation). This was problematic, because the original alpha particles only had an energy of 5MeV each. This implied that there was a tenfold increase from the alpha particles in creating the unknown radiation, and hence violated the law of conservation of energy.

When Chadwick learned of this experiment and the strange results, he proposed the existence of a neutron. By going through the calculations of energy and momentum using his hypothesis that the unknown radiation were neutrons, he discovered that he could account for the energy difference and there was no longer a violation of the conservation laws. Thus, this experiment provided strong evidence in support of Chadwick’s proposal of the existence of the neutron.

Fermi’s first encounter with nuclear fission

Enrico Fermi and his collaborators in Rome discovered that if various elements are bombarded by neutrons, new radioactive elements are produced. Fermi had predicted that the neutron, being uncharged, would be a useful nuclear projectile. Unlike the proton or alpha particle, it experiences no electrostatic repulsion or attraction with other subatomic particles. Because there is no electrostatic barrier, even very slow neutrons can penetrate and interact with even the most massive, strongly positively charged nucleus.

Fermi successfully used this method on a series of elements. When he used uranium-235 as the target of slow neutron bombardment, he obtained puzzling radioactive substances that could not be identified.

Fermi’s colleagues were inclined to believe that he had actually made a new, ‘transuranic’ element of atomic number 93. That is, during bombardment, the nucleus of uranium had captured a neutron, which beta-decayed into a proton, increasing the atomic number. Fermi did not make this claim, for he was not certain what had occurred, but he in fact had observed the first artificial nuclear fission.

Fermi’s discovery of the neutrino

Energy of electrons from beta decay

All alpha particles emitted from a particular radioactive species had the same energy but beta particles seemed to be emitted with a range of energies. This was problematic because even beta decays from the same type of nucleus, resulting in the same type of nucleus, released electrons with different kinetic energies.

To explain this, Pauli suggested that beta decay may be accompanied with the release of yet another type of neutral particle to carry away the missing energy that caused the seemingly random distribution of electron kinetic energies. Following from this, Fermi accounted for the range of kinetic energies by suggesting that a very light neutral particle, called the neutrino, was released with each beta decay. However, detection of the neutrino was extremely difficult, since the particle was extremely light, travelled at near c, and had no electric charge.

Today, we now know that in fact an antineutrino is released with each beta-minus decay:

${}_{0}^{1}n\to {}_{1}^{1}p+{}_{-1}^{0}e+{}_{0}^{0}\bar{v}$

And a neutrino is released with each beta-plus decay:

${}_{1}^{1}p\to {}_{0}^{1}n+{}_{1}^{0}\bar{e}+{}_{0}^{0}v$

The Strong Nuclear Force

Electrostatic and Gravitational forces

The force of gravitation between two bodies of mass is given by Newton’s Law of Universal Gravitation:

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

Similarly, the force of electrostatic attraction between two oppositely charged particles is given by:

$F_{E}=\frac{1}{4_{0}}\times \frac{q_{1}q_{2}}{r^{2}}$

Therefore, the ratio of the gravitational force to the electrostatic force between two protons is given by:

$\frac{F_{G}}{F_{E}}=\frac{Gm_{1}m_{2}}{r^{2}}\times \frac{4_{0}r^{2}}{q_{1}q_{2}}$

Substituting in the values from the data sheet gives the gravitational force to be smaller than the electrostatic force by a factor of $8.1\times 10^{-37}$ . Clearly, the attractive force of gravity between nucleons is so small as to be insignificant when compared to the electrostatic repulsion between protons. There must be another force present in the nucleus to hold the protons together.

The strong nuclear force

Because the gravitational attraction between nucleons is insignificant when compared to the electrostatic repulsion between the protons, it is clear that there must be another force present in the nucleus that holds the nucleons together. This force is called the strong nuclear force.

The properties of the strong nuclear force include:

An independence of charge and a similar force between proton-proton, neutron-neutron and proton-neutron.
A very strong attractive force, much stronger than the electrostatic repulsion between protons.
A very short range force acting over a distance of only about 10^-15 m. Every proton in a nucleus repels every other proton but the strong nuclear force exists only between a nucleon and its nearest neighbours. This is indicated by the almost uniform density of nuclear matter and also by the nearly uniform binding energy per nucleon.
A favouring of the binding of pairs of nucleons with opposite spins and pairs of pairs with each pair having a total spin of zero. (This helps account for the exceptional stability of two protons and two neutrons in an alpha particle).

Note that the strong nuclear force actually technically refers to the force holding quarks together within each nucleon (proton or neutron). The residual nuclear force is the force that holds protons and neutrons together in an atom. Also note that the strong nuclear force is not a force which obeys the inverse square law. It is the only force which curiously does not obey the inverse square law.

Mass Defect

The actual mass of a nucleus is always less than the sum of the masses of the constituents of the nucleus. This means that a helium nucleus with 2 neutrons and 2 protons has less mass than the combined mass of 2 neutrons and 2 protons measured separately. This implies that there is ‘missing mass’ and this missing mass is called mass defect.

The mass defect of an atom represents the energy that was released when the constituents became bound together by the strong nuclear force. It is also the same amount of energy needed to convert an atom back into its separate constituent protons and neutrons. As a result, this is also called binding energy.

The mass defect is calculated by taking the difference between the mass of the nucleus and the sum of the masses of its constituents. Einstein has shown that this mass loss has an energy equivalent, B, the binding energy of the nucleus. There are two ways to do mass defect calculations – using SI units and using AMU / eV.

Using SI units

Using atomic mass units / electron volts

$E=mc^2$ Where

E is the binding energy, in J
m is the mass defect, in kg
c is the speed of light, in m/s

$E=931.5\times m$ Where

E is the binding energy, in MeV (million electron volts)
m is the mass defect, in atomic mass units

Evidently, it can be more convenient to first express variables in terms of atomic mass units and MeV before doing mass defect calculations, but it all depends on the question you are given. E.g. if the question gives numbers to you in MeV and AMU, you should use the second equation, but if the numbers are in SI units, then use the first equation.

Fermi’s nuclear chain reaction

In 1942, in Chicago, Enrico Fermi had built the world’s first atomic reactor. He demonstrated that a chain reaction could indeed be established in uranium and, it could be controlled.

On 2 December 1942 in a squash court, Fermi and his team set up the first functioning nuclear reactor, involving a fission chain reaction. They used natural uranium as their nuclear fuel (containing only 0.7% fissile U-235, the rest being the non-fissile U-238). Fermi had known that the neutrons fissioned uranium nuclei more efficiently when they were traveling more slowing. This was because slower moving particles had a longer de Broglie wavelength, and this made them easier to be captured by large atomic nuclei.

As a result, Fermi incorporated into his atomic pile blocks of graphite alternating with the uranium rods. When the neutrons collided with the light carbon atoms in the graphite, they were slowed down. The graphite acted as the moderator, because its role was to slow down energetic neutrons.

Fermi knew that cadmium had the ability to absorb neutrons harmlessly, so he had cadmium rods inserted into his pile as a control. As these rods were slowly partially withdrawn from the pile, the pile started to heat up. What in fact was occurring was a controlled chain reaction. The control rods were withdrawn just enough for the reaction to be self-sustaining and give off some heat, but not enough for the reaction to go out of control.

Through this experiment, Fermi had shown that controlled nuclear energy production was a possibility.

The Manhattan Project

The Manhattan Project was one of the most significant scientific undertakings of the 20^th century because of the dramatic impacts it had on society. The project was led by US scientists, and included efforts from the UK and Canada to produce nuclear weaponry, which were eventually successful and resulted in the use of nuclear weapons on Japan in 1945.

The project led to many discoveries that formed the basis of our modern knowledge of nuclear technology. From these discoveries, we have developed a greater understanding of nuclear processes and their potential applications to power generation, radioisotope manufacture and particle physics.

Positive impacts

Negative impacts

Nuclear reactors: The project funded to develop the required technology incorporating fuel rods, moderator and control rods, to produce a sustained, controlled nuclear fission reaction and develop the first nuclear reactor.
Discovery of plutonium: It was within the fuel rods of this reactor that the element plutonium, used as fuel for the second bomb, was first produced.
Green energy: Nuclear power is a practical solution to replacing burning fossil fuels for our main power needs, as climate change concerns become stronger in the coming decades.
Nuclear medicine: The findings along with the development of nuclear reactors led to the advent of nuclear medicine. E.g . radioisotopes are now used to treat malignant cancers, or as tracers in medical imaging.
Isotope separation: only 0.7% of uranium is the isotope U-235 which was needed for a uranium fission bomb, thus scientists had to develop methods to separate it from natural uranium. These methods have enabled many other natural isotopes to be separated, e.g. O-18, C-13.
Ended the war: the use of the atomic bombs on Japan in August 1945 prompted their unconditional surrender, marking the end of WWII. The war may have continued for longer, leading to more deaths and economic damage if it were not for the project.
Greater understanding of nuclear physics: In general, the main legacy of the Manhattan Project was in starting an era of research into nuclear physics. Much of what we know today about nuclear processes may not have been known if it were not for the project.

Deaths from atomic bombs: The bombs may have resulted in Japan’s surrender, however many innocent people were killed and many more wounded. Hiroshima and Nagasaki needed rebuilding. Radiation affected unborn children and damaged the reproductive organs of many survivors leading to many miscarriages and an increase in deformities in newborns. Radiation illness caused many deaths in subsequent decades.
Radioactive fallout: Radioactive fallout settled on plants and entered the food chain affecting animals including humans, for many years as β-emitters concentrated in their organs causing tissue damage and cancers. This led to the signing of a world treaty in 1963 banning atmospheric testing of nuclear weapons.
Nuclear waste: Nuclear power plants generate nuclear waste, such as spent fuel rods and radioactive ‘heavy water’ which remains radioactive for millions of years. This requires them to be buried deep in the ground to prevent their radiation from affecting humans, plants and animals.
Nuclear proliferation: There is a constant fear of nuclear weapons falling into the wrong hands (e.g. terrorists or rogue nations like North Korea – which has already happened).
The Cold War: The project led to the ‘Cold War’ and an arms race initially between the USA and USSR, but later involving other nations. This caused significant economic damage, as many of each nations’ economic resources were diverted in producing and stockpiling nuclear weapons, instead of benefitting humans in other ways. The Cold War also gave rise to the real risk of all-out nuclear war that could have been much more devastating than any world war we’ve witnessed so far.

Assessment

In answering an assessment question on the Manhattan Project (e.g. “assess the impact of the Manhattan Project on world history”), you will need to come to a conclusion about whether the impact has overall been positive or negative, and back your conclusion with arguments. Depending on whether you predominantly include positive or negative arguments in your response, your concluding assessment of the impact on society should reflect your arguments. There is no right or wrong answer in arguing that the project was positive or negative.

Nuclear Chain reactions

Nuclear reactions occur as chain reactions and can be used to generate power, or provide the destructive force behind powerful nuclear bombs.

Controlled fission reactions

Uncontrolled fission reactions

The essence of a controlled fission reaction is that each fission that occurs should cause one subsequent fission of another nucleus. Such a controlled reaction is called a critical reaction and the amount of fissionable material of given purity that can provide that one-for-one capture probability is called the critical mass.

An uncontrolled fission involves a process whereby more than one nucleus is caused to undergo fission as a result of the fission of a single nucleus. Obviously this leads to an exponentially growing rate of reaction which increases extremely rapidly, causing a violent explosion of energy. This type of reaction is produced in an atomic bomb. For such a reaction to occur, the amount of fissionable material of a given purity brought together in one lump must exceed the critical mass.

Uncontrolled fission reactions in nuclear bombs

For a nuclear fission explosion to occur, a critical mass of suitable fissile material/fuel is needed e.g. U-235 or Pu-239. When neutron induced fission takes place, 2-3 neutrons are released per fission. This is accompanied by a very large energy release.

If the fissile material is larger than its critical mass, an uncontrolled chain reaction is able to develop. In such reactions, most neutrons from each fission reaction go on to produce further fission reactions. The number of fissions and their associated energies quickly increase and if unchecked the result is a violent explosion. This is produced in an atomic bomb.

To explode such a bomb, the sub-critical masses of the fuel elements are brought together by a chemical explosion (using a gun-type device or an implosive device, as shown above), so that the amount of fissionable material brought together exceeds the critical mass in a short period of time.

Controlled fission reactions in nuclear power plants

Fission reactions that are controlled (such that reaction rate is carefully monitored and kept constant) occurs in a nuclear reactor. A nuclear reactor is designed so that the chain reaction is sustained at an equilibrium rate and does not expand. This control of the fission reaction is achieved by:

Using fuel in the form of rods separated from one another so that they are near the critical mass density
Using control rods made from a neutron absorbing material (cadmium) to control the number of neutrons available in the reactor so that the chain reaction cannot expand beyond the required rate
Using a moderator (e.g. heavy water, graphite or liquid metals) which slows high energy neutrons down to the lower energies needed for fission to occur.

The level of activity in the reactor can be controlled by the number of fuel or control rods that are inserted into the reactor core. A fluid (e.g. water, or liquid metal) is circulated through the reactor to remove the heat generated that may be used directly or indirectly to run turbines to produce electricity. A vast array of remote sensors and instruments continuously monitor the operation of the reactor to ensure safety.

Fission Reactor

A fission reactor uses a controlled nuclear reaction to generate electricity. In a nuclear reactor, heat from the reaction is used to produce steam which turns a turbine.

The fuel

Most reactors use enriched uranium (uranium with a high portion of U-235), or plutonium-239. Fuel is usually uranium oxide pellets inside steel fuel rods that can be inserted or removed to help control the rate of reaction. Each fuel rod contains a sub-critical mass of fissionable material. When several of these rods are arranged vertically in the reactor core at suitable close distances in a geometric array, the effect is that a critical density of fissionable material is achieved.

Moderator

A moderator, either heavy water (H₂O with deuterium or tritium instead of hydrogen-1), graphite blocks or liquid metal (as above), is used to fill the space between the fuel rods. The purpose of the moderator is to slow the neutrons down through multiple collisions. Slow neutron will be captured by atoms of the fissionable material and cause those atoms to undergo fission.

The fact that low speed neutrons rather than high-speed neutrons are more likely to be captured by a uranium-235 nucleus is associated with the de Broglie wavelength of the neutron. The moderator must contain light atoms because neutrons lose more energy when colliding with light atoms (by giving some of its kinetic energy in moving the atoms in the moderator). On hitting a heavy atom like lead, it would bounce off at almost the same speed (think of a metal ball hitting concrete, it bounces well).

Control rods

Control rods consisting of cadmium or boron are also placed in the reactor, such that they can be moved in and out to control the reaction. The control rods absorb excess neutrons to prevent the reaction from taking place too quickly. When they are lowered, more neutrons are absorbed and the reaction slows, and when pulled out the reaction rate increases.

As a safety feature of all nuclear reactors, control rods are placed above the fuel rods so that if power is ever cut to the nuclear reactor, the control rods would fall to the bottom by gravity, blocking all fuel rods and quickly stopping the nuclear reaction.

Coolant / heat exchange fluid

To allow the heat produced by the core to be used to turn a turbine, the moderator liquid also acts as a heat exchange fluid that carries the heat away from the reactor to produce steam, which drives the turbine. The reactor shown above is the type which has a primary fluid loop of liquid sodium which heats a secondary loop of water, producing steam to turn the turbine.

Managing safety

Spent fuel rods that have been depleted in the reactor are extracted and processed or stored. They are extremely radioactive, making them very difficult to dispose of. The reactor is surrounded by multiple layers of shielding. There is a graphite shield that reflects neutrons back into the core, followed by a thermal shield to prevent unwanted heat loss from the core, a pressure vessel surrounding the core to isolate and contain everything inside the core, and lastly a biological shield of about 3 meters of concrete mixed with lead pellets, to absorb gamma rays and neutrons.

Applications of radioisotopes

Medicine

Agriculture

Engineering

Tc-99m: Technetium-99m is a very commonly used radioisotope in medicine. This radioisotope has a short half-life of about six hours which minimises harm to the body. It decays from Te-99m to Te-99 by emitting a gamma ray, which can be detected by PET and other medical scanners.
Te-99m is injected into patients’ bodies as a liquid solution where it will flow through the bloodstream, releasing gamma radiation as it does. This radiation gives a clear picture of the tissue structures inside the patient’s body.

P-32: a phosphate solution containing radioactive P-32 is injected into the root system of a plant.
Chemically it behaves identically to P-31 that plants normally use in their biological processes, and its movement can be detected by a Geiger counter.
By observing the movement of P-32 through the plant, scientists can determine the metabolic rate of plants and determine whether certain factors can affect this rate. E.g. this may be useful in researching the effect of certain fertilisers on certain crops.

Co-60: Cobalt-60 is used to detect stress fractures in metals, particularly in aircraft. Stress fractures occur when metals are repeatedly exposed to strong forces, such as those experienced by the wings of an aircraft. Small fractures can form which can eventually result in catastrophic failure. These factures are extremely hard to detect, because they can occur inside a solid piece of metal, and are often extremely small.
By placing cobalt-60 on one side of the metal, and a gamma detector on the other side (often photographic film), the cracks can be identified easily and non-destructively.

Neutron scattering

In the same way that an electron microscope uses electrons to probe materials, neutrons too can be used in microscopes. Neutron scattering has been used for research in fields such as geology, environmental science, biology and biotechnology, engineering, materials science etc.

The properties of neutron give it several advantages to be used as a probe:

Neutrons are neutral and can therefore penetrate deeply into matter with almost no attenuation (loss of signal quality). They can approach the nucleus of atoms without electrostatic repulsion. Neutrons are a non-destructive probe and so they can be used to study the properties of materials without damaging them.
As a moving particle, neutrons exhibit wave properties and their de Broglie wavelength is similar to the spacing between atoms in an atomic lattice. Therefore the diffraction patterns obtained from collisions with atomic nuclei enable the investigation of crystal structures and microstructures of materials (just like what the Braggs did with X-rays).
Neutrons interact with moving atoms inside matter and with their nuclei, and the strength of interaction varies for different nuclei. This allows scientists to probe and measure and properties of matter and of isotopes.
The energy of neutrons is similar to the vibrational energy of atoms in the lattice of solids- so they can be used to study the motion of atoms in the molecule of solids.

Types of Particle Accelerators

Linear accelerators

Charged particles are fired through a long evacuated tube. They pass through one cylindrical drift tube and are then accelerated by its electric field as they pass through a gap before encountering another drift tube. This process is repeated and the particles increase their energy.

The alternating accelerating potential has to keep in step with the particles and this requires the drift tubes to become longer and longer. Eventually it becomes impractical to add extra stages to a linear accelerator, so their maximum speeds are inherently limited. Generally, linear accelerators are the cheapest to build and use, but produce medium power ions at low volumes.

Cyclotrons

A cyclotron consists of two large dipole magnets (the ‘D’ regions) designed to produce a semi-circular region of uniform magnetic field, pointing uniformly downward. The applied high frequency voltage produces an electric field across the gap between the two D regions and because it is an alternating voltage, the electric field is reversed just after a particle has passed through it.

Particles injected into the magnetic field region of a D traces out a semicircular path until they reach the gap where the electric field accelerates the particles. As the velocity of the particles increases, so does the radius of their semicircular path in each of the D. Thus the particles gain energy as they spiral outward from the centre. When the particles reach the limit of the magnetic field strength, they are deflected into a target via the deflection plate.

Cyclotrons are ideal at producing high volume streams of particles, but each particle’s energy is lower compared to those accelerated by large linear accelerators. Cyclotrons are useful in the mass production of radioisotopes used by medicine and in industry.

Synchrotron

The main accelerators today are synchrotrons. Synchrotrons keep the particles in a path of constant radius by using magnetic fields. As the particles gain energy, the magnetic fields are increased to maintain the same path. Many powerful magnets are required around this path. A disadvantage of a synchrotron is that a ‘batch’ of particles must complete their journey through the accelerator before another batch can enter (unlike cyclotrons, which produce a continuous stream). However, the advantages of the synchrotron, in terms of energy that can be achieved, far outweighs this disadvantage.

Use of particle accelerators

Particle physicists have been able to use particle accelerators successfully in two ways to investigate the fine structure of matter:

Accelerators are used to accelerate subatomic particles to speeds approaching the speed of light. When particles hit each other at high energies, they break into their constituent ingredients. Through these experiments, physicists have been able to probe into the nature of all matter and have discovered many new particle types as a result.
Particle beams can be focused to increase the probability of collision and interactions with specific targets. Thus high energy particle beams can be aimed at target atoms or another high energy beam to produce short lived more massive particles.

Over time, particle accelerators have improved. They are being built much larger than before and can now propel particles to much higher energies than previously (e.g. in late 2009, CERN conducted its first experiment). Improved detectors are used to detect the products of events occurring. As a result, particle physicists have been able to use particle accelerators to infer the existence of over 200 types of sub-atomic particles, some with exceedingly short life times, and investigate the fundamental forces binding the sub-atomic particles. This provided physicists with the experimental evidence that led to the development of the standard model of matter.

The Standard Model of matter

The standard model of matter is a model developed to help classify and organise the numerous subatomic particles discovered by physicists. The model consists of:

Fermions (matter)
- Leptons
- Quarks
Bosons (forces)
- Photons
- Gluons
- Intermediate vector bosons
- Gravitons

Fermions are matter particles and bosons are force particles, responsible for all forces (bosons are particles which transmit forces, e.g. electromagnetic, strong or weak nuclear force).

Fermions

There are 24 types of fermions: 6 leptons and 6 quarks. Each fermion has a corresponding anti-particle, so there are 6 anti-leptons and 6 anti-quarks (all listed below but it is not necessary to know all of them by memory). Quarks and leptons can be neatly classified into three generations (see table below). Note that all visible matter consists of first-generation quarks and leptons.

Leptons

Leptons exist as single particles and only experience the weak nuclear force. Some leptons you already encountered in this course are the: electron, position, neutrino, anti-neutrino and muon.

Quarks

Quarks come in 6 ‘flavours’ and group together to form bigger particles called hadrons (quarks never exist as single particles, as they are strongly bound to each other via the strong nuclear force). These experience strong nuclear force and there are two types:

Mesons
Baryons (protons, neutrons)

Mesons are composed of two quarks (a quark and an anti-quark).

Baryons are composed of three quarks. Protons are baryons which are made up of two up quarks and one down quark (uud), which results in a charge of +1. Neutrons are baryons which are made up of one up quark and two down quarks (udd), which is the combination necessary to produce a neutral particle.

Note that the vast majority of matter in the universe is made of generation I quarks ( and only). Generation II and III quarks are inherently unstable, and only exist for a very short time before decaying.

A full taxonomy of fermions and their main properties is shown in the table below (no need to memorise for HSC).

	Quarks				Leptons
Generation	Name	Symbol	Rest mass (MeV)	Electric charge	Name	Symbol	Rest mass (MeV)	Electric charge
I	Up	u	5	$+\frac{2}{3}$	Electron neutrino	$v_e$	approx 0	0
I	Down	d	7	$-\frac{1}{3}$	Electron	$e^-$	0.511	-1
II	Charm	c	1,500	$+\frac{2}{3}$	Muon neutrino	$v_{\mu }$	approx 0	0
II	Strange	s	150	$-\frac{1}{3}$	Muon	$\mu ^{-}$	105.7	-1
III	Top	t	170,000	$+\frac{2}{3}$	Tau neutrino	$v_{\tau }$	<35	0
III	Bottom	b	5,000	$-\frac{1}{3}$	Tau	$\tau ^{-}$	1784	-1

Bosons

Bosons are ‘force’ particles – they are the particles behind interactions between fermions (matter particles). There are four kinds of bosons, the force carriers: (again, not necessary to memorise)

Photons
Gluons
Intermediate vector bosons
Graviton (predicted, but not yet detected)

Electromagnetism	Weak nuclear force
Photons are responsible for the electromagnetic force which binds together charged particles, atoms and molecules. For example, we have learnt that light has a particle nature, with energy $E=hf$ . In fact, all electromagnetic fields (electric and magnetic fields) and the forces they exert are done through the transmission of photons.	The intermediate vector bosons, also known as the W⁺, W⁻ and Z° bosons, are responsible for weak nuclear force. This force is responsible for radioactivity and nuclear decay. That is, this force drives unstable isotopes to emit radiation (alpha or beta particles, or gamma rays).
Strong nuclear force	Gravity
Gluons are responsible for the strong nuclear force which holds together neutrons and protons in nuclei, and also holds together the quarks in hadrons. The reason why quarks stick together to form protons and neutrons are due to gluons continuously travelling between quarks, transmitting the strong nuclear force between them. Gluons are also responsible for the fact protons can be packed closely together in atomic nuclei.	Lastly, the yet to be discovered graviton is responsible for gravity which acts over very long distances and causes all masses to attract each other. However, the graviton is currently purely theoretical (predicted by the mathematics of the standard model) and has yet to be experimentally observed.

HSC Physics – Quanta to Quarks notes – course summary