Particle Physics – The Standard Model, Fusion and Fission
… E = mc² this is the equivalence of energy and mass. So mass can be converted into energy, and energy can be converted into mass!
… The unified atomic mass unit, u is a more precise version of the ‘mass number’ of an element. 1u = 1.661×10^-27 kg = 931.5MeV
… The mass defect in atomic mass units, u, can be calculated using:
Δm = Z(Mp) + (A-Z)Mn – Mnuc
Where
Z is the proton number
Mp is the mass of a proton in u
A is the atomic mass
Mn is the mass of a neutron
Mnuc is the mass of the resulting nucleus.
This more simply put as:
mass defect = (added up masses of individual protons and neutrons) – (mass of the actual nucleus)
Mass defect for a helium nucleus:
The mass of a proton is 1.6726486 x 10^-27 Kg and the mass of a neutron is 1.6749544 x 10^-27 Kg.
A helium nucleus has 2 of each so the total mass of the particles needed to build it = 6.695206 x 10^-27 Kg
The actual mass of a helium nucleus is actually 6.64477 x 10 -27 Kg. Notice how it is slightly less than expected due to the energy released when the nucleus formed.
Then you can use E = mc² to calculate the energy released due to the mass defect when a nucleus is formed.
… The equivalence of energy and mass is described by the equation E = mc²
As m = E/c², the mass of a particle can be expressed in units of eV/c². Note that c is not actually a unit in itself, but is a fundamental constant with units of m/s.
… Particle physicists measure the mass of particles in units of eV/c² or GeV/c². To convert GeV/c² to mass in kg, you will need to multiply by 10^9 (Giga) and 1.6×10^-19 (to get J) and then divide by c², (the speed of light, c = 3 x 10^8 m/s).
… If a particle has a mass of 938 MeV/c², then the particle has an equivalent REST energy of 938 MeV. This way of writing mass is useful because it is then easy to see the equivalent rest-energy of a particle.
m = E/c²
938 MeV/c² = E/c² … These are units of mass
And now the c² terms cancel:
938 MeV = E … Units of energy
Note that the particle may also have kinetic energy which will add to this rest energy.
… The total energy of a particle, E(tot) = γmc²
This includes both the rest mass (mc²) and the kinetic energy of the particle (½mv²). The way to show this is by expanding the relativistic factor γ using binomial expansionγ:
γ = 1/√(1-v²/c²) = 1 + v²/2c² +… so we can see where the ½ comes in for classical kinetic energy.
Here’s a useful resource explaining this in more depth:
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/
… The relativistic kinetic energy of a particle, Ek = (γ − 1)mc²
… An example interaction involving mass and energy featuring two colliding protons:
p + p → p + 7π+ + 7π- + K+ + Λ
mass of p = 938 MeV/c²
mass of π+ and π- = 140 MeV/c²
mass of K+ = 494 MeV/c²
mass of Λ = 1115 MeV/c² (a lambda particle)
The kinetic energy of the protons can be calculated as follows:
Using the conservation of energy in MeV:
938 + 938 + Ek = 938 + 14(140) + 494 + 1115 (All these numbers have units of MeV – see above note*)
Where Ek is the total kinetic energy of the colliding protons.
Ek = 2631 MeV
So if the the kinetic energy is shared between the two protons, each requires a minimum of 1315.5 MeV for this interaction to occur.
… If one of the protons was stationary and the other had all of the 1315.5 MeV of kinetic energy for the collision, then the total momentum before is a large value.
As total momentum before = total momentum afterwards, the particles that are produced must also have large velocities.
In this case, most of the energy of the collision needs to go into the kinetic energy of the products, which greatly decreases the energy available for new particle creation.
For example, for a 500GeV particle, 96% of the energy is ‘wasted’ in this way.
If the two particles are collided together at get same speeds, then the initial momentum = 0. All that kinetic energy is then available for particle creation because the new particles could be stationary (or at least have very low speeds in opposite directions) so that momentum after = 0.
… Energy can be measured in units of eV (electron volts). 1eV is the kinetic energy gained by accelerating an electron through a potential difference of 1 volt.
Using E = VQ, this means that 1eV = 1.6×10^-19 J.
… 1 MeV = 1×10^6 eV
… The diameter of an atom ranges from about 0.1 to 0.5 nanometers (1 × 10^−10 m to 5 × 10^−10 m).
… The strong force holds nucleons (protons and neutrons) together in the nucleus. This force is highly attractive at very short ranges between 1-3 femtometres and so can counter the repulsion of the electromagnetic force caused by the positive protons. The strong force is repulsive at <1fm.
… When nucleons are pulled together by the strong force, strong bonds are formed and this process releases energy. As the nucleus has lost energy, its mass will be LESS than the mass of the individual nucleons before they came together. This ‘mass defect’ will be given by m = E/c²
… Electromagnetic waves exist in tiny packets called photons. We briefly discussed the idea that one photon can be absorbed by an orbiting electron, exciting the electron up into higher energy levels (and sometimes exciting it out of the atom – this ionises the atom!)
… Radioactive decay occurs in unstable atomic nuclei, i.e. nuclei that don’t have enough binding ener gy to hold the nucleus together due to an excess of either protons or neutrons.
… Beta Decay of a nucleus and how conservation of energy predicted the existence of a new sub atomic particle (the neutrino).
The energy of emitted beta particles was found to be a continuous distribution up to a maximum value, rather than a single energy (which you would expect from decays happening in a particular type of nucleus). This meant that some other particle must have been emitted with the beta particle… The existence of the neutrino was postulated in 1934… and finally observed in 1956!
Conservation of momentum experiments also suggested that another particle must be emitted along with a beta particle.
… Alpha particles are often emitted only at one energy from a particular nuclide. However, if there are multiple possible energy levels of the resulting nucleus (having lost the alpha particle), then alpha particles may be emitted at several discrete energies.
… N (neutron number) vs Z (proton number) curve and decay types.
Within the nucleus there is a delicate balance between the electromagnetic force, which pushes the protons apart and the strong force, which keeps the nucleons together:
1) Massive nuclei (N>80) that lie BELOW the line of stability usually decay by alpha emission.
By emitting an alpha particle the nucleus loses 2 protons and 2 neutrons. As the line of stability curves above the line N = Z (neutron number, N, on the vertical axis), the new nucleus moves closer to the line of stability. Check out:
http://ch302.cm.utexas.edu/svg302/Table_isotopes_en.svg
2) If a nucleus lies below the line of stability (i.e. It has too many protons in proportion to neutrons), then it is likely to decay via β+ emission or by electron capture to bring it closer to the line of stability.
3) If a nucleus lies above the line of stability (i.e. It has too many neutrons in proportion to protons), then it is likely to decay via β- emission to bring it closer to the line of stability.
… Decay chains can be modelled using maths equations involving the atomic mass (A) and the proton number (Z). For example, there are several decays from Uranium-238 to Lead-206, so we can say:
For A:
238 – (number of alpha decays)x4 = 206
and for Z:
92 – (number of alpha decays)x2 + (number of beta decays)x1 = 82
Note that there is a ‘+’ in the second equation because in beta decay, the proton number increases by 1.
… Matter and antimatter – for example, an electron and a positron, proton and an anti-proton.
… Annihilation of matter with anti-matter produces a pair of photons which move away in opposite directions (so that momentum is c…?…)
… Positron emission tomography (PET) is a gamma imaging technique that uses radiotracers that emit positrons (antimatter electrons).
In PET the gamma rays used for imaging are produced when a positron meets an electron inside the patient’s body, an encounter that annihilates both electron and positron and produces two gamma rays travelling in opposite directions. By mapping gamma rays that arrive at the same time the PET system is able to produce an image with high spatial resolution.
Here’s a useful video about PET scans from Imperial College: https://youtu.be/yrTy03O0gWw
… “Pair production” is when a photon of high enough energy (given by E = hf) produces a pair of matter and antimatter particles. This process happens nearby a massive nucleus as the photon needs to first interact with the nucleus’ strong electric field to ‘steal’ momentum. Here’s a link to the full explanation: https://physics.stackexchange.com/questions/229198/why-is-a-nearby-nucleus-required-for-pair-creation
… A cloud chamber can be used to observe the paths and interactions of charged particles as they leave a trail of condensed mist. See https://en.m.wikipedia.org/wiki/Cloud_chamber
Here is how to make your own! http://www.symmetrymagazine.org/article/january-2015/how-to-build-your-own-particle-detector
And here is how modern particle physicists detect particles: https://youtu.be/-d6sKfPfYTU
… Bubble chambers work on a similar principle to cloud chambers to observe particle tracks.
Note that:
Particle mass can be inferred from radius of curvature (in the magnetic field)
Particle charge sign can be inferred from the direction of rotation (in the magnetic field)
Particle life span can be inferred from the path length (relative to the other path lengths).
… Charged particles move in a SPIRAL when moving through a magnetic field at right-angles to their direction. The spiral shows that the particle is losing energy as it accelerates towards the centre of its rotation (circular motion).
The main reason that the particle loses energy is that it is ionizing the bubble chamber (or cloud chamber) medium.
However, it may also be because any charged particle that is accelerated (i.e. has its direction changed) will emit photons called synchrotron radiation.
… Dark energy… Dark matter. Here is a fascinating article from NASA that you might like to read: http://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy/
And a useful video: https://youtu.be/Fx9D-8RN4VY
… Here is a useful video about the Standard Model from CERN: https://youtu.be/V0KjXsGRvoA
… And a longer video called “the universe in a nutshell”: https://youtu.be/0NbBjNiw4tk
… The Binding energy of a nucleus is the energy required to separate the individual protons and neutrons from the nucleus.
We use the average quantity ‘binding energy per nucleon’ to describe how tightly the nucleus is held together.
… Fission of a large nucleus such as U-235 results in daughter nuclei which are bound more tightly together (higher BEPN) and therefore release energy which can be used to heat water, make steam, drive turbines etc.
… The critical condition in a nuclear reactor is maintained by controlling the neutrons that are absorbed into uranium nuclei via boron control rods. The critical condition is when one neutron goes on to create one new fission event.
… The moderator (e.g. water or graphite) slows down fast neutrons via elastic collisions. The moderator should not absorb neutrons and should have a low atomic mass (a high mass would mean that neutrons would simply bounce off the moderator at nearly the same speed).
… The purpose of the coolant in a fission reactor (water or carbon dioxide) is to take thermal energy away from the reactor core and into a heat exchanger, where water is boiled to form steam at high pressure to turn turbines etc.
Pressurised water reactors operate at 350°C and high pressure so that the water does not boil.
… Fusion of two light nuclei also can result in a nucleus which has a higher BEPN (binding energy per nucleon) and so will release energy as it forms. Fusion energy is not quite available yet, but soon!
… The radius of a nucleus ranges from about 2-12 femto metres (1 fm = 1 x 10^-15 m)
The radius of an ATOM ranges from about 0.1-0.5 nano metres (1nm = 1 x 10^-9 m)
This means that a typical nucleus is about 100,000 smaller than an atom!
… The radius of a typical nucleus can be estimated using Rutherford scattering to calculate the distance of closest approach of an alpha particle to a nucleus.
For this calculation, the alpha particle’s initial kinetic energy will be converted into electrical potential energy (at the point of closest approach):
Ek = kqQ/r where k = 1/4πεo
… This type of alpha scattering gives a good estimate of the upper limit for a nuclear radius.
… However, alpha particles are hadrons, therefore, when they get close to the nucleus they are affected by the strong nuclear force and the mathematics do not account for this.
… Also, the gold nucleus will recoil as the alpha particle approaches, which will affect the rebound velocity of the alpha particle.
… A simple model of a nucleus:
If v = volume of a nucleon, and A is the number of nucleons, the new:
Approximate volume of a nucleus = Av
If the nucleus is spherical:
(4/3)πr³ = Av
r = (ro)A^⅓
ro is an experimental constant of value 1.2 ×10^-15 m = 1.2fm. This constant can be determined experimentally by shooting high energy alpha particles at a nucleus (they get very close) and using the transfer of kinetic energy into potential energy:
Ek = Ep
½mv² = kqQ/r
Here Q is the charge of the nucleus and q is the charge of the alpha particle.
Radius of the nucleus via election diffraction
… Electrons with a de Broglie wavelength of 10^-15 (1 fm) are required to investigate the size of a nucleus (which is also about this radius).
… The diffraction pattern forms a central bright spot with dimmer concentric circles around it.
… From this pattern, a graph of intensity against diffraction angle can be used to find the diffraction angle of the first minimum.
… The electron diffraction pattern can be analysed using
sinθ = 1.22λ/2R
Where:
R = nuclear radius
λ = de Broglie wavelength
θ = angle to the first diffraction MINIMUM.
Electron diffraction occurs as the electron ‘waves’ pass by the nucleus (as waves can diffract around objects as well as through gaps). This results in a circular diffraction pattern.
Note that this effect is single slit diffraction, not double slit (or diffraction grating) diffraction, hence θ = angle to the first diffraction MINIMUM.
… Advantages include:
- Electron diffraction is much more accurate than the closest approach method.
- Gives a direct measurement of the radius of a nucleus.
- Electrons are leptons; therefore, they will not interact with nucleons in the nucleus through the strong nuclear force as an alpha particle would.
… Nuclear density can be calculated using
density = mass/volume = A(mass of a nucleon) / (4/3)πR³
… It turns out that a teaspoon of nuclear matter would have a mass of about 10 million tonnes! So:
atoms are mostly empty space
Nearly all of the mass of an atom is concentrated in its nucleus
The nucleus is tiny in comparison to the atom (think London eye and a grain of rice!)
The Four Fundamental Forces/Interactions
… What are the four fundamental forces in order of increasing strength?
… At what ranges do these forces operate?
… The range of the four fundamental forces:
- The strong force is 1-3 femtometres
- The weak force is about 10^-18 m which is about 0.1% the size of a proton!
- The electromagnetic and gravity forces have an infinite range
… The four fundamental forces (interactions) are ‘mediated’ by exchange particles called Gauge Bosons. The exchange particle associated with each force is:
Electromagnetic force = Virtual photon
Gravitational force = Graviton (not yet detected!)
Strong nuclear force (holds the nucleus together) = Gluon
The Weak force interaction is responsible for quark decay and is mediated by the W+, W- or Z bosons. The weak force can also act on both hadrons and leptons.
Quark confinement
… The quarks in a given hadron madly exchange gluons. For this reason, physicists talk about the “colour-force field” which consists of the gluons holding the bunch of quarks together.
If one of the quarks in a given hadron is pulled away from its neighbours, the colour-force field “stretches” between that quark and its neighbours. In so doing, more and more energy is added to the color-force field as the quarks are pulled apart.
At some point, it is energetically cheaper for the colour-force field to “snap” into a new quark-antiquark pair. In so doing, energy is conserved because the energy of the colour-force field is converted into the mass of the new quarks, and the colour-force field can “relax” back to an unstretched state.
Here is more information about the process: http://particleadventure.org/quark_confinement.html
… Gluons mediate the strong force between quarks. Pions mediate the nuclear force or nucleon-nucleon interaction or residual strong force (think of the pions carrying the gluons).
For AQA, you’ll need to state that pions as the exchange particle of the strong nuclear force.
… Proton-electron capture (W+ boson from the proton) vs. Proton – electron collision (W- boson from the electron).
The exchange particles W+ and W- have the same effect, but one comes from the proton and E other from the electron!
… The Heisenberg Uncertainty principle tells us that a ‘quantum’ of energy can exist for a very short time, provided that the product of energy and time is LESS than the value of the Plank constant, 6.6×10^(-34) Js.
This also applies to matter, since energy and matter are equivalent. Virtual particles can be produced for very short times without breaking the law of conservation of energy!
Feynman Diagrams
… Particle interactions can be represented using a Feynman Diagram. Usually time is represented on the vertical axis and space on the horizontal axis (but it can be the other way around!). Here is a useful video about how Feynman diagrams work: https://youtu.be/HaWhWeBxQRQ
Note that ‘Fermions’ is the group name given to particles which have half integer ‘spin’ (a type of conservation property like charge, baryon number, lepton number). Fermions include quarks, leptons, as well as composite particles like protons, neutrons.
Also, antiparticles are shown by arrows that appear to go backwards in time!
The total charge into any node of a Feynman diagram must equal the total charge out of the node.
Atomic energy levels
… If electrons in a material are excited by heating or passing an electric current through it, some of the material’s electrons will absorb quanta of energy and be excited to higher energy levels. If those electrons then fall back down to their ground state energy level, they will emit a photon of light.
This is how line emission spectra form, which are characteristic of the elections ‘jumps’ in that particular element.
… Absorption spectra: Very hot objects radiate a continuous spectrum of light (UV, visible, IR). However, dark bands in this continuous spectrum are observed when light from the hot body passes through a gas (for example, the atmosphere around a star).
These absorption spectra dark bands correspond to the line emission spectra of the gas and are due to photons of the light being absorbed at particular frequencies.
… Orbiting electrons can be ‘excited’ to higher energy levels if they absorb energy from a photon or a collision (e.g. with another electron).
… The ground state energy level is n = 1. For example, in hydrogen the ground state energy level is at -13.6eV. This means that 13.6eV are needed to remove an electron (ionise the atom) at this level.
… Excited electrons can fall down (de-excite) to a lower energy level and release a photon in the process. The energy of the photon will be the difference between the energies of the two levels.
This means that, for example, a photon of X-ray radiation could excite an electron, which could then de-excite in several stages, releasing several photons of different frequencies (energies). This is the principle used in XRF analysis which can identify different metals.
XRF analysis works by firing X-rays or gamma rays into a material. The electrons of a particular element in the material will be ‘excited’ up to higher energy orbitals and then fall back down to several other energy levels, releasing photons of radiation. These photons will have different energies and therefore different frequencies. There resulting spectrum of emitted radiation provides a characteristic ‘fingerprint’ enabling identification of the element!
… It’s worth noting that visible light has wavelengths between 400nm (violet) – 700nm (red)