Year 2 Sem 1/LI Particles and Nuclei/Lecture 01 – TODO
In this lecture:
- Introduction to key relations.
- Scales and natural units.
- The standard model, interactions and force carriers.
- Yukwa's argument.
Recap
Conversions between mass and energy, i.e. when we collide particles we can convert kinetic energy into "rest energy", i.e. particle mass. When a heavy particle undergoes a decay, it's rest energy is converted into the kinetic energy of the decay products.
$$ E_0 = mc^2 $$ Where $E_0$ is rest energy.
For a particle in motion we modify this to be: $$ E^2 = p^2 c^2 + m^2 c^4 $$
Units
Masses
It makes our lives much easier to pick sensible units for a problem. For example, $m_e = 9.109 \times 10^{-31}$kg. It's therefore, as it's a bit icky, not ideal to use this in kilograms and we may talk in multiples of masses of fundamental particles instead.
Energies
We often in particle physics accelerate particles in an electric field. The energy gained by an electron accelerating from rest is $E = qV$, which for $1$V gives $E = 1.602 \times 10^{-19}$J. We define this (the electron-volt, eV) as our fundamental unit of energy.
We modify the mass of the electron as: $$ m_e = 511 \times 10^3 eV = 511keV $$
Which is much nicer, our mass units are now eV/$c^2$ and momentum $eV/c$.
Natural Units
In this course we adopt natural units, using the convention $c=1$ to collapse mass, energy and momentum into the same dimensions. We also set $c = e = \hbar = 1$. We can only consider factors of $c$ when we're dealing with SI units, i.e. if we need to give an answer in metres or if explicitly asked in an exam question.
Scales
Typically scales for this course are fm, $10^{-15}$m aka the fermi. Wavelengths of visible light are typically on the order of hundreds of nanometres, so visible light isn't practical for investigation of particles/nuclei.
History
- 1909: Alpha particles fired in the Rutherford experiment demonstrated the existence of the nucleus.
- 1967: Electrons on hydrogen demonstrated existence for charge and mass inside the proton, now known to be quarks.
Composite Particles
Atoms, nuclei, nucleons etc are all composite particles, and exist in a measurable region of space.
By QM, if we have a finite $\Delta x$ we cannot have a zero $\Delta p$, therefore a bound particle must not be at rest. For an electron in an atom, we have: $$ \Delta x \approx 10^5 pm $$
And substituting this into the H.U.P, we have a minimum $\Delta p $ of $1$keV.
First Generation Fundamental Particles
All normal matter is made up of quarks, leptons (+ combinations thereof etc). We start by looking at quarks:
Quarks
Quarks make up the nucleons, and include (in the first generation):
- up, $u$, $m_u \sim 2$MeV, $Q = +2/3$.
- down $d$, $m_d \sim 5$MeV, $Q = -1/3$
Leptons
- electron, $e$, $m_e = 511$keV, $Q = 1$.
- electron neutrino, $\nu_e$, $m_\nu < 1eV$, $Q=0$.
These are the lightest fundamental particles which have mass.
We have further generations of fundamental particles in the standard model:
Each particle has a corresponding antiparticle, i.e. the electron has an antielectron (positron). When we have a charged particle, the antiparticle has an opposing sign.
Fermions and Bosons
We have two overarching classifications of particles, classified by their intrinsic quantum spin. We have two types of angular momentum, orbital (for a body in motion around another) and spin (inherent to an object):
$$ \vec{L}_\text{orbital} = \vec{r} \times \vec{p} \qquad \vec{L}_\text{spin} = I \omega $$
In quantum mechanics, angular momentum (and spin) are quantised into discrete values. Quantum spin has the following values: $$ 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots $$
Bosons are particles which have integer spin, and fermions are particles with half-integer spin.
Pauli Exclusion Principle
No two fermions of the same type may occupy the same quantum state
Force Carriers
We have our classic and very familiar fundamental forces:
- Gravity: Very very weak, long range.
- Weak: Very weak, short range.
- EM: Strong, long range.
- Strong: Very strong, short range.