View source for Year 2 Sem 1/LI Particles and Nuclei/Lecture 01 – TODO

<lecture number="1" title="TODO" date="2026-10-01" />

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:
[[File:LI_Particles_and_Nuclei_-_Lecture_01_–_TODO_-_20261001-112753-974.png|center|frameless|900px|The Standard Model of Particle Physics]]

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 ===
{{Box|body=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. Mediated by the W+-, Z bosons.
* '''EM''': Strong, long range. Mediated by the photon.
* '''Strong''': Very strong, short range. Mediated by the gluon.

These forces are modelled as working via particle exchange of their mediating particle. We have a version of the uncertainty we can apply:
$$
\Delta E \Delta t \qeq \frac{\hbar}{2}
$$

Effectively, an exchange particle can provide energy provided it acts for a short enough time.

We also have the nuclear force which is not fundamental (combination of protons repelling by the EM force while simultaneously being bound by the strong force). This composite force is strong and yet short range.

== Yukwa Argument ==
The Yukwa Argument suggests that these short range forces can be explained by the exchange of a massive particle between the bodies on which the force acts. The creation of a particle with mass greater than the energy transferred would appear to violate conservation of energy.

However, if the particle exists for a very short space of time, we have a little allowance from the modified H.U.P shown above. The shorter the timeframe the particle exists for, the more energy it is permitted to transfer. If the exchange period is short and the range is short, it doesn't violate CoE.

=== Example: Strong Nuclear Force ===
The strong nuclear force acts on ranges approximate a femtometre. Hence:
$$
\Delta x \sim \text{fm}
$$
And assuming $v \sim c$:
$$
c \Delta t \sim \Delta x
$$

Substitution into our modified formula gives ....

$$
\Delta E \sim 100MeV
$$

So this is the allowed energy of our exchange particle. This explains how the EM force is infinite range, as a massless carrier ''can'' result in an infinite range. The exception is the gluon, which is massless for a short-range force, but we'll come back to that.