Title: ETH_Sem3 / Physik_III / Woche6

Q: What is a blackbody and how can we approximate one?
A: A blackbody is an object that absorbs all the light that hits it.
We can approximate one by considering the inside of a box and making a tiny hole in the side to be able to observe the radiation.

Q: What is blackbody radiation?
A: An object at a certain temperature emits light (or at least electromagnetic waves). Blackbody radiation is the spectrum of frequencies emitted by an ideal blackbody at a given temperature.

Q: What is a mode when talking about electromagnetic radiation?
A: An independant solution of the wave equation and boundary conditions. They describe the field pattern of the EM-waves and are analogous to modes of vibration in mechanical systems.

Q: Rayleigh-Jeans Law
\[\rho(\omega)d\omega = [...] = [...]\]What does it represent and what is the problem with it?
A: \langle \varepsilon \rangle\cdot \frac{g(\omega)d\omega}{V}; k_BT \frac{\omega^2}{\pi^2c^3}d\omega
Back Extra: It represents the energy density form all modes with frequency between \(\omega\) and \(d\omega\).
The problem is the "ultraviolet catastrophe": since the frequency of a mode can be infinite, it's energy can too! This leads to infinite energy.

Q: Solution to the ultraviolet catastrophe problem
A: The energy of each mode is not some continuous function, but can only take discrete values: non-negative integer multiples of \(\hbar\omega = h\nu\).

Q: Planck distribution
\[\rho(\omega)d\omega = [...]\]Note that as \(\omega\to\infty\), \(\rho(\omega)\to [...]\)
A: \frac{\hbar\omega^3}{\pi^2c^3}\frac{d\omega}{e^{\beta\hbar\omega}-1}; 0

Q: What is Wien's displacement law?
A: The maximum of \(\rho(\omega)\) occurs at a value of \(\hbar\omega \approx 2.822 k_BT\).
Formally:\[\lambda_{peak} = \frac bT\]where \(b = 2.897771955\ldots\times10^{−3}\; \text{m⋅K}\) is Wien's displacement constant.

Q: Stefan-Boltzmann Law
The total energy density for blackbody radiation is given by\[U(T) = \int_0^\infty \frac{\hbar\omega^3}{\pi^2c^3}\frac{d\omega}{e^{\beta\hbar\omega} } = [...]\]Where \[a = [...]\]
A: aT^4; \frac{\pi^2k_B^4}{15\hbar^3c^3} = \text{const.}

Q: The power radiated per unit area of the blackbody surface is given by:\[j(T) = [...]= [...]\]Where \(\sigma = ac/4 = 5.67\times10^{-8}\;\text{Wm}^{-2}\text{K}^{-4}\) is the [...].
A: U(T)\cdot\frac{c}{4}; \sigma T^4; Stefan's constant

Q: What are atoms made up of?
A: Protons and neutrons in the nucleus and electrons (probably) around it.

| Charge | Mass | Is elementary particle ? | Size (approx. radius)
Proton | \(1.6\times 10^{-19}\;C = 1\,\text e\) | \(1.66\times 10^{-27}\; \text{kg}\) | No | \(8.4\times 10^{-16}\;\text{m}\)
Neutron | \(0\) | \(1.66\times 10^{-27}\; \text{kg}\) | No | \(8.4\times 10^{-16}\; \text{m}\)
Electron | \(-1.6\times 10^{-19}\;C = -1\,\text e\) | \(9.11\times 10^{-31}\;\text{kg}\) | Yes | Unknown,\(< 10^{-18}\;\text{m}\)

Q: Notation for atoms\[_Z^AG\]
A: \(G\): Element name
\(Z\): Atomic number, number of protons in the nucleus
\(A\): Atomic mass, number of protons and neutrons in the nucleus

Q: How does a mass spectrometer look like?
Answer for backside:
It allows us to determine \(\frac mq\) of particles, thus letting us identify them.
A: What is this and what does it allow us to do?

Q: How is the atomic mass unit defined?
A: \[1\;\text{u} = \frac{1}{12}(\text{mass of }^{12}_6C)\]

Van der Waals equation
\[(p+\frac{a}{V^2})(V-b) = Nk_BT\]\(a\) accounts for interactions between atoms.
\(b\) accounts for the size of the atoms themselves.

Q: What is the idea behind x-ray diffraction?
A: To look at interference of x-rays reflecting off atoms in a crystal to determine the distance between atoms.
(Similarly to how the intensity pattern from a grating depends on the spacing between the reflective strips.)

Q: What is scattering as a measurement technique?
A: Launch a beam of atoms (or other particles) at a gas (or other particles, even solids or liquids) and see how many make it through and how many bounce off.

Q: What is the scattering cross-section \(\sigma\)?
A: The area around a particle where another (specific) particle cannot pass through without being deflected.

Q: Scattering with atoms through a gas
After a distance \(x\), the number of remaining particles is \[N(x) = [...]\]Where \(n\) is [...] and \(\sigma\) [...].
A: N(0) e^{-n\sigma x}; the density of the target or sample gas; the scattering cross-section

Q: How big is (approximately) an atom's diameter?
A: A few Angstroms (\(10^{-10}\;\text{m})\)

Q: What are the two conclusions from the electron scattering experiment?
(A beam of electrons was shot at a gas and the distance the electrons travelled before bouncing off (a few cm) was measured.)
A: 1. Most of the inside of the atom is empty space.
2. Rather than being stuck in particular locations in the atom, the electrons are moving around.

Q: What was the Rutherford experiment and what was it meant to do?
A: He shot \(\alpha\)-particles (Helium-nuclei) at a thin gold foil and measured how many \(\alpha\)-particles got deflected into each angle (relative to the incoming beam).
It was meant to probe the size of the nucleus, since \(\alpha\)-particles are much heavier than electrons and wouldn't be deflected by them.

Q: What was the atom model before Rutherford?
A: The plum pudding model: electrons are uniformly scattered in a cloud of positive charges.

Q: What was the result of the Rutherford experiment?
A: The positive charges are concentrated in a very small area inside the atom, the nucleus.
It's diameter is less than \(4\times 10^{-15}\; \text m\).