====== 1.ii.1 What is a Blackbody? ======

Heating metal causes it to radiate heat and light. The color/frequency of radiation changes as we increase temperature.

The following video shows how the color of the radiation changes as a block of iron is heated up.  When current is passed through the heating element it heats up, but also becomes magnetized like a solenoid because it is a coil.  A block of magnetized iron is used that floats above the heating element so that you can see its color more clearly, but any block of metal would radiate in a similar way.

{{ :blackbody.mp4 |A metal block demonstrating blackbody radiation}}

**Video credit**: [[http://evolvingsciences.com/Why%20do%20things%20change%20colour%20when%20heated%20.html|Evolving Sciences]]

A //**blackbody**// is an idealized object that absorbs all of the radiation that hits it (none is reflected so it looks black in ordinary lighting).

A metal cavity with a small hole in it is an approximate blackbody.

{{ :cavityblackbody.png?direct&600 |Why a metal cavity behaves approximately like a blackbody}}

The following YouTube video shows what happens when you heat up a metal cavity.

{{ youtube>-_xHPp-10NU }}

====== 1.ii.2 The Blackbody Spectrum ======

In thermal equilibrium, a blackbody emits a characteristic spectrum of radiation that only depends on its temperature, e.g. the temperature of the cavity walls for an approximate blackbody made from a metal cavity.

This simulation shows how the spectrum depends on temperature: 

[[https://phet.colorado.edu/sims/html/blackbody-spectrum/latest/blackbody-spectrum_en.html|PhET Blackbody Spectrum Simulation]]

Have a play with it.   Note that infra-red radiation causes heat.  How does the efficiency of a light bulb compare with the efficiency of the sun at producing visible light?  Why does the light from a light bulb look yellower than the light from the sun?

For those who do not like interactive things, here is a graph of the blackbody spectrum for various temperatures:

{{ :blackbodyspectrum.jpg?direct&600 |The blackbody spectrum for various temperatures}}
**Image Credit**: [[https://openstax.org/books/university-physics-volume-3/pages/6-1-blackbody-radiation|Samuel J. Ling, Jeff Sanny, William Moebs, University Physics Volume 3 (OpenStax, 2016)]]

Access for free at [[https://openstax.org/books/university-physics-volume-3/pages/1-introduction|https://openstax.org/books/university-physics-volume-3/pages/1-introduction]]

The blackbody spectrum has the following features:

  * There is a well-defined, continuous energy distribution that only depends on the temperature of the cavity walls.
  * **Wien’s displacement law**: There is a maximum at a particular wavelength:
  \[\boxed{\lambda_{\max} \propto \frac{1}{T}.}\]
  * **Stefan-Boltzmann Law**: In 1879, Stefan found experimentally that the total intensity (power per unit surface area obeys)
  \[\boxed{I = a\sigma T^4,}\]
  where
    * $a$ is a constant that depends on the material
    * $a=1$ for a blackbody
    * $a<1$ for a non blackbody
    * $\sigma = 5.67 \times 10^{-8}\,\text{Wm}^{-2}\text{K}^{-1}$
    * Boltzmann derived this theoretically in 1884.

====== 1.ii.3 Attempts to derive the blackbody spectrum ======

===== Wien's Law =====

  * Wien extended the Stefan-Boltzmann law to obtain
  \[\boxed{u(\nu,T) = A\nu^3 e^{(-\beta\nu/T)}.}\]
  * This only works for large frequencies $\nu$.
  * $A$ and $\beta$ and  are free parameters that have to be fixed by experiment.

===== Rayleigh-Jeans Law =====

  * Rayleigh considered the statistical mechanics of radiation and derived
  \[\boxed{u(\nu,T) = \frac{8\pi \nu^2}{c^3} k T.}\]
  * This only works for small $\nu$.
  * Ultraviolet catastrophe: Total intensity $\int_{-\infty}^{\infty} u(\nu,T)\,\mathrm{d}\nu$ diverges for high frequency.  Implies that the cavity contains infinite energy

===== Planck Distribution =====

  * Planck obtained
  \[\boxed{u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1},}\]
  which agrees with experiment for
  \[\boxed{h=6.626\times 10^{-34}\,\mathrm{m}^2\mathrm{kg}\mathrm{s}^{-1}.}\]
  * $h$ is called //**Planck’s constant**//: A new fundamental constant of nature introduced by quantum theory.

The following graph compares the three distributions.

[[https://commons.wikimedia.org/wiki/File:Mplwp_blackbody_nu_planck-wien-rj_5800K.svg|{{ https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Mplwp_blackbody_nu_planck-wien-rj_5800K.svg/512px-Mplwp_blackbody_nu_planck-wien-rj_5800K.svg.png |Geek3, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0 , via Wikimedia Commons}}]] 
**Image Credit**: [[https://commons.wikimedia.org/wiki/File:Mplwp_blackbody_nu_planck-wien-rj_5800K.svg|Wikimedia Commons]]

====== 1.ii.4 The Quantum Postulate ======

To derive his distribution, Planck assumed that the energy of the radiation emitted at frequency $\nu$ by the oscillating electrons in the walls of the cavity could only come in integer multiples of $h\nu$.

\[\boxed{E(\nu)=nh\nu\qquad\qquad\mbox{for}\qquad\qquad n=0,1,2,3,\cdots.}\]

This is the //**quantum postulate**//.

Since not everyone taking this class has studied statistical mechanics yet, the derivation of the Planck distribution from the quantum postulate is beyond the scope of this course, but see [[http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf|this link]] for a derivation.

====== 1.ii.5 Wien's Displacement Law ======

  * Wien’s displacement law says that the maximum energy density of the blackbody spectrum is at $\lambda_{\max} = b/T$ for some constant $b$.
  * We can derive this by finding the maximum of the Planck distribution expressed as a function of wavelength $\tilde{u}(\lambda, T)$.  Expressing the Planck distribution in terms of wavelength is one of the in-class activities.
  * Setting $\frac{\partial \tilde{u}}{\partial \lambda}=0$ yields a transcendental equation (i.e. an equation that has no closed-form solution so we have to solve it numerically).  Solving this numerically gives
  \[\boxed{\lambda_{\max} = \frac{b}{T},}\]
  with
  \[\boxed{b = \frac{hc}{4.965 k}.}\]

{{:question-mark.png?nolink&50 |}}
====== In Class Activities ======

  - Derive the Rayleigh-Jeans law $u(\nu,T) = \frac{8\pi \nu^2}{c^3} k T$ from the Planck formula $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$ for small $\nu$.
  - Derive Wien’s law $u(\nu,T) = A\nu^3 e^{(-\beta\nu/T)}$ from the Planck formula $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$ for large $\nu$.  What are  $A$ and $\beta$ in terms of $h$, $c$, and $k$?
  - The energy density for frequencies between $\nu$ and $\nu + \mathrm{d}\nu$ is $u(\nu, T)\mathrm{d}\nu$, where $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$.   Derive the energy density $\tilde{u}(\lambda,T)$ as a function of wavelength using $\lambda = \frac{c}{\nu}$.