Blackbody Radiation

• Some observations in physics experiments at the beginning of the 20$^{th}$ century could not be explained by classical theories (Blackbody Radiation, The Photoelectric Effect, The Hydrogen Atom, $\ldots$).
• Among them, the phenomenon called Blackbody Radiation has a special place.
• By definition, a blackbody is an object that absorbs any heat radiation falling on it.
• Rayleigh and Jean proposed that infinitesimal amounts of energy were continuously added to the system when the frequency was increased.
• Classical physics assumed that energy emitted by atomic oscillations could have any continuous value.
• If we try and sum the energies at each frequency we find that there is an infinite energy in this system!
• This paradox was called the ULTRAVIOLET CATASTROPHE.

Figure 3.1: Variation of energy density with wavelength/frequency in blackbody radiation.

• Spectrum of the radiation inside the blackbody can be measured experimentally.
  $$dU=u(\lambda,T)d\lambda$$

• The experimentally observed curve $u(\lambda,T)$ at T=5000 K is shown in the upper figure.
• It has not been possible to explain this observed spectrum of blackbody radiation with classical theories (in the classical limit of large $\lambda$, Rayleigh-Jeans Law).

• Later, in 1901, Planck was able to come up with the formula that fully explained this curve, with an assumption that predicted the quantum nature of light:

  $$u(\lambda,T)=\frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda k_BT}-1}$$ (3.1)

  Here $c$ is the speed of light and $h=6.63\times10^{-34}~ joule.s$ becomes a new constant in physics by the name of Planck's constant.
• Planck's assumption was as follows: The energy was quantized and could be emitted or absorbed only in integral multiples of a small unit of energy, known as a quantum.
• The energy distribution of a radiation with a frequency $\nu=hc/\lambda$ is a multiple of an amount of $h\nu$:
  $$E=nh\nu~~(n=1,2,3,\ldots)$$

• Blackbody radiation helped to move our understanding in physics from a classical approach to a quantum one.

• Some notable features of the Planck distribution are:
  1. Total radiant energy (ie the area under the curve in the Figure)
     $$\int_0^\infty u(\lambda,T)d\lambda=\sigma T^4$$
     is proportional to the 4$^{th}$ power of the temperature T. This is called the Stefan-Boltzmann formula.
  2. There is a simple relation between the wavelength $\lambda_{max}$ at which each curve has a maximum value and the equilibrium temperature T
     $$\lambda_{max}T=0.0029~m.K$$
     This equation is called the Wien's displacement law.
• Here we will obtain the Wien's displacement law by numerical method.

• To find the maximum wavelength, it is sufficient to take the derivative of Equation 3.1 with respect to the wavelength $\lambda$ and set it to zero. However, it is convenient to do this based on the dimensionless variable $x=hc/k_BT\lambda$:
  $$u(x)=A\frac{x^5}{e^x-1}~~(x=hc/k_BT\lambda~and~A=8\pi (k_BT)^5/(hc)^4)$$
  $$\frac{du}{dx}=A\frac{5x^4(e^x-1)-x^5e^x}{(e^x-1)^2}=0$$
  
• If this equation is solved, $x_{max}$ and then $\lambda_{max}$ can be found. However, Equation 3.2 has no analytical solution.
• We can find the answer with the numerical root finding methods.