Blackbody Radiation

https://youtu.be/F_b3Trc6jz0

Blackbody Radiation #

Any heated solid emits radiation in a continuous spectrum. Some experimental observations:

• Hotter the body, higher the frequency, lower the wavelength of radiation
  • red hot → white hot
• Frequency of radiation is independent of the object being heated. It depends only on the temperature. (atleast at high temperatures)

Kirchoff’s Theorem #

He studied the radiation from heated bodies and proved the following based on thermodynamic arguments $$e(v)dv=J(v,t)A(v)dv$$

• e(v)dv: power emitted per unit area in frequency range [v,v+dv]
• A(v)dv: Fraction of incident power absorbed per unit area. =1 for blackbody
• J(v,t): Universal function

Energy is absorbed, temperature of the material increases, and it radiates a part of the energy back out.

Stefan’s Law #

Based on experimental studies, Stefan established that the total power per unit area emitted by a blackbody summed over all the frequencies is given by $$e_{total}=\int_0^\infty e(v)dv=\sigma T^4$$

• Stefan-Boltzmann constant σ: $5.67\cdot 10^{-8}\frac{W}{m^2 K^4}$
  • Boltzmann later derived it from thermodynamics+maxwell’s equations

Wien’s displacement law #

There is a single frequency where the emitted power is maximum. the corresponding wavelength $\lambda_{max}$

Wien’s Law

Wien’s Exponential Law #

Universal function J(v,T) can be related to

• u(v,T): energy per unit volume per unit frequency. energy density per unit frequency

$$J(v,T)=\frac{c}{4}u(v,T)$$

Instead of first thermodynamic principles, Wien took a shortcut by using and argued that $$u(v,T)=Av^3e^{\frac{-Bv}{T}}$$

• A and B are universal constants

it fits well for lower wavelengths and experimental data disagrees with it at higher wavelengths

Spectroscopists in Berlin have measured the energy density at far infrared (higher wavelengths) and they found that there’s a linear relationship between emission and temperature (But Wien’s wasn’t) ^ae59cc

Raleigh-Jeans Law #

They derived another expression for u(v,T) based on the assumptions:

1. The walls of the blackbody consist of a large number (Nₐ) of oscillators
2. Different oscillators oscillate with different frequencies and they emit and absorb radiation as predicted by Maxwell’s theory.
3. The oscillators are in thermal equilibrium with the radiation surrounding them.
4. By equipartition theorem, each oscillator has average energy kT (half from kinetic, half from potential)
   1. Probability for an oscillator at temperature T to have energy E: $P(E)=e^{-\frac{E}{kT}}$ and $\braket{E}=\frac{\int_0^\infty EP(E)dE}{\int_0^\infty P(E)dE}=kT$
   2. $\braket{E}=kT$

By assumption of equilibrium $n(v)dv=\frac{8\pi v^2}{c^3}dv$

• n(v): Number of oscillators per unit volume in the range [v,v+dv] $$u(v,T)dv = kTn(v)dv = kT\frac{8\pi v^2}{c^3}dv$$

Energy density is proportional to temperature as we saw in the experiment! but for small wavelengths it diverges which is incorrect: called “Ultraviolet Catastrophe”

Low wavelength: Wien Higher wavelengths: Raleigh-Jeans

Plank’s Hypothesis #

He assumed that the walls of the blackbody emits radiation in only certain steps called “Quanta” Energy: units of hv: nhv h=6.63*10^-34 Js Now P(E)=$$\propto e^{-\frac{E}{kT}}=e^{-\frac{nhv}{kT}}$$

So now $$ \braket{E}=\frac{\sum_0^\infty EP(E)}{\sum_0^\infty P(E)}=\frac{\sum_0^\infty nhv\cdot e^{-\frac{-nhv}{kT}}}{\sum_0^\infty e^{-\frac{nhv}{kT}}}=\frac{hv}{e^{\frac{hv}{kT}}-1} $$

^4d79b6

Substituting this <E> in u(v) $$ u(v,T)=\braket{E}n(v)=\frac{hv}{e^{\frac{hv}{kT}}-1}\cdot \frac{8\pi v^2}{c^3} $$ When hv/kT«1 we get back Rayleigh-Jeans Law

Expressions #

Kirchoff’s Theorem: $$e(v)dv=J(v,t)A(v)dv$$ Stefan’s Law: $$e_{total}=\int_0^\infty e(v)dv=\sigma T^4$$ Wien’s Displacement Law: $$\lambda_{max} = \frac{2.9\cdot 10^{-3}}{T} \frac{(meters * kelvin)}{kelvin}$$ Wien’s Exponential Law: $$J(v,T)=\frac{c}{4}u(v,T)$$ $$u(v,T)=Av^3e^{\frac{-Bv}{T}}$$

$n(v)dv=\frac{8\pi v^2}{c^3}$

• e(v)dv: power emitted per unit area in frequency range [v,v+dv]
• A(v)dv: Fraction of incident power absorbed per unit area
• J(v,t): Universal function
• u(v,T): energy density per unit frequency
• n(v): Number of oscillators per unit volume in the range [v,v+dv]

More on Planck’s Radiation Formula #

Energy per unit volume per unit frequency #

$$u(v,T)dv=\frac{hv}{e^{\frac{hv}{kT}}-1}\cdot \frac{8\pi v^2}{c^3}dv$$

Energy per unit volume per unit wavelength #

$v=\frac{c}{\lambda}$ $dv=-\frac{c}{\lambda ^2}d\lambda$

substituting these two:

$$ \begin{align} u(\lambda ,T)d\lambda = \frac{hc}{\lambda e^{\frac{hc}{\lambda kT}}-1} \cdot \frac{8\pi c^2}{c^3 \lambda ^2} \frac{c}{\lambda ^2}d\lambda\\ =\frac{hc}{\lambda(e^{\frac{hc}{\lambda kT}}-1)} \cdot \frac{8\pi}{\lambda ^4}d\lambda \end{align} $$