Radiation Laws

Max Planck found the first indications of the quantization of electromagnetic waves when he formulated the Planck's radiation law named after him. Historically, it is a further development of the Rayleigh-Jeans law, which describes the intensity of the radiated power of a black body as a function of frequency or wavelength. The derivation of this law should be briefly outlined at this point: We start with the wave equation for the electric field in three dimensions: $$\frac{\partial ^2 E}{\partial x^2} + \frac{\partial ^2 E}{\partial y^2} + \frac{\partial ^2 E}{\partial z^2} = \frac{1}{c^2}\frac{\partial E}{\partial t}$$ For each direction in space in a cubic, black body with side length $L$, the condition of a standing wave must be met and the field vector must vanish at the walls. This leads to the following solution of the wave equation: $$E = E_0\sin\frac{n_1\pi x}{L}\sin\frac{n_2\pi y}{L}\sin\frac{n_3\pi z}{L}$$ The constants $n_i$ are called modes. If this function, which is dependent on $x$, $y$ and $z$, is inserted into the wave equation, the condition for the modes is: $$\left(\frac{n_1\pi}{L}\right)^2+\left(\frac{n_2\pi}{L}\right)^2+\left(\frac{n_3\pi}{L }\right)^2 = \left(\frac{2\pi}{\lambda }\right)^2$$ If you multiply this equation by $L$ and then divide it by $\pi$, you get the following simplification: $$n_1^2 + n_2^2 + n_3^2 = \frac{4 L^2}{\lambda^2}$$ The next step is to find out for which $n_i$ this condition is fulfilled. For this purpose, as a simplification, consider the number of possible combinations on a three-dimensional grid of a sphere. The volume of all $n_i$ is then calculated with the radius $$R = \sqrt{n_1^2 + n_2^2 + n_3^2}$$ according to $$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi\left(n_1^2+n_2^2+n_3^2\right)^\frac{3}{ 2}$$ However, since the $n_i$ can only assume positive values, only 1/8 of the entire sphere may be considered. Due to the two different directions of polarization of electromagnetic waves, a factor of 2 must also be added. Finally, this leads to the following result for the number of all modes: $$N = \frac{2}{8}V = \frac{8\pi L^3}{3\lambda^3}$$ Now we consider the number of modes per wavelength, i.e. the derivative of $N$ to $\lambda$: $$\frac{\mathrm{d} N}{\mathrm{d} \lambda } = -\frac{8\pi L^3}{\lambda ^4}$$ This is often also referred to as mode density. If we now assume an infinitely large volume, we are more interested in the mode density per volume, whereby we again use the approximation of a cubic volume: $$\frac{1}{L^3} \frac{\mathrm{d} N}{\mathrm{d} \lambda } = -\frac{8\pi }{\lambda ^4}$$ Due to energy conservation and the law of uniform distribution, each mode must have the mean energy $k_\mathrm{B}T$: $$\frac{\mathrm{d} u}{\mathrm{d} \lambda } = \frac{1}{L^3} \frac{\mathrm{d} E}{\mathrm{d} \lambda } = -k_\mathrm{B}T\frac{1 }{L^3}\frac{\mathrm{d} N}{\mathrm{d} \lambda }$$ Here $u$ denotes the energy density of the individual modes. Substituting $\mathrm{d} N/\mathrm{d} \lambda$ into the equation then yields: $$\frac{\mathrm{d} u}{\mathrm{d} \lambda } = \frac{8\pi k_\mathrm{B}T}{\lambda ^4}$$ The radiated power per wavelength can be calculated by halving the energy density per wavelength $\mathrm{d} u/\mathrm{d} \lambda$ and multiplying it by the volume per unit time: $$p = \frac{1}{2}\frac{\mathrm{d} u}{\mathrm{d} \lambda }\frac{Ax}{t} = \frac{1}{2}\frac{\mathrm{d} u}{\mathrm{d} \lambda}Ac$$ In the last step, the distance $x$ was replaced by the product $ct$ so that the time can be shortened. In addition, the radiant power is not the same in all directions, but varies with the cosine of the polar angle $\theta$. If one average over all angles, one obtains the specific radiance B from the quotient of the average radiant power divided by the radiating area: $$M(\lambda) = \frac{\bar{p}}{A} = \frac{\mathrm{d} u}{\mathrm{d} \lambda }\frac{c}{4}$$ After another simplification one obtains the so-called Rayleigh-Jeans law: $$\boxed{M(\lambda) = \frac{2c\pi}{\lambda^4}k_BT}$$ This can theoretically be used to calculate the spectral emissivity of a blackbody with temperature $T$ per wavelength interval $\mathrm{d} \lambda$. to calculate. What is interesting here is the strong dependency (fourth power!) on the wavelength. Unfortunately, this law of Rayleigh and Jeans has the crucial disadvantage that the radiated power continues to increase for smaller wavelengths. This effect is commonly known as ultraviolet catastrophe. Since there is no compensating factor that counteracts this effect, this would result in infinitely large energy radiation for high frequencies, which, however, contradicts everyday experience. Max Planck then had the idea of ​​quantifying the release of energy and introduced a constant $h$ like "help!", which later became known as Planck's Constant. Another interpretation of this important constant, as well as the numerical value, is covered in the next section. Planck's modification of the Rayleigh-Jeans law then led to Planck's radiation law, which can be written as follows: $$\boxed{M(\lambda) = \frac{2 \pi h c^2}{\lambda^5} \frac{1}{\exp\left( \frac{h c}{\lambda k_\mathrm{B} T} \right) - 1}}$$ By approximating the exponential function in the denominator, some algebraic transformations can be used to show that Planck's radiation law merges into the previously derived Rayleigh-Jeans law. One can see very clearly the steep slope of the graph that follows the Rayleigh-Jeans law. Only for large wavelengths do they show similar behavior. The maximum radiated power is around 500 nm. The maximum sensitivity of our eyes at exactly this wavelength can be traced back to this. The spectral distribution, which can be calculated using Planck's radiation law, agrees excellently with the experimentally observed spectra of hot bodies. Although it was only derived for ideal black bodies, the law can be applied to almost all bodies by multiplying it by a material- and surface-dependent emission coefficient $\varepsilon$. If this formula is also integrated over all wavelengths, the well-known Stefan-Boltzmann law can be derived with a few algebraic transformations, which describe the radiated power of a body as a function of temperature.