Rydberg constant - Aetherwisp

The Rydberg constant $R$ is a constant that appears in nuclear and atomic physics, particularly in the Bohr model of the atom. In terms of other fundamental constants, it is given by

R_{\infty}=\frac{m_{e}}{4\pi c\hbar^{3}}\left(\frac{e^{2}}{4\pi\epsilon_{0}}\right)^{2}=109737\text{ cm}^{-1}

in the approximation of an infinitely massive atomic nucleus. If we set the nuclear mass to some finite constant $M$ and define the reduced mass as $\mu=m_{e}M/(m_{e}+M)$ , a more precise value for the constant is given by

R_{\mu}=\frac{\mu}{m_{e}}R_{\infty}

Depending on the form it is given in, the Rydberg constant can be interpreted in one of a few ways. In inverse length ( $\text{cm}^{-1}$ ), it represents the inverse wavelength (i.e. wavenumber) that all spectral lines of the hydrogen atom are a multiple of. The inverse of $R$ in this form is known as the Lyman limit:

\frac{1}{R}\simeq 911\ \mathring{\mathrm{A}}

In this sense, it can also be given as a frequency by multiplying by the Speed of light $c$ :

\bar{R}=cR=3.288\cdot10^{15}\text{ Hz}

Recognizing that $hcR$ is an energy (because of the Planck formula) we can write the Rydberg energy

E_\text{Ryd}=hcR\simeq13.6\text{ eV}

This represents the depth of the Potential well in which an electron resides in the ground state of the hydrogen atom. In other words, it is the ionization energy of hydrogen in the ground state.

Experimental measurements on the atom yield values of $R=109677\text{ cm}^{-1}$ , almost identical to the theoretical value, confirming that the Bohr model is fairly accurate, despite being classical.