Rydberg constant

The Rydberg constant, often denoted as R_∞, originally defined empirically in terms of the spectrum of hydrogen, is given a theoretical value by the Bohr theory of the atom as (in SI units):

$R_{\infty }={\frac {m_{e}e^{4}}{4\pi \hbar ^{3}c_{0}}}\ \left({\frac {\mu _{0}c_{0}^{2}}{4\pi }}\right)^{2}\ .$

The best value (in 2005) was:

R_∞/(hc₀) = 10 973 731.568 525 (8) m⁻¹,

where h = Planck's constant and c₀ = SI units defined value for the speed of light in vacuum.

Background

The Rydberg constant is the common scaling factor for all hydrogen transitions. Its measurement has proven often to be a testing ground for theoretical results. The original introduction of this constant by JR Rydberg in 1889 was through a formula for the wavelengths associated with alkali metal transitions, which can be formulated for the hydrogen atom as:

${\frac {1}{\lambda }}=R\ \left({\frac {1}{p^{2}}}-{\frac {1}{n^{2}}}\right)\ ,$

where R is a constant and n and p are any integers, with n>p. The case for p=2 is called the Balmer series and for p=1 the Lyman series. The first value for the Rydberg constant was found using this formula, and was totally empirical.

In 1913 Neils Bohr developed a theory of the atom predicting the hydrogen atom to have energy levels (in SI units):

$E_{n}=-{\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}n^{2}}}\ ,$

with e the electron charge, m the electron mass, ε₀ the electric constant, h Planck's constant, and n the so-called principal quantum number. According to this model, the wavelength λ of a transition of an electron moving from state n to state p is then:

${\frac {1}{\lambda }}={\frac {me^{4}}{8\varepsilon _{0}^{2}h^{3}c_{0}}}\left({\frac {1}{p^{2}}}-{\frac {1}{n^{2}}}\right)\ ,$

with c₀ the SI defined valued for the speed of light in vacuum. Thus, a theoretical expression for the Rydberg constant is obtained.