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):

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

where ℏ is the reduced Planck's constant h/(2π), μ₀ is the magnetic constant, e is the elementary charge, m_e is the electron mass, c₀ is the SI units defined value for the speed of light in vacuum, and α is the fine structure constant:

$\alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c_{0}}}\ ,$

where ε₀ is the electric constant.

The best value for R_∞ at the moment is:

R_∞ = 10 973 731.568 539(55) m⁻¹.

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:

$R_{\infty }={\frac {me^{4}}{8\varepsilon _{0}^{2}h^{3}c_{0}}}\ ,$

which is found to be the same as the expressions above using the relations:

$\hbar ={\frac {h}{2\pi }};\ \ \ \mu _{0}\varepsilon _{0}={\frac {1}{c_{0}^{2}}}\ .$

Today's theories of the hydrogen atom take into account many complexities not found in the Bohr theory of the atom. These include relativistic effects, the finite nuclear mass, and corrections due to quantum electrodynamics. However, these effects are relatively small, about 10⁻⁵, so the Bohr model gives a pretty accurate idea of the hydrogen spectrum, although the value of R in the Rydberg formula for the hydrogen spectrum is slightly different from the value R_∞ called the Rydberg constant today.