An observation on the Balmer–Bohr formula

Every student of physics and chemistry, and many of general-level astronomy, will encounter the Bohr model for the hydrogen atom during their classes. One of the most famous results of this theory is an expression for the wavelength of light emitted when an electron transits from one permitted orbit to another. If the initial orbit is n_initial and the final one n_final (n_initial > n_final ), the emission wavelength is given by the formula

$$\begin{align}{\lambda _{i \to f}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} & = {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {\frac{{8\varepsilon _o^2{h^3}c}}{{{e^4}{\mu _H}}}} \right){\mkern 1mu} {\left( {\frac{1}{{n_{final}^2}} - \frac{1}{{n_{initial}^2}}} \right)^{ - 1}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \nonumber\\ & = {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {911.76{\mkern 1mu} {\mkern 1mu} \mathop {\rm{A}}\limits^ \circ } \right){\left( {\frac{1}{{n_{final}^2}} - \frac{1}{{n_{initial}^2}}} \right)^{ - 1}}{\mkern 1mu} .\end{align} \tag{ 1 }$$

The symbols in the prefactor have their usual meanings: _o is the permittivity of free space, h is Planck's constant, c is the speed of light, e is the electronic charge, and μ_H is the so-called reduced mass of hydrogen,

$$\begin{equation}{\mu _H}\,\,\,\, = \,\,\left[ {\frac{{{m_e}{} {m_p}}}{{{m_p} + {m_e}}}} \right]\,,\end{equation} \tag{ 2 }$$

where m_p and m_e are respectively the masses of the proton and electron. The reduced mass accounts for the motion of the proton and electron around their mutual centre of mass. The inverse of the prefactor is the Rydberg constant for hydrogen.

Equation (1) is now known as the Balmer–Bohr formula, a generalization of an empirical expression for the visible-wavelength spectral lines of hydrogen found by Johann Balmer in 1885. The visible-wavelength lines, what we now call the Balmer lines, correspond to transitions down to n_final = 2 from higher-n orbits. Precise modern values for the physical constants and the resulting reduced mass and Rydberg constant are listed in table 1.

Table 1. Physical constants.

Name	Symbol	Value	Units
Speed of light	c	2.997 924 58 × 108	m s−1
Planck's constant	h	6.626 070 15 × 10–34	J s
Electron charge	e	1.602 176 634 × 10–19	C
Permittivity constant	o	8.854 187 812 8 × 10–12	F m−1
Electron mass	me	9.109 383 701 5 × 10–31	kg
Proton mass	mp	1.672 621 923 7 × 10–27	kg
Neutron mass	mn	1.674 927 498 × 10–27	kg
Rydberg constant	RH	10 967 758.3	m−1
Reduced mass	μH	9.104 425 278 × 10–31	kg

Source: National Institute of Standards and Technology Recommended Values of the Fundamental Physical Constants: 2018. https://physics.nist.gov/cuu/pdf/wall_2018.pdf

While recently programming a spreadsheet to compute transition wavelengths for class use, I was dismayed to find my results for the Balmer series coming in 1–2 Å greater than standard reference values; see table 2. At first I thought I had entered one or more of the physical constants incorrectly, but this proved not to be so. The number of significant figures used for the constants was such that the problem could not have been round-off error.

A little searching soon revealed the culprit: published and online wavelength tabulations typically quote optical region wavelengths for what would be measured in air, while those given by equation (1) would be for a vacuum. The index of refraction of air, n_air ∼ 1.0003, is only marginally greater than that of a vacuum, but sufficient to cause the difference. What I needed to do was to convert my vacuum wavelengths to air wavelengths via λ_air = λ_vac /n_air , but such things are never as simple as they seem: the index of refraction of air is a function of pressure, humidity, carbon dioxide content, and wavelength itself. Fortunately, the United States National Institute of Standards and Technology maintains an online app which calculates the index of refraction for a given combination of vacuum wavelength and atmospheric conditions [1].

For the range of wavelengths of interest here, this app gives n_air ∼ 1.00027 to 1.00028 for a temperature of 20 °C, standard pressure (101.325 kPa), 50% relative humidity, and CO₂ content 450 ppm. Adopting 1.000275 as an effective average value, the last column in table 2 lists the corresponding air wavelengths, which agree with their textbook counterparts within 0.1 Å. There will always be minor discrepancies with the Balmer–Bohr formula because it is non-relativistic, and spectral lines actually have fine structure; there are no single wavelengths for these lines.

This issue will be well-known to spectroscopists and atomic physicists, but probably not to non-specialist instructors and ambitious students who want to check the numbers for themselves. Even with a long-familiar result, there is always something new to learn.