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Discharge spectra and the Bohr model 2026-10-18

In On the Constitution of Atoms and Molecules, Part I (1913), pp. 8–9, Niels Bohr interprets existing hydrogen discharge and astronomical spectra. Electrical discharge in a vacuum tube removes Electrons; recombination supplies spectral lines. This is a theoretical interpretation, without a reported spectrograph of his own.^[ref]

An actual optical obstacle appears in Theodore Lyman's measurements: hydrogen discharge light entered an evacuated spectrograph directly through a slit, avoiding a transmitting window. A concave reflection diffraction grating dispersed the light onto photographic film. Pumping preserved the vacuum; a second slit supplied a known spectral displacement for wavelength calibration. Gas contamination and vacuum leaks limited the measurement. This apparatus resolved ultraviolet spectral lines; it did not image orbits.^[ref]

For a separate original calibration exercise, suppose an idealized spectrograph returns synthetic vacuum line positions

x_{n} = λ_{n \to 2} + b

for

n = 3, 4, 5, 6

, with chosen additive offset

b = 0.200 nm

. Assume its dispersion scale is already known and only this offset is uncertain. Compare the finite-mass prediction with the fixed-nucleus prediction

λ_{n}^{(\infty)} = (μ / m_{e}) λ_{n \to 2}

. Calibrate each model using only the

3 \to 2

line:

b = x_{3} - λ_{3}^{model}

. Then hold that offset fixed and test the other lines with residuals

d_{n} = x_{n} - b - λ_{n}^{model} .

(1)

For the finite-mass model every

d_{n} = 0

. The fixed-nucleus model has

b = 0.55733 nm

and residuals

d_{4} = - 0.09264 nm

d_{5} = - 0.12100 nm

d_{6} = - 0.13400 nm

. Algebraically

d_{n} = (1 - μ / m_{e}) (λ_{n} - λ_{3})

: one additive calibration cannot absorb a multiplicative change of the wavelengths. Allowing an unknown multiplicative dispersion factor as well would make the two models degenerate in this exercise, so independent scale calibration matters.

Figure 1.
Balmer series predictions tested after calibrating one line
. Original synthetic calibration exercise with chosen $b = 0.200 nm$ . Left: finite-mass vacuum predictions and deliberately offset synthetic positions, displayed on a separate local wavelength axis for each line. Right: model residuals after anchoring the $3 \to 2$ line; the remaining three lines test the prediction. Symbols are exact synthetic inputs and calculated results, not historical observations, error bars or an intensity trace. **Details**.

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Hydrogen transition wavelengths in the Bohr model 2026-10-18

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For emission take integers

n_{i} > n_{f} \geq 1

. Applying conservation of energy to the model's internal energy difference, while neglecting the much smaller whole-atom photon recoil, gives

Δ E = E_{n_{i}} - E_{n_{f}} = B_{H} (\frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}}), B_{H} = \frac{μ e ^{4}}{2 ( 4 π ϵ _{0} ) ^{2} ℏ ^{2}},

(1)

ν = \frac{Δ E}{h}, λ = \frac{c}{ν}, \frac{1}{λ} = R_{H} (\frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}}), R_{H} = \frac{B _{H}}{h c} .

(2)

Thus Johannes Robert Rydberg's Rydberg formula is a difference of two energy levels expressed as a spectroscopic wavenumber. The same constants determine every line; there is no new parameter for each electronic transition. With the chosen inputs,

R_{H} = 1.096775834 \times 1 0^{7} m^{- 1}

. All wavelengths below are calculated in vacuum, not air, and are rounded model predictions rather than historical measurements.

Electronic transition $n_{i} \to n_{f}$	Series	Photon energy in electronvolts	Vacuum wavelength in $nm$	$2 \to 1$	Lyman series	$10.19872$	$121.568$	$3 \to 1$	Lyman series	$12.08737$	$102.573$	$3 \to 2$	Balmer series	$1.88865$	$656.470$	$4 \to 2$	Balmer series	$2.54968$	$486.274$	$5 \to 2$	Balmer series	$2.85564$	$434.173$	$6 \to 2$	Balmer series	$3.02184$	$410.294$