Figure references are to the second edition of Modern Physics by Serway, Moses and Moyer (Saunders, 1989).

Before discussing Niels Bohr's contribution to our understanding of the atom, we need to go back to the 19th century and consider developments in the science of optical spectroscopy. We have seen that the spectrum of electromagnetic radiation emitted by a hot solid contains of a continuous range of wavelengths, whose intensity distribution is given by Planck's formula. In the case of a hot gas, however, whose atoms are much farther apart from each other than in a solid, the emission spectrum consists of a series of bright lines superimposed on a dark background. Such line spectra can be observed by passing the light from a gas-discharge lamp through a spectrometer, containing a glass prism or a diffraction grating as the light-dispersing element. The pattern of lines (their wavelengths and relative intensities) is characteristic of the type of gas emitting the light (Fig. 3.14). By the year 1860, such spectroscopy was sufficiently developed that two new elements (cesium and rubidium) were discovered by analysing the light emitted by minerals which had been vapourized in a flame.

However, close observation of the spectrum of sunlight shows that it contains dark lines superimposed on a smooth background, as originally reported by Joseph Fraunhofer in 1814 (he labelled the prominent lines A, B, C, D etc., but modern spectroscopy shows that there are many other lines in between; see Fig. 3.17). We now know that the smooth background arises from the fact that the radiant energy originates from nuclear reactions in the sun's interior, which is at a high pressure and high temperature. Under these conditions, the bright emission lines broaden and overlap (due to Doppler and pressure broadening) to give a continuous wavelength distribution, similar to that emitted by a heated solid. To reach us, however, the radiation must pass through cooler layers of the sun's atmosphere, where the pressure is lower and the atoms are further apart, such that they absorb certain wavelengths selectively, producing the dark Fraunhofer lines.

In 1859, Gustav Kirchhoff showed that Fraunhofer's D-lines originated from the presence of sodium vapour in the sun's atmosphere; they occurred at the same wavelengths as the emission lines from a sodium source and were enhanced when the sunlight was passed through a flame containing sodium atoms (Fig. 3.18). More recently, the elemental composition of many stars has been determined by attaching an optical spectrometer to an astronomical telescope.

In 1885, J.J. Balmer (a Swiss schoolteacher) published a formula which accurately predicted (to better than 0.1%) the wavelength positions of lines observed in the emission spectrum of hydrogen:

(lambda) = C n^2 / (n^2 - m^2 ) .................................... (1)

where C = 364.56 nm, m = 2 and the integer n takes values of 3, 4, 5 and 6 for lines in the visible region and higher values for further lines in the ultraviolet; see Fig. 3.19. Balmer went on to speculate that other series of lines might exist in the ultraviolet (uv) and infrared (IR) regions of the spectrum, corresponding to different values of the integer m . This prediction has proven correct (see Table 3.1 and Fig. 3.23), so Eq.(2) can be rewritten in the more general form:

hf = (m^2 hc/C) (1/m^2 - 1/n^2 ) ........................................ (2)

where f represents the frequency of the light emitted and hf is the corresponding photon energy.

Niels Bohr spent most of his life in Copenhagen but after receiving his doctorate (in 1911) he visited Rutherford's laboratory in Manchester (England) and learned about the successes and problems associated with Rutherford's nuclear model of the atom. He became convinced that the new quantum theory of light might provide a clue to the understanding of atomic structure, and when a friend suggested that Balmer's formula for the spectral lines of hydrogen might be relevant: "the whole thing was immediately clear to me" - as he said later.

Bohr's theory retains the classical idea that electrons move in circular orbits about a central nucleus. But since the electron does not continuously radiate energy, Bohr concluded that Maxwell's classical theory of electromagnetism does not apply on an atomic scale. He took inspiration from the work of Planck and proposed that the atomic electrons are restricted to certain orbits (known as stationary states, since they are stable and not evolving into something different) for which the energy of the electron is quantised. Applying Einstein's photon model of electromagnetic radiation, he further assumed that when an electron changes its orbit (from an initial state of energy E_f to a final state of energy E_f) a single photon is released with an energy hf , given by:

hf = E_i - E_f ....................................... (3)

In other words, the frequency f of the emitted radiation is determined by the separation of the energy levels of the two "stationary" states and is not necessarily equal to the classical frequency of the orbital motion. This prediction about the frequency of the emitted radiation allows Bohr's model to be tested experimentally.

Recognizing that Planck's constant has the dimensions of angular momentum (J.s = kg m^2 /s), Bohr proposed that the angular momentum of an electron (assumed to be in a circular orbit of radius r and tangential speed v ) is given by:

m v r = n (hbar) ......................................... (4)

where h/2 (pi) (called h-bar) is often given its own special symbol. and n is an positive integer, known as the principal quantum number.

Since the Bohr model is based on the assumption of circular orbits and electrostatic attraction of electrons to a central nucleus, we can use these classical concepts (in combination with Bohr's additional assumptions) to find the radii of the stationary states in the hydrogen atom (the simplest case, since there is only a single atomic electron). We can eliminate v from Eq.(4) by using the fact that the electrostatic force of attraction supplies the centripetal acceleration v^2 /r to keep the electron in its orbit:

k e^2 / r^2 = m v^2 / r ................................ (5)

where k = 1/(4 pi epsilon0) = 9.00 x 10^9 m^2/C^2 is the Coulomb constant. Substituting v = n(h/2 (pi)) / mr gives m (n(h/2 (pi)) /mr)^2 = k e^2 / r , leading to:

r = [1/(km)] (n(h/2 (pi))/e)^2 = n^2 a₀ ............................... (6)

where a0 = (h/2 (pi))^2 / (k m e^2) = 52.9 x 10^-12 m is the radius of the innermost allowed orbit (corresponding to n=1) and is known as the (first) Bohr radius. Eq.(6) shows that larger values of n correspond to orbits of larger radius. 

Likewise we can obtain an expression for the orbital velocity:

v = n(h/2 (pi)) / mr = v₁ / n ........................................... (7)

where v₁ = [ (h/2 (pi)] / m a₀) = 2.19 x 10^6 m/s = c/137. Since the highest velocity (corresponding to n=1) is less than 1 percent of the speed of light in vacuum, we are justified (to a good approximation) in using taking m to be the electron rest mass in our analysis. Eq.(7) also shows that the orbital velocity decreases for higher n (larger r). In fact the orbital period T is proportional to n^3 or to R^3/2 , just as for planets orbiting the sun (Kepler's Third Law).

Assuming that the nucleus remains stationary, the energy of the atom comprises the kinetic energy K and the potential energy U of the orbiting electron. Since v<<c as noted above, we can to a good approximation use the usual nonrelativistic expression for K, giving:

E = K + U = (1/2) m v^2 - k e^2 / r ......................... (8)

but since K = mv^2 / 2 = k e^2 / 2r from Eq.(5), the total energy is also:

E = - k e^2 / (2r)

Substituting for r gives:

E = - (k e^2 / 2a₀) (1/n^2 ) = - R / n^2 .................................... (9)

where R = 13.6 eV is known as the Rydberg energy.

The lowest energy state (E = 13.6 eV for hydrogen) corresponds to n = 1 and is called the ground state of the neutral atom; this is the state in which the atom normally exists, when energy is not being supplied externally. The highest energy state would be n = ¥ , with r = ¥ (zero potential energy since infinite separation of the electron from the nucleus), v= 0 (zero kinetic energy of the electron) and therefore E = 0 . Therefore it requires an energy of 13.6 eV , the ionization energy of hydrogen, to remove the electron from the vicinity of the nucleus and turn a neutral hydrogen atom into a positive ion and a (stationary) free electron. Smaller amounts of energy could put the ground-state atom into an excited state of higher internal energy.

More generally, the atom could be excited from an initial state of quantum number n_i to a final state of quantum number nf by supplying an energy E_f-E_i; if this energy is supplied by absorption of a photon of electromagnetic radiation (energy hf), the photoabsorption process can be represented by:

hf = E_f - E_i = -R(1/n_f^2 - 1/n_i^2 ) = R (1/n_i^2 - 1/n_f^2 ) ............................. (10)

Conversely, if the initial state is an excited state of energy E_i and the final state corresponds to a lower energy E_f, the atom may create a photon of frequency f and the photoemission process is represented by:

hf = E_i - E_f = -R(1/n_i^2 - 1/n_f^2 ) = R (1/n_f^2 - 1/n_i^2 ) ............................. (11)

This equation is seen to be identical to the generalised form of Balmer's equation for the hydrogen emission lines, Eq.(2). When an electron makes a downward transition (Fig. 3.23), the value of n_f determines which series the emission line belongs to (n_f = 2 for the Balmer series, etc.) and the values of n_i (>n_f) determines the precise wavelength of the line in this series; see Table 3.1 and Fig. 3.23.

Likewise, Eq.(10) provides an accurate prediction for the positions of the dark lines in an absorption spectrum, which (as seen from the similarity between the two equations) occur at the same wavelengths as the emission lines. The Bohr model therefore provides a simple and accurate account of both the absorption and the emission of radiation from a hydrogen atom.

The Bohr model can be extended to atoms other than hydrogen by retaining the electronic charge as -e but replacing the nuclear charge by +Ze in the equation for electrostatic force. Eq.(9) then becomes:

E = - R (Z^2 / n^2 ) ................................ (12)

According to Eq.(12), the ionization potential for helium (Z=2) should be 54.4 eV, a factor of 4 higher than that of hydrogen. However, the measured value is only 24.6 eV. This discrepancy arises because we have neglected the electrostatic interaction between the two orbiting electrons. Although the discrepancy becomes less with increasing atomic number, more advanced wave-mechanical models must be used to predict the properties of multielectron atoms. However, the Bohr model does apply to the case of some ionised atoms (such as He+ and Li++) in which all but one of the electrons have been removed. For example, the Bohr formula correctly accounts for some of the lines in the spectra of certain stars, in terms of energy levels of the remaining electron in ionised helium.

Bohr was led to his idea of angular momentum quantisation, Eq.(4), through consideration of the correspondence principle. For large values of n (corresponding to a large orbits, so that larger scale classical physics might be expected to apply) it can be shown that Eq.(9) leads to an energy difference D E between adjacent orbits (quantum numbers n and n+1) given approximately by h f_e , where f_e is the rotation frequency of the orbiting electron. So in this case, radiation of frequency f_e is emitted, as would have been predicted on the basis of Maxwell's classical theory of electromagnetism. The subsequent development of a wave-mechanical quantum theory of the electron provides further justification for Eq.(9).

Despite it success, Bohr's theory was widely criticised for its mixture of classical and quantum concepts. Nevertheless, an Institute for Theoretical Physics was created in Copenhagen and Bohr remained there as its director, contributing to the further development of the quantum theory of matter (including structure of the atomic nucleus) until his death in 1962.