Quantized Atomic Orbits

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Within the unfolding timeline, we have seen that the fundamental idea of quantized energy as introduced by Planck, in 1900, in the context of blackbody radiation, which is then followed up, in 1905, by Einstein's paper concerning the photoelectric effect. However, at this point in time, the scientific establishment is not ready to embrace the idea of quantization, as in many ways, as it has become increasingly preoccupied with the issue of the atomic structure. While the history of the atom can be traced back to Greek Sciencein 400BC, the idea is essentially limited to a philosophical concept of ’indivisibility’. In the 17th century, the idea of the atom is resurrected by Robert Boyle, but is again limited to the idea of ’indivisibility’ of chemical processes. However, in 1898, J.J. Thompson discovers the electron, which he considers to be a negative charged component of the atom, which he assumes are evenly distributed throughout a sphere of positively charged material. This arrangement becomes known as the ‘plum puddingmodel of the atom, but is significant in that the ’indivisibility’ of the atom has now been replaced by some form of sub-structure. By 1909, Ernest Rutherford has demonstrated that the structure of the atom is mostly empty space with a small positively charged nucleus, which contains most of the mass, orbited by negative charged electrons. Initially, it was thought that the electrons might orbit the nucleus, analogous to a microscopically small planetary system. However, this idea was challenged almost from the outset, as an orbiting electron would be subject to angular acceleration and therefore, in accordance with Maxwell’s accepted equations, must radiate energy. The net results being that an orbiting electron would lose energy and spiral into the nucleus under charge attraction between the positive nucleus and the negative electron. So, at this point, science is left pondering on the question:

What process will explain the structure of the atom?

One of the people who would take up the challenge posed by the question above was Niels Bohr, who in 1912 spent 6 months working for Ernest Rutherford, before returning to work in the physics department of the Copenhagen University. Given that the topic of the ‘Bohr Model of the Atomhas already been outlined in the context of the development of the classical particle model; the scope of the remaining discussion will primarily focus on the development of Bohr’s idea of ‘quantized atomic orbits’. On his return to Copenhagen, in 1912, Bohr was struggling to resolve the fact that classical physics placed no restrictions on the electron orbits and Maxwell’s equations suggested that the angular acceleration of such an orbit must lose energy and spiral into the nucleus. As a first step, he considered restricting the electrons to ‘special’ orbits, which for reasons unknown might not radiate energy. As such, Bohr was also moving towards the concept of quantized orbits, which he initially described in terms of ‘stationary states’, even though he could not justify the assumption on which they were based. At this point, late in 1912, Bohr came across a paper by John Nicholson, who was primarily a mathematician with an interest in the atomic structure. While Bohr disagreed with the overall model being forwarded by Nicholson, he was taken by the idea that the angular momentum of the electron orbits within this model were constrained in integer multiples of [h/2π]. Without repeating all the logic of the ‘Bohr Model’, the radius and the energy of the quantized orbits of hydrogen were calculated from the equations of angular momentum [L]:

[1]      1

Following this line of reasoning, Bohr was also able to formulate an expression predicting the energy levels of each of the ‘stationary states’ of hydrogen:

[2]      2

Having established a hypothesis, later in 1913, Bohr is told of the work of Johann Balmer, who has devised a formula for the spectral lines of hydrogen, which will provide another vital clue as to the quantized nature of atomic orbits. Balmer had actually devised his formula for the observed wavelengths [λ] of the spectral lines of hydrogen in 1884, which took the following form consisting of 2 integers [m,n] and a constant [b]:

[3]      3

On seeing Balmer’s formula, Bohr recognized that he could correlate [3] with [2] via equations established by Maxwell and Planck:

[4]      4

From this point, the Bohr model can be visualized as shown in the following diagram. Here we see the energy associated with the spectral lines corresponding to electron transition between a higher to lower stationary states. In so doing, the energy difference between the two levels results in the emission of a ‘photon’ of a given wavelength. By the same token, an electron transition from a lower to high stationary state would require the absorption of a photon of a given wavelength.

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It might be recognized at this point, the diagram above might also be used in support of the photoelectric effect. However, in this context, the energy of the photon absorbed has to be sufficient to move the electron outside the orbital radii of the atom.

While Bohr would publish a trilogy of papers entitled ‘On the Constitution of Atoms and Molecules’ in 1913, the model would also raise as many questions as answers. One of the most fundamental departures from classical physics was the lack of any clarity regarding the position of the electron during a transition between two stationary states. It is also worth pointing out that while Bohr had accepted the quantization of the atom, he still did not believe that electromagnetic radiation emitted or absorbed was quantized in the form of Einstein's idea of a discrete photon at this point.

Note: Given 100 years of research and hindsight, it is now known that the Bohr model is only an approximation of the simplest atom, i.e. hydrogen. Of course, within the current chronological timeline, it is clear that this model was of fundamental importance in the transition from classical to quantum physics. While Bohr initially used Balmer’s spectral lines in support of his model, advances in spectral analysis, even as early at 1915, had raised doubts as further details of the ‘atomic fine structure’ emerged. For example, when the spectral lines of the hydrogen spectrum are examined at higher resolution, they are found to be closely-spaced doublets, which are not explained by Bohr's initial model. In 1915, Arnold Sommerfeld forwarded a modification of the Bohr model, which had only taken into account circular orbits via the integer [n]. Sommerfeld’s modification recognized that an elliptical orbital would require two integers [n,k], which then predicted the doublets in the hydrogen spectra now being observed. However, as early as 1897, Pieter Zeeman had discovered that magnetic fields also caused spectral lines to split into yet finer structure. Therefore, Sommerfeld added yet another integer [m], which was initially described in terms of the orientation or plane of the orbit. Later, it was recognized that the spectra measured in an external magnetic field could still not be fully predicted with just [n, k, and m]. A solution to this problem was suggested, in 1925, by George Uhlenbeck and Samuel Goudsmit by the introduction of a fourth quantum number defined in term of binary spin number. Today, the concepts of physical electron orbits have been replaced by the concept of orbitals, which are essentially a mathematical function that describes the wave-like behaviour of an electron or pair of electrons in an atom. In this context, the function simply specifies the probability of finding an electron in a given region around the atom's nucleus – see ‘Atomic Shells’ for more details.