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4-3: The Bohr Model of Atoms

The Rutherford model of the nuclear atom had been beautifully successful in explaining the alpha particle scattering. It was however completely powerless to explain the stability of atoms and to derive the atomic spectra. As seen in the previous sections, it brought about a serious difficulty which cannot be overcome within the classical theory.

Discussing with Rutherford in UK, N. H. D. Bohr (Denmark, 1885 - 1962) learned the Rutherford model and extended this model to the so-called Bohr model.
Bohr followed Rutherford and thought that, in a hydrogen atom, an electron revolves about the proton at such speed that the Coulomb force between the electron and proton becomes just equal to the centrifugal force to keep this orbital motion.
As discussed on the preceding page, this model was unstable with respect to the classical theory. Moreover, this model had one more difficulty. It was known that all hydrogen atoms are equal to each other and have the equal radius, but there was nothing in this model to assure this equality. Thus, if the Rutherford model was to survive, it was necessary to find a radical remedy that would bring both the stability and the constant radius.
To prepare the remedy, taking a hint from Balmer's formula of the hydrogen spectra, Bohr added the hypotheses stated below to the Rutherford model (1913). This model is sometimes called the Bohr-Rutherford model of the nuclear atom or Bohr's quantum theory, which was the first unified theory to be able to explain the structure of the hydrogen atom including both the stability and the fixed radius.

[Bohr's Quantum Theory]
Bohr considered that the electrons revolve around the atomic nucleus obeying the classical theory (Newtonian mechanics and Maxwellian electromagnetism). He added the following hypotheses to the Rutherford model:
(1) An atom can exist only in special states with discrete (step-like) values of energy. (Electrons in an atom can exist only in certain special orbits.) Bohr called these special states (or orbits) stationary states.
(2) When an atom makes a transition from one stationary state to another, it emits or absorbs radiation whose frequency , is given by the frequency condition

where E' and E'' are the energies of these stationary states.
(3) In the stationary states, the electrons move obeying the classical theory. Among the possible motions obeying the classical theory, only those satisfying the following quantum condition are realized:

where p is the momentum of the electron and q is its coordinate variable. The integration should be done along the orbit over one period of the cyclic motion.
The theory constructed on the basis of the above hypotheses is called Bohr's quantum theory or sometimes the old quantum theory.

[The Hydrogen Atom]
Let us apply Bohr's quantum theory to the hydrogen atom.
Suppose that an electron (mass = m) revolves around the proton being at rest at the center under a Coulomb attractive force. This motion of the electron is described by the Newtonian equation of motion. The orbit of the electron in this case is elliptic in general. Now, let us assume it to be a circle for simplicity. In this case the absolute value of the momentum of the electron, p, is constant. Then the quantum condition is written

where a is the radius of the orbit.
From the balance of the centrifugal force and the Coulomb force, we have

Combining Eqs. (3) and (4), we obtain the radius a of the orbit as

Since the energy of the electron, E, is the sum of the kinetic and potential energies, we have

Let us write this En.
The allowed energies of the hydrogen atom are the discrete ones , given by Eq. (6). The state of n = 1 is the lowest energy state called the ground state. The radius of the orbit in the ground state, a0, is especially called the Bohr radius, whose value is

This Bohr radius is thought to be the radius of an ordinary hydrogen atom, and there can exist no hydrogen atom with a radius smaller than this.
When the hydrogen atom jumps (makes a transition) from one stationary state with the energy En to another with Ek, it emits or absorbs a light whose frequency , is given by the frequency condition Eq. (1). Using the energy Eq. (6), we therefore have

This is just the same as Balmer's Formula or Rydberg's formula which was obtained empirically. Therefore, the value of the Rydberg constant is calculated as

This result is exactly fit to the experimental value.
Thus, Bohr's quantum theory could beautifully reproduce the structure of the hydrogen atom.

[The Energy Levels and the Spectra of Hydrogen]
According to Bohr's quantum theory, the energy of the hydrogen atom is given by Eq. (6). The state of n = 1 is the ground state which has the lowest energy. The states of n = 2, 3, ... are the excited states. These energy levels are graphed in the following figure in units of eV, where the energy E = 0 corresponds to the ionized state of hydrogen with the free electron at rest. (The ionized state means the state in which the hydrogen nucleus, i.e. proton, and the electron are completely separated.)

The Energy Levels of Hydrogen
The state of n = 1 is the ground state, and n = 2, 3, ... are the excited states. The energies are in units of eV, being so measured that the completely ionized state is made to be of 0 eV.

The detailed experimental data of the spectrum of hydrogen are shown in the following figure, where the Lymann, Balmer, Paschen and Brackett series are represented. As seen in the above figure, these groups of spectral lines are considered to correspond to the radiations emitted when the transitions occur from various states to some specified state.

[The Confirmation of the Stationary States]
In Bohr's quantum theory (or the Bohr model of atoms), Bohr postulated that an atom can exist only in the stationary states with discrete (step-like) values of energy. This postulation is quite difficult to be understood with the classical theory. However, the spectrum of the hydrogen atom could be reproduced beautifully. This means that the idea of the stationary states appears "correct".
Therefore people wanted to confirm the existence of the stationary states directly by experiment. This is the famous Franck-Hertz experiment, which was carried out by J. Franck (Germany, USA, 1882 - 1964) and G. Hertz (Germany, 1887 - 1975).
This Hertz is a researcher different from H. R. Hertz (Germany, 1857 - 94) who was famous for the research on electromagnetic waves in the 19th century.

A schematic drawing of the apparatus used by Franck and Hertz is shown in the following Fig. (A). Low-pressure mercury vapor is introduced in a tube, in which a filament F is heated up and electrons are emitted from it. The emitted electrons are accelerated toward the grid G by the electric potential between the filament F and the grid G, and after passing through the grid have a kinetic energy eV. Then the electrons are subjected to a small retarding potential before being collected on the plate P.

If an electron loses no energy in the course from the filament to the grid, it will arrive at the plate with the energy and contribute to the current measured by the ammeter A. However, if eV is greater than the excitation energy of the gas atom from the ground state to the first excited state, some of the electrons will lose a part of the kinetic energy on colliding with a gas atom. Then they have insufficient energy left to overcome the retarding potential Consequently, as the potential V is increased gradually, we can find a quick drop of the current measured with the ammeter A, when the energy eV is equal to the excitation energy of the gas atom. If the potential V is then increased again, the current should rise, and the second drop would appear when eV is equal to twice the excitation energy.
In 1914, Franck and Hertz demonstrated such energy absorption whose experimental data is shown in the above Fig. (B). As the potential V between the filament and the grid is raised, the current gradually increases, but suddenly decreases at V = 4.9V. After that, the current increases again and the second drop appears at V = 9.8V, and the similar phenomena are repeated.
This means that the excitation energy of the mercury atom should be 4.9 eV. It was confirmed that this excitation energy was just equal to that obtained from the spectra of mercury.
Franck and Hertz obtained similar experimental results using not only mercury but also neon, argon, krypton and so on. These results would be the experimental proofs of the Bohr's postulation of the stationary states.
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