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4-1: The Difficulty of the Rutherford Model
of the Nuclear Atom

As discussed in Part 2, Rutherford proposed the Rutherford Model of the Nuclear Atom, with which the experimental data of the alpha particle scattering by atoms was beautifully explained.
However, as mentioned shortly on the page, 2-7: Summary of Part 2, the Rutherford model of the nuclear atom was not able to explain the stability of atoms and the atomic spectra standing on the viewpoint of the classical theory. Thus the classical theory came to a deadlock at this point.

[The Difficulty in the Stability of Atoms]
It was considered that, in the Rutherford model of the nuclear atom, a heavy atomic nucleus with the electric charge +Ze, where Z is the atomic number, locates at the center of an atom and Z light electrons are moving around the nucleus. Let us explain that this atomic structure is quite unstable in the classical theory.
According to Maxwellian electromagnetism, a charged particle (an electron at present) moving with an acceleration radiates electromagnetic waves and loses energy as

Here e is the elementary charge (the charge of an electron), is the permeability of free space, and c is the speed of light in vacuum. The minus sign on the right-hand side implies to lose the energy.
Let us think, for simplicity, of the hydrogen atom in which an electron circulating about a proton. Since the mass of a proton is enough heavy in comparison with that of an electron, the proton is assumed to be at rest at the origin of the coordinate frame and the electron is moving along a circle of the radius r around the proton. The Newtonian equation of motion in the present case is written

where the quantity on the right-hand side is the strength of the Coulomb force between the proton and the electron.
The energy that the electron loses in unit time is obtained from Eqs. (1) and (2) as

On the other hand, the energy of the electron is written

We have therefore

The speed of the electron would slow down, because it loses energy. Hence the radius of its circular motion decreases gradually. Namely, the radius r is a decreasing function of time. Writing

and inserting Eqs. (3) and (5) into Eq. (6), we have

Let us suppose that the radius of a hydrogen atom were at a time. It would be 0 after some time T because of the loss of energy of the electron. We can get the time T by integrating Eq. (7) with respect to t as

Thus, the electron in the Rutherford model must rapidly lose its energy and spiral down inward and collapse on the proton in an extremely short time. However, the real hydrogen atom seems quite stable and does not collapse spontaneously.
Thus, the Rutherford model of the nuclear atom brings about a difficulty under the classical theory.

[The Difficulty in the Atomic Spectra]
A particular distribution of light (electromagnetic radiation) against frequency or wavelength is called a spectrum. For example, when we disperse white light with a prism or a diffraction grating, we have a display of colours (violet, blue, green, yellow, orange and red). Such a display is called a spectrum. It is said that Newton observed the details of spectra of light with a prism for the first time (1666).
The lights from discharge tubes or arcs across various kinds of electrodes with a high voltage potential being applied represent line spectra which consist of a numbers of bright sharp lines characteristic of the gases involved in the tube or the matters of the electrodes. Some examples will be shown on the next page.
Each kind of atom radiates the lights of the frequencies specified to the atom. However, the Rutherford atomic model standing on the classical theory cannot give these specified spectra. This is another difficulty in the Rutherford model.
Let us study in the next section what we can get from the atomic spectra.
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