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2-5: Rutherford's Atomic Model

As discussed in the preceding page, we should consider that Thomson's raisin bread model (plum pudding model) was unsuccessful, because it could not explain the results of the alpha particle scattering experiment of Geiger and Marsden.
Let us look back upon the experimental results; we can summarize them as follows:

[The Results of Geiger and Marsden's experiment]
(1) Almost all the incident alpha particles go straight and are scarcely scattered.
(2) Only occasionally such a large-angle scattering through an angle greater than 90 degrees or near 180 degrees occurs.
(3) The scattering rate (or probability) depends on the atomic weight of the target; the more the atomic weight, the larger the probability.

[The Rutherford Model of the Nuclear Atom]
Considering the results of Geiger and Marsden's experiment and the failure of Thomson's atomic model, E. Rutherford (UK, 1871 - 1937) proposed a model in which the electric charge +Ze in an atom is not distributed over the whole area of the atom but concentrates in a small area (1911).
He thought as follows: The charge +Ze is localized to be a cluster or a group and the alpha particle is scattered by Coulomb's repulsive force, (C. A. de Coulomb: France, 1736 - 1806). Namely, his idea is that the large-angle scattering of the alpha particle is brought about by a single scattering due to Coulomb's repulsive force between the charge of the alpha particle, +2e, and that of the cluster, +Ze. The cluster is called atomic nucleus or simply nucleus. Rutherford's atomic model is often called Rutherford model or sometimes Rutherford model of the nuclear atom.
A schematic image of the Rutherford model is shown in the following figure, where a black big ball at the center is the nucleus and small red points moving around the nucleus are electrons.

The Rutherford Model of the Nuclear Atom
A black big ball at the center is the nucleus and small red points moving around the nucleus are electrons.

In 1903, H. Nagaoka (Japan, 1865 - 1950) proposed a ``saturnian model'', which was referred to in Rutherford's paper. You should note that the Nagaoka's model had been proposed several years before Rutherford's model.

[Rutherford Scattering]
Rutherford investigated whether the results of the experiment of the alpha particle scattering could be explained well by the Rutherford model of the nuclear atom. He derived the famous Rutherford scattering formula standing on the viewpoint of the Rutherford model and he found that the results of this formula fit well to the experimental data (1911). Thus the firm position of the Rutherford model was established.
Rutherford assumed that the total positive charge in an atom, +Ze, concentrates on the central point of the atom, i.e., the nucleus, and the incident alpha particle is scattered with a repulsive Coulomb force exerted by this nuclear point charge. (See the following figure.)

This type of scattering by the Coulomb force is usually called Coulomb scattering or Rutherford scattering. (See the above figure.)
The detailed formulation on Rutherford scattering will be given in the following other page. If you feel it difficult to understand, skip it, and accept the final Rutherford scattering formula.
2-5-A: Rutherford Scattering

As shown in the above figure, the distance between the incident line and the nucleus is the impact parameter b.
If b = 0, it is head-on collision, where the incident alpha particle would recoil in the direction of 180 degrees. If b is small, the trajectory would be curved largely. If b is large and the particle goes along a path far from the nucleus, then the repulsive force exerted by the nucleus is weak and the trajectory would not curve so much.
Namely, the trajectory is determined by the impact parameter only for a constant incident speed v0, This feature are shown in the following graph which is drawn by using the numerical data obtained from the solution discussed on the page (2-5-A).

[Angular Distribution of Rutherford Scattering]
As seen in the above figure, if the incident line is close to the nucleus of the target atom or if the impact parameter is small, the trajectory of the alpha particle would be largely curved and it would be scattered backward. In contrast, if the impact parameter is large, then the alpha particle goes through far from the nucleus and the trajectory does not curve so much.
In the realistic experiment, the number of the incident alpha particles in a unit area is almost constant independently of the impact parameter. The number of the incident particles in a unit area and a unit time, N, is called the intensity of the incident alpha rays. We may consider that the intensity N of the incident alpha rays is constant.
It is called the angular distribution what rate among the many incident particles is scattered in the direction of a certain angle. Strictly speaking, the angular distribution or differential scattering cross section is defined by dividing the number of the particles scattered in a unit solid angle of a certain scattering angle in a unit time by the intensity N.
Rutherford obtained the differential scattering cross section on the basis of Newtonian mechanics (1911). The details of deriving the cross section are explained on the page 2-5-A The result is shown in the following formula,

This Rutherford's formula was well fit to experiment. Thus the validity of Rutherford's model of the nuclear atom has been established.

As stated above, Rutherford derived the angular distribution of the alpha particle scattering on the basis of Newtonian mechanics. Newtonian mechanics is sometimes called classical mechanics, and it was later clarified that Newtonian mechanics is not always correct in the microscopic world, i.e. the world of atoms and atomic nuclei. It has been made clear some years later that the correct theory to describe such a microscopic world is quantum mechanics.
It has however been clarified that, as long as Rutherford scattering is concerned, both classical mechanics and quantum mechanics give the same angular distribution.

[The Size of the Charge of Atomic Nucleus]
The angular distribution derived by Rutherford was very successful not only in the angular dependence but in a by-product. As seen in the above Rutherford formula, contains a factor Z. By measuring this scattering probability precisely, we can obtain the value of Z.
The quantity Z is the number of the electrons surrounding the nucleus in an atom. Before Rutherford's formula, the value of Z had only been known to be about a half of the atomic weight. However, precise measurements of the angular distribution of Rutherford scattering have shown the precise value of the charge of the atomic nucleus and also the number of electrons in an atom. Consequently, it was made clear that Z is exactly equal to the atomic number.
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