Editing Rutherford scattering experiments (section)

===Cumulative effect===
Applying Thomson's equations described above to an alpha particle colliding with a gold atom, using the following values:

* ''q<sub>g</sub>'' = positive charge of the gold atom = {{val|79|u=''q''<sub>e</sub>}} = {{val|1.26|e=-17|u=C}}
* ''q<sub>a</sub>'' = charge of the alpha particle = {{val|2|u=''q''<sub>e</sub>}} = {{val|3.20|e=-19|u=C}}
* ''q<sub>e</sub>'' = [[elementary charge]] = {{val|1.602|e=-19|u=C}}
* ''R'' = radius of the gold atom = {{val|1.44|e=-10|u=m}}
* ''v'' = speed of the alpha particle = {{val|1.53|e=7|u=m/s}}
* ''m'' = mass of the alpha particle = {{val|6.64|e=-27|u=kg}}
* ''k'' = [[Coulomb constant]] = {{val|8.987|e=9|u=N·m<sup>2</sup>/C<sup>2</sup>}}
* ''N'' = number of electrons in the gold atom = 79

gives the average partial angle by which the alpha particle should be deflected by the atomic electrons as:

<math display="block">\bar\theta_1 = \frac{16}{5} \cdot \frac{k q_\text{a} q_\text{e}}{mv^2 R} \sqrt{\frac{3N}{2}} \approx 0.00007 \text{ radians or } 0.004 \text{ degrees}</math>

and the average partial deflection caused by the positive sphere is:

<math display="block">\bar\theta_2 = \frac{\pi}{4} \cdot \frac{k q_\text{a} q_\text{g}}{mv^2 R} \approx 0.00013 \text{ radians or 0.007 degrees}</math>

The net deflection for a single atomic collision is:

<math display="block">\bar\theta = \sqrt{\bar\theta_1^2 + \bar\theta_2^2} \approx 0.008 \text{ degrees}</math>

On average the positive sphere and the electrons alike provide very little deflection in a single collision. Thomson's model combined many single-scattering events from the atom's electrons and a positive sphere. Each collision may increase or decrease the total scattering angle. Only very rarely would a series of collisions all line up in the same direction. The result is similar to the standard statistical problem called a [[random walk]]. If the average deflection angle of the alpha particle in a single collision with an atom is <math>\bar{\theta}</math>, then the average deflection after ''n'' collisions is

<math display="block">\bar\theta_n = \bar{\theta}\sqrt{n}</math>

The probability that an alpha particle will be deflected by a total of more than 90° after ''n'' deflections is given by:

<math display="block">e^{-(90 / \bar\theta_n)^2}</math>

where ''e'' is [[Euler's number]] (≈2.71828...). A gold foil with a thickness of 1.5 micrometers would be about 10,000 atoms thick. If the average deflection per atom is 0.008°, the average deflection after 10,000 collisions would be 0.8°. The probability of an alpha particle being deflected by more than 90° will be<ref name=BeiserPerspectives>Beiser (1969). [https://archive.org/details/perspectivesofmo0000arth/page/102/mode/2up ''Perspectives of Modern Physics''], p. 109</ref>{{rp|109}}

<math display="block">e^{-(90 / 0.8)^2} \approx e^{-12656} \approx 10^{-5946}</math>

While in Thomson's [[plum pudding model]] it is mathematically possible that an alpha particle could be deflected by more than 90° after 10,000 collisions, the probability of such an event is so low as to be undetectable. Geiger and Marsden should not have detected any alpha particles coming back in [[Rutherford scattering experiments#Alpha particle reflection: the 1909 experiment|the experiment they performed in 1909]], and yet they did.