Editing Rutherford scattering experiments (section)

==Why the plum pudding model was wrong==
J. J. Thomson himself didn't study alpha particle scattering, but he did study [[beta particle]] scattering. In his 1910 paper "On the Scattering of rapidly moving Electrified Particles", Thomson presented several equations that modelled how beta particles scatter in a collision with an atom per the [[plum pudding model]].<ref name=ThomsonScattering1910/><ref name=Heilbron1968/>{{rp|277}}

The average partial deflection caused by the atomic electrons on an incoming beta particle is

<math display="block">\bar\theta_1 = \frac{16}{5} \cdot \frac{k q_\text{e} q_\text{e}}{m v^2 R} \sqrt{\frac{3N}{2}}</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{e} q_\text{g}}{m v^2 R}</math>

where ''q''<sub>e</sub> is the elementary charge, ''q''<sub>g</sub> is the positive charge of the atom, ''m'' and ''v'' are the mass and velocity of the incoming particle, ''N'' is the number of electrons in the atom, and ''R'' is the radius of the atom.

The net deflection is given by

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

Thomson did not explain how he developed these equations, but this section will provide an educated guess.<ref>Heilbron (1968). p. 278</ref> At the same time, it will adapt these equations to alpha particle scattering on the assumption that alpha particles are point charges much like beta particles. These equations will then be used to show that Thomson's model was inconsistent with the experimental results of Geiger and Marsden.

===Partial deflection by the positive sphere===
Consider an alpha particle passing by a sphere of pure positive charge (no electrons) with a radius ''R''. The sphere is so much heavier than the alpha particle that we do not account for recoil. Its position is fixed. The alpha particle passes just close enough to graze the edge of the sphere, which is where the electric field of the sphere is strongest.

[[File:Thomson model alpha particle scattering 4.svg|center|thumb|upright=2|'''Figure 4''']]

[[#Single scattering by a heavy nucleus|An earlier section of this article]] presented an equation which models how an incoming charged particle is deflected by another charged particle at a fixed position (ie infinite mass).

<math display="block">\theta = 2 \arctan {\frac{k q_1 q_2}{m v^2 b}}</math>

This equation can be used to calculate the deflection angle in the special case in Figure 4 by setting the impact parameter ''b'' to the same value as the radius of the sphere ''R''. So long as the alpha particle does not penetrate the sphere, there is no difference between a sphere of charge and a point charge, a mathematical result known as the [[Shell theorem]].

* ''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}}
* ''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>}}

<math display="block">\theta_2 = 2 \arctan {\frac{k q_\text{a} q_\text{g}}{m v^2 R}} \approx 0.02  \text{ degrees}</math>

This shows that the largest possible deflection will be very small, to the point that the path of the alpha particle passing through the positive sphere of a gold atom is almost a straight line.  Therefore in computing the average deflection, which will be smaller still, we will treat the particle's path through the sphere as a [[chord (geometry)|chord]] of length ''L''.

[[File:Thomson model alpha particle scattering 3.svg|center|thumb|upright=2|'''Figure 5''']]

Inside a sphere of uniformly distributed positive charge, the force exerted on the alpha particle at any point along its path through the sphere is<ref>{{cite web | url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesph.html | title=Electric Field, Spherical Geometry }}</ref><ref name=BeiserPerspectives/>{{rp|106}}

<math display="block">F = \frac{k q_\text{a} q_\text{g}}{r^2} \cdot \frac{r^3}{R^3}</math>

The lateral component of this force is

<math display="block">F_\text{y} = \frac{k q_\text{a} q_\text{g}}{r^2} \cdot \frac{r^3}{R^3} \cdot \cos\varphi = \frac{b k q_\text{a} q_\text{g}}{R^3}</math>

The lateral change in momentum ''p''<sub>y</sub> is therefore

<math display="block">\Delta p_\text{y} = F_\text{y} t =\frac{b k q_\text{a} q_\text{g}}{R^3} \cdot \frac{L}{v}</math>

The deflection angle <math>\theta_2</math> is given by

<math display="block">\tan\theta_2 = \frac{\Delta p_\text{y}}{p_\text{x}} = \frac{b k q_\text{a} q_\text{g}}{R^3} \cdot \frac{L}{v} \cdot \frac{1}{mv}</math>

where ''p''<sub>x</sub> is the average horizontal momentum, which is first reduced then restored as horizontal force changes direction as the alpha particle goes across the sphere. Since the deflection is very small, <math>\tan\theta_2</math> can be treated as equal to <math>\theta_2</math>. The chord length <math>L = 2 \sqrt{R^2 - b^2}</math>, per [[Pythagorean theorem]].

The average deflection angle <math>\bar\theta_2</math> sums the angle for values of ''b'' and ''L'' across the entire sphere and divides by the cross-section of the sphere:

<math display="block">\bar\theta_2 = \frac{1}{\pi R^2} \int_0^R \frac{b k q_\text{a} q_\text{g}}{R^3} \cdot \frac{2\sqrt{R^2 - b^2}}{v} \cdot \frac{1}{mv} \cdot 2\pi b \cdot \mathrm{d}b</math>

<math display="block">= \frac{\pi}{4} \cdot \frac{k q_\text{a} q_\text{g}}{mv^2R}</math>

This matches Thomson's formula in his 1910 paper.

===Partial deflection by the electrons===
Consider an alpha particle passing through an atom of radius ''R'' along a path of length ''L''. The effect of the positive sphere is ignored so as to isolate the effect of the atomic electrons. As with the positive sphere, deflection by the electrons is expected to be very small, to the point that the path is practically a straight line.

[[File:Thomson model electron scattering.svg|center|thumb|upright=2|'''Figure 6''']]

[[#Single scattering by a heavy nucleus|An earlier section of this article]] presented an equation which models how an incoming charged particle is deflected by another charged particle at a fixed position.

<math display="block">\theta = 2 \arctan {\frac{k q_1 q_2}{m v^2 b}}</math>

''m'' and ''v'' are the mass and velocity of the incoming particle and ''b'' is the impact parameter.

For the electrons within an arbitrary distance ''s'' of the alpha particle's path, their mean distance will be {{sfrac|s|2}}. Therefore, the average deflection per electron will be

<math display="block">2 \arctan \frac{k q_\text{a} q_\text{e}}{mv^2 \tfrac{s}{2}} \approx \frac{4k q_\text{a} q_\text{e}}{mv^2 s}</math>

where ''q''<sub>e</sub> is the [[elementary charge]]. The average net deflection by all the electrons within this arbitrary cylinder of effect around the alpha particle's path is

<math display="block">\theta_1 = \frac{4k q_\text{a} q_\text{e}}{mv^2 s} \sqrt{N_0 \pi s^2 L}</math>

where ''N''<sub>0</sub> is the number of electrons per unit volume and <math>\pi s^2 L</math> is the volume of this cylinder.

[[File:LbR relationship.svg|border|right|thumb|'''Figure 7''']]

Treating ''L'' as a straight line,  <math>L = 2\sqrt{R^2 - b^2}</math> where ''b'' is the distance of this line from the centre. The mean of <math>\sqrt{L}</math> is therefore

<math display="block">\frac{1}{\pi R^2} \int_0^R \sqrt{2 \sqrt{R^2 - b^2}} \cdot 2\pi b \cdot \mathrm{d}b = \frac{4}{5} \sqrt{2R}</math>

To obtain the mean deflection <math>\bar{\theta}_1</math>, replace <math>\sqrt{L}</math> in the equation for <math>\theta_1</math>:

<math display="block">\bar{\theta}_1 = \frac{4k q_\text{a} q_\text{e}}{mv^2 s} \sqrt{N_0 \pi s^2} \cdot \frac{4}{5} \sqrt{2R}</math>

<math display="block">= \frac{16}{5} \cdot \frac{k q_\text{a} q_\text{e}}{m v^2 R} \sqrt{\frac{3N}{2}}</math>

where ''N'' is the number of electrons in the atom, equal to <math>N_0 \tfrac{4}{3} \pi R^3</math>.

===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.