Editing Rutherford scattering experiments (section)

===Single scattering by a heavy nucleus {{anchor|Single scattering from a heavy nucleus}} ===
From his results for a head on collision, Rutherford knows that alpha particle scattering occurs close to the centre of an atom, at a radius 10,000 times smaller than the atom. The electrons have negligible effect. He begins by assuming no energy loss in the collision, that is he ignores the recoil of the target atom. He will revisit each of these issues later in his paper.<ref name="Rutherford 1911"/>{{rp|672}}
Under these conditions, the alpha particle and atom interact through a [[Classical central-force problem|central force]], a physical problem studied first by [[Isaac Newton]].<ref>{{Cite journal |last=Speiser |first=David |date=1996 |title=The Kepler Problem from Newton to Johann Bernoulli |url=http://link.springer.com/10.1007/BF02327155 |journal=Archive for History of Exact Sciences |language=en |volume=50 |issue=2 |pages=103–116 |doi=10.1007/BF02327155 |bibcode=1996AHES...50..103S |issn=0003-9519|url-access=subscription }}</ref> 
A central force only acts along a line between the particles and when the force varies with the inverse square, like [[Coulomb's law|Coulomb force]] in this case, a detailed theory was developed under the name of the [[Kepler problem]].<ref name=Goldstein1st/>{{rp|76}} The well-known solutions to the Kepler problem are called [[orbits]] and unbound orbits are [[hyperbolas]].
Thus Rutherford proposed that the alpha particle will take a [[hyperbolic trajectory]] in the repulsive force near the centre of the atom as shown in Figure 2.

[[File:Rutherford scattering geometry 2.svg|thumb|center|upright=2|'''Figure 2.''' The geometry of Rutherford's scattering formula, based on a diagram in his 1911 paper. The alpha particle is the green dot and moves along the green path, which is a hyperbola with O as its centre and S as its external focus. The atomic nucleus is located at S. A is the [[apsis]], the point of closest approach. ''b'' is the impact parameter, the lateral distance between the alpha particle's initial trajectory and the nucleus.]]

To apply the hyperbolic trajectory solutions to the alpha particle problem, Rutherford expresses the parameters of the hyperbola in terms of the scattering geometry and energies. He starts with [[conservation of angular momentum]]. When the particle of mass <math>m</math> and initial velocity <math>v_0</math> is far from the atom, its angular momentum around the centre of the atom will be <math>m b v_0</math> where <math>b</math> is the [[impact parameter]], which is the lateral distance between the alpha particle's path and the atom. At the point of closest approach, labeled A in Figure 2, the angular momentum will be <math>m r_\text{A} v_\text{A}</math>. Therefore<ref name=Heilbron1968/>{{rp|270}}

<math display="block">m b v_0 = m r_\text{A} v_\text{A}</math>

<math display="block">v_\text{A} = \frac{b v_0}{r_\text{A}}</math>

Rutherford also applies the law of [[conservation of energy]] between the same two points:

<math display="block">\tfrac{1}{2}m v_0^2 = \tfrac{1}{2} m v_\text{A}^2 + \frac{k q_\text{a} q_\text{g}}{r_\text{A}}</math>

The left hand side and the first term on the right hand side are the kinetic energies of the particle at the two points; the last term is the potential energy due to the Coulomb force between the alpha particle and atom at the point of closest approach (A). ''q''<sub>a</sub> is the charge of the alpha particle, ''q''<sub>g</sub> is the charge of the nucleus, and ''k'' is the [[Coulomb constant]].{{refn|These equations are in [[SI]] units Rutherford<ref name=Rutherford1911/>{{rp|673}} uses [[cgs units]]}}

The energy equation can then be rearranged thus:

<math display="block"> v_\text{A}^2 =  v_0^2 \left (1 - \frac{k q_\text{a} q_\text{g}}{\tfrac{1}{2} m v_0^2 r_\text{A}} \right)</math>

For convenience, the non-geometric physical variables in this equation<ref name="Rutherford 1911"/>{{rp|674}} can be contained in a variable <math>r_\text{min}</math>, which is the point of closest approach in a head-on collision scenario<ref name="Rutherford 1911"/>{{rp|671|q=It will be seen that b is an important quantity in later calculations}} which was explored in a previous section of this article:

<math display="block">r_\text{min} = \frac{k q_\text{a} q_\text{g}}{\tfrac{1}{2} m v_0^2}</math>

This allows Rutherford simplify the energy equation to:

<math display="block"> v_\text{A}^2 =  v_0^2 \left (1 - \frac{r_\text{min}}{r_\text{A}} \right)</math>

This leaves two simultaneous equations for <math>v_\text{A}^2</math>, the first derived from the conservation of momentum equation and the second from the conservation of energy equation. Eliminating <math>v_\text{A}</math> and <math>v_0</math> gives at a new formula for <math>r_\text{min}</math>:

<math display="block">v_\text{A}^2 =  \frac{b^2v_0^2}{r_\text{A}^2} = v_0^2 \left (1 - \frac{r_\text{min}}{r_\text{A}} \right)</math>

<math display="block">r_\text{min} = r_\text{A} - \frac{b^2}{r_\text{A}}</math>

The next step is to find a formula for <math>r_\text{A}</math>.  From Figure 2, <math>r_\text{A} </math> is the sum of two distances related to the hyperbola, SO and OA.  Using the following logic, these distances can be expressed in terms of angle <math>\Phi</math> and impact parameter <math>b</math>.

[[File:Hyperbel-def-ass-e 2.svg|right|thumb|upright=2|'''Figure 3''' The basic components of the geometry of [[hyperbola]]s.]]
The [[Eccentricity (mathematics)|eccentricity]] of a hyperbola is a value that describes the hyperbola's shape. It can be calculated by dividing the focal distance by the length of the semi-major axis, which per Figure 2 is {{sfrac|SO|OA}}. As can be seen in Figure 3, the [[Hyperbola#Polar coordinate eccentricity|eccentricity is also equal to <math>\sec\Phi</math>]], where <math>\Phi</math> is the angle between the major axis and the asymptote.<ref name=Casey1885>Casey, John, (1885) [https://archive.org/details/cu31924001520455/page/n219/mode/2up "A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples"]</ref>{{rp|219}} Therefore:

<math display="block">\frac{\text{SO}}{\text{OA}} = \sec\Phi</math>

As can be deduced from Figure 2, the focal distance SO is

<math display="block"> \text{SO} = b \cdot \csc\Phi</math>

and therefore

<math display="block"> \text{OA} = \frac{\text{SO}}{\sec\Phi} = b \cdot \cot\Phi</math>

With these formulas for SO and OA, the distance <math>r_\text{A}</math> can be written in terms of <math>\Phi</math> and simplified using a trigonometric identity known as a [[List of trigonometric identities#Half-angle formulae|half-angle formula]]:<ref name=Rutherford1911/>{{rp|673}}

<math display="block">r_\text{A} = \text{SO} + \text{OA}</math>

<math display="block">= b \cdot \csc\Phi + b \cdot \cot\Phi</math>

<math display="block">= b \cdot \cot\frac{\Phi}{2}</math>

Applying a trigonometric identity known as the [[List of trigonometric identities#Double-angle formulae|cotangent double angle formula]] and the previous equation for <math>r_\text{A}</math> gives a  simpler relationship between the physical and geometric variables:

<math display="block">r_\text{min} = r_\text{A} - \frac{b^2}{r_\text{A}}</math>

<math display="block">= b\cdot\cot\frac{\Phi}{2} - \frac{b^2}{b\cdot\cot\frac{\Phi}{2}}</math>

<math display="block">= b \frac{\cot^2\frac{\Phi}{2} - 1}{\cot\frac{\Phi}{2}}</math>

<math display="block">= 2 b \cdot \cot \Phi</math>

The scattering angle of the particle is <math>\theta = \pi - 2 \Phi</math> and therefore <math>\Phi = \tfrac{\pi - \theta}{2}</math>. With the help of a trigonometric identity known as a [[List of trigonometric identities#Reflections|reflection formula]], the relationship between ''θ'' and ''b'' can be resolved to:<ref name="Rutherford 1911"/>{{rp|673}}

<math display="block">r_\text{min} = 2b \cdot \cot \frac{\pi - \theta}{2}</math>

<math display="block">= 2b\cdot\tan \frac{\theta}{2}</math>

<math display="block"> \cot\frac{\theta}{2} = \frac{2b}{r_\text{min}}</math>

which can be rearranged to give

<math display="block">\theta = 2 \arctan \frac{r_\text{min}}{2b} = 2 \arctan \frac{k q_\text{a} q_\text{g}}{m v_0^2 b}</math>
[[File:RutherfordHyperbolas.png|thumb|right|Hyperbolic trajectories of alpha particles scattering from a [[Gold]] nucleus (modern radius shown as gray circle) as described in Rutherford's 1911 paper]]

Rutherford gives some illustrative values as shown in this table:<ref name="Rutherford 1911"/>{{rp|673}}
{| class="wikitable"
|+ Rutherford's angle of deviation table
|-
| <math>\tfrac{b}{r_\text{min}}</math> || 10 || 5 || 2 || 1 || 0.5 || 0.25 || 0.125
|-
|  <math>\theta</math>  || 5.7° || 11.4° || 28° || 53° || 90° || 127° || 152°
|}

Rutherford's approach to this scattering problem remains a standard treatment in textbooks<ref name=HandFinch3rd>{{Cite book |last1=Hand |first1=Louis N. |url=https://www.cambridge.org/highereducation/books/analytical-mechanics/7145C0BF476388E6841B44064515909E |title=Analytical Mechanics |last2=Finch |first2=Janet D. |date=1998-11-13 |language=en |doi=10.1017/cbo9780511801662|isbn=978-0-521-57572-0 }}</ref>{{rp|151}}<ref>{{Cite book |last1=Fowles |first1=Grant R. |title=Analytical mechanics |last2=Cassiday |first2=George L. |date=1993 |publisher=Saunders College Pub |isbn=978-0-03-096022-2 |edition=5th |series=Saunders golden sunburst series |location=Fort Worth}}</ref>{{rp|240}}<ref>{{Cite journal |last1=Webber |first1=B.R. |last2=Davis |first2=E.A. |date=February 2012 |title=Commentary on 'The scattering of α and β particles by matter and the structure of the atom' by E. Rutherford (''Philosophical Magazine'' 21 (1911) 669–688) |url=http://www.tandfonline.com/doi/abs/10.1080/14786435.2011.614643 |journal=Philosophical Magazine |language=en |volume=92 |issue=4 |pages=399–405 |doi=10.1080/14786435.2011.614643 |bibcode=2012PMag...92..399W |issn=1478-6435|url-access=subscription }}</ref>{{rp|400|q=Rutherford's theory of scattering, assuming an inverse square law of electrostatic repulsion from a central fixed charge, is now textbook physics. }} on [[classical mechanics]].
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