Editing Rutherford scattering experiments (section)

===Intensity vs angle===
[[File:ScatteringDiagram.svg|thumb|Geometry of differential scattering cross-section]]
To compare to experiments the relationship between impact parameter and scattering angle needs to be converted to probability versus angle.  The scattering cross section gives the relative intensity by  angles:<ref name=Goldstein1st/>{{rp|81}}
<math display="block">\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}(\Omega) \mathrm{d} \Omega = \frac{\text{number of particles scattered into solid angle } \mathrm{d} \Omega \text{ per unit time}}{\text{incident intensity}}</math>

In classical mechanics, the scattering angle <math>\theta</math> is uniquely determined the initial kinetic energy of the incoming particles and the impact parameter {{math|''b''}}.<ref name=Goldstein1st/>{{rp|82}} Therefore, the number of particles scattered into an angle between <math>\theta</math> and <math>\theta + \mathrm{d}\theta</math> must be the same as the number of particles with associated impact parameters between {{math|''b''}} and {{math|''b'' + ''db''}}. For an incident intensity {{math|''I''}}, this implies:

<math display="block">2\pi I b \cdot \left|\mathrm{d}b\right| =-2 \pi \cdot \sigma (\theta) \cdot I \cdot \sin(\theta) \cdot \mathrm{d}\theta </math> 

Thus the cross section depends on scattering angle as:

<math display="block">\sigma (\theta) = - \frac{b}{\sin\theta} \cdot \frac{\mathrm{d}b}{\mathrm{d}\theta} </math> 

Using the impact parameter as a function of angle, {{math|''b''(''θ'')}}, from the single scattering result above produces the Rutherford scattering cross section:<ref name=Goldstein1st/>{{rp|84}}

[[File:Rutherford's scattering equation illustrated.svg|thumb]]
<math display="block">
s = \frac {Xnt\cdot \csc^4{\frac{\phi}{2}}}{16r^2} \cdot {\left(\frac {2 k q_\text{n} q_\text{a}}{mv^2}\right)}^2
</math>

*''s'' = the number of alpha particles falling on unit area at an angle of deflection ''Φ''
*''r'' = distance from point of incidence of α rays on scattering material
*''X'' = total number of particles falling on the scattering material
*''n'' = number of atoms in a unit volume of the material
*''t'' = thickness of the foil
*''q''<sub>n</sub> = positive charge of the atomic nucleus
*''q''<sub>a</sub> = positive charge of the alpha particles
*''m'' = mass of an alpha particle
*''v'' = velocity of the alpha particle

[[File:RutherfordCrosssection2Scales.png|thumb|Rutherford scattering cross-section is strongly peaked around zero degrees, and yet has nonzero values out to 180 degrees.]]
This formula predicted the results that Geiger measured in the coming year. The scattering probability into small angles greatly exceeds the probability in to larger angles, reflecting the tiny nucleus surrounded by empty space. However, for rare close encounters, large angle scattering occurs with just a single target.<ref>Karplus, Martin, and Richard Needham Porter. "Atoms and molecules; an introduction for students of physical chemistry." Atoms and molecules; an introduction for students of physical chemistry (1970).</ref>{{rp|19}}

At the end of his development of the cross section formula, Rutherford emphasises that the results apply to single scattering and thus require measurements with thin foils. For thin foils the degree of scattering is proportional to the foil thickness in agreement with Geiger's measurements.<ref name="Rutherford 1911"/>