Editing Rutherford scattering experiments (section)

===Maximum nuclear size estimate===
Rutherford begins his analysis by considering a head-on collision between the alpha particle and atom. This will establish the minimum distance between them, a value which will be used throughout his calculations.<ref name="Rutherford 1911"/>{{rp|670}}

Assuming there are no external forces and that initially the alpha particles are far from the nucleus, the [[inverse-square law]] between the charges on the alpha particle and nucleus gives the potential energy gained by the particle as it approaches the nucleus. For head-on collisions between alpha particles and the nucleus, all the [[kinetic energy]] of the alpha particle is turned into [[potential energy]] and the particle stops and turns back.<ref name="BelyaevRoss2021"/>{{rp|5}} 
[[File:Rutherford model rest distance.svg|thumb|center|upright=2|Schematic view of a head-on collision between an alpha particle and an atom. The radius of the atom is on the order of 10<sup>−10</sup>&nbsp;m and the minimum stopping distance is on the order of 10<sup>−14</sup>&nbsp;m.]]

Where the particle stops at a distance <math>r_{\text{min}}</math> from the centre, the potential energy matches the original kinetic energy:<ref>[https://archive.org/details/in.ernet.dli.2015.60140/page/n647/mode/2up?q=appendix "Electrons (+ and -), Protons, Photons, Neutrons, Mesotrons and Cosmic Rays"] 
By Robert Andrews Millikan. Revised edition. Pp. x+642. (Chicago: University of Chicago Press; London: Cambridge University Press, 1947.)</ref>{{rp|620}}<ref name=Cooper1970>Cooper,&nbsp;L.&nbsp;N.&nbsp;(1970).&nbsp;[https://archive.org/details/introductiontom00coop/page/320/mode/2up "An Introduction to the Meaning and Structure of Physics"].&nbsp;Japan:&nbsp;Harper & Row.</ref>{{rp|320}}

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

where

<math display="block">k = \frac{1}{4\pi \epsilon_0}</math>

Rearranging:<ref name="Rutherford 1911"/>{{rp|671|q=It will be seen that b is an important quantity in later calculations}} 
<math display="block">r_\text{min} = k \frac{2 q_\text{a} q_\text{g}}{mv^2}</math>

For an alpha particle:
* {{math|''m''}} (mass) = {{val|6.64424e-27|u=kg}} = {{val|3.7273e9|u=eV/''c''<sup>2</sup>}}
* {{math|''q''<sub>a</sub>}} (for the alpha particle) = 2 × {{val|1.6e-19|u=C}} = {{val|3.2e-19|u=C}}
* {{math|''q''<sub>g</sub>}} (for gold) = 79 × {{val|1.6e-19|u=C}} = {{val|1.27e-17|u=C}}
* {{math|''v''}} (initial velocity) = {{val|2e7|u=m/s}} (for this example)
The distance from the alpha particle to the centre of the nucleus ({{math|''r''<sub>min</sub>}}) at this point is an upper limit for the nuclear radius.
Substituting these in gives the value of about {{val|2.7e-14|u=m}}, or 27&nbsp;[[femtometre|fm]]. (The true radius is about 7.3&nbsp;fm.)  The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27&nbsp;fm of the nuclear centre, as noted, when the  radius of the nucleus of a gold atom is 7.3&nbsp;fm.
[[File:RutherfordConcentrated.png|thumb|Figure 1. Potential energy diagram for Rutherford's atom model illustrating concentration in the nucleus.]]
Rutherford's 1911 paper<ref name="Rutherford 1911"/> started with a slightly different formula suitable for head-on collision with a sphere of positive charge:

<math display="block">\frac{1}{2}mv^2 = NeE \cdot \left (\frac{1}{b} - \frac{3}{2R} + \frac{b^2}{2R^3} \right )</math>

In Rutherford's notation, ''e'' is the [[elementary charge]], ''N'' is the charge number of the nucleus (now also known as the [[atomic number]]), and ''E'' is the charge of an alpha particle. The convention in Rutherford's time was to measure charge in [[electrostatic units]], distance in centimeters, force in [[dyne]]s, and energy in [[erg]]s. The modern convention is to measure charge in [[coulomb]]s, distance in meters, force in newtons, and energy in [[joule]]s. Using coulombs requires using the [[Coulomb constant]] (''k'') in the equation. Rutherford used ''b'' as the turning point distance (called ''r''<sub>min</sub> above) and ''R'' is the radius of the atom. The first term is the Coulomb repulsion used above. This form assumes the alpha particle could penetrate the positive charge. At the time of Rutherford's paper, Thomson's [[plum pudding model]] proposed a positive charge with the radius of an atom, thousands of times larger than the ''r''<sub>min</sub> found above. Figure 1 shows how concentrated this potential is compared to the size of the atom.
Many of Rutherford's results are expressed in terms of this turning point distance ''r''<sub>min</sub>, simplifying the results and limiting the need for units to this calculation of turning point.
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