Editing Rutherford scattering experiments (section)

=== Target recoil ===
Rutherford's analysis assumed that alpha particle trajectories turned at the centre of the atom but the exit velocity was not reduced.<ref name=GilibertiLovisetti/>{{rp|253}} This is equivalent to assuming that the concentrated charge at the centre had infinite mass or was anchored in place.  Rutherford discusses the limitations of this assumption by comparing scattering from lighter atoms like aluminium with heavier atoms like gold. If the concentrated charge is lighter it will recoil from the interaction, gaining momentum while the alpha particle loses momentum and consequently slows down.<ref name=Rutherford1911/>{{rp|676|q=It is seen that the reduction of velocity of the alpha particle becomes marked on this theory for encounters with the lighter atoms.}}

Modern treatments analyze this type of Coulomb scattering in the [[centre of mass]] reference frame.  The six coordinates of the two particles (also called "bodies") are converted into three relative coordinates between the two particles and three centre-of-mass coordinates moving in space (called the lab frame). The interaction only occurs in the relative coordinates, giving an equivalent one-body problem<ref name=Goldstein1st/>{{rp|58}} just as Rutherford solved, but with different interpretations for the mass and scattering angle.

Rather than the mass of the alpha particle, the more accurate formula including recoil uses [[reduced mass]]:<ref name=Goldstein1st/>{{rp|80}}

<math display="block">\mu = \cfrac{m_1 m_2}{m_1 + m_2}.</math> 

For Rutherford's alpha particle scattering from gold, with mass of 197, the reduced mass is very close to the mass of the alpha particle:

<math display="block">\mu_\text{Au} = \cfrac{4\times 197}{4 + 197} = 3.92 \approx 4</math>

For lighter aluminium, with mass 27, the effect is greater:

<math display="block">\mu_\text{Al} = \cfrac{4\times 27}{4 + 27} = 3.48 </math>

a 13% difference in mass.  Rutherford notes this difference and suggests experiments be performed with lighter atoms.<ref name="Rutherford 1911"/>{{rp|677}}

The second effect is a change in scattering angle. The angle in the relative coordinate system or centre of mass frame needs to be converted to an angle in the lab frame.<ref name=Goldstein1st>Goldstein,&nbsp;Herbert.&nbsp;Classical Mechanics.&nbsp;United States,&nbsp;Addison-Wesley,&nbsp;1950.</ref>{{rp|85}}  In the lab frame, denoted by a subscript L, the scattering angle for a general central potential is

<math display="block">\tan \Theta_\text{L} = \frac{\sin\Theta}{\cos\Theta + \frac{m_1}{m_2}}</math>  

For a heavy particle like gold used by Rutherford, the factor <math>\tfrac{m_1}{m_2} = \tfrac{4}{197} \approx 0.02</math> can be neglected at almost all angles. Then the lab and relative angles are the same, <math>\Theta_\text{L} \approx \Theta</math>.

The change in scattering angle alters the formula for differential cross-section needed for comparison to experiment. In general the calculation is complex. For the case of alpha-particle scattering from gold atoms, this effect on the cross section is quite small.<ref name=Goldstein1st/>{{rp|88}}