Editing Rutherford scattering experiments (section)

== An alternative method to find the scattering angle ==
This section presents an alternative method to find the relation between the impact parameter and deflection angle in a single-atom encounter, using a force-centric approach as opposed to the energy-centric one that Rutherford used.

The scattering geometry is shown in this diagram<ref>{{cite web
 |title=Impact Parameter for Nuclear Scattering
 |website=[[HyperPhysics]]
 |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/impar.html
 |publisher=[[Georgia State University]]
 |access-date=2024-05-30
 |ref={{harvid|Hyperphysics}}
}}</ref><ref name=Beiser1969>Beiser,&nbsp;A.&nbsp;(1969).&nbsp;[https://archive.org/details/perspectivesofmo0000arth/page/106/mode/2up "Perspectives of Modern Physics"].&nbsp;Japan:&nbsp;McGraw-Hill.</ref>{{rp|106}}

[[File:Rutherford scattering single alpha particle.svg|center|thumb|upright=3]]

The [[impact parameter]] ''b'' is the distance between the alpha particle's initial trajectory and a parallel line that goes through the nucleus. Smaller values of ''b'' bring the particle closer to the atom so it feels more deflection force resulting in a larger deflection angle ''θ''.<ref name=Goldstein1st/>{{rp|82}} The goal is to find the relationship between ''b'' and the deflection angle.

The alpha particle's path is a hyperbola and the net change in momentum <math>\Delta\vec{P}</math> runs along the axis of symmetry.   From the geometry in the diagram and the magnitude of the initial and final momentum vectors, <math>|\vec{P}_\text{i}| = |\vec{P}_\text{f}| = mv</math>, the magnitude of <math>\Delta\vec{P}</math> can be related to the deflection angle:<ref name=Beiser1969/>{{rp|111}}

<math display="block">\Delta P = 2mv \cdot \sin\frac{\theta}{2}</math>

A second formula for <math>\Delta P</math> involving ''b'' will give the relationship to the deflection angle. 
The net change in momentum can also be found by adding small increments to momentum all along the trajectory using the integral

<math display="block">\Delta P = \int\limits_{0} ^ {\infty} \frac{kq_\text{a} q_\text{g}}{r^2} \cdot \cos\varphi \cdot\mathrm  \mathrm dt</math>

where <math>r</math> is the distance between the alpha particle and the centre of the nucleus and <math>\varphi</math> is its angle from the axis of symmetry. These two are the polar coordinates of the alpha particle at time <math>t</math>. ''q''<sub>a</sub> is the charge of the alpha particle, ''q''<sub>g</sub> is the charge of the atomic nucleus, and ''k'' is the [[Coulomb constant]]. The Coulomb force exerted along the line between the alpha particle and the atom is <math>\tfrac{kq_\text{a} q_\text{g}}{r^2}</math> and the factor <math>\cos\varphi</math> gives that part of the force causing deflection.

The polar coordinates ''r'' and ''φ'' depend on ''t'' in the integral, but they must be related to each other as they both vary as the particle moves. Changing the variable and limits of integration from ''t'' to ''φ'' makes this connection explicit:<ref name=Beiser1969/>{{rp|112}}

<math display="block">\Delta P = \int\limits_{-\frac{\pi - \theta}{2}} ^ {\frac{\pi - \theta}{2}} \frac{kq_\text{a} q_\text{g}}{r^2} \cdot \cos\varphi \cdot \frac{\mathrm dt}{\mathrm d\varphi} \cdot \mathrm d\varphi</math>

The factor <math>\tfrac{\mathrm dt}{\mathrm d\varphi} = \tfrac{1}{\omega}</math> is the reciprocal of the angular velocity the particle. Since the force is only along the line between the particle and the atom, the [[angular momentum]], which is proportional to the angular velocity, is constant:

<math display="block">mvb = mr^2 \omega = mr^2\frac{\mathrm d\varphi}{\mathrm dt}</math>

This law of conservation of [[angular momentum]] gives a formula for <math>\tfrac{\mathrm dt}{\mathrm d\varphi}</math>:

<math display="block">\frac{\mathrm dt}{\mathrm d\varphi} =  \frac{r^2}{vb}</math>

Replacing <math>\tfrac{\mathrm dt}{\mathrm d\varphi}</math> in the integral for Δ''P'' simultaneously eliminates the dependence on ''r'':

<math display="block">
\Delta P = \int\limits_{-\frac{\pi - \theta}{2}} ^ {\frac{\pi - \theta}{2}} \frac{kq_\text{a} q_\text{g}}{vb} \cdot \cos\varphi \cdot \mathrm d\varphi
</math>

<math display="block">
= \frac{kq_\text{a} q_\text{g}}{vb} \left ( \sin\left[\frac{\pi - \theta}{2}\right] - \sin\left[-\frac{\pi - \theta}{2}\right] \right )
</math>

Applying the [[List of trigonometric identities#Reflections, shifts, and periodicity|trigonometric identities]] <math>\sin(\tfrac{\pi}{2} - \theta) = \cos\theta</math> and <math>\sin(\theta \pm \tfrac{\pi}{2}) = \pm\cos\theta</math> to simplify this result gives the second formula for <math>\Delta P</math>:

<math display="block">\Delta P = \frac{kq_\text{a} q_\text{g}}{vb} \cdot 2\cos{\frac{\theta}{2}}</math>

We now have two equations for <math>\Delta P</math>, which  we can solve for ''θ'':

<math display="block">\Delta P = \frac{kq_\text{a} q_\text{g}}{vb} \cdot 2\cos{\frac{\theta}{2}} = 2mv\cdot\sin\frac{\theta}{2}</math>

<math display="block">\theta = 2\arctan \frac{kq_\text{a} q_\text{g}}{mv^2b}</math>

Using the following values, we will examine an example where an alpha particle passes through a gold atom:

* ''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}}
* ''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>}}

When the alpha particle passes close to the nucleus barely missing it, such that the impact parameter ''b'' is equal to the radius of a gold nucleus ({{val|7|e=-15|u=m}}), the estimated deflection angle ''θ'' will be 2.56 radians (147°). If the alpha particle grazes the edge of the atom, with ''b'' therefore being equal to {{val|1.44|e=-10|u=m}}, the estimated deflection is a tiny 0.0003 radians (0.02°).<ref name=Beiser1969/>{{rp|109}}<ref>{{cite web
 |title=Determining the Impact Parameter 
 |website=[[HyperPhysics]]
 |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/rutsca2.html#c5
 |publisher=[[Georgia State University]]
 |access-date=2024-07-05
 |ref={{harvid|Hyperphysics}}
}}</ref>