Nuclear Scattering

Electrons accelerated to close to the speed of light have wave-like properties such as the ability to diffract and have a de Broglie wavelength equal to:

Where:
- h = Planck's constant
- m = mass of an electron (kg)
- v = speed of the electrons (m s⁻¹)
When beams of neutrons or electrons are directed at a nucleus they will diffract around it
The pattern formed by this diffraction has a predictable minimum which forms at an angle θ to the original direction according to the equation

$\sin θ = \frac{λ}{b}$

The diffraction pattern forms a central bright spot with dimmer concentric circles around it
From this pattern, a graph of intensity against diffraction angle can be used to find the diffraction angle of the first minimum

The graph of intensity against angle obtained through electron diffraction is as follows:

$Electron Diffraction Graph, downloadable AS & A Level Physics revision notes$

The first minimum of the intensity-angle graph can be used to determine nuclear radius

Using this, the size of the atomic nucleus, R, can be determined using:

$\sin θ = \frac{λ}{2 R}$

Where:
- θ = angle of the first minimum (degrees)
- λ = de Broglie wavelength (m)
- R = radius of the nucleus (m)

$Electron Diffraction Method, downloadable AS & A Level Physics revision notes$

Geometry of electron diffraction

Worked example

The graph shows how the relative intensity of the scattered electrons varies with angle due to diffraction by the oxygen-16 nuclei. The angle is measured from the original direction of the beam. $Worked Example - Electron Diffraction Intensity GraphWorked Example - Electron Diffraction Intensity Graph, downloadable AS & A Level Physics revision notes$

The de Broglie wavelength λ of each electron in the beam is 4.22 × 10⁻¹⁵ m.

Calculate the radius of an oxygen-16 nucleus using information from the graph.

Step 1: Identify the first minimum from the graph $WE - Electron Diffraction Intensity Graph Answer, downloadable AS & A Level Physics revision notes$

- Angle of first minimum, θ = 42°

Step 2: Write out the equation relating the angle, wavelength, and nuclear radius

$\sin θ = \frac{λ}{2 R}$

Step 3: Calculate the nuclear radius, R

$R = \frac{λ}{2 \sin θ} = \frac{4.22 \times 10^{- 15}}{2 \times \sin (42)}$ = 3.15 × 10⁻¹⁵ m

Examiner Tip

In the data booklet, you may notice the relation between angle and diameter appears a couple of times - in the Resolution topic, you are given the full equation for calculating the Rayleigh criterion

$θ = 1.22 \frac{λ}{b}$

However, in this topic, you are given the approximated version

$\sin θ \approx \frac{λ}{D}$

This means you only need to use that version in exam questions about nuclear scattering, omitting the factor of 1.22 is acceptable and likely expected unless asked!

Nuclear Scattering (DP IB Physics: HL)

Revision Note