Spark image

High energy electron scattering and nuclear size

The scattering of high energy electrons by nucleons (protons and neutrons) can reveal the internal structure of these particles. It was thought that nucleons were fundamental particles but the results of the electron scattering experiments suggest the existence of point like particles within the proton.

At the Stanford Linear Accelerator (SLAC) electrons are accelerated along the two mile accelerator to an energy of 6 GeV (6 000 MeV or 6 000 000 000 eV). This is an energy of 1.6x10-19x6x109 J = 9.6x10-10 J.

What is the point of this?

As you know the diffraction of particles depends on their wavelength and so the more energetic the particle the shorter the wavelength and the finer the detail that can be observed. So very high energies are needed to probe the very internal structure of a proton or neutron.

Fermi proposed the equation for the radius of a nucleus (r) in terms of the radius of the hydrogen nucleus (ro) and the nucleon number of the nucleus (A)

Fermi equation for nuclear radius:      r = roA1/3

The constant ro = 1x10- 15 m = 1 Fm

Example problem
Calculate the radii of the following nuclei:
(a) carbon 12 (A = 12)
(b) gold 197 (A = 197)

(a) r = 10-15x121/3 = 2.3x10-15 m = 2.3 Fm
(b) r = 10-15x1971/3 = 5.8x10-15 m = 5.8 Fm

From the Fermi equation (r = roA1/3) we know that the "diameter" of a nucleon is about 10-15 m and so if we want to have any chance of "seeing" inside a proton we need to use something with a wavelength of this order or even smaller.

At the enormous energies produced by SLAC (Stanford Linear Accelerator Centre) the electrons have a very short wavelength which we can calculate using the de Broglie equation. Of course at such an energy the mass of the electron will be greater than that of the electron at rest and this can be calculated using the special relativity equations.

The rest mass of the electron = 9.1x10-31 kg and so using classical mechanics and ignoring relativity for the moment:
The electron energy = eV = mv2 So 1.6x10-19 x 6x109 = x 9x10-31 x v2 and so v = 4.6x1010 ms-1, clearly impossible since it is greater than the speed of light.

Using the mass-energy equation: E = mc2 to give mass (m) = E/c2, and that for momentum (p) = mv we have:
Momentum (p) = mv = [E/c2]v but at these enormous energies the velocity of the electron is close to that of light and so v =c.

Therefore: p = E/c

Then using the de Broglie equation we can find the wavelength of the electron at this energy.
Electron wavelength (l) = h/p = hc/E = [6.63x10-34 x 3x108]/9.6x10-10 = 2.1x10-16 m

This wavelength is less than the "diameter" of a nucleon and so these very high energy electrons can be used to probe the "structure" of the proton or neutron.

When these experiments were carried out it was found that the scattering of electrons from a nucleon was not uniform as it would have been if nucleons were just a single particle. Just like the alpha scattering from gold it showed evidence for smaller particles within the nucleon these were named Quarks.

© Keith Gibbs 2011