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 shorted 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 = ½ mv2
So 1.6x10-19 x 6x109 = ½ 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
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Then using the de Broglie equation we can find the wavelength of the electron at this
energy.
Electron wavelength() = 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.
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