|PHY304||Particle Physics||Dr C N Booth|
For problems on this topic, see below.
Consider the scattering of an electron by a nucleus, as discussed in the Nuclear Physics course in the determination of nuclear size. If the electron has a low energy (compared with the separation of nuclear energy levels) the nucleus is left unexcited, and the scattering is elastic.
For elastic scattering |pi| = |pf| = p.
At low energies, the centre of mass frame is almost the same as the laboratory frame, since the mass of the electron is very much less than that of the nucleus. (Note that this is only true for low energy electrons.) Then
Now let us consider the probability of scattering into a given region of solid angle, dΩ. This is dσ/dΩ. As described in the introductory notes, such a transition rate is given by Fermi's Golden Rule
For a spin-less electron scattering from a point nuclear
is given by the classical Rutherford scattering cross section.
In reality, the electron has
and the form is known as the Mott formula, but we will not pursue this.
Instead we will consider the effect on spin-less electrons of having a
finite sized nuclear charge.
Mfi can be calculated
using the "Born approximation" - this assumes that a single scattering
occurs, and that the initial and final state electrons can be described
by plane waves.
We will also ignore the recoil of the nucleus, which
we have already indicated is a reasonable approximation in the low energy
limit, especially for a heavy nucleus.
The scattering amplitude (or matrix element) is then given by
|If we express the nuclear charge density as
with ∫ρ(r)d3r = 1,
then the potential energy of the electron at r is
(A) then becomes .
To simplify this, we can write R = r
− r', and note that for a given r',
We can now consider some special cases:
i) For a point-like nucleus, the charge is a δ-function at r' = 0, and the term indicated . We now have Rutherford scattering.
The modification due to the finite size of the nucleus is known as the form factor, . It can be seen that this is just the Fourier transform of the charge distribution.
NOTE: The Fourier relationship between scattered amplitude and
spatial distribution of the scatterer is general, e.g. optical diffraction,
X-ray scattering, etc.
|ii) Spherically symmetric charge distribution ρ(r')
We can choose spherical co-ordinates as shown in the figure, with the z-axis
parallel to q. Then the volume element
In the spin-less case, we must have symmetry in φ, so integrating we obtain
In principle, the measured dσ/dΩ can be used to determine F(q), and then the inverse Fourier transform used to obtain ρ(r).
However, to do this requires knowledge of F(q)
over the complete range of q, which is impractical (at large
is very small and difficult to determine accurately). In practise,
a model for ρ(r)
is assumed, described by a small number of parameters, which are then adjusted
to best fit the measured values of F(q).
|Note (from the Fourier relationship) that a broad spatial distribution leads to a narrow distribution in q.|
|e.g. in the Rutherford experiment,||large atoms → small scattering displacements|
|small nuclei → large (but rare) displacements.|
Form Factor = Fourier transform of charge distribution
For practice in calculating and interpreting form factors, see Homework 1.