Ph.D. thesis (pdf) - dirac

52 Experimental techniques and observables
∂
section, 2 σ
∂Ω∂E
. It is given by the number of out coming neutrons (or photons) per
energy interval per solid angle per flux of the incoming probe. The cross section
is a function of the transferred energy and angle at which the probe is scattered.
The basic idea is illustrated in figure 4.1. Scattering where there is no transfer of
energy is called elastic scattering. Scattering where there is an exchange of energy
between the sample and the probe is called inelastic scattering. Contributions to
the inelastic scattering which have their maximum at zero energy transfer are called
quasi-elastic scattering.
The transfer in momentum is given by
and the transfer in energy is given by 5
Q = Q out − Q in (4.3.1)
ω = E out − E in (4.3.2)
The relation between the scattering angle, 2θ, and the transfer of momentum is for
elastic scattering given by Q = 2Q in sin(θ), while the general relation is
Q = (Q 2 in + Q 2 out − 2Q in Q out cos(2θ)) (4.3.3)
Neutrons do not interact with the electrons but only with the nucleus of the atoms 6 .
Elastic scattering: Q in =Q out
Inelastic scattering: Q in ≠ Q out
Q out
Q out
Q
Q
2θ
Q in
Q in
2θ
Figure 4.1: Illustration of the principle of a scattering experiment. The scattering
is called elastic if there is no transfer of energy between the probe and the sample.
The interaction is extremely short ranged as compared to the distances we are
interested in. The corresponding potential is therefore described by a Dirac delta
function, the Fermi pseudo-potential V (r) = bδ(R − r) where R is the position of
5 In this chapter we refer to ω as a quantity of dimension inverse time. However, the difference
between angular velocity and energy is just a . The actual measurement is a measure of the
transferred energy, and we measure ω units of energy (meV) in chapters 6 to 8.
6 Ignoring the magnetic interaction between the neutron and the electron, because it plays no
role in the type of systems we consider.