very short-range nuclear interactions. The energy for typical “cold” neutrons (% meV)
differs strongly from that of typical X-rays (% keV), i.e., by factor of % 10
6
.
The consequence is first of all that the scattering amplitudes are completely
different in X-ray and neutron scattering, i.e., neutrons and X-rays “see” matter
differently. As a consequence of their interaction with matter, neutrons can distinguish between isotopes and render even light elements visible. This can be
exploited very efficiently to perform “contrast variation” studies, which are one
of the main strengths of neutron scattering, in particular for organic matter with
abundant hydrogen content. Secondly, because of their energy, neutrons are more
suitable to detect slower motions in inelastic scattering experiments. Moreover, a
photon carries no magnetic dipole moment whereas neutrons do, making neutron
scattering also very useful for probing magnetic structures. The large energy of
X-rays also causes difficulties for very high doses (from high flux sources). If the
energy dissipation is slower than the impact rate, X-rays are able to provoke
chemical changes as a consequence of free radical production, etc. However,
this is only the case at high brilliance sources, such as synchrotrons, although
there are ways to avoid or minimize these effects. For laboratory sources this is
not an issue.
Despite the much higher energy, X-rays penetrate the material much less than
neutrons due to their strong interactions with electrons. On the other hand, neutrons
are only weakly scattering and, combined with the relatively low flux available at
reactor sources (typically of the order of % 10
8 neutrons/(cm
2 s), this makes SANS
an intensity-limited technique. Synchrotron sources, however, easily deliver
% 10
12 –10
14 photons/s on the sample and thus improves the statistics issue dramatically. This opens up many exciting applications, as we will see later in Sect. 5.1,
one of which is extremely fast time-resolved measurements.
Scattering Contrast and Scattering Intensity
In a scattering experiment, the intensity is most conveniently measured as the
scattering cross-section, Σ per unit scattering volume divided by the solid angle Ω.
This quantity is referred to as the macroscopic differential cross-section dΣ/dΩ(Q),
which is measured as a function of the momentum transfer, Q ¼ k f À k i . Here, k is
the wave vector with modulus |k| ¼ k ¼ 2π/λ and λ is the wavelength.
Assuming that the scattering process is completely elastic, i.e., λ ¼ λ i ¼ λ f , the
modulus of Q can simply be cast into the following expression:
Q ¼ 4π
sinðθÞ
λ
(57)
where 2θ is the scattering angle.
Within the assumptions usually valid for small angle scattering (Born approximation, Thomson scattering, single scattering events, etc.), the amplitude of scattering is given by:
84
R. Lund et al.
differs strongly from that of typical X-rays (% keV), i.e., by factor of % 10
6
.
The consequence is first of all that the scattering amplitudes are completely
different in X-ray and neutron scattering, i.e., neutrons and X-rays “see” matter
differently. As a consequence of their interaction with matter, neutrons can distinguish between isotopes and render even light elements visible. This can be
exploited very efficiently to perform “contrast variation” studies, which are one
of the main strengths of neutron scattering, in particular for organic matter with
abundant hydrogen content. Secondly, because of their energy, neutrons are more
suitable to detect slower motions in inelastic scattering experiments. Moreover, a
photon carries no magnetic dipole moment whereas neutrons do, making neutron
scattering also very useful for probing magnetic structures. The large energy of
X-rays also causes difficulties for very high doses (from high flux sources). If the
energy dissipation is slower than the impact rate, X-rays are able to provoke
chemical changes as a consequence of free radical production, etc. However,
this is only the case at high brilliance sources, such as synchrotrons, although
there are ways to avoid or minimize these effects. For laboratory sources this is
not an issue.
Despite the much higher energy, X-rays penetrate the material much less than
neutrons due to their strong interactions with electrons. On the other hand, neutrons
are only weakly scattering and, combined with the relatively low flux available at
reactor sources (typically of the order of % 10
8 neutrons/(cm
2 s), this makes SANS
an intensity-limited technique. Synchrotron sources, however, easily deliver
% 10
12 –10
14 photons/s on the sample and thus improves the statistics issue dramatically. This opens up many exciting applications, as we will see later in Sect. 5.1,
one of which is extremely fast time-resolved measurements.
Scattering Contrast and Scattering Intensity
In a scattering experiment, the intensity is most conveniently measured as the
scattering cross-section, Σ per unit scattering volume divided by the solid angle Ω.
This quantity is referred to as the macroscopic differential cross-section dΣ/dΩ(Q),
which is measured as a function of the momentum transfer, Q ¼ k f À k i . Here, k is
the wave vector with modulus |k| ¼ k ¼ 2π/λ and λ is the wavelength.
Assuming that the scattering process is completely elastic, i.e., λ ¼ λ i ¼ λ f , the
modulus of Q can simply be cast into the following expression:
Q ¼ 4π
sinðθÞ
λ
(57)
where 2θ is the scattering angle.
Within the assumptions usually valid for small angle scattering (Born approximation, Thomson scattering, single scattering events, etc.), the amplitude of scattering is given by:
84
R. Lund et al.