2.1 Computational Methods
51
constant along the wave vector that is being interpolated. In an attempt to ensure
unbiased sampling of wave vectors in the initial case, wave vector sampling is often
done using a Monkhorst-Pack grid.
2.2 Experimental Methods
2.2.1 X-Ray Diffraction
The most definitive way to analyse the structure of a solid is via diffraction-based
techniques. Most commonly, diffraction techniques use X-ray radiation, although
neutron- and electron-based techniques are also well-known. An excellent introduction to the field of X-ray diffraction and crystallography can be found in Reference
[58].
X-rays are a form of electromagnetic radiation, with wavelength ~0.1–100 Å.
Their interaction with matter results from interactions with the electrons of the material, and the intensity of their scattering is therefore proportional to the electron
density. While X-rays are less sensitive to atom type (and insensitive to isotopes),
they scatter more strongly than neutrons. Thus, for most purposes, X-ray diffraction tends to be favoured due to faster collection times and higher quality data as
compared to neutron diffraction.
An additional benefit to X-ray diffraction is its availability, with laboratory Xray sources now commonplace. In a laboratory source, X-rays are generated by
accelerating an electron into an anode of a characteristic material. Upon collision,
the kinetic energy of the electron is sufficient to eject a core-level electron from the
anode, leaving an unstable vacant core state. An electron in a higher energy orbital
therefore drops into the vacant state, and the excess energy emitted as an X-ray. Due
to the quantized structure of atomic orbitals, the X-ray energy is characteristic of a
particular anode. Most common laboratory sources use a Cu anode (K α = 1.54056
Å), although others can be used. All diffraction data reported in this work is based
on monochromatic Cu radiation using a Bruker D2 phase diffractometer (flat plate
geometry) in the School of Chemistry, University of Edinburgh.
The “discovery” of X-rays is often accredited to German physicist Wilhelm
Röntgen in 1895 [59]. It was not until 1912 that von Laue first theorised that due to
the similar size of X-ray wavelengths and inter-atomic spacings, that X-rays could
scatter from the periodic arrays presented by crystals [60–63]. William and Lawrence
Bragg subsequently simplified the models proposed by von Laue and developed the
now famous Bragg’s Law [61, 64]. This led the Father and Son to demonstrate the
capabilities of X-ray diffraction in 1913 with the structural solution of NaCl, KCl,
KBr and KI [64], and crystallography was born.
The model proposed by the Braggs is best represented pictorially, Fig. 2.4. Diffraction is taken to occur from a set of theoretical planes, with interplanar spacing d, that
run through the real space primitive cell. In order to observe diffraction, scattered
51
constant along the wave vector that is being interpolated. In an attempt to ensure
unbiased sampling of wave vectors in the initial case, wave vector sampling is often
done using a Monkhorst-Pack grid.
2.2 Experimental Methods
2.2.1 X-Ray Diffraction
The most definitive way to analyse the structure of a solid is via diffraction-based
techniques. Most commonly, diffraction techniques use X-ray radiation, although
neutron- and electron-based techniques are also well-known. An excellent introduction to the field of X-ray diffraction and crystallography can be found in Reference
[58].
X-rays are a form of electromagnetic radiation, with wavelength ~0.1–100 Å.
Their interaction with matter results from interactions with the electrons of the material, and the intensity of their scattering is therefore proportional to the electron
density. While X-rays are less sensitive to atom type (and insensitive to isotopes),
they scatter more strongly than neutrons. Thus, for most purposes, X-ray diffraction tends to be favoured due to faster collection times and higher quality data as
compared to neutron diffraction.
An additional benefit to X-ray diffraction is its availability, with laboratory Xray sources now commonplace. In a laboratory source, X-rays are generated by
accelerating an electron into an anode of a characteristic material. Upon collision,
the kinetic energy of the electron is sufficient to eject a core-level electron from the
anode, leaving an unstable vacant core state. An electron in a higher energy orbital
therefore drops into the vacant state, and the excess energy emitted as an X-ray. Due
to the quantized structure of atomic orbitals, the X-ray energy is characteristic of a
particular anode. Most common laboratory sources use a Cu anode (K α = 1.54056
Å), although others can be used. All diffraction data reported in this work is based
on monochromatic Cu radiation using a Bruker D2 phase diffractometer (flat plate
geometry) in the School of Chemistry, University of Edinburgh.
The “discovery” of X-rays is often accredited to German physicist Wilhelm
Röntgen in 1895 [59]. It was not until 1912 that von Laue first theorised that due to
the similar size of X-ray wavelengths and inter-atomic spacings, that X-rays could
scatter from the periodic arrays presented by crystals [60–63]. William and Lawrence
Bragg subsequently simplified the models proposed by von Laue and developed the
now famous Bragg’s Law [61, 64]. This led the Father and Son to demonstrate the
capabilities of X-ray diffraction in 1913 with the structural solution of NaCl, KCl,
KBr and KI [64], and crystallography was born.
The model proposed by the Braggs is best represented pictorially, Fig. 2.4. Diffraction is taken to occur from a set of theoretical planes, with interplanar spacing d, that
run through the real space primitive cell. In order to observe diffraction, scattered