2.2 Single Crystal X-ray Diffraction
2.2.1 Data Integration
To get from the individual frames with different angles that are collected in data collection to a single dataset with defined HKL values, the data must be integrated. During this process, a crystal structure is chosen by the operator of the program, using the data from the Ewald sphere to determine the symmetries and then using these to use the Laue symmetry to reject any outlying reflections. The data rejection can be skipped if it is important to output the entire dataset (such as when trying to determine multiple phases from one dataset). The unit cell parameters are calculated from the data during the data integration step by selecting peaks and constructing the lattice based on the integration considerations.
A given Bragg peak will be collected in multiple frames (the redundancy) and these are integrated to give a single set of data when all the individual data frames are integrated. The degree to which these data sets agree is given by:
Rint= P hkl|Fhkl(o)2 − hF 2 hkl(o)i| P hklFhkl(o)2 (2.10) which means that the intensities of a given peak,F2
hkl(o), is related to all instances
of the same peak via an average. Since Rint ∝ |Fhkl(o)2 − hFhkl(o)2 i|, a perfect data integration would result in Rint = 0, so the lower Rint the better. Rint gives a
measure of the consistency of the data set.
All single crystal data integration in this thesis was conducted using the program CRYSALIS PRO from Rigaku Oxford Diffraction. This process is also known as data reduction.
2.2.1.1 Absorption Correction
Single crystal diffraction requires a sufficient diffracting volume to extract high enough scattering intensity, which pushes the ideal crystal size up. At the same time, however, the absorption of the crystal means that the maximum size is strongly limited by the chemical composition of the crystal; the higher the absorption of the materials in the crystal, the smaller the crystal has to be. Because of these two competing requirements for crystal size, there exists an ideal size for a crystal, maximising the diffraction volume while minimising the effects of absorption.
Beer’s Law [13] states that: I I0
=e−µx (2.11)
whereI0 is the incident intensity,I is the diffracted intensity,µis the linear absorption coefficient andxis the total linear distance travelled through the crystal (both incident and diffracted distance).
Assuming a roughly spherical crystal, to ignore the effects of anisotropic size, thenI0 ∝x3, and thusI ∝x3e−µx. The maximum intensity for I can be found by setting its first derivative equal to 0:
dI dx = 3x
2e−µx−x3µe−µx=x2e−µx(3−µx) (2.12)
therefore, a minimum is found whenx= 0 or whenx= 3/µ. Given that the crystal is going to be of non-zero size, this means that the ideal size scale of the crystal will bex= 3/µ. The materials discussed in this thesis are all based on at least a partially bismuth A-site, with the highest amount being 100% bismuth, and the lowest being 50% bismuth, therefore the absorption in all samples is very high, limiting single crystal sizes to the range 20µm - 100µm. This is a small size for single crystals, making preparation more challenging.
As mentioned, this does neglect size anisotropy, which needs to be accounted for in any absorption corrections carried out on a non-spherical sample. The anisotropic path length in a crystal results in a reflection dependant absorption, hence measured intensities, resulting in inaccurate data integration ; the higher µ, the larger an effect the size anisotropy would otherwise have. To this end, these
differences in the transmission factor were calculated in this thesis by the analystical method of Clark and Reid [14].
The analytical method for calculating the transmission factor involves splitting the crystal into Howell’s Polyhedra [15], ensuring that for each
polyhedra only one face is illuminated and that the x-ray intensity is either absorbed or diffracted out of another face.
This method, unlike some of the others available such as the Empirical method, provides an exact solution. The limiting factor on this method was originally computing factor, but with improvements to the algorithm for
identifying the Howell’s polyhedra [16] and increases in analytical computer power, these limitations have been overcome. The method in CRYSALIS PRO is applied mostly automatically, but still requires that the faces of the crystal be drawn on to a sequence of still images of the crystal as it is rotated, resulting in a 3D representation of the crystal.
With the absorption in mind, one of the advantages of synchrotron single crystal XRD measurements is that the energy of the beam is tunable, meaning that absorption edges in the material can be avoided. In the BFO-KBT crystals, the absorption in the laboratory was calculated to be µ = 62.90 mm−1, while the absorption at the synchrtron wasµ = 5.52 mm−1. The absorption corrections are completed as part of the data integration step.