Chapter 2 – Experimental Techniques
2.3 Neutron Diffraction
An alternative to using X-rays as a source of scattering radiation is to use neutrons. Neutron and X-ray diffraction are complementary techniques in analysing the crystal structure of a material. The fundamental difference between the two is that neutrons are scattered by the atomic nuclei while X-rays are scattered by the electron clouds. Hence, neutron scattering factors vary comparatively little with atomic number, and irregularly with atomic number, unlike X-ray scattering factors, which increase with atomic number (as the electron density increases). So the neutron scattering by light elements, like Na and O, is similar to heavier elements, like Bi and La, whereas X-rays are scattered much more by the heavier elements. This scattering property of neutrons is very useful for structural determination for oxide materials like perovskites, where an accurate determination of the oxygen positions gives important information on the octahedral tilting. The oxygen positions can be difficult to obtain from X-ray diffraction if a much heavier element is present.
Another advantage of using neutrons for diffraction is that there is no angular dependence of the intensity of the diffracted beam, whereas the intensity of the peaks in X-ray diffraction decreases with increasing incident angle. Therefore, for a given λ and 2θ range more Bragg peaks can be measured accurately. However, one drawback is that some isotopes will absorb neutrons, making detection of these
49
isotopes more difficult and producing radioactive samples. In most cases the radioactive species produced are short-lived and requiring only a short amount of time before they return to normal levels. Another disadvantage of neutrons is that they are scattered more weakly than X-rays so in general larger sample quantities are needed.
2.3.1 Instruments used
Neutron diffraction was carried out at station D2B, which is a high-resolution two-axis diffractometer, at the Institut Laue-Langevin (ILL), in Grenoble, France. The research facility houses a high-flux nuclear reactor, which operates at a thermal power of 58.4 MW using a single fuel-element of 235U. The fuel-element sits in the
centre of a heavy water filled moderating tank of diameter 2.5 m.
Approximately 1 g of sample was placed into thin walled vanadium canisters. Vanadium was used due to its very low neutron scattering length. The canister position was adjusted so that the stainless steel top was clear of the beam and only the lower region lay in the beam path. A wavelength of 1.594 Å was selected with a Ge crystal monochromator. A complete scan at room temperature between 0 < 2θ < 160o typically takes 6 hours.
50
2.3.2 The Rietveld method
The Rietveld method, which was developed in the late 1960s, is widely used to refine crystal structures from X-ray or neutron diffraction patterns.3-4 For most
powder diffraction patterns, overlap occurs between Bragg reflections, especially for lower-symmetry materials due to the polycrystalline nature of a powder sample. The Rietveld method overcomes this difficulty due to peak overlaps by calculating the expected intensity for each individual step in the diffraction pattern. So instead of analysing each individual reflection, this method performs a curve fitting procedure by fitting the observed intensity, Yi(obs), of each equally spaced steps i over the entire pattern, with the calculated background intensity (Yi(bkg)) and the sum of the contribution of reflections close to the i powder pattern step (Yi(Bragg)):
(2.6)
The Rietveld method uses a least-squares technique to minimise the residual Sy between the observed intensity Yi(obs) and the calculated intensity Yi(calc) for all the steps i.e. until an optimum fit is obtained between the whole observed powder diffraction pattern and the entire calculated one:
(2.7)
where wi is the statistical weight that equals 1/Yi(obs), and Yi(calc) is the intensity of each step which can be calculated by a mathematical expression that includes the factors related to both the structure and the diffractometer used. So certain information is needed beforehand to calculate Yi(calc) such as the space group, unit cell lattice parameters, atomic positions and instrumental details. This is obtained by
51
building a model appropriate for the likely crystal chemistry of the material to be studied.
The indicators of the quality of the least-squares refinement between the calculated and observed patterns is estimated in the residual R-factors as defined by the following (Table 2.4):
Table 2.4 The numerical criteria of fit for the Rietveld method5
Parameter Expression R-pattern R-weighted pattern R-expected R-Bragg Goodness-of-fit
where M is the number of steps in the diffraction pattern, P is the number of refined parameters, and Ik(obs) and Ik(calc) are the observed and calculated intensities of the kth
Bragg reflection.
R-pattern and R-weighted pattern are measures of how well the calculated pattern fits the data, based on the residual at each step. R-expected is the value for the theoretically best fit that can be produced from the model and data, determined by the number of steps in the data and number of parameters refined. Goodness-of-fit is the ratio of the R-weighted pattern and R-expected and gives a measure of how well the data have been fitted. R-Bragg relates only to the goodness-of-fit of the
52
Bragg reflections in the structure being modelled, meaning any second phases present will be disregarded.
A good fit with the refined structure model will be shown by a low residual value. The R-weighted pattern, Rwp is the one most commonly considered as it
constains Sy which is the quantity to be minimised by the least-square refinements. The goodness-of-fit, χ2, which is directly proportional to Sy is also typically regarded
and ideally is 1.
Refinements of the powder XRD and neutron diffraction data were performed using the TOPAS Academic software.6 A text editor software, jEdit, was
also used to create and modify the input files needed to start a refinement in TOPAS. Detector zero error and sample height corrections were allowed to refine, and the Lorentz-Polarisation factor and axial divergence corrections were applied but constrained to theoretical values. The peak shape function used in all refinements was the modified Thompson-Cox-Hastings pseudo-Voight (TCHZ). In some cases, instead of a full Rietveld refinement, a Pawley peak intensity fit was used to refine lattice parameters accurately.
53