Powder X-Ray diffraction

X-Ray diffraction is the most commonly used analytical technique for gaining structural information about crystalline and polycrystalline, but not amorphous, materials. When an X-ray is incident on, and interacts with, an atom’s electrons Rayleigh scattering occurs. In a crystalline structure, the regular repeat distance between atoms is in the angstrom region, similar to the wavelength (λ) of X-rays and thus their electron clouds can be used as a diffraction grating for Rayleigh scattering of a monochromatic source. Bragg’s law is defined in (1) where dhkl is the distance between two adjacent parallel planes and  is the Bragg angle between the incident wave and the atomic planes. When the increased path length of diffracted incident waves (Figure 2.3) is equal to a multiple of the X-ray’s wavelength, , then constructive interference will occur, whereas when this is not the case then destructive interference will occur, with the overall result of these reflections leading to a diffraction pattern. When the diffracted X-rays interfere constructively, then

Braggs’ Law is been satisfied, whereas if the diffracted X-rays interfere destructively then Braggs’

Law is not been satisfied.

nλ = 2d_hklsin (1)

If a sample is crystalline and single crystals can be grown then single crystal X-ray diffraction may be employed to determine the samples exact crystal structure (excluding hydrogen atoms, which are approximated). During single crystal ray diffraction, a sample is placed in a monochromatic X-ray beam and oriented through multiple angles in 3-dimensional space to obtain the full diffraction pattern, enabling structure determination. For an ideal polycrystalline material, every orientation of the crystal will be present equally in a sample and so full 3-dimensional data collection is not possible as there is a loss in reciprocal space from 3-dimensional to 1-dimensional. Consequently, powder X-ray diffraction is utilised to record diffraction patterns for polycrystalline materials that can be then refined using methods such as Rietveld refinement,¹³⁰ which uses a least squares approach, to identify the crystal structure. Some polycrystalline materials, which have crystals with large aspect ratios, where one dimension is significantly larger than the other dimensions, may lead to a preferred orientation in powder samples due to the crystallite anisotropy. This subsequently broadens the peaks in the diffraction pattern and without rectification, will prevent accurate analysis of crystallite size.¹³¹ A transmission-mode diffraction analysis may be able to overcome this, as described below. When X-rays are incident on a sample with only some crystals aligned in the correct orientation then cones of diffracted rays are produced. Using a detector perpendicular to the incident X-ray beam can detect these cones and build up a more detailed diffraction pattern in 3-D, rather than reflection mode which only orientates within a single plane at varying angles (Figure 2.4), however data collection time is much greater.

Figure 2.3 Schematic illustrating monochromatic rays incident on adjacent parallel planes at an incident angle  having a difference in path length equal to 2d_hkl sin.

Figure 2.4 Comparison of conventional (black) and transmission (blue) x-ray diffraction. In conventional XRD reflection mode is used with the X-ray beam and detector orientated in a diffractometer plane, whereas in transmission XRD a detector is used which can detect in the z-plane. Taken from He et al..¹³²

Naturally occurring LDHs can be ordered enough to form single crystals,¹³³ however synthetic LDHs are invariably polycrystalline and thus powder X-ray diffraction must be employed rather than single crystal X-ray diffraction. The instruments used in this X-ray diffraction analysis, diffractometers, can be used in reflection and transmission modes. The X-ray diffraction in this thesis only uses the more common reflection mode. This work was carried out using an instrument at the Department of Chemistry, University of Cambridge.

Dried LDH (in an oven at 80 °C overnight and re-equilibrated in atmosphere) and calcined MMO (at 500 °C for 3 hours) materials described herein were finely ground with an agate pestle

and mortar, and then pressed onto a ground glass sample holder. Following this preparation the materials were characterised by powder X-ray diffraction (PXRD) using a Philips X’pert PW3710 diffractometer with Cu Kα radiation (λ = 1.5418 Ǻ) at 15 °C. Cu Kβ radiation was removed through the use of a Ni filter. Diffraction patterns were acquired using a 2θ range of 3.00 to 80.00 degrees in 0.02 degree increments, with each step being held for 20.00 seconds. The divergence slit, which removes X-rays with large divergence angles, was fixed at 0.5 º and the anti-scatter slit, which is to ensure that only X-rays corresponding to the divergence slit arrive at the detector, was fixed at 1 º. The X’Pert HighScore Plus software package¹³⁴ was used to interpret the XRD data collected and indexing was attempted using the DICVOL automatic indexing algorithm,¹ with the data obtained shown in Appendix 1 and Appendix 2. The DICVOL method locates the unit cell parameters from the radial d-spacing by searching solutions in the parameter space for crystal systems of decreasing symmetry order. DICVOL indexing was carried out first through a search and match from the obtained peak list for each sample, then refining this cell, which was limited to rhombohedral/hexagonal systems. Despite the disordered nature of LDHs, various information can be obtained from the PXRD pattern, notably the cell parameters. The c parameter can be used to give the interlayer spacing and the a parameter can be used to find the divalent/trivalent cation ratio (vide infra). Both parameters are unaffected by stacking disorder, however the a parameter is indirectly connected with the stacking disorder since it accounts for the Al³⁺ substitution in the LDH lattice, which can contribute to lack of order in the stacking. LDHs are of hexagonal/rhombohedral symmetry where the a parameter equals the b parameter. This determination allows the cell to be defined. A typical LDH PXRD pattern is shown in Figure 2.5.

For rhombohedral systems the c parameter can be calculated from the (00l) reflections (Figure 2.6a) using equation (2). The interlayer spacing is equal to one third of this value when taking into consideration a three-layer repeat. The thickness of one hydroxide layer is 4.8 Å, thus subtracting this from the interlayer spacing, the gallery height can be determined.

The lattice parameter a0 can be determined using the (110) reflection (equation (3)) (Figure 2.6b), found around 60  2 for Cu Kα radiation (equating to ~3 Å), since it is related to the average cation-cation distance for the hydroxide layer in LDHs. Equation (3) is derived from the fact that rhombohedral lattices can be split up into equilateral triangles, as shown in Figure 2.7, with lengths which are equal to the unit cell a parameter, being bisected by the d₁₁₀ plane, giving the average cation-cation distance.

Figure 2.5 Powder X-ray diffraction pattern of MgAl(CO₃) layered double hydroxide of Mg:Al 2:1, prepared by co-precipitation as described in section 3, illustrating the characteristic peaks of LDHs.

( ) (

n 2n

2

00n 002n)

d d

c 

 (2)

2

0 (110)

a  d ⁽³⁾

a) b)

Figure 2.6 a) 003 and b) 110 planes shown in silver for a layered double hydroxide 3R polytype.¹³³ Colour scheme: M(II) and M(III) ions purple; oxygen red; hydrogen white; carbon grey.

Figure 2.7 Schematic of a repeating rhombohedral unit cell. Splitting the yellow rhombohedron into equilateral triangles gives the relationship oa=az=oz. The new length oz is bisected by the 110 plane and so, through symmetry, leads to the relationship a=2d₁₁₀.

As M³⁺ exhibits a different cation radius to M²⁺ this results in a different cation-cation distance.

Through empirical measurement of the range of cation ratios M^2+:M³⁺ a ratio relationship has been proposed by Kaneyoshi and Jones¹³⁵ (equation (4)) which can be utilised for nitrate and carbonate containing LDHs.

The crystallite size in the a and c directions can be determined from the Scherrer equation (5). K is a constant (~1) which varies with the breadth taking method, λ is the incident wavelength of X-rays, β_hkl is full width at half maximum height (FWHM) for a specific peak and hkl is the reflection angle at the specific peak centre. The Scherrer equation is only applicable to nano-scale particles and gives a lower limit for crystallite size. It follows that large line broadening occurs for small crystallites, but further line broadening in LDHs is uneven due to anisotropic bonding from structural disorder and preferred orientation during sample preparation.

.cos reflections can be broadened. Polytypes do not affect the 00l or hk0 peaks since they are related to the cationic sheet and interlayer spacing. Turbostratic disorder, where basal planes are randomly orientated and misaligned, along the c-axis, can affect the 113 and 0kl reflections to a certain degree.¹³⁶ 00l peaks can also be broadened by interstratification where there is an irregular repeat of two different interlayers.

Impurities and by-products present from synthesis may also identified by PXRD to confirm if the LDH is a pure-phase. MgO or similar compounds may be present and identified, however amorphous aluminium oxides will not be observed in the PXRD as it has no atoms with regular spacing to diffract X-rays in such a way that a pattern forms.