2 Introduction to characterisation techniques
2.3 X-ray diffraction
2.3.3 Bragg’s law
Peaks in XRD patterns are called reflections and they derive from the constructive interference of X-ray beams diffracted by electrons in atoms. In the case of two adjacent planes of a lattice irradiated with two parallel rays of the same wavelength, as shown in Figure 2.8, constructive interference and thus diffraction can be observed only if the path-length difference of the two parallel incoming rays between the two adjacent lattice planes equals an integer number of wavelengths, satisfying Bragg’s law
nλ= 2dsinθ , (2.10)
where λ is the wavelength experimentally used, d is the spacing between two adjacent lattice planes, θ is the incident angle and n is an integer number, usually taken as unity since higher order diffraction maxima can be represented by diffraction from a set of planes with 1/n separation.
In a crystalline material, each of the d-spacings, obtained from a set of lattice parameters and h, k and l indices, as, for example, shown in Equations 2.8 and 2.9, can be combined with Bragg’s law resulting in diffraction peaks at specific diffraction angles. The position of these peaks in an experimental diffraction pattern will then be related to the lattice parameters, Miller indices and experimental X-ray wavelength, providing information about the size of the repeat unit, the d-spacings, of the system analysed.45-46
2.3.4 Powder X-ray diffraction
Generally, the basic components for a XRD experiment are: a monochromatic source of X-ray radiation, the sample under investigation and a detector to collect the diffracted X-rays. Experimentally, XRD patterns can be acquired with different geometries both on single-crystal samples and powder samples. In this thesis, only powder samples have been analysed and the specific experimental geometries used for the characterisation of zeolite and MOF powders are described in this section.
In a powder sample for X-ray characterisation an infinite number of randomly oriented crystals are present with lattice planes equally randomly oriented. Some crystallites, for each set of lattice planes, will therefore be oriented at the Bragg angle with respect to the incident beam resulting in diffracted X-rays, which can be subsequently detected. PXRD experiments can be carried out in reflection or transmission, requiring different geometries and resulting in different experimental advantages and disadvantages. Transmission PXRD experiments can be acquired using the Debye-Scherrer geometry, as shown in Figure 2.9, where the generated X- ray radiation is applied perpendicular to the powder sample, which is packed in a capillary and rotated to avoid any preferential orientation of the crystallites. Given the random orientations of all the crystallites in the sample, it is possible that some of these will be correctly oriented to satisfy Bragg’s law for a set of crystal planes with, for example, a d1 spacing.
A B 1 1’ 2 2’ d Crystal plane Crystal plane Incident X-rays Diffracted X-rays
Two such crystallites are schematically illustrated in Figure 2.9 and, considering the whole sample, a cone of rays is diffracted at a 2θ1 angle with respect to the incoming X-ray radiation. At the same time, other crystallites will also be in the correct orientation to satisfy Bragg’s law, and hence to diffract, for a number of different d- spacings, such as the d2 spacing shown for another couple of crystallites in Figure 2.9, and the diffracted cones produced are then detected. The advantages of this technique are that very limited amounts of sample are needed and air- or moisture- sensitive samples can be easily analysed by sealing the capillary to avoid any contact with air or water during data acquisition. However, the small amounts of sample used result in low sensitivity and for some samples the packing stage can prove challenging.
Reflection PXRD experiments have been acquired using the Bragg-Brentano geometry where large amounts of sample are packed in a disk with X-ray radiation
21 Incident X-ray beam 21 Powder sample in capillary 22 22 Debye-Scherrer diffraction cones d 2 diffracted cone d1 diffracted cone Detector 2
Figure 2.9 Debye-Scherrer geometry for transmission experiments showing the incoming X- ray beam perpendicularly incident upon the randomly oriented crystallites present in the sample packed in the rotating capillary. Diffracted cones are then detected by a movable detector as a function of 2θ with respect to the incoming X-ray beam.
diffracted beams are then detected by a movable detector as a function of 2θ with respect to the incoming X-ray beam. Scanning of the desired 2θ range can be achieved either by moving at the same time the X-ray tube and the detector by θ with a static sample holder or by keeping the X-ray generation tube stationary and turning the sample holder in sync with the detector by θ and 2θ, respectively. The packing of the powder in the disk requires more sample compared to the capillary described above and could result in unwanted preferential orientation of the crystallites, which can be, at least partially, removed by rotating the disk. As a result of the greater amount of solid used, this technique usually yields patterns characterised by higher intensities and better signal-to-noise ratios compared to the transmission capillary setup, for equivalent amounts of acquisition time.44,46-48
Goniometer circle
X-ray tube Detector
2
Powder sample container
Figure 2.10 Bragg-Brentano geometry for reflection experiments showing the incoming X- ray beam incident at angle θ upon the randomly oriented crystallites present in the sample packed in the rotating disk. Diffracted X-rays are then detected by a movable detector as a function of 2θ. In the schematic setup shown the scan over the selected 2θ range is achieved by moving the X-ray tube and the detector at the same time by θ.
2.3.5 Rietveld and Pawley refinements
A powder pattern represents the fingerprint of a crystalline material since the position and intensity of the recorded peaks are related to the size and shape of the unit cell and the position and number of electrons of the atoms in the cell, respectively. Diffraction data provide, therefore, a wealth of crystallographic information ranging from routine identification of materials, analyses of mixtures and phases, determination of lattice parameters up to, where possible, full structure resolution. For this last purpose atomic positions need to be determined and so peak intensities need to be considered using different methods depending on the starting degree of knowledge of the structure. In this thesis, Rietveld and Pawley refinements from powder diffraction data have been carried out in collaboration with Dr Samuel Morris for the determination of cation ratios and unit cell variations during the breathing cycle in MIL-53 MOFs.
The Rietveld approach was introduced49 in the middle of the 1960s by the Dutch physicist Hugo M. Rietveld for the accurate determination of crystal structures from neutron diffraction data, and later X-ray data, and is the most widely used full profile refinement technique today. The basis of the Rietveld approach is that all structural and experimental parameters are refined with the non-linear least squares method by fitting the observed data to a calculated profile deriving from an adequate structural model. Therefore, in this method, intensities are treated in all calculations as functions of the geometrical and structural parameters of the compound under analysis, as opposed to the Pawley method described later in this section, where intensities are treated as free least squares variables. This basic idea should result in a fully refined structure yielding a calculated diffraction pattern characterised by little, or ideally no, difference with the experimental pattern. The Rietveld method can thus be considered a powerful structural optimisation technique, requiring, however, an approximate initial knowledge of the structure. During a refinement the function Φ is minimised using the non-linear least squares method:
Φ= w! Y!!"#−Y!!"#! ! , ! !!! (2.11)
where n is the total number of experimental points in the powder diffractogram and
Y!!"# and Y
!!"#! are the observed and calculated intensities, respectively, of the ith data
point. The weight, wi, of the ith data point, assuming that the recorded intensity is only affected by statistical errors without interference from the background, can be given as:
w! = Y!!"# !!. (2.12)
The Pawley method,50 also based on non-linear least squares minimisation of the difference between the observed and the calculated profile, is a refinement procedure introduced by Pawley in 1981 to obtain the best possible unit cell from a powder diffraction pattern starting from suggested parameters obtained from a unit cell analysis program. In this method, knowledge of the crystal structure is not required and, for this reason, each reflection peak in the diffraction pattern has a variable intensity, not being constrained by the crystal structure. If a complete structure solution is desired, the relevant structural and experimental parameters obtained through this method can then be used as initial approximate values for a Rietveld refinement.
In both methods, the least squares solution can be considered found when the best fit between the calculated Y!!"#! and experimental Y
!!"# intensities is obtained. Therefore,
to quantify the quality of the fits the following factors have been introduced: - the residual factor Rp
R! = Y! !"#−Y !!"#! ! !!! Y!!"# ! !!! ×100%, (2.13)
- the weighted profile residual Rwp
R!" = w! Y!!"#−Y !!"#! ! ! !!! w! Y!!"# ! ! !!! !/! × 100%, (2.14)
- the expected profile residual Rexp
R!"# = n−p w! Y!!"# ! ! !!! !/! ×100% , (2.15)
- the goodness of the fit χ2
χ! = w! Y!!"#−Y!!"#! ! ! !!! n−p = R!" R!"# ! , (2.16)
where n is the total number of experimental points in the powder diffractogram, Y!!"# and Y!!"#! are the observed and calculated intensities, respectively, of the ith data point, wi is the weight of the ith data point and p is the number of free least squares parameters. In general, a better fit is related to lower values of all residuals, but there aren’t exact numerical values to define the acceptability of the fit and the relation between Rwp and Rexp, established in the goodness of the fit, is probably the best evaluation method of the quality of the fit. Since the expected residual, Rexp, involves an ideal model defined and limited by the quality of the data, the value of Rwp in the best fit should tend to Rexp, and therefore χ2 should approach unity.47,51-52