2.1: Basic Crystallography75,76
As indicated in Chapter 1, zeolites and zeotypes are crystalline solids with highly ordered and regular arrangements of pores running through an oxide framework. Indeed, perhaps their most endearing characteristic is the symmetry exhibited in their framework and pore systems alike. The structure of crystalline materials, such as zeolites, can be described in terms of a repeat unit (or ‘unit cell’), the symmetry associated with that unit, and additional symmetry associated with the distribution of atoms within it. In this way the location of only a (relatively) few atoms are required to generate the entire topology via application of symmetry elements (these atoms represent the ‘asymmetric unit’).
The unit cell is a parallel-sided, three-dimensional box and defines the smallest unit of structure that can generate the entire structure through translation alone. Although it may not be uniquely defined, it is customary to select a unit cell which reflects the overall symmetry of the structure, and which, therefore, exhibits the highest symmetry possible. Seven different cell-types or ‘crystal systems’ exist (Table 2.1). These are defined, formally, by their minimum symmetry requirements, and are described in terms of the resulting parallelepid dimensions (a, b, c) and inter-axis angles (α, β, γ). The unit cell connects eight identical points of the structure (or ‘lattice points’). A cell containing one such point at each corner only is termed primitive (P), while face-centred cells contain an additional identical point in the middle of one set of opposite faces (A, B, C) or at the centre of all faces (F). A body-centred cell (I) contains an identical lattice point in the centre of the parallelepipid.
Crystal system Unit cell Essential symmetry requirements
Cubic a=b=c
α=β=γ= 90o
Four three-fold axes
Tetragonal a=b≠ c
α=β=γ= 90o
One four-fold axis or one inverse four-fold axis
Hexagonal a=b≠ c
α=β= 90o,γ= 120o
One six-fold axis or one inverse six-fold axis
Trigonal a=b=c
α=β=γ≠ 90o
One three-fold axis
Orthorhombic a≠ b≠ c
α=β=γ= 90o
Three orthogonal two-fold axes or planes
Monoclinic a≠ b≠ c
α=γ= 90o,β≠ 90o
One two-fold axis and/or one symmetry plane Triclinic a≠ b≠ c
α≠ β≠ γ≠ 90o
None
Table 2.1: The seven crystal systems.
By combining lattice type (i.e. P, C, I etc.) with point symmetry elements (i.e. symmetry around a point; inversion centre, rotation axes, mirror planes), and space symmetry elements (i.e. movement from point to point through space; screw axes, glide planes), 230 ‘space groups’ are generated. Viewed differently, there are 230 ways of arranging objects in space. Table 2.2 lists space group symmetry elements and their symbols. Space group symbols, therefore, contain information on the lattice type and the relationship of symmetry elements with the principal axes of the cell. For example,
Pmmn is an orthorhombic space group describing a primitive unit cell with mirror planes perpendicular to the a- and b-axes, and an n glide plane perpendicular to the c-
Symmetry element Symbol Centre of inversion -1 Mirror plane m (≡ -2) Glide planes a, b, c, d, n Rotation axis 2, 3, 4, 6 Screw axis 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65 Inversion axis -3, -4, -6
Table 2.2: Space group symmetry elements and their symbols (note, the minus sign in inversion operators is usually written above the number).
Miller Indices
Miller indices, named after the British mineralogist William H Miller, are employed to define sets of planes within a crystal and are quoted as three numbers in parentheses (hkl), wherea/h, b/k, c/ldefines the intercept of the plane with the three unit cell edges
a, b, cwithin the unit cell. For example, the (002) plane will be parallel to thea-andb-
axes and intercept thec-axis at ½ its unit cell length. (002), therefore represents a set of parallel planes separated by ½c and parallel to a and b. Indices quoted in square brackets define a ‘zone axis’. A zone is a collection of planes whose intersections are all parallel. The zone axis is parallel to the intersection of the planes, or, viewed differently, is perpendicular to the plane defined by the three indices within the square brackets – i.e. the [001] zone axis is equivalent to thec-axis.
2.2: Investigating Long-Range Order
2.2.1: X-Ray Diffraction76,77
X-ray diffraction (XRD) is perhaps the most widely employed technique for the characterisation of microporous materials, providing information on long-range order. Employed routinely for both everyday phase identification and in-depth structure analysis and solution, XRD is a powerful technique in the analysis of crystalline
the wavelength of the radiation involved and the separation of potential diffracting objects; diffraction will occur if the radiation wavelength is of the same magnitude as the object separation. X-radiation (a form of electromagnetic radiation lying between ultra violet and gamma rays) is particularly well suited, therefore, to the investigation of atomic arrangements within materials due to the similarity between its wavelength (typically in the order of 1Å (1x10-10m)) and inter-atomic distances.
The resulting diffraction pattern (usually given as a plot of intensity vs angle (2θ) for
powder samples) is a product of the interference (constructive and destructive) of the diffracted waves and can be regarded as an indirect ‘image’ of the atomic arrangement within the structure. The collection of good quality XRD data and its analysis is therefore key to successful structure analysis. The following sections introduce the key principles of the technique – from X-ray generation to data collection and analysis – with particular focus on the powder technique (i.e. analysis of polycrystalline materials).
X-Ray Sources
In a standard laboratory X-ray diffractometer, X-rays are produced in an ‘X-ray tube’ (Figure 2.1). In this device, electrons are emitted from an electrically heated tungsten cathode and accelerated towards a target anode, which can be made of several metals but is most commonly Cu (other common examples are Fe, Mo, Co, Cr). The impact of the electron with the target metal results in sudden and unpredictable deceleration, and the emission of ‘white radiation’. White radiation, which possesses a continuous envelope of wavelengths, is undesirable in common X-ray diffraction experiments and is therefore removed by monochromators (discussed later).
In addition to this phenomenon, the impacting electron may eject an electron from the 1s (K) shell, thus creating a vacancy into which a 2p (L) or 3p (M) electron may drop
(governed by the selection rule Δl = ± 1, e.g. p → s). In doing so, X-rays of energy
equivalent to ΔE (inter-shell energy) are emitted. Since the exact wavelength of these X- rays is dependent upon the energy gap between electron shells, which in turn is dependent upon the identity of the metal element involved, X-rays generated in this manner possess wavelengths characteristic of the metal target (Table 2.3). Figure 2.2 illustrates the generation of an X-ray photon via the ejection of a K-shell electron and its
replacement by an L-shell electron. X-rays generated from a transition from L→ K are denoted Kα, while M → K transitions are denoted Kβ. These are further subdivided
according to the total angular momentum quantum number j. For example, 2p3/2(L)→ 1s1/2 (K) = Kα1 and 2p1/2 (L) → 1s1/2 (K) = Kα2. Kα1 and Kα2 wavelengths are often sufficiently different to allow selection of one or the other by use of monochromators. However, Kβlines tend to overlap and are therefore less easily separated. Kαand KβX-
rays form sharp peaks at well-defined wavelengths superimposed on the broad background ‘hump’ of white radiation in the X-ray emission spectrum of the target metal (Figure 2.2).
Figure 2.1: Schematic illustration of a laboratory X-ray tube.
Wavelength (Å) Metal Kα1 Kα2 Kβ Cr 2.28975(3) 2.293652(2) 2.08491(3) Fe 1.93608(1) 1.94002(1) 1.75664(3) Co 1.78900(1) 1.79289(1) 1.62082(3) Cu 1.5405929(5) 1.54441(2) 1.39225(1) Mo 0.7093171(4) 0.71361(1) 0.63230(1)
Table 2.3: Typical X-ray wavelengths generated from common anode materials.
X-rays X-rays Cathode Anode Electron beam Be window Water
Figure 2.2: Schematic illustration of the generation of KαX-radiation, and typical X-
ray emission spectra for Mo and Cu.
A second source of X-rays is the synchrotron radiation source.78 Acceleration of an electron beam in a ‘storage ring’ (tens of metres in diameter) results in the emission of radiation as cones tangential to the circumference of the beam path. The desired wavelength is selected using monochromators and directed to the various stations situated round the ring (Figure 2.3). Station 9.1 at the Synchrotron Radiation Source (SRS), Daresbury Laboratory and the materials beam line ID31 at the ESRF, Grenoble were used for powder diffraction for some samples in this work. Synchrotron X-rays are more intense than laboratory sources and offer a wide range of tunable wavelengths. Furthermore, there is the opportunity of using much lower wavelengths (Table 2.4), which have lower absorption coefficients, for example.
Station Typical operating wavelength (Å)
SRS 9.1 0.4-4.5
ESRF ID31 0.3-0.9
Table 2.4: Typical operating wavelengths of stations used in this work.
Incident electron Emitted X-ray M L K
Figure 2.3: Diagrammatic representation of the ESRF synchrotron facility.79
Beam Conditioning
Prior to striking the sample the as-emitted X-ray beam must be conditioned in two principal ways. To permit ready analysis of the diffraction data the beam must be monochromatic – i.e. of single wavelength. This is achieved by a monochromator, often a crystal of Si or Ge. Following Bragg’s law (see later), different wavelengths diffract at particular angles from a given set of planes within a crystal allowing selection of the desired wavelength by appropriate manipulation of the angle at which the crystal is set to the incident beam (Figure 2.4). In this way only the desired wavelength (usually Kα1 in the case of work discussed in this thesis from laboratory diffractometers) is permitted to exit the monochromator.
whereby X-rays with parallel (or nearly parallel) propagation vectors are selected is called collimation and is achieved by a series of slits (Figure 2.5). The monochromatic and collimated beam is now directed to the sample.
Figure 2.4: Schematic illustration of a crystal monochromator. Adapted from ref. 77.
Figure 2.5: Schematic illustration of beam collimation achieved via a series of slits. The coordinate systems illustrate the type of divergence corrected at each step. Adapted
from ref. 77. Polychromatic beam Monochromatic beam Monochromator slit Monochromator crystal Long wavelengths Short wavelengths Divergence slits Soller slits Divergent beam Collimated beam In-plane divergence Propagation vector Axial divergence Propagation vector
Diffraction of X-rays: Bragg’s Law and the Ewald Sphere
W L Bragg derived a law of diffraction describing the process, mathematically, as ‘reflection’ from atomic planes within the crystal, which can be defined by their Miller indices. The resulting equation and its derivation are illustrated in Figure 2.6. The path difference for two waves both before and after reflection is indicated in Figure 2.6 as
distance AB; AB = dsinθ. The total path difference is therefore
2AB = 2dsinθ eqn. 2.1
and constructive interference occurs when 2AB = nλ, where n is an integer and λ is the
wavelength of the diffracting beam.
Therefore, at fixed wavelength the angular position of a diffraction maximum yields information on the lattice plane d-spacing; and conversely, different wavelengths will ‘reflect’ at different angles from any given set of planes (the basis of a crystal monochromator discussed previously). Miller indices are employed to index the diffraction peaks resulting from a particular set of planes.
AB =dsinθ
Total path difference = 2AB = 2dsinθ For constructive interference 2AB = nλ = 2dsinθ
Figure 2.6: Derivation of Bragg’s Law from consideration of two parallel reflecting beams. Incident wavefront Reflected wavefront (hkl) (hkl) θ θ 2θ A A dhkl B
2.7). If the origin of the reciprocal lattice is centred where the incident (or ‘direct’) beam exits the sphere, only those planes (hkl) whose reciprocal lattice point sits on the sphere surface (remembering this is a three dimensional model) will satisfy Bragg’s Law and produce a diffraction spot/peak.
Single Crystal Analysis
Single crystal XRD is by far the most preferred method for structural analysis. However, single (i.e. non-twinned etc.) crystals of sufficient size are only occasionally encountered in zeolitic materials research, thus necessitating the use of powder diffraction. Nevertheless, the experimental set-up for single crystal analysis provides a bases for the understanding of powder diffraction and is therefore briefly summarised here.
Figure 2.7: A two-dimensional representation of the equatorial plane of the Ewald sphere (equivalent to a plan view of Figure 2.8).
A crystal is selected and mounted on a fine glass capillary and fixed at a particular orientation to the X-ray beam. Data collection is conducted and measures those ‘reflections’ that satisfy the Bragg equation, or which sit on the Ewald sphere as described above. Whether a particular reciprocal lattice point sits on the Ewald sphere surface depends greatly on the angle at which the crystal is oriented towards the beam.
0 Incident beam Crystal at centre of sphere Origin of the reciprocal lattice Diffracted beam hkl d*hkl 2θ 1/λ
Therefore, the crystal is re-oriented after each data acquisition step to bring different reflections into contact with the Ewald sphere. Ultimately this permits the construction of a three-dimensional electron density map (see later) – i.e. three-dimensionality is maintained from structure to diffraction pattern, and structure solution is therefore (relatively) straightforward.
Powder Diffraction
Most samples to be analysed in work discussed in this thesis are polycrystalline – i.e. of many tiny crystallites, too small to be analysed by single crystal analysis (or exhibiting twinning/intergrowth). A small sample of powder, placed in the X-ray beam, therefore contains many crystallites at random orientation to the beam. The consequence of this is that for any set of (hkl) planes a diffraction beam exits the sample in all possible directions. Since the angle (2θ) between the direct and ‘reflected’ beam must be constant for a given set of planes, a cone is produced around the direct beam, the surface of which is formed by the accumulation of all diffracted beams from this set of planes.
Consequently, rather than discrete spots on the Ewald sphere, diffraction maxima are smeared into concentric rings about the direct beam (Figure 2.8). In a typical powder diffractometer, the diffracted X-ray beams are recorded by a detector, which scans
through an arc from 2θ = 1-3o to up to 90o, following the equatorial line of the Ewald sphere. In doing so, the ring base of each cone is traversed once and indicated in the powder diffractogram as a peak, the height of which (intensity) reflects the intensity (brightness) of the ring observed had a photographic film been employed for data collection. As a result powder diffractograms represent three-dimensional information on a two-dimensional scale, and information is therefore lost due to peak overlap and the summing of intensities.
Samples analysed during studies discussed in later chapters were either mounted in the sample holder as thin films (between plastic discs), or in capillaries (Figure 2.9). In both cases the sample is rotated to help ensure random sample orientation.
Figure 2.8: Schematic illustration of the diffraction of an X-ray beam from one set of planes in a powdered sample.
Figure 2.9: Two X-ray diffractometers employed in this work: (a) for thin film samples and (b) for capillaries. (1 – sample holder, 2 – X-ray ‘source’, 3 – beam stop, 4 –
detector, arrow indicates direction of incident X-ray beam).
Analysis of Powder Diffraction Patterns
As mentioned above, a diffraction peak represents the constructive interference of diffracted beams from a set of crystal planes. The position (in 2θ) of the diffraction peak is characteristic of the plane separation (d-spacing) and this can be related to the unit
2θ Powder sample Incident beam Direct beam (2θ = 0) One diffracted beam Path of detector 2θ = 90 Detector Diffraction cone (a) (b) 1 2 3 4 2 3 4 1
cell parameters for the material studied and the Miller indices of the planes involved. For example, in an orthorhombic system:
1/dhkl2=h2/a2+k2/b2+l2/c2 eqn. 2.2
This equation can be readily simplified for tetragonal (a = b) and cubic (a = b = c) systems. More complex equations exist for the other systems. Therefore, through analysis of peak position, peaks (or the pattern) are ‘indexed’ and the unit cell may be established. This process is conducted by algorithms accommodated in computer software, which generate a list of possibilities. A factor of merit (FOM) is given with each to aid interpretation by the user. Pattern indexing, although automated, is by no means trivial, however, and depends on accurate peak position picking and therefore data quality and resolution, and the absence of impurity peaks.
Higher resolution can be achieved from synchrotron radiation as a result of the collimated nature of synchrotron beams.78,79 The tangentially emitted radiation
possesses a vertical divergence (θv) given by:
θv≈mc2/E≈5x10-4/E eqn. 2.3 where m = electron mass, c = speed of light, E = energy of the ring (GeV)
At the SRS Daresbury Laboratory, which has an operating energy of 2GeV, the vertical divergence is approximately 0.25milliradians. This gives a vertical beam thickness of approximately 0.6cm at a 25m working distance (Table 2.5).
Source Operating E (GeV) θv(mrad) Beam thickness at 25m (cm)
SRS 2.00 0.250 0.6
ESRF 6.03 0.083 0.2
Figure 2.10: Powder XRD patterns of aluminosilicate TNU-9 recorded on (a) STOE STADI/P diffractometer (1.54056Å), (b) Station 9.1 SRS Daresbury (0.995559Å) and (c)
Station ID31 ESRF (0.80124Å). 2Theta 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 R e la ti ve In te n s it y (% )
D:\stewart\XRD\htnu9\hthu9long.raw / htnu9 longge
6.8 7.2 7.6 8.0 8.4 8.8 9.2 9.6 (a)
(b)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 3 5 7 9 11 13 d spacing (Angstroms) d el ta d /d ESRF Daresbury Stoe
The improvement in resolution is clearly illustrated in Figure 2.10: note, for instance, the improved base-line separation of the five principal peaks at low angle (insets) in the synchrotron data, as well as a better signal-to-noise ratio.
The resolution of a diffractometer is often expressed in one of three ways80: Angular Full Width at Half Maximum (FWHM) as a function of 2θ
FWHM as ΔQ (sinθ1-sinθ2/λ) as a function of Q (sinθ/λ) Δd/das a function ofd.
Figures 2.11 and 2.12 illustrate the differences in resolution of the three instruments employed in this work. In this case, angular Full Width at Half Maximum (FWHM) as a
function of 2θ would not be appropriate as the data were recorded at three different
wavelengths. Therefore, plots of ΔQ vs Q and Δd/d vs d are used. Clearly, synchrotron data is of superior resolution, particularly at low angle.
0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.03 0.05 0.07 0.09 0.11 0.13 0.15 Q (Angstroms-1) d el ta Q (A n g st ro m s -1 ) Daresbury ESRF Stoe Figure 2.12: Plot ofΔQ vs Q
The absence of certain reflections in the diffraction pattern (known as systematic absences) arises from lattice centring and symmetry elements present in the structure being analysed. For example, a body centred cubic crystal will exhibit diffraction maxima only from planes with Miller indices (hkl) where h+k+l= even. Consequently, an analysis of systematic absences can yield the space group of the structure.
While the peak positions and absences yield information on the unit cell parameters and