X-ray Diffraction Experiments 1 Introduction

Solid matter can be broken down into two broad categories – amorphous and crystalline. Amorphous materials account for approximately 5% of solids and crystalline materials about 95%. Diffraction patterns of thousands of inorganic materials are now in databases, and, therefore, carrying out diffraction experiments on unknown samples allows a search-and-match to be performed, to identify the unknown crystalline material. The distinction between amorphous and crystalline materials is highly important – amorphous materials have a more random arrangement of atoms and are not periodic structures like crystals. X-ray diffraction experiments work by exploiting the inherent periodicity of crystals.

2.4.2. Generation of X-rays

Electrons are accelerated through high voltages in an electron gun, where they collide with a metal, typically copper. Upon collision, the electrons are slowed or stopped, with the subsequent release of energy with a range of wavelengths, sometimes referred to as ‘white,’ or Bremsstrahlung radiation. When the collision occurs, the incident high-energy electrons are able to ionise an electron from the 1s subshell of the metal (Cu) target. An outer electron will ‘drop down’ from the 2p or 3p subshell to fill this vacancy. This causes the release of energy corresponding to an X-ray.

However, the 2p ! 1s and 3p ! 1s transitions result in the generation of X-rays with

a/2 c/3 b a c b a c b (a) (b) (c) a b

Figure 2.5: Examples of Miller indices showing the (a) 200, (b) (300) and (c) (213) planes.

different wavelengths. The 3p ! 1s transition results in a X-ray wavelength of 1.3922 Å, whereas the 2p ! 1s transition results in wavelengths of 1.54059 and 1.54433 Å, with an average value of 1.5418 Å. Note, this wavelength is that of X-rays commonly used in “laboratory” X-ray diffraction, used in this thesis. These emissions give rise to an X-ray emission spectrum, as shown in Fig. 2.6. Ideally, only one wavelength of radiation is used in the experiment, and for a Cu source, Ni metal is used as a filter. Alternatively, a monochromator may be used to provide radiation of one wavelength.

2.4.3. Scattering of X-rays1,2

X-ray diffraction experiments may be performed using a powdered sample or a single crystal. All diffraction work presented in this thesis will focus on powder diffraction, and as such, single crystal diffraction methods will not be discussed. In powder diffraction, millions of small crystals are present, and each of these has a slightly different orientation, resulting in a full range of orientations being observed relative to the incident X-ray beam. For some, the angle of orientation will be equal to the Bragg angle, resulting in Bragg’s law being satisfied and constructive interference in the diffraction experiment being observed. A moving detector is able to detect the diffracted X-rays and record these digitally.

The X-rays can be envisaged as diffracting in a cone-like shape, i.e., they are divergent. Each of these cones will have some intensity, I, of X-rays hitting the detector, and this signal is recorded as a Bragg peak in the diffraction pattern, of intensity, I. Sometimes, these cones will have some degree of overlap due to their close proximity and the result is partially resolved peaks being present in the pattern.

In diffraction patterns, two physical properties are of particular importance. Firstly, the d-spacing, obtained from the position of the Bragg peak, and secondly, the intensities of the peaks. In similar samples, similar d-spacings may be observed and, therefore, peaks may appear in similar places in the diffraction patterns but nonetheless, the intensities of the peaks may be different. Quantitative measurements of peak intensities can aid in the determination of unknown crystal structures.

The total intensity scattered is equal to the sum of all the individual intensities of the

peaks: I1 + I2 + I3 …. The scattering factor, or form factor, fn, of an atom is

proportional to the atomic/electron number of the atom, and as such, heavier elements have greater scattering factors. As a result, light atoms, such as H and Li, do not scatter X-rays well, and X-ray diffraction patterns are typically ‘dominated’ by heavier elements when they are present, making light atom location difficult to

determine. Additionally, atoms with similar atomic number are also difficult to

distinguish as they have similar scattering factors and therefore result in peaks with similar intensities. The atoms in the sample will only scatter quite weakly at high

Figure 2.6: Schematic (not to scale) representation of an X-ray emission spectrum from an electron gun with a Cu target showing ‘white’ radiation where low intensity X-rays of many wavelengths are produced, along with higher intensity X-rays of

specific wavelengths, denoted Kβ, Kα1 and Kα2.

Wavelength Intensity Kβ Kα2 Kα1 ʻWhiteʼ radiation

angles, and as such, the intensities of these peaks are poor. Important structural information, such as anisotropic displacement parameters, is often obtained from the high-angle region, and as such, this information cannot be accurately determined if the intensities are weak. Fig. 2.7 shows the scattering factors of two nuclei as a function of λ/2θ. These specific problems associated with using X-rays can be readily overcome by using neutron diffraction, as outlined in Section 2.6.

For each observed reflection in the diffraction pattern, which corresponds to a particular set of (hkl) values, a structure factor, Fhkl, is calculated to describe how the plane diffracted the incident X-rays. It describes both the amplitude and phase of the diffracted wave. The structure factor may also be used to determine if a reflection will be systematically absent, i.e., it has a value of zero. The amplitude of the diffracted wave has both sine and cosine components. The structure factor is given by,

Fhkl!= !fn cos 2π hx!+!ky!+!lz !+!i nfn sin⁡{2π hx!+!ky!+!lz }, (2.6) where fn is the form factor of atom n, and Fhkl is the structure factor for a particular plane.

Figure 2.7: Schematic representation of the change in scattering factors for two nuclei over a range of incident angles.

λ/2θ

Scattering factor

, f Ca2+

The sine and cosine parts may be combined to derive the simplified expression

Fhkl!!= fn exp i 2π hx!+!ky!+!lz !

n

!. (2.7)

By using Equations 2.2-2.4, dhkl may be calculated from the lattice parameters, which

in turn, can be used to calculate sin θ/λ by using Bragg’s law. The form factor for a

given atom, fn, may then be determined using the value of sin (θ/λ) - the form factors

for all atoms at different values of sin θ/λ are given in the International

Crystallography Tables.7 The value of Fhkl is summed over all the atoms in the unit

cell to obtain the final structure factor. The structure factor is closely related to the intensity of the observed reflection, as given by

Fhkl2 !∝!Ihkl . (2.8)

This is an abbreviated term, used for convenience.

2.5. Synchrotron X-rays