Powder x-ray diffraction

X-ray diffraction (XRD) employs electromagnetic radiation (monochromatic X-rays) to characterise the structure of crystalline solids (powders or crystals) by exploiting the scattering power of matter.17

The monochromatic X-rays are produced by accelerating a beam of electrons which will then collide with a metal target, usually copper. When an accelerated electron hits the metal it can ionise one electron from the 1s orbital of a Cu atom. In order to fill the electron vacancy a second electron from the 2p orbital will take its place moving to the lower energy level. By doing so it will release the excess of energy in the form of X-ray.17 Depending on the metal and which electron will fill the gap, the X-ray will have different wavelengths. The most frequent transition, and the one of interest, for copper is the 2p to 1s, the Kα radiation, characterised by a wavelength λ of 1.5405 Å. A less frequent and therefore less intense is the 3p to 1s transition, Kβ with λ = 1.3922 Å. They are simultaneously produced in the process and the Kβ removed with a filter (usually a nickel foil) to obtain a monochromatic radiation.17

Once the monochromatic X-ray is obtained it strikes the sample and it is scattered by the electrons of the atoms. This can happen if the wavelength of the radiation is close in value to the interatomic distances of the material, usually few Å.17

Bragg’s law is used to explain the relation between the wavelength, λ, of the incident beam, the scattered angle, θ, and the interatomic distance, d (Figure 2.2).

If a crystal is considered as made of layers (planes), when the x-ray reaches its surface it will be reflected with an angle θ. In some cases the angle of the reflected beam will be the same as that of the incident beam. Some others will be transmitted by the first layer to be reflected by the second layer and so forth. In order to have constructive interferences between the two reflected beams they have to be in phase, therefore the additional distance (A͞B + B͞C) that the second beam will have to travel has to be equal to nλ. From this condition and applying trigonometry to Figure 2.2 the Bragg’s law can be derived (Eq. 2.3).17 θ sin BC AB d Eq. 2.1 θ sin 2 BC AB  d Eq. 2.2 θ sin 2d nλ Eq. 2.3

where n is a whole number (1, 2, 3…) and d is the distance between adjacent planes and it is called the d-spacing.17 When this condition is verified the reflected X-rays are in phase, in all the other cases they are out of phase and destructive interferences occur bringing to cancellation of the radiation.17

In a powder diffraction experiment the crystals in the sample are distributed randomly in order to assume different orientations so that some of the planes will be able to satisfy Bragg’s law.17 _{These will diffract the beam with an angle θ and will be detected and} counted by a detector. The output is an XRD pattern which is characteristic and unique for the material. Figure 2.3 illustrate as example the XRD pattern for a silicon standard

Figure 2.2 Bragg’s law. Schematic representation of the interaction between a x-ray and the plane of a crystal. θ: Angle of the incident x-ray.

d θ A B C θ θ

used to calibrate the diffractometer. Each peak correspond to a Bragg’s reflection of the beam and their position represents the angle of the reflected beam.17

In an XRD pattern the peak position depends on the size and shape of the unit cell of a material, the d-spacing are related to the scattered angle by the Bragg’s law (Eq. 2.3). Unit cells with the same shape but different size will generate two similar patterns with a shift in the peak positions. The intensities of the peaks are dictated by the type of atoms in the unit cell. This is because the scattering power of an atom is proportional to the atomic number (Z).17 Therefore heavier atoms like Pt, will scatter more than lighter atoms like C, N or H. Furthermore, light atoms give origin to inelastic scattering, i.e. the X-rays lose or gain energy after being scattered and they are not in phase anymore. This may result in destructive interferences and a background signal in the XRD pattern.17

For a highly crystalline material the peaks in the pattern are very sharp (Figure 2.3), however in some cases these can be broader. Reasons for this can be found in the crystallite size and/or level of disorder in the material.

In the first case, the X-ray beams will be diffracted with the angles that satisfy the Bragg’s law but also with angles that will put the beams out of phase. In this case there will be destructive interferences.93 _{The extreme case is when n = 1/2 and the beams cancel out} completely. However if n < 1/2, for example n = 1/4, the sum will not be zero, but just a

Figure 2.3 XRD pattern of a silicon standard. XRD pattern of the silicon standard used to calibrate the diffractometer. The peaks position is characteristic of the material and it is used to verify the calibration of the instrument by comparing the acquired spectra with the PDF card # 27-1402. 25 50 75 100 2 / degree Inte nsit y / a rb. unit

reduced amplitude beam, which will still be recorded at a slightly different angle.93 If more planes are considered and each beam is out of phase of a quarter, the first and the third beam will cancel out. If n is even smaller it will take more planes before a completely destructive interference occurs. In a big crystal the total cancellation of the out of phase beams has more probability to happen that in a small crystal where the number of planes will be lower.93 Therefore in the latter case beams with reduced amplitude and slightly different angles will be detected and will produce a broadening of the peak in the XRD pattern. The equation that express the relation between the width of the peak and the size of the crystallite is the Scherrer formula (Eq. 2.4):93-95

θ cos k D





Where D is the crystallite size (Å), λ is the wavelength of the X-ray (Å), k is the Scherrer constant associated to the shape of the crystallite (usually for unknown shape, k = 0.9), β is the FWHM (full width at half maximum) of the peak after subtracting the instrumental line broadening (rad) and θ is the scattered angle (rad).

The second reason for the broadening of a peak is the presence of crystallites with slightly different d-spacing and therefore slightly different θ. This is called strain, a deformation of the crystallites as a consequence of a stress. If the change is the same for all the crystallites (uniform strain) a shift of the peak (or peaks, depending which planes are effected by the deformation) is produced. If the change is different, then a similar effect to that of particle size is produced and many peaks of lower intensity will be generated around the main one resulting in a broadening.93

Both, particle size and strain can have the same effect on the peaks. The Scherrer equation only accounts for size. A Williamson-Hall plot however can separate the two factors. This is particularly useful in the case of nanoparticles where the effect of the smaller size can mask the effect of the strain. The equation to be used in this case is:96, 97

D k θ sin 4 θ cos      Eq. 2.5

where all the variables are the same as defined above, with the addition of ε which is the strain due to the imperfections in the crystals. This equation is an example of y = ax + c

function, therefore, by plotting βcosθ vs. 4sinθ for the peaks in the pattern, the particle size will be given by the intercept with the y axis and the stain by the slope of the line. The X-ray diffractometer employed for this work is an Empyrean PANalytical series 2 diffractometer with a Cu Kα radiation source (λ = 1.5406 Å). The patterns were analysed with the PANalytical software DataViewer. XRD patterns in the present document are shown as square rout of normalised intensities. Theoretical pattern were generated with Fullprof suite and Highscore Plus was employed for indexing of the pattern.