Handbook of Modern Coating Technologies

Interaction of X-rays with crystalline materials

As a meticulous experimentor with a sound theoretical base, Roentgen performed a number of experiments to ascertain the properties of X-rays immediately following his initial observa­tions of these penetrating but invisible rays. However, owing to their different kind of interac­tion with matter—for they could not be reflected or refracted using optical mirrors or lenses, nothing could be concluded about their exact character. At the same time, it was thought that X-rays could be another form of electromagnetic radiation, whose wavelength lies in the range of 0.1—0.2 nm and that they could interact with the periodic structure of crystals to produce the effect of diffraction—similar to the phenomenon of diffraction of visible light by diffraction grating, which was well known at that time.

It was Peter Paul Ewald's doctoral work on scattering of electromagnetic radiation [35] that inspired German physicist Max von Laue to perform experiments involving diffraction of X-rays by crystals. Laue thought that if X-rays were electromagnetic in character and if crystals were made-up of periodic arrays of atoms with spacing comparable to the wavelength of X-rays, inter­action of X-rays with crystals should give rise to the formation of diffraction patterns. He was soon able to successfully record the first diffraction pattern on a photographic plate.

Till the dawn of 20th century, the knowledge about crystal structure was gathered through macroscopic measurements of angles between various facets of single crystals, while their elemental make-up was determined through chemical techniques. Their interaction with X-rays to produce diffraction patterns opened a whole new domain of possibilities in the study of materials. Laue's experiments were analyzed by English Physicist, Sir William Henry Bragg and his son William Lawrence Bragg. At a rather young age of 22 years, W.L. Bragg was able to explain the phenomenon of diffraction of X-rays by crystals, following which he proceeded to determine the structures of NaCl, KCl, KBr, and KI.

• Diffraction of X-rays by crystals

Diffraction of X-rays is essentially a phenomenon involving their scattering in all directions by individual atoms. When a beam of X-rays falls on a crystal, it causes oscillations of elec­trons, which in-turn emit X-rays of same frequency as the incident beam. This emission occurs in all directions and is termed as scattering of the incident beam. The radiation thus scattered from individual atoms undergoes systematic constructive or destructive interfer­ence in certain select directions owing to regular arrangement of atoms within the crystal, thereby leading to the formation of a diffraction pattern. The conditions for constructive or destructive interference occur due to introduction of path-difference between the wavelets diffracted by different sets of atoms within the crystal. The phenomenon can be easily under­stood by considering Bragg's law.

Fig. 3—16 shows schematic of a crystal being exposed to X-rays. It should be noted that the interaction of X-rays with matter is rather weak and so, only a small fraction of the inci­dent beam will be scattered by the atoms. First, we consider the topmost plane of the crystal, where the incident X-rays make an angle 6_i with the plane. Note that in XRD parlance, the

angles are defined from the surface tangent—unlike surface normal, which is the norm in optics. As said earlier, each atom will scatter the incident beam in all possible directions. Accordingly the rays 1 and 2 will be scattered in all directions by atoms C and B, respec­tively. Two such directions, namely, 1', 2' and 1", 2" are shown in the figure. The difference in the paths traversed by rays 1 — C — 1' and 2 — B — 2' is given by:

Path Difference = CM — BN = BCcose, — BCcosd_e        (3.7)

The rays 1⁰ and 2⁰ would undergo constructive interference if they are in phase, which would happen if the path-difference is zero. This condition would be satisfied for в_е = e_t and will result in formation of strong emergent beam. The resultant intensity of emergent beam would diminish as 6_e deviates from d_t. Thus elastic scattering from regularly spaced atoms lying on a planar crys­tal surface leads to a kind of "reflection” where the angles made by incident and emergent beams with the crystal surface are equal. Accordingly the symbol в will be used henceforth in place of e_t or в_е.

Since X-rays interact rather weakly with the atoms, a major portion of the incident beam passes-on to the layers beneath the surface. Accordingly the resulting diffraction pattern incorporates cumulative effects of not only the surface layer, but up to a certain depth below the surface. Considering interference between rays 1⁰ and 3⁰ reflected from the two topmost planes (planes I and II), path-difference works out to be PD + DQ or 2dsine. According to Bragg's condition, constructive interference will take place whenever path-difference is an integral multiple of wavelength, A. Mathematically:

nA = 2dsine                                               (3.8)

or A = 2dsine for n = 1, i.e., for first order reflection from planes I and II

Similarly Bragg's condition for constructive interference between rays 1⁰ and 4 reflected by planes I and III respectively, works out to be:

nA = RF 1FS = 4dsin#

or A = 2dsin#                                                                             (3.9)

for n = 2, i.e. for second-order reflection from planes I and III

Thus second-order reflection from planes I and III coincides with the first-order reflection from planes I and II. In general, higher order reflections from successive parallel planes will coincide with the first-order reflections from the first two planes of the family. For the same reason, the second-order diffraction for (100) planes will superimpose over the first-order diffraction from (200) planes and so on. Accordingly n is always taken to be unity and the Bragg's condition is therefore written as:

A = 2dsin0                                               (3.10)

To make the discussion more complete, it needs to be considered that second or higher order constructive interference between the rays reflected from planes I and II can occur if path-difference equals nA, where n $ 2. However, the typical wavelength of X-rays used for diffraction analysis lies in the range of 0.1—0.2 nm, which is comparable to the interatomic distance. Hence, higher order diffraction peaks may not occur at all while using X-rays, since that may require sin0 in Eq. (3.8) to be greater than 1, which is not possible. However, if the wavelength were small—as is the case with electron diffraction, it would be possible to "fit- in” one or more wavelengths within the path-difference. Hence, higher order peaks are quite uncommon in XRD, while they are a norm in electron diffraction.

• Ewald sphere

The preceding discussion implies that wavelength of incident X-rays puts a limit on the mini­mum spacing of planes for which diffraction (to be precise, constructive interference) can occur. Since planes with small interplanar spacing have large Miller indices, it means that strong (first order) diffraction peaks can be recorded only up to certain limit of Miller indi­ces. This condition can be easily worked-out through Ewald sphere construction.

Fig. 3—17 shows perspective projection of a reciprocal lattice corresponding to a simple cubic lattice in direct space. As noted earlier, the reciprocal of simple cubic lattice is again a simple cubic and so, makes it easier to understand the concept. We take S₀ as a unit vector in the direction of incident beam and Si as unit vectors along various diffracted beams. Then vector S₀/A lies parallel to the incident radiation, while its magnitude is reciprocal of the wavelength. Placing the tip of this vector at a lattice point (henceforth termed as origin) in reciprocal space, Ewald sphere of radius 1/A is constructed at the tail of S₀/A.

Lattice points lying on the Ewald sphere in reciprocal space (namely, 122 and 122 in Fig. 3—18A and B) correspond to the sets of planes, which would result in diffraction under the given conditions of wavelength and direction of incident X-rays, while the vectors Si/A and S₂/A drawn to these lattice points depict the directions of corresponding diffracted beams.

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If the crystal is slowly rotated about the origin, the vector S₀/A will change its direction relative to the lattice, while its tip remains hinged at the origin. During the course of rotation, the surface of Ewald sphere will successively intersect different lattice points. Likewise, if the crystal is rotated about all three axes in space, lattice points corresponding to all such sets of
planes, for which diffraction is possible, will intersect the Ewald sphere at some instance of time or the other. The set of all such lattice points in reciprocal space that can intersect the Ewald sphere can be identified by constructing a limiting sphere centered at the tip of S₀/A (refer Fig. 3—17). The radius of this limiting sphere equals diameter of the Ewald sphere. Only the hkl points lying inside or on the surface of the limiting sphere can produce diffrac­tion when the crystal is suitably oriented.

Since radius of Ewald sphere (S0/A) will increase on decreasing the wavelength of inci­dent X-rays, it follows that the Ewald sphere corresponding to a shorter wavelength can intersect with a larger number of reciprocal space lattice points, while that corresponding to a longer wavelength will contain fewer lattice points. In other words, shorter wavelengths can produce diffraction patterns corresponding to planes having higher Miller indices (i.e., smaller d-spacing), while longer wavelengths would be able to form diffraction patterns only for planes of lower indices.

• Extinction of X-rays

Extinction refers to an increase in absorption by a single crystal under conditions suitable for Bragg reflection. The absorption occurs not just for the transmitted beam, but also affects the reflected beam. It was in 1914 that Bragg [79], while passing an X-ray beam through a thin flake of diamond and subsequently through a crystal of rock salt (so oriented, as to reflect the incident beam), both the reflected beams being detected by an ionization chamber, observed a drastic reduction in the intensity of transmitted beam (as reflected by the rock salt crystal) whenever the condition for reflection was met for the thin diamond flake. The factors responsible for this phenomenon are explained below.

As said earlier in the section concerning absorption of X-rays, the transmitted beam gets reduced in intensity both due to Bragg reflection as well as absorption through various ther­mal and nonthermal processes. For perfect crystals, the moment crystal gets so oriented, as to satisfy the condition for Bragg reflection, the absorption increases by around 100 times. At such high levels of absorption, the transmitted beam does not penetrate far into the crystal and so, the lower layers do not interact with the incident X-rays. The effect is known as pri­mary extinction and it not only affects the transmitted beam, but also the reflected beam.

Primary extinction occurs because the rays undergo a phase shift of n/2 on reflection by an array of atoms. Likewise, a doubly reflected ray will have a phase shift of n. So, even though the doubly reflected ray would be parallel to the transmitted beam, but would destructively interfere with it. Similarly a triple-reflected ray will reduce the intensity of reflected beam. Obviously the number of rays decreases with each reflection and so would be their annihilation effect. The transmitted beam therefore loses energy not just by reflec­tion of a part thereof, but also by destructive interference with the out of phase, doubly reflected beam. Same phenomenon of destructive interference serves to adversely affect the reflective power of the crystal. Furthermore since the primary beam does not reach lower layers of the crystal, the reflections too are the result of contributions from the outermost skin layers of the crystal up to a depth of around 1 pm.

• Determination of crystal structure and lattice parameter

It follows from the discussion in previous subsections that for a known crystal structure, one can easily determine the directions corresponding to constructive interference of radiation, when the crystal is irradiated by a collimated beam of monochromatic X-rays. However, the converse of this problem is not as straightforward—especially for complex crystals (namely, proteins), since placement of diffraction dots on a photographic film alone does not provide complete information about the crystal structure. For crystals, whose basis comprises of two or more atoms, in addition to knowing the diffraction directions, it becomes important to quantify the beam intensities as well, namely, using a diffractometer.

For complex crystals, structure determination might require rigorous analysis of diffrac­tion data, along with the results of chemical tests and macroscopic observations. On the other hand, the structure of simple crystals with high symmetry can be determined relatively easily. It also helps develop an understanding of the basic procedure, which is essential for analyzing diffraction data pertaining to structures with higher degree of complexity. The structure of a cubic crystal with monoatomic basis can be assessed by substituting the expression for lattice spacing in the Bragg's equation as under:

A² = 4d²sin² в

The above expression holds for all Bravais lattices belonging to the cubic system, namely, SC, BCC, and FCC. It is possible to write down all possible combinations of h² 1 k² 1 l² and arrange them in ascending order, which would also be the ascending order for angle, в. Since A and a are constants, values of sin²e will follow the same ratios as (h² 1 k² 1Z²). Occurrence of reflection for all combinations of (h² 1 k² 1Z²) is feasible only for the simple cubic lattice, while reflections in certain directions get canceled for BCC and FCC lattices, thereby facilitating distinction between the three lattices of the cubic system. For example, in a monoatomic BCC lattice, the corner atoms lie on the (100) planes, while the body-centered atoms lie on the (200) planes. The count of atoms per unit area on a (200) plane is same as that for a (100) plane. Accordingly if the first- order reflection from two adjacent (100) planes is having a path-difference of A, its path- difference from (200) planes will be A/2, thereby leading to complete cancellation of intensity through primary extinction. Following similar reasoning, it is possible to frame extinction rules for all possible diffraction directions pertaining to a given lattice system. The rules pertaining to cubic system are given in Table 3—3 to serve as an aid to under­standing the concept.

The lattice structure can be determined by finding ratios of sin²e corresponding to the recorded в values of diffraction peaks and comparing them with the ratios mentioned in Table 3—3. An example is provided in Section 3.10.1.1.

• Estimation of crystallite size

For d-spacings corresponding to a path-difference, which is only slightly different from an integral multiple of wavelength, the photons "reflected” by successive planes lying beneath the top layer will be progressively out of phase against the ones reflected by the topmost few planes. In such a scenario, the planes corresponding to 180 degrees phase shift would be lying at considerable depth below the outermost plane. However, sufficiently deep planes may become nonexistent for crystallites of small sizes. As a result, complete cancellation of the scattered beam will not take place corresponding to directions for which the path- difference varies only slightly from an integral multiple of wavelength. This will result in a relatively fuzzy diffraction pattern, which, from the perspective of diffractometers, is termed as line broadening.

An estimate of the crystallite size (thickness, t) is given by the historical equation devel­oped by Scherrer in his paper concerning determination of major and the inner structure of colloidal particles by means of X-rays [47], which solves to give:

• 9A

Bcos6_B

Here, A depicts the wavelength (in any suitable units of length) of monochromatic X-rays used in the diffraction experiment, B refers to full width at half maximum (FWHM) of the given diffraction peak (in radians) occurring at the location 26_B. It is apparent that B would work out to be broader for smaller values of t, that is, for smaller-sized crystallites and vice versa.