Principles of X-ray Diffraction Diffraction effects are observed when electromagnetic radiation impinges on periodic structures with geometrical variations on the length scale of the wavelength of the radiation. The interatomic distances in crystals and molecules amount to 0. 15–0. 4 nm which correspond in the electromagnetic spectrum with the wavelength of x-rays having photon energies between 3 and 8 keV. Accordingly, phenomena like constructive and destructive interference should become observable when crystalline and molecular structures are exposed to x-rays.

In the following sections, firstly, the geometrical constraints that have to be obeyed for x-ray interference to be observed are introduced. Secondly, the results are exemplified by introducing the ? /2? scan, which is a major x-ray scattering technique in thin-film analysis. Thirdly, the ? /2? diffraction pattern is used to outline the factors that determine the intensity of x-ray ref lections. We will thereby rely on numerous analogies to classical optics and frequently use will be made of the fact that the scattering of radiation has to proceed coherently, i. . the phase information has to be sustained for an interference to be observed. In addition, the three coordinate systems as related to the crystal {ci}, to the sample or specimen {si} and to the laboratory {li} that have to be considered in diffraction are introduced. Two instrumental sections (Instrumental Boxes 1 and 2) related to the ? /2? diffractometer and the generation of x-rays by x-ray tubes supplement the chapter. One-elemental metals and thin films composed of them will serve as the material systems for which the derived principles are demonstrated.

A brief presentation of one-elemental structures is given in Structure Box 1. 1. 1 The Basic Phenomenon Before the geometrical constraints for x-ray interference are derived the interactions between x-rays and matter have to be considered. There are three different types of interaction in the relevant energy range. In the first, electrons may be liberated from their bound atomic states in the process of photoionization. Since energy and momentum are transferred from the incoming radiation to the excited electron, photoionization falls into the group of inelastic scattering processes.

In Thin Film Analysis by X-Ray Scattering. M. Birkholz Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31052-5 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 2 2 1 Principles of X-ray Diffraction addition, there exists a second kind of inelastic scattering that the incoming x-ray beams may undergo, which is termed Compton scattering. Also in this process energy is transferred to an electron, which proceeds, however, without releasing the electron from the atom. Finally, x-rays may be scattered elastically by electrons, which is named Thomson scattering.

In this latter process the electron oscillates like a Hertz dipole at the frequency of the incoming beam and becomes a source of dipole radiation. The wavelength ? of x-rays is conserved for Thomson scattering in contrast to the two inelastic scattering processes mentioned above. It is the Thomson component in the scattering of x-rays that is made use of in structural investigations by x-ray diffraction. Figure 1. 1 illustrates the process of elastic scattering for a single free electron of charge e, mass m and at position R0.

The incoming beam is accounted for by a plane wave E0exp(–iK0R0), where E0 is the electrical field vector and K0 the wave vector. The dependence of the field on time will be neglected throughout. The wave vectors K0 and K describe the direction of the incoming and exiting beam and both are of magnitude 2? /?. They play an important role in the geometry of the scattering process and the plane defined by them is denoted as the scattering plane. The angle between K and the prolonged direction of K0 is the scattering angle that will be abbreviated by 2? as is general use in x-ray diffraction.

We may also define it by the two wave vectors according to 2? = arccos K ,K 0 KK 0 (1. 1) The formula is explicitly given here, because the definition of angles by two adjoining vectors will be made use of frequently. The oscillating charge e will emit radiation of the same wavelength ? as the primary beam. In fact, a phase shift of 180° occurs with the scattering, but since this shift equally arises for every scattered wave it has no effect on the interference pattern in which we are interested and will be neglected. If the amplitude of the scattered wave E(R) is considered at a distance R we may write according to Hertz and Thomson

E (R ) = E 0 1 e2 sin ? (E 0 , R )exp(? iKR ) 4?? 0R mc 2 (1. 2) Figure 1. 1 Scattering of x-rays by a single electron. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 3 1. 1 The Basic Phenomenon 3 where ? 0 and c are the vacuum permittivity and velocity of light. The field vector E and wave vector K are oriented perpendicular to each other as is usual for electromagnetic waves. The sin term is of significance when the state of polarization is considered for which two extreme cases may arise. In one case, the exciting field E0 is confined to the scattering plane and in the second case it is normally oriented.

In classical optics these two cases are named ? polarization and ? polarization. The field vectors in both cases will be denoted by E? and E?. The angle between E? and R is always 90° and the sin term will equal unity. For the case of ? polarization, however, it may be expressed by virtue of the scattering angle according to sin? (E0, R) = |cos2? |. If the character C abbreviates the sin term it may be written ? 1 ? -polarization ? (1. 3) C=? cos 2? ? -polarization ? ? Since the intensity is obtained from the sum of the square of both field vectors the expression ? 1 ? ? e2 ? 2 2 2 ? 4?? R ? mc 2 ? E? + E? cos 2? ? ? ? ? 0 2 2 ( ) (1. 4) is obtained. In a nonpolarized beam both polarization states will have the same probability of occurring, 2 2 E? = E? = I0 / 2 and it is finally arrived at the intensity of the scattered beam at distance R I(R ) = I0 re2 1 + cos2 2? 2 R2 (1. 5) Here, use has been made of the notion of the classical radius of the electron, re = e2/(4?? 0mc2), that amounts to 2. 82 ? 10–15 m. The intensity of the scattering is seen to scale with the inverse of R2 as might have been expected. It can also be seen that I(R) scales with the ratio of squares of re over R.

Since distances R of the order of 10–1 m are realized in typical laboratory setups the probability of observing the scattering by a single electron tends to zero. The situation substantially improves if the number of scattering objects is of the same order of magnitude as Loschmidt’s number NL – as usually is the case in experiments. It also becomes evident from this equation as to why the scattering from atomic nuclei has not been considered in the derivation. In fact, the equation would also hold for the scattering from atomic nuclei, but it can be seen from Eq. (1. ) that the nuclei component will only yield a less than 10– 6 smaller intensity compared to an electron. The difference is simply due to the mass difference, which is at least larger by a factor of 1836 for any atomic species. The scattering of x-rays by nuclei may, therefore, confidently be neglected. From the viewpoint of x-ray scattering an atom can thus be modeled by the number of Z electrons, which it contains according to its rank in the periodic table. In terms of the Thompson scattering model Zre may be written in Eq. (1. 3) instead of re in order to describe the scattering from an atom, 01_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 4 4 1 Principles of X-ray Diffraction since the primary beam is then equally scattered by all electrons. In addition, it will be assumed temporarily that all electrons are confined to the origin of the atom. The consequences that follow from a refinement of the model by assuming a spatially extended charge distribution will be postponed to a later section. Hence, we have a first quantitative description for the x-ray elastic scattering from an atom. In the next step consideration is given to what the scattering will look like if it occurs or a whole group of atoms that are arranged in a periodically ordered array like a crystal lattice. Figure 1. 2 visualizes such an experiment where the crystal is irradiated with monochromatic x-rays of wavelength ?. In the special case considered here, each atom is surrounded by six neighbor atoms at distance a and the angle between two atomic bonds is always 90° or multiples of it. Atomic positions can then be described by the lattice vector rn1n2n3 = n1ac1 + n2ac2 + n3ac3 with c1, c2 and c3 being the unit vectors of the three orthogonal directions in space.

The c i axes are the unit vectors of the crystal coordinate system {c i}, which is assigned to the crystal. For some properties of the crystal this coordinate system will turn out to be extremely useful and the notion will be used throughout the book. The shape of the crystal is assumed to be that of a parallelepiped as is accounted for by the inequalities 0 ? ni = Ni – 1 for i = 1, 2, 3. Each node of adjacent cubes is thus occupied by an atom. Such a structure is called simple cubic in crystallography. Only a single element crystallizes in this structure, which is polonium exhibiting an interatomic distance of a = 0. 3359 nm.

Although this metal has only very few applications, the case shall be considered here in detail, because of its clarity and simplicity. It will now be calculated at which points in space interferences of x-rays might be observed that arise due to the scattering at the crystal lattice. The task is to quantify the strength of the scattered fields at a point R when elastic scattering occurs according to Eq. (1. 5) at all atoms. The reference point of R is chosen such that it starts at the origin of the crystal lattice r000. This means that we relate the phase difference in the summation of all scattered fields to their phase at r000.

This choice is arbitrary and any other lattice point might have been equally selected. The wave vector of the primary beam K0 is assumed to be parallel to the [100] direction of the crystal. The scattering plane defined by K0 and K may coincide with one of the (010) planes. The wavefronts of the incoming plane waves which are the planes of constant phase are then oriented parallel to (100) planes. An atom on the position rn1n2n3 would then cause a scattering intensity to be measured at R of the strength E 0 exp(? iK 0rn n 1 2 n3 ) Zre R ? rn n n sin ? (E 0 , R ? rn n 1 2 n3 )exp(? iK (R ? rn n 1 2 n3 )) (1. 6) 1 2 3

This expression differs from Eq. (1. 2) essentially by the fact that R – rn1n2n3 occurs instead of R, and for n1 = n2 = n3 = 0 it becomes equal to Eq. (1. 2). The solution of our task would simply consist in a summation over all fields scattered by the number of N1 ? N2 ? N3 atoms comprising the crystal. However, the physics of the solution will become more transparent when an important approximation is made. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 5 1. 1 The Basic Phenomenon 5 Figure 1. 2 Scattering of x-rays by a crystallite of simple cubic structure. It will be assumed that the interatomic distances rn1n2n3 (? 0–10 m) are much smaller than the distances to the point of the intensity measurement R – rn1n2n3 (? 10–1 m). The denominator in Eq. (1. 6) and in the sin term R – rn1n2n3 may then be replaced by R without introducing a large error. This substitution, however, is not allowed in the exponent of the last factor, since the interatomic distances are of the order of the wavelength and every phase shift according Krn1n2n3 = 2? rn1n2n3/? has to be fully taken into account in the summation procedure. If these rules are applied the sin term may be replaced by the polarization factor C and the sum over all scattered fields reads E0

Zre C exp ? iKR R ( ) ? exp ( ? i(K ? K 0 )rn n n ) n1n2n3 1 2 3 (1. 7) 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 6 6 1 Principles of X-ray Diffraction All terms independent of the lattice vector rn1n2n3 could be placed in front of the summation symbol. The approximation of which we have made use of is named Fraunhofer diffraction, which is always a useful approach when the distances between scattering objects are much smaller than the distance to the measurement point. In contrast to this approach stands the so-called Fresnel diffraction, for which interference phenomena are investigated very close to the scattering objects.

The case of Fresnel diffraction will not be of interest here. We have achieved a significant progress in solving our task by applying the Fraunhofer approximation and arriving at Eq. (1. 7). It can be seen that the scattered field scales with two factors, where the first has the appearance of a spherical wave while the second is a sum over exponentials of vector products of wave vectors and lattice vectors. In order to improve our understanding of the summation over so many scattering centers the geometry is shown in the lower part of Fig. 1. 2.

A closer look at the figure reveals that the phase shift for two waves (a) scattered at r000 and (b) scattered at rn1n2n3 comprises two components due to K0rn1n2n3 and to Krn1n2n3. The strength of the total scattered field of Eq. (1. 7) thus sensitively depends on the spatial orientation of the wave vectors K0 and K with respect to the crystal reference frame {ci}. Because a single phase shift depends on the vector product between the lattice vector and the wave vector difference K – K0 the latter quantity is recognized as a physical quantity of its own significance and is named the scattering vector

Q = K – K0 (1. 8) The scattering vector has the dimensionality of an inverse length, while its direction points along the bisection of incoming and scattered beam. The geometry is demonstrated in Fig. 1. 3 and a closer inspection tells that the relation |Q| = 4? sin? /? holds for the scattering vector magnitude. This relation will be made use of extensively throughout the book and the reader should be fully aware of its derivation from Fig. 1. 3. It should be realized that |Q| depends on both (a) the geometry of the scattering process via ? and (b) the wavelength ? f the probing x-ray beam. The physical meaning of Q in a mechanical analogy is that of a momentum transfer. By analogy with the kinetic theory of gases the x-ray photon Figure 1. 3 Geometry of scattering vector construction. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 7 1. 1 The Basic Phenomenon 7 is compared to a gas molecule that strikes the wall and is repelled. The direction of momentum transfer follows from the difference vector between the particle’s momentum before and after the event, p – p0, while the strength of transferred momentum derives from |p – p0|.

In the case considered here the mechanical momentum p just has to be replaced by the wave vector K of the x-ray photon. This analogy explains why the scattering vector Q is also named the vector of momentum transfer. It has to be emphasized that the scattering vector Q is a physical quantity fully under the control of the experimentalist. The orientation of the incident beam (K0) and the position of the detector (K) decide the direction in which the momentum transfer (Q) of x-rays proceeds. And the choice of wavelength determines the amplitude of momentum transfer to which the sample is subjected.

From these considerations it is possible to understand the collection of a diffraction pattern as a way of scanning the sample’s structure by scattering vector variation. If the summation factor of Eq. (1. 7) is expanded into three individual terms and the geometry of the simple cubic lattice is used it is found that the field amplitude of the scattered beam is proportional to N1 ? 1 N 2 ? 1 N 3 ? 1 n1 = 0 n2 = 0 n3 = 0 ? ? ? ? exp ( ? iQ ? n1ac 1 + n2ac 2 + n3ac 3 ? ) ? (1. 9) where the scattering vector Q has already been inserted instead of K – K0.

This expression can be converted by evaluating each of the three terms by the formula of the geometric sum. In order to arrive at the intensity the resultant product has to be multiplied by the complex conjugate and we obtain the so-called interference function ?(Q) = sin2 N1aQc 1 / 2 ) ? sin2 (N2aQc 2 / 2) ? sin2 (N3aQc 3 / 2) sin2 ( aQc 3 / 2) sin2 ( aQc 1 / 2) sin2 ( aQc 2 / 2) ( (1. 10) that describes the distribution of scattered intensity in the space around the crystallite. For large values of N1, N2 and N3 the three factors in ? Q) only differ from zero if the arguments in the sin2 function of the denominator become integral multiples of ?. Let us name these integers h, k and l in the following. The necessary condition to realize the highest intensity at R accordingly is aQc 1 = 2? h ? (Q) > max ? aQc 2 = 2? k aQc 3 = 2? l (1. 11) Here, the integers h, k, l may adopt any value between –? and +?. The meaning of these integers compares to that of a diffraction order as known in optics from diffraction gratings. The hkl triple specifies which order one is dealing with when the primary beam coincides with zero order 000.

However, the situation with a crystalline lattice is more complex, because a crystal represents a three-dimen- 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 8 8 1 Principles of X-ray Diffraction sional grating and three integral numbers instead of only one indicate the order of a diffracted beam. The set of Eqs. (1. 11) are the Laue conditions for the special case of cubic crystals that were derived by M. von Laue to describe the relation between lattice vectors rn1n2n3 and scattering vector Q for crystals of arbitrary symmetry at the position of constructive interference. The severe condition that is posed by Eq. 1. 11) to observe any measurable intensity is illustrated in Fig. 1. 4. The plot shows the course of the function sin2 Nx/sin2 x, for N = 15, which is the one-dimensional analogue of Eq. (1. 10). It can be seen that the function is close to zero for almost any value of x except for x = ? h, with h being an integer. At these positions the sin2 Nx/sin2 x function sharply peaks and only at these points and in their vicinity can measurable intensity be observed. The sharpness of the peak rises with increasing N and a moderate value of N has been chosen to make the satellite peaks visible.

It should be noted that in the case of diffraction by a crystal the three equations of Eq. (1. 11) have to be obeyed simultaneously to raise I(R) to measurable values. As a further property of interest it has to be mentioned that sin2Nx/x2 may equally be used instead of sin2 Nx/sin2 x for N o x. This property will enable some analytical manipulations of the interference function, which would otherwise be possible only on a numerical basis. In order to gain further insight into the significance of the condition for observable intensity, we will investigate the Laue conditions with respect to the magnitude of the scattering vector.

The magnitude of Q at I(R) > max can be obtained from the three conditional Eqs. (1. 11) by multiplying by the inverse cell parameter 1/a, adding the squares and taking the square root. This yields as condition for maximum intensity I(R) > max ? Q h2 + k2 + l2 = a 2? (1. 12) Figure 1. 4 Course of the function sin2 Nx/sin2 x for N = 15. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 9 1. 1 The Basic Phenomenon 9 which can be rewritten by inserting the magnitude of the scattering vector, |Q| = 4? sin? /? , known from geometrical considerations I(R) > max ? 2 a h2 + k2 + l2 sin? = ? 1. 13) This is an interesting result that may be read with a different interpretation of the hkl integer triple. The high degree of order and periodicity in a crystal can be envisioned by selecting sets of crystallographic lattice planes that are occupied by the atoms comprising the crystal. The planes are all parallel to each other and intersect the axes of the crystallographic unit cell. Any set of lattice planes can be indexed by an integer triple hkl with the meaning that a/h, a/k and a/l now specify the points of intersection of the lattice planes with the unit cell edges.

This system of geometrical ordering of atoms on crystallographic planes is well known to be indicated by the so-called Miller indices hkl. As an example, the lattice planes with Miller indices (110) and (111) are displayed in Fig. 1. 5 for the simple cubic lattice. Figure 1. 5 Lattice planes with Miller indices (110) and (111) in a simple cubic lattice. The distance between two adjacent planes is given by the interplanar spacing dhkl with the indices specifying the Miller indices of the appropriate lattice planes.

For cubic lattices it is found by simple geometric consideration that the interplanar spacing depends on the unit cell parameter a and the Miller indices according to a dhkl = (1. 15) h2 + k2 + l2 Keeping this meaning of integer triples in mind, Eq. (1. 13) tells us that to observe maximum intensity in the diffraction pattern of a simple cubic crystal the equation 2dhkl sin? B = ? (1. 15) has to be obeyed. The equation is called Bragg equation and was applied by W. H. Bragg and W. L. Bragg in 1913 to describe the position of x-ray scattering peaks in angular space.

The constraint I(R) > max has now been omitted, since it is implicitly included in using ? B instead of ? which stands for the position of the maximum. In honor of the discoverers of this equation the peak maximum position has been named the Bragg angle ? B and the interference peak measured in the ref lection mode is termed the Bragg ref lection. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 10 10 1 Principles of X-ray Diffraction The Laue conditions and the Bragg equation are equivalent in that they both describe the relation between the lattice vectors and the scattering vector for an x-ray ref lection to occur.

Besides deriving it from the Laue condition, the Bragg equation may be obtained geometrically, which is visualized in Fig. 1. 6. A set of crystallographic lattice planes with distances dhkl is irradiated by plane wave x-rays impinging on the lattice planes at an angle ?. The relative phase shift of the wave depends on the configuration of atoms as is seen for the two darker atoms in the top plane and one plane beneath. The phase shift comprises of two shares, ? 1 and ? 2, the sum of which equals 2dsin? for any arbitrary angle ?. Constructive interference for the ref lected wave, however, can only be achieved when the phase shift 2dsin? s a multiple of the wavelength. Therefore, Bragg’s equation is often written in the more popular form 2dsin? B = n? , where the integer n has the meaning of a ref lection order. Because we are dealing with three-dimensional lattices that act as diffraction gratings, the form given in Eq. (1. 14) is preferred. It should be emphasized that the Bragg equation (Eq. (1. 14)) is valid for any lattice structure, not only the simple cubic one. The generalization is easily performed by just inserting the interplanar spacing dhkl of the crystal lattice under investigation. Table 1. gives the relation of dhkl and the unit cell parameters for different crystal classes. Figure 1. 6 Visualization of the Bragg equation. Maximum scattered intensity is only observed when the phase shifts add to a multiple of the incident wavelength ?. Having arrived at this point it can be stated that we have identified the positions in space where constructive interference for the scattering of x-rays at a crystal lattice may be observed. It has been shown that measurable intensities only occur for certain orientations of the vector of momentum transfer Q with respect to the crystal coordinate system {c i }.

Various assumptions were made that were rather crude when the course of the intensity of Bragg ref lections is of interest. It has been assumed, for instance, that the atom’s electrons are confined to the center of mass of the atom. In addition, thermal vibrations, absorption by the specimen, etc. , were neglected. More realistic models will replace these assumptions in the following. However, before doing so it should be checked how our first derivations compare with the measurement of a thin metal film and how diffraction patterns may be measured. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 11 1. 2 The ? 2? Scan Table 1. 1 Interplanar spacings dhkl for different crystal systems and their dependency on Miller indices hkl. Parameters a, b and c give the lengths of the crystallographic unit cell, while ? , ? and ? specify the angles between them. 11 Crystal system Cubic Tetragonal Constraints a=b=c ? = ? = ? = 90° a=b ? = ? = ? = 90° 1 = 2 dhkl h2 + k 2 + l 2 a2 h2 + k 2 l 2 + 2 a2 c h2 k 2 l 2 + + a2 b 2 c 2 4 h 2 + hk + k 2 l 2 + 2 3 a2 c (h 2 + k 2 + l 2 )sin2 ? + 2(hk + hl + kl )(cos2 ? ? cos? ) a 2 (1 ? 3 cos2 ? + 2 cos3 ? ) 2hl cos ? h2 k2 l2 + + ? a2 sin 2 ? b 2 c 2 sin 2 ? ac sin 2 ? Exercise 4

Orthorhombic Hexagonal ? = ? = ? = 90° a=b ? = ? = 90° ? = 120° a=b=c ? =? =? Trigonal/ Rhombohedral Monoclinic Triclinic ? = ? = 90° None 1. 2 The ? /2? Scan An often-used instrument for measuring the Bragg ref lection of a thin film is the ? /2? diffractometer. Let us introduce its operation principle by considering the results obtained with the question in mind as to how x-ray scattering experiments are preferably facilitated. What we are interested in is the measurement of Bragg ref lections, i. e. their position, shape, intensity, etc. , in order to derive microstructural information from them.

The intensity variation that is associated with the ref lection is included in the interference function like the one given in Eq. (1. 10), while the scattered intensity depends on the distance from the sample to the detection system R. We therefore should configure the instrument such that we can scan the space around the sample by keeping the sample–detector distance R constant. This measure ensures that any intensity variation observed is due to the interference function and is not caused by a dependency on R. The detector should accordingly move on a sphere of constant radius R with the sample in the center of it.

In addition, the sphere reduces to a hemisphere above the sample, since we are only interested in the surface layer and data collection will be performed in ref lection mode. The geometry is shown in Fig. 1. 7. Because the scattering of x-rays depends sensitively on the orientation of the crystal with respect to the scattering vector, we carefully have to define the various coordinate systems with which we are dealing. A sample reference frame {s i } is introduced for this purpose that is oriented with s1 and s2 in the plane of the thin film, while s3 is equivalent to the surface normal. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 12 2 1 Principles of X-ray Diffraction Figure 1. 7 Sample reference frame {si } and hemisphere above it. The working principle of a ? /2? scan is visualized in Fig. 1. 8 in the hemisphere of the sample reference frame. The sample is positioned in the center of the instrument and the probing x-ray beam is directed to the sample surface at an angle ?. At the same angle the detector monitors the scattered radiation. The sample coordinate vectors s1 and s3 lie in the scattering plane defined by K0 and K. During the scan the angle of the incoming and exiting beam are continuously varied, but they remain equal throughout the whole scan: ? n = ? out. Note that the angle convention is different from the one used in optics: in x-ray diffraction the angles of incoming and exiting beam are always specified with respect to the surface plane, while they are related to the surface normal in optics. The ? /2? scan can also be understood as a variation of the exit angle when this is determined with respect to the extended incoming beam and this angle is 2? for all points in such a scan. This is the reason for naming the measurement procedure a ? /2? scan. The quantity measured throughout the scan is the intensity scattered into the detector.

The results are typically presented as a function of I(2? ) type. Figure 1. 8 Schematic representation of a ? /2? scan from the viewpoint of the sample reference frame {si}. These ? /2? scans are extensively used for the investigation of polycrystalline samples. The measurement of polycrystals is somewhat easier than that of single crystals due to the fact that, among other reasons, the scattered intensity for constant scattering angle is distributed on a circle rather than focused to a few points in space. Interestingly, in a ? /2? scan the scattering vector Q is always parallel to 01_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 13 1. 2 The ? /2? Scan 13 the substrate normal s3. This fact is evident from Fig. 1. 8 and the graphical definition of Q in Fig. 1. 3. Due to this geometrical constraint only those lattice planes hkl that are oriented parallel to the surface plane can contribute to a Bragg ref lection. The selective perception of certain subsets of crystallites in a ? /2? scan is visualized in Fig. 1. 9. If various ref lections hkl are measured they all stem from distinct subsets of crystallites – except they are of harmonic order, i. e. h? k? ? = n(hkl). Figure 1. 9 Selection principle for exclusive measurement of sur- face-parallel lattice planes in a ? /2? scan. In order to demonstrate the principles developed so far, the simulation of a ? /2? scan of a 500 nm thin Al film is shown in Fig. 1. 10. The simulation was calculated for the characteristic radiation of a copper x-ray tube having ? (Cu K? ) = 0. 154 nm (see Instrumental Box 1 for further information). Various interesting features are realized from this plot, which displays eight Bragg ref lections in the scattering angle range from 25° to 125°.

The ref lections may be assigned to their Miller indices when use is made of the Bragg equation and the unit cell parameter of the Al lattice, a = 0. 4049 nm. For this purpose the d values of the 2? B ref lex positions have been calculated according to the Bragg equation d = ? /(2sin? B) and checked for the solution of (a/d)2 = h2 + k2 + l2. It is seen that various ref lections like 111 and 200 are observed, but other peaks like 100, 110, etc. , are missing. This phenomenon has to be understood in the sense of destructive interference, which is caused by the structure of the Al lattice, which is distinct from the simple cubic lattice.

It has to be noted that a splitting of peaks into an ? 1 peak and an ? 2 peak cannot be observed, although the feature was included in the simulation. The absence is explained from the broadness of the Bragg peaks causing a severe overlap between both peaks such that they remain unresolved. Broad ref lections are caused by small grain sizes and crystal lattice faults that are often observed in thin polycrystalline films and are discussed in more detail in Chapter 3. Moreover, the diffraction pattern exhibits a pronounced decrease of scattered intensity with increasing scatter– ing angle.

Therefore, the diffraction pattern is also shown in the inset with a vI ordinate in order to emphasize the smaller peaks. The square-root intensity plot is an often-used presentation mode. It is concluded that the basic features of Section 1. 1 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 14 14 1 Principles of X-ray Diffraction are in accordance with the simulated measurement of a thin Al film, but some aspects remain to be clarified. Figure 1. 10 Simulation of a ? /2? scan of a 500 nm thin Al film measured with Cu K? radiation.

The inset shows the same pat– tern with a v I ordinate. 1. 3 Intensity of Bragg Ref lections The necessary refinement of the expression for the intensity of a Bragg ref lection is now developed. For this purpose the finding will be used that was made by deriving the Bragg equation and the Laue conditions. It has been realized that the amplitude of the total scattered field from a charge distribution in the Fraunhofer approximation is characterized by a phase factor exp(–iQrn1n2n3) comprising the scattering vector Q and the distance rn1n2n3 between all pairs of point charges.

This result may be generalized by subjecting the sum in Eq. (1. 9) to a continuous limit. Instead of writing a discrete distance vector rn1n2n3 the continuous variable r is used and it is argued that the scattered field depends as ? ? e(r )exp(? iQr )dr (1. 16) on the electronic charge distribution ? e(r) of the scattering object. The integration has to be performed over the volume dr to which the scattering electrons are confined. Because ? e has the dimensionality of an inverse volume the integration yields a dimensionless quantity, which is in accordance with our starting point.

This new expression can now be applied to the scattering objects in which we are interested, i. e. atoms and crystallographic unit cells, to check whether the provisional intensity function is improved. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 15 1. 3 Intensity of Bragg Ref lections 15 Instrumental Box 1: ?/2? Diffractometer The basic measurement geometry of by far the most frequently used x-ray diffraction instrument is depicted in Fig. i1. 1. The sample should preferably exhibit a plane or flattened surface. The angle of both the incoming and the exiting beam is ? with respect to the specimen surface.

A vast number of organic and inorganic powder samples have been measured with these instruments from which the naming of powder diffractometer is understood. Its measurement geometry may also be applied to the investigation of thin films, especially if the layer is polycrystalline and has been deposited on a flat substrate, as is often the case. Figure i1. 1 Schematic representation of ? /2? diffraction in Bragg–Brentano geometry. The diffraction pattern is collected by varying the incidence angle of the incoming xray beam by ? and the scattering angle by 2? while measuring the scattered intensity I(2? as a function of the latter. Two angles have thus to be varied during a ? /2? scan and various types of powder diffractometers are in use. For one set of instruments the x-ray source remains fixed while the sample is rotated around ? and the detector moves by 2?. For other systems the sample is fixed while both the x-ray source and the detector rotate by ? simultaneously, but clockwise and anticlockwise, respectively. The rotations are performed by a so-called goniometer, which is the central part of a diffractometer. A goniometer of a powder diffractometer comprises at least two circles or – equally – two axes of rotation.

Typically the sample is mounted on the rotational axis, while the detector and/or x-ray source move along the periphery, but both axes of rotation coincide. In most laboratory ? /2? diffractometers the goniometer radius, which is the sample-to-detector distance, is in the range 150–450 mm. Highly precise goniometers with 0. 001° 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 16 16 1 Principles of X-ray Diffraction precision and even lower on both the ? and the 2? circles are commercially available. The collected diffraction pattern I(2? ) consists of two sets of data: a vector of 2? positions and a second vector with the appropriate intensities Ii. The step size ? 2? i between two adjacent 2? i should be chosen in accordance with the intended purpose of the data. For chemical phase analysis (Chapter 2) the full width of half the maximum of the tallest Bragg peak in the pattern should be covered by at least 5 to 7 measurement points. However, for a microstructural analysis (Chapter 3) in excess of 10 points should be measured on the same scale. The appropriate value of ? 2? i will also depend on the slit configuration of the diffractometer. The preset integration time of the detector per step in 2? should allow the integral intensity of the smallest peak of interest to exceed the noise fluctuations ? (I) by a factor of 3 or 5, etc. , according to the required level of statistical significance. The control of the x-rays beam bundle suffers from the constraint that lenses and other refractive elements are not as easily available as those used for visible light. For this reason the beam conditioning in ? /2? diffractometers is mostly performed by slits and apertures and may be termed shadow-casting optics. In addition, powder diffractometers have to deal with the divergent beam characteristic that is emitted by an x-ray tube.

Most systems operate in the so-called Bragg–Brentano or parafocusing mode. In this configuration a focusing circle is defined as positioned tangentially to the sample surface (see Fig. i1. 1). The focusing condition in the Bragg–Brentano geometry is obeyed when the x-ray source and detector are positioned on the goniometer circle where it intersects the focusing circle. True focusing would indeed occur only for a sample that is bent to the radius of the focusing circle RFC. Since RFC differs for various scattering angles 2? , true focusing cannot be obtained in a ? /2? can and the arrangement is thus termed parafocusing geometry. In a ? /2? scan the scattering vector Q is always parallel to the substrate normal. It is, however, evident from the above considerations and from Fig. i1. 1 that this is strictly valid only for the central beam, while slight deviations from the parallel orientation occur for the divergent parts of the beam. If the most divergent rays deviate by ±? from the central beam their scattering vector is tilted by ? from the sample normal – at least for those scattering events that are received by the detector.

In many configurations of diffractometer optics it suffices to consider only the central beam. The analysis and interpretation of x-ray diffraction measurements necessitates distinguishing three different reference frames that are assigned to the laboratory, the sample and the crystallites and symbolized by {li}, {si} and {ci}, respectively. The unit vectors in each system are denoted by li, si or ci, with i ranging from 1 to 3 for the three orthogonal directions. Transformations between these coordinate systems are frequently used, for which unitary transformation matrices aij are defined with superscripts LS, SC, CL, etc. indicating the initial and the final reference frame. The relations are visualized in Fig. i1. 2. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 17 1. 3 Intensity of Bragg Ref lections 17 Figure i1. 2 The three reference frames used in x-ray diffraction and the appropriate transformation matrices between them. 1. 3. 1 Atomic Form Factors Formula (1. 16) can be applied to atoms by inserting the square of electronic wavefunctions for the charge density ? e(r). Before the results of this procedure are presented let us first investigate what might be expected from basic physical considerations.

For this purpose the electrons may temporarily be imagined in the atomic model of Bohr to move in circular orbits around the nuclei. If the scattering from any two arbitrary electrons from this atom could be obtained it is evident that the scattering may occur for many different distance vectors r being associated with a large variation of phase shifts –iQr. The orbital smearing of the electron density will thus lead to a cessation of coherency and a reduction in the coherently scattered intensity. This reduction will be stronger the larger Q becomes, because it is the scalar product Qr that determines the phase shift.

The ansatz is made that the scattering of an atom depends on the shape of the electron density function or on its form, and we thus define an atomic form factor f by f = ?at ? e(r )exp(? iQr )dr (1. 17) In the limit of Q = 0 the integration just runs over the charge distribution and yields the number of electrons of the atom Z. For Q ? 0 the form factors are rea- 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 18 18 1 Principles of X-ray Diffraction sonably presented as a function of |Q| or sin? /?. Atomic form factors have been calculated with various quantum mechanical methods of increasing sophistication.

A compilation of values for all chemical elements and some important ions is given in Ref. [1]. Moreover, very often an approximation of f in the form of the model function f = ? a j exp ( ? b j sin2 ? / ? 2 ) + c j j =1 4 (1. 18) is used. By this approach a precision of 10– 6 is achieved for the form factors and only nine coefficients have to be given for any atom or ion to model the whole sin? /? range. The coefficients aj, bj and cj are also tabulated in Ref. [1]. It is concluded that point charges Ze have to be substituted by fe in all the foregoing expressions in order to deal correctly with the extension of atomic charge distributions.

For some metallic atoms the atomic form factors as calculated by Eq. (1. 18) are displayed in Fig. 1. 11. For low scattering angles they can be seen to reach values close to the atomic number Z, but a steep decrease with increasing sin? /? is clearly seen for all of them. It should be noted that the intensity scales with the square of the atomic form factor and that an even stronger decrease will occur for f 2. For the example of Nb the form factor for the fivefold ion Nb5+ is also given. It can be seen that a difference between atoms and their ions is only significant for f values at low sin? ? , which is a general tendency for all atoms and ions, not just for Nb. Figure 1. 11 Atomic form factors of Be, Al, Cu, [email protected], Nb, Nb5+ and Ag. For some investigations the inelastic scattering of x-rays cannot be neglected and the concept of the atomic form factors will then have to be extended by including real and imaginary anomalous scattering factors, f ? and f ? , that have to be added to the atomic form factors f given above. In most cases, anomalous scattering factors f ? and f ? are small when compared with f. Numerical values for f ? nd f ? are given in the Ref. [1]. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 19 1. 3 Intensity of Bragg Ref lections 19 1. 3. 2 Structure Factor The crystallographic unit cell is the smallest unit by which the periodic order in the crystal is repeated. In the simple cubic lattice that has been considered to derive the Bragg and Laue equations there is only one atom per unit cell. The scattered intensity was found to scale with the square of the charge of this atom – or the form factor as should be said now – and the interference function, see Eq. (1. 10).

For more complex structures the integration has to be extended over the total charge distribution of the unit cell (uc) rather than over a single atom. This quantity is denoted as the structure factor F that is given by F= ?uc ? e(r )exp(? iQr )dr (1. 19) We will symbolize it consistently by a bold letter, since it is a complex quantity. The expression for the structure factor may be simplified by recalling that the unit cell comprises N atoms, numbered by n from 1 to N. It is thus possible to decompose the structure factor into single shares due to the individual atoms (at)

F= n =1 ? ? at ? e(r )exp ( ? iQ(r ? rn )) dr N (1. 20) and the integration just has to be performed over the charge distributions of individual atoms. These values are known: they are given by the atomic form factors fi of the nth atom. Accordingly, the structure factor can be written F= n =1 ? f n exp (iQrn ) N (1. 21) The product of the scattering factor with the positions rn of the N various atoms in the unit cell thus has to be evaluated. The latter are specified by their fractional coordinates (xn, yn, zn) that read for the cubic cell rn = xnac1 + ynac2 + znac3.

We know that ref lection intensity may only be observed when the Laue conditions are simultaneously obeyed which may be applied to simplify the phase factor by Q(x n ac 1 + yn ac 2 + zn ac 3 ) = hx + ky + lz (1. 22) Only if this equation is obeyed does measurable intensity from interfering x-rays enter into the detector and the scattering of the crystal scales with F (hkl ) = n =1 ? f n exp ? 2? i(hxn + kyn + lzn )? ? ? N (1. 23) The structure factor thus depends on the Miller indices of the ref lection under consideration, the positions of the atoms in the unit cell and the atomic scattering factor.

In monoatomic lattices the form factor is the same for all atoms and can be placed in front of the sum. For the simple cubic structure N = 1 and x = y = z = 0 and thus F = f for all hkl and ref lections are observed for each order; however, for 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 20 20 1 Principles of X-ray Diffraction more complicated structures the full structure factor has to be investigated. Although the derivative has only been given for the cubic lattice it has to be emphasized that the expression for the structure factor, Eq. (1. 23), is valid for crystals of arbitrary symmetry.

The majority of one-elemental metals are found in either the face-centered cubic (fcc), the body-centered cubic (bcc) or the hexagonal close-packed (hcp) structure. The relative arrangement of atoms in theses lattices is presented in Structure Box 1. Aluminum, for instance, crystallizes in the fcc structure. In this case the Bragg equation might be obeyed for certain lattice planes hkl, but for some combinations of hkl the phase shift in the x-rays scattered by neighboring atoms may amount to ? or odd multiples of it. The scattered beams then interfere destructively and the ref lections for these lattice planes are not extincted.

In the fcc structure, for instance, destructive interference occurs for hkl = 100, 110, etc. The extinction conditions can be derived for any crystal lattice by performing the same summation procedure that has been performed for the simple cubic lattice in the first section and it is an instructive exercise to do so (Exercise 7). One will then arrive at conditions comparable to Eq. (1. 11) which predict under which orientation of Q towards {si} ref lections might be observed. A simpler approach instead is the calculation of the structure factor. Inserting the fractional coordinates of all four atoms of the fcc structure in Eq. 1. 23) yields the result ? 1 + exp(i? (h + k)) + ? ?4 f all hkl even/odd Ffcc (hkl ) = f ? ?=? hkl mixed ? exp(i? (h + l )) + exp(i? (k + l ))? ? 0 (1. 24) The expression is seen to vanish for certain hkl and the lower equation is thus denoted as an extinction condition. It means that Bragg ref lections are only observed for the fcc lattice if all Miller indices are either even or odd. For mixed triples destructive inference occurs and these ref lections are systematically absent. It is evident from the ? /2? scan in Fig. 1. 10 that this pattern in fact is in accordance with the extinction conditions of the fcc structure.

Mathematically speaking, the Bragg equation is a necessary but not a sufficient condition for x-ray ref lections to arise. The structure factor of the bcc lattice can be obtained in the same way and results in ? 2 f Fbcc (hkl ) = f ? 1 + exp(i? (h + k + l ))? = ? ? ? ?0 h + k + l = 2n otherwise (1. 25) The extinction condition now derives from a sum over Miller indices and reads that the sum must yield an even number for the ref lection to occur. Only if this condition is obeyed is the interference nondestructive and can be detected at the position predicted by Bragg’s equation (Eq. 1. 15)). The structure factor for the hcp structure can be derived as an exercise from the atom coordinates in the unit cell. The structure factor Fh from a ref lection h is of central importance in x-ray diffraction, because it relates the position of the atoms in the unit cell to the intensity of a ref lection. Here, the Miller index triple hkl has been abbreviated by the subscript h which will be used very often in the following. The intensity scales with the product of F and its complex conjugate F*. As can be seen from the examples of the 001_Kapitel_1 23. 09. 2005 14:56 Uhr

Seite 21 1. 3 Intensity of Bragg Ref lections 21 Structure Box 1: Elementary Metals The simple cubic structure that is used in this chapter to derive the basic formulas of xray diffraction only rarely occurs in nature. It is instead observed that one-elemental crystal lattices often take the face-centered cubic (fcc) or the body-centered cubic (bcc) structure. Figure s1. 1 displays both of them. The interatomic distances are fully specified by the unit cell edge a. Each atom is surrounded by eight neighbor atoms in the bcc structure or twelve in the fcc structure. Besides he cubic structures, various metals are found to crystallize in the hexagonal close-packed (hcp) structure, also shown in Fig. s1. 1. This structure has two degrees of freedom, namely the interatomic distance a in the base plane and the distance between two of the planes, c/2. Accordingly, two distinct interatomic distances r1 and r2 occur between next neighbors in the hcp structure (Exercise 1. 12). An inspection of the periodic system reveals that the majority of elements assume one of these three basic structures under thermodynamic standard conditions (298 K, 101. 6 kPa). Figure s1. Crystallographic unit cells of the most frequently occurring structures of one-elemental metals: (a) face-centered cubic, (b) body-centered cubic and (c) hexagonal close-packed structure. In both the fcc and hcp structures the atoms are arranged according to the model of close-packed spheres. It is thereby assumed that the atoms can be modeled by rigid spheres that all exhibit the same radius rat. The value of rat is chosen such that atomic neighbors are in contact via their surfaces. According to this scheme the atomic radius can be calculated and is found to be rat,fcc = a 2 / 4 in the fcc structure.

In the hcp structure the condition can only be obeyed when r1 = r2 holds, which leads to rat,hcp = a or equivalently c/a = 8 / 3 = 1. 633. The c/a parameter in general serves to define the ideal hcp structure, which is a close-packed one. In one-elemental metals with the hexagonal Mg structure the c/a ratio is always found to be very close to this ideal value (see Table s1. 1). Although these structures deviate slightly from the ideal hcp structure, they are often considered hcp structured anyway and the value of c/a is specified additionally.

If the volume of atomic spheres in both the fcc and the hcp structure is calculated and normalized with respect to the unit cell volume Vuc, a value of 0. 74 results for the volume 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 22 22 1 Principles of X-ray Diffraction Table s1. 1 The three most common crystal structures of one- elemental metals. Unit cell edges under standard conditions are given. Structure Strukturbericht designation Copper structure (A1) Tungsten structure (A2) Magnesium structure (A3) Space group – Fm3m (225) – Im3m (229) P63/mmc (194) Atomic positions 000 0–0- –0 000 000 –2 3 1? 3 Examples with lattice parameters a and c (nm) Al: 0. 4049; Ni: 0. 3524 Cu: 0. 3615; Ag: 0. 4086 Cr: 0. 2884; Fe: 0. 2866 Nb: 0. 3307; W: 0. 3165 Mg: 0. 3209, 0. 5210 Ti: 0. 2950, 0. 4879 Zn: 0. 2665, 0. 4947 Zr: 0. 3231, 0. 5147 Face-centered cubic (fcc) Body-centered cubic (bcc) Hexagonal closepacked (hcp) ratio. This is the largest value of spatial filling that might be achieved by the packing of spheres all having the same diameter. It should be noted that crystallographic lattice planes in the hcp and also in other hexagonal structures are indexed by four Miller indices (hkil), where always i = –(h + k) holds.

This indexing results from the usage of three unit vectors in the basal plane of hexagonal unit cells. In a widely used abbreviation a period is simply inserted for the third index: (hk. l). One immediately realizes from the occurrence of both types of Miller index symbol that a hexagonal structure is being considered. There exists an interesting relation between the close-packed fcc and hcp structures. The relation becomes evident when all atoms in the fcc lattice are decomposed into atomic (111) planes and compared with the (00. 1) planes in the hcp structure (see Fig. s1. 2). The coordination within the plane is the same, i. e. ach sphere is surrounded by six neighbors to yield the highest packaging density of spheres within the plane. Looking from above on the plane stacking reveals that there exist three distinct positions where atoms might become situated, which are named A, B and C. In each plane atoms are positioned at A, B or C. It turns out that the stacking of planes may be accounted for by the sequences …ABCABC… in the fcc structure, but by …ABABAB… in the hcp structure. Therefore, both structures just differ by the vertical stacking sequence of fully occupied atom planes. Figure s1. 2 Stacking of close-packed planes in (a) fcc and (b) hcp structures. 01_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 23 1. 3 Intensity of Bragg Ref lections 23 fcc and bcc lattice the magnitude of F maximally equals the number of atoms in the unit cell multiplied by their atomic form factor. This situation is rarely observed for more complicated structures, because the scattering of the different groups of atoms often causes a partial destructive interference. This fact is demonstrated from the structure factors of technologically relevant compounds that are found in the various structure boxes of subsequent chapters. The structure factor has the mathematical form of a discrete Fourier transform.

The reverse transformation from the intensity of observed ref lections would thus allow the determination of the atomic positions in the unit cell. However, the intensity scales with the product of the structure factor and its complex conjugate, FhFh*, which is associated with a severe loss of information. If the structure factor is plotted in the Euler plane of complex numbers it may be characterized by its magnitude |Fh| and its phase ? h. In this picture the information loss can be envisaged as a loss of phase information, which is the well-known phase problem in the structure determination by x-ray diffraction.

Regarding the effect of thermal vibrations the same arguments apply as given above to justify the reduction in coherency by the spatial extension of electronic charge distribution. It is well known that the atoms in a solid oscillate at their equilibrium positions rn . Temperature vibrations entail a reduction of phase coherence in the scattered beam and thus reduce the measured intensity. The phenomenon – – can quantitatively be accounted for by the mean quadratic deviation u2 of the atom from its average position rn . The atomic form factors f have then to be replaced by the temperature-dependent expression f T = f exp ? ? u 2 sin2 ? / ? 2 ( ) (1. 26) Again, it can be seen that the scattering amplitude is exponentially damped with increasing scattering angle and that the damping coefficient scales with the square –– of momentum transfer 4? sin? /?. The 8? u2 factor is often abbreviated by the sym— – – bol B in the literature. Typically the average displacements of atoms v u2 at room temperature are in the range between 0. 005 and 0. 03 nm, which translates into a few percent to more than 10% of the bond length. In the fcc structured Cu lattice, — –– for instance, v u2 amounts to about 6% of dCu-Cu.

The effect of the temperature vibrations can be seen from Fig. 1. 11, where in addition to the zero-temperature f also the atomic form factor of Cu in the Cu lattice is shown. It is evident that the scattering strength may be significantly reduced by thermal vibration, which holds in particular for high scattering angles. These results are applied to the structure factor simply by replacing the form factor with its temperature-dependent value Fhkl (T ) = n =1 ? f n exp ( ? Bn (T )sin2 ? / ? 2 ) exp (2? i(hxn + kyn + lzn )) N (1. 27)

Because the thermal vibration amplitudes increase with increasing temperature the damping of Bragg ref lections will also increase. This causes the Bragg ref lection to sink into a background of diffuse scattered intensity when the temperature 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 24 24 1 Principles of X-ray Diffraction is increased. This is in contrast to many spectroscopic techniques, where the observed peaks broaden at elevated temperature. 1. 3. 3 Multiplicity The multiplicity specifies the number of equivalent lattice planes that may all cause ref lections at the same ? B position.

The phenomenon is visualized in Fig. 1. 12, for the laboratory in reference frame {li}. In this coordinate system the position of the incoming beam is set constant with its direction pointing along the {li} system unit vector l1. While K0 is fixed, K moves on a circle during a ? /2? scan as does the substrate normal s3. The figure displays the position during the scan when ? is at the Bragg angle of Al (111). In the case where the sample comprises a single Al crystal of (111) orientation three further ref lections would equally be excited at the intersection of the 2? 111 cone with the {li} sphere.

The ref lections would be caused by -the equally probable scattering of the incoming x-ray beam at lattice planes (111), –(111) and (111) that all exhibit the same interplanar spacing d111 to obey the Bragg equation. In the case of a polycrystalline sample being measured, however, the intensity would look totally different. Because of the random orientation of crystallites the intensity of all equivalent (111) planes would be equally distributed on a cone of opening angle 4? rather than being concentrated in a few singular spots. The intensity would be smeared out over a ring shown as a grey line in Fig. . 12. There are m111 = 8 equivalent (111) planes, but only m200 = 6 for (200) and it is evident that the multiplicity mh will enter the expression of a Bragg ref lection intensity as a scaling factor. Figure 1. 12 Scattering in the laboratory reference frame {li } for a 111 reflection from an Al single crystal of [111] orientation and a polycrystalline Al powder sample. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 25 1. 3 Intensity of Bragg Ref lections 25 1. 3. 4 Geometry Factor The spreading of the Bragg peak over a circular segment of the {li} sphere as discussed above introduces a further ? ependency into the diffraction pattern of a ? /2? scan. The effect is visualized in Fig. 1. 13 where the set of all diffracted intensity for scattering angle 2? is symbolized by a cone of opening angle 4?. The circumferences of the intensity rings scale with sin2? causing a dilution of intensity by 1/sin2?. There also arise a variety of scattering vectors Q that lie on a cone. The scattered intensity will scale with their density, which is sin(? /2 ? ?) = cos?. The geometry factor is the product of both density functions and it is finally obtained as G = cos? /sin2? = 1/(2sin? . Figure 1. 13 Scattering in the laboratory reference frame {li } to derive the geometry factor G. 1. 3. 5 Preferred Orientation (Texture) For a powder sample it may generally be assumed that all grain orientations occur with the same probability, i. e. that the distribution function of grain orientations is isotropic. It is a characteristic structural feature of thin polycrystalline films that certain crystallographic lattice planes can occur with a greater probability than others. This phenomenon is termed preferred orientation or texture. It is evident from Fig. 1. 2 that a texture might have a significant inf luence on the diffraction pattern, where density-enhanced lattice planes will be associated with an increase of the corresponding Bragg ref lection intensity Ih. The intensity then has to be scaled with the density of crystallite orientations that are indicated by the texture factors Th. For a random orientation Th = 1 holds for all of them. The measurement of texture and the determination of orientation distribution functions are outlined in detail in Chapter 5. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 26 26 1 Principles of X-ray Diffraction 1. 3. 6 Polarization Factor

The x-ray radiation emitted from a laboratory x-ray tube is of random polarization. Therefore, the scattering by a polycrystalline sample has to be decomposed into a ? component and a ? component. These considerations have already been outlined for the scattering by a single electron and they equally apply to the case considered here. In the case that I? = I? = I0/2 is valid on the average, the polarization factor takes the form C2 = 1 + cos2 2? 2 (1. 28) and it is by this factor that the intensity received by the detector has to be scaled. The geometry factor may be different for measurement configurations other than the ? 2? scan. The experimentalist should check this point carefully if integral intensities have to be analyzed quantitatively. The dependency of the geometry fac— tor G and the polarization factor C 2 are both shown in Fig. 1. 14 as a function of scattering angle 2?. Also the Lorentz factor L is shown that will be derived later and the — product GC 2L of all three factors. The product function is seen to exhibit a pro— nounced minimum close to 2? = 120°. Up to this point GC 2L continuously decreases, but recovers for high scattering angles close to2? max = 180°. Figure 1. 4 Geometry factor G, polarization factor C2 and — Lorentz factor L as a function of 2?. Also the product of the three factors is shown. Note the logarithmic ordinate scale. 1. 3. 7 Absorption Factor During their transit through matter x-rays suffer from an attenuation of intensity caused by their absorption. The Lambert–Beer law, well known from optics, can describe the absorption effect. The intensity I0 that enters into the sample will be exponentially damped to an amount I0exp(–2µ ) after a path of 2 . The parameter µ 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 27 1. 3 Intensity of Bragg Ref lections 7 Figure 1. 15 Schematic representation of the absorption effect for a thin-film sample in a ? /2? scan. is named the linear attenuation coefficient and depends on the wavelength of the radiation used, the chemical composition of the sample and its density. The inverse of µ would give a penetration depth for normal incidence ? 1/e = 1/µ that specifies the path length for which the intensity I0 drops to 1/e of its initial value. The dimensions of the attenuation coefficient are m–1 or µm–1. Often, the value of the mass absorption coefficient µm is listed in various tables that can be converted into µ = ? m by multiplication with the mass density ?. For many substances attenuation coefficients of the order of 105 to 107 m–1 are obtained for Cu K? or comparable wavelengths. This corresponds to penetration depths ? 1/e of 0. 1 to 10 µm and thus is in the range of a typical layer thickness. It can be concluded that absorption effects might significantly affect Bragg ref lections of thin films. The dominant effect the absorption factor has on a diffraction pattern is the variation of the scattered intensity. Its derivation is shown in Fig. 1. 15 for the case of a ? /2? scan.

For any x-ray beam that has traveled through a sample to become scattered into the detector the primary intensity has been reduced by the factor exp(–2µ ). The reduction of intensity of the total x-ray beam is the sum over all possible paths of the beam within the limits of 0 to max max ? 0 exp(? 2µ )d (1. 29) The path 2 that is traversed by the x-ray beam may be expressed by the depth variable z for which = z/sin? holds. Then is substituted by z/sin? , d by dz/sin? and the integration is performed from 0 up to the thickness t of the film. Here, z = 0 accounts for the surface of the film and z = t for the film–substrate interface.

The solution of the integral yields ? ? 2µ t ? ? ? 1 ? ?1 ? exp ? ? ? 2µ ? ? sin? ? ? ?? ? (1. 30) In the limit of an infinitely thick sample, t > ? which is equivalent to t o 1/µ, the result 1/(2µ) is obtained. In the following the absorption factor is denoted by 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 28 28 1 Principles of X-ray Diffraction the ratio of the absorption for a sample of finite thickness with respect to an infinitely thick sample A= ?0 ? 0 t ? (1. 31) The application of this procedure results in the absorption factor for the ? /2? configuration ? ? 2µ t ? ? A? 2? = ? 1 ? xp ? ? ? sin? ? ? ?? ? (1. 32) The subscript ? /2? has been added in order to indicate the measurement geometry. We will become acquainted with various A factors in the following chapters for different diffractometer configurations. The A factor is also termed the thickness factor and it is seen to cause the measured intensity to cease as a function of increasing scattering angle 2?. In Fig. 1. 16 the A? 2? factor is displayed for thin Al and Nb films of 500 nm and 1 µm thickness measured with Cu K? radiation, where use has been made of the material parameters Al: µm = 486. 7 m2 kg–1, ? 2700 kg m–3 ? µ = 1. 31 ? 106 m–1 Nb: µm = 1492 m2 kg–1, ? = 8550 kg m–3 ? µ = 1.

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