Laue conditions

To discuss diffraction for the more general case, where the condition of specular reflection is given up, it is convenient to introduce the so-called wave vector. The direction of the wave vector k is that of the propagating wave and its length is given by 

1 William H. Bragg, 1862-1942, professor of mathematics and physics and his son W. Lawrence Bragg, 1890-1971, professor of physics, received together the Nobel Prize for Physics in 1915. 

Instead of the scattering angle 8 we use the so-called scattering vector q. The scattering vector is the difference between the incoming wave vector ki and the outgoing wave vector kf. 

Let us consider the two lattice atoms, one at the origin and the other one at the position R = from the origin, where the a are the primitive vectors of the lattice and m  

The total path difference for the two waves scattered in the direction of kf by the two atoms is given by A = -j R (ki — kf) = jfRq. Constructive interference occurs for 

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The diffraction is essentially that of a grating. As illustrated in Fig. VIII-7, the Laue condition for incidence normal to the surface is... 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

Electrons diffract from a crystal under the Laue condition k — kg=G, with G = ha +kb +lc. Each diffracted beam is defined by a reciprocal lattice vector. Diffracted beams seen in an electron diffraction pattern are these close to the intersection of the Ewald sphere and the reciprocal lattice. A quantitative understanding of electron diffraction geometry can be obtained based on these two principles. 

The Bragg condition defines a cone of angles,, normal to the (hkl) planes. Alternatively, we use the Laue condition to specify the Bragg diffraction ... 

This Laue condition is a little less restrictive than the Bragg law, in that we no longer have the condition that K g = K 1=1/, bnt we still expect strong diffraction only when we are near the Bragg condition. Ewald proposed, and Bloch showed that waves that exist in a crystal must have the periodicity of the lattice, that is, the solutions shonld look like... 

Far from the Laue condition the absorption shows the normal photoelectric absorption, as would be measured (with allowance for density) in a liquid or gas of the same atomic species. Close to the Laue condition, the absorption is quantified by the imaginary parts of the susceptibilities, leading to imaginary components of the wavevectors. These imaginary components are always normal to the crystal surface and hence the planes of constant attenuation are parallel to the surface. The attenuation coefficient (n) normal to the surface is given by (n)=-4 lm(K o) (4.28)... 

Equations (1) are the von Laue conditions, which apply to the reflection of a plane wave in a crystal. Because of eqs. (1), the momentum normal to the surface changes abruptly from hk to the negative of this value when k terminates on a face of the BZ (Bragg reflection). At a general point in the BZ the wave vector k + bm cannot be distinguished from the equivalent wave vector k, and consequently... 

How are the Laue condition and the Bragg condition connected In Fig. A.3 the wave vectors of the incident and outgoing radiation and the scattering vector are drawn for the Bragg reflection of Fig. A.l. We can conclude that for specular reflection, the scattering vector lattice plane. Its length is given by... 

The length of each vector bj is 2it/dj, where dj is the distance between the the lattice planes perpendicular to bj. (This is ensured by the numerators a2 x a3 in Eq. A.9.) Therefore, each of these primitive reciprocal lattice vectors fulfills the Laue condition Eq. (A.7) and is a possible scattering vector for constructive interference. 

The formalism of the reciprocal lattice and the Ewald construction can be applied to the diffraction at surfaces. As an example, we consider how the diffraction pattern of a LEED experiment (see Fig. 8.21) results from the surface structure. The most simple case is an experiment where the electron beam hits the crystal surface perpendicularly as shown in Fig. A.5. Since we do not have a Laue condition to fulfill in the direction normal to the surface, we get rods vertical to the surface instead of single points. All intersecting points between these rods and the Ewald sphere will lead to diffraction peaks. Therefore, we always observe diffraction... 

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

Eqn. 7.27 defines the so-called Laue conditions for diffraction. In these, a, b, and c are the distances between scattering centres, which correspond to the length of the unit cells in the three dimensions. The Miller indices h, k and l are whole numbers... 

Consider now a quasi-two dimensional crystal of finite thickness. The basic cell vector a, perpendicular to the surface is chosen equal to this thickness. This crystal is handled by setting = 1. The diffraction is then still sharply peaked in both directions parallel to the surface, but the Laue condition on Qj (= Q ) is relaxed, and the intensity is continuous in the out-of-plane direction the reciprocal space is made of rods perpendicular to the surface plane. If we still define / by Q.a, = 2M, I is now taken as a continuous variable since intensity is present for non-integer values of /. The intensity is now given by ... 

As the crystal is rotated, so is its crystal lattice and its reciprocal lattice. If, during the rotation of the crystal a reciprocal lattice point touches the circumference of the Ewald circle (the surface of the Ewald sphere), Bragg s Law and the Laue conditions are satisfied. The resnlt js a Bragg reflection in the direction CP, with values of h, k, and 1 corresponding both to hkl values for the reciprocal lattice point and for the crystal lattice planes. 

These three equations are the Laue conditions, which can be expressed in the reciprocal lattice. Consider an arbitrary vector f in the reciprocal lattice with coordinates (h,k,l). The dot product f. a = ha. a + kb. a -i- Ic. a is equal to h. The same relation can be written for each of the vectors b and c, and therefore the Laue conditions impose that the scattering vector S must be a vector of the reciprocal lattice. ... 

This section will enable us to return to the Laue conditions which we showed with the help of the kinematic theory of diffraction. These conditions express the fact that the waves scattered by each of the points on a row have to be in phase. 

When iVfl, N, and are large, this function is sharply peaked (at Bragg points) when the three Laue conditions are satisfied ... 

We can also use the formalism presented previously to treat diffraction from a two-dimensional solid. In essence, we consider the diffracted intensity that would arise from a crystal that is only one layer thick, since we only need to satisfy two of the Laue conditions. In this case, the expression for... 

Again, we take as a point of departure diffraction from a three-dimensional crystal and relax one of the Laue conditions q 2nl). The expression for the diffracted intensity reduces to... 

Here Eo and EH are the complex amplitudes associated with the incident and diffracted X-ray plane-waves, K0 and Kh are the respective complex wave vectors inside the crystal, and v is the X-ray frequency. The two wave vectors are coupled according to the Laue condition, with... 

Figure 2. X-ray standing wave field formed in a crystal and above its surface by the interference of incident and Bragg-diffracted monochromatic X-ray plane waves. The inset shows the reciprocal space diagram for the Laue condition described by Equation (4).

Figure 6.4 Ewald Sphere Construction that demonstrates how the complete X-ray scattering pattern (all vertices of reciprocal lattice) can be visualised by adjusting i) the wavelength (X) of X-ray beam So incident upon a crystal mounted at position M ii) the orientation of the crystal relative to beam Sq. Each vertex of the reciprocal lattice corresponds with a different hW-reflection. A given hW-reflection may be visualised only when scattered beam s cuts the surface of the Ewald sphere at a position P coincident with a corresponding reciprocal lattice vertex. In principle, X and crystal orientation at M may be adjusted to visualise the vast majority of vertices of a reciprocal lattice and hence the vast majority if not all of the hW-reflections possible from a given mounted crystal (Laue Condition).

Determination of X-ray diffraction patterns for parent and isomorpous crystals. The spacing of the indexed reflections is inversely proportional to the lengths of the crystal unit cell. Since the relative spacing between reflections is needed for determining the lengths of the unit cell axes, it is important to obtain an undistorted diffraction pattern. Each reflection in the diffraction pattern can be assigned to a unique set of Miller indices from the Laue conditions for diffraction. The pattern is indexed in term of the Miller indices. 

The diffraction will occur provided that the change in the wave vector equals to a vector of the reciprocal lattice. This is called the Laue condition. 

Diffraction of X-rays is the basic technique for obtaining information on the atomic structure of crystalline soHds and is one of the key standard laboratory techniques. XRD is based on the interference of X-ray waves elastically scattered by a series of atoms orientated along a particular direction in a crystal characterized by a vector A. The waves scattered by two atoms a and b interfere constructively with each other when the path difference PQR is equal to an integer number of wavelengths PQR = W. This condition is valid for orientations K of the scattered waves which satisfy the Laue condition ... 

A short diversion into bulk (3D) crystallography is in order here. Consider a 3D crystal, built by repetition of identical unit cells on a 3D lattice defined by the primitive vectors a, b and c. Due to constructive interference of the scattering from all the unit cells of the crystal, Bragg diffraction occurs at points q = qi,u satisfying the three Laue conditions... 

Returning to monolayers, diffraction can occur if the molecules are packed with 2D- crystalline order of sufficiently long range. With a crystalline repeat of a unit cell along only two primitive vectors a and b in the monolayer plane), and no repeat out of the plane, only two Laue conditions apply ... 

When these three conditions are simultaneously satisfied, the entire diffraction phenomenon may be equivalent to a planer reflection and gives rise to Bragg s law. It will be shown in this chapter that the wavelengths for which the Laue conditions are satisfied are not the characteristic radiation but general radiation. 

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