Diffraction Ewald sphere

Figure 11.12 RHEED diffraction Ewald sphere eonstruetions for the on-Bragg and off-Bragg conditions. Note that in the on-Bragg eondition there is no difference in the diffracted intensity for the rough and smooth surfaces while for the off-Bragg condition alternate layers of the film interfere destructively. Therefore there is strong contrast for the intensity from rough and smooth surfaces. Refer to Figure 4.6 and related discussion for background on this construction.

Because of the much shorter wavelength of elecuon beams, the Ewald sphere becomes practically planar in elecU on diffraction, and diffraction spots are expected in this case which would only appear in X-ray diffraction if the specimen were rotated. 

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... 

The notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

Figure 3.4 X-ray beam passing through the Ewald sphere and diffracted by planes in a single crystal produces reflection spots. (Adapted with permission from Figure 1.13 of Drenth, J. Principles of Protein X-ray Crystallography, 2nd ed., Springer-Verlag, New York, 1999. Copyright 1999 Springer-Verlag, New York.)...

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

What happens if the set of (hkl) lattice planes is not exactly at the Bragg orientation As shown on figure 2b, the position of the two spots is hardly affected but the intensity of the diffracted beam is strongly modified. This behavior can be explained by means of the Ewald sphere construction. 

Geometrically, electron diffraction patterns of crystals can be approximated as sections of the reciprocal lattice, since the Ewald sphere can be regarded as a plane (i.e. the radius of the Ewald sphere, 1/2, is much larger than the lengths of low-index reciprocal lattice vectors). 

Figure 2. Fonnation of ring and oblique texture patterns, a - several randomly rotated artificial crystallites and its Fourier transform (inset) b - reciprocal space with rings and zero tilt Ewald sphere c - 60° tilt of the Ewald sphere (reflection centers lie on the ellipse) d - the diffraction pattern as it is seen on the image plane.

Oblique texture patterns have almost perfect 2mm symmetry and thus the whole set of diffraction spots is represented by the reflections in one quadrant. The arcs are exactly symmetrically placed relative to the major axis, being sections of the same spherical band in reciprocal space. The reflections on the lower half of the pattern are sections of reciprocal lattice rings, which are Friedel partners and thus equivalent to those giving the reflections of the upper half assuming a flat surface of the Ewald sphere. Actually, if the curvature of the Ewald sphere is taken into account, the upper and lower parts of a texture pattern will differ slightly. 

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. 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the... 

Figure 4.6 A high resolution experiment. In (a) the crystal is not yet aligned to the Bragg position and no diffracted beam occurs. In (b) the incident beam s been rotated so that the Ewald sphere falls on the 004 relp, and a diffracted beam ensues. The Ewald sphere is to scale for CuK and the reciprocal lattice is to scale for silicon...

Figure 4.8 The powder difTraction experiment, (a) Reciprocal space notation. The Ewald sphere is fixed, and the lattice is rotated about all angles about the origin. Only the rotations about [100] are shown in this two-dimensional section. Intersections with the Ewald sphere define the diffracting conditions, (b) The corresponding diffracted beams in real space...

Figure 4.9 The Laue back-reflection method. The direction of the incident beam and the reciprocal lattice are fixed in space, and wavelengths are selected out of the beam. The Ewald sphere may be any diameter between the short and long wavelength cutoffs. The larger circle shows an intermediate wavelength and the diffracted beams that result at this wavelength...

Figure 4.11 The kinematic dispersion surface. The circles centred on the origin O and the relp H, with radius (1/ vacuum )(1+ 2) represent, in the plane shown, the allowable wavevectors in the crystal far from the diffracting condition. A section of the Ewald sphere is shown...

Figure 7.8 A scattering map in reciprocal space. Equal intensity contours are shown schematically, and the Ewald sphere is represented as a plane near reciprocal lattice points 0 and h. The dynamical diffraction from the specimen is displaced slightly from the relp and from the centre of the diffuse scatter by the refractive index effect...

As will become apparent, it is important to place the photographic plate as close to the specimen as possible. With rotating anode generators, care should be taken not to allow the full power of the beam to fall on the plate when stationary as this leads to an unsightly overexposed vertical line on the topograph. The presence of a horizontal stripe on the recorded topograph is often due to the presence of a second reciprocal lattice point lying on the Ewald sphere. It can be removed by a small rotation of the crystal about the diffraction vector as if to take a stereo pair. 

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... 

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