Ewald circle

In the Ewald construction (Figure 3.17), a circle with a radius proportional to 1/A and centered at C, called the Ewald circle, is drawn. In three dimensions it is referred to as the Ewald sphere or the sphere of reflection. The reciprocal lattice, drawn on the same scale as that of the Ewald sphere, is then placed with its origin centered at 0. The crystal, centered at C, can be physically oriented so that the required reciprocal lattice point can be made to intersect the surface of the Ewald sphere. 

FIGURE 3.17. The construction of an Ewald sphere of reflection, illustrated in two dimensions (the Ewald circle), (a) Bragg s Law and the formation of a Bragg reflection hkl. The crystal lattice planes hkl are shown, (b) Construction of an Ewald circle, radius 1/A, with the crystal at the center C and Q-C-0 as the incident beam direction. 

FIGURE 3.17, (c) Orientation of the reciprocal lattice with its origin (hkl = 000) at 0. If one reciprocal lattice point hkl touches the surface of the Ewald circle (sphere), a Bragg reflection hkl will be formed. 

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. 

Figure 1.8 Geometrical construction of the Ewald circle. The 0 marks the origin of reciprocal space. The vectors are defined in the text.

Fig. 3.1. Ewald circle showing the relation of the angles between the directions of incident, reflected, and scattered light to the reciprocal lattice parameters p and o.

If polychromatic radiation is being used, as in the Laue method, the Ewald circle represents the short wavelength limit (highest energy photon). Now all planes whose representative points lie inside the circle represent possible reflections and the symmetry of the reflections is the symmetry of the reciprocal lattice relative to the incident radiation. Thus the Laue method is useful for determining the orientation of the axes in a single crystal. 

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 A.5 Ewald construction for surface diffraction, a) a side view of the reciprocal lattice at the surface. Constructive interference occurs for all intersection points of the vertical rods with the Ewald sphere. This is equivalent to the condition when the component qj of the scattering vector parallel to the surface is identical to a reciprocal lattice vector of the surface lattice, b) the top view of the reciprocal surface lattice. The circle is the projection of the Ewald sphere. If we disregarding the radiation scattered into the crystal, the number of lattice points within the circle (corresponding to the intersections of the rods with the Ewald sphere) is identical to the maximum number of observed diffraction peaks.

Ewald s42 sphere of reflections in reciprocal space explains when and in which direction diffraction will occur. A vector k is drawn from the "origin of the reciprocal lattice" O (e.g., the center of the crystal) parallel to the incident X-ray beam, to "hit" a reciprocal lattice point A. If the vector G (or k) represents the distance between two reciprocal lattice points A and B, then in the direction O to B a scattered wave (vector k or S) will appear. Ewald drew a circle (in 2D) or a sphere (in 3D), called the sphere of reflection of radius 2%/X, around the point O diffraction occurs when this sphere intersects a reciprocal lattice point (Figs. 8.5 and 8.6). As the crystal and/ or the detector are moved, the reciprocal lattice points which cross the Ewald sphere satisfy Eq. (8.3.2) or (8.3.3), and a diffracted beam is formed in direction k. ... 

FIGURE 7.9. Use of the Ewald sphere to predict which Bragg reflections will bf-observed in a particular X-ray diffraction experiment (see Figure 3.17, Chapter 3). (ai Variation in the orientation of the crystal (by oscillation or rotation about (f>). The two limits of oscillation are shown, (b) Variation in the wavelength of the radiation used, cis in a Laue photograph, with a stationary crystal. The limits for two wavelengths are shown by the two circles. In both cases all Bragg reflections in the shaded area will be observed (where the surface of the Ewald sphere intercepts reciprocal lattice points). 

Figure 2.31. The origin of the powder diffraction eone as the result of the infinite number of the completely randomly oriented identical reciprocal lattice vectors, d hki, forming a circle with their ends placed on the surface of the Ewald s sphere, thus producing the powder diffraction cone and the corresponding Debye ring on the flat screen (film or area detector). The detector is perpendicular to both the direction of the incident beam and cone axis, and the radius of the Debye ring in this geometry is proportional to tan20.

Figure 12 The Ewald construction drawn for the reflection (-2 2 0). The crystal is located at the origin O and the endpoint of the vector s lies at a lattice point of the reciprocal lattice (gray). The radius of the circle is A 1.

These spherical shells intersect the surface of the Ewald sphere in circles. Figure 1.10 shows a two-dimensional projection. Diffracted beams emanate... 

Figure 1.10 Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

Here I y is the reflection intensity measured as a function of a scan variable s z is the direction normal to the Ewald sphere at the reflection position. Integrating over. s for a typical four-circle diffractometer and approximating sine and cosine values for the small angular range of a reflection leads to the following formalism ... 

Fig. 3 Ewald construction. The white half-circle indicates the Ewald sphere in two dimensions. The points of intersection between the reciprocal lattice rods and the Ewald sphere form the set of reciprocal lattice points (bright) which obey Bragg s law and appear as diffraction spots in the diffraction pattern. Zero-, first- and second-order Laue zone are indicated. Eor electron diffraction in TEM, the ratio between the radius of the Ewald sphere and the reciprocal lattice unit is larger than visualized in the figure. (View this art in color at www.dekker. com.)...

Figure 20. Cylindrical reciprocal lattice generated by rotation of reciprocal lattice rows around c (left) and its elliptical intersection with the Ewald sphere (right) which, in the case of electron diffraction (only small Bragg angles are possible), can be approximated by a plane. At right, the effect of a non perfect planarity of the sample is shown by substituting the circles of the left figure with tori the intersection of a toms with the Ewald sphere is an arc (Fig. 22). Modified after Zvyagin...

Fig. 18. Values of g(k) for a Stockmayer fluid at p 0.8, r = 1.35, and —0.741. The curve is the QHNC theory, and the solid and open circles are the Ewald results of Pollock and Alder for A—256 and 500, respectively. The open square is the Ewald (A — 500) result for the Kirkwood g-factor.

Fig. 3,19. The Ewald construction. The circle represents the intersection of the Ewald sphere and a plane of the reciprocal lattice passing through the origin and containing the lattice points hkl) with hU- - kV- - /W= 0 U, V, Wbeing coprime integers). The primitive translation of the direct lattice lUVW] is normal to this plane (Section 1.4.3). The reader can imagine other planes of this series which obey the relation hU- -kV - IW— n, n 0, above and below the plane of the figure...

In order to determine which lattice planes give rise to Bragg diffraction, a geometrical construct known as an Ewald sphere is used. This is simply an application of the law of conservation of momentum, in which an incident wave, k, impinges on the crystal. The Ewald sphere (or circle in two-dimensional) shows which reciprocal lattice points, (each denoting a set of planes) which satisfy Bragg s Law for diffraction of the incident beam. A specific diffraction pattern is recorded for any k vector and lattice orientation - usually projected onto a two-dimensional film or CCD camera. One may construct an Ewald sphere as follows (Figure 2.44) ... 

Fig. 3.6 D k) (cf. eq. [3.14]) obtained from the dynamical data for Nd, = 30 (filled circles), 40 (filled triangles) and 60 (filled diamonds). Instead of trying to perform the limit r 0 the maximum value of D k, t) (see eq. [3.17]) was taken. Hence, the data should be viewed as an upper limit to the actual initial decay rate. For comparison, the data resulting from the static evaluation with Ewald sums are also included with corresponding open symbob (from Ref. 61).

Figure 8.2.9 Ewald construction for elastic and inelastic scattering of a particle with initial and final wave vectors kj and kf. Six crystal truncation rods perpendicular to the crystal surface are indicated as solid lines, (a) Five conditions of elastic Bragg reflection are marked as gray dashed arrows, which are defined via the intersection of the Ewald sphere with the reciprocal lattice rods. Spheres for different final energies are marked as dashed circles with a constant increase in energy between the circles. An enlargement is shown in (b), which emphasizes the relation between both the energy loss AE and the momentum transfer hAq (short solid arrows) due to phonon excitation or annihilation, to allow scattering into a defined detector direction given by the long dashed arrow.

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