Sphere of reflection

The parameters reported by Zintl Hauke were taken as the starting point of the parameter determination. Using these parameters, structure factors were calculated for all of the planes in the sphere of reflection. The atomic form factors of James Brindley (1935) were used. (Subsequent calculations made with two... 

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. 

Fig. 81. The condition for reflection in terms of the reciprocal lattice. Keflection occurs when a reciprocal lattice point P touches the surface of the sphere of reflection.

Fto. 82. Reciprocal lattice passing through sphere of reflection as it rotates. 

Since the directions of reflected rays are obtained by joining the centre of the sphere to points on its surface, the crystal itself may be regarded as rotating in the centre of the sphere of reflection, while the reciprocal lattice of this same crystal rotates about a different point—-the point where the beam emerges from the sphere. If this seems odd, it must be remembered that the reciprocal lattice is a geometrical fiction and should not be expected to behave other than oddly the fact is, the reciprocal lattice is concerned with directions its magnitude and the location of its origin are immaterial. 

Fig. 83. Sphere of reflection surrounded by cylindrical film of unit radius, a. Elevation. b. Plan.

For the purpose of visualizing the geometry of the reciprocal lattice in terms of the actual camera dimensions, it is perhaps useful to multiply the dimensions of the reciprocal lattice and of the sphere of reflection... 

What the crystal does, the film, representing a layer of the reciprocal lattice, must do also, and so its normal performs the same processing motion the film thus, so to speak, rolls around and through the sphere of reflection, picking up diffracted beams on the way, as it cannot fail to do if the film slavishly repeats the movement of the crystal. (The... 

If, as in Fig. 108 a, the c axis of the crystal is displaced from the axis of rotation in the plane normal to the beam (for the mean position of the crystal), the zero layer (hk0) of the reciprocal lattice is tilted in this same direction, and its plane cuts the sphere of reflection in the circle AD. During the 15° oscillation a number of hkO points pass through the surface of the sphere, and thus X-rays reflected by these hkO planes of the crystal strike the film at corresponding points on the flattened-out film (Fig. 108 b) the spots fall on a curve BAD, whose distance from the equator is a maximum at a Bragg angle 0 = 45° and zero at 6 = 90°. If, on the other hand, the displacement of the c axis is in the plane containing the beam (Fig. 108 c), the spots on the film fall on a curve whose maximum distance from the equator is at 6 = 90° (Fig. 108 d). When the displacement of the c axis has components in both directions,... 

Figure 4.11 Sphere of reflection. When reciprocal lattice point 012 intersects the sphere, ray R emerges from the crystal as reflection 012.

As the crystal is rotated in the X-ray beam, various reciprocal-lattice points come into contact with this sphere, each producing a beam in the direction of a line from the center of the sphere of reflection through the reciprocal-latticepoint that is in contact with the sphere. The reflection produced when reciprocal-lattice point Pfrki contacts the sphere is called the hkl reflection and, according to Bragg s model, is caused by reflection from the set of equivalent, parallel, real-space planes (hkl). 

If the sphere of reflection has a radius of 1/A, then any reciprocal-lattice point within a distance 2/X of the origin can be rotated into contact with the sphere of reflection (Fig. 4.12). 

Figure 4.12 Limiting sphere. All reciprocal-lattice points within the limiting sphere of radius 2/X can be rotated through the sphere of reflection.

A monochromatic (single-wavelength) source of X rays is desirable for crystallography because the diameter of the sphere of reflection is 1/X, and a source producing two distinct wavelengths of radiation gives two spheres of... 

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. ... 

Another useful mathematical image in reciprocal space is the sphere of reflection, introduced by Paul Ewald and named after him. The Ewald sphere (Figure 6(a)) has the diameter of 2/A, coinciding with the direction of the incident beam (wavelength A). The crystal (in direct space) is placed at the... 

Figure 6 (a) The Ewald construction illustrated, (b) Limiting sphere and sphere of reflection... 

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. 

Ewald sphere, sphere of reflection A geometrical construction used for predicting conditions for diffraction by a crystal in terms of its reciprocal lattice rather than its crystal lattice. It is a sphere, of radius 1/A (for a reciprocal lattice with dimensions d = X/d). The diameter of this Ewald sphere lies in the direction of the incident beam. The reciprocal lattice is placed with its origin at the point where the incident beam emerges from the sphere. Whenever a reciprocal lattice point touches the surface of the Ewald sphere, a Bragg reflection with the indices of that reciprocal lattice point will result. Thus, if we know the orientation of the crystal, and hence of its reciprocal lattice, with respect to the incident beam, it is possible to predict which reciprocal lattice points are in the surface of this sphere and hence which planes in the crystal are in a reflecting position. 

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