The Ewald sphere

How do we know the relationship between the crystal and the diffraction pattern that we will obtain from it Often it is easier to think of the diffraction experiment with respect to a reciprocal lattice rather than the crystal lattice planes. The reciprocal lattice is a real physical property of a crystal, and rotation of the crystal will cause a rotation of the reciprocal lattice. 

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. 

In Ewald s construction, when the magnitude of the wavelength of the X rays is constant, the vectors describing the incident beam CO and the diffracted beam CP are equal in length and proportional to the reciprocal of the wavelength. The vectors CO and CP form an isosceles triangle in which the base OP is parallel to the diffraction vector. 

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. 

Ewald [EWA 17] suggested a geometric representation in the reciprocal lattice of the results we have just shown. The Ewald sphere is defined as a sphere centered in O, the origin of the direct and diffracted wave vectors, and with radius fX. Vectors kg and k are co-linear to vectors Sg and S, respectively. 

Ewald s condition can be expressed as follows in order for n order diffraction to occur on a family of planes with index (hkl), the n node of the reciprocal lattice s row [hkl] has to be on the Ewald sphere, the corresponding diffracted beam then passes through that point 

When the crystal is in a random position, the point of the reciprocal lattice that corresponds to a given family of planes is not on the Ewald sphere, and additionally, it is generally possible that no point is on that sphere, in which case there is no diffracted beam. In order for a given family of planes to be able to difhact, the corresponding point has to be inside a sphere with radins 2fk, centered in C, the tip of the vector ko - This sphere is called the resolirdon sphere. This can also be written  

We showed in section 1.2.2.2 that the intensity diffracted by a crystal is expressed as  

A certain diffracted intensity is observed when the angle of incidence is close to the Bragg angle of the chosen family of planes. The angnlar range for which this 

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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 ... 

The tangent plane approximation is valid the curvature of the Ewald sphere is negligible at small scattering angles. 

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.

If the tilt of the texture is in the y z -plane (due to cylindrical symmetry it does not matter in which direction we tilt the sample, so the Xc coordinate of the centre of the Ewald sphere can be assigned to zero) then the Ewald sphere equation can be written as ... 

Then the coordinates of the intersection of the Ewald sphere and the /-reflection sphere of radius i/are ... 

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. 

On an image with a scale factor of 500 to 800 this additional component leads to significant shift of the reflection position (6 to 9 pixels). The shift of the reflection centre caused by the Ewald sphere curvature is seen easily while analyzing the positions of reflections lying on the minor... 

This difference in j-coordinate can be explained by the shift of the centre in y-direction caused by the curvature of the Ewald sphere. The curvature of the Ewald sphere does not affect the position of the centre in the four-clicks method, but leads to a shift of the ellipse centre when calculating this centre from a single ellipse. Thus the four-clicks method is better for the centre calculation. 

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 deviation of the electron beam from the Bragg condition is measured by the distance from the reciprocal lattice vector to the Ewald sphere along the zone axis direction, which approximately is defined by... 

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 3.7 The Ewald sphere used to construct the direction of the scattered beam. The sphere has radius 1/X. The origin of the reciprocal lattice is O.The incident X-ray beam is labeled Sq and the scattered beam is labeled s. (Adapted with kind permission of Springer Science and Business Media from Figure 4.19 of reference 11. Copyright 1999, Springer-Verlag, New York.)...

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