18 34.3 The Reciprocal Lattice The inverse of the intersections of a plane with the unit cell axes is used to find the Miller indices of the plane. The inverse of the d-spacing between planes appears in expressions for the diffraction angles, Eqs. 34.2.18 and 34.2.20. The appearance of the inverse of unit cell dimensions is a repeating pattern in diffraction experiments. The reason is that X-rays diffract from planes of atoms in the crystal. X-rays experience a different view of the crystal lattice than is depicted in direct representations of the lattice structure, Figure 34.1.2. The way X-rays interact with the lattice is best represented using the concept of the reciprocal lattice. In the real lattice, which is also called the direct lattice, lattice points are occupied by atoms, ions, or molecules. On the other hand, points in the reciprocal lattice correspond to planes within the crystal. The symmetry of the reciprocal lattice is directly displayed in the diffraction pattern. Distance within the reciprocal lattice is given by the inverse of the corresponding plane spacing. The reciprocal lattice is a map of the planes in the direct lattice. Each point in the reciprocal lattice represents a family of planes in the direct lattice. By convention, properties of the reciprocal lattice are shown with an asterisk, *. The reciprocal lattice has the repeat spacings, a*, b*, and c*, as given by the inverse of the direct lattice unit cell dimensions: a*= 1/a b* = 1/b c* = 1/c 34.3.1 The coordinates of a reciprocal lattice point correspond to the Miller indices of the corresponding plane. The direction of the vector from the origin to the reciprocal lattice point is perpendicular to the corresponding planes in the direct lattice. Consider as an example the family of (2,1)- planes depicted in the two-dimensional unit cell in Figure 34.3.1a. – 1 ,2 0,2 1,2 2,2 – 1,1 0,1 1,1 2,1 * dhkl = 1/ – dhkl 1,0 1,0 2,0 O – – – – b 1 ,1 0,1 1,1 2,1 dhkl b* – – – – 1,1 0,2 1,2 2,2 a* a (2,1) (a). (b). Figure 34.3.1: Distances in the reciprocal lattice are the inverse of the d-spacing in the direct lattice. The origin of the reciprocal lattice is chosen as the point labeled “O”. Points in the reciprocal lattice are labeled with the Miller indices of the corresponding diffraction plane. The point representing the (2,1)-planes in the direct lattice is labeled as (2,1) in the reciprocal lattice, Figure 34.3.1b. The distance of the reciprocal lattice point from the reciprocal lattice origin is the inverse of the corresponding d-spacing in the direct lattice: * 1 dhkl = /dhkl 34.3.2 The Reciprocal Lattice has the Same Symmetry as the Real Lattice: If the direct lattice is orthorhombic, then the reciprocal lattice is orthorhombic, Figure 34.3.2. If the direct lattice is hexagonal then the reciprocal lattice is hexagonal. The symmetry of the reciprocal lattice is 19 reflected in the diffraction pattern. However, if the orthorhombic lattice is tall and narrow, then the reciprocal lattice is wide and short, because of the inversion of the unit cell dimensions. For monoclinic lattices the reciprocal lattice angle is 180– , with the direct lattice angle. Orthorhombic Monoclinic * b * * b a * b a b a a Hexagonal b* b a* a Figure 34.3.2: The reciprocal lattice has the same symmetry as the direct lattice. In each example the direct lattice is on the left and the corresponding reciprocal lattice is on the right. The Ewald Sphere Maps the Reciprocal Lattice: The Ewald construction gives a simple visual approach for understanding the formation of the diffraction pattern from the reciprocal lattice, Figure 34.3.3. The Ewald sphere is centered on the real lattice at point C and has a radius of 1/. F Detector P OP * sin = dhkl = 1/dhkl OX 1/d sin = hkl X C 2 2/ O D 2dhkl sin = 1/ M Figure 34.3.3: Reflections occur at angles that correspond to an intersection of the Ewald sphere with a reciprocal lattice point. The radius of the Ewald sphere is 1/. The X-ray beam reflects from the real lattice at C. The incident beam intersects the Ewald sphere at X and O. 20 The X-ray beam line intersects the Ewald sphere at points X and O. The origin of the reciprocal lattice is placed at point O. The angle of incidence of the X-rays with the real and reciprocal lattices is . As the real lattice is rotated the reciprocal lattice also rotates by the same angle. A reflection occurs if a reciprocal lattice point intersects the Ewald sphere. The reflection angle at 2 corresponds to the line from the crystal origin to the reciprocal lattice point, CP. The reciprocal lattice point that corresponds to the d-spacing of the planes that give the reflection is at * dhkl = 1/dhkl, Eq. 34.3.2. How does the Ewald construction give the reflection angle? Consider the right triangle with hypotenuse OX with length equal to the diameter of the Ewald sphere 2(1/). The side-opposite, * OP, has the length of the reciprocal lattice distance dhkl = 1/dhkl. The side-adjacent is XP. The interior angle of this right-triangle is . The sine of the angle is the ratio of the side-opposite to the hypotenuse: OP 1/dhkl sin = = 34.3.5 OX 2/ Rearranging this last equation gives Bragg’s Law: 2dhkl sin = (34.2.3) 34.3.6 In other words, the Ewald construction is a clever geometrical representation that gives the reflection angles as the crystal is rotated about its axes. The reflection angles are determined by the lattice symmetry, while the intensities of the reflections are determined by the composition of the unit cell, which we discuss next. Planar Array Cameras in X-ray Diffraction: The camera used in a typical single crystal X-ray diffractometer is a planar multi-element solid-state array, similar to but much larger than the solid-state detector in a cell phone digital camera. Each individual detector, or pixel, in the array can detect individual X-rays. An array detector allows the intensity of multiple reflections to be acquired simultaneously. The typical size of the camera is 10x14 cm. A computer reads out the X-ray intensity of each individual pixel in a series of exposures, or frames. The crystal is rotated to orient the crystal so as to collect the diffraction pattern over the full sphere of reflection in multiple frames. Example 34.3.1: Determining Lattice Parameters from Reciprocal Lattice Projections Two two-dimensional projections of the reciprocal lattice are required to find all three unit cell dimensions, Figure 34.3.4. The magnification of the observed reciprocal lattice is determined by the crystal-detector distance, M or line CD in Figure 34.3.4. The distance between the crystal and * the reflection spot corresponding to dhkl on the detector is CF. Comparing corresponding sides of triangles COP and CDF gives:2 * OP DF dhkl DF = or = 34.3.7 OC CD 1/ M 1 M Solving for the direct lattice spacing gives: d = * = 34.3.8 dhkl DF where DF is the measured distance on the reciprocal lattice projection. 21 The measured distances between spots three rows or columns apart on the reciprocal lattice projections for 2-dimethylsufuranylidene-1,3-indanedione are shown on Figure 34.3.4. The detector distance is 70.7 mm. A Mo X-ray source was used with = 0.7107 Å. Calculate the unit cell dimensions. +h to right, k = 0, +l up h = 0, +k up, +l to right 8.2 mm 16.7 mm * c * 25.5 mm b * a c* 8.2 mm Figure 34.3.4: Reciprocal lattice projections. The intensity at each point is proportional to the spot size. The missing reflection that is obscured by the beam stop is shown as “”, which corresponds to the direction of the incident X-ray beam. Answer: The reciprocal lattice spacing in the a* direction, taken from the (h,0,l) projection, is 25.5 mm/3 = 5.57 mm. This single reciprocal lattice spacing corresponds to d100 = a in a lattice with all 90 angles. Using Eq. 34.3.8: 1 70.7 mm (0.7107 Å) a = * = = 5.91 Å dhkl 8.50 mm The reciprocal lattice spacing in the b* direction, taken from the (0,k,l) projection, is 16.7 mm/3 = 5.57 mm. Using Eq. 34.3.8: 1 70.7 mm (0.7107 Å) b = * = = 9.02 Å dhkl 5.57 mm The reciprocal lattice spacing in the c* direction taken from the (h,0,l) or (0,k,l) projections, which should give the same result, is 8.2 mm/3 = 2.73 mm. Using Eq. 34.3.8: 1 70.7 mm (0.7107 Å) c = * = = 18.4 Å dhkl 2.73 mm More accurate values result if a greater numbers of rows or columns than four are measured in the reciprocal lattice image. 34.4 Molecular Structure is Determined from Scattering Intensities The Inverse Fourier Transform Determines the Electron Density: X-rays scatter primarily from the electrons in an ion or molecule. For a spherical atom or ion the scattering power is determined by the atomic structure factor, fi, which is proportional to the number of electrons in the atom or ion.