18

34.3 The Reciprocal

The inverse of the intersections of a plane with the axes is used to find the Miller indices of the plane. The inverse of the d-spacing between planes appears in expressions for the 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 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 , fi, which is proportional to the number of electrons in the atom or ion. The intensity of a reflection is then determined by the structure factor, 22

Eq. 34.2.29. This relationship is applicable for spherical atoms or ions. However, if the lattice is composed of polyatomic ions or molecules, the lattice positions are not occupied by spherical objects. Instead we must consider the electron density as a function of position within the unit cell, (x,y,z), Figure 34.4.1. For notational simplicity, the electron density at position (x, y, z) is often simply denoted as (r). Eq. 34.2.29 must then be generalized to give the reflection intensity from this electron density distribution:

2i (hx + ky + lz) Fhkl = (r) e dx dy dz 34.4.1

Figure 34.4.1: Electron density map of the six-membered ring portion of 2-dimethylsufuranylidene-1,3-indanedione. Atom positions are not directly determined. The electron density contours are truncated at high values of the electron density.

The summation in Eq. 34.2.29 must be replaced by an integral over the unit cell volume because the scattering centers can no longer be considered as localized particles. Comparison with Eq. 27.3.8 shows that this relationship is the three-dimensional Fourier transform of the electron density distribution. The structure factors are determined by collection of the reflection * intensities, Ihkl = Fhkl Fhkl. The electron density distribution is determined by calculating the inverse Fourier transform of the structure factors. Since the Miller indices are integers, the integrals in the inverse transform reduce to summations:

   –2i (hx + ky + lz) (r) =     Fhkl e 34.4.2 h = – k = – l = –

The process of molecular structure determination then is accomplished by collection of the structure factors of the reflections with Miller indices (h,k,l) and the calculation of the electron density using the inverse Fourier transform. In practical application an infinite number of reflections are not available. As a result, the resolution of the resulting electron density map, Figure 34.4.1, is determined by the number of reflections that are used in the inverse Fourier transform. Increasing the number of detected reflections increases the special resolution of the resulting electron density map. Notice that atom positions are not determined in this process. 23

Once the electron density distribution is determined, a least squares procedure is used to find atom elemental identities and positions that best reproduce the experimentally determined electron distribution. Hydrogen atoms have a small atomic structure factor. As a result, the positions of hydrogen atoms are poorly defined in X-ray based electron density maps. The positions of the hydrogen atoms are usually inferred using molecular mechanics calculations. As a result, the positions of hydrogens in the final structure solution are largely uncertain. is sensitive to hydrogen atom positions. Correspondingly, if precise hydrogen locations are needed then neutron diffraction must be used.

The Phase Problem: One important difficulty in X-ray structure determination is that the reflection structure factors, Fhkl, are not determined directly. Each structure factor is a complex number, which is given by a magnitude and phase angle. In other words, reflected rays arrive at the detector with different relative phases, because of differing path lengths within the unit cell. The knowledge of the different path lengths is required for the faithful spatial reproduction of the * contents of the unit cell. The experimental intensity of a reflection, Ihkl = Fhkl Fhkl, does not determine the phase of the reflection, rather only the magnitude of the reflection. The resulting ambiguity is called the “phase problem.” In small molecule diffraction, several useful iterative methods have been developed for solving the phase problem, which we must leave to your further study.

X-ray Diffraction and Molecular Motion: Thermal Ellipsoids: Just as in spectroscopy, the width of a reflection is a measure of molecular motion. The least squares procedure that fits atom positions to the experimentally determined electron density also calculates the uncertainties of the atom positions. The result of the least squares fitting is displayed with the uncertainties of the non-hydrogen atom positions given as thermal ellipsoids, Figure 34.4.2.

O

S

O

Figure 34.4.2: Least squares fit of atoms to the electron density, Figure 34.4.1, gives atom positions and uncertainties. Positional uncertainties are displayed as thermal ellipsoids. Large thermal ellipsoids correspond to large amplitude motion. Thermal ellipsoids cannot be determined for hydrogens. Hydrogen positions are usually determined by molecular mechanics.

The principle axes of the thermal ellipsoids give the positional uncertainties. The minimum sizes of the thermal ellipsoids for a given molecular structure are determined by the spatial resolution 24 resulting from the number of reflections available in the inverse Fourier transform. Thermal ellipsoids larger than the minimum correspond to either large amplitude vibrational motion or disorder within the crystal. Large amplitude vibrations can have an important effect on the properties of a crystal. For example, large amplitude motions may allow small molecules to diffuse into the crystal. Large amplitude motions also increase the heat capacity of the solid substance. Extremely large amplitude vibrational motion results in disorder within the crystal that in some instances prevents the position of the corresponding atom from being determined.

31.7 Summary – Looking Ahead X-ray diffraction is the primary basis of molecular structure determination in inorganic chemistry, organic chemistry, biochemistry, and structural biology. An extensive data base of X- ray derived small molecule crystal structures is maintained by the Cambridge Data Center, CCDC, which is a primary resource of molecular structure and information. An extensive data base of X-ray and NMR derived protein and nucleic acid structures is available from the RCSB Protein data bank. These structures are commonly called “pdb” files. One disadvantage of X-ray crystallography is the requirement of stable well-formed crystals. An extensive NSF program for the determination of protein crystal structures is underway. Research areas in this program include the development of high-throughput parallel methods for producing pure proteins using molecular cloning techniques and then forming crystals suitable for diffraction studies. Synchrotron light sources are the preferred X-ray source for protein crystallography, because of the higher X-ray beam intensity as compared to the metal target X-ray tubes used in small molecule studies. The determination of X-ray crystal structures has clearly been identified as an important international research priority. The resulting molecular structures then become the basis of structure-function studies designed to conquer many of the challenges that we face as a society.