Diffraction from a crystal

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. 

Figure 12.4 SGX-CAT automated scoring, (a) Diffraction from a crystal of a protein kinase/ligand complex. Automated crystal scoring suggested that a dataset should be acguired from this sample. Standard visual inspection would almost certainly reject such a crystal.

Diffraction from a Crystal Lattice. An x-ray diffraction pattern from a single crystal has sharp peaks and regions of zero intensity between these peaks. An analogy is a diffraction pattern obtained by passing visible light through a series of equally spaced holes of uniform size (Fig. 5). In this case, each hole is analogous to a unit ceU in the crystal. 

Using diffraction from a crystal monochromator (any source, including neutrons). 

In the first three chapters of this book, we considered the fundamentals of crystallographic symmetry, the phenomenon of diffraction from a crystal lattice, and the basics of a powder diffraction experiment. Familiarity with these broad subjects is essential in understanding how waves are scattered by crystalline matter, how structural information is encoded into a three-dimensional distribution of discrete intensity maxima, and how it is convoluted with numerous instrumental and specimen-dependent functions when projected along one direction and measured as the scattered intensity V versus the Bragg angle 20. We already learned that this knowledge can be applied to the structural characterization of materials as it gives us the ability to decode a one-dimensional snapshot of a reciprocal lattice and therefore, to reconstruct a three-dimensional distribution of atoms in an infinite crystal lattice by means of a forward Fourier transformation. 

When we consider diffraction from a crystal, we will further find it useful and convenient to consider a point lattice in even more simplified terms. We can introduce this simplification by organizing the lattice points, distributed in a periodic manner through three-dimensional space, into easily characterized families of two-dimensional planes. This is accomplished by defining all families of planes of equal interplanar spacing that include all of the points in the lattice. Examples are shown in Figure 3.17. Every plane of the family need not contain... 

In crystallographic problems this situation arises frequently because diffraction from a crystal can be thought of as involving waves scattered from an almost infinite number of... 

To this point we have been interested in the scattered waves, or X rays from atoms that combine to yield the observed diffraction from a crystal. Because the waves all have the same wavelength, we could ignore frequency in our discussions. In X-ray crystallography, however, we are equally interested in understanding how the waves diffracted by a crystal can be transformed and summed, in a symmetrical process, to produce the electron density in a unit cell. 

Two basic questions we encounter in understanding X-ray crystallography are how do crystals diffract X rays, and how can we describe that process in mathematical terms These questions may be addressed in a number of ways, but the approach we choose here is divide and conquer. Diffraction from a crystal can be deconstructed into two problems, each worked out separately, and finally, their solutions combined. 

To this point we have focused single-mindedly on understanding and writing an expression that describes the diffraction from a crystal, its Fourier transform, if we know the atomic coordinates xj, yj, zj in the crystallographic unit cell. The reader may be impatient by now as the real life objective is to do the opposite, to define the xj, yj, zj coordinates of the atoms when we can measure the F making up the diffraction pattern. It is now time to go the other way, but understanding the meaning of the F makes that task easier. 

Ideally, a Bragg diffraction peak is a line without width as shown in Figure 2.19b. In reality, diffraction from a crystal specimen produces a peak with a certain width as shown in Figure 2.19a. The peak width can result from instrumental factors and the size effect of the crystals. Small crystals cause the peak to be widened due to incompletely destructive interference. This phenomenon can be understood by examining the case illustrated in Figure 2.20. 

X rays of wavelength 0.154 nm are diffracted from a crystal at an angle of 14.17°. Assuming that n = 1, calculate the distance (in pm) between layers in the crystal. 

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

X-ray diffractions from a crystal can only occur in directions defined by the reciprocal lattice points and with intensities that are governed by a structure factor of the form... 

The de Broglie wavelength associated with the golf ball is too small to be experimentally detected, as it is many orders of magnitude smaller than an atomic nucleus. However, the matter wave associated with the electron can be detected experimentally (diffraction from a crystal), as it is the same order of magnitude as the size of an atom. Clearly, the de Broglie wavelength (matter wave) associated with a particle is inversely proportional to the mass of the particle (provided the velocity is constant). 

FIGURE 11.42 Diffraction from a Crystal When X-rays strike parallel planes of atoms in a crystal, constructive interference occurs if the difference in path length between beams reflected from adjacent planes is an integral number of wavelengths. 

At room temperature, thermal neutrons have energies near 0.03 eV, velocities of 2 x 10 cm/sec, and wavelengths of 2 A. All known sources are characterized by being continuous spectrum sources with a spectrum resembling a Maxwellian distribution. Most diffraction techniques utilize a monochromatic beam, and for the neutron case this means a selection of a monochromatic band of radiation by diffraction from a crystal, or by some alternative mechanical system. This has two implications (a) a large fraction of the neutron intensity... 

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