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X Ray Crystallography

Page history last edited by Eric Chen 10 years, 5 months ago

X-Ray Crystallography



     There are only two concepts one needs to understand x-ray crystallography: Electromagnetic radiation and basic crystal structure. In fact, both of these concepts have been explored before in high school chemistry! Electromagnetic radiation, or more colloquially called light, is a type of wave. Like all waves, they consist of some oscillating quantity. Water waves oscillate in height, sounds waves oscillate in air density, a rope has a vibrating height, etc. As proposed by James Clerk Maxwell in 1865, light consists of oscillating electric and magnetic fields. Really though, it suffices to think of light as a wave to understand x-ray crystallography1. Like all waves, electromagnetic waves have maximum amplitude, a wavelength (λ), a frequency (ν), and a speed (c). These all are directly analogous to other types of waves. The speed of light is a constant, defined to be c= 2.99792458•108m/s. Because c= λν, knowing the wavelength or the frequency is enough to describe different waves of light.

     There are two features of waves that are important to x-ray crystallography: Interference and diffraction. When waves interact, the superposition principle predicts that the areas of overlap will have amplitude equal to the sum of the interfering waves. It is that simple: Where waves overlap, add the amplitudes. For example, if two waves are in phase, the amplitudes will add up to form a bigger wave. If they are out of phase, the resulting waves will have amplitude equal to zero.

     Diffraction is the bending of light around small objects. This is a phenomenon of all waves and is as simple as it sounds2. If an object is in the way of an advancing wave front, the waves will bend around the object. This makes sense in the context of all waves: Water waves will bend around a rock, sound waves will pass through a doorway and fill another room, etc. In future sections, we will explore how this concept applies to X-ray crystallography in that X-rays ben

d around electron clouds.

     Aside from electromagnetic radiation, the only other concept necessary to understanding x-ray crystallography is basic crystal structure. In solid molecules, the atoms are arranged in a specific pattern. This pattern varies with the size and the charge of the atoms (or ions involved), but each compound has only one arrangement of atoms. This arrangement is called a crystal lattice. On a final note, a unit cell is the smallest group of atoms that when repeated, make up the crystal.



Didn't want to read all of that? Here's what you need to know:

Light is a wave and exhibits wave-like properties (like constructive/destructive interference and diffraction), while crystals are made up of a well- defined lattice, or pattern of repeating unit cells. 



     X-ray crystallography began, logically, with the discovery of x-rays in 1895 by Wilhelm Röntgen. X-rays are a type of electromagnetic radiation, specified by a wavelength between 0.10 and 10.0 nanometers. By 1912, Paul Ewald had documented the optical properties of crystalline lattices. Max von Laue realized that the distance between the layers in a crystal lattice is similar enough in scale to the wavelength of x-rays that the atoms in the lattice could provide a “diffraction grating” for the x-rays, and noting how x-rays diffract off of the atoms in a crystal could provide a way to determine the crystal’s structure. In 1914, William Henry Bragg and his son William Lawrence Bragg used x-ray crystallography to figure out the structure of sodium chloride. Immediately, the technique was recognized as being extremely powerful, even though it took many years (until the 1950’s) for all of the problems (mostly mathematical) associated with it to be resolved.


Experimental Process

The actual process of performing x-ray crystallography consists of three basic steps. The first step, obtaining a crystal, is often the most difficult. The crystal must be large enough (0.1-0.5 mm) and growing crystals can be a time consuming process. The crystal must also be pure, meaning that there are no internal imperfections or twinning (when two identical crystals grow together).

Once the crystal is obtained, it is mounted in front of an x-ray source (a generator, for example). The crystal is slowly rotated and x-rays are passed through it. When the x-rays pass through the crystal they are diffracted by the electrons in the atoms of the crystal and detected by an x-ray detector.



Why are x-rays used over other forms of radiation?

Creation of a molecule’s image from a crystal is similar to creating an image from a lens. Analogously, the wavelength of light used must correspond to the size of the structures that are being visualized. Typical bond distances are on the order of angstroms (0 to 5 Å, or 0 to 0.5 nm). Correspondingly, the wavelength of x-rays (0.01-10 nm) is of the appropriate size for this distance. The wavelengths must be of a similar magnitude to the bond length of atoms in order to produce an interpretable diffraction pattern.

X-ray crystallography will produce unique data for tens of thousands of diffraction spots because of the unique angles and spacing of the atoms in the structure. The diffraction pattern obtained does not resemble any particular kind of molecular structure, so each of the diffraction spots must go through several refinement steps to obtain the structure of the crystal.

X-rays are diffracted around the fluctuating electric fields created by the electrons surrounding the atoms. The resulting diffraction pattern is then used to produce an electron density map. The map can be used to determine the location of atoms relative to each other, bond lengths, and bond angles. It is important to note that crystals with more electrons scatter electrons more effectively. This is because simple molecules/at

oms do not have the electron density or complexity to produce scattering detectable over background noise. The electron density map is then used to determine the structure of the crystal. A summary of the steps for determining the crystal structure is located to the left..




The electron density pattern is only a measure of the number of received x-ray photons at a given location, not an actual measure of electron densities. In order to obtain the electron density map, the diffraction pattern is analyzed by a computer using an inverse Fourier transform. The specific details involved in Fourier transformations are beyond the scope of the course. (If for some reason you feel a burning desire to learn about Fourier transforms, you can read more about them here.) For simplicity’s sake, it is easiest to think of the desired electron density as a mathematical function and the diffraction pattern as the Fourier transform of that function. In order to interpret the diffraction pattern, two things are needed: the amplitude and the phase of the diffracted waves.

Because the diffraction pattern is the received number of x-ray photons in each spot, it is a measure of the intensity. From basic wave principles, we know that the intensity is the amplitude of the wave squared, an

d thus, amplitude can be determined.

There are significant problems, however, when trying to determine the phase of the diffracted waves. Because what is measured in the experiment is essentially a count of the number of X-ray photons in each spot, there is no practical way of measuring the relative phase angles for the different diffracted spots experimentally. To deduce the phase indirectly, there are three approaches.

The first is isomorphous replacement, where a crystal that is nearly identical to the one being studied (except that a few atoms have been replaced or added) is scanned. If these atoms are "heavy" (i.e. they have a large atomic number), they will perturb the diffraction pattern. It is possible to deduce the positions of the few heavy atoms and from that to deduce possible values for the phase angles.

The second techniques is multiple-wavelength anomalous dispersion, which uses just one crystal that contains atoms called anomalous scatterers. By changing the wavelength of the X-rays, the degree to which the anomalous scatterers perturb the diffraction pattern changes, which gives the same kind of information as isomorphous replacement.

A third technique is called molecular replacement. This is used if the researchers already have some idea of the structure of the molecule being analyzed (i.e. it is derived from another compound). In this technique, you replace a unit of the crystal with the model compound and then compute guesses for the phases.

Applications of X-Ray Crystallography

Currently, X-ray crystallography is the most accurate and precise method available for chemical structure determination (although some NMR technologies are coming close). However, due to the nature of diffraction and need for uniform samples, only crystalline compounds can be used. “Growing” or designing these samples is relatively easy for simple and/or small compounds, but as the complexity increases (proteins, viruses, DNA), creating crystal lattices to analyze becomes more difficult. The crystal structure ensures uniform distribution and orientation of one molecule relative to the molecules surrounding it, which is crucial to ensure the diffraction pattern is only determined by the bond orientation withinthe molecules. Non-crystallized samples on the other hand have molecules in random relative orientations, which would produce jumbled, and essentially useless diffraction patterns.

X-ray crystallography has applications in almost every area of science that requires comprehensive knowledge of chemical structure. Because structure and function are affiliated qualities in biological and physical situations, determining the physical structure of a compound is crucial to conducting further research. For example, many pharmaceutical compounds are generated synthetically in labs due to their limited or non-existent) natural abundance. However, the exact structure and connectivity of the compound must be known before pharmacists can attempt to replicate it. One important note is that X-ray crystallography does notdetermine which atoms are present, but rather determines their relative connectivity (bond lengths and bond angles). Nevertheless, since other techniques exist for determining chemical composition, X-Ray Crystallography remains an invaluable tool in chemical research.




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