I. Diffraction

A. Properties of Waves: See Handout: Chapter 4 from Principles of Protein X-ray Crystallography by Drenth (Springer-Verlag, 1994) pp74-78.

B. Scattering of a wave by a single object (an electron for X-rays) is described by 2 unit vectors so and s which point in the directions of incident and scattered waves respectively.

Fig. 1 (a) ------------------------------------- (b)

Fig. 1a shows the basic geometry of a diffraction experiment with one diffracting object. Fig. 1b shows diffraction from two objects separated by r. The diffraction pattern is the set of all diffracted waves (over all possible angles, q) each of which arises from diffraction of the incident wave by all parts of the object. Each diffracted ray is described by a scattering vector as shown in Fig. 2. S is the coordinant of the diffraction pattern. A physical interpretation will be given later.

C. For diffraction from two points both with unit scattering potential shown in Fig. 1b,

1. The diffracted waves from the two points, one at the origin 0 and the other at the point r from the origin have different pathlengths given by r•s - r•so = (s - so)•r or in wavelengths,
(can you derive this from the definition of a dot vector product?). The diffraction pattern is represented by, F(S) = 1e2prS which is the sum of the two waves each of which has an amplitude normalized to 1 representing equal scattering power at the two points. Note that S has units of reciprocal distance; if l is measured in Å then S will have units of Å-1.

Fig. 2

2. Note that F(S) will be large when S•r is an integral number of wavelengths as the two waves will interfere constructively and will be small when S•r is half integral as the waves will interfere destructively. For points which are close together r is small and S (and hence 2q) must be large for constructive interference to occur. Thus, features which are close together (higher resolution features) produce diffracted waves at high angles (large values of S) while features which are farther apart (low resolution features) produce diffracted waves at lower angles (smaller S).

3. Two points closer together than l do not produce diffracted waves with large F(S) because no angle, 2q, will produce a pathlength difference which is an integral number of wavelengths.

D. For a real structure, the scattering power is not unity but varies from point to point; this is represented by a function, r(r), which might be the electron density at each point in space (probability of finding an electron) where r represents the coordinants of each point in vector notation (i.e. r = x,y,z). We can then divide the whole object into very small volume elements, dr, and the value of the diffraction pattern at any point S is the sum of waves diffracted from all volume elements.

1. This is written in integral form:
This is exactly the form of a Fourier Transform (it is called the Fourier Integral); thus, the diffraction pattern of an object is the Fourier Transform of the object.

2. There is also an inverse Fourier Transform relationship between F(S) and r(r):

Here the - sign in the exponential indicates an Inverse Fourier Transform and the integral is normalized (scaled) by dividing by the total volume of the object, V.

3. Like FT spectroscopy, we can express structures in 2 different coordinant systems or domains.

Real Space .........+FT ------------ Frequencey Space/Domain

r(r) ..................<------------ -FT...................F(S)

Analogous to nmr FID --------.................-----Analogous to nmr spectrum
.......................................................time domain (interferogram)

Unlike the case with spectroscopy, we are more comfortable in Real Space. But it is often useful to collect and analyze data in reciprocal (frequency) space.

II. Application to Imaging/Microscopy:

A. Image Formation: The object diffracts radiation which falls on it; diffraction for one value of S is diagrammed in Fig. 3. Diffracted waves, the Fourier transform of the object, are collected by the lens and recombined to form an image, this is a second Fourier Transformation from the frequency domain back to real space. The distance from the lens to the image, i, is determined by the distance from the object to the lens, o, and by the focal length of the lens, f, by the "Lensmakers Formula"

 

Figure 3

B. Resolution:

1. Note that the lens will only recombine those diffracted waves which fall within its aperture; i.e. those which actually reach the lens. Waves diffracted at high angles--corresponding to the highest resolution details--fall outside of the lens aperture as shown by the dashed lines in Fig. 3 and aren't recombined to form the image. Thus, the image is not identical to the object but is a lower resolution version. The resolution of the image is determined by the aperture of the lens. The angle of diffraction is determined by the wavelength, l, and by the spacing between points in the object, r. Note: l is the wavelength in whatever medium it travels and = n lvacuum

2. Resolution limits described in I.C.2., IC.3. and II.B.1 are combined in the Abbe Equation (Ernst Abbe) which says that the maximum resolution, d, of an optical system is:

where n is the index of refraction of the media surrounding the lens; n sin(2q) is the Numerical Aperture or N.A. The figure below gives an example of how resolution of an image is determined by the Diffraction Pattern (or Fourier Transform) The first image is a picture of the Parthenon with its Fourier Transform (Diffraction Pattern) to the right. Blow are two images reconstructed from the Fourier Transform, the first using only information near the ceter of the transform (analogous to a small numerical aperature) and the second using all of the information in the transform (alogous to a large numerical aperture).

III. Crystals and Symmetry

A. Convolution Theorem:

1. Convolution: take one function, f(r), and put it down at every point of a second function, g(r); f(r)*g(r). Here * is the convolution operator.

2. Convolution Theorem states: FT [f(r) * g(r)] = F(S) • G(S); that is, the Fourier transform of one function convoluted with another is the same as the Fourier transform of the first multiplied by the Fourier transform of the second (here we will set a convention; real space functions and their coordinant symbols are expressed in lower case, e.g. f, g, and r, and their Fourier transforms and frequency space coordinants are expressed in upper case, e.g. F, G, and S). The converse is also true: F(S) * G(S) = FT [f(r)•g(r)]. So what!!!

B. A crystal is a convolution of one function (a motif) with another (a lattice)

1. Motif is any object; e.g. a protein molecule, skunk etc.

2. Lattice -- an array of regularly spaced mathematical points

3. Lattice * Motif = Crystal

Figure 4 is an example of a 2-dimensional crystal formed by convoluting a motif with a 2-dimensional lattice. This is a small crystal owing to space limitations; normally crystals are very large in terms of numbers of unit cells and are considered to be infinite in extent. A 3-dimensional crystal would result from convolution of a motif with a 3-dimensional lattice. The Unit Cell of a crystal is the smallest unit from which the entire crystal can be generated by translations alone. In Fig. 4 the unit cell is a single skunk, but Figure 6 shows a more complicated unit cell.

Fig. 4

C. Fourier Transform of a Crystal (Diffraction Pattern)

1. FT (motif) -- a continuous function -- no sharp discontinuities

2. FT (lattice) -- another lattice with spacings which are reciprocal of those of the original lattice


Fig. 5

Note again that the coordinants of Reciprocal space are inverse distance; for example, if a and b are measured in , the dimensions of the reciprocal lattice, will be -1.

3. FT(crystal) = FT[(motif) * (lattice)] = FT(motif) . FT(lattice) = FT(motif) . (Reciprocal Lattice). Thus, the continuous Fourier transform of the motif is sampled at the points of the Reciprocal lattice; the Fourier transform of the crystal is only non-zero at the points of the Reciprocal lattice. Examples: Fig. 17-4 from Eisenberg and Crothers.

D. Symmetry: The motif of a crystal may be symmetric and exhibit one or more types of Point Group Symmetry such as 2-, 3-, 4-, 5-, 6-fold rotation axes, screw rotation axes (rotation followed by translation), mirror planes, or inversion centers; biological macromolecules are inherently chiral (they contain asymmetric carbon atoms) however and cannot contain mirror planes or inversion centers.

The Crystal in Fig. 6 shows two types of symmetry in each unit cell (outlined in dashed lines). Filled ellipses show positions of 2-fold rotation axes relating upside down skunks to rightside up skunks. The second type of symmetry is shown by single-headed arrows; these indicate positions of 2-fold screw axes which relate one set of skunk dimers at the corners of the unit cell to another set of skunk dimers at the centers of the unit cell. The symmetry operation in this case is 2-fold rotation about the axis followed by translation along the axis by 1/2 of a unit cell. Different types of point groups symmetry are given symbols: 2 for a 2-fold axis and 21for a 2-fold screw axis in this example. The Space Group of a crystal is a set of symbols which describe the type of lattice followed by the point group symmetry of the unit cell. In Fig. 6 the Space Group is P21212 in which P means the lattice is primitive (as opposed to, for example, face-centered or body-centered).

E. A Real Example: Optical Diffraction Pattern of an Electron Micrograph of a Cytochrome Oxidase Crystal

This is an electron micrograph of a flat crystal of cytochrome oxidase dimers negatively stained with uranyl acetate.

This is an Fourier Transform (actually an optical diffraction parttern) of the cytochrome oxidase crystal image shown above. Note that, except for background noise, the diffraction pattern has spots only at the points of a lattice. These spots are commonly called "Reflections, and each can be specified or indexed on the Reciprocal Lattice by specifying their inteter coordinants. Compare this "Reciprocal Lattice" with the original crystal lattice outlined in the filtered image of the crystal shown at the bottom. Note that the spacings of the real crystal latice are the reciprocal of the "Reciprocal Lattice"; i.e. in the real lattice the longest spacing is along the vertical axis (y-axis) while in the Reciprocal lattice the longest spacing is in the horizontal axis (the h-axis which corresponds to the x-axis in the crystal). Each