**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 **s _{o}** and

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•s _{o}** = (

(can you derive this from the definition of a dot vector product?). The diffraction pattern is represented by,

**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, d**r**, 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.

.......................................................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.

**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**"

**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
l_{vacuum}

**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).

**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.

**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

Note again that the coordinants of Reciprocal space are inverse
distance; for example, if **a** and **b** are measured in Å,
the dimensions of the reciprocal lattice, will be Å-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
**P2 _{1}2_{1}2** 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 **