The Phase Problem --> Two Kinds of Waves

The Crystallography experiment: X-rays illuminate the crystal, diffract into discrete rays which leave the crystal. The rays are captured (instead of entering a lens), and the image is produced by computing the recombination of those scattered rays (instead of recombining naturally when a lens focuses the rays).

This recombination of the scattered rays is computationally simulated using equations.

The Phase Problem restated

DATA: what can be measured and what is missing...

The only devices available to measure radiation respond only to the energy and not to the relative phases of the x-rays.

In a microscope every point on the lens receives light from every point on the subject, and the lens combines the light so that every point on the image receives light from every point on the lens. The physical combination of all these light rays depends on each ray’s magnitude and phase.

In the x-ray case, we can measure the intensity of each scattered ray at the place where the lens would have been. Intensity = energy = (amplitude)^{2}. (The intensity of an x-ray is just its number of photons per unit time.)

**I = F ^{2}**

in standard crystallographic notation. But we also must know the relative phases of those rays in order to compute what the image should look like, and unfortunately film, scintillation tubes, area detectors, CCD’s, etc. do not record anything about the relative phases of the various rays. (That is, there is no way to tell when the crest of a wave passes, and for wavelengths of about 1Å going past at the speed of light in a machine made of real materials at room temperature that would be a real trick indeed!) The only way to get relative phases is to use interference effects with a reference wave, and that is exactly what one does!

To summarize the phase problem:

We need to know the phases of the scattered x-rays in order to perform the computation step to get the image, but there is no way of measuring the phases directly. The solution to this problem is the main work of crystallography.

Two situations to describe with waves, using one kind of equation:

(Amplitude factor) • (Phase factor) the general equation of a wave

two varieties:

the expression for a real (light) x-ray, of wavelength set by the real physical experimental conditions.

The scattered waves in:

- a visible light microscope, real light waves that go through the lens system and combine to form the image.
- an x-ray diffraction experiment where the "waves" that form the image are calculated and expressed as the results of a Fourier Transformation. These calculated waves have wavelengths set by the relationships of the Fourier mathematics. For the special case of diffraction by a crystal, the wave lengths are integral fractions of the dimensions of the unit cell (a block of space that repeats by translations to fill all space).

The electron density in the constructed model of a crystal is the sum of these second kind of waves. Not only are the wavelengths of these density waves different from each other, each wave is going in its own particular direction. The wavelengths are integral fractions of unit cell dimensions (i.e. 1,2,3,... complete cycles within the bounds of the unit cell), thus they are standing waves. So for crystals, we will need to learn how to combine standing waves in a box (the unit cell) to build up a density-like image.

Representing waves, the phase clock (with radius = amplitude):

φ factor (e^{iφ}); e^{iφ} = cos(φ) + *i*sin(φ)

exponential form convenient to talk about; cos() & sin() form (real and imaginary components) sometimes more convenient for computation.

bottom 2, Light vs x-ray "microscopy"