# Xray Crystallography/Foundations

## X-Rays

### Wavelength

Why do we use X-rays to view molecules? X-rays are electromagnetic radiation just like visible light, but have a far shorter wavelength. X-rays have a wavelength of around 1Å ( 1 Angstrom is 1x10-10meters ) compared to 350-700nm for the visible light range. This shorter wavelength is useful because it allows the resolution of atomic detail, this is because you cannot see details of a size less than half of the wavelength you are using. However there is a downside to using short wavelengths, energy is inversely proportional to wavelength, this means X-ray photons are more damaging to the biological sample.

### Energy

Derivation: keV to Å:
 Planck's formula; $E = hv$ velocity of a wave, for EM radiation, this is velocity of light, c. $c = v\lambda$ $E = \frac{hc}{\lambda}$ This is the energy per photon at a given wavelength. $V = \frac{E}{e} = \frac{hc}{e\lambda}$ Energy can be converted into a potential difference by dividing by charge, in this case by the charge of an electron.

X-rays are part of the electromagnetic spectrum, with a magnetic and electric component to the wave. Since they have wave-particle duality, they can also be thought of as particulate in some cases, and wave-like in others. Electromagnetic radiation has an inverse relationship between wavelength and energy.

$E = \frac{hc}{\lambda}$

where E = Energy, h = Planck's constant, c = speed of light, λ = wavelength.

It is often convenient to convert wavelengths into energies, in the field of X-ray crystallography the most useful units is keV or kilo electron-Volts. A quick and simple conversion is;

$E(keV) = \frac{12.4}{\lambda{}(\AA{})}$

#### X-Ray Window

X-rays are in the section of the electromagnetic spectrum known as the X-ray window. This means that they are not absorbed by air or water. It also means that they can be very penetrating. Care should be taken around X-ray beams, never put your hand directly in the path of the beam. Lead can be used to stop X-rays. 'Soft' X-rays have a wavelength of around 2Å, these are absorbed readily by air and water and as such are not very penetrating. X-rays of wavelength shorter than 0.2Å are termed $\gamma$-rays and are very penetrating, but do not interact greatly with matter. This leaves a usable range of wavelengths from 0.5-1.6Å suitable for X-ray crystallography giving sufficient penetration of sample and enough interaction to have scattering.

### Interactions

Of the X-rays incident on a crystal, only around 2% will interact, the rest will be collected by the beam stop. Of the 2% that interact with the crystal; 8% will interact by (elastic) coherent Thompson scattering giving the observed diffraction pattern, another 8% interacts through (inelastic) incoherent Compton scattering which gives rise to the background noise on the diffraction pattern, however, a worrying 84% interact through the photo-electric effect giving rise to radiation damage by the production of free radicals (Murray J and Rudiño-Piñera E, 2005).

### Why Use Crystals

We use crystals because of the high power of the X-ray beam. X-ray beams are highly damaging especially to biological samples. By using crystals, the damage is distributed among the individual molecules, however the final image generated is a time and space averaged image of electron density for all the proteins in the unit cell.

### Lenses

Wave-particle Duality:
 The relationship between wave and particle is given by the de Broglie relationship; $\lambda = \frac{h}{p}$ where p is the momentum of the particle, and h is Planck's constant.

So why can't we just use an X-ray microscope?

The main problem with an X-ray microscope is that it is not possible to manufacture a lens capable of focusing X-rays. The refractive index of all materials to X-rays is close to 1. It is possible to use zone-plates which use diffraction rather than refraction to focus light to a wavelength of 50Å, but this is not suitable for high resolution work.

To overcome this problem we use a computer to do the work of the lens for us. In light microscopy the incident light is diffracted by the object onto the lens, the lens then takes the diffraction pattern and recombines the diffracted beams onto a focused spot generating the image. In X-ray crystallography, we are stuck with the diffraction image which can be recorded by either an image plate or a CCD, this is an image of Bragg peaks where planes within the crystal interfere constructively.

The only problem of this is that the phase information is lost. Diffraction image recording equipment can only measure the amplitude of the Bragg peak, not the phase information. The phase information can be thought of as the time of arrival of the incident wave or the depth through the unit cell.