4.1: Background

X-ray Crystallography is a scientific method used to determine the arrangement of atoms of a crystalline solid in three dimensional space. This technique takes advantage of the interatomic spacing of most crystalline solids by employing them as a diffraction gradient for x-ray light, which has wavelengths on the order of 1 angstrom (10^-8 cm).

Introduction

In 1895, Wilhelm Rontgen discovered X-rays. The nature of X-rays, whether they were particles or electromagnetic radiation, was a topic of debate until 1912. If the wave idea was correct, researchers knew that the wavelength of this light would need to be on the order of 1 Angstrom (Å) (10^-8 cm). Diffraction and measurement of such small wavelengths would require a gradient with spacing on the same order of magnitude as the light.

In 1912, Max von Laue, at the University of Munich in Germany, postulated that atoms in a crystal lattice had a regular, periodic structure with interatomic distances on the order of 1 Å. Without having any evidence to support his claim on the periodic arrangements of atoms in a lattice, he further postulated that the crystalline structure can be used to diffract X-rays, much like a gradient in an infrared spectrometer can diffract infrared light. His postulate was based on the following assumptions: the atomic lattice of a crystal is periodic, X-rays are electromagnetic radiation, and the interatomic distance of a crystal are on the same order of magnitude as X-ray light. Laue's predictions were confirmed when two researchers: Friedrich and Knipping, successfully photographed the diffraction pattern associated with the X-ray radiation of crystalline CuSO₄ • 5 H₂O. The science of X-ray crystallography was born.

The arrangement of the atoms needs to be in an ordered, periodic structure in order for them to diffract the X-ray beams. A series of mathematical calculations is then used to produce a diffraction pattern that is characteristic to the particular arrangement of atoms in that crystal. X-ray crystallography remains to this day the primary tool used by researchers in characterizing the structure and bonding of organometallic compounds.

Diffraction

Diffraction is a phenomena that occurs when light encounters an obstacle. The waves of light can either bend around the obstacle, or in the case of a slit, can travel through the slits. The resulting diffraction pattern will show areas of constructive interference, where two waves interact in phase, and destructive interference, where two waves interact out of phase. Calculation of the phase difference can be explained by examining Figure 1 below.

In the figure below, two parallel waves, BD and AH are striking a gradient at an angle \(θ_o\). The incident wave BD travels farther than AH by a distance of CD before reaching the gradient. The scattered wave (depicted below the gradient) HF, travels father than the scattered wave DE by a distance of HG. So the total path difference between path AHGF and BCDE is CD - HG. To observe a wave of high intensity (one created through constructive interference), the difference CD - HG must equal to an integer number of wavelengths to be observed at the angle psi, \(CD - HG = n\lambda\), where \(\lambda\) is the wavelength of the light. Applying some basic trigonometric properties, the following two equations can be shown about the lines:

\[CD = x \cos(θ o)\]

and

\[HG = x \cos (θ) \]

where \(x\) is the distance between the points where the diffraction repeats. Combining the two equations,

\[x(\cos θ_o - \cos θ) = n \lambda\]

Bragg's Law

Diffraction of an X-ray beam, occurs when the light interacts with the electron cloud surrounding the atoms of the crystalline solid. Due to the periodic crystalline structure of a solid, it is possible to describe it as a series of planes with an equal interplaner distance. As an X-ray's beam hits the surface of the crystal at an angle, θ, some of the light will be diffracted at that same angle away from the solid (Figure 2). The remainder of the light will travel into the crystal and some of that light will interact with the second plane of atoms. Some of the light will be diffracted at an angle θ, and the remainder will travel deeper into the solid. This process will repeat for the many planes in the crystal. The X-ray beams travel different pathlengths before hitting the various planes of the crystal, so after diffraction, the beams will interact constructively only if the path length difference is equal to an integer number of wavelengths (just like in the normal diffraction case above). In the figure below, the difference in path lengths of the beam striking the first plane and the beam striking the second plane is equal to BG + GF. So, the two diffracted beams will constructively interfere (be in phase) only if \(BG + GF = n \lambda\). Basic trigonometry will tell us that the two segments are equal to one another with the interplaner distance times the sine of the angle θ. So we get:

\[ BG = BC = d \sin \theta \label{1}\]

Thus,

\[ 2d \sin \theta = n \lambda \label{2}\]

This equation is known as Bragg's Law, discovered in 1912. {C}{C}Bragg's Law relates the distance between two planes in a crystal and the angle of reflection to the X-ray wavelength. The X-rays that are diffracted off the crystal have to be in-phase in order to signal. Only certain angles that satisfy the following condition will register:

\[ \sin \theta = \dfrac{n \lambda}{2d} \label{3} \]

For historical reasons, the resulting diffraction spectrum is represented as intensity vs. \(2θ\).

Instrument Components

The main components of an X-ray instrument are similar to those of many optical spectroscopic instruments. These include a source, a device to select and restrict the wavelengths used for measurement, a holder for the sample, a detector, and a signal converter and readout. However, for X-ray diffraction; only a source, sample holder, and signal converter/readout are required.

The Source

X-ray tubes provides a means for generating X-ray radiation in most analytical instruments. An evacuated tube houses a tungsten filament which acts as a cathode opposite to a much larger, water cooled anode made of copper with a metal plate on it. The metal plate can be made of any of the following metals: chromium, tungsten, copper, rhodium, silver, cobalt, and iron. A high voltage is passed through the filament and high energy electrons are produced. The machine needs some way of controlling the intensity and wavelength of the resulting light. The intensity of the light can be controlled by adjusting the amount of current passing through the filament; essentially acting as a temperature control. The wavelength of the light is controlled by setting the proper accelerating voltage of the electrons. The voltage placed across the system will determine the energy of the electrons traveling towards the anode. X-rays are produced when the electrons hit the target metal. Because the energy of light is inversely proportional to wavelength (\(E=hc=h(1/\lambda\)), controlling the energy, controls the wavelength of the X-ray beam.

X-ray Filter

Monochromators and filters are used to produce monochromatic X-ray light. This narrow wavelength range is essential for diffraction calculations. For instance, a zirconium filter can be used to cut out unwanted wavelengths from a molybdenum metal target (see figure 4). The molybdenum target will produce X-rays with two wavelengths. A zirconium filter can be used to absorb the unwanted emission with wavelength K_β, while allowing the desired wavelength, K_α to pass through.

Signal Converter

In x-ray diffraction, the detector is a transducer that counts the number of photons that collide into it. This photon counter gives a digital readout in number of photons per unit time.

References

1. Skoog, D . A.; Holler, F. J.; Stanley R. C.; Principles of Instrumental Analysis; Thomson Brooks/Cole: Belmont CA, 2007.
2. Sands, D. E.; Introduction to Crystallography; Dover Publications, Inc.; New York, 1975
3. Drenth, Jan. Principles of Protein x-ray Crystallography, 3rd edition. 2007, Springer Science + Business Media, LLC. pg. 14.
4. Rhodes, Gale. Crystallography Made Crystal Clear, 3rd edition. 2006, Elsevier Inc. pg. 33, 55 - 57.
5. Actual experimentation done of APS Kinase D63N Penicillium Chrysogenum.

Contributors and Attributions

• Roman Kazantsev (UC Davis) and Michelle Towles (UC Davis)