• Evaluating the diffraction pattern
• Understanding Weissenberg diagrams

For his first experiments, Max von Laue (1879-1960 (Nobel Prize in Physics in 1914) used continuous radiation (with all possible wavelengths) to impact on a stationary crystal. With this procedure the crystal generates a set of diffracted beams that show the internal symmetry of the crystal. In these circumstances, and taking into account Bragg's Law, the experimental constants are the interplanar spacing d and the crystal position referred to the incident beam. The variables are the wavelength λ and the integer number n:

Thus, for the same interplanar spacing d, the diffraction pattern will contain the diffracted beams corresponding to the first order of diffraction (n=1) of a certain wavelength, the second order (n=2) of half the wavelength (λ/2), the third order (n=3) with wavelength λ/3, etc. Therefore, the Laue diagram is simply a stereographic projection of the crystal. See also the Java simulation offered through this link.

Laue diagram of a crystal 

There are two different geometries in the Laue method, depending on the crystal position with regard to the photographic plate: transmission or reflection:

Camera developed by K. Weissenberg in 1924

Scheme and example of a Weissenberg camera. This camera type was used in crystallographic laboratories until about 1975.

  

Scheme explaining the production of a Weissenberg diagram of the rotation or oscillation variety. When the reciprocal points, belonging to the same reciprocal plane, touch the surface of  Ewald's sphere, they produce diffracted beams arranged in cones.

The problem with spot overlap was solved by Weissenberg by adding a translation mechanism to the camera, in such a way that the cylinder containing the film could be moved in a "back-and-forth" mode (in the direction parallel to the axis of rotation) coupled with the crystal rotation. At the same time, he introduced two internal cylinders (as is shown in the left figure, and also below). In this way, only one of the diffracted cones (those from a reciprocal layer) is "filtered" and therefore allowed to reach the photographic film. Thus, a single reciprocal plane (a 2-dimensional array of reciprocal points) is distributed on the film surface (two dimensions) and therefore the overlap effect is avoided.

However, as a consequence of the back and forth translation of the camera during the rotation of the crystal, a deformation is originated in the distribution of the spots (diffraction intensities)

Left: Details of the Weissenberg camera used to collect a cone of diffracted beams. Two internal cylinders showing a slit, through which a cone of diffracted beams is allowed to reach the photographic film. The outer cylinder, containing the film, moves back-and-forth while the crystal rotates, and so the spots that in the previous diagram type were in a line (see above) are now distributed on the film surface (see the figure on the right).
Right:: Weissenberg diagram showing the reciprocal plane of indices hk2 of the copper metaborate. 

Two schematic views showing the principle on which the precession camera is based.  μ is the precession angle around which the reciprocal plane and the photographic film move. During this movement the reciprocal plane and the film are always kept parallel.

The camera designed for this purpose and the appearance of a precession diagram showing the diffraction pattern of an inorganic crystal are shown in the figures below.

The precession method has been used successfully for many years, even for protein crystals:

Originally, the methods of rotating the crystal with a wide rotation angle were very successfully used. However, when it was applied to crystals with larger direct cells (ie small reciprocal cells), the collecting time increased. Therefore, these methods were replaced by methods using small oscillation angles, allowing multiple parts of different reciprocal planes to be collected at once. Collecting this type of diagrams at different starting positions of the crystal is sufficient to obtain enough data in a reasonable time. The geometry of collection is described in the figures shown below. Nowadays, with rotating anode generators, synchrotrons, and area detectors (image plate or CCD, see below), this is the method widely used, especially for proteins.

Outline of the geometrical conditions for diffraction in the oscillation method. The crystal, and therefore its reciprocal lattice, oscillate in a small angle around an axis (perpendicular to the plane of the figure) which  passes through the center. In the figure on the right, the reciprocal area that passes through diffraction conditions, within Ewald's sphere (with radius 2.sin 90/λ),  is denoted in yellow. The maximum resolution which can be obtained in the experiment is given by 2.sen θ_max/λ).

When the reciprocal lattice is oscillated  in a small angle around the rotation axis, small areas of different reciprocal planes will cross the surface of  Ewald's sphere, reaching  diffraction condition. Thus, the detector screen will show diffraction spots from the different reciprocal planes forming small "lunes" on the diagram (figure on the right).  A "lune" is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent. 

The introduction of digital computers in the late 1970s led to the design of the so-called automatic four-circle diffractometers. These goniometers, with very precise mechanics and by means of three rotation axes, allow crystal samples to be brought to any orientation in space, fulfilling Ewald's requirements to produce diffraction. Once the crystal is oriented, a fourth axis of rotation, which supports the electronic detector, is placed in the right position to collect the diffracted beam. All these movements can be programmed in an automatic mode, with minimal operator intervention.

Two different goniometric geometries have been used very successfully for many years. In the Eulerian goniometer (see the figure below) the crystal is oriented through the three Euler angles (three circles): Φ represents the rotation axis around the goniometer head (where the crystal is mounted), χ allows the crystal to roll over the closed circle, and ω allows the full goniometer to rotate around a vertical axis. The fourth circle represents the rotation of the detector, 2θ, which is coaxial with ω. This geometry has the advantage of a high mechanical stability, but presents some restrictions for external devices (for instance, low or high temperature devices) to access the crystal.

An alternative to the Eulerian geometry is the so-called Kappa geometry, which does not have an equivalent to the closed χ circle. The role of the Eulerian χ rotation is fulfilled by means of two new axes: κ (kappa) and ω_κ (see the figure below), in such a way that with a combination of both new angles one can obtain Eulerian χ angles in the range -90 to +90 degrees. The main advantage of this Kappa geometry is the wide accessibility to the crystal. The angles Φ and 2θ are identical to those in Eulerian geometry:

Scheme and appearance of a four-circle goniometer with Kappa geometry

The detection system widely used during many years for both geometries (Euler and Kappa) was based on small-area counters or point detectors. With these detectors the intensity of the diffracted beams must be measured individually, one after the other, and therefore all angles had to be changed automatically according to previously calculated values. Typical measurement times for such detector systems are around 1 minute per reflection.

One of the point detectors more widely used for many years is the scintillation counter, whose scheme is shown below:

Scheme of a scintillation counter

As an alternative to the point detectors, the development of electronic technology has led to the emergence of so-called area detectors which allow the detection of many diffraction beams simultaneously, thereby saving time in the experiment. This technology is particularly useful for proteins and generally for any material that can deteriorate over its exposure to X-rays, since the detection of every collected image (with several hundreds of reflections) is done in a minimum time, on the order of minutes (or seconds if the X-ray source is a synchrotron).

One of the area detectors most commonly used is based on the so-called CCD's (Charge Coupled Device) whose scheme is shown below:

Schematic view of a CCD with its main components. The X-ray converter, in the figure shown as Phosphor, can also be made with other materials, such as GdOS, etc.  The CCD converts X-ray photons at high speed, but its disadvantage is that it operates at very low temperatures (around -70 C). Image taken from ADSC Products

CCD-type detectors are usually mounted on Kappa goniometers and their use is widespread in the field of protein crystallography, with rotating anode generators or synchrotron sources.

Another type of detector widely used today, especially in protein crystallography, are the Image Plate Scanners, which are usually mounted on a relatively rudimentary goniometer, whose only freedom is a rotation axis parallel to the crystal mounting axis. The sensor itself is a circular plate of material sensitive to X-rays. After exposure, a laser is used to scan the plate and read out the intensities.

Area detectors

XALOC, the beamline for macromolecular crystallography (left) at the Spanish synchrotron ALBA (right)

Left: Diffraction image of a protein, obtained with the oscillation method in an Image Plate Scanner. During the exposure time (approx. 5 minutes with a rotating anode generator, or approx. 5 seconds at a synchrotron facility) the crystal rotates about 0.5 degrees around the mounting axis. The read-out of the image takes about 20 seconds (depending on the area of the image plate). This could also be the appearance of an image taken with a CCD detector. However, with a CCD the exposure time would be shorter.

Right: A set of consecutive diffraction images obtained with an Image Plate Scanner or a CCD detector. After several images two concentric dark circles appear, corresponding to an infinite number of  reciprocal points. They correspond to two consecutive diffraction orders of randomly oriented ice microcrystals that appear due to some defect of the cryoprotector or to some humidity of the cold nitrogen used to cool down the sample. Images are taken from Janet Smith Lab. See also the example published by Aritra Pal and Georg Sheldrick.

In all of these described experimental methodologies (except for the Laue method), the radiation used is usually monochromatic (or nearly monochromatic), which is to say, radiation with a single wavelength. Monochromatic radiations are usually obtained with the so-called monochromators, a system composed by single crystals which, based on Bragg's Law, are able to "filter" the polychromatic input radiation and select only one of its wavelengths (color), as shown below:

Scheme of a monochromator. A polychromatic radiation (white) coming from the left is "reflected", according to Bragg's Law, "filtering" the input radiation that is reflected again on a secondary crystal. Image taken from ESRF.

At present, in crystallographic laboratories or even in the synchrotron lines, the traditional monochromators are being replaced by new optical components that have demonstrated superior efficacy. These components, usually known as "focusing mirrors", can be based on the following phenomena:

• total reflection (mirrors, capillaries and wave guides), 
• refraction (refraction lenses) and
• diffraction (crystal systems based on monochromators, multilayer materials, etc.)

• the photon leaves the source where X-rays are produced,
• goes through the various optical elements that channel it in the right direction (mirrors, slits and collimators)
• diffracts inside the single crystal, and
• finally generates the diffraction spots on a detector

Cooling system using dry liquid nitrogen. Image taken from Oxford Cryosystems

Left: Detail of a mounted crystal using a loop filled with an antifreeze matrix
Right: Checking the position of the crystal in the goniometric optical center. Video courtesy of  Ed Berry

Cryo-protection system mounted on a goniometer

The nitrogen flow at -170 º C (coming through the upper tube) cools the crystal mounted on the goniometer head. The collimator of the X-ray beam points toward the crystal from the left of the image. Note the slight steam generated by the cold nitrogen when mixed with air humidity.
 

Visually analyzing the quality of the diffraction pattern

In summary, all of these methodologies can be used to obtain a data collection, consisting of three Miller indices and an intensity for each diffracted beam, which is to say, the largest number of reciprocal points of the reciprocal lattice.

This implies evaluating both the geometry and the intensities of the whole diffraction pattern.