Diffraction Is Special Type Of

The first neutron diffraction experiment was in 1945 by Ernest O. Wollan (Figure (PageIndex{1})) using the Graphite Reactor at Oak Ridge. Along with Clifford Shull (Figure (PageIndex{1})) they outlined the principles of the technique. However, the concept that neutrons would diffract like X-rays was first proposed by Dana Mitchell and Philip Powers. They proposed that neutrons have a wave like structure, which is explained by the de Broglie equation, ref{1}, where λ is the wavelength of the source usually measured in Å, h is Planck’s constant, v is the velocity of the neutron, and finally m represents the mass of the neutron.

The bending of light waves around the corners of an obstacle and spreading of light waves into geometrical shadow is called diffraction. Fraunhofer Diffraction and Fresnel Diffraction are two Types of Diffraction of Light. Bending of Light around the corners of Window is an example of Diffraction.

1. Diffraction is the phenomenon of spreading of the light wave when it passes through a slit or any small gap. But it is noteworthy here that the wavelength of the wave must be comparable with the dimensions of the slit. As in the case of the large opening, the light wave will not bend at the edges.
2. As we explained in a previous atom, diffraction is defined as the bending of a wave around the edges of an opening or obstacle. Diffraction is a phenomenon all wave types can experience. It is explained by the Huygens-Fresnel Principle, and the principal of superposition of waves.

[ lambda = h/mv label{1} ]

The great majority of materials that are studied by diffraction methods are composed of crystals. X-rays where the first type of source tested with crystals in order to determine their structural characteristics. Crystals are said to be perfect structures although some of them show defects on their structure. Crystals are composed of atoms, ions or molecules, which are arranged, in a uniform repeating pattern. The basic concept to understand about crystals is that they are composed of an array of points, which are called lattice points, and the motif, which represents the body part of the crystal. Crystals are composed of a series of unit cells. A unit cell is the repeating portion of the crystal. Usually there are another eight unit cells surrounding each unit cell. Unit cells can be categorized as primitive, which have only one lattice point. This means that the unit cell will only have lattice points on the corners of the cell. This point is going to be shared with eight other unit cells. Whereas in a non primitive cell there will also be point in the corners of the cell but in addition there will be lattice points in the faces or the interior of the cell, which similarly will be shared by other cells. The only primitive cell known is the simple crystal system and for nonprimitive cells there are known face-centered cubic, base centered cubic and body centered cubic.

Crystals can be categorized depending on the arrangement of lattice points; this will generate different types of shapes. There are known seven crystal systems, which are cubic, tetragonal, orthorhombic, rhombohedral, hexagonal, monoclinic and triclinic. All of these have different angles and the axes are equally the same or different in others. Each of these type of systems have different bravais lattice.

Braggs Law

Braggs Law was first derived by physicist Sir W.H. Bragg (Figure (PageIndex{2})) and his son W. L Bragg (Figure (PageIndex{3})) in 1913.

It has been used to determine the spacing of planes and angles formed between these planes and the incident beam that had been applied to the crystal examined. Intense scattered X-rays are produced when X-rays with a set wavelength are executed to a crystal. These scattered X-rays will interfere constructively due the equality in the differences between the travel path and the integral number of the wavelength. Since crystals have repeating units patterns, diffraction can be seen in terms of reflection from the planes of the crystals. The incident beam, the diffracted beam and normal plane to diffraction need to lie in the same geometric plane. The angle, which the incident beam forms when it hits the plane of the crystal, is called 2θ. Figure (PageIndex{4}) shows a schematic representation of how the incident beam hits the plane of the crystal and is reflected at the same angle 2θ, which the incident beam hits. Bragg’s Law is mathematically expressed, ref{2}, where,n= integer order of reflection, λ= wavelength, d= plane spacing.

[ nlambda = 2d sin theta label{2} ]

Bragg’s Law is essential in determining the structure of an unknown crystal. Usually the wavelength is known and the angle of the incident beam can be measured. Having these two known values, the plane spacing of the layer of atoms or ions can be obtained. All reflections collected can be used to determine the structure of the unknown crystal material.

Bragg’s Law applies similarly to neutron diffraction. The same relationship is used the only difference being is that instead of using X-rays as the source, neutrons that are ejected and hit the crystal are being examined.

Neutron Diffraction

Neutrons have been studied for the determination of crystalline structures. The study of materials by neutron radiation has many advantages against the normally used such as X-rays and electrons. Neutrons are scattered by the nucleus of the atoms rather than X-rays, which are scattered by the electrons of the atoms. These generates several differences between them such as that scattering of X-rays highly depend on the atomic number of the atoms whereas neutrons depend on the properties of the nucleus. These lead to a greater and accurately identification of the unknown sample examined if neutron source is being used. The nucleus of every atom and even from isotopes of the same element is completely different. They all have different characteristics, which make neutron diffraction a great technique for identification of materials, which have similar elemental composition. In contrast, X-rays will not give an exact solution if similar characteristics are known between materials. Since the diffraction will be similar for adjacent atoms further analysis needs to be done in order to determine the structure of the unknown. Also, if the sample contains light elements such as hydrogen, it is almost impossible to determine the exact location of each of them just by X-ray diffraction or any other technique. Neutron diffraction can tell the number of light elements and the exact position of them present in the structure.

Neutron Inventors

Neutrons were first discovered by James Chadwick in 1932 Figure (PageIndex{5}) when he showed that there were uncharged particles in the radiation he was using. These particles had a similar mass of the protons but did not have the same characteristics as them. Chadwick followed some of the predictions of Rutherford who first worked in this unknown field. Later, Elsasser designed the first neutron diffraction in 1936 and the ones responsible for the actual constructing were Halban and Preiswerk. This was first constructed for powders but later Mitchell and Powers developed and demonstrated the single crystal system. All experiments realized in early years were developed using radium and beryllium sources. The neutron flux from these was not sufficient for the characterization of materials. Then, years passed and neutron reactors had to be constructed in order to increase the flux of neutrons to be able to realize a complete characterization the material being examined.

Between mid and late 40s neutron sources began to appear in countries such as Canada, UK and some other of Europe. Later in 1951 Shull and Wollan presented a paper that discussed the scattering lengths of 60 elements and isotopes, which generated a broad opening of neutron diffraction for the structural information that can be obtained from neutron diffraction.

Neutron Sources

The first source of neutrons for early experiments was gathered from radium and beryllium sources. The problem with this, as already mentioned, was that the flux was not enough to perform huge experiments such as the determination of the structure of an unknown material. Nuclear reactors started to emerge in early 50s and these had a great impact in the scientific field. In the 1960s neutron reactors were constructed depending on the desired flux required for the production of neutron beams. In USA the first one constructed was the High Flux Beam Reactor (HFBR). Later, this was followed by one at Oak Ridge Laboratory (HFIR) (Figure (PageIndex{6})), which also was intended for isotope production and a couple of years later the ILL was built. This last one is the most powerful so far and it was built by collaboration between Germany and France. These nuclear reactors greatly increased the flux and so far there has not been constructed any other better reactor. It has been discussed that probably the best solution to look for greater flux is to look for other approaches for the production of neutrons such as accelerator driven sources. These could greatly increase the flux of neutrons and in addition other possible experiments could be executed. The key point in these devices is spallation, which increases the number of neutrons executed from a single proton and the energy released is minimal. Currently, there are several of these around the world but investigations continue searching for the best approach of the ejection of neutrons.

Neutron Detectors

Although neutrons are great particles for determining complete structures of materials they have some disadvantages. These particles experiment a reasonably weak scattering when looking especially to soft materials. This is a huge concern because there can be problems associated with the scattering of the particles which can lead to a misunderstanding in the analysis of the structure of the material.

Neutrons are particles that have the ability to penetrate through the surface of the material being examined. This is primarily due to the nuclear interaction produced from the particles and the nucleus from the material. This interaction is much greater that the one performed from the electrons, which it is only an electrostatic interaction. Also, it cannot be omitted the interaction that occurs between the electrons and the magnetic moment of the neutrons. All of these interactions discussed are of great advantage for the determination of the structure since neutrons interacts with every single nucleus in the material. The problem comes when the material is being analyzed because neutrons being uncharged materials make them difficult to detect them. For this reason, neutrons need to be reacted in order to generate charged particles, ions. Some of the reactions uusually used for the detection of neutrons are:

[ n + ^{3}He rightarrow ^{3}H + ^{1}H + 0.764 MeV label{3} ]

[ n + ^{10}B rightarrow ^{7}Li + ^{4}He + gamma + 2.3 MeV label{4} ]

[ n + ^{6}Li rightarrow ^{4}He + ^{3}H + 4.79 MeV label{5} ]

The first two reactions apply when the detection is performed in a gas environment whereas the third one is carried out in a solid. In each of these reaction there is a large cross section, which makes them ideal for neutron capture. The neutron detection hugely depends on the velocity of the particles. As velocity increases, shorter wavelengths are produced and the less efficient the detection becomes. The particles that are executed to the material need to be as close as possible in order to have an accurate signal from the detector. These signal needs to be quickly transduced and the detector should be ready to take the next measurement.

In gas detectors the cylinder is filled up with either 3He or BF3. The electrons produced by the secondary ionization interact with the positively charged anode wire. One disadvantage of this detector is that it cannot be attained a desired thickness since it is very difficult to have a fixed thickness with a gas. In contrast, in scintillator detectors since detection is developed in a solid, any thickness can be obtained. The thinner the thickness of the solid the more efficient the results obtained become. Usually the absorber is 6Li and the substrate, which detects the products, is phosphor, which exhibits luminescence. This emission of light produced from the phosphor results from the excitation of this when the ions pass thorough the scintillator. Then the signal produced is collected and transduced to an electrical signal in order to tell that a neutron has been detected.

Neutron Scattering

One of the greatest features of neutron scattering is that neutrons are scattered by every single atomic nucleus in the material whereas in X-ray studies, these are scattered by the electron density. In addition, neutron can be scattered by the magnetic moment of the atoms. The intensity of the scattered neutrons will be due to the wavelength at which it is executed from the source. Figure (PageIndex{7}) shows how a neutron is scattered by the target when the incident beam hits it.

The incident beam encounters the target and the scattered wave produced from the collision is detected by a detector at a defined position given by the angles θ, ϕ which are joined by the dΩ. In this scenario there is assumed that there is no transferred energy between the nucleus of the atoms and the neutron ejected, leads to an elastic scattering.

When there is an interest in calculating the diffracted intensities the cross sectional area needs to be separated into scattering and absorption respectively. In relation to the energies of these there is moderately large range for constant scattering cross section. Also, there is a wide range cross sections close to the nuclear resonance. When the energies applied are less than the resonance the scattering length and scattering cross section are moved to the negative side depending on the structure being examined. This means that there is a shift on the scattering, therefore the scattering will not be in a 180° phase. When the energies are higher that resonance it means that the cross section will be asymptotic to the nucleus area. This will be expected for spherical structures. There is also resonance scattering when there are different isotopes because each produce different nuclear energy levels.

Coherent and Incoherent Scattering

Usually in every material, atoms will be arranged differently. Therefore, neutrons when scattered will be either coherently or incoherently. It is convenient to determine the differential scattering cross section, which is given by ref{6}, where b represents the mean scattering length of the atoms, k is the scattering vector, r nis the position of the vector of the analyzed atom and lastly N is the total number of atoms in the structure.This equation can be separated in two parts, which one corresponds to the coherent scattering and the incoherent scattering as labeled below. Usually the particles scattered will be coherent which facilitates the solution of the cross section but when there is a difference in the mean scattering length, there will be a complete arrangement of the formula and these new changes (incoherent scattering) should be considered. Incoherent scattering is usually due to the isotopes and nuclear spins of the atoms in the structure.

[ dsigma /dOmega = b ^{2} Sigma e^{(ik.r_{n})} ^{2} + N b-b^2 label{6} ]

Coherent Exp: [ b ^{2} Sigma e^{(ik.r_{n})} ^{2} ]

Incoherent Exp: [ N b-b ^{2} ]

The ability to distinguish atoms with similar atomic number or isotopes is proportional to the square of their corresponding scattering lengths. There are already known several coherent scattering lengths of some atoms which are very similar to each other. Therefore, it makes even easier to identify by neutrons the structure of a sample. Also neutrons can find ions of light elements because they can locate very low atomic number elements such as hydrogen. Due to the negative scattering that hydrogen develops it increases the contrast leading to a better identification of it, although it has a very large incoherent scattering which causes electrons to be removed from the incident beam applied.

Magnetic Scattering

As previously mentioned one of the greatest features about neutron diffraction is that neutrons because of their magnetic moment can interact with either the orbital or the spin magnetic moment of the material examined. Not all every single element in the periodic table can exhibit a magnetic moment. The only elements that show a magnetic moment are those, which have unpaired electrons spins. When neutrons hit the solid this produces a scattering from the magnetic moment vector as well as the scattering vector from the neutron itself. Below Figure (PageIndex{8}) shows the different vectors produced when the incident beam hits the solid.

When looking at magnetic scattering it needs to be considered the coherent magnetic diffraction peaks where the magnetic contribution to the differential cross section is p2q2 for an unpolarized incident beam. Therefore the magnetic structure amplitude will be given by ref{9}, where qn is the magnetic interaction vector, pn is the magnetic scattering length and the rest of the terms are used to know the position of the atoms in the unit cell. When this term Fmag is squared, the result is the intensity of magnetic contribution from the peak analyzed. This equation only applies to those elements which have atoms that develop a magnetic moment.

[ F_{text{mag}} = Sigma p_{n}q_{n} e^

label{9} ]

Magnetic diffraction becomes very important due to its d-spacing dependence. Due to the greater effect produced from the electrons in magnetic scattering the forward scattering has a greater strength than the backward scattering. There can also be developed similar as in X-ray, interference between the atoms which makes structure factor also be considered. These interference effects could be produced by the wide range in difference between the electron distribution and the wavelength of the thermal neutrons. This factor quickly decreases as compared to X-rays because the beam only interacts with the outer electrons of the atoms.

Sample Preparation and Environment

In neutron diffraction there is not a unique protocol of factors that should be considered such as temperature, electric field and pressure to name a few. Depending on the type of material and data that has been looked the parameters are assigned. There can be reached very high temperatures such as 1800K or it can go as low as 4K. Usually to get to these extreme temperatures a special furnace capable of reaching these temperatures needs to be used. For example, one of the most common used is the He refrigerator when working with very low temperatures. For high temperatures, there are used furnaces with a heating element cylinder such as vanadium (V), niobium (Nb), tantalum (Ta) or tungsten (W) that is attached to copper bars which hold the sample. Figure (PageIndex{9}) shows the design for the vacuum furnaces used for the analysis. The metal that works best at the desired temperature range will be the one chosen as the heating element. The metal that is commonly used is vanadium because it prevents the contribution of other factors such as coherent scattering. Although with this metal this type of scattering is almost completely reduced. Other important factor about this furnaces is that the material been examined should not decompose under vacuum conditions. The crystal needs to be as stable as possible when it is being analyzed. When samples are not able to persist at a vacuum environment, they are heated in the presence of several gases such as nitrogen or argon.

Usually in order to prepare the samples that are being examined in neutron diffraction it is needed large crystals rather small ones as the one needed for X-ray studies. This one of the main disadvantages of this instrument. Most experiments are carried out using a four-circle diffractometer. The main reason being is because several experiment are carried out using very low temperatures and in order to achieve a good spectra it is needed the He refrigerator. First, the crystal being analyzed is mounted on a quartz slide, which needs to be a couple millimeters in size. Then, it is inserted into the sample holder, which is chosen depending on the temperatures wanted to be reached. In addition, neutrons can also analyze powder samples and in order to prepare the sample for these they need to be completely rendered into very fine powders and then inserted into the quartz slide similarly to the crystal structures. The main concern with this method is that when samples are grounded into powders the structure of the sample being examined can be altered.

Summary

Neutron diffraction is a great technique used for complete characterization of molecules involving light elements and also very useful for the ones that have different isotopes in the structure. Due to the fact that neutrons interact with the nucleus of the atoms rather than with the outer electrons of the atoms such as X-rays, it leads to a more reliable data. In addition, due to the magnetic properties of the neutrons there can be characterized magnetic compounds due to the magnetic moment that neutrons develop. There are several disadvantages as well, one of the most critical is that there needs to be a good amount of sample in order to be analyzed by this technique. Also, great amounts of energy are needed to produce large amounts of neutrons. There are several powerful neutron sources that have been developed in order to conduct studies of largest molecules and a smaller quantity of sample. However, there is still the need of devices which can produce a great amount of flux to analyze more sophisticated samples. Neutron diffraction has been widely studied due to the fact that it works together with X-rays studies for the characterization of crystalline samples. The properties and advantages of this technique can greatly increased if some of the disadvantages are solved. For example, the study of molecules which exhibit some type of molecular force can be characterized. This will be because neutrons can precisely locate hydrogen atoms in a sample. Neutrons have gives a better answer to the chemical interactions that are present in every single molecule, whereas X-rays help to give an idea of the macromolecular structure of the samples being examined.

6. Experimental diffraction
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Regardless of the huge improvements that have occurred for X-ray generation, the techniques used to measure the intensities and angles of diffraction patterns have evolved over time. In the first diffraction experiment, Friedrich and Knipping (1912) used a film sensitive to X-rays, but even in the same year, Bragg used a ionization chamber mounted on a rotating arm that, in general, could more accurately determine angles and intensities. However, the film technique had the advantage of being able to collect many diffracted beams at the same time, and thus during the first years of structural Crystallography (from 1920 to 1970) an extensive use of photographic methods was made. Among them the following techniques should be highlighted:
Laue, Weissenberg, precession and oscillation.
Since the mid-1970's, photographic methods have been gradually replaced by goniometers coupled with point detectors which subsequently have been replaced by area detectors.

The Laue method

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:

n λ = 2 dhkl sin θnh,nk,nl

Thus, for the same interplanar spacing d, the diffraction pattern will containthe 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.

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

Left: The Laue method in transmission mode
Right: The Laue method in reflection mode

The Weissenberg method

The Weissenberg method is based on a camera with the same name, developed in 1924 by the Austrian scientist Karl Weissenberg (1893-1976). In order to understand Weissenberg’s contribution to X-ray crystallography one should read the two following articles that some years ago were offered the British Society of Rheology: 'Weissenberg’s Influence on Crystallography' (by H. Lipson) (use this link in case of problems) and 'Karl Weissenberg and the development of X-ray crystallography' (by M.J. Buerger).
The camera consists of a metalic cylinder that contains a film sensitive to X-rays. The crystal is mounted on a shaft (coaxial with the cylinder) that rotates. According to Ewald's model, the reciprocal points will intersect the surface of Ewald's sphere and diffracted beams will be produced.
The diffracted beams generate black spots on the photographic film, which when removed from the metalic cylinder, appears as shown below.

Left: Scheme and example of a Weissenberg camera. This camera type was used in crystallographic laboratories until about 1975.
Right: Camera developed by K. Weissenberg in 1924

Two types of diffraction diagrams can be easily obtained with the Weissenberg cameras, depending on the amount of crystal rotation:

Diffraction Is Special Type Of Waves

oscillation diagrams (rotation of approx. +/-20 degrees) or full rotation diagrams (360 degrees) respectively. Oscillation diagrams are used to center the crystal, that is, to ensure that the rotation of axis coincides exactly with a direct axis, which is equivalent to saying that reciprocal planes (which by geometric construction are perpendicular to a direct axis ) generate lines of spots on the photographic film. Once centering is achieved, the full rotation diagrams are used to evaluate the direct axis of the crystal, which coincides with the spacing between the dot lines on the diagram.

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.

As shown in the diagram above, each horizontal line of points represents a reciprocal plane perpendicular to the axis of rotation as projected on the photographic plate. The figure on the left shows the real appearance of a Weissenberg diagram of this type, rotation-oscillation.
As explained below, the distance between the horizontal spot lines provides information on the crystal repetition period in the vertical direction of the film.
These diagrams were also used to align mounted crystals... This technique requires that the crystal rotation axis is coincident with an axis of its direct lattice, so that the reciprocal planes are collected as lines of spots as is shown on the left.
The crystal must be mounted in such a way that the rotation axis coincides with a direct axis of the unit cell. Thus, by definition of the reciprocal lattice, there will be reciprocal planes perpendicular to that axis. The reciprocal points (lying on these reciprocal planes) rotate when the crystal rotates and (after passing through the Ewald sphere) produce diffracted beams that arranged in cones, touch the cylindrical film and appear as aligned spots (photograph on the left).
It seems obvious that these diagrams immediately provide information about the repetition period of the direct lattice in the direction perpendicular to the horizontal lines (reciprocal planes). However, those reciprocal planes (two dimensional arrays of reciprocal points) are represented as projections (one dimension) on the film and therefore a strong spot overlapping is to be expected.

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)

The appearance of such a diagram, which produces a geometrical deformation of the collected reciprocal plane, is shown below. Taking into account this deformation, one can easily identify every spot of the selected reciprocal plane and measure its intensity. To select the remaining reciprocal planes one just has to shift the internal cylinders and collect their corresponding diffracted beams (arranged in cones).

Diffraction Is Special Type Of Energy

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.

The precession method

The precession method was developed by Martin J. Buerger (1903-1986) at the beginning of the 1940's as a very clever alternative to collect diffracted intensities without distorting the geometry of the reciprocal planes.

As in the Weissenberg technique, precession methodology is also based on a moving crystal, but here the crystal moves (and so does the coupled reciprocal lattice) as the planets do, and hence its name. In this case the film is placed on a planar casette that moves following the crystal movements.
In the precession method the crystal has to be oriented so that the reciprocal plane to be collected is perpendicular to the X-rays' direct beam, ie a direct axis coincides with the direction of the X-rays' incident.

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.

Left: Scheme and appearance of a precession camera
Right: Precession diagram of a perovskite showing cubic symmetry

Precession diagrams are much simpler to interpret than those of Weissenberg, as they show the reciprocal planes without any distortion. They show a single reciprocal plane on a photographic plate (picture above) when a circular slit is placed between the crystal and the photographic film. As in the case of Weissenberg diagrams, we can readily measure distances and diffraction intensities. However, with these diagrams it is much easier to observe the symmetry of the reciprocal space.
The only disadvantage of the precession
precession method is a consequence of the film, which is flat instead of cylindrical, and therefore the explored solid angle is smaller than in the Weissenberg case.

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

Left: Precession diagram of a lysozyme crystal. One can easily distinguish a four-fold symmetry axis perpendicular to the diagram. According to the relationships between direct and reciprocal lattices, if the axes of the unit cell are large (as in this case), the separation between reciprocal points is small.
Right: Precession diagram of a simple organic compound, showing mm symmetry (two mirror planes perpendicular to the diagram). Note that the distances between reciprocal points is much larger (smaller direct unit cell axes) than in the case of proteins (see the figure on the left).

The oscillation method

Originally, the methods of rotating crystal with a wide rotation angle were very successfully used. However, when it was applied to crystals with larger direct cells (ie small reciprocal cellls), the collecting time increased. Therefore, these methods were replaced by methods using small oscillation angles, allowing multiple parts of differents 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 θmax).

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.

Four-circle goniometers

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.

Left: Scheme and appearance of a four-circle goniometer with Eulerian geometry
Right: Rotations in a four-circle goniometer with Eulerian geometry

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 ωκ (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 Φand 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:

Area detectors

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 fromADSC 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.

Left: Goniometer with Kappa geometry and CCD detector (Image taken from Bruker-AXS)
Right: Details of a Kappa goniometer (in this case with a fixed κ angle)

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.

Left: Image Plate Scanner. (image taken from Marxperts)

The latest technology involves the use of area detectors based on CMOS (complementary metal-oxide semiconductor) technology that has very short readout time, allowing for increased frame rates during the data collection.

Area detectors

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

Diffraction Is A Special Type Of Polarization

In summary, a complete data collection with this type of detectors consists of multiple images such as the ones shown below. The collected images are subsequently analyzed in order to obtain the crystal unit cell data, symmetry (space group) and intensities of the diffraction pattern (reciprocal space). This process is explained in more detail in another section.

Left: Diffraction image of a protein, obtained with the oscilaltion 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 difraction orders of randomly oriented ice microcristals 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) comming 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.)
It can also be very instructive to look at this animated diagram showing the path of each X-ray photon in a given diffraction system:
• 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
The original video can be seen in https://vimeo.com/52155723

In order to get the largest and best collection of diffraction data, crystal samples are usually maintaned at a very low temperature (about 100 K, that is, about -170 C) using a dry nitrogen stream. At low temperatures, crystals (and especially those of macromolecules) are more stable and resist the effects of X-ray radiation much better. At the same time, the low temperature further reduces the atomic thermal vibration factors, facilitating their subsequent location within the crystal structure.

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

To mount the crystals on the goniometer head, in front of the cold nitrogen stream, crystallographers use special loops (like the one depicted in the left figure) which fix the crystal in a matrix transparent to X-rays.
This is especially useful for protein crystals, where the matrix also acts as cryo-protectant (anti-freeze). The molecules of the cryo-protectant spread through the crystal channels replacing the water molecules with the cryo-protectant ones, thus avoiding crystal rupture due to frozen water.

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

In any case, the crystal center must be coincident with the optical center of the goniometer, where the X-ray beam is also passing through. In this way, when the crystal rotates, it will always be centered on that point, and in any of its positions will be bathed by the X-ray beam.

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.

All these data, crystal unit cell dimensions, crystal symmetry (space group) and intensities associated with the reciprocal points (diffraction pattern), will allow us to 'see' the internal structure of the crystal, but this issue will be shown in another chapter...

Next chapter: Structural resolution