Single crystal X-ray diffraction (SCXRD) is a non-destructive analysis method fundamental to crystallography. It was the first analysis method based on X-ray diffraction (XRD) developed^[1] and one of the most widely used XRD techniques along with powder XRD.

In SCXRD, a beam of X-rays is directed onto a sample consisting of a single crystal, and the intensity of diffracted light is measured. A two-dimensional diffraction pattern is obtained from the measurement with intensity maxima visible as spots. An example of a single crystal diffraction pattern is presented in Fig. 1. When characterising samples with SCXRD, patterns measured from different angular positions are used to establish an electron density map of the lattice. This provides direct atomic level information of the sample structure^[2][3]. By contrast, the diffraction pattern obtained from powder XRD is generally condensed to a one-dimensional diffractogram, and the sample is either analysed by comparing the intensity peaks to existing reference materials (phase identification) or the crystal structure is solved by using Rietveld refinement.

SCXRD has unique significance in structural characterization and it is the primary characterization method for solving crystal structures. The sample scope is wide, and both organic and inorganic crystalline materials can be analysed^[3]. A limitation of SCXRD is that the requirements for the sample quality are strict, and there may be difficulties obtaining good quality single crystals for the analysis^[3]. Additionally, the data collection is relatively slow and generally requires between 24 to 72 hours^[4].

Figure 1. The diffraction pattern of icosahedrite. (Figure: Paul Steinhardt via Wikimedia Commons, CC BY-SA 4.0.)

Principle

Diffraction of X-rays on single crystals

When a beam of X-rays passes through matter, the radiation interacts with the atoms in the crystal lattice, which results in the scattering of incident X-rays^[2]. The ordered environment of the crystal lattice causes constructive and destructive interference among the scattered radiation. This is due to the distances between the scattering centres being of a similar order of magnitude to the wavelength of X-rays, 0.1-10 Å.

The exact conditions for constructive interference from a crystal lattice are described by the Bragg’s law

where \( n \)  is an integer, \( \lambda \)  is the radiation wavelength, \( d \)  is the distance between the lattice planes, and \( \theta \)  is the incident angle of the X-rays^[3]. Essentially, the total path difference \( 2d \sin \theta \)  must be a multiple of the wavelength for constructive interference to take place. At other angles, destructive interference occurs. As only certain angles satisfy the Bragg’s law, the crystal must be rotated during SCXRD measurements to angular positions, where the diffraction conditions are met^[3]. In powder XRD, varying the incident angle \( \theta \)  is enough, as the sample contains millions of randomly oriented crystals, and the diffraction conditions are practically always fulfilled.

Electron density and diffraction measurements

Diffraction patterns of single crystal diffraction measurements show the reciprocal lattice of the material^[1]. Lattice planes are represented by the Miller indices h, k, and l, which define a lattice plane in reciprocal space. Like the intensity peaks in a powder XRD diffractogram, each spot in the measured diffraction pattern originates from a lattice plane. A simulated diffraction pattern of Ti₃Al showing the Miller indices of each diffraction spot is provided in Fig. 2.

A comparison between the lattice and the electron density map of sodium chloride is shown in Fig. 3. The aim of SCXRD measurements is to obtain the electron density map, a depiction of the direct lattice, through the diffraction pattern, a depiction of the reciprocal lattice. Since the reciprocal lattice is the Fourier transform of the direct lattice, the electron density map can be accessed by an inverse Fourier transform on the diffraction pattern^[5].

In practice, a fundamental obstacle in constructing the electron density map is the loss of the radiation phase during measurement. The phase of the diffracted radiation carries structural information crucial to producing an accurate electron density map^[6]. However, it is not recorded in the diffraction measurements. Various methods for phase recovery are currently employed^[6].

Figure 2. A simulated diffraction pattern of Ti₃Al (hexagonal, space group P63/mmc). (Figure: Contributor2016 via Wikimedia Commons, CC BY-SA 4.0.)

Figure 3. The lattice (A) and the electron density map (B) of NaCl. The contours in the electron density map show regions of equal electron density. (Figure: Anonymous Publisher via Principles of General Chemistry, vol. 1, CC BY-NC-SA 3.0. Modified by Ellen Järvinen.)

Method

Instrumentation

A close-up image of a typical single crystal diffractometer is provided in Fig. 4. The general instrumentation consists of a detector (A), a goniometer (B), and an X-ray source (C). In addition to the general components, the instrument in the image has a cooling device installed (D), which allows for below-room-temperature measurements.

The sample is placed onto the mounting pin at the head of the goniometer^[3]. During the measurement, the goniometer rotates the sample to angular positions, where the Bragg's law is fulfilled. In single crystal measurements, X-ray tubes with a copper or a molybdenum target are usually used as radiation sources^[1]. The X-ray beam is additionally monochromated or filtered and collimated before directing it onto the sample^[1][7]. Area detectors, such as charge-coupled devices (CCDs) and imaging plates, are common in modern SCXRD instruments^[1][7].

Figure 4. A close-up image of a SCXRD instrument with the detector (A), the goniometer (B), the X-ray source (C), and the cooling device (D) labelled. (Figure: Leiem via Wikimedia Commons, CC BY-SA 4.0. Modified by Ellen Järvinen.)

Analysis procedure

SCXRD analysis requires a high-quality single crystal sample^[3]. The crystals may be produced, for instance, by precipitation from solution or by sublimation. Prior to analysis, the samples should be checked for cracks and for twinning, where multiple crystals have grown fused together. The diameter of the crystal should be within the diameter of the X-ray beam, and larger crystals may be cut to a more suitable size^[1][3].

Regarding the experimental conditions, SCXRD analyses at room temperature are possible. However, a lower temperature improves the quality of analysis results, as a decrease in temperature also decreases the thermal motion of the atoms, and the atomic positions can be defined more accurately^[1]. Furthermore, the intensities of higher order reflections are weakened less in lower temperatures^[1]. To give an example, a commercial cooling device based on low-temperature nitrogen gas enables measurements at around 80 K at the lowest^[8].

Prior to the full data collection run, a few preliminary frames may be recorded and examined^[5]. This is done to ensure good crystal quality and favourable diffraction properties. The data collection can then be carried out.

Data processing

Before the structure of the material can be determined, the data must be refined. The intensities obtained in the measurement are extracted from the raw data^[3]. The data set is corrected for X-ray absorption, detector and instrument characteristics, and polarization, and redundant measurement data is merged. The measurement angle information is additionally reduced to a set of Miller indices.

Once the data is refined, the electron density map can be constructed. As mentioned, this requires employing a phase recovery method. So called direct methods, for instance, are widely used in current SCXRD software^[1]. Direct methods establish a statistical relationship between the phases of small sets of strong reflections that have related Miller indices. The statistical relationship results from two properties of electron density \( ρ \) ^[9]:

• \( ρ \) is real and non-negative.

• \( ρ \)  is large within near-spherical regions located throughout the unit cell and close to zero elsewhere.

After resolving the phases, the electron density can be calculated at each point in the three-dimensional structure^[3].

Finally, the structure of the sample can be solved from the electron density map by assigning elements to density maxima. After the initial solving stage, the structure is further refined by adjusting the atomic coordinates and vibrational parameters. A least-square method is often used for the refinement^[3].

Further reading

More comprehensive mathematical background of constructing electron density maps is provided in the following works

• Girolami, G. S. – X-Ray Crystallography^[9]
• Blake, A. J., et al. – Crystal Structure Analysis – Principles and Practice^[5].

Further information on phase retrieval methods can be found in the following works

• Taylor, G. – The Phase Problem^[6]
• Girolami, G. S. – X-Ray Crystallography^[9].

References