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This is a pretty basic question but I would just like some confirmation of what I suspect is true.

As I understand it, the basic idea behind x-ray crystallography is that we take our crystal and hit it with an incident x-ray beam. Then we rotate the crystal, modifying $\theta$. From Bragg's Law, $$n\lambda = 2d_{hkl}sin\theta$$ we can then calculate $d$ at $\theta$ values where there is a diffracted beam. Ultimately we end up with a list of $d$ values.

With $d_{hkl}$ values, from equations such as $$\frac{1}{d_{hkl}^{2}}=\frac{(h^{2}+k^{2}+l^{2})}{a^{2}}$$ which applies to cubic crystal systems, we can find the lattice parameters.

However, we do not know what crystal system the sample is. That information can be found from the intensities of the diffracted beams.

Is this idea correct? I'm trying to read through Richard Tilley's Crystals and Crystal Structures text but the explanation is a bit confusing to me.

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Roughly, first of all you have to make indexing of the diffraction peaaks in the pattern. So if you are sure that your sample is a single phase (no secondary phases) you can apply the second equation that you posted. If all the peaks can be indexed with that equation, you have a cubic structure; if not, you have to consider a system wil lower symmetry (tetragonal, othorhombic,...). Now indexing is carried out using specific softwares (Dicvol, Treor,...).

After that you identified the crystal system, you have to identify the space group and in this case the intensities play a major role. However, intensities strongly depends also on the atomic species constituting the crystal structure and on their concentration, that is on the distribution of the atomic species in the different Wyckoff sites of the selected space group.

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The asymmetric unit is the smallest part of a crystal structure from which the complete structure can be built by applying the space group symmetry. The asymmetric unit has no crystallographic symmetry.

The intensities of the diffracted beams are determined by the contents of the asymmetric unit. For example changing the contents of the asymmetric unit while leaving it's size and shape the same will change the intensity of the diffracted beams but not their positions.

The positions of the diffracted beams are determined by the space group symmetry. These positions are used to determine the space group symmetry, but give no information about the contents of the asymmetric unit.

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From the Wikipedia article on X-ray crystallography:

X-ray cyrstallography is a tool used for identifying the atomic and molecular structure of a crystal, in which the crystalline atoms cause a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their disorder and various other information.

From the section "X-ray diffraction":

Crystals are regular arrays of atoms, and X-rays can be considered waves of electromagnetic radiation. Atoms scatter X-ray waves, primarily through the atoms' electrons. Just as an ocean wave striking a lighthouse produces secondary circular waves emanating from the lighthouse, so an X-ray striking an electron produces secondary spherical waves emanating from the electron.

This phenomenon is known as elastic scattering, and the electron (or lighthouse) is known as the scatterer. A regular array of scatterers produces a regular array of spherical waves. Although these waves cancel one another out in most directions through destructive interference, they add constructively in a few specific directions, determined by Bragg's law: $$2d \sin \theta = n \lambda$$

Here $d$ is the spacing between diffracting planes, $\theta$ is the incident angle, $n$ is any integer, and $\lambda$ is the wavelength of the beam. These specific directions appear as spots on the diffraction pattern called reflections. Thus, X-ray diffraction results from an electromagnetic wave (the X-ray) impinging on a regular array of scatterers (the repeating arrangement of atoms within the crystal).

About the different investigations in the crystalline arrangement of atoms some details are available below.

From the section "Crystal symmetry, unit cell, and image scaling":

The recorded series of two-dimensional diffraction patterns, each corresponding to a different crystal orientation, is converted into a three-dimensional model of the electron density; the conversion uses the mathematical technique of Fourier transforms, which is explained below. Each spot corresponds to a different type of variation in the electron density; the crystallographer must determine which variation corresponds to which spot (indexing), the relative strengths of the spots in different images (merging and scaling) and how the variations should be combined to yield the total electron density (phasing).

Data processing begins with indexing the reflections. This means identifying the dimensions of the unit cell and which image peak corresponds to which position in reciprocal space. A byproduct of indexing is to determine the symmetry of the crystal, i.e., its space group. Some space groups can be eliminated from the beginning. For example, reflection symmetries cannot be observed in chiral molecules; thus, only 65 space groups of 230 possible are allowed for protein molecules which are almost always chiral.

Indexing is generally accomplished using an autoindexing routine. Having assigned symmetry, the data is then integrated. This converts the hundreds of images containing the thousands of reflections into a single file, consisting of (at the very least) records of the Miller index of each reflection, and an intensity for each reflection (at this state the file often also includes error estimates and measures of partiality (what part of a given reflection was recorded on that image)).

A full data set may consist of hundreds of separate images taken at different orientations of the crystal. The first step is to merge and scale these various images, that is, to identify which peaks appear in two or more images (merging) and to scale the relative images so that they have a consistent intensity scale. Optimizing the intensity scale is critical because the relative intensity of the peaks is the key information from which the structure is determined.

The repetitive technique of crystallographic data collection and the often high symmetry of crystalline materials cause the diffractometer to record many symmetry-equivalent reflections multiple times. This allows calculating the symmetry-related R-factor, a reliability index based upon how similar are the measured intensities of symmetry-equivalent reflections,[clarification needed] thus assessing the quality of the data.

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    $\begingroup$ Well yes, I understand the concept of x-ray crystallography... the question was a bit subtler than that. It was about what information about the crystal is contained in what parameters. $\endgroup$
    – spinodal
    Commented Feb 25, 2016 at 4:40

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