X-ray Crystal Structure Analysis

What is necessary for crystal structure analysis

The diffraction intensities of many Bragg reflections are used to analyze the crystal structure. The number of data points and the accuracy of the data are important for the crystal structure analysis. The information about the chemical composition, density, space group or point group is also helpful.


Powder diffraction and single-crystal diffraction have pros and cons as follows.

X-ray Sources

X-ray Tube

X-rays are emitted from a vacuum tube equipped with a cathode and an anode. The cathode emits electrons into the vacuum and the anode collects the electrons. If the accelerate voltage of electrons exceeds tens of kilovolts, x-ray is emitted from the anode. The typical voltage is between 30 and 50 keV, and the current is between 20 and 40 mA.
To obtain a higher power x-ray, the target position is changed. The current can be increased to about 200 meV by using a rotating anode tube.


An electromagnetic wave is radiated forward when electrons with a speed close to the speed of light are bent.

X-ray Wavelength

A shorter wavelength x-ray has an advantage for measuring more Bragg points. Also, the need for absorption correction is reduced. On the other hand, when the wavelength is too short, the diffraction intensity becomes weak. Some reflections start to be superimposed with each other.
If the x-ray wavelength is in the vicinity of the absorption edge of the element contained in the sample, the atomic form factor deviates from the Thomson scattering. The component is dependent much on the wavelength but less on the scattering angle.

X-ray Detectors

The ionization chamber is often used for the measurement of the incident intensity, because it transmits most of the x rays. The chamber is filled with a specific type of gas and a high-voltage electric field is applied. The gas is partly ionized by x-ray. The amount of the ions are measured as electric current.
There are several types of x-ray detectors; scintillators, imaging plates, and semiconductors.

Energy resolution

The photon energy can be resolved by counting-type detectors, because the signal level (amount of luminscence or current) depends on the photon energy of x-rays. For a precise energy resolution, Bragg reflection of a crystal is used. The meV energy resolution is possible by using the back scattering of a silicon crystal in a conrolled atmosphere.

Polarization Device

Since the refractive index for x-ray of any substance is close to one, the control and selection of x-ray polarization is not easy. A synchrotron is a good source of polarized x-ray. Bragg reflection with 2θ of nearly 90 degrees is widely used for the polarization selection. X-ray kinetic scattering of a perfect crystal is used for a phase plate.

Analysis of Reflection Intensities

The x-ray reflection intensities are determined by many factors like atomic arrangement, electron density distribution, x-ray photon energy and polarization, measurement configuration, sample condition, and so on.

Thomson Scattering

Resonant Scattering

Temperature Factor

Polarization Factor

Thomson scattering is considered as dipole radiation by electrons oscillating in the electric field of incident x-rays. The scattering amplitude is hence proportional to the scalar product of the unit polarization vectors of the incident and scattered x-rays.

The polarization of x-rays generated from the x-ray tube is completely random. The polarization factor for scattering intensity is
When an energy analyzer is used instead of a filter to remove Kβ, another polarization factor is added as
Here, 2θA is the Bragg scattering angle of the analyzer crystal.

The synchrotron radiation is usually polarized parallel to the floor. Therefore, the polarization factor depends on the scattering direction.

Lorentz Factor

In the single crystal measurement, the sample is rotated about an axis perpendicular to the incident beam. Although the rotation speed is constant, the sweep rate in the reciprocal lattice space is not constant. In the case of 2θ-θ sweep or θ sweep, the correction factor for the intensity is 1/sinθcosθ.

In the case of the powder diffraction, the probability that the specific reciprocal lattice vector of a microcrystal forms an angle of π/2−θ with the incident X-ray is proportional to cos θ. The probability that the scattered x-ray irradiates the detector is proportional to 1/sin2θ. Therefore, the correction factor is 1/sin2θcosθ.

The abovementioned factor is called Lorentz factor.

Preferred Orientation

In the case of needle-like or plate-like crystals, the microcrystal grains are usually not randomly oriented. Intensity data should be corrected by considering the preferred orientation.

Taka-hisa Arima