The Basic Principles and Applications of XRD

The Testing Principle of XRD

What are X-rays?

In 1895, the German physicist W.K. Roentgen first discovered the existence of X-rays, hence they are also called Roentgen rays. X-rays are essentially electromagnetic waves with extremely short wavelengths (approximately 10^8 to 10^12 meters) and high energy, exhibiting wave-particle duality. In modern scientific research, X-rays with wavelengths less than 0.1 nm are referred to as hard X-rays, mainly used for material inspection; while X-rays with wavelengths between 0.25 to 0.05 nm are called soft X-rays, typically utilized for crystal structure analysis.

Figure 1: The first X-ray photograph

The Diffraction Behavior of X-rays in Crystals

Due to their high energy, when X-rays penetrate into a crystal, the atoms within the crystal are forced to undergo periodic motion under the influence of the X-rays. As a result, they emit secondary waves on a per-atom basis, with frequencies identical to those of the incident X-rays. This process is known as X-ray scattering (Figure 2). Considering the periodic arrangement of atoms in the crystal lattice, there exists a fixed phase relationship between these scattered spherical waves, leading to interference in space. This interference results in reinforcement of spherical waves in certain scattering directions and cancellation in others, giving rise to the phenomenon of diffraction. Consequently, X-ray diffraction in crystals is essentially the result of interference among a large number of scattered waves in space.

Figure 2: The Interaction between X-rays and Atoms in Crystals

The Association between X-ray Diffraction and Material Structure

For amorphous materials, the lack of long-range ordered atomic arrangements results in only short-range order within a few atomic ranges, leading to diffuse scattering patterns in XRD spectra. However, for crystalline materials, where atoms are arranged in three-dimensional space with long-range order, XRD diffraction patterns exhibit enhanced peaks only at specific positions (resulting from X-ray diffraction enhancement). The diffraction patterns generated by crystals reflect the distribution of atoms within the crystal.

In summary, the characteristics of a diffraction pattern can be considered to consist of two aspects:

• the spatial distribution of diffraction lines, determined by the size, shape, and orientation of the unit cell;
• the intensity of diffraction lines, dependent on the types of atoms and their positions within the unit cell. These diffraction patterns act as fingerprints of the crystal, and by identifying the positions and intensities of diffraction patterns in space, qualitative and quantitative relationships can be established between X-ray diffraction and crystal structure.

(1) The Bragg equation is the most important fundamental formula in diffraction analysis and forms the cornerstone of XRD theory. It succinctly elucidates the fundamental essence of diffraction and reveals the intrinsic relationship between diffraction and crystal structure. As shown in Figure 3, when X-rays irradiate a crystal, the optical path difference between adjacent crystal planes is 2dsinθ. If the optical path difference is equal to an integer multiple (n) of the X-ray wavelength, the diffraction intensity of X-rays will be enhanced, while it remains unchanged or decreases elsewhere.

nλ=2dsinθ, (n=1, 2, 3…..)

Here, λ, d, θ respectively represent the wavelength of the X-ray, the interplanar spacing of the crystal lattice, and the angle between the incident X-ray and the corresponding crystal plane.

Figure 3: The reflection of X-rays on multiple atomic planes within a crystal

Clearly, through the Bragg equation, one can use known X-ray wavelengths to determine the interplanar spacing d of a crystal, thereby obtaining information about the crystal structure, which is known as structural analysis. Alternatively, known crystal interplanar spacings can be used to measure the wavelength of unknown X-rays, which is known as X-ray spectroscopy.

(2) The Scherrer formula (Scherrer equation) serves as the theoretical basis for measuring crystallite size in XRD. It primarily describes the relationship between crystallite size and the width of diffraction peaks. Smaller crystallites result in broader diffraction peaks, whereas larger ones produce narrower peaks.

D = Kλ / (B cosθ)

Here, D, K, λ, B, and θ represent the average thickness of crystallites perpendicular to the plane of the crystal lattice, Scherrer constant, X-ray wavelength, measured sample diffraction peak width at half maximum (in radians), and diffraction angle, respectively. The value of the Scherrer constant (K) is generally determined by B: when B is the full width at half maximum of the diffraction peak, K = 0.89; when B is the integral of the area of the diffraction peak at half maximum, K = 1. Since crystallite sizes within the material are not entirely uniform, this method calculates the average size of crystallites of varying sizes.

XRD Diffractometer

Instrument Introduction

The Bragg experimental setup (Figure 4) serves as the prototype for modern X-ray diffractometers (Figure 5). As shown in Figure 4, the core components of an XRD diffractometer are the X-ray source generator and the X-ray detector. When incident X-rays irradiate the sample surface, an X-ray detector is positioned in the direction that satisfies the diffraction law, simultaneously recording the intensity and the diffraction angle θ (the angle between the incident beam and the reflecting surface). To ensure that the X-ray detection device remains in the position of the reflection line, the X-ray detection device and the sample stage must rotate synchronously at a 2:1 angular velocity ratio. Therefore, the crystal planes undergoing X-ray diffraction are always parallel to the sample surface.

It should be noted that since the light source generated by the X-ray generator contains X-rays with various wavelengths (Kα, Kβ, continuous spectrum), if all these wavelengths participate in diffraction, the resulting diffraction peaks will be chaotic. Additionally, when a single X-ray is irradiated onto the sample surface, it may also excite characteristic X-rays from the sample, affecting the test results. Therefore, modern X-ray diffractometers often incorporate monochromators or filters between the sample and the X-ray detector to ensure measurement accuracy and obtain high-quality diffraction patterns.

Figure 4: Bragg experimental setup schematic

Figure 5: XRD Powder Diffractometer

Precautions for Instrument Use

In contrast to the operation of the instrument and sample preparation, most students may be more concerned about the measurement method and experimental parameters selection during XRD testing. First is the selection of X-rays. The choice of target material in the X-ray generator has the greatest impact on the wavelength of X-rays. Commonly used target materials include Cu, Co, Fe, Cr, Mo, and W . Since X-rays generated by certain target materials can cause strong fluorescence absorption in some samples, choosing the appropriate target material is the first step to obtaining high-quality data. The most commonly used Cu target is suitable for almost all samples except those containing Cu and Fe, with high stability and good compatibility; Co and Fe targets are suitable for testing Fe-containing samples using a monochromator (Co) or filter (Fe); Cr target also has excellent compatibility and can test most samples, including those containing iron; Mo target is suitable for quantitative analysis of austenite due to its short wavelength; W target has the characteristic of strong continuous X-rays and is commonly used for Laue photography of single crystals.

Next is the selection of measurement parameters, including measurement mode, scan rate, and scan range. The measurement mode includes continuous scan and step scan. The former is suitable for qualitative analysis and trace detection, while the latter is suitable for calculating lattice parameters, crystallinity, analyzing microstrain, and Rietveld refinement. The general range of scan rates is from 0.001° to 8°/min. Similarly, select different scan rates according to the testing requirements; 1° to 8°/min is suitable for qualitative and general quantitative analysis, while 0.001° to 1°/min is suitable for quantitative calculation. The XRD test range is generally between 2° and 150°, with qualitative analysis typically taken from 2° to 90°. Trace detection, quantitative analysis, and lattice parameter calculation require ensuring the integrity of the main diffraction region of the sample under test. Crystallinity and Rietveld refinement are generally measured between 2° and 150°.

Typical Applications of XRD

The typical applications of XRD can be divided into qualitative and quantitative analyses, with the commonly used XRD analyses falling into the following five categories :

(1) Phase identification;

(2) Determination of lattice parameters;

(3) Crystal orientation analysis;

(4) Calculation of grain size;

(5) Quantitative phase analysis.

Below, we will explain each of these applications in detail with practical examples.

Phase Identification Analysis

Phase analysis is the most common application of XRD testing. Each phase's diffraction pattern is unique, much like everyone's fingerprints. By comparing the obtained XRD spectrum with the standard cards in the database, we can determine its structure. Based on the comparison between the XRD spectrum and the standard spectrum, we can obtain the following information:

• Whether the sample is amorphous or crystalline, where amorphous samples exhibit broad peaks without fine spectral features, while crystalline samples show rich spectral characteristics;
• The phase composition of the measured sample, whether it is a pure phase or a mixture;
• Determination of whether the unit cell is expanding or contracting.

As shown in Figure 6, the XRD peaks of the test sample are sharp and prominent, coinciding in height with the standard card for zinc blende ZnSe, with no apparent impurity peaks present. Therefore, the test sample can be determined to be high-purity ZnSe. Figure 8 provides diffraction patterns of different crystalline states of similar substances. It is evident in Figure 7a that there is only one strong peak and one weak peak, indicating a significant grain orientation in the material, suggesting that the sample may be a single crystal. In contrast, in Figure 7b, distinct diffraction peaks are detected at all standard positions, indicating that the grain orientation in the material is isotropic, suggesting that the sample may be polycrystalline. Figure 8 presents XRD spectra of zinc sulfide samples doped with selenium at different doping levels, showing that as the selenium doping level increases, the positions of the diffraction peaks shift gradually to the left. Combining with the Bragg diffraction equation, it can be inferred that the diffraction angle decreases, indicating an increase in the interplanar spacing.

Figure 6: XRD spectra of ZnSe

Figure 7: XRD spectra of single crystal and polycrystalline of the same substance

Figure 8: XRD spectra of FeS at different Se doping levels

Quantitative Calculations

In addition to qualitative analysis, XRD can also perform quantitative calculations. Common applications of quantitative calculations include:

• Calculating the average grain size of the sample using the Scherrer formula;
• Determining the relative crystallinity of the sample;
• Utilizing Rietveld refinement for full spectrum analysis to determine lattice constants, analyze stress and strain, and obtain bond length and angle information;
• Using the K-value method or Rietveld refinement for quantitative determination of the content of different phases in the crystal.