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# X-ray diffraction (XRD)

X-ray diffraction is used for the investigation of crystalline materials. All crystalline materials have one thing in common: their components (atoms, ions or molecules) are arranged in a regular manner. This is a necessary requirement for XRD as diffraction can only occur, if X-rays are scattered by a periodic array of particles with long-range order.

## Introduction X-ray diffraction (XRD)

The scattered X-rays from the sample interfere with each other either constructively or destructively. This means that detectors can read-out a signal only at angles where constructive interference occurs. This is schematically shown in the next picture.

The dots in the graph correspond to the building blocks of a crystalline material. Due to the crystalline nature, the atoms are arranged periodically. The incident X-ray beam is scattered at different planes of the material. The resulting diffracted X-rays therefore have a different optical path length to travel. The magnitude of this path length only depends on the distance between the crystal planes and the incident angle of the X-ray beam. This is summarized in the famous Bragg – Equation:

$n\lambda =2d\sin \theta$

In words this equation can be described as follows: constructive interference occurs only if the path difference (given by  2d sinθ) is a multiple (n=1,2,..) of the used wavelength of the X-ray beam. As the wavelength in XRD experiments is known and the angles at which constructive interference occurs are measured, the use of the Bragg equation allows determining the distance between the lattice planes of the material.

The result of the measurement is a so called diffractogramm. This is a plot of X-ray intensity on the y-axis versus the angle 2θ (2θ is defined as the angle between the incident and the diffracted beam) on the x-axis.

Obtainable information from a diffractogramm:

• Qualitative Analysis: Every crystalline material produces a specific diffractogramm. This can be seen as the fingerprint of the sample. Comparison of the obtained data with databases results in the identification of the material.
• Quantitative Analysis: If the sample is not a pure substance, but consists of several components, it is also possible to calculate the relative amounts of the individual phases.
• Unit Cell Lattice Parameters: As mentioned above crystalline materials are regularly arranged. The smallest building block of such a regular arrangement is the unit cell of the material. XRD allows characterizing the dimensions of this unit cell. This can be especially interesting under non-ambient conditions.
• Crystallite Size and Strain: The crystallite size of the powder has an influence on the width of the obtained peaks. Crystallites that are smaller than ~120 nm give broader peaks. This can be used to extract information about the size of the crystallites. Microstrain in the sample also results in peak broadening. Also this effect can be quantified by XRD measurements.

More sophisticated methods of X-ray diffraction can be used to extract much more information from the samples. But these will not be further discussed here.

## Introduction non-ambient X-ray diffraction

So far it has been explained what kind of information can be gained from X-ray diffraction experiments under ambient conditions. In non-ambient X-ray diffraction the sample is influenced by external parameters during the experiments.

The most important parameters are listed below:

• Temperature
• Pressure
• Relative Humidity
• Gas Environment
• Electrical fields
• Magnetic fields

These parameters result in a variety of material changes that can be investigated in-situ. Some of them are listed below:

• Material formation / structure changes
Application fields: alloys, building materials, drug APIs, catalysts, minerals,…
• Heat treatment & annealing
Application fields: alloy, ceramics, polymers,…
• Calcination & Sintering
Application fields: catalysts, zeolites,…
• (De)Hydration processes
Application fields: building materials, pharmaceuticals, food industry,…
• Material changes during operation:
Application fields: catalysts, refractory materials, alloys,…

The investigation of these processes is not only interesting for the scientific community, but also for many technological processes used in industry.

## Principal instrumental design aspects

The principal components of a non-ambient heating attachment are shown in Figure 2. On the rear side of the instrument a mechanical interface for a specific diffractometer is located. To keep the temperature load on the diffractometer at a minimum at all times, the housing of the instruments is cooled with tap water. Connection can be easily done by the use of quick coupling connections. The X-ray window of the heating attachment allows the X-ray beam to efficiently pass the instrument. Window materials like Kapton, Graphite, PEEK or Beryllium are used in most cases.

The round shape of the instrument ensures that the beam path through the window is constant at all angles. Different additional elements like a viewing window, connectors to the vacuum system, gas connectors or cooling hoses for the introduction of liquid nitrogen can also be present. An exemplary picture of the interior design of an instrument is shown in Figure 3.

In the centre of the instrument the sample holder is located. The temperature sensor is usually mounted inside the sample holder. This ensures that the temperature measurement is repeatable (no influence of the sample on the temperature measurement) and that chemical compatibility issues (reaction between sample and temperature sensor) do not play a role. Right above the sample holder a so-called beam knife that allows reducing the background scattering at low angles 2θ, can be mounted.

### Comparison of heater types

In principal there are two types of heaters: direct heaters and environmental heaters. Direct heaters are non-ambient XRD attachments that either have a sample holder that is placed on a resistance heater (Figure 4, A) or are so-called strip heaters, where the sample is directly placed on a resistively heated heating strip (Figure 4, B). The advantage of the former is that a cryostat, that also allows cooling of the sample, can be introduced. In contrast to that, very high temperatures (up to 2300°C) can be achieved with strip heaters.

The second important type of heater is the so-called environmental heater (Figure 4, C). In contrast to direct heaters, environmental heaters are heating the sample from all sides. This ensures a very homogeneous temperature distribution around the sample. Moreover, due to the design of the instrument it is possible to spin the samples by using sample spinners. This helps to improve the counting statistics and minimizes the influence of preferred orientation of the powder which would otherwise result in deviations from the expected relative peak intensities.

The heating type does not only influence the temperature distribution around the sample, but also has a critical influence on the temperature accuracy of the instrument as the temperature is measured inside the sample holders.

For a direct heater, the sample is placed on a sample holder or a heating strip and heat is transferred from the resistively heated surface from below. This results in a temperature deviation between the measured temperature of the sample holder and the temperature on the surface of the sample. The deviation is highly dependent on the thermal properties of the sample. Critical parameters that influence this temperature deviation are for example the heat conductivity, emission properties and the thickness of the sample.

In contrast to that, environmental heaters transfer the heat homogenously from all sides to the sample. Due to this homogeneous temperature distribution inside the instrument, the temperature deviation between the temperature sensor and the sample surface is minimized.

## Temperature validation

In general, the use of non-ambient XRD attachments needs a calibration of the temperature read-out due to unavoidable differences between displayed temperature and real temperature on the surface of the sample. (As mentioned before, the temperature is usually measured in the sample holder and not directly on the surface of the sample.) The term “calibration” can only be applied, if certified standard materials are available. However, no such materials exist for non-ambient X-ray diffraction (NA-XRD). There are standard materials for different other thermal analysis methods like Dynamic Scanning Calorimetry (DSC), but as the experimental conditions differ quite significantly, those can´t be applied for NA-XRD. For XRD, commonly accepted reference materials with “well-known” thermal properties are used instead. Therefore the term “temperature validation” instead of “temperature calibration” is used.

A temperature validation for non-ambient XRD determines the relation between the temperature of the temperature sensor in the heating attachment and the true temperature on the surface of the sample. Two methods for temperature validation are used in NA-XRD:

• Validation by the use of phase transformations
• Validation by the use of thermal lattice expansion

### Validation by phase transformation

This method makes use of substances with known temperature of crystallographic transformations (so called phase transitions, which describe the change from one crystallographic form into a second crystallographic form of the same material) or melting points and compares the tabulated values from the literature with the measured values.

#### Requirements for reference materials

• Well known transformation temperature
• Fast phase transformation
• Reversible phase transformationNo reaction with sample holder

#### Procedure

• Find best region (ROI: region of interest) of peak pattern to monitor the phase transformation
• Define a short 2θscan over the ROI
• Heat the sample in steps and run a scan each temperature
• Plot the peak intensity vs. the displayed temperature and determine the transformation temperature
• Compare it with the value from literature
Material Transformation Temperature / °C
KNO₃ 130
KCIO₄ 299.5
Ag₂SO₄ 430
SiO₂ 573
K₂SO₄ 583
K₂CrO₄ 665
BaCO₃ 810
SrCO₃ 925

#### Example

In the graph below (Figure 5) the results for KNO3, measured in a heating attachment for XRD, are shown.

The measured phase transition temperature from the orthorhombic to the trigonal modification of KNO3 was 129°C. According to the literature, the phase transition temperature for this phase transition is 130°C. This means that the deviation between displayed and real temperature is only 1°C.

#### Validation over a wider temperature range

One disadvantage of the validation by using phase transformations is that a single substance has only a limited number of phase transformations. In order to have a more complete picture of the thermal behaviour of the heating attachment, several substances have to be measured. (Depending on the temperature range, this can be quite time consuming.)

An example of such an extensive validation is shown in Figure 6.

• Monitoring of phase transitions and melting points is not sensitive to sample position or data quality and requires only very simple data evaluation
• The method is most feasible for validation over a small temperature range. To obtain a validation curve for the complete temperature range, measurement of thermal expansion is preferable

### Validation by thermal lattice expansion

This method can be applied, if a material with accurately known thermal expansion behaviour is available. The thermal expansion curve of a certain material relates the size increase of a lattice parameter to the applied temperature.

The thermal expansion curve for the c-axis of corundum looks, for example, as follows[1]:

$$\frac { \Delta L }{ { L }_{ 0 } } =-0,192+5,927\cdot { 10 }^{ -4 }T+2,142\cdot { 10 }^{ -7 }{ T }^{ 2 }-2,207\cdot { 10 }^{ -11 }{ T }^{ 3 }$$

∆L=L(T)- L0
L(T) … length of c-axis at temperature T
L0 … length of c-axis at 298 K
T … temperature in K

Reversely, the determination of the lattice parameter allows calculating the temperature. As axis dimensions are available from Rietveld refinement, the true temperature on the surface of the sample can be determined.

#### Requirements for reference material

• Accurately known thermal expansion curves
• Large thermal expansion coefficient (reduces the error of the calculations)
• Stable lattice parameters
• No chemical reaction during the heating process (e.g. oxidation)

#### Procedure

• Run large range 2θ scans at 25°C (reference) and the temperatures of interest
• Determine the lattice parameters with Rietveld refinement
• Calculate the relative lattice expansion a(T)/a(25°C)
• Use the inverted expansion curve to calculate the true sample temperature and compare it to the displayed scan temperature

• The complete temperature range can be calibrated with one standard sample
• Method applicable to very high temperatures
• Good data quality required for the refinement
• Considerable data evaluation required
• Inaccurate at low temperatures (<400°C) due to small lattice expansion

Results of this type of temperature validation will be shown in chapter 4.3.

### Temperature validation data for different heater types

Generally there are three types of heat transfer: Conduction, convection and radiation.

Conduction is the transfer of heat by thermal motion of the atoms within a material. The ability of a certain material to transfer heat is quantified by its thermal conductivity. As mentioned before, the thermal conductivity of the sample is especially important for direct heaters as it greatly influences the temperature deviation in these instruments.

Convection is the transport of heat by fluids (gases and liquids). Therefore this type of heat transfer is highly influenced by the pressure and the type of gas present. In general low-molecular weight gases have a much higher thermal conductivity compared to heavy weighted gases.

Radiation is the transfer of heat by electromagnetic waves. This type of heat transfer is independent of a medium, but the amount of transferred heat is highly dependent on the temperature (T to the power of 4 dependence).

#### Influence of gas atmosphere

The influence of different gas atmospheres on the temperature accuracy is discussed for a strip heater. The following conditions were evaluated: He (atmospheric pressure), N2 (atmospheric pressure) and vacuum. It is expected that the temperature deviation between the measured temperature of the temperature sensor (spot-welded to the bottom side of the heating strip) and the temperature on the sample surface should be highest under vacuum. The reason is that convection, which would help to transfer heat from the strip to the sample, is completely missing. As the thermal conductivity of He (5.193 KJ/kgK) is much higher compared to N2 (1.040 KJ/kgK), He should give the best results.

In the following figure (Figure 7) the results on temperature accuracy, shown from the measurements of the thermal expansion of MgO are shown.

It can be easily seen in Figure 7 that the assumptions from above are confirmed.

One can observe the same trends for environmental heaters, but in that case the absolute deviations are much smaller due to the better temperature homogeneity in environmental heaters. In this case the thermal expansion of the c-axis of Al2O3 was used for the validation (Figure 8).

#### Influence of sample thickness

One big advantage of environmental heaters is that the thermal properties of the sample are not that important for the temperature deviation. This is especially true for the sample thickness. For direct heaters the sample thickness has a big impact on the temperature deviation as can be seen from Figure 9. In this example the thermal expansion of Al2O3 was measured with a strip heater (=direct heater) under helium atmosphere.

It can be clearly seen that the thickness of the sample has a significant influence on the temperature deviation. Therefore for direct heaters the best way of sample preparation is as follows: the sample should be applied as thin as possible, but still covering the complete heating strip. Otherwise additional signals from the heating strip are visible in the diffractogramm and make data interpretation more difficult.

## Thermal height expansion

Almost all materials expand upon heating. This effect has to be accounted for in non-ambient XRD attachments as otherwise the sample would not stay at the centre of the goniometer, which would result in peak shifts. The way this correction is done is schematically shown in the next picture (Figure 10).

After installation of the XRD attachment, the height of the sample holder is adjusted (left picture) at room temperature (ZT=25°C). (The easiest way to align the instrument height is described in 5.1.) Upon heating the sample holder will expand which means that the sample is above the initial value for the correct height (middle picture). To keep the sample at the prealigned position at elevated temperatures, the height of the instrument is reduced (right picture). This can be done either manually on the adapter of the instrument or in a more comfortable way automatically with a so-called Z-stage (see also chapter 5.1 for further information). In general it is highly recommended to use a Z-stage for non-ambient experiments as this allows performing fully automated temperature batches without the need of any user interaction throughout the experiment.

The thermal expansion is heavily dependent on the type of instrument (direct heater or environmental heater), the atmosphere inside the instrument and the pressure. For accurate experiments it is recommended to calibrate the thermal expansion of the sample holder under exactly the same conditions that will also be used for the sample measurement.

### Calibration of thermal height expansion

The height of the sample holder (and therefore also the sample) is a function of the actual temperature.

In practice this function can be determined by measuring the thermal expansion of the sample holder at different temperatures. The most comfortable way to measure the thermal expansion of the sample holder is by using the primary beam of the diffractometer. The procedure is described schematically in the next picture.

It all starts with exactly the same procedure that is used in general to align the chamber height. This is done in the following way:

• The angles 2θ (angle between incident and diffracted beam) and ω (angle between incident beam and sample) are set to zero (use a beam attenuator to prevent damage of the detector in the primary beam!).
• Lower the height of the chamber until no parts are blocking the primary beam.
• Use this intensity as your reference intensity value I0
• Move the chamber upwards until the intensity of the beam is only 50% of I0 (=I1/2). The corresponding height of the instrument at this intensity corresponds to the aligned (=correct) height of the sample holder (ZT=25°C).
• Increase the temperature stepwise and repeat the search for I1/2. This gives you a correlation table of sample temperature vs. sample holder height.

Depending on the used diffractometer and the non-ambient equipment, these values can be stored in the program that adjusts the height of the instrument. It is sufficient to use temperature steps of 100°C. All values in between are interpolated. For all following experiments the actual temperature value of the heater is read-out and the height of the instrument is adjusted automatically.

The following picture shows an example for the thermal expansion of the sample holder of an environmental heater, measured in air and vacuum.

In this case data have been measured up to 1200°C. The expansion of the sample holder at 1200°C is nearly 490 µm in air, whereas it is 540 µm in vacuum.

## Applications

Due to the variety of non-ambient XRD attachments also the application fields are widespread and span from simple observation of phase transitions to highly complex studies of, for example, catalytic reactions. One has to keep in mind that all materials have different chemical, physical or mechanical properties under non-ambient conditions compared to those at standard conditions. As a consequence studying these differences is necessary to fully understand the material of interest.

### Mineralogy

Mineralogical applications of non-ambient X-ray diffraction comprise studies of phases (different minerals or solid-solutions in mineral groups) yielding in phase diagrams, which show the stability of phases depending on e.g. temperature, pressure or composition. A lot of phase changes are also dependent on the presence of fluids (e.g. de/hydration of zeolites and clays).

NA-XRD is also an important technique for material research which uses minerals as raw materials for e.g. sensor devices, ferroelectrics / piezoelectrics or permanent magnets.

Furthermore cements and especially their hydration/dehydration reactions are frequently studied with non-ambient diffraction. During hardening of cements several mineral reactions occur. The type of reactions and the composition of the final cement product strongly depend on humidity and also temperature.

### Metallurgy

Non-ambient X-ray diffraction in metallurgy is focussed on development, formation and processing of metals, steels and alloys.

Hardening and strengthening of the alloys is often dependent on the heating and cooling rates. Non-ambient diffraction allows studying structural and textural as well as compositional changes during such heating and cooling processes. On the other hand, sintering of metal powders can be easily observed and optimized with non-ambient XRD.

### Pharmacological industry

Non-ambient X-ray diffraction plays an important role in all production steps of the pharmaceutical industry:
drug (API) development, storage and stability testing, quality control as well as crystallisation studies. Understanding the stability conditions and properties of different phases is essential for the optimization of product processes and storage conditions. In some cases amorphous phases are less stable and crystallize during contact with air and / or humidity, which makes in-situ X-ray diffraction a powerful tool to evaluate the crystallisation behaviour of drugs.

### Food industry

Numerous compositions and structures make food a complex system. Hence qualitative and quantitative studies, among them non-ambient investigations, are necessary to determine storage, freezing or heating behaviour and stability. Non-ambient diffraction is a suitable technique as in many processing steps heating and cooling of food is required and changing properties can be studied in-situ.

Crystallisation and melting behaviour of oils and fats for instance do have strong influence on the properties of fat raw materials and of final products (e.g. desired texture properties of chocolate or ice-cream). Further examples are studies of shelf life (e.g. in refrigerators) or gelatinization of starch. Also the behaviour of cryoprotectants in frozen and dry food production needs profound investigation under non-ambient conditions.

### Ceramics

Studying phase transitions at elevated or low temperatures is a necessity for developing new ceramic and ceramic-composite materials. The demand for new materials is given due to the rapid progress in different application fields using ceramics as raw materials or final product (refractory industry, superconductors, automotive and aircraft, bioceramics, just to mention a few).

For example phase transitions of ceramics do have major influence on their mechanical strength (hardness or toughness). On the other hand ceramics in refractory applications have to be highly temperature resistant. Non-ambient diffraction is suitable for in-situ investigations of the thermal properties of ceramic materials.

### Chemical industry

Use of non-ambient diffraction in the chemical industry is divers and covers the investigation of nanomaterials, catalysts, batteries, piezoelectric and magnetic properties of materials, hydrogen storage and fuel cells, air sensitive samples, calcination and sintering and many more.

One very recent topic of interest in material science is research of storage behaviour of gases like H2 and CO2. In-situ studies of adsorption and desorption of such gases in suitable carrier materials is of key interest, for example, for fuel cells. Even if gases like H2 cannot be directly measured with XRD, the change of the unit cell dimensions during the gas storage process provides insights in the usability of the material under investigation. Therefore the ability to use different gases, pressures and temperatures provides a vast play ground in this field of research.

Metal organic frameworks (MOF) for example play a major role in adsorption, gas separation or catalysis. Fluid-induced changes of the MOFs can be investigated under controlled combination of atmosphere and temperature.

### Thin films

A growing application field lies in development and production of thin films. Chemical and physical properties of the thin film materials strongly depend on temperatures (e.g. temperature dependent strain and stress, thermal decomposition, annealing processes, microstructures as dislocations and defects), which makes non-ambient diffraction a suitable technique for their analysis.

Strain and stress in thin films do have major influence on hardness, toughness or adhesion to the substrate. Furthermore thermal straining is responsible for mechanical faults in micro- and optoelectric devices. Understanding the origin of strain and stress is therefore necessary for development of new thin film materials and appropriate production processes. On the other hand different annealing temperatures may influence mechanical and optical properties as well as the thickness of the films.

Your free guide to non-ambient X-ray diffraction

## References

1. Touloukian, Y.S., (1977). Thermophysical Properties of Matter, Vol 13, Thermal Expansion of Nonmetallic Solids