**5. X-ray diffraction**

About 95% of all solid materials can be described as crystalline. In 1919 A. W. Hull published a paper titled "A New Method of Chemical Analysis". Here he pointed out that "… every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the others. "

The X-ray diffraction pattern of a pure substance is, therefore, like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases. It is a versatile, non-destructive technique that reveals detailed information about the chemical composition and crystallographic structure of natural and manufactured materials.

Today about 50,000 inorganic and 25,000 organic single components, crystalline phases, diffraction patterns have been collected and stored on different databases. The main use of powder XRD is to identify components in a sample by a search/match procedure. Furthermore, the areas under the peak are related to the amount of each phase present in the sample.

Many solid state reactions produce polycristalline powders, therefore, for a high percentage of perovskites, the powder XRD is one of the main tools to characterize the material. Thus, it is indispensable to have a diffractometer in the laboratory.

Typically, a diffractometer consists of a tube of X-ray (source of radiation), a monochromator to choose the wavelength (for example, in the case of an anode of copper, the K2 component could be eliminated by using a primary monochromator), slits to adjust the shape of the beam, a sample and a detector. In a more complicated apparatus also a goniometer can be used for fine adjustment of the sample and the detector positions. When an area detector is used to monitor the diffracted radiation a beamstop is usually needed to stop the intense primary beam that has not been diffracted by the sample. Otherwise the detector might be damaged. Usually the beamstop can be completely impenetrable to the Xray or it may be semitransparent. The use of semitransparent beamstop allows the possibility to determine how much the sample absorbs the radiation using the intensity observed through the beamstop. Figure 9a) shows an X-ray powder diffractometer commonly seen in laboratories. Figure 9b) shows a detail of its geometry.

Ideally, every possible crystalline orientation is represented very equally in a powdered sample. The resulting orientational averaging causes the three-dimensional reciprocal space that is studied in single crystal diffraction to be projected onto a single dimension.

Bragg's law gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or

About 95% of all solid materials can be described as crystalline. In 1919 A. W. Hull published a paper titled "A New Method of Chemical Analysis". Here he pointed out that "… every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the

The X-ray diffraction pattern of a pure substance is, therefore, like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases. It is a versatile, non-destructive technique that reveals detailed information about the chemical composition and crystallographic structure

Today about 50,000 inorganic and 25,000 organic single components, crystalline phases, diffraction patterns have been collected and stored on different databases. The main use of powder XRD is to identify components in a sample by a search/match procedure. Furthermore, the areas under the peak are related to the amount of each phase present in the

Many solid state reactions produce polycristalline powders, therefore, for a high percentage of perovskites, the powder XRD is one of the main tools to characterize the material. Thus, it

Typically, a diffractometer consists of a tube of X-ray (source of radiation), a monochromator to choose the wavelength (for example, in the case of an anode of copper, the K2 component could be eliminated by using a primary monochromator), slits to adjust the shape of the beam, a sample and a detector. In a more complicated apparatus also a goniometer can be used for fine adjustment of the sample and the detector positions. When an area detector is used to monitor the diffracted radiation a beamstop is usually needed to stop the intense primary beam that has not been diffracted by the sample. Otherwise the detector might be damaged. Usually the beamstop can be completely impenetrable to the Xray or it may be semitransparent. The use of semitransparent beamstop allows the possibility to determine how much the sample absorbs the radiation using the intensity observed through the beamstop. Figure 9a) shows an X-ray powder diffractometer

Ideally, every possible crystalline orientation is represented very equally in a powdered sample. The resulting orientational averaging causes the three-dimensional reciprocal space

Bragg's law gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or

**5. X-ray diffraction** 

of natural and manufactured materials.

is indispensable to have a diffractometer in the laboratory.

commonly seen in laboratories. Figure 9b) shows a detail of its geometry.

that is studied in single crystal diffraction to be projected onto a single dimension.

others. "

sample.

destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. This analysis is called Bragg diffraction.

a)

Fig. 9. X-ray powder diffractometer.

Structural Characterization of New Perovskites 119

Compound *a* (Å) 2 *R*p *R*wp *R*exp

Ba2In1.50Co0.50O5 4.2277(2) 5.02 13.2 18.1 8.07

Ba2In1.30Co0.70O5 4.1751(1) 3.58 16.5 21.0 11.1

Ba2In0.30Co1.70O5 4.1191(2) 2.28 9.93 13.1 8.65

In order to carry out a Rietveld refinement to fit the corresponding theoretical profile – model – a statistically acceptable pattern is necessary. The goodness of the pattern depends on the sample and the measurement parameters. There are several programs suitable to do

Fig. 11. X-ray diffraction pattern refined by Rietveld method for Ba2In0.3Co1.7O5, showing the observed intensities (circles), the calculated ones (continuous line), the Bragg positions allowed by the space group (vertical lines) and the difference pattern between the observed

A synchrotron is a particular type of cyclic particle accelerator in which the magnetic field (to turn the particles so they circulate) and the electric field (to accelerate the particles) arecarefully synchronized with the travelling particle beam. In essence, it is a X-ray source with variable wavelength. Thus, it allows going deeper in the X-ray diffraction analysis of the perovskites or any other material. There is a good number of synchrotron facilities all over the world. ESRF in France and Diamond Light Source in the UK constitute two

Table 2. Cell parameters and reliability factors obtained from the Rietveld refinement.

Rietveld fittings, e.g. Fulprof and GSAS, among others.

and calculated ones (bottom of the figure).

**5.1 Synchrotron** 

examples (Figure 12).

Ba2InCoO5 4.1623(2) 3.67 12.4 15.3 8.01

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation (known as Bragg peaks). The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.

W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter *d*. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively.

The interference is constructive when the phase shift is a multiple of 2π; this condition can be expressed by Bragg's law (Equation 1),

 *n*=2*d*sen (1)

where *n* is an integer, *λ* is the wavelength of the incident wave, *d* is the spacing between the planes in the atomic lattice, and *θ* is the angle between the incident ray and the scattering planes (Figure 10).

Fig. 10. Bragg diffraction.

The Rietveld method allows us to characterize the polycrystalline materials by a least squares approach to refine a theoretical line profile until it matches the measured profile shown in the pattern. The introduction of this technique was a significant step forward in the diffraction analysis of powder samples as, unlike other techniques at that time, it was able to deal reliably with strongly overlapping reflections.

The Rietveld refinement considers the fitting of a series of parameters linked, most of them, to the structure of the perovskite (or material in general). Some of them are the unit cell parameters, and the profile ones. It is important to reach a certain grade of reliability during the fitting process, and this can be followed by checking the so-called agreement factors (or reliability factors). Table 2 shows an example of lattice parameters and agreement factors obtained after the Rietveld refinement of the cubic perovskite-type solid solution Ba2In2 xCoxO5 (0.50≤x≤1.70). In Figure 11 an example of a Rietveld refined pattern, exactly for Ba2In0.3Co1.7O5, can be observed.

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation (known as Bragg peaks). The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes. Both neutron and X-ray wavelengths are comparable with inter-atomic distances

W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter *d*. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered

The interference is constructive when the phase shift is a multiple of 2π; this condition can

 *n*=2*d*sen (1) where *n* is an integer, *λ* is the wavelength of the incident wave, *d* is the spacing between the planes in the atomic lattice, and *θ* is the angle between the incident ray and the scattering

The Rietveld method allows us to characterize the polycrystalline materials by a least squares approach to refine a theoretical line profile until it matches the measured profile shown in the pattern. The introduction of this technique was a significant step forward in the diffraction analysis of powder samples as, unlike other techniques at that time, it was

The Rietveld refinement considers the fitting of a series of parameters linked, most of them, to the structure of the perovskite (or material in general). Some of them are the unit cell parameters, and the profile ones. It is important to reach a certain grade of reliability during the fitting process, and this can be followed by checking the so-called agreement factors (or reliability factors). Table 2 shows an example of lattice parameters and agreement factors obtained after the Rietveld refinement of the cubic perovskite-type solid solution Ba2In2 xCoxO5 (0.50≤x≤1.70). In Figure 11 an example of a Rietveld refined pattern, exactly for

(~150 pm) and thus are an excellent probe for this length scale.

able to deal reliably with strongly overlapping reflections.

be expressed by Bragg's law (Equation 1),

constructively.

planes (Figure 10).

Fig. 10. Bragg diffraction.

Ba2In0.3Co1.7O5, can be observed.


Table 2. Cell parameters and reliability factors obtained from the Rietveld refinement.

In order to carry out a Rietveld refinement to fit the corresponding theoretical profile – model – a statistically acceptable pattern is necessary. The goodness of the pattern depends on the sample and the measurement parameters. There are several programs suitable to do Rietveld fittings, e.g. Fulprof and GSAS, among others.

Fig. 11. X-ray diffraction pattern refined by Rietveld method for Ba2In0.3Co1.7O5, showing the observed intensities (circles), the calculated ones (continuous line), the Bragg positions allowed by the space group (vertical lines) and the difference pattern between the observed and calculated ones (bottom of the figure).

#### **5.1 Synchrotron**

A synchrotron is a particular type of cyclic particle accelerator in which the magnetic field (to turn the particles so they circulate) and the electric field (to accelerate the particles) arecarefully synchronized with the travelling particle beam. In essence, it is a X-ray source with variable wavelength. Thus, it allows going deeper in the X-ray diffraction analysis of the perovskites or any other material. There is a good number of synchrotron facilities all over the world. ESRF in France and Diamond Light Source in the UK constitute two examples (Figure 12).

Structural Characterization of New Perovskites 121

material can be derived. This can easily be converted to the stress field experienced by the compound. This has been used to analyze stresses in aerospace and automotive components to give just two examples. There is a good number of facilities all over the world offering a neutron source. Among them, we could mention, for instance, ISIS in the UK, and ILL in

Figure 13 shows a picture of the reactor hall at ILL in Grenoble, France. ILL (Institut Laue-Langevin) is one the most important centres in the world to carry out neutron experiments.

The original form of electron microscope, the transmission electron microscope (TEM) uses a high voltage electron beam to create an image. The electron beam is accelerated by an anode with respect to the cathode, focused by electrostatic and electromagnetic lenses, and transmitted through the sample that is in part transparent to the electrons and in part scatters them out of the beam. When it emerges from the sample, the electron beam carries information about the structure of the sample that is magnified by the objective lens system of the microscope. Hardware correction of spherical aberration for the high-resolution transmission electron microscopy (HRTEM) has allowed the production of images with resolution below 0.5 Å at magnifications above 50 million times. This possibility of having direct images of the atomic arrangement in the structure has made the HRTEM an important

The perovskite is a mineral series composed of calcium titanate. Many transition metal oxides show that very versatile perovskite structure. The rich variety of physical properties

Fig. 13. Inside the reactor hall at ILL in Grenoble (France).

tool for nano-technologies research and development.

**8. Conclusion** 

**7. High-resolution transmission electron diffraction** 

France.

Fig. 12. Diamond Light Source (UK).
