**2. Rietveld refinement**

It was 1969, 27 copies of a 162 Kilobytes program were sent to different institutes all over the world. The program was accompaniment of paper published in Journal of Applied Crystallography titled "A Profile refinement Method for Nuclear and Magnetic Structures" by Hugo Rietveld. Within a span of a decade 200 structures were refined from powder diffraction data [26–28]. The method we all know as Reitveld refinement method, made possible to refine whole profile with parameters including half-width, zero shift, cell parameters [29], asymmetry correction [30, 31], preferred orientation correction [32, 33], overall scale factor, overall isotropic temperature factor, fractional coordinates of the atoms, atomic isotropic temperature, occupation numbers and the components of the magnetic vectors of each atom. The algorithm this program followed is summed up in **Figure 1**. In subsequent versions of the method, Rietveld introduced residual values (R values), allowing for a quantitative judgment of the refinement quality. Most of the findings and equations, which Rietveld published, are still used nowadays in their original form [14, 20, 21, 34–37].

current studies, antiquities and general information. This chapter attempts to accommodate most of these and present them in a more, newbie, newcomer friendly way. The chapter will follow a linear path from sample preparation, data collection to final results and conclusions accompanied by various current chal-

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

*DOI: http://dx.doi.org/10.5772/intechopen.96065*

*The algorithm of whole profile refinement program developed by Hugo M Rietveld [reconstructed from the IUcr*

In order to understand different phases of sample preparation, we first need to define and understand the term "sample". The term "sample" encompasses a much broader meaning in scientific community with or without any restriction on size, quantity, quality etc. A sample may be a rather large portion of material, or a very tiny amount. A specimen on the other hand is the representative diminutive piece of a sample. Although there is a thin line of distinction between a sample and a specimen in X-ray diffraction, the term sample preparation generally means to

The material, phase purity, homogeneity, density gradient etc. of a sample from which a specimen is taken are to be considered in advance. For a phase pure sample or mostly pure, a specimen is a good representative of the sample, so is the case with multiphase but homogenous samples. However a specimen from a multiphase and

lenges, precautionary and explanatory notes.

prepare a specimen from a larger sample [43, 44].

**3. Sample preparation**

*newsletter no. 26, Dec 2001].*

**Figure 1.**

**253**

In 1994, International Union of Crystallography (IUCr) constituted a commission on powder diffraction with the purpose of diving into the status of the world of scientific community in general and crystallographic community in particular and focus on the practical aspects of data collection, refinement software, data interpretations, future endeavors etc. [38]. The commission proposed certain protocols and few guidelines for data collection, background contribution, peak-shape function, refinement of profile parameters, Fourier analysis, refinement of structural parameters, geometric restraints, estimated standard deviation, interpretation of R values and some common problems with their possible solutions. Although the Hill and Cranswick [38–41] commission on powder diffraction formulated a set of general guidelines that encompassed the recommendations with some explanatory and cautionary notes regarding Rietveld refinement their application in the aspect for a newcomer are not totally encompassed [42, 43]. The reason is not the ineffectiveness, obsoleteness or incomprehension but rather the scattered nature of

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics DOI: http://dx.doi.org/10.5772/intechopen.96065*

### **Figure 1.**

few months, Braggs determined structure of NaCl, KCl, KBr, CaF2, Cu2O, ZnS, NaNO3, some calcites and diamond from their respective single crystals [10].

x-ray and neutron diffraction data.

**2. Rietveld refinement**

*Advanced Ceramic Materials*

form [14, 20, 21, 34–37].

**252**

The year 1914, Max von Laue was awarded Noble prize for his discovery of the diffraction of X-rays by crystals [11, 12] followed by 1915 prize for their services in the analysis of crystal structure by means of X-rays to W H and W L Bragg [6] itself concatenates the importance of crystal structure determination. In following years, Debye and Scherrer extended the theory from single crystal to powder diffraction, presenting the complete theory of powder diffraction patterns and crystal structures used today (*squared sums of hkl ordered triplets*) [13–16]. Although Scherrer, Debye and Hull solved structures of many materials, it was not until modern computational boom that new, more complex and low symmetry system could be solved via powder diffraction pattern [17–23]. In the quest of achieving a suitable pathway for attaining a solution of powder X-ray diffraction many niche-limited attempts like maximum likelihood method [24, 25], anomalous dispersion, maximum entropy method, line profile fitting [26] etc. were made abundantly in 1950s and 60s. Hugo Rietveld in 1960s came up with one such method, employing least square iteration principle to statistically estimate the weighted contribution of every point on a powder XRD pattern [27]. The method now known as Rietveld refinement was the first step towards full profile whole powder pattern fitting method for

It was 1969, 27 copies of a 162 Kilobytes program were sent to different institutes all over the world. The program was accompaniment of paper published in Journal of Applied Crystallography titled "A Profile refinement Method for Nuclear and Magnetic Structures" by Hugo Rietveld. Within a span of a decade 200 structures were refined from powder diffraction data [26–28]. The method we all know as Reitveld refinement method, made possible to refine whole profile with parameters including half-width, zero shift, cell parameters [29], asymmetry correction [30, 31], preferred orientation correction [32, 33], overall scale factor, overall isotropic temperature factor, fractional coordinates of the atoms, atomic isotropic temperature, occupation numbers and the components of the magnetic vectors of each atom. The algorithm this program followed is summed up in **Figure 1**. In subsequent versions of the method, Rietveld introduced residual values (R values), allowing for a quantitative judgment of the refinement quality. Most of the findings and equations, which Rietveld published, are still used nowadays in their original

In 1994, International Union of Crystallography (IUCr) constituted a commission on powder diffraction with the purpose of diving into the status of the world of scientific community in general and crystallographic community in particular and focus on the practical aspects of data collection, refinement software, data interpretations, future endeavors etc. [38]. The commission proposed certain protocols and few guidelines for data collection, background contribution, peak-shape function, refinement of profile parameters, Fourier analysis, refinement of structural parameters, geometric restraints, estimated standard deviation, interpretation of R values and some common problems with their possible solutions. Although the Hill and Cranswick [38–41] commission on powder diffraction formulated a set of general guidelines that encompassed the recommendations with some explanatory and cautionary notes regarding Rietveld refinement their application in the aspect for a newcomer are not totally encompassed [42, 43]. The reason is not the ineffectiveness, obsoleteness or incomprehension but rather the scattered nature of

*The algorithm of whole profile refinement program developed by Hugo M Rietveld [reconstructed from the IUcr newsletter no. 26, Dec 2001].*

current studies, antiquities and general information. This chapter attempts to accommodate most of these and present them in a more, newbie, newcomer friendly way. The chapter will follow a linear path from sample preparation, data collection to final results and conclusions accompanied by various current challenges, precautionary and explanatory notes.

## **3. Sample preparation**

In order to understand different phases of sample preparation, we first need to define and understand the term "sample". The term "sample" encompasses a much broader meaning in scientific community with or without any restriction on size, quantity, quality etc. A sample may be a rather large portion of material, or a very tiny amount. A specimen on the other hand is the representative diminutive piece of a sample. Although there is a thin line of distinction between a sample and a specimen in X-ray diffraction, the term sample preparation generally means to prepare a specimen from a larger sample [43, 44].

The material, phase purity, homogeneity, density gradient etc. of a sample from which a specimen is taken are to be considered in advance. For a phase pure sample or mostly pure, a specimen is a good representative of the sample, so is the case with multiphase but homogenous samples. However a specimen from a multiphase and

### *Advanced Ceramic Materials*

inhomogeneous sample may not be a good representative of the sample itself. The sample may consist of several phases, known or unknown, and may also include amorphous material. Depending on the technique and radiation, it may be small or large (neutron diffraction), it may be flat (Bragg–Brentano geometry), or cylindrical (Debye–Scherrer technique). In case of multiphase sample or amorphous contributions specimen should be taken with considerable representation of the sample such that during the refinement process quantitative contribution of each phase can be estimated more precisely.

successful data collection are diffractometer geometry, instrument alignment, calibration, the radiation, the wavelength, slit size, necessary counting time and most

It's important that the incident beam should always be kept on sample (specimen) such that the diffracting volume remains constant. In Bragg–Brentano configuration, the use of wide divergence slits must be accounted by a correction term. Introduction of this correction term is quite plainly geometry dependent, therefore, sample holder geometry has to be taken into consideration and an update to correction term should be applied. Most of the instruments correct this by using rotating circular sample, however, this does not always correct for low angle intensities. A more modern approach to this problem is the use of automated variable divergence slits which operate as a function of 2θ. At lower angles smaller slits are used, and at higher angles wider ones. A flat sample for Bragg–Brentano geometry is essential to ensure that focusing circle is always tangential to the sample surface. It is however practically more challenging to achieve surfaces with low roughness. At lower angles, the effect is negligible as the incident beam area is large, but at higher angles, as beam width decreases, the surface roughness can cause problems in collected data. A more common approach to this problem is to spend more time at collecting data at higher angles. Most of the modern diffractometers are equipped with such algorithms and generally adjust automatically as a function of θ. Next time you perform XRD measurements on your sample and fell the higher 2θ data

importantly the alignment and positioning of incident beam [46].

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

*DOI: http://dx.doi.org/10.5772/intechopen.96065*

collection are getting on your nerves. Remember, it is for the best [47].

be measured to ensure statistical viability of data.

i. The constant-volume assumption

**255**

all the following assumptions will be invalid [38, 48–50].

ii. The intensities measured at higher angles

Time is also an important factor to consider while data collection [48]. It is necessary to record suitable counts; therefore more time should be spent between each 2θ step. It is also necessary to record the data at suitable intervals (step size) to ensure recording of good profile and peak-broadening. As a rule of thumb, there should be at least 5 data points collected across a given peak. The maximum 2θ should always be kept to as low as you can go, however at least 50 2θ degrees should

Sample transparency is yet another problem. The assumption for XRD in reflection geometry is satisfied only when the sample is infinitely thick. If the sample contains only light elements, this condition might not get fulfilled at all, therefore

iii. The focusing circle adjustment etc. On the other hand, heavily absorbing samples can also be a problem, because the incident beam cannot penetrate the whole sample. The solution in the later case is much simpler than former one. Sample in later case may have to be diluted with a light-

Preferred-orientation effects can be very difficult to eliminate, especially for flat

powder specimens. If the intensities show a strong hkl dependence (e.g. all hk0 reflections are strong and all h00 weak), preferred orientation of the crystallites should be suspected. Rietveld refinement can be done with many programs which are based on March model allowing a specific crystallographic vector based refinement of preferred-orientation parameter [32]. The elimination (or minimization) of the problem experimentally is to be preferred due to the crude nature of such models. Grain and particle morphology can also play a major role in preferential orientation. For large crystallite size the randomness of orientation of sample gets diminished i.e.

element material (e.g. diamond powder or glass beads).

In the length of this chapter the term "sample preparation" will be used to define collection of specimen, cleaning or remolding, mounting it on sample holder and all the processing necessary to prepare the diffracting material to its mounting on goniometer.

### **3.1 Precautionary/explanatory notes**

Following few precautions are integral parts of sample preparation process


### **3.2 Current challenges**

Despite the advances in current instrumentation and techniques we will not be able to obtain a 100% representative specimen from any sample, particularly powder samples. Grain size distribution, preferred orientation, inhomogeneous grain boundaries, defects and other microscopic differences will always act against it [45].

As world dives more and more into the nanoscale world, the sample thickness poses a problem with 1D and 2D materials.

Sample geometry can also not be obtained with certainty with nanoscale samples, especially with nano-morphologies and surface rough samples. A sample of 50–100 nm thickness and spiky morphology, with each spike of let us say 20 nm thickness and 50 nm length, will have so rough surface that there will be roughly 50% of thickness change while moving from one spike to another.

Another challenge will be the porosity of the samples. In nanomaterial samples the surface area to volume ratio increases leading to apparent amorphicity in actually crystalline samples.

## **4. Data collection**

In order to perform a successful Rietveld refinement, it is essential that the powder diffraction data be collected appropriately. If relative intensities or the 2θ values (d-spacing) are recorded incorrectly, no amount of time spent on refinement will lead to any sensible results. The factors to be considered for effective and

### *Challenges in Rietveld Refinement and Structure Visualization in Ceramics DOI: http://dx.doi.org/10.5772/intechopen.96065*

successful data collection are diffractometer geometry, instrument alignment, calibration, the radiation, the wavelength, slit size, necessary counting time and most importantly the alignment and positioning of incident beam [46].

It's important that the incident beam should always be kept on sample (specimen) such that the diffracting volume remains constant. In Bragg–Brentano configuration, the use of wide divergence slits must be accounted by a correction term. Introduction of this correction term is quite plainly geometry dependent, therefore, sample holder geometry has to be taken into consideration and an update to correction term should be applied. Most of the instruments correct this by using rotating circular sample, however, this does not always correct for low angle intensities. A more modern approach to this problem is the use of automated variable divergence slits which operate as a function of 2θ. At lower angles smaller slits are used, and at higher angles wider ones. A flat sample for Bragg–Brentano geometry is essential to ensure that focusing circle is always tangential to the sample surface. It is however practically more challenging to achieve surfaces with low roughness. At lower angles, the effect is negligible as the incident beam area is large, but at higher angles, as beam width decreases, the surface roughness can cause problems in collected data. A more common approach to this problem is to spend more time at collecting data at higher angles. Most of the modern diffractometers are equipped with such algorithms and generally adjust automatically as a function of θ. Next time you perform XRD measurements on your sample and fell the higher 2θ data collection are getting on your nerves. Remember, it is for the best [47].

Time is also an important factor to consider while data collection [48]. It is necessary to record suitable counts; therefore more time should be spent between each 2θ step. It is also necessary to record the data at suitable intervals (step size) to ensure recording of good profile and peak-broadening. As a rule of thumb, there should be at least 5 data points collected across a given peak. The maximum 2θ should always be kept to as low as you can go, however at least 50 2θ degrees should be measured to ensure statistical viability of data.

Sample transparency is yet another problem. The assumption for XRD in reflection geometry is satisfied only when the sample is infinitely thick. If the sample contains only light elements, this condition might not get fulfilled at all, therefore all the following assumptions will be invalid [38, 48–50].


Preferred-orientation effects can be very difficult to eliminate, especially for flat powder specimens. If the intensities show a strong hkl dependence (e.g. all hk0 reflections are strong and all h00 weak), preferred orientation of the crystallites should be suspected. Rietveld refinement can be done with many programs which are based on March model allowing a specific crystallographic vector based refinement of preferred-orientation parameter [32]. The elimination (or minimization) of the problem experimentally is to be preferred due to the crude nature of such models. Grain and particle morphology can also play a major role in preferential orientation. For large crystallite size the randomness of orientation of sample gets diminished i.e.

inhomogeneous sample may not be a good representative of the sample itself. The sample may consist of several phases, known or unknown, and may also include amorphous material. Depending on the technique and radiation, it may be small or large (neutron diffraction), it may be flat (Bragg–Brentano geometry), or cylindrical (Debye–Scherrer technique). In case of multiphase sample or amorphous contributions specimen should be taken with considerable representation of the sample such that during the refinement process quantitative contribution of each phase can

In the length of this chapter the term "sample preparation" will be used to define collection of specimen, cleaning or remolding, mounting it on sample holder and all the processing necessary to prepare the diffracting material to its mounting on

Following few precautions are integral parts of sample preparation process

Despite the advances in current instrumentation and techniques we will not be able to obtain a 100% representative specimen from any sample, particularly powder samples. Grain size distribution, preferred orientation, inhomogeneous grain boundaries, defects and other microscopic differences will always act against it [45].

As world dives more and more into the nanoscale world, the sample thickness

Sample geometry can also not be obtained with certainty with nanoscale samples, especially with nano-morphologies and surface rough samples. A sample of 50–100 nm thickness and spiky morphology, with each spike of let us say 20 nm thickness and 50 nm length, will have so rough surface that there will be roughly

Another challenge will be the porosity of the samples. In nanomaterial samples

the surface area to volume ratio increases leading to apparent amorphicity in

In order to perform a successful Rietveld refinement, it is essential that the powder diffraction data be collected appropriately. If relative intensities or the 2θ values (d-spacing) are recorded incorrectly, no amount of time spent on refinement will lead to any sensible results. The factors to be considered for effective and

be estimated more precisely.

*Advanced Ceramic Materials*

• Sample geometry

• Sample thickness

**3.2 Current challenges**

actually crystalline samples.

**4. Data collection**

**254**

**3.1 Precautionary/explanatory notes**

• Sample homogeneity/representative specimen

• Crystalline/Amorphous nature of sample

poses a problem with 1D and 2D materials.

• Hygroscopic, gas absorbing nature and porosity of material

• Phase purity or at least the idea of chemical composition.

50% of thickness change while moving from one spike to another.

goniometer.

not all crystallite orientations are equally represented, creating a problem. In the underrepresented specimen, the preferred orientation parameter cannot be corrected at the refinement stage. Therefore the sample rotation method is strongly recommended in such cases. In smaller particle sizes, line-broadening effects due to crystallite size begin to become apparent which evidently decreases the intensity of peaks. The presence of large crystallites within such samples will cause the peaks from smaller particle size to be relatively very low or even reduced to background. In such cases also, the correction to preferred orientation parameter cannot be applied.

Another parameter to be considered in the diffractometer is to keep background to maximum peak ratio as low as possible.

Monochromatic radiation is to be preferred for all XRD measurements. Although longer data-acquisition times are required with monochromatic radiation, its use is particularly advantageous both in number of lines and the background observed.

Any temptation to smooth the diffraction data before doing a Rietveld refinement must be resisted. Smoothing introduces point-to-point correlations which will give falsely lowered estimated standard deviations in the refinement process.

The wavelength and zero offset should be calibrated with a reference material. The Si SRM 640b standard gives significantly broadened peaks, whereas the NIST LaB6 standard SRM 660 gives close to instrumental resolution and is probably a better choice.

**4.2 Challenges**

*(b) data collected after 2 hr. grinding.*

**Figure 3.**

**257**

**5. Background contribution**

The continuous motion of either or both detector and source arms of goniometer

*SnO2: (a) the observation of preferred orientation due to poor particle distribution while sample preparation*

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

*DOI: http://dx.doi.org/10.5772/intechopen.96065*

As discussed in previous section, the data collection should essentially be optimized to obtain least background. However, in practicality there are many possible unavoidable, yet necessary and characteristic reasons where background cannot be

minimized after a certain degree without degradation of peak data quality. Although for pure phase materials, the background essentially remains negligible, till the particle or crystallite sizes are greater than 100 nm and grain boundaries are insignificant. For multiphase materials, the relative intensity difference between the peaks of different phases due to preferred orientation, crystallite size difference, peak broadening, quantitative presence, and sometimes amorphous phase do make

background contributions a part of the X-ray reflection geometry [51].

Basically, the background contributions are dealt in two different ways in a powder diffraction pattern. Background can be modeled by an empirical/semiempirical polynomial function with several refinable parameters or it can be estimated and at the end subtracted by a linearly interpolated set of points. Background subtraction although seems inelegant, is more sophisticated in circumstances where polynomial function cannot describe the background well. The normal procedure for background estimation should be an initial estimation using polynomial function, followed by (if required) linear interpolation and subtraction. This method is supposed to both preserve the estimates of standard deviations and correct for the background contribution optimally. It should also be noted that if a polynomial function does not describe the background well, no amount of refinement of its

and the recording transit time of the cameras are one of the bigger challenges modern x-ray diffractometers face. Although the introduction of step size has essentially eliminated this problem, there are still concerns regarding too close and two far step sizes. Both can affect the peak geometry and background contribution in more effectual way. Too close step sizes, lower symmetry phases/peak splitting are bad combinations. Wide step sizes and nano-materials/GI mode/multiphase samples are also bad combinations. The time dependence of step size choice and effective counting times are the current limiting factors for diffractometers.

The example of over, normal and under collection of data is shown in **Figure 2 (a), (b)** and **(c)** respectively, while the presence of preferred orientation and normal XRD pattern of SnO2 are shown in **Figure 3a** and **b** respectively.

### **4.1 Precautions and explanations**

Specimen should be chosen in such a way that it represents the sample in every possible way (or at least nearly every way).

Uniform surface and thickness should be maintained across the sample.

In case of suspected preferred orientation, it should be a practice to repeat the experiment with newly prepared sample or specimen.

Many materials undergo phase transformation on exposure to humidity, Carbon Monoxide etc. In such cases, care should be taken to minimize the exposure.

Leveling of sample holder is essential to get an initial 2θ estimate.

In case of grazing incidence (GI) mode especially the background to peak height ratio is generally very low, therefore data collection is trickier. In order to minimize external errors thin film surfaces should be cleaned of any debris. Dust or other organic residues can sometimes reduce the quality of data by either hindering the path of beam or decreasing the intensity of peak recordings. This in some extreme cases can lead to inferences like oriented films, amorphous growth or preferred orientation errors.

**Figure 2.** *The data (a) over collection (b) under collection (c) normal collection conditions for XRD data.*

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics DOI: http://dx.doi.org/10.5772/intechopen.96065*

**Figure 3.**

not all crystallite orientations are equally represented, creating a problem. In the underrepresented specimen, the preferred orientation parameter cannot be corrected

recommended in such cases. In smaller particle sizes, line-broadening effects due to crystallite size begin to become apparent which evidently decreases the intensity of peaks. The presence of large crystallites within such samples will cause the peaks from smaller particle size to be relatively very low or even reduced to background. In such cases also, the correction to preferred orientation parameter cannot be applied. Another parameter to be considered in the diffractometer is to keep background

Monochromatic radiation is to be preferred for all XRD measurements. Although longer data-acquisition times are required with monochromatic radiation, its use is particularly advantageous both in number of lines and the background observed. Any temptation to smooth the diffraction data before doing a Rietveld refinement must be resisted. Smoothing introduces point-to-point correlations which will give falsely lowered estimated standard deviations in the refinement process.

The wavelength and zero offset should be calibrated with a reference material. The Si SRM 640b standard gives significantly broadened peaks, whereas the NIST LaB6 standard SRM 660 gives close to instrumental resolution and is probably a

Specimen should be chosen in such a way that it represents the sample in every

Many materials undergo phase transformation on exposure to humidity, Carbon

In case of grazing incidence (GI) mode especially the background to peak height ratio is generally very low, therefore data collection is trickier. In order to minimize external errors thin film surfaces should be cleaned of any debris. Dust or other organic residues can sometimes reduce the quality of data by either hindering the path of beam or decreasing the intensity of peak recordings. This in some extreme cases can lead to inferences like oriented films, amorphous growth or preferred

Uniform surface and thickness should be maintained across the sample. In case of suspected preferred orientation, it should be a practice to repeat the

Monoxide etc. In such cases, care should be taken to minimize the exposure. Leveling of sample holder is essential to get an initial 2θ estimate.

*The data (a) over collection (b) under collection (c) normal collection conditions for XRD data.*

The example of over, normal and under collection of data is shown in **Figure 2 (a), (b)** and **(c)** respectively, while the presence of preferred orientation and normal XRD pattern of SnO2 are shown in **Figure 3a** and **b** respectively.

at the refinement stage. Therefore the sample rotation method is strongly

to maximum peak ratio as low as possible.

*Advanced Ceramic Materials*

**4.1 Precautions and explanations**

possible way (or at least nearly every way).

experiment with newly prepared sample or specimen.

better choice.

orientation errors.

**Figure 2.**

**256**

*SnO2: (a) the observation of preferred orientation due to poor particle distribution while sample preparation (b) data collected after 2 hr. grinding.*

### **4.2 Challenges**

The continuous motion of either or both detector and source arms of goniometer and the recording transit time of the cameras are one of the bigger challenges modern x-ray diffractometers face. Although the introduction of step size has essentially eliminated this problem, there are still concerns regarding too close and two far step sizes. Both can affect the peak geometry and background contribution in more effectual way. Too close step sizes, lower symmetry phases/peak splitting are bad combinations. Wide step sizes and nano-materials/GI mode/multiphase samples are also bad combinations. The time dependence of step size choice and effective counting times are the current limiting factors for diffractometers.

### **5. Background contribution**

As discussed in previous section, the data collection should essentially be optimized to obtain least background. However, in practicality there are many possible unavoidable, yet necessary and characteristic reasons where background cannot be minimized after a certain degree without degradation of peak data quality. Although for pure phase materials, the background essentially remains negligible, till the particle or crystallite sizes are greater than 100 nm and grain boundaries are insignificant. For multiphase materials, the relative intensity difference between the peaks of different phases due to preferred orientation, crystallite size difference, peak broadening, quantitative presence, and sometimes amorphous phase do make background contributions a part of the X-ray reflection geometry [51].

Basically, the background contributions are dealt in two different ways in a powder diffraction pattern. Background can be modeled by an empirical/semiempirical polynomial function with several refinable parameters or it can be estimated and at the end subtracted by a linearly interpolated set of points. Background subtraction although seems inelegant, is more sophisticated in circumstances where polynomial function cannot describe the background well. The normal procedure for background estimation should be an initial estimation using polynomial function, followed by (if required) linear interpolation and subtraction. This method is supposed to both preserve the estimates of standard deviations and correct for the background contribution optimally. It should also be noted that if a polynomial function does not describe the background well, no amount of refinement of its

coefficients or increase in its order can fix the problem. In such cases for a complete and satisfactory refinement process, the estimation of background should be skipped and linear interpolation and subtraction procedure should be followed. While background is generally eliminated in refinement process, the peak base shapes are essentially a part of background and therefore at higher 2θ, more care should be taken in estimating the background. This is why background fitting using linear interpolation by cubic-splines should be generally avoided. The asymmetric peak shape especially at higher 2θ (where peak intensities are generally low) and non-careful background estimation or subtraction can affect the relative intensity of peaks and therefore degrade the overall refinement quality.

complete set. During refinement and background estimation/subtraction, precau-

We are essentially in a nano-technological world right now and most of the materials applications around us have transitioned from bulk to micro to nanoscale. The complexities associated with the nanoscale XRD have also risen noticeably [52–55]. Nanoscale background contribution, irregular peak shapes, non-correctable preferred orientation/asymmetry parameters, sometimes odd combination of Lorentzian and Gaussian peak parameters. The porosity and reduced dimensionality (especially, 1D,

The peak shape is one of the most important parameters in Rietveld refinement due to its dependence on crystallite/domain size, stress/strain, defects/vacancies, source/geometry, slit-size/detector resolution and 2θ/hkl indices [55]. An accurate description of the shapes of the peaks in a powder pattern is critical to the success of a Rietveld refinement. Poor description can lead to unsatisfactory refinement results, false minima and divergence. Peak shape analysis/function is the most complex parameter in Rietveld refinement, with dimensions into the space of unattainable and non-realistic. It is therefore essential for a working algorithm to make some assumptions/compromises on peak shape and sometimes neglect the otherwise essential aspect of peak shape. For x-ray and constant wavelength neutron data, the use of pseudo-Voigt approximated peak function is widely used. The pseudo-Voigt function is essentially a combination of Lorentzian and Gaussian peak

tions need to be taken for segregating peak bases from background.

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

2D materials) are very difficult to characterize via normal XRD procedures.

**5.2 Current challenges**

*DOI: http://dx.doi.org/10.5772/intechopen.96065*

**6. Peak-shape function**

function in a linear mode [30, 31, 56–61].

*<sup>p</sup>* ffiffiffiffi

Where *<sup>G</sup>* <sup>¼</sup> *<sup>e</sup>*�*x*2*=*2*p*<sup>2</sup>

With

shown in **Figure 5**.

**259**

with m = 1 for symmetry.

Voigt function is mathematically defined as

The pseudo-Voigt function is described as

*V x*ð Þ¼ , *p*, *y*

ðþ<sup>∞</sup> �∞

<sup>2</sup>*<sup>π</sup>* <sup>p</sup> is Gaussian function and *<sup>L</sup>* <sup>¼</sup> *<sup>y</sup>*

0<*η*<1

*η* is the full width half maximum parameter and the ratio of Gaussian and

The graphical representation of the pseudo-Voigt function with variable *η* is

Pearson VII peak-shape function (**Figure 6**) is used alternatively where the exponent m (Eq. 1) varies differently, but the same trends in line shape are

observed. Although the Gaussian and Lorentzian components of Voigt function can be devolved into meaningful physical interpretations of stress/strain, microstructure and line broadening effects, no such interpretation can be drawn from Pearson

Lorentzian functions *η=*ð Þ 1 � *n* determines the mixing of these functions.

*G x*ð Þ , *p L x* � *x*<sup>0</sup> ð Þ , *y dx*<sup>0</sup> (1)

*Vp*ð Þ¼ *x*, *f ηL x*ð Þþ , *f* ð Þ 1 � *η G x*ð Þ , *f* (2)

*<sup>π</sup> <sup>x</sup>*2þ*y*<sup>2</sup> ð Þ*<sup>m</sup>* is Lorentzian function

**Figure 4(a), (b)** and **(c)** respectively show contribution of amorphous, nanoscale and micrometer-scale phase towards background in LaMnO3 samples.

### **5.1 Precautions/explanations**

More time spent on measurement less significant background. This is somewhat misleading the background does not actually change with increased time spent per step. It is the increase in the number and intensity of counts per peak that increases which visibly smoothens the background. The precautions for background contribution during data collection have been discussed previously are almost entirely

### **Figure 4.**

*XRD pattern for (a) mostly amorphous, (b) nanoscale and (c) micrometer-scale phase of LaMnO3. The hump visible in (a) is a characteristic of amorphous phase, while the noisy background in (b) is characteristic of nanoscale phase due to low intensity counts.*

complete set. During refinement and background estimation/subtraction, precautions need to be taken for segregating peak bases from background.
