**9. Refinement procedure**

It is difficult to cover all the details of a full refinement, but an approximate strategy can be described. It is generally advised to begin the structural refinement first with the positions of the heavier atoms and then extend the refinement to positions of lighter atoms. It should however be always kept in mind that the statistical minima can sometimes attribute unrealistic positions to the atoms. All atomic positions, with constraints in place, can be refined simultaneously upon convergence. The scale, the thermal and the occupancy parameters are more


**Table 1.**

*The chi (goodness of fit), and other Rietveld reliability factors (explanation of each factor in "R-factor" section ahead).*

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

sensitive to the background correction due to their correlated nature. Positional parameters are somewhat independent of background. In order to reduce the number of thermal parameters to be refined in early stage, it is advisable to constrain the thermal parameters of similar atoms. Chemical constraints should be applied to maintain the physical sense of occupancy parameters. Refining a single structure using two independent data-sets e.g. x-rays and neutron diffraction the parameter correlation can be minimized. However, the experimental conditions for data collections such as pressure, temperature etc. in each case should be as similar as possible. Refinement of the profile parameters along with the structural parameters is also advisable. The structural model should be refined to convergence while care should be taken to retain the physical and chemical sense wherever applicable. Mere convergence with even a single parameter not making physical or chemical sense is all the efforts wasted. It is therefore necessary to always follow a certain procedure/ pathway of refinement or at least at the earlier stages of refinement. The likely procedure of refinement pathway is given in **Figure 11**.

Because powder diffraction data are a one-dimensional projection of threedimensional data, the inherent loss of information is always a problem. To partly compensate for this loss geometric information (bond distances and/or angles) taken from related structures is more appropriate method. The purpose of these constraints is to increase the number of observations by added geometric conditions. Another way to implement restraints is to follow rigid body model, this however results in decrease in the number of observations and complicating the structural model. The use of geometric restrains not only increases the number of observations but allows more parameters to be refined, while keeping the geometry of the structural model sensible. The set of geometric restraints can be treated as separate data set, with same rules of quantity minimization in the refinement. The geometric data set can be represented as:

$$\mathcal{S} = \mathcal{S}\_{\mathcal{Y}} + \mathcal{c}\_{w}\mathcal{S}\_{G} \tag{4}$$

where *Sy* is the weighted difference between the observed *[y (obs)]* and calculated *[y(calc)]* diffraction patterns,

$$\mathcal{S}\_{\mathcal{Y}} = \sum\_{i} w\_{i} \left[ \mathcal{y}\_{i}(obs) - \mathcal{y}\_{i}(calc) \right]^{2} \tag{5}$$

SG is the weighted difference between the prescribed *[G(obs)]* and calculated *[G(calc)]* geometric restraints,

$$S\_G = \sum w[G(obs) - G(calc)]^2\tag{6}$$

and *cw* is a factor that allows a weighting of the geometric observations 'data-set' with respect to the diffraction data-set.

Geometric restraints can enhance a refinement considerably, allowing otherwise impossibly complex structures to be refined successfully. However care must to choose the bond distance and angles in order to accommodate the appropriate polyhedral geometry. It is imperative that the final structure model should fit both the geometric and the X-ray data satisfactorily.

### **9.1 Quantitative refinement**

The methodology involved in qualitative and quantitative Rietveld refinements have been discussed at length by many authors [26, 27, 49, 72]. The theory behind

in later sections. It is important to know the source of errors in the refinement procedure for a effective and concise results. The most common error that occurs is due to the noisy data. The noisier the data the more refinement is needed for background parameter, this can sometimes lead to convolution of peak bases into background especially at higher angles. Zero shift and sometimes step size can also cause a range of errors to creep in. it is therefore a common procedure to first correct the data for zero shift and choose a more incredulous step size at the time of data collection. Apart from these, we need to look out for most of the other errors while the refinement process is underway. Sometimes lower estimated standard deviations can result from false minima observed due to unavailability of suitable

*The zoomed in view of peak at around 57o from Figure 9 to visualize the goodness of fit.*

56.0 56.7 57.4 58.1 58.8

2θ

structural model or unrealistic positional parameters (**Table 1**) [42, 43].

It is difficult to cover all the details of a full refinement, but an approximate strategy can be described. It is generally advised to begin the structural refinement first with the positions of the heavier atoms and then extend the refinement to positions of lighter atoms. It should however be always kept in mind that the statistical minima can sometimes attribute unrealistic positions to the atoms. All atomic positions, with constraints in place, can be refined simultaneously upon convergence. The scale, the thermal and the occupancy parameters are more

**χ<sup>2</sup> Rwp Rexp RF RBragg** 1.25 2.65 2.11 2.29 3.58

*The chi (goodness of fit), and other Rietveld reliability factors (explanation of each factor in "R-factor" section*

**9. Refinement procedure**

**Figure 10.**

*Advanced Ceramic Materials*

**Table 1.**

*ahead).*

**264**

The detailed discussion of the mathematical and physical interpretations of these quantities can be found abundantly in literature, particularly in the cited works

In a mixture, the intensity of *hkl* reflection originating from a particular phase (*α*)

*=*2*μ<sup>m</sup>*

Where *C<sup>α</sup>* is the volume fraction of *α* phase with *μ<sup>m</sup>* as linear absorption coefficient In terms of weight fractions, which is statistically more convenient, the equation

> *<sup>K</sup> <sup>ρ</sup><sup>m</sup>* 2*μ<sup>m</sup>*

> > *<sup>K</sup> <sup>ρ</sup><sup>m</sup>* 2*μ<sup>m</sup>*

*W<sup>α</sup>* þ *W<sup>β</sup>*

*Sαρα* þ *Sβρβ*

� �*R<sup>α</sup>*,*hkl* (8)

*R<sup>α</sup>*,*hkl* (9)

(10)

(11)

(12)

(13)

*I<sup>α</sup>*,*hkl* ¼ *CαK* <sup>1</sup>

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

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

*<sup>I</sup><sup>α</sup>*,*hkl* <sup>¼</sup> *<sup>W</sup><sup>α</sup> ρα*

> *<sup>S</sup><sup>α</sup>* <sup>¼</sup> *<sup>W</sup><sup>α</sup> ρα*

For second phase (*β*), the weight fraction can be done similarly while the net

*<sup>W</sup><sup>α</sup>* <sup>¼</sup> *<sup>W</sup><sup>α</sup>*

The equation can be solved by replacing weight fractions by equation above

The numerical way of observing the quality/goodness of fit, although not as prudent as graphical visualization of difference plots, provides a good, intuitive numerical estimate. This is usually done in terms of agreement indices also called Residual values or Retiveld refinement indices or Rietveld discrepancy indices or R

The weighted profile *R* values (*Rwp*) is most straight forward an follows directly from the square root of minimized quantity, scaled using weighted intensities and is

( )1

where *yi(obs)* is the observed intensity, *yi(calc)* the calculated intensity, and *wi*

*=* X *i*

*wi yi* ð Þ *obs* � �<sup>2</sup> *=*2

*<sup>W</sup><sup>α</sup>* <sup>¼</sup> *<sup>S</sup>αS<sup>β</sup>*

As scale parameters are refined we will get estimated weight fraction

Now the scale factor for alpha phase can be written as

contribution per phase can be sought from the equation below

[72, 82–84].

is written as

can be written as

contribution of each phase.

values [26, 27, 85–87] which are expressed as.

*Rwp* <sup>¼</sup> <sup>X</sup>

*<sup>t</sup>*<sup>h</sup> step.

*i*

*wi yi*

ð Þ� *obs yi* ð Þ *calc* � �<sup>2</sup>

*9.2.1 The weighted-profile* R*-value*

**9.2 R values**

defined as:

the weight at *i*

**267**

### **Figure 11.**

*The procedure typically followed during the refinement of XRD data via Rietveld method. The green boxes are optional calculations. The variation of procedure is necessary in cases with amorphous phases, anomalous reflections or sample induced asymmetry.*

Rietveld quantitative analysis is identical to that implemented in most conventional quantitative analyses [34, 73–76]. The integrated intensity of X-rays diffracted by a randomly oriented infinitely thick [40, 76–81] polycrystalline sample in flat-plate geometry can be written for a particular reflection as:

$$I\_{hkl} = K \left( \mathbb{1}\_{2\mu} \right) R\_{hkl} \tag{7}$$

Where *K* and *Rhkl* are the *hkl* invariant and variant parameters.

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

The detailed discussion of the mathematical and physical interpretations of these quantities can be found abundantly in literature, particularly in the cited works [72, 82–84].

In a mixture, the intensity of *hkl* reflection originating from a particular phase (*α*) is written as

$$I\_{a,bkl} = \mathbb{C}\_a K \left( \mathbb{1}\_{\mathfrak{I}\_m} \right) \mathbb{R}\_{a,bkl} \tag{8}$$

Where *C<sup>α</sup>* is the volume fraction of *α* phase with *μ<sup>m</sup>* as linear absorption coefficient In terms of weight fractions, which is statistically more convenient, the equation can be written as

$$I\_{a,bkl} = \frac{W\_a}{\rho\_a} K \frac{\rho\_m}{2\mu\_m} R\_{a,bkl} \tag{9}$$

Now the scale factor for alpha phase can be written as

$$S\_a = \frac{W\_a}{\rho\_a} K \frac{\rho\_m}{2\mu\_m} \tag{10}$$

For second phase (*β*), the weight fraction can be done similarly while the net contribution per phase can be sought from the equation below

$$W\_a = \frac{W\_a}{W\_a + W\_\beta} \tag{11}$$

The equation can be solved by replacing weight fractions by equation above

$$\mathcal{W}\_a = \frac{\mathbb{S}\_a \mathbb{S}\_\beta}{\mathbb{S}\_a \rho\_a + \mathbb{S}\_\beta \rho\_\beta} \tag{12}$$

As scale parameters are refined we will get estimated weight fraction contribution of each phase.

### **9.2 R values**

The numerical way of observing the quality/goodness of fit, although not as prudent as graphical visualization of difference plots, provides a good, intuitive numerical estimate. This is usually done in terms of agreement indices also called Residual values or Retiveld refinement indices or Rietveld discrepancy indices or R values [26, 27, 85–87] which are expressed as.

### *9.2.1 The weighted-profile* R*-value*

The weighted profile *R* values (*Rwp*) is most straight forward an follows directly from the square root of minimized quantity, scaled using weighted intensities and is defined as:

$$R\_{wp} = \left\{ \sum\_{i} w\_i \left[ y\_i(obs) - y\_i(calc) \right]^2 / \sum\_{i} w\_i \left[ y\_i(obs) \right]^2 \right\}^{\natural\_2} \tag{13}$$

where *yi(obs)* is the observed intensity, *yi(calc)* the calculated intensity, and *wi* the weight at *i <sup>t</sup>*<sup>h</sup> step.

Rietveld quantitative analysis is identical to that implemented in most conventional quantitative analyses [34, 73–76]. The integrated intensity of X-rays diffracted by a randomly oriented infinitely thick [40, 76–81] polycrystalline sample in flat-plate

*The procedure typically followed during the refinement of XRD data via Rietveld method. The green boxes are optional calculations. The variation of procedure is necessary in cases with amorphous phases, anomalous*

Phase contribution percentage

Quantitative analysis

Geometric Constraints

Scale factor correction

Lattice Parameter refinement

Zero Shift correction

Background correction

Peak shape, asymmetric and Orientation correction

Refine Atomic coordinates, anisotropic temp……

> Geometric and strain calculations

Individual phase weighted contribution

Background, Peak Shape,

Preparation of XRD data

Pre-Refinement Profile and Structure Model

*=*2*μ*

*Rhkl* (7)

Micro-strain and grain size analysis

WH, Debye Scherrer analysis….

*Ihkl* ¼ *K* <sup>1</sup>

Where *K* and *Rhkl* are the *hkl* invariant and variant parameters.

Qualitative analysis

Structure visualization

geometry can be written for a particular reflection as:

Post-Refinement

*reflections or sample induced asymmetry.*

**Figure 11.**

**266**

Refinement

*Advanced Ceramic Materials*

The numerator in Eq. (13) is the expression that is minimized during a Rietveld refinement procedure. Thus the inclusion or exclusion of background can have dramatic effect on the refinement. If the background has been excluded, and thus subtracted prior to refinement then, *yi(obs)* is the net intensity. However, the inclusion of background means the refinement of background pramaters. In such cases, *yi(obs)* includes both background and net intensity. Therefore, *yi(obs)* and *yi(calc)* both will likely include the background contribution. In the latter case when dealing with a high background to peak intensity ratio, most of intensity will be attributed to background, resulting in lowered value of *Rwp*. Therefore it is recommended to subtract background in such cases. *Rwp* for laboratory X-ray data are large �10%. This is primarily due to the level of the background. In any publication, the type of agreement index used must be clearly specified. Ideally, the final *Rwp* should approach the statistically expected *R* value or *Rexp.*

### *9.2.2 The expected R-value*

*Rexp* reflects both the quality of data and refinement and is expressed as

$$R\_{\exp} = \left\{ (N - P) / \sum\_{i} w\_i \left[ \mathbb{y}\_i (obs) \right]^2 \right\}^{\mathbb{M}} \tag{14}$$

where *N* is the number of observations and *P* the number of parameters.

However, the ratio between the *Rw*<sup>p</sup> and *Rexp*, called goodness of fit (*χ 2* ), which is quoted quite often in the literature, should approach 1.

$$\mathbf{G}^2 = \chi^2 = R\_{wp}/R\_{exp} \tag{15}$$

*RB* <sup>¼</sup> <sup>X</sup> *hkl*

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

*hkl, m* is multiplicity.

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

ii. Does the structural model make chemical sense?

iii. Are inter-atomic distances and angles realistic?

NMR etc. characterizations?

**9.3 Common problems during refinement**

more general nature and arise in many cases.

The background does not seem to fit well

ii. Try background subtraction

iii. Try combination of (i) and (ii)

The peak shapes are not suitably fitting

ii. Use a different peak-shape function

University of Ioannina.

**269**

Where *Ihkl* <sup>¼</sup> *mF*<sup>2</sup>

refinement are

j j *Ihkl*ð Þ� *obs Ihkl*ð Þ *calc =*

*R* values are useful indicators for the evaluation of a refinement, especially

in the case of small improvements to the model which are not generally visible in difference plots. However, care should be taken while evaluating the *R* values as they are prone to over-interpretation. The most important questions that need to be asked for judging the quality of a Rietveld

i. Is the fit between observed data and calculated pattern good?

iv. Are the results from the refinement consistent with results from Raman, IR

Each structure refinement has its own idiosyncrasies and will present problems that require imaginative and selective solutions. However, some problems are of a

The most frequent source of difficulty in a Rietveld refinement is error in the input file. Most of these errors if occurring due to format or syntax can be corrected by conversion of files into suitable format using software like PowDLL from

i. Try a different background function, increase the number co-efficient,

i. Check the difference plot and match with the **Figures 12(a)**-**(c)** to see if one of the characteristic difference profiles is shown. The respective profile

iv. Line broadening and shifting along with *2θ* dependence of FWHM can

change from linear to polynomial or vice versa [19]

parameter should be reset or further refined [20, 21]

iii. Perform asymmetry correction to the peak-shape function.

indicate microstructure contributions [89–91].

X *hkl*

j j *Ihkl*ð Þ *obs* (17)

Most of the statistical errors in these R values can occur either due to undercollection or over –collection of data. The ratio will be less than one if data is under collected as *Rexp* will be much higher than *Rwp*. In case of over-collection the ratio will be greater than 1. It is always recommended to have over-collected rather than undercollected data. As estimated standard deviations [88] an also alter the ratio, there are other *R* values like *RF* and *RBragg* which will improve the conclusivity of the data.

### *9.2.3 The structure factor* R *value*

An *R* value based on structure factors, *Fhkl*, can also be calculated by distributing the intensities of the overlapping reflections according to the structural model.

$$R\_F = \sum\_{hkl} |F\_{hkl}(obs) - F\_{hkl}(calc)| / \sum\_{hkl} |F\_{hkl}(obs)| \tag{16}$$

*RF* a derivative of structure factors is essentially biased towards the structural model. It can however give a clear indication of the reliability of structural refinement. Although not used actively while reporting the refinement of structure, it should necessarily decrease as the structural model improves in the course of the refinement.

### *9.2.4 The Bragg intensity* R *value*

The Bragg-intensity *R* value (*RB*) is essentially the structure factor *RF* but in terms of Intensity *Ihkl*:

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

$$R\_B = \sum\_{hkl} |I\_{hkl}(obs) - I\_{hkl}(calc)| / \sum\_{hkl} |I\_{hkl}(obs)| \tag{17}$$

Where *Ihkl* <sup>¼</sup> *mF*<sup>2</sup> *hkl, m* is multiplicity.

The numerator in Eq. (13) is the expression that is minimized during a Rietveld refinement procedure. Thus the inclusion or exclusion of background can have dramatic effect on the refinement. If the background has been excluded, and thus subtracted prior to refinement then, *yi(obs)* is the net intensity. However, the inclusion of background means the refinement of background pramaters. In such cases, *yi(obs)* includes both background and net intensity. Therefore, *yi(obs)* and *yi(calc)* both will likely include the background contribution. In the latter case when dealing with a high background to peak intensity ratio, most of intensity will be attributed to background, resulting in lowered value of *Rwp*. Therefore it is

recommended to subtract background in such cases. *Rwp* for laboratory X-ray data are large �10%. This is primarily due to the level of the background. In any publication, the type of agreement index used must be clearly specified. Ideally, the

*Rexp* reflects both the quality of data and refinement and is expressed as

where *N* is the number of observations and *P* the number of parameters. However, the ratio between the *Rw*<sup>p</sup> and *Rexp*, called goodness of fit (*χ*

Most of the statistical errors in these R values can occur either due to undercollection or over –collection of data. The ratio will be less than one if data is under collected as *Rexp* will be much higher than *Rwp*. In case of over-collection the ratio will be greater than 1. It is always recommended to have over-collected rather than undercollected data. As estimated standard deviations [88] an also alter the ratio, there are other *R* values like *RF* and *RBragg* which will improve the conclusivity of the data.

An *R* value based on structure factors, *Fhkl*, can also be calculated by distributing

X *hkl*

j j *Fhkl*ð Þ *obs* (16)

the intensities of the overlapping reflections according to the structural model.

j j *Fhkl*ð Þ� *obs Fhkl*ð Þ *calc =*

*RF* a derivative of structure factors is essentially biased towards the structural model. It can however give a clear indication of the reliability of structural refinement. Although not used actively while reporting the refinement of structure, it should necessarily decrease as the structural model improves in the course of the refinement.

The Bragg-intensity *R* value (*RB*) is essentially the structure factor *RF* but in

X *i*

( )1

*wi yi* ð Þ *obs* � �<sup>2</sup> *=*2

*<sup>G</sup>*<sup>2</sup> <sup>¼</sup> *<sup>χ</sup>*<sup>2</sup> <sup>¼</sup> *Rwp=Rexp* (15)

(14)

), which

*2*

final *Rwp* should approach the statistically expected *R* value or *Rexp.*

*Rexp* ¼ ð Þ *N* � *P =*

is quoted quite often in the literature, should approach 1.

*9.2.2 The expected R-value*

*Advanced Ceramic Materials*

*9.2.3 The structure factor* R *value*

*9.2.4 The Bragg intensity* R *value*

terms of Intensity *Ihkl*:

**268**

*RF* <sup>¼</sup> <sup>X</sup> *hkl*

*R* values are useful indicators for the evaluation of a refinement, especially in the case of small improvements to the model which are not generally visible in difference plots. However, care should be taken while evaluating the *R* values as they are prone to over-interpretation. The most important questions that need to be asked for judging the quality of a Rietveld refinement are


### **9.3 Common problems during refinement**

Each structure refinement has its own idiosyncrasies and will present problems that require imaginative and selective solutions. However, some problems are of a more general nature and arise in many cases.

The most frequent source of difficulty in a Rietveld refinement is error in the input file. Most of these errors if occurring due to format or syntax can be corrected by conversion of files into suitable format using software like PowDLL from University of Ioannina.

The background does not seem to fit well


The peak shapes are not suitably fitting


c. Is background refinement realistic and sensible?

iii. Check for oscillations in the parameter shifts and apply damping factors as.

iv. Do not refine two parameters with high correlation together. Sometimes

viii. Fix thermal (atomic displacement) parameters at certain sensible values

x. The number of parameters being refined is higher than what data can provide

The final structure is not chemically sensible (unrealistic inter-atomic distances)

ii. Delete the offending atoms and try relocating them using Fourier maps

v. Fix atomic thermal displacements and fractional coordinates in the

Refinement converged but there are few peaks which are not fitted well

The most important aspect of a Rietveld analysis is the refinement of structure. The actual structure of the sample can be calculated taking into consideration the

ii. Apply absorption correction parameter if data permits [95]?

i. Check for Lorentz–polarization correction

iii. Are atomic fractional co-ordinates correct?

iv. Is there preferred orientation in the sample?

Most modern refinement software perform this automatically

the high correlation is an indication of wrong space group

vii. If geometric restraints are already in use, are they correct?

i. Use restraints to keep inter-atomic distance sensible.

d. Is the scale factor correct?

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

v. Refine fewer parameters initially

vi. Add geometric restraints

ix. Use a different space group.

[54, 59, 92, 93]

beginning

**10. Structural visualization**

**271**

iii. Change restraints [94]

iv. Change the space group

ii. Has structural model been completely described?

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

### **Figure 12.**

*(a) a good peak fit (b) Observed intensities are higher than calculated and (c) Observed intensities are lower than calculated (in both cases, possibly any of these might require to be reset or further refinement, (i) scale factor, (ii) preferred orientation, (iii) lattice parameters).*

The peak positions in the calculated and observed patterns do not match


The tails of the peaks in the calculated pattern are cut off prematurely

i. Increase the peak range used in the calculation

The relative intensities of a few reflections are high with very few low peaks

i. This is usually indicative of rock in dust problem concerned with poor particle statistics. The only solution is to recollect the data after proper sample preparation

There multiple un-indexed peaks in the diffraction pattern


The refinement does not converge

	- a. Are the observed peak shapes well defined by peak shape function?
	- b. Is there any mismatch between peak positions?

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


The final structure is not chemically sensible (unrealistic inter-atomic distances)


Refinement converged but there are few peaks which are not fitted well


### **10. Structural visualization**

The most important aspect of a Rietveld analysis is the refinement of structure. The actual structure of the sample can be calculated taking into consideration the

The peak positions in the calculated and observed patterns do not match

3.0 3.5 4.0 4.5 5.0 3.0 3.5 4.0 4.5 5.0 3.0 3.5 4.0 4.5 5.0

*(a) a good peak fit (b) Observed intensities are higher than calculated and (c) Observed intensities are lower than calculated (in both cases, possibly any of these might require to be reset or further refinement, (i) scale*

(a) (b) (c)

iii. Determine the unit-cell parameters via independent indexing methods

The relative intensities of a few reflections are high with very few low peaks

i. This is usually indicative of rock in dust problem concerned with poor particle statistics. The only solution is to recollect the data after proper

ii. Check whether the infinite sample thickness condition was fulfilled during

a. Are the observed peak shapes well defined by peak shape function?

i. Look at the observed/calculated profiles carefully and check these.

b. Is there any mismatch between peak positions?

The tails of the peaks in the calculated pattern are cut off prematurely

i. Check if unit cell parameters are correct

i. Increase the peak range used in the calculation

There multiple un-indexed peaks in the diffraction pattern

ii. Perform Zero shift refinement

*factor, (ii) preferred orientation, (iii) lattice parameters).*

**Figure 12.**

*Advanced Ceramic Materials*

sample preparation

data collection

**270**

i. Check for sample impurity

The refinement does not converge

iii. Check for peaks from sample holder

lattice parameter variations, the microstructure, stress strain contribution and other contributions. Effectively most of the currently available software for Rietveld refinement can easily generate the refined structure file. Visualization of structure at higher resolution has become easy with enhanced computational power. However the presentation of the structure is not standardized and most of the time the axial orientation is not mentioned. Although, it is not essentially a problem for the readers, the standard representation of the structure should be preferred. In cases where a non-standard representation is used, mention of plane, axial orientation, etc. should be clearly mentioned [90, 96]. The non standard representation of the structure can sometimes lead to wrong conclusions as shown in **Figure 13** for ZnO.

Apart from the problems discussed above, the tetrahedral and octahedral geometry should be visualized carefully (**Figure 14**). The actual polyhedral tilting, rotations or other geometric variations can be truly visualized only after symmetrised unit cell representation [97–99]. The **Figure 15(a)** and **(b)** show ZnO structure in symmetrised and non-symmetrised form. The difference in visualization is quite amazing [34, 100, 101].

**11. Recommended software packages**

*meaningful in case of primitive lattice (Table 2).*

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

1.X-ray diffraction data visualization:

b. Panalytical X'Pert Highscore

d. Cyrstal Impact Match [103]

2. Inter-Conversion of XRD data between different formats

*The non-standard viewing and primitive unit cell of wurtzite ZnO ( -Zn and -O). The atomic*

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

*arrangement is not quite effectively legible and physically meaningful in non-standard viewing, while it is quite*

a. Winplotr [102]

c. X-Powder

**Figure 15.**

e. PowderPlot

a. X-powder

c. Winplotr

c. X-Powder

a. EXPO2014

4. Indexing

**273**

b. PowDLL [104]

3. Search and Match with database

b. Cyrstal Impact Match

a. PCPDFWIN from ICDD

d. Panalytical X'Pert Highscore [105]

*(a) The standard view and (b) c\* axial view of hexagonal ZnO ( -Zn and -O) unit cell with non-standardized atomic positions.*

### **Figure 14.**

*The c\* axial view of hexagonal ZnO unit cell with standardized atomic positions. The transformation of structure to represent the hexagonal arrangement of -Zn and -O atoms is effectively visible within a single unit cell.*

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

### **Figure 15.**

lattice parameter variations, the microstructure, stress strain contribution and other contributions. Effectively most of the currently available software for Rietveld refinement can easily generate the refined structure file. Visualization of structure at higher resolution has become easy with enhanced computational power. However the presentation of the structure is not standardized and most of the time the axial orientation is not mentioned. Although, it is not essentially a problem for the readers, the standard representation of the structure should be preferred. In cases where a non-standard representation is used, mention of plane, axial orientation, etc. should be clearly mentioned [90, 96]. The non standard representation of the structure can sometimes lead to wrong conclusions as shown

Apart from the problems discussed above, the tetrahedral and octahedral geometry should be visualized carefully (**Figure 14**). The actual polyhedral tilting, rotations or other geometric variations can be truly visualized only after symmetrised unit cell representation [97–99]. The **Figure 15(a)** and **(b)** show ZnO structure in symmetrised and non-symmetrised form. The difference in visualization is quite

*(a) The standard view and (b) c\* axial view of hexagonal ZnO ( -Zn and -O) unit cell with*

*The c\* axial view of hexagonal ZnO unit cell with standardized atomic positions. The transformation of structure to represent the hexagonal arrangement of -Zn and -O atoms is effectively visible within a single unit cell.*

(a) (b)

in **Figure 13** for ZnO.

*Advanced Ceramic Materials*

amazing [34, 100, 101].

**Figure 13.**

**Figure 14.**

**272**

*non-standardized atomic positions.*

*The non-standard viewing and primitive unit cell of wurtzite ZnO ( -Zn and -O). The atomic arrangement is not quite effectively legible and physically meaningful in non-standard viewing, while it is quite meaningful in case of primitive lattice (Table 2).*
