Challenges in Rietveld Refinement and Structure Visualization in Ceramics

The most common and basic characterization in the field of material science is the almighty X-ray diffraction (XRD). In every institute, every research report and every manuscript, concerning material properties, the X-ray diffraction pattern is essentially found. Although the basis of these works relies on the fact that X-ray diffraction pattern was found to be matching with some structure in a database, the in depth significance of the various characteristic diffraction manifestations of various physical characters are rarely discussed. Most of the researchers (especially beginners) are either not aware of the prowess of X-ray based characterizations, or have not been introduced to it properly or may be sometimes they are not interested in its results at all. The decreased interest (later) in the results from such studies might be for not being productive enough for time spending or non-effectiveness in justifying the motivation of the work. The former two are more related to the availability and accessibility of study material for the development of core concepts. Most of the institutes always do not have access to the span-wide scientific literature and the researchers joining these institutions are partly affected. In this context the effective open-access and free availability of intech-open, it is prudent to at least attempt to accumulate, assimilated and aggregate the concepts related to X-ray diffraction in a single package. The chapter is an attempt in the path of this route.


Introduction
Much has been written and learnt about powder diffraction in last two decades. The journey that began in 1910 with the Bragg father-son duo publishing their first paper on crystal structure determination using ionization spectrometer, a century later there are still perks and connives that have not been widely explored [1][2][3]. The meticulous solution to the single crystal NaCl structure by the Braggs was achieved by solving symmetry equations for thousands of positions within a unit cell of unknown symmetry, without the help of modern computational prowess [3][4][5][6][7][8]. As Mike Glazer put it in very powerful words, "It was the gifted mind of Lawrence Bragg seeing symmetries in space and numbers that enabled them to reach a solution much quickly than anticipated" [3,5]. In addition, W L Bragg's consideration of diffraction from crystals as merely reflections from crystal planes, simplified the theory around the structure determination considerably [9]. In just few months, Braggs determined structure of NaCl, KCl, KBr, CaF2, Cu2O, ZnS, NaNO3, some calcites and diamond from their respective single crystals [10].
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][14][15][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][18][19][20][21][22][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 x-ray and neutron diffraction data.

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][27][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][35][36][37].
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][39][40][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 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.

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 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.
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.

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.

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 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][49][50].
i. The constant-volume assumption ii. The intensities measured at higher angles 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 lightelement material (e.g. diamond powder or glass beads).
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. 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.
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 SnO 2 are shown in Figure 3a and b respectively.

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.

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.

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. Figure 4(a), (b) and (c) respectively show contribution of amorphous, nanoscale and micrometer-scale phase towards background in LaMnO 3 samples.

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

Current challenges
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][53][54][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, 2D materials) are very difficult to characterize via normal XRD procedures.

Peak-shape function
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 function in a linear mode [30,31,[56][57][58][59][60][61].
Voigt function is mathematically defined as is Gaussian function and L ¼ y π x 2 þy 2 ðÞ m is Lorentzian function with m = 1 for symmetry.
The pseudo-Voigt function is described as With 0 < η < 1 η is the full width half maximum parameter and the ratio of Gaussian and Lorentzian functions η= 1 À n ðÞ determines the mixing of these functions. The graphical representation of the pseudo-Voigt function with variable η is shown in Figure 5.
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 VII function. Another advantage of pseudo-Voigt peak function over other functions is the separation of sample and instrument contributions.
It is also imperative to point out that pseudo-Voigt peak fitting accounts for peak base asymmetry more rigorously while Pearson VII is more inclined towards the peak  centre and Intensity offset. The property of these peak functions can be employed more efficiently by empirical evaluation of the peak shapes. Although both the functions provide similar results when the variation of peak shapes with 2θ is accounted.
The variation in chi-square fitting (χ 2 $ 0.01) of peak and base in both pseudo-Voigt and Pearson VII function can be visualized in Figure 7(a)-(d).

Profile parameters
The profile parameters include every detail that a structural model packs in, except (background, peak shape and FWHM). Although FWHM is considered a part of profile in XRD, it is necessarily a variant under peak shape function. Therefore most of the available programs for Rietveld refinement list it under profile section. Practically, Clubbing of the asymmetry parameter, preferred orientation parameter and FWHM together due to their interdependence makes more sense. The structural model which is available should be complete otherwise the calculated profile will significantly deviate. The incorrect profile parameters during refinement process generally leads to refinement of FWHM, peak asymmetry, zero shift, etc. In such cases, it is more prudent to use methods that are structure independent. Le-Bail [23,39,[62][63][64][65], Pawley [66,67] etc. are suitable for obtaining initial values of profile parameters and extract a list of integrated intensities. The integrated intensities can then be used to calculate electron scattering densities and possible structure determination. In addition, the initial parameters can be refined to obtain more agreeable profile parameters. The information like crystallite size, defect concentration, microstrain etc. which can be extracted from XRD are derived from the profile

Variation of fitting between two different shapes of peaks (triangles and circles) using (a), (b) pseudo-Voigt (red) and (c), (d) Pearson VII function (blue).
parameters. Although, the independence of profile parameters and peak shape is questionable due to their correlated nature, to make physical sense from the variation of either, the profile parameter needs a separate part in X-ray diffraction. The asymmetry in peak profile is another feature that arises from convolution of closely spaced multiple peaks. The two common and prevalent reasons for such conditions are; the evolution of microstrain and the defects. However the contributions from the instrument and the sample holder cannot be undermined. The interplay between FWHM, Peak position, shape and width is best visualized graphically (Figure 8).

Precautionary/explanatory notes
Although the basic idea about chemical composition, cell volume and density are needed to obtain a solution to an unknown phase, search using Le Bail, Pawley or ITO, DICVOL, TEROR, EXPO can always be widened to obtain initial profile parameters. However lower symmetry crystal systems like monoclinic and triclinic should not be included unnecessarily. These programs are likely to give multiple solutions to single set of reflection and it remains up to the user's judgment in these cases to choose a suitable solution. The multiple solutions are more prominent when lower symmetry systems are included, and sometimes the search criteria need to be adjusted to remove unrealistic solutions. In general, unrealistic solutions tend to possess either of the characteristics listed below or their combination:

Rietveld refinement: procedure and guidelines
After getting a complete structural model, suitable unit-cell parameters, the sufficient profile parameters and agreeable background, the Rietveld refinement of structural parameters can be started [68][69][70][71]. Refinement is usually done in sets of two to five cycles at a time but for effective refinements in simpler crystal systems hundreds of cycles will be required and thousands for complex systems. While the refinement is underway, we can monitor the progress either graphically or numerically [49]. While the numerical parameters can give us a statistical idea about the refinement, the most useful information about the profile fit is best seen graphically. However the parameter shifts are much more rigorously visualized numerically. Introduction of Reliability factors or R values by Rietveld enabled us to visualize profile fit between observed and calculated patterns more effective, although the graphical inspections still retain their superiority. The difference plot is also a good indication of the quality of profile fit, however the actual difference between the observed and calculated profiles and the origination of the deviation is not always quite evident from it. Figure 9 shows the full profile Rietveld refinement of LaMnO 3 (couple more example of Rietveld refinement are given at the end of the chapter) while Figure 10 shows the observed and calculated profile for a certain peak along with difference plots, while the corresponding R values for the whole profile are listed in the table. The increased R-values can be due to insufficient structural model, or inaccurate profile parameter. The atomic coordinates and positional parameters can cause changes in relative peak intensities at both high and low angles. The refinement of all the parameters has to be either done simultaneously or in a particular order to avoid numerical and statistical errors. The order and the explanation of the various refinable parameters and reliability factors will be done 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 structural model or unrealistic positional parameters (Table 1) [42,43].

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  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: where S y is the weighted difference between the observed [y (obs)] and calculated [y(calc)] diffraction patterns, S G is the weighted difference between the prescribed [G(obs)] and calculated [G(calc)] geometric restraints, and c w 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.

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 Rietveld quantitative analysis is identical to that implemented in most conventional quantitative analyses [34,[73][74][75][76]. The integrated intensity of X-rays diffracted by a randomly oriented infinitely thick [40,[76][77][78][79][80][81] polycrystalline sample in flat-plate geometry can be written for a particular reflection as: Where K and R hkl are the hkl invariant and variant parameters. 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][83][84].
In a mixture, the intensity of hkl reflection originating from a particular phase (α) is written as Where C α is the volume fraction of α phase with μ m as linear absorption coefficient In terms of weight fractions, which is statistically more convenient, the equation can be written as Now the scale factor for alpha phase can be written as For second phase (β), the weight fraction can be done similarly while the net contribution per phase can be sought from the equation below The equation can be solved by replacing weight fractions by equation above As scale parameters are refined we will get estimated weight fraction contribution of each phase.

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][86][87] which are expressed as.

The weighted-profile R-value
The weighted profile R values (R wp ) is most straight forward an follows directly from the square root of minimized quantity, scaled using weighted intensities and is defined as: where y i (obs) is the observed intensity, y i (calc) the calculated intensity, and w i the weight at i th step.
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, y i (obs) is the net intensity. However, the inclusion of background means the refinement of background pramaters. In such cases, y i (obs) includes both background and net intensity. Therefore, y i (obs) and y i (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 R wp . Therefore it is recommended to subtract background in such cases. R wp 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 R wp should approach the statistically expected R value or R exp .

The expected R-value
R exp 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 R wp and R exp , called goodness of fit (χ 2 ), which is quoted quite often in the literature, should approach 1.
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 R exp will be much higher than R wp . 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 R F and R Bragg which will improve the conclusivity of the data.

The structure factor R value
An R value based on structure factors, F hkl , can also be calculated by distributing the intensities of the overlapping reflections according to the structural model. R F 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
The Bragg-intensity R value (R B ) is essentially the structure factor R F but in terms of Intensity I hkl : ðÞ À I hkl calc ðÞ jj = X hkl I hkl obs ðÞ jj (17) Where I hkl ¼ mF 2 hkl ,mis multiplicity. 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 i. Is the fit between observed data and calculated pattern good?
ii. Does the structural model make chemical sense?
iii. Are inter-atomic distances and angles realistic? iv. Are the results from the refinement consistent with results from Raman, IR NMR etc. characterizations?

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 i. Try a different background function, increase the number co-efficient, change from linear to polynomial or vice versa [19] ii. Try background subtraction iii. Try combination of (i) and (ii) The peak shapes are not suitably fitting 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 parameter should be reset or further refined [20,21] ii. Use a different peak-shape function iii. Perform asymmetry correction to the peak-shape function.
iv. Line broadening and shifting along with 2θ dependence of FWHM can indicate microstructure contributions [89][90][91].  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) i. Use restraints to keep inter-atomic distance sensible.
ii. Delete the offending atoms and try relocating them using Fourier maps [54,59,92,93] iii. Change restraints [94] iv. Change the space group v. Fix atomic thermal displacements and fractional coordinates in the beginning Refinement converged but there are few peaks which are not fitted well i. Check for Lorentz-polarization correction ii. Apply absorption correction parameter if data permits [95]?
iii. Are atomic fractional co-ordinates correct?
iv. Is there preferred orientation in the sample?

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 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][98][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].

Rietveld refinement examples
The case of Rietveld refinement of perovskite LaMnO3 along with various refinement parameters is given in "Retiveld Refinement … ." section, while two     additional cases of Wurtzite ZnO and double-perovskite La 2 FeCoO 6 are given here (Figures 16 and 17, Tables 2 and 3): a. ZnO b. La 2 FeCoO 6