**8. 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 [68–71]. 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 LaMnO3 (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

### **Figure 9.**

*Typical Rietveld refinement plot (LaMnO3) with observed (black circles), calculated (red), difference (blue) and Bragg positions (black bars).*

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,

*The interplay of peak width, shape and its effect on FWHM with (a) symmetric profile (b) asymmetric profile*

20 22 24 26 2θ

FWHM=1.486

(a) (b) (c)

20 22 24 26

FWHM=1.981

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

Peak position, shape and width is best visualized graphically (**Figure 8**).

*without significant peak shift and (c) asymmetric profile resulting in significant peak shifting.*

possess either of the characteristics listed below or their combination:

iii. The fractional atomic coordinates are unrealistic

ii. One or more of the cell parameters in extremely large/small<sup>1</sup>

iv. The number of atoms per unit cell are either very high or low

<sup>1</sup> The case of rhombohedral symmetry which is generally expressed in hexagonal axes format should be treated individually. It is normal in some materials which crystallize in rhombohedral symmetry to have large 'c/a' ratio when expressed in hexagonal axes. Few of the examples are Telurides, Selenides and lannonites.

**7.1 Precautionary/explanatory notes**

20 22 24 26

FWHM=0.997

*Advanced Ceramic Materials*

0.0

**Figure 8.**

 5.0x105

1.0x106

Intensity counts

1.5x106

i. Very large/small cell volume

v. Atomic overlapping

**262**

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

*Challenges in Rietveld Refinement and Structure Visualization in Ceramics*

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

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

where *Sy* is the weighted difference between the observed *[y (obs)]* and calcu-

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

and *cw* is a factor that allows a weighting of the geometric observations '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 methodology involved in qualitative and quantitative Rietveld refinements have been discussed at length by many authors [26, 27, 49, 72]. The theory behind

ð Þ� *obs yi*

*wi yi*

*Sy* <sup>¼</sup> <sup>X</sup> *i*

*S* ¼ *Sy* þ *cwSG* (4)

ð Þ *calc* � �<sup>2</sup> (5)

*SG* <sup>¼</sup> <sup>X</sup>*w G obs* ½ � ð Þ� *G calc* ð Þ <sup>2</sup> (6)

procedure of refinement pathway is given in **Figure 11**.

geometric data set can be represented as:

lated *[y(calc)]* diffraction patterns,

*[G(calc)]* geometric restraints,

**9.1 Quantitative refinement**

**265**

with respect to the diffraction data-set.

the geometric and the X-ray data satisfactorily.

**Figure 10.** *The zoomed in view of peak at around 57o from Figure 9 to visualize the goodness of fit.*

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].
