Hall plot

Transmission electron micrographs (TEM) of submicrometer-size particles show faceted particles, and selected area electron diffraction (SAED) patterns of isolated particles show that they are formed by a small number of crystallites (Fig. 9.2.14a), This result is consistent with the mean size of the crystallites, which can be inferred from the x-ray diffraction lines broadening analysis using a William-son-Hall plot (35) in order to take into account the contribution of microstrains to the line broadening. Over the whole composition range, the mean crystallite size is in the range 40-60 nm for particles with a mean diameter in the range 200-300 nm (Table 9.2.5) (33). 

Table 9.2.5 Mean Crystallite Size (D) and Mierostrains Coefficient (t ) Inferred from Williamson-Hall Plot for Co,Ni, Powders with Almost the Same Particle Size (dm)...

Table 9.2.6 Mean Crystallite Size ( >) and Microstrains Coefficient (rj) Inferred from Williamson-Hall Plot for Powders in the FeCoNi System...

The common application of the Scherrer formula in catalyst structure determination is a crude approximation to microstructural analysis. Strain and particle size give rise to the same effects, namely, line broadening, but fortunately causing different variations with diffraction angle, as shown by Equations (3) and (4). The methodologies implied by Wil-liamson-Hall plots and Warren-Averbach profile analyses provide access to the strain and size parameters in the commonly encountered case that both phenomena contribute to an experimental line broadening. 

Figure 4 Williamson-Hall plot of the two samples of Ndo.sS ro.5Mn03 indicating a larger compositional variation of sample 1...

Figure 6 Williamson-Hall plot of SrBi2Ta209 showing how the different strain influences the dielectric properties...

If the second term is considered unchanged with time, it may be dropped when plotting the integral of wellhead pressures with respect to time versus the cumulative injection. This plot, known as the Hall plot, was originally developed by Hall (1963) to analyze waterflood performance. According to Eq. 5.48, the slope of the Hall plot is... 

If an injection well is stimulated (s becomes smaller), the slope decreases if a well is damaged (s becomes larger), the slope increases. When the Hall plot is applied to a polymer injection well, if we assume s does not change, the slope increases because of higher polymer solution viscosity. Buell et al. (1990) pointed out that injection data must be plotted in the form of Eq. 5.47 to make valid quantitative calculations, and Eq. 5.49 is not appropriate when multiple fluid banks with significantly different properties exist in the reservoir. However, in practice, we may still use Eq. 5.48 in polymer injection for approximation. 

To discuss the usefulness of the Hall plot, let us first look at a simple polymer injection case. There are only two zones polymer and oil. Bo = B = 1, and s = 0. The slope is... 

The residual permeability reduction factors using the Hall plot in the test area were 1.5 to 2.5. 

Buell, R.S., Kazemi, H., Poettmann, P.H., 1990. Analyzing injectivity of polymer solutions with the Hall plot. SPERE (February), 41-46. 

The two broadening effects can be separated by using a Williamson-Hall plot [174], which combines Equations (4.7) and (4.8) to give ... 

If we plot (P cos 0)/X according to (sin 0)/X, we get a straight line with a y-intercept equal to the inverse of the size and a slope equal to the value of the microstrains. This plot is known as the Williamson-Hall plot. 

Figure 6.3. Williamson-Hall plot obtained by using peak by peak fitting of the diffraction pattern produced by the ESRF with nanocrystallized cerium oxide...

Figure 6.4. Williamson-Hall plots obtained by using the Langford method based on diffraction peak fitting of cerium oxide...

Figure 6.5. Williamson-Hall plot obtainedfrom the diffraction pattern of a non-annealed titanium oxide aerogel...

Figure 7.25. Williamson-Hall plots produced for the cross-sections of the (0006) and (00012) maps...

Figure 8.5 Williamson-Hall plots of all (001) reflections for the Pc films of the samples A to F (see Table 8.1). In aU cases the reflections of the (3-phase were evaluated, except for the sample F, for which the reflections of the a-phase were used. The FWHM values are corrected for experimental broadening.

The sinO/X-PcosO/X plots (= Hall plot) of the samples are shown in Fig.23.2 The following equation can be induced using Scherrer equation ... 

Fig. 23.2 The sin0/A-cos0/A plot (Hall plot) of the prepared samples, rj values were 1.45 x 10" (sample-1), 0.00 x 10" (sample-2) and 0.45 x 10" (sample-3), respectively...

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