Debye-Scherrer formula

The Debye-Scherrer formula enables the thickness of a crystallite to be calculated from the peak widths ... 

The powder XRD analysis of a series of InAs nanocrystal sizes is shown in Figure 3.22, where the peak positions are seen to index well with the bulk InAs lattice reflections. Information on the domain size of the sample can be obtained from the width of the lattice reflections, using the Debye-Scherrer formula for spherical particles [52] ... 

The TEM, high-resolution TEM and FESEM images of ZnO nanoparticles and FESEM images of PS/ZnO nanocomposite films (Figure 12.9) show the surface morphology. The average particle diameters determined from the XRD peaks using the Debye-Scherrer formula are found to be close to those based on TEM and FESEM analysis. 

The plot of p cosO versus sin0 is a straight line with slope rj and intercept kX/D. This expression is a combination of the Debye-Scherrer formula for size broadening and the Stokes and Wilson expression for strain broadening. The value of t] is the strain in the crystallites, and the value of D represents the size of the crystallites. The constant k is typically close to unity and ranges from 0.8 to 1.39. The Scherrer equation is mainly used for crystallites up to 1000 A (100 nm). 

It is observed that XRD pattern consists of 18 to 20 prominent peaks. The average crystallite size of YP04 Eu was determined from XRD pattern using Debye-Scherrer formula. From all peaks, the average crystallite size of YPO4 by using this formula is estimated to be 67.29 nm [15]. 

It was well known that Williamson-Hall method is more accurate method to calculate crystallite size as compared to Debye-Scherrer method. The Fig. 2 represents the Williamson-Hall (W-H) plot for YP04 Eu nanostructure phosphor. As shown in Fig. 2, the Y- intercept is 0.0027 taking X as 0.154 nm, the grain size was found to be around 62 nm. The calculated particle size was in good concurrence with the Debye-Scherrer data. The small variation in the size of grains calculated by Debye-Scherrer and W-H method was due to the fact that in Debye-Scherrer formula strain component was assumed to be zero and the diffraction peak broadening was assumed to be due to reduced grain size only. 

The crystallite size was calculated from XRD patterns using the Debye-Scherrer formula [7], described by the Eq. (7) ... 

As a simple example, we will consider an intermediate phase which occurs in the cadmium-tellurium system. Chemical analysis of the specimen, which appeared essentially one phase under the microscope, showed it to contain 46.6 weight percent Cd and 53.4 weight percent Te. This is equivalent to 49.8 atomic percent Cd and can be represented by the formula CdTe. The specimen was reduced to powder and a diffraction pattern obtained with a Debye-Scherrer camera and CuKa. radiation. 

R. Brill has carried out a series of experiments which indicate very clearly the influence of particle size on the sharpness of the Debye-Scherrer lines for iron powder. There is quite a powerful effect, the evaluation of which would be very profitable. Scherrer has already established in the work previously mentioned that the particle sizes of colloidal gold calculated by the use of his formula agree very satisfactorily with those obtained by the Zsigmondy method of direct counting. Later, Brill compared the different relations available for the computation with one another and obtained the data contained in Table 72. This includes a comparison of the relation originally given by Scherrer with the Laue formula modified by Brill and Pelzer. Various preparations of very fine-grained iron, which are of importance in the technique of catalysis in the ammonia process, served as material. 

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