Condensed Matter II: Particle Size Broadening Benjamen P. Reed & Liam S. Howard IMAPS, Aberystwyth University March 19, 2014 Abstract Particles of 355µm silicon oxide(quartz)were subjected to a ball milling process to reduce their average particle size to a fine nanometre range. Samples of 355µm and fine quartz were then characterised using x-ray di ractometry and the Bragg peak broadening was recorded. Information about the peaks positions and full-width half-maximums were substituted into the Scherrer equation to calculate the average particle size of the fine quartz, and hence demonstrate the e ect of particle size broadening in x-ray di raction patterns. Steps were also taken to correct the Scherrer analysis for instrumental broadening. By assuming that the fine quartz particles were spherical in shape, an average particle size of 146.36nm was calculated. It was shown however that the particles are oblate in shape with a larger average crystallite size in the directions perpendicular to the (110) and (11 2) lattice planes.
1 Introduction & Background Theory Characterisation of crystalline materials in the laboratory can be achieved using x-ray di ractometry, a process where specific x-ray wavelength photons are directed at a sample, and the resulting scattered radiation is recorded for analysis. By detecting the radiation intensity at varying angles in a one-dimensional (1D) plane, a di raction pattern can be produced that provides a lot of information regarding the sample s crystallographic structure, i.e. unit cell dimensions, placement of atomic constituents, and the space group of the lattice. Another useful piece of information that a di ractogram reveals, is the size of the crystallites or particles in the sample. Peaks in the 1D di ractogram are derived from the constructive interference of specific orientated plane reflections in the sample, i.e. when the photons satisfy the Bragg condition. The Bragg equation relates the angle of the scattered radiation to the wavelength of the incident electromagnetic photons, and the spacing between the di racting lattice planes, such that... n =2d sin( )...where n is the order of the scattered radiation, is the incident radiation wavelength, d is the di raction plane spacing, and is the angle of the incident radiation. The sharpness of a di raction peak is dependant on the number of photons being di racted by that specific orientation of lattice plane. Clearly, the more lattice planes there are of that orientation, the larger the number of constructively interfering photons, and therefore a sharper, more intense peak will result (Kittel, 1996). Figure 1 illustrates this concept. Figure 1: A larger crystallite (a) will have more lattices planes in a particular direction than a smaller crystallite (b). More lattice planes in a specific direction will give rise to sharper, more intense peaks (a), as opposed to broader, less intense peaks (b). 1
This peak broadening, termed particle size broadening, is due to number of di racting lattice planes can hence be used in reverse to determine the mean size of the ordered crystalline domains or particles. The Scherrer equation uses this concept to provide an approximation of crystallite size using the position and broadening of the peak, such that... = K cos( )...where is the mean size of the ordered domain or crystallite; K is a dimensionless shape factor with typical values close to unity; is the incident radiation wavelength; is the fullwidth at half-maximum (FWHM) of the peak; and is angular position of the peak. The Scherrer equation can only be applied to crystallites with an average size less than 0.1µm, i.e. nanoparticles; for crystallites much larger than this value, the Bragg peak broadening is too small to quantify accurately with this method (Monshi et al.,2012). When measuring Bragg peak broadening, it is important to identify all the possible sources of broadening. Instrumental broadening instr takes into account e ects from sample transparency, wavelength dispersion, and detector resolution among others. Most of these causes can be ignored if the di ractometry equipment is calibrated correctly, however some broadening will still remain through other means, e.g. when the particles being analysed are too large. In the case where the Scherrer formula is applied to peaks from a sample with microparticles, the x-rays cannot penetrate through to the opposite side of the particles. With fewer lattice planes contributing to peak construction, broadening is observed which can be assigned as the instrumental broadening in this case (Bish & Post, 1989). To calculate an accurate average crystallite size, it is imperative that the instrumental broadening is corrected using the equation... particle = sample instr...where sample is the observed peak FWHMs of the sample di raction pattern, and particle is the actual FWHM due to particle size broadening. Substituting this equation into the Scherrer equation provides corrected values of the average crystallite size, such that... = K ( sample instr )cos( ). This report details an experiment that was conducted to demonstrate the e ect of particle size broadening on di raction patterns, and to form a connection between this phenomenon and the average crystallite sizes of a test sample. Silicon oxide, commonly known as quartz, was used for this purpose for its well-defined Bragg peaks, and it exists in a powdered form which could be ball-milled to reduce the average particle size considerably. 2
2 Experimental Methods This experiment compares the crystalline properties of two samples of quartz. In order to exhibit particle size broadening, the two samples must contain particles on noticeably di erent length scales. The first sample contains particles of 355µm quartz. Toproduceasamplewith areducedparticlesize,the355µm quartzwasrunthroughaballmillparticlegrinder. The 355µm quartz sample and a ceramic sphere (the ball) were placed inside a ceramic capsule that breaks away into a cylindrical ring and two end pieces cushioned by a pair of rubber inner-rings. This capsule was clamped securely into place, with an empty dummy capsule on the opposite arm to balance the ball mill. The mill has a variety of modes of rotation and vibration which are quickly cycled through whilst it is active. This rapid motion causes the ball within the capsule to collide randomly with the crystal particles; this chaotic grinding motion reduces the coarse crystal particles to a fine powder with average particle sizes in the nanometre range. The 355µm quartzparticlesweremeasuredout(3.074g)andgroundintheballmillfor a total of 195 minutes. The quartz was ground into a fine powder that was extricated from the capsule before being prepared to be used in the di ractometer. The 355µm quartzand fine quartz particles were placed onto Perspex sample holders, with care taken not to induce macrostrain or a preferential direction in the samples. Each sample was then placed into a Bruker D8 Advance x-ray di ractometer, and a coarse di ractogram was recorded (i.e. 20-80 at 50 per minute). Information about the Bragg peaks was ascertained using the program Di rac.suite EVA, which was able to identify the peaks, determine their position in 2, and calculate the FWHMs. 3 Results Table 1: Comparison of FWHMs of 335µm & fine Silicon Oxide (Quartz) hkl Angle (2 ) d-spacing FWHM ( sample ) FWHM ( instr ) FWHM ( particle ) 100 20.86 4.25 A 0.213 0.130 0.083 011 26.63 3.34 A 0.203 0.102 0.101 110 36.54 2.46 A 0.173 0.127 0.046 102 40.28 2.38 A 0.165 0.085 0.080 11 1 42.43 2.28 A 0.205 0.083 0.122 200 45.75 2.15 A 0.209 0.088 0.121 201 50.12 2.01 A 0.212 0.092 0.120 11 2 54.87 1.88 A 0.173 0.090 0.083 Table 2: Average crystallite size calculation using the Scherrer formula hkl sample instr sample instr 100 44.30 nm 72.63 nm 113.81 nm 011 48.62 nm 96.78 nm 108.72 nm 110 63.49 nm 86.50 nm 239.00 nm 102 70.09 nm 136.03 nm 144.60 nm 11 1 58.31 nm 143.99 nm 98.00 nm 200 60.50 nm 143.68 nm 104.50 nm 201 64.91 nm 149.55 nm 114.70 nm 11 2 88.65 nm 170.35 nm 176.30 nm 3
4 Figure 2: Di raction pattern of 355µm Silicon Oxide (Quartz) demonstrating instrumental broadening.
5 Figure 3: Di raction pattern for fine nanoparticle Silicon Oxide (Quartz) demonstrating particle size broadening.
4 Discussion Figure 2 shows the di raction pattern for the unmilled sample of silicon oxide. This sample had a known average particle size of 355µm, and it was assumed that the particles were spherical in shape. As expected, there is some degree of peak broadening as the particles have a finite size and cannot contribute enough lattice planes to produce an infinitely narrow peak. As discussed in section 1, this is termed instrumental broadening, and by using the FWHM value of each peak, this broadening can be quantified. Figure 3 shows the di raction pattern for the fine nanoparticle silicon oxide, and clearly shows some degree of extra peak broadening in addition to the instrumental broadening from figure 2. This extra broadening is a direct result of the milling process, as the original microparticles have been ground into a finer powder consisting of nanoparticles. Figure 1 demonstrates how nanoparticles contribute to a broader, less intense peak in a di raction pattern. In order to confirm this observation, the Scherrer equation was applied to each peak separately to ascertain a value of the average crystallite size in that specific orientation (N.B. a shape factor K =1hasbeenassumedinallcases). Itwasexpectedthatthe values for the peaks in figure 2 would be contradictory to experimental observation, i.e. the values would be much smaller than 355µm. As explained in section 1, the Scherrer equation can only be used for particles whose average crystallite size is less than 0.1µm. Therefore, the values given by the Scherrer equation in figure 2 can be considered a control to gauge instrumental broadening rather than determine crystallite sizes; it is more useful when it is used to eliminate the contribution by instrumental broadening to total peak broadening. Table 2 shows the calculated values of the average crystallite sizes for the peaks in figures 2 and 3. It also provides the actual particle size after instrumental broadening corrections have been made. Most of the crystal orientations are of equitable size, however the (110) and (11 2) orientations have larger values, suggesting that the milled particles are not spherical but rather resemble oblate spheroids. Were it to be assumed the milled particles were still spherical, then they would have an average particle size of approximately 146.36nm, considerably smaller than the original quartz particles. This estimation complements the observations made in figure 3, where clear signs of peak broadening can be seen. 5 Conclusions An experiment has been conducted to investigate the e ect of particle size broadening in 1D di raction patterns. Particles of silicon oxide, commonly known as quartz, were analysed with x-ray di ractometry to ascertain quantitative values of instrumental broadening for correction. Another sample of quartz that had undergone a ball-milling process to reduce the average particle size, was also analysed and a noticeable broadening of the original Bragg peaks was observed. By using the instrumental broadening correction and the Scherrer equation, the average particle size of the fine nanoparticle sample was calculated to be 146.36 nm assuming the particles were spherical in shape. It was however shown that the average particle size changed depending on the Bragg peak being analysed, and the (110) and (11 2) orientations gave rise to larger average particle sizes. It was suggested that the particles were therefore oblate in shape. The qualitative observations of the di raction patterns and the mathematical approaches used provide an apt demonstration of particle size broadening analysis. 6
Acknowledgements The authors would like to extend their gratitude to their module coordinator, Dr. Rudolf Winter for his advice with the data analysis and for sharing his knowledge of condensed matter physics. References Bish, D. L., & Post, J. E. 1989. Modern Powder Di raction. Mineralogical Society of America. Kittel, C. 1996. Introduction to Solid State Physics. 7th edn. New York: John Wiley & Sons, Inc. Monshi, A., Foroughi, M. R., & Monshi, M. R. 2012. Modified Scherrer Equation to Estimate More Accurately Nano-Crystallite Size Using XRD. World Journal of Nano Science and Engineering, 2, 154 160. 7