Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 145-147

Section Quantitative analysis

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Quantitative analysis

| top | pdf | Crystallinity

| top | pdf |

The crystallinity of a material influences many of its characteristics, including mechanical strength, opacity and thermal properties. Crystallinity measurement provides valuable information for both materials research and quality control in materials processing. The diffraction pattern from a material containing both amorphous and crystalline solids has a broad feature from the amorphous phase and sharp peaks from the crystalline phase. The weight percentage of the crystalline phases in a material containing both crystalline and amorphous phases can be determined by X-ray diffraction (Chung & Scott, 1973[link]; Alexander, 1985[link]; Murthy & Barton, 2000[link]; Kasai & Kakudo, 2005[link]). Assuming that the X-ray scattering intensity from each phase in such a material is proportional to its weight percentage, and that the scattering intensities from all phases can be measured within a given 2θ range, the per cent crystallinity is given by[{x_{\rm pc}} = 100\% {{{I_{\rm crystal}}} \over {{I_{\rm crystal}} + {I_{\rm amorphous}}}},\eqno(2.5.90)]where xpc is the per cent crystallinity, Icrystal is the integrated intensity of all crystalline peaks and Iamorphous is the integrated intensity of the amorphous scattering. The accuracy of the measured per cent crystallinity depends on the integrated diffraction profile. Since most crystalline samples have a preferred orientation, it is very difficult to obtain a consistent measurement of crystallinity with a conventional diffractometer. Fig. 2.5.28[link] shows a 2D diffraction frame collected from an oriented polycrystalline sample. The diffraction is in transmission mode with the X-ray beam perpendicular to the plate sample surface. Fig. 2.5.28[link](a) shows a diffraction profile integrated from a horizontal region analogous to a profile collected with a conventional diffractometer. Only one peak from the crystalline phase can be observed in the profile. It is also possible that a different peak or no peak is measured if the sample is loaded in other orientations. Fig. 2.5.28[link](b) is the diffraction profile integrated from the region covering all peaks from the crystalline phase over almost all azimuthal angles. A total of four peaks from the crystalline phase are observed. This shows that a 2D-XRD system can measure per cent crystallinity more accurately and with more consistent results (Pople et al., 1997[link]; Bruker, 2000[link]) than a conventional system.

[Figure 2.5.28]

Figure 2.5.28 | top | pdf |

2D diffraction pattern from an oriented polycrystalline polymer sample. (a) Diffraction profile integrated from a horizontal region analogous to a profile collected with point detector. (b) Diffraction profile integrated from all parts of the 2D frame. Crystallite size

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The size of the crystallites in a polycrystalline material has a significant effect on many of its properties, such as its thermal, mechanical, electrical, magnetic and chemical properties. X-ray diffraction has been used for crystallite-size measurement for many years. Most methods are based on diffraction-line broadening and line-profile analysis (Wilson, 1971[link]; Klug & Alexander, 1974[link]; Ungár, 2000[link]). Another approach to crystallite-size measurement is based on the spotty diffraction rings collected with two-dimensional detectors when a small X-ray beam is used (Cullity, 1978[link]; He, 2009[link]). Line-profile analysis is based on the diffraction profile in the 2θ direction, while crystallite-size analysis with a spotty 2D diffraction pattern is based on the diffraction profile in the γ direction. The latter may be referred to as γ-profile analysis.

Fig. 2.5.29[link](a) shows a diffraction profile collected from gold nanoparticles and regular gold metal. The 2θ profile from the gold nanoparticles is significantly broader than the profile from regular gold metal. The crystallite size can be calculated by measuring the broadening and using the Scherrer equation:[B = {{C\lambda } \over {t\cos \theta }},\eqno(2.5.91)]where λ is the X-ray wavelength (in Å), B is the full width at half maximum (FWHM) of the peak (in radians) corrected for instrumental broadening and strain broadening, θ is the Bragg angle, C is a factor, typically from 0.9 to 1.0, depending on the crystallite shape (Klug & Alexander, 1974[link]), and t is the crystallite size (also in Å). This equation shows an inverse relationship between crystallite size and peak-profile width. The wider the peak is, the smaller the crystallites. The 2θ diffraction profiles can be obtained either by using a conventional diffractometer with a point or line detector, or by γ integration from a diffraction pattern collected with 2D detector. When a 2D detector is used, a long sample-to-detector distance should be used to maximize the resolution. A small beam size and low convergence should also be used to reduce instrument broadening.

[Figure 2.5.29]

Figure 2.5.29 | top | pdf |

Crystallite-size analysis: (a) 2θ profile of a gold nanoparticle (grey) and regular gold metal (black); (b) γ profile of LaB6; (c) measurement range.

Fig. 2.5.29[link](b) shows a frame collected from an SRM660a (LaB6) sample with a 2D-XRD system. The spotty diffraction rings are observed with average crystallite size of 3.5 µm. The number of spots in each diffraction ring is determined by the crystallite size and diffraction volume. Introducing a scaling factor covering all the numeric constants, the incident-beam divergence and the calibration factor for the instrument, we obtain an equation for the crystallite size as measured in reflection mode:[d = k\left\{p_{hkl}b^2\arcsin [\cos \theta \sin (\Delta \gamma /2)] \over \mu N_s \right\}^{1/3},\eqno(2.5.92)]where d is the diameter of the crystallite particles, phkl is the multiplicity of the diffracting planes, b is the size of the incident beam (i.e. its diameter), Δγ is the γ range of the diffraction ring, μ is the linear absorption coefficient and Ns is the number of spots within Δγ. For transmission mode, we have[d = k\left\{ {p_{hkl}b^2t\arcsin [\cos \theta \sin (\Delta \gamma /2)] \over N_s} \right\}^{1/3},\eqno(2.5.93)]where t is the sample thickness. In transmission mode with the incident beam perpendicular to the sample surface, the linear absorption coefficient affects the relative scattering intensity, but not the actual sampling volume. In other words, all the sample volume irradiated by the incident beam contributes to the diffraction. Therefore, it is reasonable to ignore the absorption effect [\exp (\mu t)] for crystallite-size analysis as long as the sample is thin enough for transmission-mode diffraction. The effective sampling volume reaches a maximum for transmission-mode diffraction when [t = 1/\mu ].

For both reflection and transmission,[k = \left({3\beta \over 8\pi } \right)^{1/3},\eqno(2.5.94)]where β is the divergence of the incident beam. Without knowing the precise instrumental broadening, k can be treated as a calibration factor determined from the 2D diffraction pattern of a known standard. Since only a limited number of spots along the diffraction ring can be resolved, it can be seen from equation (2.5.94)[link] that a smaller X-ray beam size and low-multiplicity peak should be used if a smaller crystallite size is to be determined.

Fig. 2.5.29[link](c) shows the measurement ranges of 2θ-profile and γ-profile analysis. The 2θ-profile analysis is suitable for crystallite sizes below 100 nm (1000 Å), while γ-profile analysis is suitable for crystallite sizes from as large as tens of µm down to 100 nm with a small X-ray beam size. By increasing the effective diffraction volume by translating the sample during data collection or multiple sample integration (or integrating data from multiple samples), the measurement range can be increased up to millimetres. Multiple sample integration can deal with large crystallite sizes without recalibration. The new calibration factor is given as[k_n = n^{1/3}k,\eqno(2.5.95)]where n is the number of targets that are integrated. The number of crystallites can be counted by the number of intersections of the γ profile with a threshold line. Every two intersections of the γ profile with this horizontal line represents a crystallite. In order to cancel out the effects of preferred orientation and other material and instrumental factors on the overall intensity fluctuation along the γ profile, one can use a trend line fitted to the γ profile by a second-order polynomial. It is always necessary to calibrate the system with a known standard, preferably with a comparable sample geometry and crystallite size. For reflection mode, it is critical to have a standard with a comparable linear absorption coefficient so as to have similar penetration.


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