Page 296 - Book Hosokawa Nanoparticle Technology Handbook

5.2 CRYSTAL STRUCTURE FUNDAMENTALS
XRD method to study on nanoparticles. The range of by internal strain in the crystallites and the spectral
crystallite size that can be evaluated by powder XRD width of the source X-ray. Integral breadth is defined by
method depends on the accuracy of the instrument the division of the integrated intensity by the peak inten-
and analytical methods. Since the guaranteed preci- sity. The ratio of the integral breadth to the full width at
sion of a commercial X-ray diffractometer is better half maximum (FWHM) of the Lorentzian (Cauchy-
2 1
than 0.01°, evaluation of line broadening larger than distribution) function: f (x) 1 [1+ ( x/ ) ] is
Lor
0.1° will sufficiently be evaluated. The broadening of equal to ( /2). Since the following relation is derived
0.1° roughly corresponds to the crystallite size for from equation (5.2.2),
frequently used CuK of the wavelength 0.15 nm as

follows, (180° 0.15nm) (0.10° ) 90nm. It cos sin , (5.2.3)
Y
X
means that crystallite size under 100 nm can quan-
titatively be evaluated with a usual powder diffrac- the parameters and are estimated as the intersect
Y
X
tometer. On the other hand, it is often difficult to and slope of a straight line fitted to the plot of cos
distinguish a diffraction peak from the neighbor versus sin evaluated from the experimental data
peak or from background scattering, when the width (Williamson–Hall plot). When the integral breadth is
of the peak is larger than several degrees, which cor- used to specify the diffraction peak width, the
responds to the crystallite size of several nm. Therefore, volume-weighted average size (the ratio of the aver-
we can conclude that the powder XRD method is espe- age of the fourth power of the diameter D to the aver-
cially suitable for evaluating crystallite size ranging age of the third power of the diameter) for spherical
from several to several tens nanometers, while the crystallites can be calculated from the value of by
X
methodology of XRD-based evaluation of crystallite using the following equation,
size is now aiming at the size of several hundreds
nanometers [2]. D 4 4
D
Since the experimentally observed width and shape D 3 3 (5.2.4)
V
of diffraction peaks are affected by the instrument and X
measurement conditions, the detailed experimental where radian is used as the unit of the angle.
parameters should properly be taken into account, if Recently, it has been reported that evaluation of the
accurate evaluation of crystallite size is needed. The broadness of statistical distribution of crystallite size
basic resolution of the instrument is evaluated by can be evaluated as well as the average crystallite size
measuring the powder diffraction profile of a well- by fitting a theoretical peak profile calculated by
crystallized sample as standard material. The effect of assuming some distribution models to the experimen-
the instrument can be removed from the observed data tal diffraction peak profile [2, 9, 10]. When the spher-
by Fourier deconvolution of an empirically obtained ical or ellipsoidal shape of crystallites is assumed, the
instrumental function in the Stokes’ method [3], diffraction peak profile of a crystallite with the diam-
where part of the sample is annealed at high tempera- eter D along the diffraction vector is given by the fol-
tures and used as the standard sample to determine the lowing equation,
instrumental resolution. More advanced methods,
where the effects of instruments are predicted by a D 4
priori calculation based on the geometry of the optics p sphere s (; ) ( sD) (5.2.5)
D
2
and diffraction condition [4, 5], and a Fourier-based 6
deconvolution method, applicable straight to the
2
wide-angle diffraction data have been proposed [6]. () 3 ⎡ 1 2 sin x 4 sin (x 2 ) ⎤ ⎥ (5.2.6)
x
There may be intrinsic difficulty in the evaluation x 2 ⎢ ⎣ x x 2 ⎦
of crystallite size by powder diffraction method, when
the observed diffraction peak profile is strongly for the deviation of the diffraction vector from the peak
affected by strain or fault in the crystallites. However, value, s = 2 (sin – sin )/ . The theoretical peak pro-
0
the effects of smallness of the crystallites and the file of the spherical or ellipsoidal crystallites with the
internal strain can be separated by using the method size distribution modeled by the log-normal distribution
of Warren–Averbach [7] or Williamson–Hall [8]. In (median m and logarithmic standard deviation ):
the Williamson–Hall method, the following depend-
ence of the integral breadth on the diffraction angle 1 ⎡ (ln D ln m) ⎤
2
⎢
D m,)
is assumed, f LN (; exp ⎥ (5.2.7)
D
2 ⎣ 2 2 ⎦

X tan (5.2.2) is given by
Y
cos
where the first term of the above equation means the

s m, )
s D f
broadening caused by finite size of crystallites, and the P SLN (; ∫ p sphere (; ) LN ( D m, ) dD (5.2.8)
;
second term can be attributed to the broadening caused 0
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