STRUCTURE 41 The widths of the Bragg peaks of the X-ray scan of Fig 3.4 can be analyzed to

provide information on the average grain size of the sample. Since the widths arise from combinations or convolutions of grain size, microcrystalline strain, and instrumental broadening effects, it is necessary to correct for the instrumental broadening and to sort out the strain components to determine the average grain size. The assumption was made of spherical grains with the diameter D related to the volume by the expression

(3.4)

and several ways of making the linewidth corrections provided average grain size values between 10 to somewhat larger than expected from the TEM histogram of Fig. 3.3. Thus X-ray diffraction can estimate average grain sizes, but a transmission electron microscope is needed to determine the actual distribution of grain sizes shown in Fig. 3.3.

An alternate approach for obtaining the angles 8 that satisfy the Bragg condition (3.2) in a powder sample is the method sketched in Fig. 3.5. It employs a monochromatic X-ray beam incident on a powder sample generally contained in a very fine-walled glass tube. The tube can be rotated to smooth out the recorded diffraction pattern. The conical pattern of X rays emerging for each angle 28, with 8 satisfying the Bragg condition is incident on the film strip in arcs, as shown. It is clear from the figure that the Bragg angle has the value 8 = where is the distance between the two corresponding reflections on the film and R is the radius of the film cylinder. Thus a single exposure of the powder to the X-ray beam provides all the Bragg angles at the same time. The Debye-Schemer powder technique is often used for sample identification. To facilitate the identification,

diffracted rays ....

0 0

Figure 3.5. powder diffraction technique, showing a sketch of the apparatus (top), an X-ray beam trajectory for the Bragg angle (lower left), and images of arcs of the diffraction beam cone on the film plate (lower right). (From G. Burns, Solid State Physics, Academic Press, Boston, 1985, p. 81

the diffraction results for over 20,000 compounds are available to researchers in a JCPDS powder diffraction card file. This method has been widely used to obtain the structures of powders of nanoparticles.

X-ray crystallography is helpful for studying a series of isomorphic crystals, that is, crystals with the same crystal structure but different lattice constants, such as the solid solution series or where x can take on the range of values 0 x 1. For these cubic crystals the lattice constant a will depend on x since indium (In) is larger than gallium (Ga), and antimony (Sb) is larger than arsenic (As), as the data in Table B.l indicate. For this case Vegard’s law, Eq. (2.8) of Section 2.1.4, is a good approximation for estimating the value of a if x is known, or the value of x if a is known.

3.2.3. Particle Size Determination

In the previous section we discussed determining the sizes of grains in polycrystal- line materials via X-ray These grains can range from nanoparticles with size distributions such as that sketched in Fig. 3.3 to much larger micrometer-sized particles, held together tightly to form the polycrystalline material. This is the bulk or clustered grain limit. The opposite limit is that of grains or nanoparticles dispersed in a matrix so that the distances between them are greater than their average diameters or dimensions. It is of interest to know how to measure the sizes, or ranges of sizes, of these dispersed particles.

The most straightforward way to determine the size of a micrometer-sized grain is to look at it in a microscope, and for nanosized particles a transmission electron microscope (TEM), to be discussed in Section 3.3.1, serves this purpose. Figure 3.6 shows a TEM micrograph of polyaniline particles with diameters close to dispersed in a polymer matrix.

Another method for determining the sizes of particles is by measuring how they scatter light. The extent of the scattering depends on the relationship between the particle size d and the wavelength of the light, and it also depends on the polarization of the incident light beam. For example, the scattering of white light, which contains wavelengths in the range from 400 nrn for blue to 750 nrn for red, off the nitrogen and oxygen molecules in the atmosphere with respective sizes d = 0.11 and 0.12 nm, explains why the light reflected from the sky during the day appears blue, and that transmitted by the atmosphere at sunrise and sunset appears red.

Particle size determinations are made using a monochromatic (single-wavelength) laser beam scattered at a particular angle (usually for parallel and perpendicular polarizations. The detected intensities can provide the particle size, the particle concentration, and the index of refraction. The Rayleigh-Gans theory is used to interpret the data for particles with sizes d less than which corresponds to the case for nanoparticles measured by optical wavelengths. The example of a laser beam nanoparticle determination shown in Fig. 3.7 shows an organic solvent dispersion with sizes ranging from 9 to 30 nrn, peaking at 12 This method is applicable for use with nanoparticles that have diameters above 2 nm, and for smaller nanoparticles other methods must be used.

3.7. Laser Doppler measurement of size distribution of conductive polymer