PROPERTIES
3.2. STRUCTURE 1 Atomic Structures
To understand a nanomaterial we must, first, learn about its structure, meaning that we must determine the types of atoms that constitute its building blocks and how these atoms are arranged relative to each other. Most nanostructures are crystalline, meaning that their thousands of atoms have a regular arrangement in space on what is called a lattice, as explained in Section 2.1.2 (of Chapter 2). This lattice can be described by assigning the positions of the atoms in a unit cell, so the overall lattice arises the continual replication of this unit cell throughout space. Figure 2.1 presents sketches of the unit cells of the four crystal systems in two dimensions, and the characteristics of the parameters a , b, and for these systems are listed in the four top rows of Table 3.1. There are 17 possible types of crystal structures called space groups, meaning 17 possible arrangements of atoms in unit cells in two dimensions, and these are divided between the four crystal systems in the manner indicated in column 4 of the table. Of particular interest is the most efficient way to arrange identical atoms on a surface, and this corresponds to the hexagonal system shown in Fig.
In three dimensions the situation is much more complicated, and some particular cases were described in Chapter 2. There are now three lattice constants a , b, and for the three dimensions z, with the respective angles a, and between them
(a is between b and etc.). There are seven crystal systems in three dimensions with a total of 230 space groups divided among the systems in the manner indicated in column 4 of Table 3.1. The objective of a crystal structure analysis is to distinguish the symmetry and space group, to determine the values of the lattice constants and angles, and to identify the positions of the atoms in the unit cell.
Certain special cases of crystal structures are important for nanocrystals, such as those involving simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) unit cells, as shown in Fig. 2.3. Another important structural arrange- ment is formed by stacking planar hexagonal layers in the manner sketched in Fig. which for a monatomic (single-atom) crystal provides the highest density or closest-packed arrangement of identical spheres. If the third layer is placed directly above the first layer, the fourth directly above the second, and so on, in an A-B-A-B-
. . .
type sequence, the hexagonal close-packed (HCP) structure results.3.2. STRUCTURE
Table 3.1. Crystal systems, and associated number of space groups, in two and three dimensionsa
Dimension System Conditions Space Groups
3 3 3 Oblique Rectangular Square Hexagonal Triclinic Monoclinic Orthorhombic Tetragonal Trigonal Hexagonal Cubic (or a = b, 120) a f b , 2 3 2 13 59 68 25 21 36
are 17 two-dimensional space groups and 230 three-dimensional space groups.
If, on the other hand, this stacking is carried out by placing the third layer in a third position and the fourth layer above the first, and so forth, the result is an
. . .
sequence, and the structure is FCC, as explained in Chapter 2 . The.
latter arrangement is more commonly found in nanocrystals.Some properties of nanostructures depend on their crystal structure, while other properties such as catalytic reactivity and adsorption energies depend on the type of exposed surface. Epitaxial films prepared from FCC or HCP crystals generally grow with the planar close-packed atomic arrangement just discussed. Face-centered cubic crystals tend to expose surfaces with this same hexagonal two-dimensional atomic array.
3.2.2.
To determine the structure of a crystal, and thereby ascertain the positions of its atoms in the lattice, a collimated beam of X rays, electrons, or neutrons is directed at the crystal, and the angles at which the beam is diffracted are measured. We will explain the method in terms of X rays, but much of what we say carries over to the other two radiation sources. The wavelength of the rays expressed in nanometers
is related to the X-ray energy E expressed in the units kiloelectronvolts through the expression
1.240
nm
E
Ordinarily the beam is fixed in direction and the crystal is rotated through a broad range of angles to record the X-ray spectrum, which is also called a diffractometer recording or X-ray-diffraction scan. Each detected X-ray signal corresponds to a coherent reflection, called a Bragg reflection, successive planes of the crystal for which Bragg's law is satisfied
2d sin =
as shown in Fig. 3.1, where d is the spacing between the planes, 6' is the angle that the X-ray beam makes with respect to the plane, is the wavelength of the X rays, and =
.
. . is an integer that usually has the value = 1.Each crystallographic plane has three indices and for a cubic crystal they are ratios of the points at which the planes intercept the Cartesian coordinate axes x, The distance d between parallel crystallographic planes with indices for a simple cubic lattice of lattice constant a has the particularly simple form
a d =
+
+
d sin
Figure 3.1. Reflection of X-ray beam incident at the angle two parallel planes separated by the distance d . The difference in pathlength for the two planes is indicated. (From C. P. Poole Jr., The Physics Handbook, Wiley, New York, 1998, p. 333.)
3.2. STRUCTURE
.
Figure 3.2. Two-dimensional cubic lattice showing projections of pairs of (1 10) and (1 20) planes (perpendicular to the surface) with the distances between them indicated.
so higher index planes have larger Bragg angles 8. Figure 3.2 shows the spacing d for and 120 planes, where the index = 0 corresponds to planes that are parallel to the direction. It is clear from this figure that planes with higher indices are closer together, in accordance with Eq. so they have larger Bragg angles from Eq. The amplitudes of the X-ray lines from different crystal- lographic planes also depend on the indices hkl, with some planes having zero amplitude, and these relative amplitudes help in identifying the structure type. For
example, for a body-centered monatomic lattice the only planes that produce observed diffraction peaks are those for which h k
+
I = an even integer, and for a face-centered cubic lattice the only observed diffraction lines either have all odd integers or all even integers.To obtain a complete crystal structure, X-ray spectra are recorded for rotations around three mutually perpendicular planes of the crystal. This provides compre- hensive information on the various crystallographic planes of the lattice. The next step in the analysis is to convert these data on the planes to a knowledge of the positions of the atoms in the unit cell. This can be done by a mathematical procedure called Fourier transformation. Carrying out this procedure permits us to identify which one of the 230 crystallographic space groups corresponds to the structure, together with providing the lengths of the lattice constants of the unit cell, and the values of the angles between them. In addition, the coordinates of the positions of each atom in the unit cell can be deduced.
As an example of an X-ray diffraction structure determination, consider the case of nanocrystalline titanium nitride prepared by chemical vapor deposition with the grain size distribution shown in Fig. 3.3. The X-ray diffraction scan, with the various lines labeled according to their crystallographic planes, is shown in Fig. 3.4. The fact that all the planes have either all odd or all even indices identifies the structure as face-centered cubic. The data show that has the FCC structure sketched in Fig. with the lattice constant a =
TEM histogram Lognormal fit
0 5 10 15 20
Grain size (nm)
Figure 3.3. Histogram of grain size distribution in nanocrystalline determined from a TEM micrograph. The fit parameters for the dashed curve are = 5.8 nm and = 1.71 nm. [From C. E. Krill et al., in Nalwa Vol. 2, Chapter 5, p. 207.1
Figure 3.4. X-ray diffraction scan of nanocrystalline with the grain size distribution shown in Fig. 3.3. Molybdenum radiation was used with the wavelength = 0.07093 nm calculated from Eq. (3.1). The X-ray lines are labeled with their respective crystallographic plane indices Note that these indices are either all even or all odd, as expected for a FCC structure. The nonindexed weak line near = 15” is due to an unidentified impurity. [From C. E. Krill et al., in Nalwa 2, Chapter 3, p. 200.1
3.2. STRUCTURE 41