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average particle size (run)

diameter d and length £ has the volume V = tuPL/A. The limit L d corresponds to the shape of a disk with the area A ~ nd2/2, including both sides, to give A/V ~ 2/d. In like manner, a long cylinder or wire of diameter d and length L » d has A ~ 2nrL, and A/V ~ 4/d. Figure 9.2 provides sketches of these figures. Using the units square meters per gram, m2/g, for these various geometries we obtain the expressions

S(r)

6 x 103 pd

sphere of diameter d

(10.6a)

S(r)

6 x 103 pa

cube of side a

(10.6b)

S(r)

2 x 103 pL

thin disk, L «. d

(10.6c)

pd

long cylinder or wire d L

(10.6d)

where the length parameters a, d, and L are expressed in nanometers, and the density p has the units g/cm3. In Eq. (10.6c) the area of the side of the disk is neglected, and in Eq. (10.6d) the areas of two ends of the wire are disregarded. Similar expressions can be written for distortions of the cube [Eq. (10.6b)] into the quantum-well and quantum-wire configurations of Fig. 9.1.

The densities of types III-V and II-VI semiconductors, from Table B.5, are in the range from 2.42 to 8.25 g/cm3, with GaAs having the typical value p = 5.32 g/cm3. Using this density we calculated the specific surface areas of the nanostructures represented by Eqs. (f0.6a), (10.6c), and (10.6d), for various values of the size parameters d and L, and the results are presented in Table 10.1. The specific surface areas for the smallest structures listed in the table correspond to quantum dots (column 2, sphere), quantum wires (column 3, cylinder), and quantum wells (column 4, disk), as discussed in Chapter 9. Their specific surface areas are within the range typical of commercial catalysts.

The data tabulated in Table 10.1 represent minimum specific surface areas in the sense that for a particular mass, or for a particular volume, a spherical shape has the lowest possible area, and for a particular linear mass density, or mass per unit length, a wire of circular cross section has the minimum possible area. It is of interest to examine how the specific surface area depends on die shape. Consider a cube of side a with the same volume as a sphere of radius r so a = (47t/3)1/3r. With the aid of Eqs. (10.6a) and (10.6b) we obtain for this case ■^cub = l-245'sph, so a cube has 24% more specific surface than a sphere with the same volume.

To obtain a more general expression for the shape dependence of the area: volume ratio, we consider a cylinder of diameter D and length L with the

Table 10.1. Specific surface areas of GaAs spheres, long cylinders (wires) and thin disks as a function of their size*

Surface Area (m2/g)

Table 10.1. Specific surface areas of GaAs spheres, long cylinders (wires) and thin disks as a function of their size*

Surface Area (m2/g)

Size (nm)

Sphere

Wire

Disk

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