2p8 1f14

Figure 4.5. A comparison of the energy levels of the hydrogen atom and those of the jellium model of a cluster. The electronic magic numbers of the atoms are 2,10,18, and 36 for He, Ne, Ar, and Kr, respectively (the Kr energy levels are not shown on the figure) and 2,18, and 40 for the clusters. [Adapted from B. K. Rao et al., J. Cluster Sei. 10, 477 (1999).]

where r is the distance of the electron from die nucleus and p is the radius of the first Bohr orbit. This comes from solving the Schrodinger equation for an electron having an electrostatic interaction with a positive nucleus given by e/r. The equation of the hydrogen atom is one of the few exactly solvable problems in physics, and is one of die best understood systems in the Universe. In the case of a molecule such as the H2+ ion, molecular orbital theory assumes that the wavefunction of the electron around the two H nuclei can be described as a linear combination of the wavefunction of the isolated H atoms. Thus the wavefunction of the electrons in the ground state will have the form,

The Schrödinger equation for the molecular ion is

The symbol V 2 denotes a double differentiation operation. The last two terms in the brackets are the electrostatic attraction of the electron to the two positive nuclei, which are at distances ra and rb from the electron. For die hydrogen molecule, which has two electrons, a term for the electrostatic repulsion of the two electrons would be added. The Schrodinger equation is solved with this linear combination [Eq. (4.2)] of wavefunctions. When there are many atoms in the molecule and many electrons, the problem becomes complex, and many approximations are used to obtain the solution. Density functional theory represents one approximation. With the development of large fast computer capability and new theoretical approaches, it is possible using molecular orbital theory to determine the geometric and electronic structures of large molecules with a high degree of accuracy. The calculations can find the structure with the lowest energy, which will be die equilibrium geometry. These molecular orbital methods with some modification have been applied to metal nanoparticles.

4.2.3. Geometric Structure

Generally the crystal structure of large nanoparticles is the same as the bulk structure with somewhat different lattice parameters. X-ray diffraction studies of 80-nm aluminum particles have shown that it has the face-centered cubic (FCC) unit cell shown in Fig. 4.6a, which is the structure of the unit cell of bulk aluminum. However, in some instances it has been shown that small particles having diameters of <5 nm may have different structures. For example, it has been shown that 3-5-nm gold particles have an icosahedral structure rather than the bulk FCC structure. It is of interest to consider an aluminum cluster of 13 atoms because this is a magic number. Figure 4.6b shows three possible arrangements of atoms for the cluster. On the basis of criteria of maximizing the number of bonds and minimizing the number of atoms on the surface, as well as the fact that die structure of bulk aluminum is FCC, one might expect the structure of die particle to be FCC. However, molecular fee hep icos fee hep icos

Figure 4.6. (a) The unit cell of bulk aluminum; (b) three possible structures of Al13: a face-centered cubic structure (FCC), an hexagonal close-packed structure (HCP), and an icosahedral (ICOS) structure.

orbital calculations based on the density functional method predict that the icosahedral form has a lower energy than the other forms, suggesting the possibility of a structural change. There are no experimental measurements of the Al13 structure to verify this prediction. The experimental determination of the structure of small metal nanoparticles is difficult, and there are not many structural data available. In the late 1970s and early 1980s, G. D. Stien was able to determine the structure of Bijv, Pbjv, In;« and Agy nanoparticles. The particles were made using an oven to vaporize the metal and a supersonic expansion of an inert gas to promote cluster formation. Deviations from the face-centered cubic structure were observed for clusters smaller thaii 8 nm in diameter. Indium clusters undergo a change of structure when the size is smaller than 5.5 nm. Above 6.5 nm, a diameter corresponding to about 6000 atoms, the clusters have a face-centered tetragonal structure with a c/a ratio of 1.075. In a tetragonal unit cell the edges of the cell are perpendicular, the long axis is denoted by c, and the two short axes by a. Below ~6.5 nm the c/a ratio begins to decrease, and at 5 nm c/a — 1, meaning that the structure is face-centered cubic. Figure 4.7 is a plot of c/a versus the diameter of indium nanoparticles. It needs to be pointed out that the structure of isolated nanoparticles may differ from that of ligand-stabilized structures. Ligand stabilization refers to associating nonme-tal ion groups with metal atoms or ions. The structure of these kinds of nanos-tructured materials is discussed in Chapter 10. A different structure can result in a

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