Bonding and energy levels

1.5 Bonding and energy levels

If one imagines atoms being brought together uniformly to form, for example, a metallic structure, then when the distance between neighboring atoms approaches the interatomic value the outer elec- trons are no longer localized around individual atoms. Once the outer electrons can no longer be considered to be attached to individual atoms but have become free to move throughout the metal then, because of the Pauli Exclusion Principle, these electrons cannot retain the same set of quantum numbers that they had when they were part of the atoms. As a consequence, the free electrons can no longer have more than two electrons of opposite spin with a particular energy. The energies of the free electrons are distributed over a range which increases as the atoms are brought together to form the metal. If the atoms when brought together are to form a stable metallic structure, it is necessary that the mean energy of the free electrons shall be lower than the energy of the electron level in the free atom from which they are derived. Figure 1.4 shows the broadening of an atomic electron level as the atoms are brought together, and also the attendant lowering of energy of the electrons. It is the extent of the lowering in mean energy of the outer electrons that governs the stability of a metal. The equilib- rium spacing between the atoms in a metal is that for which any further decrease in the atomic spacing would lead to an increase in the repulsive interaction of the positive ions as they are forced into closer contact with each other, which would be greater than the attendant decrease in mean electron energy.

In a metallic structure, the free electrons must, therefore, be thought of as occupying a series of discrete energy levels at very close intervals. Each atomic level which splits into a band contains the same number of energy levels as the number N of atoms in the piece of metal. As previously stated, only two electrons of opposite spin can occupy any one level, so that a band can contain a

14 Physical Metallurgy and Advanced Materials

Higher level unoccupied in the free atom

Energy Valency

level

Crystal spacing

Levels of core electrons Decreasing interatomic spacing

Figure 1.4 Broadening of atomic energy levels in a metal.

y g N(E)

Figure 1.5 (a) Density of energy levels plotted against energy. (b) Filling of energy levels by electrons at absolute zero. At ordinary temperatures some of the electrons are thermally excited to higher levels than that corresponding to E max , as shown by the broken curve in (a).

maximum of 2N electrons. Clearly, in the lowest energy state of the metal all the lower energy levels are occupied.

The energy gap between successive levels is not constant but decreases as the energy of the levels increases. This is usually expressed in terms of the density of electronic states N (E) as a function of the energy E. The quantity N (E)dE gives the number of energy levels in a small energy interval dE, and for free electrons is a parabolic function of the energy, as shown in Figure 1.5.

Because only two electrons can occupy each level, the energy of an electron occupying a low energy level cannot be increased unless it is given sufficient energy to allow it to jump to an empty level at

Atoms and atomic arrangements 15 the top of the band. The energy 4 width of these bands is commonly about 5 or 6 eV and, therefore,

considerable energy would have to be put into the metal to excite a low-lying electron. Such energies do not occur at normal temperatures, and only those electrons with energies close to that of the top of the band (known as the Fermi level and surface) can be excited, and therefore only a small number of the free electrons in a metal can take part in thermal processes. The energy of the Fermi level E F

depends on the number of electrons N per unit volume V, and is given by (h 2 / 8m)(3N /πV ) 2/3 . The electron in a metallic band must be thought of as moving continuously through the structure with an energy depending on which level of the band it occupies. In quantum mechanical terms, this motion of the electron can be considered in terms of a wave with a wavelength which is determined by the energy of the electron according to de Broglie’s relationship λ = h/mv, where h is Planck’s constant and m and v are, respectively, the mass and velocity of the moving electron. The greater the energy of the electron, the higher will be its momentum mv, and hence the smaller will be the wavelength of the wave function in terms of which its motion can be described. Because the movement of an electron has this wave-like aspect, moving electrons can give rise, like optical waves, to diffraction effects. This property of electrons is used in electron microscopy (Chapter 4).

Worked example

State the equation for the energy of an electron which is confined to a one-dimensional infinite potential well of width L.

with the first excited state for L = 0.2 nm. (Planck’s constant = 6.62 × 10 −34 J s, electron mass m = 9.1 × 10 −31 kg.)

Solution

Electron energy is E =

8mL 2

, where n is the quantum number.

E(n = 2) = 4 × E(n = 1) = 4 × 9.4 = 37.6 eV

= E(n = 2) − E(n = 1) = 28.2 eV.