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Band Theory, Metal

Energy Bands Band Theory About 80% of the free elements at room temperature exists in the form of metal. The conditions to form metal are vacant valence orbitals and low ionization energies. Similar to the splitting of energy levels when two or more atoms come close to each other; (See Figure 12-15.) energy levels broadened to a band (many closely spaced energy levels) for an aggregate

Figure 13-03a Energy Bands
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Figure 13-03b Band Theory
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of many atoms as shown in Figure 13-03a. Due to the "Exclusion Principle" or equivalently the "Fermi-Dirac statistics", the valence electrons from the sodium atoms cannot occupy the same energy level of each others, they fill up the energy bands up to half of the 3s band at 0oK (because the 3s level is only half filled, it can accommodate 2 electrons but there's only 1 in sodium atom), at an energy called Fermi energy Ef. Figure 13-03b shows that if there is empty levels available in the energy band, the valence electrons will be able to roam among the space in between the atoms by absorbing energy from the environment when the temperature is above 0oK. With a few exceptions, metals have a silvery-white color because they reflect all frequencies of light. They have high electrical and thermal conductivity and all metals can be drawn into wires or hammered into sheets without shattering -- that is, they are ductile and malleable. All these attributes are the result of mobile, non-rigid electron gas within the lattice. Most metals (except gold, silver, platinum, and diamond) do not occur as free elements in the Earth's crust. They are usually found in chemical combination with other elements as mineral ores. Figure 13-03b shows that in an insulator, the valence band is full and the next empty energy band is separated by a large energy gap. Conduction cannot occur unless some of the electrons in the valence band are promoted to the conduction band. Energy needed to promote a few electrons might be provided by heating the solid to a very high temperature or by shining X rays on it. No solid can remain a good insulator while it is exposed to X rays. A semiconductor has a smaller energy gap. Electrons can be promoted to the conduction band as a result of irradiation such as the conversion of sunlight to electricity by means of a silicon cell.

See "Band Structure in term of electron energy E and wave number (linear momentum) k".

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