Sodium energy bands

Sodium, energy bands as a function of atomic spacing, 179-180 sp hybrid orbital, features, 144 sp hybrid orbital, features, 144 sp hybrid orbital, features, 144-146 Spectroscopic constants asynunetric rotors, 229 diatomic molecules, 227 electronic excitation, see Electronic excitation... 

Figure 4.6 The filling of energy bands in a metal (sodium) and in an insulator (sodium chloride).

For atomic (gas) sodium (Na), the electronic configuration is ls 2s 2p 3s, leading to filled electronic energy levels Is, 2s and 2p, while the 3s level is half-filled. The other excited levels, 3p, 4s..., are empty. In the solid state (the left-hand side in Figure 4.6), these atomic energy levels are shifted and split into energy bands bands Is, 2s and 2p are fully occupied, while the 3s (/ = 0) band, the conduction band, is half-filled, so that a large number N 21 + l)/2 = N/2) of empty 3s excited levels is still available. As a result, electrons are easily excited into empty levels by an applied electric field, and so become free electrons. This aspect confers the typical metallic character to solid sodium. 

Fig. 2-7. Frontier energy bands partially occupied by electrons in metallic sodium, copper, and iron.

Figure 1.15 Energy band diagram for a sodium lattice. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc. After J. C. Slater, Phys. Rev., 45, 794 (1934).

The energy of the bottom of the sodium conduction band, denoted by 2, is determined by imposing the bonding boundary condition across the Wigner-Seitz sphere of radius, Kws, namely... 

For nonmetallic substances, the electrons cannot move as freely as in the case of metals because their energy bands are essentially completely full or empty. The electrical conductivity in nonmetallic materials is dominated by another mechanism, i.e., the defect mechanism, instead of electron conduction. In ionic crystals such as salts (e.g., sodium chloride), two types of ions, cations and anions, are driven to move by the electrical force qE once an electrical field is applied. The ions can move only by the defect mechanism that is, they exchange position with a vacancy of the same type. At the room temperature, the fraction of vacancies for salt is very small (of the order of 10-17) with low exchange frequency (of the order of 1 Hz) so that electrical conductivity is extremely low. Although impurities and high temperature can affect electrical conductivity by a large factor, nonmetallic materials generally have very low electrical conductivity and these substances are widely used as electrical insulators. 

Ferraz, A.C., Takahashi, E.K. and Leite, J.R. (1982). Application of the variational cellular method to periodic structures. Energy band of sodium, Phys. Rev. B 26, 690-700. 

Fig. XXIX-10.—Energy bands as a function of internuclear distance. The graph is drawn for metallic sodium, showing the bands that go into the 2a, 3s, and 3p levels of the atom at infinite separation. The 2s level is an x-ray level, and 3s is the valence electron level. The energy gap which appears between 3s and 3p in the figure, at distances of less than 6A, is really filled with bands from higher atomic levels.

Each atom in a bar of sodium has the same outer 3s orbital containing one electron. The individual orbitals of the atoms in the bar overlap, creating a huge number of molecular orbitals. These groups of closely spaced energy states are called energy bands. Molecular orbitals within those bands, however, must obey the Pauli exclusion principle. So each one of this huge number of... 

Elements with a valence-shell configuration 5, such as beryllium and mug nesium, might be expected to have completely filled bands and thus behave as nonmetals. However, the nearby /i-orbitals likewise form a band which overlaps the upper part of the 5-band to give a continuous conduction band with an abundance of unoccupied orbitals. Transition metals can also contribute their cf-orbitals to the conduction bands. Fig. 12.6 is a detailed plot of the band structure of metallic sodium, which shows how combinations of s, p and d energy bands can overlap. 

Figure 12.6 Energy bands in sodium as rum.non of atomic spacing. After J. C. Slater, Phys. Rev. 45 794 (1934). Dotted line rcpicscuis cqudibtium sp.tcing.

Fig. 2-11. Energy bands of sodium as a function of internuclear distance. r0 represents the actual equilibrium distance. [Reproduced by permission from J. C. Slater, Introduction to Chemical Physics, McGraw-Hill Book Co., 1939.]...

Figure 2.25 The development of energy bands from atomic orbitals for sodium metal. Isolated atoms (left-hand side) have sharp energy levels. In a solid, these are broadened into energy bands. Outermost orbitals broaden more than inner orbitals...

Figure 2.38 Energy bands of isolated atoms in sodium chloride (NaCl). As the spacing between the atoms, r, decreases, first ionisation occurs and then energy bands develop. These are narrow and widely separated, as expected from the ionic model of bonding...

Conductors (metals) have partially filled bands with no excitation gaps or the conductivity band is overlapped with the valency band, so that electrons can conduct electricity. In sofids from the monovalent alkali atom like sodium, the band containing valent electrons behaves like a conductor Only half of the levels of the isolated 3 s allowed band of sodium are filled because a sodium atom has a single electron in the 3s level, whereas the excursion principle allows such a level to accommodate two electrons. Hence, electrons in this solid can easily acquire a small amoimt of additional energy. Thus any applied electric field will be effective in giving energy to electrons, and the solid will be a conductor Conductors are also foimd in cases where valence band and conduction band overlap. 

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