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Outline of Crystal Field Theory Bums, R 6 1 Introduction 1.3 Background reading 2 Bethe, H. (1929) Splitting of terms in crystals. (Termsaufspaltung in Kristallen.) Ann. Phys., 3, 133-206. [Trans[.: Cons]lltants Bureau, New York.] Outline of crystal field theory Bums, R. G., Clark, R. H. & Fyfe, W. S. (1964) Crystal field theory and applications to problems in geochemistry. In Chemistry of the Earth's Crust, Proceedings of the Vemadsky Centennial Symposium. (A. P. Vinogradov, ed.; Science Press, Moscow), 2, 88-106. [Trans/.: Israel Progr. Sci. Transl., Jerusalem, pp. 93-112 (1967).] Bums, R. .~· & Fyfe, W. S. (1967a) Crystal field theory and the geochemistry of transition elements. In Researches in Geochemistry. (P. H. Abelson, ed.; J. Wiley & Son, New York), 2, 259-85. Bums, R. G. (1985) Thermodynamic data from crystal field spectra. In Macroscopic to Microscopic. Atomic Environments to Mineral Thermodynamics. (S. W. Kieffer & A. Navrotsky, eds; Mineral. Soc. Amer. Publ.), Rev. Mineral., 14, 277-316. Orgel, L. E. (1952) The effects of crystal fields on the properties of transition metal Crystal field theory gives a survey of the effects of electric fields of ions. J. Chern. Soc., pp. 4756-61. definite symmetries on an atom in a crystal structure. Williams, R. J.P. (1959) Deposition of trace elements in a basic magma. Nature, 184, - -A direct physical confirmation should be obtainable by 44. analysis of the spectra of crystals. H. A. Bethe, Annalen der Physik, 3, 206 (1929) 2.1 Introduction Crystal field theory describes the origins and consequences of interactions of the surroundings on the orbital energy levels of a transition metal ion. These interactions are electrostatic fields originating from the negatively charged anions or dipolar groups, which are collectively termed ligands and are treated as point negative charges situated on a lattice about the transition metal ion. This is a gross simplification, of course, because sizes of anions or ligands such as 0 2-, OH-, H 0, SO/-. etc., are much larger than corresponding ionic radii 2 of cations (Appendix 3). Two effects of the crystalline field are the symmetry and the intensity of the electrostatic field produced by the ligands. The changes induced on the central transition metal ion depend on the type, positions and symmetry of the surrounding ligands. 2.2 Orbitals The position and energy of each electron surrounding the nucleus of an atom are described by a wave function, which represents a solution to the Schrodinger wave equation. These wave functions express the spatial distribu­ tion of electron density about the nucleus, and are thus related to the probabil­ ity of finding the electron at a particular point at an instant of time. The wave function for each electron, IJ'(r,O,cp), may be written as the product of four sep­ arate functions,· three of which depend on the polar coordinates of the electron 7 8 2 Outline of crystal field theory Orbitals 9 z where En is the energy of an electron with charge e; Z is the charge on the nucleus; f.l is the reduced mass of the electron (m) and the nucleus (M), with f.l = Mmi(M + m); his Planck's constant; and n is the principal quantum number relating to the average distance of the electron from the nucleus. ' ' The principal quantum number, n, can have all integral values from 1 to ' infinity. It is a measure of the energy of an electron. Thus, n = 1 is the lowest ' ' energy orbital level. As n increases, the energies of the orbitals increase, becoming less negative. In the limit of n = oo, the energy of the electron becomes zero. The electron is no longer bound to the nucleus, so that the atom becomes ionized. There is then a continuum of states with zero bonding energy and an arbitrary kinetic energy. The ionization energy corresponds to the energy difference between then= 1 and n = oo levels. / / / X / / 2.2.2 Azimuthal quantum number l The azimuthal quantum number or orbital angular momentum quantum num­ X ber, l, is related to the shape of an orbital and occurs in solutions to the EJ( 8) Figure 2.1 Polar coordinates of an electron in space. function of the wave equation. It may be regarded as representing the angular momentum of an electron rotating in an orbit, the magnitude of which corre­ illustrated in fig. 2.1. These three functions include the radial function, R(r), sponds to [l(l + 1)] 112 units of h/2n. Again, l can have only integral values, but which depends only on the radial distance, r, of the electron from the nucleus, its maximum value is limited by the value of n associated with the orbital. and two angular functionsEJ(8) and<P(</>) which depend only on the angles 8 and Thus, l has the values 0, 1, 2, ... , (n-1). In the first shell, for example, n = 1 and 'P , lf>. The fourth function, the spin function, 8 is independent of the spatial coor­ there exists only one wave function with 1 = 0. In the second shell n = 2, and dinates r, ()and <f>. Thus, the overall wave function of an electron may be written there are wave functions with l = 0 and l = 1. Similarly, in the third shell n = 3 as the product so that 1can take the values 0, 1, and 2; and so on. 'P(r,8,<f>) = R(r) EJ(()) <P(<f>) 'Ps · (2.1) Letter symbols are given to the orbitals according to the value of las fol­ lows: Surfaces may be drawn to enclose the amplitude of the angular wave function. for l = 0, 1, 2, 3, 4, .. These boundary surfaces are the atomic orbitals, and lobes of each orbital have the letter symbol is: s, p, d, f, g, .. (2.3) either positive or negative signs resulting as mathematical solutions to the Schrodinger wave equation. Thus, s orbitals are states with l = 0 and zero orbital angular momentum. The p The overall wave function and each of its components are expressed in orbitals with l = 1 have orbital angular momenta of -Y2 units of h/2n, while the terms of certain parameters called quantum numbers, four of which are desig­ d orbitals with l = 2 have .Y6 h/2nunits of orbital angular momenta. nated n, l, m1 and ms. 2.2.3 Magnetic quantum number m 2.2.1 Principal quantum number n 1 The quantum number, m , originating from the EJ( fJ) and <PC</>) functions of the Solutions to the R(r) portion of the SchrOdinger equation are expressed as 1 Schrodinger wave equation, indicates how the orbital angular momentum is 1 2n2JL 2Z 2e4 oriented relative to some fixed direction, particularly in a magnetic field. Thus, En =- n2 h2 (2.2) m1 roughly characterizes the directions of maximum extension of the electron 10 2 Outline of crystal field theory Shape and symmetry of the orbitals 11 cloud in space. The orbiting, negatively charged electron generates a magnetic field. In the presence of an applied magnetic field along one axis, designated by 2.2.5 Spin-orbit coupling convention as the z axis, interactions between the two magnetic fields cause the The interaction of the two magnetic fields, one produced by an electron spin­ orbital angular momentum vector to be constrained in specific directions. ning around its axis and the other by its rotation in an orbital around the These directions are quantized such that they can have only integral values nucleus, is termed spin-orbit coupling. States with opposed magnetic fields along the z axis. The quantum number m1 occurs in solutions to both the 8(0) associated with spin and rotation are slightly more stable than those with and tP( ¢>)functions of the SchrOdinger equation and can take all integral values aligned magnetic fields. Effects of spin-orbit coupling become increasingly from +l to -l, or (2! + 1) values in all. For example, for a given principal quan­ important with rising atomic number. They are thus more noticable in com­ tum number n and with l = 0, there is only one possible orbital, namely one pounds of Co and Ni than they are in V- and Cr-bearing phases; spin-orbit with m1 = 0. Thus, there is only one s orbital associated with each principal interactions are particularly significant in elements of the second and third quantum shell. These are designated as ls, 2s, 3s, 4s, etc., orbitals. For a given transition series. Spin-orbit coupling influences ele::tron paramagnetic reso­ nand with l = 1, there are three possible values for m1: -1, 0 and +1. Hence, nance spectra, and also contribute features to visible-region spectra of trans­ there are three different kinds of p orbitals for each principal quantum number ition metal compounds, particularly of Co2+ and Ni2+-bearing phases. (except n = 1) and they constitute the 2p, 3p, 4p, 5p, etc., orbitals. Similarly, there are five different d orbitals, since l = 2 and m1 can take the values 2, 1, 0, -1, and -2. Thus, the 3d, 4d, 5d, etc., shells each contain five d orbitals. Thef 2.3 Shape and symmetry of the orbitals orbitals, designated as 4f, 5f, etc., occur in sets of 7; g orbitals, starting with 5g, All s orbitals are spherically symmetrical because their angular wave functions occur in sets of 9; and so on. The energies of orbitals with the same n and l val­ are independent of (J and If>. They are given the group theory symmetry notation ues but different values of m1 are the same, except in the presence of a strong a1g; here, symbol a indicates single degeneracy, that is, there is ones orbital per electric or magnetic field.
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