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Studies Absorption Or Emission Transitions Between Electronic States of Atoms Or Molecules Chapter III: Atomic spectroscopy Electronic spectroscopy: studies absorption or emission transitions between electronic states of atoms or molecules. Atoms only electronic degrees of freedom (apart translation and nuclear spin), whereas molecules have, in addition, vibrational and rotational degrees of freedom. III.1 The periodic table For the hydrogen atom, and for the hydrogen-like ions (He+, Li++ ,…) , with a single electron in the field of a nucleus with charge +Ze, the hamiltonian is: (III.1) analogous to Equation (I.30) for the hydrogen atom, where the terms are explained. For a polyelectronic atom the hamiltonian becomes: (III.2) where the summation is over all electrons i. The first two terms are simply sums of terms like those for one electron in Equation (III.1). The third term is new and represents the Coulomb repulsions between all possible pairs of electrons distant of rij . The second term represents the Coulomb attraction between each electron and the nucleus at a distance ri . Because of the electron-electron repulsion term in Equation (III.2) the Hamiltonian cannot be broken down into a sum of contributions from each electron and the Schrödinger equation is no longer be solved exactly. Various approximate methods of solution have been devised of which the self-consistent field (SCF) method is one of the most useful. In it the Hamiltonian has the form: (III.3) He approximated the contributions to the potential energy due to electron repulsions as a sum of contributions from individual electrons. The Schrödinger equation is then soluble. Important effect of electron repulsions: removes the degeneracy of those orbitals. In the H atom, we only had degenerate n=1, 2, 3,… orbitals (Figure I.1). For multielectron atoms, we now have 2s, 2p and 3s, 3p, 3d, etc.., e.g., Figure III.1. Orbital energies Ei vary with principal quantum number n, and with angular momentum quantum number. Energy Ei of a particular orbital increases with Z. For ex., ionization energy of 1s orbital= 13.6 eV for H, 870.4 eV for Ne. Electrons in atoms may be fed into the orbitals in Figure III.1 in order of increasing energy until all are used up, to give what is referred to as the ground configuration of the atom. This feeding in of electrons in order of increasing orbital energy follows the aufbau or building-up principle, but must also obey the Pauli exclusion principle. This states that no two electrons can have the same set of quantum numbers n, l. ml = l, l-1, …, -l, i.e. it can take (2l+1) values; ms = ±½ , so each orbital can accommodate 2(2l+1) electrons: ns has 2, np has 6, nd has 10 electrons, etc. Orbital: refers to a particular set of values of n and l values Shell: refers to orbitals having same value of n. Shells with n = 1, 2, 3, 4, . are labelled K, L, M, N, ... Configurations and States: A configuration describes the way in which the electrons are distributed among various orbitals and it may give rise to more than one state as a consequence of how the angular momenta sum up (see below). E.g., the ground configuration 1s22s22p2 of the C atom gives rise to three electronic states of different energies. The ground configurations of all atoms are tabulated in all the relevant textbooks: - alkali metals, Li, Na, K, Rb and Cs, have an outer ns1 configuration consistent with their being monovalent. - He, alkaline earth metals, Be, Mg, Ca, Sr and Ba, all have an outer ns2 configuration and are divalent. - Noble (inert or rare) gases, Ne, Ar, Kr, Xe and Rn, all have an outer np6 configuration, the filled orbital (or sub-shell, as it is sometimes called) conferring chemical inertness, and the filled K shell in He has a similar effect. - The first transition series: Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn, is characterized by the filling up of the 3d orbital. Figure III.1 shows that the 3d and 4s orbitals are very similar in energy, but their separation changes along the series. The result is that, although there is a preference by most of these elements for having two electrons in the 4s orbital, Cu has a . .3d10 4s1 ground configuration because of the innate stability associated with a filled 3d orbital. There is also some stability associated with a half-filled orbital (2p3, 3 d 5, etc.) resulting in the …3d54s1 ground configuration of Cr. Filling up the 4f orbital is a feature of the lanthanides. The 4f and 5d orbitals are of similar energy so that occasionally, as in La, Ce and Gd, one electron goes into 5d rather than 4f. Similarly, in the actinides, Ac to No, the 5f subshell is filled in competition with 6d. III. 2 Vector representation of momenta and vector coupling approximations III.2.1 Angular momenta and magnetic moments The electron orbital angular momentum is a vector, the direction of which is determined by the right-hand screw rule (figure III.2). Each electron in an atom has two possible kinds of angular momenta: orbital and spin, whose magnitude are (see Equation I.44): (III.5) where l=0, 1, 2,…(n-1) and (III.6) With s= ½ For an electron having orbital and spin angular momentum there is a quantum number j due to the tototal angular momentum such that: (III.7) where j Since s=1/2 only, j is not a very useful quantum number for one-electron atoms, unless we are concerned with the fine detail of their spectra, but the analogous quantum number J , in polyelectronic atoms, is very important. A charge –e circulating in an orbit is equivalent to a current flowing in a wire, it therefore causes a magnetic moment. That due to orbital motion l is a vector which is opposed to the corresponding angular momentum vector l (Figure III.2a). The classical picture of an electron spinning on its own axis indicates that there is a magnetic moment s associated with this angular momentum, and it is opposed to s. The magnetic moments l and s can be regarded as tiny bar magnets. For each electron they may be parallel, as in Figure III.2a, or opposed (Figure III.2b). III.2.2. Coupling of angular momenta It follows from the interaction between the magnetic moments due to orbital and spin angular momenta of the same electron. This interaction is the coupling of the angular momenta and the greater the magnetic moments, the stronger the coupling. However, some couplings are so weak that they may be neglected. Coupling between two vectors a and b produces a resultant vector c, as shown in Figure III.3a. If the vectors represent angular momenta, a and b precess around c (Figure III.3b), the rate of precession increasing with the strength of coupling. In practice, c precesses about an arbitrary direction in space and, when an electric or magnetic field (Stark or Zeeman effect) is introduced, the effects of space quantization (§ I.3.2) may be observed. The strength of coupling between the spin and orbital motions of the electrons, referred to a spin–orbit coupling, depends on the atom concerned. The spin of one electron can interact with (a) the spins of the other electrons, (b) its own orbital motion and (c) the orbital motions of the other electrons. This last is called spin- other-orbit interaction and is normally too small to be taken into account. Interactions (a) and (b) are more important and the methods of treating them involve two types of approximation representing two extremes of coupling. One approximation assumes that coupling between spin momenta is sufficiently small to be neglected, as also is the coupling between orbital momenta, whereas coupling between the spin of an electron and its own angular momentum, to give a resultant total angular momentum j, is assumed to be strong and the coupling between the j’s for all the electrons to be less strong but appreciable. This coupling treatment is known as the jj-coupling approximation but its usefulness is limited mainly to a few states of heavy atoms. A second approximation neglects coupling between the spin of an electron and its orbital momentum but assumes that coupling between orbital momenta is strong and that between spin momenta relatively weak but appreciable. This represents the opposite extreme to the jj-coupling approximation. It is known as the Russell–Saunders coupling approximation and serves as a useful basis for describing most states of most atoms and is the only one we shall consider in detail. III.3 Russell–Saunders coupling approximation (a) Non-equivalent electrons: Non-equivalent electrons are electrons that have different values of either n or l, ex. 3p13d1 or 3p14p1. Coupling of angular momenta of non-equivalent electrons is more straightforward than for equivalent electrons. First we consider, the strong coupling between orbital angular momenta, referred to as ll coupling, using a particular example. Consider just two non-equivalent electrons in an atom (e.g. the helium atom in the highly excited configuration 2p13d1). We shall label the 2p and 3d electrons ‘1’ and ‘2’, respectively, so that l 1 =1 and l 1 =2. The l1 and l2 vectors representing these orbital angular momenta are of magnitudes 21/2 and 61/2, respectively (see Equation III.5). These vectors couple, as shown in Figure III.3(a), to give a resultant L of magnitude (III.8) However, the values of the total orbital angular momentum quantum number, L, are limited; or, in other words, the relative orientations of l1 and l2 are limited.
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