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. The orientations which they can take up are governed by the values that the quantum number L can take. L is associated with the total orbital angular momentum for the two electrons and is restricted to the values

(III.9) In the present case L = 3, 2 or 1 and the magnitude of L is 121/2, 61/2 or 21/2. These are illustrated by the vector diagrams of Figure III.4(a). The terms of the atom are labelled S, P, D, F, G, . . . corresponding to L = 0, 1, 2, 3, 4, etc., analogous to the labeling of one-electron orbitals s, p, d, f, g, . . . according to the value of l.

It follows that the 2p1 3d1 conﬁguration gives rise to P, D and F terms. In a similar way the coupling of a third vector to any of the L in Figure III.4(a) will give the terms arising from three non-equivalent electrons, and so on.

A ﬁlled sub-shell such as 2p6 or 3d10, has L = 0. Indeed, space quantization of the total orbital angular momentum produces 2L+1 components with ML= L, L-1,…,-L, analogous to space quantization of l. In a ﬁlled sub-shell , where the sum is over all electrons in the sub-shell. Since this sum (ml )i 0 i corresponds to ML, it implies that L=0. Therefore the excited configurations: C 1s22s22p13d1 Si 1s22s22p63s23p13d1 give P, D and F terms. The coupling between the spin momenta is referred to as ss coupling. The results of coupling of the s vectors can be obtained in a similar way to ll coupling with the difference that, since s is always ½ the vector for each electron is always of magnitude (31/2/2) according to Equation (III.6). The two s vectors can only take up orientations relative to each other such that the resultant S is of magnitude

(III.10) where S, the total spin quantum number, is restricted to the values:

In the case of two electrons this means that S= 0 or 1 only. The vector sums, giving resultant S vectors of magnitude 0 and 21/2 are illustrated in Figure III.4(b). The labels for the terms indicate the value of S by having 2S+1 is 1 as a pre- superscript to the S, P, D, .. . label. The value of 2S+1 is known as the multiplicity and is the number of values that MS can take: these are

Since, for two electrons, S = 0 or 1 the value of 2S +1 is 1 or 3 and the resulting terms are called singlet or triplet, respectively. Just as L=0 for a ﬁlled orbital, S=0.

Therefore, excited conﬁgurations of C and Si give: 1 P, 3P, 1D, 3D, 1F and 3F terms.

Also noble gases (all occupied orbitals are ﬁlled), have only 1 S term from ground conﬁgurations. Table III.1 lists the terms arising from various combinations of two non-equivalent electrons.

LS coupling and is due to spin–orbit interaction and arises from coupling between the resultant orbital and resultant spin momenta. This interaction is caused by the positive charge Ze on the nucleus and is proportional to Z4. The coupling between L and S gives the total angular momentum vector J , with magnitude

(III.11) (III.12) from which it follows that if L > S, J can take 2S+1 values, but if L < S, J can take 2L+1 values.

Example: Consider LS coupling in a 3 D term. Since S=1 and L=2, then J= 3, 2 or 1 (FigureIII.4c). The value of J is attached to the term symbol as a post-subscript so that

3 the three components of 3D are 3 D3, 3 D2 and D1 . Total number of states arising from the C or Si excited configurations comprises: 1 3 3 3 1 3 3 3 1 3 3 3 P1 , P0 , P1 , P2 , D2 , D1 , D2 , D3 , F3 , F2 , F3 , F4 Electron configurations (see table III.1): gross but useful approximation in which the electrons have been fed into orbitals whose energies have been calculated neglecting the last term in Equation (III.2). Nearly all configurations (all those with at least one unfilled orbital) give rise to more than one term or state. Term: describe what arises from an approximate treatment of an electron configuration, State: describes something that is observable experimentally.

For example, we can say that the 1s2 2s2 2p1 3d1 conﬁguration of C gives rise to a 3P 3 3 3 term which, when spin–orbit coupling is taken into account, splits into P1, P2 and P3 states. If nucleus possesses spin angular momentum, these states are further split but splitting due to nuclear spin is small (hyperfine structure).

(b) Equivalent electrons: to apply the Russell–Saunders coupling scheme for two, or more, equivalent electrons (i.e. with the same n and l), we shall use the example of two equivalent p electrons, as in the ground conﬁguration of carbon:

C: 1s2 2s2 2p2 Again, for the ﬁlled orbitals L=0 and S=0, so we have to consider only the 2p electrons. Since n=2 and l=1 for both electrons, Pauli exclusion principle applies (the two electrons must have different values of either ml or ms).

We label the 2p electrons as 1 and 2, we have l1=1 and (ml)1=1, 0 or -1 (same for electron

2). Pauli exclusion principle excludes simultaneous identical values for the ml and ms numbers. This implies ﬁfteen allowed combinations of values (table III.2).

The values of ML and MS are given in Table III.2: Highest value of ML is 2, indicating that this is also the highest value of L and that there 1 is a D term. Since ML =2 is associated with only MS =0, it must be a D term. The term accounts for 5 combinations (see table III.2). In the remaining combinations, the highest value of L is 1 and, since this is associated with MS = 1, 0 and -1, there is 3 a P term. This accounts for a further nine combinations, leaving only ML = 0, MS=0, which implies a 1 S term. It is interesting to note that of the 1 S, 3 S, 1 P, 3 P, 1 D and 3 D terms which arise from two non-equivalent p electrons, as in the 1s22s22p13p1 configuration of the carbon atom, only 1S, 3P and 1D are allowed for two equivalent p electrons: the Pauli exclusion principle forbids the other three. Terms arising from three equivalent p electrons and also from various equivalent d electrons can be derived using the same methods, but this can be a very lengthy operation. The results are given in Table III.2. In deriving the terms arising from non-equivalent or equivalent electrons there is a very useful rule that, in this respect, a vacancy in a sub-shell behaves like an electron. For example, the ground configurations of C and O: C 1s22s22p2 O 1s22s22p4 give rise to terms: 1S, 3P, 1D, as do the excited configurations of C and Ne: C 1s22s22p13d1 Ne 1s22s22p53d1 Give rise to terms 1,3P, 1,3D, 1,3F.

Hund rules (enable us to determine which of the terms arising from equivalent electrons lies lowest in energy):

1. Of the terms arising from equivalent electrons those with the highest multiplicity lie lowest in energy. 2. Of these, the lowest is that with the highest value of L.

It follows that, for the ground conﬁgurations of both C and O, the 3P term is the lowest in energy. The ground conﬁguration of Ti is

Ti: KL3s23p63d24s2

Of the terms arising from the d2 conﬁguration given in Table III.1, Hund rules indicate

3F term as lowest in energy. The splitting of a term by spin–orbit interaction is proportional to J and a multiplet results:

(III.13) EJ = energy corresponding to J. If A >0, component with smallest J lies lowest in energy, multiplet is “normal”. If A < 0 , multiplet is “inverted”. 3. Normal multiplets arise from equivalent electrons when a partially ﬁlled orbital is less than half full. 4. Inverted multiplets arise from equivalent electrons when a partially ﬁlled orbital is more than half full.

It follows that the lowest energy term of Ti, 3F , is split by spin–orbit coupling 3 into a normal multiplet and therefore the ground state is F2 . Similarly, the lowest energy term of C, 3P, splits into a normal multiplet resulting in a 3P0 ground state, 3 whereas that of O, with an inverted multiplet, is P2. Atoms with a ground conﬁguration in which an orbital is exactly half-ﬁlled always have an S ground state (table III.1). Since such states have only one component the problem of a normal or inverted multiplet does not arise.

For excited terms split by spin–orbit interaction there are no general rules regarding normal or inverted multiplets. For example, in He, excited states form mostly inverted multiplets whereas in the alkaline earth metals, Be, Mg, Ca, .. . , they are mostly normal.

III.1.3 Spectra of alkali metal atoms Alkali metal atoms all have one valence electron in an outer (ns) orbital, where: n=2, 3, 4, 5, 6 for Li, Na, K, Rb and Cs. If we consider only orbital changes involving this electron, behaviour expected to resemble that of H atom. The core, consisting of the nucleus of charge +Ze and filled orbitals containing (Z-1) electrons, has a net charge of +e and therefore, effect on valence electron similar to that of H atom nucleus. Whereas emission spectrum of H shows only the Balmer series (see Figure I.1) in the visible region, alkali metals show at least three: principal series, sharp series and diffuse series (Figure III.5). A part of a fourth series, called the fundamental series, can sometimes be observed. All such series converge smoothly to high energy (low wavelength) in a way that resembles the series in H. Figure III.6 shows an energy level diagram (so-called Grotrian diagram), for lithium for which the ground configuration is 1s22s1. The lowest energy level corresponds to this configuration. The higher energy levels are labeled according to the orbital to which the valence electron has been promoted: for example the level labeled 4p corresponds to the configuration 1s24p1. The relatively large energy separation between configurations in which the valence electron is in orbitals differing only in the value of l (e.g. 1s23s1, 1s23p1,

1s23d1) is characteristic of all atoms except hydrogen (and one-electron ions). The selection rules governing the promotion of the electron to an excited orbital, and also its falling back from an excited orbital, are

These selection rules lead to the sharp, principal, diffuse and fundamental series, shown in Figures III.5 and III.6, in which the promoted electron is in an s, p, d and f orbital, respectively. Some excited conﬁgurations of lithium (with promotion of only the valence electron), are given in Table III.3, which also lists the states arising from these conﬁgurations. Similar states can easily be derived for other alkali metals.

Spin–orbit coupling splits apart the two components of the 2P, 2D, 2F , .. . terms. The splitting decreases with L and n and increases with atomic number (it is not large enough to appear in Figure III.6 for Li). 2 2 In sodium, pairs of P1/2, P3/2 states result from the promotion of the 3s valence electron to any np orbital with n > 2. It is convenient to label the states with this value of 2 2 n, as n P1/2 and n P3/2, the n label being helpful for states that arise when only one electron is promoted and the unpromoted electrons are either in filled orbitals or in an s orbital. The n label can be used, therefore, for H, alkali atoms, helium and alkaline earths. In other atoms it is usual to precede the state symbols by the configuration of the 1 electrons in unfilled orbitals, as in the 2p3p S0 state of carbon.

The splitting of the n2P1/2 , n2P3/2 states for a given alkali atom decreases with n, 2 2 and for a given n, with L, i.e for the 3 D3/2 , 3 D5/2 states it is much smaller than for

32P1/2 , 32P3/2. These 2P and 2D multiplets are normal, i.e. state with lowest J is lowest in energy. The selection rule is

Consequence: principal series consists of pairs of 2P1/2-2S1/2, 2P3/2-2S1/2 transitions

(Figure III.7a). The pairs are known as simple doublets. The first member of this series in sodium appears in the yellow region of the spectrum with components at 589.592 nm and 588.995 nm called the sodium D lines. 2 2 The 3 P1/2, 3 P3/2 excited states involved in the sodium D lines are the lowest energy excited states of the atom. Consequently, in a vapour discharge at a pressure that is sufficiently high for collisional deactivation of excited states to occur readily, a majority of atoms find themselves in these states before emission of radiation has taken place.

III.1.4 Spectra of helium and the alkaline earth metal atoms Emission spectrum of a helium gas discharge in the visible and near-ultraviolet regions shows two series of lines (one with single lines, the other with double lines) converging smoothly to high energy. This is due to two sets of energy levels, one corresponding to the single lines and the other to the double lines, while no transitions between the two sets of levels are observed. In 1925 it became clear that, when account is taken of electron spin, the two forms are really singlet helium and triplet helium. For hydrogen and the alkali metal atoms in their ground conﬁgurations, or excited conﬁgurations involving promotion of the valence electron, there is only one electron with an unpaired spin. For this electron ms=±/2 and corresponding electron spin part of total wave function is conventionally given the symbol or . All states arising are doublet states. For helium, need to look at consequences of electron spin, since it is the prototype of all atoms and molecules having easily accessible states with two different multiplicities. If spin–orbit coupling small (case for He), total electronic wave function e can be factorized into orbital and spin part: (III.14)

Spin part is derived by labeling the electrons 1 and 2 that, in general, each can have an a or b spin wave function giving four possible combinations:

Because first 2 are neither antisymmetric nor symmetric to exchange of electrons, which is equivalent to exchange of the labels 1 and 2, they must be replaced by linear combinations giving

(III.15) And

(III.16)

where factors 21/2 are normalization constants. Wave function e in III.4 is antisymmetric to electron exchange and is that of a singlet state, whereas the three in Equation (III.16) are symmetric and are those of a triplet state.

Turning to orbital part of e we consider the electrons in two different atomic orbitals a 1 1 and b as, e.g., 1s 2p conﬁguration of He There are two ways of placing electrons 1 and 2 in these orbitals giving wave functions a(1) b(2) and a(2) b(1) but, once again, we have to use, instead, the linear combinations

(III.17)

(III.18) These wave functions are symmetric and antisymmetric, respectively, to electron exchange. The most general statement of the Pauli principle for electrons and other fermions is that total wave function must be antisymmetric to electron (or fermion) exchange. For bosons it must be symmetric to exchange. For He, therefore, the singlet spin wave function of III.15 can combine only with orbital wave function of Equation (III.17) giving, for singlet states,

(III.18) Similarly, for triplet states,

(III.19)

For ground conﬁguration, 1s2, orbital wave function is given by

(III.20) which is symmetric to electron exchange. This conﬁguration leads, therefore, to only a singlet term, whereas each excited conﬁguration arising from the promotion of one electron gives rise to a singlet and a triplet. Each triplet term lies lower in energy than the corresponding singlet. The Grotrian diagram in Figure III.9 gives energy levels for all the terms arising from the promotion of one electron in helium to an excited orbital. Selection rules are (III.21) Latter rule is rigidly obeyed in the observed spectrum of He. Because of this spin 3 selection rule, atoms which get into the lowest triplet state, 2 S1, do not easily revert to 1 the ground 1 S0 state: transition is forbidden by both the orbital and spin selection rules. The lowest triplet state is therefore metastable. In a typical discharge it has a lifetime of the order of 1 ms. 1 First excited singlet state, 2 S0 , is also metastable in the sense that a transition to the ground state is forbidden by the l selection rule but, because the transition is not 3 spin forbidden, this state is not so long-lived as the 2 S1 metastable state.

Owing to the effects of spin–orbit coupling all the triplet terms, except 3S, are split into three components. For example, in the case of a 3P term, with L=1 and S=1, J can take values 2, 1, 0 (Equation III.12). The splitting of triplet terms of helium is unusual in two respects. First, multiplets may be inverted and, second, the splittings of the multiplet components do not obey the splitting rule of Equation (III.13). For this reason we shall discuss ﬁne structure due to spin–orbit coupling in the context of the alkaline earth atomic spectra where multiplets are usually normal and also obey Equation (III.13). In other respects, the alkaline earth metal spectra resemble that of helium quite closely. All the atoms have an ns2 valence orbital conﬁguration and, when one of these electrons is promoted, give a series of singlet and triplet states.

The ﬁne structure of a 3P-3S transition of an alkaline earth metal is illustrated in Figure III.9(a). The J selection rule results in a simple triplet. (The very small separation of 3 3 2 P1 and 2 P2 in helium accounts for the early description of the low- resolution spectrum of triplet helium as consisting of ‘doublets’.)

A 3D-3P transition, shown in Figure III.9(b), has six components. As with doublet states the multiplet splitting decreases rapidly with L so the resulting six lines in the spectrum appear, at medium resolution, as a triplet. For this reason the ﬁne structure is often called a compound triplet.

III.1.6 Spectra of other polyelectronic atoms So far we considered only H, He, alkali metals and alkaline earth metals but selection rules and general principles discussed here can be extended to any other atom. An obvious difference between the emission spectra of most atoms and those we have considered so far is their complexity: many lines and no obvious series. Extreme case: spectrum of iron. However complex the atom, we can use the Russell–Saunders coupling approximation (or jj coupling, if necessary) to derive states arising from any conﬁguration. The four general selection rules that apply to transitions between these states are as follows. • Rule 1: Previously we have considered the promotion of only one electron, for which l =± 1 applies, but general rule given here involves the total orbital angular momentum quantum number L and applies to the promotion of any number of electrons. • Rule 2:

Here, ‘even’ and ‘odd’ refer to the arithmetic sum l i over all the electrons and i this selection rule is called the Laporte rule. Important consequence: transitions are forbidden between states arising from the same conﬁguration. • Rule 3:

This rule is the same for all atoms. • Rule 4:

This applies only to atoms with a small nuclear charge. In atoms with a large nuclear charge it breaks down so that factorization of e no longer applies because of spin– orbit interaction and states can no longer be described accurately as singlet, doublet, etc.

Table III.1: Terms arising from some conﬁgurations of non-equivalent and equivalent electrons

Table III.2: Derivation of terms arising from two equivalent p electrons

Table III.3: Configurations and states of the Lithium atom

Figure III.1: Orbital energies, Ei , typical of a polyelectronic atom

Figure III.2: Vectors l and s and magnetic moments l and s associated with orbital and spin angular momenta when the motions are (a) in the same direction and (b) in opposite directions

Figure III.3: (a) Addition of two vectors a and b to give c. (b) Precession of a and b around c

Figure III.4: Russell–Saunders coupling of (a) orbital angular momenta l1 and l2 , (b) spin angular momenta s1 and s2 and (c) total orbital and total spin angular momenta, L and S, of a p and a d electron

Figure III.5: Four series in the emission spectrum of lithium

Figure III.6: Grotrian diagram for lithium

Figure III.7: Grotrian diagram for helium. The scale is too small to show splittings due to spin–orbit coupling

Figure III. 8: Grotrian diagramme of Helium. The SO splitting is too small to be seen on this energy scale

Figure III.9: (a) a simple triplet; (b) a compound triplet in spectrum of an alkaline earth metal atom.