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Interstellar Medium (ISM)
Week 2 March 26 (Thursday), 2020
updated 04/13, 08:56
선광일 (Kwangil Seon) KASI / UST 2
Atomic Structure, Spectroscopy 3 References
• Books for atomic/molecular structure and spectroscopy - Astronomical Spectroscopy [Jonathan Tennyson] - Physics of the Interstellar and Intergalactic Medium [Bruce T. Draine] ⇒ see https://www.astro.princeton.edu/~draine/ for errata - Astrophysics of the Diffuse Universe [Michael A. Dopita & Ralph S. Sutherland] ⇒ many typos - Physics and Chemistry of the Interstellar Medium [Sun Kwok] - Atomic Spectrocopy and Radiative Processes [Egidio Landi Degl’Innocenti] 4 Hydrogen Atom: Schrödinger Equation
• Momentum operator ℏ p = ∇ i • Hamiltonian operator p2 ℏ2 H = + V(r) = − ∇2 + V(r) 2m 2m • The time-dependent Schrödinger equation for a system with Hamiltonian H:
∂Ψ ∂Ψ (r, t) ℏ2 iℏ = HΨ iℏ = − ∇2Ψ (r, t) + V(r)Ψ (r, t) ∂t ∂t 2m The time and space parts of the wave function can be separated:
Ψ (r, t) = ψ(r)eiEt/ℏ
• Then, the time-independent Schrödinger equation is obtained:
ℏ2 Hψ (r) = Eψ(r) ∇2ψ (r) + V(r)ψ (r) = Eψ(r) 2m 5
• Expectation value of an operator
3 F = ⇤F d x
h
(r, ✓, )=R (r)Y (✓, )
radial function: 2 1/2 2Z ~ ˚ 3 ⇢ = r, a0 2 =0.529A (Bohr radius), 2Z (n l 1)! ⇢/2 l 2l+1 na0 ⌘ mec R (r)= e ⇢ L (⇢) nl na 2n (n + l)! 3 n+l L2l+1 = associated Laguerre polynomial " 0 #
3/2 Z ρ 1 R1,0 = 2e− Y0,0 = a0 4π ! " ! 3/2 Z ρ 3 R2,0 = 2(1 ρ)e− Y1,0 = cos θ 2a0 − 4π ! " ! 3/2 Z 2 ρ 3 iφ R2,1 = ρ e− Y1, 1 = sin θ e± 2a √ ± ∓ 8π ! 0 " 3 ! 3/2 Z 2 2 ρ 5 2 R3,0 = 2 1 2ρ + ρ e− Y2,0 = 3cos θ 1 3a0 − 3 16π − ! " ! " ! 3/2 " # Z 4√2 1 ρ 15 iφ R = ρ 1 ρ e− Y2, 1 = sin θ cos θ e± 3,1 ± 3a0 3 − 2 ∓ 8π ! " ! " ! 3/2 Z 2√2 2 ρ 15 2 2iφ − Y2, 2 = sin θ e± R3,2 = ρ e ± 3a0 3√5 32π ! " 2!π π ∞ 2 2 2 Normalisation: R r dr =1 Normalisation: Yl,m sin θ dθ dφ =1 n,l | | #0 $0 $0 Z Here, ρ ≡ r These show a general a feature of hydrogenicna0 wavefunctions, namely 12 12 that the radial functions for l =0haveafinitevalueattheorigin,i.e.repeated application of the lowering operator: This eigenfunction has magnetic quantum number l (l m)=m. the power series in ρ starts at the zeroth power. Thus electrons with − − l m l ilφ l =0(calleds-electrons)haveafiniteprobabilityofbeingfoundatthe Yl,m (l ) − sin θ e . (2.11) ∝ − position of the nucleus and this has important consequences in atomic Atomic Physics [Foot] physics. To understand the properties of atoms, it is important to know what Inserting E from eqn 2.20 into eqn 2.17 gives the scaled coordinate | | the wavefunctions look like. The angular distribution needs to be mul- Z r tiplied by the radial distribution, calculated in the next section, to give ρ = , (2.21) n a0 the square of the wavefunction as where the atomic number has been incorporated by the replacement 2 2 2 2 2 e /4π#0 Ze /4π#0 (as in Chapter 1). There are some important prop- ψ (r, θ, φ) = Rn,l (r) Yl,m (θ, φ) . (2.12) erties of→ the radial wavefunctions that require a general form of the| | | | solution and for future reference we state these results. The probability 2 This is the probability distribution of the electron, or e ψ can be in- density of electrons with l =0attheoriginis terpreted as the electronic charge distribution. Many atomic− | | properties, 3 2 1 Z however, depend mainly on the form of the angular distribution and ψn,l=0 (0) = . (2.22) 2 2 | | π na0 Fig. 2.1 shows some plots of Yl,m .ThefunctionY0,0 is spherically ! " | 2 | | | symmetric. The function Y has two lobes along the z-axis. The For electrons with l =0theexpectationvalueof1/r3 is | 1,0| " squared modulus of the other two eigenfunctions of l =1isproportional to sin23 θ.AsshowninFig.2.1(c),thereisacorrespondencebetween 1 ∞ 1 2 2 1 Z 3 = 3 Rn,l (r) r dr = 1 these distributions. (2.23) and the circular motion of the electron around the r 0 r l l + (l +1) na0 $ % # 2 !z-axis" that we found as the normal modes in the classical theory of the These results have been written in a& form' that is easyZeeman to remember; effect (in Chapter 1).13 This can be seen in Cartesian coordi- 13Stationary states in quantum 3 they must both depend on 1/a0 in order to have the correctnates where dimensions mechanics correspond to the time- and the dependence on Z follows from the scaling of the Schr¨odinger z averaged classical motion. In this Y1,0 , case both directions of circular mo- ∝ r tion about the x-axis give the same x +iy distribution. Y , (2.13) 1,1 ∝ r x iy Y1, 1 − . − ∝ r 7
• The spherical harmonics are eigenfunctions of the orbital angular momentum operator
ℏ L2Y = l(l + 1) 2Y ,LY = m Y L = r × p = r × ∇
L = l(l + 1) , L = m
(r, ✓, ) 2d3x = R2 (r) Y (✓, ) 2r2 sin ✓drd✓d |
l 2l +1 Y (✓, ) 2 = | lm | 4⇡
m
is the electron orbital angular momentum quantum number. The actual angular momentum is given by n[l(l + 1)]½. l can take the values 0, 1, 2, ... , n - 1. By convention, the values of l are usually designated by letters (see Table 3.1). m is the magnetic quantum number , so called because it determines the behaviour of the energy levels in the presence of a magnetic field. mn is the projection of the electron orbital angular momentum , given by l , along the z-axis of the system. It can take (2l + 1) values -l, -l + 1, ... , 0, ... , l - 1, l.
2 s 1=0 p 1=1 p 1=1 m=O m=O m=1 0 0 B m N -2 d 1=2 d 1=2 d 1=2 m=O ~ + m= 0 -
-2 -2 0 -2 0 -2 0 2 X 8 Fig. 3.2. Wavefunctions for the angular motions of the hydrogen atom which are given in terms of spherical harmonics , Yi, m(0, ). Plots are for the x- z plane except for the l = 2, m = 2 plot for which the x--y plane is shown. The plots give the absolute value • Each bound state of the hydrogen atom is characterized by a set of four quantum numbers of the spherical harmonic as a distance from the origin for each angle; the signs indicate ( ) n, thel, m , signms of the wavefunction in each region. (S.A. Morgan, private communication.) - n = 1, 2, 3,⋯ : principal quantum number - l = 0, 1, 2,⋯, n − 1 : orbital angular momentum quantum number ‣ By convention,Table the3.1. values Letter of designationsl are usually fordesignated orbital angular by letters. momentum quantum number l.
0 1 2 3 4 5 6 7 8 s p d f g h k 1
- m = − l, − l + 1,⋯,0,⋯, l − 1, l : magnetic quantum number. ‣ It determines the behavior of the energy levels in the presence of a magnetic field. ‣ This is the projection of the electron orbital angular momentum along the z-axis of the system.
• Spin 1 - The electron possesses an intrinsic angular momentum with the magnitude of |s| = . 2 1 - There are two states, m = ± , for the spin. s 2
n−1 • Degeneracy for a given n: 2 2 × ∑ (2l + 1) = 2n l=0
N N 1 N ~2 Ze2 e2 2 + (r , r , , r )=E (r , r , , r ) 2 2mri r r r 3 1 2 ··· N 1 2 ··· N i=1 i i=1 j=1 i j X ✓ ◆ X X | | 4 5 where is the coordinate of the th electron, with its origin at the nucleus. ri i
- The first term contains a kinetic energy operator for the motion of each electron and the Coulomb attraction between the electron and the nucleus.
- The second term contains the electron-electron Coulomb repulsion term.
- The equation is not analytically solvable, even for the simplest case, the helium atom for which N = 2. 10 Complex Atoms : Central Field Approximation
• Central field approximation (or orbital approximation): - We assume that each electron moves in the potential of the nucleus plus the averaged potential due to the other N - 1 electrons. - Within this model, the Schrödinger equation can be separated into N single electron equations: 2 2 2 ~ 2 Ze e i + Vi(ri) i(ri)=Ei i(ri)whereVi(ri)= + 2mr ri ri rj
- It provides a useful classification of atomic states and also a starting point. - It is standard to use the hydrogen atom orbital labels, n, l and m, to label the orbitals. This is called the configuration of electrons. 11 Central Field Approximation: Electron Configuration
• The configuration is the distribution of electrons of an atom in atomic orbitals. - The configuration of an atomic system is defined by specifying the nl values of all the electron orbitals: nlx means x electrons in the orbital defined by n and l. Each orbital labelled actually consists of orbitals with different values, each with two possible values of . - nl 2l + 1 m ms Thus the nl orbital can hold a maximum 2(2l + 1) electrons.
• shells, subshells: - Shells correspond with the principal quantum numbers (1, 2, 3, ...). They are labeled alphabetically with letters used in the X-ray notation (K, L, M, ...). - Each shell is composed of one or more subshells. The first (K) shell has one subshell, called “1s”; The second (L) shell has two subshells, called “2s” and “2p”.
• open shell configuration, closed shell configuration: - open shell = shell that is not completely filled with electrons: For instance, the ground state configuration of carbon, which has six electrons: 1s22s22p2 - closed shell = shell of which orbitals are fully occupied: For example, the ground state configuration of neon atom, which has ten electrons: 1s22s22p6
• Active electrons: As a result of the Pauli Principle, closed shells and sub-shells have both L = 0 and S = 0. This means that it is only necessary to consider ‘active’ electrons, those in open or partially-filled shells.
• Equivalent and Nonequivalent Electrons - Nonequivalent electrons are those differing in either n or l values, whereas equivalent electrons have the same n and l values. 12 Indistinguishable Particles & Parity
• Consider a system with two identical particles. It is not the wave function but the probability distribution that is physically observable. The probability distribution cannot be altered by interchanging the particles. This means that
(a ,b ) 2 = (a ,b ) 2 |
- Pauli exclusion principle: The Pauli exclusion principle is summarized as “No two electrons can occupy the same spin-orbital.” - This exclusion provides the degeneracy pressure which holds up the gravitational collapse of white dwarfs and neutron stars.
• Parity of the wave function - The parity of the wave function is determined by how the wave function behaves upon inversion. The square of the wave function, i.e., the probability distribution of the electrons, must be unchanged by the inversion operation. (r , r , , r )= ( r , r , , r )
• Energy ordering for configuration: The orbitals of complex atoms follow a revised energy ordering: - For a H atom, the energy of the individual E(1s) Atom z Configuration • Periodic Table hydrogen H 1 ls helium He 2 1s2 - The subshell structure of elements up to lithium Li 3 K 2s beryllium Be 4 K 2s2 argon ( ) is filled up in a naturally boron B 5 K 2s22p Z = 18 carbon C 6 K 2s22p2 straightforward manner, first according to nitrogen N 7 K 2s22p 3 oxygen 0 8 K 2s22p 4 and then according to . fluorine F 9 K 2s22p 5 n l neon Ne 10 K 2s22p 6 sodium Na 11 K L 3s - The subshell is all occupied in argon magnesium Mg 12 K L 3s2 3p aluminium Al 13 K L 3s23p (Ar; noble gas) with a closed subshell 6. silicon Si 14 K L 3s23p2 3p phosphorus p 15 K L 3s23p 3 The next element potassium (K; Z = 19), sulphur s 16 K L 3s23p 4 chlorine Cl 17 K L 3s23p 5 begins by filling in the 4s, instead of 3d. argon Ar 18 K L 3s23p 6 potassium K 19 K L 3s23p 6 4s calcium Ca 20 K L 3s23p 6 4s2 similar optical properties. The spectra of alkali metals will be discussed in detail in Chapter 6. Electronically excited states of atoms usually arise when one of the outermost electrons jumps to a higher orbital. These excited states can be written as configurations. Such excited states for helium might include ls2s, ls2p or ls3s, for example. It should be noted that each of the configurations actually gives rise to more than one excited state. States with two electrons simultaneously excited are possible but are less important. For many systems, all of these states are unstable. They have sufficient energy to autoionise by spontaneously ejecting an electron. For example, the lowest two-electron excited state of helium has the config- uration 2s2 • This state spontaneously decays to the ls ground state of the He+ ion and a free electron. 14 Lifting Degeneracy in Configuration: Angular Momentum Coupling, Terms • There are two coupling schemes or ways of summing the individual electron angular momentum to give the total angular momentum. • L-S coupling (Russell-Saunders coupling): - The orbital and spin angular momenta are added separately to give the total angular momentum L and the total spin angular momentum S. These are then added to give J. L = ∑ li, S = ∑ si → J = L + S i i - The configurations split into terms with particular values of L and S. • j-j coupling An alternative scheme is to consider the total angular momentum for each electron by combining - ji and and then coupling these ’s together to give the total angular momentum. li si j ji = li + si → J = ∑ ji i • Why two coupling schemes? - They give the same results for J. - For light atoms (lighter than iron), the values of L and S are approximately conserved quantities, and the L-S coupling scheme is the most appropriate. - For heavy atoms (beyond iron), L and S are no longer conserved quantities and j-j coupling is more appropriate. 15 Addition of Two Angular Momenta - In classical mechanics, adding vector a and vector b gives a vector c, whose length must lie in the range |a − b| ≤ c ≤ a + b Here, a, b, c are the lengths of their respective vectors. c = |a−b| c = a+b - In quantum mechanics, a similar rule applies except that the results are quantized. The allowed values of the quantized angular momentum, c, span the range from the sum to the difference of a and b in steps of one: c = |a − b|, |a − b| + 1, ⋯ , a + b − 1, a + b together to give - For example, add the two angular momenta L1 = 2 and L2 = 3 L = L1 + L2. The result is L = 1, 2, 3, 4, 5. 16 Spin-Orbit interaction, Levels, Fine Structure Splitting • The fact that the remaining spin-orbit splitting is much smaller makes the LS coupling scheme a very useful one. • Fine-structure splitting: Relativistic effects couple electron orbital angular momentum and electron spin to give the so-called fine structure in the energy levels. Inclusion of relativistic effects splits the terms into levels according to their J value. • When an electron moves around the nucleus with a non relativistic velocity v, the r electric field exerting on the electron will be E = Ze 3 . (Note that the nucleus has a v B0 = B = (B E)= E Here, the magnetic field is perpendicular to the ? ? ⇥ c ⇥ electron’s orbital plane. Ze v r = ⇥ c r3 Ze = 3 ` (where ` r p = m r v is the electron’s orbital angular momentum of electron) • Then, the interaction energy is Ze2 U H = ⇠ (s ` ) so i i · i H = ⇠ (S L)whereS = s , L = ` so · i i 1 2 2 2 Hso = ⇠ J L S where J is the total angular momentum of all electrons An important point to note is that the physical exis- in the atom with multiplicity or degeneracy 2J 1. The tence of atomic energy states, as given by the number + total J follows the vector sum: for two electrons, the val- of total J-states, must remain the same, regardless of ues of J range from j j to j j .Thestatesare the coupling schemes. Therefore, the total number of J- | 1 + 2| | 1 − 2| denoted as ( j j ) .Forexample,fora(pd)configuration levels j J or L S J is the same in i 2 J i i = + = j (1 1/2) 1/2, 3/2, and j (2 1/2) 3/2, 5/2; intermediate coupling or jj-coupling. A general discus- 1 ± = 2 ± = " # $ the states are designated as (1/23/2)2,1,(1/25/2)3,2, sion of the relativistic effects and fine structure is given in (3/23/2)3,2,1,0, (3/25/2)4,3,2,1 (note that the total dis- Section 2.13 crete J =0–4).TheJ-state also includes the parity and is expressed as Jπ or, more completely, as (2S 1)Lπ . + J 2.8 Hund’s rules The latter designation relates the fine-structure level to the parent LS term. For each LS term there can be several The physical reason for the variations in subshell structure fine-structure levels. The total angular magnetic quantum of the ground configuration of an atom is the electron– number Jm runs from J to J. electron interaction. It determines the energies of the − Fine-structure levels can be further split into hyperfine ground and excited states. We need to consider both the structure when nuclear spin I is added vectorially to J to direct and the exchange potentials in calculating these yield the quantum state J I F. Figure 2.2 shows energies. Before we describe the atomic theory to ascer- + = the schematics of energy levels beginning with a given tain these energies, it is useful to state some empirical electronic configuration. rules. The most common is the Hund’s rules that governs For cases where LS coupling is increasingly invalid the spin multiplicity (2S 1),andorbitalL and total J + because of the importance of relativistic effects, but the angular momenta, in that order. departure from pure LS coupling is not too severe and The S-rule states that an LS term with the high- full consideration of relativistic effects is not necessary, est spin multiplicity (2S 1) is the lowest in energy. + an intermediate coupling scheme designated as LSJ is This rule is related to the exchange effect, whereby employed. (We discuss later the physical approximations electrons with like spin spatially avoid one another, associated with relativistic effects and appropriate cou- and therefore see less electron–electron repulsion (the pling schemes.) In intermediate coupling notation, the exchange potential, like the attractive nuclear potential, angular momenta l and s of an interacting electron are has a negative sign in the Hamiltonian relative to the added to the total orbital and spin angular momenta, J1 direct electron–electron potential, which is positive). For of all other electrons in the following manner, example, atoms and ions with open subshell np3 ground configuration (N I,OII,PI,SII) have the ground state 4 o 2 o 2 o J1 li si , K J1 l, J K s, S ,lowerthantheotherterms D , P of the ground 18 = + = + = + configuration. !i !i (2.63) The L-rule states that for states of the same spin mul- tiplicity the one with the larger total L lies lower, again • Electronic configuration and energy level splitting where s 1/2. The multiplicity is again 2J 1, and the owing to less electron repulsion for higher orbital angu- - Configurations ⇒ Terms ⇒ Fine Structure (Spin-Orbit Interaction) ⇒ Hyperfine = + lar momentum electrons that are farther away from the Structuretotal (Interaction angular magnetic with Nuclear quantum Spin) number Jm runs from J − 3 2 o to J. nucleus. Hence, in the example of np above, the D term lies lower than the 2Po.Anotherexampleisthe ground configuration of O III, which is C-like 2p2 with Configuration Term structure Fine structure Hyperfine 3 1 1 structure the three LS terms P, D, Sinthatenergyorder.Amore complex example is Fe II, with the ground 3p63d64s and the first excited 3p63d7 configurations. The LS terms in L = I1 + I2 6 6 4 {ni Ii} energy order within each configuration are 3d 4s ( D, D) S = s1 + s2 and 3d7 (4F, 4P). But the two configurations overlap and the actual observed energies of these four terms lie in the J = L + S order 6D, 4F, 4D, 4P. F = J + I The J-rule refers to fine-structure levels L S J. + = For less than half-filled subshells, the lowest J-level lies LS terms LSJ levels lowest, but for more than half-filled subshells it is the FIGURE 2.2 Electronic configuration and energy level splittings. reverse, that is, the highest J-level lies lowest in energy. [Pradhan & Nahar] Atomic Astrophysics and Spectroscopy 19 Spectroscopic Notation Electron conÖguration and Spectroscopic notation • Spectroscopic Notation 2.1 Atomic Spectra 21 Spectral Notation 25+ 1 = Total Term 5pin Multiplicity: Term Parity: 5 is vector sum of electron spins (± 1/2 each) o for odd, nothing for even Inner full shells sum to 0 L = Total Term Orbital The Number of levels in Angular Momentum: a term is the smaller of Vector sum of contributing (25+ 1) or (2L+ 1) nsi npi ndk ... electron orbitals. ) Inner full shells sum to o. - A state with S = 0 is a ‘singlet’ as 2S+1 = 1. Electronic Configuration: ‣ J = L (singlet) the electrons and their orbitals / = Total Level Angular Momentum: (i.e. 1 s2 2s2 3p 1) Vector sum of Land 5 of a particular - A state with S = 1/2 is a ‘doublet’ as 2S+1 = 2 n =1, 2, 3, 4, 5 K, L, M, N, O, level in a term. ··· ! ··· ` =0, 1, 2, 3, 4 s, p, d, f, g, ‣ J = L − 1/2, L + 1/2 (doublet if L ≥ 1) Fig. ···2.4. Spectral! notation··· for an atomic term comprised of one or more levels. L 2 20Figure 4.1 Energy-level diagram for the ground configuration of the 2p ions N II and O III. (Fine-structure splitting is exaggerated for clarity.) Forbidden transitions connecting these levels are shown, with wavelengths in vacuo. Table 9.1 Neutral doma - [Draine, Physics of the ISM and IGM] K L Table 4.1 Terms for ns and np Subshells Atom hi N 0 P Q Ground Atom K ---L hf N 0 Ground _____ Ground Terms / Levels Is 282p 383p3d 484p4d 58 level 4f 585p5d5j 6a6p6d 78 l6Vd configuration (in order of increasing energy) Examples Ag 47 1 12 H11 'St ...ns S1/2 H I, He II, C IV, N V, O VI He 2 2 ___ 'So Cd 48 2 21 ...ns S0 He I, C III, N IV, O V Li 321 In 49 21 12o Sn 50 22 ...np P1/2,3/2 C II, N III, O IV Be4 22 'So 23 1 1 B 6 221 'Pt Sb 51 23 ...np P0,1,2 , D2 , S0 C I, N II, O III, Ne V, S III 34o 2 o 2 o C 6 222 3P0 Te 52 24 ...np S3/2 , D3/2,5/2 , P1/2,3/2 N I, O II, Ne IV, S II, Ar IV 'SP, 43 1 1 N 7 223 ...np P2,1,0 , D2 , S0 O I, Ne III, Mg V, Ar III 0 8 224 3p1 52o ...np P3/2,1/2 Ne II, Na III, Mg IV, Ar IV F 9 226 TI 61 10 IS0 ...np S0 Ne I, Na II, Mg III, Ar III Ne 2 2 6 %I1 226 1 zs,---- Mg 12 2 'So [Kowk, Physics and Chemistry of the ISM] A1 13 21 2Pg Si 14 10 2 2 3P0 P 16 23 's?k Pm 61 5 2 4.6 Hyperfine Structure: Interaction with Nuclear Spin S 16 Necore 2 4 3pz Sm 62 3 6 2 C1 17 25 'Pp, Eu 63 8 I 2 Gd 64 78 1 2 ALL__~_._____~ 26 IS0 If the nucleus has nonzero spin, it will have a nonzero magnetic moment. If the 2s, ~- K 19 226_____ 26 1 Tb 65 ' 9 2 'H!I nucleus has a magnetic moment, then fine-structure levels with nonzero electronic 10 2 Ce 20 2 'So Dy 66 ' 6J8 angular momentum can themselves be split due to interaction of the electrons with Sc 21 12 =Dll Ho 67 11 2 'IPI the magnetic field produced by the nucleus. This “hyperfine” splitting is typically Ti 22 22 3F2 Er 68 * 12 2 3H'3 V 23 18 32 'FI, 13 2 6 Tm 69 ' 'F% of order 10− eV. Hyperfine splitting is usually difficult to observe in optical spec- Cr 24 51 7s3 Yb 70 c? 14 2 'SO Mn 25 Acore 5 2 %i Lu 71 14 1 2 tra due to Doppler broadening, but it needs to be taken into account if precise -. -~~ ___._ __-- Fe 20 62 'D4 Hf 72- 14262-2- 3F2 Co 27 72 'Far Ta 73 3 3 'FII Ni 28 82-~______3F, w 74 ; 4 0d0 Cu29 2 26 2610 1 'SI Re 75 - 46+22 6 %I Zn 30 2 'So 0s 76 6 OD, Ce 31 21 'Pp Ir 77 72 'Fit Ge 32 28 22 -__Pt 78 ___ 91 3D3. Blue:A0 33 No fine structure in23 the ground4s3 state.Au 79 fi 14 2 6 10 1 se 34 24 3P, Hg 80 e ___ 2 'SO Br 36 26 w, TI 81 21 aP? Kr 36 26 'So- Pb 82 48+32 2 2 3p0 -7 -7 2 26 26m-6 1 Bi 83 23 '% 8r 38 ____ 2 'SO Po 84 % 24 3p, Y 39 1 2 'Drt At 86 26 Zr 40 2 2 'Fa Rn 86 20 '8, Nb 41 36 4 1 OD, Fr 87 14 2 6 10 2 6 1 asI Mo 42 61 '53 Ra 88 2 '80 To 43 Kr cow 6 2 'Sa+ Ac 89 40f32 1 2 sD,, Ru 44 71 OF6 Th 90 22 "a Rh 46 8 1 'Fa, Pa 91 2 1 2 Pd 46 'XS) io 'So U 92 3 1 2 "Lp 246 21 Energy ordering for Terms and Levels • Energy ordering: Hund’s rules 1 Note that H = ξ (J2 − L2 − S2) so 2 - (1) S-rule: For a given configuration, the state with the maximum spin multiplicity is lowest in energy. The electrons repel each other, and therefore their mutual electrostatic energy is positive. The farther away the electrons get, the lower will be the contribution of the electrostatic energy to the total energy. - (2) L-rule: For a given configuration and spin multiplicity, the state with the maximum orbital angular momentum is the lowest in energy. - (3) J-rule: The lowest energy is obtained for lowest value of J in the normal case and for highest J value in the inverted case. - The normal case is a shell which is less than half filled. The inverted case is a shell which is more than half full such as the ground state of atomic oxygen. 3 3 3 2 2 2 P0 < P 1 < P 2 for carbon (1s 2s 2p ) 3P < 3P < 3P for oxygen (1s22s22p4) - If the shell is exactly half-filled (e.g., p3), the energy order doe not obey a simple rule. • The Hund’s rules are only applicable within L-S coupling. They are only rigorous for ground states. However, they are almost always useful for determining the energy ordering of excited states. The rules show increasing deviations with higher nuclear charge. Helium Spectra 81 Table 5.1. Selection rules for atomic spectra. Rules 1, 2 and 3 must always be obeyed. For electric dipole transitions, intercombination lines violate rule 4 and forbidden lines violate rule 5 and/or 6. Electric quadrupole and 22 magnetic dipole transitions are also described as forbidden. Electric dipole Selection Electric quadrupole RulesMagnetic dipole 1. (ilJ1 ) one= 0, electron ± 1 jumpsilJ = 0, ± 1, ±2 ilJ = 0, ± 1 • Selection Rules Not J = 0- 0 Not J = 0 - 0, ½- ½0, - 1 Not J = 0- 0 (2) Δn any 2. ilMJ = 0, ± l ilMJ = 0, ± l, ± 2 selectionilMJ rule = for0, ±configuration l (3) Δl = ± 1 3. Parity changes Parity unchanged Parity unchanged (4) parity change 4. ilS = 0 ilS = 0 ilS = 0 5. ( One5) Δ electronS = 0 jumps One or no electron jumpsintercombination No electron jumps line if (iln6) ΔanyL = 0, ± 1 (exceptiln anyL = 0 − 0) only thisiln rule = 0is violated. (ill=7) Δ±1J = 0, ± 1 (exceptill= J0,= ± 0 2− 0) ill= 0 6. (ilL8) Δ= F0, = ± 0, 1 ± 1 (exceptilL = F0,= ± 0 1,− ±0 )2 ilL = 0 Not L = 0- 0 Not L = 0- 0, 0 - l It is only rarely necessary to consider this. - Allowed = Electric Dipole : Transitions which satisfy all the above selection rules are referred to as allowed transitions. These transitions are strong and have a typical lifetime of ∼ 10−8 s. Allowed transitions are Photonsdenoted dowithout not change square spin, brackets. so transitions usually occur between terms with the same spin state,e.g., C as IV expressed 1548, 1550 by the Å rule ll.S = 0. However, rela- tivistic effects mix spin states, particularly for high Z atoms or ions. As a - Photons do82 notAstronomical change spin, Spectroscopy so transitions (Third usually Edition) occur between terms with the same spin state (ΔS = 0). However,result of relativistic effects effects, mix one spin can states, get (weak) particularly spin changing for high transitions;Z atoms and ions. As a result, one can getthese (weak) are spin called changing intercombination transitions. lines. These Intercombination are called intercombination lines are denoted (semi-forbidden or intersystem)is completely transitions forbidden or lines. byThey both have dipole a typical and quadrupolelifetime of selection−3 s. rules.An intercombination This transition by one square bracket, for example: ∼ 10 is denotedtransition with a single is not right observed. bracket. Electric dipole transitionscm] 2s2 1 whichs - 2s2p violate 3 P 0 at the 1908. propensity 7 A. (Δ S rules= 1 ) 5 and/ or 6 - If any oneare of the called rules forbidden 1-4, 6-8 are transitions. violated, These they are are called labelled forbidden by square transitions brackets. or For lines. They have a This transition is important because the c2+ 2s2p 3P 0 state is metastable, typical lifetimeexample, of ∼ 1 − 103 s. A forbidden transition is denoted with two square brackets. i.e. it has no allowed radiative decay so that this transition determines the 1906.7 A [Cm] 2s2 1S0 -2s2p 3P~, ( , ) lifetime of this state. Actually, the situation is more subtle Δ thanS = this.1 Δ J The= 2 - Resonance line denotes the longest wavelength, dipole-allowed transition arising from the ground state of 3 P 0 term splits into322.57 three A levels: [Cm] 3 P 2s02, 31So-2p3sP1 and 3 P:2-1 The electric dipole a particular atom or ion. Pr 1 areintercombination both forbidden line lines at of 1908.7 c2+. AThe is actually former is Sa0 - magnetic3P'j'. It has transition an A value while 1 theof latter 114 s- is. an electric dipole transition involving the movement of two The transition 1S - 3 P~, which occurs at 1906. 7 is completely for- electrons. Forbidden transitions0 are generally weaker thanA, intercombination bidden by dipole selection rules as ll.J = 2. It only occurs via a very weak lines. magnetic quadrupole transition. The 1906.7 A line is 105 times weaker than It is also possible to get transitions driven by higher electric multi- the already-weak line at 1908. 7 A;it has an A value of 0.0052 s- 1 . These two poles or magnetic moments. The only important ones of these are electric lines can be used to give information on the electron density, as discussed quadrupole and magnetic dipole transitions. The selection rules for these in Sec. 7.1. Finally, the transition 1S0 - 3 P0 is a J = 0-0 transition, which transitions are also given in Table 5.1. Even when all the rules are satisfied, electric quadrupole and magnetic dipole transitions are both much weaker than the allowed electric dipole transitions. They are thus also referred to as forbidden transitions. Typical lifetimes, that is inverse Einstein A coefficients, for allowed decays via each mechanism are Tdipole ~ 10- 8s, Tmagnetic ~ 10- 3s, Tquadrupole ~ ls. These timescales mean that states only decay by forbidden transitions when there are no decay routes via allowed transitions. Finally, it should be noted that even the rigorous selection rules given above can be modified when nuclear spin effects are taken into considera- tion. These result in rigorous selection rules for electric dipole transitions based on the final angular momentum. In particular: l:l.F must be O or ± 1 with F = 0 +-+0 forbidden. It is only very rarely necessary to consider this. 5.3. Observing Forbidden Lines States decaying only via forbidden lines live for a long time on an atomic, if not an astronomical, timescale. Such states are called metastable states. 23 Forbidden Lines • Forbidden lines are often difficult to study in the laboratory as collision-free conditions are needed to observe metastable states. - In this context, it must be remembered that laboratory ultrahigh vacuums are significantly denser than so-called dense interstellar molecular clouds. - Even in the best vacuum on Earth, frequent collisions knock the electrons out of these orbits (metastable states) before they have a chance to emit the forbidden lines. - In astrophysics, low density environments are common. In these environments, the time between collisions is very long and an atom in an excited state has enough time to radiate even when it is metastable. - Forbidden lines of nitrogen ([N II] at 654.8 and 658.4 nm), sulfur ([S II] at 671.6 and 673.1 nm), and oxygen ([O II] at 372.7 nm, and [O III] at 495.9 and 500.7 nm) are commonly observed in astrophysical plasmas. These lines are important to the energy balance of planetary nebulae and H II regions. - The forbidden 21-cm hydrogen line is particularly important for radio astronomy as it allows very cold neutral hydrogen gas to be seen. - Since metastable states are rather common, forbidden transitions account for a significant percentage of the photons emitted by the ultra-low density gas in Universe. - Forbidden lines can account for up to 90% of the total visual brightness of objects such as emission nebulae. 24 History: Nebulium? 112 Astronomical Spectroscopy {Third Edition) • In 1918, extensive studies of the emission spectra of nebulae found a series of lines which had not been n. observed in the laboratory. - Particularly strong were features at 4959Å and 5007Å. For :c. 101 -=., = s .,-. :i::. a long time, this pair could not be identified and these lines 1.D. =: were attributed to a new element, ‘nebulium’. = 0. u - In 1927, Ira Bowen (1898-1973) discovered that the lines were not really due to a new chemical element but instead forbidden lines from doubly ionized oxygen [O III]. 4000 5000 >. (A) - He realized that in the diffuse conditions found in nebulae, Fig. 7.1. Optical Optical spectra spectra of NGCof NGC 6153 6153,from 3540 Liu to et 7400 al. A,(2000, obtained MNRAS) for a deep atoms and ions could survive a long time without (IO-minute) exposure using the ESO 1.52 m telescope in Chile. The two spectra plotted are (a) obtained by uniformly scanning the long-slit across the entire nebula, and undergoing collisions. Indeed, under typical nebula (b) taken with a fixed slit centred on the central star. [Reproduced from X.-W. Liu et al., Mon. Not. R. Astron. Soc. 312 , 585 (2000).] conditions the mean time between collisions is in the range 10-10,000 secs. This means that there is sufficient 7.1. Nebulium time for excited, metastable states to decay via weak, In 1918, extensive [Ostudies III], of [O the emissionII], [N spectraII], etc: of nebulae found a series forbidden line emissions. of lines which hadWe not beenuse observed a pair in theof laboratory.square Particularly strong were features at 4959brackets A and 5007 for A. Fora forbiddena long time, this line. pair could not be - The forbidden lines could not be observed in the laboratory identified and, inspired by Lockyer's success with helium, these lines were where it was not possible to produce collision-free attributed to a new element, 'nebulium'. conditions over this long timeframe. Ten years after their original observation, Ira Bowen (1898- 1973) found the true explanation. Bowen realisedallowed that intransition the diffuse conditions found in nebulae, atoms and ions could survive a longmetastable time without undergoingstate collisions. Indeed , under typical nebula conditions the mean time between - Other ‘nebulium’ lines turned out to be forbidden lines collisions is in the range 10- 10000 s. This means that there is sufficient time originating from singly ionized oxygen [O II] and nitrogen [N for excited, metastable states to decay viaforbidden weak , forbidden transition line emissions. II]. These lines could not be observed in the laboratory where it was not possible to produce collision-free conditions over this long timeframe. 25 Notations • Notations for Spectral Emission Lines and for Ions - There is a considerable confusion about the difference between these two ways of referring to a spectrum or ion, for example, C III or C+2. These have very definite different physical meanings. However, in many cases, they are used interchangeably. - C+2 is a baryon and C III is a set of photons. - C+2 refers to carbon with two electrons removed, so that is doubly ionized, with a net charge of +2. - C III is the spectrum produced by carbon with two electrons removed. The C III spectrum will be produced by impact excitation of C+2 or by recombination of C+3. So, depending on how the spectrum is formed. C III may be emitted by C+2 or C+3. +2 +2 +2 collisional excitation: C recombination: C - There is no ambiguity in absorption line studies - only C+2 can produce a C III absorption line. This had caused many people to think that C III refers to the matter rather than the spectrum. - But this notation is ambiguous in the case of emission lines. May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03 Atomic Hydrogen 45 Table 3.5. Fine structure effects in the hydrogen atom: splitting of the nl orbitals due to fine structure effect for l = 0,1, 2, 3. The result- ing levels are labelled using H atom, and the more general spectro- scopic notation of terms and levels (see Sec. 4.8). Configuration ls j Hatom Term Level 1 1 2 2 ns0 ns 1 n S n S 1 2 2 2 2 1 1 3 2 o 2 o 2 o np12 2 , 2 np 1 , np 3 n P n P 1 , n P 3 2 2 2 2 1 3 5 2 2 2 nd2 , nd 3 , nd 5 n D n D 3 , n D 5 2 2 2 2 2 2 2 1 5 7 2 o 2 o 2 o nf32 2 , 2 nf 5 , nf 7 n F n F 5 , n F 7 2 2 2 2 For hydrogen, s = 1 so that, except for the l = 0case,j = l 1 (see 2 ± 2 Table 3.5). This table labels the resulting levels with the common H-atom 26 notation nlj,wherel is given by its letter designations, s, p, d, etc., and (2S+1) Hydrogen Atomby spectroscopic : Fine Structure notation for which labels of the LJ are used. A full discussion of spectroscopic notation can be found in Sec. 4.8. • Fine structure of the hydrogen atom Table 3.5 shows the fine structure levels of the H atom. This table shows that the states withRelativistic principal QM quantum (Dirac’s eq) number n = 2giveriseto 2 three fine-structure levels. In spectroscopic notation, these levels are 2 S 1 , configuration L S J term level 2 2 o 2 o 22P3/2 2 2 P 12 and 2 P 3 . ns 0 1/2 1/2 S 2 S1/2 2 22S1/2 = 22P1/2 So far the discussion on H-atom levels has assumed that all those with np 1 1/2 1/2, 3/2 2P o 2P o , 2P o the1 same/2 3 principal/2 quantum number, n, have the same energy. In other 2 2 2 nd 2 1/2 3/2, 5/2 D words,D3/2, theD5/2 energy does not depend on l or j.Thisisnotcorrect:inclusion 2 o of2 relativistico 2 o (or magnetic) effects split these levels according to the total nf 3 1/2 5/2, 7/2 F F 5/2, D7/2 • Hyperfine Structure in the H atom - Coupling the nuclear spin I to the total electron angular momentum J gives the final angular momentum F. For hydrogen this means 1 F = J + I = J = 21 cm [Kwok] Physics and Chemistry of the ISM, [Bernath] Spectra of atoms and Molecules 28 Hydrogen Atom : Allowed Transitions • Selection Rules - Transitions are governed by selection rulesMay which 17, determine 2005 14:40 whether WSPC/SPI-B267: they can occur. Astronomical Spectroscopy ch03 Δn any selection rule for configuration For H-atom, l and L are Δl = ± 1 equivalent since there is ΔS = 0 For H atom, this is always satisfied as S = 1/2 for all states. only one electron. ΔL = 0, ± 1 (not L = 0 − 0) 48 Astronomical Spectroscopy ± occur. Specifically, the transitions 2s – 3p, 2p – 3s and 2p – 3d are allowed ΔJ = 0, 1 (not J = 0 − 0) For Hα transitions: whereas the transitions 2s – 3s, 2p – 3p and 2s – 3d are not allowed. If fine structure effectsNot all are Hα considered, transitions which then the selection rules can give further constraints.correspond Considering to n only= 2 the− 3 H areα transitions designated allowed above, the selectionallowed. rule on j shows that 2s 1 –3p1 is allowed; 2 2 –3p3 is allowed; 2 2p 1 –3d5 is not allowed; ( ) 2 2 ΔJ = 2 –3s1 is allowed; 2 –3d3 is allowed; 2 2p 3 –3s1 is allowed; 2 2 –3d3 is allowed; 2 –3d5 is allowed . 2 The transition between 2s − 1s 3.16 Hydrogen inis Nebulae not allowed (Δl = 0). Hydrogen atom emissions in H II regions and planetary nebulae are very similar but the latter are generally brighter, which means that more weak line emissions can be observed. In particular, lines belonging to the Balmer series are often seen strongly in emission. Indeed, the characteristic red colour seen in many nebulae comes from Hα. Balmer or other spectral series are obtained from excited atoms spon- taneously emitting photons. Every excited state has a half-life τ ,similarto that encountered in radioactive decay, which is related to the strength of emission. Thus excited states which decay only by weak line emission are long-lived and those which decay via strong transitions are short-lived. However, most excited states can emit to more than one other state. The lifetime of excited state i is given by 1 − τi = ∑ Aij ,(3.27) ! j " where Aij is the Einstein A coefficient (see Sec. 2.2). Lifetimes for allowed atomic transitions are short, perhaps a few times 9 10− s. Table 3.6 gives some examples for the H atom. A glaring exception in Table 3.6 is the lifetime of the 2s level of H. This state has a lifetime of May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03 May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03 48 Astronomical Spectroscopy occur. Specifically, the transitions 2s – 3p, 2p – 3s and 2p – 3d are allowed whereas the transitions 2s – 3s, 2p – 3p and 2s – 3d are not allowed. 48 Astronomical Spectroscopy If fine structure effects are considered, then the selection rules can occur. Specifically, the transitions 2s – 3p, 2p – 3sgive and further 2p – 3d constraints. are allowed Considering only the Hα transitions designated whereas the transitions 2s – 3s, 2p – 3p and 2s – 3dallowed are not above, allowed. the selection rule on j shows that If fine structure effects are considered, then the selection rules can 2s 1 –3p1 is allowed; give further constraints. Considering only the Hα transitions designated 2 2 allowed above, the selection rule on j shows that –3p3 is allowed; 2 2p 1 –3d5 is not allowed; 2s 1 –3p1 is allowed; 2 2 2 2 –3s1 is allowed; –3p3 is allowed; 2 2 –3d3 is allowed; 2p 1 –3d5 Mayis not allowed; 17, 2005 14:402 WSPC/SPI-B267: Astronomical Spectroscopy ch03 2 2 2p 3 –3s1 is allowed; –3s1 is allowed; 2 2 2 –3d3 is allowed; –3d3 is allowed; 2 2 –3d5 is allowed . 2p 3 –3s1 is allowed; 2 2 2 –3d3 is allowed; 2 –3d5 is allowed . 2 3.16 Hydrogen in Nebulae Hydrogen atom emissions in H II regions and planetary nebulae are very 3.16 Hydrogen in Nebulae similar but the latter are generally brighter, which meansAtomic that more weak Hydrogen 49 line emissions can be observed. In particular, lines belonging to the Balmer Hydrogen atom emissions in H II regions and planetaryseries are nebulae often seen are very strongly in emission. Indeed, the characteristic red similar but the latter are generally brighter, whichcolour means seen that in more manyTable weak nebulae 3.6. comes from Lifetimes, Hα. τ ,fordecaybyspontaneousemissionfor line emissions can be observed. In particular, lines belonging to the Balmer Balmer or other spectral series are obtained from excited atoms spon- series are often seen strongly in emission. Indeed, the characteristiclow-lying red excited states of the hydrogen atom. taneously emitting photons. Every excited state has a half-life τ ,similarto colour seen in many nebulae comes from Hα. that encountered in radioactive decay, which is related to the strength of Balmer or other spectral series are obtained from excited atoms spon- emission. Thus excited states which decay only by weak line emission are taneously emitting photons. Every excited state has a half-life τ ,similarto May 17, 2005 14:40 WSPC/SPI-B267:long-lived Astronomical and those SpectroscopyLevel which decay 2s via strong ch03 transitions 2p are short-lived. 3s 3p 3d that encountered29 in radioactive decay, which is related to the strength of However, most excited states can emit to more than one other9 state. 7 9 7 emission. Thus excited states which decay only by weak line emissionτ /s are 0.14 1.6 10 1.6 10 5.4 10 2.3 10 The lifetime of excited state is given by − − − − long-lived and those which decay via strong transitions are short-lived. i × × × × However, most excited states can emit to more than one other state. 1 − The lifetime• Hydrogen: of excited state lifetimei is given of by excited states τ = ,(3.27) Atomic Hydrogen i ∑ Aij 49 ! j " 1 − Table 3.6.τ = Lifetimes,A τ ,fordecaybyspontaneousemissionfor,(3.27) i ∑ ij where Aij is the Einstein A coefficient (see Sec. 2.2). low-lying excited! j states" of the hydrogen atom. 2s Lifetimes for allowed atomic transitions are short, perhaps a few times 2 (2 S1/2) ⌫ Problems 3.1 Give an expression for the energy levels of the hydrogen atom in terms 1 of the Rydberg constant RH.AssumingavalueRH = 109677.58 cm− , derive a wavenumber for the Lyα transition of atomic hydrogen 1 1 in cm− .ExplainwhytheRydbergconstant,R = 109737.31cm− , ∞ is more appropriate than RH for a heavy one-electron atom. Hence obtain an estimate for the wavenumber of the Lyα transition of 30 Helium Atom : Energy Level Diagram • Helium (Grotrian diagram) Complex Atoms 71 unchanged by this operation. Neglecting spin, this means (4.18) Even parity states are given by+'¢ and odd parity states are given by-'¢. In practice, the parity of all terms and levels arising from a particular configuration can be determined simply by summing the orbital angular momentum quantum numbers for each of the electrons. With this simple rule, the parity is given by (4.19) As closed shells and sub-shells have an even number of electrons, it is again only necessary to explicitly consider the active electrons. Thus for the O III configuration ls2 2s2 2p3d, it is only necessary to con- sider the sum l(2p) = 1 and l(3d) = 2. This gives (-1)1+ 2 = -1, which explains why all the terms and levels arising from this configuration were all labelled odd above. The parity of a configuration is important since it leads to a rigorous dipole selection rule known as the Laporte rule. The Laporte rule states: All electric dipole transitions connect states of opposite parity. In other words (strong) transitions can only link configurations with even to those with odd parity, and vice versa. 4.10. Terms and Levels in Complex Atoms A single configuration can lead to several terms. These terms have different 31 energies. It is worth considering a few examples. LS Spectroscopic Terms: Example (1) Helium Example 1: The helium atom. 72(1) Astronomical The ground Spectroscopy state is ls 2 (Third• Edition) This is a closed shell, with L = 0 and S = 0, hence it gives rise to a (2)72 TheAstronomical first excited Spectroscopy configuration (Third1 Edition) is ls2s. 1 72 Astronomicalsingle, even Spectroscopy parity term (ThirdS, and Edition) level S720 . Astronomical Spectroscopy (Third Edition) S L J This has li = b = 0 and hence L = 0, but s1 = s2 = ½giving both 72(2) Astronomical The first excited Spectroscopy0 configuration 0 (Third Edition) is 0 ls2s. 1 1 (2)(2) S The The = 0 first first (singlet) excited excited or configuration configurationS = 1 (triplet) is is states. ls2s. ls2s. The energy ordering of atomic S → S0 This has li = b = 0 and hence L = 0, but s1 = s2 = ½giving both s1 s2 statesThisThis has has is givenlili == bb by ==Hund's 00 and and rules.hence hence Hund'sLL == 0,(2)0, butfirst but TheS = firsts10 rule (singlet) == excited governss2 or == configuration S ½½= giving giving ordering1 (triplet) both isboth states. ls2s. of The energy ordering of atomic termsSS == 0 0 with (singlet) (singlet) different or or SS = spin= 11 (triplet)(triplet) multiplicities: states. states. The TheThisstates energy energy has is li given = ordering ordering b by = Hund's 0 and of of rules. hence atomicatomic Hund'sL = 0, first but rules1 = governs s2 = ½ orderinggiving of both S terms= 0 (singlet) with different or S = spin 1 (triplet) multiplicities: states. The energy ordering of atomic statesstates is is given given by by Hund'sHund's rules. rules. Hund'sHund's first first rule rule governs governs ordering ordering of of states is given by Hund's rules. Hund's first rule governs ordering of For a given configuration, the state with theFor a maximum given configuration, spin the state with the maximum spin S termstermsL with with differentJ different spin spin multiplicities: multiplicities: terms with different spin multiplicities: multiplicity is lowest in energy. multiplicity is lowest in energy. 0 0 0 1 1 S → S0 For a given configuration, the state with the maximum spin ForFor aa givengiven configuration, configuration, the the statestate with withSo the the the 3 S maximumterm maximum (3S1 level) spin spin is lower in energy than the 1S term (1S0 1 So 0 the 3 S term1 (3S3 level)3 is lower in energy than the multiplicity1S term (1S is lowest in energy. S1multiplicitymultiplicity → S1 is is lowest lowest ininlevel). energy. energy. In practice, the splitting between0 these terms is 0.80 eV. level). In practice, the splitting between (3) theseSo The the next terms3 S excited term is (3S0.80 configuration1 level) eV. is loweris ls2p, in which energy has than odd parity. the 1 S term (1S0 3 1 (3) TheSoSo the the next 3 SS excited term term (3S(3S configuration11 level)level) is is lower lower is ls2p, in in which energy energylevel).This has hasIn than than practice, li odd = the the0 parity. and the 1bS S splitting = term term 1, giving (1S between(1S L00 = 1; these again terms s1 = iss2 0.80= ½ eV.,giving both S = 0 and S = 1 terms. level).level). InIn practice,practice, the the splitting splitting between between(3) these these The next terms terms excited is is configuration0.800.80 eV.eV. is ls2p, which has odd parity. This has li = 0 and b = 1, giving L = 1;Following again s1 the rule= s2 above, = ½ the ,giving3 P 0 term is lower in energy than the 1 P 0 This has li = 0 and b = 1, giving L = 1; again s1 = s2 = ½,giving (3)(3) both The The next nextS 0 excited excited and S configuration configuration1 terms. is is ls2p, ls2p, which whichterm, has has in this odd odd case parity. parity. by 0.25 eV. The 3P 0 is also split into three levels: = = both S 0 and S 1 terms. 3pg, 3p1= and 3p~-= 3 0 s1 s2 1 0 FollowingThisThis has has lili the == rule 00 and and above, bb == the 1,1, giving givingP term LL == is Following1;lower1; again again in the energys1 rule== s2 above, than == ½ the½ the ,giving,giving3 P 0 Pterm is lower in energy than the 1 P 0 Figure 5.2 depicts the energy levels of helium in a form known as a Grotrian 3 0 3 0 S term,bothbothL SS in = = this 00 andJ and case SS by== 11 0.25 terms.terms. eV. The P isterm, also in split this case into by three 0.25 eV. levels: The P is also split into three levels: diagram.3pg, 3p1 The and layout 3p~- of these diagrams is discussed in Sec. 5.4. 3pg,FollowingFollowing 3p1 and the the 3p~- rule rule above, above, the the 33PP 00 termterm is is lowerlower in in energy energy than than the the 11 P P00 0 1 1 1 o 1 o P → P1 Example 2: The carbon atom. 33 Figure00 5.2 depicts the energy levels of helium in a form known as a Grotrian 1 term,term,1 in in this this0, 1, 2 case case by by 0.25 0.25 eV. eV. The The PP isis also also split split into into three three levels: levels: Figure 5.2 depicts the energy3 o levels3 oof helium3 o Start in3 a by oform considering known the as excited a Grotrian state configuration ls2 2s2 2p3p. P → P0 < P1 diagram.< P2 The layout of these diagrams is discussed in Sec. 5.4. 3pg,3pg, 3p13p1 andand 3p~-3p~- It is only necessary to consider the outer two electrons for which: diagram. The layout of these diagrams is discussedExample 2: in The Sec. carbon 5.4. atom. li = 1, FigureFigure 5.25.2 depictsdepicts the the energy energy levels levels ofof heliumheliumStart in in byaa formform considering known known the as as excited aa GrotrianGrotrian state configuration ls2 2s2 2p3p. Example 2: The carbon atom. b = 1, diagram.diagram. The The layout layout of of thesethese diagrams diagrams is is discusseddiscussedIt is only necessary in in Sec. Sec. to 5.4. 5.4. consider the outer two electrons for which: Start by considering the excited state configurationThese give L ls =2 2s0, 2 1,2p3p. 2 and S = 0, 1, which give rise to the following terms, 1 3 li 3= 1,1 3 ItExampleExample is only necessary 2: 2: TheThe carbon carbon to consider atom. atom. the outer twoall electrons with even parity: for which:S, S,1P, P, D and D. b = 1, 2 2 2 StartStart by by considering considering the the excited excited state state configuration configurationNow consider ls ls 2 the22s2s ground-state222p3p.2p3p. configuration of carbon ls 2s 2p . li = 1, TheseThis configurationgive L = 0, 1, also 2 and has Sli == 0,1, 1,s1 which= ½and give l2 = rise 1, tos2 the= ½ following- terms, It is only necessary to consider the outer two electrons for which: It is only necessary to consider the outer twoHowever, electrons the Pauli for Principle 1 which:3 restricts 3 1 which terms3 are allowed. For example, 1, all with even parity: S, S,1P, P, D and D. b = the term 3D includes the state: lili == 1,1, Now consider the ground-state configuration of carbon ls2 2s2 2p 2 . These give L 0, 1, 2 and S 0, 1, which give rise to the(li following= 1, m1 = +1, terms, s1 = ½, S1z = +½) = = This configuration also has li = 1, s1 = ½and l2 = 1, s2 = ½- bb == 1,1, all with even parity: 1S, 3S,1P, 3P, 1 D and However,3D. the Pauli (b Principle = 1, m2 restricts= +1, s2 which = ½ , terms S2z = are+½ allowed.) For example, TheseThese give give LL == 0,0, 1, 1, 22 andand SS == 0,0, 1, 1, whichwhich the give give term rise rise 3D to to includes the the following following the state: terms, terms, Now consider the ground-state configuration of carbon ls2 2s2 2p 2 . allall with with even even parity: parity: 11S,S, 33S,1S,1P, P, 33P,P, 11 D D and and 33D.D. (li = 1, m1 = +1, s1 = ½, S1z = +½) This configuration also has li = 1, s1 = ½and l2 = 1, s2 = ½- (b = 1, m2 = +1, s2 = ½, S2z = +½) 2 2 2 However,NowNow consider consider the Pauli the the ground-state ground-state Principle restricts configuration configuration which terms of of carboncarbon are allowed. ls ls22s2s2 2p2p For2 .. example, theThisThis term configuration configuration 3D includes also also the has has state: lili == 1,1, s1s1 == ½½and and l2l2 == 1,1, s2s2 == ½½-- However,However, the the Pauli Pauli Principle Principle restricts restricts which which terms terms are are allowed. allowed. For For example,example, (li = 1, m1 = +1, s1 = ½, S1z = +½) thethe term term 33DD includes includes the the state: state: (b = 1, m2 = +1, s2 = ½, S2z = +½) (li(li == 1,1, m1m1 == +1,+1, s1s1 == ½½,, S1zS1z == ++½½)) (b(b == 1,1, m2m2 == +1,+1, s2s2 == ½½,, S2zS2z == ++½½)) Complex Atoms 67 Complex Atoms 67 These are then added to give J These are then added to give J I. =L. + .s_. (4.15) I. =L. + .s_. (4.15) It is useful to remember that, as a result of the Pauli Principle, closed shells It is useful to remember that, as a result of the Pauli Principle, closed shells and sub-shells, such as ls2 or 2p6 , have both L = 0 and S = 0. This means and sub-shells, such as ls2 or 2p6 , have both L = 0 and S = 0. This means that it is only necessary to consider 'active' electrons, those in open or that it is only necessary to consider 'active' electrons, those in open or partially filled shells. In most cases, this means only one or two electrons. 32 partially filled shells. In most cases, this means only one or two electrons. When more than two angular momenta need to be added together, they ExampleWhen more than (2) two angular Excited momenta Configuration need to be added together, of they O III should be added in pairs. The result is independent of the order in which should be added in pairs. The result is independent of the order in which the additionDoubly is performed. Ionized Oxygen, O III the addition is performed. Worked Example: Consider O III with the configuration: ls22s22p3d. Worked Example: Consider O III with the configuration: ls22s22p3d. ls 2 and 2s2 are closed, so they contribute no angular momentum. ls 2 and 2s2 are closed, so they contribute no angular momentum. For the 2p electron li = 1 and s1 = ½; For the 2p electron li = 1 and s1 = ½; for the 3d electron l2 = 2 and s2 = ½. for the 3d electron l2 = 2 and s2 = ½. L. = Ii + I2 ::::} L = 1, 2, 3; L. = Ii + I2 ::::} L = 1, 2, 3; .s..= .fa + .§.2 ::::} s = 0, 1. .s..= .fa + .§.2 ::::} s = 0, 1. Combining these using all possible combinations of Land S, and the rules Combining these using all possible combinations of Land S, and the rules of vector addition, gives: of vector addition, gives: L s J Level L s J Level .J..=L.+.S..::::}1 0 1 lpl .J..=L.+.S..::::}1 0 1 lpl 1 1 0, 1, 2 3p 0, 3p 1, 3p 2 1 1 0, 1, 2 3p 0, 3p 1, 3p 2 2 0 2 102 2 0 2 102 2 1 1, 2, 3 3D'j'' 3D2, 3D3 2 1 1, 2, 3 3D'j'' 3D2, 3D3 3 0 3 iF3 3 0 3 iF3 3 1 2, 3,4 3F2, 3F3, 3F~. 3 1 2, 3,4 3F2, 3F3, 3F~. Each state of an atom or ion is characterised by a unique combination of L, Each state of anIn atom total, or 6 ionterms is characterised and 12 levels. by a unique combination of L, Sand J, known as a 'level', the notation for which is explained in Sec. 4.8. Sand J, known as a 'level', the notation for which is explained in Sec. 4.8. Thus 12 levels arise from the configuration ls 22s22p3d. Note that although Thus 12 levels arise from the configuration ls 22s22p3d. Note that although some values of J appear several times, they all correspond to distinct states some values of J appear several times, they all correspond to distinct states of the ion and it is important to retain them all. of the ion and it is important to retain them all. May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch06 90 Astronomical Spectroscopy reproduced in Fig. 6.6 which implies that the lines are optically thin in the Sun. The sodium D lines are also observed in absorption against starlight in the interstellar medium. The D lines are usually very saturated in such spectra. 100 Astronomical Spectroscopy {Third Edition) A general formulation is fairly complicated and only specific answers will be quoted here. Tables giving the ratios for all cases can be found in Allen's Astrophysical Quantities (see further reading). Using these tables, the relative strength of the triplet 3 2 D.!i.;i. 3 2'2 2 P1 .i.. transition discussed above is 2'2 This33 corresponds to a ratio of 5:2:1. Any spectrum observed in optically-thin conditions should show these ratios. If the spectrum is optically thick, then a ratio closer to 1:1:1 will be observed. Thus,Example the intensity ratios (3) between Alkali Atoms these transitions gives direct information on the optical depth at which the spectrum• Alkali is formed. atoms: Lithium, sodium, potassium and rubidium all have ground state Fig. 6.6. A solar spectrum reflected from the Moon just before a lunar electronic structures which eclipseconsist taken of one at electron the University in an s of orbital London outside Observatory. a (S.J. Boyle, private 6.4. Astronomical Sodium Spectra closed shell. communication.) The • sodiumSodium resonance (Na) line, : NaI Sodium 3s - 3p, is has prominent Z = in11 absorption and a ground in state configuration of the solar spectrum2 2 (see6 Fig.1. 6.6), and is known as the sodium D line. For an S- P doublet,1s 2s the2p intrinsic3s ratio of line intensities is always 2:1. When 2 o 3 P3/2 g = 4 unsaturated, or optically thin, the strength of the absorption by the doublet 2 o g = 2 3 P1/2 1.0 .8 .6 2 3 S1/2 g = 2 D21 DD21 g = 2J+1 .4 II I Na D Hex 5896 Å 5890 Å 600 650 Wavelength (nm) Na D lines: Fig. 6.6. A solarA solar spectrum spectrum reflected reflected from the from Moon the just Moon beforeFig. just a lunar 6.7.before eclipse The taken sodium at D lines, g = 2J + 1, give the statistical weight of each level. the University of London Observatory. (S.J. Boyle, private communication.) 2 2 a lunar eclipse taken at the University of London D1 5896 Aline:˚ 3 S1/2 3 P 1/2 Observatory (S.J. Boyle). 2 2 D2 5890 Aline:˚ 3 S1/2 3 P 3/2 2 o 2 • Ca II (potassium-like calcium) 8498.0 Aline:˚ 4 P 3/2 3 D3/2 o o H 3968.47 Aline:˚ 42S 4 2P 8542.1 Aline:˚ 42P 3 2D 1/2 1/2 3/2 5/2 2 2 o 2 o 2 K 3933.66 Aline:˚ 4 S1/2 4 P 3/2 8662.1 Aline:˚ 4 P 3 D3/2 o 1037.6 Aline:˚ 22S 2 2P 1/2 1/2 2 2 o 1031.9 Aline:˚ 2 S1/2 2 P 3/2 4p4d electron configuration J = L S , ,L+ S [Kwok] Physics and Chemistry of the ISM [Leighton] Principles of Modern Physics 36 Example (5) npn’p npn’p electron configuration The dashed levels are missing if the two electrons are equivalent (n = n’) The levels for n = n’ will be explained in the next slides. S = s s , ,s + s | 1 2| ··· 1 2 L = ` ` , , ` + ` [Kwok] Physics and Chemistry of the ISM [Leighton] Principles of Modern Physics 37 LS Terms: (1) Nonequivalent Electrons, 2p3p • Equivalent and Nonequivalent Electrons Nonequivalent electrons are those differing in either n or l values, whereas equivalent electrons have the same n and l values. • Consider the combination of two p electrons. (1) 2p3p - Two electrons are nonequivalent. In this nonequivalent case, all possible spectroscopic combinations are available. S =0, 1,L=0, 1, 2 1S, 1P,1D, 3S, 3P,3D ! 1S , 1P , 1D , 3S , 3P , 3D ! 0 1 2 1 0,1,2 1,2,3 There are two possible states for spin of each electron, and three states ms = ± 1/2 for orbital angular momentum of each electron. ml = − 1,0,1 Thus, we expect that there will be 22 × 32 = 36 distinguishable states. 38 (2) Equivalent Electrons, 2p2 (2) 2p2 - Two electrons are equivalent. Then, all the 36 states are not available. Some are ruled out by Pauli exclusion principle, and some are ruled out because they are not distinguishable from others. Table 1 Label ml mS a +1 +1/2 • The first step is to make a table to label b 0 +1/2 the states for a single electron (e.g., a, b, c -1 +1/2 c, d, e, f), as shown in Table 1. d +1 -1/2 e 0 -1/2 f -1 -1/2 39 Table 2 States ML MS • Next step is to make a table for the 1 ab 1 1 combination of (ML, MS) of two electrons, 2 ac 0 1 as shown in Table 2. Here, ML = ml1 + ml2, 3 ad 2 0 and MS = ms1 + ms2. 4 ae 1 0 5 af 0 0 - According to Pauli’s exclusion principle, the 6 bc -1 1 states that have two identical states (aa, bb, 7 bd 1 0 cc, dd, ee, and ff) are not allowed. 8 be 0 0 - Notice also that “ab” and “ba” states are 9 bf -1 -1 identical and thus the “ba” state is ignored. 10 cd 0 0 11 ce -1 0 Other identical combinations are also 12 cf -2 0 ignored. 13 de 1 -1 14 df 0 -1 15 ef -1 -1 • Following the above two rules, we construct Table 2 which contains 15 distinguishable states. 40 Table 3 • In addition to the above two rules, we can States ML MS Term 1 Term 2 Term 3 recognize that every “negative” values have ab 1 1 3P always their “positive” counterparts. ac 0 1 3P ad 2 0 1D - Therefore, it is more convenient to remove ae 1 0 1D the states with negative values. This gives us af 0 0 1D Table 3, which contains only 8 states. bd 1 0 3P be 0 0 3P cd 0 0 1S • Now, we pick the states starting with the largest ML and then the largest MS. 1 - (ad) ML = 2 and MS = 0: The presence of the ML = 2, MS = 0 indicates that a D term is among the possible terms. To this term we must further assign states with ML = 1,0 and MS = 0 (ae, af). What is left? - (ab) ML = 1 and MS = 1: This is the next largest values. The combination ML = 1, MS = 1 indicates that a 3P term is among the possible terms. To this term we must further assign states with ML = 1,0 and MS = 1,0 (ac, bd, be). What is left? 1 - (cd) ML = 0 and MS = 0: This is the only remaining combination. This implies that a S term is among the possible terms. - Finally, we obtain 3 terms 1D, 3P, and 1S. - The 3 terms are split into 5 levels : 1D2, 3P0,1,2, and 1S0. 41 (3) Equivalent Electrons, 2p3 (3) 2p3 - Three electrons are equivalent. - According to Pauli’s exclusion principle, any states that include two identical states (aaa, aab, aac, add, bbc, bbd, etc) are not allowed. - We have only seven states that have non-negative values, as shown in Table 4. Table 1 Table 4 States ML MS Term 1 Term 2 Term 3 Label ml mS abc 0 3/2 4S a +1 +1/2 2 abd 2 1/2 D b 0 +1/2 2 abe 1 1/2 D c -1 +1/2 abf 0 1/2 2D d +1 -1/2 acd 1 1/2 2P e 0 -1/2 ace 0 1/2 2P f -1 -1/2 bcd 0 1/2 4S • Now, we pick the states starting with the largest ML and then the largest MS. 2 - (abd) ML = 2 and MS = 1/2: This indicates that a D term is among the possible terms. To this term we must further assign states with ML = 1,0 and MS = 1/2 (abe, abf). What is left? 2 - (acd) ML = 1 and MS = 1/2: This indicates the presence of a P term. To this term we must further assign states with ML = 0 and MS = 1/2 (ace). What is left? 4 - (abc) ML = 0 and MS = 3/2: This indicates the presence of a S term. - Finally, we obtain three terms (2D, 2P, and 4S) and five levels : 2D3/2, 5/2, 2P1/2,3/2, and 4S3/2. 42 (4) Another method for 2p2 When we have only two electrons, we can use the Pauli principle to obtain the terms. This method is much simpler than the above method. However, this method is not easy to apply to the case of three electrons. Recall that the Pauli principle states that the total eigenfunction must be antisymmetric with respect to the exchange of two particles. Therefore, we can have only two distinct cases: (a) symmetric function for the spin + antisymmetric function for the orbital angular momentum ⇒ 3P (b) antisymmetric function for the spin + symmetric function for the orbital angular momentum ⇒ 1S, 1D 1 1 1 3 3 3 Note that among the six terms 1P is antisymmetric for both spin and orbital angular momenta 3S is symmetric for both spin and orbital angular momenta 3 S =0 s1 =1/2,s2 = 1/2 : Product of two spin functions are antisymmetric w.r.t. the exchange ! S =1 s1 =1/2,s2 =1/2 : Product of two spin functions are symmetric. ! : The first wavefunction is antisymmetric and the second one is symmetric. L L • 1 electron resonance line, Ly� ground state [Draine] Physics of the Interstellar and Intergalactic Medium 484 APPENDIX E 44 3 electrons • 3 electrons (Lithium-like ions) resonant doublet lines 1550.8 1548.2 ground state [Draine] Physics of the Interstellar and Intergalactic Medium 45 ENERGY-LEVEL DIAGRAMS 485 4 electrons • 4 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden ground state [Draine] Physics of the Interstellar and Intergalactic Medium 46 • 5 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden 486 electrons 5 PEDXE APPENDIX 2 2 2 forbidden P3/2 P3/2 P3/2 2 2 2 lines P1/2 P1/2 P1/2 [Draine] Physics of the Interstellar and Intergalactic Medium 47 • 6 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden NRYLVLDIAGRAMS ENERGY-LEVEL electrons 6 forbidden lines 487 [Draine] Physics of the Interstellar and Intergalactic Medium 48 • 7 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden 488 electrons 7 forbidden PEDXE APPENDIX lines [Draine] Physics of the Interstellar and Intergalactic Medium 49 • 8 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden NRYLVLDIAGRAMS ENERGY-LEVEL electrons 8 forbidden lines 489 [Draine] Physics of the Interstellar and Intergalactic Medium 50 • 9 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden 490 electrons 9 forbidden E APPENDIX lines [Draine] Physics of the Interstellar and Intergalactic Medium 51 • 11 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden NRYLVLDIAGRAMS ENERGY-LEVEL 1electrons 11 resonant doublet lines ground state 491 [Draine] Physics of the Interstellar and Intergalactic Medium 52 • 12 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden 492 2electrons 12 resonant doublet lines PEDXE APPENDIX ground state [Draine] Physics of the Interstellar and Intergalactic Medium 53 • 13 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden NRYLVLDIAGRAMS ENERGY-LEVEL 3electrons 13 forbidden 493 lines [Draine] Physics of the Interstellar and Intergalactic Medium 54 • 14 electrons Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden 494 4electrons 14 forbidden E APPENDIX lines [Draine] Physics of the Interstellar and Intergalactic Medium errata˙p1 2019.5.14.1706 c 2012,2013,2014,2015,2016,2017,2018,2019 B.T.Draine 55 23 Upward heavy: resonance, Upward Dashed: intercombination • 15 electrons using latest values from NIST Atomic Spectra Database (ver. 5.1 [Online])]: Downward solid: forbidden forbidden lines [Draine] Physics of the Interstellar and Intergalactic Medium Appendix F, Table F.2, p. 497, typo: the first transition listed for S III: change • 3P –1P 3P –3P 0 0 ! 0 1 noted 2016.10.03 by C.D. Kreisch. Appendix F, Table F.3, p. 498: updated electron collision strengths for O I: • Ion ` u ⌦ Note u` 3 3 0.4861+0.0054 ln T4 OI P2 P1 0.0105 T4 a 3 3 0.4507 0.0066 ln T4 ” P2 P0 0.00459 T4 a 3 3 0.4709 0.1396 ln T4 ” P1 P0 0.00015 T4 a 3 1 0.945 0.001 ln T4 ” PJ D2 0.0312(2J +1) T4 b 3 1 1.000 0.135 ln T4 ” PJ S0 0.00353(2J +1) T4 b 1 1 0.662 0.089 ln T4 ” D S 0.0893 T b 2 0 4 ... a fit to Bell et al. (1998) b fit to Zatsarriny & Tayal (2003) noted 2015.02.27 Appendix F, Table F.6, p. 501: The table title should be • “Rate Coefficients for ... Deexcitation...” rather than “... Excitation...”. noted 2015.07.03 Appendix F, Table F.6, p. 501: incorrect powers of 10 in lines 5 and 6: • 3 3 10 0.115+0.057 ln T2 k for `–u = P – P should read 1.26 10 T u` 0 1 ⇥ 2 56 Homework88 Astronomical Spectroscopy (Third Edition) 1. What is the ground-state configuration, term and '.!50level of thew- beryllium atom,z Be? One of the outerw z electrons in Be is promoted to the 3rd orbital. What3:JO terms and levels can this configuration have? 19 2. Symbols for particular levels of three different atomsC 150 are written as 1 , 0 and 3 . Explain in each 8" LOO X+YD 1 D3/2 P3/2 case why the symbol must be wrong. 50 3. Give the spectroscopic terms arising from the followingo'.?L.'.? configurations,'.?LA using L-S coupling. Include parity '.?'.?.S and J values. Give your arguments in detail for deriving these results (a) 2s2 Fig. 5.4. Helium-like oxygen, 0 VII, triplet X-ray spectra from the coronae of the stars Procyon (left) and Capella (right). See the text for an explanation of the lab els for eac h (b) 2p3s line. Spectra were recorded using the satellite Chandra. [Adapted from S.M. Kahn et al., Philos. Trans. R. Soc. Land ., Ser A 360 , 1923 (2002) .] (c) 3p4p (d) 2p43p 4. The lithium atom, Li, has three electrons. Consider the -.,..-,--- 23p0 2 following configurations of Li: (a) 1s 2p, (b) 1s2s3s, (c) 1630A . By considering the configuration only, state 1s2p3p Intercombination which of the three sets of transitions between the configuration (a), (b) and (c) are allowed and forbidden 22.IA transitions? Forbidden 22.sA 5. The right figure shows the term diagram for helium-like oxygen, O VII, showing transitions from the 1s2l states. Explain why 22.1Å line is an intercombination line and why 22.8 is a forbidden line. Å Fig. 5.5 . Term diagram for helium-like oxygen, 0 VII, showing transitions from the ls21 states . Careful monitoring of the relative intensities of these lines can give infor- mation both on the temperature and density of the environment and the ionisation mechanism involved. In particular, the critical density nc (see Sec. 2.6) of the transitions increases strongly with the nuclear charge, Z. So, for example, nc of Si XIII is about 10000 times greater than nc of the isoelectronic C v. This means that a large range of electron densities can be studied by monitoring a variety of helium-like ions. Figure 5.5 gives a simple Grotrian diagram for O VII which is helium-like , and one of the species commonly observed. Its spectrum is also observed in solar flares. The O VII resonance line lies at 21.6 A, which is in the X-ray region of the spectrum. At these wavelengths, the transitions depicted in