KAIST Astrophysics (PH481) - Part 1 - Kwang-Il Seon's Home Page · 2020-04-27 · • Why two...

KAIST Astrophysics (PH481) - Part 1

Week 3b Sep. 18 (Wed), 2019

Kwang-il Seon (선광일)Korea Astronomy & Space Science Institute (KASI)

University of Science and Technology (UST)

1

Complex Atoms : Schrödinger Equation• The time-independent Schrödinger equation for an atom with � electrons and

nuclear charge (atomic number) � .

where � is the coordinate of the � th electron, with its origin at the nucleus.

- 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 � .

NZ

ri i

N = 2

2

2

4NX

i=1

✓� ~

2

2mr2i �

Ze2

r

◆+

N�1X

i=1

NX

j=1

e2

|ri � rj |

3

5 (r1, r2, · · · , rN ) = E (r1, r2, · · · , rN )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

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 Schrodinger equation can be separated into � single electron equations:

- The solutions of the above equations are known as orbitals. The total wave function would be written as

- Within this approximation, each atomic orbital can be written as the product of a radial and an angular function, as to H atom.‣ The angular part is independent of the other electrons and is therefore simply a spherical

harmonic.‣ However, the radial function is different from that for H atom.

- It provides a useful classification of atomic states and also a starting point.- It is standard to use the hydrogen atom orbital labels, � , � and � , to label the orbitals. This

is called the configuration of electrons.

N

n l m

3

(r1, r2, · · · , rN ) = �1(r1)�2(r2) · · ·�N (rN )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

� ~

2

2mr2i + Vi(ri)

��i(ri) = Ei�i(ri) where Vi(ri) = �

Ze2

ri+X

j 6=i

⌧e2

|ri � rj |

�

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

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 � values of all the electron orbitals: � means � electrons in the orbital defined by � and � .

- Each orbital labelled � actually consists of orbitals with � different � values, each with two possible values of � . Thus the � orbital can hold a maximum � 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: �- closed shell = shell of which orbitals are fully occupied: For example, the ground state configuration of neon atom,

which has ten electrons: �

• 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 � or � values, whereas equivalent electrons have the same � and � values.

nl nlx xn l

nl 2l + 1 m msnl 2(2l + 1)

1s22s22p2

1s22s22p6

n l nl

4

• 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

- 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.

- Even parity states are given by “+” sign and odd parity states are given by “—" sign.- The parity arising from a particular configuration can be determined simply by summing the orbital

angular momentum quantum numbers for each of the electrons.

- As closed shells and sub-shells have an even number of electrons, it is only necessary to explicitly consider the active electrons.

Indistinguishable Particles & Parity 5

| (a1, b2)|2 = | (a2, b1)|2AAACJnicbZBLSwMxFIUz9VXra3zs3ASLUEHKzCjoplBw47KCfUCnDpk004ZmHiR3hNL217jxr7hxURFx508xfYC2eiBwON8NyT1+IrgCy/o0Miura+sb2c3c1vbO7p65f1BTcSopq9JYxLLhE8UEj1gVOAjWSCQjoS9Y3e/dTHj9kUnF4+ge+glrhaQT8YBTAjryzJIrWABDt6J4gXj2OfY958yVvNOF4YODS3iBOxNu/3DPzFtFayr819hzky8fBVNVPHPstmOahiwCKohSTdtKoDUgEjgVbJRzU8USQnukw5raRiRkqjWYrjnCpzpp4yCW+kSAp+nvGwMSKtUPfT0ZEuiqZTYJ/2PNFILr1oBHSQosorOHglRgiPGkM9zmklEQfW0IlVz/FdMukYSCbjanS7CXV/5rak7Rvig6d7qNSzRTFh2jE1RANrpCZXSLKqiKKHpCL2iM3oxn49V4Nz5moxljfucQLcj4+ga7a6ZY

(r1, r2, · · · , rN ) = ± (�r1,�r2, · · · ,�rN )AAACXXicbVHLSgMxFM2MVtuxatWFCzfBIlSwZaYWdCMU3LiSCvYBnVIyaaYNzTxI7ghl6E+6042/YvpYdNpeCJx7zj3k5sSLBVdg2z+GeXCYOzrOF6yT4unZeenisqOiRFLWppGIZM8jigkesjZwEKwXS0YCT7CuN31d6N0vJhWPwk+YxWwQkHHIfU4JaGpYAreleMUNCEw8P5XzofOAN7q67ugoApVh3+9f3DjAS2s1663uN1cz7mGpbNfsZeFd4KxBGa2rNSx9u6OIJgELgQqiVN+xYxikRAKngs0tN1EsJnRKxqyvYUgCpgbpMp05vtPMCPuR1CcEvGQ3HSkJlJoFnp5cLKm2tQW5T+sn4D8PUh7GCbCQri7yE4Ehwouo8YhLRkHMNCBUcr0rphMiCQX9IZYOwdl+8i7o1GvOY63+0Sg3G+s48ugG3aIKctATaqI31EJtRNGvgYyCYRl/Zs4smmerUdNYe65Qpszrf8eys6U=

parity = (−1)l1+l2+⋯+lN

Energy ordering & Periodic Table• Energy ordering for configuration:

- For a H atom, the energy of the individual orbitals is determined only by principal quantum number � .

- For complex atoms, the degeneracy on the orbital angular momentum quantum number � is lifted.

- Electrons in low � orbits ‘penetrate’, i.e., get inside orbitals with lower � -values. Penetration by the low � electrons means that they spend some of their time nearer the nucleus experiencing an enhanced Coulomb attraction. This lowers their energy relative to higher � orbitals which penetrate less or not at all.

• Periodic Table- The subshell structure of elements up to

argon (� ) is filled up in a naturally straightforward manner, first according to � and then according to � .

- The � subshell is all occupied in argon (Ar; noble gas) with a closed subshell � . The next element potassium (K; � ), begins by filling in the � , instead of � .

n

ll

nl

l

Z = 18

n l3p

3p6Z = 19

4s 3d

6

E(1s) < E(2s) < E(2p) < E(3s) < E(3p) < E(3d) ' E(4s) · · ·AAACHnicbZDLSsNAFIYn9VbrLV52bgaL0G5K0lZ04aIgBZcVbCs0oUwmk3bo5OLMRCihT+LGV3HjQhHBlb6Nk6QLbf1hho//nMPM+Z2IUSEN41srrKyurW8UN0tb2zu7e/r+QU+EMceki0MW8jsHCcJoQLqSSkbuIk6Q7zDSdyZXab3/QLigYXArpxGxfTQKqEcxksoa6mftiimql+1KPb+j9G5k3MjZrVqC+uQetitNUbWwG0ox1MtGzcgEl8GcQ7l15GXqDPVPyw1x7JNAYoaEGJhGJO0EcUkxI7OSFQsSITxBIzJQGCCfCDvJ1pvBU+W40Au5OoGEmft7IkG+EFPfUZ0+kmOxWEvN/2qDWHoXdkKDKJYkwPlDXsygDGGaFXQpJ1iyqQKEOVV/hXiMOMJSJVpSIZiLKy9Dr14zG7X6jUqjCXIVwTE4ARVggnPQAtegA7oAg0fwDF7Bm/akvWjv2kfeWtDmM4fgj7SvH6XRoDM=

The orbitals of complex atoms follow a revised energy ordering:

64 Astronomical Spectroscopy (Third Edition)

Table 4.1. Atomic configurations of the first 20 elements in the periodic table. Z is the atomic number which corresponds to the charge on the nucleus.

Atom z Configuration hydrogen H 1 ls helium He 2 1s2 lithium Li 3 K 2s beryllium Be 4 K 2s2 boron B 5 K 2s22p carbon C 6 K 2s22p2 nitrogen N 7 K 2s22p 3 oxygen 0 8 K 2s22p 4 fluorine F 9 K 2s22p 5 neon Ne 10 K 2s22p 6 sodium Na 11 K L 3s magnesium Mg 12 K L 3s2 aluminium Al 13 K L 3s23p silicon Si 14 K L 3s23p2 phosphorus p 15 K L 3s23p 3 sulphur s 16 K L 3s23p 4 chlorine Cl 17 K L 3s23p 5 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.

Angular Momentum• 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 � and the total spin angular momentum � . These are then added to give � .

- The configurations split into terms with particular values of � and � .

• j-j coupling- An alternative scheme is to consider the total angular momentum � for each electron by combining

� and � and then coupling these � ’s together to give the total angular momentum.

• 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.

LS J

L S

jili si j

7

L = ∑i

li, S = ∑i

si → J = L + S

ji = li + si → J = ∑i

ji

• Electronic configuration and energy level splitting- Configurations ⇒ Terms ⇒ Fine Structure (Spin-Orbit Interaction) ⇒ Hyperfine

Structure (Interaction with Nuclear Spin)

8

26 Atomic structure

where J is the total angular momentum of all electronsin the atom with multiplicity or degeneracy 2J + 1. Thetotal J follows the vector sum: for two electrons, the val-ues of J range from | j1 + j2| to | j1 − j2|. The states aredenoted as ( ji j2)J . For example, for a (pd) configurationj1(1 ± 1/2) = 1/2, 3/2, and j2(2 ± 1/2) = 3/2, 5/2;the states are designated as (1/2 3/2)2,1, (1/2 5/2)3,2,(3/2 3/2)3,2,1,0, (3/2 5/2)4,3,2,1 (note that the total dis-crete J = 0 – 4). The J -state also includes the parityand is expressed as Jπ or, more completely, as (2S+1)LπJ .The latter designation relates the fine-structure level to theparent L S term. For each L S term there can be severalfine-structure levels. The total angular magnetic quantumnumber Jm runs from −J to J .

Fine-structure levels can be further split into hyperfinestructure when nuclear spin I is added vectorially to J toyield the quantum state J + I = F. Figure 2.2 showsthe schematics of energy levels beginning with a givenelectronic configuration.

For cases where L S coupling is increasingly invalidbecause of the importance of relativistic effects, but thedeparture from pure L S coupling is not too severe andfull consideration of relativistic effects is not necessary,an intermediate coupling scheme designated as L S J isemployed. (We discuss later the physical approximationsassociated with relativistic effects and appropriate cou-pling schemes.) In intermediate coupling notation, theangular momenta l and s of an interacting electron areadded to the total orbital and spin angular momenta, J1of all other electrons in the following manner,

J 1 =∑

i

l i +∑

i

si , K = J 1 + l, J = K + s,

(2.63)

where s = 1/2. The multiplicity is again 2J + 1, and thetotal angular magnetic quantum number Jm runs from−Jto J .

L = I1 + I2S = s1 + s2

{ni Ii}

J = L + S

F = J + I

LS terms LSJ levels

Configuration Term structure Fine structure Hyperfinestructure

FIGURE 2.2 Electronic configuration and energy level splittings.

An important point to note is that the physical exis-tence of atomic energy states, as given by the numberof total J -states, must remain the same, regardless ofthe coupling schemes. Therefore, the total number of J -levels

(∑i ji = J or L + S = J

)is the same in

intermediate coupling or j j-coupling. A general discus-sion of the relativistic effects and fine structure is given inSection 2.13

2.8 Hund’s rules

The physical reason for the variations in subshell structureof the ground configuration of an atom is the electron–electron interaction. It determines the energies of theground and excited states. We need to consider both thedirect and the exchange potentials in calculating theseenergies. Before we describe the atomic theory to ascer-tain these energies, it is useful to state some empiricalrules. The most common is the Hund’s rules that governsthe spin multiplicity (2S + 1), and orbital L and total Jangular momenta, in that order.

The S-rule states that an L S term with the high-est spin multiplicity (2S + 1) is the lowest in energy.This rule is related to the exchange effect, wherebyelectrons with like spin spatially avoid one another,and therefore see less electron–electron repulsion (theexchange potential, like the attractive nuclear potential,has a negative sign in the Hamiltonian relative to thedirect electron–electron potential, which is positive). Forexample, atoms and ions with open subshell np3 groundconfiguration (N I, O II, P I, S II) have the ground state4So, lower than the other terms 2Do, 2Po of the groundconfiguration.

The L-rule states that for states of the same spin mul-tiplicity the one with the larger total L lies lower, againowing to less electron repulsion for higher orbital angu-lar momentum electrons that are farther away from thenucleus. Hence, in the example of np3 above, the 2Do

term lies lower than the 2Po. Another example is theground configuration of O III, which is C-like 2p2 withthe three L S terms 3P, 1D, 1S in that energy order. A morecomplex example is Fe II, with the ground 3p63d64s andthe first excited 3p63d7 configurations. The L S terms inenergy order within each configuration are 3d64s (6D, 4D)and 3d7 (4F, 4P). But the two configurations overlap andthe actual observed energies of these four terms lie in theorder 6D, 4F, 4D, 4P.

The J-rule refers to fine-structure levels L + S = J .For less than half-filled subshells, the lowest J -level lieslowest, but for more than half-filled subshells it is thereverse, that is, the highest J -level lies lowest in energy.

[Pradhan & Nahar] Atomic Astrophysics and Spectroscopy

Spectroscopic Notation• Spectroscopic Notation

- A state with S = 0 is a ‘singlet’ as 2S+1 = 1.- A state with S = 1/2 is a ‘doublet’ as 2S+1 = 2- One with S = 1 is a ‘triplet’ as 2S+1 = 3

9

Electron conÖguration and Spectroscopic notation

Spectral Notation

25+ 1 = Total Term 5pin Multiplicity: 5 is vector sum of electron spins (± 1 /2 each) Inner full shells sum to 0

The Number of levels in a term is the smaller of (25+ 1) or (2L+ 1)

Electronic Configuration: the electrons and their orbitals (i.e. 1 s2 2s2 3p 1)

nsi npi ndk ... )

2.1 Atomic Spectra 21

Term Parity: o for odd, nothing for even

L = Total Term Orbital Angular Momentum: Vector sum of contributing electron orbitals. Inner full shells sum to o.

/ = Total Level Angular Momentum: Vector sum of Land 5 of a particular level in a term.

Fig. 2.4. Spectral notation for an atomic term comprised of one or more levels.

that they have principal quantum number n, the term is a doublet (although in this particular case the lower value of j would be negative, so one of the levels of the doublet cannot exist), the orbital angular momentum is zero (S state) and the total angular momentum is 1/2. Now consider the orbital angular momentum l = 1. The possible levels have j = 1/2,3/2 and n = 1,2,3 .... In spectroscopic notation, these will be n 2P1/ 2 or n 2P3/ 2 levels. Continuing to higher orbital angular momentum, l = 2, j = 3/2,5/2 and n = 1,2,3 ... , so these are n 2D3/2 or n 2D5/2 levels, and so on to higher l states.

Now, since the Pauli exclusion principle states that no two electrons can have identical quantum numbers, then the n = 1 state can be occupied by only two electrons (l = O,S= ±1/2). This forms a closed shell of configura-tion 1s2. The leading number is the shell number, (here the first shell), s refers to the angular momentum state of the electrons occupying this shell, and the superscript 2 refers to the number of electrons present. Thus the electron configuration of the normal state of magnesium (Z = 12) would be: 1s22s22p63s2. Ionized magnesium, (Mg+ or MgII), which is isoelectronic with sodium and has one optically active electron in its outer shell will have the ground state, defined by the electron configuration and the ground term; 1s22s22p63s2 Sl/2·

While one optically active electron allows us only one way of forming the total angular momentum by combining the spin and the orbital angular momentum, with two or more electrons life gets much more complicated. Consider the case of two electrons. For light atoms, as first shown by Russell & Saunders (1925), the angular momentum vectors are coupled by electrostatic

n = 1, 2, 3, 4, 5 · · · ! K,L,M,N,O, · · ·` = 0, 1, 2, 3, 4 · · · ! s, p, d, f, g, · · ·L = 0, 1, 2, 3, 4 · · · ! S, P,D, F,G, · · ·

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

sharp, principal, diffuse, fundamental,…

10

Table 9.1 Neutral doma

- K

Atom K L hf N 0 Ground --- Is 282p 383p3d 484p4d 58 level

H 1 1 'St He 2 2 ___ 'So Li 3 2 1 B e 4 2 2 ' S o B 6 2 2 1 'Pt C 6 2 2 2 3P0 N 7 2 2 3 'SP, 0 8 2 2 4 3p1 F 9 2 2 6 T I Ne 10 2 2 6 IS0

zs,---- Mg 12 2 'So A1 13 2 1 2Pg Si 14 10 2 2 3P0

S 16 Necore 2 4 3pz C1 17 2 5 'Pp,

2 6 IS0 2 s , ~-

A L L _ _ ~ _ . _ _ _ _ _ ~

Ce 20 2 'So Sc 21 1 2 =Dll Ti 22 2 2 3F2 V 23 18 3 2 'FI, Cr 24 5 1 7s3 Mn 25 A c o r e 5 2 %i Fe 20 6 2 'D4 Co 27 7 2 'Far

8 2 3F, Ni 28 C u 2 9 2 2 6 2 6 1 0 1 'SI Zn 30 2 'So Ce 31 2 1 'Pp Ge 32 28 2 2 A0 33 2 3 4 s 3 se 34 2 4 3P, Br 36 2 6 w, Kr 36 2 6 'So-

% I 1 2 2 6 1

P 16 2 3 's?k

K 19 2 2 6 2 6 1 _____

- ~ _ _ _ _ _ _

-7 2 2 6 2 6 m - 6 1 8r 38 ____ 2 ' S O Y 39 1 2 'Drt Zr 40 2 2 'Fa Nb 41 36 4 1 OD,

To 43 Kr cow 6 2 'Sa+

Rh 46 8 1 'Fa,

6 1 '53

Ru 44 7 1 OF6

io 'So

Mo 42

Pd 46

L Atom hi N 0 P

_ _ _ _ _ 4f 585p5d5j 6a6p6d

Ag 47 1 Cd 48 2 In 49 2 1 Sn 50 2 2 Sb 51 2 3 Te 52 2 4

Q Ground

78 l 6 V d

Pm 61 Sm 62 3 Eu 63 8 Gd 64 Tb 65 ' 'H!I Dy 66 ' 6J8 Tm 69 ' 'F% Yb 70 c? 'SO

Hf 72- 3F2 Ta 73 3 'FII w 74 ; 0d0 Re 75 - %I 0 s 76 OD, Ir 7 7 'Fit -__ Pt 78 ___ 9 1 3D3. Au 79 fi 14 2 6 10 1 Hg 80 e ___ TI 81 2 1 aP? Pb 82 Bi 83 Po 84 %

Ho 67 ' I P I Er 68 * 3H'3

__-- Lu 71 -. -~~

2 'SO

A t 86 Rn 86 Fr 87 Ra 88 Ac 89 Th 90 Pa 91 U 92

5 6

7 8 1 9

10 1 1 12 13 14

I

14 1 1 4 2 6 2

3

___._

4 46+22 6

6

2 2 2 2 2 2 2 2 2 2 2

-2-

7 2

3p0 2 3 '% 2 4 3p,

2 0 '8 ,

48+32 2 2

2 6

14 2 6 10 2 6 1 a s I 2 '80

2 2 "a 2 1 2 'XS) 3 1 2 "Lp

40f32 1 2 sD,,

246

[Kowk, Physics and Chemistry of the ISM]

Blue: No fine structure in the ground state.

36 CHAPTER 4

Figure 4.1 Energy-level diagram for the ground configuration of the 2p2 ions N IIand O III. (Fine-structure splitting is exaggerated for clarity.) Forbidden transitionsconnecting these levels are shown, with wavelengths in vacuo.

Table 4.1 Terms for ns and np Subshells

Ground Termsconfiguration (in order of increasing energy) Examples

...ns1 2S1/2 H I, He II, C IV, N V, O VI

...ns2 1S0 He I, C III, N IV, O V

...np1 2P o1/2,3/2 C II, N III, O IV

...np2 3P0,1,2 , 1D2 , 1S0 C I, N II, O III, Ne V, S III

...np3 4S o3/2 ,2D o3/2,5/2 ,

2P o1/2,3/2 N I, O II, Ne IV, S II, Ar IV...np4 3P2,1,0 , 1D2 , 1S0 O I, Ne III, Mg V, Ar III...np5 2P o3/2,1/2 Ne II, Na III, Mg IV, Ar IV...np6 1S0 Ne I, Na II, Mg III, Ar III

4.6 Hyperfine Structure: Interaction with Nuclear Spin

If the nucleus has nonzero spin, it will have a nonzero magnetic moment. If thenucleus has a magnetic moment, then fine-structure levels with nonzero electronicangular momentum can themselves be split due to interaction of the electrons withthe magnetic field produced by the nucleus. This “hyperfine” splitting is typicallyof order 10−6 eV. Hyperfine splitting is usually difficult to observe in optical spec-tra due to Doppler broadening, but it needs to be taken into account if precise

[Draine, Physics of the ISM and IGM]

/ Levels

Energy ordering for Terms and Levels• Energy ordering: Hund’s rules

- (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.

• 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.

11

3P0 <3P 1 <

3P 2 for carbon (1s22s22p2)

3P2 <3P 1 <

3P 0 for oxygen (1s22s22p4)

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

• Selection Rules

- 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 � s. Allowed transitions are denoted without square brackets.

- Photons do not change spin, so transitions usually occur between terms with the same spin state ( � ). However, relativistic effects mix spin states, particularly for high Z atoms and ions. As a result, one can get (weak) spin changing transitions. These are called intercombination (semi-forbidden or intersystem) transitions or lines. They have a typical lifetime of � s. An intercombination transition is denoted with a single right bracket.

- If any one of the rules 1-4, 6-8 are violated, they are called forbidden transitions or lines. They have a typical lifetime of � s. A forbidden transition is denoted with two square brackets.

- Resonance line denotes the longest wavelength, dipole-allowed transition arising from the ground state of a particular atom or ion.

∼ 10−8

ΔS = 0

∼ 10−3

∼ 1 − 103

Selection Rules 12

e.g., C IV 1548, 1550 Å

(1) one electron jumps(2) Δn any(3) Δl = ± 1(4) parity change(5) ΔS = 0(6) ΔL = 0, ± 1 (except L = 0 − 0)(7) ΔJ = 0, ± 1 (except J = 0 − 0)(8) ΔF = 0, ± 1 (except F = 0 − 0)

selection rule for configuration

intercombination line if only this rule is violated.

It is only rarely necessary to consider this.

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 magnetic dipole transitions are also described as forbidden.

Electric dipole Electric quadrupole Magnetic dipole

1. ilJ = 0, ± 1 ilJ = 0, ± 1, ±2 ilJ = 0, ± 1 Not J = 0- 0 Not J = 0 - 0, ½ - ½, 0 - 1 Not J = 0- 0

2. ilMJ = 0, ± l ilMJ = 0, ± l, ± 2 ilMJ = 0, ± l 3. Parity changes Parity unchanged Parity unchanged 4. ilS = 0 ilS = 0 ilS = 0 5. One electron jumps One or no electron jumps No electron jumps

iln any iln any iln = 0 ill= ±1 ill= 0, ± 2 ill= 0

6. ilL = 0, ± 1 ilL = 0, ± 1, ± 2 ilL = 0 Not L = 0- 0 Not L = 0- 0, 0 - l

Photons do not change spin, so transitions usually occur between terms with the same spin state, as expressed by the rule ll.S = 0. However, rela-tivistic effects mix spin states, particularly for high Z atoms or ions. As a result of relativistic effects, one can get (weak) spin changing transitions; these are called intercombination lines. Intercombination lines are denoted by one square bracket, for example:

cm] 2s2 1s - 2s2p 3 P 0 at 1908. 7 A. This transition is important because the c2+ 2s2p 3P 0 state is metastable, i.e. it has no allowed radiative decay so that this transition determines the lifetime of this state. Actually, the situation is more subtle than this. The 3 P 0 term splits into three levels: 3 P0, 3 P1 and 3 P:2- The electric dipole intercombination line at 1908.7 A is actually 1S0 - 3P'j'. It has an A value of 114 s- 1 .

The transition 1S0 - 3P~, which occurs at 1906. 7 A, is completely for-bidden by dipole selection rules as ll.J = 2. It only occurs via a very weak magnetic quadrupole transition. The 1906.7 A line is 105 times weaker than the already-weak line at 1908. 7 A; it has an A value of 0.0052 s- 1 . These two lines can be used to give information on the electron density, as discussed in Sec. 7.1. Finally, the transition 1S0 - 3P0 is a J = 0-0 transition, which


is completely forbidden by both dipole and quadrupole selection rules. This transition is not observed.

Electric dipole transitions which violate the propensity rules 5 and/ or 6 are called forbidden transitions. These are labelled by square brackets. For example,

1906.7 A [Cm] 2s2 1S0 -2s2p 3P~, 322.57 A [Cm] 2s2 1So-2p3s 1Pr

are both forbidden lines of c2+. The former is a magnetic transition while the latter is an electric dipole transition involving the movement of two electrons. Forbidden transitions are generally weaker than intercombination lines.

It is also possible to get transitions driven by higher electric multi-poles or magnetic moments. The only important ones of these are electric quadrupole and magnetic dipole transitions. The selection rules for these 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.

( � )ΔS = 1

( � , � )ΔS = 1 ΔJ = 2

• Hyperfine Structure in the H atom- Coupling the nuclear spin � to the total electron angular momentum � gives the final angular

momentum � . For hydrogen this meansI J

F

Hydrogen Atom : Hyperfine Structure 13

F = J + I = J ± 12

AAACAnicbVDLSsNAFJ3UV62vqCtxM1gUQShJLeimUBBEu6pgH9CEMplO2qEzSZiZCCUEN/6KGxeKuPUr3Pk3TtsstPXAhcM593LvPV7EqFSW9W3klpZXVtfy64WNza3tHXN3ryXDWGDSxCELRcdDkjAakKaiipFOJAjiHiNtb3Q18dsPREgaBvdqHBGXo0FAfYqR0lLPPLiGJ1VYP7ut1p2IQ8cXCCd2mpTTnlm0StYUcJHYGSmCDI2e+eX0QxxzEijMkJRd24qUmyChKGYkLTixJBHCIzQgXU0DxIl0k+kLKTzWSh/6odAVKDhVf08kiEs55p7u5EgN5bw3Ef/zurHyL92EBlGsSIBni/yYQRXCSR6wTwXBio01QVhQfSvEQ6RTUDq1gg7Bnn95kbTKJfu8VL6rFGuVLI48OARH4BTY4ALUwA1ogCbA4BE8g1fwZjwZL8a78TFrzRnZzD74A+PzB0mJlW0=

= 21 cm

[Kwok] Physics and Chemistry of the ISM, [Bernath] Spectra of atoms and Molecules

Hydrogen Atom : Allowed Transitions• Selection Rules

- Transitions are governed by selection rules which determine whether they can occur.

14

Δn anyΔl = ± 1ΔS = 0ΔL = 0, ± 1 (not L = 0 − 0)ΔJ = 0, ± 1 (not J = 0 − 0)

For H atom, this is always satisfied as � for all states.S = 1/2

For H � transitions:

Not all H� transitions which correspond to � are allowed.

α

αn = 2 − 3

The transition between � is not allowed ( � ).

2s − 1sΔl = 0

selection rule for configuration For H-atom, � and � are equivalent since there is only one electron.

l L

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 allowedwhereas the transitions 2s – 3s, 2p – 3p and 2s – 3d are not allowed.

If fine structure effects are considered, then the selection rules cangive further constraints. Considering only the Hα transitions designatedallowed above, the selection rule on j shows that

2s 12

– 3p 12

is allowed;– 3p 3

2is allowed;

2p 12

– 3d 52

is not allowed;– 3s 1

2is allowed;

– 3d 32

is allowed;2p 3

2– 3s 1

2is allowed;

– 3d 32

is allowed;– 3d 5

2is allowed .

3.16 Hydrogen in NebulaeHydrogen atom emissions in H II regions and planetary nebulae are verysimilar but the latter are generally brighter, which means that more weakline emissions can be observed. In particular, lines belonging to the Balmerseries are often seen strongly in emission. Indeed, the characteristic redcolour 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 , similar tothat encountered in radioactive decay, which is related to the strength ofemission. Thus excited states which decay only by weak line emission arelong-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

τi =

(

∑j

Ai j

)−1, (3.27)

where Ai j is the Einstein A coefficient (see Sec. 2.2).Lifetimes for allowed atomic transitions are short, perhaps a few times

10−9 s. Table 3.6 gives some examples for the H atom. A glaring exceptionin Table 3.6 is the lifetime of the 2s level of H. This state has a lifetime of

( � )ΔJ = 2

• Hydrogen: lifetime of excited states

- Lifetimes for allowed transitions are short, a few times 10-9 s.- However, the lifetime for the (2s) 22S1/2 level is ~ 0.14 s, which is

108 times longer than the 2p states. (The level is called to be metastable.)

• Two-photon continuum radiation- In low-density environments (e.g., interstellar medium), an

electron in the 22S1/2 level can jumps to a virtual p state, which lies between � and � levels. The electron then jumps from this virtual state to the ground state, in the process emitting two photons with total frequency � .

- Since this virtual p state can occur anywhere between � and � , continuum emission longward of Ly𝛼 will result.

- Because the radiative lifetime of the 2s level is long. we need to consider the possibility for collisions with electrons and protons to depopulate 2s level before a spontaneous decay occurs. However, the critical density at which deexcitation by electron and proton collision is equal to the radiative decay rate is � . In the interstellar medium, the radiative decay is in general faster than the collisional depopulation process.

n = 1 n = 2

ν1 + ν2 = νLyαn = 1

n = 2

ncrit ≈ 1880 cm−3

15May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03

Atomic Hydrogen 49

Table 3.6. Lifetimes, τ , for decay by spontaneous emission forlow-lying excited states of the hydrogen atom.

Level 2s 2p 3s 3p 3dτ/s 0.14 1.6× 10−9 1.6 × 10−7 5.4× 10−9 2.3× 10−7

1s

2s

Fig. 3.23. Decay of the metastable 2s state of hydrogen giving two continuumphotons.

∼ 0.14 s, i.e. it lives 108 times longer than the 2p state. This is because thetransition 2s → 1s is strongly forbidden. The 2s state is metastable whichmeans that on the atomic scale, it is long-lived.

So how does the 2s state decay? By the process of two-photon emis-sion, which is an inefficient process and in this case has an Einstein Acoefficient of 7 s−1 which can be compared to A(2p → 1s) = 6.3×108 s−1.The combined energy of the photons emitted must correspond to theenergy difference E(2s) −E(1s) but the photons themselves can take anyenergy within this constraint (see Fig. 3.23). The photons thus appear ascontinuous emission radiation. Indeed the two-photon decay of the H 2sstate is responsible for approximately one half the continuum emissionobserved from H II regions.

Problems3.1 Give an expression for the energy levels of the hydrogen atom in terms

of the Rydberg constant RH. Assuming a value RH = 109677.58 cm−1,derive a wavenumber for the Lyα transition of atomic hydrogenin cm−1. Explain why the Rydberg constant, R∞ = 109737.31cm−1,is more appropriate than RH for a heavy one-electron atom. Henceobtain an estimate for the wavenumber of the Lyα transition of





2s 12

– 3p 12

is allowed;– 3p 3

2is allowed;

2p 12

– 3d 52


2is allowed;

– 3d 32

is allowed;2p 3

2– 3s 1

2is allowed;

– 3d 32

is allowed;– 3d 5

2is allowed .




τi =

(

∑j

Ai j

)−1, (3.27)







2s 12

– 3p 12

is allowed;– 3p 3

2is allowed;

2p 12

– 3d 52


2is allowed;

– 3d 32

is allowed;2p 3

2– 3s 1

2is allowed;

– 3d 32

is allowed;– 3d 5

2is allowed .




τi =

(

∑j

Ai j

)−1, (3.27)




Atomic Hydrogen 49

Table 3.6. Lifetimes, τ , for decay by spontaneous emission forlow-lying excited states of the hydrogen atom.

Level 2s 2p 3s 3p 3dτ/s 0.14 1.6× 10−9 1.6 × 10−7 5.4× 10−9 2.3× 10−7

1s

2s

Fig. 3.23. Decay of the metastable 2s state of hydrogen giving two continuumphotons.

∼ 0.14 s, i.e. it lives 108 times longer than the 2p state. This is because thetransition 2s → 1s is strongly forbidden. The 2s state is metastable whichmeans that on the atomic scale, it is long-lived.

So how does the 2s state decay? By the process of two-photon emis-sion, which is an inefficient process and in this case has an Einstein Acoefficient of 7 s−1 which can be compared to A(2p → 1s) = 6.3×108 s−1.The combined energy of the photons emitted must correspond to theenergy difference E(2s) −E(1s) but the photons themselves can take anyenergy within this constraint (see Fig. 3.23). The photons thus appear ascontinuous emission radiation. Indeed the two-photon decay of the H 2sstate is responsible for approximately one half the continuum emissionobserved from H II regions.

Problems3.1 Give an expression for the energy levels of the hydrogen atom in terms

of the Rydberg constant RH. Assuming a value RH = 109677.58 cm−1,derive a wavenumber for the Lyα transition of atomic hydrogenin cm−1. Explain why the Rydberg constant, R∞ = 109737.31cm−1,is more appropriate than RH for a heavy one-electron atom. Henceobtain an estimate for the wavenumber of the Lyα transition of

⌫1AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHgxWMF0xbaUDbbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LRTFMMYjqSPOKMGiv5fZkNvEGl6tbcBcg68QpShQKtQeWrP0xYFqM0TFCte56bmiCnynAmcFbuZxpTyiZ0hD1LJY1RB/ni2Bm5tMqQRImyJQ1ZqL8nchprPY1D2xlTM9ar3lz8z+tlJroNci7TzKBky0VRJohJyPxzMuQKmRFTSyhT3N5K2JgqyozNp2xD8FZfXiftes27rtUfGtVmo4ijBOdwAVfgwQ004R5a4AMDDs/wCm+OdF6cd+dj2brhFDNn8AfO5w+C+Y5x

⌫2AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHgxWMF0xbaUDbbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LRTFMMYjqSPOKMGiv5fZkN6oNK1a25C5B14hWkCgVag8pXf5iwLEZpmKBa9zw3NUFOleFM4KzczzSmlE3oCHuWShqjDvLFsTNyaZUhiRJlSxqyUH9P5DTWehqHtjOmZqxXvbn4n9fLTHQb5FymmUHJlouiTBCTkPnnZMgVMiOmllCmuL2VsDFVlBmbT9mG4K2+vE7a9Zp3Xas/NKrNRhFHCc7hAq7Agxtowj20wAcGHJ7hFd4c6bw4787HsnXDKWbO4A+czx+EfY5y

virtual pAAAB+nicbVDLTgJBEJzFF+Jr0aOXicTEE9lFEj2SePGIiTwSIGR26IUJs4/M9KJk5VO8eNAYr36JN//GAfagYCWdVKq6093lxVJodJxvK7exubW9k98t7O0fHB7ZxeOmjhLFocEjGam2xzRIEUIDBUpoxwpY4EloeeObud+agNIiCu9xGkMvYMNQ+IIzNFLfLnYRHjGdCIUJk3RG475dcsrOAnSduBkpkQz1vv3VHUQ8CSBELpnWHdeJsZcyhYJLmBW6iYaY8TEbQsfQkAWge+ni9Bk9N8qA+pEyFSJdqL8nUhZoPQ080xkwHOlVby7+53US9K97qQjjBCHky0V+IilGdJ4DHQgFHOXUEMaVMLdSPmKKcTRpFUwI7urL66RZKbuX5cpdtVSrZnHkySk5IxfEJVekRm5JnTQIJw/kmbySN+vJerHerY9la87KZk7IH1ifP4JQlBs=

(12S1/2)AAAB9XicbVDJTgJBEK3BDXFDPeqhIzHBC84MJngk8eIRoywJDKSn6YEOPUu6ezRkwn940IPEePXil3jSv7FZDgq+pJKX96pSVc+NOJPKNL+N1Mrq2vpGejOztb2zu5fdP6jJMBaEVknIQ9FwsaScBbSqmOK0EQmKfZfTuju4mvj1eyokC4M7NYyo4+NewDxGsNJSO28lbXt020msc3t01snmzII5BVom1pzkysdfpY9x8anSyX62uiGJfRoowrGUTcuMlJNgoRjhdJRpxZJGmAxwjzY1DbBPpZNMrx6hU610kRcKXYFCU/X3RIJ9KYe+qzt9rPpy0ZuI/3nNWHmXTsKCKFY0ILNFXsyRCtEkAtRlghLFh5pgIpi+FZE+FpgoHVRGh2AtvrxManbBKhbsG53GBcyQhiM4gTxYUIIyXEMFqkBAwCO8wNh4MJ6NV+Nt1poy5jOH8AfG+w9tJpTO

(22S1/2)AAAB9XicbVDJTgJBEK3BDXFDPeqhIzHBC84MJngk8eIRoywJDKSn6YEOPUu6ezRkwn940IPEePXil3jSv7FZDgq+pJKX96pSVc+NOJPKNL+N1Mrq2vpGejOztb2zu5fdP6jJMBaEVknIQ9FwsaScBbSqmOK0EQmKfZfTuju4mvj1eyokC4M7NYyo4+NewDxGsNJSO28nbXt020msc3t01snmzII5BVom1pzkysdfpY9x8anSyX62uiGJfRoowrGUTcuMlJNgoRjhdJRpxZJGmAxwjzY1DbBPpZNMrx6hU610kRcKXYFCU/X3RIJ9KYe+qzt9rPpy0ZuI/3nNWHmXTsKCKFY0ILNFXsyRCtEkAtRlghLFh5pgIpi+FZE+FpgoHVRGh2AtvrxManbBKhbsG53GBcyQhiM4gTxYUIIyXEMFqkBAwCO8wNh4MJ6NV+Nt1poy5jOH8AfG+w9utpTP

Two-photon continuum emission.The short wavelength limit is 1216Å.

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.

- 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.

16

C+2 + e� ! C+2⇤ + e� ! C+2 + e� + h⌫AAACMHicbZDLSgMxFIYz9VbrrerSTbAIYmmZqYJuRKGiLhWsCr2RSU/b0ExmSDJKGcY3ctNH0Y2CIm59AhcuTFsFtf4Q+Pm/czg5xw04U9q2H63E2PjE5FRyOjUzOze/kF5cOld+KCmUqM99eekSBZwJKGmmOVwGEojncrhwO8U+v7gCqZgvznQ3gKpHWoI1GSXaRPX0UbEWZQtxFmpRLr6pSNZqayKlf30zABsxzmKo5fAo+iZZ3K6IsJ7O2Hl7IDxqnC+T2fs42D3sdd5P6um7SsOnoQdCU06UKjt2oKsRkZpRDnGqEioICO2QFpSNFcQDVY0GC8d4zSQN3PSleULjQfqzIyKeUl3PNZUe0W31l/XD/1g51M2dasREEGoQdDioGXKsfdy/Hm4wCVTzrjGESmb+immbSEK1uXHKHMH5u/KoOS/knc184dTO7G+hoZJoBa2ideSgbbSPjtEJKiGKbtE9ekLPVs96sF6s12FpwvrqWUa/ZL19ApQ1q8M=

C+3 + e� ! C+2 + h⌫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

collisional excitation:recombination:

Example : Helium 17

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 energies. It is worth considering a few examples.

Example 1: The helium atom.

(1) The ground state is ls 2 • This is a closed shell, with L = 0 and S = 0, hence it gives rise to a single, even parity term 1S, and level 1S0 . 72 Astronomical Spectroscopy (Third Edition)

(2) The first excited configuration is ls2s. This has li = b = 0 and hence L = 0, but s1 = s2 = ½ giving both S = 0 (singlet) or S = 1 (triplet) states. The energy ordering of atomic states is given by Hund's rules. Hund's first rule governs ordering of terms with different spin multiplicities:

For a given configuration, the state with the maximum spin multiplicity is lowest in energy.

So the 3 S term (3S1 level) is lower in energy than the 1S term (1S0 level). In practice, the splitting between these terms is 0.80 eV.

(3) The next excited configuration is ls2p, which has odd parity. This has li = 0 and b = 1, giving L = 1; again s1 = s2 = ½, giving both S = 0 and S = 1 terms. Following the rule above, the 3P 0 term is lower in energy than the 1 P 0

term, in this case by 0.25 eV. The 3P 0 is also split into three levels: 3pg, 3p1 and 3p~-

Figure 5.2 depicts the energy levels of helium in a form known as a Grotrian diagram. The layout of these diagrams is discussed in Sec. 5.4.

Example 2: The carbon atom.

Start by considering the excited state configuration ls2 2s2 2p3p. It is only necessary to consider the outer two electrons for which:

li = 1, b = 1,

These give L = 0, 1, 2 and S = 0, 1, which give rise to the following terms, all with even parity: 1S, 3S,1 P, 3P, 1 D and 3D.

Now consider the ground-state configuration of carbon ls22s22p2 . This configuration also has li = 1, s1 = ½ and l2 = 1, s2 = ½-However, the Pauli Principle restricts which terms are allowed. For example, the term 3D includes the state:

(li = 1, m1 = +1, s1 = ½, S1z = +½) (b = 1, m2 = +1, s2 = ½, S2z = +½)










li = 1, b = 1,



(li = 1, m1 = +1, s1 = ½, S1z = +½) (b = 1, m2 = +1, s2 = ½, S2z = +½)










li = 1, b = 1,



(li = 1, m1 = +1, s1 = ½, S1z = +½) (b = 1, m2 = +1, s2 = ½, S2z = +½)

1Po → 1Po13Po → 3Po0 < 3P

o1 < 3P

o2

1S → 1S03S → 3S1










li = 1, b = 1,



(li = 1, m1 = +1, s1 = ½, S1z = +½) (b = 1, m2 = +1, s2 = ½, S2z = +½)










li = 1, b = 1,



(li = 1, m1 = +1, s1 = ½, S1z = +½) (b = 1, m2 = +1, s2 = ½, S2z = +½)

S L J

0 0 0

S L J

0 0 0

1 0 1

S L J

0 1 1

1 1 0, 1, 2

1S → 1S0

Example : Doubly Ionized Oxygen, O III 18

Complex Atoms 67

These are then added to give J

I. =L. + .s_. ( 4.15)

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 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. When more than two angular momenta need to be added together, they should be added in pairs. The result is independent of the order in which the addition is performed.

Worked Example: Consider O III with the configuration: ls22s22p3d. ls 2 and 2s2 are closed, so they contribute no angular momentum.

For the 2p electron li = 1 and s1 = ½; for the 3d electron l2 = 2 and s2 = ½. L. = Ii + I2 ::::} L = 1, 2, 3; .s.. = .fa + .§.2 ::::} s = 0, 1. Combining these using all possible combinations of Land S, and the rules of vector addition, gives:

L s J Level .J..=L.+.S..::::} 1 0 1 lpl

1 1 0, 1, 2 3p 0, 3p 1, 3p 2 2 0 2 102

2 1 1, 2, 3 3D'j'' 3D2, 3D3

3 0 3 iF3

3 1 2, 3,4 3F2, 3F3, 3F~.

Each state of an atom or ion is characterised by a unique combination of L, 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 some values of J appear several times, they all correspond to distinct states of the ion and it is important to retain them all.

Complex Atoms 67

These are then added to give J

I. =L. + .s_. ( 4.15)

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 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. When more than two angular momenta need to be added together, they should be added in pairs. The result is independent of the order in which the addition is performed.

Worked Example: Consider O III with the configuration: ls22s22p3d. ls 2 and 2s2 are closed, so they contribute no angular momentum.

For the 2p electron li = 1 and s1 = ½; for the 3d electron l2 = 2 and s2 = ½. L. = Ii + I2 ::::} L = 1, 2, 3; .s.. = .fa + .§.2 ::::} s = 0, 1. Combining these using all possible combinations of Land S, and the rules of vector addition, gives:

L s J Level .J..=L.+.S..::::} 1 0 1 lpl

1 1 0, 1, 2 3p 0, 3p 1, 3p 2 2 0 2 102

2 1 1, 2, 3 3D'j'' 3D2, 3D3

3 0 3 iF3

3 1 2, 3,4 3F2, 3F3, 3F~.

Each state of an atom or ion is characterised by a unique combination of L, 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 some values of J appear several times, they all correspond to distinct states of the ion and it is important to retain them all.

In total, 6 terms and 12 levels.

Example : Alkali Atoms• Alkali atoms: Lithium, sodium, potassium and rubidium all have ground state

electronic structures which consist of one electron in an � orbital outside a closed shell.

• Sodium (Na) : Sodium has � and a ground state configuration of � .

s

Z = 111s22s22p63s1

19

Na D lines:



reproduced in Fig. 6.6 which implies that the lines are optically thin in theSun. The sodium D lines are also observed in absorption against starlightin the interstellar medium. The D lines are usually very saturated in suchspectra.

Fig. 6.6. A solar spectrum reflected from the Moon just before a lunareclipse taken at the University of London Observatory. (S.J. Boyle, privatecommunication.)

D2 D1

g = 4

g = 2

g = 2

32P3/2

5896 Å 5890 Å

o

32P1/2

32S1/2

o

Fig. 6.7. The sodium D lines, g = 2J + 1, give the statistical weight of each level.

g = 2J+1D1 D2

D1 5896 Å line : 32S1/2 � 3 2P 1/2

D2 5890 Å line : 32S1/2 � 3 2P 3/2

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

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 2D.!i. ;i. 3 2'2

2P1 .i.. transition discussed above is 2'2

This 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, the intensity ratios between these transitions gives direct information on the optical depth at which the spectrum is formed.

6.4. Astronomical Sodium Spectra

The sodium resonance line, NaI 3s - 3p, is prominent in absorption in the solar spectrum (see Fig. 6.6), and is known as the sodium D line. For an S- P doublet, the intrinsic ratio of line intensities is always 2:1. When unsaturated, or optically thin, the strength of the absorption by the doublet

1.0

.8

.6

.4 II Na D

I Hex

600 650

Wavelength (nm)

Fig. 6.6. A solar spectrum reflected from the Moon just before a lunar eclipse taken at the University of London Observatory. (S.J. Boyle, private communication.)

A solar spectrum reflected from the Moon just before a lunar eclipse taken at the University of London Observatory (S.J. Boyle).

• Ca II (potassium-like calcium)

• Mg II (sodium-like magnesium)

• C IV (lithium-like carbon)

• N V (lithium-like nitrogen)

• O VI (lithium-like oxygen)

20

8498.0 Å line : 4 2Po3/2 � 3 2D3/2

8542.1 Å line : 4 2Po3/2 � 3 2D5/2

8662.1 Å line : 4 2Po1/2 � 3 2D3/2

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

1550.8 Å line : 2 2S1/2 � 2 2Po1/2

1548.2 Å line : 2 2S1/2 � 2 2Po3/2

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

1240.4 Å line : 3 2S1/2 � 4 2Po1/2

1239.9 Å line : 3 2S1/2 � 4 2Po3/2

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

H 3968.47 Å line : 4 2S1/2 � 4 2Po1/2

K 3933.66 Å line : 4 2S1/2 � 4 2Po3/2

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

2802.7 Å line : 3 2S1/2 � 3 2Po1/2

2795.5 Å line : 3 2S1/2 � 3 2Po3/2

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

(Note that 2Po1/2 � 2D5/2 is forbidden because �J = 2.)

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

1037.6 Å line : 2 2S1/2 � 2 2Po1/2

1031.9 Å line : 2 2S1/2 � 2 2Po3/2

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

1242.8 Å line : 2 2S1/2 � 2 2Po1/2

1238.8 Å line : 2 2S1/2 � 2 2Po3/2

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

• 3 electrons (Lithium-like ions)

21484 APPENDIX E

3 electrons

1550

.8

1548

.2

ground state

resonance lines

• 5 electrons

22

486A

PP

EN

DIX

E

5electrons

forbiddenlines

Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden

2P3/22P1/2

2P3/22P1/2

2P3/22P1/2

• 6 electrons

23

EN

ER

GY-LE

VE

LD

IAGR

AM

S487

6electrons

forbiddenlines

Upward heavy: resonance, Upward Dashed: intercombination Downward solid: forbidden

• 11 electrons

24

EN

ER

GY-LE

VE

LD

IAGR

AM

S491

11electrons

ground state

resonance lines

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.

25

Atomic Processes - Photoionization• Interstellar medium (ISM) is transparent to � photons, but is very

opaque to ionizing photons. In fact, the ISM does not become transparent until � .- Sources of ionizing photons include massive, hot young stars, hot white dwarfs, and

supernova remnant shocks.

• From the Outer Shells- Photoionization is the inverse process to radiative recombination.

- If the incoming photon has sufficient energy, it may leave the ionized species in an excited state.

hν < 13.6 eV

hν ∼ 1 keV

26

Ai+ + h⌫ ! A(i+1)+ + e� +�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

Ai+ + h⌫ ! A(i+1)+⇤ + e� +�E

A(i+1)+⇤ ! A(i+1)+ + h⌫1 + h⌫2 + · · ·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

Collisional Ionization & Recombination• Direct collisional ionization: The process whereby an electron strikes an ion A

(with charge � ), with sufficient energy to strip out a bound electron:

• Radiative recombination- Radiative recombination is the process of capture of an electron by an ion where the

excess energy is radiated away in a photon.- The electron is captured into an excited state. The recombined but still excited ion

radiates several photons in a radiative cascade, as it returns to the ground state:

- The photon in the first line represents a recombination continuum ( � ) photon. However, photons (h𝜈1, h𝜈2, h𝜈3) represent quantized transitions and are therefore termed recombination lines.

i+

hν

27

Ai+ + e� ! A(i+1)+ + 2e� ��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

Ai+ + e� ! A(i�1)+⇤ + h⌫

A(i�1)+⇤ ! A(i�1)+ + h⌫1 + h⌫2 + h⌫3 + · · ·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

(recombination continuum)

(recombination lines)

Collisional Excitation• Under the conditions of very low density and weak radiation fields, as in the Interstellar Medium.

- The vast majority of the atoms reside in the ground state.collisional excitation timescale >> radiative decay time scaleThis condition will remain true even if the excited state has a radiative lifetime of several second, which is frequently the case for the forbidden transitions observed in ionized astrophysical plasmas.

- Flux of an emission line � Flux number of collisions � Product of the number densities of the two colliding species.

- If the energy gap between the ground state and the excited state E12 is much larger than the mean energy of the colliding species (~ T), then, because there are few very energetic collisions, relatively few collisional excitations can occur. Therefore, the resulting emission line will be very much weaker than when E12 < kT.This gives us the possibility of measuring temperature from the relative strengths of lines coming from excited levels at different energies above the ground state.

• At high enough densities,- The collisional timescales are shorter than the spontaneous radiative decay timescale.- The population in any upper level is set by the balance between collisional excitation, and the

collisional de-excitation out of these levels, and are governed by the Boltzmann equilibrium.

∝ ∝

28

Ionization Equilibrium• Collisional Ionization Equilibrium (CIE) or coronal equilibrium

- dynamic balance at a given temperature between collisional ionization from the ground states of the various atoms and ions, and the process of recombination from the higher ionization stages.

- In this equilibrium, effectively, all ions are in their ground state.

• Photoionization Equilibrium:- dynamic balance between photo-ionization and the process of recombination.

• Ionization balance under conditions of local thermodynamic equilibrium (LTE)- The ionization equilibrium in LTE is described by the Saha equation.- In LTE, the excited states are all populated according to Boltzmann’s law.

29

Emission Line Mechanisms• There are three types of mechanism to create emission lines.

• Collisional Excitation: electron collisions largely populate low-lying excited states. Excitations depend on the electron temperature.

• Radiative Recombination: the cascade of emissions following the recombination of an electron with an ion leads to emissions from more highly-excited states than collisional excitations.

• Optical Pumping: this is the process of excitation from a lower level � by absorption of a (continuum or line) photon to an upper level � .- Resonance fluorescence (Bowen fluorescence): (H and He) resonance-line photons

can pump other species when there is an accidental coincidence with a resonance line, resulting emission in specific longer-wavelength lines as part of the radiative decay cascade from the pumped level.

ℓu

30

Homework

1. What is the ground-state configuration, term and level of the beryllium atom, Be? One of the outer electrons in Be is promoted to the 3rd orbital. What terms and levels can this configuration have?

2. Symbols for particular levels of three different atoms are written as � , � and � . Explain in each case why the symbol must be wrong.

1D1 0D3/2 3P3/2

31


'.!50 w- z w z 3:JO

19 150 C " X+Y 8 LOO

50

o'.?L.'.? '.?LA '.?'.?.S

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 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) .]

Intercombination

22.IA

-.,..-,--- 23p0

1630A

Forbidden

22.sA

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

Deadline: 2019, Sep. 25 (Wed)

3. The lithium atom, Li, has three electrons. Consider the following configurations of Li: (a) � , (b) � , (c) � . By considering the configuration only, state which of the three sets of transitions between the configuration (a), (b) and (c) are allowed and forbidden transitions?

4. The figure shows the term diagram for helium-like oxygen, O VII, showing transitions from the � states. Explain why 22.1� line is an intercombination line and why 22.8� is a forbidden line.

1s22p 1s2s3s 1s2p3p

1s2l ÅÅ

KAIST Astrophysics (PH481) - Part 1 - Kwang-Il Seon's Home Page · 2020-04-27 · • Why two...

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