SpectroscopySpectroscopy

• Spectral lines• The Fraunhofer spectrum• Charlotte Moore Sitterly (Allen!)

– Multiplet table– Rowland table

• Formalism of spectroscopy

1990 Bruce Gold Medal1937 AJC Prize

Quantum Numbers of Atomic Quantum Numbers of Atomic StatesStates

• Principal quantum number n defines the energy level• Azimuthal quantum number l

– states with l=0 called s states– states with l=1 called p states– states with l=2 called d states– states with l=3 called f states

• “orbits” of s states become more eccentric as n increases • Electron transitions take place between adjacent angular

momentum states (i.e. l=1)– “sharp series” lines from p to higher s states– “principal series” lines from s to higher p states– “diffuse series” lines from p to higher d states– “fundamental series” lines from d to higher f states

• The first line(s) of the principal series (s to p) are called resonance lines since it involves the ground level

• In alkali metals, the p, d, and f energy levels are doubled (e.g. the Na D lines) due to the coupling between the magnetic moment of the orbital motion and the spin of the electron (the quantum number s, which can be +1/2 or –1/2

Spectroscopic NotationSpectroscopic Notation• The total angular momentum quantum number is j = l +S*

– For s states, j=1/2– For p states, j=1/2 or j=3/2

• Electron levels are designated by the notation “n2(L)J”

• n is the total quantum number• The superscript 2 indicates the levels are doubled• L is the azimuthal quantum number (S,P,D,F)• J denotes the angular momentum quantum number

• For the sodium ground level is 3s2S1/2

• The two lowest p levels are 3p2P1/2 and 3p2P 3/2

• The Na D lines are described

• 3s2S½ - 3p2P3/2 5889.953 and 3s2S½ - 3p2P1/2 5895.923

* This is a different S than the s state!

More Spectroscopic VocabularyMore Spectroscopic Vocabulary

• The Pauli exclusion principle requires that two s-electrons in the same state must have opposite spin

• Therefore S=0 and these are called “singlet” states

• The ground state of He is a singlet state – 1S0

– The superscript 1 means singlet– The subscript 0 means J=0

• In the first excited state of He, one electron is in the 1s state, and the second can be in either the 2s or the 2p state.

• Depending on how the electrons’ spins are aligned, these states can either be singlets or triplets

• Electrons can only jump between singlet states or between triplet states

It goes on and on and on…It goes on and on and on…• The state of the electrons is described with a term for each

electron above the closed shell.• For carbon atoms, “1s22s22p2”says there are

– 2 electrons in the 1s state– 2 electrons in the 2s state– 2 electrons in the 2p state

• Allowed and forbidden transitions– Transitions with l=1 and J=1 and 0 are allowed (except

between J=0 and J=0)– Other transitions are forbidden– For some electron states there are no allowed transitions to

lower energy states. Such levels are called metastable– The situation is more complex in atoms with more electrons– A multiplet is the whole group of transitions between two

states, say 3P-3D

Grotrian Grotrian Diagram for Diagram for

HeHe

• Struve and Wurm 1938, ApJ

Grotrian Grotrian Diagram for Diagram for

Li ILi I

NIST

Grotrian Grotrian Diagram for Diagram for

O IO INIST

Fraunhofer LinesFraunhofer LinesLines Due To ... Wavelengths (Å)

A - (band) O2 (telluric) 7594 - 7621

B - (band) O2 (telluric) 6867 - 6884

C H-alpha 6563

a - (band) O2 (telluric) 6276 - 6287

D - 1, 2 Na 5896 & 5890

E Fe 5270

b - 1, 2 Mg 5184 & 5173

c Fe 4958

F H-beta 4861

d Fe 4668

e Fe 4384

f H-gamma 4340

G (band) Fe & Ca , CH 4308

g Ca 4227

h H-delta 4102

H Ca 3968

K Ca 3934

Spectral Line FormationSpectral Line Formation

• Classical picture of radiation• Intrinsic vs. extrinsic broadening

mechanisms• Line absorption coefficient• Radiative transfer in spectral lines

Spectral Line Formation-Line Spectral Line Formation-Line Absorption CoefficientAbsorption Coefficient

• Radiation damping (atomic absorptions and emissions aren’t perfectly monochromatic – uncertainty principle)

• Thermal broadening from random kinetic motion

• Collisional broadening – perturbations from neighboring atoms/ions/electrons)

• Hyperfine structure• Zeeman effect

Classical Picture of RadiationClassical Picture of Radiation

• Photons are sinusoidal variations of electro-magnetic fields

• When a photon passes by an electron in an atom, the changing fields cause the electron to oscillate

• Treat the electron as a classical harmonic oscillator:

mass x acceleration =external force – restoring force – dissipative

• E&M is useful! (well…)

Atomic Absorption CoefficientAtomic Absorption Coefficient

• N0 is the number of bound electrons per unit volume

• the quantity -0 is the frequency separation from the nominal line center

• the quantity e is the dielectric constant (=1 in free space)• and g/m is the classical damping constant

220

20

)4()(

4

mc

N

The atomic absorption coefficient includes atomic data (f, , ) and the state of the gas (N0), and is a function of frequency. The equation expresses the natural broadening of a spectral line.

The Classical Damping ConstantThe Classical Damping Constant

• For a classical harmonic oscillator,• The shape of the spectral line depends on the size of the

classical damping constant• For -0 >> /4, the line falls off as (-0)-2

• Accelerating electric charges radiate.

• and

• is the classical damping constant ( is in cm)

220

20

)4()(

4

mc

N

Wmcdt

dW3

222

3

8

teWW 0

123

222

sec2223.0

3

8

mc

The mean lifetime is also defined as T=1/, where T=4.52

The Classical The Classical Damping Line ProfileDamping Line Profile

220

20

)4()(

4

mc

N

The The ClassicalClassical Line Profile Line Profile• Look at a thin atmospheric layer between 2 (the

deeper layer) and 1

• The line profile is proportional to

• At line center =0, and • Half the maximum depth occurs at (-0)=/4• In terms of wavelength

• Very small – and the same for ALL lines!

)1)(()()( 112 xIeII x

)()()( 112 xIII

mcN24

Amc

cc000118.0

3

2

4 2

2

22

122

1

An example…An example…

• The Na D lines have a wavelength of 5.9x10-5 cm. = 6.4 x 107 sec-1

• The absorption coefficient per gram of Na atoms at a distance of 2A from line center can be calculated:

0- = 1.7 x 1011 sec-1 andN = 1/ = 2.6 x 1022 atoms gm-1

• Then = 3.7 x 104 f• and f=2/3, so

= 2.5 x 104 per gram of neutral sodium

220

20

)4()(

4

mc

N

123

222

sec2223.0

3

8

mc

The Abundance of SodiumThe Abundance of Sodium

• In the Sun, the Na D lines are about 1% deep at a distance of 2A from line center

• Use a simple one-layer model of depth x (the Schuster-Schwarzschild model)

• Or x=0.01, and x=4x10-7 gm cm-2 (recall that Na=2.5 x 104 per gram of neutral sodium at a distance of 2A from line center)

• the quantityx is a column density

99.00

xeI

I

Line Absorption with QMLine Absorption with QM

• Replace with !• Broadening depends on lifetime of level• Levels with long lifetimes are sharp• Levels with short lifetimes are fuzzy• QM damping constants for resonance

lines may be close to the classical damping constant

• QM damping constants for other Fraunhofer lines may be 5,10, or even 50 times bigger than the classical damping constant