Spectroscopy Spectral lines The Fraunhofer spectrum Charlotte Moore Sitterly (Allen!) –Multiplet table –Rowland table Formalism of spectroscopy

Quantum Numbers of Atomic States 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 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 “n 2 (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 3s 2 S 1/2 The two lowest p levels are 3p 2 P 1/2 and 3p 2 P 3/2 The Na D lines are described 3s 2 S ½ - 3p 2 P 3/2  and 3s 2 S ½ - 3p 2 P 1/ * This is a different S than the s state!

More 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 – 1 S 0 –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…. The state of the electrons is described with a term for each electron above the closed shell. For carbon atoms, “1s 2 2s 2 2p 2 ”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 3 P- 3 D

Grotrian Diagram for He Struve and Wurm 1938, ApJ

Spectral Line Formation Classical picture of radiation Intrinsic vs. extrinsic broadening mechanisms Line absorption coefficient Radiative transfer in spectral lines

Spectral Line Formation-Line Absorption 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 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!

Atomic Absorption Coefficient N 0 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 The atomic absorption coefficient includes atomic data (f, ,  ) and the state of the gas (N 0 ), and is a function of frequency. The equation expresses the natural broadening of a spectral line.

The 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) The mean lifetime is also defined as T=1/ , where T=4.5 2

Line 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

The Classical 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!

The Classical Damping Line Profile

An example… The Na D lines have a wavelength of 5.9x10 -5 cm.  = 6.4 x 10 7 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 sec -1 and N = 1/  = 2.6 x atoms gm -1 Then  = 3.7 x 10 4 f and f=2/3, so  = 2.5 x 10 4 per gram of neutral sodium

The 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