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
•
•
“orbits” of s states become more eccentric as n increases
Electron transitions take place between adjacent angular
momentum states (i.e. Dl=1)
•
•
–
–
–
–
states with
states with
states with
states with
l=0 called s states
l=1 called p states
l=2 called d states
l=3 called f states
–
–
–
–
“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 “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 l5889.953 and 3s2S½ - 3p2P1/2 l5895.923
* 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 – 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….
• 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 Dl=1 and DJ=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
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 0e 2
g 4
n  =
mc (n n 0 ) 2  (g 4 ) 2
• N0 is the number of bound electrons per unit volume
• the quantity n-n0 is the frequency separation from the
nominal line center
• the quantity e is the dielectric constant (e=1 in free space)
• and g=g/m is the classical damping constant
The atomic absorption coefficient includes atomic
data (f, e, g) 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 Constant
N 0e 2
g 4
n  =
mc (n n 0 ) 2  (g 4 ) 2
• For a classical harmonic oscillator,
• The shape of the spectral line depends on the size of the
classical damping constant
• For n-n0 >> g/4, the line falls off as (n-n0)-2
• Accelerating electric charges radiate.
dW
8 2n 2e 2
=
W
3
dt
3mc
• and
8 2n 2e 2 0.2223 1
g=
=
sec
3
2
3mc
l
W = W0 e
gt
The mean lifetime is
also defined as T=1/g,
where T=4.5l2
• is the classical damping constant (l is in cm)
Line Absorption with QM
•
•
•
•
•
Replace g with G!
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 t2 (the
deeper layer) and t1
In (t 2 ) = In (t 1 )e n Dx  In (t 1 )(1   n Dx)
In (t 2 )  In (t 1 )  n DxIn (t 1 )
•
•
•
•
The line profile is proportional to n
4e 2 N
At line center n=n0, and n  = mcg
Half the maximum depth occurs at (n-n0)=g/4
In terms of wavelength
Dl 1 =
2
c
n
2
Dn 1
2
c g
2e 2
= 2
=
= 0.000118 A
2
n 4 3mc
• 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.
g = 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:
 n0-n = 1.7 x 1011 sec-1 and
N = 1/m = 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
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)
I
= e x = 0.99
I0
• Or x=0.01, and x=4x10-7 gm cm-2