1990 Bruce Gold Medal
1937 AJC Prize
Spectroscopy
• Spectral lines
• The Fraunhofer spectrum
• Charlotte Moore Sitterly
– 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
– 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
Grotrian
Diagram for
Li I
NIST
NIST
Grotrian
Diagram for
OI
Fraunhofer Lines
Lines
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 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! (well…)
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)
2
N
e
g 4
The Classical Damping n  = 0
2
2
mc
(
n

n
)

(
g
4

)
0
Line Profile
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!
An example…
N 0e 2
g 4
n  =
mc (n n 0 ) 2  (g 4 ) 2
8 2n 2e 2 0.2223 1
g=
=
sec
3
2
3mc
l
• 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:
Dn0-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 (recall
that Na=2.5 x 104 per gram of neutral sodium
at a distance of 2A from line center)
• the quantity x is a column density
Natural Broadening
• From Heisenberg's uncertainty principle: The electron in an
excited state is only there for a short time, so its energy
cannot have a precise value.
• Since energy levels are "fuzzy," atoms can absorb photons
with slightly different energy, with the probability of
absorption declining as the difference in the photon's
energy from the "true" energy of the transition increases.
• The FWHM of natural broadening for a transition with an
average waiting time of Dto is given by
2
(Dl1 / 2 ) =
l 1
c Dto
• A typical value of (Dl)1/2 = 2 x 10-4 A. Natural broadening is
usually very small.
• The profile of a naturally broadened linen is given by a
dispersion profile (also called a damping profile, a Lorentzian
profile, a Cauchy curve, and the Witch of Agnesi!) of the
form (in terms of frequency)
In 
• where g is the "damping constant."
g
(n n 0 ) 2  g 2
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
Add Quantum Mechanics
• Define the oscillator strength, f:
• related to the atomic transition

probability Bul:


0
dn =
e 2
mc
 dn = hnB
0
lu
mc
mc3 g u
7 Blu
15 2 g u
f = 2 hnBul = 7.5 x10
=
Aul = 1.9 x10 l
Aul
2 2
e
l 2e n g l
gl
• f-values usually tabulated as gf-values.
• theoretically calculated
• laboratory measurements
• solar
f
Collisional Broadening
• Perturbations by discrete encounters
• Change in energy approximated by a power law of the form
DE = constant x r-n
•
•
•
•
•
•
(r is the separation between the atom and the perturber)
Perturbations by static ion fields (linear Stark effect
broadening) (n=2)
Self-broadening - collisions with neutral atoms of the same
kind (resonance broadening, n=3)
if perturbed atom or ion has an inner core of electrons (i.e.
with a dipole moment) (quadratic Stark effect, n=4)
Collisions with atoms of another kind (neutral hydrogen atoms)
(van der Waals, n=6)
Assume adiabatic encounters (electron doesn’t change level)
Non-adiabatic (electron changes level) collisions also possible
Pressure Broadening – DE=constant x r-n
n
Type
Lines Affected Perturber
2
Linear Stark
Hydrogen
Protons, e-
3
Self broadening or
resonance
broadening
Common
species
Atoms of the
same type
4
Quadratic Stark
Most, esp. in
hot stars
Ions, e-
6
Van der Waals
Most, esp. in
cool stars
Neutral
hydrogen
Approaches to Collisional Broadening
• Statistical effects of many particles (pressure broadening)
– Usually applies to the wings, less important in the core
• Some lines can be described fully by one or the other
• Know your lines!
• The functional form for collisional damping is the same as
for radiation damping, but Grad is replaced with Gcoll
e 2
g 4 2
n =
f
mc (Dn 2  (g 4 ) 2
• Collisional broadening is also described with a dispersion
function
• Collisional damping is sometimes 10’s of times larger than
radiation damping
log gamma
Damping Coefs for Na D
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
Na D 5890 A
Natural
van der Waals
Stark
-4
-3
-2
-1
Log tau
0
1
2
7
log g 6  19.6  log C6 ( H )  log Pg  log T
5
10
2
5
log g 4  19.4  log C4  log Pe  log T
3
6
2
Doppler Broadening
•
•
Two components contribute to the intrinsic Doppler broadening
of spectral lines:
– Thermal broadening
– Turbulence – the dreaded microturbulence!
Thermal broadening is controlled by the thermal velocity
distribution (and the shape of the line profile)
2
dN (vr )  m  3
=
 e
NTotal  2kT 
•
 mvr 2

 2 kT





dvr
where vr is the line of sight velocity component
The Doppler width associated with the velocity v0 (where the
variance v02=2kT/m) is
v0
l  2kT 
DlD = l = 

c
c m 
1
2
= 4.3x10 l (T m )
and l is the wavelength of line center
7
1
2
More Doppler Broadening
•
Combining these we get the thermal broadening line profile:
In
I total
•
n
m 
e
2kT
n 2 2 kT
At line center, n=n0, and this reduces to
In
I total
•
=
c
mc 2 (n n 0 ) 2
=
c
n
m
2kT
Where the line reaches half its maximum depth, the total width is
2l0
2Dl1 2 =
c
2kT ln 2
m
Thermal + Turbulence
• The average speed of an atom in a gas due to thermal
motion - Maxwell Boltzmann distribution. The most
probably speed is given by
vmp = 2kt / m
• Moving atoms are Doppler shifted, and individual atoms
will absorb light at slightly different wavelengths
because of the Doppler shift.
• Spectral lines are also Doppler broadened by turbulent
motions in the gas. The combination of these two
effects produces a Doppler-broadened profile:
(Dl )1/ 2
2l  2kT
2
=
 vturb  ln 2

c  m

• Typical values for Dl1/2 are a few tenths of an Angstrom.
The line depth for Doppler broadening decreases
exponentially from the line center.
Combining the Natural, Collisional and Thermal
Broadening Coefficients
•
•
The combined broadening coefficient is just the convolution of all
of the individual broadening coefficients
The natural, Stark, and van der Waals broadening coefficients all
have the form of a dispersion profile:
b
n = a
Dn 2  b 2
•
With damping constants (grad, g2, g4, g6) one simply adds them up to
get the total damping constant:
g total 4 2
n =
f
mc Dn 2  (g total 4 ) 2
e 2
•
The thermal profile is a Gaussian profile:
1
n = 1 2
e
 Dn D
 Dn

 Dn D



The Voigt Profile
• The convolution of a dispersion profile and a Gaussian profile
is known as a Voigt profile.
V (Dn , Dn D , g ) = 

0
g 4
1
e
2
2
12
(Dn  Dn 1 )  (g 4 )  Dn D
2
 Dn

 Dn D



2
dn 1
• Voigt functions are tabulated for use in computation
• In general, the shapes of spectra lines are defined in terms
of Voigt profiles
• Voigt functions are dominated by Doppler broadening at small
Dl, and by radiation or collisional broadening at large Dl
• For weak lines, it’s the Doppler core that dominates.
• In solar-type stars, collisions dominate g, so one needs to
know the damping constant and the pressure to compute the
line absorption coefficient
• For strong lines, we need to know the damping parameters to
interpret the line.
Calculating Voigt Profiles
V (u, a) 
1
DvD
l2 / c
H (u, a) =
H (u, a)

DlD 
• Tabulated as the Hjerting function H(u,a)
• u=Dl/DlD
• a=(l2/4c)/DlD =(g/4)DnD
• Hjertung functions are expanded as:
H(u,a)=H0(u) + aH1(u) + a2H2(u) + a3H3(u) +…
• or, the absorption coefficient is
e 2 l2  f
=
H (u , a )
2
mc
DlD
Plot a Damped Profile
1.2
Line Strength
1
0.8
0.6
0.4
Line Profiles
0.2
Natural + Thermal
Natural + Thermal + Collisional
0
-5
-4
-3
-2
-1
0
1
Doppler Widths
2
3
4
5