Natural BroadeningNatural 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 to is given by

• A typical value of ()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)

• where is the "damping constant."

otc

1)(

2

2/1

220 )(

I

The Classical Damping The Classical Damping ConstantConstant

• 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

Add Quantum Add Quantum MechanicsMechanics

• Define the oscillator strength, f:

• related to the atomic transition probability Bul:

fmc

ed

2

0

luBhd

0

ull

uul

l

uluul A

g

gxA

g

g

e

mcBxBh

e

mcf 215

22

37

2109.1

2105.7

• f-values usually tabulated as gf-values.

• theoretically calculated• laboratory measurements• solar

Collisional BroadeningCollisional Broadening• Perturbations by discrete encounters• Change in energy approximated by a power law of the form

E = constant x r-n

• 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

Approaches to Collisional Approaches to Collisional BroadeningBroadening

• 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 rad is replaced with coll

• Collisional broadening is also described with a dispersion function

• Collisional damping is sometimes 10’s of times larger than radiation damping

22

22

)4((

4

fmc

e

Damping Coefs for Na DDamping Coefs for Na D

Na D 5890 A

55.5

66.5

77.5

88.5

99.510

-4 -3 -2 -1 0 1 2Log tau

log

ga

mm

a

Naturalvan der WaalsStark

TPHC g log10

7log)(log

5

26.19log 66

TPC e log6

5loglog

3

24.19log 44

Doppler BroadeningDoppler 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)

where vr is the line of sight velocity component• The Doppler width associated with the velocity v0 (where the

variance v02=2kT/m) is

and is the wavelength of line center

r

kT

mv

Total

r dvekT

m

N

vdNr

2

32 2

2

)(

217

21

0 )(103.42 Txm

kT

cc

vD

More Doppler More Doppler BroadeningBroadening

• Combining these we get the thermal broadening line profile:

• At line center, =0, and this reduces to

• Where the line reaches half its maximum depth, the total width is

m

kT

c

2ln222 0

21

kT

mc

total

ekT

mc

I

I 2

)(2

20

2

2

kT

mc

I

I

total

2

Thermal + TurbulenceThermal + Turbulence• The average speed of an atom in a gas due to thermal

motion - Maxwell Boltzmann distribution. The most probably speed is given by

• 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:

• Typical values for 1/2 are a few tenths of an Angstrom. The line depth for Doppler broadening decreases exponentially from the line center.

mktvmp /2

2ln22

)( 22/1

turbvm

kT

c

Combining the Natural, Combining the Natural, Collisional and Thermal Collisional and Thermal Broadening CoefficientsBroadening 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:

• With damping constants (rad, 2, 4, 6) one simply adds them up to get the total damping constant:

• The thermal profile is a Gaussian profile:

22 b

ba

22

22

)4(

4

total

totalfmc

e

De

D

21

1

The Voigt ProfileThe Voigt Profile• The convolution of a dispersion profile and a Gaussian

profile is known as a Voigt profile.

• Voigt functions are tabulated for use on computations• In general, the shapes of spectra lines are defined in

terms of Voigt profiles• Voigt functions are dominated by Doppler broadening at

small , and by radiation or collisional broadening at large

• For weak lines, it’s the Doppler core that dominates.• In solar-type stars, collisions dominate , 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.

1210 221

22

1

)4()(

4),,(

deV D

DD

Calculating Voigt ProfilesCalculating Voigt Profiles

• Tabulated as the Hjerting function H(u,a)• u=/D

• a=(2/4c)/D =(/4)D

• Hjertung functions are expanded as:

H(u,a)=H0(u) + aH1(u) + a2H2(u) + a3H3(u) +…

• or, the absorption coefficient is

),(/

),(1

),(2

auHc

auHv

auVDD

),(2

22

auHf

mc

e

D

Line ProfilesLine Profiles

Line Profiles

0

0.2

0.4

0.6

0.8

1

1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5

Doppler Widths

Lin

e S

tren

gth

Natural + Thermal

Natural + Thermal + Collisional

Line Profiles

0

0.2

0.4

0.6

0.8

1

1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5

Doppler Widths

Line

Str

engt

h

Natural + Thermal

Natural + Thermal + Collisional

Plot a Damped ProfilePlot a Damped Profile