Atmospheric Spectroscopy

A look at Absorption and Emission Spectra of Greenhouse Gases

Our Atmosphere

Diagram taken from http://csep10.phys.utk/astr161/lect/earth/atmosphere.html

Composition of the Atmosphere

N2 = 78.1%

O2 = 20.9%

H20 = 0-2%Ar + other inert gases = 0.936%

CO2 = 370ppm (0.037%)

CH4 = 1.7 ppm

N20 = 0.35 ppm

O3 = 10^-8+ other trace gases

Earth’s Radiation Budget

Electromagnetic Spectrum

• Over 99% of solar radiation is in the UV, visible, and near infrared bands

• Over 99% of radiation emitted by Earth and the atmosphere is in the thermal IR band (4 -50 µm)

Near Infrared

Thermal Infrared

Electromagnetic Spectrum

• Over 99% of solar radiation is in the UV, visible, and near infrared bands

• Over 99% of radiation emitted by Earth and the atmosphere is in the thermal IR band (4 -50 µm)

Near Infrared

Thermal Infrared

Diagram modified from www.spitzer.caltech.edu/Media/guides/ir.shtml

Blackbody Radiation Curves for Solar and Terrestrial Temperatures

• Without greenhouse gases the temperature of the Earth’s surface would be approximately 15 degrees Fahrenheit colder than it is today

• This is due to the fact that certain trace gases in the atmosphere absorb radiation in the infrared spectrum (wavelengths emitted by the Earth) and re-emit some of this radiation back down to Earth

Diagram taken from Peixoto and Oort (1992)

What are the Major Greenhouse Gases?

N2 = 78.1%

O2 = 20.9%

H20 = 0-2%Ar + other inert gases = 0.936%

CO2 = 370ppm

CH4 = 1.7 ppm

N20 = 0.35 ppm

O3 = 10^-8+ other trace gases

Molecular Absorption

• The total energy of a molecule can be seen as the sum of the kinetic, electronic, vibrational, and rotational energies of a molecule

• Electronic energy α => visible/ultraviolet

• Vibrational energy α => thermal/near infrared

• Rotational energy α => microwave/far infrared

• Vibrational transitions (higher energy) are usually followed by rotational transitions (lower energy) and we thus see groups of lines that comprise a vibration-rotation band

2

2

ma

h

mMa2

2

2

2

Ma

Energy level diagram of CO2 molecules showing relative energy spacing of electronic, vibrational, and rotational energy levels

electronic

vibrational

rotational

Vibrational Transitions of a Diatomic Molecule

• The molecular bond can be treated as a spring and thus a harmonic oscillator potential can be approximated for the molecule

• Evib = v(v+1/2) and v = (1/2π)(k/µ)1/2

• However, polyatomic molecules are more complicated due to their more complex structure

• For polyatomic molecules, any allowed vibrational motion can be expressed as the superposition of a finite amount of vibrational normal modes, each which has its own set of energy levels

Vibrational Transitions of Polyatomic Molecules

• Any molecule has 3N degrees of freedom, where N is the number of atoms in the molecule. – Translational Degrees of Freedom: 3

Specifies center of mass of the molecule– Rotational DOF: 2 (linear), 3(nonlinear)

Describes orientation of the molecule about its center of mass– Vibrational DOF: 3N-5 (linear), 3N-6 (nonlinear)

Describes relative positions of the nuclei

• Vibrational DOF represent maximum number of vibrational modes of a molecule (due to degeneracies and selection rules)

Harmonic Oscillator Approximation for Polyatomic Molecules

• Evib = G(v1,v2,…) = ∑ vj(vj’+1/2) • where vj’= 0,1,2,… are the vibrational quantum numbers

vj = (1/2π)(k/µ)1/2 is the frequency of vibrationand k is the bond force constant

• Selection rules: Δvj = ±1• This means that in the motion of a polyatomic molecule = motion of Nvib harmonic

oscillators, each with their own fundamental frequency vj => normal modes

• Vibrational state of triatomic molecule represented by (v1v2v3)– v1 = symmetric stretch mode, v2 = bending mode, v3 = asymmetric stretch mode– Stretching modes of vibration occur at higher energy than bending modes

• If dipole moment doesn’t change during normal mode motion, that normal mode is infrared inactive.

• Number of IR active normal modes determines number of absorption bands in IR spectrum

• Higher order vibrational transitions lead to frequencies slightly displaced from the fundamental and of much less intensity due to smaller population at higher energy levels.

Rotational Transitions of Polyatomic Molecules

• Approximate as rigid network of N atoms (rigid rotator approximation)• Rotation of a rigid body is dependent on its principle moments of inertia

Ixx = ∑ mj [(yj-ycm)2 + (z-zcm)2] • A set of coordinates can always be found where the products of inertia (Ixy, etc)

vanish. The moments of inertia around these coordinates are the principle moments of inertia.

• Spacing between rotational lines described by rotational constants:

A = h / (8 π2 c IA) B = h / (8 π2 c IB) C = h / (8 π2 c IC)where by convention IA > IB > IC

• If IA = 0, IB = IC => linear (CO2)• If IA = IB = IC => spherical top (CH4)• If IA = IB ≠ IC => symmetric top • If IA ≠ IB ≠ IC => asymmetric top (H20, O3, N20)

• Due to the selection rule ΔJ = 0, ±1, the rotational band is divided into P (ΔJ = -1), Q (ΔJ =0), and R (ΔJ = +1) branches

• A pure rotational transition (Δv=0) can only occur if molecule has permanent dipole moment

Linear Molecules

• Ia = 0, Ib = Ic.Erot = BJ(J+1)

• Centrifugal Distortion Correction for polyatomic molecules (less rigid than diatomic molecules)

= -D[J(J+1)]2 + higher terms

Spherical Tops

• IA = IB = IC

• Quantum mechanics can solve the energy of a spherical top exactly

• Result: Erot(J,K) = F(J,K) = BJ(J+1) J = 0,1,2… degeneracy: gJ = (2J+1)2

• Selection rule: ΔJ = 0, ±1• The symmetry of these molecules requires that they do

not have permanent dipole moments. This means they have no pure rotational transitions.

• Centrifugal Distortion Correction: -D[J(J+1)]2

Symmetric tops

• Quantum mechanics can also solve symmetric tops• Ia = Ib < Ic => oblate symmetric top (pancake shaped)• Ia < Ib = Ic => prolate symmetric top (cigar shaped)• Oblate sym top: Erot(J,K) = F(J,K) = [BJ(J+1) + (C-B)K2]

degeneracy: gJK = 2J+1 J = 0,1,2… K = 0,±1,±2... ±Jwhere J = total rotational angular momentum of molecule K = component of rotational ang. momentum along the

symmetry axisProlate sym top:

Erot(J,K) = F(J,K) = [BJ(J+1) + (A-B)K2]

For the sym. top molecules with permanent dipole moments, these dipole moments are usually directed along the axis of symmetry. The following selection rules are assigned for these molecules:

ΔJ = 0 ,±1 ΔK = 0 for K ≠ 0ΔJ = ±1 ΔK = 0 for K = 0

Where ΔJ = +1 corresponds to absorption and ΔJ = -1 to emission

Asymmetric Tops

• IA ≠ IB ≠ IC

• Schrodinger eqn has no general solution for asymmetric tops

• The complex structure of asymmetric does not allow for a simple expression of their energy levels. Because of this, the rotational

spectra of asymmetric tops do not have a well-defined pattern.

Summary of Tuesday• Atmosphere is composed primarily of N2 and O2 with concentrations

in the ppm of greenhouse gases (aside from H20 which varies from 0-2%)

• These GHG (H20, CO2, CH4, O3, N20) have huge impact on the Earth’s energy budget, effectively increasing temperature of Earth’s surface by ~15 degrees Fahrenheit.

• GHG absorb largely in the infrared region which indicates vibrational and rotational transitions of the molecules upon absorption of a photon

• Vibrational energy levels are greater than rotational by a factor of √(m/M)

• Vibrational transitions described by fundamental (normal) modes which are determined by number of vibrational degrees of freedom of that molecule: 3N -5 for linear, 3N-6 for nonlinear. Superposition of these normal modes can describe any allowed vibrational state.

• Ex) for triatomic molecule, vibrational state represented by (v1v2v3) where v1 = symmetric stretch mode, v2 = bending mode, v3 = asymmetric stretch mode

• Rotational energy levels determined by principle moments of inertia- divides molecules into four catagories (linear, spherical top, symmetric top, assymetric top). Each has own energy eigenvalues and selection rules.

Rovibrational Energy• Vibrational and rotational transitions usually occur simultaneously splitting up

vibrational absorption lines into a family of closely spaced lines• Rotational energy also dependent on direction of oscillation of dipole moment

relative to axis of symmetry– When oscillates parallel, ΔJ = 0 transition is forbidden and only P and R

branches are seen– When oscillates perpendicular, P, Q and R branches are all seen

• The rotational constant is not the same in different vibrational states due to a slight change in bond-length, and so rotational lines are not evenly spaced in a vibrational band

Diagram taken from Patel (1968)

Rovibrational transitions in a CO2 molecule

The Primary Greenhouse Gases

H20

• Most important IR absorber• Asymmetric top → Nonlinear, triatomic molecule has complex line

structure, no simple pattern• 3 Vibrational fundamental modes

• Higher order vibrational transitions (Δv >1) give weak absorption bands at shorter wavelengths in the shortwave bands

• 2H isotope (0.03% in atm) and 18O (0.2%) adds new (weak) lines to vibrational spectrum

• 3 rotational modes (J1, J2, J3) • Overtones and combinations of rotational and vibrational transitions

lead to several more weak absorption bands in the NIR

symmetric stretch v1 = 2.74 μm

asymmetric stretch v3 = 2.66 μm

bend v2 = 6.25 μm

H

o

H o

Absorption Spectrum of H2O

v3=2.66 μm

v1=2.74 μmv2=6.25 μm

CO2• Linear → no permanent dipole moment, no pure rotational spectrum• Fundamental modes:

• v3 vibration is a parallel band (dipole moment oscillates parallel to symmetric axis), transition ΔJ = 0 is forbidden, no Q branch, greater total intensity than v2 fundamental

• v2 vibration is perpendicular band, has P, Q, and R branch• v3 fundamental strongest vibrational band but v2 fundamental most

effective due to “matching” of vibrational frequencies with solar and terrestrial Planck emission functions

• 13C isotope (1% of C in atm) and 17/18O isotope (0.2%) cause a weak splitting of rotational and vibrational lines in the CO2 spectrum

symmetric stretch v1 = 7.5 μm => IR inactive

asymmetric stretch v3 = 4.3 μm

bend

v2 = 15 μm

bend v2

o c o

IR Absorption Spectrum of CO2

v3 v2

Diagram modified from Peixoto and Oort (1992)

O3• Ozone is primarily present in the stratosphere aside from

anthropogenic ozone pollution which exists in the troposphere• Asymmetric top → similar absorption spectrum to H20 due to similar

configuration (nonlinear, triatomic)• Strong rotational spectrum of random spaced lines• Fundamental vibrational modes

– 14.3 μm band masked by CO2 15 μm band– Strong v3 band and moderately strong v1 band are close in frequency,

often seen as one band at 9.6 μm – 9.6 μm band sits in middle of 8-12 μm H20 window and near peak of

terrestrial Planck function– Strong 4.7 μm band but near edge of Planck functions

symmetric stretch

v1 = 9.01 μm asymmetric stretch

v3 = 9.6 μm

bend

v2 = 14.3 μm

oo

o o

IR Absorption Spectrum of O3

v2v1/v3

Diagram taken from Peixoto and Oort (1992)

CH4

• Spherical top

• 5 atoms, 3(5) – 6 = 9 fundamental modes of vibration

• Due to symmetry of molecule, 5 modes are degenerate, only v3 and v4 fundamentals are IR active

• No permanent dipole moment => No pure rotational spectrum

• Fundamental modes

C

H

HHH

C C C

v1 v2 v3 = 3.3 µm v4 = 7.7 µm

IR Absorption Spectrum of CH4

v3 v4

Diagram taken from Peixoto and Oort (1992)

N2O

• Linear, asymmetric molecule (has permanent dipole moment)• Has rotational spectrum and 3 fundamentals• Absorption band at 7.8 μm broadens and strengthens methane’s 7.6

μm band.• 4.5 μm band less significant b/c at edge of Planck function.

• Fundamental modes:

symmetric stretch v1 = 7.8 μm

asymmetric stretch v3 = 4.5 μm

bend

v2 = 17.0 μm

bend v2

N NO

IR Absorption Spectrum of N2O

v3=4.5 µm v1=7.8 µm

v2=17 µm

Diagram taken from Peixoto and Oort (1992)

Total IR Absorption Spectrum for the

Atmosphere

Visible

Diagram taken from Peixoto and Oort (1992)

References• Bukowinski, Mark. University of California, Berkeley. 21 April 2005.• Lenoble, Jacqueline. Atmospheric Radiative Transfer. Hampton,

Virginia: A. DEEPAK Publishing, 1993. 73-91, 286-299.• McQuarrie, Donald A., and John Simon. Physical Chemistry.

Sausalito, California: University Science Books, 1997. 504-527.• Patel, C.K.N. “High Power Carbon Dioxide Lasers.” Scientific

American. 1968. 26-30.• Peraiah, Annamaneni. An Introduction to Radiative Transfer.

Cambridge, United Kingdom: Cambridge University Press, 2002. 9-15.

• Petty, Grant W. A First Course in Atmospheric Radiation. Madison, Wisconsin: Sundog Publishing, 2004. 62-66, 168-272.

• Thomas, Gary E., and Knut Stamnes. Radiative Transfer in the Atmosphere and Oceans. Cambridge, United Kingdom: Cambridge University Press, 1999. 110-120.