cernicharo_cospar_intro_spectro_mod.pdf

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Leuven, 16/March/2010

INTRODUCTION TO R DIO STRONOMY FROM

MILLIMETER TO SUBMILLIMETER

Molecular Spectroscopy

What are the mechanisms

of emitting spectral l inesfor molecules? Which kindof transitions exist?


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G. Herzberg, "Atomic Spectra and Atomic Structure"G. Herzberg, "Spectra of Diatomic Molecules"

G. Herzberg, " Infrared and Raman Spectra"G. Herzberg, "Electronic Spectra of Polyatomic Molecules"G. Herzberg, "The Spectra and Structures of Simple Free Radicals

An Introduction to Molecular Spectroscopy"

G. M. Barrow, " Introduction to Molecular Spectroscopy"H. B. Dunford, "Elements of Diatomic Molecular Spectra"R. N. Dixon, "Spectroscopy and Structure"W G Richards and P R Scott " Structure and Spectra of Molecules" ,.C N Banwell and E M McCash, "Fundamentals of Molecular

Spectroscopy",.J M Hollas, "Modern Spectroscopy" , 3rd ed, Wiley, 1996.J. M. Brown, Molecular Spectroscopy , Oxford University Press,

1998.

I.N. Levine Molecular Spectroscopy , John Wiley, 1975D. Emerson Interpreting Astronomical Spectra


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Formation of molecules and

molecular stability


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Electronicisodensity

contours:


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The dependency of the electronic energy of the systemas a function of the internuclear distance has the

following shape:

0.85 A

2.7 eV

(values excluding the interaction

between electrons and the factthat the two protons interact with

the electrons)

Experimental values:

0.74 A and 4.7 eV


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------------------------------------------------------E(HA+HB)

IMPORTANT:When two H atoms coll ide they have some

extra kinetic energy

H2 cannot be formedin gas phase in the ISM !!!

3-body collisionsare required

(3rd body will carry outthe excess of energy)


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ONCE A MOLECULE IS FORMED THEINTERNAL ENERGY IS MUCH DIVERSIFIED

Electronic: Energy of the electronic orbitals

Vibrational: Energy of the vibrations of thenuclei around the equilibrium positionRotational: Energy associated to the

rotation of an electric dipole.Other: unpaired electron spins or nuclear spins

can couple with the angular momentum of theelectric dipole rotation. Internal magnetic dipolescan couple with external magnetic fields, etc

MOST OF THESE ARE TREATED AS PERTURBATIONS OFTHE ROTATIONAL ENERGY


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Energy, Frequency (E=h

Wavelength (=c/

Molecular Transitions

Rotational Vibrational Electronic

There is a hierarchy in the

energies needed for transitions

However, the quantum mechanical problem isrelatively simple only for very small molecules.


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As the movement of the nuclei is much slower than those of the

electrons we could consider that the electronic energy of the

molecule is independent of the vibration and rotation.

(Born-Oppenheimer approximation)

Problem: finding the potential

energy of the molecule, U( r ),describing in a reasonable way

the variation of energy as a

function of the internuclear

Distance for a given electronic

state. The solution of the

Schrdinger equation will

depend on this potential.

Vibrational Energy


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Morse Potential :

The wave equation is then :

Making the following changes:


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We obtain :

For A = 0, i.e., J=0 it is possible to find an analitycal

solution. In the general case the solution is given by:Centrifugal distortion

Vibrational terms Undisturbedrotation

Harmonic

oscillatoranharmonicity

Higher order corrections

Where:


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There is another potential proposed by Dunham.It is represented by a series in (r- re), where re is

the equilibrium distance

Where=(r-re)/re and Be =h/82

re2

The solution can be given as

Where the terms Ylj are the Dunham coefficients


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and Obviously, there is a

relation between the

Dunham coefficients andthose obtained with the

Morse potential

these terms can be calculated easily.


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IN SUMMARY: The empirical expression used in spectroscopy

to fit the ro-vibrational spectra of diatomic molecules is

INVERSE PROBLEM: From spectroscopic measurementsit Is possible to fit spectroscopic constants and form themit is possible to derive important information, through the

Morse or Dunham expression for the potential energy, onthe dissociation energy, equilibrium distance, etc.

This is the main subject of papers in journals such as theJournal of Molecular Spectroscopy or the Journal of Molecular Structure


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A few facts


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Electronic transitions will connect the lowest

vibrational level of the ground electronic

state with several vibrational states of theupper electronic state. The most probable

transition is the one having the best overlap

in the vibrational eigen functions.

Franck-Condon Principle


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For simple molecules we can separatethe vibrational and rotational parts of the

wave function as another Born-Oppenheimer aproximation


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SIMPLE CASE: ROTATIONAL SPECTRUMOF DIATOMIC OR LINEAR MOLECULES

From a classical point of view of quantum theory a moleculehas a rotational energy proportional to the square of the

angular momentum, J2. Hence the energy levels should have a

dependency on J given by

E(J)

J(J+1)

The frequencies of the transitions could be given by

(J=>J-1)

J

The angular momentum of the molecule is given by I, where I

is the momentum of inertia of the molecule,

I =

r2 , : reduded mass, r: separation between the nuclei.


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In this classical approximation for the energy of a rigid molecule

the energies can be written as

E(J) = B J(J+1)

and the frequencies as

= 2 B Ju

The constant B, the rotational constant,is given by

B = ( h / 8

2 I ) [frequency units]

1cm-1= 29979.2459 MHz

30 GHz

1.4388 K

The heavier the molecule, the smaller B; the rotational

spectrum will be tighter.


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ROTATIONAL SPECTRUM OF CARBON MONOXIDE


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EXAMPLE

Be(CO)=57897.5 MHz

Be(CO)=505379.05/I(CO)/r

e(CO)2

I(CO) = 6.856208642 u.a.

re(CO)=1.1283

m(13C)=13.00335483; m(12C)=12; m(O)=15.99491464

Be(13CO) = (12CO)/(13CO)*Be(

12CO) =0.95591388*57897.5 = 55345.02 MHz

Experimental = 55344.9 MHz


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ROTATIONAL SPECTRUM OF HC3N (linear)

ROTATIONAL SPECTRUM OF HC5N (linear)

SpectrumGettingtighter

However when the spectral accuracy of the observations is large


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However, when the spectral accuracy of the observations is large

it is observed that the frequencies of successive rotational

transitions depart from the last expression.

WHY ?

Because when molecules are rotating the nuclei are submitted tocentrifugal forces which increase the distance between them and

increase the momentum of inertia.

Errors with respect to the simple approximation increase:

For lighter molecules.

For larger rotational numbers.

THIS IS JUST ONE OF MANY DISTORTION EFFECTSTHEY GET INCREASINGLY COMPLICATED AS THE NUMBER

OF ATOMS IN THE MOLECULE INCREASES


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Selection rules J=1


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By selecting the appropriate molecule

we can trace different physicalconditions.

Molecules with low dipole moment, as

CO, are easily excited through collisions

with H2, even for low volume densities.

Under some assumptions thesemolecules could trace the kinetic

temperature of the gas.

High dipole moment molecules could be

used as tracer of the gas volume density

(n(H2)).


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IN ASTROPHYSICS: By selecting the appropriate

molecule we can trace different physical parametersand/or gas submitted to different physical conditions.

Molecules with low dipole

moment, as CO, are easily

excited through collisions with

H2

, even for low volume

densities. Under some

assumptions these molecules

could trace the kinetic

temperature of the gas.

High dipole moment molecules

could be used as tracers of the

gas volume density [n(H2)].


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Energy leveldiagram of theCO molecule

Well, thewhole thing

can get verycomplicated

Radioastronomy

with CO


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Frank Houwing's Lecture Notes Web site!

www.anu.edu.au/Physics/houwing/LectureNotes/ Phys3034_MolSpec/MolSpec_L2_OH.pdf


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More on rotational spectroscopy


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More on rotational spectroscopy Diatomic or linear molecules are just a simple case (J)

In general a molecule will have three principal axiswith moment of inertia IA, IB, IC. Here we can distinguish:

Spherical tops: IA=IB=IC Example: Amonium ion NH4+

Symmetric tops: IA=IB


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