1 Lecture 5 Lecture 5 By Tom Wilson
Dec 26, 2015
22
Dust clouds in TaurusDust clouds in Taurus
Regions shown in gray are those where very few stars are seen (Becvar atlas)
33
The Discovery of Molecular The Discovery of Molecular CloudsClouds
-The discussion began in 1960’s when it -The discussion began in 1960’s when it was noticed that dark clouds had a was noticed that dark clouds had a deficiency of HI emission, based on an Adeficiency of HI emission, based on an Avv to HI relation obtained from studies of to HI relation obtained from studies of diffuse clouds. diffuse clouds.
Some thought this could be due to high Some thought this could be due to high optical depth in HI, or self absorption, optical depth in HI, or self absorption, etc. etc.
Barrett et al. found intense OH linesBarrett et al. found intense OH lines
55
HI Self Absorption in Dark HI Self Absorption in Dark CloudsClouds
The broader HI emission is from a warm background
66
Molecular CloudsMolecular Clouds
The discovery of thermal OH emission from these The discovery of thermal OH emission from these clouds, and then NHclouds, and then NH33, H, H22O and HO and H22CO were found in CO were found in warmer clouds. This made it likely that the gas in warmer clouds. This made it likely that the gas in these clouds was mostly Hthese clouds was mostly H2 2
The final piece of evidence was the discovery of The final piece of evidence was the discovery of COCO
Still were problems with the abundance (even the Still were problems with the abundance (even the very presence) of complex molecules. very presence) of complex molecules.
77
--Make this molecule on grains, since must Make this molecule on grains, since must conserve energy and momentumconserve energy and momentum
-The exact production rate is known to a -The exact production rate is known to a factor of 2factor of 2
-Once made, H-Once made, H2 2 enters the gas phaseenters the gas phase
- H- H22 is dissociated by 11 eV photons is dissociated by 11 eV photons
- HH2 2 is destroyed by spectral line radiation, is destroyed by spectral line radiation, so is self shielded, and also shielded by so is self shielded, and also shielded by dust dust
- HH2 2 combines with Hcombines with H++ to form H to form H33++ which which
promotes promotes ion-moleculeion-molecule chemistry chemistry
The Life Cycle of HThe Life Cycle of H22
99
Trapeziumstars, which ionize the HII region
Bar, anEdge-onPhotonDominatedRegion
Photo From HST(O’Dell)
The molecular cloudin Orion is extended north-south,behind the HII region
OrionHot CoreA “chemistry factory” in the sky
1010
Methyl alcohol
Methyl formate
Dimethyl ether
Structures of a few interstellar molecules
These are asymmetric
top molecules which have many transitions in the
millimeter range
in warm sources
1212
Sketch of a Photon Dominated RegionSketch of a Photon Dominated Region
From Hollenbach & Tielens 1999
1313
Lecture5 page 2Lecture5 page 2
For molecules, the electronic states are given analogous assignments:
S
PP
DD
2 electron systems
3S = “triplet S” parallel electron spins
1S = “singlet S” auto-parallel electron spins
1414
MOLECULES
Vastly more complex than hydrogen atoms. The main assumption in a discussion of molecules is Born-Oppenheimer approximation:
W W W WT O T E L V IB R O T
WEL (electronic energies): transitions with energies of a few eV
WVIB (vibrational energies): transitions have energies of 0.1 to 0.01 eV
WROT (rotational energies): transitions have energies of 0.001 eV
To first order there no mixed terms in Hamiltonian, this means that the wavefunctions are product of:
Another way to say this, is that the electrons move much fasten than the rotation of the nuclei. Thus have a negative bowl of potential energy from electrons. In this, the nuclei rotate.
(electronic) * (vibration) * (rotation)
What is the energy equivalent of 1 eV in degrees Kelvin?What is the energy equivalent of 1 eV in degrees Kelvin?
1616
Approximations for P(r) P r D eea r re( ) 1
2
Often can use 1st order expansion P r a D r re e( ) 2 2
Since for most radio lines, one is deep in potential well
ROTATIONAL SPECTRA OF DIATOMIC MOLECULES
The Hamiltonian isH O
J
Oee
1
2 22
2
╫
╫
O m r m r m r
r r r mm m
m m
J O
e A A B B e
e A BA B
A B
e
2 2 2
╫
╫
1717
,..1,01 JJJJ
Demand
Assumed perfectly rigid
Calculate the Be for 12C16O and 13C16O using the nuclear masses in
Atomic mass units (AMU), rounded to integers. Compare the values and calculate the line frequencies using h=EROT
)(2
122
1
2
2
definitionO
Bwhere
JJBO
JJ
O
JE
ee
eee
ROT
╫ ╫
╫
1919
So E h B J J
h E E JH E J h B J jR O T e
R O T e
( )
( ) ( ) ( ) , , . . .
1
2 1 0 1
This is correct if molecule is perfectly rigid. If stretching, have an extra term
h D J J ( )12
2 1 4 1 3B J D Je e( ) ( )
Where D is > 0 and D / B =10-5
EXAMPLE: For 12C 16O, Be = 57.6360 GHz and De = 0.185 MHz
Use this relation given above to calculate the J = 1 – 0, 2 – 1, 3 – 2 line frequencies
2020
For linear molecules, find J
J
JJ J
2
02 1
2 31
From before A G H z J 1 1 6 5 1 0 11 3 2.
And
For total population of CO, under LTE, get
z J ek T
h Bh B k T
h B J J
k T
J ee
e
2 11
0
( )
n to ta l
z
Jn J e
T
Jn J e
h B J J
k T
h B J J
k Te e
( ) ( ).
( )( ) ( )
2 1
2 2
2 1
1 1 1
Work out the Einstein A coefficients for CS, SiO, HCO+, N2H+.
Then calculate the critical density, n*, where A = n* < v > = 10-10 n*.
Compare to n* for CO J = 1 → 0
exTk
h
l
u
l
u
Vu
l
ul
GHzl
eg
g
N
NbydefinedisT
TkhifdVTg
g
AN
331094.1
2121
SYMMETRIC AND ASYMMETRIC TOPS
is constant, and projection on ax axis
1.- SYMMETRIC TOPS – projection on molecule axis of symmetry is also a constant.
Definition: O O O O Ox y z II
So
HJ
O
J
O
J
O
J
OJ
O O
call J K
x y z
IIz
II
z
2 2 2 22
2
2 2 2 2
1
2
1
2
╫╫ ╫ ╫ ╫
╫ ╫ ╫ ╫ ╫ ╫
J 2
2222
CH3CN: Prolate symmetric Top
NH3: Oblate symmetric Top
Usually one defines
4 41
OB
OC
II╫ ╫
Then for energy eigenvalues
And the energy above the ground state is
2221 KKJJJ
)(1
1 2
JWJWh
BChKJJBhW e
CBABCKJB
BAKJB
)()1(2
)()1(22
2
2323
Ammonia Ammonia
Quantum numbersQuantum numbers
are (J,K),are (J,K),
where where
J is total angularJ is total angular
momentum,momentum,
K is componentK is component
along moleculealong molecule
axis axis
2424
Note that J J JF IN A L IN IT IA L P H O T O N
So 0 → 0 is forbidden, but J J is a llow ed for JF IN IT IA L 1
An extra feature of NH3 is inversion doubling. This is caused by the
Q. M. tunneling of N through the plane of the 3 hydrogen atoms
GHzatKJ
BCKJB
5.572)0,0()0,1(),(
12 2
2525
The inversion doubling gives rise to many lines in the centimeter range from 20 GHz to 40 GHz. These are (J,K) → (J,K) lines
Ammonia is an OblateOblate symmetric top. CH3CN and CH3C2H are ProlateProlate symmetric tops. For prolate, “C” becomes “A”.
Dipole moment along axis of symmetry of molecule. So no radiative transition involving a change in K quantum number
2.- ASYMMETRIC TOPS
Most common type of polyatomic molecule: H2CO, H2O,…
Now, only is a constant. Describe energy levels by the approximation that the asymmetric top is somewhere between a PROLATE and OBLATE symmetric top. That is between K values Ka and Kc. Levels are described by the quantum numbers
Each of the K values is less than or equal to the J value
J 2
J K Ka c
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H C O
A G H z
B G H z
C G H z2
2 8 2 1
3 8 3
3 4 0
.
.
.
Almost a symmetric top molecule, with dipole moment along A axis. In an allowed radiative transition, no charge in Ka. There will be a change in J an Kc.
So 101 → 000 (n = 72 GHz)
Can also have 110 → 111 or 110 ← 111 (6 cm, or 4.8 GHz)
H2CO is nearly a prolate symmetric top
H2O very asymmetric top, with dipole moment along B axis
FormaldehydeFormaldehyde
2929
EXAMPLE: H2CO
A rotation about the A axis interchanges the 2 hydrogens. This must have an anti-symmetric wavefunction since these are FERMI particles.
So (space) * (spin) = anti symmetric
Definite relation of spin and (-1)Ka : if spins are parallel, ↑↑, then is ortho. If so, the parity of the space wavefunction must be anti-symmetric
↑↓ If spin is para, then the space wavefunction must be symmetric
A rotation about B, A, and C, must leave the molecule in the initial state. So the product of parities of the 3 rotations must be symmetric. Then for ortho: the space part is anti symmetric in B since (-1)Ka is the symmetry. Then have Ka Kc = (even)(odd).
Transitions for ortho: Since J must change in an allowed radiation transition, must connect in upper and lower states, Kc must change must change between states. between states.
Ka Kc==(e)(o)-(e)(e)
(e)(e)-(e)(o)
For para: the space symmetry of A is even, so wavefunctions must be odd. So para transitions for H2CO are (o)(o)-(o)(e)
(o)(e)-(o)(o)
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Allowed radiative transitions are dipole. Thus parity of initial and final states must be different:
r rf i
For i and f states the parity must be different
r: over all space r changes sign, but product be positive, so the partiy of the initial and final wavefunctions must be opposite.
Symmetry of spatial part of wavefunctions is crucial: look at H2CO
If proton spins parallel, have OrthoOrtho: the spin wavefunction is symmetric. Rotation of 180° about A axis interchanges protons. These are FERMIFERMI particles as total symmetry must be odd. Thus since the spin wavefunction is even so space wavefunction must be odd. Since the spin wavefunction varies as (-1)Ka. So when Ka = 0, the spin wavefunction is
symmetric. When the spins ↑↓ this is a para state; the Ka = 1. For parallel spins, ↑↑, ortho states, Ka must be 0, 2, 4, 6, etc…
3131
EXAMPLE: H2O
A rotation about the B axis interchanges the 2 hydrogens. This must have an anti-symmetric wavefunction since these are FERMIFERMI particles.
So (space) * (spin) = anti symmetric
Definite relation of spin and (-1)Kb : ↑↑ orthoortho space: anti-symmetric
↑↓ parapara space: symmetric
A rotation about B, A, and C, leave the molecule in the initial state. So the parities of the 3 rotations must be symmetric then orthoortho: space anti symmetric in B since (-1)Ka is the symmetry have Ka Kc = (even)(odd) or (odd)(even)
OrthoOrtho: Since J must change in an allowed radiation transition, must connect in upper and lower states (o)(e)-(e)(o)
(e)(o)-(o)(e)
ParaPara: the symmetry of B is even, so A and C wavefunctions must be both even or both odd. So para para transitions for H2O are (o)(o)-(e)(e)
(e)(e)-(o)(o)
3333
MethanolMethanol Not just more complexNot just more complex Has different notationHas different notation The CHThe CH33 and OH arm can rotate relative to each and OH arm can rotate relative to each
other, giving rise to ‘internal rotation’other, giving rise to ‘internal rotation’
3434
J, TotalJ, Total
Angular Angular
momentummomentum
AngularAngular
MomentumMomentum
AlongAlong
Axis of Axis of
Symmetry Symmetry
3535
OH MoleculeOH Molecule
Not shown areNot shown are
The Far IR The Far IR
TransitionsTransitions
connecting connecting
rotationally rotationally
excited states excited states
to groundto ground
Sketch is to showSketch is to show
The effect of The effect of
Doubling Doubling
84 micron lines84 micron lines
120 micron lines120 micron lines
3636
How convert line intensity into a column density?
Use
ex
ex
GHz
Tk
h
l
u
l
u
ulGHzul
B
Tul
GHz
u
ll
eg
g
N
N
A
dVT
eAg
gN
2311
108.4
3
10165.1
1
1
(sec)5.93 2
General General
Relation Relation
exTk
h
l
u
l
u
Vu
l
ul
GHzl
eg
g
N
NbydefinedisT
TkhifdVTg
g
AN
331094.1
T is TT is Texex
Emission lines:Emission lines:
TTLL=T=Texex(1-e(1-e).).
If If <<1, T<<1, TLL=T=Texex
3737
J
J
J
2
02 1
2 3
JK
JK
J K
J JJ K J K
K
J JJ K
2
02
2 2
2
02
1
1 2 31
10 0
, ,
,
AS u b
JG H z x
1 1 6 5 1 02 1
11 3 2.
( , )
'
Asymmetric Tops: T&S Table IV or Journal of Physical and Chemical Ref. Data for S(u,b)Asymmetric Tops: T&S Table IV or Journal of Physical and Chemical Ref. Data for S(u,b)
In General:In General:
Ammonia:
CO:
3838
LVG MODEL
Assumes close coupling of molecule and radiation field. Also
V Vr
rSobo lev
0
0
This makes the radiative transfer a local problem.
Qualitative result:A
Aue
ue
Instead of an escape from cloud, after a collision the photon is reabsorbed many times. Only happens if > 1
4040
The story of X
The clumps are more dense. Each is assumes to be Virially stable, and then entire cloud is stable Virially.
From LVG,
Virial theorem:
T C O n H n( ) ( )
M m R pc V km s
VM m
R pc
n R
Rn
SU N
SU N
( ) ( ) ( )
( )
( )
2 5 0
2 5 0
43
2 5 0
1 2 1
2
1 2
3
So T dV n n or n H LB ( )2
4141
Need more data to get a secure total column density
1. Could measure lots of lines of a species, make “ Boltzmann Plot”
N J etJ
E
k Tex
2 1
2. For CO, a number of models of excitation. From Mauersberger, one is
Another is
N H X T C O J dV T C O J dVM B M B( ) ( , ) . ( , )22 01 0 2 3 1 0 1 0
based on large velocity gradient model
N(H2): based on “clumpy cloud” model and a virialized cloud
)(1065.2)2( 1821 OdVCTHN MB