MICROWAVE MICROWAVE SPECTROSCOPYSPECTROSCOPY

Prof. V. Krishnakumar

Professor and Head

Department of Physics

Periyar University

Salem – 636 011, India

Summary of information from microwave spectroscopySummary of information from microwave spectroscopy

It is mainly used to get information about gas It is mainly used to get information about gas molecules, such asmolecules, such as1. Accurate bond lengths and angles.1. Accurate bond lengths and angles.2. Electric dipole moments.2. Electric dipole moments.3. Centrifugal distortion constants.3. Centrifugal distortion constants. It can also be used to study relaxation times, It can also be used to study relaxation times, dielectric constants, dipole moments in liquids and dielectric constants, dipole moments in liquids and solutions, and potential energy barriers to rotation.solutions, and potential energy barriers to rotation.

In some cases, we can get information about the In some cases, we can get information about the mechanism of chemical reactions, such as the mechanism of chemical reactions, such as the decomposition:decomposition: 1515NHNH44

1414NONO33 →→ 1515NN1414NONO  The requirements to get a microwave spectrum are:The requirements to get a microwave spectrum are:Substance must have electric dipole moment (or Substance must have electric dipole moment (or magnetic dipole moment) magnetic dipole moment) Its vapour pressure > 10Its vapour pressure > 10-3-3 mmHg.  mmHg. 

Characteristics of microwave spectroscopy, Characteristics of microwave spectroscopy, compared with other techniques are:compared with other techniques are:It has a high resolving power. It has a high resolving power. It analyses the WHOLE molecule (not like nmr, or ir It analyses the WHOLE molecule (not like nmr, or ir spectra, which fingerprint selected parts).spectra, which fingerprint selected parts).It detects isotopic species, and conformational It detects isotopic species, and conformational isomers.isomers.Only a few ng of gas are required.Only a few ng of gas are required.It is a non-destructive technique.It is a non-destructive technique.It can be used remotely, such as for interstellar It can be used remotely, such as for interstellar analyses.analyses.The spectra of large molecules are very complex. The spectra of large molecules are very complex. AbsoluteAbsolute absorbance is difficult to measure. NBS has absorbance is difficult to measure. NBS has list of microwave spectra for qualitative analysislist of microwave spectra for qualitative analysis ..

Basic conceptsBasic concepts Rotational energies of molecules are quantized (i.e. Rotational energies of molecules are quantized (i.e. only have definite energies) only have definite energies) E = hE = hννE, energy in J; h Planck’s constant, Js; E, energy in J; h Planck’s constant, Js; νν rotational rotational frequency, Hz.frequency, Hz.  The range of rotational frequencies is about 8x10The range of rotational frequencies is about 8x101010 - - 4x104x1011 11 Hz, which corresponds to wavelengths, Hz, which corresponds to wavelengths, λλ ~ ~ 0.75 - 3.75 mm. These wavelengths fall in the 0.75 - 3.75 mm. These wavelengths fall in the microwave region of the electromagnetic spectrum. microwave region of the electromagnetic spectrum. 

By absorption of microwave radiation, transitions can By absorption of microwave radiation, transitions can occur between rotational or inversion energy levels occur between rotational or inversion energy levels of molecules. of molecules. N.B. Molecule must have N.B. Molecule must have permanent dipole momentpermanent dipole moment (D.M.) if it has a rotational spectrum.(D.M.) if it has a rotational spectrum.

DIRECTION DIRECTION OF DIPOLE OF DIPOLE VERTICAL VERTICAL COMPONENT COMPONENT OF DIPOLE OF DIPOLE (along z)(along z)

Rotation of a Rotation of a polar diatomic polar diatomic molecule molecule showing D. M. showing D. M. along z versus along z versus timetime

t

+

-

- + - + - +

+

+-

-

To an observer, there is a change in dipole moment To an observer, there is a change in dipole moment along z when the molecule rotates. The oscillating along z when the molecule rotates. The oscillating electric field of microwave radiation, incident upon electric field of microwave radiation, incident upon the molecule, can therefore make this rotation the molecule, can therefore make this rotation occur (i.e. the radiation is absorbed).  occur (i.e. the radiation is absorbed).  

Some definitions about rotation. Some definitions about rotation. For simplicity, we For simplicity, we consider a diatomic molecule throughout. consider a diatomic molecule throughout.     QQ P P   moment of inertia, I = moment of inertia, I = ΣΣ m miirrii

22

r = distance of atom i from rotation axis (m); m in kg.r = distance of atom i from rotation axis (m); m in kg.  Angular momentum = IAngular momentum = Iωω, where the angular frequency , where the angular frequency (radian s(radian s-1-1),), ωω = 2 = 2πνπν

Classification of molecules according to I values

1. Linear molecules

IA = 0; IB = IC

 

22. Symmetric tops

 

IA ≠ 0; IB = IC 

Q P

B

C

A

H C FH

H

H Cl

H H

3. Spherical tops3. Spherical tops  

IIAA = I = IBB = I = ICC

This is type of molecule has no rotational This is type of molecule has no rotational spectrum.spectrum. 4. Asymmetric tops4. Asymmetric tops

  IIAA ≠≠ I IBB ≠≠ I ICC

H C HH

H

C = COH

H

Rotation spectra of diatomic moleculesRotation spectra of diatomic molecules  Consider molecule with nuclear masses mConsider molecule with nuclear masses m11 and m and m22

r0

m2 m1

r2 r1

c

(C is centre of mass)(C is centre of mass)

Assume a rigid (not elastic) bondAssume a rigid (not elastic) bond

   rr00 = r = r11 + r + r22

For rotation about center of gravity, C :For rotation about center of gravity, C :

   mm11rr1 1 = m= m22rr22 ( = m( = m22 (r (r00 - r - r11) )) )

/ \/ \

21

021 mm

rmr+

=21

012 mm

rmr+

=

Moment of inertia about C:Moment of inertia about C:

IICC = m = m11rr1122 + m + m22rr22

22 = m = m22rr22rr1 1 + m+ m11rr11rr22

= r= r11rr22 (m (m11 + m + m22))

  

µµ = reduced mass, = reduced mass, 2

02

021

21 μrrmm

mmI =+

=⇒21 m

1m1

μ1 +=

More detailed derivation:More detailed derivation:

  

[ ]

[ ]

20

21

21

2121

02

21

01

2121

2

1122

1

2211

rmm

mm

mmmmrm

mmrm

mmrrm

rmrmm

rmrm

+=

+

+

+=

+=

+

=

222

211C rmrmI +=

From the Schrödinger equation:From the Schrödinger equation:

Rotational energy of level J,Rotational energy of level J,

J(J+1) JoulesJ(J+1) JoulesI8π

hE 2

2

J =

Where J is the rotational quantum number, having Where J is the rotational quantum number, having the values 0, 1 , 2….Note that J = 0 is the lowest the values 0, 1 , 2….Note that J = 0 is the lowest level, and the molecule is not rotating in this level. level, and the molecule is not rotating in this level. Now the rotational frequency is the same as the Now the rotational frequency is the same as the frequency of the microwave radiation need to cause frequency of the microwave radiation need to cause the rotation: the rotation:

νν / Hz = / Hz =

or in energy units:or in energy units: , where c is in cms , where c is in cms-1-1.. SoSo J(J+1) cm J(J+1) cm-1-1 = BJ(J+1) = BJ(J+1) cmcm-1-1

B is called the rotational constant for a given B is called the rotational constant for a given molecule. Its units are cmmolecule. Its units are cm-1-1, since J is just a quantum , since J is just a quantum number (label).number (label).

hΔE

hcΔE/cmν 1 =−

Ic8πhE 2J =

Appearance of microwave spectrumAppearance of microwave spectrum  Microwave absorption lines should appear atMicrowave absorption lines should appear atJ = 0J = 0 →→ J = 1 : J = 1 : = 2B - 0 = 2B cm = 2B - 0 = 2B cm-1-1

J = 1J = 1 →→ J = 2 : J = 2 : = = = 4B cm = 4B cm-1-1

  Or generally:Or generally:J J →→ J + 1 J + 1 = B(J+1)(J+2) - BJ(J+1) = B(J+1)(J+2) - BJ(J+1)

= 2B(J+1) cm= 2B(J+1) cm-1-1

Note that the selection rule is Note that the selection rule is ∆∆J = J = ±±1, where + 1, where + applies to absorption and - to emission.applies to absorption and - to emission.

ν

νν

This diagram shows the rotational energy levels of This diagram shows the rotational energy levels of a diatomic molecule. Fill in the ???a diatomic molecule. Fill in the ???

Here are some data for carbon monoxide:Here are some data for carbon monoxide:

EnergyEnergy

42B42B

30B30B

?B?B

?B?B

6B6B2B2B0B0B

J levelJ level

J = 6J = 6

J = 5J = 5

J = 4J = 4

J = 3J = 3

J = 2J = 2J = 1J = 1J = 0J = 0

1414

13 13

12 12

11 11

10 10

9 9

8 8

7 7

6 6

5 5

4 4

3 3

2 2

1 1

0 0

00

100100

200200

400400

F(J)

cm

F(J)

cm

-1-1

.14 .14

.18 .18

.23 .23

.29 .29

.36 .36

.43 .43

.51 .51

.59 .59

.67 .67

.75 .75

.81 .81

.89 .89

.95 .95

.98 .98

1.0 1.0

29 29

27 27

25 25

23 23

21 21

19 19

17 17

15 15

13 13

11 11

9 9

7 7

5 5

3 3

1 1

4.0 4.0

4.9 4.9

5.8 5.8

6.6 6.6

7.5 7.5

8.1 8.1

8.6 8.6

8.9 8.9

8.8 8.8

8.3 8.3

7.5 7.5

6.3 6.3

4.7 4.7

2.9 2.9

1.0 1.0

53.8

50.0

46.1

42.3

38.4

34.6

30.8

26.9

23.119.215.411.57.69

ν (cm-1) J 2J+1 eJ 2J+1 e-E/kT-E/kT N NJJ/N/Noo

F(J) = energy of levels as a function of J.F(J) = energy of levels as a function of J.2J+1 = degeneracy of J level.2J+1 = degeneracy of J level.ee-E/kT-E/kT = Boltzmann temperature factor. = Boltzmann temperature factor.NNJJ//N//Noo = population of level J compared with = population of level J compared with level O.level O. = transition wavenumber= transition wavenumber

Rotational Energy Levels of CORotational Energy Levels of CO

ν

This is part of the rotational (far infrared) This is part of the rotational (far infrared) spectrum of CO. You can see that the separation, spectrum of CO. You can see that the separation, 2B, is roughly 4 cm2B, is roughly 4 cm-1-1. Assign the lines.. Assign the lines.

15 20 25 30 35 40ν (cm-

1)1212CC1616O (major species)O (major species)1313CC1616O and O and 1212CC1818O linesO lines

%

% tr

ansm

issi

ontr

ansm

issi

on

ApplicationApplication  The measurement of a microwave spectrum The measurement of a microwave spectrum enables us to determine bond lengths and angles enables us to determine bond lengths and angles accurately for gaseous molecules.accurately for gaseous molecules.  Example for CO:Example for CO:

(J=0 (J=0 →→ J=1) for J=1) for 1212CC1616O is at 3.84235 cmO is at 3.84235 cm-1-1..

C = 12.0000 ; C = 12.0000 ; O = 15.9994 O = 15.9994 amuamu1 amu = 1 atomic mass unit = 1.6605402 x 101 amu = 1 atomic mass unit = 1.6605402 x 10-27-27 kg kgh = 6.6260755 x10h = 6.6260755 x10-34 -34 JsJsc = 2.99792458 x 10c = 2.99792458 x 101010 cm s cm s-1-1

Find Find r(Cr(CO)O)

  

= = µµrr22

B = 1.921175 cmB = 1.921175 cm-1-1 ; ; µµ = 1.1386378 x 10 = 1.1386378 x 10-26-26 kg kg

⇒⇒ = 1.131 x 10= 1.131 x 10-10-10 m m

⇒⇒ 0.1131 nm0.1131 nm

Answer: C-O bondlength is 0.1131 nm. Answer: C-O bondlength is 0.1131 nm.

246

2 kgmB

102.7992774Bc8π

hI−×==

μIr =

Intensities of rotation spectral lines Intensities of rotation spectral lines   

Now we understand the locations (positions) of Now we understand the locations (positions) of lines in the microwave spectrum, we can see which lines in the microwave spectrum, we can see which lines are strongest.lines are strongest.  JJ BJ(J+1) BJ(J+1)  J=0J=0 0 0

Intensity depends upon initial state population.Intensity depends upon initial state population.

Greater initial state population gives stronger Greater initial state population gives stronger spectral lines.This population depends upon spectral lines.This population depends upon temperature, T.temperature, T.

kk = Boltzmann’s constant, 1.380658 x 10 = Boltzmann’s constant, 1.380658 x 10-23-23 J K J K-1-1

((k k = R/N)= R/N)

  

We conclude that the population is smaller for We conclude that the population is smaller for higher J states.higher J states.

−=

−∝

kTνhcexp

kTEexp

NN J

0

J

cmK1.52034khc =

−∝

Tν1.52034e

NN

o

J

Intensity also depends on degeneracy of initial state.

(degeneracy = existence of 2 or more energy states having exactly the same energy)

Each level J is (2J+1) degenerate

⇒ population is greater for higher J states.

To summarize: Total relative population at energy EJ α (2J+1) exp (-EJ / kT) & maximum population

occurs at nearest integral J value to :

 

Look at the values of NJ/N0 in the figure, slide #27.

2hcBkT

21J +−=

Plot of population of rotational energy levels versus Plot of population of rotational energy levels versus value of J. value of J.

B = 5cmB = 5cm-1-1

B = 10cmB = 10cm-1-1

max. pop.max. pop.

J0

Po

p P

op

αα (2

J +

1)

e (

-BJ(

J +

1)h

c/kT

) (

2J

+ 1

) e

( -B

J(J

+ 1

)hc/

kT)

At maximum population value, At maximum population value, slope = 0: Putting x = hc/slope = 0: Putting x = hc/kkTT

Slope = 0 at maximumSlope = 0 at maximumWhat is J value?What is J value?

J = 0 J = NJ = 0 J = NJ J →→

(2J

+ 1)

e –x

BJ(J

+1) →→

( )[ ][ ] [ ][ ]21)xB(2Je0

2e1)(2JxBe1)(2J0

0e12JdJdslope

21)xBJ(J

1)xBJ(J1)xBJ(J

1)xBJ(J

++−=++⋅−+=

=+=

+−

+−+−

+−

So:So:

2hcBkT

21J

2xB1

21J

021)xB(2J 2

+−=

+−=

=++−

Effect of isotopesEffect of isotopes  

FromFrom 1212CC1616OO →→ 1313CC1616O, mass increases, B O, mass increases, B decreases (decreases (∝∝ 1/ 1/II), so energy levels lower.), so energy levels lower.

2B 4B 8B 12B2B 4B 8B 12B

cmcm-1-1 spectrum spectrum

J = 6J = 6

55

44

33221100

1212COCO1313COCO

EnergyEnergylevelslevels

Comparison of rotational energy levels of Comparison of rotational energy levels of 1212COCO and and 1313COCO  

Can determine: Can determine:

(i) isotopic masses accurately, to within 0.02% of (i) isotopic masses accurately, to within 0.02% of other methods for atoms in gaseous molecules; other methods for atoms in gaseous molecules;

(ii) isotopic abundances from the absorption relative (ii) isotopic abundances from the absorption relative intensities. intensities.

Example:Example:

for for 1212CO CO J=0 J=0 →→ J=1 J=1 atat 3.84235 cm3.84235 cm-1-1

for for 1313COCO 3.67337 cm 3.67337 cm-1-1

Given : Given : 1212C = 12.0000 ;C = 12.0000 ; O = 15.9994O = 15.9994 amu amu

What is isotopic mass of What is isotopic mass of 1313C ?C ?

B(B(1212CO) = 1.921175 cmCO) = 1.921175 cm-1-1

B(B(1313CO) = 1.836685 cmCO) = 1.836685 cm-1-1

Now Now

⇒⇒ ((1313C) = 13.0006 amuC) = 13.0006 amu

μ1

I1B ∝∝

1.046001.8366851.921175

CO)μ(CO)μ(

12

13

==⇒

15.99941215.999412

15.9994C)(15.9994C)(1.046 13

13

×+×

+×=⇒

Refinements to theory for diatomic moleculesRefinements to theory for diatomic molecules

Rotation spectrum of hydrogen fluoride in the far IR Rotation spectrum of hydrogen fluoride in the far IR region region 

JJ00 41.0841.08 41.1141.1111 82.1982.19 82.1882.18 41.1141.11 20.5620.56 0.09290.092922 123.15123.15 123.14123.14 40.9640.96 20.4820.48 0.09310.093133 164.00164.00 163.94163.94 40.8540.85 20.4320.43 0.09320.093244 204.62204.62 204.55204.55 40.6240.62 20.3120.31 0.09350.093555 244.93244.93 244.89244.89 40.3140.31 20.1620.16 0.09380.093866 285.01285.01 284.93284.93 40.0840.08 20.0420.04 0.09410.094177 324.65324.65 324.61324.61 39.6439.64 19.8219.82 0.09460.094688 363.93363.93 363.89363.89 39.2839.28 19.6419.64 0.09510.095199 402.82402.82 402.70402.70 38.8938.89 19.4519.45 0.09550.09551010 441.13441.13 441.00441.00 38.3138.31 19.1619.16 0.09630.09631111 478.94478.94 478.74478.74 37.8137.81 18.9118.91 0.09690.0969

)(cmν 1obs

− )(cmν 1calc.

− )(cmνΔ 1obs

− ν(1/2)ΔB = r(nm)

note:note: r r increases with increases with JJ because the bond is not rigid because the bond is not rigid but elastic.but elastic.

H-F atoms are pushed apart at higher rotational H-F atoms are pushed apart at higher rotational speed by centrifugal force.speed by centrifugal force.  

For an elastic bond : For an elastic bond :

wherewhere k k is the bond force constant (Nm is the bond force constant (Nm-1-1). Smaller ). Smaller kk, , less rigid bond.less rigid bond.

Note also that Note also that r r and and BB vary during a vibration. vary during a vibration.

μcν4πk 222=

We can refine the theory by adding a correction We can refine the theory by adding a correction term, containing the centrifugal distortion constant, term, containing the centrifugal distortion constant, D, which corrects for the fact that the bond is not D, which corrects for the fact that the bond is not rigid. Assuming harmonic forces:rigid. Assuming harmonic forces:

EEJJ = BJ(J+1) - DJ = BJ(J+1) - DJ22(J+1)(J+1)22 cm cm-1-1

where is bond stretch wavenumber. where is bond stretch wavenumber.

  i) can find J values of lines in a spectrum - fitting 3 i) can find J values of lines in a spectrum - fitting 3 lines gives 3 unknowns: J, B, D.lines gives 3 unknowns: J, B, D.

ii) We can estimate from the small correction ii) We can estimate from the small correction term, D.term, D.

1224

3

cmkcrI32π

hD −= 2vib

3

ν4B=

vibν

vibν

Polyatomic moleculesPolyatomic molecules  Things get much more complicated, but the general Things get much more complicated, but the general principles are the same.principles are the same.  e.g. OCSe.g. OCS HCHC≡≡CClCCl IIcc = I = IBB; I; IA A = 0= 0  * I greater than for diatomic molecule, * I greater than for diatomic molecule, ∴∴ B smaller; B smaller; lines more closely spaced.lines more closely spaced.  * Remember that the molecule must have D.M. for * Remember that the molecule must have D.M. for microwave spectrum.microwave spectrum.

Microwave spectrum of carbon oxysulphide

J J →→ J+1 J+1 B(cmB(cm-1-1)) 0 0 →→ 1 1 …… 2 2 ×× 0.4055 0.4055 0.20270.20271 1 →→ 2 2 0.81090.8109 0.40540.4054 0.20270.20272 2 →→ 3 3 1.21631.2163 0.40540.4054 0.20270.2027 CalculateCalculate3 3 →→ 4 4 1.62171.6217 0.40540.4054 0.20270.2027 IIBB

4 4 →→ 5 5 2.02712.0271 0.40550.4055 0.20270.2027

)(cmobsν 1−

N atoms N atoms →→ N-1 bond lengths, so for OCS must N-1 bond lengths, so for OCS must determine rdetermine rCOCO, r, rCSCS : that is, two bondlengths are : that is, two bondlengths are unknown - not just 1 as in a diatomic molecule.unknown - not just 1 as in a diatomic molecule.

νΔ

∴∴ need 2 values for need 2 values for IIBB - the second can come from - the second can come from an isotopically substituted molecule, which has same an isotopically substituted molecule, which has same bondlength (almost), but different mass.bondlength (almost), but different mass.  e.g.e.g. 1616OCOC3434S, S, 1818OCOC3434S ….S ….

O O C C S S

mmoorroo + m + mCCrrCC = m = mSSrrSS

I = mI = moorroo

22 + m + mCCrrCC22 + m + mSSrrSS

22

  

In accurate work isotopic bondlengths differ, due to In accurate work isotopic bondlengths differ, due to differences in zero point energy. differences in zero point energy.

r0 rcrs

centre of gravity

Microwave instrumentationMicrowave instrumentation

Schematic diagram Schematic diagram of a microwaveof a microwave

spectrometerspectrometer

    S: source Klystron oscillator (few mW). S: source Klystron oscillator (few mW).

This is monochromatic, but can be tuned This is monochromatic, but can be tuned mechanically or electronically. By using several mechanically or electronically. By using several klystrons we can cover the spectral range 1000 Mc/s klystrons we can cover the spectral range 1000 Mc/s (30 cm) - 37500 Mc/s (8 mm)(30 cm) - 37500 Mc/s (8 mm)

More recent instruments use solid-state microwave More recent instruments use solid-state microwave sources.sources.

MICA WINDOWS

SAMPLE

VACUUM

S D

The waveguides are hollow metallic conductors The waveguides are hollow metallic conductors through which the energy propogates. through which the energy propogates.    WM: wavemeter measures WM: wavemeter measures λλ (or (or νν).).  Vacuum - prevents atmospheric (HVacuum - prevents atmospheric (H22O) absorption.O) absorption.  Sample - 0.01 mm Hg pressure adequate, so liquids, and Sample - 0.01 mm Hg pressure adequate, so liquids, and even some solids, as well as gases may be studied.even some solids, as well as gases may be studied.  D: Detector. Radio receiver or crystal detector.D: Detector. Radio receiver or crystal detector.  Output: absorption vs frequency.Output: absorption vs frequency.

Special cases of microwave absorptionSpecial cases of microwave absorption

a) Inversion spectrum of NHa) Inversion spectrum of NH33

Pyramidal molecules Pyramidal molecules not only rotate, but not only rotate, but can turn inside out can turn inside out (i.e. invert) because (i.e. invert) because this has a low this has a low potential barrier.potential barrier.

Rotation-inversion levels of NHRotation-inversion levels of NH3 3 : Each level is : Each level is split into two (+,-), which show the orientation of split into two (+,-), which show the orientation of the molecule. Inversion energy (~ 23000 MHz the molecule. Inversion energy (~ 23000 MHz ×× hh Js) depends slightly Js) depends slightly onon rotational energy. rotational energy. More generally, this type of phenomenon is useful in More generally, this type of phenomenon is useful in studying the interconversion of conformers.studying the interconversion of conformers.  b)b) Microwave spectrum of OMicrowave spectrum of O22

OO22 has no permanent dipole moment, but in has no permanent dipole moment, but in the electronic ground state has 2 unpaired the electronic ground state has 2 unpaired electrons with parallel spins:electrons with parallel spins:

OO22 11σσgg2 2 2 2σσuu

22 3 3σσgg22 1 1ππuu

44 2 2ππgg22