Hippler Spectroscopy

Front 1

Equation for the speed of light

Back 1

Speed of light = wavelength x frequency

c = λv

Front 2

What is the range of visible light on the electromagnetic spectrum?

Back 2

400-800nm

Front 3

What is the wavenumber ()?

Back 3

1/wavelength: number of waves per unit distance (nm-1)

Front 4

Define frequency

Back 4

The number of peaks that pass a stationary point in a given amount of time

Front 5

Define wavelength

Back 5

On a periodic curve the distance between two troughs or peaks

Front 6

Light is made of photons, each of which carries energy. What is the equation for the energy of a photon?

Express in terms of wavenumber

Back 6

E=hv

But we know c=v λ

E = hc/ λ

E= hc

Front 7

What are the three fundamental processes where light interacts with a molecule?

Back 7

Light absorption: photon energy is absorbed by a molecule, the molecule is promoted from a lower energy ground state to a higher energy excited state level, where ΔE=hv

Spontaneous emission: If a molecule is in an excited state, it may after a while lose its excitation releasing energy in a photon with ΔE=hv

Induced emission: If a molecule is in an excited state and there is already a photon present with the right energy ΔE=hv, then it may be stimulated to lose its excitation releasing the energy in a photon with ΔE=hv. Essentially ditches one photon for another releasing energy.

Front 8

Explain Beer-Lambert law

Back 8

Light passes through an absorbing sample of concentration c, and absorption length l in m.

The initial light intensity I0 is weakened as it passes through the sample to give I.

I=I0exp(-σCl) where σ is the molecular absorption cross-section in m^2

Absorbance A = ln(I0/I)

Front 9

What is ɛ

Back 9

The decadic molar extinction coefficient in m^2 mol^-1 = σmol/ln(10) or σNa/ln(10)

Front 10

What leads to the appearance of an absorption peak?

Back 10

If the frequency of light is tuned, then at the position ΔE=hv0 corresponding to an energy transition in a molecule, light will be absorbed: at this position in the absorption spectrum we expect an absorption peak

Front 11

What are the reasons for line broadening?

Back 11

1) A spectrometer only has a limited instrumental resolution, meaning that the light that falls on the detector is not truly monochromatic with v0 but has a distribution around v0 because of the resolving power of the instrument.

2) May not be a single line but actually be thousands of lines giving the impression of one broad line - 'spectral congestion'

3) Doppler effect:

Light may shift frequency because of Maxwell-Boltzman distribution of different velocities at a temperature T there is a spread of frequencies around v0

4) Lifetime broadening :

According to Heisenberg's uncertainty principle if the excited state has a limited lifetime then its energy is not perfectly defined. Distribution of excited state energies and therefore a distribution of frequencies around v0 in the spectrum.

Front 12

What can be determined by measuring the Doppler line shape?

Back 12

The temperature T

Δv / v0 = √T/m

Where m = mass in u, Δv=full width at half maximum

Front 13

What can be deduced from a Lorentzian line shape?

Back 13

If the excited state shows an exponential decay time law (e.g. 1st order kinetics) then there is a Lorentzian line shape (like a bell shape but broader foot) with =1/2πΔv

is the mean lifetime of the excited state

Front 14

All information about a molecule is contained in the wavefunction Ψ, which is the solution to the Shrodinger equation ĤΨ = EΨ.

hard to solve as Ĥ and Ψ depend on many variables.

How are these variables separated?

Back 14

1)External centre-of-mass motion of the molecule (translation) is approximately separated from the internal motion (rotation, vibration, electronic excitation)

2) Rotation of a molecule does not influence vibrations greatly

3) Electrons are very light and fast, nuclei are slow and heavy. Movement of electrons and nuclei thus separate, only position matters. These positions adjust according to the electrostatic mean forces but this adjustment is not influences by the movement.

Front 15

How can Ĥ be separated into a sum of contributions, which depend on different degrees of freedom

Back 15

Ĥtotal = Ĥ(el) + Ĥ(vib) + Ĥ(rot)

Front 16

What does this tell you about Ψ?

And about E?

Back 16

Ψ is a product of Ĥ,

Ψ = ΨelΨrotΨvib

where

ΨelĤel = EelΨel

ΨvibĤvib = Evib Ψvib

Ĥrot Ψrot = Erot Ψrot

E= Eel + Evib + Erot

Front 17

What is the order in magnitude in energy change for rotational excitation, electronic transitions and vibrational transitions??

Back 17

ΔErot<<ΔEvib<<ΔEel

Front 18

What are the 'handles' for the electromagnetic field that lead to light absorption?

Back 18

Electric charge distribution - electric dipole moment, electric quadrupole moment

Magnetic moments - magnetic dipole moment, magnetic quadrupole moment

Front 19

What is the transition probability?

Back 19

The probability of a particular spectroscopic transition to take place

Front 20

What two things does the probability of an atom or molecule to transit from one energy level to another depend on?

Back 20

The nature of the initial and final state wavefunctions and the strength of the photon interaction with an eigenstate

Front 21

What is the equation for the transitional probability

Back 21

M⃗ 21=∫Ψ2μ⃗ Ψ1dτ

Front 22

What is μ⃗ ?

Back 22

The operator for a transition, where

μ⃗ = Σ(qi)r The electric dipole moment: where q is the ith charge of the nucleus

μ⃗ =Σ(γi)Li The magnetic dipole moment operator: where γ is the gyromagnetic ratio of the ith particle with angular momentum L

Front 23

What has a stronger interaction - electric or magnetic dipole moments?

Back 23

Electric

Front 24

What are the selection rules?

Back 24

Selection rules determine whether or not electronic transitions are allowed.

If the matrix is not = 0, then transition from Ψ1 to Ψ2 is allowed, if it =0 it is forbidden.

Front 25

How can atoms be described in terms of quantum numbers?

Back 25

By their primary quantum number n

Angular momentum quantum number L

Spin quantum number S

Total angular momentum quantum number J

Front 26

What are the selection rules for electronic transitions in terms of quantum numbers?

Back 26

Total spin cannot change, ΔS=0

Change in total orbital angular momentum can be ΔL=0, ±1, but L=0 ↔ L=0 transition is not allowed

Change in total angular momentum can be ΔJ=0, ±1, but J=0 ↔ J=0 transition is not allowed

Initial and final wavefunctions must change in parity.

Front 27

What is the transition matrix element?

Back 27

Transition matrix element is just the name for an integral, the integral over initial wavefunction x transition moment operator x final wavefunction. If this integral is 0, then the transition from initial to final is not allowed, if >0 then allowed. The transition moment operator is usually just the electric dipole moment of the molecule, polarity x distance.

Front 28

What is orbital angular momentum?

Back 28

The component of angular momentum of an electron or a nucleon in a nucleus arising from its orbital motion rather than its spin

Front 29

What is the spin quantum number?

Back 29

The Spin Quantum Number (s) is a value (of 1/2) that describes the angular momentum of an electron. An electron spins around an axis and has both angular momentum and orbital angular momentum.

Front 30

What is parity?

Back 30

Parity is what happens when you inverse all the coordinates. i.e. x--> -x, y--> -y and z--> -z

Front 31

Describe parity in terms of wavefunctions

Back 31

A wavefunction has even parity (∏ =1) if Ψ-->Ψ

A wavefunction has off parity ( ∏ = -1) if Ψ--> - Ψ

Front 32

Explain why the parities of Ψi and Ψf must be different for an electric dipole transition to be allowed

Back 32

The electriv dipole moment operator has odd parity r-> = -r->

The matrix element is a number, under inversion it remains the same number: the matrix element has therefore either even parity or is 0

Front 33

What is Raman spectroscopy?

Back 33

The phenomenon of Raman spectroscopy is the scattering of light. Raman scattering is most easily seen as the change in frequency for a small percentage of the intensity in a monochromatic beam as the result of coupling between the incident radiation and vibrational energy levels of molecules. A vibrational mode will be Raman active only when it changes the polariazbility of the molecule.

Front 34

What are stokes and antistokes lines?

Back 34

they give a direct measure of vibrational energy of a molecule.

Intense monochromatic light is scattered from the sample. This will give the incident frequency vi. There will be additional components, Stokes frequency Vs and Antistokes frequency Vas. Antistokes have a higher frequency, than the incident, stokes lower. The difference in Δhv is related to the molecular energy levels of scattering species.

Front 35

What is the spontaneous Raman effect?

Back 35

Normal 'spontaneous' emission, where the photon energy exactly matches the electronically excited upper level.

If light is intense photon energy does not need to match upper level exactly.

For Stokes lines the photon terminates at a higher energy than before, red shifted.

Front 36

How do anti-Stokes lines from?

Back 36

From a transition from an excited level to ground state transition. Finishes at lower energy than before, blue shifted.

Front 37

Explain when and why a molecule is Raman active

Back 37

If the rotation/vibration changes polarizability:

Light scattering is connected to the polarizability of a molecule. The oscillating electric field of incoming light distorts the charge distribution of the molecule and causes it to change frequency to the same as that of the driving light field. The molecule will emit light of this frequency.

If the molecule rotates or vibrates and the polarization is modified; the polarization contains frequency of the driving light field and the rotation/vibration. It will therefore emit light of the driving light field and at frequencies plus/minus the rotation/vibration (Raman scattering, Anti-stokes and stokes lines)

Front 38

At room temperature many ground-state levels are thermally populated, from which different spectral transitions may start. This leads to spectral congestion and complicated spectra. What is a possible solution to this?

Back 38

Supersonic jet cooling - adiabatic expansion of a gas pulse into a vacuum; cools molecules down to few K and simplifies spectra

Front 39

What is the idea behind supersonic jet cooling?

Back 39

Before the nozzle there is broad thermal velocity distribution at 300K.

The nozzle opens and closes allowing only a small velocity slice to escape into the vacuum.

By collisions between molecules, this velocity slice assumes a new thermal equilibrium, say 5K

Front 40

How is wavelength of light measured?

Back 40

Not by a prism or grating but by a Fourier transformation

Front 41

What is a Fourier transformation?

How does it work?

Back 41

Measures wavelength of light.

There is a white light source that's split in a beam splitter into two parts, with both beams being reflected back by two different mirrors into a detector. One of the mirrors is moving (displacement x) which changes the interface pattern (inteferogramm) (cosine pattern), I(x)

Front 42

In a Fourier transformation what is the cosine patter or a monochromatic light source with wave number

Back 42

Cosine pattern: I(x) =[1+cos(2πx)]

Front 43

What effect does moving the mirror give in a Fourier transformation?

Back 43

Sin wave ---> one frequency peak (monochromatic)

Mix of waves ---> frequency + intensities of white light

Front 44

Explain why the wavenumbers will be superimposed on a detector interferogramm when a white light source is shone in a Fourier transformation?

Back 44

Each wavelength (cosine) has its own wavenumber, white light contains loads of wavenumbers hence they will all superimpose.

Front 45

What does a Fourier transformation give overall?

What is the advantage of this?

Back 45

An intensity distribution I() as a function of wavenumber

All light intensity is used at once for measurement

Front 46

What does LIDAR stand for?

Back 46

Light Detection And Ranging

Front 47

What is the idea behind LIDAR?

Back 47

An intense laser light pulse (10ns typically) is sent into the sky. Some of the light is scattered back (e.g. Rayleigh scattering, scattering on water droplets or smoke particles etc.), and is detected with a sensitive telescope or photomultiplier as a function of time.

From the time difference between sending the laser pule and receiving the scattered light it can be calculated at which distance along the laser beam path the light has been scattered, and from the intensity of the scattered light the concentration of scattering particles (clouds) at a particular location can be derived. Gives the ceoncentration profiles along the laser beam

Front 48

What does DIAL stand for?

Back 48

Differential Absorption LIDAR

Front 49

What is the idea behind DIAL?

Back 49

Consider the black light scattering of LIDAR as a light source. Its absorption on the way back to the receiver (photomultiplier) indicates the concentrations of absorbing species along the beam path.

From time difference between sending light pulse/ receiving back scattered light (with absorption losses), the concentration profile of absorbing species can be obtained after a deconvolution

Front 50

What are the hierarchy of energy levels within a molecule?

Back 50

Small steps: Rotational excitation (rotational transitions - microwave spectroscopy)

Medium steps: Vibrational energy levels (vibrational transitions - IR spectroscopy)

Big energy steps: Electronic excitation (electronic transitions - UV-vis spectroscopy

Front 51

What are the three steps for setting up the quantum mechanical shrodinger equation?

Back 51

1) Write down the classical energy E

2) Replace all measurable quantities by operators; this gives the Hamiltonian operator Ĥ

3) Shrodinger equation ΨĤ= EΨ; solve for allowd energy levels E, and for stationary Ψ

Front 52

What is the consequence of a linear, rigid rotator?

Back 52

The distance between atoms does not change by the rotation

Front 53

What is the equation for the classical energy of a rigid, linear rotation?

Back 53

E = (1/2)Iω^2 = ( J^2 )/ 2I

Where:

J = angular momentum

I = moment of inertia

ω = angular velocity

Front 54

What is the inertia (I) for a linear molecule?

Back 54

Σmiri^2

Where:

ri is the distance of the ith atom with mass mi from the centre of mass

Front 55

What is the intertia for a special case diatomic molecule?

Back 55

I = μr^2

Where r is the distance between the two atoms and μ is the reduced mass

Front 56

How can you manipulate the expression for classical energy ( J^2 )/ 2I to get the Hamiltonian operator?

Back 56

Substitute angular momentum J for the corresponding operator J^

Ĥ = ((J^)^2) / 2I

Front 57

Rotational energy levels are normally given in cm^-1. how can this be converted into joules?

Back 57

By multiplying by hc

Front 58

What are the rotational energy levels called and how are they denoted?

Back 58

Rotational term values, denoted by F

Front 59

What is the equation for the rotational term values F as a function of J?

Back 59

F(J) = BJ(J+1)

Where F is the rotational energy in cm^-1

B is the rotational constant in cm^-1

J is the angular momentum quantum number, and for a normal number is an integer, but for a radical may also contain the unpaired electron spin e.g. 0.5, 1.5 etc.

Front 60

What is the equation for calculating the rotational constant (cm^-1) B?

What can be deduced from it?

Back 60

B = h/8π^2c x 1/I

Note that B depends inversely on the moment of inertia, thus if B is know I can be determined, and therefore so can the intramolecular distance r

Front 61

What are the two important things that affect non-rigid rotators?

Back 61

Vibrations and centrifugal distortion

Front 62

What is the effect of vibration on a non-rigid rotator (e.g. a diatmoic molecule?)

Back 62

Bond length r changes periodically, which in turn changes moment of inertia I and the rotational constant B. However, typically hundreds of vibrational period before a molecule completes one rotation, therefore only an average rotation is seen.

Each vibrational state has its own effective (average) Bv

Front 63

What is a good approximation of Bv for a vibrational state of a non-rigid rotator?

Back 63

Bv = Be - a (v+ 1/2) + ..

Where v is the vibrational quantum number

Be is the rotational constant without rotation

And ae is a constant (usually positive)

Front 64

What is centrifugal distortion?

Back 64

Due to the rotation centrifugal forces push atoms apart --> R becomes larger therefore B becomes smaller.

Front 65

What is the equation for the rotational term value for a non-rigid rotator taking into account the effect of vibration and centrifugal distortion?

Back 65

Fv (J) = BvJ (J+1) - Dv J^2 (J+1)^2

where Dv is the centrifugal distortion constant

Front 66

What graph gives a straight line with slope -Dv (centrifugal distortion constant) and intercept Bv (average rotational constant)

Back 66

Fv/ J(J+1) vs J(J+1)

Front 67

How are transitions (light absorptions) induced?

Back 67

The electromagnetic field shakes the electric charge distribution of a molecule by its periodic electric field; the electric dipole moment of the molecule is the main handle for the electromagnetic field

Front 68

What are the selection rules for IR/microwave absorption spectroscopy?

Back 68

If transitional probability (transitional matrix element) equals zero --> transition electric dipole forbidden.

if TME equals anything but 0, electric dipole allowed

Front 69

What are the two problems with calculating the TME?

What are their solutions?

Back 69

1) Atomic distances ri change periodically during vibrations =>μ⃗ = μ⃗ (ri)

As there are many vibrations in one rotation of the molecule these effects can be averaged over the vibrational periods as done for Bv.

Can use an effective, averaged electric dipole operator: μ⃗ v = [Ψv(ri) |μ⃗ (ri)| Ψv (ri)]

2) μ⃗ depends on 'lab coordinates' x,y,z because the direction of light propagation and the polarisation of light all refer to the light/source or the laser which is sitting in the lab. The wavefunction however, depends on internal molecular coordinates x', y' and z' which rotates all the times with regard to the lab.

Therefore have to rotate μ⃗ into internal, molecular coordinates by multiplying with a rotational matrix D with time-dependent orientation of the molecule relaive to the lab

=μ⃗ (internal) [Ψi|D|Ψf]

Front 70

When is the transition matrix element not equal to zero?

I.e. what are the selection rules for IR/microwave absorption rotational spectroscopy

Back 70

1) The molecule must have μ⃗ (internal) not = 0

(therefore no pure rotational transitions for homonuclear diatomic molecules like N2, O2)

2) ΔJ = +/-1 (That is the rotational quantum number has to change by 1)

3) For an electric dipole transition to be allowed the parity must change

Front 71

What must be noted about the selection rule

ΔJ = +/-1?

Back 71

This is only valid for pure rotational transitions. For transitions where the vibrational state is also changing (rovibrational transitions) ΔJ may also be 0.

Front 72

Name transitions where:

ΔJ = -1

ΔJ = 0

ΔJ = +1

Back 72

P lines (e.g. from J=3 to J=2): P(J)

Q lines: Q(J)

R lines: R(J) lines

J in the parenthesis donates the J of the ground state

Front 73

What does R(4) mean?

Back 73

CHECK: Transition is from J=4 to J=5

Front 74

What can we deduce from the analysis of a rotational spectrum?

Back 74

we can determine B, from which we can determine the bond length R

Front 75

What are the selection rules for Raman rotational spectra?

Back 75

Raman transitions are like two one-photon transitions. Selection rules are of two consecutive one-photon transitions.

1) ΔJ = 0, +/-2 (when ΔJ=+2 called S lines, when =-2 called O lines)

2) For an electric dipole transition to be allowed the parity of the wave function must not change

3) A permanent dipole moment is not required; the polarizability must change during a rotation

This allows observation of rotational molecules (e.g. N2 or o2) that are not normally observed by absorption spectroscopy

Front 76

Why are there no rotational levels with even J in a rotatational Raman spectrum for CO2?

Back 76

The wavefunctions of the O atoms cannot change in any respect. When the molecule is rotated by 180 degrees, the rotational wavefunction will change sign by (-1). Therefore only even J values are permissible and the Raman spectrum shows only alternate lines

Front 77

What is the Pauli exclusion principle in terms of bosons?

Back 77

When two identical bosons are exchanged, the overall wavefunction must remain unchanged in every respect, including sign

Front 78

In homonuclear diatomic molecules the 2 nuclei are indistinguishable. What are the consequences of this?

Back 78

If two nuclei are exhancged we have the same physical situation: Ψ^2 ---> Ψ^2

This gives two possibilities:

1) Ψ---> -Ψ (Antisymmetric, Fermions; half integral spin)

2) Ψ ---> +Ψ (Symmetric, Bosons; integral spin particles)

Front 79

Why are there 3 times as many ways of achieving a state with odd J than even J with a corresponding 3:1 intensity alteration in their rotational Raman spectra for H2?

Back 79

Hydrogen nuclei have a spin of I=1/2

Number of ways of achieving odd J = (I+1)/I = 3

Gives a ratio of 3:1, therefore 3 ways of achieving odd J and one for achieving even J

Front 80

Justify why there is a 3:1 ratio of odd J to even J for Hydrogen nuclei with spin I=1/2

Back 80

As the spin is equal to 1/2 the spins can either be paired to give I(total) =0 or parallel to give I(total) = 1.

If they are parallel possible spin states are: a(1)a(2), b(1)b(2) and a(1)b(2) + b(1)a(2).

Where nuclei 1 and 2 have spin up (a) and spin down (b) combinations.

For a(1)a(2) and b(1)b(2) when the molecule rotates 180 degrees rotational wavefunction is unchanged, hence only odd values for J allowed.

When a(1)b(2) + b(1)a(2) is rotated 180 the wavefunctions stay the same as all that changes is the order of terms. J still odd.

When the spin is paired wavfunction is a(1)b(2) - b(1)a(2). Therefore for the overall wavfunction to change sign the rotational wavefunction must NOT change sign. Only even J allowed.

Front 81

What are Hydrogens showing the paired and parallel spins called respectively?

Back 81

Paired(one up one down) = para-hydrogen

Parallel = ortho-hydrogen

Front 82

Why does ortho-hydrogen continue to rotate at very low temperatures and have an effective zero-point energy?

Back 82

Because J cannot be equal to zero as the rotational wavefunction must change sign so total wavefunction changes sign.

Front 83

What is the consequence of Hydrogen having a 3:1 ratio of even to odd J?

Back 83

The rotational spectrum of H2 has line intensities which alternate with the weight 1:3 depending on whether the transitions start from an even J or odd J, as Para-Hydrogen more statistically likely

Front 84

Why do no J=even exist of O2?

Back 84

16O has spin I=0, it is a boson (as opposed to a fermion where I=1/2)

Therefore only one possible orientation in space.

Electronic ground state of O2 is unsymmetrical Ψ(elec) = -1

for bosons overal wavefunction must be symmetric so:

Therefore Ψrot must equal -1 so overall wavefunction can be symmetric. Therefore only J odd can exist

Front 85

What can you automatically deduce is you see a strange alternating rotational Raman spectrum?

Back 85

Answer is something to do with Pauli principle

Front 86

What can be deduced about the Raman spectrum if there are two identical, indistinguishable nuclei

Back 86

There will be in general 'strange' intensity alternations of rotational lines, since different J levels have different nuclear spin statistics/weights, some J levels may even be missing. This is a consequence of the Pauli principle

Front 87

What is different for the angular moment vector J-> for linear and non-linear molecules?

Back 87

For linear molecules J-> = I ω->

Where I takes into account multiplication only, angular momentum always has same direction and angular velocity.

For non-linear molecules J-> = jw-> (j is moment of intertia in a matrix)

where j takes into account multiplication and rotation, angular momentum does not have same direction and angular velocity.

Front 88

What is the effect on the intertia of a molecule if it is non-linear?

Back 88

Has three principle axes A,B and C. Gives IA, IB and IC

Front 89

Why is the soltution for

F(J)= BJ(J+1) not straightforward for non-linear molecules?

Back 89

Because there are three rotational constants; A, B and C

A=h(8π^2c) * 1/IA

B=h(8π^2c) * 1/IB

C=h(8π^2c) * 1/IC

Front 90

What are the three different types of non-linear rotators and what are their differences?

Back 90

Spherical tops: All I are equal IA=IB+IC

- e.g. molecules with 2 or more 3/4-fold symmetry axes CH4, SiH4, SF6

Symmetric tops: 2 I are equal (there is one unique axis and two degenerate axes)

-e.g. molecules with one 3/4-fold symmetry axes

IA=IB (<IC) oblate top - benzene, CF3H

(I<A)IB+IC prolate top (egg): CH3F

Asymmetric tops: All I are different

Molecule may have 2-fold symmetry axis, e.g. H2O, O3

Front 91

What is the difference between a prolate top and an oblate symmetrical tops?

Back 91

An oblate top has a unique rotational axis with a greater inertia than the degenerate axes.

A prolate top has a unique rotational axis that has a lower inertia than the degenerate axes.

Front 92

What are the selection rules for symmetric tops?

Back 92

μ permanent

ΔJ = +/-1

Change in parity

ΔK = 0 (K is the projection quantum number of angular momentum J along the principle axis

Front 93

Which is the principle axis for a symmetrical top?

Back 93

The one with different I, known as the figure axis

Front 94

What is a rigid rotor?

How are they classified?

Back 94

A body that does not distort under the stress of rotation.

Classified by noting the number of equal principles of inertia

Front 95

Define moment of intertia of a molecule

Back 95

The mass of each atom multiplied by the square of its distance from the rotational axis passing through the centre of mass of the molecule

I = Σmixi^2 : where xi= perpendicular distance of atom i from axis of rotation

Front 96

Where can I find a good full explanation about rigid rotators?

Back 96

Pages 449 - 454 of Atkins

Front 97

Why do spherical tops such as CH4 and SF6 have no pure rotational spectrum?

Back 97

Due to their high symmetry, leads to no permanent dipole moment μ(permanent), therefore fails selection rules

Front 98

What is rotational spectroscopy used for?

Back 98

Can analyse line positions to obtain rotational constants which contain distances of atoms in molecules

Front 99

What are the two components of classical vibrational energy E?

Back 99

Kinetic energy

Potential energy as a function of r bond distance)

E = p^2/2μ + V(R)