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IB Spectroscopy Techniques

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2019/03/15 178 Share

Overall picture

Scale

Type Energy scale / cm1cm1 EM wave
Electronic 10000 UV/VIS, X-ray
Vibration 1000 IR
Rotation 1 Microwave

Requirements

  1. Rotation (Microwave) Spectra

    • Need permanent dipole
    • x, y or z transforms as totally symmetric representation
  2. Infrared

    • Need change in dipole upon vibration
    • Corresponding normal modes transform like x, y or z
  3. Raman

    • Need change in polarisability
    • Complementary to IR
      1. Rotation: detectable except spherical caps (eg CHX4CHX4, SFX6SFX6)
      2. Vibration: corresponding normal modes transform like any coordinate squared term

Unit conversion

Just remember: wavenumber = 1/λ1/λ, everything follows

Energy E=hν=hc/λ=h˜c˜νE=hν=hc/λ=h~c~ν in Joule

But quantity in Joule is not equal to quantity in cm1cm1.

Rotation

Derived from the Schrödinger equation for rigid rotors.

Key quantities

Quantity Formula
Energy E=hBJ(J+1)E=hBJ(J+1)
Rotational constant B=22IB=22I in J
˜B=h8π2˜cI~B=h8π2~cI in cm1cm1
Degeneracy 2J+12J+1
  • JJ gives the amount of angular momentum
  • MJMJ specifies the direction of anuglar momentum

Form of spectrum

Intensity

Population of levels

Generally, population obeys Boltzmann distribution:

NJ=N0exp(h˜c˜BJ(J+1)kT)NJ=N0exp(h~c~BJ(J+1)kT)

This combined with degeneracy,

population(2J+1)exp(EJkT)population(2J+1)exp(EJkT)

Factors affecting intensity

  1. As stimulated emission can happen at the same time as absorption, the intensity depends on population differences (instead of population of the lower state).
  2. Transition dipole moment <ψ1|μ|ψ2><ψ1|μ|ψ2>.

Max populated level

Obtained by differentiation of the population expression. Result is JmaxkT2h˜c˜B12JmaxkT2h~c~B12.

Transition requirements

  1. Must have permanent dipole
  2. During transition JJ must change by ±1±1. This is due to the photon having 1 unit of angular momentum.

Polyatomic linear

Same form, just more complicated in calculating II.

Isotopic substitution

Substituting an isotope changes - obviously- the mass, but does not change bond length significantly. This is because:

  • The potential energy curve does not change
  • Only the ZPE changes (slightly), hence gives a very slightly different <r><r>, or <1/r2><1/r2> if calculating II.

Hence, the bond lengths can be calculated by solving simultaneous equations.

Rotation Raman

Raman

  • The electric field of the photon induces a dipole in the molecule
  • The molecule promoted to a virtual state
  • The molecule re-emits a photon

If:

hν=hvE=ERayleigh scattering{hν<hνE>EStoke scatteringhν>hνE<EantiStoke scattering

For Rayleigh, since photons are indistinguishable, we don’t know whether a photon is absorbed and re-emitted or straight through.

Selection Rule

ΔJ=0,±2

Example spectrum

Vibration spectrum

Harmonic oscillator

Key quantities

Quantity Formula
μ m1m2m1+m2
ν 12πkμ in Hz
ϵ (v+12)˜ω in cm1

Selection rule

Δν=±1

Anharmonic Oscillator

The Morse potential takes the form:

VM(x)=De[1exp(βx)]2
Quantity Formula
kM - spring constant 2Deβ2 obtained by Taylor expansion
Ev (v+12)ω(v+12)2ωxe in Joules - this is exact
ϵv (v+12)˜ω(v+12)2˜ωxe in cm1
xe β22μω
De Depth of well
D0 DeE0 Dissociation energy
β Steepness of well

The Ev curve is parabolic - the points beyond the energy start to decrease in energy, which are non-physical. vmax can be obtained by differentiating energy w.r.t v.

vmax=12xe12˜De=˜ω4xe

Transitions

  1. Must have change in dipole during vibration
  2. Selection rule is less strict, Δv=±1,±2,±3.... The curve is no longer symmetric, so integrals no longer vanishing.

Important transitions

  • Fundamental - 01 - strong
  • 1st overtone - 02 - weak
  • Hot band - 12 - weak, temperature dependent

Vibrating rotor

For a simultaneous change of both vibrational energy and rotational energy in a diatomic molecule:

Δv=±1,±2,±3...ΔJ=±1

Notations

Rot state for lower vib state J

Rot state for upper vib state J

Lines for ΔJ -2 -1 0 1 2
O P Q R S

Combination difference

As v changes, <r> (also <1/r2>) changes, I changes, hence B is different for upper and lower vibration states.

Nevertheless they can be solved by combination difference method - using transitions of the same upper state or same lower state.

  • The larger v, the more molecule vibrates, the smaller <1/r2>, the smaller B.
˜Bv=˜Be˜α(v+12)

where ˜Be is the hypothetical rotation constant with the equilibrium bond length at the bottom of well.

Graphical method

e.g.

˜RJ˜PJ=2˜B1(2J+1)

Polyatomic

Perpendicular and parallel vibrations

  • Stretching changes the dipole moment parallel to the principal axis
  • Bending changes the dipole moment perpendicular to the principal axis

For perpendicular vibrations, the ΔJ=±1 need not hold! This is due to vibrational angular momentum - when the two degenerate perpendicular modes combine in phase, a circular motion along the principal axis results.

Hence in linear, bending modes, Q branch is seen. Also, B1>B0 in bending modes.

Lack of Q branch in any one mode indicates the molecule is linear.

Electronic spectroscopy

Different electronic states give different PE curves.

Selection rules

The total transition dipole moment can be separated into electronic and vibrational parts.

Using the independence of the two parts, and the orthogonality of electronic wavefunctions, the expression reduces to

|<ψE|μE|ψE><ψV|ψV>|2

the second term is called the Franck Condon factor, which measures the overlap between the two vib states.

  • If the bond length changes, the FC factor is the greatest for v>>0; if the bond length is roughly the same, 0-0 transition is the most intense.

  • The bond length change is predicted by the bond-order of the electronic states.

Photoelectron spectroscopy

Conservation of energy

For atoms,

KE=hνIi

For molecules,

KE=hνIΔEvibΔErot

But note that removing an electron changes the energy of all the remaining electrons, hence not exactly the orbital energy.

Form of atomic spectrum

  • Peaks are labeled with the term symbols of the resulting ions.
  • Within a term, the relative energies of levels are according to Hund’s 3rd rule.
  • The degeneracies 2J+1 is a main contributor of relative intensities of peaks.

Spin-Orbit coupling

The size of spin-orbit coupling is seen on the PES spectrum. Two main factors

  1. The greater the nuclear charge Z, the larger size;
  2. The closer the electron to the nucleus, the larger the interaction.

Molecular spectrum

  • A progression is seen for each peak, representing the different vibration states in the resulting electronic state.
  • Within each progression, ˜ω and ˜ωxe can be determined, for that electronic state.

Bond order

We can deduce the change in bond order - or how bonding/antibonding is the orbital that the electron is removed from, through two features.

  1. Number of intense peaks. According to F-C factor, as in electronic spectra.
  2. The ω value of the resulting bond vibration. The stronger the resulting bond, the larger the gap between peaks in the progression.

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Author:2LalA猪

Permalink:https://butteraddict.fun/2019/03/IB-Spectroscopy/

Published on:15th March 2019, 10:30 am

Updated on:9th June 2020, 3:20 pm

License:CC BY 4.0

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    CATALOG
    1. 1. Overall picture
    2. 2. Rotation
    3. 3. Rotation Raman
    4. 4. Vibration spectrum
    5. 5. Vibrating rotor
    6. 6. Polyatomic
    7. 7. Electronic spectroscopy
    8. 8. Photoelectron spectroscopy