Concept 2. A. Description of light-matter interaction B. Quantitatities in spectroscopy

Transcription

1 Concept 2 A. Description of light-matter interaction B. Quantitatities in spectroscopy Dipole approximation Rabi oscillations Einstein kinetics in two-level system B. Absorption: quantitative description

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3 Description of light-matter interaction: Classically, the behaviour of a particle of charge q in an electromagnetic field (E (r; t); B (r; t)) is described by the Lorentz force: Schematic representation of monochromatic wave as oscyllating E and B fields

4 Formal description A quantum system under the inuence of external time-dependent electromagnetic fields (which we call external perturbations) is described by a time-dependent Hamilton operator H(t) The interaction of a molecular system with an electromagnetic wave can be described by the interaction of the electric and magnetic dipole moments with the electric and magnetic fields, respectively.

5 Representation of dipole moment of molecule and E, B components the electromagnetic wave Example:

6 Relevant interaction regimes dipole approximation The electric dipole interaction is the dominant interaction in the microwave, infrared, visible and ultraviolet ranges of the electromagnetic spectrum The magnetic dipole interaction is used in spectroscopies probing the magnetic moments resulting from the electron or nuclear spins such as EPR and NMR. At short wavelength, i.e., for X- and -rays, the dipole approximation breaks down because λ<< d.

7 Basic description of the interaction E 1 ψ 1 E 1 -E 2 =hv 10 Transition probability is obtained by solving the time-dependant Schrödinger equation: E 0 ψ 2 Ψ ih = [ H + V '] Ψ t H hamiltonian of the system V ' represents small perturbation = μe cos( ωt) Ψ = a ( t) ψ + a ( t) ψ 2 a 2 1, a 2 2 are probabilities to find the system( molecule) in the ground or excited states

8 Rabi frequency, Rabi oscyllations Solution of the Schrödinger equation define: - ω 12 - resonance frequency, Rabi frequency E 1 -E 2 =hv 10 =h/2π ω 10, the system is coherently cycled between the ground and excited states by the electromagnetic radiation. At resonance the system is completely inverted after a time t=π/ω 10. Ia 1 (t)i 2 ~ sin 2 (ω 10 t/2) Ia 0 (t)i 2 ~ cos 2 (ω 10 t/2)

9 - the integral M 01 =M 10 ; it the transition dipole moment and is the most critical factor in determining selection rules and line intensities - M 01 =M 10 = <ψ 1 Ι μ Ι ψ 2 > - the probability for finding the system in the excited state after a time t is given: P 1 0 = a 1 2 = ω Δ sin 2 Δt ( ) = 2 μ h sin 2 [( ω ω ( ω ω ) ) t 2 / 2] Lorentzian shape

10 Population? The atoms molecules can be found in various of their stationary states. At equlibrium the populations of the upper and lower states are related according to the Boltzmann distribution law: N a /N b = (g a /g b )exp (-ΔE ab /kt) (If the energy levels are degenerated)

11 Population and absorption Absorption disturbes equlibrium by increasing N a and decreasing N b. A. M+hν M* induced absorption Removing source of radiation, the equlibrium will be restored over the period of time by loss of the excitation energy. B. M* +hv M+2hν induced emission C. M* M+hν spontaneous emission

12 Einstein description A. Einstein, Physik Z. 18, 121 (1917)

13 Einstein descriptions The rate equations for processes 1-3 are: A system in thermal equilibrium with the radiation field at temperature T, where N 1, N 2 are the populations per unit volume in states 1 and 2, ρ(ν) radiation energy density. A, B are the important rate constants.

14 Einstein coefficients and spectroscopic transitions In the electric dipole approximation, the radiative transitions between the quantum states 1 and 2, or more generally between states i and f (i for initial, f for final) are correlated as follow: B fi =B if

15 Example 1: basis for development of laser The Einstein theory provide basis for development of lasers. These represent one of the main tools of research in spectroscopy and also in many areas of science and technology. 1950's: development of masers (microwave amplification by stimulated emission of radiation; NH 3 maser). 1960's: development of lasers (light amplication by stimulated emission of radiation).

16 Laser:

17 Example 2: Development of quantitative spectroscopic theory The mathematical description given by Einstein can be well correlatated with the measuring quantitities in spectroscopic experiment like the absorption measurements:

18 Quantities This subsection provides a short summary of the physical quantities used in the literature to quantify the interaction of light with matter. The measurement of the strength of this interaction can be used to determine the concentration of particles in a given sample (analytical chemistry) or to model the radiative processes taking place in a certain system (e.g. atmospheric chemistry).

19 Development of the analytical description The generic spectroscopic experiment can be described by the following scheme: I 0 and I represent the flow of photons per unit surface and time (i.e. the intensity of the light beam) before and after traversing the sample of concentration c i

20 The number of absorbed photons per unit length dx is proportional to the concentration c i of absorbing particles and to the intensity: where σ fi is the absorption cross section associated with the transition from state i to state f and depends on the frequency. Lambert-Beer law

21 Expressions for the absorption Used in the gas phase studies σ fi is absorption cross section and must have the unit of m 2 or, as often used, cm 2. can be interpreted as the surface of a completely absorbing disc which has the same absorption effect as the atom/molecule under investigation, ci number of species/cm 3 Used in the liquid and solid state studies ε fi the molar extinction coeffcient (molar absorptivity) usually expressed in dm 3 /(cm mol) and c i,m the concentration in mol/dm 3 ε=2.61x10 20 σ link between the cross section and molal extinction coeficient Sometimes one uses: α=σν the attenuation coefficient

22 Experimental vs. Theoretical coefficients Absorption can be viewed as a bimolecular collision between a photon and an atom/molecule (see script for details) general, c ph (ν if ) depends on the light source, and c ph ( ν fi ) represents the concentration of photons with frequency lying in the interval ν and ν+ δν

23 Concentration of photons as a function of the frequency

24 σ dependence on ν

25 Development of integral absorption cross section G G fi - the integral absorption cross section

26 Development of integral absorption cross section G If the line is narrow, the function 1/ν does not vary significantly over the line profile and can be approximated by 1/ν fi;max Mesured Absorption Band S fi the line strength

27 Experimental procedure to characterize the transition between i and f:

28 Comments

29 Linewidths characteristic properties of experimental absorption bands

30 Spectral intensities and band (line) widths

31 Intensities The maximum value of the absorbance corresponds to maximum value of the absorption cross section or molar absorption coefficient, ε max σ max or ε max an approximate measure of the total absorption intensity a more accurate measure is the area under the absorption curve

32 Line widths of transitions A transition between two states a and b is seen as reproducible line shape Contributing factors: Instrumental Other including homogeneous and inhomogeneous line broadening Natural broadening Doppler broadening Pressure broadening Wall collision broadening Power saturation broadening Modulation broadening

33 1. Natural line broadening Excited states which are populated in excess of their Boltzmann equlibrium decay as a function of time: t dn dt a = ln 2 k kn N k 1 / 2 = = = a a ( t) = N 0.693τ a (0)exp( kt) k- rate constant t 1/2 -half-life time τ=1/k, relaxation time, life time of the excited species

34 Natural line broadening The uncertainty ΔE of the energy of a state with life time t is given by the Heisenberg formel: τδe h It follows that the state would be a truly stationary only if the lifetime were infinite. τ is not infinite and the state is represented by energies smeared over a range ΔE. Contribution to τ due only to spontaneous emission is called the natural lifetime of the state

35 Lorentzian shape

36 Lifetime and absorption coefficient The natural lifetime τ is related to the A ab Einstein coefficient: Typical lifetimes: A = 1 τ = 64π ab M 3 fi 4πε 0 3hc 4 ν 3 2 Electronic state: 10 ns ---ΔE~ J Δv~ ~ cm -1, or 16 MHz Vibrational state: Δv~ 20 khz The natural line broadening is small relative to most other contributions. It contributes in an identical way by each atom or molecule and is an example of homogeneous broadening Rotational state: Δv~ 10-4, 10-5 Hz

37 2. Doppler broadening The frequency of the radiation absorbed during a transition in an atom or molecule differs according to the direction of motion relative to the source of radiation: ν a = ν res ϑa 1 c The distribution of velocities υ a in a gas phase has been deduced by Maxwell and results in broadening of a spectral line to a half width at half maximum given by Δν = ν 2kT ln 2 1/ [ ] c m 2 The Doppler effect is smaller for heavier molecules and can be overcome, by lowering the temperature. Molecular beams Lamb dip spectroscopy

38 Doppler broadening The lineshape, due only to the Doppler effect, is expressed by the equation α α max = p p c ν ( ) ν o 2 ν ν 0 exp[ ln 2 ( Δν ) 2 ] Gaussian shape α=a/l=εc - p,p c the absorption coefficient; pressure and pc is the pressure at which pressure broadening becomes significant; ν, ν 0 are frequencies, the latter is the frequency corresponding to the maximum absorption coefficient a max Because atoms/molecules with different velocities absorb and emit at different frequencies: inhomogeneous line broadening.

39 Gaussian shape

40 Doppler frequency shift

41 3. Pressure broadening If τ is the mean time between collisions in a gaseous sample and each collision results in a transision between two states a and b, there is a line broadening Δν: Δν=(2πτ) 1 The line shape (Lorentz, Debye, van Vleck & Weisskopf) At pressures of the order of 1 atm this high frequency tail on microwave transitions is readily observed. Quantitatively, it cannot be reproduced by the Van Vleck Weisskopf theory which neglects the effect of collisions between more than two atoms

42 Examples of the spectral bands

43 4. Wall collision broadening Collisions between the atoms or molecules of the sample and the walls of the cell broaden a transition in the same way as interparticle collisions. This effect, however can be eliminated: - large cell Difficulties can arise using microwave spectroscopy

44 5. Power saturation If the power of the incident radiation is high, the populations in state a and b becoming nearly equal. This is called saturation Under saturation conditions the absorption coefficient a is dependant on the intensity I of incident radiation α = α0(1 2I N a b ktt ) hν a 0 absorption coefficient at low intensity t= ½ t ab, t ab is probability per unit time that a molecule will transfer from state a to b by collisions Initially, this effects have played role in low frequency spectroscopies, (microwave spectroscopy). They are now apparent also in other regimes (laser spectroscopy)

45 Gaussian shape

46 Liquid Phase: broadening due to interaction with the environment

47 Other informations from lineshapes Radiationless processes: Photodissociation The rate of these processes is convoluted with other and one would need Additional experiments to clarify this contributions k = A + k + k + k [ M ] + ab radless photo coll...

48 Summary line broadening In most gas phase spectroscopy, line broadening is due to Doppler broadening and pressure broadening In microwave and mm-wave spectroscopy a typical half width at half maximum intensity is 1-10 khz due to pressure broadening at a pressure of only 1mTorr compared to about 10 khz due to Doppler broadening When inhomogeneous and homogeneous line broadening occurs, the line shape is a combination of gaussian and lorentzian Voigt curve.

49 Summary Description of light-matter interaction apart to understanding of the process, provide means for the quantitative description. Attenuation of light can be quantified and related to parameters derived from theory. Spectral band shapes can yield information of the conditions at which the molecule is present: velocity, pressure, decomposition stage.