Chapter 2. Electromagnetic Radiation and its Interaction with Atoms and Molecules 2.1 Electromagnetic Radiation Ey = A sin(2πνt - kx) Eq. (2.1) Hz = A sin(2πνt - kx) where, k = 2π/λ wbt 1 2.2 Absorption and Emission of Radiation ΔE = En - Em = hν = hcν~ wbt Eq. (2.2) 2 There are 3 processes which occur in this two-state system: Eq. (2.3) 1. induced absorption M + hν Æ M* 2. spontaneous emission Eq. (2.4) M* Æ M* + hν 3. induced emission M* + hν Æ M + 2 hν Eq. (2.5) For an induced absorption process (from state m to state n) Rate of change of population Nn of state n dN n dt = N m Bmn ρ (ν~ ) Eq. (2.6) where Bmn is Einstein coefficient for induced absorption ρ (ν~ ) is spectral radiation density 3 ~ 8πhcν ~ ρ (ν ) = exp(hcν~ / kT ) − 1 wbt Eq. (2.7) 3 For an induced emission process (from state n to state m) dN n ~) Eq. (2.8) − N B ρ ( ν = n nm dt Note: Einstein coefficient for induced emission Bnm = Bmn For a spontaneous emission process (from state n to various lower states) dN n dt = − N n Anm Eq. (2.9) where Amn is Einstein coefficient for spontaneous emission In the presence of radiation, all 3 processes occur dN n dt wbt ~) ( N N ) B ρ ( ν =0= − N n Anm m n nm Eq. (2.10) 4 At equilibrium, Nn and Nm are related through the Boltzmann distribution law Nn gn ∆E ∆E = exp(− ) = exp(− ) Nm gm kT kT Eq. (2.11) gn and gm are the degrees of degeneracy. Recall that E = En – Em = hν = hc ν~ Assuming gn = gm, putting Eqs (2.11) and (2.7) into Eq. (2.10), one gets Anm = 8πhcν~ 3 Bnm Eq. (2.12) This indicates that as ν~ increases, spontaneous emission increases rapidly relative to induced emission wbt 5 Sample calculation 2.1 For temperatures of 25 and 1000ºC, calculate the ratio of molecules in a typical excited rotational, vibrational, and electronic energy level to that in the lowest energy level, assuming that the levels are 30, 1000, and 40000 cm-1, respectively, above the lowest energy level. (Three significant figures are sufficient. Assume that, for the excited rotational level, the rotational quantum number J is 4 and remember that each level is (2J+1)-fold degenerate. Assume that the vibrational and electronic energy levels are non-degenerate.) Possible solutions: Form Eq. (2.11), we consider rotational level J =4, Nn/Nm = NJ/N0 = [(2J+1)/1]•exp(-hcν~/kT) = 9•exp[-(6.626 x 10-34J•s x 2.998 x 1010 cm•s-1 x 30 cm-1)/(1.382 x 10-23 J•K-1 x 298 K)] = 7.79 at 25ºC (298 K); wbt Nn/Nm = 8.70 at 1000ºC (1273 K). 6 For vibrational level v, Nv/N0 = exp[-(6.626 x 10-34J•s x 2.998 x 1010 cm•s-1 x 1000 cm-1)/(1.382 x 10-23 J•K-1 x 298 K)] = 8.01 x 10-3 at 25ºC (298 K); Nv/N0 = 0.323 at 1000ºC (1273 K). For electronic level e, Ne/N0 = exp[-(6.626 x 10-34J•s x 2.998 x 1010 cm•s-1 x 40000 cm-1)/(1.382 x 10-23 J•K-1 x 298 K)] = 1.40 x 10-84 at 25ºC (298 K); Ne/N0 = 2.35 x 10-20 at 1000ºC (1273 K). wbt 7 Bnm (Einstein coefficient for induced processes) can be related to the wavefunctionsΨm and Ψn through the transition moment Rnm Rnm = ∗ ψ ∫ n µψ m d τ Eq. (2.13) µ is the electric dipole moment operator µ = ∑ qi ri Eq. (2.14) i where qi and ri are the charge and position vector of the ith particle (electron or nucleus) R nm 2 Bnm = wbt is the transition probability and is related to Bnm by 8π 3 nm R (4πε 0 )3h 2 2 Eq. (2.15) 8 2.3 Line Width An absorption line is not infinitely narrow. A typical line exhibits a half-width at half maximum (HWHM) of ∆ν. wbt 9 2.3.1 Natural Line Broadening If the species M* in state n decays to the lower state by a first-order process dN n = kNn Eq. (2.22) dt where k is the 1st-order rate constant τ 1 k =τ Eq. (2.23) is the life time of state n If spontaneous emission is the only process by which M* decays, then k = Anm Eq. (2.24) wbt 10 Heisenberg uncertainty principle: τ∆Ε ≥ ħ, where ħ = h / 2π Eq. (2.25) Since τ is finite, all energy levels are smeared out to some Extent with resulting line broadening. To relate Anm (Einstein coefficient for spontaneous emission) 2 to the transition probability R nm , one combines Eqs. (2.12) and (2.15) and gets 64π 4ν 3 nm 2 Anm = ( 4πε )3hc 3 R 0 Thus, Eq. (2.25) becomes 32π 3ν 3 nm 2 ∆ν ≥ R 3 ( 4πε 0 )3hc wbt Eq. (2.26) Eq. (2.27) 11 Since ∆ν is dependent on ν3, ∆ν is much larger for an excited electronic state than an excited rotational state. ∆ν is ~ 30 MHz for an excited electronic state ~ 10-4 – 10-5 Hz for an excited rotational state Eq. (2.27) illustrates the natural homogeneous line broadening, which results in a Lorentzian line shape. In general, the natural line broadening is very small compared to other causes of spectral line broadening. wbt 12 Gaussian and Lorentzian lineshapes of the same FWHM wbt 13 2.3.2 Doppler Broadening If a molecule is traveling towards the detector with a velocity υa, then the observed frequency νa is related to the actual frequency ν in a stationary molecule by νa = ⎛ υa ⎞ -1 ν ⎜1 − ⎟ c⎠ ⎝ Eq. (2.28) Because the velocities of molecules exhibit a Maxwell distribution, it gives rise to an inhomogeneous broadening with a line width Δν Δν = ν c 1/2 ⎛ 2kT ln 2 ⎞ ⎜ ⎟ m ⎝ ⎠ Eq. (2.29) This Doppler broadening results in a Gaussian line shape. wbt 14 2.3.3 Pressure Broadening If a gas-phase spectroscopic experiment is performed with a cell, molecular collisions may take place, leading to so-called pressure broadening with a line width Assume that τ is the mean time between collisions and each collision results in a transition, the Heisenberg uncertainty principle (Eq. 2.25) gives Δν = (2πτ)-1 Eq. (2.29) This pressure broadening causes a Lorentzian line shape, similar to that of homogeneous natural line broadening. wbt 15 2.3.4 Removal of Line Broadening 2.3.4.1 Effusive Molecular Beams An effusive molecular beam is produced by pass the gas at a high pressure (1-3 atm) through a tiny hole into a vacuum (~10-6 Torr) system. This isentropic expansion process quenches the internal rotations and vibrations of molecules. It results in a narrow spectral line. Since the rotational cooling is more efficient than the vibrational cooling, scientists often referred to as a rationally cooled process. Pressure broadening of spectral lines is removed in an effusive beam. If observations are made perpendicular to the direction of the beam, Doppler broadening is considerably reduced because the velocity component in the direction of observation is very small. wbt 16 Fluorescence excitation spectrum of NO2: (a) conventional gas cell, room temp. 0.04 Torr, (b) molecular beam, (c) seeded molecular beam, 5% NO2 in Ar. Ref.: Smalley, et al. J. Chem. Phys. 61, 4363 (1974). wbt 17 Ref.: Smalley, et al. J. Chem. Phys. 64, 3266 (1976). wbt 18 Ref.: Smalley, et al. J. Chem. Phys. 64, 3266 (1976). wbt 19 2.3.4.2 Lamb Dip Spectroscopy If a narrow light source (e.g. laser) is tuned to a frequency νa which is higher than the resonance frequency νres at the line center, only molecules like 1 and 2 which have a velocity component va away from the source in Fig. 2.6 absorb radiation. wbt 20 On the return journey of the radiation back to the detector, molecules like 6 and 7 which have a velocity component –va away from the source absorbed at –va. When the radiation is tuned to νa , the number of molecules in the lower state m of the transition with velocity component va away from the source is depleted. This is referred to as hole burning. Molecules like 3, 4, and 5 which have zero velocity component away from the source absorb radiation at vres regardless the radiation travels towards or aways from R, resulting in “saturation”. The result is a dip in the absorbance curve as seen in Fig. 2.5 and is known as a Lamb dip which marks an accurate value of vres. wbt 21

1/--pages