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Problem sets 09/20/2008 1) True or False Rayleigh scattering is 16 times stronger at 400nm than at 1600nm? T/F The intensity of Raman scattered light is proportional to the incoming light intensity, and about 10-6 times that of Rayleigh scattered light? T/F Elastic Raman scattered light provides a useful signal for analytical purposes? T/F Anti-Stokes lines in vibrational Raman Spectroscopy are generally about 10 times stronger than the Stokes lines? T/F The intensity of Anti-Stokes lines in rotational Raman Spectroscopy is comparable to that of the Stokes lines? T/F The Raman shift describes the difference in energy of an incoming and scattered photon; it is defined positive if the scattered photon is of higher energy than the incoming photon? T/F The first vibrational overtone band above the vibrational ground state is characterized by the vibrational quantum number v=2? T/F The integral strength of a transition line only depends on the square of the transition dipole moment and the line position? T/F Natural lifetime broadening gives rise to a Lorentzian line shape? T/F Pressure broadening gives rise to a Gaussian line shape? T/F Doppler broadening give rise to a Lorentzian line shape? T/F A Voigt line shape is a convolution of a Gaussian and Lorentzian line shape? T/F At atmospheric pressure, the primary broadening mechanism determining the width of an absorption line in the ultraviolet spectral range is Doppler broadening? T/F At atmospheric pressure, the primary broadening mechanism determining the width of an absorption line in the far infrared spectral range is Doppler broadening? T/F For comparable integral absorption cross section, a Lorentzian shaped transition line has a higher peak absorption cross section than a Gaussian shaped transition line? T/F O(2) transitions form part of any IR absorption spectrum? T/F P(2) transitions form part of any IR absorption spectrum? T/F S(2) transitions form part of any Raman spectrum? T/F O(0) transitions form part of any Raman spectrum? T/F S(0) transitions form part of any Raman spectrum? T/F 2) Selection rules of IR absorption spectroscopy and Raman Spectroscopy. 2a) The IR absorption spectrum of HCl gives rise to a P, Q, and R branch. What are the selection rules that govern each of these branches in the mid IR and far IR spectral region? 2b) List the quantum numbers that characterize the ground and the excited state of the P(2), and R(3) transition in the mid IR absorption spectrum of HCl. 2c) Why can there be no P(0) transition, but there is a R(0) transition? 2d) The Raman spectrum of O2 gives rise to an O, Q and S branch. What are the selection rules for each of these branches in the mid IR and far IR spectral region? 2e) List the quantum numbers that characterize the ground and the excited state of the O(1), Q(2) and S(5) transition in the mid IR absorption spectrum of HCl. 2f) Why is there a S(0) transition, but no O(0) and no O(1) transition? 3) The infrared spectrum of N2O has three fundamental bands. 3a) Assuming that the structure of N2O is linear, explain how this spectrum allows you to distinguish between N-N-O and N-O-N. 3b) Sketch the approximate atomic motions of the normal modes. 4) A 10-cm long cell containing 10 mTorr of H35Cl is prepared at 296K. Consider the P(2) transition of the fundamental band at 2843.6245 cm-1. The HITRAN database lists the pressure-broadening parameter for self broadening (broadening upon collisions of H35Cl molecules) as 0.2395 cm-1 / atm HWHM, the squared transition dipole moment 102 = 23.95 D2, and the line intensity S = 3.692 10-19 cm/molecule. Note: HITRAN uses half width at half maximum (HWHM), not FWHM, for the pressure-broadening. 4a) Determine the FWHM for pressure broadening, Doppler broadening, and the natural linewidth. What are the relative contributions to the total FWHM? 4b) Is the line shape Lorentzian, or Gaussian ? (Note: Use IGOR to calculate the absorption cross section across the lineshape as a function of wavenumber.) 4c) What will be the transmission of the cell at line center, i.e., at 2843.6245 cm-1 ? 4d) If the same cell is then filled with air to a total pressure of 1 atm, repeat (a) and (b) given that the air-broadening coefficient (HWHM) for HCl is 0.0799 cm-1 / atm. 5) Several of lines in the v=0 to v=1 transitions of H35Cl have the following wavenumbers (IR absorption spectrum): 5a) Use these data to determine the band origin 0 [cm-1]. 5b) Calculate and Be. (Note: Use IGOR to derive these numbers from a suitable linear regression fit to the data, and determine the level of confidence that is attainable based on the error of the regression only). 5c) Determine the equilibrium internuclear separation re to as many significant figures as the data justify. 6) The HCl molecule has a B0 value of 10.4 cm-1. 6a) What are the J values of the levels with maximum population at 300K and 2000K ? 6b) Graph the populations of the J levels as a function of J for 300K and 2000K. Note: Use IGOR to calculate the population of J-levels as a function of temperature, and graph the population as a function of J. 7) From the following separations of vibrational levels in the ground electronic state of CO, obtain values of we, wexe and also an approximate value for the dissociation energy. Note: see equation 6.18 in Hollas, and use IGOR to derive the results in the worked Example 6.1.