LIBRARY Midfigan Sate “F” University This is to certify that the thesis entitled Lineshapes in Infrared Laser Stark Spectroscopy presented by Scott Sandholm has been accepted towards fulfillment of the requirements for Masters Jegree in Chemistry fimwk Major professor 7 Ma 1979 Date y 0-7639 ._. WWWWWMWWE 3 1293 00998 9348 OVERDUE FINES ARE 25¢ PER‘ DAY \ PER ITEM Return to book drop to remove this checkout from your record. LINESHAPES IN INFRARED LASER STARK SPECTROSCOPY By Scott Sandholm A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemistry 1979 ABSTRACT LINESHAPES IN INFRARED LASER STARK SPECTROSCOPY By Scott Sandholm The Doppler-broadened lineshapes of selected transi- tions in the v3 band of CH3F have been investigated. The transitions studied are among those that can be brought into resonance with the CO2 laser lines near 1050 cm"1 by application of an electric field. The laser Stark spectrometer used in the investigation and the necessary vibration-rotation theory are described. Relative in- tensities of transitions occurring at different electric fields, but with the same laser line, were found to fol— low predicted values to approximately 5%. The derivatives of the frequencies of the transitions with respect to electric field have been estimated from linewidths with an average deviation from predicted values of 2.3%. In addition, the relative intensities of a given transition appear to be a linear function of pressure to within a mean deviation of H.7%. The effect of modulation broaden- ing and sample pressure on the appearance of the lineshape have also been investigated. To Linda ii ACKNOWLEDGMENT Special thanks are due Dr. R. H. Schwendeman for the many useful discussions and his moral support through all the difficult hours in the laboratory, as well as in preparing this Thesis. I would also like to thank Dr. Erik Bjarnov for his assistance in the laboratory and especially for the modification of the computer program GDFITB. I am grateful for the numerous helpful discus- sions with John Leckey and Keith Peterson. Also, financial aid from the National Science Foundation is gratefully acknowledged. iii TABLE OF CONTENTS Chapter Page LIST OF TABLES. . . . . . . . . . . . . . . . . . . v_ LIST OF FIGURES . . . . . . . . . . . . . . . . . . vii LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . ix I. INTRODUCTION. . . . . . . . . . . . . . II. EXPERIMENTAL. . . . . . . . . . . 2.1. The Laser . 2.2. The Sample Cell . . . . . . «a ~q .t I:- H 2.3. The Laser Stark Spectrometer. 2.“. Computer Programs . . . . . . . . . . . 11 III. THEORY. . . . . . . . . . . . . . . . . . . . I“ 3.1. The Stark Effect. . . . . . . . . . . . 1A 3.2. Lineshapes. . . . . . . . . . . . . . . 20 IV. RESULTS AND CONCLUSIONS . . . . . . . . . . . 27 n.1, Introduction. . . . . . . . . . . . . . 27 u.2. The Gaussian Lineshape. . . . . . . . . 31 “.3. Modulation Broadening . . . . . . . . . 32 H.A. The Effect of Sample Pressure . . . . . 38 n.5, Determination of the Laser Stark Slopes. . . . . . . . . . . . . . 41 “.6. Relative Intensities. . . . . . . . . . A6 H.7. The Dependence of Absorption on Sample Pressure. . . . . . . . . . 50 “.8. Discussion. . . . . . . . . . . . . . . 50 REFERENCES 0 I O O O O O O O C O O O O O O O O O O O 5” iv Table II III IV VI LIST OF TABLES Page Laser Stark Transitions in CH3F that were Studied in this Investi- gation. . . . . . . . . . . . . . . . . . 28 Experimental Resonant Voltages Obtained from Least Squares Fits of Experimental Data to Theoretical Lineshapes and Derived Values of the Cell Spacing. . . . . . . . . . . . . 29 Variation of the Apparent Doppler Width of the R(l,l), m=0+l Transi- tion with the Voltage Applied to the Piezoelectric Translator. . . . . . . 36 Variation of the Apparent Doppler Width of the R(l,0), m=0+tl Transi- tion with the Voltage Applied to the Piezoelectric Translator. . . . . . . 37 The Effect of Moderate Pressure Changes on the Apparent Doppler Width for the R(l,l), m=0+l Transition. . . . . . . . . . . . . . . . » 39 The Effect of Moderate Pressure Variations on the Apparent Doppler Width of the P(2,1), m=l+0 Transition. . D0 Table VII VIII Page Laser Stark Slopes Determined from Experimental Data for the Transitions Investigated. . . . . . . . . . . . . . . HA Linear Least Squares Parameters for the Relative Intensity Data Depicted Graphically in Figures (6) and (7). . . . “9 vi Figure LIST OF FIGURES A block diagram of the laser Stark spectrometer used in these experi- ments . . . . . . . . . . . . . . . . . A theoretical plot of absorption frequency as a function of electric field for the P(2,l) and P(2,0) transitions in methyl fluoride where Am=tl selection rules are assumed A photograph of an oscilloscope plot of the data file for the R(l,l) m=-l+0 transition in methyl fluoride. . A photograph of an oscilloscope plot of the difference between experimentally observed data and the best theoretical Gaussian line- shape for the R(l,l), =-l+0 transi- tion in methyl fluoride magnified 1A0 times . . . . . . . . . . . . . . . A graph of the square root of the exponential parameter as a function of the scaled Stark slope for de- termination of the Doppler width. . . . vii Page 18 33 3A "3 Figure Page A graph of the signal amplitude (pre-exponential) as a function of theoretical intensity. . . . . . . . . A7 A graph of the area of a transition as a function of the theoretical intensity . . . . . . . . . . . . . . . . 48 A graph of the relative intensity as a function of pressure for the P(2,l), m=1+0 transition at three different Stark modulation ampli- tudes . . . . . . . . . . . . . . . . . . 51 viii NRC PZT :11) OPS K ABBREVIATIONS National Research Council. Piezoelectric translator. Hamiltonian operator. Transition dipole moment from state i to state f. The molecular dipole moment component along the F space fixed axis (the direction of the electric field vector). Total angular momentum operator. Total angular momentum quantum number. Projection of J onto the space fixed Z axis. Projection of J onto the molecule fixed z axis. The P(22) laser line where the transition J"+J' is from J'=22 to J"=2l. Symmetric top transition J+1+J, k+k. Symmetric top transition J-1+J, k+k. Frequency of radiation in the laboratory frame of reference. The Doppler frequency shift. Boltzmann's constant. The Lorentzian half-width at half-height. The Doppler half-width at half-height; i.e., the Doppler width. Peak absorption coefficient. Integrated or absolute absorption coefficient. Operational Power Supply. IkI. ImI. ix I. INTRODUCTION Since the mid—1960's laser spectroscopy experiments in the infrared region of the electromagnetic spectrum have been performed in which structural parameters of numerous molecules have been determined.(1'u) Laser Stark spectroscopy, one of the laser techniques developed during this time, has become a useful tool for obtaining structural information. In this relatively new form of spectroscopy, the absorption of a sample irradiated by a fixed frequency laser is measured as a function of applied electric field. Rotational constants, dipole moments, centrifugal distortion constants, and Coriolis coupling constants have been obtained by laser Stark experiments. This thesis describes an investigation to determine whether measurements of the lineshapes of laser Stark transitions would be of assistance in the assignment of the transitions. The difficult process of assignment is the first step in the analysis of a laser Stark spectrum. The basic technique for analyzing the lineshapes of laser Stark transitions is quite simple in principle, but not necessarily so in practice. The intensities of the transi- tions are recorded as a function of electric field and fit to a Gaussian lineshape by a least squares fitting program. The lineshape is Gaussian as a result of the Doppler ef- fect, which is the main broadening process for gases at low pressure in the infrared region of the electromagnetic spectrum. By comparing the pre-exponentials (the transition amplitudes) from the least squares fit, it is possible to compare the relative intensities of nonoverlapping transi- tions that fall on the same laser line, but at different electric fields. At present, relative intensities of transitions on different laser lines are not compared be— cause of the difficulty in obtaining the same output power and optical alignment for the different laser lines. The Stark slopes (change in frequency per change in electric field) of the various transitions are determined from the exponential constants and the theoretical Doppler width. From the slopes, zero field frequencies may be estimated. The relative intensity and the Stark slope data are not obtained in conventional laser Stark experiments. This information along with the electric fields and laser fre- quencies of the transitions from a conventional laser Stark experiment should help alleviate some of the difficulties in assigning energy levels. This information should also facilitate the calculation of precise rotational constants and dipole moments. It is also possible that the relative intensities and Stark slopes will be useful in applications of laser Stark spectroscopy for chemical analysis. Methyl fluoride was selected as the initial sample for the lineshape investigation because its spectrum consists of many strong absorption lines that may be observed at convenient electric fields with easily obtainable 002 laser lines. In addition, methyl fluoride had been thor- oughly studied by laser Stark techniques at the National Research Council of Canada in 197“ by Freund and assoc- iates.(2) Several of the transitions assigned by the NRC group were selected for lineshape studies performed in our laboratory. II. EXPERIMENTAL 2.1. The Laser The CO2 laser is a good example of a fixed-frequency laser and is a common source of radiation power for laser experiments in the infrared region. There are several ad- vantages to a fixed-frequency laser compared to a tunable laser.(3) Since the laser frequencies have been determined very accurately by beating the laser output against har- monics of lower frequencies, time-consuming calibration pro- cedures and the expense of calibration equipment are not necessary. Also, the fixed frequency 002 laser is more monochromatic and has higher power than presently available tunable infrared lasers. The main disadvantage of a fixed frequency laser is the fact that absorption occurs only at those frequencies corresponding to quantum-mechanically allowed transitions. Therefore, absorption of radiation from a fixed frequency laser is obtained only if a fortunate coincidence occurs or if the molecular transition frequencies can be tuned by an electric or magnetic field. In 1965 Patel discovered that a 002 molecular laser would function with a mixture of 002 and N2 gas, where the 002 molecules are excited by collisions of the second kind with vibrationally excited N2 molecules. Since the N2 molecule has no permanent dipole moment, it cannot relax back to the ground state through electric dipole radiation. Therefore if N2 is excited to its v=l vibrational level in the ground electronic state, it must relax through col- lisions with the walls or with other molecules. The 002 molecule, whose OO°l (2:) level is close in energy to the v=l level of N2, becomes the important collision partner. When a population inversion occurs between the OO°1 level and either the 10°O or O2°O level, both of lower energy in the 002 molecule, the system undergoes laser action.(5-8) The infrared CO2 laser that was used in these experi- ments had a laser cavity of approximately A.25 meters and rested in an Invar frame. The water-cooled plasma of the laser was 2.25 meters long. The plasma was water-cooled because the efficiency of operation of the 002 laser was very dependent on the temperature of the walls. Therefore, cooling with liquid nitrogen or dry ice would be expected (9) to improve the laser efficiency. A continuously flow- ing mixture of 002, N2, and He gases was used as the active gas in the laser at a total pressure between 8 and 13 torr, measured at the outlet of the plasma tube. The par— tial pressures of the gases, as well as the total pressure of the gas mixture, varied, depending on the laser line, but was typically 1-2 torr for 002, 1-2 torr for N2 and about 8 torr for He. The ends of the laser tube had NaCl windows mounted at the Brewster angle, requiring the laser to emit a plane-polarized light beam. One end of the laser cavity had a plane grating ruled with 150 lines per millimeter and blazed at 10 pm. The grating could be rotated to select different C02 vibration-rotation transi- tions for laser action. At the other end of the cavity‘ was a partially-transmitting, dielectric-coated, germanium mirror mounted on a voltage-controlled piezoelectric trans- lator (PZT). Either an 80% or 95% reflecting mirror could be used, depending on the strength of the laser line. The laser was stabilized on a particular laser line by a feed- back loop to the PZT which varied the length of the cavity. There were two ways to monitor the fluctuations in the laser output: by monitoring the current in the plasma, or by monitoring the laser output directly. Monitoring the current was more convenient, but monitoring the laser output by means of a detector was more reliable. A model 80-21H Lansing Lock-In Stabilizer provided the DC bias voltage for the PZT. When the Model 80-21“ was operated in the stabilizer mode, a 520 Hz sinusoidal voltage caused the PZT to expand and contract, which produced a 520 Hz fluc- tuation in the laser output unless the laser was oscillat- ing at the top of its laser gain curve. The phase and amplitude of the fluctuations were monitored by the Lansing Stabilizer and this information was used to provide a correction signal for the DC bias to maintain the laser frequency at its value at the peak of the laser gain ' curve. By this means variations in laser frequency could be reduced to a few megahertz. 2.2. The Sample Cell The sample cell for the laser Stark experiments con- sisted of a 15 cm diameter Pyrex pipe tee which housed two nickel plates, NO x 5 x 2 cm, with rounded edges and broad faces that had been ground flat to within 10.5 um. The two nickel plates rested in a cradle and were separated by five optically polished quartz spacers which had a nominal thickness of 1 mm. To produce the desired electric field, one of the plates was connected to an operational power supply (Kepco Model 2000) and the other plate was connected to a Fluke Model AloB power supply. The sample cell was connected to a vacuum line for evacuation and sample introduction, and was fitted with a Hastings Model VT-5B vacuum gauge for rough pressure measurements. For more precise measurement, a MKS Baratron Pressure meter, type 77, was connected to the vacuum line. The sample pres- sure was typically 3 to 5 mtorr. 2.3. The Laser Stark Spectrometer Figure l is a block diagram of the laser Stark spec- trometer. The electric field across the sample was varied by driving the operational power supply (OPS) by the triangular wave output from a Model 112 Wavetek Signal Generator, or by stepping the OPS through its scan by the output from a digital-to-analog converter controlled by Figure l. A block diagram of the laser Stark spectrom— eter used in these experiments. a». <\ — o , mwkaZOU % - 5&3. , «35.523 . $385.: u0 . ”5.55ng $4.50) 915. 84w...“- ._.. mo : mo“. 809.5 >JQQDm N39“. >.I m ALL. mun—54..”— 10 a PDP—8E computer. Voltage marker signals were produced by a zero crossover detector in a voltage comparator cir- cuit. The OPS voltage was continuously compared to the DC voltage of a Fluke Model N12B power supply. When the vol- tages were equal a marker was generated. Then the electric field at the marker was calculated by adding the voltages of the two Fluke power supplies and dividing by the cell spacing. Alternately, a Keithley Model 172 Autoranging Digital Multimeter was connected to the output of the OPS so the voltage to the plates could be continuously moni- tored. For modulation purposes, the output of a Hewlett— Packard Model 651A audio oscillator was stepped up by an audio transformer and then mixed with the output of the OPS. A second output from the audio oscillator was used as a reference signal for a Keithley Model 8H0 Autoloc Amplifier for phase-sensitive detection. By choosing the right mirror configuration, the plane polarized radiation could be sent through the sample cell with the plane of polarization either parallel or perpen- dicular to that of the Stark field, allowing for detection of AM=0 or AM=¢1 transitions. In either case, the signal was detected by a Santa Barbara Research Model AO7A2 Hg- Cd-Te photoconductive detector or by a Barnes Engineering Pb-Sn-Te photovoltaic detector. The lO-kHz absorption signal at the detector was proportional to the derivative of the lineshape with respect to the field because of the 11 small amplitude Stark modulation. The detector output was amplified by a lO-kHz tuned preamplifier, processed by the phase-sensitive detector, and sent to an oscilloscope, an analog-to-digital converter attached to the PDP-8E computer, or on occasion to a chart recorder. The data that were collected on magnetic tape by the computer were later fitted to the theoretical expression for the deriva- tive of a Gaussian lineshape. 2.A. Computer Programs There were two main computer programs used in these experiments, BOXA and GDFITB. Program BOXA steps the out- put of a digital-to-analog converter from 0-10 volts, reads the output of an analog-to-digital converter after each step and records the readings on magnetic tape. Pro- gram GDFITB reads data from a magnetic tape and analyzes the data by fitting them to a theoretical lineshape. BOXA was originally written by R. H. Schwendeman for use with a boxcar integrator, from which it takes its name. The input of BOXA includes the number of sweeps, number of channels, delay number, integration number, and timing number. The number of sweeps is the number of times the computer sweeps the Stark field from the highest to the lowest electric field. Five or ten sweeps were used in these experiments, the latter if the signal-to- noise ratio was low. The number of channels (always 200 12 in these experiments) is the number of steps in each sweep. The integration number was always chosen to be five and is the number of times the computer reads the analog-to— digital converter at each channel. The delay number, chosen to be five, is the number of centiseconds the computer waits after changing channels before reading the voltage. The timing number, also five, is the time in centiseconds between readings at a single channel. The GDFITB program is a Gaussian derivative line— fitting program which is a modification by E. BJarnov of a program originally written by R. H. Schwendeman and R. Creswell. The program reads a data file from a magnetic tape and performs a least—squares fit to the theoretical expression for the derivative of a Gaussian function. The fitting parameters of the theoretical expression are printed along with the standard deviations, variance-covariance matrix, and correlation matrix. The reason for fitting a derivative lineshape stems from the applied experimental technique. The lO—kHz modu- lation that was added to the ramp voltage generator output was a relatively small amplitude AC signal which slightly varies the static electric field in the sample cell. The electric field was varied slowly enough that the molecular absorption could follow the variation without loss of-co- herence. When the field variation was a small fraction of the transition field width, the amplitude of the lO-kHz 13 oscillation at the detector was proportional to the deriva- tive of the absorption with respect to the electric field. The data could have been numerically integrated and fit to a Gaussian lineshape, or the derivative of the analytical equation could be calculated and the data fit directly. Because the derivative of the analytic equation for the lineshape can be taken exactly, one approximation is eliminated by the latter method and it is the preferred way. III. THEORY 3.1. The Stark Effect The Stark effect is the name given to the change in energy levels or spectrum of a molecule or system as a result of putting the system in an electric field.(10?ll) The effect is called first order or second order Stark effect depending upon whether the change in energy is pro- portional to the first or second power of the electric field. If the shifts in levels are much smaller than the spacing of the levels, perturbation theory can be used to give a good approximation to the shift in energy levels. This is not the case in laser Stark spectroscopy where the level shifts are calculated by diagonalizing a Hamiltonian matrix of the form, fi=fi° +fi. (1) In the dipole moment approximation the Stark portion of the Hamiltonian is fis = -fi-E where u is the dipole moment and e is the applied field. The matrix elements of the Hamil- tonian may be computed from a basis in which the matrices of the square of the total angular momentum and the com- ponents of the angular momentum along the space-fixed Z axis and the molecule-fixed z axis are all diagonal, 1H 15 P2Ika> = H2J(J+l)|ka>, (2) PZIka> = mMIka>, (3) PZIka> = khlem>.(l2’13) (A) The Hamiltonian in Equation (1) is diagonal in the quantum numbers k and m, but not in J. It turns out that the matrix for H is of infinite dimension and therefore must be trun- cated at some maximum value of J. Even though J is not a good quantum number in the presence of the field, it is traditional to label the states by their zero-field J value. Thus, the wavefunction for the eigenstate |kae> in the field is a linear combination of the wavefunctions for the states |ka> with the same k and m values but several J values. The nonvanishing contributions to the energy matrix (2) are the diagonal elements, = E3 + hBJ(J+l) + h(A-B)k2 - hDJJ2(J+l)2 - 2 A hDJkk J(J+l) - thk (5) and A _ -1J€km - mm , (6) 16 and off diagonal elements, 2 2 2 2 = - 2 J (2J-1)(2J+l) In these equations, A and B are rotational constants; DJ, DJk’ and Dk are centrifugal distortion constants; and E; is the vibrational energy of the molecule. The eigenvalues of H are the vibration—rotation energy levels of the mole- cule for a given vibrational state. The frequencies of the laser Stark spectrum are the differences in the energy for two different vibration-rotation states. Several com- puter programs for the calculation of laser Stark frequen- cies existed in our research group before this project was undertaken. A portion of the plot of frequency as a function of electric field for methyl fluoride is shown in Figure 2. In this example, the m-components of the P(2,l) and the P(2,0) transitions are illustrated. As the electric field is increased, the m-component degeneracy is removed. Note that in this example the P(22) laser line at 31328961.530 MHz is coincident with four different sample transitions at different electric fields. The value of the laser frequency and the electric field at resonance are all that are recorded for a single line in normal laser Stark experiments, but from the lineshape the slope of the m-component can be determined and from that value the zero field frequency Figure 2. 17 A theoretical plot of frequency as a function of electric field for the P(2,l) and P(2,0) transitions of methyl fluoride where Am=il selection rules are assumed. The dashed line represents the frequency that corresponds to the P(22) line of the CO2 laser. 18 3l336 r “-0 4.2 3: 335 *— 0"" 3| 334 I P(ZJ) 352.0) 3|333 P-Z 3: 332 - """2 GHz 3|33l —- 3|330- 3|329r—_— ._....._..._..... _____ ._ P(22) 3l328 r 3| 327 - 3l326 '— 1 I l l 1 J l 5 IO :5\ 20 25 so 35\ «:0B (KV/cm) 19 can be estimated. The Stark slope may be calculated theoretically by starting with Equation (1) fi = fi° - 5-2, (8) from which it follows that aH/ae = -uz, (9) where “Z is the component of U in the direction of field 6. Therefore, the derivative of any energy level W(ka) with respect to s will be given by the expectation value of ’“Z’ or [3K 32 6:80 = - . (10) It is possible to calculate the expectation values from the transformation coefficients generated during the diagonalization of the Hamiltonian matrix. 20 3.2. Lineshapes In the laboratory frame of reference the molecules in a sample cell are moving randomly in all directions, but the source of radiation is stationary. Some of the mole— cules have a velocity component in the direction of the radiation while others are moving toward the source. Be- cause of their motion, the molecules see a shifted source radiation frequency. This is known as the Doppler effect, and the absorption frequency of the molecular transition is a Doppler—shifted frequency when measured in the laboratory frame of reference. If the frequency of radiation in the laboratory frame of reference is represented by wt, then the frequency in the moving frame is v v w = w£(l - a) = “A - “A? =“W,' w m (11) s’ where V is the velocity component of the molecule in the direction of the beam, 0 is the velocity of the radiation, and ms is the Doppler frequency shift. The probability that a molecule will have a velocity component (v) in the direction of the beam is given by a one-dimensional Maxwell—Boltzmann distribution, p(v) = (M/21rkBT)1/2 exp(-Mv2/2kBT), (12) where M is the mass of the molecule, kB is Boltzmann's 21 constant, and T is the absolute temperature. The prob- ability that a molecule will have a velocity between Vi and vf along the path of the beam is given by p(vi+vf) = (M/zflkBT)l/2JC:% exp(-Mv2/2kBT)dv. (13) In the limiting case where vi = -m and vf,= w the probabil- ity is unity. Now consider the absorption of laser radiation by gaseous molecules. The absolute or integrated line in- tensities can be calculated by integrating the absorption coefficient over all frequencies. P~=me ydwg. (1“) The steady state absorption coefficient for a Lorentzian lineshape at low radiation power is 2 AwL (15) Y (m ) = Yo Equation (15) may be derived from the optical Bloch equa- tions.(lu'l6) The half—width at half-height for the ab- sorption by the molecules at a single velocity is AwL and the peak absorption coefficient is yo, which occurs at wl-ws=wo. The peak absorption coefficient is defined by 22 2 ifANo MC Aw = unwou Yo (16) L where ANO is the initial population difference between the two levels, and ”if is the transition dipole moment matrix element, uif = (J'k'm'IUFIJ"k"m">, (17) where “F is the molecular dipole moment along the space fixed axis F,(12) the direction of the electric vector of the applied radiation. The absorption coefficient for a large number of molecules with the velocity distribution of Equation (12) is given by Y 3- (21) B A 2kBTw2 The absolute or integrated line intensity can now be calculated from Equation (21), w Nwzw u2 AN 2 M 1/2 0 if 0 -Mc 2 2“(ET w ““2 2kBTw§ 2 ° 1 and it follows that the integrated absorption coefficient is 2N 2 -Mc 2 2 2 2 A o 1m w 11 AN 0° 2k Tm Only the transition dipole moment “if’ the population difference AND, and the frequency w will change when com- 2 paring two transitions. Therefore, the ratio of the inte- grated line intensities will be a ratio of the product of the square of the transition dipole moments and ANO, since the frequencies will cancel when the two transitions are on the same laser line. From Equation (21), a ratio of the absorption coefficients 7 will yield a ratio of ”IfANo if the transitions are on the same laser line also. Therefore, relative intensities can be calculated as the ratio of the absorption coefficients. By rewriting Equation (21) as (w ) = A eXpE- 2( - )2] (2A) Y 2 q wk. (00 a where A is the product of several constants with the square of the transition dipole moment and q2 = (Mcz/ZkBTwzz), the half-width at half-height of the lineshape, (wi'wo)1/2’ may be obtained from 2 1/2 = exp[—:——2- (mg-wofi/Z], (25) 25 or £n2 =- Mc2 (w -w 2 (26) 2k Tm: 2 ° 1/2 B A Therefore 2k Amp 2 (wn‘wo)1/2 = ( T£n2 B )1/2 fig C 9 (27) where AwD is called the Doppler width. The Doppler width is proportional to the frequency of the radiation and the square root of the temperature, but inversely proportional to the square root of the mass of the molecule. Then, T' 1/2 AwD = wk (MT) (3.5815x10-7) , (28) where M' is in atomic mass units and T is the absolute (l6) temperature. The zero field frequency, w in Equation (2“), may be 0 expanded in a Taylor series in electric field and truncated after the first derivative,(19) as follows: 3—‘") (s-eo) . (29) If e is chosen to be the field at which mg equals the 0 laser frequency w£, then Equation (8) for the laser Stark absorption coefficient becomes, 26 - 3w 2 2 2 Y(w£) - A exp [-(5E)O(e-eo) q 3 . (30) The electric field is defined by e = V/d , (31) where V is the voltage and d is the cell spacing (the dis- tance between the two plates in the sample cell). Then if 2 8w 2 2 3 q (Ego/a B . (32) it follows that Y(w£) = A exp [-B(v-vo)2] . (33) From the definition of q and Equation (32), it can be shown that the Stark slope of a transition may be estimated from the experimental value of B, as follows: w (3w/3€)0 = 7é-(2kBTB/M)l/2d . (3A) IV. RESULTS AND CONCLUSIONS A.l. Introduction The laser Stark spectrum of methyl fluoride was assigned and the rotation and centrifugal distortion constants de- termined by Freund and co-workers in 197A.(2) In their calculations of the frequency of the v3 band (C-F stretch) a Hamiltonian matrix truncated at Jmax = J + 5 was used. The laser frequencies and resonant electric fields calcu- lated by dividing their reported resonant voltages by their mean cell spacing for the transitions used in this work are given in Table I. From these known electric fields of the transitions and from the experimental resonant vol- tage measurements in this work, the cell spacing in the cell used here was determined by Equation (31). The transition voltage is taken as the voltage that cor- responds to the maximum amplitude of the transition in the least squares fit lineshape. The transition voltage and cell spacing data are listed in Table II. The following sections describe the results of experi- ments to test several questions concerning the validity of lineshape analysis in infrared laser Stark spectroscopy. The questions tested are as follows: 1. How well do the observed lineshapes match the Gaussian shape expected for Doppler—broadened spectra? 27 28 Table I. Laser Stark Transitions in CH3F that were Studied in this Investigation.a CO2 Laser Electric Field/ Line (V/cm) Assignmentb m1+m"c P(l2) 21109.8 R(2,2) 1+2 " 2992A.7 " 0+1 " 37991.7 R(2,l) 1+2 " A9909.7 R(2,2) -l+0 " 57360.5 R(2,1) 0+1 P(lA) 1A863.6 R(l,l) 0+1 " 32680.8 " —1+0 " uu255.8 " 2+1 " 53A77.5 R(l,0) 0+11 P(22) 9738.2 P(2,l) 1+0 " 23697.A " 0+-1 " 26A73.5 " 1‘2 " A266A.A P(2,0) 1+0 aData taken from S. M. Freund, G. Duxbury, M. R6mheld, J. T. Tiedje and T. Oka, J. Mol. Spec. :8, 38-57 (197A). bR(J,k) indicates a transition from state J,k in the ground vibrational state to state J+1, k in the upper vibrational state. P(J,k) indicates a transition from state J,k to state J-l,k. cThe m' and m" are the m-quantum numbers for the upper and lower states, respectively. 29 Table II. Experimental Resonant Voltages Obtained from Least Squares Fits of Experimental Data to Theoretical Lineshapes and Derived Values of : the Cell Spacing. Transitiona m'+m"b V/voltsc d/cmd R(2,2) 1+2 2172.97 0.102937 " " 2172.A2 0.102910 " " 2172.68 0.102923 " " 2173.85 0.102978 " " 2173.93 0.102982 " " 2173.78 0.102975 R(2,2) 0+1 3080.79 0.102951 " " 3080.36 0.102937 " " 3080.58 0.1029AA " " 3082.08 0.102995 " " 3082.11 0.102996 " " 3082.17 0.102998 R(2,l) 1+2 3908.35 0.10287A " " 3907.57 0.102853 " " 3908.16 0.102869 " " 3909.60 0.102907 " " 3909.67 0.102909 " " 3909.85 0.102913 R(2,2) -l+0 5139.A6 0.102975 " " 5139.85 0.102983 " " 5139.83 0.102983 " " 51A1.22 0.103010 " " 51A1.A8 0.103016 " " 51A1.A9 0.103016 R(2,l) 0+1 590A.35 0.10293A "" " 590A.8l 0.1029A2 " " 5905.05 0.1029A6 " " 5907.35 0.102986 " " 5907.38 0.102987 " " 5907.19 0.10298A R(1,1) 0+1 1528.60 0.1028A2 " " 1528.58 0.1028A0 “ " 1528.57 0.1028A0 30 Table II. Continued. Transitiona m'+m"b V/voltsc d/cmd R(l,l) -1+0 336A.u6 0.1029A9 " " 336A.52 0.102951 " " 3369.79 0.102959 R(l,l) 2+1 “556.30 0.10295“ " " “558.96 0.102303 " " “557.99 0.102991 R(l,0) 0+il 5505.A3 0.102999 " " 550”.“2 0.102930 " " 5505.39 0.1029A8 aSee footnote b, Table I. bSee footnote c, Table I. cResonant voltage obtained from a least squares fit of the lineshape to a derivative of a Gaussian function. dThe cell spacings (d) are obtained by combining the resonant voltages in this table with the resonant voltages and mean gell spacing in Reference (2). The mean cell spacing, d=0.1029A710.000097 cm, where the uncertainty is two standard deviations. 31 2. What is the effect of changing sample pressure on the observed lineshape? 3. Are the transitions broadened by either the modula— tion of the laser used for laser frequency stabiliza- tion or by the modulation of the electric field used for observation of the spectra? A. Can the linewidths obtained be used with the ex~ pected Doppler width to determine the derivative of the frequency of the transition with respect to the electric fields? 5. Do the amplitudes of the transitions obtained from the fits of the lineshapes for several transitions observed with a single laser line vary according to the theoretical intensities of the transitions? 6. Are the amplitudes of a single transition obtained from fits of lineshapes at several pressures pro- portional to the sample pressure? In the remaining sections of this chapter the results of the examination of these questions will be discussed. A.2. The Gaussian Lineshape An important question to be answered before any results are presented or conclusions drawn is how well the data fit the derivative Gaussian lineshape. The fact that it is a derivative lineshape is an artifact of the applied 32 experimental technique (Section 2.“). The equation that is fit is the voltage derivative of a form of Equation (33) to which empirical slope (D) and base line (B) terms have been added, as follows: F(V) = A(V—C)exp[-B(V-C)2] + D(V-C) + E. (35) An oscilloscope plot of a data set for the R(l,l) m=-l+0 transition is shown in the photograph of Figure (3) and a corresponding plot of the difference between the observed and calculated values is shown in the photograph of Figure (“). The scatter in Figure (“) shows a random pattern and therefore indicates that there are no obvious systematic deviations from the Gaussian lineshape. The fact that the standard deviations of the observed minus calculated data points is only 0.1-2.0% of the difference between the maxi— mum and the minimum relative intensity values is also an indication that the data fit the Gaussian lineshape quite well. “.3. Modulation Broadening In the laser Stark spectrometer there are two possible sources of modulation broadening: one from the laser fre- quency modulation that is used to stabilize the laser, and the second from the 10 kHz voltage that is added to the Stark field to modulate the absorption. Both are important 33 Figure 3. A photograph of an oscilloscope plot of a data file for the R(l,l) m=-l+0 transition in methyl fluoride. Figure “. 3“ A photograph of an oscilloscope plot of the difference between an experimentally observed data set and the best theoretical Gaussian lineshape for the R(l,l)m=-l+0 transition in methyl fluoride magnified l“0 times. 35 points of concern and will be discussed here. The purpose of modulating the laser frequency is to keep the laser functioning at the frequency that is at the peak of the laser gain curve. A laser gain curve is a graph of power output as a function of frequency or wave- length. If the modulation amplitude becomes too large, the monochromaticity of the laser is lost. For example, a modulation voltage of 13.9 volts peak-to-peak applied to the piezoelectric translator used in this investigation results in a 1 MHz frequency variation. This is similar to the effect of opening the slits in a conventional mono- chromator. In Tables III and IV the measured Doppler width and the laser frequency modulation voltage are listed for several data sets for the R(l,l), m=0+l, and the R(l,0), m=0+tl, transitions. The Doppler width appears to increase with increasing modulation voltage, but for the various modulation amplitudes shown the increase is not much larger than the random error. To reduce the broadening from this effect the modulation voltage was kept as low as possible (5-“8 volts peak-to-peak) consistent with maintaining stabilization of the laser. The 10 kHz modulation signal that is added to the Stark field is the second source of modulation broadening. The modulation amplitude for the Stark slope and relative intenSity experiments described in Sections “.5 and “.6 was either “.1 or “.6 volts, peak—to—peak. This small 36 Table III. Variation of the Apparent Doppler Width of the R(l,l), m=0+l Transition with the Voltage Applied to the Piezoelectric Translator. Voltage/voltsa B/(volts)'2b AvD/MHzc 10 0.009171(3o)d “359(9)d 20 0.009“88(27) A2.86(8) 30 0.009556(29) u2.71(8) uo 0.00951A(28) A2.80(8) 50 0.009500(27) A2.83(8) 60 0.009553(26) u2.71(8) 90 0.009A22(30) u3.01(9) 100 0.009A6u(27) u2.91(8) aPeak-to-peak amplitude of the sinusoidal voltage applied to the PZT. bExponential parameter in the fit of the lineshape (Equa- tion (35)) cAvD=[dv/de)/d] (Ln2)l/2 from Equations (26) and (32) and the definition ofB q. The standard deviation in the last digit quoted is in parentheses. d 37 Table IV. Variation of the Apparent Doppler Width of the R(l,0), m=0+il Transition with the Vol- tage Applied to the Piezoelectric Translator. Voltage/voltsa B/(volts)’2b AvD/MHzc 10 0.003A91(A)d 36.50ame 20 0.003513(u) 36.38(u) 30 0.003“9l(3) 36.“9(3) uo 0.003“93(3) 36.u9(3) 50 0.003“9“(3) 36.50(3) 60 0.003“86(“) 36.52(u) 70 0.003“82(“) 36.5“(“) 80 0.003505(3) 36.“2(3) 100 0.003A73(3) 36.59(3) aSee footnote a, Table III. bSee footnote b, Table III. The values for B are the averages of from 3 to 5 values. 0See footnote 0, Table III. dSee footnote d, Table III. The standard deviations were 2 n 01 1/2 calculated by using the equation, Gave = ( E _fi_9 eSee footnote d, Table III. 38 voltage may be as much as 0.5% of the Stark field, as in the case of the P(2,l), m=l+0 transition. More recently, experiments were performed at slightly higher pressures and smaller amplitude modulation voltages (1.“ and 0.73 volts peak-to-peak). Slightly narrower lines (GAvD 30.3 MHz) resulted from this decrease in modulation vol- tage (see Table VI). Up to a point, higher modulation voltages are favored, because they lead to larger signals and greater signal-to-noise ratio. However, from these results it is apparent that the smallest possible modula— tion voltage should be used if accurate line-width informa- tion is desired. “.“. The Effect of Sample Pressure As the pressure of the sample is increased, the line- shape should change and become pressure broadened. As the pressure increases, the molecules experience more collisions per unit time, and collision-broadening is the dominant mode of pressure-dependent broadening. The pressure broadening parameterscfi‘the transitions studied in this work are not known, but rotational linewidths in CH3F are of the order of 17 MHz/torr.(20) Tables V and VI show the relationship between AvD and pressure. Presumably at higher pressures AvD will increase, but since the experiments were performed at such low pressures, pressure broadening should be 39 Table V. The Effect of Moderate Pressure Changes on the Apparent Doppler Width for the R(l,l), m=0+l Transition. Pressure/mtorra B/(volts)-2b AvD/MHzc 3.A7 0.01u062(73)d 35.21(11)d 3.A9 0.01“136(62) 35.11(9) 3.52 0.0l“033(“8) 35.2“(8) 3.5“ 0.01“370(1“) 3“.83(3) 3.58 0.01u372(13) 3A.82(3) 9.52 0.01“9“3(10) 3“.15(3) 9.51 0.01“97l(9) 3u.12(3) 9.51 0.015052(12) 3“.02(3) 9.51 0.015007(15) 3“.08(3) 11.23 0.015210(l“) 33.85(3) 11.30 0.01509“(11) 33.98(3) 11.38 0.015180(10) 33.88(3) ll.“5 0.01505A(10) 3A.03(3) aPressure in mtorr, as measured with the MKS Baratron Gauge. bSee Table III, footnote b. CSee Table III, footnote c. dSee Table III, footnote d. “0 Table VI. The Effect of Moderate Pressure Variations on the Apparent Doppler Width of the P(2,l), m=l+0 Transition. Stark Pressure/ Modulation/ _2b d mtorra Volts B/(volts) AvD/MHz 50.u5 0.73 0.015221(A9)e 33.80(7)f uu.77 0.73 0.015383(36) 33.62(6) 36.56 0.73 0.015233(60) 33.78(8) 30.76 0.73 0.015231(“3) 33.79(5) 27.30 0.73 0.015333(“2) 33.67<6) 22.81 0.73 0.015267(A3) 33.75(6) 19.81 0.73 0.015378(A2) 33.62(6) 19.92 1.“ 0.015192(3l) 33.83(5) 16.77 1.u 0.015uou(59) 33.59(8) 12.92 1.A 0.015505(38) 33.A9(6) 10.13 1.“ 0.0153A9(35) 33.65(5) 6.80 1.“ 0.01A990(36) 3u.06(6) 3.00 1.“ 0.015010(“8) 3A.03(7) aSee footnote a, Table V. The values reported are the average of three trials. bThe peak-to—peak modulation voltage applied to the Stark cell for signal detection. 0See footnote b, Table III. The values are the average of three trials. See footnote c, Table III. eSee footnote d, Table IV. fSee footnote d, Table III. d “l insignificant. The pressure of the gas sample was kept between two and five millitorr for the laser Stark s10pe and relative intensity experiments. The pressure was kept low to avoid electric discharge in the sample cell at high electric fields. From the tables it appears that a pressure of 10-13 millitorr would have been a better pressure to conduct relative intensity and Stark slope experiments, since there appears to be a narrower Doppler width in this pres- sure range. This may be explained by considering the signal-to-noise ratio and the laser stability. The signal- to-noise ratio should be higher at moderately higher pres- sure. Therefore, if the scatter were signal-to-noise dependent, the scatter would be less at moderately higher pressures. It appears that the scatter is independent of the sample pressure; therefore, the scatter is prob- ably a result of fluctuations in the laser output. “.5. Determination of the Laser Stark Slopes The half-width at half-maximum of the laser Stark lineshape was investigated in order to determine whether the slope of a laser Stark transition (derivative of the frequency with respect to the Stark electric field) could be resolved from a study of lineshapes. In Chapter III the effective electric field width (B'l/2) of a laser Stark transition is predicted to be a function of the “2 Doppler width and Stark slope (Equations (32) and (27) and the definition of q). The Doppler width depends only on the mass of the molecule, the frequency of the transi- tion, and the temperature. Therefore, it is possible to calculate it for a given transition in a given molecule. According to Equations (27) and (32) and the definition 1/2 is proportional to (dv/de)(2n2)1/2/d. There- B1/2 of q, B fore, from a plot of as a function of the scaled Stark slope, the Doppler width can be determined as the reciprocal of the slope (Figure 5). The predicted Doppler width at 2“°C for CH3F is 33.39 MHz and the experimental value of 3“.5“ MHz is only 3.“% wider than the theoretical value. From the theoretical value for the Doppler width, calculated values for the laser Stark slopes have been determined and are listed in Table VII. All of the entries in Table VII show a smaller absolute slope than their corresponding theoretical values (from 0.17 to 19 kHz per unit electric field (volts/cm) smaller). If the effective electric field linewidth is larger than it should be because of some broadening process, then the effective exponential parameter B will be smaller than it should be. This wider effective electric field width is then the cause of the smaller absolute slope. Nevertheless, it is clear that for CH3F, at least, the measured linewidths could have been used to predict the Stark slope of the “3 .npofiz abandon on» no cofiumcfienopop Lou oQOHm xnmum omfimow can no cofiuocsu w on nmuoEmnma fimfiucocoaxm on» no poop mpmsum on» mo nawpw < .m msswfim 53sz 38.8 9W3 _ Qn 04V QM ON 0.. O O _ _ _ _ _ Aflfia «ESQ QNX. 35a Ran 35a 83a mZJ mwmdn. JOmE>m 000xo+o é? -N—N-mn l to C) l I O. (D. " O |—(s“°/\) 7% (8) I N. ““ Table VII. Laser Stark Slopes Determined from Experi- mental Data for the Transitions Investigated. Transé- b Bl/Z-J. dv dv d tion m'+m" (volts) (5;) Exp.0 (5;) Theory R(l,l) 0+1 0.1205 0.u970 0.516222(0.019)e R(l,l) -1+0 0.0685 0.2825 0.287156(0.00A7) " 0.067“ 0.2780 " (0.0092) R(l,l) 2+1 0.0u03 0.1662 0.177035(0.011) " 0.0A19 0.1728 " (0.00u2) R(l,0) 0+il 0.0628 0.2590 0.2666“7(0.0076) R(2,2) 1+2 0.1130 0.A668 0.A78959(0.012) " 0.1121 0.A631 " (0.016) R(2,2) 0+1 0.0828 0.3u21 0.352071(0.010) " 0.0828 0.3“21 "" (0.010) R(2,2) -1+0 0.0563 0.2326 0.237892(0.0053) " 0.0553 0.2285 " (0.009u) R(2,l) 0+1 0.0A72 0.1950 0.202120(0.0071) " 0.0u78 0.1975 " (0.00u6) P(2,l) 1+0 0.1230 -0.5038 -0.515571(-0.012) " 0.1217 -0.A985 " (—0.017) " 0.1222 —0.5006 " (-0.015) P(2,l) 0+—1 0.0611 -0.2503 -0.250673(-0.00037) " 0.0612 —0.2505 " (-0.00017) " 0.0606 -0.2u82 " {-0.0025) P(2,l) 1+2 0.0A65 -0.1905 -0.19A7Au(-0.00A2) " 0.0A70 -0.1925 " (-0.0022) " 0.0u66 -0.1909 " (-0.0038) P(2,0) 11+0 0.0529 -0.2167 —0.217AA5(-0.00075) " 0.0527 -0.2159 " (-0.0015) " 0.0526 -0.2155 " (-0.0019) “5 Table VII. Continued. aSee Table I, footnote b. bSee Table I, footnote 0. cThe units are MHz/(volt/cm) and the values were calculated by Equation (3“). dThe units are MHz/(volt/cm). The values were determined by the computer program LSINT. 8The values in parenthesis are (dv/de) Theory-(dv/de) exp. “6 transitions to within a few percent. An apparent excep- tion, for which we have no explanation at present, is the R(2,l), m=l+2, transition. “.6. Relative Intensities Figure (6) represents the relationship between theo- retical intensities and the pre-exponential (A in Equation (2“)) from fitted data, while Figure (7) shows the cor- relation between theoretical intensities and the calcu— lated relative areas of the absorption curves. The rela- tive areas have been calculated by multiplying each ampli- 'tude by the corresponding full width at half maximum. All of the lines on both graphs have been fit to a linear equation whose parameters can be found in Table VIII. From a comparison of the correlation coefficients Figure (6) shows a better relationship between predicted and experimental intensities than does Figure (7); i.e., the data are more highly correlated to a straight line when the amplitude instead of the area is compared. However, differences in correlation coefficients may not be real, as they are very small. Nevertheless, there appears to be no reason to go through the extra trouble of calculating areas . “7 6.0 — SYMBOL LASER RUN lJNE NO P04) P04) P02) P02) P(22) P92) P(22) 5.0 — 4£)- O+- 3.0 ‘- l 1 l I o .04 .08 .12 .l6 .20 .24 .28 INTENSITY Figure 6. A graph of the signal (pre-exponential) as a function of theoretical intensity. “8 4£)—' 3-5 " SYMBOL LASER RUN LINE NO 0 Fm4) I 4 F04) 2 3.2 F- 0 P02) I x P02) 2 v P(22) I + P92) 2 2-8 ’ o P(22) 3 <1 2.4 - + {3 Q N .. 2.0 -— v <1 III 0: <1 l.6 — / I, 0 0 L2”— V -8 " b I o c) .4 -— o / I I I I I I 0 .04 .08 .l 2 .I 6 .2 O .2 4 INTENSITY Figure 7. A graph of the area of a transition as a func- tion of the theoretical intensity. “9 Table VIII. Linear Least Squares Parameters for the Relative Intensity Data Depicted Graphically in Figures (6) and (7). Laser Run Correlation Line Fig. No. Slope Intercept Coefficient P(l“) 6 1 0.1536 -0.0077“ 0.993 " " 2 0.0“91 -0.00367 1.000 P(l2) " 1 0.0362 -0.00131 0.995 " " 2 0.0375 -0.00303 0.990 P(22) " 1 0.0950 -0.0109 0.997 " " 2 0.0785 -0.0106 0.999 " " 3 0.0750 -0.0112 0.999 P(l“) 7 1 0.1777 -0.00382 0.99“ " " 2 0.0600 -0.00736 1.000 P(l2) " l 0.0“““ -0.00335 0.981 " " 2 0.0“57 +0.0051“ 0.979 P(22) " 1 0.1150 -0.007“6 0.996 " " 2 0.0970 -0.0121 0.998 " " 3 0.0916 —0.0119 0.998 50 “.7. The Dependence of Absorption on Sample Pressure The Beer-Lambert-Bouguer law relates the path length of the radiation, the molar absorptivity, and the concen— tration to the absorption. The molar absorptivity and path length are constants for a given gas, transition, and absorption cell. According to the ideal gas law the concentration of a gaseous sample is proportional to its partial pressure. Therefore, there should be a linear relationship between the pre-exponential in the fitting equation (Equation (2“)) and the pressure. Some typical results of pressure dependence of the pre-exponentials at three different levels of modulation amplitude are shown in Figure (8). The large fluctuation between points is believed to be due to the variation in the absolute power output of the laser. However, the mean intensities at the various pressures (also plotted) follow the straight lines rather well. “.8. Discussion There are at least three applications of the results of this study: gas mixture analysis, laser Stark slope estimation, and measurement of pressure broadening param- eters. In the analysis of gas mixtures, intensity appears to be a linear function of pressure for a given laser line with a mean deviation of average measurements of 51 4.6 VOLTS P-P l8"- 0 '6 _ L4 VOLT P-P o o V x O _. OJB\KRIS l4 . P42 0 o l.2 -— D ‘ ° A o o LO- 0 .8 h' o o 0 £5“- 0 Data Point X Average A+- .2 o I I I I I I I O 5 IO IS 20 25 30 35 PRESSURE (MICRONS) Figure 8. A graph of relative intensity as a function of pressure for the P(2,l) m=l+0 transition at three different Stark modulation ampli- tudes. 52 “.7%. Relative intensities of transitions occurring at different fields with the same laser line follow predicted values to about 5.3%. In the transitions studied here, where the laser Stark slopes were estimated from line widths, the deviation from predicted values were about 2.3%. Thus laser Stark slope estimation should prove very useful in assignment of laser Stark transitions and the next step would be an application to such assignment. The third application is the measurement of pressure broadening by the study of lineshapes at higher pressures. In order to obtain significant pressure broadening, pres- sures will have to be of the order of one torr or larger, and consequently resonant fields must be low. This will require either near coincidence of laser frequency and zero field absorption frequency or a tunable laser. There are several improvements to be made and future experiments to be carried out. In terms of experimental improvements, a better method of laser frequency stabiliza- tion and some method for compensation for fluctuation of laser power are needed. For the latter, the relative laser power output for each recorded data point could be measured, and the ratio of the spectral intensity to relative power stored. This would be analogous to a double beam spectrophotometer. 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