&:“:’a£1" c .v . . . 1:3 ’ . 3 hfi‘“ « ‘ 2‘2 ' _ ~%._ ,?.r .512 .. .. ’ .....‘9xr-v ‘.—. .<.‘.o.-n-0> coavv r—‘ - -- u— ijIM In: an I 0:. ~21 ' '- c- t‘ " 'gfigkr ”44.2.: An ‘.. o... ofl monk mums 02.5.5.5 En. _ u \\ N {II— haaam $30.. 1.— . T mmNjidbm _ w H. 12 computer-driven Operational power supply (Kepco Model 2000) is mixed with a 10 kHz sine wave voltage and connected to the other plate. The sine wave voltage provides low-am- plitude modulation of the Stark field. The separation of the plates (0.29894 cm in the experiments described here) was determined spectroscopically by measuring the resonant voltage of the saturation dip in the Q(l,l) transition in the v3 band of CH3F; a 9P(32) CO2 laser line was used. The resonant voltage was divided by M9) to Obtain the resonant field reported by Frend 23 21° the cell spacing. Sample pressures are measured with two different capacitance manometers (MKS Model 77 and Model 220B Baratrons); in the present experiments the pressure ranged up to ml torr. After passing through the cell the laser radiation is monitored by a Pb-Sn-Te photovoltaic detector (Barnes Engineering). The detector output is processed by a phase-sensitive detector and is stored in a Digital Equipment Corporation PDP-8E computer as well as being displayed on an oscilloscope. The computer also controls the voltage of the operational power supply. 3.2. The Sample of CH3§§_ The acetonitrile was obtained from Mr. Walter Cleland at MSU who had purified a reagent grade sample by double distillation,(50’51) first in CaH2 to remove any water and second in phosphorus pentoxide to remove the last traces 13 of water and any acidic impurities. As already mentioned, acetonitrile was chosen as an initial sample because the (J,k,m) = (2,:l,il) + (1,11,11) transition in the v4 band resonates with the P(20) N20 laser line at low elec- (31) This allows one to tric field (1009.25 volts/cm). reach relatively high pressures without an electrical dis- charge in the sample cell. 3.3. Software For this study, four major line shape fitting programs were used: a Voigt profile program and three programs essential to the Dicke narrowing aspects of this research. The two major contributors to the software development were Drs. E. Bjarnov and R. H. Schwendeman; Bjarnov for the Voigt fitting program VGTFIT and Schwendeman for the Dicke narrowing programs DNRFIT (for the soft collision model), DNHFIT (for the hard collision model), and DNGFIT (the Gaussian fitting program). The hard collision model is appropriate when the velocity of the molecule under study after a collision is independent of its velocity before the collision,(53) and the soft collision model is proper when several col— lisions are required to produce a substantial change in the direction of travel. The DNGFIT program was used in the lower pressure region (below 50 millitorr) to determine an experimental Doppler width. In this program the l4 narrowing (Dicke) and the broadening (Lorentzian) param- eters were held constant and the Doppler width allowed to vary. All of the fitting programs read a data file and per- form a least squares adjustment of several parameters to fit the data to the derivative of the appropriate theo- retical equation. The derived fitting parameters are printed along with the standard errors, variance-covariance matrix, and correlation matrix. The derivative func- tions are a result of the experimental method used to obtain the data (phase sensitive detection of a small amplitude modulated signal). All of the calculations and fittings were performed in the MSU CYBER 750 computer. Another program indispensable to the data acquisition in these experiments is BOXM. This program for the PDP8/E computer is virtually the same as the BOXA program used in earlier laser Stark experiments from this laboratory.(52) The new name is a result of changes required to implement a new computer interface designed and built by Mr. Martin Rabb at MSU. The BOXM program increments a digital to analog converter that controls the operational power supply attached to one plate in the sample cell. After each incre- ment in electric field, the program delays for a predeter- mined period of time and then reads the spectrometer Output by means of an analog to digital converter (A/D). The A/D is read an optional number of times with a preselected 15 time period between readings. The field sweep may be repeated a selected number of times and all of the A/D readings at each field setting are averaged. The stored data are continuously displayed on an oscilloscope during acquisition. After acquisition, the stored data may be filed on a flexible disc for permanent storage and later data treatment. CHAPTER IV THEORY There is a wealth of information hidden in spectral line shapes or profiles. The amplitude of absorption as a function of frequency can lead to information about the concentration of a sample as well as to the transition frequency. More relevant to this research are the pres- sure broadening parameters, which are useful in the design of optical resonant transfer lasers, as well as for measure- ments of chemical abundance in the atmosphere or outer space or for the determination of motion in the inter- stellar media. Also, information pertaining to relaxation processes and collision dynamics in the molecular system is obtained from line shape investigations. In order to determine this kind of information from spectral lines, an understanding of what gives rise to the various line shapes is required. If a molecule could be represented by a stationary "free particle", its spectra would have a natural line width from spontaneous emission and uncertainty broadening (a result Of the Heisenberg uncertainty principle). If the "free particle" is allowed to interact with its 16 17 surrounding through collisions, but yet frozen in space (everything is allowed to have translational motion ex- cept for the molecule undergoing the transition), the width of the line would grow. Collisional or pressure broadening may take place by several mechanisms - self or resonant broadening (collisions with like molecules), foreign gas broadening (collisions with unlike molecules). or wall broadening (collisions with the sample cell or container walls). All of the processes mentioned so far result in homogeneously broadened line shapes, because each molecule is allowed to absorb radiation over the entire line profile. The exponential decay of the dipole cor- relation for these collision relaxation processes result in Lorentzian line shapes. If the "free particle" is allowed to have transla- tional motion, the DOppler effect, beam transit effects, and Dicke or collisional narrowing can contribute to the Spectral line shape. The Doppler effect can be traced back to 1842 when Christian Doppler of Prague first noted a velocity dependent frequency shift. Velocity shifts are a very important part of infrared line shapes in gaseous samples. Dicke or collisional narrowing is the result of the molecule experiencing multiple collisions before travel- (53) If the narrowing mechanism ing a distance of l/2n. is not taken into consideration when interpreting the results some experiments show a reduced effective Doppler 18 width. The beam transit effects result from excited or hot molecules moving out of the beam and being replaced in the beam by ground state or cold molecules, a pseudo relaxation process. Beam transit effects and wall broaden- ing are calculated by the kinetic theory of gases, after assuming that the resulting line shape is the same as that due to intermolecular collisions. Modulation broadening, another potential contributor to line broadening, occurs when the experimental modula- tion amplitude is too large. Recall that the small ampli- tude modulation is the reason for the derivative line shapes. A derivative of a function F is defined by, F(x+h)-F(X) h . (l) F'(x) 5 lim h+o Then, the modulation amplitude being too large is analog- ous to the derivative not being evaluated in the limit as h + o. If one could freeze the gas molecules in place to eliminate any translational motion (i.e., remove the Doppler contribution to the line shape), the spectral line from a laser Stark experiment would fit a Lorentzian equation. Then the steady state absorption coefficient at low radia- tion power would be 19 2 Yo AwL 2 (mi-mo) + Aw YLWL) = 2 ’ (2) L which can be shown to result from the steady state solution (12,54,55) of the optical Bloch equations. The peak ab- sorption coefficient Yo occurs at w£=wo and is defined by 2 41TwouifANo Y0 = I (3) h C AwL where the initial population difference between the two levels is AND and “if is the transition dipole moment matrix element, _ ' U ' I! H II uif _ . (4) Here pF is the molecular dipole moment along the space fixed axis P(SG) in the direction of the applied radiation electric vector. The half width at half height for ab— sorption by the molecule at a single velocity is AwL. This is represented by, -1 Am = (21rT2) , (5) L where T2 is the relaxation time for the induced electric field polarization. 20 If the molecules are allowed to have translational motion, they will move randomly in all directions with an isotrOpic velocity distribution (i.e., a Maxwell-Boltz- mann distribution).(57) The DOppler motion will contribute to the line shape in such a way that the profile will no longer be Lorentzian. If the Doppler and pressure broad- ening mechanisms are independent, and if the Lorentzian pro- file is folded into the Gaussian distribution of the DOp- pler broadened line,a new line shape is generated. This convoluted line shape, first studied by Voigt(32) in 1912, fails to give a simple analytic result for the integrated absorption coefficient; it can, however, be written as an integral, as follows cw -M s 2 2 m exp[ ( ) 1 = ( M )1/2 Yo c AwL ZkBT um, d Y 2'1IkBT mg -.. ( _ _ )2+A2 “’s ° wt mo ms wL (6) In this expression, in addition to the quantities already defined, M is the molecular mass Of the absorbing molecule, kB is the Boltzmann constant, T is the absolute tempera- ture, and ms is the Doppler shift in frequency (ms = w£(v/c) where v is the component of the velocity of the molecule in the direction of the radiation). A study of the low pressure line shape of CH3F vibra— tion—rotation transitions showed that in the low pressure 21 limit (AwD >> AwL) the expected limiting Gaussian shape of the Voigt equation is obtained experimentally.(52) At intermediate pressures and under certain circumstances, the spectral line shape can be deconvoluted into its Guassian and Lorentzian components. From the Lorentzian moiety, it can be shown that the half width at half maximum should be linear in pressure with a slightly positive inter- cept. The intercept is not expected to be situated at the origin because of the beam transit and wall broadening relaxation processes. The lepe of this line would be the pressure broadening parameter for the particular transition and the molecular system. The parameter would be a self broadening or a foreign gas broadening parameter, or a mixture of the two, depending on the nature of the col- lision partners. The Voigt equation used in this investigation is a derivative of Equation (6) with respect to the electric field, 2A A("L 8x m (x+y)exp(-y?/02) )o dy + S(e) = ——7r-—-(-- n3 20 as [(X+Y)24-Awi]2 + B(€'€o) + C I (7) where 22 3w 32w2 _ - _ .2 _ I 0 - 2 X—wtwo—(Beo(€ so)+2(3€2) (e co) , (8) and 2k T w = B 1/2 2 _ 1/2 The constant 0 is referred to as the reduced DOppler width; AwD is the ordinary Doppler half width at half maximum. Five parameters are fit in a least squares calculation: the amplitude (A), the center field (so), the Lorentz width (AwL), a lepe (B), and a background (C). The spectrometer scans through the electric field 5, which is measured in volts per centimeter. The derivatives (Boo/Be)o and (32w0/382)0 are obtained from the known spectroscopic parameters and dipole moments of the sample. (34) on what In 1952, R. H. Dicke published an article has come to be called Dicke or collisional narrowing. Quantum mechanically the Doppler effect is treated as a recoil momentum transferred from the photon to the ab— (58) a Mdssbauer effect in gases. If sorber molecule, the pressure is such that the collision time is short in comparison to the time between collisions, then the momentum can be absorbed by a single molecule. This is the case where one would expect a Voigt profile to fit the spectral data satisfactorly. If the molecule experiences several velocity changing collisions before the molecular dipole 23 undergoes a single cycle while following the radiation, then the effective DOppler shift is an average of the shifts produced by the velocities between the collisions. If time wavelength divided by 2n is comparable to the mean free path between collisions, the spectral line is ex- pected to be narrowed. In other words, the molecule ex- periences several collisions before traveling a wave- length. Since Dicke's original work, the theory has been ex- (59) panded and modified by Galatry and by Rautian and (53) (60'61) and there have been Sobel'man among others several models devised to deal with this phenomenon. Two limiting models have been considered in this investiga- tion: the "soft" or "weak" collision model and the "hard" or "strong" collision model. In the soft collision model the spectral line shape has been shown to be(35’39) _ °° _ _ 2 2 18cm) — (Io/1r) j; exp[ ML“: (0 /BSC) X {eXp('Bscfi-1-BSCT}]COS[(mg-wo)T]dT . (10) After substitution from Equation (8) and differentiation with respect to electric field, an expression for the ' spectrometer signal is obtained as follows: 24 5(5) = 37%- (9—5)Of TeXp[-y(T)]Sin(XT)dT + B(e-eo) + c. O e (11) In these expressions, O2 Y(T) = 'AwLT - 282 [exp(-BSCT)-1-BSCT] , (12) sc Bsc is the narrowing parameter,(53'59) and A, B, C, x, o, and AwL have the same definitions as in the Voigt equation. In the hard collision model the line shape function is,(35'53’59’61) Ihc(w) = (Io/n) Re[F/(l-thF)] . (13) where F = (“l/Zol’l [00 “MHz/“2) d: (14) .. IehC+AwL+i(T-x)] ' so that the spectrometer signal is, 5(a) = % Re [ ._Q 2] + th(€-€o) + c . (15) (l-BhCF) and Q= (5’13) = 'i (91) jm e’EPHZ/OZ) . (16) 36 o Tr172O as o __ [AwL+th+l(T-X)] 25 Again, the previous definitions of A, B, C, AwL, x, and 0 hold, and she is the hard collision model narrowing parameter. A classic example of a hard collision in foreign gas broadening is the kind in which a small light molecule is being observed while colliding with a large heavy molecule; for a soft collision the situation is just the reverse. To close this section on the line broadening processes, it is appropriate to present a quantitative explanation of the wall broadening and beam transit effects, since they are used to help determinetflmzintercepts in the de- termination Of both the pressure broadening parameter and the narrowing parameter. The beam transit effect can be considered a subset of the wall broadening relaxation processes, only with beam transit relaxation the walls are not solid objects, but rather the imaginary boundary of the laser beam. Then, if the line shape for the wall col- lisions is assumed to be the same as for intermolecular collisions (not a bad assumption since both processes involve collisional de-excitation). a line broadening parameter similar to Equation (5) can be assumed, where the relaxation time is the mean free time from the kinetic theory of gases. The wall broadening relaxation parameter then is,(62) 1/2 RT ) . (17) 8h M 26 where R is the gas constant, T is the absolute temperature, M is the molecular mass, V is the irradiated volume, and A is the surface area surrounding the irradiated volume. CHAPTER V RESULTS AND CONCLUSIONS 5.1. Treatment of the Raw Data All of the data were temperature corrected to 298 K (the data were collected at room temperature). A short explanation of the method employed for the correction is given in this section. The derived half width at half height AwL is a function of the temperature (T), the narrowing parameter (8), and the Doppler width AwD. There- fore, dAwL = (Bde) +(3AwL)(§§)4-(BAwL)(3AwD) (18) dT 3T 38 ST BAwD 3T ‘ Also, dAwL AwL(298K) = AwL(TExp ) + ( dT ) dT . (19) If dT 2 AT = 298 - TExp. , (20) then 27 28 dAwL ALUL(298K) = AwL (TExp.) + (w) (298-TEXP.)' (21) so that BAwL BAwL 38 ADJL(298K) = AwL(TEXP.) + {(T) + (———- L)(-—- T) + BAwL BAwD + (836;)(* BT D)} (298‘ TExp.) . (22) From Equation (9), -7 AwW(T)1/2 (3.5815 x 10 ) . (23) D To obtain expressions for (BB/ST) and (BAwL/BT) it was assumed that and AwL = kL/T . (25) Then, 298-T BAw Aw _ Exp. L D AwL(298K) — AwL(TExp.) + ( 298 ){(§XEB)(—§—) BAwL - b—-—)B - AwL (T )} . (26) Exp. 29 The iterative manner in which 8 is calculated generates a wealth of data on the functional relationship between B and AwL from which the (BAwL/BB) can be determined. The derivative (BAwL/BAwD) required calculations of the pressure broadening parameter for slightly different Doppler widths. A linear interpolation of the data generated a satisfactory value for the derivative. All of the tem- perature corrected data are listed in Table (I). 5.2. Experimental Determination of the Doppler Width At very low pressure there is not a significant number of collisions to cause much pressure broadening and at high pressures there is much more pressure broadening than in homogeneous broadening. It then follows that a better estimate of the pressure broadening can be obtained from the data taken at higher pressure, and a better measure- ment of the Doppler width can be obtained from the low pressure data. The experimental Doppler width was determined by first using the theoretical Doppler width with the data above 300 millitorr to determine the pressure broadening and Dicke narrowing parameters. Then, the pressure broadening parameter was held constant and the low pressure data (below 50 millitorr) were used to derive the Doppler width param- eter. The experimental Doppler width was used and held constant in the fit of all of the data to determine a 30 4.88:0.09 Table I. Acetonitrile Laser Stark Pressure Broadened Half Width at Half Maximum for the Soft Collision, Hard Collision and Voigt Profile Models (T=298K). Pressure AvL/MHz AvL/Msz AvL/MHz —EEE?E3— (soft)b'C (Hard)b' (Voigt)b'e 1299. 90.55:0.73 89.90:0.75 87.78:0.82 1194. 82.81i0.46 82.19:0.47 79.95:0.49 973.56 67.09i0.33 66.3710.28 63.82:0.22 945.60 65.40:0.13 64.8110.12 62.26:0.06 918.36 63.34:0.l7 62.25:0.l6 60.10:0.05 878.21 60.8910.20 60.23:0.20 S7.73:0.28 845.37 57.81:0.40 57.19:0.40 54.72:0.42 839.79 58.2310.38 57.54:0.34 55.11:0.36 805.18 56.26:0.05 55.62:0.05 53.15:0.06 801.37 55.16:0.27 54.50:0.28 52.09:0.30 770.25 53.6510.12 53.0110.12 50.5410.13 720.07 49.81:0.09 49.22:o.10 46.56i0.12 674.10 46.94:0.ll 46.31:0.ll 43.79:0.17 617.48 43.21i0.20 42.52:0.21 40.1510.20 565.69 39.0510.13 38.5510.13 35.97:0.l3 564.37 39.26:0.07 38.72:0.07 36.21:0.09 542.11 38.0710.16 37.5110.16 35.02:0.l6 514.16 35.88:0.lO 35.36:0.ll 32.88:0.lO 500.30 34.8910.15 34.44:0.l4 31.78:0.14 496.38 34.74:o.35 34.24:0.35 31.86:0.50 484.65 33.9210.15 33.4010.15 30.80:0.15 460.37 32.38:0.29 32.00:0.29 29.39:0.29 454.71 31.36:0.08 30.92:0.09 28.45:0.09 438.64 30.56io.21 30.00:0.21 27.44:0.26 420.03 29.00i0.l4 29.75:0.l4 36.12:0.15 419.78 29.29:0.l9 28.85:0.l9 26.29:0.21 398.45 27.80:0.24 27.42:0.24 24.83:0.25 383.87 26.62i0.06 26.30:0.06 23.74:0.07 377.84 26.4710.15 26.0910.15 23.5710.16 357.99 24.87:0.l9 24.5310.18 22.03:0.21 353.44 24.33:0.07 24.05:0.07 21.53:0.07 331.39 23.23:0.10 23.0110.10 20.54:0.lO 128.24 9.15:0.14 9.19:0.13 7.56:0.15 109.50 7.53:0.10 7.56:0.09 6.16:0.10 97.79 6.49:0.10 6.55:0.10 5.20:0.12 93.68 6.32:0.11 6.43:0.11 5.10:0.11 87.18 5.64:0.11 5.55:0.11 4.62:0.08 76.31 5.28:0.13 5.4110.12 4.37:0.13 72.89 5.01:0.09 4.03:0.09 31 Table I. Continued. aPressure measured with a Baratron capacitance monometer (MKS type 77 or type 2203). bThe error limits are the standard errors from the least square fits. CTemperature-corrected data obtained from fits of Equation (11). In the numerical integration by Simpson's rule, 40 points were included with AT=0.006 us. dTemperature-corrected data obtained from fits to Equation (15). In the numerical integration by Simpson's rule, 40 points were included with AT=2.5 MHz. eTemperature-corrected data obtained from fits to equation (7). In the numerical integration by Simpson's rule 40 points were included with Ay=2.5 MHz. 32 pressure broadening and collisional narrowing parameter (Table II). It was assumed that any effect of modulation Table II. Line Shape Parameters. CH3CN: (J,K,M) = (2,i1,i1) + (1,i1,il) Model AvL 8 Soft Collision 69.4:0.7 MHz/torr 17.2i2 MHz/torr Hard Collision 68.6:0.7 MHz/torr 11.512 MHz/torr — broadening would be taken into account by using the experi- mental Doppler width. Theoretical calculations of the effect of modulation broadening on the value of the pres— sure broadening parameter obtained, where the modulation amplitude was approximately that used in the experiments, proved this to be the case. The theoretical Doppler width at 298K is 26.7 MHz, whereas the experimental Doppler width was found to be 27.0 i 0.26 MHz (:20). 5.3. Determination of the Narrowing Parameter It was stated earlier that the beam transit and wall broadening effects were used in the determination of the narrowing parameter. But, in order to calculate the wall broadening, the irradiated volume, as well as the 33 corresponding surface area, had to be determined. It was assumed that the geometry of the volume was that of a cylinder whose height was equal to the length of the Stark plates (40 cm) and whose diameter was approximately equal to the cell spacing (0.3 cm). The radius of the cylinder is based on the fact that the laser beam was collimated with two iris diaphragms with 3 mm openings. The wall broadening parameter obtained by using the kinetic theory of gas model is on the order of 15 kHz at 298K. The narrow- ing parameter (8) was varied by interpolation or extrapola- tion until a straight line with a reasonable intercept, was obtained for the plot of AwL gs. pressure. 5.4. Final Results The temperature-corrected data for the models used in this investigation are listed in Table (I). A graph of the pressure broadening YE- pressure based on the soft col- lision model and on the Voigt profile are shown in Figure 2. The slope of the plot, the collisional broadening parameter for the soft collision model, is 69.4:0.7 MHz/ torr; the intercept is 21:94 kHz. The slope of the plot of the Voigt profile data is 67.7:0.7 MHz/torr with an intercept of -l.77iO.l7 MHz. The temperature-corrected Am '3 for the hard col- L lision model and the Voigt profile are represented in Figure 2. 34 Plot of the pressure broadening gs pressure for the JKM = 211+1ll transition in the v4 band of CH3CN; soft collision model assumed. Solid line is the best straight line through the circles; slope = 69.4 MHz/torr, intercept = 0.021 MHz. 1 00 90 35 IIIiIIIIIITTIIIIIIIVIIIIII_+ C—o Dicke Narrowed Lineshope . '— (Soft Collision Model) I I—" . . * T F + Vougt Profile _ __ .* - __ ‘4. —1 — § — I —— '6 __ )— + ——1 q. r— + — _ .s - 4- __. 4’E’ .— up '— ‘f — __ ‘F’ __ r- - __ .4 h- -—-( llllllllnlllllllllllllnllllln .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 PRESSURE (torr) Figure 2 36 graphical form by Figure 3. The collisional broadening parameter for the hard collision model is 68.6:0.7 MHz/ torr with an intercept of 25:90 kHz. The large standard errors in the intercept in each case reflect back into uncertainty in determination of the narrowing parameter. The Dicke narrowing parameter for the soft collision model is 17.2:l.7 MHz/torr, while that for the hard collision model is 11.5:l.2 MHz/torr. The uncertainties given above for the intercepts is one stan- dard error, as calculated from the linear regression analysis. The errors given for the slopes are mainly the result of an estimated 1% error in the pressure measurement. The error limits for the narrowing parameters were handled in a much more subjective manner. The value reported is 10% of the narrowing parameter and was based on a judgment of the sensitivity of the determined value of B to the un- certainty in the intercept. It was noted during the course of the determination of the narrowing parameter that a very small change in 8 creates a rather large change in the intercept. Inclusion of the collisional narrowing in the fitting of the Observed laser Stark line shape has produced two positive results: First, the plot of pressure broadening gs. pressure is now a straight line with a reasonable intercept. Second, a value of the collisional narrowing parameter can be estimated. This does not solve all of Figure 3. 37 Plot of the pressure broadening gs pressure for the JKM = 211+lll transition in the v4 band of CH3 Solid line is the best straight line through the circles; slope = 68.6 MHz/torr, intercept = 0.025 MHz. CN; hard collision model assumed. 1 00 38 _lllllTl‘ll|lllll1lllITll|l1j 9° ”’0 Dicke Narrowed Lineshope.. 8C) __ 3C) - 20 - 1C) - (Hord Coleion Model) 4' + Voigt Proffle l.lll.l.l.|lllIllllllllllllfi l 0.2 0.4 0.6 0.8 1.0 1.2 1.4 PRESSURE (torr) Figure 3 39 the problems, but rather creates a small dilemma: the data fit the hard collision model just as well as the soft collision model, and so do not allow a choice to be made between them. It would not be appropriate to close without comment- ing on two other possible explanations considered for the curvature in the Voigt line: power broadening and dimeriza- tion. Power broadening is a possible explanation for the curvature, but corrections for power broadening would lead to a line with an even more negative intercept. Concern- ing the dimerization a calculation based on an estimated (63) indicates that value of the dimerization constant there would be less than 0.1% dimer in the sample cell at one torr pressure. The effect of this small amount of dimer is at least an order of magnitude less than the effect of the error in the pressure measurement. The narrowing parameter (8) has been interpreted as the reciprocal of the mean time between velocity-changing collisions. If it is assumed that every kinetic theory collision is hard enough to cause a change in direction, then the kinetic theory collision diameter can provide an estimate of the lower limit to 8. Based on the Stock- mayer potential, the acetonitrile collision diameter has been estimated to be :4 A,(63) which leads to Bk.t. = 1.4 MHz/torr. The kinetic theory narrowing parameter (Bk t ) is approximately ten times smaller than the determined values. CHAPTER VI INTRODUCTION TO LEVEL CROSSING AND ANTICROSSING Level crossing is a more general form of the Hanle ef— fect(64) depolarization of fluorescence radiation in a magnetic field), which dates back to 1924. In 1959, Cole- (65) grove and co-workers carried out the first application of level crossing to fine structure determination, in a 'study of the effect of magnetic field on fluorescence from He. Since then, many molecules have been studied both by level crossing experiments (Xe,(66) ,(67) (53-70) F(71,72) H2 CH4, and CD3I(74)) and avoided crossing experi- (74) (75) (76,77) 3’ A variety of experimental techniques and spectral regions CH3 ments (CD31, Li, PCP and the CN radical(78)) have been used for both level crossing and anticrossing (67’75) Zeeman effect (68,74,76) measurements: Zeeman fluorescence, (69) on absorption, Stark effects on infrared laser (72) and microwave spectroscopy: and radio frequency induced crossings.(70'7l) The theory of level crossing and avoided crossing has been considered by a number of authors;(75'76'78’79) most were concerned with the effect on detection of fluores- cence, rather than on detection of absorption. In the 40 41 study of the effect of level crossing or anticrossing on fluorescence induced by weak radiation, saturation effects may be ignored. By contrast, level crossing or anticrossing is a non-linear effect in direct absorption and therefore (79) presented a saturation effects are crucial. Shimoda theory of the effect of level crossing on a single mode laser field; Sakai and Katayama<74> developed perturbation theories for the effect of crossing and anticrossing on direct absorption; and Amano and Schwendeman<76) extended the earlier theories to a higher order of approximation. An extension of the theory of Amano and Schwendeman was used in this investigation. Level crossing or anticrossing occurs when two levels become degenerate as a result of the shifting of the molecular energy levels by an electric or magnetic field. Both terms refer to saturation phenomena and which effect occurs depends on whether the crossing levels are con- nected by a non-zero matrix element of the Hamiltonian for the system. Consider a three level system in which the two levels that become degenerate are lower in energy than the third level (Figure 4). If the matrix element connect- ing the two levels that cross is zero, and the transition dipole moments connecting the crossing levels to the third level are nonzero, a level crossing is expected. If the matrix element between the two levels that cross is non- zero and one of the transition dipole moments connecting 42 Figure 4. A diagram of a simple three level crossing crossing and level anticrossing. 43 LEVEL CROSSING AND ANTICROSSING LEVEL CROSSING LEVEL ANTICROSSING Int. Int. 44 the crossing levels to the third level is non-zero, a level anticrossing or Von Neuman avoided crossing is expected. The term level crossing is almost self explanatory in terms of the effect on the molecular energy levels. As the two levels become degenerate in the Stark field, there is no interaction between the two levels because the matrix element between them is zero. The wave functions do not mix and the energy levels cross. The anticrossing or avoided crossing term is a little more ambiguous. The matrix element between the two levels that cross is non- zero in the avoided crossing; therefore, as the levels proceed toward degeneracy, a mixing of the states occurs and the crossing is prevented by the interaction between the two levels. In the simple case of a level crossing in an infrared laser Stark experiment the laser pumps molecules out of the lower two levels, sampling from different velocity groups. As the Stark field is swept through the level crossing, the velocity groups corresponding to the transitions change. At the electric field corresponding to the crossing, both transitions are pumping out of the same molecular velocity group and the effect of the additional saturation can be detected as a dip in the absorption. In the case of an avoided crossing in infrared laser Stark spectrOSCOpy only one of the lower state to upper state transitions is al- lowed when the field is far from the value at which the 45 levels would cross, if they could. As the Stark field is swept through the crossing region, a mixing of the anti- crossing wavefunction occurs. At the electric field of closest approach for the avoided crossing there is a maximum in the intensity borrowing. Because the two tran- sitions sample different velocity groups, saturation is (reduced and there is an increase in the signal strength. CHAPTER VII EXPERIMENTAL DETAILS OF THE 13CH3F ZERO FIELD LEVEL CROSSING 7.1. The Spectrometer The laser Stark spectrometer is virtually the same as that used in the Dicke narrowing experiments (Figure 1) and only the modifications necessary to do the zero field level crossing will be discussed. In all of the other laser Stark experiments the electric field was swept from some positive lower dc value to a higher positive dc value. In the zero field level crossing experiment it is essential that the electric field be swept through zero field. To accomplish this the polarity of the output of the power supply (Fluke 4103) applied to one plate in the sample cell was reversed so that its voltage Opposed that of the output of the operational power supply applied to the other plate. The Stark electric field voltage ranged from :5 to :20 volts depending on the width of the level crossing effect. 46 47 7.2. Stabilization The previous laser Stark experiments used a dither stabilization method in which the piezoelectric transla- tor (PZT), which supports the partially transmitting mirror, was modulated at 520 Hz. The resulting modulation of the length of the laser cavity was used to lock the laser fre- quency to the value at the tOp of the laser gain profile (through phase sensitive detection). For the zero field level crossing experiments the laser was locked to the minimum of a saturation dip (Lamb dip) in the fluorescence from a C02 gas sample. A detector (Judson model JlOD InSb) was mounted over a Ban window in the side of a brass tube for the detection of the CO2 fluorescence. The tube, with ZnSe Brewster angle windows, was placed within the laser cavity. The inside of the tube was polished to enhance the fluorescence reaching the detector. The laser was frequency modulated by application of a sine wave voltage to the PZT. A derivative lineshape of the fluorescence signal was produced by a phase sensitive detector and used with a feedback loop to the PZT to con- trol the frequency of the laser. This method of frequency stabilization was tested and compared to the dither laser gain method by repeated measurements of a CH3F laser Stark Lamp dip; the results are reported in a later section. 48 7.3. Laser Power Measurements Because the zero field level crossing signal is a saturation phenomenon, it is necessary to properly measure the laser power in order to account for the power broaden- ing in the detected signal. A laser power meter (Scien- tech Incorporated Model 36001) was used to monitor the laser output. The laser power entering the Stark cell was measured at the beginning and at the end of each day at a point approximately 75 cm from the mirror used to par- tially focus the laser radiation into the sample cell. During the day, the laser power reflected from a beam splitter was continuously monitored. It was assumed that if the laser power, as measured off the beam splitter, did not fluctuate, the power was also constant in the Stark cell. The laser power measured before the Stark cell ranged from 17 mW to 37 mW from day to day, but only varied :2.5 mW within a day. 7.4. Computer Programs As in the other laser Stark experiments, the BOXM program was used for data acquisition. Typical input parameters were as follows: 200 channels with 3 sweeps: integration number (number of A/D converter readings at each field) from 10 to 30, depending on the signal strength; timing number (time between readings of the A/D converter 49 at each field) either 5, 10 or 20 centiseconds, always chosen to be at least 3 times as large as the time constant on the lock-in amplifier; delay number (additional time before the first A/D converter reading at each field) 5 or 10 centiseconds. A program called LDFIT3 was used for fitting the spectral data. This is a PDP-8 program which fits the background to a polynomial and the level crossing to a derivative of a Lorentzian line shape. The amplitude, halfwidth at half maximum, and center electric field of the Lorentzian as well as the coefficients of linear and cubic terms in the polynomial were varied in the fitting. A program LDFITZ, which fit the signal to a Gaussian deriva- tive was found to be a poorer way to fit the background than the polynomial. A program LDFIT, which fits the back- ground to a linear equation with a slope and intercept failed miserably for the zero field level crossing. Modulation broadening effects were investigated theo- retically by generating modulation-broadened zero field level crossing signals on the PDP-8. These curves, which were based on the polynomial plus a Lorentzian line shape, were fit with the LDFIT3 program. Theoretical line shapes were generated exactly by using the program LVXZ on the CYBER 750 computer, and fit to the polynomial minus a Lorentzian. The input required for the program LVX2 includes the Stark lepes 50 for all the different m states, the laser electric field amplitude, and a pressure broadening parameter. By vary- ing the laser electric field factor and the pressure broadening parameter, data corresponding to the experimental line shapes were generated. 7.5. The Sample The sample of carbon 13 fluoromethane was obtained from Merck and Company, Incorporated. The sample had 90 atom % isotopic carbon 13 and was used as received. CHAPTER VIII THEORY 8.1. Solution of the Density Matrix Equation of Motion The lineshape theory of the level crossing effect begins with the density matrix equation of motion(12'80) ihb = [HT.o] , (27) where the total Hamiltonian HT is, HT = H0 + HR + HC (28) and HR = - u° ER . (29) R is the contribution to the Hamiltonian C In Equation (28), H from the radiation, H represents the collisional relaxa- tion terms, and H0 is the molecular Hamiltonian in a static Stark electric field a. In the three level system dis- cussed earlier the crossing levels are labeled 1 and 2, and the upper level is labeled 3. The crossing Stark 51 52 field is denoted by EC and the corresponding transition energy by hwc. Then by the previous explanation of cross- ing and avoided crossing, H12 = 0 or H12 # 0, respectively. The elements for the density matrix equation of motion, in a basis in which H12 = O, are ‘ - :1 R _ R _ _ 0 p11 ‘ 11[H13p31 913H31] Y11(911 911) ' (3°) ' — :1 R _ R _ _ O p22 ‘ r1[H23932 023H32] Y22(922 922) . (31) _:sR R _ R_ R _ _o 033 ‘ 11[H31013+H32923 p31H13 932323] Y33(033 033’ (32) ° _:_i_R _ R0 _ 0_ p21 ‘ 11[H23p31 p23H31+H22921 plell] Y21921 ' (33) ' — :1 R R o _ o _ R _ p31 ‘ 11[H31911*H32921+H33p31 p32H11 p33H31] Y32932 ' (34) _ :i R R O _ o _ R _ ' 11[H31012+H32922+H33p32 p32H22 p33H32] Y32932 ' (35) 32 The phenomenological relaxation times Yll' y22, 733, 721, Y31: and 732 have been introduced to represent the col- lisional relaxation effects of HC. The applied laser radiator is assumed to be of the form, ER ER cos(m£t - kz) , (36) 53 then HR - k fi"-ufi€2C°s(w2t- z). (37) If _ o _ 0 ”£1 ' (Hff Hii)/h ' (38) u13 = “31 # 0 , (39) and “23 = U32 # O I (40) then the density matrix elements become is - __£ _ _ _ 0 p11 ’ h [“31‘031 plancoswzt Y11(911 911’ ' (41) is p =—-‘£[u (o -p )]coswt-y (p -p°> (42) 22 h 32 32 23 2 22 22 22 ' 033 = ‘If”[“31‘931’pl3)+“23(932-923)]C°swlt O Y33 33 2 31 13 2 23 32 33 33 33 ' (52) ix ix . _ _. 32— _ 31- _ p21 ' 1“21921 + ‘7T'931 “‘2"923 Y21921 ' (53) . . . ix . a - wt . - -lwt _ _. - -lwt 31 _ -th p31e ”lwp3le “ 1“319319 '+ 2 (911 "33)e ix . . 32 -1wt _ - -1wt + 2 0219 7319319 I (54) . . . . ix . - -lwt . - -lmt _ _. - -lwt 31 -lwt p32e 1“"3329 1“329329 *’ 2 p21e ix . . 32 -1wt _ - -1wt + .—-——2 (p22-p33)e Y320326 . (55) Then, it follows that iX 1 1X32 ; _ . _ 3 - p31 ‘ 1(‘*”‘*’3l)"31 + ”7F’(911’p33) + ‘Tf’pzl ' Y31931 ' (56) 56 . ix ' _ - _ ‘ __§3 _ p32 ‘ 1“” ”32’932 + 2 (922 933’ + ix _ ‘Tf‘pzl ' Y32932 ° 31 (57) Now, the density matrix elements are separated into pure imaginary and pure real components by letting pa = 011 ' p33 ' 0b = p22 ‘ p33 ' _ _ O ADi ' 0in. 0ii ' 531 = w ‘ ”31 ' 32 = w ‘ w32 ' I ' II pfi ‘ pfi + lpfi ' and recalling that Then, it follows that 0a = p11 ‘ p33 = '2X3lp31 ' YllApl (58) (59) (60) (61) (62) (63) (64) ’ X32932 + Y33Ap3 ' (65) 57 pb = p22‘933 = ‘2x32932 ' Y22A92 ' x31931 + Y33““3 ' (66) 5 = -iw (0. +10" ) + 1333(51 +15" ) - ::§§l(" -15" ) 21 21 21 21 2 31 31 2 p32 32 ' (67) z - - ix31 _ . ' . u . I ° u 031 ‘ 1531(p3l+1p31’ + _7T—pa + 1X32‘921+1921) - Y31(p§1+1p§l) I (68) . ix ix - - ° -' '-" __23 __£1 I _' n p32 ‘ 1532‘932+1p32’ + 2 ‘h>*' 2 (921 1921) Clearly, x x 'g _ n g 32 -n _ 31 -u _ I 021 ‘ leDZl ‘2‘ p31 2 p32 Y21921 ' (7°) x x 0" _ _ ' 32 -' _ 31" _ u p21 ‘ ‘021921 + ‘2‘ p31 “2'932 Y21921 ' ‘71) '1: ._. '0 -n X23 "n (72) p31 31931 2 21 Y31°31 ' . x x “n _ -I _;£ _§£ I -n 031 531931 + 2 pa + 2 p21 Y31931 ' (73’ I _ X31 _ I _ u n _ l 032 532932 + ‘2‘ p21 Y32932 ' (74) - x x _" _ —' 32 31 . _ -II p32 ‘ 632932 + 2 0b + 2 p21 Y32932 ° (75) 58 Next, pa and ob may be rewritten as pa = '2X31531 ' X32532 ' (Y112Y33) (A91+A93) (Y112Y33) (Aol-Ao3) . (76) and pb = '2x32532 ' x315§1 ' (Y222Y33) (A92+Ap3) - (Y222Y33)(Aoz-Ap3) . (77) But Yii-Yff z 0 and Api+Apf z 0 so that the product of these two terms is very small. Then, it is convenient to let (Yll + 133”2 = Ya (78) which represents the collisional relaxation of the popula- tion difference of levels one and three, and to let which correspondingly represents the relaxation of the population difference of levels two and three. 'Finally, the density matrix equation may be written in matrix form as 59 (80) 2t“ ll 2: II." + 20 where the transposes of L and C are _ -| -II II -n - (pa p31 p31 p21 021 pb 032 p32) ' (81’ and g = (yapa o o o o prg o 0) , (82) and the A matrix is a 0 2X31 0 0 0 0 X32 0 Y3l 631 0 X32/2 0 0 0 -X31/2 631 y31 -X32/2 0 0 0 0 0 0 X32/2 YZl -w21 0 0 X31/2 0 -X32/2 0 w21 721 0 X13/2 0 0 X31 0 0 0 Yb 0 2X32 0 0 0 0 -X31/2 0 y32 632 O 0 0 -X31/2 0 -X32/2 -632 y32 60 The steady state solution to Equation (80) is 2 = 5‘1 9 . <84) and the following is an outline of a method for solving this linear system. (A) First use rows 4 and 5 to solve for p51 and p31 and substitute back into the matrix. This reduces the 8 x 8 matrix to a 6 x 6 matrix. (B) Use rows 1 and 4 of this reduced system to solve for pa and ob. After substitution, the system is reduced to a 4 x 4 matrix. (C) The new 4 x 4 matrix can be reduced to a~2 x 2 matrix by solving for the real parts of B31 and 532' (D) Solve the 2 x 2 system for 5"31 and 532. The DOppler effect is an important contribution in infrared lineshape investigations; it quite often is the major contributor to the breadth of a laser Stark transi- tion. The density matrix elements are Doppler averaged as follows = [mW(v)pfidv (85) where 61 W(v) = exp(-v2/u2)/nRu , (86) with u2 = 2kBT/M . (87) In these equations, v is the velocity of the molecule in the direction of the radiation, T is the absolute tempera- ture, M is the molecular mass, and kB is the Boltzmann constant. In this investigation the calculation of the Dop- pler averaged density matrix elements was carried out by numerical integration in a frequency vector space rather than a velocity coordinate system. The Doppler frequency shift is w = w v/c . (88) Therefore, dv = (c/wg) dws (89) and the Doppler averaging is (pr> = / W(U)S)pfi de (90) where 62 W(wS) = exp(-w82/02)//?o (91) with o = AvD/VinZ == uwg/c ; (92) AvD is the usual Doppler half-width at half-height. 8.2. Absorption Coefficient The polarization (P) can be written as the sum of an in-phase (with respect to the laser radiation) component (PC) and an in-quadrature component (PS), P = PC cos(w£t-kz) - PS sin(w£t-kz) . (93) The absorption coefficient (a) is proportional to the in- quadrature component of the polarization, a = -(4nw/c)(PS/s£) , (94) The polarization for an N particle gas with mean dipole moment

is denoted as P = N . (95) Then, since 63 = tr(fi8) , (96) P = Ntr(fi 5) . (97) and it follows that for the three level system defined pre- viously, the polarization is P = N[ul3(+) + u23(+)]o (98) Recall that the density matrix elements are related by _. —I --u -iwt and _ * _ .(-l '1-" )eiwt (100) 013 ‘ p31 ' p31 031 ° Therefore, the in-quadrature component of the polarization is P = 2n[u135§1 + u23532] , (101) and the absorption coefficient is a = -(8an/C€£) (“13(031> + u23) . (102) 64 32 for. After velocity averaging to obtain <5 In the solution to the linear system 5 and 551 are solved II > - II 32 and <931>' these can be substituted into the absorption coefficient equation to give the lineshape for the simple level cross- ing system. 8.3. Multi Overlapping Level Crossings In the zero field level crossing studied in this investi- gation there are more than just three levels involved and the energy relationship between the crossing pair and the third level differs for differing sets of levels. But, the biggest problem is what to do with states that are multiply pumped (i.e., states that are simultaneously involved in more than one level crossing). As a particular example consider the case in Figures 5 and 6. Note that each of the states in the lower level is the third level for a crossing pair in the upper level and also one member of a crossing pair with the third state in the upper level. Clearly, the problem is not as simple as the theory developed so far might have indicated. And, if the laser radiation pumps off-resonance, still another parameter must be considered. The following assumptions were made in order to make the theory more tractable. First, since there is no way to separate the individual relaxation parameters, they are all treated as being equal; more eloquently stated, 65 Figure 5. A diagram of the multilevel zero field level crossing for the l3CH3F R(4,3) transition. 66 m musmflm wad .- m- n- lei. - m- n. v- m- _ H . :2 was as (m .2 oznmomo .msmj SurvoEN 67 13 Figure 6. A diagram of the multilevel CH3F R(4,3) level crossing in the Stark electric field. 68 ZERO-FIELD LEVEL CROSSING (J =5" 4) [11+ ’IV ”V 4 3 2 I _ O -| ._I '2 ‘3 '4 0 Figure 6 m “'5 4—4 J; 43 5 Amsil Ak=() AJ-+l -4 '3 '2 .Ifl 0 I4 2 3 4 8 H’ 69 an effective relaxation parameter is measured. The prob- lem of a state subjected to multiple pumping is handled in a more mundane fashion; the total crossing effect is treated as a sum of separate three-level systems. The last problem then is the off-resonance pumping. The dif- ficulty arises not in the generation of the theoretical line- shapes but in the analysis of the measured line shapes. The off—resonance generates a background line shape that is the sum of all of the transitions. Each transition is off-resonance in the radiation electric field by a dif- ferent amount, depending on the Stark slopes. Thus, some effort was required to determine a suitable form for the background on which the level crossing is superimposed. 8.4. Optics 8.4.a. Introduction It is necessary to know the laser electric field am- plitude (s1) in order to calculate the power broadening factor (Xfi). In order to calculate the laser electric field emplitude inside the Stark cell it is necessary to know the beam waist, the divergence of the laser beam, the focal length as well as the location of all the mirrors and optics, and the size of the openings in any diaphragms and their locations. A measure of the laser power and the knowledge of the laser mode, with a TEM mode preferable, 00 70 are also required. The physical dimensions of the laser determine the beam waist and the terminal Optics of the laser determine the spot size on the output mirror and the beam divergence. 8.4.b. The Beam Waist, Divergence, and Focal Lengshs The geometry of the electric field distribution, s£(x,y), of the TEM mode of a transverse laser beam is 00 approximately described by the Gaussian expression,(80) 2 2 s£(x.y) = so exp (- X_:ZZ_) . (103) W or r2 s£(r) = 80 exp (-—-2-) . (104) W Here, w is the beam waist, or spot size radius, and can be thought of as a measure of the standard deviation of the amplitude of the laser beam. The beam waist at the sur- face of the partially transmitting mirror at one end of the laser cavity is given by(80131) W4 = (AR/n)2/[(R/d) - 1], (105) when one mirror is flat and the other concave. The assump- tion here is that the grating on the laser can be represented 71 approximately as a flat mirror. The spot size then is a function of the radius of the curved mirror (R), the spacing between the mirrors (d), and the wavelength (A) of the radiation. The laser beam will diverge as it leaves the laser cavity and the spot size can be calculated at some distance 2 from the output mirror as follows: _ z(f—R) wz — w[l + ——§f_—] (106) where the focal length (f) is related to the index of refraction (n) of the material that the mirror is made of by f = R/(n-l) . (107) Then, it can be shown that wz = w[l + z(2-n)/R] . (108) The size of the diaphragms in the laser spectrometer (Figure 1) can be adjusted to limit the laser power going into the Stark cell. The radius of the laser beam (s) going into the Stark cell will be given by a similar equa- tion as that of the Spot size at some distance from the partially transmitting mirror, because the laser beam 72 will be diverging from the same source. Therefore, 52 = sd[l + z(2-n)/R] , (109) where sd is the size of the Opening in a diaphragm and s2 is the laser beam radius at some distance 2 from the diaphragm. The mirror that focuses the laser beam into the Stark cell is a concave mirror with a four meter radius of curva- ture. Once the beam strikes this mirror it no longer diverges at the rate determined by the output mirror, but instead converges according to 5b = 8m (l-zm/fm) , (110) where sm is the radius of the beam on the mirror, fm is the focal length of the mirror, and 3b is the beam radius at a distance zm from the mirror. Then, as it leaves this last mirror, the beam waist will also converge at the same rate as the restricted beam. Therefore, the spot size or beam waist at some distance zm from the focusing mirror is wzm = wm (l - zm/fm) , (111) where wm is the beam waist at the mirror obtained from Equation (108). 73 8.4.c. Laser Power and Electric Field Amplitude To determine the electric field amplitude (sfl) of the laser, the laser power (P) is measured at some distance 2 m from the focusing mirror. The quantities P and s£ are related by EECA Pg = 8" (112) where c is the speed of light and A is the cross-sectional 2 area of the beam. In this equation P and s1 are average values. In conclusion, the average power is measured from which the average electric field amplitude can be calculated, since the cross-sectional area of the beam can be de- termined once the limiting Spot size has been calculated. 8.5. Transition Dipole Moments The transition dipole moments (ufi) are determined as follows,(82) * “fi = fo u dgdg (113) ywi where the integration is over the normal coordinate space Q and the rotational space a. The y component of the dipole ~ moment is needed here because the direction of the Stark 74 field has been chosen to be the z direction and the laser electric field is polarized perpendicular to the Stark field. The total wave function (W) for each state is the product of a rotational and a vibrational wave function, VR . . . (114) 3W3 The initial state vibrational wave function (8:) is (I) . v— 1 - £¢Vm(Qm) . (115) where vm = 0 for all m in the transition studied here. Similarly, for the final state vibrational wave function, = 11 (Q ) . (116) If n ¢vn n where vn = 0 for all n except v3 = l. The rotational wave functions are (I)? = wR (JIIKIIMII) (117) and I? = IRIJ'K'M') . (118) The molecular dipole moment (my) is 75 (119) 'E II pacose a + (lbcoseyb + uccose Y yc ' or u = ua + ub + uc . (120) where the a, b and c refer to the principal molecular frame of reference coordinates and the x, y, and z the laboratory frame of reference coordinates; in both vector spaces the origin is at the center of mass. The dipole moments ua, ub, and “c may be expanded in a Maclaurin series of the normal coordinates (Qk), as follows: . (121) Then, the transition dipole moment (ufi) is C + (uylfi . (122) _ a b in which Bu 9 = * R . . . o g 3 R II II n . E wvéQ£)w (J K M )dgag . (123) Finally, Equation (123) reduces to 76 g _ O R I I R II II II I II (“y)fi — ugfw (J K M')coseygw (J K M )dgég 'Y + Bu * J R I I I R II ll " i (aoklofwvi(Qi)inonifw (J K M )coseygw (J K M )de. (124) For the R(4,3) transition in the v3 band of 13CH3F, the vibrational integral multiplying u; vanishes for all g and the only non-zero (Bug/30k)o is (Bug/303)o. Therefore, an _ __g R R “fi - (303)fwv3=10ewv3=odgfw (sgiM11)coseyaw (4fi1M)de (125) and<67) 1 5 II _ Q I _ do = (———) , (126) v3—l 3 v3—0 2y3 in which — 4 2 /h (127) Y3 — 1T V3 . Also, 2 (:g—a 2 = 12,3532 . (128) e n 3 where F3 is the area of the absorption band for the transi- tion of frequency v3 with degeneracy d3 = l. Also,(56) 77 wa(J+l,k,M:l)coseya¢R(J,k,M) = 2_ 2 k + + % L(J+l) K 1 [(J_M+l) (J-M+2)l (129) + 8 2(J+1)[(2J+1)(2J+3)] 8.6. Wall Broadening from the previous section on laser power and beam di- vergence the dimensions of the beam inside the Stark cell can be determined. The geometry of the beam in the cell then is a frustum of a right circular cone with the base of smaller radius at the far end of the Stark plates and the altitude equal to the length of the Stark plates. The wall broadening relaxation parameter can be determined by(82) _ A RT 5 ' Aww b. — V'(—_§') . (130) 81rM where T is the absolute temperature, M is the molecular mass of the sample, and R is the gas constant. The total sur- face area of the frustum (A) is(83) 2 _ 2 2 2 8 ~ A — n{Rl + R2 + (R1+R2)[(Rl-R2) +h J } . (131) and the corresponding volume (V) is(68) 78 -_1_ 2 V — 3irh(R 1+R 2 2+R1R2) . (132) where h is the altitude, R1 is the radius of the larger base, and R2 is the radius of the smaller base. CHAPTER IX RESULTS AND CONCLUSIONS 9.1. Stabilization Test Results In all of the previous COz/NZO laser experiments per- formed in our laboratory, the laser was stabilized by a dither technique. The piezoelectric translator (PZT), which supports the partially transmitting mirror, is forced to oscillate at 520 Hz by superimposing a sine wave voltage on the high DC voltage that is used to expand or contract the translator. The laser output power is detected and processed by a lock in amplifier which provides a feedback signal to the PZT. The signal adjusts the cavity length to keep the laser frequency at the top of the laser gain profile. In the present experiments a new stabilization technique (at least to our laboratory) was used. The laser was locked to the minimum in the saturation dip in the fluorescence from CO2 in a sample cell in the laser cav- (84-86) A comparison of the quality of the two methods ity. will be discussed here. Both laser stabilization methods were used to measure the laser Stark Lamp dip of a CH3F Q(l,l) -1+0 transition. In both cases several measurements were made in a single 79 80 day, as well as from day to day. The data were collected on the PDP8/E cOmputer with the BOXM program and the LDFIT program was used to fit the line shapes. What was of in- terest in these experiments was the reproducibility of the center frequency or center Stark electric field, of the selected transition. Also, free running laser data were taken for comparison. In the fluorescence stabilization method the center Stark voltage of 1325.95 volts has a standard error in the fit of 0.09 volts within a day and 0.19 volts from day to day, while the dither stabilization method has a standard error in the center voltage (1322.59 volts) of 0.91 volts within a day. The free running laser Lamb dip data has a standard error in the center voltage of 1.02 volts within a day and 2.16 volts standard error in the data from day to day. Figures 7, 8, and 9 are oscilloscope pictures of the Q(l,l) -l+0 transition, one for each of the stabilization methods. It was noted in the course of analyzing the data that the dither stabilization also locks the laser to a center voltage m3.4 volts lower in field than the fluores- cence technique. In conclusion, the laser Stark center voltage is reproducible to a center position with ten times greater accuracy when the fluorescent stabilization is used rather than the dither method; also, the signal to noise is better (see Figures 7, 8, and 9). 81 Figure 7. A photograph of an oscilloscope plot of a data file for the Q(l,l) m = -l+0 Lamb dip transition in methyl fluoride under the ex- perimental conditions of fluorescence stabiliza- tion. Figure 8. A photograph of an oscilloscope trace of a Lamb dip data file for the Q(l,l) m = -l+0 transition in methyl fluoride under the experimental con- dition where the laser was locked on tOp of the laser gain profile. Figure 9. A photograph of an oscilloscope trace of a Lamb dip data file for the Q(l,l) m = -l+0 transition in methyl fluoride under the experimental condi- tions of a free running laser (unlocked). 82 I .J / Figure 7 3 l .1. 0" 0 °'\« . 2.. :5 \‘-‘N~\ I Figure 8 : \‘:‘ : 1‘ ..* - “W 83 9.2. Modulation Broadening When small amplitude modulation techniques are used where the experimental information is dependent on the line shape, it is very important that the modulation does not distort the line shape. In order to determine whether modulation broadening is contributing to the line shape in the present experiments, some computer experiments were performed in which theoretical modulation-broadened line shapes were generated and fit. The modulation amplitudes and the parameters used in the calculated line shapes were comparable to those obtained for the experimental data. The computer program used to generate the modulation broadened data is a PDP-8 program called MODLVX, an abbre- viation for modulation broadened level crossing. A com- parison of these results of the theoretically generated data with that of the fits of experimental data taken under conditions of comparable pressure broadening is represented in Figure 10. In this figure, the derived value of the pres- sure broadening parameter is plotted against the modulation amplitude in volts. It should be noted that the modulation broadening for the highest modulation amplitude is still less than the scatter in the experimental data. Also, the total length of the ordinate shown is only m8% of the modulation broadened value of the half width. It was. concluded that modulation broadening is not a significant contribution to the half width in these experiments. 84 Figure 10. A graphical representation of the effects of modulation broadening on the pressure broaden- ing parameter. 85 ca ousmflm A 2.9, V “53:828. 20:52.02 £9. and 3.0 ond omd one 8.0 luHJfiJ—-_qu1ud4fi1-—_uu-—--q—-udd—_--—~.dq—1_ddfiqujddW—-l H x ..o « m.om~ co o.=..oeeoe “ W Leo..:.:_ m~.o a. na.¢m .6 otamaote A mod .1. aco:_ucoo .oacoE_..oaxu *7 I U HI 1.. .I... mwd n u .I Xdu Mn v I. . H. m 0 x x (u omo I O x i 1 xxx X J—V i 1 O X 4V 4% l . rl. X ll #00 T l l X ..1 n I m . n. -r .H 8.0, V -- I nwo r 1 W I ¢Qo n M W ¢ mwo Iprb——FIEPILF——I—P-—InIp—phwk—Pbppphppp—L_Ih—bb_——_—__—_——bL OZ_ZMQIn/a (p2>0ut/b <8295/: <82>%/d Date (mW) (mW) (Volts/cm) (Volts/cm) Oct. 17.5 1.0 30.0 15.9 Oct. 33.9 10.0 41.7 27.0 Oct. 21.6 8.0 33.2 22.8 Oct. 25.4 7.5 36.1 23.4 Oct. 29.1 11.0 38.6 26.6 aAn average laser power before the Stark cell. bAn average laser power after the Stark cell. cr.m.s. electric field amplitude in the center of the Stark cell, assuming no losses at the windows. dr.m.s. electric field amplitude in the center of the Stark cell assuming half the laser loss in passing through the cell occurs at the first window. Figure 11. 92 Plot of the pressure broadening gs. pressure for the R(4,3) zero field level crossing in the 0 band of 13CH F. The dotted line represents tge linear leasf squares fit of the experimenta data and the dashed line the linear least squares fit of the theoretical data. 93 Ha whomflm A to. 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Eu\n:o> on .o 5...; 3:25 .16.... ism o - I l as I J I 0.w 5 Montana... 1 1 com... 02:05”. «mg... ”1.: ..o 20:02:... < - - m< .393 .35.. z<.~ezumo._ u...— ..o zimm .msam .umxmewonm .umzwum .mucmaflummxo mmwnem .xmemac womaummea .mm .3 .xusooz .o .n .mfluuoz .q .o .comumwm .m .m .Hmuumn .mo n w.on.nH oom N.N+N.m mommnm mafia ~.5H com H.~+H.m mm>mz “mumsflaaflz N.ea com o.~+o.m ammo monom m>m3ouoflz om III o.o+o.a mmmu mcofiumuomnm unmflmcmue ma III H.H+H.N mmmo Ummmnm mafia w>mzouoflz om oom o.o+o.a mmmu mocmcommu cannon m>m30uoflz m.ow~.ma III m.v+m.m opwumnucfl pm>a0mwu mafia m.ow~.mH III m.m+m.v Awmov mmmUma cacaumpsc Hmoflumo m.ma In: m.¢+m.m Awomv mmmomH uncommouo Hm>mH cameo oumm a.ono.mfl mam m.e+m.m Awoae mMmUMH ponumz Annou\nmzv Amy mo.~Mh masooaoz \ucmfloflmmwou .meB cofluflmcmne mcflcmpMOHm musmmmum .mmmu How wumumfimnmm mcflcwpmonm musmmmum mo comfiummeoo .HH> THEME 104 .xmmmav mmamnomam .me ..msnm .amco .n .xmsm .m .n .cmnoo .m .m .esmncuflm .om .Amemav ammuhmm .oe ..uumq .msnm .Emso .mxomz .m .ccmuqu .6 .xnmeuoao .mm .ommH ofiso .msn IESHOU .wmoomouuommm “masomaoz co EsflmomEMm summ .mxomz .m .uHsmum .m .b .uumsom .mm .xmemflv eaoa .mM .>mm .msnm .mcuoo .mcum3cm .EMflHHflow .omsceucoo .HH> wanna CHAPTER X FINAL COMMENTS In this chapter the two types of experiments performed in this investigation are compared. The strong as well as the weak points of line shape studies by both laser Stark spectroscopy and level crossing spectroscopy will be pointed out. Then a possible experiment that would have as many strong points as both techniques with fewer weak points than either will be described. The measurement of a single laser Stark line shape is a relatively easy experiment, but in most cases it can be quite difficult to acquire a set of line shapes at reason— ably different sample pressures. This is due to the fact that at moderate to high Stark fields electrical discharges in the sample cell cauSe extreme difficulties. That same Stark field can be beneficial in removing the degeneracy of a level, allowing only one transition to be studied, except for the unfortunate situation where overlapping transitions are encountered. Laser Stark line shapes can, and in most cases are, studied at low enough laser power that power broadening or saturation of the transition is not a significant contribution to a profile. Unfortunately, 105 106 separation of the inhomogeneous contribution from the homogeneous broadening mechanisms in the line shapes can cause problems, and the ever present modulation, that may cause modulation broadening, must be considered. In level crossing and avoided crossing, which are satura- tion phenomena, there is no Doppler contribution to the line shape that must be removed. But rather, the DOppler broaden- ing is a major contributor to the background on which the desired signal is superimposed. Unfortunately, this back- ground is not linear. Since there is no Doppler contribu- tion to the recorded profile, the level crossing signal is usually narrower than a laser Stark line shape, allowing for a smaller sweep width and higher resolution. In zero field level crossing experiments, like the one of concern here, there is no problem with a Stark field breakdown, but rather the difficult problems of power broadening and overlapping transitions have to be taken into consideration. Proper handling, measuring, and accounting for the laser electric field amplitude are essential. A major drawback for level crossing and especially zero field level crossing experiments is that they are rare. Transitions whose resonant frequencies are coincident with laser lines are not as common as laser Stark transitions. Again, as in the laser Stark line shape experiment, modulation broadening can be a serious problem. A more ideal experiment then would be one in which all 107 of the strong points of both methods, with none of the weak points of either (except for the ubiquitous modulation broadening), are present. 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