My...» _, ‘ ,A " rm.‘.‘.. ., ‘ ‘ ELECTRIC FIELD lNDUCED SPECTRA 0F LENEAR AND AXIAL“ S‘t’teEMEYRSC MQLECULES Thesis for the Uegree 05 Ph D. MICHEGAN SYATE UNIVERSITY PETER LEE WELLSGN I369 THEbI: ‘ I I?! ‘-..' l L. - LIBRARY Michigan State University This is to certify that the thesis entitled Electric Field Induced Spectra of Linear and Axially Symmetric MO 1e cu le S presented by Peter Lee Willson has been accepted towards fulfillment of the requirements for WW Major professor Date July 8, 1969 0-169 ABSTRACT ELECTRIC FIELD INDUCED SPECTRA OF LINEAR AND AXIALLY SYMMETRIC MOLECULES BY Peter Lee Willson Selection rules and line strengths have been calculated for electric field induced spectra (EFS) of axially symmetric and linear molecules. A suitable optical arrangement and appropriate experi- mental techniques have been developed. Experimental observations of EFS of the linear molecules HCN and DCN and of the axially symmetric molecules CH 1, CH C1, CH F, and CH CN were carried out and 3 3 3 3 3 found to exhibit many of the predicted features. Br, CH ELECTRIC FIELD INDUCED SPECTRA OF LINEAR AND AXIALLY SYMMETRIC MOLECULES By Peter Lee Willson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1969 / 3 To My Wife Ingrid and Daughter Susanne ii ACKNOWLEDGMENT I wish to express my sincere gratitude to Dr. T.H. Edwards for his guidance, encouragement, and numerous helpful suggestions throughout the course of this research. His constant availability was deeply appreciated. I would also like to thank Drs. P.M. Parker and C.D. Hause for their encouragement and help. The many useful suggestions, eSpecially concerning experimental portions of this work, of my co-workers Richard Peterson, Richard Blank, and Lamar Bullock have also been greatly appreciated. I am very much in- debted to the National Aeronautics and Space Administration for the financial support given me in the form of a fellowship. Finally, I want to express my heart-felt thanks to my wife, Ingrid, for her patience, understanding, and constant encouragement and to my parents whose timely help made this all possible. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES Chapter I II III IV VI INTRODUCTION THEORY OF ELECTRIC FIELD INDUCED SPECTRA EXPERIMENTAL TECHNIQUE ELECTRIC FIELD INDUCED SPECTRA OF HCN ELECTRIC FIELD INDUCED SPECTRA OF AXIALLY SYMMETRIC MOLECULES CONCLUSIONS BIBLIOGRAPHY iv Page vi 44 56 83 95 97 Table (2. (2. (2. (2 (2. (2. (2. (4. (5. 1) 2) 3) .4) 5) 6) 7) 1) LIST OF TABLES Character Table for C . 3v Direct Product Multiplication Table for C3v' EFS Selection Rules for C Symmetry Molecules 3v Line Strengths for AK = 0 Electric Field Induced Transitions. Line Strengths for AK Induced Transitions. ‘il Electric Field Line Strengths for AK =‘:2 Electric Field Induced Transitions. Line Strengths of EFS Type (c) Transitions EFS Frequencies in v and v + v - v of HCN. 3 2 3 2 Measured Ratio of v4 and Ground State Electric Dipole Moments for CH3F. Page 21 21 25 26 28 31 39 74 94 ’4‘ 1‘ Figure (2. (2. (2. (2 (3. (3. (3. (4. (4. (4. (4 (4. (4. (4. (4. (4. (5 (5. 1) 2) 3) .4) 1) 2) 3) 1) 2) 3) .4) 5) 6) 7) 8) 9) .1) 2) LIST OF FIGURES Angular Momentum Components of Axially Symmetric Molecules Relationship Between Space-Fixed Coordinates and Molecule-Fixed Coordinates Symmetry of Doubly Excited Degenerate C Vibration 3v Electric Field Induced Line Shapes General EFS Experimental Arrangement Stark Cell Fore-Optical System Normal Vibrational Modes of HCN + - EFS of v3 and v2 v3 v2 Bands of HCN Overlapping in P(l) Transition in v3 of HCN Transitions in H - H Band of Linear Molecules Energy Level Diagram of U - H Transitions in a Linear Molecule Transitions in H - 2 Band of Linear Molecule EFS of v2 + v3 and 2v2 + v3 - v2 Bands of HCN Overlapping in EFS R(O) Transition in v2 + v3 Band of HCN Tracing of Low Pressure HCN v2 + v3 Band Induced Q Branch d I v1 and v4 Ban 5 of CH3 v1 and v4 Absorption Bands of CH3I EFS of vi Page 10 15 24 41 45 47 52 57 61 66 68 71 76 79 8O 80 86 87 CHAPTER I INTRODUCTION This thesis is concerned with the theory and experimental observation of electric field induced Spectra (EFS) of linear and axially symmetric molecules in the near infrared region. Qualitative features of the theory and method of experimental observation are outlined in this chapter. A short description of the Stark effect is included because of its influence on electric field induced line shapes and line strengths, and finally the Kerr effect is discussed for the sake of completness. Chapter 11 contains a detailed development of the theory of electric field induced spectra for axially symmetric molecules with linear molecules included as a Special case. The remainder of this work is devoted to a description of the experiments performed and the interpretation of EFS obtained for the linear molecules HCN and DCN and for the axrally symmetric Br. and CH:I. molecules CH 3 .F, CH 3 3 ELECTRIC FIELD INDUCED SPECTRA In 1932 E.U. Condon (1) proposed production of infrared molecular Spectra by application of an electric field. He pointed out that the resulting Spectra should exhibit selection rules differing from those of field-free absorption spectra. Infrared vibrational absorption Spectra arise from changes in the electric dipole moment due to excitation of one or more 1 quanta of vibrational energy. The probability of a radiative transition occurring between two vibrational states is proportional to the square of the matrix element of the electric dipole moment between the two vibrational states. If the molecule is placed in an electric field, another contribution to the electric dipole moment is induced which is related linearly to the vector components of the applied field through the molecular polarizability tensor. The probability that a different radiative transition may occur while the molecule is in an electric field is pr0portiona1 to the square of the matrix element of the induced dipole moment. Because of the linear relationship between the induced dipole moment and the field, the intensities of the new radiative transitions are prOportional to the square of the magnitude of the electric field. Inducing an electric dipole moment in a molecule has three important consequences. First, if a molecule has no dipole moment in any vibrational state (such as the homopolar diatomic molecules H2, D2, 02, etc.), an otherwise unattainable infrared absorption spectrum may be induced. Crawford and Mac Donald (2) and Terhune and Peters (3) produced infrared H spectra by this method. 2 Secondly, molecules either possessing a permanent dipole moment or having a dipole moment in particular states due to vibration (such as 002) have possible radiative transitions that are not otherwise allowed because in this case selection rules are de- termined by matrix elements of the polarizability tensor instead of by the dipole moment vector. Finally, the intensities of even those transitions allowed in the absence of the field are modified by the presence of the field. The usual technique of detecting non-induced absorption spectra is to interrupt the absorption signal periodically with a mechanical light chopper to provide a pulsed signal for phase sensitive detection and amplification. On the other hand, for EFS, the absorption coefficient is proportional to the square of the applied electric field, an ac electric field produces an "internal" modulation at twice the applied field frequency. Omission of the mechanical light chopper and application of an ac electric field, therefore, produces a signal at the detector made up of a large dc component corresponding to field-free absorption and a small ac component at twice the field frequency corresponding to induced intensity changes. Phase sensitive ac detection and amplification of this "modulated dc" signal eliminates the large dc component and leaves only that part of the absorption signal arising from the field. The ac signal is the difference between the signal due to absorption with the field on and the signal due to absorption with the field off. This "difference Spectrum” technique of detection of electric field induced spectra was used in the experimental portion of this work. STARK EFFECT The Stark effect describes the changes in rotational energy of a molecule due to the interaction of the molecular dipole moment with the applied field. The interaction Hamiltonian is given by: = _ 1.1 Hint u F cos 9 ( ) where u is the electric dipole moment, F is the magnitude of the electric field, and e is the angle between F and A. In the absence of an external field, the molecular electric dipole moments have random spatial orientations and the rotational energy levels have an essential degeneracy. The application of an electric field defines a spatial direction along which the dipole moments tend to align through interaction with the field thereby lifting the rotational degeneracy. Therefore, a transition between two previously degenerate rotational energy levels would exhibit fine structure in the presence of an electric field if sufficient resolving power were available. If the wave functions for a molecule in the presence of an electric field were known explicitly, the change in energy of any rotational state could be calculated exactly. Unfortunately, this is never the case and approximate methods similar to those described below must be employed. The most common treatment of Stark energy shifts for polar axially symmetric molecules is the weak field approximation. In this approximation, the interaction Hamiltonian is assumed to be small in comparison to the separation of rotational energy levels and quantum mechanical perturbation theory is invoked, c.f. (4). To second order in the field, the change in energy is given by perturbation theory as: 2 A, = Les. +12F2{o2 - K502 -M) 'J(J+1) 2th J3(2J _ 1) (2.] + 1) _ [(J + 1)?- — xzito + 1)2 - M21} (1.2) (J + l)3(2J + l) (2.] + 3) where A is the permanent electric dipole moment, F is the magnitude of the electric field, B is the inverse of the principal moment of inertia about an axis perpendicular to the molecular symmetry axis in units of cm- , h is Planck’s constant, and J, K, and M are the rotational quantum numbers (J is the total rotational angular momentum quantum number, K is the projection of J along the molecular symmetry axis, and M is the "magnetic" quantum number - the projection of J along the direction defined by the electric field). For a linear molecule possessing a per- manent electric dipole moment, K = O, and no first order Stark effect is possible. The above formula (1.2) for a linear molecule simplifies to AB = L12 F2 {JO + 1) - 3M2_} (1.3) 2thJU + 1) (ZJ - 1) (ZJ + 3) An exact solution of the Stark effect problem is available, but is not in closed form. This method is attributed to W.E. Lamb and a detailed discussion is given by Maker (5). The energy expression is in the form of an infinite continued fraction. Shirley (6) has numerically evaluated thlS expression for the few rotational states through J = 4. KERR EFFECT The Kerr effect refers to the optical birefrigence exhibited by gases, liquids, and solids in the presence of an electric field (electro-optical birefrigence). The observable in Kerr effect experiments is the phase difference between electric vector com- ponents parallel and perpendicular to the electric field, FZ, of a light beam initially plane polarized at an angle of 450 with reSpect to FZ after passing through a medium of length L. The theoretical details of this effect are given by Buckingham (7) and Dows and Buckingham (8). The following discussion is taken from those papers. The difference in phase 6 of the electric vector components parallel and perpendicular to the field direction of initially plane polarized light after passage through the gas sample is given by: 6 = 2nL(nZ - nX)/l = ZWLBF (1.4) 2 where L is the path length, l is the wave length of the radiation in vacuo, (nZ - nx) is the difference between refractive indices of the medium parallel and perpendicular to the field, respectively, and B is the Kerr "constant". The Kerr constant exhibits con- siderable'dispersion. For frequencies of incident radiation far removed from those of strong absorption bands of non-isotropic polar molecules, the dominating process in the dispersion of the Kerr constant B is the tendency for the molecular dipole moments to align along the field direction. Dispersion of the Kerr constant in this case is strongly temperature dependent (1-2) and its measurement leads to a knowledge of the anisotropy, (QH - 0'). of the polarizability tensor aij' The Kerr constant for frequencies of incident radiation near those of strong absorption bands, on the other hand, is temperature independent and the dominating contribution to dis- persion of the "constant" B is due to distortion of the molecule by the electric field. Quantitative measurements of the diSpersion in this instance would provide values of excited state electric dipole moments. Qualitative observation of the dispersion in this region would provide information on the direction of the transition j dipole moment with respect to the molecular axes as the increase or decrease of the Kerr constant with increasing frequency in the vicinity of an absorption line is governed by the direction of the transition moment. The phase difference 6 is proportional to the square of the magnitude of the applied electric field (1.4) and the same ac experimental detection techniques could be used for Kerr effect experiments as are used for electric field induced spectra. Some method of taking into account the polarizing effects of a long narrow Stark cell would be necessary for successful Kerr effect data interpretation. GMHERII THEORY OF ELECTRIC FIELD INDUCED SPECTRA The theory of electric field induced spectra will now be con- sidered in detail. The line strengthsof transitions not allowed in the absence of an external field and the corrections to the line strengths of transitions allowed in the absence of the field will be calculated for axially symmetric molecules and particularized for linear molecules as needed. The calculated line strengths are given in Tables 2.4 through 2.6. Selection rules obeyed by the rotational quantum numbers will be derived qualitatively and then rigorously. Finally, expected line shapes will be described. LINE STRENGTHS As a starting point, the integrated absorption coefficient k(v) of an absorption line will be needed; Fcr a transition from state m to state m’, this coefficient is k’ 'm1 - Nm . my . w)m .- h E (2i) . v , 2.+l mmE m where Nm is the population of the mgh state, vmml is the frequency of the absorption line; and h is Planckgs constant. 82? is the Einstein coefficient of induced absorption, which can be calculated using time-dependent perturbation theory. For absorption of radiation polarized in the Spacewfixed 5 direction (X Y, or Z) by a gas of axially symmetric moleculesS the Einstein B coefficient is given by T',V',R' T,V,R (2'2) 3 _ 8n | hzc E where the unprimed quantities refer to the lower state and the primed quantities refer to the upper state. The electronic and vibrational quantum numbers are symbolized, respectively, by T and v. R is used to abbreviate the rotational quantum numbers J, K, and M; where J is the total rotational angular momentum quantum number, K is the quantum number of projection of the total angular momentum along the molecular symmetry axis, and M is the quantum number of the projection of the angular momentum along the Spatial axis defined by the electric field. is the ”s electric dipole moment operator along the space-fixed § (X,Y,Z) direction and is the transition moment for radiation polarized in the Space-fixed g direction. The angular momenta are shown graphically in Figure (2.1). The problem of calculating intensities of transitions allowed in the presence of an external electric field is therefore one of evaluating the line strengths ‘ 2 in a suitably chosen basis set of perturbed eigenfunctions. Using rigid rotator harmonic oscillator wave functions as the zero-order basis and ’“ze as the perturbation, the perturbed eigenfunctions (to second order in the magnitude FZ of the electric field applied along the space-fixed Z-direction) have the form 10 0 Figure 2.1 Angular Momentum Components of Axially Symmetric Molecules. P 3 ,/JZJ+I$ total angular momentum. projection of P on molecular symmetry axis. projection of P along field defined spatial Z axis. 11 Z: <¢II’VII,RH|pz‘T’v’R> YT ,V,R(FZ) YT,V,R + F2 n n u Pawn" A YTII,VH’RH T ,V sR 11ch V R , 9 2 I Z. <‘f",V",R"‘uz‘T,V,R> + F 3—— TV j..— Z n u n In ”I I" 2 2 an’vfif’nun Tusvu R l R , 3 T ,v s T ,V R h C T,V,R VT,V,R ' YT"',V"',R"' 2 .4 l P 2 I ‘‘ y (2 3) 2 Z ¢",v" R" h2c2(v:":zv;’R")2 l’v’R ' where the prime on the summation indicates exclusion of the state T,v,R. Using the perturbed wave functions (2.3), the transition moment has the form: = <-r,v,R|ug|1-' ,v' ,R')o +<'r,v,R‘ug"r',v',R'>F (2.4) 2 where <7,v,R|ug‘T',v',R'>o is the transition moment of transitions allowed in the absence of the field (unperturbed transition moment) and F includes the correction to the transition 2 moment due to the field for transitions allowed in the absence of the field, as well as the transition moment of transitions not allowed in the absence of the field. This work is concerned with the second term of (2.4) which is given by: 12 F = z 5" <1,v.R‘uz|T",V",R"> F ___, ' Z n n n T‘fiV" ,R" T ,v ,R ' thT,v,R Z: <1.’V’R‘uglT":V",R"><'T",V",R"‘uz|T I ,v' ,Rl> + F2 n n n TII.VII.R" (2'5) T 9" sR hcv¢l’vl’RI to lowest order in F2. The doubly primed quantities refer to the intermediate states through which the unprimed (lower) and primed (upper) states are coupled by the perturbation, -szz. Dows and Buckingham (9) have shown that the terms in the sums of equation (2.5) can be separated into three types, viz., = + + 2.6 ‘”§‘ Y; 5; l; ( ’ with the following conditions imposed on the terms: Part yg: T",V" = ¢,v in first sum of (2.5) T",v" = T',v' in second sum of (2.5) Part 6g: T" = T in first sum of (2.5) r" = T' in second sum of (2.5) Part kg: T" f r in first sum of (2.5) T" f T' in second sum of (2.5). Part y; results from the mixing of rotational wavefunctions within the same vibronic state (vibrational and electronic quantum numbers are unchanged) through the perturbation; part a; arises from the mixing of rotational wavefunctions belonging to different vibrational states but within the same electronic state; and part lg results from the mixing of rotational wavefunctions belonging to different 13 electronic states through the perturbation. Because of the relative magnitudes of the energy denominators, >>6g>>>xg in all cases V; where the Born-Oppenheimer approximation is valid. For polar molecules, therefore, Y; will predominate, whereas for non-polar molecules, 1; is the only non-zero term. The explicit forms of yg, 6g, and Kg were given by Dows and Buckingham (9). TI RH . , #R hcvR z! } (2 7) 2' (TsVsRIl-lvz|TsvsRHXTsVsRn‘HE‘T ' 9". 9R.) u I RW R *R hCVRI <1.,v,R|uz‘T,VII’RHXT,VII’RH‘u ‘TI,VI,RI> 5; . Fz / v" RII g v"*v R hcv ’ v,R +2 2 <7, v, £5“. ,v", R"><1" ,v", R"‘u.zl'r' ,v "R > v" I u VwaRfl— (2.8) fV R hcvv V',R' <5, 3V:Rluz| Tll’v"’Rflx,rll,vll’Rll‘ug|TI ’VJ ,RJ) :3 F2 _ t TIZIfi VII R TII’VIT’RjI hch” v, R 2' Z (‘1', v ."R‘ugh ,v", R',v‘><'r" " ,'R"‘uz|ur ,v' ,'R > + ' ' TI v" R“ " (2'9) Tlfi V",R" th¢:',:v R. From these expressions (2.7), (2.8), and (2.9) the selection rules obeyed by the rotational quantum numbers for transitions in the presence of an external electric field may be deduced. kg is an expression for the matrix elements of the polarizability tensor of an axially symmetric molecule (10). Therefore, in the case of a non-polar molecule where 1g is the only surviving term, the selection rules obeyed by the rotational quantum numbers in electric field 14 induced spectra are the same as those of Raman scattering. In this case, electric field induced spectra can be thought of as Raman spectra in the limit where the frequency of incident radiation approaches zero, as B.U. Condon (1) pointed out. Since y; and a; are the same functions of the rotational matrix elements as 1;, electric field induced spectra obey the selection rules of the Raman effect in general, but the intensities are different due to the energy difference denominators in v; and 6;. x; is the transition moment for the Raman effect and the line strengths for this effect have been given by Placzek and Teller (11). In all cases, evaluation of the matrix elements may be accomplished by expressing the dipole moment operators in terms of molecular-fixed coordinates: . . Z 2. ”A a-x,y,zaA°'u" ( 10) where A ==X.Y,Z (space-fixed coordinates), and the ‘Aa are the direction cosines between the molecule-fixed coordinates (lower case subscript) and space-fixed coordinates, whose matrix elements are well known (cf. Strandberg (12) or Townes and Schawlow (4)). The relationship between the two coordinate systems is shown in Figure (2.2). POLARXHDLBCULBS When the relevant transition moment y; for molecules having a permanent dipole moment (which must lie along the (2) symmetry axis) is written in terms of molecule-fixed coordinates, each summation in equation (2.7) becomes a summation over nine terms. _J‘ (a. Figure 2.2. 15 Relationship between space-fixed coordinates X,Y, and Z and molecule-fixed coordinates x,y, and z. 16 A convenient separation of yg into two terms characteristic of the type of transition involved is possible. a 3 2.11 vg Hyg +‘iYg ( ) flyg includes all terms containing ”2 and p;, and Lyg includes all terms containing ”x = “y and u; = u;, where the “a and p; are dipole moment expectation values, viz., .0 «mums.» and .4 maximum». Finally, the fly; and Ly: can then be further separated according to the selection rules on K, which may be determined by inspection of the direction cosine matrix elements, so that = 1‘ KA'=K K'gKil . ‘hg ( Vg) + (ME) (2 12) K’=K K'=K+1 K'=K+2 = . _. - .13 .L'Yg (mg) + (ix/g) + (.L‘Yg) (2 ) where . K’=K (I ) = F 'T.v r',v'> u ‘2: -;7 n ‘Yg Z < 1112‘ { z R"#R hcv: ' “NZ ' g“ E" z‘ } (2.14) z R"#R' hcvR. (“Yg)x =x11 a Fz «’V‘H‘x ‘1’ ,v '>{fl.zz 22‘ RM R"fR hcvR ' <11" a R' > +p' 'Z | z“ 1“" fi—E—iL—} (2.15) 'R"#R' hcv RI and l7 K'=K, + (iY ) K-2 = 5 I 'R a + a R' R" a + a 3') F2 {~ ”is |( Ex Zy)| '>: |( g; Ex)l X 11'ch 11ch +2 ux} (2.16) RniéR' thR. '= (1y§)K Kil = ' FZ {uxz Zx Z! R? :2 R"#R hcv R ' +u'z | 5‘ L. z" 2’ }. (2.17) x R"#R' hcvR, The selection rules on J, the total angular momentum, in equations (2.14) through (2.17) may be seen by inspection of the direction cosine matrix elements to be AJ = 0, 1:1, 1:2 (with the obvious restriction that J and J' must remain non-negative and K s J) as in the Raman effect. The conditions under which the various AK transitions can take place and the line strengths appropriate to these conditions are obtained using symmetry considerations in the section which follows. If individual Stark effect transitions (transitions between lower and upper state N ‘values) are resolvable, the electric field induced line strengths for the various transitions possible in polar axially symmetric molecules are given by individual terms such as J'.K',M', ',I K. 2 . (H °r ”Batman; v '| (H °r Haul . (2.18) 18 In the near infra-red region, however, Stark effect frequency shifts are seldom as large as the line width due to pressure broadening (13), especially under our experimental conditions where the Stark cell was pressurized to 1/2 atm. with SF to prevent electrical break- 6 down. Thus, to a very high degree of approximation, a molecuiat absotpu'an fine in the phaence 06 an (ax/twat etemc field can be aha/Lacteuzed by a line suength, (1 came Mequency, and a Line shape whose origin and form are related to the amount of Stark broadening it suffers. The effect of Stark splitting on line shape is discussed later. Under the assumption of small Stark splitting, the M degeneracy can be treated as remaining and equation (2.18) must be summed over M and M”; the lower and upper state degen- eracies, respectively. Listed for C3V symmetry molecules in tables (2.4 through 2.6) are my calculated values of the allowed (\‘ or if )Jl’KU’T ,v = 2|(\\ or iY ) K” 2 E J.K.T.v MM g l . (2.19) the line strengths of transrtions not allowed in the absence of an external electric field to lowest order in the field and the lowest order correction to the line strengths of transitions allowed in the absence of the field. The tables may be used for electronic transitions in their given form. For rotation-vibration Spectra r' = w and for pure rotation Spectra r” = T and v' = v. A partial check of the results for the (Hfg) was obtained by comparison with the results of Dows and Buckingham (9) for diatomic molecules by setting K = O in the tabulated expressions, In addition, the quantum dependence of the (Hfg) and (if ), for the § most part, could be checked with the Raman line strengths given by 19 Placzek and Teller (11) because in the Raman effect, as in electric field Spectra, a sum over M and M2 of the product of two direction cosine matrix elements occurs. The energy difference denominators in the electric field induced intensities prevent com- parison with the AJ = 0 Raman values. Complete agreement in quantum dependence was found in all cases which could be compared. SELECTION RULES The discussion of electric field induced spectra will now be restricted to fundamental and overtone vibration-rotation bands (T = W') in molecules possessing C3v symmetry; Specifically, those molecules having the form XYZ The following discussion 3. uses elementary group theoretical arguments to predict the selection rules tabulated in Table (2.3). The line strengths, (F2), are proportional to the square of the product of a diagonal matrix element, , and an off Bl B ‘Tv'>. A useful theorem from group j theory can be applied to these matrix elements to determine the diagonal matrix element, = 0 provided ilMlevn.a.i. * O the direct product representation F X PS X Fa, does not contain a . F 1’ the trivial representation (14). In a simpler form this rule * states that the direct product representation Fa X Pa, must transform, or have component that transforms, like F for the B) matrix element to be non-zero (15). 20 Application of the above theorem requires knowledge of the transformation properties of the three components of the dipole moment operator, the transformation properties of the normal vibrations, and the direct product multiplication table of the irreducible representations of the group under consideration. Table (2.1) is the character table for the C point group. 3v From the definition of the components of the dipole moment operator; x i 1 i y i z 1 i i where ei is the charge of the particle having coordinates x1,y1, and 21 and the character table for 03v, it is clear that ux,uy, and “2 have the same behavior with respect to symmetry operations as the translations TX, Ty, and T2, respectively. Therefore, ”x and uy transform according to E, the doubly degenerate irreducible representation of CBv’ and “Z transforms according to Al, the totally symmetric irreducible representation of 03v (16). The degenerate normal vibrational modes of molecules possess- ing C symmetry must transform according to E, the only two 3v dimensional irreducible representation of c3v' There are two choices according to which the non-degenerate vibrations could transform - A1 and A2. Since A2 is the irreducible representation of rotations about the symmetry axis, R2, the only true A2 type vibrations are torsional modes. Therefore, molecules of the form XYZ will have no true vibrational modes that transform according 3 to A2 and all non-degenerate vibrational modes must transform according to A1 (16). 21 C3v E 203 35v 2 2 x + yz, z 2 A1 1 1 1 Rz A2 1 1 -1 2 2 X ' y a X)’ xsy E 2 -1 O xz, yz Rx, Ry TABLE 2.1 Character Table for C 3v A2 E A A2 E A A1 E E E A1 + A2 + E TABLE 2.2 Direct Product Multiplication Table for C3v 22 The direct product multiplication table for 03v can be determined using the character table, Table (2.1), and is shown in Table (2.2). Application of the above theorem to the diagonal matrix elements ”a = and u; = , shows that uz and u; are never necessarily zero, whereas, ux is always zero and H; can only be non-zero if v' refers to an upper vibrational state that transforms, or has a component that transforms, according to E. Simple physical arguments yield the same result. The average value of the dipole moment vector in the ground state is the component along the molecular symmetry axis. ~u;, on the other hand, can only be non-zero in upper vibrational States that induce a dipole moment perpendicular to the molecular symmetry axis. In non-degenerate fundamental and overtone vibrational modes, both v and v' transform according to A1; therefore, only the diagonal and off diagonal matrix elements of uz are non-zero. Examination of equations (2.14) through (2.17) shows that only (2.14) satisfies this criterion and hence AK = 0 transitions are the only possibility in non-degenerate fundamental and over- tone bands. Singly and multiply excited degenerate vibrations have an associated vibrational angular momentum L that must be considered in conjunction with other symmetry considerations in any selection rule calculation. For the degenerate vibration Vi’ the associated vibrational angular momentum Li can take on values 23 Li = jvi, i-(vi - 2), iflvi - 4),... 0 or + 1; ' where V1 is the number of vibrational quanta of the v EE-mode i excited (Herzberg p. 81, 16). Amat's rule relating AK and AL for C symmetry is given by (17) BV AK.-jZ:AL1 = ifip l where p is an integer and ALi .‘Li because L of the rovibronic ground state is zero. In a degenerate fundamental vibration, as previously mentioned, u;; uz; and u; are non-zero and L has values L - id. Consider- ation of the transformation prOperties of the matrix element shows that ' 0, whereas, is not necessarily zero because v transforms according to A1 and v' transforms according to 3. Examination of equations (2.16) through (2.17) shows that (2.15), AK I‘il, and (2.16). AK = 0,‘:2, both satisfy the above conditions. Application of Amat's rule for this case gives the following results: for AK 3 :2, AL = w%'AK; for AK =‘il, AL ' AK; and for AK ' 0, the rule cannot be satisfied. Therefore, the selection rules obeyed by degenerate fundamentals are AK - _+_2 and AK - 11 with the line strengths given by equations (2.16) and (2.15), respectively. A doubly excited degenerate vibrational state transforms according to the symmetric direct product representation, and 3 refer to the parallel and E X E I A +-E, where A 1 l 24 perpendicular components of the band, respectively (15). Therefore, none of the matrix elements _, uz, u;’ and u; are necessarily zero and AK = O, :l,‘:2 transitions are allowed. It must now be determined which AK transitions take place to the E level and which take place to the A1 level as well as whether the intensities of the AK =‘:1 transitions are given by either or both of equations (2.15) and (2.17) and whether the intensities of the AK = 0 transitions are given by either or both of equations (2.14) and (2.16). A doubly excited degenerate vibrational state has vibrational angular momentum components of L = i2, 0; the level that transforms according to E has L =‘:2 associated with it, while the level that transforms as A has L = 0 associated with it as shown in Figure (2.3). E,{,=_-t2 symmetric E X E = A1 + E Degenerate Vibrational State A1 A - 0 A1 d = O Rovibronic Ground State Figure 2.3 Symmetry of Doubly Excited Degenerate C3 Vibration. v The AK = 0 transition can only occur between the ground rovibronic state and A1 state of the doubly excited degenerate vibrational state because Amat’s rule required that AL = 0 for AK = 0. There- fore the AK = 0 transitions can only occur in the parallel com- ponent of the band with line strengths given by (2.14). The AK =.il transitions can only occur between the rovibronic ground state and the E state of the degenerate overtone because Amat's 25 rule requires that AL ='i2 for AK = :1. Therefore, the line strengths of the AK = :1 transition are given by equation (2.15). Finally, the AK = i2 transitions from the ground state must terminate at the E level because Amat’s rule requires AL =‘i2 for AK = :2. The resulting selection rules are summarized in Table (2.3). Type of vibration AK AL Transition moment Line strength iven by equation in Table Non-degenerate fun- damentals and over- AK=0 A£=AK (2.14) 2.4 tones Degenerate Funda- AKéil AL=AK (2.15) 2.5 mentals 1 AKéiZ AL=-EAK4 (2.16) 2.6 Degenerate Overtones a. Parallel AK=O AL=AK (2.14) 2.4 b. Perpendicular AKéil AL=-2AK (2°15) 2'5 AKéiZ AL=AK (2.16) 2.6 E F.S. selection rules for C3v symmetry molecules. AJ = O, :1, :2 in all cases. Table ('2. 3) LINEAR_MOLECULES The Sayvetz (18) conditions may be employed to Specialize all of the expressions given for symmetric tOp molecules in this chapter to the case of linear polar molecules. For non-degenerate fundamentals and overtones of a linear molecule, the transition moment is given by equation (2.14) with K set equal to zero and the line strengths are given in Table (2.4) with K set equal to 26 TABLE (2.4) Line Strengths for AK=O Electric Field Induced Transitions Ao Transitions Forbidden in Absence of Field 2 , J=0,K=O,r',va _ Z , , 2 11z u z 2 (“rZ)J,K,r,v “‘36-|l FEE + GB 2 c 2 J+2,K,r',v“_ FZ .1 . , , 2 u z 11z (“rZ)J,K,r,v - §fi-|‘Trvlhle IV >l ‘ th'(J+2) th(J+l) [(J+l)2-K2][(J+2)2-K2] X (J+1)(J+2)(2J+3) F 2 u u' 7- - v v 2 ("Fz)g fi'E’; IV = Z l| z ._ z ’ ' ’ 30 thJ th'(J-l) 2 2 2 2 X (J -K )[(J-1) —K 1 J(J-1)(2J-l) J=O,K=O,I',V’ _ (“rX)J,K,t,V ‘ 0 J+2,K,I',V‘ _ 3 J+2,K,r',v' (”FX)J,K,I,V ‘ Z(IIFZ)J,K,I,V J 2,K,T ,v' _ 3 J-2,K,T',V' (”PX)J,K,T',V' " 1(IIFZ)J,K,I,V 27 B. Correction to Transitions Allowed in Absence of Field 2< . ..) 2 2 ("I )J,K,.',ve . f‘zl “my. N 33, + Z J#O,K,T,V 60h2C2 B B9 (4J2+1)(J2-K2) 4(J2-K2)[(J+1)2-K21 X ——~ .. J‘xZJ-l) J2(J+1)2 -_. (4J2+8J+5)[(J+1)2-K2] + M.. (J+l)“(2J+3) L___ 2' I v 2 2 2_ 2 w. )W F. bvluzlr M awn-m «M K] Z J'K'T'V 60h2c2 (J+1)5 U U x .MJL__ _.JL_ B(J+2) B'J 2'. I LIZ _ 2 2_2 (”F )J~1,K,T“,v' FZ |.rv|uz|r ,v | (J+l)(J 1)K (J K ) Z JIK'IVV 60h2C2 J5 . _ I 12 B(J-l) B'(J+l) J+1,K,r',v' _ 3 J+l,K,r',V' (”FX)J,K,T,V ‘ 1("rz)J,K,r,v J-1,K,r',v' _ 3 J+1,K,T',V' (“FX)J,K,I,V ‘ Il”rz)J,K,r,v 1h 28 TABLE (2.5) Line Strengths for AK=il Electric Field Induced Transitions Ao Transitions Forbidden in Absence of Field J+2 K+1 :' v" F22 “2 (ler)J KIT-V. ' = -—|l2 ’ ’ ’ 60 thJ _ “oz ‘12 X (JZ-K2)(JIK-1)(J¥K-2) th'(J-11J J(J—1)(2J-l) (”F )J+2'KtlyT°,VD = 3(“f )J+21Kil,T',V' X J'K'T'v I Z JyKrTrV V J-2’Kil'tv'vu _ 3' J-ZVKil'T'IV' (IIIX)J;K,T9V — z(|er)J,K,I'V 29 Ba Corrections to Transitions Allowed in Absence of Field a F 2 + I I (Hfz)gfg:§:i'r 'V = ~£~l|2 (J;K)(JiK+l) 120 (2J+1) 2 ~ 1 u' 2 (éJ +1) ' z (Jix) — z (JiK-l) Jb(2J-1) th th' 4 {—112 11.2 Hz + —7-————— i (JiK) - . (J;K-1) ———(J;K+1) J2(J+1)2 'th th' th uuz _ (4J2+8J+5) uz M'z 2 - (JiK+2) + ' ~——th;K+l)- (J:K+2) th' _J (J+l)5(2J+3) th th' F 2 i v ' . (Hrz)g+i,§ i,r ,v = ____§__.] 2 ' 'T'V 240 Md2 x _ 2 + + + I“ I J(J l)(J_K)(J_K+l) X l I z (JiK)4(6J-l) (2J+l) J5(2J-1) LFZ 8pzp'z p'22 ~ .j + ———-——(J;K-l) 2 - _. sir )J,2,Ki2,r',v' 2 F2 l“""L‘xlt 'V ' (J+K)(J*K 1) Z J,K,T,V 240 h2c2 (2J+l) (mummy x ml “x J3(2J-1) L2BJ-(A-B)(li2K) .4 I ‘2 I _ + L x ! _ 4 _A_ MX ZB'J+(A'-B')(3i2K[J J(J+1) L2BJ-(A-B)(li2K) h' .1 f_ u + X xl-e x ZB'J+(A'-B')(312KLJ L2B(J+l)+(A-B)(li2K) ‘1 + n X i + (4J2+8J+5) 2B'(J+l)-(A'-B')(3i2K:J (J+l)3(2J+3) u' 2 X + x ZB'(J+l)-(A'-B')(3i2K) l l U l L2B(J+l)+(A-B)(li2K) F 2 J+lyKi2"f"v' _ : ‘ (LIZ)J,K,I,V — [,1 v Irv,vv>|2 _£l£:£l (J+l)3 u ' 2 j x _ u I L_2B(J+1)+(A-B)(liZK)](J+—2) [ZB'(J+1)+(A'-B')(3i2K)]J x (J;K)(J1K+1)(J1K+2)(JtK+3) 32 F 2 _ + .I I .. ' 2 “x _ p x [_DBJ-(A—B)(li2K)](J-l) [2B'J-(A'-B')(3i2K)](J+l) X x (JxK)(J:K-l)(J¥K+1)(J:K-2) F 2 J+2 Ki2 ' ' ' "' 30 h2c2 . 72 F LI bx X — LEB(J+1)+(A-B)(li2K) ZB'(J+2)+(A'-B‘)(3i2KLJ (JiK+l)(JiK+2)(JiK+3)(JiK+4) (J+l)(J+2)(2J+3) F 2 J—2 Kiz r' v' z - J,K:T.V' ' :‘TS‘TW "’" C . "I 2 px _ p x x L2BJ-(A-B)(li2K) ZB'(J—l)-(A'-B')(3i2K) (J:K)(J:K-1)(J$K-2)(J¥K-3) J(J-l)(2J+l) X 2!< I I, 2 (if )J>2,Ki2,r',v' = FZ ' I'Vl“xIT 'V I (J;K)(J;K-l) X J,K,T,V 120 hZCZ (JiK+1)(JiK+2) x J(J+1) 1 p2(6J-1) - 8pq(J-1) (2J+l) J“(2J-l) 33 7 r 7 + q2(6J-l) + ~2—3—-—- ,2ps - 3(Pt + qs) + th .1 J‘ (J+l)l L. and l + l {52(6J+7) - 8st(J+2) + t2(6J+7) l (J+1)“(2J+3) where p = “X zBJ-(A-B)(1t2K) p'x q: 2B'J+(A'-B')(3i2K) “x s: ZB(J+1)+(A-B)(1i2K) 13' t = X ZB'(J+l)—(A'—B')(3i2K) J+l,KiZ,I',V' _ 3 J+1,Ki2,r ,v' (irX)J,K,t,v " 4(JFZ)J,K,I,V J-1,K12,r',v' _ 3. J-1,Kt2,r',v' (iFX)J,K,*,V " 4LJFZ)J,K,T,V J+2,Ki2,r',v' _ 3 J+2,Kiz,r',v' (iIX)J,K,I,V ‘ 4(ifz)J,K,r,v J-2 Ki2 r' v' 3 J-2 Kiz v' v' 1* I II =__ I I~I (i X)J,K,I,v 4(JFZ)J,K,T,V 34 zero. Transitions involving the degenerate vibrational states of linear molecules cannot be treated by the above non-degenerate perturbation theory because degenerate states with the same 1i.L within the same vibrational state interact through the electric field. Degenerate perturbation theory is applied to this problem in Chapter IV in connection with the interpretation of observed electric field induced spectra of HCN. LINE SHAPES The following development of the theory of electric field induced Spectra by Bridge, Haner, and Dows (l3) parallels the perturbation treatment already presented and gives additional information on expected line shapes. In the absence of an external field_absorption line shapes are due to three processes. First, the natural line breadth results from the finite lifetime of energy states in conjunction with the uncertainty principle. Secondly, the random motion of molecules in a gas causes absorption of a distribution of frequencies due to the Doppler effect that can be described by a Gaussian function. Finally, interactions between molecules- pressure broadening- causes absorption with a Lorentzian distribution of frequencies. In addition. an external electric field removes the upper and lower state M degeneracies and even though the transitions may not be resolvable, a distribution of frequencies about the central frequency will be present. We will characterize electric field induced absorption lines by their center frequency, an integrated line strength, and a line shape that arises from the 35 field splittings in addition to the other broadening effects described above. Assuming a line shape function 3 and retaining the previous weak field assumption so that the absorption coefficient k(v) is a sum over the upper and lower state M degeneracies, the absorption coefficient becomes k(v,FZ) = [thK/(21+l)il BJKM(FZ)S[v - (FZ)] , (2.20) M,M' vOJKM where the only previously undefined symbol is v , the frequency OJKM of the absorption line including field effects. The form of S can be taken as independent of M so long as the assumption of weak electric fields is valid. Under the assumption of weak fields the absorption coefficient k(v,FZ) and the Einstein B coefficient may each be expanded in powers of the field. kw F > = k (v) -+ sz (v) + FAR (v) +... (2.21) " Z 0‘ Z 2 Z 4 =8 fF 2 + +. 2.22 BJKM(FZ) ‘OJKM ZBIJKM FZBZJKM ( ) Odd powers of the field drop out of the absorption coefficient ka,Fz) on summing over M and M' because n n A 1 g = A - . nJKMJ'K'M‘FZ nJK-MJ'K'M'( F2) This is apparent because absorption cannot depend upon the sense of the field. The BOJKM in (2.22) are the field independent Einstein B coefficients for induced absorption and the BZJKM are pro- . 3 J'K'M' portional (by a factor of Sn /hc) to the (1")JKM calculated earlier. The are discussed qualitatively below. A similar BlJKM 36 (F ) occurring in the argument of expansion of the frequency VOJKM Z the line shape function yields: . o 2 F = ‘ '+ f + ... . VOJKM( z) VOJKM lJKM F2 f2JKM Fz + ’ (2 23) th . . where f are the n—— order Stark splittings. nJKM If the Stark frequency shifts are small compared to the line width A, the line shape function can be expanded in powers of the shift. . . _ - O I o . Slv - VOJKM(FZ)} _ Slv - v0} + S {v-- vol[v - vo]/A + SII{v - v:}[(v - VO)/A]2 +..., (2.24) where the coefficients Sn(v) are, by Taylor's theorem, n n -l dn 3 (v) = (-A) (n!) ‘ ——E S(v) dv Combining expansions (2.22), (2.23), and (2.24) and comparing coefficients of F; with expansion (2.21) makes the following identifications possible. - . w . .1 . .. _ 0 ko(») - [thK/(2J+l)]MMJ BOJKM S{v v0} _ . ‘ . _ , 2 II _ o k2(v) - [thK/(2J+1)]{ ' BOJKM(f1/A) .8 {v v0} MM [3 (f /A) + B (f /A)lSI{v - v0} . OJKM‘ 2 lJKM 1 , - 0 MM - O + , BZJKM Si" ' VJ} MM . , 4 k4(u) = [huJK/(2J+1)] M [Bo(fl/A) sIV{v - v:} +...]. (2.25) 37 I“ k2(v) in equation (2.25) is made up of three terms: b k2(v) = k:(v) + k2(v) + k:(v) , (2.26) where a \ 0 k2(\)) = [thK/(2J+1)ide;1' BZJKM s{\) - v0} kb(\)) = [hv /(2J+l)']z [B (f /A) + E (f /A)]Slf\) - v0} 2 JK m' OJKM 2 lJKM 1 - o kC - - h /(2J+l \ (f/ 2 II 0 2“) _ l VJK )Zqu. BOJKM 1 A) S {v ' "01 These three terms correspond to three different types of electric a field induced spectra (EFS)-type (a) EFS in which k2 dominates b the other two terms, type (b) EFS when k2 dominates the other two terms, and type (c) EFS when k: dominates k: and kg. The type of EFS observed is determined by the molecular parameters involved rather than being the choice of the experimenter. k:(u) is the absorption coefficient of a line having the same shape S(u - v2) as an absorption line in the absence of the field but a different line strength B Therefore, a type (a) 2JKM' EFS difference spectrum will be a spectrum showing the same line shapes as a field free absorption Spectrum but with a different intensity distribution and different selection rules. The line strengths of type (a) EFS are those calculated previously in the chapter, given in Tables 2.4 through 2.6. k:(v) contains two terms - one involving first order Stark splitting f and the other involving second order Stark Splitting 1 £2. The second order Stark effect term has the same line strength 38 as a field-free absorption line so a type (b) EFS difference spectrum will appear as the difference between two lines of the same in- tensity, one slightly shifted in frequency from the other due to second order Stark splitting. The first order Stark effect term describes the fact that some intensity is transferred from the M-» M' transition to the -M a -M' transition. I have not calculated the BlJKM' k;(v) is directly interpretable in terms of the first order Stark effect. A type (c) EFS difference spectrum will appear as the difference between two absorption lines of the same strength BOJKM one of which is broadened with respect to the other by first order Stark splitting. The line strengths for k;(v) have been calculated for perpendicular bands (13) and are given in Table (2.7). If the line shape is assumed Lorentzian, then the three line shapes are (from Taylor's theorem): type (a) EFS S{v - v2} = (imam/(x2 + 1)), (2 27) type (b) EFS 2x I. o S v - v } = (1/fiA) : (2-28) { o (x2 + 1)2 type (c) EFS slltv - v2} = (l/nA)((3X2 - 1)/3>. (2.29) where x = (v - v:)/Aa These are shown graphically in Figure (2.4) and are in agreement Transition 39 TABLE 2.7 Line Strengths of EFS Type (c) Transitions Light Polarized Parallel to the Electric Field Line Strength |<—r, vhix \T',v'>‘2 J(J + 2)(J : K + DU 3: K + 2) . ‘ J-oJ-+1 22A2 + KnKil 60hc (2J 1)(J+1)3 . 2 (Kilmz Kuz X J+2 'J J _. J ‘<:),v\u,x ‘7',v'>\2 (3.14 + 3.1 - 1)[J(J + 1) - MK: 1)] he222 3 K-aKil AJ(J+1)2(2J+1) , 2 X{(K:1)uZ-Kuz} J—oJ-l ‘<1',v‘ux ‘m',v'>|2 (J2-1)(J;K)(J:K-l) stil 6OhC'22A2J3(2J+1) 1 2 X(K:1)u.z-E_u.z J-l J+l 40 B. Light Polarized Perpendicular to the Electric Field Transition Line Strength 112 (J 1 K + no 1 K + 2) J .. J + 1 2 K _. K + 1 120h2c A2 (J + 1)3(2J + 1) 4J 4- S 2 v2 ' X{ J+2 (K:1)uz ' 8K(Ki 1>uzuz 4J + 3 2 2 +————_—— _ J K ,2} 2 J .. J 1| [J(J + 1) - 1((K i 1)] K .. K + 1 120h2c2A2 J3(2.J + 1) (J + 1)2 2 . ' 2 x {(ZJ + ZJ + 1)[(1< i' 1)uz - K “‘21 I + 10 K(K : l)u,zp,z} . I y! 2 _ , I. K - 1) J —» J - 1 \<"2V\U'X\" ~V >l (J +K)(J x .. x + 1 120112352 J3(2J + 1) 4.1-1 2,2 2 , x{ J _ 1 (1<_+ 1) uz - 8K(K: nuzuz I. ' ..‘ o 41 EFS Type (a) -—-—-—--’———--- | EFS Type (b) ///"\\\\ EFS Type (c) -—_-—-__—_-—- < Figure (2.4) Electric Field Induced Line Shapes. 42 with the qualitative arguments presented above. In practice, the line shape function is a much more compli- cated function than Lorentzian or Gaussian (c.f., 19 or 20) and monochromator slit effects must be considered. The intensity of the absorption line is transformed by the spectrometer according to i 3 _ 1 (v ,FZ) - f I(v,FZ)B(v - v')dv, (2.30) where the slit function B(v - v') is the fractional intensity of light of frequency v transmitted by the spectrometer set to frequency v’ and I(v,FZ) is given for a Beer's law absorber by I(v,FZ) = 10(v) exp [-k(v,FZ)L], (2.31) where L is the absorption path length and' I°(v) is the intensity of the incident radiation of frequency v. VALIDITY OF PERTURBATION EXPANSION The perturbation and power series expansion treatments of electric field induced spectra both require the perturbing field to be weak. This requirement means that the perturbation energy uF must be small compared to the separation of rotational energy levels, i.e., for a polar axially symmetric molecu1e__uF < 2thJ for the Jgh'rotational level, B being the rotational constant in units of cm-l. For methyl fluoride with -p g 2 Debye and B 3 l cm-l, the upper limit on the field is ZOOJ esu (6OJ KV/cm) so that a field of 30 Kv/cm is small for practically all rotational levels. Ea- 43 For high electric fields and in the case of molecules posses- sing large electric dipole moments, the perturbation expansion does not converge. In these cases, Iamb's method (see Chapter I) of find- ing an exact solution for the Stark energy levels could be used. With very accurate values of Stark energy changes, accurate numerical wave functions good for all values of electric field could be found for use in intensity calculations. A considerable amount of computer time and storage space would be required for this problem in all cases except for very low values of J. CHAPTER III EXPERIMENTAL TECHNIQUE The apparatus and frequency measuring techniques used in the experimental portion of this thesis are described in this chapter. Two experimental stages for the detection and measurement of electric field induced Spectra were used in the course of this work. The original system was assembled to determine whether EFS could be observed in the near infrared region. As soon as induced Spectra were obtained using the original system, improvements were made in the design of the fore-optical system to increase energy and im- prove resolution. Both experimental arrangements are described below. Because induced intensity changes are proportional to the square of the electric field and are orders of magnitude smaller than zero-field absorption Signals, it was decided to use phase sensitive ac electronic techniques. An ac electric field was used to modulate the light transmitted by the cell (rather than a mechanical light chopper) and amplification was done at twice the electric field frequency because of the F22 dependence of the line strength. A block diagram of the apparatus is Shown in Figure (3.1). The general experimental arrangement was the following. Light from the carbon rod source (21) was focused by a Can lens at the entrance of the parallel plate, electrically capacitive, "Stark cell" described in detail below. The light was modulated 44 45 ucuauwuuuud Huucoaauonxu mun uuvuooom w uuaaaaaa< _ _ umanaon “unease Au.mv ousmam poumHHHomo _ _ _ _ zocmsvouh — 0Hp3< nouuouon umeuowmauua .>.m mama vuusom moauao Haou spasm AHV nonmeouno ouch socoz 46 by the 45 Hz. high voltage applied to the hot electrode of the Stark cell and then focused by fore optics on the entrance Slit of the monochromator (described in detail by Aubel (22) with modifications described by Keck (21)). The diSpersed light from the monochromator was then focused on a lead sulfide detector cooled to liquid nitrogen temperature. The signal from the lead Sulfide detector was phase sensitive detected (Aubel (22)) using a 90 Hz. reference signal, amplified, and finally recorded by one pen of a Leeds and Northrup dual pen recorder. Electrical energy for the high voltage and the reference signal was provided by a Behlman Engineering audio oscillator having three 120 volt outputs. The 45 Hz. reference voltage had its frequency doubled by a lab built full wave rectifier and was fed into the phase sensitive detector through a phase shift- ing network providing phase changes throughout 360 degrees. INITIAL EXPERIMENT Because electric field induced Signals are small, Stark cell design is important. In order to observe induced transitions, the cell must provide electric fields of the order of tens of Kv/cm, transmit a reasonable fraction of incident radiation, and provide an absorption path length sizable enough for appreciable absorption to occur. Professor C. W. Peter's group at the University of Michigan faced identical problems in their Stark effect experiments and our cell is essentially a c0py of theirs (5). The original Stark cell was constructed as Shown in Figure (3.2). Two slabs, 2"x36"x2", of selected Libbey Ownes Ford 'Mirro- Pane', a front surface chromium coated mirror, were separated by 5" wide .0075" strips of insulating Du Pont 'Mylar' and stainless “g_'_ ' .r- _'..' 11.2 I :8 seam 3.3 25»: ungmuouugt chocog 3* n :0“ x o-N 4 Joey m 0959 loq“Il/I\ 5&3 :8. l nouwam and»: 48 i: steel Lufkin gauge stock. The gauge stock also served to make '* electrical contact to the mirror surfaces. With the Spacers in place, the cell was placed between two pieces of channel iron loaded with lead bricks. Pressure was uniformly distributed along the edges of the cell by placing rubber strips between the glass and the iron rails. Ren Plastics RP 201 green epoxy was used to cement along the edges. Later cells were cemented with RP 1240 because it is easier to handle. The cell was placed in a housing consisting of a piece of brass S band wave guide with removable endplates and CaF2 end windows. Sample pressures were measured by a Welch Scientific Co. Dubrovin gauge connected to the housing through a glass manifold. The manifold also served to introduce and remove sample gases from the Stark cell. One electrode lead (gauge stock) was soldered directly to the grounded brass housing. The high voltage was connected to the cell through a hermetic seal. The exposed gauge stock was covered with corona dope to prevent electrical breakdown of the gas at these places. The brass housing and accessories remained basically unchanged throughout the exper- imental portion of this work. All six Stark cells constructed during this portion of the work suffered electrical breakdown after five to eight hours of operation and had to be discarded. In the first system, a 12 cm focal length CaF2 lens was used to form an image of the exit aperture of the Stark cell at an inter- mediate focus of the monochromator. Because the optical elements used in the preliminary investigation were those on hand in the laboratory, matching the f/ number of the light beam to the mono- chromator was impossible. Weak but interesting electric field 49 Br, CH I, and CH CN were observed with this n. 3 3 3 apparatus although no frequency measurements were attempted at that induced transitions in CH time. FINAL ARRANGEMENT The largest defect in the original experiment was the lack of energy available in the monochromator. In order to overcome this difficulty, it was decided to increase the separation of the Stark cell plates to about 2 mm. as a method of increasing trans- mission efficiency and make a careful study of the optical pro- perties of such a cell to see if it could be conveniently matched to the monochromator. However, wider separation of the plates decreases the breakdown voltage of a gas at low sample pressures; but this problem can be minimized with a heavy inert buffer gas Such as SF6 (23). Essentially, this is a trade of possible res- olution, because of the pressure broadening, for energy in the monochromator. It was observed experimentally with plate separations on the order of 2 mm. that light diverged horizontally from the 2513 end of the Stark cell at approximately f/8 and diverged vertically from the entrance end of the cell at approximately f/20 using f/4 source Optics. The horizontal f/ number was not well defined. The beam was observed to have a non-uniform intensity distribution in the horizontal plane due to multiple reflections from the cell walls. The appearance of the pattern resembled an interference pattern with poorly defined edges and was highly dependent upon alignment. 50 The optical problem was to fill the monochromator, correcting for the astigmatism (equal to the length of the cell), and simulta- neously match both horizontal and vertical f/ numbers to that of the monochromator. My solution to this problem for a Stark cell of arbitrary length L is the following two lens system. A spherical lens placed at its focal distance from the exit end of the cell to simultaneously match the vertical f/ number of the cell to the monochromator and collimate the horizontally divergent portion of the bean. With these two conditions plus the fact that the difference between the object distances for the horizontal and vertical portions of the beam is the length of the cell L, the required focal length f of the spherical lens may be calculated from the thin lens formula as f = mL. m is the ratio of the f/ number of the monochromator to the cell's vertical f/ number, Viz., the magnification required to match the cell's vertical f/ number to that of the monochromator. The second lens, beyond the Spherical lens, is cylindrical with an f/ number equal to that of the monochromator and a focal length determined by the width of the collimated beam. The cylindrical lens, while having little effect on the vertical portion of the beam, focuses the collimated protion of the beam at the point where the Spherical lens focused the vertical portion of the beam. Although our monochromator is f/S, it was more convenient to focus the emergenent light at an intermediate focus in the regular fore-optical system at f/lO. The 100 cm length of our Stark cell and the demagnificaticn of the vertical portion of the beam from 51 f/20 to f/10 determined the focal length of the spherical lens to be 50 cm. The measured horizontal dimension of 6.5 cm, together with the required f/lO, of the collimated beam determined the focal length of the cylindrical lens to be 65 cm. The Optical system is Shown in Figure (3.3). Details of the shape calculations for the two CaF lenses 2 used in this system are presented below. LENS CALCULATIONS The radii of curvature of the lenses in an optical system should be chosen so that spherical aberration and coma introduced by one element are minimized or cancelled by succeeding optical elements. This can be done by calculating the position factors of the elements involved (determined by image and object distances) and then choosing the best shape factor for each element which, in turn, fixes the radii of curvature of the lenses in the system. Cancellation of spherical aberration is only possible for a combination of two or more lenses of opposite sign, viz., con- verging and diverging lenses. Coma, on the other hand, may be made positive, negative, or zero for a single lens by proper choice of shape factor (24). The parameters of our optical system were not sufficiently free to allow cancellation of Spherical aberration; but the calculations for minimization of this effect were performed. Fortunately, the best shape factor for minimization of Spherical aberration is very close to that for minimization of coma. After performing the necessary calculations, it was learned that con- verging cylindrical lenses other than plano-convex are excessively expensive. Plano-convex is, fortunately, close to the best lens Euumhm HQUwuuc-wuom Am.mV muswwm 52 mod, 02m 83 3.2.5 as. .8228 =8 {am 53 shape for focusing collimated light. The following lens calculations are those for the cost-constrained lens system. The horizontal (collimated) portion of the beam in the fore- optical system was measured to be approximately 6.5 cm. wide. There- fore: an f/lO converging cylindrical lens of focal length 65 cm. was required. Since the index of refraction of CaF is approximately 2 1.42 in the near infrared and one radius of curvature r2 of this lens Was fixed by cost to be coD the other radius of curvature r1 is given by the lens makers' formula (24) . l l 1/f = (n - 1) (— - -—).. (3-1) r r 1 2 to be 27.30 cm. The spherical lens: on the other hand, has only its focal length (50 cm.) and the position factor p9 given by (24) _ s' - s P 3—3:: (3'2) fixed: where s' and s are the image and object distances, respectively. The position factor for this lens is of course different for the horizontal and vertical beams. The best shape factor q: to minimize spherical aberration, and hence the radii of curvature of this lens must be a compromise between the best values for the horizontal and vertical beams individually. It is desirable that the horizontal portion of the beam be as free of spherical aberration and coma as possible because this part of the beam determines how many rulings on the grating of the monochromator are covered. The horizontal beam has object and image distances of s = 50 cm. and s' = m, respectively. 54 Therefore its position factor from (3.2) is p = +1.0. The best shape factor for reducing spherical aberration in a lens is given by (24): -2(n2 1) q = n + 2 4P . (3.3) For the horizontal beam, the best shape factor is q = -0.594. The vertical portion of the beam, on the other hand, has object and image distances of s = 150 cm. and s' = 75 cm., respectively. The vertical beam has a position factor of p = -l/3 and therefore a best shape factor of q = +0.198. The quantity Ls, which is a measure of the axial difference in focal length, is defined by (24): 1 l LS=§T"‘-T_a (3-4) s h p or 2 3 h 1 n+2 2 . 2 n = + + - —' LS 8f3 n(n-l) [9'1 q 4 (n l)pq + (3n + 2)(n l)p + n-l , where s' is the image distance for an oblique ray traversing the h lens at a distance h from the axis and s'p is the image distance of the paraxial ray. The quantity LS was plotted for the horizontal and vertical beams as a function of the shape factor to find a compromise shape factor. q was chosen to be q = -0.50. The radii of curvature of the spherical lens are then given by (24) 2f n-l 2f n-l = ' = . 3.5 r1 q+1 and r2 q_1 ( ) With q = -0.SOy f = 50.0 cm., and n = 1.42; the radii of curvature are r1 = 84.0 cm. and r2 = -29.0 cm. - the negative sign indicates surfaces curved in opposite directions. It is important that the 55 lenses face in the proper direction when in use. The two lenses were purchased from John Unertl Optical Co. The two lenses were mounted on a large two axis cross slide that allowed accurate independent adjustments of each lens parallel and perpendicular to the light beam. The brass Stark cell housing was mounted on a specially designed three axis cross slide and connected to the fore-optical vacuum chamber (the shoe box) through an evacuated bellows. This arrangement allowed the walls of the fore- optical vacuum chamber to flex under atmospheric pressure without affecting alignment as well as providing easy external adjustments in focus and position. FREQUENCY CALIBRATION Frequency measurements of EFS signals were made in the following manner. Absorption lines of an appropriate molecule or molecules whose frequencies bracket the frequency region of the EFS signal and whose frequencies are well enough known to be suitable as standards, are recorded before and after the EFS signal. The grating is never allowed to stop. Simultaneously, visible light Edser-Butler interference fringes having constant frequency spacing are recorded throughout the region of interest on the second pen of the dual pen recorder. The fringes provide a linear interpolation scale for frequency determination of the EFS lines lying between the standard frequencies. The details of this method are given by Olman (25) and by Rao, Humphreys, and Rank (26). CHAPTER IV ELECTRIC FIELD INDUCED SPECTRA OF HCN The linear molecule HCN was chosen as an experimental test of the new optical system because of its large permanent electric dipole moment (3 Debye), well spaced lines, and L-type doubling. The result- ing induced Spectra proved so interesting that a major part of the experimental effort was directed toward this molecule. Linear Molecules--General Features The three normal vibrational modes of a linear tri-atomic molecule are shown in Figure (4.1). Two of these, v1 and v3 is two-fold degenerate, are non-degenerate while the bending mode, v2, e.g., the molecule can bend in the plane of the paper or perpendicular to the plane of the paper. The rotation-vibration absorption spectra of non-degenerate fundamental and overtone bands of linear molecules are extremely simple. Approximately evenly spaced R (AJ = +1) and P (AJ = -1) lines march out from the band origin where no Q (AJ 0) transitions are possible. In terms of axially symmetric molecules, the quantum number K, the projection of the total angular momentum along the molecular symmetry axis, is zero so that in an electric field only second order Stark effect is possible. Transitions involving the two-fold degenerate v2 vibration are somewhat more complex both in the appearance of absorption bands and in the effects that an electric field may produce. 56 57 H C N {a o '< w’L {/‘L V1 e wk v ”2 v 69L mm {x} \)2 < 9 {k 40 Figure (4.1) - Normal Vibrational Modes of HCN. 58 v2 has a fairly low frequency (712 cm-1 for HCN) so that this state is appreciably populated at room temperature. Consequently, many fundamental and overtone rotation-vibration bands are accompanied by "hot bands", viz., transitions for which v2 is the lower vibra- tional state rather than the vibrational ground state. The v 2 vibration being essentially a two dimensional har- monic oscillator has an associated angular momentum. If v2 quanta of the v2 vibration are excited, a (v2 + l)-fold degeneracy is present and angular momenta (in units of h/2n) of L = :Jvz - 2), id? - 4),...,‘:l or 0 are possible. If v = l, for example, L 2 2 can be ':1 corresponding to an angular momentum of rotation in either direction about the moelcular symmetry axis. Obviously, the total angular momentum quantum number must be greater than or equal to L in a degenerate vibration. The (v2 + l)-fold degeneracy remains only to lowest order for the following reasons (16). A linear molecule excited to a degenerate vibrational state is formally analogous to a slightly asymmetric rotor. The angular momentum L is analogous to K in the limiting case symmetric top. Therefore in the symmetric top limit, states of the same magnitude of L are degenerate. Further small splittings of states having the same magnitude of L (commonly referred to as L-type doubling) takes place through the mutual interaction of the v1 and v vibrational states. The 2+; v1 vibrational state 2 lying above the vibrational state (of symmetry U) in energy, v2 + . interacts with the positive component n of v The negative 2. component H. of v2 is unaffected. The interaction lowers the . + . energy of the statesbelonging to H with reSpect to those belonging to U- by an amount given by qJ(J + 1), where q is approximately 59 constant within a given vibrational state (q ‘ 0.007 cm.1 for HCN (4) in the fundamental). v2 For absorption spectra, the selection rules governing rotation- vibration transitions of linear molecules are (16): AL=0.:1;AJ=0.:1(J=04J=0);+~-; where the - (+9 indicates whether the rotational wave function changes sign (doesn't change sign) under the inversion operation. Q branch transitions cannot occur when AL ' 0 and L - 0. The 3300 cm“1 Region of HCN When observed with zero field, the strongest absorption in the 3300 cm"1 region of HCN is due to the fundamental, a 2+ - 2+ v3 transition. Accompanying this band at lower frequencies are the fairly strong "hot band" v2 +'v3 - v2 (11- ID and the 2v2 + v3 - 2v2 (A - A) "hot band" which is very weak at room temperature due to the Boltzmann factor. Another standard notation _,.,L , L denoting these bands 15 (J1, v2"v3)upper (V1, v2, V3) e.g., v2 +v3 - v2 is written (0, 11, 1) - (o, 11 lower’ , 0). In absorption, both bands are parallel bands (AL = 0) and have rotational selection rules AJ = 0,311 so that R, P, and. Q branches are allowed. The calibration of this region was particularly simple because the frequencies of the v3 absorption lines are well known and commonly used as standards. The procedure during a calibrated EFS run was to admit a few Torr of HCN and run an absorption spectrum of the higher PfJ) lines. The mechanical light chopper was then turned off and the absorption cell pressurized to 8 atm. with SF6 60 to prevent electrical breakdown while the grating continued to turn. The electric field was turned on and the EFS run- At the end of the EFS portion of th: run, the electric field was turned off, the pressure of the HCN and SF6 mixture reduced to 2 cm., and the absorption spectrum cf the higher RtJ) lines run. Figure (a 2) shows the observed electric field induced Spectrum in this region. Positive signal represents an increase in absorption With the field on. Experimental conditions for this spectrum are given in Table (4.1). EFS of the Band of HCN V3 The v3 fundamental exhibits several interesting aSpects of electric field induced spectra as well as allowing a ”direct” measure— ment of the band origin. The induced Q branch is clearly evident in Figure (4.2). The line shape is cf type (a) as expected from the line shape expan- sion of Chapter ll. Solely positive signal represents the difference between absorption with the field on and a lack of absorption without the field. The Q branch is made up of three partially resolved "lines”. From fcrmula (1.3), it is evident that the M = :l a M = O and M = O a M = :1 transitions will be symmetrically displaced toward higher and lower frequencies, reSpectively, while the AM = 0 transition is not displaced from the band origin. Table (4.2) shows calculated values of the relative intensities of the Q(O), Q(l), and Q(2) lines for electric field strengths near 30 vacm. It is reasonable to assume from the values in the table, that only these three lines contribute significantly to the intensity of the observed induced Q branch- The splitting of the induced Q branch is . ,4. ...." 61 N n N .Bmmoavgm 7- 7+ > can m > no men Au.sv unamam can can» omdm 2.x 8? so Se 35% a? .odaegxodv 3.”. 5... 8....0.-. .... .e w aw. alum. .;S§Alilflll _. V 62 .zua can ma+ga. o maze zom mom $502 0 ma>H mam mom 0 mazy 20$ mom m) a maze zoz mam «mo :oflurun “opomura squo 20m mo mmm Mom on om oq oq Amconuaev cu©H3 uaam nous ,u83uuouam mcoflufiwcoo amucmefleuaxm mumps we Auuowv onnmmmum vaEmm @a ma OH AEo\syw Lumcuuum wamflm oauuoowm 2;: 3.92. N)- n?r Uvam N) ZUQ zu: zo: masomaoz 63 attributable to partial resolution of the second order Stark effect components. The apparent absence of observable AM = 0 transitions in the induced Q branch prompted polarization experiments because AM = 0 transitions occur for light polarized parallel to the field while AM = :1 transitions occur for light polarized perpendicular to the field. It was found experimentally that in the 3 micron region our Stark cell transmitted about 8 times as much light energy polarized perpendicular to the field Chithe plane of the cell height) as that polarized parallel to the field. Therefore, aflfl 06 the Apectha 6h0wn in this the/bus awe amen/daffy aecon dad thh Light paid/ulna! paper:— décuzaa to the field dined/ton. Attempts to increase the resolution of the Q branch lines by decreasing the monochromator slit width were unsuccessful. The result of this operation was simply to decrease the intensity and eliminate some of the detail observed in the hot bands that are present. It was concluded that the pressure broadening due to the SF6 was sufficrent to make further resolution impossible. Weak and noisy induced AM = 0 transitions in the v3 Q branch of HCN were recorded by placing a silver chloride stacked plate polarizer in the light beam emergent from the Stark cell with the polarizer oriented to transmit light polarized parallel to the electric field. The two side peaks of the EFS Q branch were measured at frequencies of 3311.335 cm.1 and 3311.582 cm-l. The average fre- quencies of these two lines is 3311.458 cm.1 and should be the fre- quency of the band origin. The low point between the two peaks was 64 ' I” Line Transition Relative g9: Q(J) |M| a lM" Intensity Q(O) 0<~ 0 0.09 0 a 0 0.03 Mb 1 —. o 0.02 1 a 1 0.02 0 a 0 0.002 1 a 0 0.001 Q(2) 1 a 1 0.001 2 a 1 0.002 2 a 2 0.004 Table (4.2) Relative Intensities of Q Branch Transitions for HCN. (30 Kv/cm Electric Field) measured to be 3311.465 cm-l. Both of these measurements agree - * within experimental error of approximately .015 cm 1 with the calculated frequency of the band origin of 3311.4733 cm.1 (27). R(O) and P(l) in v are striking examples of type (b) 3 EFS line shapes. All AM components of R(0) are shifted toward higher frequency while all of those of P(l) are shifted toward lower frequency by second order Stark effect (see equation (1.3)). The difference between absorption with the field on and absorption with the field off produces this characteristic up-down or down-up asymmetric line shape. If all of the assumptions in the perturbation development of the line shape expansion were valid, an EFS type (b) three standard deviations 65 line shape--first derivative of a Lorentzian line shape with respect to frequency-~would have a zero at the central frequency of the ab- sorption line in the absence of a field. The zero of the EFS P(1) line was measured at a frequency of 3308.467 cm.1 compared to the value of Rank, 35. 31., (27), of 3308.5169 cm-l. The zero of R(0) was measured to be 3314.472 cm"1 compared to the value of Rank, gt. 31,, (27), of 3314.4089 cm-l. The larger discrepancy between these numbers is actually to be expected when the weak field assumption under which the line shape expansion was developed is examined for HCN. The assumption that second order Stark splitting is too small for AM type transitions to be resolved in HCN clearly doesn't hold because partial resolution was attained in the Q branch. The "smallness parameter", uF/JB, for the perturbation expansion of HCN is nearly unity for electric field strengths near 30 Kv/cm. With the electric field strengths used in this experiment. the theoretical model has been extended beyond its intended range, though the qualitative line shapes are in excellent agreement with the pre- dictions of the model. The measured zero of P(1) and the measured zero of R(O) should occur at lower and higher frequencies, respectively, than predicted by the expansion. The condition for the EFS line shape of P(1) to have a zero at the central frequency of the absorption line is shown graphically in Figure (4.3a). The induced line (shown dotted) has to have its position shifted toward lower frequency, its intensity increased, and perhaps have its envelope broadened by the field such that the induced line overlaps the absorption line (shown solid) precisely as shown in Figure (4.3a) for only then will the 66 Figure (4.3) Overlapping in P(1) Transition in v3 of HCN. (Field = 0, indicated by solid line) (a) According to theory. (b) Experimentally observed. difference between the intensities of these two lines be zero at the central frequency of the absorption line. Because of the large Stark shifts shown by HCN, P(1) would be expected to be shifted to a lower frequency than assumed in the expansion with the resulting zero of the EFS line shape also occuring at a lower frequency. This situation is shown in Figure (4.3b). The opposite effect would occur in R(O) and its EFS line shape zero would be expected at a higher frequency than theoretically predicted. For higher values of the total angular momentum J, the P(J) and R(J) lines are asymmetrically broadened by the field, viz., not ‘31_ of the AM components for either type of line are shifted toward higher or lower frequency. In addition, the Stark splittings become smaller and the increase in intensity due to the field decreases rapidly with increasing J. Consequently, those few lines that are intense enough to be observable have asymmetric line shapes resembling EFS type (c) in the sense that positive-negative-positive signal is 67 observed. No quantitive conclusions have been derived from these 4’ weaker transitions. EFS Of the v2 + v3 - v2 Band of HCN The electric field induced spectrum of the HCN v2 + v3 - v2 hot band exhibited several interesting effects arising from the L-type doubling of the upper and lower vibrational states. Figure (4.4a) shows an energy level diagram of the upper and lower vibrational states for the v2 +*v3 - v2 band including the (+) and (-) symmetry of the states with respect to inversion. The lowest member of each allowed transition is also indicated. (The L-type doubling of the J states has been exaggerated.) In the presence of an electric field, the two nearly degen- erate :; states for a given J value, having eigenfunctions 0 Y1 and Y; and unperturbed energies W0 and W0 1 2, can interact strongly through the perturbation : - O O ' an” F f Y1 u cos(e)‘1’2 Sin(e)dedtp or (4.1) .. .= _ F_.Ll_M_ Fl‘lz J(J-+1) A is the average value of the electric dipole moment in the L-type doubled state. The following results are taken from the lucid development of this problem given by Townes and Schawlow (4). Since a linear molecule excited to a degenerate vibrational state is formally equivalent to a slightly asymmetric rotor, the problem is also included in the general development of Stark effects in asymmetric rotors given by Golden and Wilson (28). 68 (a) (b) g 1:). ........ _______________ 2 717 iii ........ 1 , "131' """" P (2) 0(1) Ru) 1:). ........ . 2 ‘(U """" iii ________ T—‘T _______________ 1 Figure (4.4) Transitions in n - n Band of Linear Molecules. (a) Electric Field = 0 (b) Electric Field # 0 69 The resulting energy shift obtained by solving the two dimensional secular equation for this degenerate perturbation theory problem is: wi+w3 (WE-way 22 35 o o . . where W1 - W2 = qJ(J+l) in this case. The perturbed eigenfunctions are: re II o o a(FW1 + b(F)‘l’2 and (4.3) 430w: + a(FW‘Z’ . »E II where the coefficients a(F) and b(F) are given by: o o 2 2 2 1 o o k _ fil -w2) +41“"124r m1 "‘2’ a(F) - , o o 2 2 2V] 2 Jml - w2> + 4F ”.12 (4.4) ,7 W: -w3>2 + ~ cw: -w:> ’5 b(F) = o o 2 2 2 l 2\/(W1 - W2) + 4F “.12 If the degeneracy is nearly complete [as it is in HCN where (W: - W3) = qJ(J+l) is small)], then (4.2) may be expanded o 0 W1 +-W2 AB = —-—2—: F “12. (4.5) Thus, a "first order" Stark effect with a symmetric Splitting of the degenerate L states occurs in the degenerate vibrational states of linear molecules. If w: - w; = 0, (4.5) would be exact and be the magnitude of the result obtained by application of first order perturbation theory 70 j’ to an axially symmetric rotor in the presence of a field. For a slightly asymmetric rotor, the two solutions of (4.5) describe the splitting of the nearly degenerate symmetric rotor limiting case states .1 K by the field. Due to the mixing of the :;L states through the field, both L components for a given J in the upper and lower vibrational states of a n - H transition contain (-) as well as (+) character and the (+-« -) selection rule allowing only two transitions between the two-fold degenerate upper and lower states breaks down. Consequently, Egg; transitions are possible rather than the two allowed in the absence of the field. This is shown diagramatically in Figure (4.4b) for the lowest member of each type of allowed transition. The R(l) EFS transition shown in Figure (4.2) is a splendid example of these field effects. The symmetric positive-negative- positive shape of the induced line is characteristic of EFS type (c) line shapes resulting from symmetric displacement of the two transi- tions allowed in the absence of a field. The weak, partially resolved positive signals flanking the main "line" are induced transitions corresponding to the field-free forbidden (+-« +9 and (- o -) transitions having lower and higher frequencies than either of the field-displaced previously allowed (+-e -) transitions. The ratio of the intensity of field-free forbidden (+-« +) induced transitions to the intensity of the induced field-free allowed (+-« -) transi- tions is shown below to be proportional to a'(F)b(F) - a(F)b'(F) 2 a(F)a'(F) + b(F)b'(F) a(F) and b(F) are the coefficients in the perturbed eigenfunctions 71 given in equations (4.3) and (4.4). Primes denote upper state co- efficients.' Thus, the competitively weak induced transitions ob- served in the (,2 + v3 - v band of HCN that are not allowed in the 2 absence of a field are consistent with theoretical considerations because a'(F) and a(F) as well as b'(F) and b(F) will be nearly equal. Figure (4.5) shows the upper and lower L-type doublet states participating in the transitions under consideration. The zero- field eigenfunctions and zero-field Symmetry with respect to in- version are indicated for each of the four energy levels. A prime indicates the upper state. The example transition forbidden in the absence of a field is indicated by a broken line and the zero-field allowed transition is shown by a solid line. [0 V2 '0 Y1 0 Y2 0 Y1 Figure (4.5) Energy Level Diagram of n - n Transitions in a Linear Molecule. Solid line ' allowed transition. f _L I I I I I I I L . Broken line - forbidden transition. The eigenfunction of the upper A level of the upper vib- rational state in the presence of a field Y; is given by equation (4.3). l=_l to I ,0 Yu b (PM!1 +-a (F)?2 . As the field approaches zero, b'(F) .. o and .1"; .. Yé° as it should. Sinularly, the eigenfunction‘ of the lower level of the lower 72 vibrational state in the presence of a field Y is given by L O 0 v, amwl + b(mz. The transition moment between these two states is given by = (4.6) <{a(F)‘fc{ + b(F)Y;}'pz‘{-b' (P)‘Yio + 8' (F)Yéo}>. Only two of the four possible matrix elements on the right hand side of equation (4.6) are non-zero because of symmetry considerations. Therefore, equation (4.6) reduces to I . I O I0 a (F)b(F) dzhszl‘rz > (4.7) o o ‘8(F)b'(F) . The two matrix elements in equation (4.7) are the changes in the dipole moment in going from the lower to the upper vibrational state and should be almost identical. Thus, the field induced transition moment for this transition is proportional to [a'(P)b(F) - a(F)b'(F)] and the intensity of the induced transition is proportional to [a'(F)b(F) - a(F)b'(F)]2. The eigenfunction of the lower L level of the upper state Y' in the resence of a field is given by L P I. I I0 I 0 wt 8 (1’)?1 +b (PWZ. Vi reduces to Yio as the field approaches zero because b'(F) goes to zero as P a 0. Calculation of the transition moment for a transition between the lower state used in the preceding example and this upper state yields: 73 = a(F)a'(F) . o ,o + b(F)b' (F) <~y2\sz\w2 >. Thus, the induced transition moment of this field-free allowed transition is proportional to [a(F)a'(F) + b(F)b'(F)] and the intensity of the induced transition is proportional to [a(F)a'(F) + b(F)b'(F)]2. The ratio of the induced intensity of the field-free forbidden (+-« +) transition to the induced intensity of the zero-field allowed (+-« -) transition is therefore La'b, RRO(6>. RRO(9), P91(2), Pp1(3), Pp1<6), and Pp10). Using a triangular approximation, their areas were measured using a 6X magnifier for use as intensities in the calculations. All possible combinations of the above lines were used in formulas (5.1) to esti- mate the ratio of (p"/u). The fifteen sample values listed in Table (5.1) illustrate the results. The average value for the ratio of (u’/u) was 0.9 with a maximum deviation of 0.5. The electric dipole moment in the v3 vibrational state of HCN has been measured using Stark effect frequencies to be only 0.5 percent larger than that in the ground vibrational state (5). The 0.5 percent change can be assumed to be a typical order of magnitude value for dipole moment changes due to molecular vibration. Even the most sophisticated present day methods of intensity measurement are incapable of high enough accuracy for determination of such small changes in electric dipole moments. If highly precise intensity measurements become possible in the future, this method of determin- ing excited state electric dipole moments may prove more useful. 94 J of RRO(J) of P Transition Trans 2 2 2 3 2 6 2 7 3 2 3 6 3 7 5 2 5 3 5 6 5 7 6 3 6 6 6 7 9 7 Table 5.1 Measured Ratio of v4 J P10) ition and Ground State Electric Dipole Moments for CH F. 3 CHAPTER VI CONCLUSIONS EFS appears to be a useful new tool for infrared spectro- Scopists. Observed EFS exhibit line shapes and intensities in good qualitative agreement with theoretical predictions. In the linear molecule HCN, for example, we observed line shapes corresponding to two different types of normally forbidden transitions, second order Stark effect, first order Stark effect, and intensity enhancement due to L-type doubling. For all of the axially symmetric molecules, we found EFS a valuable tool in assigning transitions because - those transitions involving states of low angular momenta are very much stronger than those involving states of higher angular momenta. From the ratios of intensities of EFS transitions in the v4 band of CH F, we obtained an estimate of the excited state electric dipole 3 moment. Improved experimental techniques should eventually lead to more quantitative results. Suggestions for Future Development Several improvements Should be made in the present electric field induced Spectrum apparatus. First, a Princeton Applied Research (PAR) phase sensitive lock-in amplifier (or better) in conjunction with a precision reference frequency doubler should provide a Sub- stantial increase in information to noise ratio and allow operation at an optimum detection frequency. This piece of apparatus would also be of benefit to those interested primarily in absorption 95 96 '1 y it lir' spectra. Secondly, a dc amplifier to enable simultaneous recording of absorption and EFS signals would provide a useful frequency calibration method as well as greatly facilitating transition iden- tifications. Thirdly, the optimum dimensions and separation of a Stark cell for use in EFS studies should be experimentally determined. Finally, if the Spectral deconvolution studies about to be undertaken in our laboratory prove successful and experimental improvements can be made in the EFS apparatus to obtain an order of magnitude better resolution, useful information about line widths and excited state electric dipole moments could be obtained from induced spectra. BIBLIOGRAPHY 10. ll. 12. 13. 14. 15. 16. 17. 18. BIBLIOGRAPHY E.U. Condon, Phys. Rev. 43, 759 (1932). M.F. Crawford and R.E. MacDonald, Can. J. Phys. 36, 1022 (1958). R. Terhune and C. Peters, J. Mol. Spectry. 3, 138 (1959). C.H. Townes and A.L. Schawlow, "Microwave Spectroscopy". McGraw-Hill, New York, 1955. P.D. 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