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The infrared absorption spectrum of fluoroform Reesor, Thomas Richard 1952

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THE INFRARED ABSORPTION SPECTRUM OF FLUOROFORM by . THOMAS RICHARD REESOR A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR. THE. DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming t o the standard r e q u i r e d from candidates f o r the degree of MASTER OF APPLIED SCIENCE Members of the Department of P h y s i c s ' . THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1952 ABSTRACT The infrared absorption spectrum of fluoroform (CHF3) has been observed i n the region ^00 to 5550 cm-1 with a Perkin-Elmer spectrometer and, i n the higher regions, a multiple.path absorption c e l l . The t h e o r e t i -c a l shapes of a p a r a l l e l and a perpendicular band have been calculated and have been confirmed by the observed spectrum. The assignments of the fundamental frequencies by Rank, Schull and Pace^ have been confirmed and the apparent discrepancy i n the appearance of the 507.5 cm-1 band has been explained. The anharmonicity constant x&6 has been calculated to be -0.08 cm - 1. A l l of the funda-mentals except n a v e been remeasured to be: X 3033 cm-1 Pa 700.1 P4 1375 Ps- 1153 507.5 ACKNOWLEDGEMENT I wish t o thank.Dr. A. M. Crooker f o r h i s a s s i s t a n c e and.encouragement, d u r i n g the course of t h i s i n v e s t i g a t i o n . I a l s o g r e a t l y a p p r e c i a t e the a s s i s t a n c e and. c o - o p e r a t i o n of Mr. W. Ross.and. the. t e c h n i c a l a s s i s -tance of Mr. J . Lees, Mr. A. J . F r a s e r and. Mr. W. Maier i n the Physics. Shop. I am extremely g r a t e f u l f o r the f i n a n c i a l a s s i s t a n c e g i v e n by the N a t i o n a l Research C o u n c i l and the Defence Research Board. TABLE OF CONTENTS Page I . INTRODUCTION 1. Previous Work 1 2. Object and.Scope of Present Research 2 I I . THEORY - The. Symmetric Top Molecule w i t h S p e c i a l Reference to CHF-j 1. R o t a t i o n a l Levels and R o t a t i o n Spectra 3 a . R o t a t i o n a l Energy Levels 3 b . R o t a t i o n Spectra 5 2. V i b r a t i o n a l Energy Levels .and V i b r a t i o n a l Spectra; a . V i b r a t i o n a l Energy Levels 6 b . V i b r a t i o n a l Spectra..:. 9 3. I n t e r a c t i o n .of R o t a t i o n and . V i b r a t i o n a . Energy Levels 11 b . Infrared Spectra 12 I I I . EXPERIMENTAL. PROCEDURE 1. Apparatus a . The. Spectrograph 16-b . The M u l t i p l e Path Absorption C e l l 16 c . Sources 18 2. C a l i b r a t i o n 19 IV. RESULTS 21 V . CONCLUSION 26 APPENDIX: 27 BIBLIOGRAPHY 29 LIST OF FIGURES Figure., 1 F o l l o w s page 3 Figure.. 2 Follows page 10 Figures 3 Follows page 13 Figure.. 4 F o l l o w s page 15 Figure;; 5 F o l l o w s pager 16 F i g u r e . 6 F o l l o w s page e 17 F i g u r e 7 Page 18 F i g u r e .8 F o l l o w s page 2 5 F i g u r e . 9. F o l l o w s page. 27 THE INFRARED SPECTRUM OF FLUOROFORM I INTRODUCTION 1. P r e v i o u s Work The i n f r a r e d spectrum.of f l u o r o f o r m (CHF3) has been o b s e r v e d . i n the r e g i o n of the fundamentals by P r i c e 1 and i n the r e g i o n from 4340 cm" 1 to 14050 cm"*1 by p B e r n s t e i n and Herzberg. . The Raman spectrum of l i q u i d f l u o r o f o r m has been observed by Glocker and Leader^ and 4 Glocker a n d . E d g e l l . The o r i g i n a l assignment of the fundamental v i b r a t i o n f r e q u e n c i e s by P r i c e has been changed by.Rank, S c h u l l , and Pace5 who have made d e p o l a r i z a -t i o n measurements i n the, Raman spectrum to a s s i s t . i n the assignment.. The Raman l i n e a t 1117 cm""1 which i s not i n f r a r e d a c t i v e was r e a s s i g n e d as uz because of i t s i n t e n s i t y and. p o l a r i z a t i o n . Also., the o r i g i n a l , assignment by P r i c e of P3 and P6 , which he l a t e r r e v e r s e d , was con-f i r m e d . The fundamental assignments of these i n v e s t i g a t o r s a r e g i v e n i n Table I , page 2 . The r o t a t i o n a l c onstant, B, has been measured p —1 by B e r n s t e i n and Herzberg t o be 0.3451 cm and hence.the moment of i n e r t i a p e r p e n d i c u l a r to. the symmetry a x i s -40 2 I B = 8 1 . 0 8 x 10 g cm . A s s u m i n g . t e t r a h e d r a l angles f o r F-C-F ( 1 0 9 ° 2 8 ' ) , the moment of i n e r t i a p a r a l l e l t o the symmetry a x i s was c a l c u l a t e d to be I Q = 148.7 x 10 g cm' and the r o t a t i o n a l constant C = 0 .1887 cm - 1. The r o t a t i o n a l constant B has a l s o been measured i n the microwave r e g i o n by G i l l i a m , Edwards and Gardy^ to. be 0.34522 cm""1. - 2 -Table I . Fundamental .Frequencies of CHF. Glocker and Edge11 ( l i q u i d ) . P r i c e (gas) Rank, S c h u l l , and Pace ( l i q u i d ) A IS Assign.. . I n f r a r e d Assign.. &v A s s i g n . 508.1. 509.4 508. ^ C F 3 ( e ) 696.7 703.2 P>3 («) 697 936.8 1116.5 1117 Ji CF (.ai) II52.4 1160 Pf CF (e) 1209 IS2 & (a) 1376.2 1351.5 cF(e) 1376 ^.CF (e) 3062.0 "7 3035.6 P, cH(a) 3062 P, CH (a-^) 2. Object and.Scope.of Present..Research An, examina.tion.-of Table I shows t h a t there i s s t i l l , much d i s c r e p a n c y between t h e i n f r a r e d and .Raman d a t a . The most n o t a b l e d i f f e r e n c e s are the non-appearance of the. fundamental w2 i n . the. i n f r a r e d .and the l a r g e wave number d i f f e r e n c e between the Raman and i n f r a r e d v a l u e s f o r 44 and . The, purpose, o f the. present r e s e a r c h i s t o remeasure. . the fundamental, f r e q u e n c i e s . a n d to observe.whether or.not the spectrum i n the r e g i o n from 400 c m _ 1 t o 5500 cm"l i s compatible w i t h the new assignment o f 4 . , - 3 -I I THEORY r- The.Symmetric Top Molecule, w i t h S p e c i a l Reference  to C H F y # The CHF^ m o l e c u l e . ( F i g . 1) belongs t o t h a t c l a s s of molecules which has a 3?-fold a x i s of symmetry and has.--.two .moments of. ..inertia.perpendicular., t o .the. symmetry axis-, e q u a l . The m o l e c u l e . i s thus, p l a c e d i n p o i n t .group C^ v w i t h . t h r e e planes of symmetry and.one symmetry a x i s . 1. R o t a t i o n a l Levels, and R o t a t i o n S p e c t r a . C l a s s i c a l l y , .the motion of the symmetric top molecule can be v i s u a l i z e d as a r o t a t i o n of the molecule about the symmetry a x i s which.In t u r n p r o c e s s e s about another a x i s p o i n t i n g . i n the d i r e c t i o n of F, the t o t a l . an g u l a r momentum. The p r e c e s s i o n a l frequency i s |Pi/27tlg and the frequency of r o t a t i o n about the f i g u r e a x i s i s 2^(TQ •* j^j^Z* where I B and I c are the moments of i n e r t i a p e r p e n d i c u l a r . t o and about the symmetry a x i s . r e s p e c t i v e l y , and P z i s the component of P i n the d i r e c t i o n of the symmetry a x i s . In the quantum.mechanical, treatment of r o t a t i o n , the P* corresponds t o J and the corresponds to K. Thus i t i s i n t u i t i v e l y obvious t h a t K ^ J . a. R o t a t i o n a l Energy L e v e l s The r o t a t i o n a l term v a l u e s of the symmetric t i p molecule are g i v e n by # The theory of the symmetric top molecule has been covered q u i t e adequately by Herzberg.' Only r e f e r e n c e s t h a t have not been summarized by him w i l l be r e f e r r e d to i n t h i s s e c t i o n . c FIGURE I. THE CHF3 MOLECULE - 4 -F(J,K) = B J ( J + 1) + (C - B)K 2 - D j J 2 ( J 2 + l ) 2 - D J K J ( J • 1 ) K 2 - D K K* K — 0, 1, 2, • • • J ss K, K+l, K+2, • • • . .. 2 2 ' where B =. h/87ie I B , C == h/87tc3 I G and. the c o n s t a n t s (1) DJ» DJK» a n < i D K a r e d u e ' t o c e n t r i f u g a l s t r e t c h i n g of the molecule due to the. r o t a t i o n . In most cases> except f o r v e r y h i g h r e s o l u t i o n , o n l y the f i r s t two terms need be c o n s i d e r e d . symmetry depending on whether the e i g e n f u n c t i o n d e s c r i b i n g the s t a t e i s . unchanged, or changes s i g n .when a l l the. atoms are r e f l e c t e d a t the. o r i g i n . The e i g e n f u n c t i o n s are g i v e n where i j^- , <p and X are the E u l e r l a n a n g l e s , M i s the magnetic quantum.number, co r r e s p o n d i n g to the d i f f e r e n t o r i e n t a t i o n s of J i n space, and i s a c o m p l i c a t e d f u n c t i o n of The two m o d i f i c a t i o n s , p o s i t i v e and-n e g a t i v e , can be transformed one to the o t h e r by p a s s i n g through a p o t e n t i a l h i l l . I f the p o t e n t i a l h i l l i s not i n f i n i t e l y h igh, the two c o n f i g u r a t i o n s w i l l have, s l i g h t l y d i f f e r e n t , e n e r g i e s . c a u s i n g a s l i g h t s p l i t t i n g of the lines., ( i n v e r s i o n d o u b l i n g ) . In most cases t h i s s p l i t t i n g r e q u i r e s v e r y h i g h r e s o l u t i o n to measure a n d . w i l l be. n e g l e c t e d i n f u r t h e r d i s c u s s i o n . Apart, from the i n v e r s i o n d o u b l i n g , the s t a t i s t i c a l weight or degeneracy of each l e v e l i s 2J + 1 f o r K =s 0 and 2(2J + 1) f o r K ^ 0. In a d d i t i o n , t h e r e i s a s t a t i s t i c a l weight due to the s p i n of the R o t a t i o n l e v e l s have e i t h e r p o s i t i v e or n e g a t i v e by (2) i d e n t i c a l n u c l e i . F o r a molecule of p o i n t group C 3 v , the s t a t i s t i c a l weights are g i v e n by: F o r K d i v i s i b l e by 3 ( i n c l u d i n g zero) 1/3(21 + l ) ( 4 l 2 + 41 + 3) Fo r K n o t . d i v i s i b l e by 3 ... (3) 1/3(21 + l ) ( 4 l 2 + 4 1 ) where I i s the s p i n of the t h r e e i d e n t i c a l n u c l e i . Thuss the l e v e l s w i l l have a n . i n t e n s i t y a l t e r a t i o n of st r o n g , weak, weak, s t r o n g , weak, ... where the r a t i o of s t r o n g . to weak i s 1 + 3/41(1+1) ::1 or 2:1 f o r CHF^. The p o p u l a t i o n /VJK of the v a r i o u s l e v e l s i s dependent on b o t h the s t a t i s t i c a l , weight and the Boltzmann d i s t r i b u t i o n f a c t o r ~?« e'£(r-">/kT . . . (4) where $ T K i s the s t a t i s t i c a l weight. b. R o t a t i o n - S p e c t r a An i n f r a r e d r o t a t i o n spectrum can.appear o n l y i f the molecule has.a permanent dipole.moment. F o r a molecule of p o i n t group C 3 v , the. d i p o l e moment.will l i e a l o n g the symmetry axis, and the s e l e c t i o n rules, f o r J and K w i l l be A K - 0 A J = 0, +1 ... (5) In a d d i t i o n , the symmetry r u l e s w i l l "be + •*—>• - j +<//>+ — < # > — ... (6) The symmetry r u l e w i l l always, b e - f u l f i l l e d f o r non-planar molecules s i n c e the + and - l e v e l s always, o c c u r i n . p a i r s ( i n v e r s i o n d o u b l i n g ) . N e g l e c t i n g c e n t r i f u g a l s t r e t c h i n g , the I n f r a r e d a b s o r p t i o n l i n e s w i l l be g i v e n by F[(T+t),K] - FCS, K] = 2B(J+l) ... (7) The i n t e n s i t y of a l i n e i s g i v e n by the product of the t r a n s i t i o n p r o b a b i l i t y and the p o p u l a t i o n of the i n i t i a l s t a t e I c u ( T « f - K 2 d J K e ' ^ ^ r ... (8) where C i s a. constant depending ..on. the permanent, d i p o l e moment. The. i n t e n s i t i e s , of the pure r o t a t i o n l i n e s w i l l have. the. same d i s t r i b u t i o n as i n the R branch of a p a r a l l e l band shown i n F i g u r e 3» 2. V i b r a t i o n a l . Energy L e v e l s .and ..Vibrational. S p e c t r a a« V i b r a t i o n a l . Energy. L e v e l s The n a t u r a l v i b r a t i o n f r e q u e n c i e s of a molecule can, i n . p r i n c i p l e , be determined.from c l a s s i c a l mechanics i f the f o r c e c o n s t a n t s of the molecule.are known.. These f r e q u e n c i e s w i l l , be determined by the s o l u t i o n of the s e c u l a r e q u a t i o n il ^xy a KXZ "XX in KXZ ll kyy 'I™, A ll kyZ kyX k,N KyZ, H ii KZY kzz * i - k'N *ZZ 21 * X X 21 KXY a *XZ kxx~l71z^ *XZ fill Nl Kt, Nl *ZZ NZ N/J ta-where k'rs i s the r e s t o r i n g , force, on the i r p a r t i c l e i n the r d i r e c t i o n due to a u n i t displacement of the j th p a r t i c l e i n the s d i r e c t i o n , and A s 4 7 T 2 i / 2 where is i s the. n a t u r a l frequency of v i b r a t i o n . T h i s e q u a t i o n w i l l have 3N r o o t s g i v i n g r i s e to 3 N - 6 normal v i b r a t i o n f r e q u e n c i e s , where N i s the.number of atoms i n the molecule. The quantum .mechanical 1 treatment shows t h a t the molecule can e x i s t . o n l y i n s t a t e s t h a t are. sums, of h a l f -i n t e g e r m u l t i p l e s of these fundamental frequencies.. The Schroedinger e q u a t i o n of a system of N p a r t i c l e s of masses mj_ i s , i n C a r t e s i a n c o o r d i n a t e s , z i - f t r + £s+ £zya?(£-v)r*o ... ( 1 0 , T "V V. 3xf 3yf dz'J st v. . /• g T h i s can be reduced ( P a u l i n g and. Wilson ) t o the 3 N separate equations where.the \ft are the r o o t s of the s e c u l a r e q u a t i o n ( 9 ) , E = E-j. + E2 + . . . + E ^ + . . . + E -JN , and the are the normal c o o r d i n a t e s i The e i g e n f u n c t i o n s of eq u a t i o n (.11) are where Nw. i s a n o r m a l i z a t i o n constant and - H/^ ( j i s a Hermlte p o l y n o m i a l of the t h degree. The. e i g e n v a l u e s of e q u a t i o n (11) are g i v e n by £; = hi/i + OJ7 = Oj lj 2j-' (13) where v*; i s the frequency of the normal, v i b r a t i o n / and njf \& the v i b r a t i o n a l quantum number. The. t o t a l energy of the system i s # The 3 N r o o t s w i l l Include the s i x non-genuine v i b r a t i o n s which correspond t o r o t a t i o n a n d . t r a n s l a t i o n and.do not c o n t r i b u t e to the v i b r a t i o n a l , energy. - 8 -g i v i n g the term v a l u e s ( i n c m - 1 u n i t s ) G(AT „rri ...) * - eJs (AJT+4)+ CJz (*rf +4-)-h--- (15) hC where the Ui = i& are the c l a s s i c a l , v i b r a t i o n f r e q u e n c i e s c i n cm" 1. I f some of the l e v e l s a re degenerate, then the term v a l u e s w i l l be g i v e n by O (iTij nT2) •••) = Z- u)i (<Vi + ... (16) JUL where rf; i s the degree of degeneracy of the / v i b r a t i o n . I f the v i b r a t i o n s .are anharmonic, which i s . a l w a y s the case to some extent, the term v a l u e s w i l l , be g i v e n by (to second order) + 21. j** (17) where dj_, i s the. degree.of degeneracy, i s .a quantum number of degenerate, v i b r a t i o n s which assumes, the v a l u e s /if-J i/t-z, • - • lorOjUjis the v i b r a t i o n a l frequency f o r i n f i n i t e s i m a l amplitudes., and and g(t( are anharmonicity c o n s t a n t s . F o r a molecule, of p o i n t group C3 V,- a v i b r a t i o n a l s t a t e must be e i t h e r symmetric (A) or degenerate (E) w i t h r e s p e c t t o the symmetry a x i s , and symmetric.(Aj) or a n t i -symmetric (A2) w i t h r e s p e c t t o the pla n e s of symmetry. For the fundamental, v i b r a t i o n s of CHF-^, there are t h r e e , degenerate v i b r a t i o n s of s p e c i e s E and three.non-degenerate t o t a l l y symmetric, v i b r a t i o n s of s p e c i e s A^. The s e l e c t i o n r u l e s of the t r a n s i t i o n i n a b s o r p t i o n . s p e c t r a depend on the s p e c i e s (A o r E) of the i n i t i a l and f i n a l s t a t e s . The s p e c i e s of overtone and combination states, can be determined from, the f o l l o w i n g r u l e s : a l x a l = A-l. a^ x &2 = Ag a]_ x e = E 2 3 e x e .= A]_ + A2 + E (e) ==A]_ + E (e) = A]_ + Ag + E 4 • . . (e) = A x + E + E ... (18) b. V i b r a t i o n a l Spectra.:. Normal vi b r a t i o n s . , t h a t . are accompanied by a change i n the dipole.moment g i v e . r i s e t o a b s o r p t i o n l i n e s i n the I n f r a r e d and are. c a l l e d i n f r a r e d . a c t i v e . A l l of the., fundamental v i b r a t i o n s : should be bot h i n f r a r e d and Raman a c t i v e a l t h o u g h i n t e r a c t i o n of r o t a t i o n a l and v i b r a t i o n a l energy may prevent, t h i s . The observed f r e q u e n c i e s of the i n f r a r e d a b s o r p t i o n l i n e s w i l l be g i v e n by G0faj Gfa^zj-')-G(o,o,-•') = X CJ° ^ C- r - _ . . . (19) where <J» s u + x . . d . + ± c(« From,this, i t w i l l be noted t h a t the observed frequency of the. fundamental l i n e s w i l l be g i v e n by Vif = u% + xa + ga . . . (20) and the *xthovertone (^ifjT- (H-f)**? + «f(*r-t)x'< + < £ * M " ) 9 i i . . . (21) where 14{ and faff' are the a c t u a l observed t r a n s i t i o n f r e q u e n c i e s of the. fundamental and the a>tth overtone . r e s p e c t i v e l y , I t can be seen, from e q u a t i o n (21) that, the h i g h e r overtones of the degenerate v i b r a t i o n s , w i l l . b e . s p l i t because the degenerate quantum numbers £-t can take a l l the v a l u e s - 10 -Since the X4-K and $iK are u s u a l l y n e g a t i v e , the overtone l i n e s w i l l tend t o converge. The.observed combination l i n e f r e q u e n c i e s w i l l a l s o , i n . g e n e r a l be l e s s than.the sums of the observed fundamental f r e q u e n c i e s . The observed frequency of the d i f f e r e n c e lines., f o r example ^i-^s > w i l l , be e x a c t l y the wave number d i f f e r e n c e of the observed l i n e s . V, and V3 even when anharmonicity i s taken i n t o account. The i n t e n s i t i e s of the a b s o r p t i o n l i n e s , a r e , as b e f o r e , g i v e n by the. product of the. t r a n s i t i o n p r o b a b i l i t y and.the p o p u l a t i o n , o f the i n i t i a l l e v e l . As a r e s u l t of t h i s , i t can be seen.that the . d i f f e r e n c e l i n e Pt-^K w i l l have, a s m a l l e r i n t e n s i t y than the summation l i n e p + PK by a f a c t o r of G~ ^yy-" due.to the. s m a l l e r . I n i t i a l p o p u l a t i o n i n the.case of the d i f f e r e n c e l i n e . The overtone and comb i n a t i o n , l i n e s , w i l l , have a lower i n t e n s i t y due t o a lower t r a n s i t i o n p r o b a b i l i t y . The numbering of the fundamental, a b s o r p t i o n , l i n e s , of a molecule i s a c c o r d i n g t o the. s p e c i e s of the v i b r a t i o n c a u s i n g the .absorption .and .the f r e q u e n c y . The l a r g e s t t o t a l l y symmetric'frequency ( s p e c i e s k-± i n CHF^) i s c a l l e d If, , the second l a r g e s t i £ , and so on up to iSf. The l i n e with.the l a r g e s t frequency i n the next s p e c i e s (the degenerate s p e c i e s E i n CHF^) i s c a l l e d and so on u n t i l a l l , t h e fundamentals a r e l a b e l l e d . F i g u r e 2 shows the t h r e e t o t a l l y symmetric v i b r a t i o n s , and u3 , and one FIGURE 2 THE FUNDAMENTAL VIBRATIONS OF CHF3 - 11: component of each of the three degenerate v i b r a t i o n s U4, Us and V 6 f o r CHF^. 3. I n t e r a c t i o n of R o t a t i o n and V i b r a t i o n a . Energy L e v e l s The t o t a l v i b r a t i o n a l and. r o t a t i o n a l . e n e r g y of a symmetric top molecule i s g i v e n by 7"= G ( V , + FATCT,*) . . . (22) where F„ (J}K) = B^ J(J+l) + (G*r ~ B„)K* . . . (23) n e g l e c t i n g c e n t r i f u g a l , d i s t o r t i o n , terms.. The r o t a t i o n a l c o nstants C/v and B -^ are s l i g h t l y dependent on the v i b r a t i o n a l l e v e l because of the, l a r g e r average dimensions of the molecule i n h i g h l y e x c i t e d s t a t e s . To. a f i r s t a p proximation B«r*Be -'-H tifat . . . ( 2 4 ) where Be i s the r o t a t i o n a l constant f o r zero e x c i t a t i o n and of/ i s s m a l l compared to Be* F o r degenerate v i b r a t i o n s , the C o r i o l i s i n t e r a c t i o n , c a u s e s a s p l i t t i n g of the l e v e l so t h a t the term ±2C„ '£ K . . . (25) must be added to the r i g h t side..of e q u a t i o n (23). The f a c t o r £ can v a r y f r o m - 1 t o +1 and.can o n l y be c a l c u l a t e d i f the f o r c e c onstants of the, molecule .are known. I f anharmonicity i s n e g l e c t e d , the f o l l o w i n g r u l e h o l d s f o r any XYZ^ molecule: £ +Z =£{<~0.9 f o r CHF 3) ... (26) where $^  , £ and £ are the C o r i o l i s constants f o r the s i n g l y e x c i t e d v i b r a t i o n s , jss and JS& r e s p e c t i v e l y . 12 -b. I n f r a r e d S p e c t r a Pure r o t a t i o n s p e c t r a are not u s u a l l y observed i n i n f r a r e d spectroscopy because of the. l o ng wavelengths I n v o l v e d . The r o t a t i o n a l , energy i s u s u a l l y m a n i f e s t e d as f i n e s t r u c t u r e i n - t h e vibration-bands:. . In the case of CHF^, the f i n e s t r u c t u r e d i s t i n g u i s h e s between two d i s t i n c t types, of a b s o r p t i o n bands* A p a r a l l e l type of band occurs when the change of d i p o l e moment d u r i n g .the, t r a n s i t i o n i s i n the d i r e c t i o n o f the symmetry axis.. The s e l e c t i o n r u l e s f o r the rot a t i o n a l . q u a n t u m numbers, f o r a t r a n s i t i o n t h a t g i v e s r i s e t o a . p a r a l l e l band a r e : AK = 0 , AJ== 0 , ± 1 i f K ^ 0 AK = 0 , A J = ± 1 i f K = 0 . . . (27) A p e r p e n d i c u l a r . b a n d occurs when,the change of d i p o l e moment i s i n a d i r e c t i o n p e r p e n d i c u l a r t o the symmetry a x i s . The s e l e c t i o n r u l e s w i l l then be: AK = ± 1 AJ = 0, ± 1 ... (28) The symmetry s e l e c t i o n r u l e s w i l l always be f u l f i l l e d p r o v i d e d the i n v e r s i o n . d o u b l i n g i s not r e s o l v e d . The d e t a i l e d .structure of the bands can be c a l c u l a t e d by c o n s i d e r i n g the. a l l o w e d t r a n s i t i o n s and the . t h e o r e t i c a l i n t e n s i t i e s , of the i n d i v i d u a l lines.. C o n s i d e r i n g f i r s t the t r a n s i t i o n s between non-degenerate (A^) l e v e l s , t h a t i s . the p a r a l l e l bands., i t . can be seen that, f o r each value of the quantum number K there i s a complete sub-band w i t h a P brandii (AJ = - 1 ) , a. Q branch (AJ — 0)and an R branch ( z ^ J = +1). These sub-bands have o r i g i n s a t Usub + Lict- - Ct.) - (B£ - Kz . . . (29) where Va i s the frequency of the v i b r a t i o n a l t r a n s i t i o n , u i Cvrefers to the upper s t a t e and-C^ r e f e r s t o the lower s t a t e . The r o t a t i o n l i n e s w i l l t h e n , f o l l o w the formula ( A J , = +1) R branch V= PSut> + 2B(T+i) (AJ = 0) Q, branch V' =VSvl> * ]J(J+!) . . . (30) ( A J = - 1 ) P branch V- Psub - 2.8 J with, the added r e s t r i c t i o n t h a t J ^K. The v a r i a t i o n of B has been n e g l e c t e d i n . t h e P and R branch formulae of e q u a t i o n (30) because i t makes no s i g n i f i c a n t d i f f e r e n c e i n the appearance of the band. The i n t e n s i t i e s of the l i n e s w i l l be g i v e n by the Honl London formulae - F(T.K) A C I=CAJK i/-9rK e r { ' ' T ? . . . ( 3 D where C i s a constant depending,.on the v i b r a t i o n a l , t r a n s i t i o n , 9TK i s the s t a t i s t i c a l weight i n c l u d i n g the. r o t a t i o n a l , l e v e l degeneracy of 2 J + 1 f o r K = 0 and 2 ( 2 J + 1) f o r ,K ^ 0, and the. n u c l e a r s p i n i n t e n s i t y r a t i o g i v e n by e q u a t i o n ( 3 ) . The f a c t o r A J K Is g i v e n by A J = +1 ATTT - ( J + l ) 2 - K 2 2 _ v2 2 2 v x.2 * J = ° A J K = J ( J — (32) ^ J = - X A & = J ( 2 j - / l ) where J and K r e f e r t o the i n i t i a l , s t a t e i n a l l of the above formulae. The d e t a i l e d s t r u c t u r e of a p a r a l l e l band.has been c a l c u l a t e d f o r CHF-j on the b a s i s of these formulae and i s shown i n F i g u r e 3 . The v a l u e s G =---.0.189 cm" 1, B = 0.3^5 cm" 1 and [(c^.~ c£) - (B^- B^)J = .001 cm" 1 - J 50 25 25 50 J-K-0 K ° 6 K»9 K<I2 K'I5 K< 21 K« 3 M l l l l l l l " H I I -HI OIL J L L L L , i . .nil I n i l l l l l mil ILu. III! Hill I I I I I I I I I I . . . i i I n I I II J ± 1 1 1 1 1 1 1 1 1 ml 111111111 i I Ii I I I I i . II 111111 "' i i m M i n i . , milium 1 1 1 1 1 1 1 1 -16 cm FIGURE 3. SUB-BANDS AND COMPLETE PARALLEL BAND OF CHF3. The intensity of the Q bronches i nc reoses up to K = 2 4 ond then d e c r e a s e s . To ta l Q branch intensity should be greater than i nd i c a t ed . T " 3 0 0 " K . - 14 -were used i n these c a l c u l a t i o n s . The s e p a r a t i o n between the envelope peaks of the P and R branches i s 37 cm"*1 i n c o n t r a s t t o Gerhard, and Dennlson*s^ v a l u e of 33 c m - 1 c a l c u l a t e d from the formula" where logio Sfd) = Q- 7ZL and /3 = £- - / T r a n s i t i o n s between a. non-degenerate (A) and a degenerate l e v e l ( E ) g i v e r i s e to p e r p e n d i c u l a r type bands. T h i s c o n d i t i o n a p p l i e s to. the fundamentals of the degenerate v i b r a t i o n s . The o r i g i n s o f the sub"*bands f o r each value, of K a r e - g i v e n by: v.* - * • r(cW<-zp - &] * 2lot (i- a - B^JK + [(cl-Bl)-(C^-Bir)l K 2 . . . ( 3 4 ) where the.+ s i g n r e f e r s t o the t r a n s i t i o n f o r which A K = +1 and the - s i g n r e f e r s t o AK = - 1 . The P, Q and R branches of the sub-bands .are then arranged the same way as i n the p a r a l l e l , bands.. The . i n t e n s i t i e s of the l i n e s , i n the p e r p e n d i c u l a r bands w i l l be g i v e n by equ a t i o n (31) where the i n t e n s i t y f a c t o r A J K i s g i v e n by: A J = + 1 A J K = <J+>2_ 1,4 K> (J -+ 1)(2J + 1) * J = 0 A J K = * 1 ± * K) ... ( 3 5 ) # E q u a t i o n (33) was given,as AP= 5(0} -JEX i n r e f e r e n c e ( 9 ) . . The r i g h t , s i d e must be 77" g^r m u l t i p l i e d by the f a c t o r l / c t o make i t d i m e n s i o n a l l y c o r r e c t and n u m e r i c a l l y r e a s o n a b l e . 15 -where the upper s i g n r e f e r s to A K = +1, the lower s i g n r e f e r s t o A K = -1 , and. J and K r e f e r t o the i n i t i a l . s t a te. The values, f o r the K = 0 sub-band,must, be m u l t i p l i e d by-two. F i g u r e 4 shows a p e r p e n d i c u l a r band of CHF-j c a l c u l a t e d w i t h the. a i d of these formulae. The v a l u e of £ i n e q u a t i o n (34) was taken t o be zero f o r convenience. T h i s constant d i f f e r e n t from zero would, te n d to i n c r e a s e or decrease the spread o f the o r i g i n s of the sub-bands depending, on whether £ was p o s i t i v e or n e g a t i v e . I f the constant £ = - 0 . 8 , then the f a c t o r [C%,(/-£)-££] i n e q u a t i o n (34) would be approximately zero and the o r i g i n s of the sub-bands would not v a r y w i t h K. In such a case, the appearance of a p e r p e n d i c u l a r band would be approximately the same as t h a t of a p a r a l l e l band. T r a n s i t i o n s between two degenerate. (E) l e v e l s w i l l g i v e r i s e t o b o t h . p a r a l l e l a n d , p e r p e n d i c u l a r bands a c c o r d i n g t o the s p e c i e s r u l e s . g i v e n i n e q u a t i o n (18). The two bands, w i l l . i n general,have v e r y d i f f e r e n t magnitudes so t h a t o n l y one of the two components w i l l c o n t r i b u t e t o the appearance of the. a b s o r p t i o n band. FIGURE 4. SUB-BANDS AND COMPLETE PERPENDICULAR BAND OF CHF 3. The peaks of the P and R branches increase up to K = 23 and then decrease. Calculations are for a temperature of 300°K and £ =0. - 16 -I I I EXPERIMENTAL PROCEDURE 1. Apparatus a . The Spectrograph The i n f r a r e d s p e c t r o g r a p h used was,a P e r k i n -Elmer model 12-B w i t h L i F , KBr and NaCl prisms., a n d . a d . c . thermocouple d e t e c t o r . T h i s p a r t i c u l a r instrument has been . more, f u l l y d e s c r i b e d by R o s s 1 0 and. L i t t l e 1 1 . The.sample c e l l s used were a 10 cm. c e l l , a standard Perkln-Elmer meter c e l l , and a m u l t i p l e p a t h a b s o r p t i o n c e l l d e s c r i b e d i n s e c t i o n b. b . The M u l t i p l e P a t h A b s o r p t i o n C e l l . In o r d e r to measure weak a b s o r p t i o n bands, a l o n g p a t h l e n g t h - i s necessary i f the p r e s s u r e i s to be kept low enough to prevent p r e s s u r e broadening and p r e s s u r e s h i f t s of some of the bands.. The,most, p r a c t i c a b l e way of g e t t i n g a l o n g . a b s o r p t i o n p a t h without h a v i n g extremely l a r g e sample c e l l s i s by means-of a m u l t i p l e p a t h a b s o r p t i o n c e l l as f i r s t c o nceived by White^ 2. The o p t i c a l arrangement of the. c e l l , shown i n F i g u r e 5, i s the o n l y type now used which w i l l a l l o w a. l a r g e e f f e c t i v e a p e r t u r e . It. c a n be seen from F i g u r e 5 t h a t none of the l i g h t i s l o s t by s u c c e s s i v e t r a v e r s a l s . The l i g h t from the entrance s l i t i s imaged,by m i r r o r A on p o i n t 1 of m i r r o r C. M i r r o r C then images m i r r o r A on.mirror B and so on u n t i l the entrance s l i t i s f i n a l l y imaged on the e x i t s l i t . Due t o the l a r g e number of o f f - a x i s r e f l e c t i o n s bf the beam, i t was suspected t h a t the astigmatism of the f i n a l image FIGURE 5. OPTICAL ARRANGEMENT OF ABSORPTION C E L L S E T FOR 8 T R A V E R S A L S . - 17 -would be so l a r g e t h a t the i n t e n s i t y would be reduced. C a l c u l a t i o n s 1 ^ showed however, t h a t i f the entrance s l i t of the c e l l were made l o n g e r than the entrance s l i t of the spectrometer, t h e r e would be no l o s s . i n . i n t e n s i t y due ' to a s tigmatism. T h i s c a l c u l a t i o n i s given, i n the appendix-. The e n t i r e o p t i c a l system of the c e l l i s mounted on an.aluminum, channel and p l a c e d i n s i d e an e i g h t i n c h diameter pipe w i t h NaCl windows f o r entrance and e x i t beams.. F i g u r e 6 shows the instrument w i t h the. o p t i c a l p a r t taken o u t s i d e the tank. A l l the m i r r o r s have the necessary a d j u s t i n g screws f o r h o r i z o n t a l and angular a d j u s t ment. so that the m i r r o r s can.be a d j u s t e d while f r e e of the tank and then p l a c e d i n s i d e . The channel r e s t s on a t h r e e p o i n t suspension i n s i d e the,tank so t h a t any d i s t o r t i o n of the tank due to e v a c u a t i o n w i l l not a f f e c t the f o c u s of the beam. The number of t r a v e r s a l s of the beam i s changed by t u r n i n g a micrometer screw mounted on the o u t s i d e of the tank. T h i s r o t a t e s m i r r o r A by means of a c o n n e c t i n g s h a f t which,passes through the si d e of the tank. A vacuum seal, i s maintained, by-mounting an/'O" r i n g on the movable -shaft so that, the gas sample need not be d i s t u r b e d when the number of t r a v e r s a l s i s changed. T h i s f e a t u r e has been found t o be v e r y convenient i n p r a c t i c e . One e i g h t - i n c h diameter pyrex m i r r o r was p o l i s h e d and cut as shown i n Figure..7. T h i s method.of c u t t i n g ensures a cheaper and more e f f i c i e n t use of the m i r r o r s u r f a c e s than by p o l i s h i n g two separate m i r r o r s . The pyrex s u r f a c e s were coated w i t h aluminum and magnesium F I G U R E 6. T H E M U L T I P L E P A T H A B S O R P T I O N C E L L . - 18 -f l u o r i d e which gave a measured r e f l e c t i o n c o e f f i c i e n t of 0.96 i n the near i n f r a r e d . F i g u r e 7. C e l l M i r r o r s s c u t f o r most E f f i c i e n t Use. c. Sources. A g l o b a r was used.with the 10 cm and. meter c e l l s , but a b r i g h t e r source was d e s i r a b l e f o r . t h e m u l t i p l e p a t h c e l l because, of the lower aperture of the a l l o w e d beam when t h i s c e l l was used ( f / 6 compared to f / 4 . 5 of the s p e c t r o g r a p h ) . A l a r g e 2 . 5 mm.diameter Nernst glower was used w i t h ' t h i s c e l l . T h i s , source was. more s t a b l e than e i t h e r the carbon a r c or the s m a l l e r 1.3 mm diameter glower. The glower was p l a c e d i n the p o s i t i o n of the entrance s l i t , so t h a t no mechanical entrance or e x i t s l i t s were needed as the dimensions of. the glower were of the proper s i z e to a c t as an entrance s l i t . - 19 -2. C a l i b r a t i o n The method of c a l i b r a t i o n was t h a t developed by 14 Ross and L i t t l e • The c a l i b r a t i o n p o i n t s w e r e , f i t t e d t o the formula ° r- 7-0 where T i s the wave d r i v e r e a d i n g and the c a l c u l a t e d 2 constants v0 , a and. .7© have no r e a l . s i g n i f i c a n c e . Above 4000 cm" 1, f o r m u l a ' ( 3 6 ) does not f i t over a l a r g e enough r e g i o n to be u s e f u l so a c a l i b r a t i o n , curve was drawn i n , t h i s . r e g i o n . The c h i e f o b s t a c l e i n the. c a l i b r a t i o n was i n the c h o o s i n g of s u i t a b l e c a l i b r a t i o n p o i n t s . In most r e g i o n s , the accuracy of the measurements :was l i m i t e d by the c a l i b r a t i o n points; themselves i I t would, be d e s i r -a b l e t o have a c c u r a t e c a l i b r a t i o n p o i n t s l i s t e d t o 0.1 cm" 1 i n s t e a d of t o the n e a r e s t wave number as i s g i v e n i n the * Perkin-Elmer c a l i b r a t i o n d a t a . The KBr r e g i o n was, c a l i b r a t e d u s i n g the water, vapour data of R a n d a l l , Dennlson, Ginsburg and,Weber 1^, the C0 2 data of M a r t i n and B a r k e r 1 ^ , and the 1-2-4 t r l -chlorobenzene and, p o l y s t y r e n e p o i n t s of P l y I e r and P e t e r s 1 7 . A l t h o u g h the g r a t i n g s p e c t r a of water vapour had a much hig h e r d i s p e r s i o n than c o u l d be o b t a i n e d from the P e r k i n -Elmer spectrograph, the p o i n t s c o u l d . e a s i l y be. c o r r e l a t e d . T h i s c a l i b r a t i o n f o r KBr was found to be f a r b e t t e r than the c a l i b r a t i o n g i v e n i n the Perkin-Elmer manual • - 20 -The c a l i b r a t i o n p o i n t s used i n a l l r e g i o n s a re l i s t e d i n . T a b l e I I . In a d d i t i o n . t o these, some of the 17 p o i n t s s u p p l i e d by P l y I e r and P e t e r s ' were used i n a l l t h r e e r e g i o n s . The main.objection, t o these c a l i b r a t i o n p o i n t s (17) i s t h a t they are not spaced c l o s e l y enough and some of the l i n e s a re broad. Table I I C a l i b r a t i o n . P o i n t s L i F Region NaCl Region Hg A r c 1 7 . HgO " 1-2-4 - T r l c h l o r o b e n z e n e NH 3 > 18 H 2 0 1 9 co 2 > NBj " CH 4 - KBr Region HBr co 2 * 18 1-2-4 17 T r l c h l o r o b e n z e n e 1 CO Polystyrene- 1- 7 HgO , 0 G 2 1 6 - 21 -IV RESULTS In the r e g i o n from 400 fct> 5540 cm - 1, over, f o r t y a b s o r p t i o n bands were found. These bands w i t h t h e i r assignments and c a l c u l a t e d v a l u e s are l i s t e d , i n T a b l e I I I . A l l of the., bands above 1700 cm" 1 w i t h i n t e n s i t i e s .marked as v e r y v e r y weak (v.v.w.), were found by u s i n g the m u l t i p l e p a t h a b s o r p t i o n c e l l . B e f o r e : d i s c u s s l n g the a n a l y s i s of the r e s u l t s , i t i s p r o f i t a b l e t o examine the appearance of the t h e o r e t i c a l p a r a l l e l and p e r p e n d i c u l a r bands shown i n F i g u r e 3 and F i g u r e 4 . F i r s t the appearance of the p a r a l l e l band: I t s h o u l d be noted t h a t the s e p a r a t i o n of the maxima of the P and R branches i s 37 cm"*1 and the d i p on the h i g h f r e q u e n c y s i d e of the c e n t r a l , Q, branch i s deeper than.on the low frequency s i d e . A l s o , the. i n t e n s i t y of the R branch i s s l i g h t l y g r e a t e r than t h a t of the P branch. Figure. 8(b) shows a band w i t h these d e f i n i t e c h a r a c t e r i s t i c s . The. p e r p e n d i c u l a r band has l e s s d i s t i n g u i s h i n g c h a r a c t e r i s t i c s . Under low r e s o l u t i o n , i t would show up as a s i n g l e l i n e w i t h no c h a r a c t e r i s t i c s t r u c t u r e , o r a t b e s t as. a s l i g h t l y non-symmetrical band, as shown i n the t h e o r e t i c a l t r a c e of F i g u r e 4 and.also i n the 1153 cm""1 l i n e i n F i g u r e 8 ( a ) , w i t h the steeper: slope on the h i g h frequency s i d e . The p e r p e n d i c u l a r band may have an a l t o g e t h e r : d i f f e r e n t appearance depending on the v a l u e of the constant . $° as was p o i n t e d out i n S e c t i o n I I - 3 - b . - 22 -Table I I I I n f r a r e d A b s o r p t i o n Bands of CHF^ Observed cm" 1 Assignment^ Calculated:" 507.5 m. E 700.1 m. // A ----. 869 v.v.w. A + E 867.5 1015 v.w. A + E 1015 1153 v.s • _1 E 1209 m. E 1208 1375 s. E 1658 w. Pr +^6, A + E 1661 1716 v.v.w. 2P(,+p3 A + E 1715 1818 v.v.w. Pz + P3 A 1817 1849 w. Ps +P3 E 1853 1909 v.w. Pc+ZP3 E 1908 2029 v.v.w. + P&, A + 2E 2030 2076 (double) w. P 3 + P * E 2075 2161(d) v.w. P5 A + 2E 2168 2281 s. E 2270 2292 2303 ZP5 A + E 23066 2315 . on COg 2408 2415 J v.w. 2P±+2.P6, A- + E 2415 25177 v . s . P4 + Ps A + E 2528 # The s p e c i e s g i v e n i s the s p e c i e s of %r %r which i s obtained, by the m u l t i p l i c a t i o n r u l e s (18). In a l l cases.except f o r the d i f f e r e n c e bands., i t . i s .the. s p e c i e s of the upper s t a t e . _ - 23 Table I I I - Continued Observed cm"*1 Assignment C a l c u l a t e d 2692 ~ 2709 s. 2^2 + P<, E 2742 2728 • 2754 m. 2»4 A + E 2750 2778 m. E 2775 3033 v . s . // Pi A 3041 w. P+ + P5 + ^ A + 3E 3036 3183 v.w. PZ y- P3 +P+ E 3192 3207 v.v.w. P3 +P4+"S A + E 3228 3255 v.v.w. 2P4 +P* A 3E 3258 3406 5 w. , »z + 2Pr A + E 3423 3438 w. 2P4 t P3 A + E 3450 3483 v.v.w. 3P3 + P4 E 3475 3545 v.v.w. Pi +P*> E 3541 3645" 3740 3850. . w. on • HgO w. bands w. J/g +L>4 +Pf P, +Ps Pz+2P4 A A A + + E" E 3645 3733 3867 4045 ' 4056:: r v.v.w. P+2Pb A + E 4048 4171 ^  4186^ s . . M +Ps E 4186 4210 v.w. E 4241 4342 v.v.w. P, +P3 -hPt+Ps- A + E 4345 4405 (d) s. E 4408 4515 v.v.w. 3Pz + Pg E 4504 - 24 -Table. I l l - Continued Observed cm" 1 Assignment C a l c u l a t e d 4550 v.v.w. P, +3Pf, A + E, 4556 4691 v.v.w. A + E -4694 4716 v.v.w. V, rP3+2P6 A + E 4748^ 4770 v.v.w. A + E 4762 4877 v.v.w. J/, i- P3 +Ps E 4886 4987 v.v.w. v, +"3 E 5108 5185 w. t/, + P5 + Mb A + 2E 5201 53^0 w. P, + tPf A + E: 5339 5541 w. P, + P+ +• Pf A + E 5561 The. band., a t -700 .1 cm" has the appearance, of the., p a r a l l e l band shown i n F i g u r e 3 . This. i s . contrary, t o the statement, by P r i c e d t h a t the 700.1 cm" 1 band had,an i n d i f f e r e n t appearance. The 507»5 cm"l band a l s o has the appearance of a p a r a l l e l band but i n .view of the d e p o l a r -i s a t i o n measurements, of Rank, S c h u l l and Pace-', i t i s c o n s i d e r e d t o be a . p e r p e n d i c u l a r band w i t h t h e . c o r i o l i s . c onstant £ — - 0 . 8 i T h i s assignment i s c o n s l s t a n t w i t h the bands / ^ - ^ , P3 + Pt> > Zt>z+Pt, and 2J/3 + J/6 a l l of which a r e p e r p e n d i c u l a r bands but have the appearance of p a r a l l e l bands• The bands 2709, 2754 and 2778 cm" 1 are d i f f i c u l t t o a s s i g n . There i s probably some displacement of a l l t h r e e due t o Fermi resonance, and i n a d d i t i o n , the. 2778 cm" band c o u l d e q u a l l y w e l l be + Z£ *ifc or l^ + 2J/g £ because the - 25 -i n t e r a c t i o n constants, are not w e l l enough known to help decide between them. The group; of bands 3183, 3207 and 3255 cm 1 shown, i n Figure 8(d) i s a l s o d i f f i c u l t to a s s i g n because of Fermi resonance, and . .overlapping,structures. The -3033 c m " 1 band shown, i n Figure 8(c) was. i d e n t i f i e d as a p a r a l l e l band with the R branch peak at +20 c m " 1 and the P branch peak at -17 c m " 1 from the band c e n t r e . This assignment i s most probable because t h i s frequency i s c h a r a c t e r i s t i c of the 5C-H s t r e t c h i n g v i b r a t i o n i n other molecules-and,would,give r i s e to a . s t r o n g p a r a l l e l band The e x t r a - l i n e between the Q and R branches at 3041 cm""-*- i s considered to be due to an e n t i r e l y d i f f e r e n t t r a n s i t i o n . The only overtone s e r i e s found was , 2 ^ and 4 l i • The l i n e 3/^6 would be d i f f i c u l t to f i n d because i t would l i e on the H 2 0 r o t a t i o n bands which are always present i n . t h e . spectrogram. The anharmonicity term, x 6 6 would then be 3 f 6 6 = - 0 . 0 8 c m " 1 a c c o r d i n g to equation (21). The assignment of . bands...with ..higher wave numbers becomes i n c r e a s i n g l y d i f f i c u l t because t h e , r e s o l u t i o n i s u s u a l l y . n o t h i g h enough to give any observable s t r u c t u r e that might help, i n the assignment. Also the anharmonicity terms are not known w e l l enough t o c a l c u l a t e , m o r e , c l o s e l y the expected,band,,frequency. An attempt was made to c a l c u l a t e a l l . the anharmonic i t y constants but i t was found.that there were too,many I n c o n s i s t e n c i e s with the data a v a i l a b l e , ©nly 14 of the 21 double,combination bands, which would c (a) A ± Band of CHF3 (b) A II Band of CHF2 (c) The fundamental 2Jr (d) Overlapping structures FIGURE 8 ABSORPTION BANDS OF CHF3 - 26 -give the anharmonicity constants d i r e c t l y , were found. Even some of these may have been a f f e c t e d by Fermi resonance. I t was hoped that one of the; d i f f e r e n c e bands. P2-P6» pz-1/^ , or even U, - Pz could be found to e s t a b l i s h a m o r e , d i r e c t measurement f o r uz than the. Raman value- of 1117 c m " 1 f o r l i q u i d f luoroform.. As. none of. these bands were found, the. value, of 1117 c m " 1 was used, f o r Pz i n , a l l assignments. A l t o g e t h e r , u2 was assigned i n seven combina-t i o n bands, the most c e r t a i n of which was the 1818 c m " 1 band. V CONCLUSION The assignment of the fundamental v i b r a t i o n s by Rank,, S c h u l l and Pace-* has been found c o n s i s t e n t w i t h the observed spectrum. The apparent discrepancy i n the appearance, of vb as a p a r a l l e l band has been explained and the. anharmonicity constant x^g was c a l c u l a t e d to be - 0 . 0 8 cm*"l. The other.anharmonicity and. i n t e r a c t i o n constants were s u f f i c i e n t l y u n c e r t a i n as to r e q u i r e an examination under,higher d i s p e r s i o n . t o c a l c u l a t e , t h e m w i t h any d e g r e e o f c e r t a i n t y . The fundamental v i b r a t i o n l i n e s were remeasured to be: U, 3033 cm*"l yz (1117) 5 p3 700.1 P+ 1375 JJ; H 5 3 Pe 507.5 - 27 -APPENDIX The Astigmatism of a M u l t i p l e P ath A b s o r p t i o n : C e l l The number of t r a v e r s a l s , N, of the. l i g h t , beam through the l e n g t h of the c e l l i s g i v e n by N = 2d/Mr where d, M and: r are shown . i n F i g u r e 5• The astigmatism,.can be c a l c u l a t e d as f o l l o w s : For the v e r t i c a l l i n e image, _L + JL - a — - — - where S, = v e r t i c a l l i n e image d i s t a n c e , S S, r cos <p ' $ = o b j e c t distance. — r a d i u s of c u r v a t u r e , r , and, <p — o f f - a x i s a n g l e . F o r the h o r i z o n t a l l i n e image, JL -f 4r = -  0 0 5 — where S, = h o r i z o n t a l l i n e image distance', S Sz r 2 /±\ j _ J. _ 2 2 g<?5 4> ~ AS **\Sr S, " S2 r ros<fr r s2 where A S = d i s t a n c e between h o r i z o n t a l and v e r t i c a l l i n e image s• AS -*S-r - Zr cos* « 2 r £ ( ^ £ £ + £ + £ - • - ) ] ~ COS d> *. 2/ The astigmatism i s a d d i t i v e on each r e f l e c t i o n s i n c e J. 4 . - i - = 2. ... simple m i r r o r formula P * 1 r. Ap - '(•^•) (^t) - ~(At) s i n c e pz*?-=*r T h e r e f o r e the a s t i g m a t i s m , i s a d d i t i v e i f . the d i s t a n c e between.the h o r i z o n t a l and v e r t i c a l l i n e images i s s m a l l compared t o r . Th e r e f o r e * 2 r f a % - $ • • + where the <p. are the s u c c e s s i v e o f f - a x i s a n g l e s . FIGURE 9. ASTIGMATISM OF ABSORPTION CELL . - 28 -By c a l c u l a t i n g the s u c c e s s i v e a n g l e s 0t-, i t i s easy t o show t h a t X 4 *£uz+3*+-+(%f]M2- i(i+z+.... + *)Ma + 4ez t'=l * ' z where 0 — angle between source and a x i s of f i x e d m i r r o r . U s i n g M = 2 d/rN and. Q— d / 2 r (may be s l i g h t l y g r e a t e r r depending on the. geometrical, set u p), the formula can be N/, f u r t h e r s i m p l i f i e d : t o 2d* T4 5~ * - N — 2.1 The measured and c a l c u l a t e d .values of astigmatism, up t o 32 t r a v e r s a l s a re shown i n F i g u r e 9 . In low a p e r t u r e instruments the agreement would be even b e t t e r than t h a t shown. One would expect some l o s s i n i n t e n s i t y of the e x i t beam, due to the astigmatism. By u s i n g simple geometry, however^ i t can be seen t h a t i f the instrument i s a d j u s t e d t o be i n f o c u s f o r the. v e r t i c a l l i n e image;?, and the e n t r a n c e . s l i t i s made lon g e r than the e x i t s l i t . b y an amount , then t h e r e w i l l be no l o s s i n i n t e n s i t y due r t o a s t i g m a t i s m . B i b l i o g r a p h y 1. W. C. P r i c e , u n p u b l i s h e d , r e p o r t e d by B e r n s t e i n and Herzberg^. 2. H. J . B e r n s t e i n and G. Herzberg, J . Chem. Phys.,16,30,1948. 3 . G. Glocker and G. R. Leader, J . Chem. Phys., 8, 699, 1940. 4 . G. Glocker and W. F. E d g e l l , J . Chenu Phys. 9, 224, 1941. 5 . . D. H. Rank, E . R. S c h u l l and E . L. Pace, J.Chem. Phys., 18, 885, 1950. 6. 0 . R. G i l l a m , H. D. Edwards and Walter Gordy, Phys. Rev., 75, 1014, 1949. 7. G. Herzberg, M o l e c u l a r S p e c t r a and M o l e c u l a r Structure., I I I n f r a r e d and Raman S p e c t r a ;of Polyatomic M o l e c u l e s . (D. Van Nostrand, Hew York, 1945). 8. L. P a u l i n g and E . B. W i l s o n J r . , I n t r o d u c t i o n to Quantum Mechanics. (McGraw H i l l , New York, 1935). 9. S. L. Gerhard and D. M. Dennison, Phys. Rev., 43,197,1933. 10". W. L. Ross, T h e s i s f o r the degree of Master of A r t s , U. B. C , 1950. 11. D. E. L i t t l e , T h e s i s f o r the degree of Master of A r t s ? U. B. C•, 1950. 12. J . U. White, J . 0 . S. A., 32, 285, 1942. 13. IB. R. Reesor, J . 0 . S. A., 4 l , 1059, 1951. 14. W. L. Rosssand D. E. L i t t l e , J . O. S. A., 41,1006, 1951. 15. H. M. R a n d a l l , D. M. Dennison, N. G. Ginsburg and L. R. Weber, Phys. Rev., 52, 160, 1937. I 6 i P. E . M a r t i n and E . F. Barker, Phys. Rev., 41,. 291, 1932. 17. E . K. P l y l e r and C. W. P e t e r s , J . Res. Nat. Bur. S t d . , 45, 462,. 1950. 18. Perkin-Elmer C o r p o r a t i o n , I n s t r u c t i o n Manual, Model 12-C 19. W. W. S l e a t o r , A s t r o p h y s . J . , 48, 125, 1918. 

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