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Jun 1, 1977 - In measurements involving resonant absorption or emission of light, the Doppler effect furnishes a different means of access to velocity...
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8 Fourier Transform Doppler Spectroscopy: A New Tool for State-to-State Chemistry JAMES L. KINSEY

Downloaded by TUFTS UNIV on June 12, 2018 | https://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch008

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139

To achieve a complete description of the states of reactants or products in a chemical reaction, it i s necessary to specify the translational states of the participants. A somewhat coarser level of detail still might include the dependence of the rate on the magnitude of the initial or final relative v e l o c i t y . Molec­ ular beam reactive scattering experiments are the principal sources of data on translational-energy dependence. For reactants this is achieved through velocity selection and for products by the combination of velocity analysis and angular d i s t r i b u t i o n s . In measurements involving resonant absorption or emission of l i g h t , the Doppler effect furnishes a different means of access to velocity information potentially as detailed as that attained by direct velocity analysis. This approach appears especially promising in measurements employing laser-induced fluorescence. To first order, the Doppler shift v in an absorption line produced by a velocity component of magnitude w in the direction of propagation of the incident l i g h t is given by v =wλ where λ is the wavelength of the resonant radiation. Under conditions that the main source of linewidth is Doppler broadening, the line shape d i r e c t l y reflects the sample's distribution in w. Such a one-dimensional determination at first seems inadequate for the general situation, since the Doppler effect is blind to the transverse velocity components. It has recently been demon­ strated (1), however, that the set of Doppler profiles as a func­ tion of the direction of propagation of incident l i g h t can be d i r e c t l y inverted to recover the f u l l three-dimensional velocity distribution F(v) of the sample, irrespective of the nature of F(v). Let D(w;n) be the distribution in the parallel velocity com­ ponent w as exhibited by the Doppler l i n e p r o f i l e for l i g h t i n ­ cident along the direction of the unit vector n. The equivalence of D(w;n) and F(y) is established by the demonstration that the one-dimensional Fourier transform of D(w;n) is identical to the three-dimensional Fourier transform of F(v), evaluated along a line parallel to η in Fourier space ( 1_). " i . e . if D

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Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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Fourier Transform Doppler Spectroscopy

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G(k)=jd vF(v)exp(27nk.v) then JdwD(w;n)exp(27riKw)=G(nK). Hence, 3

the term, F o u r i e r - t r a n s f o r m Doppler spectroscopy (FTDS). In the conventional molecular beam experiments, angular d i s ­ t r i b u t i o n s are accomplished with a detector t h a t views only those molecules with v e l o c i t i e s d i r e c t e d i n t o a s e l e c t e d w e l l - d e f i n e d s o l i d angle. Consequently, the magnitude of the s i g n a l i s l i m ­ i t e d by the small s o l i d angle subtended ( d 2 ) no matter what means i s used to probe the d i s t r i b u t i o n i n speed. FTDS a f f o r d s an average gain over t h i s method of 4 / d f t i n the r a t e of s i g n a l a c q u i s i t i o n , s i n c e every molecule formed has some value of the v e l o c i t y component p a r a l l e i to η regardless of how η i s chosen. Estimation of the s i g n a l / n o i s e improvement i s more complex and depends on the nature of the noise as w e l l as of the v e l o c i t y d i s t r i b u t i o n , but there are no circumstances i n which FTDS has d i s f a v o r a b l e s i g n a l / n o i s e compared to conventional methods with equivalent detector c h a r a c t e r i s t i c s . C e r t a i n experimental arrangements permit s i m p l i f i e d forms of FTDS, f o r example, processes of the type A*+B-K+D where A* i s prepared by D o p p l e r - s e l e c t i v e absorption of l i g h t . In t h i s c a s e , the q u a n t i t y whose v e l o c i t y d i s t r i b u t i o n i s monitored w i l l be the r e a c t i o n r a t e R(w;n)=k(w;n)[A*(w;n)] where [A*] i s the con­ c e n t r a t i o n of A* and k i s t h e s p e c i f i c r a t e c o e f f i c i e n t . I f ^ these measurements are c a r r i e d out i n a s t a t i c gas, n e i t h e r [A ] nor k depends on the d i r e c t i o n n. The F o u r i e r - t r a n s f o r m r e l a ­ t i o n s h i p s t i l l h o l d s , but takes on a s i m p l i f i e d form: -(2ffw)- dR(w;o)/dw=F(|y|) where F gives the dependence of the r e a c t i o n r a t e on the magnitude of the vector y. For t h i s t e c h ­ nique to work, i t i s necessary that the observed process occur before the prepared A* i s t r a n s l a t i o n a l l y r e l a x e d , but t h i s i s e a s i l y guaranteed i f the r a d i a t i v e l i f e t i m e of A* i s short com­ pared to the c o l l i s i o n time. Processes of the type A*+B+C*+D where A i s s e l e c t i v e l y prepared using the Doppler e f f e c t and C* i s observed by a Dopplers e l e c t i v e method can be subjected to a s i m i l a r a n a l y s i s . Let R(wA, w ; nA, nC) be the observed r a t e where A* i s s e l e c t e d at

Downloaded by TUFTS UNIV on June 12, 2018 | https://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch008

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v e l o c i t y component wA p a r a l l e l to nA and C* i s analyzed a t component WC p a r a l l e l to nc i n a s t a t i c gas. I t i s evident t h a t t h i s r a t e can depend only on the two magnitudes wA, wC and the angle between the two l a s e r beams yAC= c o s ( n A - n c ) . The F o u r i e r transform of t h i s q u a n t i t y (with respect to both w/\ and WQ) can be i n v e r t e d to o b t a i n the s i x - d i m e n s i o n a l d i s t r i b u t i o n F ( V A , V Q ) g i v i n g the dependence r a t e on the vectors ν& and V Q . However, t h i s F can depend only on: l y l , ly >0 = angle between V A > V C _ 1

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Hence, i n p r i n c i p l e the use of two l a s e r s to measure the Doppler p r o f i l e s of both r e a c t a n t and product i n a s t a t i c gas can y i e l d the r e a c t i o n r a t e as a f u n c t i o n of i n i t i a l and f i n a l v e l o c i t y and s c a t t e r i n g angle. A g a i n , i t i s necessary t h a t the process being

Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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s t u d i e d occur with unrelaxed A*, and t h a t unrelaxed C* be d e t e c t e d . The kernel f o r i n v e r s i o n of D ( W ^ W Q ; Y ^ ) i n t o F( | ^ | , |y | , 0 ) , which r e s u l t s from i n t e g r a t i o n over redundant angles i n the s i x - d i m e n s i o n a l F o u r i e r transform, i s not e a s i l y e x p r e s s i b l e i n terms o f simple f u n c t i o n s , but i t i s s u s c e p t i b l e to numerical computation. 3

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Downloaded by TUFTS UNIV on June 12, 2018 | https://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch008

Literature Cited 1. Kinsey, J. L., J. Chem. Phys. (1977), 66, 2560-2565.

Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.