State-to-State Chemistry

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

D

100

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

8.

KiNSEY

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


2

1

c

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

A

cl

Α

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


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STATE-TO-STATE CHEMISTRY

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

c

c

A C


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


State-to-State Chemistry

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