658
J . Phys. Chem. 1989,93, 658-664
Photon Locking and Its Observation by the Probe-Echo Method: Application to Optical and Microwave Transitions R. Vreeker, M. Glasbeek, Laboratory for Physical Chemistry, University of Amsterdam, Nieuwe Achtergracht 127, 1018 WS Amsterdam, The Netherlands
and A. H.Zewail* A . A . Noyes Laboratory of Chemical Physics,+ California Institute of Technology, Pasadena, California 91 125 (Received: June 20, 1988)
In this paper we consider the coherences that are produced in a photon (or spin) locking experiment involving a molecular two-level system. New coherences, not anticipated from the Feynman-Vernon-Hellwarth representation, are predicted if the transition is inhomogeneously broadened. The influence of duration, power, and phase of the driving and probe pulses in the locking experiment on the shape of the simulated echo responses is discussed in detail, and a comparison with experiments is made.
1. Introduction Photon locking, the optical analogue of spin locking, has recently been reported for atomic and molecular two-level quantum systems.'-" In these experiments, phase-controlled optical pulse sequences are applied such that the state vector of the molecular system becomes (partially) aligned in-phase with the laser field vector. The results show that it is now possible to develop coherent averaging techniques for optical transitions, in close analogy to those well-known in solid-state NMR.5-7 With use of similar coherent excitation pulse trains in the microwave regime, spin locking was observed for a color center, in an ionic solid, in the photoexcited electron spin triplet state.s As in the optical case? the locked coherence was probed optically by an echo pulse sequence applied after the locking field is turned off. However, the spin echo responses appeared very sensitive to the duration and phase of the locking pulses. Also, for short locking times, three echoes could be observed. In the photonlocking experiment two echoes were seen. To better understand the various coherences that might be generated in these phase-locking experiments, in this paper we provide general expressions for the coherences one can expect in a typical locking experiment. The results of our analysis of the locking signals that can be expected for various experimental conditions in the optical and microwave regimes are reported, and a comparison with experiment is made. In section 2 we introduce the framework for calculating the coherences in the photon-locking experiment without taking into account the effects of power and inhomogeneous broadening. In section 3 the additional effects of power and inhomogeneous broadening are considered in some limiting cases. 2. Locking and Probe-Echo Pulses in a Two-Level System The response of a two-level system interacting with a classical external electromagnetic field through the dipolar interaction is well-known in the l i t e r a t ~ r e . ~ ,The ' ~ total Hamiltonian of the system is given by H = Ho Hi,,,
+
where Hois the Hamiltonian of the two-level system in the absence of the radiation field and Hi,, denotes the interaction between the molecular ensemble and the applied radiation field. Here we consider that the framework discussed below is applicable for both electric and magnetic transitions. The justification for the use 'Contribution No. 7801.
0022-3654/89/2093-0658%01.50/0 , I
,
of a similar formalism for describing coherence effects in optical and microwave (radio frequency) cases was detailed recently.' I Basically, it is assumed that all pulses of the radiation field (regardless of the frequency) have a well-defined polarization direction and phase, and the medium is optically dilute (thus optical density effects, self-induced transparency, etc., can be ignored12). In matrix notation, the terms in eq 1 are written as
(3)
where o,is the Rabi frequency, w is the laser/microwave frequency, and 4 is the phase for the traveling plane waves: 4 = fp(t) i . 7 (4)
+
The evolution with time of the two-level system in the rotating frame is obtained from ap*/at = i/h[p*,H*]
(5)
p* = UpU'
(6)
with
H* = UHU'
+ ih(au/at)W
(7)
(1) Sleva, E. T.; Xavier, I.; Zewail, A. H. J. Opf.SOC.Am. B 1986,3,483. (2) Sleva, E. T.; Glasbeek, M.; Zewail, A. H. J . Phys. Chem. 1986, 90, 1232. ( 3 ) Bai, Y. S.;Yodh, A. G.;Mossberg, T. W. Phys. Rev. Lett. 1985, 55, 1217. (4) Bai, Y. S.;Mossberg, T. W.; Lu, N.; Berman, P. R. Phys. Rev. Lett. 1986, 57, 1692. ( 5 ) Haeberlen, U. Advances in Magnetic Resonance; Waugh, J. S . , Ed.; Academic: New York, 1976; supplement 1. (6) Abragam, A.; Goldman, M. Nuclear Magnetism: Order and Disorder, Clarendon: Oxford, 1982. (7) Weitekamp, D. P. Adu. Mogn. Reson. 1983, 1 1 , 111. (8) Vreeker, R.; Glasbeek, M.; Sleva, E. T.; Zewail, A. H. Chem. Phys. Letf. 1986, 129, 117. (9) Abragam, A. The Prlnciples of Nuclear Magnetism: Oxford: London, 1961. (10) Burns, M. J.; Liu, W. K.; Zewail, A. H. Spectroscopy and Excitation Dynamics of Condensed Molecular Systems; Eds. Agranovich, W., Hochstrasser, R. M., Eds.; North-Holland: Amsterdam, 1983; p 301. (11) Warren, W. S.;Zewail, A. H. J . Chem. Phys. 1983, 78, 2279. (12) Loudon, R. The Quanfum Theory of Light; Oxford: London, 1973. Yariv, A. Quantum Electronics; Wiley: New York, 1975.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 659
Photon Locking and Its Observation
U = exp(iwHot/hwo)
(8)
‘ t p - - t ~
In the rotating-wave appr~ximation,’~ H * becomes H*=Yzh(-AW -iqd+
iw AW e-‘+
(b)
(a)
)
(9)
with AW = w - W O
(10)
It is noted that in our treatment relaxation effects are not incorporated. While the two-level system is being driven by the radiation field, p* develops according to p * ( d = W t ) P * ( O ) S(t)
(11)
where the propagator S ( t ) = exp(iH*t/h)
(12)
in matrix notation is given as14J5 S(r) =
(
cos y2at - i(Ao/a) sin Xar (a1/a)d+ sin
-(wl/a)e-i+ sin Xat cos Yzat + i(Aw/a) sin Xat
l/at
1
(13)
Figure 1. (a) Scheme of pulse sequence in photon-locking experiment characteristicof probe-echo method. (b) Labeling of two-level system mentioned in text. and phase 6. The Rabi frequency for the preparation and the echo pulses is wl;the Rabi frequency in the presence of the locking field is wll. Furthermore, within the two-level system the lower level (denoted la)) is considered as nonradiative and the upper level (called Ib)) is radiative. Then, the echo signal generated at t = 32, T 7’ (cf. Figure l ) , manifests itself by laser (microwave)-induced changes in the fluorescence intensity at the time t . In turn, these changes will be proportional to the population in level Ib) which is given by p*bb(t=3tp+T+T’). The latter quantity is determined from
+ +
P*(t)
=W)P*(O)
p ) %$I)
S(7 - tl)
v(r)
(16)
where
with i3 = (wIz
+ (AW)’)’/~
(14)
In the absence of the laser/microwave irradiation (wl = 0), p * ( t ) can still be found from eq 11, but now
V ( t )= W
W p )
S(#)W
p )
(17b)
In eq 17, S(tp)is obtained from eq 13 for 4 = 0, whereas S(T tl) and S(T’) are representative of free precession and obtained from eq 15. For the propagator in the time interval from tp to t , + tl we have
When the two-level system is irradiated by a multiple-pulse sequence, then to obtain the final density matrix, of course, one has to calculate the effect of the product propagator that factorizes the propagators representative of the various steps in the pulse sequence. We now consider the pulse sequence given schematically in Figure 1. Before calculating the density matrix that results from the sequence, it is convenient to first visualize the effects of the successive pulses. Ideally, the initial pulse is a 7r/2 pulse applied along the x axis of the rotating frame; the second pulse oriented along they axis when 4 = 90° in the rotating frame serves to lock the coherent-state vector component to the radiation field vector. After the locking pulse is turned off, a free-induction decay is anticipated since the resonance offset, Aw, is not identical for all the members (in the ensemble) of the noninteracting two-level systems. An x-polarized pulse at time T generates a rephasing, leading to the occurrence of a photon (spin) echo for T’ N T tl. Evidently, this echo will develop only provided coherence has persisted during the locking time, tl. Finally, in the probe pulse the echo is monitored after the application of a third x-polarized pulse, the probe pulse, which restores the initial population distribution among the two levels when applied at T~ T - tl (remember that relaxation is not considered). The probe pulse method is particularly advantageous when one of the levels is luminescent. In this case, coherence effects can be detected as incoherent light changes superposed on the spontaneous luminescence emitted in a direction perpendicular to the exciting beam. Furthermore, phase information regarding the coherent state is retained.” In the calculation given below we assume for simplicity that, in the scheme of Figure 1, the three pulses with 4 = 0 have equal duration (tp), whereas the locking pulse has a pulse length of ti (1 3) See: e.g., ref 12, Chapter 2. (14) Breiland, W. G.; Brenner, H. C.; Harris, C. B. J . Chern. Phys. 1975, 62, 3458. ( 1 5 ) Shoemaker, R. L. In Laser and Coherence Spectroscopy; Steinfeld, J. I., Ed.;Plenum: New York, 1978; p 197.
S(t1) =
cos
(
1/2i31tl - i(Aw/al)
sin 1/Zaltl -(ull/al)e-i+ sin y2airi
1
cos yzaltl+ i(Au/al) sin t/2altl
(wll/al)d+sin L/2altl
(18)
where BI = ( ~ 1 :
+ (Aw)~)’/~
(19)
The evaluation of p*bb(t) from eq 16 and so on is straightforward but very tedious. Retaining only terms dependent on T ’ , p*bb(t=3tp+T+T’) is given by
+
P*bb(3tp+T+T’) = -20 Re [a3€* exp(iAw(7 - tl 7’))a0% exp(-iAw(.r - tl - 7‘)) &3(1 - 26) X exp(iAw~’)] A B C (20)
+
+ +
where we have introduced the following definitions: S(tp,4=0)=
(; );
The evaluation of P*bb(3tp+T+T’) in eq 20 for the general case that tp, tl, Aw, wl, wI1, and 4 may attain arbitrary values leads to an extremely long and complicated expression. Thus, in section 3, we limit ourselves to the analysis of a few special cases. Finally, we note that the resonance frequency, w0, is inhomogeneously distributed, and thus the echo signal monitored after the final probe pulse will be proportional to
where g(Aw) is the spectral density function for Aw,
660 The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 3. Coherences and the Influence of Inhomogeneous Broadening
In this section E(T,T’)is considered for several limiting cases of the inhomogenous broadening function g(Aw) in eq 23. We examine first, in section 3A,the situation where the inhomogeneous broadening of the la) -,Ib) transition is much larger than the laser pulse width and wI.This situation applies for most optical systems when relatively narrow-band lasers are used. In gaseous systems, for instance, Doppler broadening usually exceeds lo2 MHz, and in solids inhomogeneous broadening due to random crystal strain, dislocations, etc., is at least 0.1-10 cm-I. On the other hand, normally w1 does not exceed =lo2 MHz in the type of experiments mentioned here. Second, in section 3B moderate inhomogenous broadening effects, on the order of the probe-radiation field interaction, are considered. Experimentally, this situation is most often encountered at microwave or radio frequencies. A . Phase Locking under Extreme Inhomogeneous Broadening Conditions. Extreme inhomogeneous broadenings are charac>> w,, where is the average of the terized by ((Aw off-resonance frequencies of the probed molecules. In this limit, g(Aw) in eq 23 can be regarded as constant, and we write
Vreeker et al. at T’ E T - t, when the power of the locking pulse (wn) is no longer equal to that of the coherence preparation and probe pulses ( w , ) . rWhereas the aforementioned echoes that occur for r’ ti could have been anticipated in a simple 7 model,16%17 we will now show that term B in eq 20 also gives rise to an additional coherence r and choosing for the effect that occurs for T’ N T . When 7’ sake of simplicity C#J = 712, evaluation of B(T,T’:Au)yields B(T,T’;Aw)= {(nb- n , ) / ( n ,
I
+ nb)]
cos (Aw(T’- 7 ) +
(a]+
z)2)1/2
E(T,T’)a Jd(Aw)
p*bb(T,s’;Aw)
(24)
Terms A , B, and C (cf. eq 20) in the integrand of eq 24, after integration, each give rise to special coherences, as discussed now. The major coherence peaks are due to term B and are observed when r’ takes values near T - tl or T , respectively. In the event that r’ N T - tl, B in eq 20 becomes
cos (Aw(T’- T )
+ (Aw - al)tl)a4
+ nb)][cos (Aw(T’ + tl)] x sin4 + cos2 + + -sin6 + + aI2a4 a2
B(T,T’;A.w)= ((nb - n , ) / ( n ,
e( ( A W )sin4 ~ +)
-,
T
1
+ sin (Aw(T’- r + t l ) + 41 X
+ tl) +, 241 X sin6 + - cos2 + sin4 + + sin (Au(T’ - r +
cos (Au(T’ - r
where
It is readily seen that upon integration of the function B ( r , f ; A w ) in frequency space all terms on the right-hand side of eq 25 contribute to the value of E(T,T’).Thus provided that the system shows appreciable inhomogeneous broadening, one need not phase shift the locking field with respect to the initial preparation pulse exactly x / 2 , but any phase shift will effect photon locking. To illustrate this more quantitatively, we present in Figure 2 some computer-simulated echo intensities as calculated on basis of eq 25 taking wll = wl,q t , = n/4,and for 4 values equal to 0, x/4, n/2,and x , respectively. It is seen from Figure 2 that regardless of the value of 4, in all instances an echo signal characteristic of the locked state is obtained. Moreover, the echo shape function appears to be very sensitive to the phase of the applied locking pulse. The effects of the duration, t,, of the coherence preparation and echo probe pulses on the shape of the echo signal obtained after locking are illustrated in Figure 3. The computer simulations were performed for 4 = n/2 and w I I = wl. Once again we find that the shape function of the echoes probing the locked state is influenced but now due to variations of the flipping angle q t p . Figure 4 illustrates what happens to the shape of the echo probed
sin (Aw(r’ - T )
+ (a,+ Aw)t,)
:Aw
U ,4 ~ ,
a5q2
+
sin 2+ sin4 -
- sin2 +(wI2 cos 2+ + WI2
a4
+
Wl2
sin 2+ sin4 - - sin2 $(q2 cos 2+
+
a4
Upon the substitution of eq 27 into the integral of eq 24, it is seen immediately that an echo is expected when T’ i= T , provided wltl 5 1, i.e., short enough locking pulses are used. Figure 5, parts a and b, illustrate the echo line shapes that can be simulated by using eq 27 when T’ E T and taking tl = t , and flipping angles of a/4 and x/2,respectively. On the other hand, when tl 103t,, the echo at T’ = T has disappeared as shown in Figure 5c. Experimentally, the occurrence of the echo at T’ = r has already been demonstrated recently for the B X transition (at 589.7 nm) of I2 gas at low pressures.2 In Figure 6 we reproduce the result for the I2 gaseous system at 30 mTorr, using the pulse scheme of Figure 1 . Two echoes are observed: the first occurs when 7‘ N T - tl (and is explained from eq 25), the second occurs for T‘ N T and is accounted for by eq 27.
-
(16) Dicke, R. H. Phys. Rev. 1954, 93, 99. (17) Feynman, R. P.; Vernon, F. L.; Hellwarth, R. W. J . Appl. Phys. 1957, 28, 49.
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 661
Photon Locking and Its Observation
.2 -1 0 1 2 ( * +T’(Ui’)
T-tL
Figure 2. Influence of phase of locking-field vector on shape of probe echo at T’ rr, T - tl as calculated from eq 24 with p * given by B of eq 25 and taking wltp = ~ / 4 wI1 , = w l r and (a) 4 = 0, (b) 4 = 7r/4, (c) 4 = ~ / 2 (d) , 4 = R.
-
In the limit of wI1 0, Le., no locking field is applied, eq 27 becomes B(T,T’;Aw) a {(nb- n , ) / ( n , nb))cos (Aw(7’ - 7))X
+
I
\
Figure 3. Influence of flipping angle of preparation and probe pulses on shape of probe echo at T’ N 7 - tl as calculated from eq 24 with p* given by B of eq 25 and taking wI1 = w1 and 4 = ~ / 2 where , ultpis given the value (a) r/4, (b) 7r/2, (c) 3n/4, (d) R , (e) 3n/2.
Equation 28 represents the echo intensity in a normal probe pulse echo experiment using three pulses of equal duration. The form of eq 28 is identical with the expression derived previously by Warren and Zewail (cf. eq A7 of ref 11). We will now examine the coherences originating from the presence of term C in eq 20. To facilitate the calculation we again limit ourselves to the special case that wI1 = wI and 4 = 7r/2. Rewriting C, we have C(T,T’;AW)= -28 Re [azoexp(iAw~’)]+ 402 Re [a% exp(iAw~’)] (29) The first term in the summation will result in a free-induction decay signal immediately after the application of the probe pulse (of length tp) at time T’ N 0. Evaluation of the second term in eq 29 is straightforward, and we obtain (ignoring again terms contributing to the free-induction decay at T’ = 0) for 7’ = tl additional coherences, that can be calculated from the expression given in Appendix A. From eq A1 it follows that an additional coherence is expected to occur when T’ = t l , provided a significant portion of the molecules is subjected to off-resonance excitation. Equation A1 also indicates that the shape of the coherence signal at T’ = tl depends in a very complicated way on laser power, duration of preparation and probe pulses, and the phase of the locking field. Figure 7a presents an example of the calculated echo shape near 7‘ N tl, choosing in eq A1 the condition that all pulses have equal duration, whereas w1 = wI1 and colt = ~ / 4 .However, when ti is increased by orders of magnitde, the echo at 7’ = tl has vanished (cf. Figure 7b). As discussed in more detail below (cf. section 3B), very recently the coherence at 7’ = tl has actually been observed experimentally for the Fz2+center, in CaO, in its
Figure 4. Influence of power of locking pulse on shape echo at 7’ = 7 - tl as calculated from eq 24 with p * given by B of eq 25 taking wltp = ~ / 2 4, = n/2,and (a) wII = O.lwl, (b) wll = 0.2wl, (c) wI1 = 0.5W1,(d) wlI = wI,(e) w I 1 = 2 w l , (f) wll = 100 wl.
photoexcited 3B1state.8 The optical analogue has not (yet) been reported.
662 The Journal of Physical Chemistry, Vol. 93, No. 2, 1989
Vreeker et ai.
-1
0
1
0
1
2t~-T
-T’(q’)
-1
2tL-T
Figure 8. Coherent response at T’ = 24 - r , as calculated from eq 24 with p* given by A of eq B1, where wII = wI,4 = r/2,ultp= r/4,and (a) wllti = r/4,(b) wlltl = 1000a/4.
0
-1
1
DJ L-
-T’(w;’)
Figure 5. Shapes of coherent responses for r’ = T as calculated from eq 24 with p* given by B of eq 27 taking w l I = wl, 4 = r/2,and (a) wItp = wlltl = r/2,(b) wltp = ulitl= a/4,(c) W i t p = a/4,wlltl = 1000r/4.
-2
0
-1
T-tL
1
\
2 -2
i-1
0
+T’(w;‘
2
1 )
Figure 9. Echoes probing photon locking at r’ = T - ti, as calculated for Gaussian-shaped transitions from eq 23,with wII = wl,4 = r/2,g(Aw) = e~p(-(Aw)~/2$), u = w i , and (i) ultp= r/4,(ii) wltp = r/3,(iii) wltp = r/2,(iv) ultp= 3r/4,(v) wltp = a,(vi) wltp = 3r/2.
0 S
O O T’(ns)
A
”
for A(7,f;Aw), taking wI = wI1, 4 = ~ 1 2u, = m, wltp = a/4, and tl = tp (Figure 8a) and tl = lo3?, (Figure 8b), respectively. As a general conclusion we find that for all coherences discussed in this section (at 7’ 7 - tl, 7, tl, and 2tl - 7, respectively) the shape functions depend critically on the experimental parameters: wl, all,t tl, and 4. This is borne out by the experimental results obtainexpreviously by us for 122and photoexcited color centers in ionic solids.8 B. Coherences under Moderate Inhomogeneous Broadening Conditions. To calculate the coherences by the sequence in Figure 1 under the condition that inhomogeneous broadening is moderate as compared to the laser (microwave) band width and wI,one cannot simply apply eq 24. Instead, the complete expression in eq 23 should be used. We have computer simulated the echo line shape near 7’ N T - tlr using a Gaussian-shaped distribution function
-
1
600
T’(ns)
Figure 6. (a) Photon echo for I2 at 30 mTorr measured with XXX(x) three-pulse echo squence of Figure 1, thus without “locking” pulse; t = 50 ns, T = 400 ns. (b) Photon echoes as measured for I2 with XYXXfx) four-pulse sequence, t1 = 200 ns; everything else as in (a).
g(Aw) = e ~ p ( - ( A w ) ~ / ( 2 u ~ ) )
-1
0
tL
1
-1
0 t~
1
-+T‘(w;’)
Figure 7. Coherent response at 7’ N fl, as calculated from eq 24 with p* given by C of eq Al, where wI1 = w , , 4 = a/2,wltp = a/4,and (a) W l l t l = */4,(b) WI1tI = 1000~/4.
Finally, we consider the coherence that might arise from term A in eq 20. Straightforward evaluation of this term produces among others the terms given in Appendix B. Terms given in eq B1 may lead to coherence when f 2tl - 7, provided tl is limited to relatively small values. In Figure 8 we present the results for this coherence as calculated from the complete expression derived
(30)
for a series of different values of the pulse duration, tp. In Figure 9, the line shapes are given as calculated for u = wl, 4 = */2, and t, values as indicated in the figure. The echo line shape is seen to be sensitive to the flipping angle, w l t p . Especially for the larger flipping angles, the echo signal shows some structure, mainly due to the influence of off-resonance pumping of members in the inhomogeneously broadened ensemble. Comparison with the results of Figure 3 shows that in the event of larger inhomogeneous broadening the structure appears earlier for the same flipping angle. Also, the signal is affected by the phase of the driving locking field. This is illustrated experimentally in Figure lob. There we have reproduced the signals presented previously in ref 8. The simulated curves were calculated by using eq 23 and 30, near T’ 7 - t l , for u = w l , wltp = 1, tl = 330 ps, and taking 6
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 663
Photon Locking and Its Observation
spin-locking experiment performed on the F22+defect the limit of moderate inhomogeneous broadening is applicable. 4. Conclusions
5
10
I
lL
-
1
15
15
10
1
T ’ W
+
T‘(ps)
Figure 10. (a) Optically detected spin coherence responses for the ID(
- 17 zero-field transition a t 1870 MHz for the F?+ defect, in CaO, in
the BI state at 1.4 K. Pulse sequence as in inserted scheme with wltp = So,wII = q , 71 = 12 ps, and 7 = 18 ps. (b) Influence of phase of locking pulse. Simulated curves were obtained by using eq 23 and 30, for u = wl, wlrp = x/2,wI1= w l , 71 = 330 w,@ = 0 and a,respectively.
5
10
T-ti
We have considered photon and spin locking experiments in inhomogeneously broadened systems. Expressions for calculating the coherences that arise in a photon locking experiment using the pulse echo method are given. The influence of inhomogeneous broadening of the involved spectral transitions on the number and the nature of the coherences is analyzed explicitly. Two limiting cases are treated: extreme and moderate inhomogeneous broadening conditions. In the extreme broadening case (applicable in the optical region) a total of four echo coherences are predicted for short locking times; when the duration of the locking pulses is much larger than that for the driving and probe pulses three out of the four coherences will vanish and only one echo (predicted by the simple FVH description) remains. For all calculated echoes the line shapes are sensitive to the assumed power of the driving and locking pulses, the duration of these pulses, and their relative phases. All these effects are associated with the presence of inhomogeneous broadening. Under moderate inhomogeneous broadening conditions, in principle, the same features are found although much less pronounced. For the microwave locking experiments performed on the F22+defect, in CaO, in the photoeexcited 3Bl state good agreement with the theory is obtained by assuming moderate broadening, whereas for the photon locking results obtained for gaseous iodine the extreme broadening limit is found to apply. These findings suggest that several new experiments can now be planned to align the polarization with the field (locking), keeping conditions of pulse phase, Rabi frequency, and duration under control.
20
15
T’( CLS)
+
Figure 11. (a) Optically detected spin echo for the F?’ center, in CaO, in the photoexcited ’B,state after spin locking. The pulse scheme is as in Figure loa, but now 71 = 330 ps and 7 = 342 ps. (b) Computed echo shape function by evaluation of eq 23 after taking p* as given by eq 25 and g(A0) as given by eq 3 1.
= 0 and T,respectively. In Figure 10b we also reproduce the experimental results as obtained for the 1 0 1- IEI transition at 1870 M H z within the 3B, state of the F22+center in CaO.* From a comparison of the calculated and measured echo shapes it is clear that indeed for the F22+system the limit of intermediate broadening effects applies. Further support is obtained from the analysis r - tl and = ~ / 2 .As shown of the echo observed when T’ T - tl in Figure 1 l a , the experimentally observed echo a t T’ shows dips on the trailing edges of the wings. From our simulations on the basis of eq 23 and 26, these dips cannot be reproduced. However, in these experiments the phase-controlled multiple-pulse sequences were applied at a repetition rate of 25 Hz, which is high on the time scale of the lifetime (210 ms) of the T, triplet sublevel.’* Consequently, the resonantly coupled triplet sublevels in the locking experiment are never emptied completely or, equivalently, within the inhomogeneously broadened T, T, spin transition a hole is permanently present. To correct for this distortion of the lineshape function, we have recalculated the echo signal assuming the line-shape function to be of the form
Acknowledgment. This work was supported in part by the John van Geunsfonds and the United States National Science Foundation (Grant No. DMR 8521191).
Appendix A C(r,r’;Aw) Term Important to Eq 29. WIL
C(T,T’;AW)0: - sin2 +((n, - nb)/(na a2
atl) + COS (Awr’ - 021)) sin 2+
a2
sin2
- (sin (Awr’ = at1)+ sin (Awr’ - ail))X sin 2+
w14 +sin2 + a4
+ (cos (AwT’- atl)-
(31)
where g(Aw) is given by eq 30 andf(Aw) is a Lorentzian with a line width (fwhm) of wl. The results are depicted in Figure 11b. The good agreement with experimental data confirms that in the
(18) Vreeker, R.;Kuzakov, S. Glasbeek, M. J . Phys. C Solid Srare Phys. 1986, 19, 1215.
w*4 +sin2 + a4
+))
-
g’(Aw) = g(Aw)(l -f(Aw))
+ nb))[(cos (Aw7’ +
sin 2+ sin2
+ + (sin (AwT’- atl)- sin (AwT’ + at,))X
J . Phys. Chem. 1989,93, 664-670
664
Appendix B
+ + w12 Aww12 ( A W ) ~ ~ -- + 2-(0 + A ~ ) t l ) (
A(T,T’;Aw)0: {(n,- n b ) / ( n a nb))[Y2cos {Au(T’
(
(s $) -
a2
sin
a3
+(
(
+ Aw)tl){
(8
A(T,+’;Aw)Term Important to Eq 20.
Aww12
2-
- -4
(AW)~W~~ 2-)
a3
+
84
sin2 a5
+x
T)
~ ~
a3
8 4
2a2
+- (Aw)2 sin2+ cos + -
cos3
a3
92
02 sin 2+ sin + a2
(
(
(Aw)4
- sin2 + cos4 + - -sin4 + ) + ’/z cos {Aw(T‘+ T ) wI4 a24
8 4
wl
--82 (cos4
- 2-
(Awl4 sin4 + ++7
(-: $) +( sin
-
Aww12 ( A W ) ~ O ~ ~
-
-
&3
AwwI2
y2 sin (Aw(+’ + +) - (Aw - 8)tl){2sin2
~
a4
2%)
8 5
+ sin 2
+ -4
83
(AW)~W,~ a4
+
4 cos2
~(Aw)’
-sin2 +
cos3
2a2
+- (A.W)’ sin2 + cos + -
AW
0 2
a3
PS
sin 2+ sin
82
(
+
wI4 sin2 + cos4 a4
)
+ )/z sin {Aw(T’+
T)
Registry No. 12, 7553-56-2; CaO, 1305-78-8.
8 4
Klnetlcs of the Thermal Decomposltion of Methoxybenzene (Anisole) J. C. Mackie, K. R. Doolan, and P. F. Nelson* Department of Physical Chemistry, University of Sydney, NSW 2006, Australia (Received: October 2, 1987; In Final Form: April 21, 1988)
The thermal decomposition of anisole vapor dilute in argon has been studied in a perfectly stirred reactor over the temperature range 850-1000 K and at total pressures of (16-120) X lo-) atm. Decomposition of anisole takes place principally by the reaction C6H50CH3 C6H50+ CH,, for which the rate constant k l was found to be (2.9 A 1.0) X 10l5exp(-64.0 0.6 kcal mol-IlRT) s-l. Phenoxy radicals thus generated may decompose unimolecularly to cyclopentadienyl radicals and CO or react with methyl radicals to form cresols. Phenol is also an important secondary product. Most of the product oxygen originally contained in anisole is found in phenolic compounds rather than in carbon monoxide.
-
Introduction The alkoxy-aryl linkage plays an important role in the thermal decomposition of brown coals and other low-rank fuels. The weak alkyl-oxygen bond’ is readily ruptured by heat, leading to the formation of the phenoxy radical (C6H50). This radical is also thought to be an important intermediate in the combustion of aromatic hydrocarbons.2 In low-pressure systems phenoxy radicals have been found to undergo a unimolecular decomposition3g4at high temperatures to form carbon monoxide and the cyclopentadienyl radical. Recently *To whom correspondence should be addressed at CSIRO Division of Fossil Fuels, PO Box 136,North Ryde, NSW 2113, Australia. 0022-365418912093-0664$01.50/0
*
Lin and Lin5s6reported a shock-tube study of the decomposition kinetics of C6H50 in which C O concentration profiles were monitored by a time-resolved laser resonant absorption technique. They used anisole as a source of phenoxy radicals which were (1) McMillen, D. F.; Golden, D. M. Annu. Rev.Phys. Chem. 1982, 33,
493.
( 2 ) Venkat, C.; Brezinsky, K.; Glassman, I. Symp. (Int.) Combust., [Proc.]
1983, 19, 143.
(3)Colussi, A. J.; Zabel, F.;Benson, S. W. Int. J . Chem. Kiner. 1977, 9, 161.
(4)Harrison, A. G.;Honnen, L. R.; Danben, H. J.; Lossing, F. P. J . Am. Chem. SOC.1960,82, 5593. ( 5 ) Lin, C.-Y.; Lin, M. C. I n r . J . Chem. Kinet. 1985, 117, 1025 (6) Lin, C-Y.; Lin, M. C. J . Phys. Chem. 1986, 90, 425. 0 1989 American Chemical Society
