J. Phys. Chem. 1984, 88, 1695-1697 amounts of Pt vapor deposited thereon. (2) Suppression of CO and H2 chemisorption on reduced, as compared to fully oxidized, titania films with Pt overlayers is interpreted in terms of two factors: (a) flatter Pt particle morphology on the reduced samples and (b) a different electronic interaction between the Pt and TiOl which, at low Pt coverages, suppresses CO and H2 chemisorption.
1695
(3) Auger parameter measurements indicate that the electronic interaction is not the result of a simple charge transfer.
Acknowledgment. This work was supported in part by the Office of Naval Research. Registry No. Pt, 7440-06-4; TiO,, 13463-67-7.
Spin Echoes in the Slow-Motion Regime A. Baram Chemical Physics Department, Weizmann Institute of Science, Rehouot, Israel 76100 (Received: July 20, 1983)
Spin-echoamplitudes and their resulting line shapes are calculated by using the slow-motion approach to the line shape problem. It is shown that reductions in the echo line shape due to irreversible relaxation process occurring at the time interval between the pulses are less marked near the characteristicfrequencies of the powder spectrum, while they are more pronounced elsewhere. The short-time evolution of the echo amplitude is represented in terms of the moment expansion. The long-time decay is shown to be dominated by a single exponent whose argument is the ground-state eigenvalue of a harmonic oscillator, with basic frequency proportional to Q - I / ~ .
Introduction Spectral absorption lines can be observed either as a steady-state absorption or as a Fourier transform of a transient signal following a pulse. For broad lines the steady-state method is slow and tedious, since it requires extensive signal averaging. In the transient experiment the very important initial part of the free induction decay (fid) is hidden by the instrumental dead time. Thus, much of the information about components of the spectrum is lost and cannot be reproduced. In particular the second moment of the absorption line may be directly determined from the initial part of the decay, while its determination by the steady-state method is difficult and tedious. In order to overcome these problems various schemes of spin echoes have been proposed.’-5 The spin-echo technique is based on the application of two pulses separated by a time interval r. The pulses are chosen in such a way that for static systems of isolated spins the decay is shifted by 27. Thus, for t > 27 the signal is identical with a fid whose time zero is ai t = 27. The technique therefore effectively results in a zero-time resolution even though the actual time resolution is finite. As a result of this property spin-echo spectroscopy is becoming a standard technique for studying molecular dynamics in liquids, glasses, membranes, and proteinsS6 In the slow-motion regime with finite correlation times the spin-echo decay can no longer be expected to be identical with the fid. Similarly Fourier transform of the echo leads to line shapes that differ from the true absorption spectrum. Obviously such distortions have to be taken into account while analyzing line shapes obtained by the echo method. Woessner et al.7 have calculated spin-echo amplitudes and line shapes for systems modulated by rotational diffusion. Their treatment follows the fast-motion expansion of the stochastic Liouville equation introduced by Fixmam8 Spiess and Sillescu9 investigated the two-site (1) Hahn, E. L. Phys. Rev. 1950, 80, 580. (2) Carr, H. Y.; Purcell, E. M. Phys. Rev. 1954, 94, 630. (3) Solomon, I. Phys. Rev. 1958, 110, 61. (4) Powles, J. G.; Mansfield, P. Phys. Letr. 1962, 2, 58. (5) Powles, J. G.; Strange, J. H.; Proc. Phys. SOC.1963, 82, 6. (6) Salikov, K. M.; Semenov, A. G.; Tsvetkov, Yu. D. “Electron Spin Echoes and Their Applications”; Nauka: Novosibirsk, 1976. (7) Woessner, D. E.; Snowden, B. S.; Meyer, G. H. J . Chem. Phys. 1969, 51, 2968. ( 8 ) Fixman, M. J . Chem. Phys. 1968, 48, 223. (9) Spiess, H. W.; Sillescu, H. J . Mugn. Reson. 1981, 42, 381.
0022-3654/84/2088-1695$01.50/0
exchange problem and its effect on line shapes obtained from spin echo. They found that the spin-echo technique produces spectra with reduced intensities for pulse separation r longer than the exchange correlation time. In this paper the slow-motion approach to the line shape problemlo is utilized to obtain analytic expressions for spin-echo amplitudes and their resulting line shapes, for systems modulated by rotational diffusion. In particular it is shown that the long-time behavior of the spin-echo amplitudes, which determines the characteristic features of the line shape function, is determined by the ground-state eigenvalue of a perturbed linear harmonic oscillator, whose basic frequency is proportional to the square root of the correlation rate. 11. General Theory The amplitude of the signal induced in the sample coil is given by (11.1)
where p(t) is the spin density matrix in a set of coordinates rotating at the resonance frequency, and the signal is normalized to unity at t = 0. The time evolution of the density matrix is given by (11.2)
where H = H ( Q ) is the anisotropic part of the Hamiltonian representing the interaction of the spin system operators with the time-dependent lattice variables Q(t). The system evolves for a time r and then a pulse R is applied. At a time r’ later the signal is
= tr [Ix(r)RZx*(r’)R-l]/tr):Z(
(11.3)
The rotation operator R is chosen in such a way that complete refocusing of the static signal occurs at r’ = r. The irreversible relaxation effects arise from the stochastic molecular reorientation processes. The processes are reflected by the time dependence of the lattice variables Q(t),which describe the tumbling of the molecular frame of reference relative to the (10) Baram, A. Mol. Phys. 1981, 44, 1009. (1.1) Kac, M. “Probability and Related Topics in Physical Sciences”; Interscience: New York, 1959, Chapter 4.
0 1984 American Chemical Society
1696 The Journal of Physical Chemistry, Vol. 88, No. 9, 1984
Baram
fixed laboratory frame. The time evolution of the signal at 0 < t < T is identical with that of a fid. The action of the pulse R is represented by the time evolution of ZX*(7'). The total signal amplitude is given by averaging eq 11.2 over all Q values P(T,T')
=
S ~ ( T , T ' , Q )dQ
(11.4)
The expectation values of the time-dependent operators ZX(7) and ZX*(7') are determined by the semiclassical Liouville equation*,"
dp(t,Q)/dt
i[p,Hj
+ I'p + 6(2)/4a
The stochastic operator r operates exclusively on 0, the variables of the lattice subspace. In the slow-motion regime the various transitions are uncoupled, and it is sufficient to consider the secular part of H. In the special case when a first-order frequency shift is identically zero it is sufficient to treat the nonsecular secondorder frequency shift.I2 Assuming modulation through rotational diffusion eq 11.5 reduces to
+
dp(t,Q)/dt = [-iwl(Q) 4- ( 1 / T ~ ) v ~ ] p ( t , Q ) 6(t)/4n
(11.6)
-1
(m t n t 1) t i(m - n )
(11.5)
[It-
(~BTR)' '*
(1I.llb)
In eq 11.1l a m and n are even, otherwise c,,(x) vanishes, and the prime denotes summation over even k values only. It is easy to is the weight coefficient of see that cnm= cmn*and c, = Crncflrn the fid. The spectrum Z(W,T) is obtained through Fourier transform off(7,r') starting at t = 27. It is convenient to shift the time and then to transform the shifted time evolution function f(~,t') I(W,T)
=
where
+
q(Q) = B[P2(6') '/2q sin2 6' cos 241
(11.7)
(11.12)
In eq 11.7 6' and 4 are the polar and azimuthal angles, respectively, specifying the direction of the external field in the body frame of reference, and 9 is the asymmetry parameter. Equation 11.6 describes the motion of a rotator in an anisotropic potential field, with the correlation time playing the role of an imaginary moment of intertia.1° In the slow-motion limit, Le., B T R >> 1, the motion of the rotator is hindered and it tends to librate about the extrema points of w,(Q), which are the characteristic frequencies of the powder spectrum. During the time scale of the experiment there is no coupling between different regions of the quasistatic spectrum, and the librators are as a result uncoupled.I0 The angular distribution function w l ( 0 ) is quadratic in its variables at the neighborhood of the extrema, and therefore the librations reduce to lowest order to harmonic oscillations about these points. Thus, the total amplitude of the fid is given by the sum of local harmonic oscillator eigenfunctions centered about the characteristic frequencies of the powder spectrum. The signal amplitude of the spin-echo sequence is the average over all orientations of the product of two fid functions.
Each of the complex Lorentzians forming the echo line shape is multiplied by a reduction factor depending on the pulse separation T . It is convenient to define the reduction factor r, as the ratio between the weight of an echo Lorentzian and the weight of a Lorentzian contributing to Z(o)
rm = ( e h ~ * T / c m * ) C c f l m e h ~ T
(11.13)
n
It is easy to generalize the treatment of the axially symmetric Hamiltonian presented here to include the q # 0 case, using the relevant eigenvalues and weight factors from ref 10. 111. Short-Time Behavior
The short-time behavior of the fid and the spin-echo amplitude is best represented in terms of the moment expansion.I3 Let w n be the nth static moment; then
For 7 = 0 the signal amplitude is the sum of an oblate branch A single secular line is not symmetric relative to the resonance and a prolate branch corresponding to the perpendicular edge and frequency and as a result p3 # 0. As long as t < T R , the decay the parallel edge, respectively is governed by the quadratic term that depends only on properties C,,(X)~[X.(X)'+X.'(X)''] + C c,,(z)e[".(z)~+~m*(~)~] f(7,~'= ) of the reversible Hamiltonian. Indeed, at the slow-motion regime m,n=O n,m the time T2* at which the fid decays to e-l times its initial value (11.9) is T R independent. At t = 27, Le., 2' = 0, the purely oscillating where terms resulting from the Hamiltonian do not contribute, by construction of the pulse sequence, tof(7,O) and as a result the function is real at this point.
X,(Z)
=
-(
9 1 1 2 ( 2 n+ 1)
+ -(2n2 1 + 2n + 1) TR
i[ B
- (:)l1;2n
+ 1)
]
(1I.lOb)
are the perturbed harmonic oscillator eigenvalues that determine the time evolution of the signal. The weight coefficients c,, are evaluated by using the generating functions for the Hermite and Laguerre polynomials (12) Baram, A. J . Phys. Chem. 1983, 87, 1676.
For an isotropic system the macroscopic observables are proportional to expectation values of the rotationally invariant state. Since this state is invariant to r, Le., rl0) = 0, the first term contributing tof(7,O) is of the form ( o l H ( Q ) r H ( Q ) l o ) ~As ~ . long as T
