J. Phys. Chem. 1981, 85,3887-3891
pared to neutral water, these short values indicate that 7, is a strong function of the OH- (or H+) concentration. In addition, the results show that 7, is a function of temperature at constant pH. Obviously the proton lifetime in liquid water depends on several interacting factors. The separation of the contributing factors would require a
3887
considerable amount of further study. Acknowledgment. This research was partially supported by the Department of Energy under Contract DE-ACO276ER01198 and the Air Force Office of Scientific Research under Grant AFOSR-81-0010.
Quadrupolar Relaxation. The Multiquantum Coherences Lawrence Werbelow” Department of Chemistry, New Mexico School of Mines, Socorro, New Mexico 8780 1
and Guy Pouzard Centre de St. Jerome, Universite de Provence, 13397 Marseille Cedex 4, France (Received: June 24, 1981)
The appropriate time evolution equations for the n quantum coherence of a spin I nucleus relaxed by quadrupolar interactions are developed and discussed. The contrasting behaviors for the extreme narrowing and nonextreme narrowing regimes for both isotropic and anisotropic spin environments are examined. The influence of rank-one-type interactions and second-order dynamic frequency shifts are also considered.
I. Introduction The recent development of various multidimensional NMR experiments has stimulated much intrigue and has resulted in numerous applications of practical imp0rtance.l But perhaps the most exciting aspect of these multipulse techniques is the ability to observe (albeit indirectly) the creation and subsequent destruction of multiquantum coheren~e.~-~ As beautifully illustrated by Wokaun and the study of intensive processes which are responsible for the loss of multiquantum coherence provides an important complement to conventional relaxation studies. For example, certain spin correlations which do not affect the observable quantities in conventional “TI”and “Tz” studies do indeed affect the relaxation behavior of the multiquantum c ~ h e r e n c e .Likewise, ~~~ the measurement of the loss of single quantum coherence (lQC), double quantum coherence (2QC), ..., and n quantum coherence (nQC) often provides linearly independent combinations of spectral density term^.^-^ This can greatly aid in the isolation and identification of the large assortment of factors responsible for effecting nuclear spin thermalization. In this paper, we examine the simplest of spin systems that exhibits multiquantum coherence-the isolated multipolar nucleus relaxed by quadrupolar interactions. Of course, for systems at thermal equilibrium, all coherences of all order vanish. However, we shall assume that multiquantum coherence can be produced and monitored (e.g., with a 90°-~-900-~1-“look” pulse-digitize ( T ~ pulse ) sequence). Our immediate interest is the quantification of the disappearance of coherence subsequent to creation. We will consider only the case where the anisotropic quadrupolar Hamiltonian is averaged (not necessarily to zero) in a time short compared to the reciprocal rigid lattice quadrupolar splitting (motional narrowing), and the case where the anisotropic quadrupolar Hamiltonian is averaged in a time short compared to the reciprocal Zeeman splitting *Visiting Professor, Physics Department, Universite de Provence, 13397 Marseille, Cedex 4,France.
(7
Jl(w) > Jz(w). This implies that 2Ql(wo) = Q1(2w0) > 2Qz(wo) = Q2(2w0). Hence, 2Ql(w0) - Q2(2w0)> 0 for anisotropic environment even in extreme narrowing. Although the magnitudes of Ql(wo)and Q2(2w0)become vanishingly small compared to Jl(wo) and J2(2w0) whenever the (Re) spectral density for a given projection is frequency independent, the preceding analysis would suggest that, if the extreme narrowing limit is not rigorously obeyed, dynamic frequency shifts can assume a greater importance for spins in ordered (anisotropic) environments. It is also worth mentioning that eq 14 demonstrates that, as Z increases, the magnitude of the maximum observable shift difference is quenched. Another extremely relevant concern related to this study is the possible influence of rank-one-type relaxation such as scalar coupling of either kind. A rank-one-type interaction merely signifies any time-dependent magnetic dipole coupling that can be accurately represented in the form, b(t).I. The need to consider rank-one-type interactions in the study of nQC relaxation has been emphasized by Bodenhausen et a1.6 The contribution to the width of the nQC component, Irn) Im - n ) , made by a rank-one-type relaxation mechanism is easily shown to be of the form
-
(1/7’2)rn,m-n
Werbelow and Pouzard
25, 1981
=
n2jo(0)+ 2(Z(I+ 1) - m(m + 1) - n2/2)j1(w0) (15)
Jr
where jo(0) = Re (b,(t)b,(O)) dt and jl(wo) = Re S t ( b+(t)b-(O))exp(-Lot) dt. Cross correlation between quadrupolar and rank-one-type interactions are not
considered in this work. The dynamic frequency shift can be ignored because all components of all coherences are affected by adiabatic contributions. For the systems that we are considering, all cases of practical importance will result when jo(0) >> jl(wo), In this limit, it is seen that the rank-one-type interaction contributes equally to every component of any given coherence-including the “narrowed” component. This adiabatic contribution increases quadratically with increasing order, n. In certain situations, this feature may preclude the use of higher order nQC studies to investigate the interaction parameters of a rank-two-type interaction such as the quadrupolar interaction. Conversely, it could prove useful to suppress the effects of rank-two-type interactions by studying these higher order coherences. Analogous features to those implied by eq 15 were discussed by Tang and Pines in their study’ of the rankone-type relaxation of the I = 3/2 momentum. (The system composed of p identical spin 1 / 2 nuclei is equivalent to the system composed of one spin I = p/2, p - 1 isochronous spins Z = p / 2 - 1, (p/2)(p- 3) isochronous spins I = p/2 - 2, ). Since the fully correlated ranktwo-type dipolar and fully correlated rank-one-type interactions do not induce cross relaxation between states corresponding to different representations (different I), the results derived in this work are equally useful to describe dipolar and rank-one-type relaxation (in the limit of complete correlation) of spin systems composed of p identical spin 1/2 nuclei. Up until this point, we have outlined the general relaxation features of the multiquantum coherence. We will conclude our discussion by providing additional comment regarding the applicability of these findings to the description of 1QC studies in isotropic environments where it is well-known that, if extreme narrowing does not obtain, simple analytic solutions describing the loss of coherence do not exist unless Z = 1or Z = 3/2. However, if J(0) >> J(wo),J(2w0), a very simple kinetic behavior results. For example, consider the Z = 5/2 spin system. The timedependent properties of the 1QC for the Z = 5/2 system are summarized in eq 16, where v1 = d5(ub/2,3/2+ u-3 2,+/2), v2 = V ’ ~ ( U ~ / ~ , I + u-.1/2,-3/2), and u3 = 3 ~ p , - ~ / 2l!.o emphasize that Ihese expressions are applicable only for isotropic environments, the projection (k)of the spectral density has been omitted.16 To a good approximation, these three sets of components decouple rapidly as the extreme narrowing approximation breaks down. (As noted previously, if extreme narrowing obtains, (Z-(t)) E v1 + v2 + v3 = (Z-(O)) exp(ioot)exp(-(25/32)43 The width of the
...
- 2OJ( 2w0) -16J(w0)
1
-18J(2w0)
(15)J. Jacobaen, H.Bildsoe, and K. Schaumburg, J. Mugn. Reson., 23, 153 (1976).
(16)P.S.Hubbard, Phys. Reu., 180,319 (1969).
J. Phys. Chem. 1981, 85, 3891-3896
narrowed component will be on the order of (16/25)(J1(w0) (7/2)J2(2wO))whereas the broadened components will have widths on the order of (24/25)J0(0) and (6/25)J0(0). The coupling between these (almost) degenerate components will be quenched by the rapid loss of coherence in the 15/2) 13/2) and 13/2) 11/2) components. It is important to note that eq 10 is not strictly valid for the isotropic phase where each of the 21 + 1- n components are coupled by secular terms. However, since these secular couplings do not contain adiabatic contributions, the behavior suggested by eq 10 is obeyed to a high degree of accuracy even for isotropic systems whenever extreme narrowing fails. (Of course, eq 7 is obeyed when extreme narrowing is valid.) Thus eq 10 and 13 will prove equally useful in the interpretation of conventional spin-spin relaxation times (1QC widths) of high-spin nuclei relaxed by quadrupolar interactions. We believe that eq 10 and 13 are timely generalizations of our earlier investigations.13J4 Before concluding, we should mention one additional feature of the 1QC. For all half-integer spins, the narrowed component, 11/2) 1-1/2), will be increasingly dominated by the J2(2w0) spectral density term as the value for I increases. This can be easily demonstrated by rewriting eq 10 in the following form:
+
-
-
-
The behavior noted for this component is in direct contrast to the behavior noted for the n = 21 narrowed coherence (cf. eq 12) where the width was increasingly dominated by Ji ( ~ 0 ) .
-
A more significant difference emerges if one considers the dynamic frequency shift of the 11/2) 1-1/2) component. As can be seen from eq 13 6W1/2.-112 =
the dynamic frequency shift is always proportional to
389 1
Ql(wo) - Q2(2w0).As I increases, the proportionality factor decreases. Furthermore, eq 14 tells us that the maximum shift difference between any two components of the 1QC is 61-2(2Q1(~o) - Q2(2wo)). Since (1/ T2)l/2,-1/2 will tend toward a value equal to J2(2w0)for large I, we see that the dynamic frequency shift will be strongly quenched for high-spin nuclei. Although this quenching is most marked for the lQC, it is a general characteristic for all orders of coherence as was suggested earlier.
IV. Summary This work has demonstrated that, whenever the loss of coherence is dominated by noninterferring quadrupolar or rank-one-type interactions, (1) analytic expressions can be developed for the widths of all components for any general multiquantum coherence and (2) similar closedform expressions for the second-order dynamic frequency shift also can be derived. These results were shown to be valid in both the motional narrowing and extreme narrowing limits for anisotropic spin environments and in the motional narrowing (but not the extreme narrowing) limit for isotropic spin environments. The two fundamental expressions presented in this work, eq 10 and 11or 13, were subsequently used to discuss a number of general properties. In particular, it was argued that (i) if Jo(0)>> Jl(wo),J2(2w0),and jo(0),then each odd (even) order coherence of every half-integer (integer) spin will have one narrowed component and 21 - n broadened components and (ii) dynamic frequency shifts become decreasingly important for higher order spins. The topics addressed in this work outline some of the basic features and general characteristics of multiquantum coherences for an arbitrary spin, I. In addition to providing practical expressions of utmost generality, these expressions lend insight into the symmetry and structures that affect the time-dependent behavior of multiquantum coherence.
Acknowledgment. This work was supported in part by grants from the National Science Foundation (CHE80001839) and the donors of the Petroleum Research Fund, administered by the America1 Chemical Society. L.W. acknowledges the continuing support of Montana State University (Bozeman, MT).
Kinetics of the Reaction of CI with ClNO and ClN02 and the Photochemistry of ClN02 H. H. Nelsont and H. S. Johnston* Department of Chemistty, University of California, and Materials and Molecular Research Division, Lawrence Berkeley Laboratoty, Berkeley, California 94720 (Received: March 12, 1981;In Final Form: August IO, 1981)
The room temperature rate constants for the reactions C1+ ClNO (2) and C1+ C1N02 (4) have been measured by the method of laser flash photolysis/resonance fluorescence. The rate constants obtained are (k2f 2a) = (1.65 f 0.32) X cm3s-l and k4 = (5.50 f 0.75) X cm3 s-l. The absorption spectrum of CINOzwas measured in the region 270-370 nm and the photodissociation channels ClN02 + hv C1+ NO2 and CINOz hv ClNO + 0 were investigated by resonance fluorescence detection of C1 or 0 atoms. The quantum yields derived are @cl= 0.93 f 0.15 and & < 0.02.
+
-
-
Introduction The photochemistry of ClNO has been studied since 1930 when Kistiakowsky’ showed that the quantum yield t Chemistry Division, Naval Research Laboratory, Washington,
DC 20375. 0022-365418112085-3891$01.25/0
for NO production was approximately equal to 2. More recent work2shows that the primary process is photolysis (1) G. B. Kistiakowsky, J. Am. Chem. SOC.,52, 102 (1930). (2) J. G. Calvert and J. N. Pitts, Jr., “Photochemistry”, Wiley, New York, 1967, and references therein.
0 1981 American
Chemical Society
