1147
RAMAN SPECTRA OF WATER
Raman Spectra and an Assignment of the Vibrational Stretching Region of Water1 by W. F. Murphy and H. J. Bernstein* Division of Chemistry, National Research Council of Canada, Ottawa, Canada
(Received October 26,1971)
Publication costs assisted by the National Research Council of Canada
The Raman spectra for some isotopic forms of liquid water (HzO,DzO, and the OH region of HOD) have been obtained in the stretching region, separated into their isotropic and anisotropic components, and decomposed into a series of symmetric bands. The number of bands, their approximate frequency positions, and their depolarization ratios were dictated on the basis of a simple model for liquid water involving two molecular species having either four or three hydrogen bonds. The results provide a self-consistent description of the observed Raman spectra. They are also qualitatively consistent with the infrared spectra and the effect of temperature on these spectra.
Introduction The structure of liquid water is a very complex problem, which has been treated from several points of view.za-c Vibrational spectrazdhave been used to support various models, and a specific region of interest has been the stretching region. We restrict ourselves here to consideration of this region for water and its deuterated analogs. A suitable model for describing the stretching region observed in the infrared and Raman spectra of these liquids should provide a rationale for the following experimental results. (1) The intensity distribution for liquid HzO (and D2O) in the infrared spectrumS is significantly different from that for the corresponding Raman s p e c t r ~ m . (2) ~ The depolarization ratio of the Raman spectrum changes across the band.s (3) There is a significant differencee between the contours observed in the infrared and Raman spectra of HOD. (4) The recent availability of improved Raman techniques led to the observation of a high-frequency shoulder in the Raman spectrum of HzO, HOD, and DzO.’ ( 5 ) Any model should explain the observed temperature dependence4!*of these bands. We shall describe here a simple model for water structure which can account for these This model combines features of previously published water as indicated The features be described in terms of vibrational stretching modes of two types of hydrogen-bonded water molecules. These are the bonded tetracoordinated species and the species of the tricoordinated type which has a free OH. The suitability of this model was demonstrated for both the isotropic and anisotropic components of the observed R~~~~ spectra.This requires not only that the total intensity be described in. terms of the model, but the intensity Of these components, that is, the depolarization ratio throughout the band.
Experimental Procedure and Data Handling The Raman spectra of liquid HzO, DzO, and 5 mol % HzO in D20 were recorded using the data system developed in this laboratory. The samples were at room temperature and in a 1 X 1 X 5 em fused silica cell. The source was a CRL argon laser operating a t 4880 A with a power of about 700 mW. A Spex 1400 double monochromater and EM1 (3256 photomultiplier were used with photon-counting detection. The two polarized components of the scattered light were measured by using a Polaroid HN-22 analyzer in the scattered beam and recording the spectra of the light polarized parallel and perpendicular to the polarization vector of the incident light, which was fixed perpendicular to the optical axis of the spectrometer. I n terms of the generally accepted scattering geometry c o n v e n t i ~ n s ,these ~ are the Y ( 2 Z ) X and Y ( 2 Y ) X components, respectively. The spectra were obtained several times and remained constant to within the statistical counting errors. The spectra were first processed using a program (1) Issued as NRCC No. 12515. (2) (a) D. Eisenberg and W. Kauzman, “The Structure and Properties of Water,” Oxford University Press, London, 1969; (b) R. A. Horne, Ed., “Structure and Processes in Water and Aqueous Solutions,” Wiley, New York, N. Y.,1971; (c) J. Lee Kavanau, “Water and Solute-Water Interactions,” Holden-Day, San Francisco, Calif., 1964; (d) a comprehensiveliterature review of the vibrational spectra of liquid water is presented in K. M. Cunningham, Ph.D. Thesis, Yale University, 1971. (3) A. N. Rusk, D. Williams, and M. R. Querry, J. Opt. SOC. Amer., 61,895 (1971); R.E.Frech, Ph.D. Thesis, University of Minnesota, 1968(4) G. E. Walrafen~J. them. Phys., 47*114 (1967). (5) J. w. S o h u h and D. F. Hornig, J . Ph@. Chem., 65,2131 (1961). J. Chem., 483 607 (1970). (6) H. Re WYSS and M. F a k (7) G.E. Walrafen, J. Chem. Phys., 52,4176 (1970). 49, 4150 (1968). (s) J. schiffer and D. F. Hornig, (9) T.C. Damen, S. P.S. Porto, and B. Tell, Phys, Rar., 142, 570 (1966). The Journal of Physical Chemistry, Vol. 76, No. 8,197%
1148
12-
W. F. MURPHY AND H.J. BERNSTEIN LIQUID
H,O
L A S I R . bBLLOA, 7 0 0 ~ 4 1 ACOUlSlTlON PERIOD. Z s e o STEP S l l l 0 2 1 SPLCIRAL SLIT W l O l M - I A W R A L L L L ?IXARIZED
.
.-.o,' , ,H'
H '
418000
VTRI
(FREE)
V T ~ I(BONDED)
'TRIlsymI ""I "TRI b y m ) "'3
CORRECTED DATA
Figure 2. Water species and related vibrational modes which contribute t o the OH stretching region.
Figure 1. Parallel polarized component of the Raman spectrum of liquid H20 in the OH stretching region; observed and corrected spectra.
which corrects the measured intensities for the dead time losses due to the finite pulse pair resolution time of the counting system, subtracts the dark count, converts the data to a linear frequency shift scale which includes correction for the periodic scanning error of our spectrometer, and finally corrects the observed intensities for the spectral sensitivity of the spectrometer system. The special importance of the sensitivity correction in this type of study is obvious from Figure 1, which shows the uncorrected input and corrected output data. Certainly any meaningful quantitative discussion of the shape of this band must be based on data which have undergone such a correction. The isotropic and anisotropic components of the scattering spectrum were derived from the parallel and perpendicular components following a procedure similar to that described by Scherer.'O Thus, if a band is completely depolarized (pa = 3/4), it will have only an anisotropic component, while a completely polarized band (pa = 0) will have only an isotropic component. A partially polarized band (0 < p s < 3/4) will have components in both spectra. Furthermore, it is expected from band contour considerations" that the bandwidth of the anisotropic component would be greater than that of the isotropic component. No further processing of the data was applied before using it in the band-fitting procedure. In particular, the data were not smoothed or otherwise treated to eliminate noise. The band decomposition was carried out using a modification of the band-fitting program developed by Jones and coworkers.12 This program was altered to allow use with Raman spectra and extended to allow selected band parameters to be held fixcd or linearly related to one another. The Journal of Physical Chemistry, Vol. 76, No. 8,1078
The Model. The model is based on the assumption that liquid water is made up of predominantly one type of water molecule which is tetracoordinated via hydrogen bonds with four other water (see Figure 2 ) . This structure corresponds to that of crystalline ice, and the bands in the vibrational spectrum of the solid are quite narrow.13 In the liquid, however, a distribution of configurations involving differences in bond lengths and angles occurs (see, e.g., ref 2c), and this results in the observation of broad bands in the vibrational spectra. For our purpose we assume this tetracoordinated species has approximatc Czv symmctry, and its spectrum to a good first approximation is independent of coupling beyond the first coordination shell.l* The stretching vibrations of this species may thus be classified as symmetric and asymmetric modes having frequencies v1 and v3, respectively. The symmetric mode is expected to have an intense Raman band which is highly polarized ( p s < 0.25), while the asymmetric mode should be weaker and virtually completely depolarized ( p s N 3/4). On the other hand, v3 is cxpected to be stronger than v1 in the infrared spectrum due to the complementary nature of infrared and Raman intensities. The anisotropic Raman spectrum (Figure 3) has a maximum a t 3455 =t 5 cm-' which is thus associated with the asymmetric stretching frequency, v 3 . It is expected then that the observed frequency for the OH vibration of tetracoordinated HOD at 3400 f 5 cm-1 (see below) will bc the average of the symmetric and asymmetric frequencies of H20 VOH(HOD) (l/z)(vi
+ va)
(1)
Using the above values for v3 and ~oH(HOD), the (10) J. R. Scherer, S. Xint and G. F. Bailey, J . Mol. Spectrosc., 39, 146 (1971). (11) A. E. Douglas and D. H. Rank, J. O p t . SOC.Amer., 38, 281 (1948); I. I. Sobelman, Izv. Akad. Nauk SSSR, Ser. Fiz., 17, 654 (1953). (12) J. Pitha and R. N. Jones, "Optimization Methods for Fitting Curves to Infrared Band Envelopes," NRC Bulletin No. 12,National Research Council of Canada, Ottawa, 1968. (13) J. E. Bertie and E. Whalley, J . Chem. Phys., 40, 1637 (1964); M.J. Taylor and E. Whalley, ibid., 40,1660 (1964). (14) G.E.Walrafen in "Hydrogen-Bonded Solvent Systems," A. K. Covington and P. Jones, Ed., Taylor and Francis, London, 1968.
RAMAN SPECTRA OF WATER
1149
symmetric frequency of HzO is estimated to be about 3345 cm-l, roughly intermediate between the two strong features observed in the isotropic spectrum. These features then could be explained by Fermi resonance between the overtone of the bending vibration a t 1640 f 5 em-’ and the symmetric stretching v i b r a t i ~ n . ~ If these features are part of a Fermi doublet, say V A and V B , then their average frequency should be the same as that for the unperturbed frequencies, l5 i.e.
f
(‘/!Z)(VlA
VIB)
=
Therefore, substituting for gives VIA
+
VIB
= 20
-
~3
(l/Z)(Vl
vl
+ 2V2)
(2)
from eq 1 into eq 2
+ 2voH(HOD) = 6625
f
25 cm-l
(3)
In the Raman spectrum of tetracoordinated hydrogenbonded HzO in the OH stretching region, one should therefore observe a partially polarized Fermi doublet whose frequencies are symmetric about 3312 cm-l. For DzO (v2 = 1210, vs = 2540, and v o ~ ( H 0 D )= 2500 em-’), parallel reasoning gives VIA VlB = 4880 f 25cm-‘. The predicted spectrum of the tetracoordinated species thus can explain several of the experimental results listed above. The two strong features in the Raman effect are due to the Fermi resonance of VI with the first overtone of v2, the difference between the infrared and Raman spectrum is due to the change in relative intensity of V I and va, and the change in depolarization ratio across the band is due to the contribution from the depolarized v3. However, on the basis of this species alone, the spectrum of HOD in the OH (and OD) region would be expected to be a single band with little difference between the infrared and Raman contours. Also, another type of OH stretching vibration is necessary to explain the high-frequency shoulder observed in all of these spectza. One choice for the source of this shoulder is the scattering due to a free OH bond, i.e., one not participating in a hydrogen bond. Since we have assumed that liquid water is mainly made up of tetracoordinated molecules, (-C2,),the free OH could occur most easily in the next most probable species, a molecule (C,) with three hydrogen bondsI4 (see Figure 2). As indicated in Figure 2, we assume that the stretching frequencies of the tricoordinated species with two bonded OH’S are close enough to those of the tetracoordinated species that their bands will be obscured by the more intense bands of the latter. Thus, the spectrum of this tricoordinated species will be neglected. The tricoordinated species with one bonded OH will then have a high-frequency band associated with the free OH vibration. In addition, it will have a bonded OH vibration at lower frequency, which, in
+
fact, provides an explanation for the difference between the infrared and Raman spectra of HOD. AS shown by Wyss and Falk,e the infrared frequency a t the band intensity maximum is different in value from that observed in the Raman spectrum, namely 3400 and 3420 em-], respectively. This behavior can be explained on the basis that this band is made up of two unresolved bands due to the bonded OH vibrations of tetra- and tricoordinated HOD molecules. This shift in band maxima between Raman and infrared is caused by the difference in the contribution of the tricoordinated species to the total intensity of the band; Le., this band is relatively weaker in the absorption spectrum. This intensity relation is justified by noting that free OH stretching modes are weaker relative to bonded OH bands in the infrared compared t o the Raman. This fact is extended to postulate that bonded OH stretching vibrations in waters with a free OH will also be weaker in the infrared compared to the bonded OH bands of tetracoordinated molecules. Thus, in the infrared band there is little intensity contribution from the tricoordinated species, giving an estimate of V O H (HOD) = 3400 cm-1 for the tetracoordinated species.l6 The frequency of the tricoordinated bonded OH band is sufficiently higher than 3400 em-l to give the observed Raman maximum a t 3420 em-’. Thus, for HOD we expect a band at 3400 cm-‘ due to the OH vibration of the tetracoordinated species, the free OH vibration a t 3620-3640 cm-’, and a third band, intermediate in frequency, due to the bonded OH vibration of the tricoordinated species. These latter two are expected to be relatively weak in the infrared. Similarly, for HzO, in addition to the bands of the tetracoordinated species, two bands due to the tricoordinated species are expected. The shoulder attributed to the free OH in HzO and the shoulder due to OH in HOD are observed at very nearly the same frequency. By analogy, the frequencies of the bonded OH in the tricoordinated species of H20 and HOD are expected to be nearly the same. These bands are also expected to be weak in the infrared. On the basis of these two species, then, the observed frequency and intensity properties of the infrared and Raman spectra of the isotopic forms of liquid water may be rationalized. The temperature behavior of these spectra will be considered in the Results and Discussion, Band-Fitting Procedure. This model was used as the basis of a band-fitting or band-decomposition procedure in an attempt to reproduce t h e Qbserved Raman profiles (Figures 3, 4, and 5). The band contours were approximated by symmetric Gaussian Lorentzian prod(15) G . Placzek, “Rayleigh and Raman Scattering,” U. 8. Atomic Energy Commission UCRL-TRANS-526(L), 1962, pp 139-145; translated from “Marx Handbuch der Radiologie,” Vol. 6, Part 11, 2nd ed, Akademie Verlagsgesellshaft, Leipzig, 1934,pp 205-374. (16) The temperature dependence of the infrared spectrum of HOD6 leads to a better estimate of VOH(HOD)= 3390 cm-1. However, this should make little difference in our results.
The Journal of Physical Chemistry,Vol. 76, No. 8,107.2
1150
W. F. MURPHY AND H. J. BERNSTEIN
RAMAN SPECTRUM
RAMAN SPECTRUM
2200
FREQUENCY ( C M - 1 )
FREQUENCY ( C M - I )
A N ( SOTROPI C RAMAN SPECTRUM AN ISOTROPl C RAMAN SPECTRUM
2200 FREQUENCY
(04-1)
FREQUENCY ( C M - 1 )
Figure 3. Isotropic (above) and anisotropic (below) components of the Raman spectrum of HzO, showing observed spectrum and results of band decomposition procedure. The intensity calibration marks are the same in both spectra.
Figure 4. Isotropic (above) and anisotropic (below) components of the Raman spectrum of D20,showing the observed spectrum and results of band decomposition procedure. The intensity calibration marks are the same in both spectra.
uct functions of the type
I(u) =
21 exp(--za(v
1
+
28(Y
-~
) z )
- x2)2
The parameters XI, x3, and x4 were varied as necessary in the least-squares procedure in order to attain a best fit. The band position, x2,was constrained to be equal for the isotropic and anisotropic components of a single band. For a set of parameters for which convergence was achieved, an important criterion for goodness of fit was the requirement that the bandwidth of the anisotropic component be greater than that for the isotropic component of a given band. The positions of the bands due to the tetracoordinated species were forced to maintain the fixed or reIative range of values estimated above. Different sets of frequency values for the two bands of the tricoordinated species were then tried until a mutually consistent set was obtained which reproduced the spectra of all the isotopic species. The additional band shown in the HOD spectrum (ca. 3000 cm-' in Figure 5) is due to D20. The band parameters were found by fitting a DzO spectrum taken under comparable experimental conditions. These values were then used to estimate the background in the HOD spectrum. Previous results7 and our own preliminary investigation indicated that there is no unexpected difference between the contours of the OH and the OD bands of HOD. We thus did not consider the OD band in detail. We are confident that it may be decomposed in a manner strictly analogous to our OH resuults.
Results and Discussion The results of the fitting procedure are plotted in Figures 3 , 4 , and 5, and listed in Table I. Note that the total calculated intensity is plotted in the figures; it The Journal of Physical Chemistry, Vol. 76,No. 8,1973
RAMAN SPECTRUM
FREQUENCY ( C M - 1 )
3500
3000 FREQUENCY ( C M - I )
Figure 5. Isotropic (above) and anisotropic (below) components of the Raman spectrum of HOD, showing the spectrum and results of band decomposition procedure. The intensity calibration marks are the same in both spectra. Bands centered below 3000 cm-l are due t o DzO.
is usually obscured by the noise of the experimental data. These results are considered to provide a satisfactory confirmation of the model. The frequency positions of the component bands have in general an estimated error of 10-20 cm-l. The best fit of the observed contour of HzO was found for V I A f V ~ B= 6615 em-' and for DzO, 4860 em-', which is well within the accuracy of the values estimated using eq 3. The ratio between corresponding frequencies for H20 and DzO is relatively constant, as expected. However, since this was a criterion for the selection of frequency estimates for the tricoordinated species, the significance of this point is diminished. The requirement on the relative bandwidths of the isotropic and anisotropic components of a partially
1151
RAMAN SPECTRA OF WATER Table I: Numerical Results of Band-Decomposition Procedure for Ramen Spectra of Isotopic Forms of Liquid Water Peak height Species
Tetra
Vibration
VIA
Component
Is0
Aniso VlB
Is0
Tetra Tri
va
Aniso Aniso
Vbonded
Is0
Tri
Vfree
Tetra
Aniso
Is0 Aniso
21
43400 1560 35700 5250 6040 8720 3180 6200 1100
Band psrametera Frequency Lorentzian 21, cm-1 xa
3215 3400 3455 3545 3635
(a) Ha0 6 x
6x
x x x x x 3x 2 x
2 5 9 9 5
lo-' 10-7 10-6 lo-' 10-23 10-20 10-10
Gaussian 24
Width, cm-1
Band area
10-6 10-6 10-6 10-4 10-4 10-4 10-4
220 245 219 255 200 140 157 100 79
1.1 x 5.0 X 8.4 X 1.4 X 1.6 X 1.3 X 5.3 x 6.6 X 9.3 x
3 x 10-6 4 x 10-6 1 x 10-4 1 x 10-4 4 x 10-6 3 x 10-4 3 x 10-4 1 x 10-4 4 x 10-4
123 108 162 161 139 92 81 93 91
3.6 X 5.2 x 3.5 x 2.3 X 9.6 X 2.5 X 2.2 x 3.1 X 1.2 x
2 x 1x 9 x 9 x 5 x 1x
223 225 176 177 77 39
2.4 X 7.0 X 8.2 X 1.5 X 1.6 X 5.8 x
2 x 2 x 4 x 4 x 4 x 1x 1x 3 x 4 x
10-6 10-6
Depolarization ratio
107 lo6
0.04
lo6
0.14
lo6 lo8
3/4
lo6
0.26
106 lo6 104
0.12
(b) DaO Tetra
VlA
Is0 Aniso
Tetra
VlB
Is0
Tetra Tri
Pa
Aniso Aniso
Vbonded
Is0
Tri
Vfree
Aniso
Is0 Aniso
25000 400 20000 1340 5000 2500 2220 2900 1340
2372 2488 2540 2610 2680
2 3 3 3 2 6 9 3 4
x 10-4 x 10-4 x 10-8 x 10-7 x 10-4 x 10-l6 x 10-7 x 10-4 x 10-12
lo6 104 106 los
0.014 0.06
lo6
a/4
lo6
0.40
106 lo6
0.25
106
(c) HOD-OH Stretch Tetra
VOH
Is0 Aniso
Tri
Vbonded
Is0 Aniso
Tri
Vtree
Is0 Aniso
9523 2640 4374 810 1930 135
3400 3530 3625
5 x 6 x 5 x 3 x 2 x 8 X
polarized band is, in general, well satisfied. This breaks down only for weaker bands which are not well defined in one or the other component. In addition, the comparison between the relative bandwidths of bands associated with different vibrations is encouraging. The free OH should have the least perturbation from its environment, and the bands attributed to it are, as expected, the narrowest. The bonded OH in the tetracoordinated species is expected to have the greatest range of environmental perturbations and thus the broadest bands, as observed. The intensity ratio of the Fermi components in the isotropic part of the spectra of H20 and DzO is not unrealistic. However, it is difficult to account for this ratio changing by a factor of 3-4 in the anisotropic part. Although a change of this type is not precluded theoretically,'s a change of this magnitude has not been reported." Also, there are other factors which might contribute to the large change in this ratio. One of these could be that the anisotropic components of these bands are weak and not well defined in the spectra; thus, the fit for these bands cannot be expected to have a high degree of accuracy. Another could be that the model assumption of identical tetracoordinated species does not adequately explain the details of the
10-6 lo-' lo-' 10-7 10-10
10-6 10-6 10-6 10-6 10-6 10-3
lo6
0.21
lo6
lo6 lo6 lo6
0.15 0.03
103
Fermi interaction. A further factor could be that the assumption of symmetrical bands which forms the basis for the band decomposition procedures may be poor in this case. Another possible objection to these results is the large values of va - v1 found from the band analysis of liquid HzO and D20. However, if the change in geometry and change in potential function cannot be predicted, the change in v3 - v1 on going from gas to liquid cannot be specified. It can be shown that if only an increase in H-0-H angle takes place, as would be expected if the molecule becomes more tetrahedral in the liquid, then v3 - v 1 will be larger in the liquid phase than in the gas. However, since unpredictable changes in cross-term potential force constants may also take place, it is not possible to predict the sign or magnitude of va - VI. The rationalization of the temperature dependence of these bands in terms of this model is quite straightforward. It has been reported4s8 that the high-frequency sides of the contours of these bands increase in (17) E.g., from the data for the Fermi doublet of CSZit can be shown that this ratio changes by about a factor of 2: W. R. Hess, H. Hacker, H. W. Sohrotter, and J. Brandmtiller, 2.Angew. Phya., 27, 233 (1989).
The Journal of Physical Chemistry, Vol. 76, N o . 8,107.2
1152
MICHAEL J. GERACE AND KARL F. KUHLMANN
intensity with respect to the low-frequency sides as the sample temperature is increased. Walrafen, in fact, reported an “isosbestic point” for HzO at 3460 13m-l.~ Since the thermodynamically less stable tricoordinated hydrogen-bonded species would be expected to increase in relative concentration with increasing temperature, their respective bands at 3545 and 3635 em-’, which are on the high-frequency sides of the contour, would be expected to increase in intensity, as reported. Thus, as can be seen from Table I, the frequency for the isosbestic point is consistent with our results. Similar arguments hold for the isosbestic point of HOD a t 3470 cm-l.ls However, the present model is not capable of supporting a quantitative discussion of the temperature effects, first, because of the simplifying assumptions discussed above and second, because the complicating effect of hot bands involving intermolecular modes of vibration was
ignored. Consideration of the effect of these hot bands in general and on the Fermi doublet in particular is felt to be beyond the level of sophistication of this model. In conclusion, we have demonstrated that the chosen model satisfies the several requirements outlined above and that the band-fitting procedure verifies its suitability for understanding the bond-stretching region of liquid water. We realize that this model is a simplified description of a very complex system, and it must be emphasized that we do not insist that this is the only model which can be used to interpret these data. Certainly, any model which is consistent with the requirements set out in the Introduction could be an acceptable alternative to this one. However, we feel that the virtue of the model considered here lies in its simplicity. (18) K. A. Hartman, J . Phy8. Chem., 70,270 (1906).
Measurement of Longitudinal Relaxation Times for Spin-Decoupled Protons by Michael J. Gerace and Karl F. Kuhlmann* Department of Chemistry, Dartmouth College, Hanover, New Hampshire
08766 (Received November 9, 1971)
Publication costs asskted by the N a t w d Scknce Foundation
The saturation recovery method has been employed to obtain the longitudinal relaxation time for spin-decoupled protons. The methyl group TIfor a solution of methanol in dimethylde sulfoxide was the same for samples decoupled either by double irradiation (0.62 i 0.02 sec) or by the addition of acid (0.64 i 0.02 sec). The measurement of the longitudinal proton relaxation time T1 for a single-lined nmr spectrum has provided information concerning molecular motion. Knowledge of relaxation times of different chemically shifted nuclei in the same molecule can render additional information, but their attainment in the presence of spin-spin coupling is usually a difficult task. Not only does each line in the spectrum have to be assigned to a particular spin state, but the initial state from which the spin relaxes must be estab1ished.l Moreover, the return rate of an individual spectral line to equilibrium is not governed by a single exponential decay.2 Some of these problems have been avoided by using pulse-Fourier-transform techniques;3 however, this method requires instrumentation which is not always available. Theoretically, T1 can be obtained from decoupled ~ p e c t r a provided ,~ the spectrum is of first order. That is, when two nonequivalent Rets of homonuclear spinThe Journal of Physical Chemistry, Vol. 76,No. 8,1079
coupled protons are decoupled via double irradiation, unambiguous relaxation times for either proton can be obtained. We have used the saturation recovery method to demonstrate the feasibility or measuring relaxation times for such a system using standard highresolution nmr equipment. An experimental method for the determination of proton relaxation times by the saturation recovery technique is reported by Van Geet and Hume.6 We have made the following modifications of the method
* Address correspondence to this author at the Department of Chemistry, Stanford University, Stanford, Calif. 94305. (1) J. E. Noggle, J. Chem. Phys., 43,3304 (1965). (2) A. Abragam, “The Principles of Nuclear Magnetism,” Oxford University Press, London, 1961,p 294. (3) R.L.Vold and J. S. Waugh, J. Chem. Phys., 48,3831 (1968). (4) K.F.Kuhlmann, D . M . Grant, and R. K . Harris, ibid., 52, 3439 (1970). (6) A. L.Van Geet and D . H. Hume, Anal. Chem., 37,983 (1965).
