resulting from a 90” pulse followed by 180” pulses. The pulse interval was 8 or 12 milliseconds. RESULTS AND DISCUSSION
To test thp method, the relaxation times of air-saturated aqueous solutions of copper sulfate wercl measured. For all measurements, the temperature was 38” f 2’ C. The results are presented in Table I. TI and T Pare equal for the copper solutions, and the relaxation times are inversely proportional to concentration except a t the very lowest concentrations, where TI and T z become comparable to the relaxation times of v, ater. The progressive saiuration data for the copper sulfate solutions are shown in Figure 1. Both peak height and peak area measurements gave the same result, but the measurement of peak height with negligible inhomogeneity was the most satisfactory. The peak height method in very inhomogeneous fields seems to be sensitive to the actual shape of the magnetic field, and if the inhomogeneity line width is not large enough, the apparent relaxation time will be high. The area method is time-consuming because the integral is recorded slowly. The relaxation time of water measured by this method (Table I) was low-. The progressive saturation method
LITERATURE CITED
Table I. Proton Spin Relaxation Times of Water and Aqueous Copper Sulfate Solutions
Concn., mM Water 2 4 16 50
100
d/T,2,sec. Prog. sat. 1 30
T P , sec. Line width
0 53
0 33 0 090 0 037 0 021
0 0 0 0 0
58 27
090 029 016
seems to be susceptible to systematic errors, and not as reproducible as the direct method or the measurement of the line width. Also useful for comparison are the data on relaxation times in nickel-ammonia and nickel-cyanide solutions obtained in a recent study of the nickel-cyanide complex system (fa). I n this, T1 was measured also by the direct method. From these results, i t is apparent that the line width measurement gives the same result as the spinecho technique. These nickel solutions exhibit different values for T I and T 2 . This somewhat unusual phenomenon has been observed before ( 3 ) . ACKNOWLEDGMENT
We are grateful for many helpful discussions with Rolf W. Arndt.
(1) Abragam, A,, “Principles of Nuclear
)lagnetism,” p. 50, Clarendon Press, Oxford, England, 1961. ( 2 ) Anderson, b’., “NRIR and EPR Spectroscopy,’’ p. 180, Third Annual Workshop, 1-arian Association, Palo Alto, Calif., Pergamon Press, New York, 1960. (3) Bernheim, R. A., Brown, T. H., Gutowsky, H. S., Woessner, I). E., J . Chem. Phus. 30.950 (1959). ( 4 ) Bloembergkn, k., Purcell, E. >I., Pound, R. V.)Phys. Rec. 73, 679(1948). (5) Carr, H. Y., Purcell, E . AI,, Ibzd., 94, 630 (1954). (6) Ernst, R., Tarian Associates. Palo Alto, Calif.; private communication. 1964. ( 7 ) Luz, Z., Rleiboom, S., J . Chem. Phys. 40, 2686 (1964). (8) Meiboom, S., Gill, D., Reo. Sei. Instr. 29, 688 (1958). (9) Xederbragt, G. W., Reilly, C. A., J . Chem. Phys. 24, 1110 (1956). (10) Pople, J. A,, Schneider, W. G., Bernstein, H. J., “High Re;olution Nuclear hlagnetic Resonance, p. 82, McGraw-Hill, New York, 1959. (11) Primas, H., Helv. Phys. 9cla 31, 17 (1958). (12) Van Geet, A. L., Hume, U. N., Inorg. Chem. 3 , 523 (1964). (13) Williams, R. B., Ann. S. I’. Acad. Scz. 70, 890 (1958).
RECEIVED for review August 4, 1964. Resubmitted February 23, 1965. Accepted May 5, 1965. Work supported in part by the Cnited States Atomic Energy Commission under Contract A T (30-1)-905.
Measurement of Proton Relaxation Times with a High Reso Iutio n Nuclea r M a gnetic Resona nce Spectrometer Direct Method ANTHONY L. V A N GEET Department of Chemistry, State University o f New York at Buffalo; Buffa/o; N. Y. DAVID N. HUME Department of Chemistry and laboratory for Nuclear Science; Massachusetts Institute o f Technology, Cambridge, Mass. Details are given of an experimental technique whereby the proton relaxation time, TI, may b e measured with a high resolution nuclear magnetic resonance spectrometer, using the direct method. The method is best suited for relaxation times between 50 milliseconds and 25 seconds. A theoretical study of the method i s made. The Bloch equations are solved under the simplifying assumption of negligible saturation (YHI TI-^). The result shows that a simple exponential signal recovery of time constant T I occurs only if the magnetic field i s sufficiently Tn-’). inhomogeneous
P
ROTOP:
nuclear magnetic resonance
(KMR) spectra are widely used in
structure determination and in qualitative and quantitative analysis of organic compounds. By comparison, much less attention has been given to relaxation time measurements (13) with this instrument. The transverse relaxation time, TB,follows from the line width, of course, but relaxation times longer than about 1 second cannot be determined accurately in this way because of limited resolution. This paper shows the longitudinal relaxation time, TI, may readily be determined by the direct method (If).
I n spite of its simplicity, the direct method has not found wide application (3, 6). I n its simplest form, the sample is introduced into the magnetic field and one observes the signal growing in. I t is not necessary to introduce the sample physically into the magnetic field. Instead, the sample may be saturated by the application of a sufficiently strong R F field, HI,so y HI >> ( T I T z ) - ” * . When the amplitude of the RF field is suddenly reduced to a nonsaturating value, the ISAIR signal (v-mode) recovers exponentially with a time constant which is essentialll- T1. VOL. 37, NO. 8, JULY 1965
983
THEORY
When a sample is introduced into a static magnetic field, the magnetic moment, M,, will grow in with the time constant, T1, the spin-lattice relaxation time. This will essentially still be the case in the presence of a sufficiently small R F field (?HI < < l / T I ) . I t is not immediately obvious that the u-mode should grow in with the same time constant. On the contrary, it is shown below that the transverse relaxation time, Tz, also affects the signal recovery. An additional complication is introduced if one uses saturation to bring the initial magnetic moment down. Even for strong saturation conditions, the u-mode (in phase with HI) is still appreciable. This could influence the result (6). In the following, these effects will be considered by investigating the Bloch equations (?) :
D
=
PVO
+
(8)
mo
Torrey (12) obtained a general solution for this case, in which the assumption a > > 1 was not made. When (j3 - a ) > > 2 , Torrey's solution reduces (6) to Equation 7. The special case a = fl may be obtained from Equation 7 by calculating - a) approaches zero: the limit when
(a
V ( , = 8) =
(1
-(Avo/@)
+ 8r)e-P'
-
[1 - (1 - mol
(Boo
+ mo)e-Sr1
x (9)
Apparently, the recovery is not purely exponential, even if uo = mo = 0. By differentiation it is seen that d v / d r = 0 a t r = 0. This follows also from dM./dr (2) Equation 1, if the initial values of F and M , are zero. No sizable recovery of where u occurs until M , has recovered apF =u iu 6 = (wo - w ) / ~ H i preciably. The solution of Equation 5 follows a t r = YHit wo = T H O once from Equation 7 by replacing u by F , uo by Fo,and p by (@ is). After a = l / r H i T i Mo = xoHo rationalization of the fractions, and P = I/rHiTz (3) reorganization, the u-mode is obtained as the real part of the resulting The initial values of u, u, and M . are expression : and W O , represented by UOMO, UOMO, respectively. u/Mo = x The instrument used, a Varian A-60 NMR spectrometer, operates by the side band modulation method (13). For this instrument, H 1 is the effective R F field. The real R F field is H 1 / J 1 ,where J1 is a 62(1 - a / f l ) ] e - a r constant. If the R F field is sufficiently weak {a@ - a) - 62(a/fl)]e-@rcos 6r (CY > > I ) , the u-term of Equation 2 will be negligibly small during the recovery a6 { 1 - a ) / p J e - B Tsin b r ] as long as ( Y T is of the order of 1. Ultimately, as CYT becomes much larger ( uo pmo/ ( a 2 6 2 ) } e - P r cos 67 than 1, dM,/dr tends to zero as M , and ( Q - &/(p2 62)}e-Pr sin 6 7 (IO) u approach their steady-state value, and the approximation breaks down. For 6 = 0, Equation 10 reduces to Equation 2 may be integrated readily Equation 7 . If 6 # 0, the decaying part if the u-term is neglected : of u apparently oscillates ("wiggles"). So far, no special assumptions have M , = Mo[(l - mo)e-arl (4) been made about UO, vo, and mo, The M , may be eliminated from Equations initial state is a stationary one. Thus, 1 and 4 : Q, uo, and mo are related by the steadystate equations dF/dr (B i6)F uo = 6 / ( P o 2 62 B o / ~ o ) (lla) M o [ l - (1 - mo)e-Qr] = 0 (5)
dF/dr
+ (a + i6)F + M , = 0 + a ( M . - Mo) - u = 0
(1)
+
+
-@
+
}+
+ (a + + +
+ +
+
A t resonance (6 = 0), the u-mode follows from Equation 5: dv/dr
+ BU + M o [ l - (1 - r n ~ ) e - ~=~0] (6)
The solution of Equation 6 is 984
ANALYTICAL CHEMISTRY
1
The constant of integration, D, is chosen so that u = uJfo a t T = 0:
uo =
mo
=
+
+ + -Bo/(Po2 + 6 2 + + + 62 +
(Bo2
Pol~ro)
(llb)
Bo/ao)
(llc)
62)/
(602
The special case a Equation 10:
=
P follows from
+ + 62) cos 6 7 + Po2
where Equations 11 have been used to eliminate ~ 0 uo, , and WI+ Equations 10 and 12 may be used to calculate the signal recovery in an inhomogeneous magnetic field. In such a field, only those protons having a Larmor frequency wo = THO close to the radio-frequency, w , will contribute significantly to the signal. Protons for which 6 >>@only make a negligible contribution. If the line width is determined entirely by the inhomogeneity of the magnetic field (AyHo> > Ti-]), the response (1) will be proportional to
J-+;vd 6. Equation 12 may be integrated exactly. The integrals containing cos 67 and sin 6s are Fourier cosine and sine transforms (4). The result is :
v=
J-Y
uds
=
-rMo (1 - e-Pr)
-
e - P r VoMo[ ( V o / r )(1
+ V O / ~ ) e - r d m a (13) ]
Mo
JUods
where
V&O
=
=
-rMod.%
(14)
represents the initial value of V , and 2 0
= 1/(1
+ 1/80?
(15)
is the saturation factor. Because of the exponential factor, e - @ r , which let disappear the second term of Equation 13 for Br>>l, the result is of interest only for values of Pr not much larger than 1. With j3>>1, this means r < < l . The initial state is a saturated one, d o When a and p are not equal, the recovery in an inhomogeneous field is
obtained by integration of Equation 10. To simplify the calculation, i t is assumed that % = uo = mo = 0. The Fourier integrals are obtained as before (4). The other integrals (10) cause no difficulty. After integration and collection of the terms containing exp ( - C Y . ) , exp (-8.1 X e w [-(8 - a ) 71, and exp ( - p 7 ) X exp (-8.1, respectively, the coefficients of the latter vanish, leaving the result
v = J-+;vd6 =
- e-.')
-7rMo(l
= - T M ~ (~ e-t/T1)
(18)
Apparently the recovery in an inhomogeneous magnetic field is purely exponential, and the time constant is TI. Because of the large inhomogeneous broadening, the F-vectors in different parts of the sample precess a t different frequencies, and get out of phase very fast. This provides an effective transverse relaxation (6) of time constant Tz*, where Tz* l :
u/Mo = -8(1 e-@'(p(]. cos{r(l
+ 82 +
P)-1
+ +
+ 82 + x + + x ( P ( 1 + 82 + 67-1 - mo + 6%) x (1
+ 62)-1/z
up)
gZ)-1
e-@r
62)1/2}
+
sin ( ~ ( 1
62)1/2]
+
+ a"--' + + @sin7 ) e - B r
+
T
+
-S(l
(20)
m-1
sin
(O
+
cos
(T
- rp)e-Pr
(21) (22)
arctg ,9
=
(p
Thus, the decaying part oscillates slowly with the angular frequency, 7 H 1 , because of nutation around HI. With p > > l , the decay is observable only for values of 7< < 1, and Equation 20 can be approximated by Equation 9. If p is not sufficiently high during the recovery, the apparent recovery will be too fast. This may be seen from the factor cos ( T - rp) = sin ( 7 ( ~ / 2 ) rp). When p > > l it follows that
+
(5
between [I by exp - PT (Z-l12 - 1 ) . With this, Equation 24 becomes
V/Mo =
where
- p)
