NOTES
4133
Acknowledgments. The author is indebted to J. R. Merrill for helpful discussions and to R. T. Edwards, Jr., and R. B. Uhlig for assistance in the experimental work.
8 = (w
Determining Nuclear Spin-Lattice Relaxation Times
p
=
V(.)
ScientificLaboratory, Ford Motor Company, Deurbwn, Michigan 481.81 (Received January 28,1967)
Perhaps the least complicated method of determining nuclear spin-lattice relaxation times (TI) in liquids is that of saturation-recovery. 1--3 This method consists of saturating a particular resonance line with a large HI field4 and following the subsequent recovery of the nuclear signal after HI has been reduced to a small, nonsaturating, value.5 It is presumed that the v mode will recover exponentially with a time constant TI. It is well known that residual saturation effects in the low HI field impose an effective upper limit on the TI values that can be determined by this method. The saturation-recovery method has been less critically examined in the range of short TI values. Experimentally, it has been observed in a number of laboratoriesaP6that there is large initial distortion in the recovery curve, due to a “transient” which seems to last about 200 msec. A typically distorted recovery curve is shown in Figure 1. The cause of this distortion has not been well understood: it has been attributed to both experimental and nuclear origins. The transient caused us considerable concern, for being uncertain of its true duration, we were unable t o gauge the accuracy of short TI values obtained from saturation-recovery measurements. It is hoped that the following discussion will prove valuable t o others who may face the same problem. Torreyz has obtained a general solution to the Bloch equations, including transient terms. We shall restrict our attention to the case where TI = Tz. This case is easy to treat mathematically, and it describes the relaxation behavior found in many simple liquids. When T1 = Tz, Torrey’s solution for the v mode becomes
+ 1 + p2 +
62
)cos (sr)
+boy*
(IC)
T = TI = Tz
(14
s = (1
l/THIT;
by J. E. Anderson and Robert Ullman
e-qvo
Ob)
The quantities ~ 0 ,vo, and mo are the initial values of u, v, and m,, respectively, for a particular value of 6. Equation 1 describes the radiofrequency absorption of a particular “spin isochromat” a distance (6) away from resonance. The experimental absorption signal is obtained from
On the Saturation-Recovery Method for
=
- Uo)/YH1
+
=
J”; 9 (6)v (6,
(2)
where g(6) is a weighting factor reflecting the Ho field inhomogeneities. Specifically, we will take g(6) = (a/.) [az a2]-’, where a = l/-yHITz* (Tz* is the apparent value of T2in the inhomogeneous field). We are interested in v ( r ) following the reduction of HI at t = 0. For this reason, it is convenient to let the symbol HI
+
-10
0
01
02
03
04
05
06
07
TIME IN SECONDS
Figure 1. Solid curve: saturation-recovery curve obtained from eq 3. T h e initial and final HI values are 5.9 and 0.030 mg, respectively; T I = Tz = 0.40 sec; Tz*= 0.04 sec. Dashed curve: exponential recovery. T h e two curves have identical asymptotic behavior.
(1) See, for example, E. R. Andrew, “Nuclear Magnetic Resonance,” Cambridge University Press, New York, N. Y., 1958,p 107 ff. (2) H. C. Torrey, Phys. Rev., 76, 1059 (1949). (3) A. L. Van Geet and D. N. Hume, Anal. Chem., 37, 983 (1965). (4) The notation used in this article has become conventional. A complete exposition is given in ref 2. (5) J. G. Powles and D. J. Neale, Proc. Phys. SOC.,77, 737 (1961), have used a variation of this saturation-recovery method which is preferable for long TI values. Following the reduction of the saturation HIfield, they shift the HOfield off resonance. The recovery of the magnetization is then monitored by periodic sweeps through resonance a t low HI levels. The absence of a continuously resonant HI greatly reduces residual saturation effects. It does not circumvent the difficulties with short TIvalues, however. (6) G. Weill (private communication, University of Strasbourg, France) has observed the transient on a Varian HA100 spectrometer.
Volume 71, Number 12 November 1967
NOTES
4134
represent the intensity of the smaller, nonsaturating radiofrequency field. In the following paragraphs, the parameters 6, s, and P also refer to values in the nonsaturating field (eq 1). We will use the symbol F to designate the ratio of the radiofrequency field intensity during saturation to that during recovery. I n this notation, the boundary conditions become
’ = 62 + vg
= -
+ F2
+ 62 + F 2
(P2
P2
62
PF
P2
mo =
6F
+
+ a2 + F 2
Owing to the form of s in eq 1, we were unable to integrate the resulting equations analytically. v(6,r) is a rapidly oscillating function of (a), so that numerical integration is also difficult. We observed, however, that the small HI values used during recovery cause 6 2 to be >>1 for all nuclei except the small number very close to resonance (commonly, those within about ~k0.3cps of resonance.) If s is approximated by 6, V ( T ) may be obtained by integration of eq Z7v0 The result is V(.)
=
---
K b
P
+K)
”[(
aK2
+
e-07{-PF(ae-LT L(a2
KZe--ar - a2e-KT a2-K2
- Le-“) - L2)
) + I]}
The quantities K and L in eq 4 stand for (1
(F2
+ P2)”‘.
+
(4)
+ P2)”* and
I n a typical experiment, F may be about loa, and 1
