Jan B. Wooten' and John Jacobus2 J. E. Gurst3 William Egan4 W. G. Rhodes5 and Ken Wagener6
Nuclear Spin-Lattice Relaxation Times from Continuous Wave NMR Spectroscopy
Manv fundamental ohvsical and chemical properties of . . . . molecules are influenced by molecular motion. Information about the dynamic behavior of molecules in the gas phase is provided by the study of rotational-vibrational spectra and nf easeous diffusion. In the liquid phase, nuclear magnetic . . resonance (nmr) provides a convenient probe of molecular motion since the nuclear magnetic relaxation times of molecules denend unon interactions which result from molecular tumbling and molecular translation. T h e rate of rotational motion obtained from nmr relaxation times can he related to macroscopic fluid properties such as viscosity and to translational or rotational diffusion coefficients. The availability of pulsed Fourier transform (FT) nmr spectrometers has spurred interest in relaxation phenomena and the study of nuclei which occur in low natural abundance such as 1°C and W . Relaxation time studies of most nuclei, however, are heyond the capabilities of the conventional continuous wave (CW) spectrometers which are availahle in many laboratories. evert he less, the general principles of relaxation can be demonstrated for certain molecules on many nroton ('H) CW instruments. An exoeriment readilv. adapt. , able to an undergraduate physical chemistry laboratory which describes these principles is reoorted herein. This experiment is valuable for introducing the nmr concepts of saturation, spin-inversion, and relaxation. It may also provide the basis for discussion of first-order rate processes and of rotational diffusion. If the available instrumentation has a variable temperature accessory, the measurement of relaxation rates as a function of temperature allows the calculation of an activation energy for relaxation from the well known Arrhenius equation. Additional introductory information on the theory of nmr can he found in standard texts and a review of the study of molecular motion in polymers has been given in this Journal ( I ) .
-
. ~ - ~ ~ ~ ,
Theory Spin 'Iz nuclei in a magnetic field of intensity H, occupy non-degenerate energy levels, commonly designated a and 0, in which the spins are aligned with and against the applied magnetic field, respectively. These states differ in energy by AE, defined by AE = yhH,
(1)
where y is the magnetogyric ratio of the nucleus. Energy transitions between the a and 0states can be induced by ap-
'Present address: National Institutes of Health, Bethesda, Maryland 20014; work done at Clemson University. %Presentaddress: Department of Chemistry, Tulane University, New Orleans, La. 70118: author to whom correspondence should be addressed. Work done at Clemson University. "University of West Florida, Pensacola 32504. National Institutes of Health, Bethesda, Maryland 20014. Whitman College, Walla Walla, Washington 99362. "kzona, he., American Enka Research, Enka, North Carolina 28728. Induced magnetization has been equated to intensity 304 1 Journal of Chemical Education
plication of energy in the radiofrequency range (at the Larmor precession frequency of the spin) such that AE
= h u = yhH,
(2)
or
The number of transitions from the lower enerev ".level (a) to the higher energy level (P) is proportional to the population of spins in the lower state (N,,). Similarly, transitions from the (3 state to the a state will be proportional to Np. Absorption intensitv" is . orooortional to the difference in . populations (N,. . . - No). Initially, a t equilibrium, t h e n state possesses excess population, the ratio of spins being calculable by the Boltzmann relation
N" = e - ~ h H o / k T
(4)
Nir
If the sample in the magnetic field is irradiated with the correct frequency, u (eqn. 31,transitions (energy absorptions) are stimulated between the a and P states. As energy is ahsorbed (and recorded as an absorption signal), the population of the a state decreases and that of the 0state increases. Since absorption intensity is proportional to the population difference, the absorption intensity will decrease under the correct conditions (see below). When the populations of the two states become equal, the absorption intensity becomes zero. This condition is defined as saturation (N, = N,d. A common source of saturation in CW nmr experiments arises from a combination of slow sweep rates and high irradiation (RF) power. At saturation, the (nuclear spin) system is no longer in equilibrium with its surroundings. The processes which tend to restore the system to equilibrium are termed relaxation processes (1.2). T h e restoration of equilibrium along the direction of the magnetic field is referred to as longitudinal relaxation, or, more commonly, spin-lattice relaxation. Spinlattice relaxation is a first-order process, following normal first-order kinetics. The relaxation time ( T I ) is defined by k'
=
1
(5)
where k l is the first-order rate constant for return to equilibrium. In practice, a resonance line can be intentionally saturated (N,, = Ns; intensity (I)= OI7 by slowly sweeping through the resonance line a t high observing (RF) field strengths. Once saturated, the RF power is rapidly reduced to a normal (non-saturating) level and the resonance line is recorded as a function of time until the equilibrium absorption intensity is regained. Treatment of the data in the normal manner ( I as a function of time (t)) would yield the relaxation time TI. This method of obtaining the relaxation time has been named the "saturation-recovery method" (3). A more efficient method of obtaining T I(on a continuous wave spectrometer) is to use the "rapid adiabatic passage method" (4). If a resonance line is rapidly swept a t high RF power levels, saturation does not occur. Rather, spin inversion
results, that is, the spin populations of the a and 0 states are reversed. The 0 state is populated in excess of the a state! If the resonance line is now sweot a t normal RF Dower levels, the initial intensity of the line is negative (I is proportional to N,. - N,, and N p > N,,). If the line is swept as a function of time, the absorption intensity will return to its normal level by first-order decay; the relaxation process can be observed from net inversion, through a null point, to complete recovery, yielding substantially more data than the saturation recovery method. The appropriate kinetic equation is
for the determination of spin-lattice relaxation times greater than -30 sec. Numerous mechanisms for spin-lattice relaxation have been recognized (2). For spin '12 nuclei, the most common mechanism is dipole-dipole relaxation,'" governed by the equation
where rij is the intranuclear separation of the nuclei i and j (here to be the same isotooic N is the , - ~ assumed - - ~ . soecies), . number of equivalent nuclei contributing to the ;elaxation, and T.. is the correlation time. For an isotro~icallvreorienting moleche r, is the time required for the molecule turn in an; direction through approximately 33". For small molecules in non-viscous liquids a t room temperature the normal range of r , is > r, > 10-l2 sec. The study of molecular motions (on both the molecular scale and segmentally) which occur in very short time spans is one of the major applications of nmr relaxation time studies. ~~~
-dl =-d-.t
I (I)
T,
where I is the instantaneous intensity of the resonance line and k = (l/T1). In integrated form eqn. 6 is t I(t1 (7)
At t = 0, assuming complete inversion, the system is removed from the equilibrium value I, by (I, - (-1,)) = 21,. At any subsequent time t ( t > O), the intensity is (I, - It). Substitution into equation 7 yields
., Thus, plots of In (I ,. - I, I veriui [ yield lines ufslope (-LITl) i ~ n dassuming , cumplt:tr inversion, intrrcept 21=." This latter metha,rl is similar to the pulsrd nmr experiment? the mort! general method tor menwring spin-latrire r ~ l ~ x a t i t m times. A wlaxed samolr (at euudihriumr is ~ u l s e du,ith a burst of RF energy, leading to spin inversion (180' pulse). After an aoorooriate .. . wait time r. a second ~ u l s e(90') rotates theresidual magnetization (along the direction of the field) into the plane containing the R F receiver which records the intensity of the residual magnetization. The intensity can thus he determined as a function of r and the data treated as described above. This particular pulsing sequence is described as (180"-r-90'-T), where T is a second wait time prior to repetition and accumulation of the signal in a computer. During the time T the system returns to equilibrium such that the accumulated spectra are (hopefully) identical, such that "noise" can be averaged out. Among the advantages of the pulsing method is the range of r that can he employed (from milliseconds upward) to determine Tl. The rapid adiabatic passage method is best suited (with conventional recorders) 1.8
Wxperimentally, complete inversion is not required; the derivation of Equation 8 requires the assumption of complete inversion. 9 Pulse nmr methods are discussed by Farrar and Becker (21. "'Farrar and Becker (21, p. 53ff. 'I Thisexperiment has heen performed on a Varian A-60A, aVarian T-fiOA, and a JEOL C-6OHL spectrometer. Similar results have been obtained on all instruments.
1
o
1
1 10
1
1
u
1
1
1
so
1 80
1
1
im
1
1
la
1
1
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20 L
80
I 108
tw,
Figure 2. Plot of the relaxatm data from Figurer 1 and 3.
1
t40
I
(see1
Figure 1. 60 MHz rapid,adiabatic passage spectra of the aldehydic proton of acetaldehyde.
Figure 3. 220 MHz rapid adiabatic passage spectra of the aldehydic proton of acetaldehyde. Volume 56,Number 5. May 1979 1 305
Experimental We shall describe the determination of the relaxation time of the aldehydic proton of acetaldehyde on a Varian A-60A spectrometer." Other molecules can be e m d o ~ e d e?E~. . henzene, provided that their T,'s are sufficiently long to Govide time to eoll&t reliable data. The acetaldehyde sample employed was the "Resolution" sample supplied by Varian. The instrument was tuned in the normal manner. For the T I measurement a sweep width of 500 Hz and a sweep time of 500 sec were employed. The RF Field setting was adjusted such that repetitive scans of the aldehyde quartet did not result in any loss of signal intensity (non-saturating conditions). The RF Field strength was increased to its maximum intensity and the aldehyde signal was swept (from low field to high field) employing the Fast Sweep mode. '2 Saturation df the aldehydic resonance of acetaldehyde will alter the spin populations of the methyl group. The relaxation of thespins of the methyl group could affect the aldehydic proton relaxation rate, that is, the decay could become multiexponential. The precision of the data collected in thls experiment dues not allvw the determination vf other than simple exponential decay. For this reason, benzene might be a "better" choice for the experiment, hut theacetaldehyde sample is more generally available. '"mong the conditions stated by Noggle and Schirmer (2, p. 10) for application of the rapid adiabatic passage technique is that the J-couplings be small enough such that a single line is observed under low resolution. Our results obtained on a "single" resonance line (220 MHz, low resolution) are the same as those obtained on the resolved signals a t 60 MHz.
Immediatelv after oassine thmueh the sirnal the normal S W e D switch wna eneaeed. RF Field settine was lvwered to its non-saturatine " ~ . the - ~ \ d u e , and the ;u,rrp i>tf;rt wns rmpluywl ~c,aurrpihe nld~hydes~anal pa51 the Pen. t h nldehyde ~ r ~ ~ n wn ls re~wrded.and i h p pnwedurr repeated until the "normal" intensity was regained. A typical "run" is depicted in Figure 1. The data were treated by plotting In (1- - b ) versus t , where t is the time a t which a particular signal was recorded (Fig. 2), relative to an arbitrav zero time. The inside (IN) and outside (OUT) lines of the aldehvdicauartet were treated sevaratelv. The value of T 1is -37 sec. The data dbtained from the A - 6 0 experiments ~ can be compared (Fig. 2) with those obtained on a Varian 220 (Fig. 3), which yields a T Iof 41.9 see.I3 If it is assumed that the acetaldehyde molecule tumbles isotropically and if it is further assumed that the average H-H distance (HCCH) in acetaldehyde is -2.8 A, the correlation time of aeetaldehyde can he roughly estimated from equation 9 to be -9.5 X lo-" ~
~
~~
~
see.
Literature Cited (11 Slichtor. W. P.,J.CHEM. EDUC., 47,193 119701. (2) Relaxation mechanisms are extensively discussed by Fsrrsr. T. C., and Reeker. E. D.. in "Pulse and Fourier Trandnrm nmr" (Academic Press. Now Yark. 1971, Chapter 41 and hy No&. J. H.. and Sehirmer, R. E.. in "The Nuclear Overhauser Eifed" 1AcadrmicPress. New Yurk. 1971,Chspter 21. (31(al Blwmklper N., Purce1l.E. M., and Pound. R. V.. Phyr Rru. 73.679 11948): lb) Puple, J. A,. Schneidor. W. G.. and Bernstoin, H. J.."High Resolution Nuclear Magnetic Resonance," McGraw-Hill Ruok Cu., New Yurk, 1959. p. 82. (4)is) Abragam.A., "ThePrinciploaof Nuclear Magnetism,"Clsrendon P-, Oxford, 1961, o. 65ff: Ib) Andemm. W. A,. in"NMRand ESRSoectnram~y,"PorxamonPrpas, New
