SS NMR Timescale

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Robert G. BryantUniversity of Minnesota, 207 Pleasant Street SE, Minneapolis, N 55455

It is common jargon to say that some process is fast or slowon the NMR time scale implying that there is in fact someprecise time scale that is particularly appropriate to NMR andwhich is understood by all. While there may have been somevalidity in this imprecise jargon when relaxation time mea-surements were ignored, and when NMR was confined toprotons, the truly multinuclear character of NMR spectros-copy and the ready access to NMR relaxation rates providedby commercial instrumentation today makes any blanketstatements about the NMR time scale imprecise at best andmore often than not completely meaningless.What are the time scales associated with NMR? A conve-nient way to look at this question is in terms of the rates ofphysically or chemically significant processes that can beextracted from a study of NMR spectra. The primaly featuresthat are easily available in any spectrum are the intensity, theresonance frequencies or the chemical shifts, the fine structureor the scalar coupling patterns, and the relaxation rates as-sociated with the lines. Each leads to a somewhat differentNMR time scale because different rates or rate constantsmay he extracted from study of these different aspects of thespectrum. The intensity of a line may be used trivially tomeasure concentration as a function of time as in any form ofspectroscopy and, since this application provides no confusionabout th e time scale for the reactions or the rates involved, itwill not he discussed further.Fundamental Background

The fundamental basis for extracting kinetic informationfrom changes induced in an NMR spectrum by a chemicalexchange event is the uncertainty principle. If a nucleus maysample more than one magnetic environment in a chemicalexchange process, then the uncertainty in the energy orequivalently the resonant frequency is related to the lifetime,r , by the familiar relation

Au = hl 2ar) 1)That is, the frequency of the resonance is precisely measurableas long as the lifetime in that state is long, but as this lifetimedecreases the uncertainty in the energy or resonance fre-quency increases and one observes a lifetime broadening. Inthe simplest case that the observed nucleus exchanges he-tween two environments with equal populations, the spectrumcollapses to a single line at the average resonance positionwhen the exchange rate, 117,is large compared with the fre-

Figure 1. The idealized NMR specha for a pair of exchanging resonancessep-arated b y a frequency Au = v shown schema tically at the extremes ofslow and fast exchange.

quency difference between the two lines 1,2).A pair of lineswill just merge to become one at t he point,r,,,~,,,,,,, = tJ? TAU)- = k- 2)

where here v is the energy difference between the two linesexpressed in Hz. An idealized representative situation issketched in Fieure 1 This relation is useful for making more

time scale. If i t is slow, resonances are well resolved for theinterchanging species, while if it is fast, resonances are aver-aged and only a single line is resolved.The Chemical Shift

When considering the time scales appropriate to averagingtwo resonance lines separated by a chemical shif t difference,eqn. 2) shows that t he two lines will he resolved if the ex-change rate is small compared with the chemical shift differ-1m r i l l Hz,Av, la.tw.e~ll ~ e w ~ine,. Only n inglt nwrn;men,illbe ,,hserved r rhr rn.hnn:v r:t t t I hr:e wnpared t u theshirr diiter rnw. Thus. t o kmru the time >C RC nnlmmri~t~ .I Ithe averaging process that causes coalescence of the NMR

Representative Chemical Shift Ranges, Coupling Constants, and Time ScalesApproximate Shifts at Time Scale Shifts at ScalarShift 1.4T Field or Rangea for 7T Field or Time Scale Scalar Coupling

Range. 6 0 MHz far 14T 300 MHz for Ranged for Coupling TimeNucleus P P ~ 'H, kHz shifts 'H, kHz 7T shifts Consts, Hz Scaleb

H 0- 10 0 0.6 0.2 5 - 0 4 rns 0- 3 0.2s-75 ps ZjHH-10 -22 rnsI3C 0- 200 0 3 0.2 5-75 s 0 15 0.2 s-15 ps 'J~ ,. 150 -1.5 rns15N 0 900 0 5.4 0.2 5-40 s 0- 27 0.2 s- ps ljN 50 -4.5 rnsF 0 300 0 17 0.2 5-33 ps 0- 85 0.2 s- 3 ps 2 ~ n F50 -4.5 rns

31P 0 700 0 17 0.2 5-13 ps 0- 85 0.2 s 3 p s ZJp-p-20 -11 rns5eCo 0-15.000 0-214 0.2 5-Ips 0-1.070 0.2 s- 0 . 2 ps J~*.N -50 -4.5 rns

r*9Hg 0- 3,000 0 32 0.2 S-7 ps 0- 160 0.2 s 1.4 ps J > ~ - ? e e ~ ~,500 -90 snken as he coalescence lifetime given by eqn. 2 for a maximum shift; eg. . 10 ppm for H, and a minimum resolvable shin of 1 Hz.

Brakenas the coa lescence lifetime obtained by substitution of J for v in eqn. 2.

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lines, one must know the chemical shift difference a t leastapproximately. Here additional care is required about blanketstatements because chemical shifts may differ greatly fromone nucleus to another as shown in the tahle.In>pesrion of column 2 d t h e tnhlv shova t h ~ the cherniwlshiit mnw s diiier cuns~drrahly rtm one nu' Ieus to another.I t must be appreciated tha t the chemical shifts actually av-eraged by an exchange event generally will be some fractionof the approximate total ranges indicated. Nevertheless, theconditions for the averaging of two resonances will clearly bewry diiirren~ P ,me is h5erv ing protun, compared withnuclei like tl~wrine r ,di:dt. u,hich h ~ \ . e uch larger chemiljilshifts. Comparison of columns 3 and 4 representing thechemical shift ranges at two common proton resonancefrenuencies. 60 and 300 MHz. voints out an addit ional prob-.>~~~ ~ ~ ~lem. Because the chemical shif t difference between two res-onances increases linearly with the strength of the applied dcmagnetic field, the exchange rate between two averagingresonances must be 5 times larger at 300 MHz than at 60 MHzto achieve a coalesced spectrum. Thus, to make spectra lookidentical on the ppm chemical shift scale, the temperature ofthe compound studied would have to be somewhat higher onthe 300 MHz svectrometer than on the 60 MHz spectrometer

priate for averaging NMR chemical shifts, it is clearly neces-sary to know the magnitude, in Hz, of the chemical shift beingaveraged.Scalar Coupling

Essentially identical arguments apply to t he averaging ofscalar couplings, and it is sufficient for present purposes toreplace u by J in eqn. (2) to provide the same sort of basisfor discussion of the time scales that will lead to averaging orcollanse of the svectrum. If the lifetime of the nucleus in aparticular environment is short compared with the reciprocalof the scalar coupling constant in this case, then the couplingwill not be directly observable, and only a single, coalesced linewill annear. Insvection of the tahle indicates that , while thescalar coupling constants do not span as wide a range of valuesas the chemical shifts and the scalar coupling constants are

and distance between the coupling nucleiRelaxation

The averaging of even large shifts generally still limits theaccessible rates of vrocesses conveniently studied to times onth e order of microieconds or longer. ~ i w e v e r , MR relaxa-tion spectroscopy extends this range considerably. There isa great variety of NMR relaxation rates that may be measuredwhich all differ somewhat in the instrumental and theoreticalframework required. However, the range of rates that may bestudied is adequately represented by consideration of therelaxation time that is most easily understood and measured,th e lonritudinal NMR relaxation time, TI , or the rate, 1lT1.l/ Tl is th e first-order rate constant that characterizes thereturn of nuclear magnetization to an equilibrium magnitudein the direction of the applied field strength following a per-turbation. The perturbation may be as simple as dropping thesample into the field in the first place and monitoring thegrowth of magnetization or monitoring the return t o equilib-rium after the magnetization has been driven away fromequilibrium by a s trong r.f. pulse 3).

The longitudinal NMR relaxation rate is driven bv fluctu-

ations in the magnetic interactions that modulate the energylevels of the nuclear spins in the sample. However, since or-

50

40IT,, s-3020

10

0

ientation of a spin in-a magnetic field requires energy, thefluctuation spectrum must have components at the resonanceor Larmor frequency of th e observed nucleus for the fluc tu-ations to be effective. The relaxation equation for the isotropicintramolecular dipole-dipole interaction is

: . . .. .. .

L

where v is the maenetoevric ratio. Planck's constant divided

0. o ioFREWENCY Mlz

Figure 2. roton longitudinal relaxa tion rate as a function of frequency for a 0.80mMa queo us MnCI solution at 286 K measured from 0.01 to 35 MHz usingafield cycling technique 4).

-- - uby 2a, r the intermoment distance, the resonance frequency,and 7 he correlation time characterizing the fluctuations inr.The restmanw frequency, of cows( dtpendz m t h r t r e n ~ t hd t h e at~i~li tvlI ninenetit, field. I f . and on the marnrlwvric~ratio, y , of the nucleus observed according to the relationI ,,, = yH 2a (4)

Depending on the nucleus observed and the field strength, thepractical resonance frequencies presently range from severalMHz to a maximum of 600 MHz for protons. The usual ex-periment is to measure the relaxation rate a t fixed frequencyas a function of temperature. Equation (3) predicts that t herate will nass through a maximum when w ~ s close to unity(0.6158 t he more precise). If the structure of the moleculeis known, the internuclear distances are determined so thatthe only unknown in eqn. (3) at th is point is the correlationtime characterizine reorientation of the intermoment vector,r. T h e timt, ir.ile of molwular event; that can he moni t


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where the correlation times 7 nd 7 may have several con-tributions 5-7). For the aqueous solution of the aquo com-D ~ X .he correlation time for the firs t term in hrackets is ther t t iticmi~l. melaritm rime t h y mrt:il c~~ nll ~l ~. x.I11Ie thati t r i n t it.al ~i ~t i t ut ~. s~ ~tle:iltl~ ( ; 3 1 2 ) 4 h MI the Ka11on.11Science Foundation (PCM-8106054).

Literature Cited(11 S e e f o r e x a m p i e B o u ey , F A , N ~ ~ I e ~ r M ~ g ~ e t i i R ~~ ~ ~ ~ ~ ~ ~ s p ~ ~ t t tP~ es s . ew York, 1969,Ch. VLI.(21 Ksplan, J I., and Fraenkel. G.."NM R nfChemiraiiy Exchanging Systems? AcademicP~ es s . ew York . 1980.13) Farrar,T.C.. and Becker,E. D.."Pul se and FourierTransform NMR,"AesdemicPress ,New York, 1971.0 Hsllenga, I < . and Kuenig, S.H. i~ihhfmisW 5,4255 (1976).151 Swift, T. J., n "NMR of Paramagnetic Moleculms,'. (Editors:I.aMar, G N., Hunucks,Jr.. W. Dew.,Ho1m.R. H.l ,AcademiePress , NeaYork, 1973, Ch. 2.(61 Dwek, R. A Nuclear Magnetic Resonance in Biochemistry: Applications to EnzymeSystema?Tho Ciarendon Press. Oxford,1973.

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