Nuclear Magnetism in. Relation to Problems of theLiquid and Solid States

E. M. PurcellLyman Laboratory of Physics, Harvard University

THE MAGNETIC BEHAVIOR OF MATTERin the bulk-paramagnetism, diamagnetism, andferromagnetism-can be traced to the electronic

structure of the atoms of which the matter is composed.The electron, in short, is the cause of ordinary mag-netism. But the electron is not the only magneticconstituent of most substances, for many atomic nu-clei have certain magnetic properties. Although themagnetic effects to which these nuclear propertiesgive rise are too feeble to be observed under mostcircumstances, it has been possible to devise techniqueswhich bring them into prominence. From these ex-periments in nuclear magnetism one learns more aboutthe nucleus, and, since the most active frontier inphysics lies at the nucleus, such information is highlyprized. In quite another direction, however, a line ofapproach is opened to certain features of the structureof liquids and solids which prove to be involved in aninteresting and unusual way in the manifestations ofnuclear magnetism.

It is the latter aspect of nuclear magnetism that Ipropose to survey in this article, after providing anintroduction to the magnetic properties of nuclei andto the phenomenon of resonance upon which the ex-perimental method depends. "Survey" is perhaps tooambitious a term, for any attempt to predict the ulti-mate extent or importance of this new field must berenounced at the outset.

NUCIEAR SPINS AND MAGNETIC MOMENTS

In addition to their more familiar properties ofmass and electric charge, many atomic nuclei possessintrinsic angular momentum, or "spin," and magneticmoment; that is to say, the nucleus behaves like aspinning top with a magnet embedded along its axis.It is natural to seek the origin of nuclear magnetism inthe motion of the positive electric charge of thenucleus; charge in motion is the same as electric cur-rent, and electric currents produce magnetic fields.We are thus led to regard the proton, for example, asa positively charged sphere spinning with a certainangular momentum and equivalent, in its magneticproperties, to an infinitesimal magnetic dipole. Al-though such a view eventually proves inadequate, itwill serve very well as a basis for our discussion.The intrinsic angular momentum of a nucleus is

specified, in units of h/27r (h = Planek's constant),SCIENCE, April 30, 1948, Vol. 107

by the number I, which is either integral or half-integral. The quantity Ih/2r, to be more explicit, isthe maximum component of the angular momentumvector measured along a selected direction. Themagnetic moment, g (the strength of the nuclear di-pole), is conveniently expressed as a multiple of a unitcalled the nuclear magneton and defined as eh/4rwMc,where e is the electronic charge; M, the mass of theproton; and c, the speed of light. Nuclear magneticmoments are, in general, of the order of magnitude ofone nuclear magneton. The intrinsic magnetic mo-ment of the electron, by contrast, is eh/4mc, wherem is the electronic mass. Thus, nuclear magneticmoments are roughly 1,000 times smaller than elec-tronic moments.The spin number, I, of a given nuclear species and,

so far as we know, the magnetic moment, [t, are per-manent, sharply defined characteristics of the nucleus,not to be altered by any means short of raising thenucleus to an excited state, a comparatively violentprocess with which we shall not be concerned.The properties of spin and magnetic moment were

first attributed to nuclei in 1924, by Pauli, to accountfor the so-called hyperfine structure of lines in thespectra of certain elements. Such a line, analyzedwith a spectrograph of high resolving power, had beenfound to consist of a very closely spaced group oflines, or multiplet-an effect which could be explainedby taking account of the interaction of the nuclearmagnet with the magnetic fields arising from themotion of the electrons in the atom. The recognitionby Goudsmit and Uhlenbeck of the intrinsic spin andmagnetic moment of the electron itself, and the sub-sequent rapid progress in the correlation of atomicspectra with atomic structure, opened the way to acomplete quantitative interpretation of hyperfine struc-ture. The magnitude of the nuclear angular momen-tum and magnetic moment could, in turn, be deducedfrom the results of hyperfine structure measurements.To determine the nuclear magnetic moment by this

means one must measure the separation in frequencyof exceedingly closely spaced spectral components.Very precise values can seldom be obtained, despitethe remarkable advances in high-resolution spectros-copy which this challenging problem has called forth.The nucleus is an extremely feeble magnet, and thisis the cause of the difficulty. The energy differences

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involved in the interaction of the nuclear magnet withthe surrounding electrons, or with an applied magneticfield, are usually not larger than 10-20 to 10-18 ergs.According to the Bohr relation, AE=hv, the corre-sponding frequencies lie in the band of ordinary radiofrequencies or at most, in a few extreme cases, in theregion of centimeter-wave lengths.The possibility is thereby suggested of measuring

such weak interactions directly, by detecting the ab-sorption or emission of waves of radio frequency.Just this has been done in the celebrated molecularbeam experiments of Rabi and his students at Colum-bia (7). By this form of radio-frequency spectros-copy,1 augmented in some cases by the more recent-although in many respects more conventional-radio-frequency methods to be described presently, the mag-netic moments of nearly 40 nuclear species have beendetermined with high precision. Hyperfine structuremeasurements in the optical region have establishedthe spins and, at least approximately, the magneticmoments of about 35 additional species. There areleft, among the stable nuclei which would be expectedto possess magnetic moment, more than 40 which arestill unmeasured.The spins and magnetic moments of several nuclear

species are listed in Table 1. Most of the entries in

TABLE 1

NUCLEAR SPINS IN UNITS OF h/2ir, AND NUCLEAR MAGNETICMOMENTS IN UNITS OF THE NUCLEAR MAGNETON, ls,

OF THE NEUTRON, THE TRITON, AND THESTABLE NUCLEI THROUGH 016*

NucleusI 11/11o Nucleus I g/go

on, 1/2 -1.910 ABeg 3/2 -1.1761Hl 1/2 2.7896 5B1O 1 0.598.1H2 1 0.8564 eB" 3/2 2.687IHS 1/2 2.9756 oC12 0 021e3 1/2 ? sC3 1/2 0.701sHe' 0 0 7Nl4 1 0.403sLi 1 0.8214 7N15 1/2 0.280aLI7 3/2 3.2535 s016 0 0

The sign of the magnetic moment refers to the polarityof the nuclear dipole with respect to the direction of theangular momentum vector.

this abbreviated table originated in the Columbiamolecular beam laboratory. The magnetic moment ofthe neutron deserves special mention. This was firstmeasured by Alvarez and Bloch, at Berkeley, by aningenious adaptation of molecular beam principles;the value listed was obtained more recently by Arnoldand Roberts, at the Argonne Laboratory, by a refine-ment of the same method. The precision with whichnuclear magnetic moments can now be compared is

1 The rapidly growing field of radio-frequency and micro-wave spectroscopy, which has applications much broader thanthe scope of the present article, has been surveyed by RobertsIn Nucleonics, October 1947.

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not fully displayed in the tabulations. For example,Bloch has recently redetermined, at Stanford, theratio of the magnetic moment of the proton 1H' to thatof the deuteron 1H2, obtaining 3.257195 ± 0.00002.The magnetic moment of the triton 1H3, a novel par-ticle still unavailable in most laboratories, is equallywell established.The table discloses no obvious systematic variation

of the magnetic moment, it, with charge and massnumber. So far as the spin is concerned, two generalrules can be stated: (1) The spin number, I, is an oddor an even multiple of A as the mass number, A, isodd or even; this is a direct consequence of the half-integral spins'of the nuclear constituents, the neutronand the proton. (2) A nucleus that contains an evennumber of protons and an even number of neutrons,such as .01s, or 38Sr88, has zero spin and moment; thisis an empirical rule to which no exception has yet beendiscovered. Further than this, the task of relatingquantitatively the magnetic moments of nuclei to thedetails of nuclear structure remains one of the largerproblems of theoretical nuclear physics. A beginninghas been made among the light nuclei (2), but be-yond the limits of Table 1, roughly, even an approxi-mate theory is lacking. The depth of the problem, aswell as the inadequacy of the spinning, charged sphereas a model of the nucleus, is suggested by the fact thatthe neutron, an uncharged "elementary" particle, hasa magnetic moment!

Fortunately, we need not concern ourselves herewith the details of nuclear structure. In the experi-ments to be described, the nucleus can be regardedmerely as a carrier of magnetic moment and angularmomentum to which the rules of quantum mechanicsapply. We shall be interested in the interaction ofthe magnetic nucleus with its surroundings and with anexternally applied magnetic field. It will not seriouslyrestrict the discussion if we confine our attention to asingle nucleus, the proton, which has, according toTable 1, the intrinsic angular momentum A (h/2fr) anda magnetic moment amounting to 2.7896 nuclearmagnetons.

REsoNANCE ABSORPTIONThe emission or absorption of light by a free atom

takes place in accordance with the resonance condi-tion which quantum theory permits us to express inthe following way: The possible states of an isolatedatom include a sequence of discrete states sharply de-fined with respect to energy; a transition of the sys-tem from one of these states to another, differing inenergy from the first by AE, is accompanied by emis-sion (or absorption) of light of the frequency v = AE/h(h = Planck's constant).

Consider, for the moment, a free proton in a mag-netic field of strength H0. Ignoring the translational

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degrees of freedom, which are here irrelevant, thissimple system will be found in one of two possiblestates which differ in the orientation of the spin axisof the proton with respect to the magnetic field. Thatis, the component, in the direction of the magneticfield, of the angular momentum vector can assumeeither of the values + h/47r and - h/47r. (For a par-ticle of spin I there are 2I + 1 possible states, or spinorientations.) The two states differ in energy, forthe component of nuclear magnetic moment in thedirection of the field is + A in one case and -, in theother. In fact, the difference in energy is just 2pfO,the work that would be required to reverse the direc-tion of a magnet of strength g which had originallybeen aligned with a field Ho. The two states will bedenoted by (+) and (-), the (-) referring to the stateof higher energy in which the nuclear magnet isdirected counter to the magnetic field. Actually, thespin axis of the nucleus lies in neither case parallel tothe direction of field Ho, but is tipped at a consider-able angle and precesses about the direction of thefield, like the axis of a tilted top. This does not com-plicate matters, for it is only the components of spinand magnetic moment parallel to H0 which concern us,and it is just these components which are specifiedby I and ,u.

tE = 2pHO -. *- _

Hot

(+) p

FIG. 1. The two energy states of a proton in a mag-netic field, Ho. The energy levels are drawn at the left.

The situation is illustrated in Fig. 1, in which theenergy level scheme appears. By exposing the systemto radiation of the proper frequency, it should be pos-

sible to bring about a transition from one of the twostates to the other, with the consequent absorption ofa quantum of energy, in a (+) -- (-) transition, or

emission of a quantum, in a (-) -- (+) transition.

In the former case, resonance absorption will haveoccurred; the latter process is usually called stimu-lated emission, although, as it is the exact counterpartof the former process, it might well be called reso-

nance emission. For a proton in a magnetic field, H0,

of 10,000 gauss, the resonance frequency, computed

SCIENCE, April 30, 1948, Vol. 107

from v = AE/h = 2pHo/h turns out to be about 41megacycles/sec. The radiation required is therefore aradio wave, or, since only the magnetic field of thewave would be effective anyway, merely an oscillatingmagnetic field such as one might make by passing aradio-frequency current through a suitable coil.

Transitions of the sort described are the basis ofthe magnetic resonance method of molecular beams.The nuclei involved are located in electrically neutralmolecules-a bare proton would be unmanageable forfairly obvious reasons. The problem is not seriouslycomplicated by the presence of the surrounding con-stituents of the molecule. The molecule itself movesthrough a highly evacuated chamber and remains ex-ceptionally free during its exposure to the radio-fre-queney oscillating field. It is therefore not surpris-ing that sharply defined resonance effects can be ob-served. It will be recalled that one of the conditionsessential for obtaining sharp lines in ordinary spec-troscopy is that the emitting atoms occupy a region ofrelatively low pressure. The frequency of atomic col-lisions sets a limit on the lifetime of an excited stateand hence (a familiar illustration of the uncertaintyprinciple) limits the precision with which the energyof such a state can be specified, with a consequentbroadening of the observed spectral line. This "col-lision broadening" becomes more pronounced as thepressure is increased until eventually the line struc-ture of the spectrum is obliterated and we observe thecontinuous emission spectrum characteristic of an in-candescent solid.

It might be expected that even lower pressures wouldbe necessary for successful radio-frequency spectros-copy, which is concerned with extremely small energydifferences. Low pressures are indeed necessarywherever atomic or molecular states are involved, apressure as low as 10-5 atmospheres being required toreduce the width of a spectral line to 100 kilocycles/sec. The nucleus, however, occupies a uniquely iso-lated and protected position in the interior of the atom.More important, the states (+) and (-) in Fig. 1 canbe perturbed only by magnetic fields, and in the major-ity of substances the electrons, which would otherwisecause strongly perturbing magnetic fields, are pairedoff in such a way that their net magnetic effect iszero. For this reason it has been possible to observesharply defined resonance absorption, due to transi-tions of the type (-) (-), in ordinary solids andliquids containing magnetic nuclei.When such a substance-a drop of water, for ex-

ample, or a lump of paraffin-is placed in a strongmagnetic field, it can be regarded in first approxima-tion as a mere collection of a large number of pro-tons, for each of which the two energy states, (+) and

are available. With the application of a weak

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radio-frequency magnetic field, oscillating at preciselythe frequency of resonance, protons which were inthe (+) state will make transitions to the (-) stateand 'vice versa. The vice versa is important; in fact,the "up" and "down" transitions are induced withequal probability, and if the protons were initiallyequally divided between the two states, no net absorp-tion of energy can occur. If the resonance effect is tobe detectable, an inequality of the population of thetwo levels must be established. This will come aboutif the magnetic nuclei can transfer energy to, and comeinto thermal equilibrium with, their surroundings, forin thermal equilibrium the lower of the two energylevels will be favored with the larger population. Atany easily attainable temperature the excess in thelower state is very slight; the ratio of the number ofnuclei in the upper state to the number in the lowerstate is the well-known Boltzmann factor, e-hV/kT which,for v = 40 Me/sec and T = 300° K, is 0.9999934. It isupon the fractionally minute surplus of nuclei in thelower state that the observed effects entirely depend.2We shall inquire later into the means by which thermalequilibrium is established, thereby discovering a rathersurprising situation. At present we shall assume thatit can be attained somehow, and turn to the detectionand interpretation of nuclear resonance absorption.The method of observation is extremely simple in

principle. Approximately 0.5 cc of water (for ex-ample), contained in a glass tube, is surrounded by acoil consisting of a few turns of copper wire. Thecoil is connected into a radio-frequency bridge circuitwhich is excited by an oscillator at, say, 40 Me/sec.The coil with its water "core" is located within thefield of a strong magnet. When the resonance condi-tion is attained by adjustment. of the frequency or ofthe strength of the magnetic field (usually the latter iseasier) the absorption of energy by the protons in thewater is manifested as a change in the apparent re-sistance of the coil. Because the effect to be detectedis extremely feeble, the apparatus is actually moreelaborate than the above description might suggest.However, a full description of the experimental tech-nique, which can be found elsewhere (5), is not neces-sary for the present discussion.The method outlined is essentially that used in the

first observation of nuclear magnetic resonance absorp-tion at Harvard in 1946 (10). At the same time andindependently, Bloch, Hansen, and Packard, at Stan-ford, demonstrated the closely related phenomenonwhich Bloch has named nuclear induction (3,4) . The

'Another consequence of the surplus in the lower state isthat the sample Is feebly magnetized; that is, It displaysstatic paramagnetic susceptibility. In a remarkable experi-ment performed In Russia In 1937, Lasarev and SchubnIkovsucceeded In detecting the paramagnetism caused by theprotons In solid hydrogen, at 21 K.

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two methods take advantage of somewhat differentmanifestations of the same fundamental process; theremay be areas of investigation where one or the otheris preferable, but both are coming into general use.The resonance absorption by protons in water is

displayed in Fig. 2, a reproduction of an oscilloscope

FIG. 2. Resonance absorption by protons in water,recorded on an oscilloscope.

trace obtained while sweeping through the resonanceline by varying the magnetic field, HE. The value ofH0 at resonance was 6,940 gauss in this experiment,the radio frequency being 29.5 Mc/sec. The totalrange covered in the picture is 2 gauss; thus, reso-nance absorption occurs within an interval of only 0.2gauss. The equivalent line width on a frequency scaleis about 1 kilocycle/see, or roughly 105 times smallerthan the width of a typical sharp line in the visiblespectrum.In other substances containing protons an absorp-

tion line is observed at precisely the same value of themagnetic field, if the radio frequency is the same.This was to be expected, for the resonance conditioninvolves only the specifically nuclear property, Mu/IHowever, the width of the line varies greatly from onesubstance to another. The proton resonance in ice,for example, is found to be 20-50 times as wide as,and correspondingly weaker than, the resonance shownin Fig. 2. In some crystals even more striking effectsare observed. The proton resonance line develops amore complicated shape, or even breaks up into sev-eral more or less well-resolved components. A fewtypical examples are sketched in Fig. 3; beginning withthe intense narrow line which is observed in water andin many other liquids. The peak intensity of the ab-sorption is inversely related to the line width, otherthings being equal; the quantitative examination ofthe broad, and therefore weak, lines illustrated inFig. 3 (b, c, d) calls for the utmost refinement in ex-perimental technique.

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These effects are not to be explained in terms ofnuclear properties alone, but are connected with theperturbing magnetic fields which originate in the im-mediate surroundings of a given hydrogen atom and

11

1 (b)1iA H ~0

/ 1 (c)~ he

(a) 1b ~ d)

t t ~~~~~Ho=IhV/2j"FIG. 3. The shape of the proton resonance absorption

line in four substances: (a) water; (b) Ice; (c)powdered gypsum, CaS04 2EHa0; (d) a single crystal ofgypsum, suitably oriented with respect to the magneticfield. The contrast between a typical liquid line (a)and the others Is actually much greater than the sketchsuggests; were the drawing to scale, (a) would beseveral times higher and narrower than shown above.

which depend on, and are modified by, the location andmotion of neighboring atoms. The, investigation thusleads back into the physics of the liquid and solidstates.

INTERPRETATION OF LINE WIDTH AND LINE STRUCTURE

The nuclei are, of course, not entirely isolated frommagnetic disturbances of local origin. The most ob-vious sources of perturbing magnetic fields are themagnetic nuclei themselves. A selected proton in adrop of water is subject not only to any field whichwe may apply with an electromagnet, but also to themagnetic fields arising from the magnetic moments ofneighboring protons in the liquid, including that ofthe other proton in the same water molecule. (Theoxygen nucleus, as Table 1 indicates, is not magnetic.)The sum of all such perturbing fields, measured at thelocation of a given proton-but, of course, excludingthe field of that proton-we shall call the "local field,"H1,,. The local field will vary in magnitude and di-rection, from point to point in the material. It mayalso vary in time. The nature of these variationsmust now be discussed.The magnetic field produced by a proton resembles

that of a short bar magnet. The strength of the field

SCIENCE, April 30, 1948, Vol. 107

decreases with the inverse cube of the distance. The"lines of force" surrounding such a magnetic dipoleare indicated in Fig. 4. The dipole can represent thez-component of the magnetic moment of a proton inthe state (-). Let us consider only the z-componentof the field produced by the dipole, a, at some nearbypoint P which might be the location of another proton.

z

FIG. 4. The magnetic field from a dipole, A, at anearby point, P. The magnitude of the component, H.,is (4/r9) (3 cos2 0-1), and the total field at P, In thez-direction, Is Ho + H.. In the cases considered here,Hecc Ho.

This component of the local field at P is given by

(Hgoc)=z (3 cos2 O- 1)where 0 is the angle shown in Fig. 4. Thus the totalmagnetic field, in the z direction, that would be ex-

perienced by another proton located at P., is Ho +

(3 Cos2 0-1). If the spin vector of the proton at theorigin had been directed downward rather than upward(as we have seen, the states (+) and (-) are almostequally probable) the field at P would have been

HO- "A (3 cog 0 -l).The "ocal field" term in each of these expressions

amounts to a few gauss, if g is the proton moment andr is a distance of the order of 10-8 em, whereas thefield, H0, is ordinarily several thousand gauss. Theeffect of the neighboring protons is thus to alterslightly, either increasing or decreasing, the total mag-netic field experienced by a selected proton, and weshould expect the resonance condition, for that par-ticular proton, to be modified accordingly to read:

v= 2(HO+ H100)/hwhere H10, represents the instantaneous sum of thefields of all its neighbors. Only the nearest neighborswill contribute significantly to H10E, because of the in-verse-cube dependence and the angular dependence ofthe dipole field.Although the picture which has been drawn is con-

siderably oversimplified, the gravest omission beingthe neglect of the effect of the precession of the nuclear

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dipole, it can serve to explain the main features of theresults shown in Fig. 4. Oddly enough, it is easiestto begin with the most complicated of the patterns,Fig. 3 (d), which has been observed and interpretedby Pake (8). The four components are observed in a

single crystal of gypsum (CaSO4 2H120) when thecrystal axis is oriented in a particular way with re-

spect to the direction of-the strong magnetic field, H0.

The only important magnetic nuclei in gypsum are

the protons in the molecules of water of hydration.The crystal structure of gypsum has been well estab-lished through the analysis of Wooster (12), but itwill be recalled that X-ray diffraction yields no directevidence of the location of protons in a crystal.Nevertheless, there is reason to assume that eachproton has one near magnetic neighbor, the other pro-

ton belonging to the HOH molecule, and that thenext nearest proton neighbor is considerably furtheraway. If these more remote neighbors are ignored,the situation resembles that in Fig. 4; we have iso-lated interacting pairs of magnetic nuclei. Angle 0in Fig. 4 is now the angle between the H-H line in an

HOH molecule and the magnetic field H0. As it turnsout, there are only two distinct H-H directions in a

gypsum crystal; that is, for a given orientation of thesingle crystal in the magnet, has the same value forhalf the molecules in the crystal. From the protonsin these molecules we should therefore expect a pairof resonance lines, symmetrically displaced to eitherside of the central position by an interval determinedby u, 6, and the interproton distance, r. (The doubletarises from the possibility of the partner being ineither the (+) or (-) state.) The remainder of themolecules, to which a different value of applies,should give rise to another pair of lines, thus fourlines in all. Each of these lines will be smeared outsomewhat by the hitherto neglected fields of the more

distant neighbors. This is just what is observed.Moreover, as the crystal is rotated in the apparatus,the separation of the components is found to vary

precisely in accordance with the factor 3 cos2 6-1.From these observations, interpreted with the aid ofa theory (8) more rigorous than we have given here,it is possible to determine not only the directions ofthe H-H lines with respect to the crystal axes, butalso, with a precision of about 1%, the distance be-tween the two protons in the HOH molecule-infor-mation which the X-ray analysis does not yield.

If the sample is a powder, rather than a single crys-

tal, all possible values of will occur at once. A sta-tistical analysis shows that the resulting superpositionof many pairs of lines should yield the double-humpedcurve which is actually observed in this case (Fig. 3,c). A measurement of the hump separation againpermits a determination of the interproton distance,

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but with less precision than before. The direction ofthe H-H lines in the crystal cannot be inferred in thiscase.When each proton is surrounded by several equally

near magnetic neighbors, no well-defined structure isobserved. One finds instead a single broad line, thewidth of which, expressed in gauss, is of the order ofmagnitude, u/t-3-hence, in general, several gauss.Such a situation is encountered in ice.We have now to account for the extremely narrow

line observed in water and other liquids. The essen-tial difference is that the geometrical configuration ofthe molecule and its immediate environment can nolonger be regarded as static. Instead, we must takeinto account the thermal motion of the molecule andits neighbors. The time during which the spatialorientation of a molecule of the liquid persists can beestimated from the molecular diameter and the viscos-ity of the liquid. In the case of polar liquids suchas water, more direct evidence is provided by the di-electric dispersion; it will be recalled that the per-sistence of molecular orientation plays a central rolein the Debye theory of polar liquids (6). In water atroom temperature, this characteristic time is of theorder of magnitude of 10-11 sec. -In a time substan-tially longer than this, not only will the H-H line in agiven molecule have tumbled into a new position, butthe configuration and identity of the nearest neighborswill very probably have changed. The local field,Hp,,, at the position of a given proton, is now a quan-tity which fluctuates in a random manner; the rapidityof fluctuation is very much higher than the resonancefrequency for nuclear absorption.

It might be supposed that this rapid Brownian mo-tion would completely obliterate the resonance phe-nomenon. On the contrary, the nucleus rides out thestorm like a well-balanced gyroscope on perfect gim-bals. In fact, a careful statistical treatment showsthat the very rapidity of the fluctuations reduces theirperturbing effect, the appropriate average value ofHz00 being very much smaller than its instantaneousmagnitude. The width thus predicted for the protonresonance in water is even smaller than that observed.The observed line width is caused by lack of perfecthomogeneity in the field of the electromagnet.

If the viscosity of the liquid is increased, as by low-ering its temperature, the local Brownian motion be-comes less rapid, the perturbations due to the local fieldbecome more effective, and the absorption line broad-ens. Proceeding to the limit in this direction, the localmotion is finally frozen out, and the line attains thewidth characteristic of a crystal. This whole range ofbehavior has been observed in glycerol, with results inquantitative agreement with the theory. Even in icea certain amount of local motion persists well below

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the freezing point, as other lines of evidence stronglyindicate (9), and the width of the proton line increasessomewhat, with decreasing temperature, in the range-50 C to -400 C.The nuclear resonance line width is thus a sensitive

indicator, in some circumstances, of internal motion ona molecular scale. By this means, Bitter and Alpertand others at the Massachusetts Institute of Technol-ogy have recently investigated the so-called x-pointtransitions in certain crystals. These transitions aremarked by anomalies in the specific heat and havebeen attributed by various investigators to the moreor less abrupt "freezing-in" of internal degrees offreedom associated with the rotation or oscillation ofmolecular groups. -Solid methane, for example, dis-plays a A-point in the neighborhood of 200 K. Alpert(1) found that the width of the proton line in methanedecreased markedly as the temperature of the solidwas elevated through the A-point and interpreted thisas evidence for the activation of a rotational ratherthan a vibrational degree of freedom. In deuteratedmethane, which displays two A-points, he found asimilar behavior at the upper X-point. Solid H~l andnatural rubber were also investigated.

NuCxAR RELAXATION

We return now to the question raised earlier: Howare the nuclear spins brought into thermal equilibriumwith the rest of the sample? It is seldom necessary toask this question about an atomic or molecular system.A few interatomic collisions, occurring within 10-1see or so, suffice to bring about equilibrium. So thor-oughly insulated is the nucleus, however, that in manysubstances several seconds are required for the popu-lations of the levels (+) and (-) in Fig. 1 to adjustthemselves into conformity with the Boltzmann factor.This time is called the thermal relaxation time of thenuclear spins. It is perhaps surprising to find a timeso long associated with any atomic process; as a mat-ter of fact, previous theoretical studies of this questionhad predicted even longer relaxation times than wereobserved. Two tasks were thus presented: (1) tomeasure the nuclear relaxation time in various sub-stances under many conditions; (2) to devise an ade-quate theoretical explanation of the phenomenon.Although the problem is by no means completelysolved, it is now possible to account rather satisfac-torily for the relaxation effects observed in liquidsand in a restricted class of solids (5). We can onlyindicate here the connection between nuclear rela'xa-tion and those features of the liquid or solid structurewhich have already been encountered in our discussionof line width.Exchange of energy-that is, "thermal contact"-

between the nuclear spins and their surroundings re-

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quires the existence at the nueleus of oscillating mag-netic fields that (a) satisfy the resonance conditionand are hence capable of inducing (+) -- (-) and(-) -> (+) transitions, and (b) originate in the ther-mal motion of the surroundings. Just such a mecha-nism is provided by the local Brownian motion which,as we have seen, causes a rapid and random variationof the local field produced by neighboring magneticnuclei. One has to turn to the theory of random proc-esses to compute the effective intensity, in the "spec-trum" of the fluctuating local field, of the particularfrequency required. One finds that the intensity issufficient to explain the effects observed, and that atheory can be constructed which successfully accountsfor the observed variation of nuclear relaxation timewith the viscosity of the liquid.

102

0

vw ALOHOL IC

0

x X GLYCER

10X

DEEBYE TIME IN SECONDS

Fso. 5. The thermal relaxation time for protons inethyl alcohol, glycerin, and ice, measured at 29 M~c/sec,plotted against the Debye relaxation time, obtainedfrom dielectric dispersion data. Note the logarithmicscales. The slope of the solid lines for alcohol and iceand the-shape of the solid curve for glycerin have been-drawn in accordance with the theory.

Again, the important parameter is the characteristictime for reorientation of a molecule and redeploymentof its neighbors, which, as noted earlier, is closely re-lated to the characteristic time in the Debye theoryof dielectric dispersion. In Fig. 5 the observed nuclearrelaxation times, in ethyl alcohol, glycerin, and ice, areplotted against the "Debye characteristic time" ob-tained from published data on dielectric dispersion.Each of the points belonging to one of the curves wasobtained at a different temperature. The curves aredrawn in accordance with theory. The agreement isas good as one could expect, and it can be said withsome confidence that the main features of the processof nuclear relaxation in these substances are under-stood.

So far as the structure of the substance is con-cerned, one learns from these curves only what couldbe learned from measurement of the resonance linewidth or, forgetting about the nucleus, from measure-

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ments of dielectric dispersion. However, the relaxa-tion time can be measured even when the true linewidth is too small to be resolved, and the method ap-plies equally well to nonpolar liquids which would notshow dielectric dispersion.One of the most interesting problems encountered

in the study of nuclear relaxation is still unsolved.In many crystals the nuclear relaxation time appearsto he much too short to be explained by any existingtheory. The most striking evidence of the inadequacyof existing theories comes from the work of B. V.Rollin and collaborators at Oxford, who have investi-gated nuclear relaxation at very low temperatures(11). They report a relaxation time of the order of5 see for Li7 nuclei in lithium fluoride, at a tempera-ture of 21 K. The mechanism which brings about sorapid an exchange of energy between the nuclear spinsand their environment, at this low temperature, is un-known. The thermal vibration of the nuclei in thecrystal lattice, even at room temperature, is not vio-lent enough to explain a relaxation time as short asthis. It is difficult to imagine any other internal de-gree of freedom in a crystal of this type.

CONCLUDING REiMARKS

Other lines of investigation, which could not betraced in detail here, have been followed far enoughto suggest interesting applications of nuclear magne-tism to problems of structure. The interaction be-tween nuclear magnetic moments and the electronicmagnetic moments of paramagnetic ions in solution,studied in a preliminary way by Bloembergen (5),depends upon the distance of closest approach of amagnetic ion to the nucleus in question, and henceupon the size and permanence of any complex sur-rounding the nucleus. R. V. Pound has very recentlyfound evidence of interaction between the interatomicelectric fields in a crystal and the electric quadrupolemoment of a nucleus-a property of some nuclei

which has been ignored in this article. It is too earlyto foresee the significance of this development.To survey a field at such an early stage in its de-

velopment is doubtless a rash venture. In the processof simplifying and condensing the statement of theproblems, one runs the risk of creating an impressionof tidiness which the actual state of affairs does notjustify. More serious, perhaps, is the danger of exag-gerating the importance of the information which thenew techniques make available and the ease with whichit can be obtained. On this last point, it should besaid that the quantitative measurement of relaxationtime and line shape is not easy. Nuclear resonanceabsorption is, after all, a feeble effect. The peak in-tensity of absorption in ice is such that a radio wave ofthe resonant frequency would be attenuated to half itsamplitude only after traversing some hundred kilo-meters of solid ice. The data recorded in Fig. 5 wereobtained from a sample weighing less than 1 gm. De-spite this difficulty, I think we can look forward tothe profitable development and extension of the fieldwhich these early experiments have opened.

References1. ALPERT, N. L. Phys. Rev., 1947, 72, 637.2. BETHE, HANS A. Introduction to nuclear theory. New

York: John Wiley, 1947. Chap. VII.3. BLOCH, F. Phys. Rev., 1946, 70, 460.4. BLOCH, P., HANSEN, W. W., and PACKARD, M. Phys.

Rev., 1946, 69, 127.5. BLOBMBERGEN, N., PURCELL, E. M., and POUND, R. V.

Phys. Rev., 1948, 73, 679.6. DEBYi, P. Polar molecules. New York: Dover, 1945.

Chap. V.7. KsLLoGG, J. M. B., and MILLMAN, S. Rev. mod. Phys.,

1946, 18, 323. (This article reviews molecular beammethods and results.)

8. PAKI, G. E. J. chem. Phys., April, 1948.9. PAULING, LINUS. The nature of the chemical bond.

Ithaca, N. Y.: Cornell Univ. Press, 1940. P. 302 if.10. PURCELL, E. M., TORREY, H. C., and POUND, R. V. Phys.

Rev., 1946, 69, 37.11. ROLLIN, B. V., HATTON, J., COOKE, A. H., and BENZIE,

R. J. Nature, Lond., 1947, 160, 457.12. WOOSTER, W. A. Z. Krist., 1936, 94, 375.

SCIENCE, April 30, 1948, Vol. 107440

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Nuclear Magnetism in Relation to Problems of the Liquid and Solid StatesE. M. Purcell

DOI: 10.1126/science.107.2783.433 (2783), 433-440.107Science 

ARTICLE TOOLS http://science.sciencemag.org/content/107/2783/433

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