Contains reports on two research projects

Bitter, F.

Kohler, R. H.

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V. NUCLEAR MAGNETIC RESONANCE AND HYPERFINE STRUCTUREProf. F. Bitter Dr. P. L. Sagalyn R. F. LaceyProf. L. C. Bradley III B. B. Aubrey E. LustigProf. J. S. Waugh R. W. Fessenden A. C. MelissinosDr. S. P. Davis R. H. Kohler S. T. Wray, Jr.A. ATOMIC-BEAM LIGHT SOURCESStudies of nuclear structure based on the nuclear perturbation of atomic energylevels may be divided into two classes; one class employs microwave techniques tomeasure the hyperfine structures directly, the other uses optical techniques in whichthe desired structures are inferred from differences in radiated optical frequencies.This report is concerned with possible improvements in the optical techniques whichgive promise of becoming an important supplement to the currently popular microwave-absorption experiments.In conventional optical experiments, the detail of the hyperfine structure is limitedboth by the Doppler broadening of the radiated line and the resolving power of the appa-ratus. This note concerns the first problem: the reduction of the Doppler broadeningof spectral lines radiated from a vapor. It should be pointed out that even if it provesdifficult to push the resolving power of interferometers as far as might be desirablefor the analysis of some very sharp lines, nevertheless the magnetic-scanning tech-niques developed in the M. I. T. Magnet Laboratory can be used to achieve a resolvingpower which is limited only by the width of the lines that are radiated or absorbed by abeam of atoms.An atomic-beam light source is placed in a magnetic field, and the light which itradiates is then passed through an atomic beam absorber. The value of the scanningfield for which the radiated frequencies are absorbed by the normal unperturbed atomcan be used to calculate the frequency differences not only between the hyperfine com-ponents of a single atom, but also between lines radiated by various isotopes. Anapparatus for this purpose is being built and will be reported on later. For certainlines, effective resolving powers of the order of 108 , or more, seem possible.In Fig. V-1 the Doppler width of lines radiated at room temperature by atoms withmass numbers in the range of 25 to 200 are plotted as a function of frequency or wave-length. It will be seen that resolving powers in the vicinity of 106 are required tomatch the Doppler widths. The ratio of the Doppler widths to the classical widths ofallowed electric dipole radiation is indicated by the straight lines and the scale on theright. It will be seen that the Doppler widths range from 15 to 150 times the naturalwidths, and therefore that resolving powers from 107 to 109 would be required to extractall the information about nuclear structures contained in these lines. For some for-bidden lines, even greater resolving powers would be needed. Since even this proce-dure will not give the detail that can be obtained on ground states with microwave(V. NUCLEAR MAGNETIC RESONANCE)170150DOPPLERCLASSICAL10050I I I I I I I50,000CM 30,000 20,000CM-' 15,000 12,500CM-25,000 AUFig. V-1. Chart showing Doppler and classical natural linewidths of spectrallines radiated from atoms of mass number A at room temperature.measurements, it seems important to extend the observations to other energy levels,particularly to the study of isotope shift and nuclear-charge distribution.While the use of atomic beams to reduce Doppler width is an old story (1), thereseems to be no thorough discussion of the main drawback of atomic-beam light sources,that is , of their low intensity. To increase the light intensity coming from a givensurface area in a specified solid angle, we may either increase the depth towhich radiating atoms are present (up to the point where self-absorption of the centerof the line broadens it) or increase the density of atoms in a small depth. Both of thesemethods are useful for different spectral lines and different optical instruments.A system of beams for a deep source and for an absorption cell is shown in Fig. V-2and Fig. V-3. In the lamp shown in Fig. V-2 a heated mercury container is thermallyinsulated from the refrigerated walls of a system of flat narrow passages. Quartzlamps illuminate the beam emerging from the slits. The absorbed and re-emittedresonance radiation is observed perpendicular to the aperture at A. A 1/200 degreeof collimation is attempted. The construction of absorber and emitter are similar, inthat thin flat mica spacers 0. 001 inch by 1 inch by 1 inch are held apart by stainlessAPERTURE FOR LIGHT BEAM WITHREDUCED DOPPLER WIDTHATOMIC BEAM IS EXCITED BY RESONANCERADIATION FROM THE SIDESREFRIGERATEDSECTIONHEATED MERCURYCONTAINER -PREVENT CLOGGINGSTAINLESS STEEL.OO001 MICA0.005 STAINLESS STEELCOPPER BAFFLE" ALTERNATE MICA AND STAINLESSSTEEL -ONE STACK,I REFRIGERATED SLITS "F"THERMAL INSULATION FORM ATOMIC BEAMSFig. V-2. Schematic diagram and photograph of atomic-beam resonance lamp.(V. NUCLEAR MAGNETIC RESONANCE)Fig. V-3. Two views of the atomic -beam absorption cell, partly assembled.Four intersecting beams are used. The light to be absorbed passesaxially through the central hole.steel pieces 0. 005 inch by 0. 25 inch by 1 inch. Baffles sop up most of the transverselymoving mercury before it reaches the narrow collimating slots. It is hoped to attaindensities in the useful portions of the beams of the order of 109 and to keep the appa-ratus in operation for many hours.In applications which do not require a great depth of beam it is necessary to raisethe atomic density to as high a value as possible. A simple argument, given below,leads to a useful conclusion. The maximum density of atoms in a beam at a distancer from an oven hole of radius R << r is proportional to the angle of collimation 4 = Ro/r,and inversely proportional to r.N (1)r 28 rrr rHere w-2 is the atomic collision cross section. The density of atoms in the oven shouldbe such that the mean free path of atoms in the oven is comparable to R . This expres-sion is only intended as an order-of-magnitude guide. For = 1/50, r = 3 X 108 cm,r = 10 cm, we find that nr = 10 11/cm3 , a number slightly smaller than the vapor pressureof mercury at -25*C. To prevent clogging in a single beam of the kind that is envisioned,a construction like that shown in Fig. V-4 may be used._ - - -- de ~cr~r~- --_ _~ __~lfY(V. NUCLEAR MAGNETIC RESONANCE)BEAMHOT UPPER APERTUREHEATED OVEN WITHHOT APERTUREFig. V-4. Proposed design for prevention of cloggingof apertures in high-density beams.EFFECTIVEOVEN 2RoFig. V-5. Definition of symbols: R is that distance from the oven openingat which LR = 10R; no is the number of atoms/cc in the oven;L is the mean free path of atoms in the oven.The argument leading to Eq. 1 is as follows. For a sufficiently small hole in anoven, the number of atoms escaping per second is simply the number that would havestruck the area of the hole when it was closed.N = n VTR2/4 atoms/sec (2)no = number of atoms/cc in the ovenV = mean velocity of atoms in the ovenR = radius of the circular hole in the ovenoKnudsen (2) has shown experimentally that Eq. 2 is satisfied for the range of pressuresin which the mean free path in the oven Lo is greater than 10 R . As the oven pressure0 0(V. NUCLEAR MAGNETIC RESONANCE)is increased, the rate of efflux increases by a factor of 2 over that predicted in Eq. 2as Lo approaches Ro. Finally, as the pressure is further increased to values ofLo = Ro/10, the rate of efflux is still approximately twice that predicted by Eq. 2. Weneglect this small variation, and assume that Eq. 2 is applicable to the ranges of ovenpressures and apertures that are of interest.In the range of oven pressures required for high-intensity beams, we shall assumeLo < Ro, and we define R as the distance from the aperture at which collisions maybe neglected, which is the point at which LR = 10 R. (See Fig. V-5.) The approximateexpressions for LR and Lo are1L R 2- 10R (3)nR T1 (4)o 2 (4)n 7wrG-oAssuming that for r = R we already have a transport velocity of the order of the meanvelocity in the oven, and a more or less uniform distribution over a hemisphere, wemay setN =n V TR /4 = n 2wrZO O o rn R 2nr = (5)and, therefore, from Eq. 3 and Eq. 4, we obtainLR 10R no 0 R (L L Ro o RR1=R 0' L0 (6)o= LIn order to check the validity of our assumptions (the assumption for r > R collisionsmay be neglected), approximate calculations of the rate at which atoms leave a cylinderof radius R and length r - R through the end face, and through the sides as the resultof collisions, were undertaken, These calculations indicate that if factors of the orderof 2 are not important, then the value for R given in Eq. 6 may be adopted.The result follows immediately if we define an effective angle of collimation ' asR/(r-R), or, since R << r, as $' = R/r. We have, from Eq. 5 and Eq. 6,(V. NUCLEAR MAGNETIC RESONANCE)2 no 2n R R8r TT L 8 Z01 R o(7)For maximum beam intensity, r should be small, and, therefore, the pressureshould be low enough so that R R = L . Equation 7 then reduces to the desiredexpression (Eq. 1).An apparatus is being built to test this expression. It is hoped to extend the inves-tigation presently to include the shaped nozzles suggested by Kantrowitz and Grey (3)and tested by Kistiakowsky and Slichter (4).F. BitterReferences1. Cf. K. W. Meissner, Revs. Modern Phys. 14, 68 (1942) for an introduction to thesubject.2. M. Knudsen, Ann. Physik 28, 999 (1909).3. A. Kantrowitz and J. Grey, Rev. Sci. Instr. 22, 328 (1951).4. G. B. Kistiakowsky and W. T. Slichter, Rev. Sci. Instr. 22, 333 (1951).B. MULTIPLE QUANTUM TRANSITIONS IN MAGNETIC-MOMENT INTERACTIONSThe following report is a summary of a paper that is being prepared for submissionfor publication.Transitions between two neighboring Zeeman levels induced upon a system by asteady magnetic field are commonly produced with an additional rf field of a frequencyequal to this separation. Such transitions, however, have also been observed when theradio frequency equals one-half, one-third, one-fourth, and so on, of this same energyseparation, and explanations for such observations have been made (1, 2, 3).Among other things, our own investigations show the following: All of these reso-nances are deduced from classical theory, which gives a very good approximation tothe complete time-dependent behavior of a Larmor system in a general rf field ofone frequency. The experimental effect of the higher-order resonances may be aslarge as that of the first-order resonance. Moreover, once the classical behavior of(V. NUCLEAR MAGNETIC RESONANCE)the system is known, the quantum mechanical result can be written immediately, andbe easily evaluated.1. Classical Formulation of the ProblemConsider a Larmor system, which is governed by the familiar equations of motion:d - .dt J =y J XX= J X Hwhere J is the total angular momentum; M is the total magnetic field, which may betime-dependent; y is the gyromagnetic ratio; and H = y M.It is convenient to employ the well-known procedure (4) of viewing J in a rotatingcoordinate system specified by w(t). The equations of motion become[dJ]o= JX(H+w)= JXHdt e[dJ]where - o is the rate of change of J relative to the rotating system, and H (thedteffective magnetic field in the rotating frame) = H + w.2. Classical Solution of Particular Problemsa. Stationary magnetic field: H = H k, - H 0o dt oFor w = -H k, J is at rest, or J precesses counterclockwise about the z-axisviewed from positive z to negative z. This motion may be written symbolically interms of a classical rotation operator, Rz(0), which, acting on a space vector, rotatesthat space vector about the z-axis by an angle, 0. Thus,J(t) = Rz(-Hot) J(0)b. Stationary magnetic field along the z -axis and rotating fieldin the xy-plane: H = H k + R (-2't) H 2 i (See ref. 4.)By setting o = -w' k, or, equivalently, by applying R (C't) to all vectors, we findH e = (H - ') k + H2 iSince the effective field is now stationary, the results of section a can be applieddirectly, to giveJ(t) = Rz(-w't) RH (-H't) (O)e(V. NUCLEAR MAGNETIC RESONANCE)whereH' = [(Ho w) 2 + H] 1/tanO - H -00 = angle between He and k.In other words, there is precession of J about He and precession of He about thez-axis. When w' = Ho, Jz changes rapidly to give a resonance. The time, T, for acomplete cycle in Jz isS2TrT = H 2c. Stationary field plus sinusoidal field along the z -axis and rotating fieldin the xy-plane: H = Ho0 k - H 1 cos(w't) k + Rz(-w't) H2 iBy suitable rotation about the z-axis, the term -H 1 cos(w't)k can be eliminatedfrom the effective magnetic field. In the new frame, the field in the xy-plane rotateswith an angular velocity that is not uniform. This field can be decomposed by Fourieranalysis into an infinite sum of fields, each rotating with a different angular velocitythat is always an integral multiple of w'. Thus there will appear fields rotating at thefrequencies w', 2w', 30', 4w', and so forth. Each field can produce a resonance if itsfrequency of rotation is near the Larmor frequency, H o . Thus there will be resonancesfor w' = H ', 2  = H , 3w' = H , and so forth.First let w = H 1 cos(w't)k, or, equivalently, apply to all vectors Rz(a), where-Ha -= sin(w't)ThenHe = Hok + R (a) R (-o't) H 2 ioo= H k + DR Rz(nw' t) H in=-ooIt can be shown thatS(H (n+l) (n+1)n \Z'/ (n+l)!(V. NUCLEAR MAGNETIC RESONANCE)D (n-) >+1-n 2W' (n-1)!Section b shows that the effect of a rotating field is appreciable if and only if thefrequency of that rotation is near the Larmor frequency, H . Thus, if we now assume0that n' = Ho, then He can be written approximately asH = Hok + R z(-nw't) D_n H iThis magnetic field has the same form as that with which section b begins, withw't replaced by nw't, and Hz replaced by D_nH2. Thus the results of that section canbe taken over directly, if the effect of the rotation Rz(a) is included. Hence,J(t)= R (-nw't) R (-a) R - (-H' t) J(0)z z H neewhereH' = (Ho - nw) 2 + (D H 2 )2n -nHtanO = Dn -n H - w'0on = angle between Hee and kAs before, the time for a complete cycle in Jz near a resonance of order n is2Trn D H 2-n 2Thus for this special case, the classical theory predicts resonances for nw' = Ho,where n is any positive integer.Many magnetic-resonance experiments measure the extent to which magnetic-moment orientation is destroyed. Such destruction occurs in the first-order resonancewhen To becomes smaller than the relaxation time of the system. Larger values ofH2 do not increase the height of the resonance. Accordingly, if T becomes smallerthan the relaxation time of the system, the first n resonances should appear with equalheights.d. General case: steady field and arbitrary rf fields of one frequencyThe general case exhibits multiple resonances classically, but it will not be dis-cussed here.(V. NUCLEAR MAGNETIC RESONANCE)3. Quantum Mechanical Formulation of the ProblemOnce a classical solution is obtained, as in the examples above, the quantum mechan-ical solution can be written immediately, and yields the same behavior as the classical.R. H. KohlerReferences1. J. Margerie and J. Brossel, Compt. rend. 241, 375 (1955).2. J. Winter, Compt. rend. 241, 375 (1955).3. J. Winter, Compt. rend. 241, 606 (1955).4. N. F. Ramsey, Nuclear Moments (John Wiley and Sons, Inc., 1953), Sec. 3D.

Nuclear Magnetic Resonance and Hyperfine Structure

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Nuclear Magnetic Resonance and Hyperfine Structure

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