Contains research objectives and reports on four research projects

Bitter, F.

Bradley, L. C., III

Fessenden, R. W.

Hull, Robert J.

Kohler, R. H.

Lacey, R. F.

Stroke, H. H.

Waugh, J. S.

DSpace@MIT

VIII. NUCLEAR MAGNETIC RESONANCE AND HYPERFINE STRUCTUREProf. F. Bitter F. W. Dobbs I. G. McWilliamsProf. L. C. Bradley III H. R. Hirsch E. C. PenskiProf. J. S. Waugh R. J. Hull G. H. PriceDr. H. H. Stroke R. H. Kohler W. W. SmithDr. J. F. Waymouth R. F. Lacey C. V. StagerB. B. Aubrey Mrs. Ilana Levitan W. T. WalterM. Ciftan K. K. Li S. T. Wray, Jr.J. H. LoehlinRESEARCH OBJECTIVES1. Nuclear Magnetic ResonanceWe are still concerned with the general problem of studying the structures and inter-nal dynamics of materials by observing perturbations of the magnetic resonances ofnuclei embedded in them. The direct interaction of nuclear magnetic moments in solidshas given us some information about crystal structures and molecular motion in crystals.We are preparing to carry out pulsed radiofrequency experiments which will provide amore powerful tool for studying these and related effects. Most of our recent researchemphasis, however, has been on the study of electronic perturbations of resonances inrapidly tumbling molecules in liquids, and particularly on the phenomenological aspectsof these interactions and their interpretation. Since it has also been found that nuclearresonances in molecules can provide a good deal of information about the problem ofrestricted internal rotation about single bonds, we are continuing studies of the ener-getics of this kind of process.J. S. Waugh2. Hyperfine StructureThe purpose of this work is to learn about nuclear structure from detailed inves-tigations of atomic energy levels. Atomic electrons actually penetrate the nucleus inthe course of their orbital motion, and the position of the various possible energy levelsis influenced by the electron-nuclear interaction.The investigations of this group are concerned with changes in the energy levelsproduced by reorientation of the nucleus with respect to the atomic angular momentum,and with accurate comparison of energy levels of isotopes, both stable and unstable.A combination of radiofrequency or microwave and optical techniques is being used.Previously reported investigations of the isotopes of mercury are being extended.F. Bitter, L. C. Bradley IIIA. NUCLEAR MAGNETIC RESONANCE1. Nuclear Spin Interaction in MoleculesIt is generally accepted that "high-resolution" nuclear resonance spectra of mole-cules, which offer a fruitful approach to studies of molecular structure, can be inter-preted on the basis of the phenomenological spin Hamiltonian-1 -.h d = V (1 - 6.) I . + A.. I. * Ii i j>i(VIII. NUCLEAR MAGNETIC RESONANCE)in which v is a reference Larmor frequency, 5. is a "chemical shift" arising from field-O 1induced electron currents, and A.. is a constant describing the coupling of nuclei i and jthrough hyperfine interactions with their common electron distribution. When I Aij ismuch less than Ivo(5 i - 6.j) for all i and j, the problem can be treated by perturbationtheory, and a particularly simple spectrum results, from which the A's and 5's can beextracted by inspection. When this is not the case, the evaluation of these parameters,which are of considerable current interest in molecular theory, may become very diffi-cult. The customary approach has been one of repeated explicit diagonalization of ' byusing trial parameters until a spectrum is obtained that matches the theoretical spec-trum. Even with as few as three interacting spins (six independent parameters) thisprocess may be very laborious unless the trial parameters are chosen by inspiration.We have been concerned with simplifying this procedure along two lines: (a) intelligentchoice of the starting representation in order to make maximum use of existing sym-metries, and (b) actual inversion of the calculation, so that the A's and 6's appear asthe output of a calculation of which the experimental spectrum is the input. The secondtreatment, applied specifically to systems of three spins, but extendible in principle tolarger systems, will be described in detail in an article to be submitted to the Journalof Chemical Physics. It may be broadly outlined as follows:a. The fact that . commutes with the total z angular momentum and the invarianceof Tr(') result in a set of interval rules for spectral lines which enables the construc-tion of the set of eigenvalues from the spectrum.b. These same facts, together with the invariance of Tr(-2) and Tr(3 ), make itpossible to obtain from the eigenvalues a semiunique set of diagonal elements of dC inthe representation in which the spins are independently quantized.c. In this representation the desired A's and 6's are then easily obtained as defi-nite linear combinations of the diagonal elements.This method has been applied to a number of representative cases, with great savingin labor over the customary procedure. A number of interesting features are displayedwhen the problem is attacked from this point of view, among them is the fact that theremay be as many as 16 independent sets of A's and 8's which are consistent with a givenset of spectral line positions.J. S. Waugh2. Hindered Internal RotationIn sufficiently unsymmetrical molecules it is, in general, possible to observe in thehigh-resolution nuclear resonance spectrum, characteristic complexities that are relatedto hindrance of rotation about a single bond (1). We have made careful measurementsof the temperature dependence of the spectrum in two cases, and we have interpretedthem on the basis of a simple model to obtain the energy difference between two adjacent(VIII. NUCLEAR MAGNETIC RESONANCE)potential minima. In CH Br-CF Br , AE is found to be 0 ± 400 cal/mole, as comparedwith a result of approximately 700 cal/mole from an equally simple treatment based onthe infrared spectrum (2). In CHCiBr-CF 2 Br , which has not been studied by any othermethod, we find that aE is 2200 + 700 cal/mole. A more sophisticated treatment awaitsa firmer understanding of the variation of spin-spin coupling constants A.. with internalconfiguration.R. W. Fessenden, J. S. WaughReferences1. J. S. Waugh, Nuclear resonance spectra of hindered internal rotors, QuarterlyProgress Report, Research Laboratory of Electronics, M.I. T., Oct. 15, 1957, p. 22.2. R. E. Kagarise, Rotational isomerism in 1, 2-dibromo-2, 2-difluorethane, J. Chem.Phys. 24, 1264L (1956).B. HYPERFINE STRUCTURE OF RADIO THALLIUMThe caption of Fig. VII-2 in Quarterly Progress Report No. 51, page 52, should haveread: "From left to right: the first, third, and sixth lines are T199 ; we tentatively assignthe second, fifth, and seventh lines to natural thallium, and the fourth line to Tl200.,,R. J. Hull, H. H. StrokeC. THE g-FACTOR OF THE 5s 5p 3 p LEVEL OF CADMIUMThe g-factor of the 5s 5p 3 p level of cadmium has been determined by comparingthe magnetic field at which Amj transitions induced by a 50-mec oscillating magneticfield occurred in cadmium with the magnetic field at which the same transitions occurredin mercury, for which the g-factor of the 6s 6p 3P1 level is known. The magnetic fieldswere compared by measuring the currents in the Helmholtz coils which produced them.The result is g = 1.4975 ± 10. The value predicted by theory (1) is g = 1.4986 ± 6.R. F. LaceyReferences1. R. F. Lacey, The relation between the g-factors and the lifetimes of excited statesof two-electron atoms, Quarterly Progress Report No. 51, Research Laboratory ofElectronics, M.I.T., Oct. 15, 1958, p. 53.D. HEAVY ABSORPTION OF A SPECTRAL LINEWe shall calculate the transmission of light from a single Doppler-broadenedspectral line passing through a gas with a Doppler-broadened absorption resonance in(VIII. NUCLEAR MAGNETIC RESONANCE)the limit of small transmission.The relative transmission, , can be expressed (1) as an integral:, ,y) = 1/2 C exp - . ex + P(x+y)2 dx (1)The following symbols will be used:[1] = the incident line[2] = the absorbing lineT = absolute temperatureM = gram molecular weightv 1 = center frequency of line [1]n = density of molecules in the absorbing gask = path length of light in the gas0- = cross section for absorption of the center of absorption resonancev - v ) C2 1/2Sv 2RTzY v2 ZRT 2TP M1 V2 M V4 = rnThe term p. exp(-x 2 ) is the Gaussian-shaped absorption coefficient; the factorexp[- exp(-x )] represents the transmission of the gas as a function of frequency. Thesecond factor, exp[-P(x+y)2], represents the Gaussian frequency distribution of the inci-dent light. The factor p measures the difference in widths of the two Gaussian lines,and y measures the difference in their center frequencies. This frequency differencecould have two causes. First, the incident line could arise from one pure isotope, andthe absorbing line from another at a slightly different frequency. Second, small fre-quency changes can be produced either on the light source or on the absorber by meansof magnetic fields.In the evaluation of p, we first consider the special case, y = 0. Then((, p) = 2 00 exp e - pxdx (2)The integrand has a maximum at(VIII. NUCLEAR MAGNETIC RESONANCE)1/2Let2 2x =a +zThen( p111/22-aexp[-p e -Z - pz] dz(a2 + z)1/2since . exp(-a ) = , and exp(-pa2) = (p/The term exp[-p e-z - pz] has a maximum for z = 0, and falls off rapidly to zero forlPzI > 1. Then, if a1/2~ rS>> 1/p, or if log[(G/pP) ] >> 1,pwe haveexp[-P e - z - pz] dzLet(P) J exp[-p e- z - pz] dz-aand-zpe -wThenP) = PPi oe-w wP- dwe w - dwIf > 1, then1 00PPP oe-w w 1 dw=So that, finally, we obtain( p 1/2r logfor ~>epFor the special case, p = 1 - that is, when the incident and absorbing lines have ther(p)) P)(9, ) ~rJ(VIII. NUCLEAR MAGNETIC RESONANCE)same Doppler widths - Eq. 8 reduces toS1 1i-(-r log j) 1/2which, for large [i, is essentially the same as the result of Chapman (2).We shall now consider the general case, y # 0.((, P, y) = 1( ( ' P, y) + (, ' p, y )Let(10)exp - e -x + P(x+y)2 dxexp -[KFirst, consider 2."the following equation:2-xThe maximum, a2, of the integrand must be determined froma 2 = + (log1/2P(a 2 + Y)If Eq. 12 is solved by iteration, it converges very rapidly.have1 /22 2 7r)00exp2-a2- exp a ) -z 2e2 2Let x =a 2 + z.+ z + 2y a2 + z)l/2(a )1/2zConsider the numerator of the integrand in the vicinity of z = 0.as follows:a 21/2za2 )Here we can expandz za + -S2a 32 8a 2(14)2For z << a 2 , the numerator becomes(9)and then=Tr(11)(12)Then, we+ 2)] dz2Ka2+ /2+ Z)(13)(VIII. NUCLEAR MAGNETIC RESONANCE)ep (a 2 + y)exp Lp a= exp - P(a 2-z (2e + p a2 + z + 2a2z+yz +zatrl2~P~~i a.+ a2 + 2(15)+ (a : za?If we leta2 +P - pthe numerator in the vicinity of z = 0 becomesexp [-(a2 + y)2] exp[-p2 e -z - pZz]As this function falls off rapidly to zero for Iz I > 1/p Z, it can be used as a good approx-imation to the numerator of the integrand, provided thatPy<< 12 34p 2 a 2which comes from the third term in Eq. 14.2As before, if p2a2 >> 1, the denominator of the integrand becomes a 2 .becomes1/2 exp[-p(a2 + y) 2]2-a 2exp I- (p 2 e -z+ p2 z)]Thus, c 2(16)As before, if expr(p 2 )( PZ)and then, we have(a) >>1, the integral in Eq. 15 becomes approximately/Z exp [-(az + y)2] r(p 2 )2a 2 (P 2)(17)2(, G , y) = ( T)(VIII. NUCLEAR MAGNETIC RESONANCE)It will be noticed that c1( (1, P, y) = 2(1 , p,-y), so that we obtain directlyGL' P,, Yz exp -(a -y) F(p) r(P(18) (18)2al (Pi1in which1/2a1 =+ logP(al - y)(19)a1 -yal1Equation 17 is valid ifPy<< 12 3 14p1 al 12 (20)P1 al 1exp (a) > 1The relative transmission 4 is, then,# = 1 t2 (21)R. H. KohlerReferences1. J. E. R. Young, The hyperfine structure of the resonance line of natural mercury,2537 A, Ph.D. Thesis, Department of Physics, M.I.T., July 1956.2. S. Chapman, Proc. Phys. Soc. (London) 51, 93 (1939).3. A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms(Cambridge University Press, London, 1939).

Nuclear Magnetic Resonance and Hyeprfine Structure

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Nuclear Magnetic Resonance and Hyeprfine Structure

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