Contains reports on six research projects

Bitter, F.

Fork, R. L.

Smith, W. W.

Stroke, H. H.

Walter, W. T.

Waymouth, J. F., Jr.

DSpace@MIT

IV. NUCLEAR MAGNETIC RESONANCE AND HYPERFINE STRUCTUREProf. F. Bitter W. D. Halverson P. G. MennittProf. J. S. Waugh E. R. Hegblom S. R. MillerDr. L. C. Bradley III F. A. Lidgeois O. RediDr. H. H. Stroke J. D. Macomber C. J. Schuler, Jr.Dr. J. F. Waymouth F. Mannis R. W. SimonT. Fohl I. G. McWilliams W. W. SmithR. L. Fork W. T. WalterA. DIELECTRIC SUSCEPTIBILITY TENSOR FOR AN ATOMIC VAPORThe dielectric susceptibility of an atomic vapor in the presence of a magnetic fieldcan be expressed as a 3 X 3 tensor. All of the properties of the medium relevant to thepropagation of electromagnetic waves through the vapor, such as absorption, dispersion,Faraday effect, and so forth, can be obtained from the tensor elements. The imaginaryparts of the tensor elements can be obtained from the absorption coefficients for rightand left circularly polarized light traveling parallel to the field, and for linearlypolarized light propagating perpendicular to the applied field. The absorption coeffi-cients can be calculated from known atomic oscillator strengths and the level struc-ture of the ground and excited atomic states. The tensor analogs of the scalarKramers-Kronig dispersion relations can then be used to calculate the remainingcomponents of the tensor.The dispersing medium exhibits an axial symmetry about the direction of the appliedmagnetic field. In this case the susceptibility tensor must be invariant to rotations aboutthe symmetry axis or field direction. This permits reduction of the tensor to the generalform:X11 -XZ1 0X = X21 X11 00 33 (1)There are, therefore, only three independent components. For purposes of calcu-lation, the tensor is rewritten asP-1 -ia 0 a = a' - ia"x = ia P-1 0 = ' - iP"0 0 y-1 y = y' - iy". (2)Maxwell's equations can then be written in the formQPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)a aS 0 a8z 8xaay00 a8za8zaay_] H xaaxHzioIf we assume that the magnetic fieldto z, and the field independent of x andwave equations.-ia 00 0 yJis parallel to z,y, the formulasEEzthe propagation vector parallelyield the following pairs of+a LE (1-a) = 0a2+ W~2E o(P+a)]H] = 0.Lz 0.1E -J (Ex ±iEy)The effective dielectric permeabilities for the two circularly polarized waves are:E =E (P-a)E_ = Eo(P+a).-0If we assume that the magnetic field is parallel to z, the propagation along x, andthe light linearly polarized along z, we obtain the wave equation+ LE Ez = 0.The effective permeability for a wave propagating along x, linearly polarized parallelto H isE I=E y.The tensor elements may now be expressed in terms of the permeabilities for theQPR No. 67xEz-iW HxHyHz(IV. NUCLEAR MAGNETIC RESONANCE)several polarizations of waves. The imaginary parts of the permeabilities correspondto the absorption coefficients for the medium. These may be taken as experimentallyknown and are sufficient to determine the susceptibility tensor.The absorption coefficients for the three arrangements of propagation vector, k,polarization, and field for the mercury S - P1 transition are:For right circularly polarized light, H parallel to k,k+ = k e-(For left circularly polarized light, H parallel to k,k =k e -2For linearly polarized light parallel to H, H perpendicular to k,k =k eTr OThe corresponding elements are1 Oal = (E -E ") - e -e2E + 2k O21 o" = (E"+E") k+ e2E + 22y " Err /E = k 0 eHere, use has been made of the relationsE +k+- EXok -oEl"k TrThe Kramers-Kronig relations connecting the real and imaginary parts of the tensorelements are:(9a)= P 00 Wvik( wo)uik I 2 20 Co -QPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)2 P Uik 8 )Vik() = P0 2 i 2 dw.ik Tr o Z(9b)Here, Xik = uik + ivik, and use may be made of (1) and (2) to obtain the following rela-tions:Ul1 = p' -1V11 = -u21 = allv21 = a'u 3 3 = y' - 1v 3 3 = -YThe real parts of the susceptibility tensor, then, are:2 P 'o a " (3) k -ka' = - "P 2 d=- o [F(w+Aw)-F(w-Aw)]0 W -w2 2 qWoP"(d)dcZZ 26.y"( )d2 2k KS-0 [F(w+Aw)+F(w-Aw)]2NJ;k o F(w).Nr_1_The complete susceptibility tensor therefore is defined asX1 -F(W+A))-F(W -Aw)-i 0(e - + ) 2 +e - ( - A ) 2)]X ko- [N-'(e- (w-2 -_ei+Aw21 2 4-LkX ko -F(w)-i -- e-Table IV-1 gives the parameters describing the medium for various arrangements offield, polarization, and direction of propagation.For small values of Aw, the expressions given in Table IV-1 may be written in termsof derivatives. Given 8F(w) _ 1 - 2wF(w), we can write the magneto rotation and themagneto double refracti n coefficients for a length of dispersing medium asmagneto double refraction coefficients for a length f of dispersing medium as-k Io 8Fk -q-; a W-k f0 [1 -ZF(w)] Aw%r 7--k f 2 k fP o F 2 = + [w+F(w)(1-2z2)] (AQPR No. 67 24Z 02r 12y' -1 =+ +P+ 1iT 00)-i(F(w+Aw)-F(w-Aw))I(IV. NUCLEAR MAGNETIC RESONANCE)Table IV- . Optical parameters.Field Orientations Absorption Refractive IndexH parallel to k 2 k 0KRight circularly k+ = k e- (  n+- I =- F(w+Aw)polarized light NFW2 k Left circularly k = k e- ( - A )  o1= F( w)polarized light - o -H perpendicular 2 -k0- oto k k =k e n - - F(w)E parallel to H TrE perpendicular ko -( +A) ( n -k [F(w )F(wto H 2 i 26Magneto Rotation(n -n )2 k 0-K k [F(w+Aw)-F(w -Aw)]Voigt Effect(n -n ) -k 0K 0= [F(w+Aw)+F(w-Aw)-2F(w)]ZqNJWThe variation in thegiven byabsorption coefficients for a small excited-state splitting, Aw, is2k 2 -wC 28k z -o e (Aw) =k e (2Zw-1)(Aw)o. 2 2 o.'Sk = 0.The only other case of interest is the absorption coefficient for linearly polarizedlight propagating along the field direction. The variation of this absorption coefficientwith small changes in Ac will be the same as that for k . This can be shown by treatingthe linearly polarized light as a sum of right and left circularly polarized beams and byusing the absorption coefficients given above for right and left circularly polarized light.The derivation of the tensor form of the Kramers-Kronig relations is given byQPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)1Gourary, and the parameter k is a reciprocal penetration depth as defined by Mitchelland Zemansky.2 The parameter = 2=  in (v-v )/AVD, where Av D is the Dopplerwidth. The splitting of the excited state is characterized byAW = 2 Nin g 2m'B H/h vD'andF(w) = e- e y dy.0This calculation was made for a 3P 1-So transition. The approach may be generalizedto include transitions between states with different multiplicities by rewriting the absorp-tion coefficient and proceeding in essentially the same manner as above.For a transition involving a ground state with levels denoted by m, and an excitedstate with levels denoted by m', the absorption coefficient would bek = kmmm, m'-(.+A )2kmm = k H e mmmm o mm'Aw = (2 1Y /AvD) [(g 1 m+g 2 m')PBH/h].Here, gl is the g-factor for the ground state, and g2 is the g-factor for the excited state.Hmm, contains the angular dependence of the transition probability and is essentiallythe square of a Clebsch-Gordan coefficient.R. L. ForkReferences1. B. Gourary, J. Appl. Phys. 28, 283 (1957).2. A. Mitchell and M. Zemansky, Resonance Radiation and Excited Atoms(Cambridge University Press, London, 1934).B. HYPERFINE STRUCTRUE AND ISOTOPE SHIFT IN RADIOACTIVE CADMIUMWe have completed our measurements I of the isotope shift between radioactive Cd 1 0 9113.-0 -1and Cd 1 1 3 . The value of the shift in the 3261-A transition is 0.033 ± 0.004 cm withthe sign of the volume-dependent isotope shift. A well-blazed, high-resolution grating, 210-inches wide, mounted in the 36-foot multiple wavelength mirror monochromator 3QPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)Table IV-2. Relative isotope shifts in cadmium. These representthe variations in <R 2 > of the nuclear charge distribution. After cor-recting for the normal mass effect, the shifts were normalized to theshift between the even isotopes 110-11Z. The increasing nuclear polar-izability as one gets into the collective region (here, larger N) is appar-ent. The staggering parameter y = odd pair I. S./1/2 (even pair I. S.).Relative StaggeringIsotope pair isotope shift parameter y108-110 1.00 0. 06108-109 0. 03110-112 1.00 0. 18110-111 0. 09112-114 0.93 0.28112-113 0. 13was used for these spectroscopic measurements. After correcting the isotope-shiftvalue of Cdl 0 9 for the normal mass effect, and using the available results for the shiftsin the stable isotopes, 4 we obtained the relative shifts given in Table IV-2. It can beobserved that the "odd-even staggering" increases with decreasing neutron number; thatis, as we proceed toward the shell-model region we again appear to have a manifesta-tion of the decreasing nuclear polarizability as a magic number (N=50) is approached.5We have previously observed such a trend in mercury isotopes. The error limits onthe stable-isotope results are considerably smaller than for the radioactive experimentand are a small fraction of the actual shifts. The former results were obtained fromthe Cd II 4416 A line for which the shifts are much larger than for Cd 3261 A. There-fore, in the observations we can rely only on the actual value of the Cd 1 0 9 shift. Moreprecise experiments will have to await the development of cooled hollow-cathode lightsources that are suitable to work with the small quantities of material that are useful,at present, in electrodeless lamps.H. H. StrokeReferences1. R. J. Hull and H. H. Stroke, Hyperfine structure and isotope shift in radio-cadmium, Quarterly Progress Report No. 62, Research Laboratory of Electronics,M.I.T., July 15, 1961, pp. 108-109.2. G. W. Stroke, J. Opt. Soc. Am. 51, 1321 (1961).3. H. H. Stroke and K. K. Y. Li, J. Opt. Soc. Am. 51, 678 (1961).4. P. Brix and A. Steudel, Z. Physik 128, 260 (1950); H. G. Kuhn andS. A. Ramsden, Proc. Roy. Soc. (London) A 237, 485 (1956).5. W. J. Tomlinson III and H. H. Stroke, Phys. Rev. Letters 8, 436 (1962).QPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)C. LEVEL CROSSINGS IN MERCURY 195 AND 195Zeeman level crossings I have been observed, and the magnetic fields at which thesecrossings occur have been measured in Hg195 (9.5-hour halflife) and its isomer Hg 1 9 5 *(40-hour halflife). Our apparatus is similar to that used by Hirsch2, 3 to measure the197 197hhyperfine structure of Hg and Hg . The principal changes in the apparatus havebeen in the use of a Harvey-Wells 12-inch magnet (Model L-128) to improve homogeneityand hence the precision of the measurements, and a new more compact magneto-opticscanning apparatus that provides a variable-frequency source of resonance radiationTable IV-3. Summary of preliminary results.Measured protonDescription of Approximate resonance frequencyIsotope crossing levels crossing field at crossing(low field representation) (gauss) (kc)Hg 19 5  F = 3/2, mF = -3/2 7603 32369. 79 ± .66(I= 1/2)andF= 1/2, mF = 1/2Hg 1 9 5  F = 15/2, m F = 15/2 7968 33925. 86 ± .84andF = 13/2, mF = 11/2in the neighborhood of 2537 A. Searching for the crossings was greatly facilitated bythe spectroscopic data on these isotopes obtained by Tomlinson and Stroke.5 The iso-topes were produced by the reaction Au197(p, 3n) Hg 9 5 in the Harvard Cyclotron byusing the procedure of Tomlinson and Stroke for the spectroscopic runs.Preliminary results for the level crossings in the 3P 1 state observed, thus far, aresummarized in Table IV-3. Errors are two standard deviations plus an estimate of thesystematic error in measuring the field.Using the position of the Hg 1 9 5 crossing and Hirsch's value 6 for gj = 1.4861 ± .0003,I calculated the magnetic dipole interaction constant for the 3P state to be A1 A19515810.82 ± .2 inc. From the spectroscopic value of the electric quadrupole interactionconstant B 5 and the one measured level crossing in Hg 9 5  I obtained A 1 9 5-2368.081 ± .17 mec. Indicated errors in A are 2o-, and represent random errors(including the effect of the uncertainty in B for Hgl 9 5 ').An effort was made to see if the Hg 1 9 5 crossing field would be shifted by changes inQPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)source lamp power, scanning field (that is, lamp frequency) or by reversing the fieldin the Harvey-Wells splitting magnet. No significant shifts were observed, but the pos-sibility of error resulting from changes in field homogeneity if the Harvey-Wells mag-net is not cycled several times through a hysteresis loop after the field is suddenlyreversed is still to be investigated.The accuracy of the value for A 9 5, could be improved by a factor of 10, or more,by a different level-crossing measurement in Hg 9  (since B*0). The intensity andapproximate position of the second most intense level crossing in this isotope have beencalculated by using computer programs run on the IBM 7090 computer at the Computa-tion Center, M.I.T. The intensity change that is due to scattering by the two crossinglevels is approximately 48 per cent; but when the incoherent light scattered by the otherZeeman levels lying within a Doppler width of the crossing point is included, it is foundthat the change in total scattered intensity at the crossing is less than 0.1 per cent. Thissmall change may, nonetheless, be observable by simultaneous application of the tech-niques of field modulation, lock-in detection, and comparison of the light scatteredthrough two different angles. A feasibility test for the proposed second crossing experi-ment is in progress. This will consist in looking for the very weak level crossings instable Hg 2 0 1 by using a natural mercury cell (that is, nonenriched sample). If this testis unsuccessful, a second measurement in Hg 1 9 5 . will be attempted by means of a stan-dard double-resonance experiment.W. W. SmithReferencesI. F. D. Colegrove, P. A. Franken, R. R. Lewis, and R. H. Sands, Phys. Rev.Letters 3, 420 (1959).2. H. R. Hirsch and C. V. Stager, J. Opt. Soc. Am. 50, 1052 (1960).3. H. R. Hirsch, Hyperfine structure in the 3p1 level of the twenty-four hourisomer of mercury 197, J. Opt. Soc. Am. 51, 1192-1202 (1961); Technical Report 372,Research Laboratory of Electronics, M.I.T., October 15, 1961.4. W. W. Smith, A compact magneto-optic scanning apparatus, Quarterly ProgressReport No. 63, Research Laboratory of Electronics, M.I.T., October 15, 1961,pp. 83-85.5. W. J. Tomlinson III and H. H. Stroke, Bull. Am. Phys. Soc. 7, 350 (1962).6. H. R. Hirsch, Ph.D. Thesis, Department of Physics, M.I.T., 1960.D. THE DETAILED ANALYSIS OF A PLASMAIn 1956, an attempt was made to describe in detail the energy flows and particle flowsin one of the best-known plasmas, the low-pressure mercury discharge in the presenceof a millimeter, or so, of rare gas.1 Recent observations by Verweij 2 have madeQPR No. 6715,000A Observations by Kenty, Easley and Barneso Observations by Verweiji VVVV150E10000 200 400Current (mamp)250200150E-6I0050600 800A Observationsby Kenty, Easley and Barneso Observations by Verweij)0 40 60Bulb Wall Temperature (OC)(b)15,00010,00050004 8 12 16 20Argon pressure (mm) 24Plots of experimental and computed electron temperature and axial fieldin a mercury-argon discharge in a 1. 5-in. diameter tube as the lamp cur-rent, bulb-wall temperature and argon pressure are separately variedaround the following standard values: lamp current, 400 ma; bulb-walltemperature, 42°C; argon pressure, 1 mm.QPR No. 67ETeFig. IV- 1._ __ ______I I I1(IV. NUCLEAR MAGNETIC RESONANCE)possible a more detailed comparison between theory and experiment. This has beenreported in detail in a lecture recently given by the writer at Cornell University.3 Thechief results of this comparison are shown in Fig. IV-1. They concern the electrontemperature T e and electric field E in a mercury discharge in a tube approximately1. 5 inches in diameter with a vapor pressure corresponding to a bulb-wall temperatureof 420 C, an argon pressure of 1 mm, and a lamp current of 400 ma as each of the lastfour variables are changed separately around the given values. Verweij shows that theobserved electron mobility can be quite well accounted for by known electronic collisioncross sections. A study of the existing data indicates that the chief uncertainty of theanalysis lies in the radial diffusion of resonance radiation.Another study of this discharge, with more justifiable assumptions regarding theexcited states of mercury, is being undertaken. Also, an analysis by J. F. Waymouth 4of the theory of Langmuir-probe measurements when the probe dimensions are compar-able to the electron mean-free path indicates the need for a review of certain aspectsof the existing data.F. BitterReferences1. J. F. Waymouth and F. Bitter, J. Appl. Phys. 27, 122 (1956); F. Bitter andJ. F. Waymouth, J. Opt. Soc. Am. 46, 882 (1956).2. W. Verweij, Physica 25, 980 (1959).3. To be published by the Material Sciences Center of Cornell University.4. This work will be published as Technical Report 406, Research Laboratory ofElectronics, M.I.T.; see also Section IV-F.E. THE MAGNETIC MOMENT OF MERCURY 197 BY MEANS OF OPTICALPUMPINGA Ph.D. thesis with this title was presented to the Department of Physics,M.I.T., June 1962, by the writer. The final numerical values were also reportedin Quarterly Progress Report No. 65, page 39.W. T. WalterF. PERTURBATION OF A PLASMA BY A PROBEOne of the fundamental conditions to be satisfied before Langmuir-probe char-acteristics can be interpreted to yield data on plasma density and temperatureis that probe size be much smaller than mean-free paths of plasma particles.Then and only then are the values of electron density and plasma potential theQPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)same as those in the absence of the probe.In several recent high-caliber probe measurements that have been published, thiscondition has been violated to a greater or lesser degree. Moreover, in the presenceof a magnetic field, when the effective mean-free path becomes comparable to the cyclo-tron radius it is especially difficult to meet this requirement.A study has been made of this problem, starting with the hypothesis that the sheathequations still apply, but that the plasma density and potential at the sheath boundaryare perturbed from their values in the absense of the probe by depletion resulting fromthe probe current. In the plasma outside the sheath boundary there is a density andpotential gradient to supply, by ambipolar diffusion to the sheath boundary, the currentsof electrons and ions extracted across the sheath by the probe.The problem has been considered for a spherical probe in the absence of a magneticfield, and for a plane disc probe normal to a magnetic field. The results for both casescan be expressed as follows:non 1 + PQe Qwhere no is the plasma density in the absence of the probe, np the plasma density atthe probe sheath boundary, Ve and Vi the voltage equivalents of electron and ion tem-peratures, AV is the perturbation of plasma potential, and P is the magnetic field fac-tor 1 + e i B , with ±e and .i the electron and ion mobilities. Here, Q is given byQ Qe + Qie / V < 0r p eVS es exp(V/V ) V 0e (Ve+V.) erp eV_i _ _ -e ~V >0i (V e+Vi) ep e V.iSrp eV.V + V Is exp(-V /Vis) V > 0.i(V e+V ) 2mi s is sQPR No. 67(IV. NUCLEAR MAGNETIC RESONANCE)Here, Vs is the potential across the sheath, assumed to be positive when the probe ispositive with respect to the plasma at the sheath boundary; Ves and Vis are electronand ion temperatures (in volts) at the sheath boundary. Under conditions for which thisanalysis is valid (Q>1), these are the same as the respective temperatures in theplasma. The radius of the spherical or disc probe is rp .A report is in preparation which gives the derivation of these equations in consider-able detail, as well as calculations for the perturbed probe current-voltage character-istics to be expected for various values of Q.J. F. WaymouthQPR No. 67

Nuclear Magnetic Resonance and Hyperfine Structure

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Nuclear Magnetic Resonance and Hyperfine Structure

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