Contains reports on three research projects.Signal Corps Contract DA36-039-sc-7489

Strandberg, Malcom W. P.

Thiele, A. H.

White, R. M.

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VI. MICROWAVE SPECTROSCOPYProf. M. W. P. Strandberg W. J. C. Grant J. D. StettlerProf. R. L. Kyhl J. G. Ingersoll Winifred F. TankDr. R. A. McFarlane E. C. Ingraham N. TepleyDr. G. J. Wolga J. D. Kierstead A. H. ThielePatricia A. Andre M. S. Lipsett M. T. UyJ. M. Andrews, Jr. D. W. Oliver S. H. WempleP. H. Carr W. J. Schwabe R. M. WhiteP. C. Clapp J. R. Shane R. M-C. WuA. CROSSOVER TRANSITIONSA theoretical and experimental investigation was undertaken to study the behavior ofa spin 3/2 paramagnetic system with trigonal field symmetry near the Zeeman-levelcrossings.The spin Hamiltonian for such a system isH = gl BSzHz g B  +(SH +SH D [S2- S2Here, the quantization axis has been chosen along the c-axis of the crystal. When theapplied de magnetic field is parallel to the c-axis, Sz is diagonal in the representa-tion qm. However, if the de field makes some angle 0 with the c-axis, the eigenfunctionthat diagonalizes S is i = E c Q . Such eigenstate mixing is particularly evident inz m mmmthe region of energy-level crossings. When two levels are so mixed it is possible tohave dipole transitions (Am=±l, 0) between them.A AThe eigenfunctions bm can be found by solving the secular determinant H-EmI = 0for E m , and then substituting these eigenvalues in H )m = E m, . However, Strandberghas shown (1) that this process is simplified by changing the quantization axis of theHamiltonian so that it makes an angle P with the c-axis. The resulting matrix is thendiagonalized by contact and rotation transformations to give the energy differencesdirectly. For the 3/2, -1/2 crossover, Strandberg obtains~E =(E + 2g B(Ho-Hor)21E3 = (E 3 1 )0  +/3 D sin23where(E 1) D sin 23,-1 0 2By applying similar operations to the rf matrix, the transition probability is foundto be proportional toThis work was supported in part by Signal Corps Contract DA36-039-sc-74895.(VI. MICROWAVE SPECTROSCOPY)22 2g B cos[+ 2gg (H-Ho ) 23 D sin23The spin system used to study these results was that of Cr in an Al20 3 lattice(ruby). (E3,-1) 0 and P for ruby are shown in Fig. VI-1 and Fig. VI-2.The crossover region was studied experimentally by observing the paramagneticresonance with low-frequency (171-mc) equipment. This equipment has beendescribed (2) elsewhere.From Figs. VI-1 and VI-2 we can get a qualitative idea of how the resonance nearthe crossover will behave. Suppose, for instance, p = 60. Then, from Fig. VI-1, wecan see that as we increase the magnetic field, we should obtain a resonance at2018 gauss and one later at 2135 gauss. Because of spin-lattice interactions theseresonances will have a finite width. Kittel and Abrahams (3) have shown that at the con-centrations used in this experiment, the frequency line shape is Lorentzian with line-width Av. If we maintain a fixed frequency and sweep the magnetic field throughAvresonance, the magnetic linewidth will be AH = - , where av/aH increases fromav/aHzero at the minima of (E3,-1) 0 to some finite value at higher and lower fields (Fig. VI-1).If we now superimpose the probability restrictions given in Fig. VI-2 on our reson-ances, the amplitude of the lower tail of the 2135 line and that of the upper tail of the2018 line will be increased. The resulting absorption will then look like that indicatedin Fig. VI-3a, in which we have plotted Pg(H-Hr) against H, for p = 60. Similarly, weobtain curves for p = 80 (Fig. VI-3b), and P = 100 (Fig. VI-3c).The experimental results are shown in Fig. VI-4. We must remember that theseexperimental lines are the derivatives of the absorption. A comparison of the theoreti-cal and experimental values for p = 80 and P = 10' are shown in Table VI-1.Table VI-1. Comparison of line positions.Theoretical ExperimentalH H Linewidth AHP o aH o aH(gauss) (mc/gauss) (gauss) (gauss) (mc)80 2065 2.29 2060 23 502105 2105 23 50100 2100 1.34 2102 33 452040 2080MAGNETIC FIELD (GAUSS)2120 2160Fig. VI-l.CO0I..0azWEnergy dependence AE 3 ,-l between the +a function of magnetic field for P = 2', 40,1.00.90.80.7 -0.6 -0.5 -0.4 - 80.3 - 60.2- 400.1 2-02000 2040 2080 2120MAGNETIC FIELD (GAUSS)3 1Sand - levels as2 20°80, 100.3 1Relative probability of + 3 ' 1 transition as a function of magnetic2 2field for several values of 3.14 x 1019171 MC2000Fig. VI-2.r r l 1 i 1I I I I I i I2000 2040 2080 2120 2160MAGNETIC FIELD (GAUSS)(a)I I I I i I2000 2040 2080 2120 2160MAGNETIC FIELD (GAUSS)(b)2000 2040 2080 2120 2160MAGNETIC FIELD (GAUSS)(c)Fig. VI-3. Theoretical shape of +p = 80; (c) for p = 100.3 132' 2 absorption: (a) for p = 6o; (b) for2' -4 1--f I-3 1Fig. VI-4. Derivative of absorption for the + 2 - 2 crossover transition:o is the angle between the c-axis and the dc magnetic field;v = 171 mc; T = 3000K; time constant, 2 sec.95(VI. MICROWAVE SPECTROSCOPY)The measured spectra reflect the validity of our matrix transformations. The experi-ment itself points out the fact that crossover transitions produce spectra quite differentfrom ordinary microwave transitions.R. M. WhiteReferences1. M. W. P. Strandberg, Crossover transitions, Research on Paramagnetic Reson-ances, Ninth Quarterly Progress Report on Signal Corps Contract No. DA36-039-sc-74895, Research Laboratory of Electronics, M. I. T., Nov. 15, 1959, pp. 1-12.2. R. M. White, Crossover Transitions in a Spin 3/2 Paramagnetic System withTrigonal Field Symmetry, S. B. Thesis, Department of Physics, M. I. T., May 1960.3. C. Kittel and E. Abrahams, Phys. Rev. 90, 238 (1953).B. RUBY LINEWIDTHSIn a previous discussion of ruby linewidth measurements (1), the interpretation ofthe procedure that was given for determining the effective chromium-ion concentrationwas not sufficiently stressed. For this reason, some discussion of the attitude thatmust be taken toward this kind of measurement will be given here.It is apparent from the discussion of the measurements that one is determining themagnitude of the maximum slope of the imaginary part of the resonant susceptibility fora substitutionally chrome-doped sapphire lattice as a function of the amount of chromiumdoping. If, then, we appeal to the usual argument that we have a Lorentz-shaped linenear resonance, a formula may be evoked that relates this maximum slope of the sus-ceptibility to quantum-mechanical numbers, such as the dipole-moment matrix elements,and the physically observed properties of the resonance, such as linewidth and effectiveconcentration of ions. However, the word "effective" should be used when we speak ofthe concentration of paramagnetic ions in a lattice diluted with diamagnetic ions, in our27case Al27. The reason is that the number that stands for the effective concentration ofions properly relates the susceptibility to the dipole moment and the linewidth, but onlyfor a purely Lorentz-shaped line. We know, in fact, that a diluted paramagnetic crystaldoes not have a Lorentz-shaped susceptibility function, except near the region of reso-nance. There will always be broad skirts whose contribution to the line strength - that is,the frequency integral of the imaginary part of the susceptibility - can be appreciable,as compared with the contribution to the line strength from the region near the resonantsusceptibility. In other words, the ion concentration given (1) before may be looked uponas a number which, if it is introduced into the conventional formulas, will allow one tocalculate the susceptibility obtainable from a ruby crystal.The fact that this concentration differs from that determined by other means, chemi-cal means, for instance, may only indicate that, in fact, the skirt contribution to the line(VI. MICROWAVE SPECTROSCOPY)strength is, indeed, appreciable and must be taken into account in any analysis that triesto relate the amount of chromium introduced substitutionally into the sapphire latticewith the maximum resonant susceptibility.A proper discussion of paramagnetic shapes is a difficult one; this topic has beenunder consideration for some time. Great strides have been made toward understandingthis phenomenon in the work of Anderson (2) and of Kittel and Abrahams (3). At thepresent time, we are trying to develop our analytical understanding of the paramagnetic -resonance line shape. We are also accumulating data on the ratio of the effective toactual ion density in other crystals that have undiluted linewidths greater than and lessthan those of ruby and sapphire.Briefly, our position seems to be somewhat as follows: If Anderson's analysis (2)is used, we are led to the conclusion that the intensity of the absorption, or the imagi-nary part of the magnetic susceptibility, is related as I c = I I(1+c), where Ic is theintensity at a concentration, and Ii is approximately the intensity with no dilution, orwith c = 1. It is seen from this relationship that the peak intensity is diminished by afactor of 2 as we dilute the paramagnetic ions. This relationship is approximate andis restricted by the assumptions expressed in its derivation. We have been able toextend this result by a simple calculation, using Anderson's results, and have computedthe slope of the intensity - the imaginary part of the susceptibility - with respect to fre-quency. When the concentration dependence of the linewidth is removed from the slope,the slope for a concentration of 1 is approximately Tr greater than the slope at vanishingconcentration. This would mean, for example, if it were applied to our previous meas-urements (1), that if we determined the appropriate Lorentz line-shape function fromthe slope (which is probably what we are doing), the effective concentration would be-Ir-1 as great as the actual concentration, and the discrepancy of a factor of approxi-mately 3. 5 between the EMR measurements and the chemical determinations would thusbe explained. But because the calculations are somewhat restricted by their assumptionsand do not describe sufficiently the actual line shape with which we are working, thisexplanation is not completely satisfying, and, in fact, may be merely fortuitous.If, indeed, our present understanding of the relationship of the effective concentra-tion of ions with the actual concentration of ions is adequate, there should be greater orless correlation between these numbers, and it would depend upon the ratio of the actuallinewidth of the diluted crystal to the expected linewidth in the undiluted crystal. Wehave recently obtained (4) a crystal of a metal organic compound, chromium acetylace-tonate, which, because of the generous proportions of the unit cell, has an undilutedlinewidth that is considerably less than that of chromium in sapphire. This type of crys-tal will be very useful in studying the concentration dependence of line shapes.Until this question of the actual line shape of a diluted paramagnetic crystal has beenanalyzed in detail, there seems to be no alternative but to compute the effective(VI. MICROWAVE SPECTROSCOPY)chromium-ion concentration, as we have done and reported (1) on the basis of a singleLorentz -shaped line.M. W. P. StrandbergReferences1. M. W. P. Strandberg, Ruby linewidth, Research on Paramagnetic Resonances,Tenth Quarterly Progress Report on Signal Corps Contract No. DA36-039-sc-74895,Research Laboratory of Electronics, M. I. T., Feb. 15, 1960, pp. 17-27.2. P. W. Anderson, Phys. Rev. 82, 342 (1951).3. C. Kittel and E. Abrahams, Dipolar broadening of magnetic lines in magneticallydiluted crystals, Phys. Rev. 90, 238 (1953).4. We are indebted to Dr. Ellis Hormatz, of Stromberg-Carlson Company,Rochester, New York, for providing us with samples of chromium acetylacetonate.C. PARAMAGNETIC RESONANCE IN THE MANGANOUS ION AT LOWFREQUENCIESThe present work represents an attempt to observe the paramagnetic -resonanceabsorption of the manganous ion at energies so low that the nuclear interaction isimportant and produces a mixing of the electron spin states. The ion is contained in asolution that is placed in a magnetic field. The rapid motion of the ions and dipoles inthe solution averages their electric field, as seen by a paramagnetic ion, to a field ofspherical symmetry over the period of a transition; thus this field produces linebroadening, but no splitting. Since the experiment is carried out at room temperature,the ion is in the ground state with respect to the rest of the electrostatic terms.The manganous ion is composed of five 3d electrons outside the argon kernel. Sincethe orbital angular momentum is zero, the Hamiltonian for the magnetic terms isE= gePH. S + gIPIH I + AI - Swhere S is the total electron spin vector, I is the nuclear angular momentum vector,H is the externally applied magnetic field, ge is the electronic g factor, P is the Bohrmagneton, gI is the nuclear g factor, PI is the nuclear magneton, and A is the experi-mentally determined nuclear-electronic coupling coefficient. Since the Hamiltonian isexpressed only in terms of angular momentum, matrix methods were used to find theeigenvalues of the Hamiltonian. The A value used has already been determined at highfields (1). The diagonalization of the Hamiltonian matrix was carried out exactly inthe (F, m F , I, S) representation by iterating a continued fraction expansion.Since for small energy splittings the populations of the energy levels are proportionalto the energy splittings, and the energy absorbed in one transition is also proportionalto the difference in the energy of the two levels, the strength of the absorption signal is(VI. MICROWAVE SPECTROSCOPY)proportional to (hv) 2 . This means that a compromise must be made between a high fre-quency, which gives a good signal-to-noise ratio, and a low frequency, at which themixing of states is pronounced and therefore gives an interesting spectrum. The pres-ent attempt was made at a frequency of 200 mec. At this frequency, no data wereobtained. The calculated energy levels predict that there are permitted transitions at1 kmc in which the mixing of states caused by the nuclear interaction is quite strong.The present spectrometer consists of a feedback oscillator that is held at a presetlevel in its linear range by a stabilizer circuit. The stabilizer controls the level bycontrolling the plate current of the oscillator tube. We found that the chopper modula-tion of the stabilizer circuit was getting into the oscillator circuit and producing modu-lation noise. This noise was cut down by the addition of decoupling networks andshielding between the stabilizer and signal circuits. The tank circuit of the oscillatorconsists of a helical coil in a cavity. The coupling is by means of probes. The coil issupported by polystyrene, foamed in place to reduce microphonics. The oscillator isequipped with built-in ports for putting in and taking out alignment signals. Isolationcapacitors had to be placed in the outer conductors of the feed lines to prevent modula-tion ground loops.The signal obtained is the derivative of the absorption of the sample as a function ofthe applied magnetic field. The applied magnetic field is modulated at an audio rate.The absorption of the sample then changes at an audio rate, and this causes the outputof the oscillator to change. The time constant of the stabilizer is sufficiently large thatthe average level of the oscillator over a 1-sec interval is held constant. The resultingsignal is then synchronously detected against the original modulation, integrated, andrecorded on a strip recorder.Attempts to obtain spectra were made with water, methanol, and acetone as solvents.Concentrations were 0. 1 and 0. 01 molar, except for the acetone solution, which wassaturated. In all cases, enough solution was put into the cavity to reduce the Q to one-half of its initial value. For example, 0. 1 cc of 0. 1 molar MnCl 2 reduces the Q of thecavity to one-half of its original value. In no case could spectra be seen above the noise.A 1-kmc oscillator is now being constructed. It is similar in all respects to the 200-mcoscillator, except that it uses a coaxial cavity at 1-kmc.A. H. ThieleReferences1. M. Tinkham, A Study of Paramagnetic Resonance Absorption in Liquids,S. M. Thesis, Department of Physics, M. I. T., 1951.

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Microwave Spectroscopy

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