Contains reports on one completed research project and two current research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E

Fox, L. R.

Kyhl, Robert L.

Strandberg, Malcom W. P.

DSpace@MIT

GENERAL PHYSICSI. MICROWAVE SPECTROSCOPYAcademic and Research StaffProf. M. W. P. StrandbergProf. R. L. KyhlJ. D. KiersteadGraduate StudentsJ. B. Barton L. R. Fox R. M. Langdon, Jr.J. Berman J. U. Free J. M. SmithA. WORK COMPLETED1. MEASUREMENT OF THE ELECTRONIC CONTRIBUTION TOULTRASONIC ATTENUATION IN A TIN FILM AT 9 GHzThis work has been completed by Robert L. Woerner and submitted as a thesis tothe Department of Physics, M. I. T. , January 1970, in partial fulfillment of the require-ments for the degree of Bachelor of Science. A summary of the thesis researchfollows.Measurements of the attenuation of coherent phonons in a tin film (6. 3 t thick)at liquid Helium temperatures were made as the film was switched between thenormal and superconducting states. The frequency of the phonons was variedbetween 9. 1 GHz and 9. 4 GHz. These relative attenuation measurements yielded-1a value for the electronic contribution to the attenuation of the phonons of 980 cm-1±15%. This agrees with the theoretical formula for the attenuation given by Pippardif it is assumed that in tin four electrons per atom contribute to the attenuation.This is consistent with the valence of tin, which is 4.M. W. P. StrandbergB. METALLIC CONDUCTIONThe equations given here form a complete set of transverse exponential solutions ofMaxwell's equations and the Boltzmann equation in an infinite ideal metal in the semi-classical low-temperature limit (small signal, no x or y dependence, spherical Fermisurface, isotropic mean-free path, all excitations at the Fermi surface). They can beused to expand the field and momentum distribution of the anomalous skin effect withThis work was supported by the Joint Services Electronics Programs (U. S. Army,U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).QPR No. 97(I. MICROWAVE SPECTROSCOPY)arbitrary surface reflection conditions.E = E exp j(wt: k z)x o zTkH = Z E exp j(wt:k z)(la)(2/h 3 ) eT(px/PF)f(p) = E exp j(t T k z)1 + jwt jkz (pz/pF o zComplex k is the solution of the transcendental equation k + jw c(kz) = 0.z z zE = E exp j(wt - kiz)x 1H = E exp j(wt-k z)(lb)L (2/h 3 ) e(px/PF)f(p) = + AT 6(1 -pz pzi px/PF) E exp j (wt-kiz)(1 + jwt)(1 - PZ/Pzi)wherejki' = (l+jt)pF/pziPzi runs from -pF to +pfF jk 2 f3 o 2(1+j 2t) (1+j /t) ( 2+jwt) 1 1c(k ) = -+ j 11+ ) I +(k 4(l+jwt) (k + kz + (k) in jk1 + jwtand22A =1 (1+jt) PF +SpF -pzi e + j P2 ziziBoth forward- and backward-wave terms are included. Equations la are well known.Equations lb do not seem to have been published. Although they offer a conceptuallysimple approach that may give insight into the anomalous skin-effect solutions, they aresomewhat singular functions without obvious orthogonality properties. This may explainwhy there has been no stampede away from the conventional approach.R. L. KyhlQPR No. 97(I. MICROWAVE SPECTROSCOPY)C. GEOMETRIC DISPERSION OF INCOHERENT PHONONPULSES IN QUARTZBolometric observations of incoherent phonons in x-cut quartz by Andrews 1 includedpulses associated with the usual archetypical longitudinal (L), fast transverse (FT), andslow transverse (ST) modes, as well as a remotely off-axis mode termed the oblique (0)mode. A plane wave in an anisotropic medium, characterized by a wave vector k, andangular frequency w may propagate in any direction with any of three phase velocities,v, given by the point on each of the three figured velocity surfaces corresponding to thedirection of k (see Fig. I-1). In the nondispersive limit (w 1012 Hz), v is indepen-dent of the magnitude of k. The acoustic power flow, or Poynting vector, S, for a waveN \iFig. I-1. Coordinate geometry showing circular bolometer apertureof diameter a, located in yz-plane at distance f from origin.The coordinate x axis is the crystallographic x axis of quartz.An example is given of detectable power, represented by thePoynting vector S, with k outside the receiving cone.of given k and v is in the direction of the gr-adient of that particular velocity surface atits intersection with k. Consequently, as k moves over a particular locus, S movesover some other locus, whose relation to the first is quantitatively different for eachvelocity surface. As has been shown in the detailed analysis of Farnell,2 waves havingtheir wave vectors lying within an elliptical cone centered on the x axis contribute toQPR No. 97BOLOMETER CONE9 (deg)-5.0 5.0FT95.09 (deg)(0)15.0 20.065.070.0Q (deg)(b)Fig. I-2. (a) Angular limits of the k excursion for the three near-axialmodes of the three velocity surfaces. The designations L,FT, and ST refer to the type of polarization of the three exactlyaxial modes. The x axis corresponds to 0 = 900; € = 00.(b) One lobe of the region corresponding to the oblique (0) mode.The other lobe is located, by symmetry, by rotation through1800 about the x axis.QPR No. 97(I. MICROWAVE SPECTROSCOPY)power flow inside a coaxial circular cone of different angular dimensions. In Fig. I-2a,we show the results of calculations mapping out the region of k excursion of the threearchetypical L, FT, and ST modes for power flow within a circular cone of half-angle 4.50 corresponding to the bolometer cone in Andrews' experiments (see Fig. I-1).In Fig. I-2b we give one lobe of another angular region, located on the same velocitysurface that corresponds to the FT region in Fig. I-2a. This also gives rise to detect-able power within the bolometer cone, and corresponds to the observed "oblique" mode.This region (both parts - the other lobe corresponds to a figure rotation of 1800 aboutthe x axis) has a relatively large total angular size compared with those in Fig. I-2a.Following Andrews' assumption that a (temperature) pulsed hot-film source ofincoherent phonons gives rise to black-body thermal "spikes" propagating in all direc-tions, independently of polarization, 3 we have calculated the power delivered throughthe aperture of the bolometer as a function of time. This was done by calculating thedelivery time of the detectable acoustic power corresponding to points on the unit spherein k space on a 0. 10 mesh. The "active" points were then tabulated against time with-3a v-3 density of states and sin 0 geometrical weighting. A histogram of the powerdelivery was then constructed by interpreting the density of the weighted, tabulatedpoints per unit time interval as the power detected at the bolometer. Figure I-3 showsthe broadening of instantaneous thermal spikes caused by the allowed excursion of kover each of the regions in Fig. 1-2. Comparison with Andrews' discussion 3 of elastic1.00 -0<u0 D 0.10ozL FT O ST4.0 5.0ARRIVAL TIME (ps)Fig. 1-3. Power pulses corresponding to a hot-point source of infinitesimalduration, being delivered through a bolometer aperture a = 3 mm,situated at I = 19 mm from the origin (see Fig. I-1). Verticalbars denote arrival times associated with the average velocity ineach k region.QPR No. 97(I. MICROWAVE SPECTROSCOPY)dispersion indicates that at generating temperatures below about 5 0 K, the geometricdispersion for the usual experimental arrangement is dominant, being in the range0. 1-0. 3 ps. The relative pulse heights are in qualitative agreement with those observedby Andrews.L. R. FoxReferences1. J. M. Andrews, Jr., "Observations of Incoherent Phonon Propagation in X-cutQuartz," Quarterly Progress Report No. 77, Research Laboratory of Electronics,M.I.T., April 15, 1965, pp. 7-15.2. G. W. Farnell, Canad. J. Phys. 39, 79 (1961).3. J. M. Andrews, Jr., "Incoherent Phonon Propagation in a Born-Karman Lattice,"Quarterly Progress Report No. 79, Research Laboratory of Electronics, M. I. T. ,October 15, 1965, pp. 7-14.QPR No. 97 L

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