Contains reports on two research projects.Lincoln Laboratory (Purchase Order DDL BB-107)United States Air Force (Contract AF19 (608)-500

Andrews, J. M., Jr.

Strandberg, Malcom W. P.

Zobrist, A.

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RADIO PHYSICSI. MICROWAVE SPECTROSCOPYProf. M. W. P. Strandberg R. Huibonhoa S. ReznekProf. R. L. Kyhl J. G. Ingersoll W. J. SchwabeDr. J. M. Andrews J. D. Kierstead J. C. ShahA. Fukumoto J. W. Mayo A. ZobristA. A COMPUTER SOLUTION FOR ALLOWED ENERGY BANDS FOR PERIODICPOTENTIALSGiven an arbitrary periodic potential of period a, we know that a solution ofSchrddinger's equations will be absolutely periodic if it satisfies periodic boundaryconditions.(x+a) = (x)d d@d- (x+a) = dr (x)Also, a solution will be half-periodic, that is, it will repeat every two periods, if it sat-isfies the following periodic boundary conditions.4(x+a) = -4(x)d d( x+a) - (x)Periodic and half-periodic solutions occur at discrete energies. Moreover, the ener-gies depend upon the initial conditions of the solution. As an example of this, considerthe solution of the wave equation for a potential consisting of a periodic delta function(a) t V .\./ (b) tt I tv IThe delta function does not affect the electron in (b),since it is at a node of the solution.Hence, the electron in (a) has higher energy.The program has been written for a PDP-1 computer controlled by teletype. A fieldof 512 registers is used to store one period of the potential. Subroutines are includedfor placing the potentials on this field. The solution, when found, is stored on anotherfield of 512 registers. The potential and solution may be displayed on the PDP-1oscilloscope.This work was supported in part by Purchase Order DDL BB- 107 with Lincoln Lab-oratory, a center for research operated by Massachusetts Institute of Technology, withthe support of the U.S. Air Force under Contract AF 19(608)-500.QPR No. 75v7 v8 v9 vl v2 v3 v4 v5 v6-i1immmmm m-m-m-m.-U.-El-El-El-El-Umm-m m E1m_mm, ---El------ INDICATES THE MAXIMUM HEIGHT OF THEPOTENTIAL FUNCTION0.01 LFig. I-1. Allowed energy bands for an electron in a one-dimensionalsolid for various potentials.QPR No. 751000.1m mvl - BARRIER1/ 4 A8.521-0.013v3 - COULOMB1 .50- 3.3671/4 vv5 - SINUSOIDALv2 - SQUARE WELL0.5925- 7.9411/4 xv4 - COULOMB(1 .50-6.7641/16v6 - TRIANGULAR3.061-1.810DELTA FUNCTIONSv = - 4682 evov = - 1560 evv = -390 ev0Fig. 1-2. The potentials used by the program. (All cellsenergy in electron volts.)are 2 A long;QPR No. 75v7:v8:v9: 1/ 256 XFig. I-3. Four periods of square well. Fig. I-4. Four periods of square well.Electron at 1. 81 ev. Electron is Electron at . 81 ev. Electron isin an allowed band. in a forbidden band.Fig. 1-5. Periodic solution for sinusoidalpotential. 1. 23 ev.Fig. 1-7. Periodic solution for Coulomb(v4). . 094 ev.Fig. I-6. Periodic solution for deltafunction (v7). .347 ev.Fig. 1-8. Half-periodic solution forsquare well. 0. 90 ev.QPR No. 75(I. MICROWAVE SPECTROSCOPY)If an electron energy is typed in and the program is started, the computer will finda solution with the end heights matched.(x+a) = (x)or.(x+a) = - (x)Upon command, the program increments the energy upwards until the end slopes arematched.d4 d4d(x+a) = (x)Thus all of the values of the periodic and half-periodic solutions can be found one by one.Two sets of initial conditions were used.4'(0) = ' '(0) = 0or(0) = o (0) = oThe energies found in this manner were taken to be edges of the allowed and forbiddenbands for an electron in a one-dimensional solid. The results are plotted in Fig. I-ifor the various potentials shown in Fig. 1-2. The energies have been chosen so that thehighest energy of the first allowed bands will coincide (except for potentials v4 and v6).The Coulomb potentials are actually the lattice sum of several neighboring Coulombpotentials.These results were confirmed by another section of the program which extended thesolution over several periods of the potential (Figs. I-3 and 1-4). Some periodic solu-tions are shown in Figs. 1-5, 1-6, 1-7, and 1-8.The program is useful for demonstrating the properties of a wave function in onedimension, and is capable of handling any symmetric potential.A. Zobrist, M. W. P. StrandbergB. INCOHERENT PHONON PROPAGATION IN ANISOTROPIC MEDIAThe usual ultrasonic phonon experiment involves piezoelectric (or magnetostrictive)transducers oriented and cut in such a way that their surfaces produce and respond towave surfaces of constant phase. Such configurations are chosen to enhance the gener-ation of coherent phonons of some desired mode, and the detectors are relatively insen-sitive to phonons whose wave vectors deviate significantly from the normal to theirsurface.The direction of travel of a phonon in an anisotropic medium is not generally colinearQPR No. 75(I. MICROWAVE SPECTROSCOPY)with its wave vector. In a crystal the specific directions in which the ultrasonic wavevector and the Poynting vector are colinear are termed pure-mode axes. For example,if the wave vector of a phonon propagating in a-quartz is not directed along a pure-modeaxis, its direction of travel can deviate up to 500 from its wave vector. 1 Such effectshave been observed in experiments where the transducer diameters are small in com-parison with their separation, yet are large in comparison with the ultrasonic wave-length. The experimental configuration is sketched in Fig. 1-9. Here, no signal isR R2d k ANISOTROPICPROPAGATIONMEDIUMXFig. 1-9. Ultrasonic propagation in an anisotropic medium. The ultrasonic Poyntingvector S makes an angle 8 with the wave vector k.detected at the receiving transducer R 1, which is directly opposite the transmittingtransducer X. However, a large signal is detected at the receiving transducer R 2 . (Theultrasonic Poynting vector S makes an angle 0 with the wave vector k.) Phase velocitiesare determined by the quotient of the component of the transducer separation in thedirection of the wave vector (i. e., the distance d in Fig. I-9) and the elapsed time. Thevelocity of the flow of ultrasonic energy, on the other hand, is greater by the factor sec 8.vE = Vp sec 6 (1)A small heating element is capable of producing simultaneously the entire phononspectrum, characteristic of some absolute temperature T, that can be propagated withina crystal. If a small bolometer is attached at another point on the crystal, all phononswhose Poynting vectors lie within the subtended solid angle can be detected, regardlessof the orientation of their wave vectors. In general, the double mode-dependence of Eq. 1complicates the theoretical analysis for arbitrary propagation directions in typicalanisotropic media. If observations were to be made within very small solid angles cen-tered about a pure-mode axis, however, most of the data could be interpreted in termsof the well-known velocities characteristic of the axis.The x axis of a-quartz is a pure-mode axis for both longitudinal and transverse pho-nons. Thus, as the solid angle subtended by a bolometer approaches zero at the x axis,QPR No. 75(I. MICROWAVE SPECTROSCOPY)three pulses should be detected whose energy velocities approach the phase velocitiescharacteristic of x-axis phonon propagation (v =5. 75, v 1 t=5. 18, v2t=3. 36 km/sec). This,however, does not rule out the existence of "extra" pulses caused by other modes whosePoynting vectors also happen to lie along the x axis. An examination of the energy-flowvector diagram for transverse waves in a-quartz, calculated by Farnell, indicates thatphonons whose wave vectors intercept the unit sphere somewhere in the region near 0 =700 and c = 100 might contribute to energy detected on the x axis.An "extra" pulse has been observed by von Gutfeld and Nethercot 3 for this orientationof a-quartz, although the angle subtended by their detector was not stated. Farnell'sdiagram shows that a detector that subtends a half-angle of 25. 50 will pick up the trans-verse mode whose wave vector intercepts the unit sphere at 0 = 900 and 4 = 400. Sincevon Gutfeld and Nethercot did not mention the angle subtended by the bolometer in theirx-cut quartz experiment, the question concerning the existence of "extra" modes propa-gating along the x-axis in a-quartz remains open.J. M. Andrews, Jr.References1. G. W. Farnell, Can. J. Phys. 39, 65 (1961).2. M. F. Markham, Brit. J. Appl. Phys., Suppl. No. 6, S56 (1957).3. R. J. von Gutfeld and A. H. Nethercot, Jr., Phys. Rev. Letters 12, 641 (1964).QPR No. 75

Microwave Spectroscopy

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Microwave Spectroscopy

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