FORCED RAY TEGH SCATTERING 1N UQUEDS Thesis for the Degree of M. S. MEC‘HEGAN STATE UNWERS‘lTY VERENCE D. MOORE 1977 ’. M; “‘"T L [B {73 A 12 ‘1’ I‘v'EiChigan State gmvcrsity 1, Iy‘ld‘" Ir"_.n..:"..:" A" 1“ w_ ABSTRACT FORCED RAYLEIGH SCATTERING IN LIQUIDS By Verence D. Moore Forced Rayleigh Scattering was used to determine the thermal diffusivity of a water-polymer solution. We found we could get consistent results to within 1% for fixed d, with our equipment. Solutions of up to 5% by weight of polymer in pure water, had the same thermal diffusivity of pure water. When we used methyl-red to color our solution, we observe two times. One time 1% which was due to the thermal diffusivity. The second time ’tl was due to some prOperty of methyl-red in a basic solution. FORCED RAYLEIGH SCATTERING IN LIQUIDS By . ._.’.\\¢ 0(- Verence D. Moore A THESIS Submitted to the College of Natural Science of Michigan State University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Department of Physics 1977 / 05 / {L1/ (/7 ACKNOWLEDGMENT I would like to express my deepest gratitude to Dr. J. A. Cowen for his patience and help in making this thesis possible. Many thanks to Mr. Daniel Edmonds for writing several of the computer problems and also for keeping the computer up and running. My special thanks is also extended to Mr. Edward Grabowski, who taught me all I know about running the computer and for building and maintaining most of the equipment. Also for all the long nights he and Mr. Edmonds spent keeping the computer alive so that I could take data. iii TABLE OF CONTENTS I. Introduction . . . . . II. The Experiment . . . . A. B. C. D. E. F. G. Introduction. . . Lasers. . . . . . The Computer. . . Detector. . . . . Furnace . . . . . Measurement of d. Theory. . . . . . III . ReSU1tS. O O O O O O 0 List of References . . . . iv OOOWQH 14 15 20 23 30 35 LIST OF TABLES Table 1. Power Output of Laser. . . . 2. RM4 Programs . . . . . . . . 3. Analyzing Programs . . . . . 4. Data at 22°C. . . . . . . . LIST OF FIGURES Figure 1. Experimental Setup. . . . . . . . . . . . . 2A. Cross-Sectional View of Furnace. . . . . . B. Cross-Sectional View of Control Block. . . C. Top View of Brass Control Block at 3. 4. 5. 6. Beam.Leve1 . . . . . . . . . . . . . . . . Geometry of the Two Intersecting Beams . . Volume Element dxdydz. . . . . . . . . . . Diffraction Geometry . . . . . . . . . . . View of Typical Signal With Heating Pulse. vi 19 21 23 27 34 I. INTRODUCTION While Dr. Cowen was in France on Sabbatical, he was introduced to the technique of using Forced Rayleigh Scattering to determine the thermal diffusivity of liquid. The purpose of me doing this experiment was twofold. First, to test the apparatus and furnace to see if they were working and also to estimate the accuracy of the measurements. Secondly, to determine if we could see any effect of rather large amounts (up to 5% by weight of a high molecular weight polymer on the thermal diffusivity of water-polymer solutions. The reason we only went up to 5% is because for solutions with greater amounts of polymer the solution can be considered a solid and we couldn't be able to get the solution into the cell. The polymer is Dow Separan AP30-high molecular weight (cv106) polyacrylamide. The interesting property of this polymer is that when it is mixed with water the solution becomes a gel. The gel's viscosity varies directly as the percent by weight of the polymer in the solution. The way the solutions were made up was that we weighed out an amount of water. Then we weighed the approximate amount of polymer to correspond to 1%, 2.5% or 5% by weight of polymer to water. When we first started we put the polymer in first then added the water, but this caused a mixing problem. At the polymer water interface there was a high concentration solution, with 2 almost pure water and polymer on either side. By heating the samples for several hours (24 to 72), we were able to get a uniform solution. Later I found that I could get a uniform solution in about 16 to 32 hours by putting in half the water first, followed by the polymer, then the rest of the water. The solution is normally clear so we had to add a dye to color the solution. The dye we used was methyl- red. The dye didn't readily mix with the solution, so heating the solution also help dissolve the dye. After we had a uniform mixture we filtered the solution, to get rid of large particles of dye. We use five micron filter paper when filtering the solution. We use a millipore filter with a syringe to filter the solution. The syringe was used to force the solution through the filter system. We then put the filtered solution into our cells. 3 II. THE EXPERIMENT A. Introduction In this experiment we are using forced Rayleigh scattering to determine the thermal diffusivity of liquids. This is done by using two lasers of different wavelengths. One wavelength is transmitted by the liquid. The other wavelength is absorbed by the liquid. This absorption of light is used to heat the sample in a small region. The heating laser beam is pulsed, heating the sample while the laser is on, and allowing it to cool while the laser is off. The heating beam, as it comes from the laser, is sent through two lenses which focus the beam. This allows us to choose the laser beam size inside the sample. The beam size we use is about one millimeter in diameter. Once the beam passes through the lenses it goes to a beam splitter. The beam splitter divides the beam into two equal intensity beams. The beam splitter consists of two mirrors; one fifty percent reflecting and the other one hundred percent reflecting. As the beam passes through the first mirror, fifty percent of the beam is reflected and fifty percent is transmitted. When the transmitted beam gets to the second mirror, one hundred percent of the beam is reflected. The beam splitter is positioned so that the two beams intersect inside of the sample. When the two beams intersect they form an interference pattern. This interference pattern causes a temperature grating in the region where the two beams meet. 4 Because of this the beam intensity in this region can be approximated as I = Iocos ky, where I0 is the maximum beam intensity inside the sample, and k = 2 F where d d is the distance between fringes. If we superimpose the second laser beam, which is transmitted by the liquid, over the two intersecting heating beams the transmitted laser beam sees a diffrac- tion grating. This grating is caused by the heat grating Because there is a temperature grating, the index of refraction is different for different areas inside this region where the heating is going on. Therefore, the effective path length the light must travel is different for different areas. Since the transmitted beam sees a diffraction grating as it passes through the sample, it forms a diffraction pattern after it leaves the sample. If we put a detector at the place where the first order’ maximum is formed, we can observe the heating and cooling of the sample. Since we expect the cooling is exponential and that the time constant is inversely proportional to the thermal diffusivity, we can measure the thermal diffusivity of the sample. Figure 1 shows our experimental setup. Zoacwmnon Wwano:.fimmmH x ¢ .9: I $75-20 Hmmmw wows mUHHdnoH ........ +.....:.Ti..l...i.\ commonOH [llllllly _ _ _ _ _ - EXPERIMENTAL SETUP FIGURE 1 B. Lasers In this experiment we are using two lasers. The first laser is Coherent Radiation - CRSOO K Krypton.Laser, used to heat the samples. The second is a small Helium- Neon laser, used as a probe. The Krypton laser has the ability to let the user choose from eight different wave- lengths in the visible; two in the infrared and one in the ultraviolet. We are only interested in the visible range. In Table l are listed the wavelengths )-in nanometers, the color, and the maximum output power, in milliwatts. The ability to change the wavelength of the heating laser is important because the heating comes from the absorption of the laser light. Our samples have different absorption curves, therefore, a wavelength which is absorbed by one sample may be transmitted by another sample. The second laser is a Spectra-Physics Model 120 Helium-Neon laser, which has a wavelength of 632.8 nanometers with a maximum output power of five milliwatts. This laser light should be transmitted by the sample (should not be absorbed). This is the light which is diffracted by the grating created by the heating laser. For the experiment to work the heating laser must be pulsed. This is done by a modulator. The one we use is a Coherent Associates Model 304 Acousto-Optic Modulation System. It is ideal for our use because it has the ability to put 75 to 80 percent of the laser power into the first order maximum. 7 TABLE 1 - POWER OUTPUT OF LASER Wavelength Power Measured Color Nanometers Milliwatts Power* Red - 1 676.4 120 25 Red - 2 647.1 500 180 Yellow 568.2 150 100 Green - 1 530.9 200 30 Green - 2 520.8 70 90 Blue - 1 482.5 30 8 Blue - 2 476.2 50 30 Blue - 3 468.9 5 5 *You only get the maximum power with all the laser mirrors clean. The modulator is controlled by a waveform generator and two pulse generators. The waveform generator is used to generate a pulse. The rise and fall time is approximately one microsecond. It is used also to trigger one of the pulse generators. The waveform generator allows you to vary the repetition rate while the pulse generator allows you to vary the heating pulse width. The second pulse generator is used as a trigger for the computer and the oscilloscOpe. It also allows you the choice of triggering on the positive or negative edge of the heating pulse. The triggering pulse generator is triggered by the heating pulse generator. We are using Tektronix Type 162 Waveform Generator and Tektronix Type 163 Pulse Generator. C. The Computer We used a PDP 8/e computer to take and analyze data. The computer is equipped with an analog to digital converter which takes the analog signal from the detector, converts' it into digital form and stores the numbers. The computer can take values which are between plus and minus one volts. The detector is nonlinear near zero volts, there- fore we don't use negative values. This is where the oscilloscope comes in. The oscillOSOOpe is used to make sure that the signal from the detector is between zero and plus one volts. It is also used to adjust the equipment until we obtain the maximum signal size. The oscilloscope allows us to measure the size of the D.C. signal from scattered light from dirt on the cell walls, or small bubbles or large particles in the sample. By large particles we mean particles whose diameter is larger than five microns. This then allows us to adjust the D.C. offset on the detector to minimize the D.C. level. We can also use the oscillosc0pe to get an approximate value for 2k, the time constant. From this we can tell which program to use in taking the data. The computer has two basic programs for taking data. The first one is RM4 which has three versions. The difference between the three versions is the time between data points. See Table 2. 9 The RM4 series of programs was written by Mr. Daniel Edmonds. TABLE 2 - RM4 PROGRAMS Program Time Between Points Data Name In.Microsecongs File RM4A5 3.6 RM4E__.DA RM4A6 7.2 RM4G__.DA RM4A7 14.4 RM4F__.DA The second data program is Basic Averager. Both programs do the same thing. The main difference is in the time between points. In basic averager we can choose any value between 30 microseconds and 4095 microseconds to be the time between points. Where in.RM4 we have the choice of 3.6 microseconds, 7.2 microseconds, and 14.4 microseconds. The other difference is in the number of sums. In RM4 you can do 2x number of sums. Where x is equal to any positive integer between 0 and 11. In Basic Averager you can take any integer number of sums up to 4095. This is because in RM4 you read in the number of sums from the switch register, while in Basic Averager the number of sums is read in from the tele- type. Basic Averager came With the computer as part of the software. The programs are designed to take the numerical value of each point every x microseconds. (x is the time you pick determined by the program you choose), and add this value to the sum of these points from previous scans. It repeats this process y times; where y is equal to the 10 number of sums you ask it to do. The program gives us the sum at each point. The reason we want to take the total sum is to eliminate noise. If you look at the signal from the detector on the oscillosc0pe you will see that it h0ps up and down. This hopping is caused by large particles inside the sample drifting through the beam. By taking a large number of sums, in some cases 2048, we can cancel out most of the noise, which gives us a nice smooth curve. Since the noise is random and the signal is always there, the randomness of the noise should cancel itself out. For analyzing the data we have five interacting analyzing programs written in Fortran IV. The programs are basically the same. They differ only in the number of parameters and the parameters themselves used in fitting the data. The programs were written by Mr. Edward Grabowski. He took the program from a book.1 The reason they are called interacting programs is because they allow you to interact with the computer. You can have the computer display the data, display the data plus the best curve superimposed, display their difference, and you can have it plot the data. It is a least-squares fitting program which allows you to vary up to ten parameters. It is called Curfit: Program 11-5. The reason for five analyzing programs is that some- times there are two times and if one time is much longer than the other then we can assume we had a single time 11 with a leping base line. This is used when you are only interested in the shorter time. If you want the longer time or both times you want a program that will fit two different times. This is the purpose for the first three programs. One to fit a single time, one to fit a single time with a sloping base line, and one to fit two times. For the programs to work properly at least one third of the data must be base line. The reason for the next two programs is caused by having the light being homodyne. In the program above we say that the light is heterodyne. By heterodyne we mean that the light causing the D.C. level of the detector has a definite phase with respect to the light causing the first order maximum. The detector is a square law device! so that: 1 )2 (E +E hetv‘r DC signal since Esignalvchos eXp(-t/’Ck) 2 IhetV‘E DC + 2EDC Eos exp(-t/’tk) + E03 2 eXp(-2t/ TR) If ED,)) EOS then: 2 Ihet3==EDC + ZEDC EOS exp(-t/1&) If there is no D.C. signal or if the D.C. signal has no definite phase relationship to the signal then we call the signal homodyne and using the same detector 2 2 . Ih0m0“"EDC i E 31g 2 2 IhOmO~EDC + E 08 eXp(-2t/’C'k) 12 ’L Thus we must be able to analyze data with two times Lk A and (dc corresponding to a mixture of heterodyne and homodyne signals. This is the reason for the fourth A A. and fifth programs. One to fit Lk and Lk with a straight A! base line and another to fit Tk and LR with a sloping base line. For a listing of the programs see Table 3. The reason we use these programs is because they can calculate’tk much more accurately than we can by plotting the data points on semilog graph paper. Also they have a subroutine that calculates the difference between the data and its calculated curve. This is helpful because most times you can't see any difference by just looking at the fit. The computer displays the data and its calculated curve simultaneously superimposed on each other so you can see how good the fit is. 13 TABLE 3 - ANALYZING PROGRAMS PROGRAM NAMES FIT TO FAST 1 A(l) + A(Z) * EXP(-X/A(3)) heterodyne; fitting one time FAST 2 A(l) + A(Z) * EXP(-X/A(3)) + A(4) * X heterodyne; fitting one time with leping base line FAST 3 A(1) + A(2) * EXP(-X/A(3)) + A(4) * EXP(-2X/A(3)) heterodyne and homodyne; fitting TR and TR .2- FAST 4 A(1) + A(2)*EXP(-X/A(3) + A(4)>'HCBHDCS >mummn0m mummm nonnHoH wHoow amt omHH memm ooDnHoH wHoow >mdmmn0m >Hcawsca FIGURE ~2A- CROSS-SECTIONAL VIEW OF FURNACE 18 Hole for Thermistor T: f . Heating tape Cooling coil 11 I b s 1 or aser beam 1% 1 FIGURE -2B- CROSS-SECTIONAL VIEW OF CONTROL BLOCK l9 FIGURE -2C- TOP VIEW OF BRASS CONTROL BLOCK AT BEAM LEVEL 20 F. Measurement of d On our beam splitter is a micrometer. The micrometer is used to vary the distance between the mirrors. We use the readings on the micrometer to determine d (spacing between fringes). Following is the theory that justifies this. The way we obtained a grating is by splitting the heating beam into two equal beams and letting them intersect inside the sample. This will cause a diffraction pattern in the region where they overlap. The spacing between the fringes, d, is equal to d = for small sin «9 /2) angles sin ( 9/2)==9/2. See Figure 3. Therefore d = _;L_ where 9 is defined as the angle between the two beams..9 But for small angles Q a x/D where x is defined as the distance between the two mirrors of the beam splitter, and D is the distance between the beam splitter and where the two beams intersect. This leads to the equation: d = QED x If we let x = Y - C, where C is equal to a constant and Y is equal to a micrometer reading which is attach to the beam splitter. C is put in the equation, because the micrometer doesn't read zero when the distance between the mirrors is zero. Therefore d = AD ‘Z-C We can measure D and we select.1. Therefore if we measure d and plot 1_ using our least square program we can get d C and Y from the program. Now if we plot d vs. AD Y - C 21 - We) FIGURE 3 - GEOMETRY OF THE TWO INTERSECTING BEAMS - - - Is the fringes 22 we get a curve that translates the dial setting on the micrometer into d (fringe spacings). This has a great advantage over the old system of measuring fringes. In the old system it was almost impossible to get the same d back after you have changed it. 'But with the new system all we have to do is set the micrometer to the setting it was previously and we have the same d value. In the old system we projected the image of the intersecting beam on a screen superimposed over a scale and counted the number of fringes for a certain length. 23 G. Theory The thermal diffusivity of a sample which has one dimensional heat flow can be derived as follows.4 The total amount of heat entering the differential face dxdz at y is given by dQ 5: -dxdz (K( %))dt See Figure 4. FIGURE 4 - VOLUME ELEMENT dxdydz where K is defined as the thermal conductivity and Q is defined as heat. To find the amount of heat leaving the element at y + dy, let _ $1.1; F(y.T) — K(dy By Taylor's series expansion: F(y + dy.T) = F(y.T) + QT dy dy _QT_<1_.QI — K dy + dy (K dy) dy This leads to the fact that dQ (y+ dy) = dxdz (K3—;+—q— (Kgl)dy) dt 24 If we add to this dE, which is defined as the total quantity of heat which goes into increasing the internal energy of the volume element: dE= CP dxdydzd £11 dt where C is defined as the specific heat, and P is defined as the density. Also if we add ng which is defined as the total amount of heat generated in the volume element. Where, ng = q" th = q" dxdydzdt where q" is the rate at which heat is being generated internally. Therefore using conservation of energy we have: dQ(y) + ng - dQ(y + dy) -dE =0 dQ(y) + ng = dQ(y + dy) + dB This leads to -dxdz (K %E) dt + q" dxdydz dt +-3‘-y < =-dxdz ((K%— T) dt+ K 91%) dy ) dt + CP dxdydz—ELI dt dt Divide by dxdydzdt which reduces to q" = -‘g_ dT dy (K dy)'+ CP dt Therefore, 2 d T dT q" = -K + CP gyz dt d T dT q" + K.-—2 - CP -— = 0 which leads to 2 dT " CP dT 2+q/K: K dt d Y where DTh =_K_ which is defined as the thermal diffusivity. CP 25 Therefore: __d2T+lqn=1_ Ell dyz K DTh dt where in our case q" = Ole, where 9 is defined as the light absorption coefficient of the sample. Ie is defined as the incident intensity. Therefore; lgl_dT_l . D dt 2 — K OI (equation 1) Th If the depth of penetration (9'1) is very large compared to d (the distance between the fringes), and if d is very small compared to the sample thickness, then at the end of the heating pulse (t = 0) there is a temperature distri- bution which varies with position. .ATT(0,y) = T (0)cos ky (equation 2) T is the temperature amplitude and K.= Zfit . Now if we assume a solution to equation 1 for time t = 0, that is I8 = 0, then if we replace T with AI? using equation 2 then: T(t,y) = T(t) cos ky = T(O) exp(-t/’tQ cos ky Now if we insert this into equation 2, we have: 2 d j; d§A.T) _ 33;: (AT) - pm dt - 0 Therefore: 2 9_ (AI) = 1 MAT) dyz DTh dt 2 g_2 (T (0) eXp(-t/‘tk) cos ky) dy %_Th—3E ('T’ (o) exp(-t/“L‘k) cos ky) 26 Now take the derivative of both sides 2 -k T (9) exp(-t/'Tk) cos ky D-1 T (0) exp(-t/‘tk) cos ky Th1%; which reduces to DTh TR which leads to’tk = l D 2 Thk =_K; and k = 24? PC d but DTh therefore: T=£§d2 k K(Fr)' The temperature decreases exponentially with a time constant of‘tk, which depends on d2, and l/DTh or d2 , P, C, and l/K. This shows that if we measure the “relaxation time of a liquid, and know what d is, we can calculate the thermal diffusivity of the liquid. And from the thermal diffusivity we can obtain the thermal conductivity of the liquid.5 When the two beams meet there is a temperature grating formed. Since the liquid has a temperature variance which can be approximated as T = T cos ky We can say it has an index of refraction that varies as A11: -n' (l + cos ky) where n' is the peak value for the change in the index of refraction divided by two. 27 This is caused by the fact that when the sample is heated its density changes which changes its index of refraction. The next question is why this change in density doesn't change Tk since DTh (the thermal diffusivity) is dependent on density. This is because heating the sample we only change the temperature a few millidegrees. Therefore, the change in the density is very small. If we have a plane wave in the region of the heat grating we will see that some of the light is refracted. See Figure 5. P(a,b) FIGURE 5 - DIFFRACTION GEOMETRY The light intensity at a point P(a,b) outside of the sample can be determined as follows: =‘fa2 + (b-y) Where D is the distance from the heat grating to point P(a,b). 28 a is the x component of D and b is the y component of D measured from y = 0. Let n = n(y) = n -n' (l + cos ky). o The phase shift through the medium: 9 (y) = 3711 t n(Y) where t = the thickness of the temperature grating. Ais the wavelength of the transmitted beam. The total phase shift of point P is: 9 total = 91-17-5— n(y) + 2*” ‘fa2 + (b-y)2 at 9 = 0 plane, let E = E0 Also lets assume D>>d. Therefore at point P; I =53 exp (i 9 total) dyl2 I em = I52 exp (123-1: n + .4.. WW «w There is no closed form solution to this equation but you can get a numerical solution. All we are interested in is the relative position of point P and we can get a good approximation of this by assuming that in the sinewave is a step function. If we let the sinewave be a step function and assume that the grating is caused by intensity variation and not phase variations our problem reduces to the one of a classical multislit diffraction grating.6 The major difference is when you take the Fourier transform of a sinewave you get a sinewave back, but the Fourier transform of a step function is a family of sinewaves. Therefore by replacing a sine function with a step function the theory will predict a first order maximum, 29 a second order maximum, a third order maximum, and so on. In our case all we should get is a first order maximum. The reason we feel that the grating is caused by an intensity variation and not a phase variation is because if it were a phase variation a change in the intensity of the heating beam should change the position of the first order maximum. We never saw this change in position. 30 III. RESULTS There were two things which we tried to do in this experiment. 1) Demonstrate that the apparatus and furnace worked and estimate the accuracy of measurement. 2) Determine if we could see the effect of rather large amounts (up to 5%) of a high molecular weight polymer on the thermal diffusivity of water-polymer solution. The apparatus including the furnace did work, it was possible to take data for d ranging from 15 microns to 120 microns, and at temperatures from room temperature to 800C. If the samples were clean and if not much local heating of the samples occur we can get consistent results to within 1% for fixed d. If we vary d, the results are consistent to several percent which is the accuracy to which we can measure d. Table 4 gives a set of results obtained on pure water, 1%, 2.5%, and 5% polymer at fixed d = 50.3 microns and T = 220C. Data was taken with RM4A6 and analyzed at FAST 2. The variation is less than 1%. This demonstrates that we did not observe any effect of the polymer on the thermal diffusivity of the solution in this concentration range. If we compute the thermal diffusivity of the solution: 1 d2 1 (50.3 x 10'4”“)2 D = = Th 41r2 “ck 41r2‘ 374 x 10‘6 sec = 1.71 x 10'3 cmZ/Sec 31 Compare it with DTh = §%— obtained from data on water in the Handbook of Chemistry and Physics. = 1.43 x 10"3 cal/sec - cm - 0K Th (.997 gm/cmz) (.998 cal/gm - OK) = 1.44 x 10'3 D cm2/sec We see that the difference is rather large, approximately 19 percent. We do not understand this large discrepancy but it may come in part from air dissolved in the water and in part from a possible error in d. The solution of polymer in water with a little methyl- red to absorb the light gave interesting results which we now think we understand. We saw two relaxation times,’ts («~.400 microseconds) and'ti (many milliseconds). '1; was dependent on d2, was independent of temperature or of the polymer concentration. ‘11 was independent of d, strongly dependent on temperature and on polymer concen- tration. After many experiments with different concen- trations of polymer and also with the dye dissolved in water whose pH we changed by adding an acid or a base, we found that the long time was apparently due to the bleaching of the dye. ’tl was very long in basic solutions and vanished in acid solutions. (The color of the solution also changed). We think that‘ti was the time for recovery of the dye and therefore it did not depend on d. The apparent dependence of'1a on polymer concentration was 32 due to the fact that the polymer was strongly basic. A 1% solution has a pH of 10.3 so that changing the concen- tration also changed the pH. We do not understand why ‘11 depends so strongly on either pH or temperature but it must be related to the change of Optical and chemical properties of this organic dye. “ts was the thermal relaxation time and as demon- strated above in Table 4 it was independent of polymer concentration and also of temperature. Figure 6 is a picture of a typical signal with heating pulse. 33 TABLE 4 - DATA AT 220C % Polymer Microseconds 0 376 1% 375 2.5% 376 5% 371 Average 374 i 2 34 FIGURE 6 VIEW OF TYPICAL SIGNAL WITH HEATING PULSE 35 LIST OF REFERENCES P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill (1969), p. 237. G. G. Hacker, III, D. M. Eshelman, and R. L. Schmidt, American Journal of Physics, 45, 310 (1977). RCA, Linear Integrated Circuits, RCA (1970), p. 288. P. J. Schneider, Conduction Heat Transfer, Addison- Wesley, (1955), p. 3. H. Eichler, G. Salje, and H. Stahl, J. Appl. Phys., 44, 5383 (1973). E. Hecht, A. Zajac, Optics, Addison-Wesley (1974), p. 343. RR mm” ”"71111111171111[ill/illlflmllfll’l