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SSD-TDR-62-127 - Volume VI

INFRARED TRANSMISSION STUDIES

FINAL REPORT, VOLUME VI

MIE SCATTERING AND ABSORPTION CROSS SECTIONS OF

fl ALUMINUM OXIDE AND MAGNESIUM OXIDE

1 Gilbert N. Plass

L '• Contract No. AF 04(695)-96(5 ~ Project No. 4479-730F

. • Task No. 447904

F--C-5 27 May 1963

Prepared for -

LAM SPACE SYSTEMS DIVISIONAIR FORCE SYSTEMS COMMAND jug 2 •° l

Los Angeles, California

AERONUTRONIC DIVISIONFORD MOTOR COMPANY Newport Beach, California

SSD-TDR-62-127 - Volume VI

INFRARED TRANSMISSION STUDIES

FINAL REPORT, VOLUME VI

MIE SCATTERING AND ABSORPTION CROSS SECTIONS OF

ALUMINUM OXIDE AND MAGNESIUM OXIDE

Gilbert N. Plass

Contract No. AF 04(695)-96Project No. 4479-730F

Task No. 447904

27 May 1963

Prepared for

SPACE SYSTEMS DIVISIONAIR FORCE SYSTEMS COMMAND

Los Angeles, California

AERONUTRONIC DIVISIONFORD MOTOR COMPANY Newport Beach, California

jK.

ABSTRACT

The scattering and absorption of electromagnetic radiationby spherical particles of aluminum oxide and magnesium oxide is cal-culated from the Mie theory. The complex index of refraction wastaken from the best available experimental data. Tables of theefficiency factors for scattering, absorption, and extinction aregiven for particle radii from 0.1 to 5 .1 and a range of wave-lengths from 0.5 to 10

•" ii

-'

CONTENTS

SECTION PAGE

I INTRODUCTION ........... ................... 1

2 MIE ABSORPTION AND SCATTERING CROSS SECTIONS 2

TABLES

TABLE PAGE

1 COMPLEX REFRACTIVE INDEX OF ALUMINUM OXIDEAND MAGNESIUM OXIDE ........ ............... 3

2 EFFICIENCY FACTORS FOR ABSORPTION AND SCATTERINGBY ALUMINUM OXIDE ........ ................. 6

3 EFFICIENCY FACTORS FOR ABSORPTION AND SCATTERINGBY MAGNESIUM OXIDE ......... ................ ii

iii

SECTION I

INTRODUCTION

Particles of aluminum oxide and magnesium oxide occur inmany different types of flames including those from solid propellantrockets. In order to understand the emissivity of these flames itis important to know the contribution of these particles.

The absorption and scattering cross sections of theseparticles were calculated from the Mie theory for a range of particleradii from 0.1 to 5 .1u and a range of wavelengths from 0.5 to 10P .The complex index of refraction was chosen after a study of the bestavailable measurements.

SECTION 2

MIE ABSORPTION AND SCATTERING CROSS SECTIONS

Measurements of the complex index of refraction of sapphire(aluminum oxide) have been made by a number of authors. The realpart of the index has been measured only at room temperature. The

values given in Table I were adopted after plotting the measurementsof Malitson, Murphy, and Rodney1 and Malitson 2 aad using the dis-persion equation given in the latter article. The imaginary partof the index has been measured at temperatures of 1000 0 C as well asat room temperature. The values given in Table 1 are obtained bycombining absorption measurements of several authors 3 , 4 made at 10000 C.

The measurements of the complex index of refraction formagnesium oxide have recently been reviewed by Bauer. 5 The valuesgiven in his figures are based on measurements of Burstein, Oberly,and Plyler 6 and on unpublished data of McAlister and a review ofBallard, McCarthy, and Wolfe.

ii. H. Malitson, F. V. Murphy, W. S. Rodney, J. Opt. Soc. Amer. 48,72 (1958).

21. H. Malitson, J. Opt. Soc. Amer. 52, 1377 (1962).3 R. D. Olt, "Synthetic Sapphire and Infrared Optical Material", Linde

Bulletin F917-A (1958).4 U. P. Oppenheim, U. Even, J. Opt. Soc. Amer. 52, 1078 (1962).5 E. Bauer, "Emissivity in the Near Infrared of Particulate Matter inSolid Propellant Rocket Flames", Aeronutronic Report (to bepublished, 1963).

6 E. Burstein, J. J. Oberly, E. K. Plyler, Proc. Ind. Acad. Sci. 38,388 (1948).

-2-

TABLE I

COMPLEX REFRACTIVE INDEX OF ALUMINUM OXIDE AND MAGNESIUM OXIDE

Aluminum Oxide Magnesium OxideWavelength nI n 2 nI n2

0.5p 1.77 10-6 1.725 5(10)-6

1 1.75 10-6 1.720 7(10)-6

2 1.74 10-6 1.705 1.2(10)-5

3 1.71 10-6 1.690 1.6(10)-5

4 1.68 10-5 1.665 2(10) -5

5 1.63 10-4 1.64 5(10)-5

6 1.54 2.2(10)-4 1.60 10-4

8 1.35 3.3(10)-4 1.51 8(10)-4

10 1.09 5(10) 1.44 9(l0)4

-3-

Only a few measurements have been made of the size distri-bution of alumina particles from solid propellant rocket flames,while none have been reported for magnesium oxide particles. Themeasurements of Sehgall indicate that the maximum of the curve whichgives the number density of alumina particles as a function of sizeoccurs at a radius of about 0.2p . However, some particles areobserved with radii out to 3. and a few particles with much largerradii. No size distribution is adopted in this report. The readermay average the final results over any desired distribution.

The index of refraction used in this calculation is forsapphire. Actually there is considerable X-ray and electron dif-fraction evidence to show that the particles are polycrystalline.The characteristic size of the crystalline domain is O.01p . Sincethe domain size is much smaller than the wavelengths of light con-sidered here and since the light beam encounters only a small numberof different domains in a particle with a typical radius of 0.2P,it may be concluded that the effects due to the polycrystalline natureof the particles is probably small. The difference in index of refrac-tion between the different crystalline directions in alumina is smallas is the difference in index between the two crystalline forms,o and Y-alumina, which occur under these conditions. Thus theelectromagnetic wave does not encounter any regions of greatlychanged index of refraction. Both the size distribution and poly-crystalline effects are considered in more detail by Bauer. 5

The scattering and absorption of an electromagnetic waveby a spherical particle can be calculated from the Mie 8 theory whenthe complex index of refraction is known. Some time ago an efficientprogram for the calculation of Mie scattering and absorption crosssections was developed by Stull and Plass. 9 The fundamental equationsof the Mie theory and the details of the calculation are given in thatpaper. More recently some improvements in the original program weremade by P. J. Wyatt which decreased the running time and increased theallowable range of the parameters.

7R. Sehgal, "An Experimental Investigation of a Gas-Particle System",Jet Prop. Lab. Tech. Rep. 32-238 (March, 1962).

8G. Mie, Ann. Physik 25, 377 (1908).

9 V. R. Stull, G. N. Plass, J. Opt. Soc. Amer. 50, 121 (1960).

-4-

The cross sections obtained in this manner from the MieTheory are given in Tables 2 and 3 for aluminum oxide and magnesiumoxide respectively. The first two columns give real (nl) andimaginary (n 2 ) parts of the index of refraction. The wavelength isgiven in the third column. The radius of the particle is given inthe fourth column. The efficiency factors, Q, for absorption,scattering, and extinction are given in the last columns. The ef-ficiency factor is defined as the cross section divided by the areaof the particle.

It appears from these results that the emissivity of aluminumand magnesium oxide is relatively small for the number and size ofparticles reported in the literature as occurring in rocket exhausts.Only the presence of a fair number of relatively large particlescould appreciably increase the emissivity to the blackbody level.Thus, the most important effect of these particles appears to be intheir ability to scatter radiation. The application of the Mie theoryresults to actual rocket motors has been considered in more detailby Bauer. 5 He also obtains approximate solutions for the radiativetransfer equations which describe the scattering of radiation inrocket exhausts by these particles.

ACKNOWLEDGMENT

It is a pleasure to acknowledge the assistance of I. Brownand R. Buley in obtaining these results from the IBM 7090 computer.

-5-

TABLE 2

EFFICIENCY FACTORS FOR ABSORPTION AND SCATTERING BY ALUMINUM OXIDE

nI n2 Wavelength Radius Qabs Qsca Qext

1.77 10- 6 0.5p 0.1l 4.74 (10)-6 1.074 1.074

0.2 1.59 (10)-5 4.68 4.68

0.4 6.08 (10)-5 2.21 2.21

0.6 4.26 (10)-5 2.78 2.78-5

0.8 5.01 (10) 2.33 2.33-5

1.1 8.85 (10) 2.09 2.09

1.6 9.94 (10)-5 2.38 2.38

2.1 1.183(10)-4 2.2"2 2.22

2.6 1.292(10)-4 2.12 2.12-4

3.1 2.14 (10) 2.22 2.22-4

5.1 2.95 (10) 2.14 2.14

1.75 10-6 i/ 0.1,m 1.32 (10)- 6 7.44 (10)-2 7.44 (10)-2

0.2 4.63 (10) 1.018 1.018-5

0.4 1.603(10) 4.68 4.68-5

0.6 3.26 (10) 3.90 3.90-5

0.8 4.66 (10) 1.813 1.813-5

1.1 4.30 (10) 2.97 2.97-5

1.6 7.20 (10) 2.25 2.25-5

2.1 7.84 (10) 1.952 1.952-5

2.6 8.50 (10) 2.48 2.48-4

3.1 1.198(10) 2.49 2.49

5.1 1.533(10)-4 2.36 2.36

-6-

TABLE 2 (Continued)


nI n 2 Wavelength Radius Qabs Qsca Qext

1.74 10-6 2p 0.11 5.09 (10)- 7 4.32 (10)-3 4.32 (10)-3

0.2 1.327(10)-6 7.27 (10)-2 7.27 (10)-20.4 4.59 (10) 9.90 (0) 9.90 (10)1

-60.6 9.21 (10) 3.12 3.12

-50.8 1.627(10) 4.68 4.68

-51.1 2.01 (10) 3.69 3.69

-51.6 4.03 (10) 1.677 1.6772.1 3.65 (10) 2.62 2.62

2.6 1.046(10)- 4 3.19 3.19-5

3.1 6.00 (10) 1.868 1.868

5.1 9.92 (10)-5 2.49 2.49

1.71 10-6 3ý 0.Me 2.82 (10)-7 7.91 (10)-4 7.91 (10)-4

0.2 7.53 (10)-7 1.299(10)-2 1.299(10) -2

0.4 2.12 (10)-6 2.16 (10)" 2.16 (10)-1

0.6 4.59 (10)- 6 9.10 (i0)-I1 9.10 (I0)-I1

0.8 9.80 (10)-6 2.55 2.55

1.1 1.496(10)-5 3.92 3.92-5

1.6 1.991(10) 4.33 4.33

2.1 3.29 (10)-5 2.51 2.51-5

2.6 3.39 (10) 1.919 1.919

3.1 5.66 (10) 2.40 2.40

5.1 5.48 (10)- 5 2.08 2.08

-7- 0

TABLE 2 (Continued)



1.68 10- 41 0.1i 1.658(10)-6 2.33 (10)-4 2.35 (10)-4

0.2 5.60 (10)-6 3.78 (10)-3 3.78 (10)-3

0.4 1.404(10)-5 6.29 (10)- 2 6.29 (10)-2

0.6 2.64 (10)-5 3.13 (10)- 3.13 ( 10)-I

0.8 4.52 (0)-Y 5 8.35 (10)-1 8.35 (10)-I

1.1 9.93 (10)-5 2.50 2.50

1.6 1.651(10)-4 4.42 4.42

2.1 2.14 (10)-4 4.71 4.71

2.6 2.44 (10)-4 3.60 3.603.1 2.79 (10)- 4 2.36 2.36

5.1 1.300(10) 3.46 3.46

1.63 10-4 5/ 0.1/ 2.17 (10)- 5 8.44 (b0)- 5 1.061(10)-4

0.2 4.72 (10)-5 1.361(10)- 3 1.408(10)-3

0.4 1.063(10)-4 2.23 (10)- 2.24 (10)- 2

0.6 1.875(10)-4 1.144(10)-1 1.145(10)-1

0.8 2.95 (10)-4 3.43 (10)- 3.44 (10)-1

1.1 5.40 (10)-4 9.58 (10)- 9.59 (10)-1

1.6 9.52 (10)- 4 2.66 2.66

2.1 1.558(10)-3 4.16 4.16

2.6 2.18 (10)-3 4.70 4.70

3.1 2.68 (10)-3 4.11 4.115.1 3.85 (10)-3 1.960 1.963

* -8-

TABLE 2 (Continued)


n1 n2 Wavelength Radius Qabs Qsca Qext

1.54 2.2(10)-4 6, 0.11A 4.42 (10)-5 3.16 (10)-5 7.58 (10)-5

0.2 9.08 (10)- 5 5.07 (10)-4 5.98 (10)-4

0.4 1.965(10)-4 8.20 (10)-3 8.39 (10)-3

0.6 3.28 (10)-4 4.17 (10)-2 4.21 (10)-2

0.8 4.91 (10)-4 1.295(10)-1 1.300(10)-1

1.1 7.94 (10)-4 4.04 (10)- 4.05 (10)-I

1.6 1.667(10)-3 1.275 1.277

2.1 2.14 (10)-3 2.33 2.33

2.6 3.23 (10)-3 3.52 3.52

3.1 3.82 (10)-3 3.90 3.90

5.1 5.88 (10)- 3 2.95 2.95

1.35 3.3(10)-4 8 r 0.ltA 4.97 (10)-5 4.70 (10)-6 5.44 (10)-5

0.2 1.140(10)- 4 7.51 (10)-5 1.891(10)-4

0.4 2.39 (10)-4 1.195(10)-3 1.434(10)-3

0.6 3.74 (10)-4 5.99 (10)-3 6.36 (10)-3

0.8 5.25 (10)- 4 1.855(10)-2 1.907(10) -2-4 -2 -

1.1 7.84 (10) 6.24 (10) 6.32 (10)2

1.6 1.274(10Y)-3 2.22 ( 10)-I 2.23 (10)-1

2.1 1.868(10)-3 4.65 (10)- 4.67 (10)-1

2.6 2.62 (10)-3 8.64 (10)-I 8.67 (10)-1-3

3.1 3.11 (10) 1.292 1.295

5.1 5.69 (10)- 3 3.06 3.07

-9- 0

TABLE 2 (Continued)


nI n 2 Wavelength Radius Qabs Qsca Qext

1.09 5 (10)-4 lo0 0.11 5.69 (10)-5 1.449(10)-7 5.71 (10)-5

0.2 1.612(10)-4 2.30 (10)-6 1.636(10)-4

0.4 3.24 (10)-4 3.63 (10)-5 3.60 (10)-4

0.6 4.91 (10)-4 1.795(10)-4 6.70 (10)-4

0.8 6.62 (10)-4 5.48 (10)-4 1.21 (10)-3

1.1 9.25 (10)-4 1.823(10)-3 2.75 (10)-3

1.6 1.382(10)-3 6.86 (10)-3 8.24 (10)-3

2.1 1.846(10)-3 1.602(10)-2 1.786(10)-2

2.6 2.31 (10)-3 2.84 (10)-2 3.07 (10)-2

3.1 2.80 (10)-3 4.35 (10)-2 4.63 (10)-2

5.1 4.78 (10)-3 1.47 (10)- 1 1.52 (10)-1

S -10-

TABLE 3

EFFICIENCY FACTORS FOR ABSORPTION AND SCATTERING BY MAGNESIUM OXIDE


-6 -51.725 5(10) 0.5p 0.1i/. 2.34 (10) 0.950 0.950

0.2 8.24 (10)-5 4.64 4.64

0.4 1.723(10)-4 1.620 1.620

0.6 3.49 (10)-4 3.55 3.55-4

0.8 2.88 (10) 2.13 2.13-4

1.1 3.36 (10) 2.10 2.10-4

1.6 4.38 (10) 2.42 2.42

2.1 5.89 (10)- 4 2.11 2.11

2.6 6.56 (10)- 4 2.20 2.20

3.1 7.59 (10)-4 2.14 2.14

5.1 1.226(10)- 3 2.17 2.17

1.72 7(10)- i6 0.i & 9.65 (10)-6 6.94 (10)-2 6.94 (10)-2-5

0.2 3.27 (10) 0.936 0.936-4

0.4 1.157(10) 4.63 4.63

0.6 2.23 (10)-4 3.99 3.99-4

0.8 2.32 (10) 1.629 1.629

1.1 3.66 (10)-4 3.23 3.23-4

1.6 4.05 (10) 2.08 2.08-4

2.1 4.27 (10) 2.29 2.29-4

2.6 5.88 (10) 2.54 2.54

3.1 9.53 (10)- 2.21 2.21-45.1 8.85 (10) 2.06 2.06

-11-

0

TABLE 3 (Continued)



1.705 1.2(10)-5 2t 0 !e 6.54 (10)-6 4.01 (10)- 3 4.01 (10)"3-5 -2 -

0.2 1.660(10) 6.70 (10) 6.70 (10)2

0.4 5.54 (10)-5 8.98 (10)-1 8.98 (10)-1-4

0.6 1.153(10) 2.96 2.96-4

0.8 1.992(10) 4.57 4.57

1.1 2.35 (10)-4 4.00 4.00-4

1.6 3.66 (10) 1.714 1.714

2.1 8.07 (10)-4 2.67 2.67-4

2.6 5.20 (10) 2.89 2.89-4

3.1 5.95 (10) 1.929 1.929

5.1 1.021(10) 2.55

1.69 1.6(10)-5 3/ 0.11A 5.33 (10)-6 7.56 (10)- 4 7.62 (10)-4

0.2 1.293(10)- 5 1.240(10)-2 1.241(10) -2

0.4 3.49 (10)-s 2.05 (10)-1 2.05 (10)-I

0.6 7.32 (10)- 8.60 (10)1 8.60 (10)1-4

0.8 1.567(10) 2.38 2.38-4

1.1 2.29 (10) 3.70 3.70-4

1.6 3.24 (10) 4.48 4.48-4

2.1 4.63 (10) 2.59 2.59-4

2.6 5.43 (10) 1.835 1.835

3.1 6.36 (i0)- 4 2.10 2.10

5.1 8.96 (10)- 4 1.923 1.924

-12- 0

TABLE 3 (Continued)


nI n 2 Wavelength Radius Qabs Qsca Qext-5 -6 -4 -4

1.665 2(10) 4 0.1i1 4.69 (10) 2.25 (10) 2.30 (10)-5 -3 -3

0.2 1.170(10) 3.65 (10) 3.66 (10)

0.4 2.83 (10)-5 6.05 (10)-2 6.05 (10)-2

0.6 5.30 (10)- 5 3.00 (10)-1 3.00 (10)-I-5 -1 -1

0.8 8.96 (10) 7.99 (I0) 7.99 (I0)

1.1 1.956(10)- 4 2.38 2.38

1.6 3.27 (10)- 4 4.30 4.30

2.1 4.32 (10)-4 4.76 4.76

2.6 5.00 (i0)-4 3.78 3.78-4

3.1 5.64 (i0) 2.50 2.50-3

5.1 1.082(10) 2.94 2.94

1.64 5(10)-5 51 0.I1p) 1.080(10)-5 8.66 (I0)-5 9.74 (I0)-5-5 -3 -3

0.2 2.31 (I0) 1.396(10) 1.419(10)-5 -2 -2

0.4 5.29 (i0) 2.29 (I0) 2.30 (I0)

0.6 9.35 (I0)-5 1.177(10)-1 1.178(10)-1-4 -1 -1

0.8 1.475(10) 3.54 (10) 3.54 (10)

1.1 2.73 (I0)-4 9.91 (i0)- 9.91 (i0)-I-4

1.6 4.76 (I0) 2.72 2.72

2.1 7.79 (10)-4 4.22 4.22-3

2.6 1.099(10) 4.75 4.75

3.1 1.393(10)-3 4.14 4.14

5.1 1.946(10)-3 2.05 2.05

-13-

TABLE 3 (Continued)


n n2 Wavelength Radius Qabs Qsca Qext

1.60 10-4 6m 0.1y 1.764(10)-5 3.76 (10)-5 5.52 (10)-5

0.2 3.98 (10)-5 6.04 (10)- 4 6.44 (10)-4

0.4 8.63 (10)-5 9.82 (10)-3 9.91 (10)-3

0.6 1.455(10)-4 5.05 (10)-2 5.06 (10)-2

0.8 2.21 (10)-4 1.580(10)-1 1.582(10)-1-4 -1 -1

1.1 3.67 (10) 4.97 (10) 4.97 (10)-4

1.6 8.43 (10) 1.661 1.662

2.1 1.020(10)-3 2.69 2.69

2.6 1.511(10)- 3 3.99 3.99

3.1 2.03 (10)-3 4.42 4.42

5.1 3.16 (10)-3 2.27 2.27

1.51 8 x 10-4 8/ 0.01 1.198(10)-4 9.08 (10)-6 1.289(10)-40.2 2.51 (10)- 4 1.455(10)- 4 3.97 (10)-4

0.4 5.25 (10)- 4 2.34 (10)- 3 2.86 (10)-3-4 -2 -2

0.6 8.37 (10) 1.190(10) 1.274(10)

0.8 1.206(10)-3 3.76 (10)-2 3.88 (10)-2

1.1 1.876(10) 1.305(10)1 1.323(10)1

1.6 3.28 (10)- 3 4.72 (10)- 4.75 (10)-I-3

2.1 5.56 (10) 1.053 1.059

2.6 7.42 (10)-3 1.935 1.942-3

3.1 9.02 (10) 2.47 2.48-2

5.1 1.655(10) 4.05 4.06

-14- li

TABLE 3 (Continued)


n1 n2 Wavelength Radius Qabs Qsca Qext

1.44 9 x 10-4 101A 0.1p 1.168(10)-4 2.89 (10)-6 1.197(10)-4

0.2 2.40 (10)-4 4.62 (10)-5 2.86 (10)-4

0.4 4.86 (10)-4 7.40 (10)-4 1.225(10)-3

0.6 7.55 (10)-4 3.74 (10)-3 4.50 (10)-3

0.8 1.055(10)-3 1.181(10)-2 1.286(10)-2

1.1 1.574(10)-3 4.16 (10)-2 4.32 (10)-2

1.6 2.62 (10)- 3 1.701(10)-1 1.727(10)-1

2.1 3.85 (10)-3 4.05 (10) 4.09 (10)-1

2.6 5.54 (10)-3 7.34 (10)- 7.40 (10)-I

3.1 7.55 (10)-3 1.267 1.274-3

5.1 8.32 (10) 1.660 1.669

-15-

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