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Introduction to Quartz Frequency Standards

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14th Piezoelectric Devices Conference and Exhibition, Sept. 15-17, 1992.

The Constants of Alpha Quartz
(1992 Update)

ROGER W. WARD

Quartzdyne, Inc.
1020 Atherton Drive
Building C
Salt Lake City, Utah 84123
(801) 266-6958

SUMMARY

Anyone performing calculations on quartz crystal devices requires numerical values for the physical constants used in his equations. However, as with any physical constant, there is no absolute value which may be assigned to a constant of quartz¾only a "best" value based upon numerous observations performed under controlled laboratory conditions on a variety of documented samples.

With time, and much effort, the quartz engineer accumulates his own list of "best" values for the frequently used constants of quartz. Still, when a new calculation involves an obscure constant, a literature search is required to find that obscure value.

Such a literature search has been conducted, and a list of "good" constants for alpha quartz is presented.

NOMENCLATURE

The word "quartz" as used herein means electronic grade crystalline Si02 at temperatures below 573°C, either natural or man-made (cultured).

Historically, the term rock crystal, low-quartz, alpha quartz, or crystalline quartz has been used for "quartz".

Modern usage in certain industries uses the term "quartz" to refer to fused quartz¾quartz which has been heated to above its melting point (1710°C). Fused quartz is non-piezoelectric and non-crystalline; hence, of no usefulness to the quartz engineer. Sosman [1], p. 43, says: "The use of the single word "quartz" to refer to vitreous silica can not be too strongly condemned. It has arisen through carelessness or ignorance and is already [1927!] causing troublesome confusion." Sosman suggests the use of "quartz-glass" or "fused quartz" for this material.

Some quartz engineers refer to cultured (man-made) quartz as "synthetic" quartz. Synthetic gives the connotation of "not real", so is to be avoided in this context, since cultured quartz is "real" quartz.

HISTORY

Ever since man first held a piece of quartz in his hand, he has been aware of one of quartz' physical constants¾its density. Since then, most physical constants of quartz have been studied and measured. There are thousands of references in the literature on the subject. Many of the measurements are of little value today since details of the experiments were often neglected¾i.e., temperature, source of the quartz, measurement standards, etc.

Quartz obtained from most locations is not useful for electronic applications, due to excessive twinning, inclusions, and fracturing. Through World War II, all quartz used was natural quartz, mostly from Brazil. Since then, the art of culturing quartz has evolved to where today cultured quartz is used almost exclusively for electronic applications.

The constants presented here may be applied to only the finest grades of cultured quartz¾those that most nearly imitate natural quartz. Devices fabricated from lower quality cultured quartz have physical constants enough different from natural quartz to produce sizeable errors when compared to otherwise identical natural quartz devices.


ACTUAL DEVICES

Due to the assumptions used in any theory, and also due to the probability that the calculated device has a different geometry (diameter, contour, electrode size) than the units measured to produce a given physical constant, and due to manufacturing tolerances (especially angular orientation) it is usually impossible to theoretically predict quartz device behavior to better than an equivalent angular orientation of about ± 1/2° for double-rotated cuts.

Some researchers have used microwave measurement techniques to determine the elastic stiffnesses, and their temperature coefficients, of bulk quartz¾often 1 inch cubes of carefully oriented quartz samples. Other researchers have called for more precise measurements, over extended temperature ranges, to be performed. (See, for example, [20].) But such a set of constants cannot predict the behavior of real-world resonators to an accuracy of better than approximately 30 arc minutes!

To better refine a theory, it is necessary (and it will always be necessary) to make a matrix of actual, real-world devices¾ all of the same physical design, except for a well controlled slight variation in orientation.

For example, EerNisse[2] in 1975 predicted the SC-cut to occur at phi=22°30'. Kusters and Leach [3] experimentally showed that for their crystal design, phi=21° 56', a variation of 34'. Kusters and Leach determined phi by a carefully controlled experiment involving a matrix of orientations about EerNisse's predicted angle, careful measurements, and computer reduction of the data to define the orientation for the zero thermal transient effect (assuming that the in-plane stress of EerNisse is the same mechanism measured by Kusters and Leach in their thermal transient tests).

Similarly, Adams et al.[4] determined the temperature coefficients of the elastic stiffnesses of quartz, using a matrix of precisely oriented, identically prepared resonators, as opposed to Bechmann's et al.[5] determination of the same coefficients using a varied assortment of crystal designs of moderate orientational precision. And yet, neither set of constants can predict the temperature behavior of the SC-cut, as actually manufactured, to better than an equivalent angular orientation of 1/2°; but both can be used to predict the existence of, and the shape of, the temperature curve for the SC-cut, and do so accurately enough to allow the experimental cuts to be selected with enough precision to "close the loop" with only one or two iterations of the actual devices! What more could one ask?

Similar experimental tests will always be required to ultimately define a desired quartz device orientation. (Unless the theory can be expanded to include the now unsolvable effects of the boundary conditions: finite diameter, contoured surfaces, film stress, mounting stress, etc.)

THE CONSTANTS

In 1927, Sosman[1] published 800 pages devoted to the physical properties of silica in its many forms, with the major emphasis on quartz. Sosman studied in detail the many measurements of each property presented in the literature and studied in his own lab. He points out errors and omissions of each researcher, and attempts to arrive at a "best" value for each constant of quartz. Hence, Sosman was used as the primary resource for this presentation.

There are many interesting historical notes included in the references, too numerous to include here; but one observation made by Dr. Virgil Bottom emphasizes the historical contribution made by Pierre Curie, the "Father of Piezoelectricity": "It is remarkable, therefore, that the Curies were able to obtain a value for d11 in quartz which is only about 7% below the best value known today. Between 1880 and 1970, no fewer than thirty independent measurements of d11 in quartz have been reported and half of these values are further from the value commonly accepted today than that given by the Curies in 1880." Dr. Bottom goes on to conclude that, ". . . it may truthfully be said of Pierre Curie that he laid the cornerstone of modern electronic communication."[6]

The constants presented in Table I are not represented to be "The" constants, or "the best" constants, but only "good" constants¾for the reasons outlined above. "The" constant only exists for a given piece of quartz of a given design. Change the design and some of the measurable constants will change. Use another piece of quartz from the same autoclave or the same vug (a cavity in which the crystals grow in nature) and the constants will change (at least to within the precision allowed by modern "state-of-the-art" measurement techniques).

No attempts have been made to "improve" upon these constants by curvefitting several sets of data, or by re-calculation. The only modification has been to convert a few constants to the same units of measurement. Where this has been done, the conversion constant is noted.

The temperature at which a measurement was made is indicated, if available. When no temperature is noted, the measurement was probably made at room temperature.

No representation is made as to completeness, accuracy, or appropriateness of any constant. Indeed, only a few of the constants found in the literature noted the error band of the values given; hence, allowing for the small differences between different sources, no error bands are indicated in Table I.

The author would appreciate receiving suggestions for the inclusion of other constants, or new or better values for the ones presented, with the intent of publishing a new list from time to time as data warrants. This is the second update of the original 1984 paper. Thanks to everyone who pointed out errors and omissions in the previous edition. Such suggestions may be sent to the author at the address above.

REFERENCES

[1] R.B. Sosman, The Properties of Silica, New York: Chemical Catalog Co., 1927. (Available from University Microfilm, 300 N. Zeeb Rd., Ann Arbor MI 48106 (313-761-4700).)

[2] E. P. EerNisse, "Quartz Resonator Frequency Shifts Arising from Electrode Stress," Proceedings 29th Annual Symposium on Frequency Control, US Army Electronics Command, Ft. Monmouth, NJ, pp. 1-4, 1975. (Copies available from Electronics Industries Association, 2001 Eye St., NW, Washington DC 20006.)

[3] J.A. Kusters and J. Leach, "Further Experimental Data on Stress and Thermal Gradient Compensated Crystals," Proceedings IEEE, Vol. 65, pp. 282-284, Feb 1977.

[4] C.A. Adams, G.M. Enslow, J.A. Kusters, and R.W. Ward, "Selected Topics in Quartz Crystal Research," Proceedings, 24th Annual Symposium on Frequency Control, US Army Electronics Command, Ft. Monmouth, NJ, pp. 55-63 (1970). (National Technical Information Service, Sills Building, 5285 Port Royal Road, Springfield, VA 22161, Accession Nr. AD746210.)

[5] R. Bechmann, A. Ballato, and T.J. Lukaszek, "Higher Order Temperature Coefficients of the Elastic Stiffnesses and Compliances of Alpha-Quartz," Proceedings IRE, Vol 50, pp. 1812-1822, Aug. 1962, p. 2451, Dec. 1962.

[6] V.E. Bottom, "The Centennial of Piezoelectricity," unpublished paper, 1980.

[7] C. Frondel, Dana's System of Mineralogy Volume II Silica Minerals, New York: John Wiley and Sons, 1962.

[8] R. A. Heising, Quartz Crystals for Electrical Circuits, New York: D. Van Nostrand, 1946. (Reprinted 1978, Electronics Industries Association, 2001 Eye St. NW, Washington DC 20006.)

[9] W.G. Cady, Piezoelectricity, New York: Dover, 1964.

[10] V.E. Bottom, "Dielectric Constants of Quartz," Journal of Applied Physics, V. 43, No. 4, Apr 1972, p. 1493.

[11] R.N. Thurston, H.J. McSkimin, and P. Andreatch, Jr., "Third Order Elastic Coefficients of Quartz," Journal of Applied Physics, V. 37, No. 1, Jan 1966, p. 276.

[12] V.E. Bottom, "Measurement of the Piezoelectric Coefficients of Quartz Using the Fabry-Perot Dilatometer," Journal of Applied Physics, V. 41, No. 10, Sept. 1970, p. 3941.

[13] G.E. Graham and F.N.D.D. Pereira, "Temperature Variations of the Piezoelectric Effect in Quartz," Journal of Applied Physics, V. 42, No. 7, June 1971, p. 3011.

[14] S.V. Kolodieva, A.A. Fotchenkov, and S.A. Linnik, "Change in the Anisotropy of Electrical Conductivity of Quartz Crystals," Soviet Physics¾Crystallography, Vol, 17, No. 3, pp. 509-511, Nov-Dec, 1972.

[15] C.H. Scholz, "Static Fatigue of Quartz," J. of Geophysical Research, Vol 77, No. 11, pp. 2104-2144, Apr 10, 1972.

[16] A. Ballato and M. Mizan, "Simplified Expressions for the Stress-Frequency Coefficients of Quartz Plates," IEEE Trans. Sonics and Ultrasonics, Vol. SU-31, No. 1, pp. 11-17, Jan 1984.

[17] H. Jair and A.S. Nowick, "Electrical Conductivity of Synthetic and Natural Quartz Crystals," Journal of Applied Physics, 53 (1), pp. 477-484, Jan. 1982.

[18] J. C. Brice, "The Lattice Constants of a-Quartz," J. of Materials Science, 15; pp. 161-167, 1980.

[19] R. C. Weast, Editor, CRC Handbook of Chemistry and Physics, 63rd Edition, CRC Press, Inc., Boca Raton, FL, 1983.

[20] J.A. Kosinski, J. Gualtieri, and A. Ballato, "Thermoelastic Coefficients of Alpha Quartz," IEEE Trans. Ultrasonics, Ferroelectrics, and Freq. Control, Vol. 39, No. 4, pp. 502-507, July, 1992.

Note: Also published as: "Thermal Expansion of Alpha Quartz", Proc. 45th Ann. Symp. on Freq. Control, pp. 22-28, 1991.

[21] J. Lamb and J. Richter, "Anisotropic Acoustic Attenuation With New Measurements for Quartz at Room Temperature," Proc. R. Soc. London, Ser. A293, pp. 479-492, 1966.

TABLE I

"GOOD" FUNDAMENTAL MATERIAL CONSTANTS FOR CRYSTALLINE QUARTZ

CONSTANT NAME VALUE
ACCOUSTIC ATTENUATION SEE REFERENCE LAMB [21]
AXIAL RATIO c/a

***************************** TEMPERATURE COEFFICIENT

1.1015 @ -250°C

1.1014 -200

1.1009 -100

1.1003 0

1.0996 100

1.0988 200

1.0979 300

1.0960 400

1.0956 500

1.0946 550

1.0940 573

1.09997 ?

1.100 20

1.10013 25

*******************************-6.14X10-6/°C @ 0°C

SOSMAN [1] p. 205,

368-370.

SEE ALSO

FRONDEL [7] P. 7,20,39

HEISING [8] P.103

CADY [9] P. 27

BRICE [18]

*********************** FRONDEL P. 39

SOSMAN P. 377

COMPOSITION SILICON 46.72%

OXYGEN 53.28% BY WEIGHT

SOSMAN P. 22,27

COMPRESSIBILITY COEFFICIENT

VOLUME, (TRUE)

2.76X10-6/kg/cm2 @ 0 kg/cm2

2.65 2039

2.53 4079

2.42 6118

2.33 8157

2.25 10197

2.18 12236

NOTE: 1 megabarye=106 dyne/cm2

= 1.1097 kg/cm2

SOSMAN P.427.

SEE ALSO P. 426- 433

CONDUCTIVITY, THERMAL PARALLEL PERPENDICULAR TEMP

-- 0.68 -252°C

0.117 0.0586 -190

0.0476 0.02409 - 78

0.0325 0.01731 0

0.0215 0.01333 100

0.029 0.016 20

cal/cm/s/°C

SOSMAN P. 419,420

FRONDEL P. 116

CURIE TEMPERATURE (ALSO KNOWN

AS LOW-HIGH INVERSION,

ALPHA-BETA INVERSION)

573.3°C (ON HEATING) SOSMAN P. 116-125

FRONDEL P. 3, 117

CADY P. 31

KEY

PRIMARY REFERENCE

SECONDARY REFERENCE

TABLE I, CONT.

CONSTANT NAME VALUE REFERENCE
DENSITY, ABSOLUTE

***************************** TEMPERATURE COEFFICIENT, TRUE

*****************************

TEMPERATURE COEFFICIENTS

2.65067 g/cm3 @ 0°C

2.64822 25

2.665 g/cm3 @ -250°C

2.664 -200

2.659 -100

2.651 0

2.641 100

2.630 200

2.616 300

2.601 400

2.581 500

2.554 573

*********************************** 12x10-6/°C @ -200°C

25.2 -100

33.6 0

40.0 100

46.6 200

54.9 300

67.4 400

100 500

141 550

***********************************T1 = -34.92X10-6/°C

T2 = -15.9X10-9/°C2

T3 = 5.30X10-12/°C 3

(APPARENTLY REFERENCED TO 25°C)

FRONDEL P. 114

CADY P. 412

SOSMAN P. 361

(ALSO P. 291-295)

***********************

SOSMAN P.291, 362, 366 SEE ALSO FRONDEL P.114

***********************BECHMANN [5]

SEE ALSO CADY P. 412

DIELECTRIC CONSTANT

*****************************

*****************************

TEMPERATURE COEFFICIENT

***************************** FIELD STRENGTH COEFFICIENT

4.6 PARALLEL TO Z-AXIS

4.60

4.5 PERPENDICULAR TO Z-AXIS

4.51

***********************************e11T = e22T = 39.97X10-12F/m

e11S-e11T = -0.76

e33T = 41.03

e33S-e33T = 0

***********************************PARALLEL:

K=4.926[1-1.10X10-3(T-10)-

2.4X10-5(T-10)2]

PERPENDICULAR:

K=4.766[1-9.9X10-4(T-10)]

FOR T=10 TO 31°C

***********************************K=0 TO 2,000 V/cm (PARALLEL)

K=0 TO 12,000 V/cm (PERPENDICULAR)

SOSMAN P. 515

BOTTOM [10]

SOSMAN

BOTTOM

SEE ALSO CADY P.414 FRONDEL P. 116

***********************BECHMANN

***********************SOSMAN P. 523

AND GRAPH P. 524

***********************

CADY P. 415

TABLE I, CONT.

CONSTANT NAME VALUE REFERENCE
ELASTIC COEFFICIENTS

THIRD ORDER

C111 = -2.10X1012dyn/cm2

C112 = -3.45

C113 = +0.12

C114 = -1.63

C123 = -2.94

C124 = -0.15

C133 = -3.12

C134 = +0.02

C144 = -1.34

C155 = -2.00

C222 = -3.32

C333 = -8.15

C344 = -1.10

C444 = -2.76

THURSTON [11]
ELECTRIC STRENGTH 4X106V/cm @ -80°C

7 @ 60

CADY P. 413
ENTROPY OF TRANSITION 1.08 e.u. CRC P. D-51
ENTROPY 0.166 cal/g/°C @ 25°C CRC P. D-85
HARDNESS, PENETRATION

(AUERBACH)

MHO

SCRATCH

30.8X103 kg/cm2 PARALLEL TO Z

22.9 PERPENDICULAR

7

667 (CORUNDUM = 1000)

SOSMAN P. 491

SOSMAN P. 494

SOSMAN P. 494

HEAT CAPACITY, TRUE 5.4X10-3cal/g @ -250°C

41.0 -200

111.2 -100

166.4 0

204.3 100

232.7 200

254.3 300

270.0 400

291.0 500

340(?) 573

(IN 20°C grams)

SOSMAN P. 314, 331

(Note: The CRC [19] equation on P. D-51 does not agree with Sosman.)

HEAT OF SOLUTION 30.29 kg-cal/formula wt in 34.6% HF SOSMAN P. 318
HEAT OF TRANSFORMATION, LATENT

(LOW-->HIGH QUARTZ)

2.5 cal/g

0.15 kg-cal/formula wt

SOSMAN P. 312
LATTICE CONSTANT "a" 4.9035 Angstroms @ 18°C

4.903 ?

4.91331 25

4.90288 25

4.91267 25

4.9127 25

4.9134 25 CULTURED

SOSMAN P. 226

HEISING P. 103

FRONDEL P. 25

CADY P. 735

CADY

BRICE [18]

BRICE

MAGNETIC SUSCEPTIBILITY (VACUUM) PARALLEL PERPENDICULAR TYPE

-1.21X10-6 -1.20X10-6 VOLUME

-0.45 -0.45 MASS

SOSMAN P. 576

TABLE I. CONT.

CONSTANT NAME VALUE REFERENCE
MAGNETO-OPTIC ROTATION

(VERDET CONSTANT)

************************** TEMPERATURE COEFFICIENT

0.15866 min @ 2194.92 angstroms, 20°C

0.04617 3612.5

0.02750 4678.15

0.02257 5085.82

0.01664 5892.9

0.01368 6438.47

************************************

w = w20[1 + 0.00011(T - 20)]

FOR T = 20 TO 100°C

SOSMAN P. 776

**********************

SOSMAN P. 777

MELTING POINT <1670°C

1710°C

FRONDEL P. 3

CRC P. D-201

PENETRATION, MODULUS OF PARALLEL PERPENDICULAR

1062 kg/cm2 859 kg/cm2

SOSMAN P. 465

PIEZOELECTRIC COEFFICIENTS

************************** PRESSURE COEFFICIENT

STRAIN

d11 = -2.30X10-12m/V

d11 = -2.27

d11 = -2.25

d11 = -2.30

d14 = 0.57X10-12m/V

d14 = 0.85

d14 = 0.67

NOTE: 1 esu/dyne = 3 X 104m/V

d11 = 2.32 X 10-12m/V @ 1.5°K

2.32 4.2

2.31 -196 °C

2.22 20

2.05 100

STRESS

e11 = 0.171C/m2

e11 = 0.180

e14 = 0.0403

e14 = 0.04

**********************************

d11 Varies by <0.1% to 3519 kg/cm2

SOSMAN P. 559

BOTTOM [12]

HEISING P. 20

CADY P. 219

SOSMAN

HEISING

CADY

GRAHAM [13]

BECHMANN

CADY P. 219, 224

BECHMANN

CADY

**********************

SOSMAN P. 559

POISSON'S RATIO S12/S11 = 0.130

S13/S11 = 0.119

CADY P. 156
RESISTIVITY PARALLEL PERPENDICULAR TEMPERATURE

0.1 X 1015 20 X 1015 20°C

0.8 X 1012 100

70 X 109 200 60 X 106 300

ohm-cm

SOSMAN P. 528-537

ALSO SEE KOLODIEVA[14] and JAIN & NOWICK [17]

TABLE I, CONT.

CONSTANT NAME VALUE REFERENCE
REFRACTIVE INDEX

***************************** TEMPERATURE COEFFICIENTS

*****************************

BIREFRINGENCE,

TEMPERATURE COEFFICIENT

ORDINARY RAY:

no2=3.4269 + 1.0654X10-2/(L2-0.010627)

+ 111.49/(L2-100.77)

ORDINARY RAY:

no2=3.53445 + 0.008067/(L2-0.0127493)

+ 0.002682/(L2-0.000974)

+ 27.2/(L2-108)

EXTRAORDINARY RAY:

ne2=3.5612557 + 0.00844614/(L2-0.0127493)

+ 0.00276113/(L2-0.000974)

+ 127.2/(L2-108)

where L=wavelength in mu

no = 1.54425 (Na @ 18°C)

ne = 1.55336

***************************************

ORDINARY RAY: -6.50 x 10-6/°C

EXTRAORDINARY RAY: -7.544

***************************************

B = Bo - (972T + 1.6T2) 10-9

FOR T = 4 TO 99°C

SOSMAN P. 588-625

FRONDEL P. 129

FRONDEL

CADY P. 723

******************

FRONDEL P. 129,

SOSMAN P. 637

******************

SOSMAN P. 684,

FRONDEL P. 131

ROTARY POWER

***************************** TEMPERATURE COEFFICIENT

201.9°/mm @ 2265.03 angstroms

95.02 3034.12

21.724 5892.9

11.589 7947.63

0.972 25000

ROTATION IS CW IN RIGHT HAND QUARTZ AND CCW IN LEFT HAND QUARTZ.

***************************************

about +1.4X10-4/°C at 20°C

(independent of wavelength)

SOSMAN P. 648

FRONDEL P. 132

******************

SOSMAN P. 689

SPECIFIC HEAT 0.1412 cal/g/°C @ -50°C

0.1664 0

0.1870 50

0.2043 100

CADY P. 411
TABLE I, CONT.

CONSTANT NAME VALUE REFERENCE
STIFFNESSES

************************** TEMPERATURE COEFFICIENTS

cD cE

c11 = 87.49 86.74X109N/m2

c13 = 11.91 11.91

c33 = 107.2 107.2

c14 = -18.09 -17.91

c44 = 57.98 57.94

c66 = 40.63 39.88

***********************************

FIRST SECOND THIRD

ij X10-6/°C X10-9/°C2 X10-12/°C3

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ

11 -48.5 -107 -70

-49.6 -107 -74

13 -550 -1150 -750

-651 -1021 -240

33 -160 - 275 -250

-192 -162 67

14 101 -48 -590

89 -19 -521

44 -177 -216 -216

-172 -261 -194

66 178 118 21

167 164 29

BECHMANN

SEE ALSO HEISING P.40ff SOSMAN p. 463,

CADY P.137-155 (GRAPHS) ALSO CADY P. 757

*************************

BECHMANN [5]

ADAMS [4]

SEE ALSO HEISING P. 55,

CADY P. 136-140,

KOSINSKI [20]

STRENGTH

COMPRESSIVE

************************** COMPRESSIVE

TENSILE

RUPTURE (BENDING)

STRENGTH CONFINING PRESS TEMP

24,000kg/cm2 1 atm 20°C

150,000 25,000 atm 400

*********************************** 24,500 kg/cm2 PARALLEL

22,400 PERPENDICULAR

1,120 PARALLEL

850 PERPENDICULAR

1,380 PARALLEL

920 PERPENDICULAR

FRONDEL P. 109

*************************

SOSMAN P. 481

SEE ALSO SCHOLZ [15]

SYMMETRY

************************** CLASS

TRIGONAL TRAPEZOHEDRAL or TRIGONAL

ENANTIOMORPHOUS HEMIHEDRAL

TRIGONAL HOLOAXIAL or

ENANTIOMORPHOUS HEMIHEDRAL

***********************************CLASS 18, SYMMETRY D3 (SCHONFLIES)

SYMMETRY 32 (HERMANN-MAUGUIN)

SOSMAN P. 183

CADY P. 19

*************************

CADY P. 19

TABLE I, CONT.

CONSTANT NAME VALUE REFERENCE
THERMAL EXPANSION

COEFFICIENT, LINEAR

(MEAN, FROM 0°C)

**************************

TEMPERATURE COEFFICIENTS

PARALLEL PERPENDICULAR TEMPERATURE

4.10X10-6/°C 8.60X10-6/°C -250°C

5.50 9.90 -200

6.08 11.82 -100

7.10 13.24 0

7.97 14.45 100

8.75 15.61 200

9.60 16.89 300

10.65 18.50 400

12.22 20.91 500

15.00 25.15 573

***************************************

FIRST SECOND THIRD

ij X10-6/°C X10-9/°C2 X10-12/°C3

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 11 13.71 6.5 -1.9

33 7.48 2.9 -1.5

NOTE: a11 = a22

SOSMAN P. 370

**********************

BECHMANN

KOSINSKI[20]

THERMOELASTIC COEFFICIENTS

(HIGHER ORDER)

ORDER a11(n) a33(n) UNIT

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ

1 13.16 6.37 10-6/°C

2 15.68 8.18 10-9/°C2

3 -7.86 6.88 10-12/°C3

REFERENCED TO 0°C

BALLATO [16]

WAVELENGTH, X-RAY

Cu Ka1

1.5374 angstroms

1.54051

HEISING P. 97

FRONDEL P. 25

VOLUME, UNIT CELL 37.40X10-24cm3 SOSMAN P. 225
YOUNG'S MODULUS 1.03X10+12 dynes/cm2 PARALLEL

0.78 PERPENDICULAR

S'33x1015 = 1269 - 841 cos23 + 543 cos43

-862 sin 33 cos3 sin3Æ cm2/dyne

NOTE: Ym = 1/s'33

CADY P. 155 (GRAPH)

FRONDEL P. 122-CORRECTED

(NOTE: EQ. IN FRONDEL HAS EXTRA TERM DUE TO TYPO AND INCORRECT POWER OF 10)

VAPOR PRESSURE 10mm @ 1732°C

40 1867

100 1969

400 2141

760 2227

CRC P. D-201
VISCOSITY SEE REFERENCE LAMB [21]

KEY

PRIMARY REFERENCE

SECONDARY REFERENCE

NOTE: PARALLEL = PARALLEL TO Z-AXIS (OPTICAL AXIS)

PERPENDICULAR = PERPENDICULAR TO Z-AXIS