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 Technical References
Introduction to Quartz Frequency Standards
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ROGER W. WARD
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.
The word "quartz" as used herein means electronic grade crystalline Si0_{2} at temperatures below 573°C, either natural or manmade (cultured).
Historically, the term rock crystal, lowquartz, 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 nonpiezoelectric and noncrystalline; 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 "quartzglass" or "fused quartz" for this material.
Some quartz engineers refer
to cultured (manmade) 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.
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.
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 doublerotated 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 realworld 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, realworld devices¾ all of the same physical design, except for a well controlled slight variation in orientation.
For example, EerNisse[2] in 1975 predicted the SCcut 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 inplane 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 SCcut, 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 SCcut, 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.)
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 d_{11} in quartz which is only about 7% below the best value known today. Between 1880 and 1970, no fewer than thirty independent measurements of d_{11} 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 "stateoftheart" measurement techniques).
No attempts have been made to "improve" upon these constants by curvefitting several sets of data, or by recalculation. 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.
[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 (3137614700).)
[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. 14, 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. 282284, 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. 5563 (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 AlphaQuartz," Proceedings IRE, Vol 50, pp. 18121822, 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 FabryPerot 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. 509511, NovDec, 1972.
[15] C.H. Scholz, "Static Fatigue of Quartz," J. of Geophysical Research, Vol 77, No. 11, pp. 21042144, Apr 10, 1972.
[16] A. Ballato and M. Mizan, "Simplified Expressions for the StressFrequency Coefficients of Quartz Plates," IEEE Trans. Sonics and Ultrasonics, Vol. SU31, No. 1, pp. 1117, Jan 1984.
[17] H. Jair and A.S. Nowick, "Electrical Conductivity of Synthetic and Natural Quartz Crystals," Journal of Applied Physics, 53 (1), pp. 477484, Jan. 1982.
[18] J. C. Brice, "The Lattice Constants of aQuartz," J. of Materials Science, 15; pp. 161167, 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. 502507, July, 1992.
Note: Also published as: "Thermal Expansion of Alpha Quartz", Proc. 45th Ann. Symp. on Freq. Control, pp. 2228, 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. 479492, 1966.
TABLE I
"GOOD" FUNDAMENTAL MATERIAL CONSTANTS FOR CRYSTALLINE QUARTZ
*****************************
TEMPERATURE COEFFICIENT
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
368370.
SEE ALSO
FRONDEL [7] P. 7,20,39
HEISING [8] P.103
CADY [9] P. 27
BRICE [18]
***********************
FRONDEL P. 39
SOSMAN P. 377
OXYGEN 53.28% BY
WEIGHT
VOLUME, (TRUE)
2.65^{ }
2039
2.53^{ }
4079
2.42^{ }
6118
2.33^{ }
8157
2.25^{ }
10197
2.18^{ }
12236
NOTE: 1 megabarye=10^{6}
dyne/cm^{2}
= 1.1097 kg/cm^{2}
SEE ALSO P. 426
433
 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
AS LOWHIGH INVERSION,
ALPHABETA INVERSION)
FRONDEL P. 3, 117
CADY P. 31
PRIMARY REFERENCE
SECONDARY REFERENCE
CONSTANT NAME
VALUE
ACCOUSTIC
ATTENUATION
SEE REFERENCE
LAMB [21]
AXIAL RATIO
c/a
1.1015
@ 250°C
SOSMAN
[1] p. 205,
COMPOSITION
SILICON
46.72%
SOSMAN
P. 22,27
COMPRESSIBILITY
COEFFICIENT
2.76X10^{6}/kg/cm^{2}
@ 0 kg/cm^{2}
SOSMAN
P.427.
CONDUCTIVITY,
THERMAL
PARALLEL
PERPENDICULAR TEMP
CURIE TEMPERATURE
(ALSO KNOWN
573.3°C
(ON HEATING)
SOSMAN
P. 116125
KEY
CONSTANT NAME  VALUE  REFERENCE 

DENSITY,
ABSOLUTE
***************************** TEMPERATURE COEFFICIENT, TRUE
***************************** TEMPERATURE COEFFICIENTS
 2.65067
g/cm^{3} @ 0°C
2.64822 25
2.665 g/cm^{3} @ 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 ***********************************T^{1} = 34.92X10^{6}/°C T^{2} = 15.9X10^{9}/°C^{2} T^{3} = 5.30X10^{12}/°C ^{3} (APPARENTLY REFERENCED TO 25°C)  FRONDEL
P. 114
CADY P. 412
SOSMAN P. 361 (ALSO P. 291295)
*********************** 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 ZAXIS
4.60
4.5 PERPENDICULAR TO ZAXIS 4.51
***********************************e_{11}^{T} = e_{22}^{T} = 39.97X10^{12}F/m
e_{11}^{S}e_{11}^{T} = 0.76
e_{33}^{T} = 41.03
e_{33}^{S}e_{33}^{T} = 0
***********************************PARALLEL: K=4.926[11.10X10^{3}(T10) 2.4X10^{5}(T10)^{2}] PERPENDICULAR: K=4.766[19.9X10^{4}(T10)] 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

CONSTANT NAME  VALUE  REFERENCE 

ELASTIC
COEFFICIENTS
THIRD ORDER  C111 =
2.10X10^{12}dyn/cm^{2}
^{C}112 = 3.45 ^{C}113 = +0.12 ^{C}114 = 1.63 ^{C}123 = 2.94 ^{C}124 = 0.15 ^{C}133 = 3.12 ^{C}134 = +0.02 ^{C}144 = 1.34 ^{C}155 = 2.00 ^{C}222 = 3.32 ^{C}333 = 8.15 ^{C}344 = 1.10 ^{C}444 = 2.76  THURSTON [11] 
ELECTRIC STRENGTH  4X10^{6}V/cm
@ 80°C
7^{ } @ 60  CADY P. 413 
ENTROPY OF TRANSITION  1.08 e.u.  CRC P. D51 
ENTROPY  0.166 cal/g/°C @ 25°C  CRC P. D85 
HARDNESS,
PENETRATION
(AUERBACH)
MHO
SCRATCH  30.8X10^{3}
kg/cm^{2} PARALLEL TO Z
22.9^{ } PERPENDICULAR
7
667 (CORUNDUM = 1000)  SOSMAN
P. 491
SOSMAN P. 494
SOSMAN P. 494 
HEAT CAPACITY, TRUE  5.4X10^{3}cal/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. D51 does not agree with Sosman.) 
HEAT OF SOLUTION  30.29 kgcal/formula wt in 34.6% HF  SOSMAN P. 318 
HEAT OF
TRANSFORMATION, LATENT
(LOW>HIGH QUARTZ)
 2.5 cal/g
0.15 kgcal/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

CONSTANT NAME  VALUE  REFERENCE 

MAGNETOOPTIC
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 = w_{20}[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. D201 
PENETRATION, MODULUS OF  PARALLEL
PERPENDICULAR
1062 kg/cm^{2} 859 kg/cm^{2} 
SOSMAN P. 465 
PIEZOELECTRIC
COEFFICIENTS
************************** PRESSURE COEFFICIENT
 STRAIN
d_{11} = 2.30X10^{12}m/V d_{11} = 2.27 d_{11} = 2.25 d_{11} = 2.30
d_{14} = 0.57X10^{12}m/V d_{14} = 0.85 d_{14} = 0.67
NOTE: 1 esu/dyne = 3 X 10^{4}m/V
d_{11} = 2.32 X 10^{12}m/V @ 1.5°K 2.32^{ } 4.2 2.31^{ } 196 °C 2.22^{ } 20 2.05^{ } 100
STRESS e_{11} = 0.171C/m^{2} e_{11} = 0.180
e_{14} = 0.0403 e_{14} = 0.04 ********************************** d_{11} Varies by <0.1% to 3519 kg/cm^{2} 
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  S_{12}/S_{11}
= 0.130
S_{13}/S_{11} = 0.119
 CADY P. 156 
RESISTIVITY  PARALLEL
PERPENDICULAR TEMPERATURE
0.1 X 10^{15} 20 X 10^{15} 20°C 0.8 X 10^{12 } 100 70 X 10^{9 } 200 60 X 10^{6 } 300 ohmcm 
SOSMAN P. 528537 ALSO SEE KOLODIEVA[14] and JAIN & NOWICK [17] 
CONSTANT NAME  VALUE  REFERENCE 

REFRACTIVE
INDEX
***************************** TEMPERATURE COEFFICIENTS
***************************** BIREFRINGENCE, TEMPERATURE COEFFICIENT  ORDINARY
RAY:
n_{o}^{2}=3.4269 + 1.0654X10^{2}/(L^{2}0.010627) + 111.49/(L^{2}100.77)
ORDINARY RAY:
n_{o}^{2}=3.53445 + 0.008067/(L^{2}0.0127493) + 0.002682/(L^{2}0.000974) + 27.2/(L^{2}108)
EXTRAORDINARY RAY:
n_{e}^{2}=3.5612557 + 0.00844614/(L^{2}0.0127493) + 0.00276113/(L^{2}0.000974) + 127.2/(L^{2}108) where L=wavelength in mu
n_{o} = 1.54425 (Na @ 18°C) n_{e} = 1.55336 *************************************** ORDINARY RAY: 6.50 x 10^{6}/°C EXTRAORDINARY RAY: 7.544 *************************************** B = B_{o}  (972T + 1.6T^{2}) 10^{9} FOR T = 4 TO 99°C 
SOSMAN P. 588625
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 
CONSTANT NAME  VALUE  REFERENCE 

STIFFNESSES
************************** TEMPERATURE COEFFICIENTS
 c^{D}
c^{E }^{
}
c_{11} = 87.49 86.74X10^{9}N/m^{2} c_{13} = 11.91 11.91 c_{33} = 107.2 107.2 c_{14} = 18.09 17.91 c_{44} = 57.98 57.94 c_{66} = 40.63 39.88 *********************************** FIRST SECOND THIRD ij X10^{6}/°C X10^{9}/°C^{2} X10^{12}/°C^{3} 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.137155 (GRAPHS) ALSO CADY P. 757
*************************
BECHMANN [5] ADAMS [4] SEE ALSO HEISING P. 55, CADY P. 136140, KOSINSKI [20]

STRENGTH
COMPRESSIVE
************************** COMPRESSIVE
TENSILE
RUPTURE (BENDING)
 STRENGTH
CONFINING PRESS TEMP
24,000kg/cm^{2} 1 atm 20°C 150,000^{ } 25,000 atm 400
*********************************** 24,500 kg/cm^{2} 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 D_{3} (SCHONFLIES) SYMMETRY 32 (HERMANNMAUGUIN)  SOSMAN
P. 183
CADY P. 19
************************* CADY P. 19 
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}/°C^{2} X10^{12}/°C^{3} ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 11 13.71 6.5 1.9 33 7.48 2.9 1.5
NOTE: a_{11} = a_{22}

SOSMAN P. 370
**********************
BECHMANN KOSINSKI[20] 
THERMOELASTIC
COEFFICIENTS
(HIGHER ORDER)  ORDER a_{11}^{(n)}
a_{33}^{(n)} UNIT
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 1 13.16 6.37 10^{6}/°C 2 15.68 8.18 10^{9}/°C^{2} 3 7.86 6.88 10^{12}/°C^{3} REFERENCED TO 0°C 
BALLATO [16] 
WAVELENGTH,
XRAY
Cu Ka_{1}  1.5374
angstroms
1.54051  HEISING
P. 97
FRONDEL P. 25 
VOLUME, UNIT CELL  37.40X10^{24}cm^{3}  SOSMAN P. 225 
YOUNG'S MODULUS  1.03X10^{+12}
dynes/cm^{2} PARALLEL
0.78^{ } PERPENDICULAR
S'_{33}x10^{15} = 1269  841 cos^{2}3 + 543 cos^{4}3 862 sin^{ 3}3 cos3 sin^{3}Æ cm^{2}/dyne
NOTE: Y_{m} = 1/s'_{33}  CADY P.
155 (GRAPH)
FRONDEL P. 122CORRECTED (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. D201 
VISCOSITY  SEE REFERENCE  LAMB [21] 
KEY

PRIMARY REFERENCE SECONDARY REFERENCE 