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 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.

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 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.)

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.

[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	REFERENCE
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

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/°C3 (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