H. J. McSkimin

Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves–Data for Silicon and Germanium Single Crystals, and for Fused Silica

Ultrasonic waves (shear or longitudinal) in the 10–30 mc range are transmitted down a fused silica rod, through a polystyrene or silicone one‐quarter wavelength seal, and into the solid specimen. Measurement of reflections within the specimen yields values for velocities of propagation and elastic constants.Data obtained over a temperature range of 78° to 300°K for silicon and germanium single crystals, and 1.6° to 300°K for fused silica are listed. For the latter, a high loss is noted, with an indicated maximum near 30°K.

Elastic Moduli of Diamond as a Function of Pressure and Temperature

Values of wave velocities and elastic moduli at 25°C were measured for hydrostatic pressures to 20 000 psi (excess over 1 atm). Variations of velocities and moduli at 1 atm were obtained over a temperature range of +50°C to −195.8°C. Adiabatic stiffness moduli (units of 1012 dyn/cm2), their pressure derivatives, and their temperature coefficients (units of 10−5/C), are shown below for 1 atm and 25°C. ModulusValuePressure derivativesTemperature coefficientc1110.79±0.055.98±0.7−1.37±0.2c12 1.24±0.053.06±0.7−5.70±1.5c44 5.78±0.022.98±0.3−1.25±0.1

Elastic Moduli of Silicon vs Hydrostatic Pressure at 25.0°C and − 195.8°C

Ultrasonic wave velocities and elastic moduli for high purity silicon (resistivity ∼400 Ω‐cm) have been measured as a function of hydrostatic pressure and temperature in the ranges 0–30 000 psi (0–2100 kg/cm2) and − 195.8° to 25°C. Variations of moduli with pressure are found to be nearly independent of temperature in the range listed. For 25°C(Δc11/Δp)=4.33, (Δc12/Δp)=4.19, (Δc44/Δp)=0.80, (ΔK/Δp)=4.24. For − 195.8°C(Δc11/Δp)=4.29, (Δc12/Δp)=4.20, (Δc44/Δp)=0.75, (ΔK/Δp)=4.23.

Third‐Order Elastic Coefficients of Quartz

All 14 third‐order elastic coefficients of quartz have been calculated from the measured transit times for small‐amplitude ultrasonic waves as functions of applied stress. Thirty‐four different experimental runs were made, ten under hydrostatic pressure, and 24 under uniaxial stress. The data permit calculation of the 14 coefficients with 20 crosschecks. Our recommended values at 25°C, based on a least‐squares fit, are as follows, all in 1012 dyn/cm2. CoefficientValueStandard errorC111−2.100.07C112−3.450.06C113+0.120.06C114−1.630.05C123−2.940.05C124−0.150.04C133−3.120.07C134+0.020.04C144−1.340.07C155−2.000.08C222−3.320.08C333−8.150.18C344−1.100.07C444−2.76 0.17.

Measurement of Third‐Order Moduli of Silicon and Germanium

Experimental techniques for determining third‐order moduli of single crystals by means of ultrasonic wave propagation are described. Results for silicon and germanium demonstrate a good degree of self‐consistency among the basic experimental data obtained with both hydrostatic and uniaxial pressure.In units of 1012 dyn/cm2 and for 25°C, the third‐order moduli are: ModulusValue for SiValue for GeC111−8.25±0.10−7.10±0.06C112−4.51±0.05−3.89±0.03C123−0.64±0.10−0.18±0.06C144+0.12±0.25−0.23±0.16C166−3.10±0.10−2.92±0.08C456−0.64±0.20−0.53±0.07 Thermodynamic definitions for the moduli (K. Brugger, Phys. Rev. 133, A1611 (1964)] have been used.

The Elastic Stiffness Moduli of Diamond

The elastic stiffness moduli of a 22‐carat Type II diamond have been measured at 25.0°C by the ultrasonic technique described by H. J. McSkimin. Basic wave velocities were obtained over a frequency range of 20–500 MHz in order to minimize errors due to diffraction and coupling effects. Moduli (in units of 1012 dyn/cm2) are as follows: c11=10.79±0.05, c12=1.24±0.05, c44=5.78±0.02. Comparison with literature values is made.

Third‐Order Elastic Moduli of Germanium

Measurements have been made for all six third‐order elastic moduli of germanium by measuring ultrasonic velocities in selected directions when directed static stresses are applied to the crystal. Three measurements are obtained by using hydrostatic pressures, three by using a static compression along the axis, and six by stressing the axis with measurements being made along the direction and the direction. Using the finite strain formulas of Murnaghan, the measured velocities are related to the three second‐order elastic moduli and the six third‐order elastic moduli for a cubic crystal. The 12 sets of measurements provide considerable overlap, and the probable errors are shown to be moderate.

Elastic Moduli of Single‐Crystal Gallium Arsenide

Velocities of propagation of high‐frequency ultrasonic waves and the adiabatic elastic moduli for single‐crystal gallium arsenide are reported. Data at 25°C were obtained in the range of 20–180 mc/sec.The elastic moduli based on a density of 5.307 g/cm3 in units of 1012 dynes/cm2 are c11 = 1.188±0.14%; c12 = 0.538±0.36%; c44 = 0.5940±0.14%.

Elastic Moduli of Cadmium Telluride

The elastic moduli of cadmium telluride at 25°C have been determined by measurement of ultrasonic wave velocities in the frequency range 40–300 Mc/sec. Preparation of the crystal specimens is described, and factors affecting the accuracy of results are considered. The question of possible piezoelectric stiffening and its effect on measured wave velocities is also considered. Based on a density of 5.854 g/cm3 as computed from the lattice constant (and taken as exact with respect to uncertainty estimates) the zero field elastic moduli are: ModulusValue d/cm2Uncertainty in %c115.351×10110.15c123.6810.2c441.9940.15

ADIABATIC ELASTIC MODULI OF SINGLE CRYSTAL ALPHA-URANIUM

The adiabatic elastic moduli at 25°C for single crystal alpha‐uranium are reported here as derived from measurements of high‐frequency ultrasonic wave velocities. A detailed description is given of the specimens used, and of the technique employed for obtaining small plates of proper orientation suitable for the problem. Although the specimens were small the technique of measurement produced results of relatively good precision, as evidenced by the numerous cross checks obtained. Best values of the moduli and the estimated probable errors are given.The elastic stiffness moduli in units of 1012 dynes/cm2 are as follows: c11=2.147 ±0.14%,c12=0.465±0.6%,c13=0.218±1.5%,c22=1.986±0.14%,c23=1.076±0.34%,c33=2.671±0.14%,c44=1.2444±0.10%,c55=0.7342±0.10%,c66=0.7433±0.10%.