The lattice parameter of high‐purity silicon is measured as a function of temperature between 300 and 1500 K, and the linear thermal expansion coefficient is accurately determined. Precise measurements are made by the high‐temperature attachment for Bond’s x‐ray method to a few parts per million. It is found that the temperature dependence of the linear thermal expansion coefficient α(t) is empirically given by α(t)=(3.725{1−exp[−5.88×10−3{(t−124)} +5.548×10−4t)×10−6 (K−1), where t is the absolute temperature ranging from 120 to 1500 K. It is shown that the lattice parameter in the above temperature range can be calculated using α(t) and the lattice parameter at 273.2 K (0.5430741 nm). Measured values of the lattice parameter and the thermal expansion coefficient for high‐purity float‐zoned (100 kΩ cm) and Czochralski‐grown (30 Ω cm) single crystals are uniformly distributed within ±1×10−5 nm and ±2×10−7 K−1 with respect to the values obtained from the above empirical formula.

Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K

Studies of the thermal expansion of solids in the temperature range 1–100 K are reviewed. Effects due to thermal vibrations of the lattice, electrons, magnetic interactions and defects are fully discussed. The survey of experimental data is intended to be comprehensive. From the large volume of theoretical work in recent years the authors have aimed to select for discussion the most significant contributions.

Thermal expansion of solids at low temperatures

Linear thermal expansion measurements have been carried out from 6 to 340 K on a high‐purity silicon sample using a linear absolute capacitance dilatometer. The accuracy of the measurements varies from ±0.01×10−8 K−1 at the lowest temperatures to ±0.1×10−8 K−1 or 0.1%, whichever is greater, near room temperature, and is sufficient to establish silicon as a thermal expansion standard for these temperatures. The agreement with previous data is satisfactory at low temperatures and excellent above room temperature where laser‐interferometry data of comparable accuracy exist. Thermal expansions calculated from ultrasonic and heat‐capacity data are preferred below 13 K where experimental problems occurred.

Linear thermal expansion measurements on silicon from 6 to 340 K

The techniques and instrument designs developed for continuously recording tiltmeters and strainmeters are reviewed. Consideration is given to short-base and long-base tiltmeters as well as to rod, wire, laser and borehole hydraulic strainmeters. Since a good design is difficult to achieve, it is necessary to select the goals an instrument must meet and be prepared to compromise on others. The importance of comparison tests is demonstrated, and it is concluded that good results cannot be gotten at low cost or without careful attention to details.

Strainmeters and tiltmeters

We investigate the thermal expansion of low thermal noise Fabry–Perot cavities made of low thermal expansion (LTE) glass spacers and fused silica (FS) mirrors. The different thermal expansion of mirror and spacer deforms the mirror. This deformation strongly contributes to the cavity’s effective coefficient of thermal expansion (CTE), decreasing the zero crossing temperature by about 20 K compared to an all-LTE glass cavity. Finite element simulations and CTE measurements show that LTE rings optically contacted to the back surface of the FS mirrors allow to tune the zero crossing temperature over a range of 30 K.

http://www.exphy.uni-duesseldorf.de/PDF/Legero%202010%20Tuning%20the%20thermal%20expansion%20properties%20of%20optical%20Reference%20Cavities%20with%20fused%20silica%20mirrors.pdf

Tuning the thermal expansion properties of optical reference cavities with fused silica mirrors

We have measured length change vs time for several low thermal expansion materials maintained in evacuated environments at constant temperature (near 300 K). Materials were two types of fused silica, Cer-Vit, ULE, Zerodur, Invar, and super Invar. ΔL/L was measured over a period of 170 days to a precision of two to three parts in 109. In addition, we have measured time-dependent changes in optical contact interfaces and have placed an upper limit on drift of optical phase shift on reflection from multilayer dielectric coatings.

https://iopscience.iop.org/article/10.1088/0026-1394/13/1/004/pdf

Dimensional Stability of Fused Silica, Invar, and Several Ultra-low Thermal Expansion Materials

A method is developed for testing the long-term dimensional stability of an iodine-stabilized He-Ne laser, using a technique whereby thermal expansion coefficients are measured by forming a Fabry-Perot etalon from the sample and monitoring the optical resonant frequencies with tunable sidebands impressed on a laser beam from a frequency-stabilized He-Ne laser. A change of 1 ppm over a 3-yr period on the part of fused silica dimensions and the differential thermal expansion of Invar LR-35 and Super Invar materials are noted. The method is of interest for the metrology of extremely stable structures such as telescopes and optical resonators.

Dimensional Stability of Fused Silica, Invar, and Several Ultralow Thermal Expansion Materials

We report absolute measurements of the thermal‐expansion coefficient of single‐crystal silicon having a precision of ±02% Five thermal‐expansion values have been determined using 19 °C temperature measurement intervals within the temperature range 34–44 °C Our value for the thermal expansion of silicon at 3522±008 °C is 2683±0005×10−6 cm/cm °C, which agrees with previous measurements by others and represents an almost fivefold increase in accuracy

Precise measurement of the thermal expansion of silicon near 40 °C

A program is described for measurement of dimensional stability of fused silica and several ultralow thermal expansion materials. Two techniques are being used: Fizeau interferometry by NBS‐Corning and Fabry‐Perot interferometry by the Optical Sciences Center, University of Arizona. In the latter measurements an accuracy of one part in 109 is sought. A by‐product of the Arizona results will be measurements of the stability of optical phase shifts on reflection from multilayer stacks and also dimensional stability of optical contacts.

Dimensional Stability of Fused Silica and Several Ultralow Expansion Materials

A technique was developed for measuring, with a precision of one part 10 to the 9th power, changes in physical dimensions delta L/L. Measurements have commenced on five materials: Heraeus-Schott Homosil (vitreous silica), Corning 7940 (vitreous silica), Corning ULE 7971 (titanium silicate), Schott Zero-Dur, and Owens-Illinois Cer-Vit C-101. The study was extended to include Universal Cyclops Invar LR-35 and Simonds-Saw Superinvar.

/pdf/measurement-of-dimensional-stability-2sbfuixy5x.pdf

M. A. Norton

Papers

Dimensional Stability of Fused Silica, Invar, and Several Ultra-low Thermal Expansion Materials

Dimensional Stability of Fused Silica, Invar, and Several Ultralow Thermal Expansion Materials

Precise measurement of the thermal expansion of silicon near 40 °C

Dimensional Stability of Fused Silica and Several Ultralow Expansion Materials

Measurement of dimensional stability