changes. In such cases it is necessary to treat the coefficient of thermal expansion as a tensor with up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by x-ray powder diffraction. The thermal expansion coefficient tensor for the materials possessing cubic symmetry (for e.g. FCC, BCC) is isotropic. Temperature dependence Thermal expansion coefficients of solids usually show little dependence on temperature (except at very low temperatures) whereas liquids can expand at different rates at different temperatures. However, there are some known exceptions: for example, cubic boron nitride exhibits significant variation of its thermal expansion coefficient over a broad range of temperatures. Another example is paraffin which in its solid form has a thermal expansion coefficient that is dependent on tempertaure. Isobaric expansion in ideal gases Since gases fill the entirety of the container which they occupy, the volumetric thermal expansion coefficient at constant pressure, , is the only one of interest. For an ideal gas, a formula can be readily obtained by differentiation of the ideal gas law, . This yields where is the pressure, is the molar volume (, with the total number of moles of gas), is the absolute temperature and is equal to the gas constant. For an isobaric thermal expansion we have , so that and the isobaric thermal expansion coefficient is: which is a strong function of temperature; doubling the temperature will halve the thermal expansion coefficient. Computation of the absolute zero From 1787 to 1802, it was determined by Jacques Charles (unpublished), John Dalton, and Joseph Louis Gay-Lussac that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100 °C. This suggested that the volume of a gas cooled at about −273 °C would reach zero. In October 1848, William Thomson, a 24 year old professor of Natural Philosophy at the University of Glasgow, published the paper On an Absolute Thermometric Scale. In a footnote Thomson calculated that "infinite cold" (absolute zero) was equivalent to −273 °C (he called the temperature in °C as the "temperature of the air thermometers" of the time). This value of "−273" was considered to be the temperature at which the ideal gas volume reaches the zero. By considering a thermal expansion linear with temperature (i.e. a constant