For crystals under external stress and temperature, a general equation to determine their period vectors (cell edge vectors) was derived based on the principles of statistical physics. This equation applies to both classical systems and quantum systems. It is consistent and can be combined with the previously derived one in the Newtonian dynamics. The existing theory for crystals under external pressure is covered as a special case. The new equation is also the mechanical equilibrium condition and the equation of state for crystals under external stress and temperature. It should be helpful in studying piezoelectric and piezomagnetic materials, since the period vectors change with external stress. For linear elastic crystals, it is the microscopic and temperature-dependent form of the generalized Hooke's law, therefore, it can be used to calculate the corresponding elastic constants, for given temperatures.