extremes (relative to room temperature, ~25 °C). These internal stress points produce undesirable frequency shifts in the crystal, decreasing its accuracy. The relationship between temperature and frequency is cubic. The cubic relationship has an inflection point near room temperature. As a consequence the AT-cut quartz crystal is most effective when operating at or near room temperature. For applications which are above room temperature, water cooling is often helpful. Stress-compensated (SC) crystals are available with a doubly rotated cut that minimizes the frequency changes due to temperature gradients when the system is operating at high temperatures, and reduces the reliance on water cooling. SC-cut crystals have an inflection point of ~92 °C. In addition to their high temperature inflection point, they also have a smoother cubic relationship and are less affected by temperature deviations from the inflection point. However, due to the more difficult manufacturing process, they are more expensive and are not widely commercially available. Electrochemical QCM The QCM can be combined with other surface-analytical instruments. The electrochemical QCM (EQCM) is particularly advanced. Using the EQCM, one determines the ratio of mass deposited at the electrode surface during an electrochemical reaction to the total charge passed through the electrode. This ratio is called the current efficiency. Quantification of dissipative processes For advanced QCMs, such as QCM-I and QCM-D, both the resonance frequency, fr, and the bandwidth, w, are available for analysis. The latter quantifies processes which withdraw energy from the oscillation. These may include damping by the holder and ohmic losses inside the electrode or the crystal. In the literature some parameters other than w itself are used to quantify bandwidth. The Q-factor (quality factor) is given by Q = fr/w. The “dissipation factor”, D, is the inverse of the Q-factor: D = Q−1 = w/fr. The half-band-half-width, Γ, is Γ = w/2. The use of Γ is motivated by a complex formulation of the equations governing the motion of the crystal. A complex resonance frequency is defined as fr* = fr + iΓ, where the imaginary part, Γ, is half the bandwidth at half maximum. Using a complex notation, one can treat shifts of frequency, Δf, and bandwidth, ΔΓ, within the same set of (complex) equations. The motional resistance of the resonator, R1, is also used as a measure of dissipation. R1 is an output parameter of some instruments based on advanced oscillator circuits. R1 usually is not strictly