1. Superconducting Transition Edge Sensors

Another type of calorimeter is the superconducting transition edge sensor (Irwin et al. (1996); Irwin (1995)). The resistive thermometer in this system is a superconducting film which is voltage biased at a point on the superconducting-normal transition. Because $\alpha$ is positive the detector operates with negative electrothermal feedback. The absorption of an X-ray photon results in a drop in current which is measured with a SQUID amplifier. The thermal resistance between the film and substrate will be due to electron-phonon decoupling for thin films at low temperatures or the Kapitza resistance (e.g.Little (1959)) for thick films at higher temperatures. The power flowing to the substrate is

P₀ = K(Tⁿ - T₀ⁿ) (71)

where the thermal conductance is

g = nK(T^{n - 1} - T₀^{n - 1}). (72)

If electron-phonon decoupling dominates the thermal resistance n will be 5 or 6. If the Kapitza resistance dominates n = 4. The effective time constant of a transition edge detector is

$$\tau_{\rm eff}^{} {\tau\over1+{\alpha\over n}}$$ (73)

For high-purity films values of $\alpha$ = 100-1000 are feasible. The availability of larger values of $\alpha$ is the only fundamental difference between transition edge detectors and calorimeters with semiconductor thermometers. The much higher values of $\alpha$ allow the transition edge detector to operate in the extreme electrothermal feedback regime where almost all of the energy deposited by an X-ray photon is removed by a decrease in Joule heating. This allows higher counting rates and better energy resolution to be attained, provided an amplifier with a wide bandwidth is used such as the two-stage SQUID amplifiers developed by Welty and Martinis (1993). The fundamental limit on the resolution of transition edge sensors is

$$\Delta \sqrt{4kT^2C\left({1\over\alpha}\right)\sqrt{n\over2}}$$ (74)