Calorimeter Group Introduction Physics Today paper The basics

The basics of x-ray calorimetry

As shown in figure 2, an x-ray calorimeter is conceptually a simple device, whose basic principles are obvious from the outset. Those principles are as follows:

• The calorimeter should operate at a low temperature so that the energy deposited is large relative to the thermodynamically unavoidable random transfer of heat across the weak link.
• The absorber should be opaque to x-rays, and yet have a low heat capacity so that a small deposition of energy is translated into a measurable temperature change.
• The absorber must thermalize well --- that is, it must reproducibly and efficiently distribute the energy of the initial photon across a thermal distribution of phonons (or electrons, depending on the thermometer).
• The thermometer must be sensitive.
• And the thermal link should be weak enough such that the time for the base temperature to be restored is the slowest time constant in the system (compared with thermalization and diffusion times), yet the link should not be so weak that the device is too slow to handle the incident x-ray flux.

How well can a calorimeter measure the energy it absorbs? Consider a calorimeter with a thermometer that translates changes in temperature into a voltage. It has a heat capacity C and a link with thermal conductance G to a heat sink. An instantaneous deposition of energy into the absorber produces a voltage pulse with an exponential decay time constant equal to C/G.

In the frequency domain, the signal is nearly independent of frequency f when f<< 1/2 , and falls off as 1/f above this corner frequency. The magnitude of the noise due to the random transfer of energy across the link to the heat sink (an elementary calculation in statistical mechanics) has the same frequency distribution as the signal. If this so-called phonon noise were the only source of noise, the signal-to-noise ratio would not depend on frequency, and the measurement error could be made arbitrarily small by employing a sufficiently large bandwidth.

In reality, a frequency-independent noise term and a finite detector response time both reduce the signal-to-noise ratio at high frequencies, effectively limiting the usable bandwidth. This state of affairs is illustrated in figure 3.

To understand the bandwidth limitations and calculate the highest energy resolution attainable, we need to consider specific implementations of the calorimeter concept, which, to date, have been based on resistive, capacitive, inductive, paramagnetic, and electron tunneling thermometers. The best energy resolution so far has been achieved with resistive thermometers --- specifically, semiconductor thermistors and superconducting transition-edge sensors.

Johnson noise and Joule heating are important effects in resistive calorimeters. An electrical analog of Brownian motion, Johnson noise is frequency-independent voltage noise that scales with the square root of temperature and resistance. Joule heating occurs because the bias current or voltage used to convert a change in resistance to an electrical signal is continually dissipating power in the thermometer. (See figure 4 for representative circuit diagrams illustrating the role of the applied bias.)

These two physical effects lead to the concept of optimal bias. Increasing the bias raises the signal and phonon noise relative to the Johnson noise, extending the useful bandwidth, until the decrease in signal-to-noise resulting from self-heating offsets that gain. Also contributing to the optimization is the increase in heat capacity with temperature.

Fifteen years ago, Harvey Moseley (the US father of quantum calorimetry) and his colleagues⁷ calculated the energy resolution attainable in an ideal calorimeter with a resistive thermometer, following the similar optimization done by John Mather for infrared bolometers.⁸ Moseley obtained an expression for the energy resolution of the form

where kis Boltzmann's constant, T_b is the temperature of the heat sink, and C_b is the heat capacity evaluated at that temperature.

Writing the reduced temperature T/T_b as t, the heat capacity and thermal conductance functions are
and

respectively, but the thermal conductance does not appear in the expression for except through its exponent . depends on the resistance through the logarithmic sensitivity , which is defined by

For simplicity, we have not included the definition of above, but, for most practical values of and , is proportional to . Minimizing the full expression in t determines the optimal biastemperature and the highest resolution for a given detector.

It is assumed that an incident photon changes the temperature so little that it is valid to neglect the temperature dependence of C, G, , and the noise terms during the pulse. External sources of noise, such as from a signal amplifier, are also assumed to be negligible compared with the intrinsic noise. Though amplifier noise requires careful attention, it can be rendered negligible for resistive thermometers by the right choice of detector resistance and by the use of low-noise junction field effect transistors (JFETs) for high-impedance devices and superconducting quantum interference devices (SQUIDs) for low-impedance devices.

Another property of resistive thermometers, electrothermal feedback (ETF), occurs because the bias power changes as the resistance changes. Consider a thermistor with a negative that is biased with a constant current I, as in the first example in figure 4. As T increases, R will fall, causing I²R, the Joule power, to drop. This loss makes the pulse recovery time faster than the value of C/G expected in the absence of feedback. In devices with large , this effect can be very important.

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