1. Calorimeters with Resistive Thermometers

X-Ray microcalorimeters have been constructed by anisotropic etching of single-crystal silicon. This fabrication technique is described by Peterson (1982). The calorimeter volume consists of a back etched panel of silicon with an X-ray absorber and an ion implanted thermistor. This is suspended on four thin silicon legs which provide the thermal link to the heat sink and carry electrical connections. These devices have attained a resolution of 7.3 eV at 6 keV (e.g.McCammon et al. (1993)).

An order of magnitude estimate of the best possible energy resolution may be obtained from the thermodynamic energy fluctuations in the detector (Moseley et al. (1984)). The effective number of phonon modes in the detector is N = C/k_B. The typical phonon mode has occupation number and rms fluctuation 1 and mean energy k_BT. Thus the mean square energy fluctuation is

$$\langle \Delta \rangle$$ (61)

It is possible in principle to obtain energy resolutions better than that in Equation 2.1 through negative electrothermal feedback (Mather (1982)) if the resistive thermometer is biased so that the temperature derivative of the power dissipation is negative:

$${dP\over dT}$$ (62)

If this condition holds the absorption of an X-ray photon causes the temperature to increase which causes the bias power dissipation to drop. This negative electrothermal feedback tends to return the calorimeter to the base temperature as the thermal energy deposited by the X-ray photon is compensated for by a drop in Joule heating. This allows thermal equilibrium to be reestablished faster than if the deposited heat had to be conducted to the thermal bath. The calorimeter thus has an effective time constant $\tau_{\rm eff}^{}$ < $\tau$. This means that, when the effects of electrothermal feedback are taken into account, microcalorimeters can be faster and allow higher count rates than those implied by the thermal time constant $\tau$. Electrothermal feedback also suppresses the Johnson noise for frequencies $\ll$ 1/$\tau$. How the detector should be biased depends on the sign of the temperature coefficient of the thermistor

$$\alpha {d\log R\over d\log T} \left(\vphantom{{T\over R}}\right. {T\over R} \left.\vphantom{{T\over R}}\right) {d R\over d T}$$ (63)

In the case of ion-implanted thermistors where current is carried by electron hopping

$$\propto \left(\vphantom{{T_0\over T}}\right. {T_0\over T} \left.\vphantom{{T_0\over T}}\right)^{\gamma}_{}$$ (64)

with $\gamma$ $\sim$ ${1\over2}$ so that $\alpha$ < 0. Thus, if the thermistor is current biased, the Joule power dissipated will be P = I²R(T) so that

$${dP\over dT} {d R\over d T} {I^2 R \over T} \alpha {P\over T} \alpha$$ (65)

If, however, $\alpha$ > 0, as in the case of the transition edge detectors discussed in Section 3.1, the detector should be voltage biased. Then P = V²/R(T) and

$${dP\over dT} {V^2\over R^2} {d R\over d T} {V^2\over TR} \alpha {P\over T} \alpha$$ (66)

If noise from the amplifier and load resistor are neglected the square of the noise equivalent power has two parts:

NEP² = NEP²_Johnson + NEP²_phonon (67)

where

$$\propto \omega^{2}_{} \tau^{2}_{}$$ (68)

is due to Johnson noise and P is the dc power dissipated in the thermometer. The second term in Equation 2.7 is due to fluctuations in heat transfer across the thermal link:

NEP²_phonon = 4k_BT²G . (69)

It can be shown that the energy resolution is

$$\Delta \xi \sqrt{k_BT_0^2C}$$ (70)

where T₀ is the temperature of the heat sink and the factor $\xi$ is of order unity for semiconductor thermometers with - $\alpha$ $\sim$ 3-8. Thus the resolution of an ideal calorimeter is on the order of the thermodynamic energy fluctuations and scales as T^5$\over$2 assuming the heat capacity scales as T³. The factor $\xi$ is independent of $\tau$ and G so the conductance of the thermal link may be chosen to optimize the counting rate without any loss in resolution. It is also a weak function of the temperature coefficient of the thermistor $\alpha$ and $\xi$ $\propto$ 1/$\sqrt{\alpha}$ as $\alpha$$\to$$\infty$. Moseley et al. (1984) calculated that a calorimeter with realistic optimized parameters could have $\xi$ = 2.56 and $\sqrt{k_BT^2C}$ = 0.15 keV. This yields a theoretical resolution of 0.45 eV rms or 1.1 eV FWHM.

It has since been found that the ultimate attainable energy resolution is worse than this. One problem is the 1/f noise common in solid state devices. Another problem is that the above estimate of the energy resolution assumed that the temperature increase due to the absorption of an X-ray photon was small compared with the base temperature. In practice, if the heat capacity of the detector is small enough, this assumption does not hold and the resulting non-linearity degrades the resolution. The resolution is also degraded by metastable energy storage in the absorber which has already been mentioned. Another problem is that the initial phonon spectrum from an absorbed X-ray is non-thermal (Stahle et al. (1993)). The thermistors operate by phonon-assisted electron hopping and the resulting sensitivity to very high energy phonons causes position dependence and degrades the resolution. It is likely that the best resolution attainable by resistive calorimeters operating at 0.1 K will be limited to $\sim$ 5 keV because of these effects.

The X-Ray Quantum Calorimeter (XQC; Cui et al. (1994)) was a sounding rocket payload consisting of an array of microcalorimeters. It successfully observed the diffuse X-ray background in June 1996. A pre-flight spectrum is shown in Figure 3.1.

Figure 5: Spectrum obtained from a single pixel of the XQC instrument before launch. The energy resolution is 9 eV.