  Transition-edge sensor X-ray microcalorimeters are usually calibrated
empirically, as the most widely-used calibration metric, optimal filtered pulse
height (OFPH), in general has an unknown dependance on photon energy,
$E_{\gamma}$. Because the calibration function can only be measured at specific
points where photons of a known energy can be produced, this unknown dependence
of OFPH on $E_{\gamma}$ leads to calibration errors and the need for
time-intensive calibration measurements and analysis. A calibration metric that
is nearly linear as a function of $E_{\gamma}$ could help alleviate these
problems. In this work, we assess the linearity of a physically motivated
calibration metric, $E_{Joule}$. We measure calibration pulses in the range 4.5
keV$<$$E_{\gamma}$$<$9.6 keV with detectors optimized for 6 keV photons to
compare the linearity properties of $E_{Joule}$ to OFPH. In these test data
sets, we find that $E_{Joule}$ fits a linear function an order of magnitude
better than OFPH. Furthermore, calibration functions using $E_{J}$, an
optimized version of $E_{Joule}$, are linear within the 2-3 eV noise of the
data.
