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Silicon diodes have traditionally been the detectors of choice for quantitative X-ray spectroscopy. Their response has been very well characterized and existing software algorithms process the spectra for accurate, quantitative analysis. But Si diodes have limited sensitivity at energies above 30 keV, while recent regulations require measurement of heavy metals such as lead and mercury, with K X-ray emissions well above 30 keV. Measuring the K lines is advantageous to reduce the interference between the L lines of these elements with each other and with the K lines of light elements. CdTe has much higher stopping power than Si, making it attractive for measuring higher energies, but the quality and reproducibility of its spectra have limited its use in the energy range of characteristic X-rays, < 100 keV. CdTe detectors have now been optimized for energies < 100 keV, yielding electronic noise of 500 eV for a 5 mm × 5 mm × 0.75 mm device. This provides adequate energy resolution for distinguishing characteristic X-ray peaks > 30 keV. The response function of these CdTe detectors differs in important ways from that of Si detectors, requiring changes to the X-ray analytical software used to process the spectra. This paper will characterize the response of the latest generation of high resolution CdTe detectors at the energies of characteristic X-rays. It will present effects important for X-ray analysis, including the peak shape arising from hole tailing, escape peaks, spectral background, linearity, and stability. Finally, this paper will present spectral processing algorithms optimized for CdTe, including peak shape and escape peak algorithms. [ABSTRACT FROM AUTHOR]