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The uncertainty of the energy measurement of the electron beam on circular electron positron collider (CEPC) must be smaller than 10 MeV to make sure the accurate measurement of the mass of the Higgs boson. In order to simplify the energy measurement system, a new method is proposed by fitting the Compton edge of the energy distribution of the gamma ray from a microwave–electron Compton scattering. With our method, the uncertainty of the energy measurement is 6 MeV for the electron energy of 120 GeV in the Higgs mode. In this system, the energy resolution of the gamma detection needs to reach 1 0 − 4 . Therefore, only the high-purity germanium (HPGe) detector can meet the critical requirement. In a head-on collision mode, the initial photons should be microwave photons with the wavelength of 3.04 centimeters. A cylindrical resonant cavity with selected T M 010 mode is used to transmit microwaves. After the microwave–electron Compton backscattering, the scattered photons and the synchrotron-radiation background transmit a shielding structure and then are detected by a HPGe detector at the end of the beam line of the synchrotron-radiation applications. The hole radius in the side wall of the cavity is about 1. 5 mm to allow the electron beam passing through. The results of the computer simulation technology (CST) software shows that the influence of the hole radius on the cavity field is negligible. The change of the resonance frequency can be easily corrected by fine-tuning the cavity size. [ABSTRACT FROM AUTHOR]