  The uncertainty of the energy measurement of the electron beam on circular
electron positron collider (CEPC) must be smaller than 10$\mathrm{MeV}$ 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$\mathrm{MeV}$ for the electron energy of
$120\mathrm{GeV}$ in the Higgs mode. In this system, the energy resolution of
the gamma detection needs to reach $10^{-4}$. 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
${TM_{010}}$ 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 $1.5\mathrm{mm}$ 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.
