Compton scattering of single-energy ?-ray results in a one-to-one relationship between the incident and

scattered photon energies. This relationship is altered when the incident beam has a definite energy distribution

because of the broadening occurring in the energy distribution upon Compton scattering. This broadening

causes a change in the spectral density of the Compton-scattered spectra. To restore the spectral density, the

energy distribution of the scattered radiation must be manifested as a function of the scattering kinematics.

Here, we propose a simple analytic way to calculate the energy spread of the scattered photons in terms of the

geometry of the scattering process and the energy spread of the incident photon beam. The predictions of the

model agree with measurements of Compton scattering of quasi-monochromatic ?-ray beams, carried out at the

High Intensity ?-ray Source (HI?S) facility, Duke University. As a benchmark of our method, we measured the

intensity profile of energy-distributed ?-ray beams by direct measurements as well as by Compton scattering.

We found that only when the spectral density of the scattered radiation is restored, the measured intensity

profile agrees with the actual profile of the incident beam. The proposed method can continuously measure

the flux of an energy-distributed ?-ray beams in real time and on a bin-by-bin basis. Such online monitoring

of ?-ray beams is indispensable for in-beam measurements and applications.