A Universal Equation for the Emission Range of X Rays from Bulk Specimens

The derivation of a universal equation to compute the range of emitted X rays is presented for homogeneous bulk materials. This equation is based on two fundamental assumptions: the φ(ρz) curve of X-ray generation is constant and the ratio of the emitted to the generated X-ray range is equal to the ratio of the emitted to the generated X-ray intensity. An excellent agreement is observed with data obtained from Monte Carlo simulations of 200,000 electron trajectories in C, Al, Cu, Ag, Au, and an Fe–B alloy with boron weight fractions equal to 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.99, performed with the electron beam energy varied from 1 to 30 keV in 1-keV steps. When the ratio of the generated X-ray range to the photon mean free path is much smaller than one, the emission X-ray range is equal to the generated X-ray range, but when this ratio is much greater than one, the emission X-ray range is constant and is given by the product of the effective photon mean free path multiplied by the sine of the take-off angle.