Intensity of Bremsstrahlung X-rays: Kramers' Law
- Practical Electron Microscopy and Database -
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This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.


Kramers' law, discovered by a Dutch physicist Hendrik Anthony Kramers, is a formula for the continuous spectral distribution of bremsstrahlung X-rays produced by an X-ray photon or electron striking a solid. The Kramer’s law can be used to subtract the background spectrum from a measured EDX spectrum (see page3918).

In the case with the radiation of an X-ray tube, this law is usually given by the relation,

         Kramers' law - X-rays ---------------------- [2522a]

         K -- A constant,
         I0(λ)dλ -- The intensity emitted over the wavelength interval dλ,
         λmin -- The shortest emitted wavelength,
         λ -- The wavelength at the middle of the interval dλ.

In electron microprobe analysis, similar to the case with an X-ray tube, the intensity of the bremsstrahlung X-rays (Ib) at an energy (Ev) is also quantified by the modified Kramers' law,

         Kramers' law - X-rays ---------------------- [2522b]

         Z -- The average atomic number of the specimen,
         E0 -- The incident beam energy,
         I -- The electron beam current,
         Ev -- The continuum photon energy.

Based on Equation 2522b, the x-ray continuum intensity decreases with increase of the photon energy, yielding zero at the energy of the incident electron beam. At low X-ray photon energies the intensity increases rapidly due to the greater probability for slight deviations in the trajectory caused by the Coulombic field in the atoms.

Note that the Kramers' algorithm presents a relatively poor approximation of the actual spectrum because it does not involve the X-ray absorption in the specimens, which is wavelength dependent and attenuate the longer wavelengths to a far greater degree than the shorter wavelengths. Furthermore, for the case of X-ray tube, the absorption of the X-ray also occurs in the tube window.




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