Analytic limits on the forms of spectra possible from optically thin collisional bremsstrahlung source models

The constraints on hard X-ray bremsstrahlung spectral forms required in order for them to correspond to physically acceptable (nonnegative) electron distributions in thin-target, thick-target, and thermal source models are discussed. The extent to which various spectra can be attributed to the different models is examined, showing that many possible spectra cannot be described by all, or in some cases, any of the models. It is shown that for any bremsstrahlung cross section, the thick-target and thermal models require that successively higher derivatives of the thin-target constraint have the appropriate sign. It is found that thermal models are the most restrictive, and that thin-target models are the least restrictive. Explicit analytic constraint expressions are derived for all three cases for the Kramers cross section and examples of acceptable and unacceptable spectra are given. Application of these criteria to the testing and exclusion of models is discussed.