the above estimate for equal to , we find the lower cut-off to the impact parameter to be about We can also use as an estimate of the cross section for large-angle collisions. Under some conditions there is a more stringent lower limit due to quantum mechanics, namely the de Broglie wavelength of the electron, where is Planck's constant. At large impact parameters, the charge of the ion is shielded by the tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The upper cut-off to the impact parameter should thus be approximately equal to the Debye length: Coulomb logarithm The integral of thus yields the logarithm of the ratio of the upper and lower cut-offs. This number is known as the Coulomb logarithm and is designated by either or . It is the factor by which small-angle collisions are more effective than large-angle collisions. For many plasmas of interest it takes on values between and . (For convenient formulas, see pages 34 and 35 of the NRL Plasma formulary.) The limits of the impact parameter integral are not sharp, but are uncertain by factors on the order of unity, leading to theoretical uncertainties on the order of . For this reason it is often justified to simply take the convenient choice . The analysis here yields the scalings and orders of magnitude. Mathematical treatment for plasmas accounting for all impact parameters An N-body treatment accounting for all impact parameters can be performed by taking into account a few simple facts. The main two ones are: (i) The above change in perpendicular velocity is the lowest order approximation in 1/b of a full Rutherford deflection. Therefore, the above perturbative theory can also be done by using this full deflection. This makes the calculation correct up to the smallest impact parameters where this full deflection must be used. (ii) The effect of Debye shielding for large impact parameters can be accommodated by using a Debye-shielded Coulomb potential (Screening effect Debye length). This cancels the above divergence at large impact parameters. The above Coulomb logarithm turns out to be modified by a constant of order unity. Historical aspects In the 1950s, transport due to collisions in non-magnetized plasmas was simultaneously studied by two groups at UC Berkeley’s Radiation Laboratory. They quoted each other’s results in their respective papers. The first reference deals with the mean-field