Form Factor $g$ In Longitudinal Space Charge Impedance

In carrying out calculations of the effect of longitudinal space charge on longitudinal motion, the transverse beam size appears in a form factor which is usually written as $g=1+2\ln (b/a)$. In fact, this expression applies to particles with vanishing betatron amplitude in a beam with uniform transverse distribution. It is argued that an average over the transverse distribution should be used instead of the value on axis. It is shown that for the realistic `binomial' family of distributions the 1 in the above expression for $g$ should be replaced by a value near 0.5 if $a$ is interpreted as twice the rms width of the beam.