so, if there are no charges in the beam region, then expanding into a Taylor series in the vicinity of the axis and leaving only the first term, taking</p>
<div class="math notranslate nohighlight">
\[\nabla \vec {E} = 0\]</div>
<div class="math notranslate nohighlight">
\[E_r = -0.5 r E'_z \approx - 0.5 r \gamma '' mc ^ 2 / q.\]</div>
<p>Then we can finally write the envelope equation for a round beam of radius <span class="math notranslate nohighlight">\(r\)</span> with a Kapchinsky-Vladimirsky distributi