0 - 5 0 x ▁larger ▁due ▁to ▁un control led ▁aber r ations ▁present ▁at ▁high ▁angles ▁of ▁t ilt . ▁ ▁For ▁example , ▁a ▁ 2   nm ▁pre cess ed ▁pro be ▁with ▁> 4 0   m rad ▁pre cess ion ▁angle ▁was ▁demonstrated ▁in ▁an ▁aber r ation - correct ed ▁N ion ▁Ult ra ST EM ▁with ▁native ▁sub - Å ▁pro be ▁( ab err ations ▁corrected ▁to ▁~ 3 5   m rad ▁half - angle ). ▁ ▁If ▁the ▁pre cess ion ▁angle ▁is ▁made ▁too ▁large , ▁further ▁comp lications ▁due ▁to ▁the ▁overlap ▁of ▁the ▁Z OL Z ▁and ▁H OL Z ▁ref lections ▁in ▁the ▁project ed ▁pattern ▁can ▁occur . ▁This ▁comp licates ▁the ▁indexing ▁of ▁the ▁diff raction ▁pattern ▁and ▁can ▁cor rupt ▁the ▁measured ▁intens ities ▁of ▁ref lections ▁near ▁the ▁overlap ▁region , ▁thereby ▁reducing ▁the ▁effect iveness ▁of ▁the ▁collected ▁pattern ▁for ▁direct ▁methods ▁calculations . ▁ ▁The oret ical ▁consider ations ▁▁ ▁For ▁a ▁curs ory ▁introduction ▁to ▁the ▁theory ▁of ▁electron ▁diff raction , ▁see ▁the ▁theory ▁section ▁of ▁the ▁electron ▁diff raction ▁w iki . ▁For ▁a ▁more ▁in ▁depth ▁but ▁understand able ▁treatment , ▁see ▁part ▁ 2 ▁of ▁Williams ▁and ▁Carter ' s ▁Trans mission ▁Electron ▁Micro sc opy ▁text ▁ ▁While ▁it ▁is ▁clear ▁that ▁pre cess ion ▁reduces ▁many ▁of ▁the ▁dynam ical ▁diff raction ▁effects ▁that ▁pl ague ▁other ▁forms ▁of ▁electron ▁diff raction , ▁the ▁resulting