, ▁because ▁ ▁for ▁trans verse ▁ph on ons , ▁only ▁interactions ▁with ▁longitud inal ▁ph on ons ▁are ▁non - zero . ▁Therefore , ▁the ▁complete ▁interaction ▁matrix ▁is ▁ ▁where ▁the ▁Kr one cker ▁delta ▁enfor ces ▁momentum ▁conservation ▁and ▁ar ises ▁from ▁assuming ▁the ▁electronic ▁wave functions ▁( final ▁state , ▁< k ' | , ▁and ▁initial ▁state , ▁| k >) ▁are ▁plane ▁waves ▁as ▁well . ▁ ▁Ac oust ic ▁Ph on ons ▁ ▁Using ▁F erm i ' s ▁golden ▁rule , ▁the ▁scatter ing ▁rate ▁for ▁low ▁energy ▁ac oust ic ▁ph on ons ▁can ▁be ▁approxim ated . ▁The ▁interaction ▁matrix ▁for ▁these ▁ph on ons ▁is ▁with ▁the ▁ph on on ▁radial ▁frequency ▁ ω q = c q , ▁volume ▁V , ▁solid ▁density ▁ ρ , ▁and ▁ph on on ▁group ▁velocity ▁c . ▁Pl ug ging ▁this ▁into ▁Eq . ▁ 6 ▁gives ▁. ▁With ▁the ▁assumptions ▁that ▁N q >> 1 , ▁ ħ ω << k T ▁and ▁g ( E ') ▁~ ▁g ( E ) ▁( which ▁generally ▁holds ▁for ▁ 3 D ▁cry st als ▁since ▁condu ction ▁electron ▁energ ies ▁are ▁generally ▁much ▁greater ▁than ▁ ħ ω ▁and ▁g ( E ) ▁la cks ▁any ▁van ▁H ove ▁singular ity ) ▁gives ▁the ▁scatter ing ▁rate : ▁▁▁▁▁▁ ▁where ▁g ( E ) ▁is ▁the ▁electronic ▁density ▁of ▁states ▁for ▁which ▁the ▁ 3 - dimensional ▁solution ▁with ▁par abol ic ▁disp ersion