▁was ▁used ▁to ▁obtain ▁the ▁final ▁answer . ▁ ▁Opt ical ▁Ph on ons ▁ ▁Typ ically , ▁ph on ons ▁in ▁the ▁optical ▁branches ▁of ▁v ibr ational ▁disp ersion ▁relationships ▁have ▁energ ies ▁on ▁the ▁order ▁of ▁or ▁greater ▁than ▁k T ▁and , ▁therefore , ▁the ▁approxim ations ▁ ħ ω << k T ▁and ▁N q >> 1 ▁cannot ▁be ▁made . ▁Yet , ▁a ▁reasonable ▁route ▁that ▁still ▁provides ▁a ▁det our ▁from ▁dealing ▁with ▁complex ▁ph on on ▁dispers ions ▁is ▁using ▁the ▁Ein stein ▁model ▁which ▁states ▁that ▁only ▁one ▁ph on on ▁mode ▁exists ▁in ▁sol ids . ▁For ▁optical ▁ph on ons , ▁this ▁approximation ▁turns ▁out ▁to ▁be ▁sufficient ▁due ▁to ▁very ▁little ▁slope ▁variation ▁in ▁ ω ( q ) ▁and , ▁thus , ▁we ▁can ▁claim ▁ ħ ω ( q ) ▁ <0xE2> <0x89> <0x85> ▁ ħ ω , ▁a ▁constant . ▁Con sequently , ▁N q ▁is ▁also ▁a ▁constant ▁( only ▁T ▁dependent ). ▁The ▁last ▁approximation , ▁g ( E ') = g ( E ± ħ ω ) ▁~ ▁g ( E ), ▁cannot ▁be ▁made ▁since ▁ ħ ω ▁~ ▁E ▁and ▁there ▁is ▁no ▁workaround ▁for ▁it , ▁but ▁the ▁added ▁complexity ▁to ▁the ▁sum ▁for ▁ τ ▁is ▁minimal . ▁▁▁ ▁. ▁The ▁sum ▁turns ▁to ▁the ▁density ▁of ▁states ▁at ▁E ' ▁and ▁the ▁B ose – E in stein ▁distribution ▁can ▁be ▁taken ▁out ▁of ▁the ▁sum ▁due ▁to ▁ ħ