1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where the equilibrium spacing of the atoms within each unit cell is b (where b

Extracted text: 1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where the equilibrium spacing of the atoms within each unit cell is b (where b<5. the="" displacements="" of="" the="" atoms="" from="" their="" equilibrium="" positions="" are="" given="" by="" u,,="" uz,="" ...="" ,="" uzn-1»="" uznı="" uzn+1,="" ---="" the="" harmonic="" forces="" between="" nearest-neighbour="" atoms="" are="" characterised="" by="" the="" altemating="" interatomic="" force="" constants="" b,="" and="" bz.="" (a)="" develop:="" (i)="" the="" equation="" of="" motion="" for="" the="" 2nth="" atom="" in="" terms="" of="" forces="" exerted="" by="" the="" (2n="" –="" 1)th="" and="" (2n="" +="" 1)th="" atoms.="" (ii)="" the="" equation="" of="" motion="" for="" the="" (2n="" +="" 1)th="" atom="" in="" terms="" of="" forces="" exerted="" by="" the="" 2nth="" and="" (2n="" +="" 2)th="" atoms.="" (b)="" using="" the="" equations="" of="" motion="" and="" assuming="" travelling="" wave="" solutions="" of="" the="" form="" uzn="Aellat-kna)" and="" uzn+1="," belwt-kna-kb)="" derive="" two="" simultaneous="" equations="" for="" a="" and="" b.="" (c)="" making="" use="" of="" the="" fact="" that="" a="" homogeneous="" system="" of="" linear="" equations="" c11x="" +="" c12y="0" c21x="" +="" c22y="0" only="" has="" a="" non-zero="" solution="" for="" x="" and="" y="" when="" c11="" c12="0," c21="" c22="" obtain="" an="" expression="" for="" w?.="" (d)="" making="" use="" of="" the="" approximation="" 14="" x="" vp2="" –="" qx²="p" -="" 2="" p="" for="" small="" x,="" determine="" the="" dispersion="" relation="" for="" the="" acoustic="" branch="" in="" the="" long-wavelength="" limit="" and="" thus="" find="" the="" group="" velocity="" of="" acoustic="" waves="" in="" the="" lattice.="" b.="">