Vibrations of an Infinitely Long Chain of CH2-Groups and the Infra-red Spectrum of Polythene

Abstract

THE vibrations of an infinitely long chain of CH₂-groups have been calculated using a valence and angle force potential. It is assumed that the carbon atoms form a zigzag line in the x,y-plane, from which the two C—H linkages branch off symmetrically with respect to this plane, with the x-axis as molecular axis. This array can be considered as a crystal lattice the elementary cell of which consists of two CH₂-groups (six particles in the cell) resulting in eighteen vibrations of the elementary cell. The potential function V has the form : where l_i is stretching of C_i–C_i+1 linkages; l_i and l_i are stretching of two C—H linkages attached to C_i; λ_i is change of the angle C_i1—C_i—C_i+1; α_i is change of the angle H_i—C_i–H_i; δ_i, δ_iâ are changes of the angle C_i1—C_i—H_i; ϵ_i and ϵ_i are changes of the angle H_i—C_i—C_i+1; Î_i is angle of twist of C—C linkages produced by a rotation of the two CH₂-groups at the ends of the bond around it; a is the equilibrium distance of the carbon atoms, b of the C–H linkages and θ the tetrahedral angle; f_C, f_H, etc., are the force constants associated with these movements.