Exercise:
The molecular vibrations of a linear triatomic molecule e.g. mathrmCO_ can be modelled as a system of three masses coupled through springs see figure. center includegraphicswidth.cm#image_path:co-molecul# center abcliste abc Derive the differential s for the longitudinal vibrations and the corresponding coefficient matrix. abc Determine the normale modes of oscillation. abc Calculate the ratio of the two nontrivial frequencies. abcliste

Solution:
abcliste abc The differential s for the three masses left to right are M ddot x_ kx_-x_ -k x_ + k x_ m ddot x_ -kx_-x_+kx_-x_ k x_ - k x_ + k x_ M ddot x_ -kx_-x_ k x_ -k x_ where M is the mass of an oxygen and m the mass of a carbon atom. The coefficient matrix is thus bf A pmatrix -frackM & frackM & frackm & -frackm & frackm & frackM & -frackM pmatrix abc The eigenvalues are the zeros of the characteristic polynomial: detbf A-lambda bf I pmatrix -frack+lambda MM & frackM & frackm & -frack+lambda mm & frackm & frackM & -frack+lambda MM pmatrix frac-k+lambda M^k+lambda m+k^k+lambda MM^ m Longrightarrow k^+k^ M lambda & quad - k^+k M lambda+M^lambda^k+lambda m k^+k^ M lambda & quad -k^-k^ M lambda-k M^ lambda^ & quad -k^ m lambda - k M m lambda^ - M^ m lambda^ -lambda k^ M+k^ m + k M^ lambda+k M m lambda + M^ m lambda^ -lambda leftk^M+m+k MM+mlambda+M^ m lambda^right The first zero is trivial: lambda_ The other two zeros follow from k^M+m+k MM+mlambda+M^ m lambda^ so lambda_ frac-k MM+mpmsqrtk^M^M+m^-k^M^mM+mM^m frac-kM+mpm ksqrtM^+Mm+m^-Mm-m^Mm frac-kM+mpm kMMm Longrightarrow lambda_ -frackM lambda_ -frackM+mMm The nontrivial angular frequencies are thus omega_ sqrtkM omega_ sqrtkM+mMm The eigenvectors are hat x_ pmatrix pmatrix hat x_ pmatrix - pmatrix hat x_ pmatrix -fracMm pmatrix The first normal mode is no vibration but a uniform motion of all three atoms. In the second normal mode the carbon atom remains at rest while the oxygen atoms oscillate in opposite directions. This explains why the mass of the carbon atom does not appear in the expression for the frequency. The third normal mode corresponds to the two oxygen atoms moving in phase while the carbon atom moves in the opposite direction. The amplitudes are such that the molecule's centre of mass remains at rest. abc The ratio of omega_ and omega_ is fracomega_omega_ sqrtfrackM+mMmfracMk ratioF sqrtfractimesM+mm resultratioS abcliste