Question

Consider a triatomic molecule, like CO2. The three atoms lie in a straight line, and we model the O atoms to be connected to the central C atom by a spring of constant k (same k for the two bonds). For this question, the atoms all oscillate in the plane of the molecule.

(a) What is the Lagrangian of the motion?

(b) What are the eigenfrequencies of the oscillations? (there are three, although one is zero)

(c) What are the ratios of the energies of the two non-zero frequencies?

(d) For the mode that oscillates with zero frequency, what is its motion? What about for the heighest frequency? It may help to find the eigenvectors.

Answer 1

Let a linear triatomic molecule Consists of tooo atoms 0, each of mass m,, placed at eithes side of thisd alom C The interaction betwren equal distamce 0 mass O and c is the elastic trca the atoms Spring of force Constant k and for beta een the 0-atoms is Zeso. that be displace ment o atoms from thers ubim pasitions, OV Let is Kinetic energy stem T-( , + 2 2 k m poteatial en er14he System k V : -K(+2222) +2 +-192-2.4 Systen Lagrau giam nction L = T-V

L= 출어 (역+)+노일을 *(912일을 . [T] K -K and [V - -K K 2k -K -k The Characteristicuati is k-t m k 2K-'mk K'm -k -K 0 k ave thyee oots The K + K m, 0 eigen frequencies th are hese oscillations

exgies of the to0 fiequedcies is Ratios of mon- 20x0 K 2 R 1-9 K "2t2m, The motion fo oscillates the mode that ith Rexo fsegye2mcy i.e o auu three molecules moves im the Same Phase Dih Same amplitude. the C and that f heighest treguency ie for K 2 m, m The two end molecules CO O ave in phase wih Same amptitude @hile is in middle 'ppesite phase double amplitude

Determinatim eigenvectr: Let aigevecns case :1 W: 0, 0 be (,21M) Then, -K 0 -k 021 2k -k K -K katka 0 +Ka 1D =0 -ka t2ka21Ka31 + Ka51 Kazy -A (say) a Hence arives by axe 1 Veda Case :2 be (an22 Then "32

12 K 2k-KW/ :0 22 -K -K 0 -Ka 22 3/M22-Ka20 a,2- M (Say) ano A2 2-2 yanvacta 2M1 2 Case 3: вз 1 eigen veca ) eiganvec given as 3 -2K 2 -KM2 -K M -2K 23 -k 23 0 -K l

- 2K к22 -0 - K KAA 013KA5 -ka23 -2K 0 V 12 Conditin TA[I] 0 2 O2M V Smpitinton On 7 M2 2.

St Tim igenvectos he T 2 m, M2 + П2 / {0s} amd 2M
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