Fig: Model of a linear symmetric triatomic

Let us consider two atoms of mass m are symmetric triatomic located on each side of an atom of mass M. All three atoms are on one straight line the equilibrium distances apart being denoted by 'b'. For simplicity, we shall first consider only vibrations along the line of molecule. And the actual complicated interatomic potential will be approximated by two springs of force constant k joining the three atoms. Let us consider three coordinates x1, x2 and x3 are making the position of the three atoms on the line.

In these coordinates, the potential energy is given by

We now introduce coordinate relative to the equilibrium position

ηi = xi - xoi

η1 = x1 - x01 , η2 = x2 - x02, η3 = x3 - x03 .............2

where x02-x01 = b = x03-x02

Using equation 2 in equation 1 we get

Now, writing in the matrix notation,

The equation of motion for small oscillation is given by

where w is given by secular equation as

Equating each term to zero, we get

For first term:

then the equation of motion can be written as

ξ = ct + D ................6

The equation 6 gives the rigid body transformation of the atoms of the molecule.

For second term

Then the equation of motion becomes

The equation 7 gives the simple harmonic oscillator of ends atom with equal amplitude,

For third term

Then the equation of motion becomes,

The equation 8 gives the oscillation of central atom as well as the oscillation of end atoms.

The equation of particular mode of equation can be written as

According to Normalisation condition

Equating LHS and RHS matrix and solving we get

Thus we have both eigen values and eigen vectors for linear triatomic molecule.

Physical Interpretation

1. For w = 0, all the atoms of the molecule produces a uniform translational motion with same amplitude and phase.

2. For w = √ k/m , the central atom remains stationery and end atoms vibrates with equal amplitude but opposite in phase.

3. For w = √ k/m (1 + 2m/M), the central atom vibrates with different phase and amplitude but in end atoms vibrates with same amplitude and in same phase.

This note is taken from Classical Mechanics, MSC physics, Nepal.

This note is a part of the Physics Repository.

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