- #1
stevenytc
- 12
- 0
Homework Statement
A diatomic molecule, when excited, can vibrate in such a way that the relative distance
between the two atoms, r(t), executes periodic oscillations. If the potential energy of the
molecule as a function of r is given by V (r), the time dependence of r(t) is identical to that
of a particle of reduced mass
μ= m1 m2/ (m1 + m2)
moving in the potential V (r). An analytic expression which is sometimes used to approximate the actual interatomic interaction is the Morse potential,
V (r) = B[1 − exp(−β(r−r0 ))]^2 − B
B is the depth of the well, r0 is the equilibrium separation, and β is a parameter which
governs how rapidly the energy rises as one moves away from the equilibrium position.
(3.a) Find the frequency of small classical vibrations about the minimum of the Morse
potential in terms of B, r0 , and β.
(3.b) For the molecule H2 , β = 1.93 Angstrom-1 , r0 = 0.74 angstrom, and B = 4.8 eV. The mass of a hydrogen atom is 1.67 × 10^ -24 grams. What is the frequency of small vibrations? The observed vibrational frequency is 1.32 × 10^14 Hz.
Homework Equations
The Attempt at a Solution
I used Taylor expansion to expand the exponent in V(r) and get, to first order,
V = Bβ^2(r-r0)^2 - B
Compare this to the energy for harmonic oscillator, I conclude that
ω = 2Bβ^2 / μ
Putting in the numbers give me a frequency of 4.17 x 10^12 Hz, which is two orders off.