Ammonia's inversion doublet arises from tunnelling of the nitrogen atom through the…

PHYSICS · Subjective · Fermi University Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra · Molecular structure and spectra

Deriving the Spectrum, Ch Polyatomic molecules: normal modes and symmetry

Problem

Ammonia's inversion doublet arises from tunnelling of the nitrogen atom through the plane of the three hydrogens, in a symmetric double-well potential with a barrier of about

2000\,\mathrm{cm^{-1}}

. The splitting depends exponentially on

\sqrt M

, where

M

is this chapter's tunnelling reduced mass,

M=\mu_{\rm inv}=3m_Xm_{\rm N}/(3m_X+m_{\rm N})

for hydrogen isotope

X

and

m_{\rm N}=14.003\,\mathrm u

; the barrier height and width are set by the electronic structure and are unchanged by isotopic substitution. Measured ground-state splittings (extrapolated to

J=0

) are

\Delta(\mathrm{NH_3})=23.8\,\mathrm{GHz}

and

\Delta(\mathrm{ND_3})=1.6\,\mathrm{GHz}

, with

m_{\rm H}=1.008\,\mathrm u

and

m_{\rm D}=2.014\,\mathrm u

. (a) Write the splitting as

\Delta=\Delta_0\exp(-\gamma\sqrt M)

, for constants

\Delta_0

and

\gamma

set by the (isotope-independent) barrier and width, and state in one sentence why the exponent is expected to carry essentially all of the mass dependence. (b) Use the

\mathrm{NH_3}

and

\mathrm{ND_3}

data to find

\gamma

and

\Delta_0

. (c) Predict the ground-state inversion splitting of tritiated ammonia,

\mathrm{NT_3}

, using

m_{\rm T}=3.016\,\mathrm u

, and express it as a fraction of the

\mathrm{NH_3}

splitting.

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