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ν 1
ν 2
OH
ν 3
CO 2
ν 1
(a)
ν 2
(b)
ν 1
CH 4
ν 3
H 2 O
Figure 3.3
(a) Vibrations of molecules, OH, H 2 O, CO 2 ,CH 4 .Allthevibrations,howevercomplex,canbe
described as the superposition of a small number of normal modes (shown here as arrows)
corresponding to different frequencies
measurable by infrared or Raman spectroscopy.
(b) For solids, the vibrations take the form of traveling waves.
ν
1 ,
ν
2 ,
...
the presence of these gases in the molecular clouds of distant galaxies as well as in the
terrestrial atmosphere.
Crystalline solids behave in yet another way ( Fig 3.3b ) : because ions are regularly dis-
tributed and interact with their neighbors in a continuous way, vibrations take the form of
traveling waves. The pseudo-particles associated with these waves are known as phonons.
Models related to the statistical theory of the heat capacity C V of solids - the increment
of energy associated with a small temperature step - assume either a single vibration fre-
quency (Einstein solid) or a frequency continuum increasing up to a cut-off value imposed
by the lattice (Debye elastic solid). The Einstein and Debye models are reasonably suc-
cessful for simple crystals, but the wealth of spectroscopic data and the advent of large
computers have now made it possible to evaluate exact isotopic properties using so-called
“ab initio” models. Models of fractionation involving liquid phases require particularly
intensive numerical computing.
Where are the isotope effects in all this? Heavy atoms react more slowly than light ones
and there fo re tend to occupy lower energy levels. Bond energy varies with
ν
and therefore
with 1/ μ
is the harmonic mean mass of the atoms that form the molecule. A
consequence of this rule is that the quantized energy levels and the zero-point frequencies
are lower for a bond involving the heavier isotopes: the molecule OD is more stable than
OH ( Fig. 3.1 ). This can be seen for the substitution of H by D in magnesium hydroxide
Mg(OH) 2 (brucite), which drastically reduces the frequency of the OH vibration ( Fig. 3.4 ).
, where
μ
 
 
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