Geoscience Reference
In-Depth Information
Bringing everything together, we get:
d
2
(
r
−
r
0
)
k
M
(
+
r
−
r
0
)
=
0
(3.6)
d
t
2
in wh
ich w
e use the property that the derivative of
r
0
is zero. Periodic functions of the form
sin
√
k
Mt
clearly satisfy this equation. The m
ost
important parameter of this solution is
the frequency of the oscillation
/
π
)
√
k
ν
=
(
1
/
2
/
μ
. The harmonic mean mass (known as
the reduced mass)
M
H
) arises because each atom actually
carries out only part of the work it would have to do if it was attached to a steady support.
As expected, heavy atoms do not jump around very quickly!
Upon formulation of black-body radiation theory, Planck laid the groundwork of quan-
tum mechanics by hypothesizing that a system undergoing a periodic movement with
frequency
μ
such as 1
/μ
=
(1
/
M
O
)
+
(1
/
ν
can only occupy the discrete energy states
E
n
=
nh
ν
, in which
h
is the Planck
10
−
34
constant equal to 6.63
From observations on the photo-
electric effect, Einstein postulated that the energy of an electromagnetic wave is actually
bundled with the properties of a particle, the photon. Photons have no mass and no electric
charge. Upon emission or absorption of photons, material shifts from one energy level to
another, but the energy carried by each photon is always
h
×
J s and
n
=
0, 1, 2,
...
.
For the harmonic oscillator, the situation is slightly more complicated by the existence
of the zero-point energy, without which, however, no fractionation of stable isotopes would
exist. The relationship between the energy of our ball-and-spring system and frequency is:
ν
n
h
1
2
E
n
=
+
ν
(3.7)
with
n
=0,1,2,
Again, moving up and down the energy levels involves the absorption
or the emission of photons with energy
h
...
. This does not happen, however, for homo-
nuclear, symmetrical diatomic molecules, such as H
2
and O
2
, which cannot change their
vibrational and rotational energy level and explains why these gases are transparent (see
ν
0 is a result of
the Heisenberg uncertainty principle: if the energy of the system was allowed to reach, as
it i
s in classical ph
ysics, its minimum value, both the position (
r
=
r
0
) and the velocity
=
(
√
2
m
) of each atom would be precisely known, which is not permitted by the
uncertainty principle. Such a limitation does not arise for rotational energy because the
position of the atom on its orbit remains undetermined at all times. Vibrational zero-point
energy is important for natural systems because at nearly up to ambient temperature, most
oscillators are precisely at that energy level.
Not all the natural systems are simple diatomic harmonic oscillators. More complex
molecules have more complex patterns of vibration. Fortunately, group theory allows us
to make the situation more tractable and demonstrates that any arbitrary vibration of a
molecule can be broken down into the superposition of simple “normal modes.” Most
molecules made of
N
atoms have 3
N
(
E
0
−
V
0
) /
−
6 normal modes of vibration, whereas this number
reduces to 3
N
5 for linear molecules.
Figure 3.3a
shows some of the normal modes
for some of the common gaseous species, OH, H
2
O, CO
2
,CH
4
. The frequencies shown
are well known to spectroscopists (infrared, Raman, ultraviolet) and are used to identify
−
