Chemistry Reference
In-Depth Information
Table 1 Debye temperatures
for some elemental solids and
simple compounds. Data are
obtained from thermal
measurements at low
temperature [
25
]
Solid
Y
D
(K)
Na
158
Si
640
Al
428
P
105
Ar
93
K
91
Fe
457
Cu
343
Cs
38
C (diamond)
2,230
KCl
235
NaCl
321
218
a
H
2
O
LiF
732
a
From [
26
]
Crystals held together by strong forces between atoms and molecules will certainly
have higher phonon frequencies (especially of external modes), and therefore higher
Y
D
.InTable
1
, we can see some known Debye temperatures obtained from experi-
mental heat capacities measurement for some elements or simple compounds. We
clearly see that Debye temperature can be quite low (such as for crystals of noble gases
and alkali metals) or relatively higher (such as for transition metals and ionic solids).
Very few data are available for most molecular crystals. The practical meaning of
recommending
T
Y
D
is that only translational and librational modes of an ideally
rigid molecule in the crystal are actually activated. If the temperature is significantly
lower than Y
D
(
T
~0.1Y
D
) then the Debye
T
3
regime is valid. For most molecular
crystals, in neutral electrostatic conditions and in the absence of strong intermolecular
interactions like hydrogen bonding, the temperature should be well below 100 K. In
reality it is enough to de-activate substantially the optic modes in order to maximize the
benefits of low temperature as we will see in the following.
<
2.2.1 Resolution
As we saw in (9), the structure factors are affected by the temperature through the
so-called Debye Waller factor
T
(S). As temperature decreases, the diffracted
intensity increases because
T
(S) grows, especially for large diffraction angles. For
a molecular crystal of an organic compound, the isotropic atomic displacement
parameter can be reduced by a factor of 2-3 from room temperature down to 100 K
(for example from 0.05 to 0.02
˚
2
), corresponding to ca. 100 times larger intensity
for a reflection at 0.5
˚
resolution. The ratio could be even higher if the temperature
is further reduced to 10 K, taking advantage of even smaller displacement
parameters. A reflection at high diffraction angle is typically very weak because
only core electrons contribute to the structure factor and, as discussed, Debye
