1.2 Bragg Scattering and Bernal–Fowler Water
9
molecules too.
7 The 3D structure of ice was deduced from general assumptions that
molecules with the structure shown in Fig. 1.2 electrostatically interact with each
other. As the interaction of X-rays with ice reduces to the interaction with oxygen
atoms (except modern small-angle X-ray diffraction studies, which are sensitive
to hydrogen atoms as well [25]), no hydrogen–oxygen configurations leaving the
positions of oxygen atoms unchanged have been rejected on the ground of X-ray
experiments.
Figure 1.7 shows the structure that Bernal and Fowler proposed for ice on the basis
of the least energy configuration of five-point tetrahedral H 2 O species. This structure
has been used for the interpretation of Laue diagrams of ice. The oxygen atoms in
this model form a hexagonal configuration, similar to that for liquid water discussed
above. One oxygen atom was assumed occupying each corner of the hexagon. The
hexagons, being assembled in a plane, form a wrinkled surface of conjugate rings.
Then planes alternate in an (a)–(b)–(a)–(b) order along the c-axis, where each (b)plane is a reflection of the (a)-plane along the same axes as the planes themselves [26].
The distance between two neighboring (a)- or (b)-planes is 7.3 Å. The distance
between any neighboring oxygen atoms is about 2.8 Å. The whole structure has
hexagonal symmetry (P6 3 /mmc structure group
8 ). The angle between the sides of the
hexagons is close to the tetrahedral angle (109.5
◦ ) and the angle between the hydrogen
atoms in the water vapor molecule (104.5
◦ ). The cavities inside the hexagonal rings
were assumed large enough to place another water molecule inside. That is why
the electrical conductivity and self-diffusion coefficient of ice along the c-axis, and
perpendicular to this axis differ several times, as has been experimentally confirmed
latter [27, 28].
The hydrogen atoms are not shown in Fig. 1.7, but they were placed by Bernal and
Fowler in fixed positions on the lines between oxygen atoms. Later Pauling adopted
this configuration, because he found that it does not contradict the residual entropy of
ice at low temperatures [29]. On the basis of statistical physics, Pauling then deduced
the following phenomenological ice rules, which although not applicable for real ice
crystals, were adopted, since they directly followed from the structure of the water
molecule proposed by Bernal and Fowler (see Fig. 1.2):
• Each oxygen atom has two hydrogen atoms attached to it at distances of about
0.95 Å, forming a water molecule, the HOH angle being about 105
◦ as in the gas
molecule.
• Each water molecule is oriented so that its two hydrogen atoms are directed approximately toward two of the four oxygen atoms which surround it tetrahedrally.
7 Modern data shows that the infrared and Raman spectra of ice and water are more complex than
those for water vapor. Apart from line shift and broadening [31], overtones, Fermi resonances, and
contributions from ionic species have been observed [32] (see Chap. 2 for details).
8 P6 3 /mmc means having a sixfold screw axis (rotation around an axis in addition to a translation
along the axis); the structure repeats itself three times during a complete 360 ◦ rotation; the “m’s”
stand for mirror planes perpendicular to the basal plane and parallel to the “c”-axis, and with the c
in “mmc” standing for the glide plane.
9
molecules too.
7 The 3D structure of ice was deduced from general assumptions that
molecules with the structure shown in Fig. 1.2 electrostatically interact with each
other. As the interaction of X-rays with ice reduces to the interaction with oxygen
atoms (except modern small-angle X-ray diffraction studies, which are sensitive
to hydrogen atoms as well [25]), no hydrogen–oxygen configurations leaving the
positions of oxygen atoms unchanged have been rejected on the ground of X-ray
experiments.
Figure 1.7 shows the structure that Bernal and Fowler proposed for ice on the basis
of the least energy configuration of five-point tetrahedral H 2 O species. This structure
has been used for the interpretation of Laue diagrams of ice. The oxygen atoms in
this model form a hexagonal configuration, similar to that for liquid water discussed
above. One oxygen atom was assumed occupying each corner of the hexagon. The
hexagons, being assembled in a plane, form a wrinkled surface of conjugate rings.
Then planes alternate in an (a)–(b)–(a)–(b) order along the c-axis, where each (b)plane is a reflection of the (a)-plane along the same axes as the planes themselves [26].
The distance between two neighboring (a)- or (b)-planes is 7.3 Å. The distance
between any neighboring oxygen atoms is about 2.8 Å. The whole structure has
hexagonal symmetry (P6 3 /mmc structure group
8 ). The angle between the sides of the
hexagons is close to the tetrahedral angle (109.5
◦ ) and the angle between the hydrogen
atoms in the water vapor molecule (104.5
◦ ). The cavities inside the hexagonal rings
were assumed large enough to place another water molecule inside. That is why
the electrical conductivity and self-diffusion coefficient of ice along the c-axis, and
perpendicular to this axis differ several times, as has been experimentally confirmed
latter [27, 28].
The hydrogen atoms are not shown in Fig. 1.7, but they were placed by Bernal and
Fowler in fixed positions on the lines between oxygen atoms. Later Pauling adopted
this configuration, because he found that it does not contradict the residual entropy of
ice at low temperatures [29]. On the basis of statistical physics, Pauling then deduced
the following phenomenological ice rules, which although not applicable for real ice
crystals, were adopted, since they directly followed from the structure of the water
molecule proposed by Bernal and Fowler (see Fig. 1.2):
• Each oxygen atom has two hydrogen atoms attached to it at distances of about
0.95 Å, forming a water molecule, the HOH angle being about 105
◦ as in the gas
molecule.
• Each water molecule is oriented so that its two hydrogen atoms are directed approximately toward two of the four oxygen atoms which surround it tetrahedrally.
7 Modern data shows that the infrared and Raman spectra of ice and water are more complex than
those for water vapor. Apart from line shift and broadening [31], overtones, Fermi resonances, and
contributions from ionic species have been observed [32] (see Chap. 2 for details).
8 P6 3 /mmc means having a sixfold screw axis (rotation around an axis in addition to a translation
along the axis); the structure repeats itself three times during a complete 360 ◦ rotation; the “m’s”
stand for mirror planes perpendicular to the basal plane and parallel to the “c”-axis, and with the c
in “mmc” standing for the glide plane.