c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come
172 LIQUIDS AND SOLIDSABCDFIGURE 11.8 A less efficient packing of marbles. This packing is less efficient than close
packing because the same number of marbles take up a greater space. To see this most clearly,
notice that the interstitial spaces are larger than they are in Fig. 11.7.With difficulty, marbles might be juggled into a less efficient but still regular
pattern like Fig. 11.8. The vertical distance between centers for 0.500 cm marbles
is now twice the radius as compared to only 0.866rfor close packing. The same
number of marbles take up more space.
Suppose we had some experimental method of determining distance between
layers DC. That would enable us to tell the difference between the packing pattern in
Figs. 11.7 and 11.8. Assuming that atoms fall into a regular array when an element
or compound crystallizes, we can picture a laminar sheet of atoms with very dense
nuclei in an array similar to that Fig. 11.7 or 11.8. If marbles were packed into a three-
dimensional box, the packing pattern determination would be very closely analogous
to the pattern determination for marbles restricted to a plane. They would pack in
a more or less regular way, and different packing patterns would be a more or less
efficient with respect to space use, just as the marbles were in the simple illustrative
case of two dimensions. Atoms, molecules, or ions pack in three-dimensional crystals
as well. We would like to have an experimental method to solve the reverse of the
problem we just solved; from an experimental value for the distance between layers
of atoms, we would like to obtain the radius of the atoms themselves. The object of
X-ray crystallographyis to determine the distances and angles between atomic centers
as we have done with marbles. The problem may be much more complicated, as in
the example of proteins, or it may be nearly as simple as the method just described
extrapolated to there dimensions, as in the case of pure metals and ionic salts.
Electromagnetic waves, from radio frequency of meter wavelengths toγrays of
λ≈ 0 .1 nm, have an electrical and a magnetic component. The radiation can be
described mathematically as two sine waves, one electricalεand the other magnetic
H, describing two oscillating vector fields oriented at right angles to one another.
Forconstructive interferencebetween two waves, the field vectors must point in the
same direction. Otherwise, the radiation of one wave dims or obliterates the other by
destructive interference.
Figure 11.9 shows that, for the radiation reflected from two adjacent horizontal
lines of atoms to be in phase (to have their arrows pointed in the same direction),
the difference in path must be an integer multiple of the wavelengthλ. The path
difference is shown as a heavy line. A path difference that is twice the wavelength
of incoming radiation will interfere constructively, and one that is three timesλ