LIQUIDS AND SOLIDS
11.4 CRYSTALS
In many crystals, atoms are arranged in a very regular three-dimensional pattern of rows and columns, echelons deep. Not surprisingly, when radiation strikes a crystal, it is reflected from planes of atoms, ions, or molecules in the way that light is reflected from a mirror. The difference is that experimentalists use penetrating X rays to detect planes within the crystal. The appearance of planes within a crystal is a consequence
CRYSTALS 171 A
B C D
FIGURE 11.7 Close packing of marbles between two sheets.
of the regular arrangement of the atoms, ions, or molecules of the crystal, which is itself a consequence of the regular forces holding it together.
Simply dropping marbles into a box, one gets the idea of a tendency of the marbles to settle into a regular structure with layers separated by a distance that can be calculated as a function of the radii of the marbles. If you shake the box gently, so that the marbles assume a more or less compact aggregate, you may notice a repeating structural unit of a cube or hexagon.
To simplify the picture by making a two-dimensional array, think of marbles dropped into the space between two clear plastic sheets separated by a space equal to the diameter of the marbles (Fig. 11.7). Now marbles are separated into alternat- ing rows. Repeating structural units may now be rectangles. Knowing the diameters (hence the radii) of the marbles enables one to find the distance between alternat- ing rows.
We have extracted an isosceles triangle ABC from the marble pattern. The altitude of the triangle DC is also the radius of the marbles. The length of a side AC is twice the radius of the marbles. That gives us a right triangle ADC. The sum of the squares of the two sides of ADC is equal to the square of the hypotenuse AC. Distance DC is equal to one marble radius r, and AC is equal to 2r so AD is
AD=
AC2−DC2=
(2r )2−r2=√ 3r2
The distance AD is the distance between horizontal lines through the centers of the alternating rows. For example, if r =0.500 cm, the distance between the horizontal lines through the centers of the marbles parallel to DC is 0.866 cm.
Distance DC permits us to take the inverse cosine cos−1of the adjacent side over the hypotenuse of angle DAC to find that it is 30◦. The remaining angle of the right triangle must be 60◦. Now that we know all the distances and angles relating the centers of the marbles, we know all that can be known about the geometry of the marble packing everywhere the pattern in Fig. 11.7 is maintained. Dropping real marbles into a real space, one may find irregularities and fissures in the structure. This is analogous to real crystal structure as well. They show the same kind of irregularities and fissures.
One more thing before going to three dimensions: If we rotate the diagram in Fig. 11.7 by 60◦, we get a pattern that is identical to the one we just analyzed. There are other lines at other angles with the same geometric relationships as those we have found.
The marble pattern has some rotational symmetries.
A
B
C D
FIGURE 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 0.866 r for 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 crystallography is 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.
For constructive interference between 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 wavelength λ. 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 times λ