Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

CRYSTALS 171

A

B

D C

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 radiusr, and AC is equal to 2rso AD is

AD=

√

AC^2 −DC^2 =

√

( 2 r)^2 −r^2 =

√

3 r^2

The distance AD is the distance between horizontal lines through the centers of the

alternating rows. For example, ifr= 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−^1 of 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 somerotational symmetries.