• The indeterminate situation arises because the plane passes through the origin. After translation, we obtain intercepts . • By inverting them, we get .

),1,1(

)101(

Stacking of (0002) planes

Figure 9-7 Hexagonal structure consisting of a three-unit cell.

[100]

Atoms in primitive cellAdditional atoms

• The third common metallic crystal structure is the hexagonal close-packed (hcp) structure ( Fig.9-7).

• For hexagonal structures, we have slightly more complicated situation.

• We represent the hexagonal structure by the arrangement shown in Figure 9-7.

• The atomic arrangement in the basal plane is shown in the

top portion of the figure. Often, we use four axes (x, y, k, z) with unit vectors to represent the structure.

• This is mathematically unnecessary, because three indices are sufficient to represent a direction in space from a known origin.

),,,( likh

• Still, the redundancy is found by some people to have its advantages and is described here. • We use the intercepts to designate the planes. • The hatched plane (prism plane) has indices.

• After determining the indices of many planes, we learn that one always has h + k = -i

1

,1

,1

1,

11

• Thus, we do not have to determine the index for the third horizontal axis. If we use only three indices, we can use a dot to designate the fourth index, as follows:

• For the directions, we can use either the three-index notation or a four-index notation. • However, with four indices, the h+k=-i rule will not apply in general, and one has to use special “tricks” to make the vector coordinates obey the rule.

)011(

• Crystallographic directions are indicated by integers in brackets: [uvw]. Reciprocals are not used in determining directions.

• For example, the direction of the line FD of Figure 9.1 is obtained by moving out from the origin a distance of ao along the x axis and moving an equal distance in the positive direction.

• The indices of this direction are then [ 110]. • A family of crystallographically equivalent directions

would be designated <uvw>. • For the cubic lattice only, a direction is always

perpendicular to the plane having the same indices.

• The notation used for a direction is [uvw].

• When we deal with a family of directions, we use the symbol <uvw>.• The following family encompasses all equivalent directions:

][],[],[],[],[

],[],[],[],[],[

uwvuvwvwuvuwwvu

vuwwvuwuvuwvuvwuvw

]100[],010[],001[

]001[],010[],100[100

]110[],011[],101[

]101[],110[],011[

]110[],101[],011[

]011[],101[],110[110

Figure 9-8 Various directionsin a cubic system.

• For cubic systems there is a set of simple relationships between a direction [uvw] and a plane (hkl) which are very useful.

1) [uvw] is normal to (hkl) when u=h;v=k;w=l. [111] is normal to (111).

2) [uvw] is parallel to (hkl), i.e., [uvw] lies in (hkl), when hu + kv + lw = 0 [112] is a direction in (111).

3) Two planes (h1k1l1) and (h2k2l2) are normal if h1h2 + k1k2 + l1l2 = 0. (100) is perpendicular to (001) and (010). (110) is perpendicular to (110)

4) Two directions u1v1w1 and u2v2w2 are normal if u1u2 + v1v2 + w1w2 = 0. [100] is perpendicular to [001]. [111] is perpendicular to [112].

5) Angles between planes (h1k1l1) and (h2k2l2) are given by

2/122

22

22

2/121

21

21

212121

)()(cos

lkhlkh

llkkhh

Figure 9-9

Example: Write the indices of the marked planes

Figure 9-9

Answer:

Figure 9-10

Example: Write the indices of the marked directions

Figure 9-10

Answer:

Figure 9-11

Example: Write the indices of the marked planes and directions

Figure 9-11

Answer:

Exercise:Sketch the 12 members of the <110> family for a cubic crystal. Indicate the four {111} planes. You may use several sketches.

These are the 12 members of the <110> family of directions for a cubic crystal.

These are the four members of the {111} family of planes for a cubic crystal.