Crystal Nomenclature--(Handout by Dr. Joseph Halbig)

In order to be able to refer to the different faces on a crystal, crystallographers assigned one of six sets of reference axes, roughly following the conventions previously discussed. It was assumed that the axes intersected at a common point, the origin, arbitrarily chosen somewhere within the interior of the crystal. Of the faces that cut all three axes, they selected the one of greatest area and called it the crystal's unit face or parametral plane. This unit face, extended if necessary, was assumed to cut each positive crystallographic axis at an equal but unknown number (n) of units.

To illustrate the choice of a unit face, consider the sulfur crystal . It occurs in crystal class 2/m 2/m 2/m, with three mutually perpendicular 2-fold axes which serve as the a, b and c axes. Both face s and p cut all three positive axes, but p is chosen as the unit face because of its greater area.

The face's intercepts on a set of axes are the distances from the origin at which the face cuts the three axes. If p is the unit face, then the intercepts, distances 0A_p, 0B_p and 0C_p, equal na_o, nb_o and nc_o.

Introduced by C.S. Weiss in 1818, these parameters given an approximate indication of a face's angular attitude to the crystallographic axes, and also serve as a symbol for the face.

Weiss parameters state the relative number of units at which a face cuts each axis. As an example, 2a:2b:1c indicates that a face intercepts the a and b axes at twice as many a_o and b_o units as it cuts the c axis in c_o units. For a unit face, the parameters would be 1a:1b:1c; that is, it cuts each of the axes in the proportion 1:1:1, or at na_o, nb_o, and nc_o units, respectively.

Once the crystallographic axes are located and a unit face is chosen, the Weiss parameters for other faces may be determined by dividing their intercepts (on each axis) by the intercepts of the unit face. Assume the intercepts for the unit face p are 7.08:8.70:16.57 cm, and for face s they are 14.95:18.34:11.65 cm. To obtain the Weiss parameters for face s, divide its intercepts by p's, thus

14.95/7.08 : 18.34/8.70 : 11.65/16.57 = 2.111a:2.108b:0.703c

Above we selected p as the unit face and assumed that p's intercepts equaled na_o, nb_o and nc_o. Consequently, the intercepts of s are 2.111, 2.108 and 0.703 times as long as p's, or 2.111na_o units, 2.108nb_o units and 0.703nc_o units.

Like any proportion, it remains unchanged by dividing by a common factor, 0.703n. This yields the proportion 3.003a:2.999b:1c, or 3a:3b:1c; the slight deviation from exactly 3.000 for a_o and b_o are attributed to experimental error. Thus the Weiss parameters for face s indicate that intercept on the a and b axes is at three times as many units as that on the c axis.

The general procedure for obtaining an unknown face's Weiss symbol is as follows,

1) divide its intercept on a given axis by that of the unit face on the same axis, and list the three ratios in the order which they refer to the a, b and c axes (or a₁, a₂ and c axes for tetragonal, and a₁, a₂ and a₃ axes for isometric), and

2) divide through by the ratio of smallest value, which results in relatively simple numbers or fractions (see later law of rational indices).

An orthorhombic sulfur crystal in which each face is labeled with Weiss parameters. These could be calculated by the above procedure. Note that if a face is parallel to an axis its intercept is considered to be at infinity from the origin.

Early in the 19th century W.H. Miller developed a system of crystal face notation which has many advantages over Weiss symbols. These symbols, called Miller indices, are simply the reciprocals of Weiss parameters, cleared of fractions, with the letters denoting the axes omitted.

A face that has the Weiss symbol 3a:3b:1c. The reciprocal is 1/3:1/3:1, and multiplying by 3 to clear fractions gives 1,1,3. Miller then eliminated the commas and set the symbol in parentheses as (113), which is read as "one-one-three". Commas are used only in the rare cases when two-digit numbers occur, such as (12,2,1).

Miller indices contain neither fractions or a common factor. For example, a (226) would be divided by common factor of 2 to yield (113).

As with Weiss parameters, Miller indices are listed in the order to which they refer to: 1) the a, b, and c axes for triclinic, monoclinic and orthorhombic, 2) the a₁, a₂ and c axes for tetragonal crystals, 3) the a₁, a₂, a₃ and c axes for hexagonal, and 4) the a₁, a₂ and a₃ axes for isometric crystals.

Note that the reciprocal of infinity is considered to equal zero.

From the Miller indices, we can visualize the approximate angular attitude of a face to the crystallographic axes by considering the reciprocals of the indices (in effect, converting them back to Weiss parameters). For example, face (210) has the reciprocals 1/2, 1, infinity, and thus cuts the first axis at 1/2 as many units as the 2nd axis, and is parallel to the 3rd axis. The face (111) has the reciprocals 1,1,1, so it cuts all three axes at equal (the same) numbers of units.

d. Determination of Miller Indices from Intercepts

The procedure is to 1) divide the unit face's intercept on each axis by that of the unknown face on the same axis (note that the numerator and denominator are reversed from that of the previous calculation to determine Weiss parameters, or in other words, the reciprocal is obtained in this first step), 2) divide the resulting intercept ratios by the smallest of them, and 3) the resultant numbers may be multiplied (or divided) by an appropriate factor, if necessary, to clear them of fractions or remove any common factor.

Using the intercepts for faces s and p in Table 3-1, step 1) gives

4.60/8.28 : 5.66/10.18 : 10.77/6.46, or 0.556:0.556:1.667

and step 2 requires division by 0.556 to give 1:1:3, or the Miller index (113). 

Step 2) above always results, after due allowance for experimental error, in simple rational numbers (that is, simple fractions, or simple integers usually less than 6 in value).

Definition: A rational number is one that can be written as a ratio of two integers, such as 10/1, 1/6, 2/3, 3/4, etc. Irrational numbers are such as pi, 2^1/2, etc.

Thus, early on crystallographers observed the rational nature of Miller's indices (or Weiss parameters) and referred to it as the law of rational indices or Hay's law. At first it was just an observed fact, but later its validity became understood once crystal lattices and structures could be studied.

Certain conventions are used when assigning symbols to Miller indices,

1) The general index (hkl) specifies a face that may bear any orientation except parallelism to the three crystallographic axes, and cuts them at different numbers of units.

2) The indices (0kl), h0l), and (hk0) signify faces that are parallel to the a, b, or c axes respectively, but may bear any orientation except parallelism to the remaining two axes. When the faces cut the remaining two axes at equal numbers of axial units, they are written usually as (011), (101), and (110) and not (0kk), (h0h) and (hh0). This is because they can be divided through by a common factor, in this case h or k.

3) The general index (hhl) indicates a face that cuts the first two axes at equal numbers of axial units and the third face at a different number of units.

4) The index (111) means the face cuts all three axes at the same number of units. Again, (hhh) is not used since it may be divided through by the common factor h.

5) The indices (001), (010), and (100) are used for faces that are parallel to two of the axes; for the same reason as 4) above, (00l), (0k0) and (h00) are not used.

e. Miller-Bravais Indices

In the hexagonal system the set of four axes is often called a Bravais axial set because Augusta Bravais introduced them in 1851. In assigning indices to faces of hexagonal crystals, Bravais followed the Miller method; hence they are called Miller-Bravais indices.

When expressed, the index consists of four numbers in order of the a₁, a₂, a₃ and c axes. Thus the index (1121), which is comparable to the Weiss symbol 2a₁: 2a₂: 1a₃: 2c, refers to a face that cuts the positive a₁ and a₂ axes at 2X as many a_o units as the +a₃ axis, and cuts the c axis at the same number of c_o units as the +a₁ and +a₂ in a_o units.

The first three numbers of Miller-Bravais indices total zero, i.e., when (hkil) is the general symbol, then h + k + i = O. As examples, note that this equation holds for the indices (111) and (101), where the minus sign is written as a "bar" over the symbol.

Some workers replace the third digit in the index by an asterisk or dot. Thus (111) becomes (11*1). The identity of the third digit is simply the algebraic sum of the first two but of opposite sign. Even more confusing are notations which eliminate the asterisk, so (111) becomes simply (111).

2. Axial Elements

The axial elements of a crystal consist of data which adequately describe it crystallographic axes: 1) data that stipulate the angles between the crystallographic axes, that is, the interaxial angles, and 2) the relative sizes of the units along the axes, that is, the axial ratio.

a. Interaxial Angles

The only interaxial angles that need to be stated are beta for monoclinic crystals, and alpha, beta and gamma for triclinic. The other systems have fixed angles of either 90^o or 60^o between axes.

b. Axial Ratios

The axial ratio for orthorhombic, monoclinic and triclinic is the ratio of lengths a_o, b_o and c_o relative to b_o, that is,

a_o/b_o : b_o/b_o : co/b_o , or a_o/b_o : 1 : c_o/b_o

Even though early crystallographers could not measure absolute lengths, they were able to determine lengths relative to each other and express this as an axial ratio.

As an example, where face p was accepted as the unit face, the intercepts were

7.08 = na_o 8.70 = nb_o 16.57 = nc_o.

Thus, the axial ratio is

na_o/nb_o : nb_o/nb_o : nc_o/nb_o = 7.08/8.70 : 1 : 16.57/8.70

= 0.814 : 1 : 1.905

This ratio expresses the fact that the a axis units are slightly over 0.8 times as long as those along the b axis, and the units along the c axis are almost twice as long as its b axis.

For the sulfur crystal, the intercepts for the unit face p were given as 4.60, 5.66, and 10.77 cm. The calculated axial ratio would be 0.814 : 1 : 1.906 which, allowing for experimental errors, is the same as before. Therefore, all orthorhombic crystals of sulfur must have the same axial ratio.

X-ray methods now available show the actual values of the units (where n=1) in orthorhombic sulfur to be a_o = 10.45Å, b_o = 12.84 Å, and c_o = 24.46Å (where Å = angstrom = 10^-8cm). These actual values yield the same axial ratio as determined by earlier crystallographers.

Prior to x-ray crystallography, axial ratios were of importance in identifying trimetric crystals. However, in isometric and dimetric crystals (tetragonal and hexagonal) because axial units of only one or two sizes occur, the axial ratio is less definitive.

In the dimetric systems the c_o unit is expressed relative to the a_o unit. In quartz, for example, x-ray measurements show a_o = 4.913 and c_o = 5.405A. The axia ratio is thus stated as

a_o/a_o : c_o/a_o = 1.000 : 1.100,

or more briefly as c_o/a_o = 1.100, or c = 1.100.

For isometric crystals the axial ratio is of no significance since it equals 1.000; that is, equal units occur along all three axes.

4. Nomenclature for Linear Directions

The axes of orthorhombic sulfur are marked in units of no particular size except their lengths are in proportion to the axial ratio of 0.814:1:1.905. These axes, so marked, now provide a three-dimensional coordinate system which permits a point in space to be specified by three numbers, u, v, and w, which are the point's coordinates in terms of units along the a, b, and c axes, respectively. Several such points are marked in the figure by ellipses or crosses.