EENS 2110 
Mineralogy 
Tulane University 
Prof. Stephen A. Nelson 
Axial Ratios, Parameters, Miller Indices 

We've now seen how crystallographic axes can be defined for the various
crystal systems. Two important points to remember are that
We also noted last time that the relative lengths of the crystallographic axes control the angular relationships between crystal faces. This is true because crystal faces can only develop along lattice points. The relative lengths of the crystallographic axes are called axial ratios, our first topic of discussion. Axial Ratios Axial ratios are defined as the relative lengths of the crystallographic axes. They are normally taken as relative to the length of the b crystallographic axis. Thus, an axial ratio is defined as follows: Axial Ratio = a/b : b/b : c/b where a is the actual length of the a crystallographic axis, b, is the actual length of the b crystallographic axis, and c is the actual length of the c crystallographic axis.
a/b : 1 : c/b (this is usually shortened to a : 1 : c)
1 : 1 : c/b (this is usually shorted to 1 : c)
1 : 1 : 1 (this is usually shorted to 1)
1 : 1 : 1: c/a (usually shortened to 1 : c) Modern crystallographers can use xrays to determine the size of the unit cell, and thus can determine the absolute value of the crystallographic axes. For example, the mineral quartz is hexagonal, with the following unit cell dimensions as determined by xray crystallography: a_{1 }= a_{2} = a_{3} = 4.913Å c = 5.405Å where Å stands for Angstroms = 10^{10} meter. Thus the axial ratio for quartz is 1 : 1 : 1 : 5.405/4.913 or 1: 1 : 1 : 1.1001 which simply says that the c axis is 1.1001 times longer than the a axes. For orthorhombic sulfur the unit cell dimensions as measured by xrays are: a = 10.47Å b = 12.87Å c = 24.39Å Thus, the axial ratio for orthorhombic sulfur is: 10.47/12.87 : 12.87/12.87 : 24.39/12.87 or 0.813 : 1 : 1.903 Because crystal faces develop along lattice points, the angular relationship between faces must depend on the relative lengths of the axes. Long before xrays were invented and absolute unit cell dimensions could be obtained, crystallographers were able to determine the axial ratios of minerals by determining the angles between crystal faces. So, for example, in 1896 the axial ratios of orthorhombic sulfur were determined to be nearly exactly the same as those reported above from xray measurements. In a later lecture we will see how we can determine axial ratios from the angular relationships between faces. First, however we must determine how we can name, or index faces of crystals and define directions within crystals.

Crystal faces can be defined by their intercepts on the crystallographic axes. For nonhexagonal crystals, there are three cases. 



Two very important points about intercepts of faces:
Because one does usually not know the dimensions of the unit cell, it
is difficult to know what number to give the intercept of a face, unless
one face is chosen arbitrarily to have intercepts of 1. Thus, the
convention is to assign the largest face that intersects all 3
crystallographic axes the parameters  1a, 1b, 1c. This face
is called the unit face. 
For example, in the orthorhombic crystal shown here, the large dark shaded face is the largest face that cuts all three axes. It is the unit face, and is therefore assigned the parameters 1a, 1b, 1c. 
Once the unit face is defined, the intercepts of the smaller face can be determined. These are 2a, 2b, 2/3c. Note that we can divide these parameters by the common factor 2, resulting in 1a,1b,1/3c. Again, this illustrates the point that moving a face parallel to itself does not change the relative intercepts. Since intercepts or parameters are relative, they do not represent the actual cutting lengths on the axes. 
By specifying the intercepts or parameters of a crystal face, we now
have a way to uniquely identify each face of a crystal. But, the
notation is cumbersome, so crystallographers have developed another way of
identifying or indexing faces. This conventional notation called
the Miller Index is our next topic of discussion.

Miller Indices The Miller Index for a crystal face is found by
For example, if the face has the parameters 1 a, 1 b, ¥ c inverting the parameters would be 1/1, 1/1, 1/ ¥ this would become 1, 1, 0 the Miller Index is written inside parentheses with no commas  thus (110)

As further examples, let's look at the crystal shown here. All of
the faces on this crystal are relatively simple. The face [labeled
(111)] that cuts all three axes at 1 unit length has the parameters 1a,
1b, 1c. Inverting these, results in 1/1, 1/1, 1/1 to give the Miller
Index (111).
The square face that cuts the positive a axis, has the parameters 1 a, ¥ b, ¥ c. Inverting these becomes 1/1, 1/¥, 1/¥ to give the Miller Index (100). The face on the back of the crystal that cuts the negative a axis has the parameters 1a, ¥ b, ¥ c. So its Miller Index is (00). Note how the negative intercept is indicated by putting a minus sign above the index. This would be read "minus one, one, one". Thus, the other 4 faces seen on this crystal would have the Miller Indices (001), (00), (010), and (00). 
Now let's look at some more complicated examples. The drawing to the right is the same orthorhombic crystal we looked at earlier. Recall that the small triangular face near the top that cuts all three axes had the parameters 1a, 1b, 1/3c. Inverting these becomes 1/1, 1/1, 3/1 to give the Miller Index for this face as (113). Similarly, the small triangular face the cuts the positive a axis and the negative b axis, would have the Miller Index (13), the similar face on the bottom of the crystal, cutting positive a, positive b, and negative c axes would have the Miller Index (11). See if you can determine the Miller Indices for the 8 faces on the back of the crystal that are not seen in this drawing. 
Note once again, that moving a face parallel to itself does not change
the parameters nor the Miller Index for that face.
To refer to a general face that intersects all three crystallographic axes where the parameters are not known, we use the notation (hkl). For a face that intersects the b and c axes with general or unknown intercepts the notation would be (0kl), for a face intersecting the a and c axis, but parallel to b the notation would be (h0l), and similarly for a face intersecting the a and b axes, but parallel to c we would use the notation (hk0). 
This Miller Index notation applies very well to crystals in the Triclinic, Monoclinic, Orthorhombic, Tetragonal, and Isometric systems, but requires some modification to be applied to the Hexagonal crystal system.
Miller Bravais Indices Since the hexagonal system has three "a" axes perpendicular to the "c" axis, both the parameters of a face and the Miller Index notation must be modified. The modified parameters and Miller Indices must reflect the presence of an additional axis. This modified notation is referred to as MillerBravais Indices, with the general notation (hkil) 
To see how this works, let's look at the dark shaded face in the hexagonal crystal shown here. This face intersects the positive a_{1} axis at 1 unit length, the negative a_{3} axis at 1 unit length, and does not intersect the a_{2} or c axes. This face thus has the parameters: 1 a_{1}, ¥ a_{2}, 1 a_{3}, ¥ c Inverting and clearing fractions gives the MillerBravais Index: (100) An important rule to remember in applying this notation in the hexagonal system, is that whatever indices are determined for h, k, and i, h + k + i = 0 
For a similar hexagonal crystal, this time with the shaded face cutting
all three axes, we would find for the shaded face in the diagram that the
parameters are 1 a_{1}, 1 a_{2}, 1/2 a_{3},
¥ c. Inverting these intercepts
gives:
1/1, 1/1, 2/1, 1/¥ resulting in a MillerBravais Index of (110) Note how the "h + k + i = 0" rule applies here! 
Although we will not cover this in detail in this lecture, the next step is to use the Miller Index notation to designate crystal forms. A crystal form is a set of crystal faces that are related to each other by symmetry. To designate a crystal form (which could imply many faces) we use the Miller Index, or MillerBravais Index notation enclosing the indices in curly braces, i.e. {hkl} or {hkil} Such notation is called a form symbol. 
As an example, look at the crystal drawing shown here. This crystal is the same orthorhombic crystal discussed above. It has two forms. The form {111} consists of the following symmetrically 8 related faces: (111), (11), (11), (1), (11), (1), (1), and (). This form is called a rhombicdipyramid. The other form is also a rhombicdipyramid, but consists of the triangular shaped faces similar to the face (113). The form symbol for this form is {113} and consists of the following 8 faces: (113), (13), (11), (1), (13), (1), (3), and (). We'll go over this in more detail in the next lecture.

Examples of questions on this material that could be asked on an exam
