3-D Order and Lattices

During the last class we derived and examined the five 2-D plane lattices and combined them with the 10 planar point groups to generate the 17 2-D plane (space) groups. Today we will examine the 14 Bravais 3-D lattices and combine them with the 32 3-D point groups to generate the 230 3-D space groups. To make this final step we will also have to introduce a new symmetry operation - the screw axis. We also need to revisit glide planes because in 3-D there is a little more to this subject than the obvious 2-D "left-right-left-right" analogy.

Three-Dimensional Lattices

Start with the five 2-D plane lattices and add one additional translation out of the plane of the paper. Depending on the lattice you start with and how special or general your choice of a 3rd translation (in both direction (angle) and magnitude (distance)), you can generate the 14 3-D lattices that are known as the Bravais lattices. Remember that every lattice point or node must have an identical environment.

Figure 3.17 on pages 125 and 126 provides simple explanations of the construction of each of the lattices.

Oblique net: Stacking an oblique net (unequal 'a' and 'b' translations in the x and y directions respectively, at no special angle to one another) at an arbitrary angle and distance 'c' (in the z direction) produces a primitive triclinic lattice.
Rectangular net: Stacking a primitive rectangular net (unequal 'a' and 'b' translations at 90 degrees) in a vertical direction (z) at an angle (x^z) not equal to 90 degrees and a random distance 'c' produces a primitive monoclinic lattice.
If the angle (x^z) is equal to 90 degrees this produces a primitive orthorhombic lattice.
Stacking this same primitive rectangular net along a direction (a special angle) that will coincide with the body diagonal of the finished unit cell (front lower left corner toward back upper right corner) and translating the planar net half this distance (a special distance) produces a body-centered orthorhombic lattice.
Centered rectangular net: Stacking a centered rectangular net (unequal 'a' and 'b' translations at 90 degrees) in a vertical direction (z) at an angle (x^z) not equal to 90 degrees and a random distance 'c' produces a centered monoclinic lattice. Conventionally this is referred to as being "c-centered" because the faces that have the centered lattice point are related by the 'c' translation. Because the beta angle (x^z) is not constrained by symmetry, it is also possible to choose a different beta angle and 'a' translation length (along a different x direction) and convert this array of lattice points to a body-centered monoclinic one.
If the angle (x^z) is equal to 90 degrees this produces a centered orthorhombic lattice. Because all the angles in the orthorhombic lattice are 90 degrees and all the translations are different there is no compelling reason for a particular choice among calling this centered lattice "a-centered" or "b-centered" or c-centered". You may also see this referred to as end-centered.
Stacking this same centered rectangular net along a direction that will coincide with the front face diagonal of the finished unit cell (front lower left corner toward front upper right corner) and translating the planar net half this distance (a special distance) produces a face-centered orthorhombic lattice. There is thus a lattice point in the centers of all the faces in the three dimensional lattice.
Square net: Stacking a square net along the (z) direction a random distance 'c' but constraining the angle to be 90 degrees produces a primitive tetragonal lattice.
Stacking this same square net along a direction that will coincide with the body diagonal of the finished unit cell (front lower left corner toward back upper right corner) and translating the planar lattice half this distance (a special distance) produces a body-centered tetragonal lattice.
Stacking a square net along the (z) direction a distance 'a' equal to the translation defining the square net ('a') and constraining the angle to be 90 degrees produces a primitive isometric lattice.
Stacking this same square net along a direction that will coincide with the body diagonal of the finished unit cell (front lower left corner toward back upper right corner) and translating the square net half this distance (a special distance) produces a body-centered isometric lattice.
It is also possible to stack square nets along a special direction (angle) and distance (translation) which will produce a face-centered isometric lattice. The direction and angle are not obvious and at this time you may simply accept that this is possible.
It is also possible to generate a face-centered tetragonal lattice analogous to the face-centered isometric one. However unlike the isometric system where all three translations must be identical, in the tetragonal system the c-axis is a different length and it is possible to choose other, shorter, translations in the face-centered tetragonal lattice which define a body-centered tetragonal lattice. Since shorter means smaller and this is a positive attribute for a unit cell, the body-centered tetragonal cell is chosen.
Hexagonal net: Stacking a hexagonal net along the (z) direction a random distance 'c' but constraining the angle to be 90 degrees produces a primitive hexagonal lattice. If this lattice is repeated 3 times by rotation about the c-axis a c-centered hexagonal lattice results.
A hexagonal lattice can also be stacked along the edge directions of a rhombohedron resulting in a rhombohedral lattice.

All 14 of these possibilities are summarized in Fig. 3.18 in your text along with the axial and angular assignments and the conventional abbreviations for the primitive (P), the c-centered (C), the body-centered (I) and the face-centered (F) lattices respectively.

Screw Axes

A screw axis is the result of combining a rotational operation with translation parallel to the axis of rotation. This is analogous to combining a translation parallel to a mirror and making a glide plane. See pages 129-132 in your text for details:

can have 2-, 3-, 4- and 6-fold screw axes
the motifs generated by the combined operation of rotation and translation define a helix
the translation distance is 1/2, 1/3, 1/4, or 1/6
screw axes can be either right-handed or left-handed
screw axis symbols consist of the rotation axis number followed by a subscript which defines the total translation by placing the subscript over the main axis symbol as a fraction: i.e. 4₁ means that the translation distance would be 1/4 of the total translation for a right-handed screw
several screw axes make up enantiomorphic pairs (right- and left-handed): 3₁ and 3₂, 4₁ and 4₃, 6₁ and 6₅, and 6₂ and 6₄.
screw axes are never expressed morphologically in crystals - all you can see is the presence of the rotation axis

Glide Planes

The glide-line operations that you have become expert at observing in 2-D patterns in the lab have their 3-D manifestation in glide planes. See pages 132-134 for details:

glide translations are named after the direction in which the translation occurs
i.e. a t/2 glide component in the 'b' direction is referred to as a 'b-glide'
diagonal glides are referred to as 'n glides' and have glide components such as a/2 + b/2
a diamond glide (d-glide) has a glide component with a translation that can be defined by a/4 + b/4 + c/4
in the space group symbols to come these glides are represented by replacing the 'm' for the mirror symmetry by either 'a', 'b', 'c', 'n', or 'd'
glide planes are never expressed morphologically in crystals - all you can see is the presence of the mirror plane

Space Groups

The combination of the 14 3-D Bravais space lattices with the 32 translation-free 3-D point groups (or crystal classes) and the two symmetry operations that involve translation (screw axes and glide planes) gives us the 230 space groups.

These are all the ways that atoms can be arranged in space in an infinite repeating pattern.
Because the translation distances in the lattice and the glide planes and the screw axes are on the order of a few angstroms they can not be seen by eye and so morphologically all 230 space groups appear to us as one of the 32 crystal classes.
All the space groups are said to be 'isogonal' with one of the 32 point groups, "the point group is the translation-free residue of a family of possible isogonal space groups."
The space group symbol is composed of a letter designating the lattice type (P, A, B, C, I, F, R) followed by the symmetry such as I2/b2/a2/m
Need to be aware of the use and meaning of abbreviated symbology in the systematic mineralogy chapters where the above I2/b2/a2/m would be reported as Ibam. The justification of this abbreviation is that mirrors (or glides) intersecting at 90 degrees create 2-fold axes and thus they don't really have to mentioned explicitly in the symbol.
You definitely need to be able to reduce complete space group designations to their isogonal point group because, again, this is what the crystal should actually look like. For example you need to recognize that P 2₁/n 2₁/m 2₁/a is an orthorhombic crystal with 2/m 2/m 2/m symmetry. In the systematic mineralogy section this would be designated Pnma.

Crystal Structure Determination

The determinations of crystal structures involves the gathering of single crystal X-ray diffraction data on a small crystal of the material. This is beyond the scope of this course - however we will be making powder X-ray diffraction measurements on ground-up samples of minerals (see Week 6). Powder X-ray techniques can provide infomation about the lattice type, symmetry, and size (length of the unit cell edges). On p. 146 of the text is an example where

the morphology of halite (isometric, 4/m bar3 2/m) is combined with
the X-ray diffraction determined cell edge of 5.64 Angstroms,
a chemical analysis showing 39.4 weight % Na and 60.6 weight % Cl,
the atomic weights of Na and Cl,
and the density = 2.165 g/cc
to calculate that the Na:Cl ratio must be 1:1 and that there must be 4 formula units per unit cell.
Combining this with the knowledge that the unit cell is face-centered allows the by now familiar structure of halite to be deduced.