3.10 Crystallographic Symmetry Elements

So far, we have not considered where atoms are located within the unit cell, but it should come as no surprise that symmetry plays a role in that, too. If you have studied point group symmetry and not crystallographic symmetry, you have learned about two types of symmetry axes, proper and improper rotation axes, which are defined slightly differently in crystallographic use. However, point group symmetry does not include two additional types of symmetry that involve translations: screw axes and glide planes. Before reviewing these four types of symmetry elements, let’s dive deeper into the notation used to describe symmetry operations and their algebra. We use the concept of an operator to describe how a duplicate atom is created from the initial position with a triplet of formulae, f _x,f _y,f _z, where each of the f _i is an algebraic expression dependent on x, y & z.

As we saw, if we take an atom at position x,y,z and we have a body center in that lattice, this means that for every atom in the unit cell, there is another identical atom displaced from that atom by a half unit cell in each direction. We can write that as an operator as ½+x,½+y,½+z. We understand this operator to mean that we create a new atom from the first with its coordinates shifted by half unit cell translations. If the coordinates of the original atom is x,y,z and the coordinates of the second atom are x’,y’,z’, then x’=½+x, y’=½+y, z’=½+z. Likewise if we consider another symmetry operator, ½+y,-x,z, then that creates a copy of that atom where x’= ½+y and y’=-x but z’=z.

Symmetry can also be written as a matrix and vector. In this notation ½+y,-x,z would be written as

\[\begin {matrix}x\prime \\y\prime \\z\prime \end {matrix}=\ \left [\begin {matrix}0&1&0\\-1&0&0\\0&0&1\end {matrix}\right ]\left [\begin {matrix}x\\y\\z\end {matrix}\right ]+\left [\begin {matrix}{1/2}\\0\\0\end {matrix}\right ]\]

Let’s now consider a mirror plane symmetry operation, where we will place that symmetry element in the x,y-plane and at z=½. What does that mirror plane do? Viewed along the c direction, perpendicular to the mirror plane, it appears to do nothing, because x and y will be unchanged by the action of the mirror plane. So, we can see the new position has the same values of x and y. Viewed from a direction in the plane of the mirror (for example, viewing along a or b), it becomes clear that the mirror plane involves a change in the z value to z’. A bit of thought makes clear that whatever the distance from the atom to the mirror plane, that distance must stay the same, but the atom will lie on the opposite side of the mirror plane. So, if our atom is at position z, then the distance to the mirror plane is —½-z— and the direction from the atom to the mirror plane will be along a vector from z to ½. For simplicity in visualization, we will assume 0 ¡ z ¡ ½, but what here is true for all values of z, assuming that a periodic lattice is present, so that adding or subtracting any multiple of one from z yields an indistinguishable value. The position on the other side of the mirror plane with the same distance to the mirror plane will be z’ = 1-z. We can see that —½-(1-z)— which is equal to —z-½—. Thus, we can say that our mirror plane in the x,y-plane and at z=½ is equivalent to a x,y,1-z operation. Since we can add or subtract any multiple of one to x, y and/or z, a simpler form for this operator is x,y,-z. Crystallographers have an even more compact notation for this, written as \(x,y,\bar {z}\), where the last term is called “z bar” and means -z. Modern software does not make \(\bar {z}\) easy to type, so I will use this notation sparingly.

There are two things we can learn from this x,y,-z operator. One is the idea of special positions, which is demonstrated by what happens if we have an atom that is placed on the mirror plane, meaning that that z=½ (note that x & y are arbitrary). We can designate this in a more compact notation, as position (x,y,½). Applying this x,y,-z operator to position (x,y,½) generates position (x,y,-½). Since adding any positive or negative multiple of 1 to any coordinate generates an equivalent position in the lattice, the positions (x,y,½) and (x,y,-½) are indistinguishable and thus while in general an atom placed at (x,y,z) generates a second atom at (x,y,-z), the position (x,y,½) is special; we do not actually generate a new position for any atom with z=½. Thus, we say that due to the presence of this mirror plane, atoms at (x,y,½) are at special positions, because if there are n atoms on one side of the mirror plane, then the mirror plane will generate a second set of n atoms on the other side of the mirror plane and thus the unit cell must contain 2n atoms, but not so for the atoms on the special position (x,y,½) because there will no duplicates for the atoms that lie on the mirror plane. The unit cell contents will thus be A_2nB_n where the B atoms have z=½ [or as we will see where z=0, as (x,y,0) is also a special position.]

The second concept is that the presence of symmetry operators in combination with each other and/or the lattice can create additional symmetry operations. To understand how one symmetry operation can modify another operation, note that -x,-y,-z is equivalent to replacing the expression for x’,y’ and z’ with those values multiplied by -1. If we combine the -x,-y,-z and x,y,z symmetry operations this yields the original symmetry operation -x,-y,-z. If we apply -x,-y,-z to symmetry operation -x,y,½+z we get x,-y,½+z [note that -1*(½-y) is y-½ but in an lattice adding 1 but the completely equivalent position, ½+z, is preferred by convention.] Let us consider what happens with a mirror plane in the x,y-plane placed at z=0. This will “reflect” any atom at position z to z’=-z. The operator form of this is also x,y,-z, which is also identical to the operator created by the mirror plane at z=½. Thus, the presence of a lattice and a mirror plane at z=½ (and at –½, 1+½,…) requires the presence of an additional set of mirror planes at z=…-2,-1,0,1,2,… Likewise, a mirror plane at z=0 cannot exist without a mirror plane at z=½.

Now let us review the four types of symmetry elements.

3.10.1 Proper rotations

Proper rotations are defined relative to a line and by the fraction of a circle that is rotated. Thus, to fully describe a proper rotation, we need to specify a direction relative to the a-, b-, and c-axes and a point that the axis travels through and finally the fraction of a circle for the rotation (example: a 3-fold axis along a that passes through y=z=¼.) The only rotations that are possible in crystals are 1-fold, 2-fold, 3-fold, 4-fold and 6-fold and the Bravais lattice dictates which of these are allowed. Thus, an N-fold rotation produces rotations of 360°/N (2p/N radians). Not all proper rotation axes can be found in any given Laue class. For example, a triclinic cell is not compatible with any rotation axis other than 1-fold. A 4-fold rotation axis interchanges axes and thus requires a minimum of two equivalent length axes (cubic and tetragonal cells), while a 3-fold axis requires a rhombohedric or hexagonal cell. A 6-fold rotation axis can only exist in a hexagonal cell. The symbol used for a N-fold proper rotation axis is simply the number 2,3,4,6 (no symbol is needed for the 1-fold)

Table 3.4 shows the coordinates for atom positions generated by a proper rotation axis along the c-axis passing through x=y=0. Note that a 1-fold rotation axis is simply the identity operation. It generates only the original position of an atom.

3.10.2 Improper rotations

Proper rotations are defined relative to a line for the rotation axis, but also a point along that axis. Thus, to fully define an improper rotation axis one needs the rotation order, a point and an axis of rotation. The way that improper rotations are applied is to first perform an inversion operation through the defined point and then apply the rotation, as was done with proper rotations. (Note that improper rotations, as used in molecular point group symmetry, are defined slightly differently and the Schoenflies labels work slightly differently; the crystallographic conventions are, of course, better.) Again, the improper rotation axes are named by the fraction of a circle that is rotated and again, only the only improper rotations that are possible are 1-fold, 2-fold, 3-fold, 4-fold and 6-fold, where only some of these are consistent with each Bravais lattice. Below is a table showing the coordinates for atom positions generated by an improper rotation axis along the c-axis and point x=y=z=0. The symbol used for a N-fold improper rotation axis is simply the number with a horizontal line above, as seen in the table. Note that the \(\bar {1}\) operation is simply an inversion center, also sometimes called a center of symmetry. This does not have an axial direction, but does have an origin location. The \(\bar {2}\) operation is simply a mirror plane. This is defined relative to a plane, but has no origin. Table 3.5 shows the coordinates for atom positions generated by an improper rotation axis along the c-axis passing through x=y=0.

3.10.3 Screw axes

So far, we have seen point symmetry in the form of proper and improper rotations and the only translational symmetry we have seen is in the form of unit cell translations. There is also a symmetry operation that combines both rotational symmetry and translation. These are called screw axes and these describe helictical symmetry in a material. A screw axis has a defined translation direction and a location in the plane perpendicular to that direction. A screw axis is labeled N_m where the first number, N, (2, 3, 4 and 6) describes the amount of the rotation, where the rotation is 360°/N (2p/N radians) so a 2₁ screw axis will cause a 180° rotation (generating 2 positions), while a 6_m screw axis will have 60° rotations. The second number, m combined with N (the first), dictates the translation amount as m/N, so the translation is a half unit cell for a 2₁ screw axis and one-sixth of a unit cell for a 6₁ screw axis. Thus, a 2₁screw axis will duplicate an atom (two atoms/cell) and a 6₁ screw axis will create five copies (six atoms/cell). The crystallographically defined screw axes are 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄ and 6₅. Screw axes can occur along the a, b or c directions or in three-fold symmetry materials (cubic and rhombohedric cells, along the body-diagonal, the 111 direction.) Thus, a full description of a screw axis will specify the type, the direction and a location of the axis, usually by giving a point that the axis travels through.

For an example, let us consider a 4₁ screw axis along the c axis through the point x=y=0. The rotation portion of the axis will transform the in-plane coordinates of the atom from (x,y) to (-y,x) and then to (-x,-y) and finally to (y,-x) while simultaneously adding a ¼ unit cell translation. Thus, if we have an atom at coordinates (x,y,z) the generated positions are (-y,x,z+¼), (-x,-y,z+½) and (y,-x,z+¾).

Note that since unit cell translations are applied in combination with screw axes, the translation of 5/6^th of a unit cell is the same as a translation of 1/6^th of a unit cell in the negative direction, thus a 6₁ and 6₅ screw axes are an enantiomeric pair, that differ only in the rotation direction, where the 6₁ screw axis produces a right-handed helix and 6₅ screw axis produces a left-handed helix. (The same is true for 3₁/3₂, 4₁/4_3, and 6₂/6₄.) Thus, the 4₃ screw axis describes a helix with the opposite rotation direction of the 4₁ axis shown before, with generated coordinates (x,y,z), (y,-x,z+¼), (-x,-y,z+½) and (-y,x,z+¾). Note that while a 4₁ and a 4₃ axes will be found in different space groups, with powder diffraction all the reflections that could be sensitive to this rotation direction (due to anomalous dispersion) are superimposed, so it is impossible for a powder diffraction experiment to differentiate the enantiomeric handedness for a material. Without some other type of data, the distinction between each pair of related symmetry elements can be ignored. Space group P6₁ and P6₅ are considered different space groups as they can accommodate objects of with different handedness, but to powder diffraction they are the same.

3.10.4 Glide planes

The last type of symmetry element found in crystals combines translation and reflection and is called a glide plane. A glide plane causes an atom to be translated by a half-unit cell in one direction and then reflected through a plane. We name the glide plane for the translation direction (as an a-glide, b-glide or c-glide), but the full description for a glide plane will also name the reflection plane by specifying the perpendicular axis (which obviously must be a different axis than the translation direction.) If we take for example a c-glide perpendicular to a, which I placed at x=0, we can see that the reflection operation will take (x,y) and transform that to (-x,y) the translation of a half-unit cell adds to z. Thus, coordinates (x,y,z) are transformed to (-x,y,z+1/2).

There are few special types of glide planes. An n-glide, which is sometimes called a diagonal glide, is similar to a a-glide, b-glide or c-glide except that the translation is applied along two axes and the glide is named for the axis normal to the reflection plane. One must specify the coordinate along this axis to know where te . As an example, in space group Pn there is a n-glide perpendicular to b at y=0. This will cause an atom to be reflected from y to -y followed by a translation along both a and c, thus the operator for this will be ½+x, -y, ½+z.

A d-glide, sometimes called a diamond glide, involves a reflection followed by a translation along either a face-diagonal direction (translating along a+b, or a+c, or b+c, or any positive or negative permutation, such as a-b, -a-b or -a+b) or a body-diagonal direction (which can be any of the 6 directions permuting +a, -a, +b, -b and +c,-c). However, the translations for a d-glide is a quarter of a unit cell unlike all other glide plane types. Note that d-glides are only found in body-centered (I) and face-centered (F) lattices. A specify a d-glide, one needs to indicate the direction of the translation as well as the location and orientation of the reflection plane. As an example, a d-glide along the 110 diagonal and perpendicular to c at z=0 will have operator ¼+x,¼+y,-z.