Space Symmetry I

1. Space Symmetry I • Definition: a crystal consists of atoms arranged in a pattern that repeats periodically in 3-D. • note: doesn’t require (or acknowledge) surface. • pattern can be 1 atom, groups of atoms, 1 molecule, groups of molecules. • Question: In what form does NaCl exist?

2. Space Symmetry I • Definition: a crystal consists of atoms arranged in a pattern that repeats periodically in 3-D. • note: doesn’t require (or acknowledge) surface. • pattern can be 1 atom, groups of atoms, 1 molecule, groups of molecules. • Question: In what form does NaCl exist? • As a crystal lattice ≡ “a regular geometrical arrangement of points or objects over an area or space.” • not necessarily ionic. • “A lattice is not a physical thing; it is simply an abstraction, a collection of points whereupon real objects may be placed.” F.A. Cotton

3. Lattice Points • As an analog of a 2-D crystal, look at an infinitely large piece of wall paper, where the pattern (which can be of any complexity) repeats periodically in both dimensions. • Imagine that you are an infinitely small person standing at a randomly chosen point on the wallpaper. You examine your surroundings. • You are blindfolded and moved in a certain distance along a straight line to a 2nd point. You look around and can’t tell that you have moved. Where are you? ☺

4. Lattice Points • As an analog of a 2-D crystal, look at an infinitely large piece of wall paper, where the pattern (which can be of any complexity) repeats periodically in both dimensions. • Imagine that you are an infinitely small person standing at a randomly chosen point on the wallpaper. You examine your surroundings. • You are blindfolded and moved in a certain distance along a straight line to a 2nd point. You look around and can’t tell that you have moved. Where are you? ☺ ☺ ☺

5. Lattice Points • What happens if you are moved again an identical distance still along that straight line? • the positions are each indistinguishable! • What type of operation is this then? Symmetry! ☺ ☺ ☺

6. Lattice Points ● ● ● ● ● • To aid in further discussion and calculations, it will be convenient to choose some points and axes of reference. • If we choose 1 point at random, then all points identical with this point will constitute a set of lattice points. • These points all have exactly the same surroundings and are identical in position relative to the repeating pattern. • NOMENCLATURE: • 1-D: Row • 2-D: Net • 3-D: Lattice (or space lattice) ● ● ● ● ● ● ● ● ● ● ● ● ● ●

7. Lattice Points & Unit Cell • If we connect the lattice points by straight lines, obtain 2-D parallelograms. • In 3-D this space is divided into parallelepipeds. • Note that any one parallelogram is a template for all of the rest. • Call this a UNIT CELL.

8. Lattice Points & Unit Cell • If we connect the lattice points by straight lines, obtain 2-D parallelograms. • In 3-D this space is divided into parallelepipeds. • Note that any one parallelogram is a template for all of the rest. • Call this a UNIT CELL. • The choice of the initial lattice point could have been anywhere. • If we know the exact atomic arrangement in one unit cell, then we, by extension, can know the arrangement of the whole crystal.

9. Primitive Unit Cells • The choice of a unit cell is not unique; any parallelogram whose edges connect lattice points is a valid unit cell. So, infinite number of ways of choosing a unit cell for a given structure. • Definition: A unit cell with lattice points only at the corners ≡ primitive.

10. Centered Unit Cell • It is permissible to have lattice points inside a unit cell. • Definition: A unit cell containing more than one lattice point ≡ centered. • We’ll talk later about why you might want this. ● ● ● ● ● ● ● ● ● ●

11. Size and Shape of a Unit Cell • May be specified by means of the lengths a, b & c of the three independent edges, and the three angles, α, β & γ between the edges. • α is the angle between b & c, • β is the angle between a & c, & • γ is the angle between a & b. • The location of a point within a unit cell is specified by three fractional coordinates x, y, & z. This point is located by starting at the origin (0, 0, 0) and moving a distance xa along the a axis, yb parallel to the b axis, and zc parallel to the c axis. This point is located at (0.5. 0.75, 0.6).

12. Size and Shape of a Unit Cell • If x, y or z = 1, then you are all the way across the unit cell. • If any are > 1, then the point is in the next unit cell. • e.g. (1.30, 0.25, 0.15) is identical to (0.30, 0.25, 0.15), since all unit cells are identical. • one of the advantages of fractional coordinates is that 2 points are equivalent if the fractional parts of their coordinates are equal. • caveat: (-0.70, 0.25, 0.15) = (0.30, 0.25, 0.15) differ by 1

13. Crystallographic Symmetry • Our ultimate goal will be to consider 3-D arrays, but it will be useful to consider 1-D and 2-D arrays first. Most of the concepts applicable to 3-D can be illustrated more simply with 1-D and 2-D arrays. • The type of array we are concerned with is obtained by repetition of some object or unit in a regular way thoughout space. Our object, motif = • We’ve defined symmetry operations as movements after which no change could be detected in the object; it is indistinguishable. • Thus far, we’ve looked at the following: E, Cn, σ, Sn, and i. • What do these have in common? At least one point of the object is unmoved by the operation. • A complete symmetry classification scheme for crystallography requires that we consider other operations as well. Before we were dealing with finite objects; now with infinite arrays.

14. Crystallographic Symmetry Operations • Translation. Shifting a motif by a defined distance in a certain direction, then doing this again and again by the same distance and direction. • This distance = unit translation. • Can be in 1-D, 2-D or 3-D (each with different unit translation and direction) • ALL crystals possess translation.

15. Crystallographic Symmetry Operations • Translation. Shifting a motif by a defined distance in a certain direction, then doing this again and again by the same distance and direction. • This distance = unit translation. • Can be in 1-D, 2-D or 3-D (each with different unit translation and direction) • ALL crystals possess translation. • Glide Plane. A combination of translation and reflection. Operation is translation by one-half unit dimension, followed by reflection in the plane. glide plane

16. One Dimensional Space Groups • A Space Group includes both point symmetry elements and translation. • There are seven One Dimensional Space Groups. • pxyz nomenclature p = primitive (i.e. one lattice point per unit cell). x = mirror plane ┴ to axis of translation? yes = m; no = 1. y = mirror plane ║ to axis of translation? yes = m glide plane along axis of translation? yes = a no = 1. z = Cn axis? n = 1; n = 2. • p111: simplest; only translation present. • p1a1: includes glide plane.

17. One Dimensional Space Groups • pm11: translation and transverse reflection. Note that second set of mirror planes are generated. Often introduction of 1 set of symmetry elements creates a second not equivalent to the first. • p1m1: translation with longitudinal reflection. • p112: two-fold rotation axis (located below the motif on the line of translation). The second C2 axis is explicitly introduced. If you had started with it, the first would have arisen automatically.

18. One Dimensional Space Groups • pma2: glide plane plus transverse reflection. C2 axis created automatically. • pmm2: translation with longitudinal and transverse reflection. Just as in point groups, the intersection of two mirrors generates a C2 axis.

19. One Dimensional Space Groups

20. One Dimensional Space Groups: Examples

21. One Dimensional Space Groups: Examples p1m1

22. One Dimensional Space Groups: Examples p1m1 p111

23. One Dimensional Space Groups: Examples p1m1 p111 p1a1

24. One Dimensional Space Groups: Examples p1m1 p111 p112 p1a1

25. One Dimensional Space Groups: Examples

26. One Dimensional Space Groups: Examples pm11

27. One Dimensional Space Groups: Examples pm11 pma2

28. One Dimensional Space Groups: Examples pm11 pma2 pmm2