3-D Symmetry

1. 3-D Symmetry The 32 3-D Point Groups The symmetry of every crystal of a mineral must conform to one of them. This includes every point within a crystal

2. Of the 32 point groups, 21 contain a centre of symmetry and 11 do not. These are called centric and acentric point groups, respectively. A centre of symmetry is present when every point is connected by a line passing through the centre to an equidistant point. The creation centre of symmetry does not require the presence of a roto-inversion axis.

3. 3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System

4. Minimum Symmetry of Crystal Systems

5. Some texts, particularly older British ones, Consider that there are seven crystal systems. The additional one is called trigonal, based on the presence of a unique 3-fold rotation axis as compared to the 6-fold axis of the hexagonal class. Although this seems logical enough in considering the crystal classes, or point groups, when one adds consideration of the atomic arrangement in crystals, problems arise. The point will be discussed at a later point.

6. Crystal morphology and symmetry

7. Crystal morphology and symmetry m Point group: 4/m 2/m 2/m

8. Crystal Morphology Crystal Faces = limiting surfaces of growth Depends in part on shape of building units and physical conditions (T, P, matrix, nature and flow direction of solutions, etc.)

9. Crystal Morphology Different planes have different atomic environments

10. Crystal Morphology The law of Bravais: The frequency with which a given face in a crystal is observed is proportional to the density of lattice nodes along that plane • Explanation: A higher number of bonds along that plane • = fewer bonds between atoms are missing on the surface

11. Crystal Morphology Because faces have direct relationship to the internal structure, they must have a direct and consistent angular relationship to each other

12. Crystal Morphology Nicholas Steno (1669): Law of Constancy of Interfacial Angles Quartz

13. Crystal Morphology How do we keep track of the faces of a crystal?

14. Crystal Morphology How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't Note: “interfacial angle” = the angle between the faces measured like this

15. Crystal Morphology How do we keep track of the faces of a crystal? Remember, face sizes may vary, but angles can't . Thus, it's the orientation and angles that are the best source of our indexing.

16. Consider an orthorhombic sulfur – the 2-fold axes with perpendicular mirror planes can be readily located by inspection, corresponding to the 2/m2/m2/m point group. These should selected as the a, b and c axes of the crystal The lengths of the axes could be easily measured in a crystal that was large enough. The shaded face and its extension to the c-axis define the lengths of the three axes: a = 7.08 mm b = 8.70 mm c = 16.57 mm If we divide each axial length by b, we have a set of axial ratios a/b: b/b: c/b = 0.813: 1 : 1.905 In reality, axial ratios for many minerals were known in the 19th Century from measurement of interfacial angles and trigonometric calculations,.

17. For orthorhombic sulfur the unit cell dimensions as measured by X-rays 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 This is in excellent agreement with the ratio set calculated based on the interfacial angles. This also demonstrates that the correct axes were selected based on the shaded face, which is called the unitface.

18. 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.

19. 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. 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. This type of designation of crystal faces is called the Weissparameters, originated by C.S. Weiss in 1818. Although the derivation is fairly simple, fractional indices can result as seen above. An improvement in this system was devised by W.H. Miller.

20. Miller Indices The Miller Index for a crystal face is found by first determining the parameters second inverting the parameters, and third clearing the fractions. 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) Notes: If an axis is not intercepted, the intercept is taken to be at infinity ∞. In algebra, 1/∞ is defined to be 0.

21. 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 1a, ∞b, ∞ c.  Inverting these becomes 1/1, 1/∞, 1/∞ to give the Miller Index (100).

22. The drawing below is the same orthorhombic crystal we looked at previously.  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 to the left of (113) that cuts the positive a axis and the negative b axis would have the Miller Index (113). The similar face on the bottom of the crystal, cutting positive a, positive b, and negative c axes would have the Miller Index (113). 

23. Miller-BravaisIndices For designation of the faces of a hexagonal crystal such as quartz, it is convenient to modify the Bravais lattice. This is done by adding a third horizontal axis at 120° to the other two. This version of the hexagonal system has three a- axes perpendicular to the c- axis,and 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 Miller-Bravais Indices, with the general notation (hkil)

24. To see how this works, let's look at the dark shaded face in the hexagonal crystal shown here.  This face intersects the positive a1 axis at 1 unit length, the negative a3 axis at 1 unit length, and does not intersect the a2 or c axes.  This face thus has the parameters: 1 a1,  ∞ a2, -1 a3,  ∞ c Inverting and clearing fractions gives the Miller-Bravais Index: (10 10) 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

25. 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 a1, 1 a2, -1/2 a3,  ∞ c.  Inverting these intercepts gives: 1/1, 1/1, -2/1, 1/∞ resulting in a Miller-Bravais Index of  (112 0) Note how the "h + k + i = 0" rule applies here.

26. Some conventions • The Miller index is enclosed in parentheses. • The numbers of the index are not separated by commas. _ • A negative number is indicated, e.g. 1, and is called “one bar”.

27. A line, such as a rotation axis, can also be Indicated with Miller indices of the form [hkl]. A line so indicated is perpendicular to the face (hkl). Thus, [001] is perpendicular to the face (001).

28. Form = a set of symmetrically equivalent faces braces indicate a form {hkl} Multiplicity of a form depends on symmetry

29. Form = a set of symmetrically equivalent faces pinacoid prism pyramid dipryamid related by a mirror or a 2-fold axis related by n-fold axis or mirrors related by n-fold axis related by n-fold axis and mirror

30. Form = a set of symmetrically equivalent faces • β-Quartz = 2 forms: • Hexagonal prism (m = 6) • Hexagonal dipyramid (m = 12)

31. _ 111 111 __ _ 111 111 Isometric (cubic) forms include Cube {100} Octahedron {111) Dodecahedron {110} (001) (010) (100) 011 101 _ 110 110 _ _ 011 101

32. All three combined: {100},{110},{111}

33. 48 (or 47, a technical argument) distinct forms are possible. The cube, octahedron and dodecahedron are three possible forms in the cubic system, for example.

34. The rhombohedral axes are very difficult to use or even visualize; however, the rhombohedral axes can be transformed to a system of axes with the same relations of lengths and angles as the hexagonal axes. These transformed axes are no longer primitive.

35. 3-D Symmetry Unit-cell axes and angles are equal to crystal axes and angles 3-fold axes always in the cube diagonals

36. Cleavage can also be described by a form symbol {110} {100} {111} Octahedral Cubic Dodecahedral {100} {010} {001} {1011} Prismatic Rhombohedral Basal

37. Zone Any group of faces parallel a common axis with the indices [uvw] h k l h k l [uvw]? p q r p q r kr - lq, lp - hr, hq - kp (010) (100) 010 010 010 010 = [001] 100 100 100 100