For this exercise, you will hand in your answers to the last page only by the end of next Friday, September 29. All other questions are a "self-test" with answers provided in a binder kept at the back of Room 211. The models must be kept in Room 211 at all times, even if you need to examine them outside of class hours. Recommended reading: Klein & Hurlbut, pp. p.46-53, 66-100 (21st edition) OR pp. 40-48, 60-96 (20th ed.).

1. Crystal forms

A form is defined as a collection of faces with identical structure and properties. Unlike most mineral specimens, the wooden and plaster models are idealized representations of crystal forms. On these models, all faces belonging to a given form are identical in size and shape.

a) Find an example of each basic type of crystal forms on one of these blocks:
 

Form	Definition	Block no.
pedion	only one face of its kind
dihedron	two faces of same shape meet along one edge
parallelohedron	pair of parallel faces (only two of that shape)
prism	3, 4, 6 or 8 faces with parallel edges
pyramid	3, 4, 6 or 8 faces meeting at a single point
dipyramid	top & bottom pyramids are mirror images
trapezohedron	top & bottom pyramids are somewhat offset
rhombohedron	top & bottom pyramids offset by 60 degrees
cube	6 mutually perpendicular faces
octahedron	8 faces perpendicular to four 3-fold axes
tetrahedron	4 faces perpendicular to four 3-fold axes

 

b) Find prisms, pyramids or dipyramids displaying these different symmetries:
 

Type of form (e.g., prism)	Symmetry	Block no.
	rhombic: 4-sided cross-section, 2 unequal angles
	tetragonal: square cross-section
	trigonal: triangular (equilateral) cross-section
	ditrigonal: 6-sided cross section, 2 unequal angles
	hexagonal: 6-sided cross section, all angles equal
	ditetragonal: 8-sided cross section

1. Crystal forms (continued):

c) Some forms are closed, i.e. each one encloses space completely. Others are open, i.e. their faces do not enclose space completely. A closed form may occur alone on a crystal but an open form must occur in combination with others (closed or open). Are these forms open or closed?
 

Form	Open	Closed
parallelohedron
pyramid
tetragonal dipyramid
hexagonal prism

d) Some forms are said to be "variable" and others "invariable". A variable form is one where the faces may display different interfacial angles. Find examples of each on blocks set aside for this question:
 

Form	Variable (V) or Invariable (I)	Blocks no.
parallelohedron
rhombic prism
tetragonal prism
tetragonal pyramid
octahedron

The external shape of crystals may show faces related by:
- a rotation axis
- a mirror plane
- a center of inversion
- a rotoinversion axis

If a center of inversion is present, every face should have an identical counterpart, parallel but inverted, on the opposite side of the crystal. The crystal is said to be centrosymmetric.

To visualize a symmetry element, imagine an axis or a plane going throughout the wooden model. Rotation axes are often parallel or perpendicular to mirror planes. A center of inversion, if present, is at the exact center of the model. When looking for a symmetry element, pick one set of faces identical in shape and size (i.e. a form) and determine which symmetry operation (rotation, reflection, inversion, rotoinversion) relates each face to the next identical one.

Block No.	Center of inversion present? Yes (Y) or No (N)

b) The next easiest symmetry element to identify is a mirror plane. If present, a mirror plane relates a top half to the bottom half of a crystal, or its left half to its right half. Some models show more than one mirror plane.

Block No.	Number of mirror planes
	Only one.
	Only two mirror planes, mutually perpendicular.
	Three mirror planes, intersecting along a single axis.

c) A rotation axis is an imaginary "rod" around which faces occur every 180, 120, 90 or 60^o.

Block No.	Number and type of rotation axes
	a single 3-fold rotation axis
	a single 4-fold rotation axis
	three 4-fold rotation axes

 
 

d) The last type of symmetry element is a combination of rotation and inversion, called a "rotoinversion". The top and bottom halves of the resulting forms (rhombohedra, scalenohedra) are identical but their faces are offset by exactly half the amount of rotation. If a prism is also present, the shape of opposite faces is inverted, revealing the inversion that is part of the operation.

What type of rotoinversion axis is present on these models?
 

Block no.	Rotoinversion axis
	One 3-fold rotoinversion axis
	One 4-fold rotoinversion axis

e) Some models show a trapezohedron, i.e. forms where the top and bottom halves are identical and pyramid-like but offset not exactly halfway. Do they show a center of inversion or a mirror plane? What operation relates the top half to the bottom half?

Any trapezohedron can occur in two versions, said to left- and right-handed or enantiomorphs. Enantiomorphs only occur for forms that have neither a center of inversion nor a mirror plane. Quartz is the most common example of enantiomorphism.

3. The six crystal systems and the 32 crystal classes.

There are only 32 possible combinations of symmetry operations, which define 32 crystal classes. The classes are further grouped into six crystal systems, based on the absence or presence of certain types of rotation axes (see opposite page).

The Herman-Mauguin symbols of the crystal classes listed on the next page are easy to interpret:

l Numbers refers to an axis of rotation (2, 3, 4 or 6-fold) or rotoinversion (bar 3, bar 4 or bar 6-fold).

In the tetragonal and hexagonal system, the first symbol (e.g. 4/m, 6, or "bar 3") refers to the axis of maximum symmetry. The crystal is usually either flattened or elongated along this axis (the c axis). The other symbols (2, m or 2/m) refer to axes (a₁ and a₂) that are perpendicular to the first axis.

In the isometric system, the first symbol (2, 4 or 4) refers to three identical axes (a₁ , a₂ , a₃) that are mutually perpendicular. The second symbol refers to four directions located exactly between the three axes a₁ , a₂ , a₃, along which 3- or 3-fold axes are found. The last symbol refers to directions at 45º between any two of the three identical axes.

3. The six crystal systems and the 32 crystal classes (continued):

12 models are set aside for this exercise. Match at least one model to each system and determine its class. Do this by looking for specific symmetry elements appearing in the Hermann-Mauguin symbol:
 

System	Symmetry	Classes	Block no.
triclinic	- no rotation axis, - no mirror plane	1 (no symmetry) bar 1 (center of inversion)
monoclinic	a single 2-fold axis or a single mirror plane, or both elements, mutually perpendicular.	2 m 2/m
orthorhombic	three 2-fold axis and/or mirror planes, mutually perpendicular	2 2 2 m m 2 2/m 2/m 2/m
tetragonal	a single 4-fold rotation or rotoinversion "bar 4"-fold axis, with or without perpendicular 2-fold axes and/or mirror planes	4, "bar 4" 4 2 2 4 m m "bar 4" 2 m 4/m 4/m 2/m 2/m
hexagonal  - certain classes are grouped by some into a trigonal or rhombohedral system	a single 3-fold or "bar 3"-fold rotoinversion axis, or a single 6-fold rotation or 6-fold rotoinversion axis, with or without perpendicular 2-fold axes and/or mirror planes	3, "bar 3" 6, "bar 6" 3 2/m, 6 m 2 3 2, 6 2 2 3 m, 6 m m,  6/m 6/m 2/m 2/m
isometric	four 3-fold rotation or "bar 3"-fold rotoinversion axes, with or without mirror planes and three 2-, 4-fold or "bar 4"-fold axes.	23 432 2/m "bar 3" "bar 4" 3 m 4/m "bar 3" 2/m

1) Select one wooden model among those provided for this exercise. Make note of the number on the sign-up list taped to the table at the back of FDA Room 211. The models must be kept in FDA Room 211 at all times (they are irreplaceable).

2) Use masking tape and labels to show the orientation of the rotation and/or rotoinversion axes and of mirror planes. Draw the appropriate symbol on the labels showing the location of axes (you should label both ends of each axis).

3) Fill in the information requested on this sheet (to be handed in with your labeled model).

HINTS: The quickest way to determine the crystal system is to identify the rotation axis of highest symmetry (except for some isometric crystals). The term centrosymmetric means having a center of inversion.

Note: the table on this answer sheet has room for as many as six forms. Your model does not necessarily display that many forms. If the same form appears more than once (with different kinds of faces), list each occurrence on separate lines. You must account for all the faces present on your model.
 

Model number:	Crystal system:
Crystal class (Hermann-Mauguin symbol):	Is this class centrosymmetric (Y/N)?

Name of form (e.g., trigonal prism)	number of faces of same kind making up the form	Is the form open or closed? Variable or invariable?
		OPE	CLO		VAR			INV
Total no. of faces on the model