Homework & Lab Assignments

by W.P. Leeman

click

leeman@rice.edu

Updated: 30 Aug 2000

**HW#1 - symmetry elements**- Klein exercise 1**HW#2 - symmetry of solid figures****HW#3 - symmetry of patterned blocks****HW#4 - Miller indices****HW#5 - stereograms**- Klein exercise 5**HW#6 - 2-D & 3-D symmetry**- Klein exercises 10, 11**HW#7 - space groups & crystal structures**- Klein exercises 12, 13**HW#8 - ionic radii****HW#9 - explain measurement of ionic radii****HW#10 - mineral chemistry****HW#11 - mineral formuli and projections****HW#12 - buckyball exercise****HW#13 - retardation effects**(practice exam question)**HW#14 - uniaxial optics**(practice exam question)**HW#15 - biaxial worksheet****HW#16 - crystal optics problem**(practice exam question)**HW#17 - student project prospectus**(abstract/outline)**HW#18 - optics grain scenarios**

**Lab#1 - Physical properties of minerals****Lab#2 - Crystal Symmetry, Miller indices****Lab#3 - Stereographic projections, interfacial angles****Lab#4 - Packing, coordination, crystal structures****Lab#5 - Mineral chemistry, electron microprobe demo****Lab#6 - X-ray diffraction, XRD demo****Lab#7 - Complete XRD exercises****Lab#8 - PRACTICAL MINERAL ID EXAM****Lab#9 - Petrographic microscope & Isotropic minerals****Lab#10 - Uniaxial minerals****Lab#11 - Biaxial minerals****Lab#12 - Minerals in thin section****Lab#13 - OPTICS LAB EXAM**

The lab exercise is to describe one specimen from the HMNS collection in terms of all the properties you can discern or infer from various reference tables, and to compile this information on a 4"x6" index card. You will be making similar cards for other unknown minerals in subsequent labs.

The main objective of this lab is to learn how to observe and how to use this information to distinguish specific minerals. In the following weeks you will be tested on what you have learned to observe using 'unknown' minerals.

Due at beginning of next week's lab: your completed index card.

**Lab 2: Crystal Symmetry, Miller indices** - This lab draws on what we have covered in class concerning recognition of crystal symmetry and indexing of crystal faces and forms.

**2.A.2**- (1) locate all symmetry elements in these models and illustrate in the diagrams, and (2) determine the point group symmetry (Hermann-Mauguin system) and crystal system (cf. hints and tables on p. 15-20).**3.A.2**- identify all distinct faces (forms) and designate with letters on the corresponding diagram.**3.A.3**- identify and accurately sketch the crystal coordinates corresponding to the respective crystal systems.**3.A.4**- show the symmetry elements (rotation axes, mirrors) with respect to the crystallographic axes.**4.A**- do the exercises corresponding to Fig. 4.7.**4.B**- using the same models as above, find appropriate Miller indices for each face.**Optional**- you can practice identifying Miller indices for other crystals using the examples in Figs 2.6 and 3.4.

Materials needed: Crystal models and corresponding clinographic projections (perspective drawings) will be provided by the TA. Copy of Klein workbook. Colored pencils to prepare illustrations.

To turn in at beginning of next week's lab: A neat copy of the completed exercises, with the requested information summarized on the diagrams provided.

**Lab 3: Stereographic projections, interfacial angles** - The purpose of this lab is to solidify the relation between the external morphology of crystals and their inherent symmetry. In essence, you will learn how to prepare stereographic projections of 'poles' to crystal faces, and from these derive the symmetry following **Klein exercises 5, 6, and 8**. The first of these exercises (**#5**) is to be completed prior to the lab as Homework #5.

For simplicity, in exercise **#6** we will use some of the crystal models that you studied in the previous lab. Exercise **#8** will illustrate the utility of stereographic projections in predicting the development of crystal forms as a consequence of specified point group symmetry; it may be completed outside of the lab. Assignments are as follows:

**5.1-6**-**(do prior to lab)**this exercise demonstrates the construction and interpretation of stereographic projections.**6.1-12**- following the methodology provided in Klein,**as a group**determine the interfacial angles for a specified crystal model, plot this information on stereographic projection, and complete all parts of this exercise.**6.1-12**- repeat the previous exercise, but this time with each person**working independently**on a different crystal model (one per person) selected by your TA.**8.1-6**- complete the specified steps for each of 4 point groups as indicated in Figs. 8.1; refer to Fig. 7.1 and Table 3.3 in Klein.

Materials needed: Crystal models, corresponding clinographic projections (perspective drawings), and contact goniometers will be provided by the TA. Bring your copy of Klein workbook, a stereonet mounted on a clipboard or hard surface (as explained in class and HW#5), and colored pencils to prepare illustrations.

To turn in at beginning of next week's lab: A neat copy of the completed exercises, with the requested information summarized on well-labelled diagrams (trace them from examples in Klein if necessary, but be sure they are **accurate**).

**17.1-5**- these exercises will be augmented by examination of three-dimensional packing and crystal structure models in the laboratory. Related aspects of symmetry are discussed in**Klein exercises 11, 12, and 13**.**18.1-2**- this module provides insights into the arrangements and linkages between structural units (or coordination groups) within representative mineral groups; the architecture of such groups influences their physical and chemical properties and provides a logical basis for their classification.

Materials needed: Structure models, materials for packing models will be provided by the TA. Bring your copy of Klein workbook, a ruler, and a set of colored pencils.

To turn in at beginning of next week's lab: Completed workbook sheets.

**Lab 5: Mineral chemistry** - This lab consists of two distinct parts, beginning with a brief demonstration of the departmental electron microprobe facility. Among other things, this facility is used to perform chemical analyses of micron size spots on mineral samples. We will then follow **Klein exercises 19, 20, 21, and 35** to illustrate how such chemical data are routinely manipulated to obtain maximum information. Assignments are as follows:

**19.1-2**- these exercises concern recalculation of typical mineral analyses to produce what is called a 'structural formula', which recasts the analysis into more intuitive molecular proportions in accordance with the actual mineral structure.**20.1-6**- these exercises deal with graphical representation of complex mineral analyses.**21.1-2**- further experience in preparing triangular graphs of chemical data.**35.1-3**- an application of all the above to make a diagram showing the projected locus of minerals in the system SiO_{2}-MgO-Al_{2}O_{3}-H_{2}O (note that this will require creation of only one diagram).

Materials needed: Bring your copy of Klein workbook, a calculator, and a set of colored pencils.

To turn in at beginning of next week's lab: Completed workbook sheets and diagrams.

**Lab 6: X-ray diffraction** - demonstration of XRD lab facility, analysis of unknown mineral.

**Lab 7: X-ray diffraction** - continue exercises. This week, in addition to hand sample identification, your assignment is to complete **Klein exercise 16**. This introduces another important application of XRD data to determine the unit cell dimensions of a relatively simple isometric mineral as explained in Klein.

**Lab 8: PRACTICAL MINERAL ID EXAM** - cumulative test of mineral identification from hand specimens; closed book but mineral index cards can be used.

**Lab 9: Petrographic microscope & Isotropic minerals** - We will assign individual petrographic microscopes and explain how they are used. **It is very important that you have read the first few chapters in Nesse before coming to lab**. You should be familiar with the general features of a petrographic microscope and the optics for isotropic materials!

**Lab 10: Uniaxial minerals** - This lab will extend what you have learned to the study of anisotropic materials, specifically uniaxial minerals. Following the lab handout your assignments are to:

- study oriented specimens of known uniaxial minerals to learn how to observe and interpret interference figures
- use this information along with refractometry to measure the refractive indices (RIs) for a known uniaxial mineral
- apply the same methods to determine the RIs and identity of an unknown uniaxial mineral

Please submit a written report on your activities and findings at the beginning of the next week's lab.

**Lab 11: Biaxial minerals** - This lab will extend what you have learned to the study of biaxial minerals. Following the lab handout your assignments are to:

- study oriented specimens of known biaxial minerals to learn how to observe and interpret their distinctive interference figures
- use this information along with refractometry to measure the refractive indices (RIs) for a known biaxial mineral
- apply the same methods to determine the RIs and identity of an unknown biaxial mineral

Please submit a written report on your activities and findings at the beginning of the next week's lab.

**Lab 12: Minerals in thin section** - In this lab you will become familiar with the optical properties of a variety of common rock-forming minerals in thin section. A simple optical method for the determination of plagioclase composition will also be introduced.

**Lab 13: OPTICS LAB EXAM** - will cover practical applications of the petrographic microscope; closed book, any necessary tables/figures will be provided.

Do **Klein exercise 1** (its fun!). This exercise involves identifying simple rotational and mirror-plane symmetry operations in a number of familiar two- and three-dimensional objects.

Construct the following solid figures using the patterns in Klein:

- cube
- octahedron
- tetrahedron
- rhombohedron
- trigonal prism

For each object, determine and list ALL symmetry elements that you can discern (cf. KH 26-32). For example, what rotation axes, mirror planes, etc. are present and how many of each are there? How many *distinct * symmetry operators are there? Make clear perspective diagrams of each object and use colored pencils to show the symmetry operators. Label clearly! Determine the appropriate point group symmetry designation for each object.

For selected patterned cubes in the class handout, repeat the previous exercise. Note that the patterns may LOWER the symmetry present, and although each object is "CUBE-shaped" they all have different symmetry. Determine the appropriate **point group symmetry designation** (Hermann-Mauguin notation) and **crystal class** for each object (cf. K 35).

Using the clinographic drawings of two ideal crystals (class handout), determine appropriate and consistent Miller indices for each of the labelled faces ({forms}). To accomplish this task you must first designate appropriate crystallographic axes (see text or workbook for examples); neatly and accurately illustrate these axes on the diagrams provided. Remember that Miller indices for the pinacoid faces are readily ascertained on the basis of the axis that they intersect. For prism, dome, or other faces you will have to determine (approximately) the axial intercepts to calculate Miller indices. For this step, make reasonably accurate drawings of sections through the crystals showing axial intercepts of the extended faces. Remembering that parallel faces have equivalent indices, construct parallel faces that intersect one of the axes at a 'unit' translation (e.g., coincident with one of the pinacoids). Please show your complete derivation of the Miller indices for each {form}.

Do **Klein exercise 5** to develop a basic understanding of how the symmetry content of an ideal crystal can be plotted quantitatively. This exercise will prepare you for Lab #3, where you will apply this stereographic plotting technique. The concepts of stereographic projection are described further in K&H and will be addressed further in lab.

To prepare for the lab, mount a copy of a stereonet on a hard surface - such as a clipboard. Before attaching in any permanent way, mount a thumbtack through the reverse side of the stereonet, so that the sharp point is located **exactly** at the center of the net. A sheet of tracing paper can be impaled on this point and readily rotated to plot specific data as explained in Klein. You might want to purchase say 10 sheets of tracing paper for use in this class. Your mounted stereonet will also be useful in structural geology and for amazing your friends!

Do **Klein exercises 10 & 11** to get an introduction to two- and three-dimensional arrangements of motifs.

The combination of mirrors/glides and rotations in 2-D results in 17 plane groups (Klein p. 127). **Exercise 10** develops the relations between these symmetry operations. Do the following parts of this exercise:

**10.1**- determine symmetry operators (**Fig. 10-7**,**a**through**f**)**10.2**- determine appropriate unit cells (note criteria on K p. 130) and show symmetry operators (**Fig. 10.8**,**a**through**f**and**j**); what is the corresponding 'plane group' for these examples?

**Exercise 11** extends our line of thought to three dimensions, incorporating the 14 Bravais lattices (K p. 152-153) to develop up to 230 space groups. Read the introductory material and recall the relations in Tables 3.2 and 3.3, then do the following parts of this exercise:

**11.A1**- follow all steps of this exercise using**Fig. 11.8**examples**a**and**d**.**11.B1**- determine the symmetry content and plane group symmetry symbol for the examples in**Fig. 10.9**.**11.B2**- follow the instructions for this exercise using the three examples on the second page of**Fig. 11.10**(K p. 171).

Do **Klein exercises 12 & 13** to familiarize yourself with graphical representations of the 230 distinct space groups (cf. Table 12.3) that are produced by application of mirrors, glides, and rotations in three dimensions. Read the introductory material, then complete the following:

**12.3**- in this example you are asked to show all symmetry elements that are compatible with the drawing in top panel of**Fig. 12.10**, and to specify the space group to which it belongs. It might be helpful for you to work the preceeding examples.

**Exercise 13** takes the space group concept further into the realm of actual crystal structures. Read the introductory material, then complete the following parts of the exercise: (the objective is to determine an appropriate space group notation in each case)

**13.1-3**- these three examples (**Figs. 13.3**and**13.4**) deal with simple stacking arrangements of spheres and will help prepare you for lab 4. Hint, for the third case, it will be useful to refer to Table 10.1**13.7**- this example (**Fig. 13.8**) concerns the mineral sanidine - a common feldspar.**13.8**- this example (**Fig. 13.9**) concerns the mineral diopside - a representative clinopyroxene.

A few words about crystal structure diagrams in the last two examples. Each figure shows two views of the mineral. The first (a) is a representation of the spatial distribution of individual atoms, with coordinates indicating percent of the 'b' translation distance above the plane of the page (note that 'b' is normal to the page for these cases).

For **sanidine [KAlSi _{3}O_{8}]**, the solid circles represent randomly mixed Si & Al atoms (in a 3:1 ratio) and open circles with numbers inside are O atoms; four O atoms are coordinated with each Si (or Al) atom (solid lines represent bonds) to form 'SiO

For

You will find it instructive to read the specific descriptions of these minerals, and especially their crystallography, in Klein & Hurlbut.

Using the in-class handout, this problem concerns application of simple principles to understanding the structure of NaCl (halite).

- Do the following calculations:
- (a) Calculate weight % of constituent oxides for labradorite (plagioclase) having the formula Ab50An50 where:
- An (anorthite) = CaAl
_{2}Si_{2}O_{8}(or CaO*Al_{2}O_{3}*2SiO_{2}) - Ab (albite) = NaAlSi
_{3}O_{8}(or 0.5Na_{2}O*0.5Al_{2}O_{3}*3SiO_{2})

- An (anorthite) = CaAl
- (b) Calculate the structural formula for this mineral using the analysis derived in (a) (i.e., cation proportions on the basis of 8 oxygens). From these cation proportions show that the feldspar can be represented by 50% An, 50% Ab. This is a check on your computations.

- (a) Calculate weight % of constituent oxides for labradorite (plagioclase) having the formula Ab50An50 where:
- Using the above information and similar calculations for Fo
_{80}(from class lecture), give the oxide wt % analysis of a rock consisting of 40 wt % olivine (Fo_{80}) and 60 wt % plagioclase (An_{50}). - Determine the formuli and identify the minerals represented by the following analyses (wt %).
- (a) Cu: 63.3, Fe: 11.1, S: 25.6
- (b) SiO
_{2}: 38.0, Al_{2}O_{3}: 21.5, FeO: 26.6, MgO: 6.3, MnO: 7.6 - (c) SiO
_{2}: 51.5, FeO: 30.8, MgO: 17.7

- (a) Cu: 63.3, Fe: 11.1, S: 25.6

Use the resources in your textbook to explain this fundamental concept in a brief essay. Discuss the sources and magnitudes of uncertainty in such measurements. Upon what assumptions is the concept of ionic radius based?

The following exercise will help you grasp the concept of **chemical systems** and **projections of mineral formuli** into a simple tetrahedral compositional diagram.

- Determine the standard chemical formuli for the following "endmember" minerals:
- enstatite
- forsterite
- periclase
- pyrope
- grossularite
- anorthite
- corundum
- kyanite
- quartz
- diopside
- wollastonite

- All of these minerals can be considered parts of a single "system" or compositional space defined by four oxide components. Make a tetrahedral diagram showing this system and plot the locations of each of the above minerals in it.

Buckyballs are not strictly minerals (why is this statement true?). Nevertheless, they are highly symmetric entities. Using the template provided, construct your own BB model and determine the symmetry operators present in this object. What rotation axes/mirrors are present, and how many of each can you find? Try to assign an appropriate point group symmetry designation to a BB; be creative.

Write a brief description of the project you have selected, giving the reasons why you choose it. Provide at least three references (other than standard mineralogy textbooks) that you will use in completing your project, and a brief outline of the content of your report.

- with answers!