Crystallography - Excecise I. Exercise - crystal lattice Cuprite, an oxide of copper, has the Lattice: a0=b0=c0=0.427 nm, ===90 and the basis: Cu: 1/4,1/4,1/4; 3/4,3/4,1/4; 3/4,1/4,3/4; 1/4,3/4,3/4. O: 0,0,0; 1/2,1/2,1/2. a) Draw a projection of the structure on x,y,0 and a perspective representation of the structure b) What is the chemical formula of this compound? What is Z? c) Calculate the shortest Cu-O distance. d) What is the density of cuprite. II. Exercise - indice 1. Give (hkl) for a few planes containing thea few lines lying in the plane (121). 2. What are the relationships between (110[110] and [110]; [211] and [211]. 3. Draw [310], [310], [310] [123], [123], [123] in the following unit cell. Cu line [211], and give [uvw] for

) and (110); (211) and (211);

O

ab

c

Projection in one axis

4. Index the directions 5. Draw (100), (110), (123) (110), (210), (211) 5. Index the planes 6. Die Abbildung ist die Projektion eines Raumgitters parallel zur a-Achse auf die b,c-Ebene. Die eingezeichneten Geraden sind Gittergeraden (...) bzw. die Spuren von Netzebenen (_ _).

a) Indizieren Sie die eingezeichneten Gittergeraden und Netzebenen;

a b

c

a b

c

1/2,1,0

1,1,1/2

0,0,1/2

a b

c 1,1/2,1/2

1,1,1

1,2/3,0 1,1/3,0

b) Geben Sie die [uvw] der Schnittgeraden beider Netzebenen an; c) Zeichnen Sie die Spuren der Netzebenen (023) und (021) in die

Projektion ein.

d) Geben Sie einige Netzebenen an, die die Gittergerade [101] enthalten und einige Gittergeraden, die in d Netzebene (-1-1-1) liegen. 8. Draw the directions [1120], [121] and planhexagonal unit cell: III. Exercise Morphology 1. Plot the poles of the faces on a stereogra

bcer es (110), (123) in the

m

a1

c

a3 a2

2. Plot the directions corresponding to the following axial systems on a stereogram 3. Plot the poles of the faces on a stereogram

note

4. What is represented by the following stereogram VI. Exercise - Principle of symmetry 1. Carry out the operations on an unsymmetrical pyramid, whose base lies into the plane of the paper.

Exercise - 14 Bravais lattices V. Exercise - 14 Bravais lattices 1. a) Determine the symmetry elements for the given plane lattices. b) Draw in the edges of the unit mesh and give the lattice parameters.

2. Given orthrohombic unit cell and ist projection on x,y,0. Draw the third symmetry element generated by the two given elements, give ist symbol and coordinates. 3. a) Draw the unit cell of each lattices as a projection on x,y,0, or in the monoclinic case, on x,0,z. use a scale of 0.1 nm= 1 cm. Monoclinic P: a0=0.55, b0=0.40, c0=0.40 nm; =105 orthorhombic P: a0=0.3, b0=0.45, c0=0.4 nm tetragonal P: a0=0.4, c0=0.3 nm hexagonal P: a0=0.4, c0=0.3 nm trigonal R: a0=0.45, c0=0.3 nm b) determine the symmetry operations of the lattices you have drawn, and plot the symmetry elements on the projection of the lattice.

VI. Exercise - Point group 1. Complete the stereographic projections in the table. Which symmetry elements are generated? What are the resultant point groups? Give the symbols for each. 2. Complete the stereographic projections in Table and give the symbols for the resultant point groups.

3. Determine the International symbol for the point groups whose symmetry elements are illustrated in the following: a) find the symmetry elements that characterise the crystal system. b) indicate the crystallographic axes a, b, c on the stereogram, bearing in mind the orientation of the symmetry directions for the crystal system. c) give the International symbol. VII. Exercise - space group 1. In the projections below of a unit cell onto x,y,0, only a single symmetry element is given. Allow this symmetry element to operate on an asymmetry point at x,y,z and give the coordinates of the equivalent point(s) generated.

m in x,y,1/2 m in x,1/4,z

a in x,y,1/4 b in ,y,z

c in x,1/4,z n in ,y,z

n in x,y,0 n in x,0,z

2 in ,y,0 21 in , 0,z 21 in 0,y,1/4 2 in , , z

2. The symmetry diagrams for the seven space groups are given below as projections on x,y,0. a) enter on each diagram a point in a general site x,y,z, and allow the symmetry to operate on it. b) give the coordinates of the points equivalent to x,y,z. c) what is the multiplicity of the general position.

VIII. Exercise - reciprocal space 1. Draw the (100)- and (001)-lattice planes of the rutile structure. Construct the a*c*- and a*b*-planes of the reciprocal lattice. Description of the crystal structure of rutile TiO2 is as follows: 2. A powder photography taken from cube-shaped crystals of KI, using Cu Ka radiation (=0.154 nm). The first nine lines, measured from the position of the direct beam, give the following 2-values: 21.80; 25.20; 36.00; 42.50; 44.45; 51.75; 56.80; 58.45; 64.65. A) index these powder lines and calculate their d-values. B) determine the lattice constant a0. C) what is the value Z? ( the density of KI is 3.13 g/cm3).

Crystal structure of rutile, TiO2, a) in a perspective drawing, b) in projection on x,y,0.