Reference no: EM132687288

PRACTICE 1 CRYSTALLOGRAPHIC POINT GROUPS OF SYMMETRY

PRACTICE 1: Point group: nomenclature, multiplicity or order, special positions, indexing equivalent points (and more)

1.1.- Represent the point group mmm. Make the stereographic projection. Indicate the multiplicity of the group or group order.
a) Applying the elements of symmetry to a face or point, obtain the number of equivalent faces of (hkl). Give the indices of all the faces obtained.
b) Applying the elements of symmetry to a face or point, obtain the number of equivalent faces of (110). Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained in section b).

1.2.- Represent the point group 222. Make the stereographic projection. Indicate the multiplicity of the group or group order.
a) Apply the elements of symmetry to (110). Give the indices of all the faces obtained.
b) Apply the elements of symmetry to (111) and obtain all the faces of the crystalline form {111}. Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.3.- Represent the point group 2/m. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to {111}. Give the indices of all the faces obtained. Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.4.- Represent the point group 3m. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to (110). Give the indices of all the faces obtained. Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.5.- Represent the point group 6mm. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry and obtain the faces of the {100}. Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.6.- Represent the point group m3. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to (110) and (111). Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.7.- Represent the point group m3m. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to (110) and (100). Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.8.- Represent the point group 4mm. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to (110) and (111). Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.9.- Represent the point group 4/m. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to (110) and (111). Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

1.10.- Represent the point group 422. Make the stereographic projection. Indicate the multiplicity of the group or group order.
Apply the elements of symmetry to (110) and (111). Give the indices of all the faces obtained.
Tell or explain the relationship between the multiplicity of the point group and the number of faces obtained.

PRACTICE 2 - CRYSTALLOGRAPHIC SPACE GROUP OF SYMMETRY

PRACTICE 2: Crystal structure: projection of the crystalline structure, elements of symmetry, Bravais lattice Type: Space groups. Nomenclature. Distance between points.

2.1.- A monoatomic substance crystallizes in the orthorhombic system with the atoms in the following positions: (x, y, z); (-x, y, z); (x, y , z); ( x , y ,z,) .
Project on the plane (001) the atomic positions:
Identify the elements of symmetry Identify the Bravais lattice type.
Identify the space group.
Which is the multiplicity of the group?

2.2.- A monoatomic substance crystallizes in the orthorhombic system with the atoms in the following positions: (x, y, z); (-x, y, z); (x, y , z); ( x , y ,z,); (x+½, y+½, z); (½-x, ½- y, z); (½+x, ½+ y , z); (1/2+ x , ½+y, z).
Project on the plane (001) the atomic positions:
Identify the elements of symmetry Identify the Bravais lattice type.
Identify the space group.
Which is the multiplicity of the group?

2.3.- The cell of PdS2 has the following dimensions: a= 5.460 Å, b=5.541 Å, c= 7.531 Å, α=β=γ=90º.
The atoms are in the following coordinates:
Pd: (0, 0, 0); ( ½, ½, 0); (½, 0, ½); (0, ½, ½)
S: (x, y, z); (½+x, ½-y, z ); ( x ,½+y, ½-z); (½-x, y , ½+z);

( x , y , z ); (½-x, ½+y, z); (x, ½-y, ½+z); (½+x, y, ½-z) where x= 0.107, y= 0.112 i z= 0.425
Project the crystalline structure on the plane (001).
Find and project the elements of symmetry on the plane (001). Identify the Bravais lattice type.
Identify the space group.
Which is the multiplicity of the space group?.
How many atoms are there in each class, inside the cell? What is the Z (formula units)?.Relationship between number of atoms of each class, Z and the multiplicity of the space group in this crystalline structure.
Calculate the density of PdS2. (Atomic mass of Pd: 106.4; S: 32.1; Avogadro number: 6.023 . 1023)
Calculate the distance between Pd(1/2,1/2,0) and the S of the first coordination sphere.

2.4.- The AgO crystallizes in the monoclinic system. The parameters of the lattice are a
= 5.85Å, b = 3.45 Å, c = 5.50 Å and β = 107º, the coordinates of the atoms are:
Ag: (0, 0, 0); (0, ½, ½); (½, 0, ½); (½,½, 0)
O: (x, y, z); ( x , y , z ); (x, ½-y, ½+z); ( x ,½+y, ½-z) where x= 0.30, y= 0.35 i z= 0.23
a) Project the crystalline structure on the plane (001)
b) Identify the elements of symmetry, the Bravais lattice type and the space group. c)Relationship between number of atoms of each class, Z and the multiplicity of the space group in this crystalline structure.

2.5.- Project the symmetry of the space group P21/c on the plane (001).
a) Give the equivalent positions to (x, y, z).
b) Find all the special positions and indicate the multiplicity in each case.

2.6.-Project the symmetry of the space group C222 on the plane (001).
a) Give the equivalent positions to (x, y, z).
b) Find all the special positions and indicate the multiplicity in each case.

2.7.- The aragonite, CaCO3, crystallizes in the orthorhombic system. The cell parameters are a = 4.94 Å, b = 7.94 Å, c = 5.72 Å. Cell atoms are in the following positions:

Ca: (0, 1/12, ½); (½,7/12 ,½); (0, 5/12, 0); (½, 11/12, 0)
C: (½, ¼, 1/6); (0, ¾, ?); (½, ¼, ?); (0, ¾, 5/6)
O: (¼, 1/6, 1/6); (¾, 1/6, 1/6); (½, 5/12, 1/6); (¼, 5/6, ?);
(¾, 5/6, ?); (0, 7/12, ?); (¼, ?, ?); (¾,?, ?);
(½, 1/12, ?); (¼, ?, 5/6); (¾, ?, 5/6); (0, 11/12, 5/6)

a) Project the crystalline structure on the plane (001)
b) Identify the space group.
c) Indicate if there are atoms in special positions. If so, justify it because its number in the cell is smaller. (Relationship between the number of atoms of each class, Z and the multiplicity of the space group in this crystalline structure.)
d) Calculate the density of the aragonite. (Atomic mass Ca: 40.08; C: 12.01; O: 16.00; Avogadro's number:6.0231023)

CRYSTALLOGRAPHIC SPACE GROUP OF SYMMETRY

PRACTICE 3: Crystal structure: projection of the crystalline structure, elements of symmetry, Bravais lattice Type: Space groups. Nomenclature. Distance between points.
3.1.-Project the symmetry of space group C2/c on the plane (001).
a) Give the equivalents positions to (x, y, z). Which is it the multiplicity.
b) Indicate the special positions and indicate the multiplicity of each one.

3.2.- AgO crystallize in the monoclinic system. The cell parameters are a= 5.85Å, b=
3.45 Å, c= 5.50 Å i β= 107º, the atoms coordinates:
Ag: (0, 0, 0); (0, ½, ½); (½, 0, ½); (½,½, 0)
O: (x, y, z); ( x , y , z ); (x, ½-y, ½+z); ( x ,½+y, ½-z)
Where x= 0.30, y= 0.35 i z= 0.23
a) Project the atoms in the cell on the plane (001)
b) Determine the Bravais lattice type, the elements of symmetry and the space group.
c) Relationship between the number of atoms of each class, the Z and the multiplicity of the space in this crystalline structure.

3.3.- The rutile TiO2 crystallize in the tetragonal system. The cell parameters are a=
4.59 Å i c= 2.96 Å. The atoms are in the followings positions:
Ti: (0, 0, 0); (½, ½, ½)
O: (x, x, 0); ( x , x , 0); (½-x, ½+x, ½); (½+x, ½-x, ½); x = 0.3

a) Project the crystalline structure on the plane (001).
b) Determine the Bravais lattice type.
c) Knowing that the TiO2 belongs to the point group 4/m2/m2/m, determine the space group.
d) Calculate the distances Ti -O. Coordination figure of Ti.

3.4.- The cuprite crystallizes in the cubic system, a = 4.296 Å. with the atoms located in the following positions:
Cu: (¼, ¼, ¼); (¾, ¾, ¼); (¾, ¼, ¾); (¼, ¾, ¾)
O: (0, 0, 0); (½, ½, ½)

a) Project the structure on the plane (001)
b) Find parallel symmetrical elements in the direction of c.
c) Find the characteristic ternary axes of the cubic system.
d) Knowing that cuprite belongs to the m3m point group, give the notation of the space group.
e) Indicate the position of the ternary axis.
f) Calculate the distances Cu -O. Cu coordinate figure.

PRACTICE 4
APPLICATION TO X-RAY DIFFRACTION
PRACTICE 4: Crystal structure: projection of the crystalline structure, elements of symmetry, Bravais lattice Type: Space groups. Nomenclature. Distance between points.
Distances between planes. dhkl list. Application to X-ray diffraction
4.1.- The cuprite crystallizes in the cubic system, a = 4,296 Å. with the atoms located in the following positions:
Cu: (¼, ¼, ¼); (¾, ¾, ¼); (¾, ¼, ¾); (¼, ¾, ¾) O: (0, 0, 0); (½, ½, ½)
a) Project the structure on the plane (001)
b) Find symmetrical elements parallel to the direction of c.
c) Find the characteristic ternary axes of the cubic system.
d) Knowing that cuprite belongs to the m3m point group, give the notation of the space group.
e) Number of atoms of Cu and O. Stoichiometry formula. Find the relationship between the number of atoms with the multiplicity of the space group.
f) Calculate the distances Cu -O. Cu coordinate figure.
g) Calculate the dhkl list of this crystalline structure, sorted from larger to smaller.

4.2.- The BaTiO3 compound has a polymorphism with the following characteristics:
a) At high temperatures the structure is cubic: a = 4.00 Å, Pm3m, Z = 1.
b) At 120oC a transformation occurs to a following tetragonal structure: a =3.99Å, c =
4.03Å, P4mm, Z=1.
c) At 0 oC a new transformation occurs in a orthorhombic structure: a = 5.669 Å, b = 3.990 Å, c = 5.682 Å, Bmm2, Z = 2.
d) Prepare the dhkl list of crystalline structure, ordered from larger to smaller.

4.3.- Prepare the dhkl list of the following crystalline structures, with space group
Fm3m, ordered from larger to smaller.
Fm3m:
NaCl (a=5.642 Å); LiCl (a=5.129 Å); NaF (a=4.623 Å); MgO (4.203 Å) Pb(a=4.939 Å); Al(a=4.041 Å) ; Pt (a=3.916 Å); Cu (a=3.608 Å)

4.4.- At 25º, the α-Fe crystallize in the cubic system (Im3m) with cell parameters a= 2.86Å. The atoms coordinates are. (0, 0, 0) i (½, ½, ½) that belongs to a I Bravais cell.
a) Prepare the dhkl list of this crystalline structure, ordered from larger to smaller
b) Make the representation of the graph of the atomic diffusion factors, fj of Fe (ordinate axis), as a function of sin θ/λλ (abscissa axis).
The atomic diffusion factors of Fe (atomic number= 26) are the followings:

sin θ/λ 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
fFe 26.00 25.30 23.68 21.85 20.09 18.40 16.77 15.29
0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20
13.84 11.47 9.71 8.47 7.60 6.99 6.51 6.12 5.79

The wavelength used in the experiments of X Ray Diffraction is Cu Kα , λ=1.542 Å.

PRACTICE 4BIS
X-RAY DIFFRACTION

PRACTICE 4BIS: X-RAY DIFFRACTION of Polycrystalline Materials. dhkl list. Identification of a compound. Identification of a mixture. Identification of Materials with the same stoichiometric formula and different crystalline structure (Polymorphs).

4BIS.1.- At 25º, the α-Fe crystallize in the cubic system (Im3m) with cell parameters a= 2.86Å. The atoms coordinates are. (0, 0, 0) i (½, ½, ½) that belongs to a I Bravais cell.
a) Prepare the dhkl list of this crystalline structure, ordered from larger to smaller
b Calculate which angle theta corresponds to each dhkl, according to the expression (Bragg's law) if the radiation used has been the Kα of the Cu, λ = 1,542 Å.
2 dhkl sinθ = λ
c) Calculate the Fhkl structure factors of reticular plans.
Indicate systematic extinctions. (Systematic extinction if Fhkl = 0

d) Make the representation of the graph of the atomic diffusion factors, fj of Fe (ordinate axis), as a function of sin θ/λ (abscissa axis).
The atomic diffusion factors of Fe (atomic number= 26) are the followings:

sin θ/λ 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
fFe 26.00 25.30 23.68 21.85 20.09 18.40 16.77 15.29
0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20
13.84 11.47 9.71 8.47 7.60 6.99 6.51 6.12 5.79

The wavelength used in the experiments of X Ray Diffraction is Cu Kα , λ=1.542 Å.

4BIS.2.- The powder diffractogram of a sample, which is a mixture of two cubic substances, results in the following list of dhkl.

3.24 3.13 2.81 2.21 1.98 1.81 1.69 1.62 1.56 1.40 Å

Knowing that a substance is cell type F.
Find the type of cell that has the other substance. The parameter a, of each one of them. Which spacings dhkl correspond to each one of the substances?.
What indexes correspond in each case?.

4BIS.3.- . The BaTiO3 compound has a polymorphism with the following characteristics:
a) At high temperatures the structure is cubic: a = 4.00 Å, Pm3m, Z = 1.
b) At 120oC a transformation occurs to a following tetragonal structure: a =3.99Å, c =
4.03Å, P4mm, Z=1.
c) At 0 oC a new transformation occurs in a orthorhombic structure: a = 5.669 Å, b = 3.990 Å, c = 5.682 Å, Bmm2, Z = 2.
d) Prepare the list of the first 5 dhkl for each crystalline structure, ordered from larger to smaller. Give the angles (2θ) if the radiation Kα of Cu is λ= 1.5406 Å.

4BIS.4.- A crystalline powder diffractogram that presents its maximum in the dhkl: 3.257, 2.821, 2.333, 2.020, 1.995, 1.701, 1.629, 1.429, 1.411, 1.262, 1.218, 1.167,
1.152, 1.010 Å ; it is a mixture of two compounds A and B that are cubic. Determine A and B, knowing:

A can be: NaCl (a=5.642 Å); LiCl (a=5.129 Å); NaF (a=4.623 Å); MgO (4.203 Å) B can be: Pb(a=4.939 Å); Al(a=4.041 Å) ; Pt (a=3.916 Å); Cu (a=3.608 Å)
Index all the peaks of the diffractogram.

Attachment:- CRYST Practices.rar