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Problems:
Bonding. Crystalline structures
Problems
- Find the Miller indices corresponding to the planes presented in the figure.
For an FCC structure, with lattice parameter a , calculate the linear density along the direction [110], the planar density on the plane drawn on figure I and the volumetric density.
(solution: ( ̅̅ ) , ( ̅ ) , ( ̅̅ ) , l = 2/ a, s =4/ a^2 3, v =4/ a^3 )
- What is the atomic mass of a metal element with FCC structure, density of 1.74 g/cm^3 and lattice constant of 4.527Å? Find the element with these characteristics.
(solution: M =24.3 g/mol, Mg)
- Draw the crystalline planes in a cubic lattice that present the following Miller indices:
a) (1 01) b) (12 1) c) (2 1 3) d) (13 3) e) (1 2 2)
f) (3 1 2) g) (1 23) h) ( 1 4 3) i) (3 13) j) (31 3)
solution:
I (^) a/
a/
II
III
- For a BCC structure determine the surface (planar) density for the planes (100), (110) and (111); the linear density in the direction [100], and the volume density.
(solution: (100)=1/ a^2 , (110)= 2/ a^2 , (111)=1/ a^2 3, [100]=1/ a , v=2/ a^3 )
- Given that aluminium has an FCC structure, the atomic radius 1.43 Å and the atomic weight 26.98 uma/at. Data: 1m = 10^10 Å; NA = 6.023 10^23 mol-1. Calculate:
a) planar density, in at./cm^2 , for the (110)plane. (solution: (110)=8.7x10^14 at/cm^2 )
b) linear density , in at./cm, for the [110] direction. (solution: [110]=3.5x10^7 at/cm)
c) weight of a bar of 20 mm in diameter and 1 m long (solution: m =851 g)
- Calculate the linear density of atoms in the [111] direction and the planar density in the(111) plane for:
a) Iron BCC (solution: [111]=2/ 3 a , (111)=1/ a^2 3)
b) Nickel FCC (solution: [111]=1/ a 3, (111)=4/ a^2 3)
- The figure shows the crystallographic directions of a hypothetical material with orthogonal structure:
a) Draw the unit cell. To which crystalline lattice does it belong?
b) Calculate the atomic weight given that the density is 5.97 g/cm3.
c) Calculate the planar density in atoms/mm2 in the planes (100) and (110), and compare them with each other.
(solution: a) face centred orthorhombic lattice, b) M =107.87 g/mol, c) (100)=6.66x10^12 at/mm^2 ,
(110)=5,2x10^12 at/mm^2 )
8.- Calculate the fraction of area occupied by atoms for the (111), (200), (220), (222), (400) and (420) planes, in the FCC structure.
(solution: (111): /2 3, (200): /4, (220): 2/8, (222): 0, (400):0, (420): /4 5)
- A pure metal undergoes a polymorphic change from BCC to FCC when it reaches 910ºC. Calculate the volume change associated with the change in structure given that the interplanar spacing d 321 for the BCC structure is 0.07565 nm, and the planar density in the FCC lattice in the plane (002) is 15.18·10^18 at./m^2.
(solution: 5.7 %)
- Three distinct crystallographic planes in the unit cell of a hypothetic metal are shown below. The circles represent atoms.
5Å
[010]
7,21Å
[101]
6,4Å
[110]
Exercise examples
Exercise 1.
a) Find the value of the Madelung constant A for Na+^ in the second neighbour approximation for FCC crystal structure of NaCl.
( solution: A =2.48 )
b) Calculate the lattice energy E l of NaCl FCC crystal using previously estimated value of Madelung constant, as well as values n =5 and r 0 =0.236 10-9^ m for the Born exponent and Na-Cl distance, respectively (parameters: 4 0 =1.12 10-10^ C^2 /Jm, e=1. 10 -19^ C, NA=6.022 10^23 mol-1).
( solution: El =-1160 kJ/mol )
c) Find the relative difference (in %) between the estimated and the experimental value of the lattice energy of NaCl is -786 kJ/mol. What can be done to improve the estimated value?
( solution: =47%, increase the number of atoms in the calculation of the Madelung constant )
d) Calculate the percentage of ionic character of NaCl if the electronegativities of Na and Cl are X Na =0.93 and X Cl=3.16, respectively.
( solution: %IC=71% )
Exercise 2.
The plane (111) and the direction [111] of a unit cell of an orthogonal metal crystal lattice are shown in the figure. Circles represent atoms.
a) Construct the unit cell. To which crystalline system this lattice belongs? What is the Bravais lattice of this crystalline structure?
( solution : , cubic , BCC )
b) Calculate the radius of the atom in the unit cell.
( solution : 1.249 Å )
c) Calculate the planar density for the plane (110) in atoms/ Å^2
( solution : 0.17 atoms/Å^2 )
d) Given that the density of the material is = 7.19 g/cm^3 , calculate the molecular weight of the metal
(NA=6.022 10^23 mol-1).
( solution : 51.97 g/mol )
