The unit cell of a compound of Sodium (Na) and oxygen (O) is illustrated in the

Question

The unit cell of a compound of Sodium (Na) and oxygen (O) is illustrated in the fig below. How many atoms of each type of elements are in the unit cell? What is the Bravais lattice of this structure? Sketch the arrangement of atoms on the (110) and (111) planes of the structure and identify each atom in your sketch. A tensile stress is applied to a single-crystal fiber of Al_2O_3. The fiber is oriented such that the applied stress sigma, the long axis of the fiber, and the c-axis of the crystal are all parallel. Three possible slip planes for Al_2O_3, and their normals, are illustrated in the figure below: plane I, parallel to the c-axis (prismatic slip); plane 2, neither parallel nor perpendicular to the c-axis (pyramidal slip); and plane 3, perpendicular to the c-axis (basal slip). On which slip plane(s) will the resolved shear stress be highest ? On which slip plane(s) will the resolved shear stress be lowest!? Justify your answer.

Explanation / Answer

Q1 : (a) NaCl is face centered cubic cell. A face centered cubic unit cell contains atoms at all the corners of the crystal lattice and at the center of all the faces of the cube. The atom present at the face-center is shared between two adjacent unit cells and only 12 of each atom belongs to a unit cell. In case of Sodium and Chlorine, it is made up of 2 elements that is Na and Cl. This consists of one atom each in sodium and chlorine therefore making it 2 atoms in total. If trying to calculate number of atoms in 1 mole of sodium chloride, Avogadro’s number of atoms that is 6.023*10^23 atoms. This gives us conclusion of 1 mole of NaCl = 6.023*10^23 atoms each of Na and Cl, together would be 12.046*10^23 atoms.

(b) Bravais Lattice is a three dimensional lattice which tiles space without any gaps or holes. For Nacl, the features are as follows:

·         Chloride ions contains chloride ions at the corners and at the center of each face of the cube.

·         Sodium ions are so located that there are six chloride ions around it. Making the sodium ions occupy all the octahedral sites.

·         As there is only one octahedral site for every chloride ion, the stoichiometry is 1 : 1.

·         For sodium ions to occupy octahedral holes and the arrangement of chloride ions to be close packed the radius ratio, rNa+/rCl-, should be equal to 0.414. The actual radius ratio 0.525 exceeds this limit. To accommodate large sodium ions, the arrangement of chloride ions has to slightly open up.

·         It is obvious from the diagram that each chloride ion is surrounded by six sodium ions which are disposed towards the corners of a regular octahedron.

·         The structure of sodium chloride consists of eight ions a unit cell, four are Na+ ions and the other four are Cl- ions.

In this structure, each corner ion is shared between eight unit cells, each ion a face of the cell by two cells, each ion on an edge by four cells and the ion inside the cell belongs entirely to that unit cell.

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