Crystal Structures MCQ Quiz - Objective Question with Answer for Crystal Structures - Download Free PDF

Explanation:

The atomic packing factor is defined as the ratio of the volume occupied by the average number of atoms in a unit cell to the volume of the unit cell.

Mathematically, Atomic Packing Factor (APF):

APF \( = \frac{{{N_{atoms}} ~\times ~{V_{atoms}}}}{{{V_{unit\;cell}}}}\)

Characteristics of various types of structures are shown in the table below:

Calculation:

No of atoms in f.c.c unit cell = 4

\(APF = \frac{{{N_{atoms}}{V_{atom}}}}{{{V_{crystal}}}} = \frac{{4\left( {\frac{4}{3}} \right)\pi {r^3}}}{{{{\left( {a } \right)}^3}}}= \frac{{4\left( {\frac{4}{3}} \right)\pi {r^3}}}{{{{\left( {{{2√2r}}{}} \right)}^3}}}\)

for FCC a = 2√2 r where a is side of the cube and r is atomic radius.

APF = 0.74

Hexagonal closed pack structure:

• In a hexagonal closedpacked structure the third layer has the same arrangement of spheres as the first layer.
• Since the structure repeats itself after every two layers the stacking for hcp may be described as "ababab".
• The atoms in a hexagonal closed packed structure efficiently occupy 74% of space while 26% is empty space.
• Ex: Mg, Zn

Coordination Number and Number of Atoms Per Unit Cell:

• The hexagonal closed packed (hcp) structure has a coordination number of 12 and contains 6 atoms per unit cell.
• The facecentered cubic (fcc) structure has a coordination number of 12 and contains 4 atoms per unit cell.
• The bodycentered cubic (bcc) structure has a coordination number of 8 and contains 2 atoms per unit cell.
• The simple cubic structure has a coordination number of 6 and contains 1 atom per unit cell.

Crystal structure of Material

FCC: Ni Cu Ag Pt Au Pb Al Austenite or Ƴiron

BCC: V Mo Ta W Ferrite or αiron δferrite or δiron

HCP: Mg Zn

Cobalt: HCP < 420°C FCC > 420°C

Chromium: HCP < 20°C BCC > 20°C

Glass: Amorphous

Important Points

In the crystal structure the atomic packing factor (APF) or packing efficiency or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a dimensionless quantity and always less than unity.

The atomic packing factor of different crystal structures is given in the table below:

Hexagonal closed pack structure:

• In a hexagonal closedpacked structure the third layer has the same arrangement of spheres as the first layer.
• Since the structure repeats itself after every two layers the stacking for hcp may be described as "ababab".
• The atoms in a hexagonal closed packed structure efficiently occupy 74% of space while 26% is empty space.
• Ex: Mg, Zn

Coordination Number and Number of Atoms Per Unit Cell:

• The hexagonal closed packed (hcp) structure has a coordination number of 12 and contains 6 atoms per unit cell.
• The facecentered cubic (fcc) structure has a coordination number of 12 and contains 4 atoms per unit cell.
• The bodycentered cubic (bcc) structure has a coordination number of 8 and contains 2 atoms per unit cell.
• The simple cubic structure has a coordination number of 6 and contains 1 atom per unit cell.

Crystal structure of Material

FCC: Ni Cu Ag Pt Au Pb Al Austenite or Ƴiron

BCC: V Mo Ta W Ferrite or αiron δferrite or δiron

HCP: Mg Zn

Cobalt: HCP < 420°C FCC > 420°C

Chromium: HCP < 20°C BCC > 20°C

Glass: Amorphous

Important Points

In the crystal structure the atomic packing factor (APF) or packing efficiency or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a dimensionless quantity and always less than unity.

The atomic packing factor of different crystal structures is given in the table below:

Explanation:

Hexagonal Close-Packed (HCP) Crystal Structure

Definition: The hexagonal close-packed (HCP) crystal structure is one of the most efficient ways of arranging spheres in a solid, allowing atoms to pack together as closely as possible. In this structure, each atom has 12 nearest neighbors, and the atoms are packed in such a way that the distances between them are minimized.

Characteristics of various types of structures are shown in the table below:

Explanation:

Hexagonal Close-Packed (HCP) Crystal Structure:

• The hexagonal close-packed (HCP) crystal structure is one of the most efficient ways of packing spheres, where each atom is surrounded by twelve others. In the HCP structure, the atoms are packed closely together in a repeating pattern that forms a hexagonal lattice. This type of structure is characterized by its high density and coordination number, which is 12.

Structure Details:

• In the HCP crystal structure, the atoms are arranged in layers. The pattern follows an ABAB stacking sequence where each atom in the 'A' layer is directly above or below an atom in another 'A' layer, while the 'B' layer atoms fit into the depressions of the 'A' layers. This results in a hexagonal unit cell with two atoms per unit cell.

Examples of HCP Structures:

• Metals like magnesium (Mg), titanium (Ti), and zinc (Zn) are common examples of elements that crystallize in the HCP structure. Additionally, certain minerals exhibit the HCP structure due to their atomic arrangement.

Explanation:

If the atoms or atom groups in the solid are represented by points and the points are connected, the resulting lattice will consist of an orderly stacking of blocks or unit cells.

• The orthorhombic unit cell is distinguished by three lines called axes of twofold symmetry about which the cell can be rotated by 180° without changing its appearance.
• This characteristic requires that the angles between any two edges of the unit cell be right angles but the edges may be any length.

Important Points

There are 7 types of crystal systems:

Explanation:

The properties of a material depend not only on the bond strength but also on the arrangement of atoms.

The two types of solid based on the arrangement of their atoms are discussed in the table below.

Explanation:

Atomic Packing Factor:

Packing factor is the fraction of the volume of a unit cell that is occupied by "hard sphere" atoms or ions.

It is the sum of the sphere volumes of all atoms within a unit cell (assuming the atomic hard - sphere model) divided by the unit cell volume (total volume).

The atomic packing factor of different crystal structures is given in the table below:

Concept:

In the FCC unit cell effective number of atoms = 8 corner atoms x (1/8) (each atom is shared by 8-unit cells) + 6 face cantered atoms x (1/2) (each shared by two-unit cells) = 4

Slip systems: The combination of a slip plane and its direction of slip is known as slip system. Each pattern of atomic arrangements results in different number of slip systems.

• For BCC crystal structure: 48
• For FCC crystal structure: 12
• For Hexagonal close packed structure: 3

Explanation:

In the FCC unit cell effective number of atoms = 8 corner atoms x (1/8) (each atom is shared by 8-unit cells) + 6 faces cantered atoms x (1/2) (each shared by two-unit cells) = 4

Slip systems: The combination of a slip plane and its direction of slip is known as a slip system. Each pattern of atomic arrangements results in a different number of slip systems.

• For the BCC crystal structure: 48
• For FCC crystal structure: 12
• For Hexagonal close packed structure: 3

Additional InformationCo-ordination Number: 

• The coordination number of an atom is the number of atoms it is touching in a unit cell.

Explanation

BCC crystal structure: BCC stands for Body-Centered Cubic. In one unit cell, there is one atom at center, 1 atom at each corner. The crystal structure is used for Brittle materials only. 

AC = Body diagonal of the unit cell

a = Side of the unit cell

\(a√ 3 = 4r\)

\(r = \frac{{√ 3 a}}{4}\)

The correct answer is Glass.

Key Points

• The properties of a material depend not only on the bond strength but also on the arrangement of atoms.
• The two types of solid based on the arrangement of their atoms are discussed in the table below.

Explanation:

Atomic packing factor

In the crystal structure, the atomic packing factor (APF) or packing efficiency or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a dimensionless quantity and always less than unity.

The atomic packing factor of different crystal structures is given in the table below:

Co-ordination number: The coordination number of an atom is the number of atoms it is touching in a unit cell.

Concept:

In the unit cell effective number of atoms = corner atoms × (1/8) + face cantered atoms × (1/2) + 1 (Inner atom)

Concept:

• Cementite is a chemical compound of carbon & Iron. It is denoted by \({F_e}_3C\).
• Iron carbide is normally classified as a “ceramic”. i.e. it is hard, brittle, and insulator.
• It has an orthorhombic crystal structure.

Some other phrases & their crystal structure: -

α – Fe; Ferrite ⇒ BCC

γ  – Fe, Austenite ⇒ FCC

\({F_{{e_3}}}C\), cementite ⇒ Orthorhombic