Tips to crack JEE mains : BCC, FCC & HCP

In this article you will get some Tips to crack JEE mains for BCC, FCC & HCP.

The crystalline structure, formed by the distribution of atoms, ions or molecules, is actually what constitutes the material base that forms the crystal. While the crystal lattice reflects the fact that the crystal is periodic and therefore determines the symmetry treated so far, the structure of the crystal not only determines its periodicity, marked by the network and by the unit cell of the same, but that determines the reason, that is to say, the material part constituted by atoms, ions and molecules that fill the mentioned unit cell.

There are three types of cubic lattices corresponding to three types of cubic close packing, as summarized in the following table. Now that the Kepler conjecture has been established,hexagonal close packing and face-centered cubic close packing, both of which have packing density of \eta = \pi /(3\sqrt{2})=0.74048  , are known to be the densest possible packings of equal spheres.

lattice type

basis vectors

packing density

simple cubic (SC)

\hat{x}, \hat{y}, \hat{z}

\frac{\pi }{6}\approx 52.3 %

face-centered cubic (FCC)

\frac{1}{2}(\hat{y} + \hat{z}), \frac{1}{2}(\hat{x} + \hat{z}), \frac{1}{2}(\hat{x} + \hat{y})

\pi /(3\sqrt{2})\approx 74.0 %

body-centered cubic (BCC)

\frac{1}{2}(-\hat{x}+\hat{y} + \hat{z}), \frac{1}{2}(-\hat{y}+\hat{x} + \hat{z}), \frac{1}{2}(-\hat{z}+\hat{y} + \hat{x})

\pi\sqrt{3}/8 \approx 68.0

Simple cubic packing consists of placing spheres centered on integer coordinates in Cartesian space.

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Arranging layers of close-packed spheres such that the spheres of every third layer overlay one another gives face-centered cubic packing. To see where the name comes from, consider packing six spheres together in the shape of an equilateral triangle and place another sphere on top to create a triangular pyramid. Now create another such grouping of seven spheres and place the two pyramids together facing in opposite directions.

If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand uniformly until running into each other, they form cubes, hexagonal prisms, and rhombic dodecahedra, respectively. In particular, if the spheres of face-centered cubic packing are expanded until they fill up the gaps, they form a solid rhombic dodecahedron, and if the spheres of hexagonal close packing are expanded, they form a second irregular dodecahedron consisting of six rhombi and six trapezoids known as the trapezo-rhombic dodecahedron. The latter can be obtained from the former by slicing in half and rotating the two halves 60 degree with respect to each other. The lengths of the short and long edges of the rotated dodecahedron have lengths 2/3 and 4/3 times the length of the rhombic faces. Both the rhombic dodecahedron and trapezo-rhombic dodecahedron are space-filling polyhedra.

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Types Of Crystal Structures

Cubic Structure Centered On The Body (BCC)

how to prepare for JEE advancedThis structure receives the designation of I and in it all the axes are equal to = b = c and their angles are 90º. The reticular parameter is a. If an atom is placed in each point of the network of cubic Bravais centered in the body, I have 8 atoms in the vertices and one in the interior. The central atom touches all the atoms of the vertices across the cubic of the cube. This number of atoms that touch the central atom are the immediate neighbors and is what is called the coordination number. Then in this structure NC = 8.

Face-centered cubic (FCC)

It is designated with the letter FCC. All the unit axes are equal and their angles are 90º, with the reticular parameter a. If we put an atom in each point of the cubic Bravais network centered on faces, we will have 6 atoms in the six centers of faces, in addition to the 8 of the vertices.

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Hexagonal Closest Packed (HCP)

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In the compact hexagonal structure the atoms occupy the vertices of a regular hexagonal prism, the centers of the bases and the centers of the alternating triangles in which the intermediate section of the prism can be decomposed. The axial lengths of this structure are the edge of the base, and the height of the prism.

The compact hexagonal structure is constructed from the network of Bravais called simple hexagonal, but associating to each node of the network not a single atom – the structure obtained in that case would not take advantage of space well – but a pair of atoms, located in positions 3 and 4, where 2 is the position of any node in the HS network.

Structures derived from the cubic structure of centered faces

The octahedral, tetrahedral and triangular positions provide regions that can be occupied by ions of opposite charge to those located in the normal atomic positions of the packaging, provided that they have the appropriate size.

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The derived structures that originate are:

Halite-like structure, NaCl: When in a cubic structure of centered faces all octahedral positions are occupied by equal atoms, but different from the original ones. The Cl and Na ions alternate in the three main directions of space.

The elementary cell is cubic with centered faces, and the structure can be described as two such networks, one of Cl- and one of Na +.

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Fluorite-like structure, CaF2: When in a cubic structure of centered faces all tetrahedral positions are filled by other atoms, identical to each other. Each Ca ++ is in the center of a cube whose vertices are occupied by F-. These, in turn, are in the center of a tetrahedron whose vertices are occupied by two Ca ++.

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The Ca ++ form a cubic cell with centered faces and the F- two other displaced cells. The structure has two types of coordination; 8 for Ca and 4 for F.

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This structure is suitable for 2: 1 stoichiometric compounds

Sphalerite type structure: The atoms that occupy the tetrahedral positions are chemically different from the one located at 000.

Structures derived from the hexagonal structure of centered faces

Analogous to compact cubic structures, octahedral, tetrahedral and triangular positions provide regions that can be occupied by ions of opposite charge to those located in normal atomic positions of the packaging, provided they are of adequate size. Thus, we can distinguish:

Niccolite-like structure, NiAs, or pyrrotin, FeS: When all coordination positions 6 in a hexagonal packaged structure are filled by atoms that are the same but different from those that constitute the basic structure.

It is the hexagonal equivalent structure of cubic ClNa.

Wurtzite type structure, SZn: It originates because the distribution of tetrahedral positions in hexagonal packaging is such that only half of them can be filled. This structural type allows the substitution of different atoms in the basic structure and even structures with atomic deficiencies appear.

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