Question

Determine, in terms of the radius of the atom, the distance between adjacent close packed planes in FCC, BCC, and HCP

Please explain them step by step, thanks in advance!

Answer 1

Ans: Atom is the smallest constituent unit of ordinary matter that constitutes a chemical element. Atoms are in plenty of different size -Atoms are mainly made of three particles:

Protons- nucleus of the atom with a +1 charge and a relative mass of one

Neutrons- nucleus of the atom with a charge of 0 and a relative mass of one

Electron- found in shells around the nucleus of the atom with a charge of -1 and relative mass of 1/1836

The atomic radius is defined as the one-half the distance between the nuclei of identical atom that are bonded together.

Atomic radius is generally defined as being the total distance from an atom’s nucleus to the outermost orbital of electron. In simpler words, it can be defined as something like  to the radius of a circle, where the center of the circle is the nucleus & the outer edge of the circle is the outermost orbital of electron.

The radius of an atom ranges from 0.05 to 0.25 nano-meters (0.5 × 10-10 m to 2.5 × 10-10 m).

Body-Centered Cubic (BCC) Structure :
The body-centered cubic unit cell has atoms at each of the 8 corners of a cube plus 1 atom in the center of the cube . Each of the corner atoms is the corner of another cube so the corner atoms are shared among 8 unit cells. It is said to have a coordination number of eight . The bcc unit cell consists of a net total of 2 atoms; 1 in the center and eight eighths from corners atoms as shown in the middle image below . The photo below highlights a unit cell in a larger section of the lattice.

the volume of atoms in a cell per the total volume of a cell is called packing factor. The bcc unit cell has a packing factor 0.68.Some of the materials that a bcc structure include lithium, sodium, potassium, chromium, barium, vanadium, alpha-iron & tungsten. Metals which have a bcc structure are usually harder and less malleable than close-packed metals such as gold. When the metal is deformed, the planes of atoms must slip over each other, & this is more difficult in the bcc structure.

Face Centered Cubic (FCC) Structure
The face centered cubic structure (Fcc) has atoms located at each of the corners and the centers of all the cubic faces (left image below). Each of the corner atoms is the corner of another cube so the corner atoms are shared among 8 unit cells. Additionally, each of its 6 face centered atoms is shared with an adjacent atom. Since 12 of its atoms are shared, it is said to have a coordination number of 12. The fcc unit cell consists of a net total of four atoms; eight eighths from corners atoms and 6 halves of the face atoms as shown in the middle image above.

in the fcc structure the atoms can pack closer together than they can in the bcc structure. The atoms from 1 layer nest themselves in the empty space between the atoms of the adjacent layer. To picture packing arrangement, imagine a box filled with a layer of balls that are aligned in columns and rows. When a few additional balls are tossed in the box, they will not balance directly on top of the balls in the first layer but instead will come to rest in the pocket created between four balls of the bottom layer. As more balls are added they will pack together to fill up all the pockets. The packing factor (the volume of atoms in a cell per the total volume of a cell) is 0.74 for fcc crystals. Some of metals that have the fcc structure include aluminum, copper, gold, iridium, lead, nickel, platinum and silver.

Hexagonal Close Packed (HCP) Structure
Another common close packed structure is the hexagonal close pack i.e HCP. The hexagonal structure of alternating layers is shifted so its atoms are aligned to the gaps of the preceding layer. The atoms from 1 layer nest themselves in the empty space between the atoms of the adjacent layer just like in the fcc structure. However, instead of being a cubic structure, the pattern is hexagonal. (See image below.) The difference between the HCP and FCC structure is discussed later in this section.

The hcp structure has 3 layers of atoms. In each the top and bottom layer, there are six atoms that arrange themselves in the shape of a hexagon and a seventh atom that sits in the middle of the hexagon. The middle layer has 3 atoms nestle in the triangular "grooves" of the top and bottom plane. Note that there are six of these "grooves" surrounding each atom in the hexagonal plane, but only three of them can be filled by atoms.
As shown in the middle image above, there are six atoms in the hcp unit cell. Each of the 12 atoms in the corners of the top and bottom layers contribute 1/6 of atom to the unit cell, the two atoms in the center of the hexagon of both the top and bottom layers each contribute ½ atom and each of the three atom in the middle layer contribute 1 atom. The image on the right above attempts to show several hcp unit cells in a larger lattice.
The coordination number of the atoms in this structure is 12. There are 6 nearest neighbors in the same close packed layer, three in the layer above and three in the layer below. The packing factor is 0.74, which is the same as the fcc unit cell. The hcp structure is very common for elemental metals and examples include beryllium, cadmium, magnesium, titanium, zinc and zirconium.