The atomic radius of Ga and As are 136 pm and 114 pm, respectively. Using hard sphere approximat

step 1 Okay, so for this question, they want us to find the radius of the atom in picometers, and they

The atomic radius of Ga and As are 136 pm and 114 pm,...

The atomic radius of Ga and As are $136 \mathrm{pm}$ and $114 \mathrm{pm}$, respectively. Using hard sphere approximation, find the lattice constant of GaAs and the volume of the primitive cell. $\left[1 \ A^{\circ}=100 \mathrm{pm}\right]$

Key Concepts

Crystal Structure (Zinc Blende Structure)

Gallium arsenide crystallizes in the zinc blende structure, which is characterized by a face-centered cubic lattice with a two-atom basis. This structure dictates the spatial arrangement of the different atoms and directly influences how the individual atomic radii contribute to the overall lattice parameters of the material.

Atomic Radius

The atomic radius is a measure of the size of an atom, typically defined as the distance from the nucleus to the outermost electron shell. In the context of crystallography, the atomic radius is used in models like the hard sphere approximation to determine how atoms fit together in a lattice, which in turn affects the lattice constant and packing efficiency.

Lattice Constant

The lattice constant is the fixed distance between equivalent points in a crystal lattice. It defines the periodicity of the atomic arrangement in the solid and is a fundamental parameter for understanding the dimensions and structure of crystalline materials, particularly in solid-state physics and materials science.

Hard Sphere Approximation

The hard sphere approximation is a simplifying model where atoms are considered as rigid, non-deformable spheres that touch each other without overlapping. This concept helps in estimating geometric parameters of a crystal structure, such as the lattice constant, by assuming that the atoms pack together exactly as these spheres would.

Primitive Cell

The primitive cell is the smallest volume in the crystal lattice that, when translated through space via lattice vectors, can fill the entire crystal without overlapping. It contains the minimum number of atoms needed to represent the symmetry and structure of the entire lattice and is crucial for calculations involving density and electronic properties.

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