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Chapter 12: Problem 20

Of the seven three-dimensional primitive lattices, (a) which one has a unit cell where the \(a\) and \(b\) lattice vectors form a base that is an arbitrary parallelogram (like the unit cell of a two-dimensional oblique lattice), while the \(c\) lattice vector is perpendicular to the other two? (b) What is the lattice if the \(a\) and \(b\) lattice vectors form a base that corresponds to the two-dimensional hexagonal unit cell and the \(c\) lattice vector is perpendicular to the other two?

Short Answer

Expert verified
(a) The lattice with an arbitrary parallelogram base formed by the $a$ and $b$ lattice vectors and a perpendicular $c$ lattice vector is the Simple Orthorhombic lattice (P). (b) The lattice with a 2D hexagonal base formed by the $a$ and $b$ lattice vectors and a perpendicular $c$ lattice vector is the Simple Hexagonal lattice (P).

Step by step solution

01

(a) Lattice with arbitrary parallelogram base and c vector perpendicular

To find the lattice that meets the conditions mentioned in (a), we must look for a lattice in which the angles between the a and b vectors can be arbitrary, and the c vector needs to be perpendicular to both the a and b vectors. Among the seven lattices listed above, the simple orthorhombic lattice (P) meets these conditions. In this lattice, the a and b vectors form an arbitrary parallelogram, and the c vector is perpendicular to both the a and b lattice vectors. So, the lattice mentioned in (a) is the Simple Orthorhombic lattice (P).
02

(b) Lattice with 2D hexagonal base and c vector perpendicular

To find the lattice that meets the conditions mentioned in (b), we must look for a lattice in which the a and b vectors form a base corresponding to the two-dimensional hexagonal unit cell, and the c vector needs to be perpendicular to both the a and b vectors. Among the seven lattices listed above, the simple hexagonal lattice (P) meets these conditions. In this lattice, the a and b vectors create a hexagonal structure (120-degree angles between vectors), and the c vector is perpendicular to both the a and b lattice vectors. So, the lattice mentioned in (b) is the Simple Hexagonal lattice (P).

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