Relation between direct and reciprocal lattice
Understanding the Relationship Between Direct and Reciprocal Lattice
Introduction to Direct and Reciprocal Lattice
- The discussion begins with an overview of the relationship between direct and reciprocal lattices, highlighting their differences.
- Every crystal structure is associated with two types of lattices: direct (real) lattice and reciprocal lattice. The direct lattice represents the geometric pattern of a crystal where atomic sites are depicted as points.
Characteristics of Direct Lattice
- In a non-periodic lattice, points are not equivalent, while in a periodic lattice, separation between atoms remains consistent throughout the structure.
- The direct lattice is characterized by its microscopic structure, which maps atomic arrangements with separations on the order of a few angstroms.
Characteristics of Reciprocal Lattice
- The reciprocal lattice serves as a map for diffraction patterns from crystals; it reflects how waves propagate through the medium.
- The smallest volume unit in reciprocal space is known as the primitive unit cell or Brillouin zone. This contrasts with the various unit cells found in direct lattices.
Geometric Relationships
- To construct the Wigner-Seitz cell in reciprocal space, lines are drawn connecting each lattice point to nearby points; planes are then drawn at midpoints normal to these lines.
- An animated image illustrates how X-rays diffracted by periodic atomic structures create regular light patterns that represent reciprocal lattices.
Deriving Relations Between Direct and Reciprocal Lattices
- The derivation starts by examining relationships between crystallographic axes (a, b, c), focusing on primitive unit cells defined by specific interplanar spacings.
- Volume calculations for primitive unit cells involve determining base area multiplied by height; this leads to vector relations involving cross products.
Mathematical Formulations
- Key equations relate volume and area using vector forms; these include expressions for calculating magnitudes of reciprocal vectors based on interplanar spacing.
Understanding Reciprocal Lattice Vectors
Definition and Construction of c* Vector
- The c* vector is defined as perpendicular to the a-b plane, represented by the normal along the c-axis. It is expressed mathematically as c^* = m cdot (a times b) / (a cdot (b times c)), where m is a scale factor in X-ray diffraction.
Relationships Between Lattice Vectors
- When m is set to 1, three equations emerge:
- a^* = (b times c) / (a cdot (b times c))
- b^* = (c times a) / (a cdot (b times c))
- c^* = (a times b) / (a cdot (b times c))
- These equations illustrate the relationship between reciprocal lattice vectors and direct lattice vectors.
Geometrical Representation
- A figure illustrates the geometrical construction linking direct unit cells with reciprocal unit cells. The normals drawn to planes such as BC, AC, and AB form another unit cell in reciprocal space.
Conditions for Reciprocal Lattice Vectors
- Specific conditions are outlined for reciprocal lattice vectors:
- a^* is orthogonal to both b and c: thus, a^* cdot b = 0, and a^* cdot c = 0.
- Similar conditions apply for vectors b^* and c^*.
Dot Product Relations
- The dot product relationships among these vectors are established:
- For example, the dot products yield results like:
- b^* cdot a = 0
- Each vector's self-dot product equals one: e.g., a^* cdot a^* = 1.
Proof of Relationships
- The proof involves substituting reciprocal lattice vectors into their definitions. For instance: