Maximum possible numbers of two dimensional and three dimensional lattices are respectively

Options

A. 5 and 14
B. 7 and 14
C. 14 and 4
D. 5 and 13

Answer

To determine the maximum possible numbers of two-dimensional and three-dimensional lattices, we need to identify the Bravais lattices in each dimension.

### Step-by-Step Solution:

1. **Understanding Lattices**:
- A lattice is a regular arrangement of points in space. In two dimensions, we can visualize this as a flat grid, while in three dimensions, it extends into space.

2. **Two-Dimensional Lattices**:
- In two dimensions, there are five distinct types of Bravais lattices:
1. **Square Lattice**: Points are arranged in a square grid.
2. **Rectangular Lattice**: Points are arranged in a rectangular grid.
3. **Centered Rectangular Lattice**: Similar to the rectangular lattice but with additional points at the center of the rectangles.
4. **Hexagonal Lattice**: Points are arranged in a hexagonal pattern.
5. **Parallelogram Lattice**: Points are arranged in a parallelogram shape.

- Therefore, the maximum number of two-dimensional lattices is **5**.

3. **Three-Dimensional Lattices**:
- In three dimensions, there are fourteen distinct types of Bravais lattices:
1. **Simple Cubic**: Points are arranged in a cube.
2. **Body-Centered Cubic (BCC)**: Points at the corners of the cube and one in the center.
3. **Face-Centered Cubic (FCC)**: Points at the corners and the centers of each face of the cube.
4. **Simple Tetragonal**: Similar to cubic but with a different height.
5. **Body-Centered Tetragonal**: Similar to BCC but in a tetragonal shape.
6. **Simple Orthorhombic**: Points arranged in an orthorhombic shape.
7. **Base-Centered Orthorhombic**: Similar to orthorhombic but with additional points at the base.
8. **Face-Centered Orthorhombic**: Points at the corners and the centers of the faces of the orthorhombic shape.
9. **Simple Monoclinic**: Points arranged in a monoclinic shape.
10. **Base-Centered Monoclinic**: Similar to monoclinic but with additional points at the base.
11. **Rhombohedral**: Points arranged in a rhombohedral shape.
12. **Hexagonal**: Points arranged in a hexagonal prism shape.
13. **Triclinic**: Points arranged in a triclinic shape with no symmetry.
14. **Base Monoclinic**: Similar to monoclinic but with additional points at the base.

- Therefore, the maximum number of three-dimensional lattices is **14**.

4. **Final Answer**:
- The maximum possible numbers of two-dimensional and three-dimensional lattices are **5 and 14**, respectively.