Rank the crystal lattice structures in order of decreasing efficiency of space in the structure?

Face centered cubic, body centered cubic, simple cubic, hexagonal close packing.

To determine the efficiency of space in crystal lattice structures, we need to consider their packing arrangements and the number of atoms per unit cell.

The most efficient structure in terms of space is the one with the highest packing efficiency, which is a measure of how closely the atoms are packed together in the lattice.

Let's examine each crystal lattice structure and rank them in order of decreasing efficiency of space:

1. Hexagonal close packing (HCP):
HCP is the most efficient crystal lattice structure. It consists of two interlaced hexagonal layers, forming a close-packed structure. The HCP structure has a packing efficiency of 74%.

2. Face-centered cubic (FCC):
FCC is the second most efficient structure. It consists of a cubic arrangement of atoms with additional atoms at the face centers. The FCC structure has a packing efficiency of 74%.

3. Body-centered cubic (BCC):
BCC is less efficient than FCC and HCP, but more efficient than the simple cubic structure. It consists of a cubic arrangement of atoms with one additional atom placed in the center. The BCC structure has a packing efficiency of 68%.

4. Simple cubic (SC):
Simple cubic is the least efficient structure among the four. It is the basic cubic arrangement with no additional atoms. The SC structure has a packing efficiency of 52%.

Therefore, the ranking of these crystal lattice structures based on decreasing efficiency of space is as follows:

1. Hexagonal close packing (HCP)
2. Face-centered cubic (FCC)
3. Body-centered cubic (BCC)
4. Simple cubic (SC)

These rankings are based on the highest to lowest packing efficiencies of the respective lattice structures.

The order of crystal lattice structures in decreasing efficiency of space is as follows:

1. Hexagonal close packing (HCP): HCP has the highest efficiency of space among the given lattice structures. In HCP, the spheres are packed in a hexagonal arrangement, with each layer alternatingly aligned above and below the previous layer.

2. Face-centered cubic (FCC): FCC has the second highest efficiency of space. In FCC, the spheres are arranged in a cubic lattice, with additional spheres positioned at the center of each face of the cube.

3. Body-centered cubic (BCC): BCC has a slightly lower efficiency of space compared to FCC. In BCC, the spheres are arranged in a cubic lattice, with an additional sphere positioned at the center of the cube.

4. Simple cubic (SC): SC has the lowest efficiency of space among the given lattice structures. In SC, the spheres are arranged in a simple cubic lattice, with equal spacing between each sphere.

Therefore, the rank of crystal lattice structures in the decreasing efficiency of space is HCP > FCC > BCC > SC.

I think you can find the answers here.

http://departments.kings.edu/chemlab/vrml/packgeo.html