Final answer:
The chiral properties of a (9,8) carbon nanotube are derived from its chiral vector, angle, and diameter, which determine its conductivity as either metallic or semiconducting. Metamaterials possess unique optical characteristics that allow for manipulation of electromagnetic waves beyond the capabilities of natural materials.
Step-by-step explanation:
Estimating the chiral properties of a (9,8) carbon nanotube involves calculating its chiral vector, chiral angle, and tube diameter, which determine its electrical conductivity properties. The chiral vector C for a nanotube is defined as C = n·a1 + m·a2, where n and m are integers, and a1 and a2 are the unit vectors of the hexagonal graphene lattice. For a (9,8) nanotube, the chiral vector would have coordinates based on 9·a1 and 8·a2.
The chiral angle θ can be estimated from the equation cos(θ) = (2n + m) / (2√(n² + nm + m²)). Using this formula, we can determine whether a nanotube displays metallic or semiconducting behavior, as the electronic properties are dependent on the diameter and chiral angle. If the chiral angle results in a value that makes (n - m) a multiple of 3, the nanotube is metallic; otherwise, it is semiconducting.
The nanotube's diameter can be calculated using the equation d = √3 · a0 · √(n² + nm + m²) / π, where a0 is the carbon-to-carbon distance in the graphene sheet, approximately 0.142 nm. This diameter is critical in understanding the electronic properties mentioned in the statement regarding the behavior of carbon nanotubes as conductors or semiconductors.
Metamaterials are artificial materials engineered to have properties that may not be found in naturally occurring materials, particularly optical characteristics such as negative refraction, which can lead to applications like superlenses and cloaking devices. The optical characteristics of metamaterials stem from their ability to manipulate electromagnetic waves in unusual ways, typically achieved through structures smaller than the wavelength of the waves they affect.