Question

In the face centered cubic unit cell, the radius of atoms in terms of edge length \((a)\) of unit cell is: (a) \(a / 2\) (b) \(a / \sqrt{2}\) (c) \(a / 2 \sqrt{2}\) (d) \(\sqrt{3} a / 4\)

Short answer

The radius of atoms in terms of edge length \(a\) of a face-centered cubic unit cell is \(a / 2 \sqrt{2}\).

Step-by-step solution

Understand the Geometry of a Face-Centered Cubic Unit Cell

A face-centered cubic (fcc) unit cell consists of atoms located at each of the corners and at the center of each face of the cube. In an fcc unit cell, the diagonal passing through the face centers of the cube connects the centers of three atoms, with one atom at each end and one atom whose center is at the midpoint of this diagonal.

Relate the Diagonal to the Edge Length

The face diagonal can be represented in terms of the edge length (a) of the unit cell by using the Pythagorean theorem in two steps. First, considering the triangle formed by the edge of the cube and half the diagonal (since the centered atom splits the diagonal in half), we find that the half-diagonal (d/2) is the hypotenuse of a right-angled triangle with sides of length 'a' and 'a/2'. So, the relationship is a^2 + (a/2)^2 = (d/2)^2. The full diagonal is then d = a * sqrt(2).

Connect the Diagonal with the Radius of Atoms

The full diagonal through the face-centered atoms is the sum of the radii of the three atoms it passes through. Given that each atom's radius is the same, we write this as d = 4r. Substitute the expression from Step 2 into this equation to find the radius.

Solve for the Radius of the Atom

Combining the expressions from the previous steps, we have 4r = a * sqrt(2). Solving for 'r' gives r = (a * sqrt(2))/4.

Simplify the Expression

The expression for the radius can be further simplified to r = (sqrt(2)/4) * a. However, the answer choices are in terms of 'a', we can rewrite this as r = (a)/(2 * sqrt(2)), which matches one of the given answer choices.

Key concepts

Face-Centered Cubic Geometry

The face-centered cubic (fcc) structure is prevalent among elements, especially metals, due to its efficient packing and high coordination number. In this type of geometry, an atom sits at each corner of a cube and also at the center of each of its six faces. Imagine a cube with a ball snugly fitted at each corner and additional balls glued to the center of each face; the latter are shared between two adjacent cells. What's particularly noteworthy about this arrangement is that each corner atom is shared among eight unit cells, and each face-centered atom is shared between two. This sharing means that each unit cell effectively contains four whole atoms: 1/8 of each of the eight corner atoms and 1/2 of each of the six face-centered atoms.

The fcc geometry is not just space-efficient, but it also influences the material's properties, such as density and slip systems in crystalline solids. Indeed, the configuration of atoms even affects how materials like metals deform under stress, due to the sliding of planes of atoms over one another—this behavior is central to the study of materials science and engineering.

Unit Cell Edge Length

In the landscape of crystallography, the edge length of a unit cell is akin to the fundamental measurement governing the dimensions of the entire crystal lattice. For a cubic system, this edge length is denoted by 'a'. It is the distance from one corner to an adjacent corner along an edge. Think of it like the baseline of a square, which serves as a starting point for scaling the entire structure up to its full size.

The significance of the edge length in an fcc lattice is that it relates directly to the arrangement and radius of the atoms within the cell. Indeed, knowledge of the edge length allows one to calculate other important crystallographic parameters, such as the cell's volume, the atomic packing factor — which is a measure of how tightly packed the atoms are in the crystal structure — and the distance between nearest-neighbor atoms. Understanding these dimensions is crucial for determining the properties of materials, including their density, how they interact with light, and how they conduct electricity.

Pythagorean Theorem in Crystallography

The Pythagorean theorem isn’t just a staple of high school geometry; it's a powerful tool in crystallography as well. This theorem allows us to relate the face-diagonals and body-diagonals of a crystal’s unit cell to the edge length 'a'. In the context of a face-centered cubic (fcc) cell, where a diagonal is drawn across the face of the cube, intersecting the face-centered atoms, we're looking at a right-angled isosceles triangle.

Through the theorem, we can state that the square of the hypotenuse (the face-diagonal) is equal to the sum of the squares of the other two sides — in this case, the edge of the unit cell and half of its edge length. Formulated mathematically, if 'd' represents the diagonal and 'a' the edge length, the relationship is \(a^2 + (a/2)^2 = (d/2)^2\). Such geometric relationships are key in determining distances between atoms and within crystal lattices. When the dimensions are too small to measure directly, the Pythagorean theorem becomes an indispensable part of a crystallographer's toolkit to infer these measurements.