Question

Show that the angle between any two of the lines (bonds) joining a site of the diamond lattice to its four nearest neighbors is \(\cos ^{-1}(-1 / 3)=109^{\circ} 28\) '

Short answer

The angle between any two bonds in a diamond lattice structure is indeed approximately 109.47 degrees, which is equivalent to 109 degrees 28 minutes.

Step-by-step solution

Understand the Diamond Lattice

The diamond lattice is a crystal structure where each carbon atom (site) is bonded to four other carbon atoms, forming a tetrahedral shape around each carbon atom. These bonds (lines) represent the edges of the tetrahedron.

Determine the Position Vectors

To calculate the angle, we need to know the position vectors of the nearest neighbors relative to the central atom. In a tetrahedron, if we take the central atom at the origin, the position vectors to the four vertices can be taken as \[\vec{A} = \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right),\]\[\vec{B} = \left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}\right),\]\[\vec{C} = \left(\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}\right),\]and \[\vec{D} = \left(-\frac{1}{2}, \frac{1}{2}, -\frac{1}{2}\right)\]respectively (scaled to suit the bond length).

Use the Dot Product

The angle between two vectors can be found using the dot product formula:\[\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)\]where \(\theta\) is the angle between the vectors. For normalized vectors, the dot product is simply the cosine of the angle between them, since the magnitudes are 1.

Calculate the Angle Between Two Vectors

Taking the dot product of any two distinct position vectors, say \(\vec{A}\) and \(\vec{B}\):\[\vec{A} \cdot \vec{B} = \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) \cdot \left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}\right) = -\frac{1}{4} - \frac{1}{4} + \frac{1}{4} = -\frac{1}{4}\]We now find the cosine of the angle using \[\cos(\theta) = -\frac{1}{4}\]

Compute the Angle

Since the magnitude of each position vector is \(\frac{\sqrt{3}}{2}\) (from their components), the above dot product reduces to:\[\cos(\theta) = -\frac{1}{3}\]Hence, the angle \(\theta\) between the bonds is therefore:\[\theta = \cos^{-1}\left(-\frac{1}{3}\right)\]

Verify the Angle Value

Using a calculator or trigonometric tables to find \(\cos^{-1}\left(-\frac{1}{3}\right)\), we confirm that the angle is indeed approximately 109.47 degrees, which is often rounded to 109 degrees 28 minutes, agreeing with the given value of \(\cos^{-1}(-\frac{1}{3}) = 109^\circ 28'\).

Key concepts

Crystallography

Crystallography is the scientific study of crystal structures and their properties. It is a foundational discipline in materials science that involves analyzing the arrangement of atoms in a crystalline solid. A crystal lattice, such as the diamond lattice, is a three-dimensional arrangement of points in space where each point represents the position of an atom or molecule. The symmetry, spacing, and orientation of these points determine the physical properties of the material.

When studying crystal structures, we commonly examine the unit cell, which is the smallest repeating unit of the lattice that, when repeated in all three dimensions, creates the whole crystal. In the case of the diamond lattice, each unit cell exhibits a high degree of symmetry, which is characteristic of the tetrahedral coordination inherent to this crystal structure. The understanding of how atoms are bonded in a particular structure allows scientists and engineers to predict and manipulate material properties for various applications.

Tetrahedral Coordination

Tetrahedral coordination refers to a spatial configuration where a central atom is bonded to four neighboring atoms, positioned at the corners of a tetrahedron. This geometric shape is highly symmetrical and is one of the closest packings of spheres in three-dimensional space.

In a diamond lattice, each carbon atom is at the center of a tetrahedron, with bonds extending to four other carbon atoms. These tetrahedral units are the building blocks of the crystal, providing the lattice with its remarkable hardness and thermal conductivity. The angle between any two bonds in a perfect tetrahedron is approximately 109.47 degrees, known as the tetrahedral angle. The calculation of this angle using vector analysis is fundamental in understanding the spatial geometry of the tetrahedron.

Dot Product in Vector Analysis

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In geometric terms, the dot product of two vectors can be interpreted as the product of the magnitudes of the two vectors and the cosine of the angle between them.

Mathematical Formulation of the Dot Product

Given two vectors \(\vec{A}\) and \(\vec{B}\), the dot product is defined as:\[\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)\]where \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\), and \(|\vec{A}|\) and \(|\vec{B}|\) are the magnitudes of vectors \(\vec{A}\) and \(\vec{B}\) respectively. This operation is fundamental in the field of vector analysis and is widely used in physics and engineering to determine the angle between vectors in space.

Spatial Geometry

Spatial geometry, also known as three-dimensional geometry, is the study of shapes and figures in three-dimensional space. It examines positions, dimensions, shapes, and the properties of space occupied by these figures. Key concepts in spatial geometry include points, lines, planes, angles, and various three-dimensional figures such as cubes, spheres, pyramids, and tetrahedra.

Understanding spatial geometry is crucial when working with crystal structures, as the properties of these materials are often directly related to the geometric arrangement of their atoms. For example, the diamond lattice structure is directly linked to the spatial geometry of tetrahedra that carbon atoms form. Analyzing the spatial relationships and angles within such a lattice involves applying principles of geometric theory and vector analysis.