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Chapter 4: Problem 1

Show that two lattice vectors, \(\left[U_{1} V_{1} W_{1}\right]\) and \(\left[U_{2} V_{2} W_{2}\right]\), lie in a plane parallel to \((h k l)\), where \(h=V_{1} W_{2}-V_{2} W_{1}\), \(k=W_{1} U_{2}-W_{2} U_{1}\) and \(l=U_{1} V_{2}-U_{2} W_{1} .\) Similarly show that the intersection of two lattice planes, \(\left(h_{1} k_{1} l_{1}\right)\) and \(\left(h_{2} k_{2} l_{2}\right)\), is a vector parallel to \([U V W]\), where \(U=\) \(k_{1} l_{2}-k_{2} l_{1}, V=l_{1} h_{2}-l_{2} h_{1}\) and \(W=h_{1} k_{2}-h_{2} k_{1}\).

Short Answer

Expert verified
Both lattice vectors lie in a plane parallel to \((h k l)\). The intersection of the planes is a vector parallel to \([U V W]\).

Step by step solution

01

Understand the Given Vectors and Plane

Understand that you have two lattice vectors \(\left[U_{1} V_{1} W_{1}\right]\) and \(\left[U_{2} V_{2} W_{2}\right]\) and a plane defined by the Miller indices \((h k l)\). The goal is to demonstrate that these vectors lie in a plane parallel to \((h k l)\).
02

Compute the Dot Product

Compute the dot product of each lattice vector with the normal to the plane. The normal vector to the plane \((h k l)\) is \((h, k, l)\). Compute \(U_{1}h + V_{1}k + W_{1}l\) and \(U_{2}h + V_{2}k + W_{2}l\).
03

Substitute the Given Values of h, k, and l

Substitute \(h = V_{1}W_{2} - V_{2}W_{1}\), \(k = W_{1}U_{2} - W_{2}U_{1}\), and \(l = U_{1}V_{2} - U_{2}W_{1}\) into the dot products calculated in the previous step.
04

Show Dot Product Results in Zero

For both vectors, demonstrate that the dot products result in zero, confirming they lie in a plane parallel to \((h k l)\):\(U_{1}(V_{1}W_{2} - V_{2}W_{1}) + V_{1}(W_{1}U_{2} - W_{2}U_{1}) + W_{1}(U_{1}V_{2} - U_{2}V_{1}) = 0\) and\(U_{2}(V_{1}W_{2} - V_{2}W_{1}) + V_{2}(W_{1}U_{2} - W_{2}U_{1}) + W_{2}(U_{1}V_{2} - U_{2}V_{1}) = 0\).
05

Understand the Given Planes

Understand that you now have two lattice planes defined by \((h_{1} k_{1} l_{1})\) and \((h_{2} k_{2} l_{2})\). The goal is to demonstrate that their intersection results in a vector parallel to \([U V W]\).
06

Compute Cross Product of Planes' Normals

Compute the cross product of the normals to these planes, \((h_{1}, k_{1}, l_{1})\) and \((h_{2}, k_{2}, l_{2})\), to find a direction vector \((h_{1}, k_{1}, l_{1}) \times (h_{2}, k_{2}, l_{2})\).
07

Substitute the Values to Find the Intersection Vector

Substitute \(U = k_{1} l_{2} - k_{2} l_{1}\), \(V = l_{1} h_{2} - l_{2} h_{1}\), and \(W = h_{1} k_{2} - h_{2} k_{1}\) into the cross product to find the intersection vector.
08

Confirm Parallelism

Verify that the intersection vector aligns with the direction vector by comparing components to confirm they are scalar multiples, ensuring the confirmation of parallelism: \([U V W]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

lattice vectors
In crystallography, lattice vectors describe the positional coordinates of points in a crystal lattice. Think of them as arrows that point from the origin of the lattice to any other point within it. These vectors are usually denoted by three components \([U, V, W]\) representing coordinates in three-dimensional space.

**Key Points about Lattice Vectors:**
  • Lattice vectors are fundamental in defining the geometry of crystal structures.
  • They are used to describe both the position of atoms and the periodicity of the crystal.
  • Parallel lattice vectors lie in the same plane, crucial for understanding crystal symmetry.
For the given exercise, understanding how these vectors relate to crystal planes can help establish how they parallel certain planes, determined by specific indices. This relationship leads us into the next important concept: Miller indices.
Miller indices
Miller indices are a notation system in crystallography to describe the orientation of planes within a crystal lattice. A plane in a lattice is defined by three indices \((h, k, l)\), which are derived from the inverse of the intercepts of the plane with the crystallographic axes.

**Important Features of Miller Indices:**
  • They help in identifying and categorizing different planes and directions in a crystal.
  • Miller indices are integral for predicting and explaining diffraction patterns.
  • The notation \((h, k, l)\) signifies the plane’s orientation relative to the unit cell axes. This is crucial for understanding material properties.
For instance, in solving the exercise, we use the Miller indices to determine the normal vector of a plane. The given indices are substituted into equations, establishing connections between lattice vectors and their orientation relative to crystallographic planes.
plane equation
The plane equation in crystallography is derived from the Miller indices and is used to describe the geometric orientation of crystal planes.

**Core Elements of Plane Equations:**
  • The equation of a plane in the form \((h, k, l)\) is \(\text{hx} + \text{ky} + \text{lz} = 0\). This form helps in locating the plane in three-dimensional space.
  • The direction vector or normal vector to the plane is \((h, k, l)\). This normal vector is perpendicular to every vector lying on the plane.
  • In the exercise, we compute the dot product between lattice vectors and the plane’s normal vector to check for parallelism.
By substituting given values and showing that this dot product equals zero, we confirm that the lattice vectors are parallel to the described plane. This approach is essential when determining relationships between different crystallographic features.

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Most popular questions from this chapter

Show that the volume of the reciprocal unit cell is equal to \((2 \pi)^{3} / V_{\text {cell }}\), where \(V_{\text {cell }}\) is the volume of the unit cell. ( Note: there are several ways of approaching this, one being the brute-force manipulation of equations, and the other is to write the components of the lattice vectors as a \(3 \times 3\) matrix, to note that the determinant of the matrix is the volume enclosed by the three vectors, and to use the mathematical result for square matrices \(\mathbf{A}\) and \(\mathbf{B}\) that \(\operatorname{det}[\mathbf{A} \times \mathbf{B}]=\operatorname{det}[\mathbf{A}] \times \operatorname{det}[\mathbf{B}] .)\)

Show that the reciprocal lattice points of a \(C\)-centred lattice are restricted to the condition \(h+k=2 n\), where \(n\) is an integer. Generalize this result for \(A\) and \(B\)-centred lattices.
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