Finding Components of Vectors in a Crystal Lattice

In summary, the problem involves finding the components of three vectors (a, b, c) whose lengths are all 10.5 Angstron and have angles of 109.5 degrees between them. One vector can be arbitrary and the other two can be determined using trigonometric functions and spherical trigonometry. The solution involves solving a spherical right triangle.
  • #1
Eduard1
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Homework Statement



Be the vectors a, b, c such as:

| a | = | b | = | c | = 10.5 Angstron

The angles between these vectors are:

alpha = beta = gamma = 109.5 degree

These vectors represent the lattice vectors of a crystal.

Find out their components (a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3).

Homework Equations





The Attempt at a Solution



So, we know the length of the vectors and the angle between them. We want to find out the components of each vector. I know one has to use the trigonometric functions, but I am not sure how. Should I project the vector a on b (to find a_2) and on c (to find a_3) ? Or how do I find the components of a ?

Thank you very much for any fast answer/hint/suggestion,
Eduard
 
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  • #2
Looks like one of the vectors can be arbitrary, so let a = {10.5, 0.0, 0.0}. Vector b can be assumed to lie in the 1-2 plane without loss of generality, so let b = {10.5 cos(109.5), 10.5 sin(109.5), 0.0}.

You can find vector c using spherical trigonometry. The most expedient way appears to be to solve the spherical right triangle consisting of the head of vector a, the head of vector c, and a point half-way between the heads of vectors a and b. Hope that helps!
 

FAQ: Finding Components of Vectors in a Crystal Lattice

What is a crystal lattice?

A crystal lattice is a repeating, three-dimensional arrangement of atoms, ions, or molecules in a crystalline solid. It is the fundamental structural unit of a crystal.

How are vectors represented in a crystal lattice?

Vectors in a crystal lattice are represented by connecting the points where the atoms, ions, or molecules are located. The direction of the vector is determined by the direction of the line connecting the points, and the magnitude is determined by the distance between the points.

What is the process for finding the components of vectors in a crystal lattice?

The process involves breaking down the vector into its components along the three crystallographic axes (a, b, and c). This can be done mathematically using trigonometric functions or graphically by constructing right triangles.

Why is it important to find the components of vectors in a crystal lattice?

Finding the components of vectors in a crystal lattice allows us to understand the physical properties and behavior of crystals. It also helps in determining the direction and magnitude of forces acting on the crystal, which is crucial in many scientific applications.

What are some common methods for finding the components of vectors in a crystal lattice?

Two common methods are the geometric method, which involves constructing right triangles and using trigonometric functions, and the analytical method, which uses mathematical equations to calculate the components. Other methods include using computer programs and X-ray diffraction techniques.

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