Equivalence between Bravais lattice definitions

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Homework Statement
Prove that consider a Bravais lattice as in infinite array of discrete points with ann arrangemente and orientation that appears exactly the same, from whicever of the points the array is viewed, definition (a) from now on, implies that is can also be described as al points with position vectors ##\vec{R}=\sum_{i=1}^{3}n_{i}\vec{a}_{i}## where ##\vec a _ 1##, ##\vec a _ 2## and ##\vec a _ 3## are any three vectors not all in the same plane, and ##n_1##, ##n_2## and ##n_3## range throuhg all integral values.
Relevant Equations
No relevant equations for this one, only mathematical reasoning is necessary.
Proof. To demonstrate that a Bravais lattice (Figure 1) can be considered as a set of points located by a linear combination of primitive vectors of the lattice with integer coefficients, a sequence of claims in increasing order of complexity can be adopted. First, what is shown for one octant of Cartesian space will be valid for the others (hence only this octant is displayed in Figure 1, which does not affect the proof's development and facilitates diagram representation).
1700443628372.png


Figure 1.
Now, let's say an observer is at position O and looks towards position A. They will perceive it at a distance, let's say ##a_{1}## (Figure 2).
1700444830632.png

Figure 2.
By definition (a), this must also be the distance at which, being at A, they will perceive B (Figure 3).

1700445018473.png

Figure 3.

For the same reason, any point in that direction from some other point will also be separated by ##a_{1}## from a neighbor thus chosen.

Consider again that the observer is at O, but this time looking towards position ##A^{\prime}## (Figure 4).
1700445528965.png
Figure 4.
If ##A^{\prime}## is separated by ##a_{2}## from O, by the previous argument, every neighbor adjacent to another in that direction will also be.
And if this time the observer looks towards position ##A^{\prime\prime}## and perceives a spacing of ##a_{3}##, by the arguments presented earlier, this will be the separation of any two adjacent neighbors in that direction.
Note that all neighbors are separated from their adjacent neighbors in the three alignment directions by distances ##a_{1}##, ##a_{2}##, and ##a_{3}##. Therefore, every position in the lattice is determined by a linear combination of integers of the vectors ##\vec{a}_{i}## whose magnitude is given by ##a_{i}##, and whose direction is the alignment between neighbors.

"So, any advice? Is this proof good, or is there still something I haven't noticed? Thank you."
 

Related to Equivalence between Bravais lattice definitions

What is a Bravais lattice?

A Bravais lattice is a set of points in space arranged in such a way that the arrangement looks the same from any point in the lattice. It is a fundamental concept in crystallography, describing the repetitive, periodic array of points in three-dimensional space.

How many types of Bravais lattices are there?

There are 14 distinct types of Bravais lattices in three-dimensional space. These are categorized into seven crystal systems: cubic, tetragonal, orthorhombic, hexagonal, trigonal (rhombohedral), monoclinic, and triclinic.

What are the different definitions of a Bravais lattice?

Bravais lattices can be defined in multiple ways, such as through the geometric arrangement of points, symmetry operations, or translational vectors. These definitions are mathematically equivalent and describe the same periodic structure, but they emphasize different aspects of the lattice.

Why is it important to understand the equivalence between different definitions of Bravais lattices?

Understanding the equivalence between different definitions of Bravais lattices is crucial for consistency in crystallographic studies and materials science. It ensures that researchers and practitioners can communicate effectively and apply the correct symmetries and properties to analyze crystal structures.

How can one prove the equivalence between different definitions of Bravais lattices?

The equivalence between different definitions of Bravais lattices can be proven mathematically by showing that the same set of points can be generated using different approaches. This involves demonstrating that the translational symmetry, geometric arrangement, and symmetry operations lead to the same periodic structure.

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