What's the difference between lattice vectors and basis vectors?

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SUMMARY

The discussion clarifies the distinction between lattice vectors and basis vectors in crystallography. A lattice vector is defined as \vec{R} = n_1\vec{a}_1 + n_2\vec{a}_2 + n_3\vec{a}_3, where \vec{a}_1, \vec{a}_2, and \vec{a}_3 are the basis vectors, which are linearly independent. While basis vectors can be any linearly-independent trio, they also serve as lattice vectors. The set of lattice vectors represents all lattice points in space, emphasizing the foundational role of basis vectors in constructing lattice vectors. For further reading, Ashcroft and Mermin is recommended as a more comprehensive source than Kittel.

PREREQUISITES
  • Understanding of crystallography concepts
  • Familiarity with vector mathematics
  • Knowledge of linear independence in vector spaces
  • Basic principles of solid-state physics
NEXT STEPS
  • Study the differences between lattice and basis vectors in detail
  • Explore crystallography textbooks, specifically Ashcroft and Mermin
  • Learn about the construction of lattice points from basis vectors
  • Investigate applications of lattice vectors in solid-state physics
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Students and professionals in physics, materials science, and crystallography, particularly those seeking to deepen their understanding of lattice structures and vector relationships in solid materials.

Raziel2701
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Google has not been very useful, and Kittel has too little on crystallography. Actually, what's a good source on crystallography?
 
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A lattice vector is \vec{R} = n_1\vec{a}_1 + n_2\vec{a}_2 + n_3\vec{a}_3 where a1, a2 and a3 are the basis vectors (n's are integers). Generally there are three basis vectors, these form a linearly independent set from which you can construct any lattice vectors. The set of lattice vectors is the set of all lattice points in space. Of course, the basis vectors are also lattice vectors. Any linearly-independent trio of lattice vectors could be chosen as basis vectors. Or it could be a pair if you have a two-dimensional lattice.

Check Ashcroft and Mermin.. I think it has a little more than Kittel.
 

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