Deduce primitive lattice vectors from position vector.

In summary, the position vector given can be written as a linear combination of the primitive lattice vectors a1, a2, and a3, where a1 = a/10 x, a2 = a/5 y, and a3 = a z. The coefficients for each vector can be found by representing the position vector as c1 a1 + c2 a2 + c3 a3, where c1 = 10n1 + 9n2 + 19n3, c2 = 6(n2 + n3), and c3 = 2n3. This approach can help in finding the primitive lattice vectors.
  • #1
maqdah
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Homework Statement


given the following position vector:

R = (10n1 + 9n2 + 19n3)(a/10) x + 6(n2+n3)(a/5) y + 2(n3)a z

where n1, n2 and n3 are integers
Find the primitive lattice vectors.

Homework Equations


any position vector of a lattice point is of the type
R= c1 a1 + c2 a2 + c3 a3;
and a position vector like the one showed above is a linear combination of the primitive lattice vectors a1, a2 and a3.

The Attempt at a Solution



I think I solved the question correctly, but my intuition tells me its wrong:

we can do the following:
a1 = a/10 X
a2 = a/5 Y
a3 = a Z

In our case, since n1,n2 and n3 are just integers:

c1 = 10n1 + 9n2 + 19n3
c2 = 6(n2 + n3)
c3 = 2n3
 
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  • #2
Any hint in the right direction is helpful?
 

FAQ: Deduce primitive lattice vectors from position vector.

1. What is a primitive lattice vector?

A primitive lattice vector is the smallest set of vectors that can be used to describe the entire lattice structure. It is the basis for the lattice and can be used to construct all other lattice vectors.

2. How do you determine primitive lattice vectors?

To determine primitive lattice vectors, you need to start with a known set of position vectors of the lattice points. Then, you can use the basis transformation matrix to convert these position vectors into primitive lattice vectors.

3. What is the basis transformation matrix?

The basis transformation matrix is a matrix that allows you to convert between different sets of vectors that describe the same lattice. It is used to transform position vectors into primitive lattice vectors and vice versa.

4. Can primitive lattice vectors be calculated for any type of lattice?

Yes, primitive lattice vectors can be calculated for any type of lattice, including simple cubic, body-centered cubic, and face-centered cubic lattices. The process for determining the primitive lattice vectors may vary slightly depending on the type of lattice, but the concept remains the same.

5. Why is it important to deduce primitive lattice vectors from position vectors?

Deducing primitive lattice vectors from position vectors is important because it allows you to fully describe the lattice structure in a more compact and efficient way. It also allows for easier calculations and analysis of lattice properties, such as lattice parameters and interatomic distances.

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