Solving Algebraic Relations of Reciprocal Lattice Vectors

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In summary, in this conversation, the speaker is struggling to prove the relation between reciprocal lattice vectors a', b', and c' and their corresponding vectors a, b, and c. They discuss using component expansion and Levi-Civita symbols, but eventually use the fact that a'×b'=b'×c'=c'×a'=1 to show that a, b, and c are proportional to b'×c', c'×a', and a'×b', respectively. They also use the identity (A×B)×(C×D)=(A·B×C)D−(A·B×D)C to prove the relation. Ultimately, they are able to prove the relation by
  • #1
dingo_d
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Homework Statement



Basically I have reciprocal lattice vectors:

[tex]a'=\frac{b\times c}{a\cdot(b\times c)}[/tex]
[tex]b'=\frac{c\times a}{a\cdot(b\times c)}[/tex]
[tex]c'=\frac{a\times b}{a\cdot(b\times c)}[/tex]

And I have to prove that these relations hold:

[tex]a=\frac{b'\times c'}{a'\cdot(b'\times c')}[/tex]
[tex]b=\frac{c'\times a'}{a'\cdot(b'\times c')}[/tex]
[tex]c=\frac{a'\times b'}{a'\cdot(b'\times c')}[/tex]

The Attempt at a Solution



I really dk where to start :\

Do I try with the direct component expansion or can I do it with Levi-Civita symobol:

[tex]a\cdot(b\times c)=\varepsilon_{ijk}a_ib_jc_k[/tex] and [tex]b\times c=\varepsilon_{ijk}b_jc_ke_i[/tex]

And then it would be:

[tex]a'=\frac{\varepsilon_{ijk}b_jc_ke_i}{\varepsilon_{ijk}a_ib_jc_k}[/tex]

but what can I do with it?
 
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  • #2
from given data u can find thata`xb`=b`xc`=c`xa`=1so they are a set of orthogonal vectors

now use this to solve the required thing
 
  • #3
Well I found in Arfken Weber that this is used in reciprocal lattice. Where it's also the information you gave.

In my problem I only have the three primed vectors [tex]a',\ b',\ c'[/tex] with the given forms and I have to proove the following. There is no mentioning of that information, not even a hint :\
 
  • #4
Note that

[tex] a\cdot b' = a \cdot c' =0,[/tex]

so we can write

[tex] a = \alpha ( b'\times c')[/tex]

for some scalar [tex]\alpha[/tex]. Find similar expressions for b and c, using the symmetry to relate the scalars. You can compute the proportionality by computing

[tex] a\cdot b\times c.[/tex]

You will need the identity

[tex] (A\times B)\times (C\times D) = (A\cdot B\times C)D - (A\cdot B\times D)C.[/tex]
 
  • #5
Thanks for the hint :)
 
  • #6
Am I doing this right?

[tex]a\cdot b'=0\Rightarrow\frac{b'\times c'}{a'\cdot(b'\times c')}\cdot b'=\frac{b'\cdot(b'\times c')}{a'\cdot(b'\times c')}=\frac{c'\codt(b'\times b')}{c'\cdot(a'\times b')}=0[/tex]

I used scalar triple product for the numerator - either [tex]b'\times b'=0[/tex] or I use the determinant and see that I have two same rows - therefore determinant is 0.

EDIT:

I've proved it! Yay for me XD

Basically you transform everything and in the end just show that a=a, b=b and c=c ^^ Thank you all for help ^^
 
Last edited:

FAQ: Solving Algebraic Relations of Reciprocal Lattice Vectors

What is the definition of reciprocal lattice vectors?

Reciprocal lattice vectors are a set of vectors that describe the periodicity of a crystal lattice in reciprocal space. They are used in crystallography to relate the positions of atoms in a crystal to the diffraction patterns produced by x-rays or electrons.

How do you solve algebraic relations of reciprocal lattice vectors?

The first step is to determine the basis vectors of the crystal lattice. These are usually given in terms of the unit cell dimensions and the angles between them. Next, use the formula for the reciprocal lattice vectors, which is the inverse of the basis vectors. Finally, solve the algebraic equations using matrix operations.

What is the importance of solving algebraic relations of reciprocal lattice vectors?

Solving algebraic relations of reciprocal lattice vectors allows us to understand the crystal structure and properties of materials. It also helps in the analysis of diffraction patterns, which can provide information about the arrangement of atoms within a crystal lattice.

Can you give an example of solving algebraic relations of reciprocal lattice vectors?

For example, if the basis vectors of a crystal lattice are a1 = (1,0,0), a2 = (0,1,0), and a3 = (0,0,1), then the reciprocal lattice vectors are b1 = (2π,0,0), b2 = (0,2π,0), and b3 = (0,0,2π). This can be solved using the matrix equation b = A-1a, where A is the matrix of basis vectors and b is the matrix of reciprocal lattice vectors.

Are there any practical applications of solving algebraic relations of reciprocal lattice vectors?

Yes, there are many practical applications in materials science and engineering. For example, it is used in the design and development of new materials, in the study of crystal defects and grain boundaries, and in the analysis of crystal structures in various industries such as semiconductors, metals, and ceramics.

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