Understanding Reciprocal Lattice Vectors and Orthogonality in Primitive Lattices

In summary, the conversation discusses the concept of reciprocal lattice vectors and their relationship with primitive lattice vectors. These vectors are used in the calculation of the electronic structure of materials. The discussion also includes formulas for calculating the reciprocal vectors and the use of matrices in this process. The conversation ends with a question about the components of these vectors.
  • #1
torehan
41
0
[tex]f(\vec{r}) = f(\vec{r}+\vec{T}) [/tex]

[tex]\vec{T}= u_{1} \vec{a_{1}} + u_{2} \vec{a_{2}}+u_{3} \vec{a_{3}} [/tex]

[tex]u_{1},u_{2},u_{3}[/tex] are integers.

[tex]f(\vec{r}+\vec{T})= \sum n_{g} e^{(i\vec{G}.(\vec{r}+\vec{R}) )}= f(\vec{r}) [/tex]

[tex]e^{i\vec{G}.\vec{R} }= 1 [/tex]
[tex] \vec{G}.\vec{R} = 2\pi m[/tex]

we call [tex] \vec{G}=h\vec{g_{1}} + k \vec{g_{2}}+l \vec{g_{3}} [/tex] reciprocal lattice vector.

but what about the primitive lattice vectors [tex]\vec{g_{1}} , \vec{g_{2}} , \vec{g_{3}}[/tex] ?

To simplify the discussion consider [tex]\vec{T_{1}} [/tex] in 1D;

[tex]\vec{T_{1}} = u_{1} \vec{a_{1}} [/tex]

[tex] \vec{G}.\vec{T} =(h\vec{g_{1}} + k \vec{g_{2}}+l \vec{g_{3}}) . ( u_{1} \vec{a_{1}}) = 2 \pi m [/tex]

Is there any definiton that indicates direct primitive lattice vectors and reciprocal primitive lattice vectors orthogonalities?
i.e

[tex]\vec{g_{1}} . \vec{a_{1}} = 2 \pi [/tex]

[tex]\vec{g_{2}} . \vec{a_{1}} = \vec{g_{3}} . \vec{a_{1}} = 0 [/tex]
 
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  • #2
g1,g2,g3 are the primitive vectors for the reciprocal lattice.
And the orthogonality condition is [tex]\vec{a}_i \cdot \vec{g}_j = 2\pi \delta_{i,j}[/tex]

A trick for calculating the reciprocal vectors is to form the matrix A where the columns are the direct lattice vectors, and the matrix G where the columns are the reciprocal lattice vectors. Then you have
[tex]G^T \cdot A = 2\pi I[/tex]
so
[tex]G = 2\pi (A^{-1})^T[/tex]
 
  • #3
I is 3x3 identity martix isn't it?
 
  • #4
yes it is
 
  • #5
It's still confusing for me..
so how can I invers a (1x3) matrix?
 
  • #6
You can get the vectors [tex]\vec{g_1}, \vec{g_2}, \vec{g_3}[/tex] without using any matrices. To do this use the following formulas:
[tex]\vec{g_1}=2\pi\frac{[\vec{a_2},\vec{a_3}]}{\vec{a_1}[\vec{a_2},\vec{a_3}]},[/tex] [tex]\vec{g_2}=2\pi\frac{[\vec{a_3},\vec{a_1}]}{\vec{a_1}[\vec{a_2},\vec{a_3}]},[/tex] [tex]\vec{g_3}=2\pi\frac{[\vec{a_1},\vec{a_2}]}{\vec{a_1}[\vec{a_2},\vec{a_3}]}.[/tex]
 
  • #7
A and G are 3x3 matrices, not 1x3 matrices. The columns of A are the vectors of your lattice:
[tex]A = \left(
\left( \! \! \begin{array}{c}\\ \vec{a}_1 \\ \, \end{array} \!\! \right)
\left( \! \! \begin{array}{c}\\ \vec{a}_2 \\ \, \end{array} \!\! \right)
\left( \! \! \begin{array}{c}\\ \vec{a}_3 \\ \, \end{array} \!\! \right) \right)[/tex]

Personally, I think this is easier than manually evaluating three separate cross products. But either way works.
 
  • #8
corydalus said:
You can get the vectors [tex]\vec{g_1}, \vec{g_2}, \vec{g_3}[/tex] without using any matrices. To do this use the following formulas:
[tex]\vec{g_1}=2\pi\frac{[\vec{a_2},\vec{a_3}]}{\vec{a_1}[\vec{a_2},\vec{a_3}]},[/tex] [tex]\vec{g_2}=2\pi\frac{[\vec{a_3},\vec{a_1}]}{\vec{a_1}[\vec{a_2},\vec{a_3}]},[/tex] [tex]\vec{g_3}=2\pi\frac{[\vec{a_1},\vec{a_2}]}{\vec{a_1}[\vec{a_2},\vec{a_3}]}.[/tex]

Thanks but the discussion is how we get these formulas.

kanato said:
A and G are 3x3 matrices, not 1x3 matrices. The columns of A are the vectors of your lattice:
[tex]A = \left(
\left( \! \! \begin{array}{c}\\ \vec{a}_1 \\ \, \end{array} \!\! \right)
\left( \! \! \begin{array}{c}\\ \vec{a}_2 \\ \, \end{array} \!\! \right)
\left( \! \! \begin{array}{c}\\ \vec{a}_3 \\ \, \end{array} \!\! \right) \right)[/tex]

Personally, I think this is easier than manually evaluating three separate cross products. But either way works.


O.K , as you said ai and gi must be vectors which has three components. The question is what are these components?
 
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What is a reciprocal lattice?

A reciprocal lattice is a mathematical representation of the periodicity in the arrangement of atoms in a crystal. It is a set of points in reciprocal space that correspond to the points in real space where the crystal's unit cell repeats itself.

How is a reciprocal lattice related to a real lattice?

A reciprocal lattice is the Fourier transform of the real lattice. This means that the reciprocal lattice captures the periodicity and symmetry of the real lattice in a different way. The two lattices are mathematically related and provide complementary information about the crystal structure.

What is the significance of the reciprocal lattice in materials science?

The reciprocal lattice is an essential tool in understanding the diffraction patterns produced by crystals. It allows scientists to determine the crystal structure, orientation, and crystallographic features of a material. It is also used in the design and characterization of new materials.

How is the reciprocal lattice of a crystal determined experimentally?

The reciprocal lattice can be experimentally determined using techniques such as X-ray diffraction, electron diffraction, and neutron diffraction. These methods involve analyzing the diffraction patterns produced by the crystal and using mathematical algorithms to calculate the reciprocal lattice.

Can the reciprocal lattice be used in other fields of science?

Yes, the concept of a reciprocal lattice is not limited to materials science. It is also used in other branches of science such as solid-state physics, chemistry, and crystallography. It is a powerful tool for understanding the atomic and molecular structures of various materials and molecules.

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