Reciprocal crystal lattice

In summary, the conversation discusses the process of converting a crystal lattice 2D representation to a 2D reciprocal lattice. This involves finding the reciprocal lattice vectors using a specific equation, with the z unit vector being used for the third vector in the 2D case. There is also mention of the unit of the reciprocal lattice being length-1, but it is not as simple as just inverting the lengths. The conversation ends with a request for assistance in preparing for a test and a mention of only finding examples for converting from reciprocal to crystal lattice.
  • #1
big man
254
1
I know this might be a really stupid question, but to convert a crystal lattice 2D representation to a 2D reciprocal lattice do you justdo you just invert the scaling. I know this is a pretty poor explanation so I will try and illustrate what I mean.

Let's say that you have a reciprocal lattice like the one below:

. 020 . 120 . 220. 010 . 110 . 210. 000 . 100 . 200

|----|
. 25 nm^-1

Is the crystal lattice just a similar drawing with the spacing inverted, that is, 4 nm? By the way the above diagram is meant to have the vertical and horizontal spacings equal so they are both 0.25 nm^-1.

I'm sorry but I just haven't really found anything on this at all and I'm preparing for my test this week. The only example question I could find was converting from reciprocal lattice to crystal lattice, but I imagine the process will be the same if you were trying to convert from crystal lattice to a reciprocal lattice.

Thanks for any assistance.
 
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  • #2
To find the reciprocal lattice vectors you use the following equation.

[tex] \mathbf{b_1} = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3} {\mathbf {a}_1 \cdot \mathbf{a}_2 \times \mathbf{a}_3} [/tex]

For a 2D case the vector [tex] \mathbf{a}_3[/tex] becomes just the z unit vector.

To get the vector [tex]\mathbf{b}_2[/tex] you just cyclically permute the numerator.

You are of course correct that the unit of the reciprocal lattice is length-1 but its a little more complicated than simply inverting the lengths.
 
  • #3


Hello,

Thank you for your question. I can provide some clarification on the concept of reciprocal crystal lattice.

Firstly, the reciprocal lattice is a mathematical representation of the crystal lattice in reciprocal space, which is essentially the Fourier transform of the crystal lattice in real space. It is used to describe the diffraction pattern of a crystal.

To answer your question, to convert a crystal lattice 2D representation to a 2D reciprocal lattice, you do not simply invert the scaling. The reciprocal lattice is a different mathematical representation and cannot be obtained by simply inverting the scaling of the crystal lattice.

To illustrate this, let's consider your example of a reciprocal lattice with a spacing of 0.25 nm^-1. The corresponding crystal lattice would have a spacing of 4 nm. However, this does not mean that the crystal lattice is simply the inverse of the reciprocal lattice. The crystal lattice is a physical representation of the atomic arrangement in the crystal, while the reciprocal lattice is a mathematical representation.

In order to convert between the two, you need to use mathematical equations and transformations. For example, to obtain the reciprocal lattice vectors, you need to take the inverse of the real space lattice vectors. This is just one example, and the exact equations and transformations will depend on the type of crystal lattice and the specific reciprocal lattice you are trying to obtain.

I hope this explanation helps. It is important to note that understanding reciprocal crystal lattice and its conversion from a crystal lattice is a complex topic and requires a solid understanding of crystallography and mathematical concepts. I would recommend consulting with your textbook or professor for more detailed explanations and examples. Good luck on your test.
 

What is a reciprocal crystal lattice?

A reciprocal crystal lattice is a mathematical representation of the arrangement of atoms or molecules in a crystal. It is a reciprocal or inverse of the direct crystal lattice and is used to describe the diffraction of X-rays or other particles when they interact with a crystal.

How is a reciprocal crystal lattice related to the direct crystal lattice?

The reciprocal crystal lattice is mathematically related to the direct crystal lattice through the use of a transformation called a Fourier transform. The positions of the atoms in the direct lattice are represented by the amplitudes of the diffraction peaks in the reciprocal lattice.

What is the significance of reciprocal crystal lattice in crystallography?

The reciprocal crystal lattice is essential in crystallography because it allows scientists to determine the structure of a crystal by analyzing the diffraction patterns produced by X-rays or other particles. It also provides information about the symmetry and properties of the crystal.

How is the reciprocal crystal lattice represented?

The reciprocal crystal lattice is typically represented by a series of points or nodes in a three-dimensional space. The positions of these points are determined by the diffraction angles and intensities measured in experiments. The reciprocal lattice is often visualized using a unit cell, just like the direct lattice.

What factors affect the shape and size of the reciprocal crystal lattice?

The shape and size of the reciprocal crystal lattice are primarily affected by the spacing of the atoms or molecules in the direct lattice. Other factors that can influence the reciprocal lattice include the type of crystal, the symmetry of the crystal, and the type of radiation used in the diffraction experiment.

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