# How to evaluate the strain-induced change in reciprocal space?

• PRB147
In summary, the conversation discusses the strain-induced change in crystal structures and the challenges of evaluating it both in real space and reciprocal space. The possibility of calculating it in high-symmetry points in quasi-momentum space is also questioned. It is mentioned that Neto and his students have previously calculated this change in a specific equation in a research paper. However, the conversation ends without a clear solution or hint for the problem.
PRB147
In real space in crystal, strain-induced change can be written as follows:
$${\bf r'}=(1+\epsilon)\cdot {\bf r}$$
But there is no way to evaluate the strain-induced change in reciprocal space.
Can one calculate the strain-induced change in high-symmetry point in quasi-momentum space?
I check almost many books, I still can not find a way.
But Neto and his students had calculated a change in Eq.(11) in PRB Vol.80, 045401 (2009).
Would anyone here give a hint?

Thank you all!

Best wishes!

PRB147 said:
In real space in crystal, strain-induced change can be written as follows:
$${\mathbf r'}=(1+\epsilon)\cdot {\mathbf r}$$
But there is no way to evaluate the strain-induced change in reciprocal space.
Can one calculate the strain-induced change in high-symmetry point in quasi-momentum space?
I check almost many books, I still can not find a way.
But Neto and his students had calculated a change in Eq.(11) in PRB Vol.80, 045401 (2009).
Would anyone here give a hint?

Thank you all!

Best wishes!

Let us denote the matrix: $${\mathbf P}=(1+\epsilon)$$. We consider that r is a column. Under the transformation P the basis vectors $$({\mathbf a}', {\mathbf b}')=({\mathbf a}, {\mathbf b}) {\mathbf P}$$. Note, the rows. The new basis vectors will have appropriate reciprocal vectors
$\begin{pmatrix} {\mathbf a}^*'\\ {\mathbf b}^*' \end{pmatrix}={\mathbf Q}\begin{pmatrix} {\mathbf a}^*\\ {\mathbf b}^* \end{pmatrix}$,

where $\mathbf Q={\mathbf P}^{-1}$.

Now we are interested how an arbitrary reciprocal vector looks like in the old reciprocal basis, but this is simply changing

a* -> a* h
b* -> b* k

in the above formula.

About the transformation in crystallography have a look in:
International Tables for Crystallography (2006). Vol. A, Chapter 5.1, pp. 78–85.

Thank you very much, read, I will learn your explanation step by step.
By the way, the meaning of h and k is the Miller indices?

Last edited:
PRB147 said:
By the way, the meaning of h and k is the Miller indices?

Yes it is.

Using this method, the Neto's result can not be recovered.
I am stuck in this problem for one month.

PRB147 said:
Using this method, the Neto's result can not be recovered.
I am stuck in this problem for one month.

There are two things here.

(i) The change of the basis from the orthogonal to the hexagonal one. Unfortunately, they do not give the matrix explicitly, but from the Fig.2 one can infers that:

A1=1/2 a1 + sqrt(3)/2 a2
B1=-1/2 a1 + sqrt(3)/2 a2,

where a1,a2 - orth. basis, A1,B1 - hexagonal. Actually this is not correct from the hex-symmetry but this is the way they used... (correct way would be to use \delta_1 and \delta_2 as a1 and a2). This matrix written by columns, as I explained before, I call P.

(i) The strain induced change of the lattice given in orth. a1,a2 system (1+e) where e is symmetric matrix with elements e11, e22 and e12.

Now you make a product (1+e) P, and then make the inverse matrix of the product. This matrix I call Q-matrix. The reciprocal lattice b1 and b2 are given by the Q-matrix elements as b1=(Q11,Q12),... in the orthogonal basis. I have got

b1= 1-e11 -e12/sqrt(3)
b2= 1//sqrt(3)- e22/sqrt(3)-e12

that corresponds to their formulas

A1=1/2 a1 + sqrt(3)/2 a2
B1=-1/2 a1 + sqrt(3)/2 a2,

where a1,a2 - orth. basis, A1,B1 - hexagonal. Actually this is not correct from the hex-

sorry, I have made a couple of misprints in my explanations with the basis vector notations (they does not affect the solution).

hex-notations: instead of B1 should be A2. Letters "b" stand for the reciprocal state basis vectors in this paper...

symmetry but this is the way they used... (correct way would be to use \delta_1 and \delta_2 as a1 and a2). This matrix written by columns, as I explained before, I call P.
again... instead of "a1 and a2" should be "A1 and A2", i.e I meant hex-basis

Thank you very much! Read, Happy New year!
With my Best Wishes!

## 1. What is strain-induced change in reciprocal space?

Strain-induced change in reciprocal space is a phenomenon in which the atomic structure of a material is altered due to the application of mechanical strain. This results in a change in the positions and intensities of diffraction peaks in the reciprocal space, which can be measured using diffraction techniques such as X-ray or neutron diffraction.

## 2. How is the strain-induced change in reciprocal space evaluated?

The strain-induced change in reciprocal space can be evaluated by comparing the diffraction patterns of the strained and unstrained samples. This involves measuring the positions and intensities of the diffraction peaks and calculating the strain tensor using specialized software or mathematical equations.

## 3. What factors can influence the strain-induced change in reciprocal space?

The strain-induced change in reciprocal space can be influenced by various factors such as the type and magnitude of the applied strain, the crystal structure of the material, and the nature of the bonding between atoms. Additionally, factors such as temperature, pressure, and the presence of defects or impurities can also affect the strain-induced changes.

## 4. Why is evaluating the strain-induced change in reciprocal space important?

Evaluating the strain-induced change in reciprocal space is important because it provides valuable information about the structural and mechanical properties of a material. This can help scientists and engineers better understand the behavior of materials under different conditions and design new materials with desired properties.

## 5. What are some applications of evaluating strain-induced change in reciprocal space?

The evaluation of strain-induced change in reciprocal space has various applications in materials science and engineering. It can be used to study the effects of strain on the microstructure and mechanical properties of materials, as well as to optimize the performance of materials in various applications such as aerospace, automotive, and biomedical industries.

Replies
1
Views
2K
Replies
61
Views
3K
Replies
2
Views
2K
Replies
6
Views
1K
Replies
2
Views
2K
Replies
0
Views
2K
Replies
1
Views
2K
Replies
20
Views
2K
Replies
4
Views
919
Replies
5
Views
2K