2D-Diamond and 2D-Center Rectangular Lattice Comparison

[Karim Habashy]

Hi All,

Using (Mirror + Translation in 2D).
I see that Diamond and Center Rectangular Lattice in 2D has the same Symmetry i.e:

1)Horizontal Mirror Plane
2)Vertical Mirror Plane
3)2-Fold Rotation Axis.

and Diamond has Smaller Area.

Then why we say Center Rectangular has higher Symmetry and we take it as the Standard Cell.

Image Attached.
Reference, Edx.org, Course:
Symmetry, Structure and Tensor Properties of Materials, Lecture 7.

Thanks.

 

[eljose79]

Thank you for bringing up this interesting observation. I would like to provide some insight into why we consider the center rectangular lattice to have higher symmetry compared to the diamond lattice.

Firstly, it is important to understand that symmetry is a fundamental concept in materials science, and it plays a crucial role in understanding the properties and behavior of materials. In 2D, symmetry is described by the presence of mirror planes, rotation axes, and translation vectors.

In the case of the diamond and center rectangular lattices, both have the same number of symmetry elements, as you have correctly identified. However, the placement and orientation of these elements differ between the two lattices. In the diamond lattice, the mirror planes and rotation axes are not aligned with the lattice vectors, whereas in the center rectangular lattice, they are aligned. This alignment of symmetry elements in the center rectangular lattice results in a higher degree of symmetry compared to the diamond lattice.

Additionally, as you have mentioned, the diamond lattice has a smaller area compared to the center rectangular lattice. This means that the diamond lattice does not fully fill the 2D space, and there are gaps between the lattice points. On the other hand, the center rectangular lattice fully fills the space, resulting in a more regular and symmetric arrangement of lattice points.

In summary, while both lattices have the same number of symmetry elements, the alignment and regularity of these elements in the center rectangular lattice give it a higher degree of symmetry. This is why it is considered the standard cell in 2D crystallography.

I hope this helps clarify the concept of symmetry in these two lattices. If you have any further questions, please do not hesitate to ask.