Bravais Lattice in Two Dimensions

In summary, in the book Applied Physics by P.K.Mittal, it is stated that there are 10 point groups in two dimensions. This means that there are 10 different symmetry operations, such as inversions, reflections and rotations, that can be performed in a 2D plane. However, when an extra dimension is added, there are more possible directions for these operations to be performed, resulting in a total of 32 point groups in three dimensions. This increase in the number of point groups is due to the extra degree of freedom in 3D. The 10 point groups in 2D can be created by performing these symmetry operations in different combinations, as shown by the examples of C1, C2, C3
  • #1
shayaan_musta
209
2
In the book Applied Physics by P.K.Mittal, on page#25 under the heading of "Bravais lattice in two dimensions", a paragraph says,
"The number of point groups in two dimensions is 10."
My 1st question is,
Then how many in three dimensions?

Paragraph further says,
"10 groups in two dimensions produce only 5 Bravais lattice."
What is meant by above statement?
 
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  • #2
I believe there are 32 point groups in 3D.
 
  • #3
Foder said:
I believe there are 32 point groups in 3D.

Thanks Foder.
Yes I agree with you. But in 2D there are 10 point groups then how will you justify this statement in a simple and easy language so that I could be able to teach other for their difficulties.
 
  • #4
3D have more point groups than 2D because of the extra degree of freedom. I mean, point groups represent symmetry operations, such as inversions, reflections and rotations. When you add an extra dimension you can perform those in new directions not parallel to the original 2D plane, allowing also more combinations of them.
 
  • #5
Yeah Foder I agree with you.
But I think you are not understanding my question.

Look, there are 10 point groups in 2D which are given below,
C1, C2, C3, C4, C6, D1, D2, D3, D4, D6

Agree?

Then I want to know that how these group can be created?

Means there was any genius who discovered that there are 10 point groups in 2D. So how he did it?
If you give one or two example then may be I would be able to understand that how there are 32 point groups in 3D.

I hope now you can understand my question.
Thanks for hard working for me. GOD bless you.
 

What is a Bravais lattice in two dimensions?

A Bravais lattice in two dimensions is a mathematical concept used to describe the arrangement of points in a two-dimensional crystal lattice. It is characterized by a set of discrete translation vectors and can be described by a unit cell, which is the smallest repeating unit of the lattice.

How many types of Bravais lattices exist in two dimensions?

There are exactly five types of Bravais lattices in two dimensions: rectangular, square, oblique, centered rectangular, and hexagonal.

How do I determine the symmetry of a Bravais lattice in two dimensions?

The symmetry of a Bravais lattice in two dimensions can be determined by examining the arrangement of points within the unit cell. If the arrangement remains the same when rotated or reflected, the lattice has rotational or reflectional symmetry, respectively.

What is the significance of Bravais lattices in two dimensions?

Bravais lattices in two dimensions play a crucial role in understanding the properties of crystals, such as their mechanical, thermal, and electrical properties. They also serve as a basis for studying more complex crystal structures.

How are Bravais lattices in two dimensions different from three-dimensional lattices?

The main difference between Bravais lattices in two dimensions and three-dimensional lattices is the number of dimensions they exist in. Two-dimensional lattices have only two dimensions, while three-dimensional lattices have three. This results in different symmetries and properties between the two types of lattices.

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