Are the vibration modes at the lattice centre odd or even?

In summary, the two vibration modes in the given image are both odd, as they involve two-dimensional and three-dimensional translations respectively. This is because they do not have an inversion center, as confirmed by the person asking the question.
  • #1
Frank Einstein
170
1

Homework Statement



Good morning. I am currently doing an solid state problem; in this part, I have to tell if these vibration modes are odd or even. I am working at the lattice centre.
The modes are these http://postimg.org/image/8nmv7ickz/.

Homework Equations


None

The Attempt at a Solution


The centre of inversion is the blue cross; I have studied only two dimensional modes, so I don't have very clear how the three dimensional translations have to be done; I think it that this is what happens: http://postimg.org/image/xd01nfxr1/
which would make both modes odd.
Can someone please confirm this?
Thanks for reading.
 
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  • #2
Yes, both of the modes are odd. The first mode involves a two-dimensional translation, with one atom moving to the left and one atom moving to the right. This is an odd vibration mode because it does not have an inversion center. The second mode involves a three-dimensional translation, with one atom moving up, one atom moving down, and one atom moving to the side. This is also an odd vibration mode because it does not have an inversion center.
 

FAQ: Are the vibration modes at the lattice centre odd or even?

1. What are odd and even vibration modes?

Odd and even vibration modes refer to the different types of vibrations that can occur in a system. Odd modes are characterized by nodes where the displacement is zero, while even modes have antinodes where the displacement is maximum.

2. How do odd and even vibration modes differ from each other?

Odd modes have an odd number of nodes, while even modes have an even number of nodes. Additionally, the shape of the displacement pattern for odd modes is symmetric about the midpoint, while even modes have an asymmetric shape.

3. How are odd and even vibration modes determined?

The determination of odd and even modes depends on the boundary conditions of the system. For a fixed-fixed system, odd modes will occur at odd multiples of half-wavelengths, while even modes will occur at even multiples of half-wavelengths. For a fixed-free system, the opposite is true.

4. What is the significance of odd and even vibration modes?

Odd and even modes play an important role in understanding the behavior of vibrating systems. They can help determine the natural frequencies of the system, which can be used to predict potential modes of failure or optimize design parameters.

5. Can a system have both odd and even vibration modes?

Yes, a system can have both odd and even modes of vibration. The number and combination of odd and even modes that occur depend on the specific characteristics and boundary conditions of the system.

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