How Is the Polyacetylene Chain Structured in a 1D Lattice Model?

[unscientific]

Homework Statement

The polyacetylene chain is a 1D chain of Carbon atoms with single bonds and double bonds in succession. Spacing for single bond is $a_s = 0.144~nm$ and spacing for double bond is $a_d = 0.136~nm$.

Describe the structure using a "lattice" and a "basis".
Sketch the dispersion plot of $\epsilon(k)$ and mark the occupied states, fermi energy and first Brillouin zone.

Homework Equations

The Attempt at a Solution

I suppose the lattice is simple 1d chain with a basis of another carbon at $a$? Since the double bond has a shorter length, it means it has a higher $k_d$.

 

[ehild]

Homework Statement

The polyacetylene chain is a 1D chain of Carbon atoms with single bonds and double bonds in succession. Spacing for single bond is $a_s = 0.144~nm$ and spacing for double bond is $a_d = 0.136~nm$.

Describe the structure using a "lattice" and a "basis".
Sketch the dispersion plot of $\epsilon(k)$ and mark the occupied states, fermi energy and first Brillouin zone.

Homework Equations

The Attempt at a Solution

I suppose the lattice is simple 1d chain with a basis of another carbon at $a$? Since the double bond has a shorter length, it means it has a higher $k_d$.

Yes, the basis consist of two carbon atoms - how far from each other? And what is the lattice parameter?
Yes, the double bond is shorter and more "rigid" than the single one so it has higher stretching force constant.

 

[unscientific]

Yes, the basis consist of two carbon atoms - how far from each other? And what is the lattice parameter?
Yes, the double bond is shorter and more "rigid" than the single one so it has higher stretching force constant.

The basis consists of 2 carbon atoms, one at (0) and one at 0.136nm away. Lattice parameter is 0.144nm

 

[ehild]

The basis consists of 2 carbon atoms, one at (0) and one at 0.136nm away. Lattice parameter is 0.144nm

No, 0.144nm is not the lattice parameter. That should be the length of the repeating unit. If you translate the atom at (0) by 0.144 nm you do not get an equivalent atom.

 

[unscientific]

No, 0.144nm is not the lattice parameter. That should be the length of the repeating unit. If you translate the atom at (0) by 0.144 nm you do not get an equivalent atom.

Good point. Lattice constant is then $a_d + a_s = 0.28nm$.

 

[ehild]

Good point. Lattice constant is then $a_d + a_s = 0.18nm$.

That is much better :)

 

[unscientific]

That is much better :)

Is the dispersion curve like this:

 

[ehild]

I never saw such dispersion curve :)

Yes, it will have two branches, but they both are centred at k=0 and end at k=±pi/a

 

[unscientific]

I never saw such dispersion curve :)

Yes, it will have two branches, but they both are centred at k=0 and end at k=±1/a

It is a 1-dimensional diatomic lattice. So total modes = 2. 1 acoustic (moving in phase), 1 optic mode (moving anti-phase).

 

[ehild]

It is a 1-dimensional diatomic lattice. So total modes = 2. 1 acoustic (moving in phase), 1 optic mode (moving anti-phase).

Yes, it is right.

 

[unscientific]

Yes, it is right.

So the dispersion curve looks something like this where $a=a_s+a_d = 0.28nm$

 

[ehild]

Yes, it can be plotted this way. But the acoustic part starts linearly, and is curved at the edge of the Brillouin zone.

 

[unscientific]

Yes, it can be plotted this way. But the acoustic part starts linearly, and is curved at the edge of the Brillouin zone.

True, as $\omega \propto sin \left( \frac{ka}{2} \right) \sim k$ for small $k$