# 1D Chain crystal structure

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1. May 31, 2015

### unscientific

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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 $k_d$.

2. Jun 1, 2015

### ehild

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.

3. Jun 2, 2015

### unscientific

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

4. Jun 2, 2015

### ehild

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.

5. Jun 2, 2015

### unscientific

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

Last edited: Jun 2, 2015
6. Jun 2, 2015

### ehild

That is much better :)

7. Jun 2, 2015

### unscientific

Is the dispersion curve like this:

8. Jun 2, 2015

### 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

Last edited: Jun 2, 2015
9. Jun 2, 2015

### unscientific

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

10. Jun 2, 2015

### ehild

Yes, it is right.

11. Jun 2, 2015

### unscientific

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

12. Jun 2, 2015

### ehild

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

13. Jun 2, 2015

### unscientific

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