Bloch's Theorem

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Bloch’s Theorem

As polyenes get longer and longer you could calculate and combine the wavefunctions for all the bonds, but this is not very efficient. Instead it is more powerful to consider the periodicity of repeated units.

Note: Polyacetylene is made from acetylene but it does not contain triple bonds. One can concentrate on the repeat unit cell (with cell length a) and the interactions with its neighbors. For example if you move over 2a units from point r you should find the same electron density at that point ja as point r. Because of the translation symmetry electron density <math>\overrightarrow{r}\,\!</math> in cell j (j equiv integer) must equal the electron density at point <math>\overrightarrow{r}\,\!</math> in the origin cell.

<math>\vert \Psi (\overrightarrow{r} + j \overrightarrow{a}) \vert^2 = \vert \Psi (\overrightarrow{r} ) \vert ^2\,\!</math>

<math>\Psi (\overrightarrow{r} + j\overrightarrow{a}) = exp (i \overrightarrow{k}j\overrightarrow{A}) \Psi (\overrightarrow{r})\,\!</math>

Where

<math>exp (i \overrightarrow{k}j\overrightarrow{a})\,\!</math> is the phase factor

The square of the wave function at r plus ja is the same as square of the wave function at r.

reminder

<math>exp (i \overrightarrow{k}j\overrightarrow{a})= cos(\overrightarrow{k}j\overrightarrow{a})+i sin(\overrightarrow{k}j\overrightarrow{a})\,\!</math>

<math>\vert exp(i \overrightarrow{k}j\overrightarrow{a})\vert ^2 = exp (i \overrightarrow{k}j\overrightarrow{a}) exp(-\overrightarrow{k}j\overrightarrow{a})\,\!</math>

<math>=(cos(\overrightarrow{k}j\overrightarrow{a}))^2 +(sin(\overrightarrow{k}j\overrightarrow{a}))^2\,\!</math>

<math>=1\,\!</math>

The phase factor takes into account that there has been a translation.

<math>\overrightarrow{k}j\overrightarrow{a} \equiv\,\!</math> argument of an exponential (no dimensions)

<math>\overrightarrow{a} \equiv\,\!</math> length of a unit cell in direct space

<math>\overrightarrow{k} \equiv \frac {1} {length}\,\!</math> defined in reciprocal space

The phase factor accounts for the translation in the original equation. The argument of an exponential has to be dimensionless. If you have something like a which is a length, then you need to have a term k that is 1/length in order to remove dimensions from the equation. A is measured in direct space that is a measure of the length of the unit cell, k is defined in reciprocal space.

Any kind of mathematical analysis that involves periodic structures (x-ray diffraction or neutron diffraction) in three dimensions will have some form of a phase factor. As a convention the phase factor is written (ikja).

The same kind of analysis can found in X-ray structure determination of crystals. They use the convention for phase factor as

<math>exp(i2 \pi \overrightarrow{k} j \overrightarrow{a})\,\!</math>

This doesn’t change things because the 2pi simply returns you to the same point on the imaginary and real axes plot.

<math>\overrightarrow{k}\,\!</math> is a wavevector that related to the wavelength of the electron.

Any object has an associated wavelength that is

for

<math>\overrightarrow{k}= \frac{ 2\pi} {\lambda}\,\!</math>

<math>\overrightarrow{\Lambda} \equiv \frac {\nu} {m\overrightarrow{v}}\,\!</math>

<math>\hbar \overrightarrow{k}\equiv\,\!</math> electron momentum

Where

<math>\mu\,\!</math> is plank’s constant

m is mass

v is velocity

<math>\hbar (\equiv \frac {\mu} {2\pi} )\,\!</math>

when

<math>\overrightarrow{k}= 0 \Rightarrow \overrightarrow{\Lambda} = \infty\,\!</math>

<math>\overrightarrow{k}= \frac {2\pi} {\overrightarrow{a}} \Rightarrow \overrightarrow{\Lambda} = \overrightarrow{a}\,\!</math>

<math>\frac {2\pi} {\overrightarrow{a}}\,\!</math> is the unit-cell in reciprocal space

Mass x velocity is the momentum.

A person moving has a very large mass compared to that of an electron. Using the equation a person’s wavelength would be 10^-25 angstroms.

But in the range of a the electron mass of 10^-27 this leads to electron wavelengths in the range of angstroms which is observable.

K is directly related to the wavelength of the electron and can have any value.