Bloch's Theorem - Revision history

Cmditradmin at 19:31, 29 December 2009

2009-12-29T19:31:40Z

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			[[category:electronic band structure]]
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Cmditradmin at 20:59, 7 December 2009

2009-12-07T20:59:03Z

← Older revision		Revision as of 13:59, 7 December 2009
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	As polyenes get longer and longer one 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. This is the purpose of Bloch's Theorem.		As polyenes get longer and longer one could calculate and combine the wave functions for all the bonds, but this is not very efficient. Instead it is more powerful to consider the periodicity of repeated units. This is the purpose of Bloch's Theorem.

	== Bloch’s Theorem ==		== Bloch’s Theorem ==
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	:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a}) = exp(0) = 1\,\!</math>		:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a}) = exp(0) = 1\,\!</math>

	At k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the ~~wavefunction~~ in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.		At k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wave function in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.

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	In case of an even unit cell number of J the phase factor is +1, while for odd numbered unit cells there a phase factor of -1. A -1 phase factor times the original ~~wavefunction~~ flips the orientation of the bonding orbital. At the single bonds (π/a) there are nodes where energy is zero. The top of valence band at k=π/a has a ~~wavefunction~~ which for a long molecule constututes the HOMO level of the molecule. There bonding factors where in the ground state there are double bonds, and antibonding factors where in the ground state there are single bonds.		In case of an even unit cell number of J the phase factor is +1, while for odd numbered unit cells there a phase factor of -1. A -1 phase factor times the original wave function flips the orientation of the bonding orbital. At the single bonds (π/a) there are nodes where energy is zero. The top of valence band at k=π/a has a wave function which for a long molecule constututes the HOMO level of the molecule. There bonding factors where in the ground state there are double bonds, and antibonding factors where in the ground state there are single bonds.

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	The π* band goes down in energy from <math>\overrightarrow{k}=0\,\!</math> to <math>\overrightarrow{k}=\frac {\pi} {\overrightarrow{a}}\,\!</math>		The π* band goes down in energy from <math>\overrightarrow{k}=0\,\!</math> to <math>\overrightarrow{k}=\frac {\pi} {\overrightarrow{a}}\,\!</math>

	The π and π* bands have the same ~~wavefunction~~ characteristics as the HOMO an LUMO levels of the small polyenes.		The π and π* bands have the same wave function characteristics as the HOMO an LUMO levels of the small polyenes.

	The bandgap ( here E<sub>g</sub>~1.5eV) is said to be a direct bandgap at <math>\overrightarrow{k}=\frac {\pi} {\overrightarrow{a}}\,\!</math>		The bandgap ( here E<sub>g</sub>~1.5eV) is said to be a direct bandgap at <math>\overrightarrow{k}=\frac {\pi} {\overrightarrow{a}}\,\!</math>

Cmditradmin: /* Bloch’s Theorem */

2009-09-15T21:59:20Z

Bloch’s Theorem

← Older revision		Revision as of 14:59, 15 September 2009
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			As polyenes get longer and longer one 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. This is the purpose of Bloch's Theorem.

	== Bloch’s Theorem ==		== Bloch’s Theorem ==
	[[Image:Repeate_unit.png\|thumb\|300px\|A polyene has repeated units with length a]]		[[Image:Repeate_unit.png\|thumb\|300px\|A polyene has repeated units with length a]]
	As polyenes get longer and longer one 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. This is the purpose of Bloch's Theorem.

	Consider 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 at point <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.		Consider 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 at point <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.

Cmditradmin at 21:41, 15 September 2009

2009-09-15T21:41:09Z

← Older revision		Revision as of 14:41, 15 September 2009
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	=== Bloch’s Theorem ===		== Bloch’s Theorem ==
	[[Image:Repeate_unit.png\|thumb\|300px\|A polyene has repeated units with length a]]		[[Image:Repeate_unit.png\|thumb\|300px\|A polyene has repeated units with length a]]
	As polyenes get longer and longer one 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. This is the purpose of Bloch's Theorem.		As polyenes get longer and longer one 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. This is the purpose of Bloch's Theorem.
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	=== Band Structure of Polyacetylene ===		== Band Structure of Polyacetylene ==
	[[Image:Bandstructure polyacet.JPG\|thumb\|300px\|Band structure of polyacetylene - energy versus wavevector of electron]]		[[Image:Bandstructure polyacet.JPG\|thumb\|300px\|Band structure of polyacetylene - energy versus wavevector of electron]]
	This is band structure of polyacetylene derived using Bloch’s Theorem. It is a plot of energy versus wavevector of the electron. This makes sense since the wavevector is related to the momentum and therefore energy of the electron. The periodic behavior will continue across the length of the chain. The energy of the π an π* bands are symmetrical and inverted. There is periodicity in both direct and reciprocal space.		This is band structure of polyacetylene derived using Bloch’s Theorem. It is a plot of energy versus wavevector of the electron. This makes sense since the wavevector is related to the momentum and therefore energy of the electron. The periodic behavior will continue across the length of the chain. The energy of the π an π* bands are symmetrical and inverted. There is periodicity in both direct and reciprocal space.
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	=== Band Gap ===		== Band Gap ==

	[[Image:piband_temperature.JPG\|thumb\|300px\|]]		[[Image:piband_temperature.JPG\|thumb\|300px\|]]

Cmditradmin: /* Band Gap */

2009-08-31T18:08:29Z

Band Gap

← Older revision		Revision as of 11:08, 31 August 2009
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	It is possible to connect the electronic properties of periodic systems and molecular orbitals of simple molecules like butadiene.		It is possible to connect the electronic properties of periodic systems and molecular orbitals of simple molecules like butadiene.

	At absolute zero temperature the π band is completely filled with electrons. Any electron with momentum of +K will have a corresponding electron with –K, so the net momentum of all the electrons is zero, so there is no net displacement of charge, so there is no current. In order to have a current there has to be electrons getting into the empty band, so that when there an applied field you can have a net displacement of charges in one direction. This is why the π* band is called the conduction band. Electrons have to jump into the conduction band for there to be current. This is considered an intrinsic system, with no input and outputs. ~~Using dopants you~~ can make the system conductive so that when electrons are externally added or removed there will be conduction.		At absolute zero temperature the π band is completely filled with electrons. Any electron with momentum of +k will have a corresponding electron with –k, so the net momentum of all the electrons is zero, so there is no net displacement of charge, so there is no current. In order to have a current there has to be electrons getting into the empty band, so that when there an applied field you can have a net displacement of charges in one direction. This is why the π* band is called the conduction band. Electrons have to jump into the conduction band for there to be current. This is considered an intrinsic system, with no input and outputs. Dopants can make the system conductive so that when electrons are externally added or removed there will be conduction.

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Cmditradmin: /* π Band */

2009-08-31T16:55:36Z

π Band

← Older revision		Revision as of 09:55, 31 August 2009
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	[[Image:piband_piover_a.JPG\|thumb\|300px\|Wavefunction at π/a]]		[[Image:piband_piover_a.JPG\|thumb\|300px\|Wavefunction at π/a]]
	Consider what happens approaching the edge of the Brillouin zone (k= π/a). In case of an even unit cell number of J the phase factor is +1, while for odd numbered unit cells there a phase factor of -1. A -1 phase factor times the original wavefunction flips the orientation of the bonding orbital. At the single bonds (π/a) there are nodes where energy is zero. The top of valence band at k=π/a has a wavefunction which for a long molecule constututes the HOMO level of the molecule. There bonding factors where in the ground state there are double bonds, and antibonding factors where in the ground state there are single bonds.		Consider what happens approaching the edge of the Brillouin zone (k= π/a).

			:<math>\rightarrow \overrightarrow{k} = \frac {\pi} {\overrightarrow{a}} : exp (i \overrightarrow{k}j \overrightarrow{a}) = exp(i \frac {\pi} {\overrightarrow{a}}j \overrightarrow{a}) = exp(i j \pi)\,\!</math>


			In case of an even unit cell number of J the phase factor is +1, while for odd numbered unit cells there a phase factor of -1. A -1 phase factor times the original wavefunction flips the orientation of the bonding orbital. At the single bonds (π/a) there are nodes where energy is zero. The top of valence band at k=π/a has a wavefunction which for a long molecule constututes the HOMO level of the molecule. There bonding factors where in the ground state there are double bonds, and antibonding factors where in the ground state there are single bonds.

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Cmditradmin: /* π Band */

2009-08-31T16:52:26Z

π Band

← Older revision		Revision as of 09:52, 31 August 2009
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	=== π Band ===		=== π Band ===
	[[Image:piband.JPG\|thumb\|300px\|Wavefunction at k=0]]		[[Image:piband.JPG\|thumb\|300px\|Wavefunction at k=0]]
	The π band within the unit cell behaves like an isolated ethylene subunit. The π level is bonding and the π* level is antibonding .		The π band within the unit cell behaves like an isolated ethylene subunit. The π level is bonding and the π* level is antibonding.



	:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a}) = exp(0) = 1\,\!</math>		:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a}) = exp(0) = 1\,\!</math>

Cmditradmin: /* π Band */

2009-08-31T16:52:15Z

π Band

← Older revision		Revision as of 09:52, 31 August 2009
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	The π band within the unit cell behaves like an isolated ethylene subunit. The π level is bonding and the π* level is antibonding .		The π band within the unit cell behaves like an isolated ethylene subunit. The π level is bonding and the π* level is antibonding .

	A k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wavefunction in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.

	:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a} = exp(0) = 1\,\!</math>
			:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a}) = exp(0) = 1\,\!</math>

			At k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wavefunction in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.

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Cmditradmin: /* π Band */

2009-08-31T16:51:24Z

π Band

← Older revision		Revision as of 09:51, 31 August 2009
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	A k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wavefunction in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.		A k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wavefunction in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.

	:<math>\rightarrow \overrightarrow{k} = 0 \, exp (i \overrightarrow{k}j \overrightarrow{a} = exp(0) = 1\,\!</math>		:<math>\rightarrow \overrightarrow{k} = 0 : exp (i \overrightarrow{k}j \overrightarrow{a} = exp(0) = 1\,\!</math>

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Cmditradmin: /* π Band */

2009-08-31T16:50:47Z

π Band

← Older revision		Revision as of 09:50, 31 August 2009
Line 133:		Line 133:

	A k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wavefunction in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.		A k = 0 the phase factor is exp(0) which is 1. At K equals zero the imaginary component goes away, the cos(0)=1. When you look at the wavefunction in any other unit cell with respect to the origin unit cell you multiply by the phase factor =1; this results in the same fully bonding wave function all along the molecule.

			:<math>\rightarrow \overrightarrow{k} = 0 \, exp (i \overrightarrow{k}j \overrightarrow{a} = exp(0) = 1\,\!</math>

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	[[Image:piband_piover_a.JPG\|thumb\|300px\|Wavefunction at π/a]]		[[Image:piband_piover_a.JPG\|thumb\|300px\|Wavefunction at π/a]]