CleanEnergyWIKI - User contributions [en]

Interchain Interactions

2009-08-27T01:02:39Z

Nsylvain: /* Transition dipole in isolation */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photochromism|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>

</tr>
</table>

=== Interchain interactions ===

This topic is an important consideration with light emitting polymer and oligomers. Photoluminescence is easily measured by shining light and measuring the emission.

In dilute solutions or inert matrices the photoluminescence quantum yield of a given conjugated polymers can be very large: up to 60-80 %

But in thin films the solvent goes away, the polymer chains get closer so they can interact and the luminescence can be nearly totally quenched.

There are a few situations where going from solution to solid states you can get increased luminescence. Silos for example can cause an aggregation in the solid state that triggers luminescent qualities. The observation of quenching led researcher to explore the cause and remedy for this.

=== Transition dipole in isolation ===
[[Image:Stilbene_excitation.png|thumb|300px|Transitions in stilbene are polarized along the x axis]]
A transition dipole is directly related to the probability of a transition occurring from one excited state to the ground, or the ground to the excited state. It is also related to the intensity of the exciting energy.

If the wave function for the S1 state is localized in a different area of space than the ground state then transition dipole will be zero. There must be wave function overlap for the wavefunction equation to be non-zero.

In stilbene the transition dipole for the S0 → S1 transition and the S1 →S0 transition is polarized mainly along the long axis (x axis) of the molecule. This is generally the case in pi conjugated systems because as you extend the length of the chain electrons are delocalized over a longer distance, just as in an electrostatic dipole a larger distance of charge separation make for a larger dipole moment. An exception is pentacine whose first optical transition is polarized at 90 degrees from the long axis of the chain. In most other optical polymers such as polyvinylene and polythiophene the optical transition is polarized along the long axis.

see Cornil <ref>J. Cornil et al., JACS 120 (1998) 1289 DOI: [http://dx.doi.org/10.1021/ja973761j 10.1021/ja973761j]</ref>

=== Two chain interaction ===
[[Image:Interchain_coord.png|thumb|300px|Molecules are represented as planes, transition dipole is along X]]

When two chains come into close proximity they there is the possibility during excitation the electron will jump from one chain to another, leaving a hole and forming a charge transfer exciton. The probability of finding h+ and e- on separate chains in S1 can be obtained from the excited-state wavefunction .

S1 → S0 intensity can go down since the transition is polarized along x.

 

=== Symmetrical interaction ===
[[Image:Interchain_distant.png|thumb|300px|At a distance greater than 8 Å there is no interaction]]
Consider first the most symmetric configuration which is perfectly cofacial.

From a distance of infinity down to 8 Å there is no significant interaction. There is no significant wavefunction overlap between the units.

*excitation is always localized on a SINGLE UNIT
*PL is not affected

This is the situation in dilute solution or inert matrices

[[Image:Interchain_close.png|thumb|300px|Below 8 Å there is band level splitting]]
When the distance (R) drops below 8 Å , as they do in thin films, the wavefunctions of frontier orbitals (homo and lumo) begin to overlap and you get a new splitting of the homo level. Wavefunctions are delocalized over the two units. They become equally spread in the zone of r~5 Å.
 

=== Exciton band formation ===
[[Image:Bandsplitting.png|thumb|400px|Split levels create forbidden transitions]]
The HOMO and LUMO may be split to different degrees. In a highly symmetrical situation the transition from the ground state to the lowest excited state (S1) is forbidden. The HOMO (au) and the LUMO (bu) is are both ungerade which means the transition is forbidden (because odd time odd is even, times odd is odd and when this is integrated over all space the result is zero).

If you were able to stack five molecules on top of each other, you could have five exciting states corresponding the mixing of the five states corresponding to the five S1 states, this is known as an exciton band. Even if a higher excited state such as S2 is achieved, Kasha’s Rule dictates that energy will relax down to lower states and the emission can only occur from the lowest excited state. S1 can not transition to S0, it is a “dark state” .The lowest state of that exciton band will be dark and luminescence is quenched.
 
[[Image:Kasha.png|thumb|300px|Parallel and antiparallel orientations of transition dipole]]
Kasha’s model from goes back fifty years and attempts to explain the emission from the lowest state. He only considered the wavefunction of one molecule interacting with that of another molecule. He did not consider mixing of wavefunctions so a charge transfer state with an electron on one molecule and a hole on another is impossible in his model. The diagram shows two molecules with transition dipoles along the main axis of each, one dipole parallel and the other anti-parallel. When the two dipoles are opposing the energy is less than the parallel situation but the oscillator strength is zero. In the parallel situation the oscillator strength is large and but the energy required to get there is higher.
 
[[Image:Bandsplitting2.png|thumb|300px|Interaction between molecules leads to band splitting]]
So when two transition dipoles interact, the two S1 states split forming a two state exciton band with S1 and S2. The lower ground state S1 is most favorable because it formed by the antiparallel orientation of the transition dipoles. The S2 has a high oscillator strength (its bright) but higher energy, S1 is lower energy but has low oscillator strength (its dark). So in the highly symmetrical situation molecule interact and illumination is quenched.
 
[[Image:Cofacial_energy.png|thumb|300px|Transition energy vs distance between molecules. Blue and red lines show shift of interacting molecules due to bandsplitting.]]
As molecules get closer and closer (approaching 4 angstroms) in cofacial configuration their wavefunctions interact and there is a splitting into S1 and S2. This results in a blue shift of the aborption because the molecule has to get up to the higher S2 state since S1 is not optically coupled with the ground state.
 

=== Geometry relaxation effects ===
[[Image:Geometry_relax.png|thumb|300px|Unequal relaxation due in excited state]]
If a conjugated system goes into an excited state there will be geometry relaxation. Considering two molecules one on top of another there will be unequal relaxation. In the this case the top molecule achieves lowest excited state (LES) geometry while the lower molecule stay in ground state (GS) geometry. Excitation into S2 trickles down to S1 state and then localizes on one of the molecules. The two molecules do not have the same geometry and the system has less symmetry. This asymmetric geometry deformations allow it to break the selection rules. Thus relaxation allows for a little bit of electronic coupling and some emission.
 
[[Image:Cofacial_energy_shift.png|thumb|300px|Emission and Absorption intensity vs energy]]
This is a simplified depiction of the spectra that are found experimentally. The black curves represent the isolated molecules. The molecules absorb at a higher energy and emit at a lower energy (emission is redshifted with respect to absorption). When there is cofacial dimer (R= 4 angstrom) there is blue shift of absorption because the you can only go to S2. If there is no geometry relaxation in the excited state there will be no emission. With relaxation on one molecule there will be reduction of symmetry and some emission that is red shifted with respect to the emission in the isolated system.

The larger the transition dipole ( meaning larger oscillator strength, larger electronic coupling) the shorter the excited state lifetime. Strong coupling means there is a high probability that the transition will occur and there will be higher quantum yield. Probability determines the time in the excited state. The longer the molecule stays in the excited state the higher the more time for the system to find vibrations that will allow it to go back to the ground state by other means such as non-radiative decay. This results in a decrease in luminescence quantum yield.

For example there is electron transfer that can take place between PPVm derivative and fullerene C60 that occurs in 35 femptoseconds. That is so short that nothing else can occur. In that case electron transfer has 100% quantum yield. In general the process that is fastest that is the one that will dominate all others.

== References ==
<references/>

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photochromism|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>

</tr>
</table>

Interchain Interactions

2009-08-27T01:00:52Z

Nsylvain: /* Transition dipole in isolation */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photochromism|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>

</tr>
</table>

=== Interchain interactions ===

This topic is an important consideration with light emitting polymer and oligomers. Photoluminescence is easily measured by shining light and measuring the emission.

In dilute solutions or inert matrices the photoluminescence quantum yield of a given conjugated polymers can be very large: up to 60-80 %

But in thin films the solvent goes away, the polymer chains get closer so they can interact and the luminescence can be nearly totally quenched.

There are a few situations where going from solution to solid states you can get increased luminescence. Silos for example can cause an aggregation in the solid state that triggers luminescent qualities. The observation of quenching led researcher to explore the cause and remedy for this.

=== Transition dipole in isolation ===
[[Image:Stilbene_excitation.png|thumb|300px|Transitions in stilbene are polarized along the x axis]]
A transition dipole is directly related to the probability of a transition occurring from one excited state to the ground, or the ground to the excited state. It is also related to the intensity of the exciting energy.

If the wave function for the S1 state is localized in a different area of space than the ground state then transition dipole will be zero. There must be wave function overlap for the wavefunction equation to be non-zero.

In stilbene the transition dipole for the S0 → S1 transition and the S1 →S0 transition is polarized mainly along the long axis (x axis) of the molecule. This is generally the case in pi conjugated systems because as you extend the length of the chain electrons are delocalized over a longer distance, just as in an electrostatic dipole a larger distance of charge separation make for a larger dipole moment. An exception is pentacine whose first optical transition is polarized at 90 degrees from the long axis of the chain. In most other optical polymers such as polyvinylene and polythiophene the optical transition is polarized along the long axis.

see Cornil <ref>J. Cornil et al., JACS 120 (1998) 1289 DOI: [http://doi.org/10.1021/ja973761j 10.1021/ja973761j]</ref>

=== Two chain interaction ===
[[Image:Interchain_coord.png|thumb|300px|Molecules are represented as planes, transition dipole is along X]]

When two chains come into close proximity they there is the possibility during excitation the electron will jump from one chain to another, leaving a hole and forming a charge transfer exciton. The probability of finding h+ and e- on separate chains in S1 can be obtained from the excited-state wavefunction .

S1 → S0 intensity can go down since the transition is polarized along x.

 

=== Symmetrical interaction ===
[[Image:Interchain_distant.png|thumb|300px|At a distance greater than 8 Å there is no interaction]]
Consider first the most symmetric configuration which is perfectly cofacial.

From a distance of infinity down to 8 Å there is no significant interaction. There is no significant wavefunction overlap between the units.

*excitation is always localized on a SINGLE UNIT
*PL is not affected

This is the situation in dilute solution or inert matrices

[[Image:Interchain_close.png|thumb|300px|Below 8 Å there is band level splitting]]
When the distance (R) drops below 8 Å , as they do in thin films, the wavefunctions of frontier orbitals (homo and lumo) begin to overlap and you get a new splitting of the homo level. Wavefunctions are delocalized over the two units. They become equally spread in the zone of r~5 Å.
 

=== Exciton band formation ===
[[Image:Bandsplitting.png|thumb|400px|Split levels create forbidden transitions]]
The HOMO and LUMO may be split to different degrees. In a highly symmetrical situation the transition from the ground state to the lowest excited state (S1) is forbidden. The HOMO (au) and the LUMO (bu) is are both ungerade which means the transition is forbidden (because odd time odd is even, times odd is odd and when this is integrated over all space the result is zero).

If you were able to stack five molecules on top of each other, you could have five exciting states corresponding the mixing of the five states corresponding to the five S1 states, this is known as an exciton band. Even if a higher excited state such as S2 is achieved, Kasha’s Rule dictates that energy will relax down to lower states and the emission can only occur from the lowest excited state. S1 can not transition to S0, it is a “dark state” .The lowest state of that exciton band will be dark and luminescence is quenched.
 
[[Image:Kasha.png|thumb|300px|Parallel and antiparallel orientations of transition dipole]]
Kasha’s model from goes back fifty years and attempts to explain the emission from the lowest state. He only considered the wavefunction of one molecule interacting with that of another molecule. He did not consider mixing of wavefunctions so a charge transfer state with an electron on one molecule and a hole on another is impossible in his model. The diagram shows two molecules with transition dipoles along the main axis of each, one dipole parallel and the other anti-parallel. When the two dipoles are opposing the energy is less than the parallel situation but the oscillator strength is zero. In the parallel situation the oscillator strength is large and but the energy required to get there is higher.
 
[[Image:Bandsplitting2.png|thumb|300px|Interaction between molecules leads to band splitting]]
So when two transition dipoles interact, the two S1 states split forming a two state exciton band with S1 and S2. The lower ground state S1 is most favorable because it formed by the antiparallel orientation of the transition dipoles. The S2 has a high oscillator strength (its bright) but higher energy, S1 is lower energy but has low oscillator strength (its dark). So in the highly symmetrical situation molecule interact and illumination is quenched.
 
[[Image:Cofacial_energy.png|thumb|300px|Transition energy vs distance between molecules. Blue and red lines show shift of interacting molecules due to bandsplitting.]]
As molecules get closer and closer (approaching 4 angstroms) in cofacial configuration their wavefunctions interact and there is a splitting into S1 and S2. This results in a blue shift of the aborption because the molecule has to get up to the higher S2 state since S1 is not optically coupled with the ground state.
 

=== Geometry relaxation effects ===
[[Image:Geometry_relax.png|thumb|300px|Unequal relaxation due in excited state]]
If a conjugated system goes into an excited state there will be geometry relaxation. Considering two molecules one on top of another there will be unequal relaxation. In the this case the top molecule achieves lowest excited state (LES) geometry while the lower molecule stay in ground state (GS) geometry. Excitation into S2 trickles down to S1 state and then localizes on one of the molecules. The two molecules do not have the same geometry and the system has less symmetry. This asymmetric geometry deformations allow it to break the selection rules. Thus relaxation allows for a little bit of electronic coupling and some emission.
 
[[Image:Cofacial_energy_shift.png|thumb|300px|Emission and Absorption intensity vs energy]]
This is a simplified depiction of the spectra that are found experimentally. The black curves represent the isolated molecules. The molecules absorb at a higher energy and emit at a lower energy (emission is redshifted with respect to absorption). When there is cofacial dimer (R= 4 angstrom) there is blue shift of absorption because the you can only go to S2. If there is no geometry relaxation in the excited state there will be no emission. With relaxation on one molecule there will be reduction of symmetry and some emission that is red shifted with respect to the emission in the isolated system.

The larger the transition dipole ( meaning larger oscillator strength, larger electronic coupling) the shorter the excited state lifetime. Strong coupling means there is a high probability that the transition will occur and there will be higher quantum yield. Probability determines the time in the excited state. The longer the molecule stays in the excited state the higher the more time for the system to find vibrations that will allow it to go back to the ground state by other means such as non-radiative decay. This results in a decrease in luminescence quantum yield.

For example there is electron transfer that can take place between PPVm derivative and fullerene C60 that occurs in 35 femptoseconds. That is so short that nothing else can occur. In that case electron transfer has 100% quantum yield. In general the process that is fastest that is the one that will dominate all others.

== References ==
<references/>

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photochromism|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>

</tr>
</table>

Interchain Interactions

2009-08-27T00:57:40Z

Nsylvain: /* References */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photochromism|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>

</tr>
</table>

=== Interchain interactions ===

This topic is an important consideration with light emitting polymer and oligomers. Photoluminescence is easily measured by shining light and measuring the emission.

In dilute solutions or inert matrices the photoluminescence quantum yield of a given conjugated polymers can be very large: up to 60-80 %

But in thin films the solvent goes away, the polymer chains get closer so they can interact and the luminescence can be nearly totally quenched.

There are a few situations where going from solution to solid states you can get increased luminescence. Silos for example can cause an aggregation in the solid state that triggers luminescent qualities. The observation of quenching led researcher to explore the cause and remedy for this.

=== Transition dipole in isolation ===
[[Image:Stilbene_excitation.png|thumb|300px|Transitions in stilbene are polarized along the x axis]]
A transition dipole is directly related to the probability of a transition occurring from one excited state to the ground, or the ground to the excited state. It is also related to the intensity of the exciting energy.

If the wave function for the S1 state is localized in a different area of space than the ground state then transition dipole will be zero. There must be wave function overlap for the wavefunction equation to be non-zero.

In stilbene the transition dipole for the S0 → S1 transition and the S1 →S0 transition is polarized mainly along the long axis (x axis) of the molecule. This is generally the case in pi conjugated systems because as you extend the length of the chain electrons are delocalized over a longer distance, just as in an electrostatic dipole a larger distance of charge separation make for a larger dipole moment. An exception is pentacine whose first optical transition is polarized at 90 degrees from the long axis of the chain. In most other optical polymers such as polyvinylene and polythiophene the optical transition is polarized along the long axis.

see Cornil <ref>J. Cornil et al., JACS 120 (1998) 1289</ref>

=== Two chain interaction ===
[[Image:Interchain_coord.png|thumb|300px|Molecules are represented as planes, transition dipole is along X]]

When two chains come into close proximity they there is the possibility during excitation the electron will jump from one chain to another, leaving a hole and forming a charge transfer exciton. The probability of finding h+ and e- on separate chains in S1 can be obtained from the excited-state wavefunction .

S1 → S0 intensity can go down since the transition is polarized along x.

 

=== Symmetrical interaction ===
[[Image:Interchain_distant.png|thumb|300px|At a distance greater than 8 Å there is no interaction]]
Consider first the most symmetric configuration which is perfectly cofacial.

From a distance of infinity down to 8 Å there is no significant interaction. There is no significant wavefunction overlap between the units.

*excitation is always localized on a SINGLE UNIT
*PL is not affected

This is the situation in dilute solution or inert matrices

[[Image:Interchain_close.png|thumb|300px|Below 8 Å there is band level splitting]]
When the distance (R) drops below 8 Å , as they do in thin films, the wavefunctions of frontier orbitals (homo and lumo) begin to overlap and you get a new splitting of the homo level. Wavefunctions are delocalized over the two units. They become equally spread in the zone of r~5 Å.
 

=== Exciton band formation ===
[[Image:Bandsplitting.png|thumb|400px|Split levels create forbidden transitions]]
The HOMO and LUMO may be split to different degrees. In a highly symmetrical situation the transition from the ground state to the lowest excited state (S1) is forbidden. The HOMO (au) and the LUMO (bu) is are both ungerade which means the transition is forbidden (because odd time odd is even, times odd is odd and when this is integrated over all space the result is zero).

If you were able to stack five molecules on top of each other, you could have five exciting states corresponding the mixing of the five states corresponding to the five S1 states, this is known as an exciton band. Even if a higher excited state such as S2 is achieved, Kasha’s Rule dictates that energy will relax down to lower states and the emission can only occur from the lowest excited state. S1 can not transition to S0, it is a “dark state” .The lowest state of that exciton band will be dark and luminescence is quenched.
 
[[Image:Kasha.png|thumb|300px|Parallel and antiparallel orientations of transition dipole]]
Kasha’s model from goes back fifty years and attempts to explain the emission from the lowest state. He only considered the wavefunction of one molecule interacting with that of another molecule. He did not consider mixing of wavefunctions so a charge transfer state with an electron on one molecule and a hole on another is impossible in his model. The diagram shows two molecules with transition dipoles along the main axis of each, one dipole parallel and the other anti-parallel. When the two dipoles are opposing the energy is less than the parallel situation but the oscillator strength is zero. In the parallel situation the oscillator strength is large and but the energy required to get there is higher.
 
[[Image:Bandsplitting2.png|thumb|300px|Interaction between molecules leads to band splitting]]
So when two transition dipoles interact, the two S1 states split forming a two state exciton band with S1 and S2. The lower ground state S1 is most favorable because it formed by the antiparallel orientation of the transition dipoles. The S2 has a high oscillator strength (its bright) but higher energy, S1 is lower energy but has low oscillator strength (its dark). So in the highly symmetrical situation molecule interact and illumination is quenched.
 
[[Image:Cofacial_energy.png|thumb|300px|Transition energy vs distance between molecules. Blue and red lines show shift of interacting molecules due to bandsplitting.]]
As molecules get closer and closer (approaching 4 angstroms) in cofacial configuration their wavefunctions interact and there is a splitting into S1 and S2. This results in a blue shift of the aborption because the molecule has to get up to the higher S2 state since S1 is not optically coupled with the ground state.
 

=== Geometry relaxation effects ===
[[Image:Geometry_relax.png|thumb|300px|Unequal relaxation due in excited state]]
If a conjugated system goes into an excited state there will be geometry relaxation. Considering two molecules one on top of another there will be unequal relaxation. In the this case the top molecule achieves lowest excited state (LES) geometry while the lower molecule stay in ground state (GS) geometry. Excitation into S2 trickles down to S1 state and then localizes on one of the molecules. The two molecules do not have the same geometry and the system has less symmetry. This asymmetric geometry deformations allow it to break the selection rules. Thus relaxation allows for a little bit of electronic coupling and some emission.
 
[[Image:Cofacial_energy_shift.png|thumb|300px|Emission and Absorption intensity vs energy]]
This is a simplified depiction of the spectra that are found experimentally. The black curves represent the isolated molecules. The molecules absorb at a higher energy and emit at a lower energy (emission is redshifted with respect to absorption). When there is cofacial dimer (R= 4 angstrom) there is blue shift of absorption because the you can only go to S2. If there is no geometry relaxation in the excited state there will be no emission. With relaxation on one molecule there will be reduction of symmetry and some emission that is red shifted with respect to the emission in the isolated system.

The larger the transition dipole ( meaning larger oscillator strength, larger electronic coupling) the shorter the excited state lifetime. Strong coupling means there is a high probability that the transition will occur and there will be higher quantum yield. Probability determines the time in the excited state. The longer the molecule stays in the excited state the higher the more time for the system to find vibrations that will allow it to go back to the ground state by other means such as non-radiative decay. This results in a decrease in luminescence quantum yield.

For example there is electron transfer that can take place between PPVm derivative and fullerene C60 that occurs in 35 femptoseconds. That is so short that nothing else can occur. In that case electron transfer has 100% quantum yield. In general the process that is fastest that is the one that will dominate all others.

== References ==
<references/>

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Photochromism|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>

</tr>
</table>

Photochromism

2009-08-27T00:57:09Z

Nsylvain: /* References */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Absorption and Emission|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Interchain Interactions| Next Topic]]</td>
</tr>
</table>

=== Photochromism ===

Photochromism is a reversible transformation of a chemical species induced in one or both directions by absorption of electromagnetic radiation between two forms, A and B, having different absorption spectra.

A common example is the sunglasses that turn dark when exposed to sunlight. In order to qualify as photochromic the different forms of the species must have different absorption spectra.

Photochromism is derived from the Greek words: phos (light) and chroma (color).
When light of wavelength λ1 is exposed to species A it goes to B and when B is exposed to light with wavelength λ 2 or in the presence of heat, B goes to A.

[[Image:Photochromic_reaction.png|thumb|300px|]]

[[Image:Photochromic_spectra.png|thumb|300px|Species A gives rise to species B with a different absorption spectrum.]]

The example spectra of A absorbs around 350 nm and B absorbs around 500 nm. One of the ways that you can control the transition is by changing the wavelength of the absorbing light, if they have different spectra. Photochromic materials are used in optical memory in which case you want A to go B and stay there. In this case it desirable that the heat driven transition from B back to A be very small.

The first report of photochromism was given by Fritzsche 1867 <ref>J. Fritzsche. Comptes Rendus Acad. Sci., Paris, 69,1035 (1867)</ref> who observed the bleaching of an orange-colored solution of tetracene in the daylight and the regeneration of the color in the dark.

Tetracene is a four ring conjugated colored molecule. When the crystals are exposed to sunlight the molecule is broken into the two smaller molecules neither of which absorbs in the visible also known as bleaching. There is transient bleaching, permanent or semi transient types of bleaching.

See Photochromism by Brown<ref>Photochromism,G. H. Brown (Ed.)(Techniques of Chemistry Vol. III), Wiley-Interscience, New York,1971 (853 pp.).</ref>

See also Photochromism, Molecules, and Systems by Dürr<ref>Photochromism, Molecules, and Systems ,H. Dürr and H. Bouas-Laurent (Eds.)(Studies in Organic Chemistry 40), Elsevier, Amsterdam, 1990(1068 pp.).</ref>

See also Wikipedia on Photochromism [http://en.wikipedia.org/wiki/Photochromism]

=== Interconversion of Isomers ===

A photochromic conversion of A to B, and B to A can be described by the equations:
:<math>
N_{A\rightarrow B} = N_A .f(\epsilon^A(\lambda), I(\lambda)). \phi_{A \rightarrow B}\,\!</math>

:<math>N_{B\rightarrow A} = N_A .f(\epsilon^B(\lambda), I(\lambda)). \phi_{B \rightarrow A}+ N_B .f’(T)\,\!</math>

Where:

:<math>f(\epsilon^X(\lambda), I(\lambda))\,\!</math> is the fraction of molecules of the species x that is excited by the light per unit time,
:<math>f(\epsilon^X(\lambda), I(\lambda))\,\!</math> is a function of the extinction coefficient of the molecule :<math>[\epsilon^X(\lambda)]\,\!</math> at wavelength l and of the light intensity :<math>[I(\lambda)]\,\!</math> at that wavelength (the absorption spectra)

:<math>\phi_{A\rightarrow B}\,\!</math> is the quantum yield the conversion of A to B

:<math>\phi_{B\rightarrow A}\,\!</math> is the quantum yield of the conversion of B to A

:<math>f'(T)\,\!</math> is the fraction of molecules B that thermally convert to A per unit time, it is a function of temperature [T] (thermal reversal)

The rate of absorption is the rate that molecules go from the ground to the excited state because of the collision with a photon. The rate law for this chemical reaction would be dependent on the concentration of A and the rate of photo flux and the probability of a photo being absorbed. In order for A to go to B it must achieve an excited state. The quantum yield predicts how often the excited state will lead to the formation of B as opposed to decaying, giving off a photo or other non-productive outcomes.

==== Interconversion – poor coupling ====
[[Image:interconv_poorcouple.png|thumb|400px|Potential energy diagram for interconversion with poor coupling]]
Consider the exited state and ground state potential surfaces. If you assume that the excited state potential well between A and B are not strongly coupled or do not interact then the photon is absorbed into the excited state, over a period of 100s of femptoseconds the molecule relaxes down to the lower vibrational levels within the excited state, and then it can go back down buy fluoresence or non-radiative decay. If the excited state of A goes directly to the ground state of B that will be a temperature independent process because it is extremely fast and because the excited state can slide easily back to the ground state of A because there is no thermal barrier. This pathway is intensity dependent.

 

==== Interconversion- strong coupling ====
[[Image:Interconv_goodcouple.png|thumb|400px|Energy diagram for interconversion with strong coupling]]

If there is strong coupling of the two potential wells in the excited state of the two species then energy moves by a thermally activated process in the excited states. The ground state of B is not forming directly, instead there is an excited state of B which then goes to the ground state by fluorescence or non-radiative decay. This pathway is temperature dependent because there is thermal barrier to convert excited A to excited B. You can control the outcome of reaction and its equilibrium by changing the temperature. However it is not dependent on intensity.
 

==== Photostationary State ====
A photostationary state occurs when under irradiation the rate of A → B equals the rate of B → A at dynamic equilibrium. This is dynamic because we are not a thermal equilibrium, we are continually adding photons. The details of the photons makes a big difference. Suppose you have a molecule A that absorbs at exactly 400nm and its counterpart B absorbs at 500nm. If you excite this molecule at 400nm and there is some quantum yield of A going to B and there little thermal conversion from B to A then the photostationary state will be 100% B. You are exciting A it goes to B but it has no way of getting back. If you excite at 500nm the the photostationary state will be 100% A. The distinction between the equilibrium and the photostationary state is that the latter depends on the wavelength of the light source, the overlap with the absorption spectra of the two molecules and the quantum yields for each of the reactions. It is the number of number of A or B over the total number of molecules.

The photostationary composition is defined as:

:<math> N_{AorB}/(N_A+N_B)\,\!</math> at dynamic equilibrium

Here

:<math>N_{A\rightarrow B}\,\!</math>= :<math>N_{B\rightarrow A}\,\!</math>

Thus:

:<math>N_A .f(\epsilon^A(l), I(\lambda)). \phi_{A\rightarrow B} = NB .f(\epsilon^B(\lambda), I(\lambda)). \phi_{B\rightarrow A}+ NB .f’(T)\,\!</math>

The rate is determined by number of molecules, the overlap of extinction coefficient and the intensity of the light at each wavelength and the quantum yield. Light sources rarely have equal intensities over all wavelengths. When the two parts of the equation are equal there is photostationary state.

=== Isosbestic Point ===

Consider the absorption spectra of a composite comprised of A and B. The absorption at λ equals the number of molecules of A times the extinction coefficient of A at λ plus the number of molecules of B plus the extinction coefficient of B at λ.

At any wavelength the absorption at a given wavelength, A(λ), of a two-component system (photochromism is a special case) is given by:

:<math>A(\lambda)= N_A \epsilon_A^{\lambda} + N_B\epsilon_B^{\lambda}\,\!</math>
[[Image:Isosbestic.png|thumb|300px|]]

If at one wavelength the extinction coefficients are equal they can be factored out:

:<math>\epsilon_{A}^{\lambda} = \epsilon_{B}^{\lambda}\,\!</math>

That wavelength defines the isosbestic point because:

:<math>A(\lambda)= \epsilon^{\lambda} (N_A + N_B)= \epsilon^{\lambda} N_{tot}\,\!</math>

The isobestic point is the point where the absorption remains constant.
Thus at the isosbestic point the absorption is independent of the relative concentration of the two forms. Chemists often use this property to follow reactions that are not necessarily photochromic reactions. Assume that A goes to B but in order to get there it had to form C.

:<math>A \Leftrightarrow C \Leftrightarrow B\,\!</math>

If C has a different absorption spectra than A or B the absorption spectra then all the curves would not cross at a single point. So when you run absorption curves and observe an isosbestic point that means there are no intermediates going from A to B that exist for an appreciable time. A very rapid formation and breakdown of C will not change the absorption spectra. If there is an isosbestic point either there are no intermediates, ie it’s a very clean reaction, or the intermediates have a very short lifetime.

=== Photochemical Reaction Fatigue ===
Photochemical fatigue has for many years limited the use of photochromic materials for optical memory. Even today rewriteable discs do not use photochromic materials, instead they utilize phase change going from one crystalline form to another that results in different reflectivity. A magnetic memory is written over hundreds of thousand of times, this is the requirement. For a photochemical to be useful it must have a very low quantum yield for any decomposition pathway. The quantum yield is independent of light intensity.

Recall that quantum yield for a process X is given by:
:<math>\phi_x \,\!</math> = # of molecules that undergo process X/# of photons absorbed

or

:<math>= k_x/ \sum k\,\!</math> for all processes leading to loss of excited-state population.

For photochromic reaction some important rates constants to consider are:

:<math>k_{I\rightarrow>II}\,\!</math> rate of photochromic transformation

:<math>k_r\,\!</math> rate of radiative decay (fluorescence)

:<math>k_{nr}\,\!</math> rate of nonradiative decay

:<math>k_{bl}\,\!</math> rate of photobleaching

=== Example Photochromic Reaction ===
[[Image:Spiropyran.png|thumb|300px|Spiropyran photochromic reaction]]
An example of a photochromic reaction involving a spiropyran which is a merocyanin. This is a conjugated molecule. On the B form the O- is donor and the C double bond N+ is an acceptor. This leads to a charge transfer absorption band that makes the molecule purple. Whereas the A form has a benzene ring with a nitrogen that absorbs in the UV. The oxygen transfers charge to the nitrogen and this probably makes the molecule somewhat yellow. Ideally you want the quantum yield for phi 1 (A going to B) equal to the quantum yield for phi 2 (B going to A) and equal to 1. Thus every time you put in photon in it goes back and forth, and you control which way it goes by the wavelength of the photon.

Ideally :

:<math>\Phi_1 = \Phi_2 >= 1\,\!</math> and :<math> \Phi_3 = 0\,\!</math>

Where:

:<math>\Phi_1\,\!</math> is the quantum yield for the forward reaction

:<math>\Phi_2\,\!</math> is the quantum yield for the reverse reaction and

:<math>\Phi_3\,\!</math> is the decomposition quantum yield

=== Photochemical Degradation ===

However, if the quantum yield for degradation (i.e., formation of C) is f3, the non-degraded fraction y after n cycles will be:

:<math>y = (1 - \phi_3)^n\,\!</math>

For very small f3, and when n is very large, the expression can be approximated as:

:<math>y = (1 -n\phi_3)\,\!</math>

Thus, for even a highly reversible reaction with f3 = 0.001 (yield = 99.9%), after 1,000 cycles, 63% of A will be lost, and after 10,000 cycles effectively all of A will be destroyed. In synthetic chemistry a 90% yield is considered very good, here we are talking about a 99.9% yield. Thus you need a reaction with decomposition quantum yield of 10-5 or 10-6 to be useful for optical memory.

=== Cis- Trans Isomerizations ===
[[Image:Stilbene_azobenzene.png|thumb|300px|Stilbene and Azobenzene]]
There has been a lot of work molecules that use pericyclic and isomerization reactions to optimize them so they can be fully reversible. One reaction that was looked at early on was the cis-trans isomerization of stilbene. Common in stilbenes, azo compounds, azines, thioindigoids, etc., as well as some photochromic biological receptors that are part of living systems.
These molecules have different absorption spectra. They can have different groups added causing the molecules to twist. If a benzene ring on stilbene were to twist 90 degrees it would no longer participate in the π conjugation. The benzene and vinyl group that are left constitute styrene. The conjugation length is greater in stilbene than in the styrene. In a particle in a box picture the box is longer with stilbene and therefore the orbitals will be closer in energy. Beta carotene is a polyene with 11 double bonds, and it that makes orange juice the color orange. As molecules get longer the gap between the homo and lumo is smaller. So if you do something to a molecule that changes the conjugation length the absorption will change. The conjugation length requires orbitals and overlap. As you go from trans-stilbene to cis-stilbene you would expect a higher energy absorption. This system can undergo an electrocyclic ring closure resulting a dihydrophinanthrine. This molecule is not aromatic but it is 2 hydrogens away from being aromatic. A general rule is if a molecule is a step away from being aromatic the molecule will find a way to get there. This is an example of photochemical fatigue. The oxygen will pull out the two hydrogen. This is thermally reversible but once a molecule gets in the aromatic state the molecule tends to stay there.

Another classic example is azobenzene which also undergoes cis-trans isomerization. The cis form is not thermally stable and it reverts to the trans form. Cis stilbene are more stable than cis azobenzenes.

*The cis form is not thermally stable with respect to the trans form, so the cis form reverts back to the trans form in the dark.

*cis-stilbenes are more stable than their azobenzenes counterparts.

*The photochromic quantum yield is temperature dependent for both trans-stilbene, trans-azobenzene.

=== Heterocyclic Cleavage ===
[[Image:Spiropyran_spirooxane.png|thumb|300px|Spiropyran and Spirooxazine have dramatic photochromic shifts but poor reversability]]
Merocyanines and spiropyrans have a heterolytic bond cleavage, the oxygen takes two electrons, the nitrogen gives up an electron and the electron does a cis-trans isomerization.

These molecules are characterized by relatively poor photochromic properties:

*low conversion quantum yields

*fast thermal back reaction

These molecules are used for various photochromic reactions but they break down after 10 ^5 repetitions. Their change in optical properties are dramatic and can be used for low repetition application such as real time holography in which an image is written but only needs to be changed a few times.

== References ==

<references/>

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Absorption and Emission|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Interchain Interactions| Next Topic]]</td>
</tr>
</table>

Absorption and Emission

2009-08-27T00:56:29Z

Nsylvain: /* Simulation of Emission */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Transition Dipole Moment|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Photochromism| Next Topic]]</td>
</tr>
</table>

=== Vibronic Progression ===

Vibronic progression can be seen in the absorption and emission spectra. Vibronic progression means that there is a coupling to vibrational modes in your polymer or oligomer as you have an excitation from the ground state to the excited state, or emission from the excited state down to the ground state.

=== Change in excited state geometry ===

[[Image:Transpolyacetylene_vib.png|thumb|300px|Changes in transpolyacetylene during excitation]]
Consider the first optically allowed excited state of the transpolyacetylene which has a 1 Bu symmetry. As the molecule moves from the ground state to that first optically excited state, there is a shift of the double bonds from one bond to the next resulting in a shift of the π bond densities from one bond to the next. This is described as the reversal of the bond length alternation pattern. If there is a single excitation on a long polyacetylene chain, that modification will be localized somewhere.

For PPV, you start with a chain where the rings are aromatic-like, with single, double, single bonds between the rings. The wave functions of the HOMO and the LUMO, and the π bonding and anti- bonding pattern dictates that as you go to the excited state and an electron leaves the HOMO and goes into the LUMO, you will shift the π-bond densities in such a way that you end up with a quinoline-like structure. In between the rings, there is a double bond, single bond, double bond, or a larger factor of that kind .
 

=== Vibrational modes ===

[[Image:Transpolyacetylene_vibstretch.png|thumb|300px|Changes in bond lengths during excitation for polyacetylene]]
Because of the π bond densities shifting there will be geometry relaxation when moving from the ground state to the excited state. There will be a coupling with vibrational modes, that leads from the ground state geometry to the excited state geometry. In the case of polyacetylene, these modes brings the top carbon closer to the right, the bottom carbon to the left, then the next top one to the right, bottom one to the left. The same pattern happens for PPV. In other words, there will be strong coupling to vibrational modes that correspond to C-C stretching. Remember the C-C stretching modes in those polymers are at the order of about 1400 - 1600 wave numbers and 0.15 - 0.2 eV.
 

=== Geometry relaxation in S1 of PPV5 ===
[[Image:C-cbondchange_exitation.png|thumb|400px|Change in bond length vs site position during excitation]]
Consider a linear representation of long PPV chain with about 5 rings in order to relate all the bond lengths directly with their modification, from the ground state to the excited state. This plot shows how the C-C bond length changes from the ground state to the S1 excited state in that five ring oligomer. From the plot, it can be inferred that the double bond increases significantly in length by .03 Å (angstrom), where as the single bond actually shortens by about .03 angstrom or more. In other words, the diagram shows the difference in bond length as you go from the optimal geometry in the ground state to the optimal geometry of the S1 state. It is clear that the vinylene double bonds increase in length and the single bonds between rings decrease in length. There is a reversal of the bond length alternation within the vinylene units. Also, within the quinoid rings, you do get an evolution of the bonds that leads you to a more quinoid structure. The geometry relaxation to form the S1 excited state in this oligomer takes place over something like 20 carbons. Polaron relaxation takes place was over about 20 carbons as does the exciton in long chains of PPV.
 

=== Transitions and vibrational levels ===
[[Image:Potentialenergy_surfacechang.png|thumb|400px|The potential energy surface of the ground state and of the excited state.The potential energy surface of the ground state varies over about 2 eV.]]

The abscissa of this graph is the deformation coordinates, which is how the system deforms; the modifications in the coordinates of you system. The lower figure is the optimal geometry of the ground state, and it is different from the optimal geometry of the S1 state that is shown above. The two potential energy surfaces are displaced with respect to one another. The larger the relaxation in the excited state, the larger the geometry of the modification in the excited state, and thus, the larger the displacement would be. If the geometry in the excited state were to be exactly the same as in the ground state, then the two potential energy curves would sit on top of one another. The vibrational levels corresponding to those C-C stretching modes are connected to the geometry relaxation in the excited state and therefore, they are connected to the electronic transition. The dashed curves show the zero vibrational level, the first vibrational level, the 2nd vibrational level and so on. At room temperature, the vibrational mode has energy of on the order of 0.15 to 0.2 electron volts. As you know, room temperature Kt is 0.025. That energy difference between the zeroth vibrational level and the first vibrational excitation is higher than the kinetic energy available at room temperature. In other words, in the ground state, basically all of the oligomers or polymers are in the zeroth vibrational level.

The curve of the dotted lines is the vibrational wave function. In the vibrational ground state, there is no node. In the first excitation state there is one node, and in the 2nd vibrational excited state there are 2 nodes, and so on. The same thing is what you would have if you were to simply look at a particle in the box. In the vibrational ground state, the largest probabilities to find the molecule or the polymer is in the middle. That is the state when the photons strike. The transition takes place faster than the geometry relaxation and thus, there is a vertical transition. Since the potential energy curves are displaced, if you start from the most probable situation in the ground state, and go up, you can have the largest overlap of the vibrational wave functions with the first or the second vibrational level. When there is a coupling of the electronic transition with vibrational modes, the intensity of your electronic transition is modulated by the overlap of the vibrational wave functions as you go up the excited state potential energy surface.

The rule of thumb is that the more the potential energy curve in the excited state is displaced, the larger the relaxation of the geometry in the excited state, and the larger the displacement, the higher the vibrational level, where there will be strong overlap between the ground state vibrational wave function of the electronic ground state and the excited vibrational wave function of the excited electronic state. However, if there were no displacement in the excited state or the two curves were perfectly superimposed on top of one another, then there will be a very strong overlap between the zero vibrational level of the bottom curve and the zero vibrational level of the excited state. But the overlap between the zero vibrational level of the bottom curve and any of the other ones above would be zero.

If the absorption into the first excited state of a pi conjugated system is extremely sharp, there is hardly any geometry relaxation in the excited state because one potential energy curve is right on top of the other. The broadness of the absorption curve indicates a large of the change in geometry in the excited state. For instance, cyanines are important in nonlinear optics. Cyanines and thalocyanines have extremely sharp optical transitions and absorptions. This is because the geometry of the ground state and the excited state are basically identical. However, with systems like oligo PPVs or other conjugated polymers or oligomers, there are geometry relaxations that are significant in the S1 state and the two potential energy surfaces are displaced. The larger the displacement, the higher the most intense transition will be going up the vibrational level scale of the top curve.

Vibronic progressions that go from 0 level below to 0 level above will be referred to as 0-0 absorption. Then you have the 0-1 absorption, 0-2 absorption, 0-3 and so on. For example, 0-2 absorption is the most intense and the geometry relaxations in the excited state will be pretty strong. For 0-1, there is a geometry relaxation but not as large as 0-2. A 0-0 absorption means there is not much geometry relaxation.

In emission first there is absorption and then the excited state comes down to the lowest vibrational level of the lowest excited state during its life time. So the emission will take place from from the 0 level. Then, depending on the overlap of the vibrational wave functions, it will come down to levels ground state levels 2,1 or 0. But in emission, the largest energy transition (or emission) will be the 0-0. Where as in absorption, since you start with the zeroth vibrational level of the ground state, the 0-0 will have the smallest energy of the transmission.

=== Stokes Shifts ===

[[Image:Abs_Emis_stokes.png|thumb|300px|Emission an absorption are offset from each other]]

Absorption starts at 0-0 and then 0-1, 0-2, 0-3, 0-4, and so on. The 0-2 absorption is the most intense. If the energy of the vibrational mode is 0.2 EV, or if 0-2 is the most intense, there is a relaxation of about 0.4 eV in the excited state because the transition from the zeroth vibrational level of the ground state to the 2nd vibrational level of the electronic excited state is most dominant. From that 2nd vibrational level, the system will trickle down to the zeroth vibrational level for a total of change of 0.4 eV.

When the 0-0 transitions in absorption and emission are on top of one another, there is no Stokes shift. When the 0-0 transitions are separated from one another, there is a Stokes shift.
For example, in the case for transpolyacetylene where the absorption is into the 1bu state, (which is actually the S2 state). From there the system trickles down to the S1 state. Then the very feeble emission that is obtained from the S1 state will be displaced, with respect to what you have in absorption. But rigorously speaking, the Stoke shift occurs only if the 0-0 transitions have different energies?. Often there is an abuse in the literature because in the literature, usually when you look at the absorption band, you can’t see all that fine structure and it makes it harder to decide where the 0-0 band is. For instance, if you consider the entire emission and absorption envelope the difference between the peaks in absorption and emission are sometimes referred to as a Stokes shift. In the case of PPV, there is no reason for any Stokes shift because the 0-0 transitions are the same energies.

A structured emission diagram gives a clue to the vibronic progression. The difference between the energies of those two peaks can usually give you the energy of the vibrational mode that is coupled to the electronic transition. However, when there are several modes that are coupled, things become quickly a bit more complicated. However, in those π conjugated systems, you can do very well with such an analysis.

=== Absorption and emission in oligophenylene vinylenes ===
[[Image:Stilbene_abs_emis.png|thumb|300px|Experimental absorption and emission spectra of oligophenylene vinylenes. 77K in inert PMMA matrix.]]
These are oligophenylene vinylenes going from two rings (n is the number of rings) to five rings. When n = 2, it is stilbene. The graph shows the redshift of the absorption as the systems becomes increasingly conjugated. Also, for the shorter species, the 0-2 transition is almost as intense as the 0-1 transition. As the chain elongates, the 0-1 becomes slightly more intense than the 0-2 but there is not much difference there. On the other hand, in the case of the emission, there is a pretty strong evolution. The stilbene 0-1 is by far the most intense where as for the five ring oligomer, the 0-0 is more intense more intense than the 0-1. This reflects something pretty simple. Think in terms of ionization: Taking away an electron from the stilbene that has 14 π electrons results in a big perturbation. Conversely, taking away an electron from a longer chain with 100 π electrons will give a smaller perturbation. The same applies for the excitation. An excitation of a very small molecule will give a pretty large predabation. As the chain elongates the perturbation becomes less important. As the chain gets longer and longer, the excitation eventually localizes over four or five rings and ends up having the same type of relaxation beyond the five or six ring system as would occur for a very long chain.

The electronic coupling between the ground state and the S1 state is the same whether you go from S0 to S1 or S1 to S0. Thus, the electronic coupling also referred to as the transition dipole or the oxidative strength is the same. A photon of the right energy will be absorbed with a very high probability. In emission, that will depend on the luminescence quantum yield. If you have a system that has a very small quantum yield then you will see very small photons come out in emission. But in terms of the intrinsic electronic coupling, transition dipole, between the electronic ground state and the electronic excited state, it is exactly the same; whether you go from S0 to S1 or S1 to S0.

Very often the emission curve will be sharper than absorption. The reason is that, unless you have a perfectly ordered system, usually there is some disorder so that the five ring chains can be slightly distorted in one way in one molecule and in a slightly different way in another molecule. Although the oligomers have exactly the same length, there will be a slight distribution of the optical transition depending on whether the chains are perfectly coplanar and therefore more conjugated, or the chains are more strongly twisted. This type of disorder leads to an absorption that is broader. The reason why is when you look at the first absorption band like at 8 EV, the perfectly coplanar chains can absorb at about 275 and other slightly distorted chains can absorb at 276, others at 277 and so on.

As soon as the energy of the photon is large enough, any molecule in the thin film will be able to absorb. Remember that the excited state has a given life time on the order of nanoseconds. During that life time, the excited state on a slightly distorted molecule can jump on a less distorted molecule and therefore, have a smaller excited state energy. Since the system is more conjugated, the band gap is smaller and the excited state energy is smaller. Therefore, what will happen is that the excitation will move across the material and will get to the most perfectly conjugated molecules because these are the ones for which the energy of the excited state is smallest. Because they emit only from the most well conjugated molecules, the emission will be from a single species where as in absorption, you can have not only the most conjugated but also the slightly distorted molecules that will also absorb. Absorption will indicate disorder in the system much more than emission will.

=== Simulation of Emission ===
[[Image:Emission_simulation_exp.png|thumb|300px|Simulation of absorption. S1 relaxation energy decreases as the chain grows.]]

The theoretical data matches the experimental data.
 
[[Image:Emission_simulation.png|thumb|400px|Simulation of emission predicts various transition energies.]]
In stilbene the relaxation energy is .27 eV. As you increase the length the relaxation comes down to .15eV. The relaxation energy as a function of 1/n shows a linear relationship. A relaxation energy of .15 eV for the S1 excited state is typical for many polymers. The 0-1 is almost as large as the 0-1 transition in simulating the emission spectrum.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Transition Dipole Moment|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Photochromism| Next Topic]]</td>
</tr>
</table>

Transition Dipole Moment

2009-08-27T00:55:53Z

Nsylvain: /* Summary */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Fluorescence Process|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Absorption and Emission| Next Topic]]</td>
</tr>
</table>

== Spectroscopy ==
[[Image:Transmittance_spectra.png|thumb|300px|A plot of (I/I0) x 100 vs. Wavelength gives the transmittance spectra]]
When measuring a spectrum, the simplest thing you can do is have a light source where you can vary the wavelength. The intensity of the light can be measured with a detector as a function of wavelength or as a function of frequency. Afterwards, by placing a sample in the detector called a spectrometer, you can take the ratio. In old spectrometers, and even still today in the new spectrometers, there are literally two paths: one for reference beam and one for sample beam. The spectrum that is referred to as a transmittance spectrum can be obtained by finding the ratio of the light that is going through the sample, dividing by the light that has been unperturbed, and multiplying that by a hundred. If the molecule does not absorb any light, then all the light goes through and there is a 100% transmittance. If the molecule absorbs 99% of the light, then there is a 1% transmittance. This is one way of doing that and that is referred to as transmittance spectrum.

For reasons of unknown origins, people look at different spectra in different modes. Typically when looking at color, chemists, not physicists, look at spectra in a mode that is referred to as absorbance. But when looking at infrared spectra we typically consider them in terms of transmittance. Absorbance and transmittance are related to one another by the Beer- Lambert law.

:<math>log \frac {l_0}{l} = \varepsilon cl = A\,\!</math>

Where
:<math>A\,\!</math> is the absorbance (a unitless quantity),

:<math>l\,\!</math> is the path length of the sample in cm,

:<math>c\,\!</math> is the concentration of the chromophore in the medium in mol-l-1

:<math>\varepsilon\,\!</math> is the molar absorptivity or extinction coefficient with units of l-mol-1cm-1.

The extinction coefficient characterizes the ability of a molecule to absorb light at a given wavelength

What this law simply says is that the log base 10 of the transmittance is equal to εcl or the absorbance. So the absorbance is simply the log of I0 over I. Keep in mind that absorbance is a unitless quantity. It turns out that absorbance can be related back to some characteristic features of the molecule and that is one of the reasons why it is used today. This formula shows that if there is a certain amount of a molecule and the path length is doubled, twice as much light will be absorbed. At least with the first order, if the concentration of a molecule is doubled, assuming that there are no intermolecular effects, twice as much light will be absorbed. Those are not molecular characteristics; they are characteristics of your sample. ε is related to how well the molecule absorbs light and it is referred to as the molar absorbtivity or the extinction coefficient. Since A (absorbance) is unitless, c is concentration given in moles per liter, and path length is typically given in cm, the units of the extinction coefficient are in: Liter mol-1 cm-1 liter or inverse molar, inverse centimeters. The extinction coefficient characterizes the ability of the molecule to absorb light of the given wavelength. It doesn’t matter whether the sample is liquid or gas. Chemist use log base 10 for the liquid phase, in gas phase spectroscopy you use an absorption cross section with a natural log. Often electrical engineers use natural logs as well.

== Oscillator Strength and Molecular Parameters ==

One important question that people ask is “What is the probability my molecule will absorb light?” One way to calculate this is by plotting your absorption spectrum not in wavelength units, but in energy units. The total absorption under the band is expressed with the integral. Note that a scale that is linear in wavelength is not linear in energy.

:<math>f = 4.32 \times 10^9 \int_{-\infty}^{\infty} \varepsilon \partial v\,\!</math>

This integration (dν) gives the '''oscillator strength''', which refers to the probability that a molecule interacting with light over a certain energy range is going to absorb that light.

:<math>f \approx 4.32 \times 10^9 \varepsilon_{max} \Delta v_{1/2}\,\!</math>

That oscillator strength can be approximated by imagining the spectrum as a triangle, and multiplying the extinction at the peak by the width of the band at the half height. This term f can be related back to the transition dipole moment which in turn can be related back to the wave function of the molecules.

==Transition dipole moment ==

If we can say how strong is an absorption band, we can integrate it and extract out the transition dipole moment and that goes right back to the wave function of the molecules, which can be calculated quantum- mechanically. This will allow a chance to think about certain points from what we know about molecular orbital theory.

:<math>f = 4.703 \times 10^{29} \bar{v} \mu^{2}_{ge}\,\!</math>

where

:<math>\bar{v}\,\!</math> is the mean absorption frequency of the band in cm–1

:<math>\mu^{2}_{ge}\,\!</math>, refers to the square of the transition dipole moment between the ground state and the excited state.

=== Dipole moment ===
In classical electrostatics, the energy due to the interaction of an electric field E and a dipole is given by the dot product of the dipole moment and the electric field vector.

:<math>energy = \vec{E} \cdot \vec{\mu}\,\!</math>

The dipole moment is related to the charge on an electron times the difference in charge between two things (so if you have two charges separated that would be two, times their distance.

:<math>\mu = e_n\Delta q_n r_n\,\!</math>

where:

:<math>e_n\,\!</math> is the elementary charge on the particle n (+ or -).

:<math>\Delta q_n\,\!</math> is the fractional charge.

:<math>r_n\,\!</math> is the distance of that particle from a reference coordinate

In order to get the total dipole moment in a system of many charges you have to sum over all those different vectors.

:<math>\mu_{tot}= \sum_{n} e_n \Delta q_n r_n\,\!</math>

You need to know the distance between the charges and the spatial distribution of the charge. Quantum mechanically the probability is related to the square of the wave function. Therefore, if we want to figure out what the dipole moment is by using quantum mechanics, instead of having those charges, we are going to keep track of those charges with the wave function. However, we will still have this r term, the distance between them; that is going to tell us what the moment is.

=== Quantum Description of Dipole Moment ===

Consider two electrons in p-orbitals. The distance is 2r between them. The main question here is “Where are the electrons and how can we keep track of them?” In quantum mechanics, the molecular orbital is a linear combination of the atomic orbitals. Therefore, the orbital has a certain description that is given by a linear combination. Take the integral of the wave function of the molecule times the sum of all the different charges, and multiplied again by the complex conjugate of the wave function. These track the position of the electrons in the molecule.

:<math>\mu = \int \Psi^*(\sum_n e_n r_n) \Psi \partial \tau\,\!</math>

Where
:<math>\Psi^*\,\!</math> is the complex conjugate of the wavefunction

:<math>r_n\,\!</math> is the position of the particle with respect to the coordinate system

:<math>e_n\,\!</math> is the charge on the nth particle

The previous formula can be given more succintly as

:<math>\mu =\int \Psi^* (R) \Psi \partial \tau\,\!</math>

Where

:<math>R\,\!</math> is the charge on the electron times that distance

=== Quantum Description of Transition Dipole Moment ===
It is also possible to obtain a transition dipole moment describing the transition moment between two different states.
The transition dipole moment between two states is the same integral as the previous dipole moment integral except now there is one in the ground state (g) and another in the excited state (e).

:<math>
\mu^{2}_{g,e} \propto \int \vert \Psi^{*}_{g} (R) \Psi_e \partial \tau \vert ^2\,\!</math>

The transition dipole moment involves different states. If they are the same states, for example, one is the ground state and the other is also in the ground state and you have an integral with ψ ground state R times complex conjugate ground state, that integral over all space will tell you the dipole moment. If the integral contains ψ excited state complex conjugate R ψ excited, that integral will tell you the excited state dipole moment. If one is in the ground and one is in the excited state, the integral will tell you the transition moment between the two states. This integral relates the wavefunction of the molecule back to the oscillator strength and the extinction coefficient.
That integral relates to the strength of the absorption band.

=== Odd and Even Functions and Transition Moments ===
[[Image:Ethylene_mo.png|thumb|300px|In ethylene the excited state must have different symmetry than the ground state.]]

It is important to understand the concept of '''even''' and '''odd''' functions. A function is considered to be even when f(x) = f(-x). An example of an even function is a parabola. An odd function is said to be odd when g(x) = - g(-x). An example of this is the cubic function (y= x3, y=x).

There are a few simple rules:
*The integral over all space for an even function is non zero.
*If you integrate an odd function over all space, that integral will come out to be identically 0.
*An even function times an even function gives an even function.
*An odd function times an odd function gives an even function.
*An even function times an odd function gives an odd function.

Suppose you have a molecule ethylene with a π and a π* orbital. Consider it’s symmetry with a mirror plane between them. The π orbital is even. The &p;* orbital is odd. The transition dipole moment operator goes as R as distance(i.e. y = x or y = R). The operator function is odd.

Can the integral of the ground state wave function times R times the excited state wave function be non-zero? Since the ground state is even, R is odd, the excited state is odd, the formula contains an even function times an odd function, which is odd, and then odd times odd, is even. Therefore it is possible to be nonzero. However, what if the excited state had the same symmetry as the excited state? For example, if you looked at a molecule like butadiene and drew out the molecular orbitals, you would find that certain transitions would be allowed and certain transitions would be forbidden just based on the symmetry.

=== Transition Moment Operator ===
[[Image:Ethylene_mo_transitiondipole.png|thumb|300px|The ground- and excited-state wavefunction must not have the same symmetry if the transition dipole moment is to be non-zero.]]
Thus, it is should clear that, since R is odd, for molecules where the ground and excited state wavefunctions are even or odd, the ground- and excited-state wavefunction must not have the same symmetry if the transition dipole moment is to be non-zero.

Consider the evolution of the wavefunction for a particle in a box upon excitation from the ground state (with no nodes) to the first excited state with one node.
 

=== Pseudo-physical description of transition dipole moment ===

[[Image:Wavefunction_boxed.png|thumb|300px|Wavefunction during transition]]
Initially the function is symmetric with respect to the axis of the one dimensional box.
In the final state it is also symmetrical, however you can envision a snapshot of the system as the light field is interacting with the wave-function wherein a node begins to develop as is shown in the middle and the wavefunction is evolving from the initial to final state.
Now consider that the electron density during process is the square of the wavefunction:

 
[[Image:Electrondensity_boxed.png|thumb|300px|Electron density during transition]]
As can be seen in the initial and final states the electron density is symmetrically distributed with respect to the axis of the box. However with the field on, the electron density is not symmetrically distributed and a transitory dipole moment can be present.

To relate back to real molecules think of each of those orbitals as a linear combination of atomic orbitals. One important factor is the symmetry. But there may be one other factor that will be just as important as symmetry. If you treat orbital 1 as a linear combination over n orbitals and orbital 2 as a linear combinations of orbitals as well, there will be a spatial over lap between the orbital in the ground state and the orbital in the excited state. If there is no spatial overlap between the ground state and excited state orbitals there will be no transition dipole moment. However, if the electrons are in the same place spatially, a large transition dipole moment will result.

Now if the wavefunctions are considered to be linear combinations of atomic orbitals then:

:<math>\mu^2 _{g,e} \propto \int \vert (\sum_i c_i \Phi_i )^{*}_{g} (R) (\sum_i \Phi_i)_e \partial \tau \vert ^2\,\!</math>

== Summary ==

*There needs to be significant spatial overlap throughout the molecule between the ground- and excited-state wavefunction, if the transition moment that couples the two states is to be large.

*Because of the position operator, R, if there is good overlap at sites that have a large distance from the origin of the coordinate system, these terms will have enhanced contributions to the transition dipole moment.

*Molecules with good spatial overlap between two states that are not of the same parity (symmetry) will have large transition dipole moments and strong (allowed) transitions in the electronic spectrum

*Conversely, if any of the above criteria are not met, then the transitions will be weak.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Fluorescence Process|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Absorption and Emission| Next Topic]]</td>
</tr>
</table>

Changes in Absorption Spectra

2009-08-27T00:55:11Z

Nsylvain: /* Solvatochromism */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Introduction to Absorption|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Jablonksi Diagram| Next Topic]]</td>
</tr>
</table>

=== Terminology for absorption shifts ===

[[Image:Bathochromic.png|thumb|300px|]]
Changes in chemical structure or the environment lead to changes in the absorption spectrum of molecules and materials. There are several terms that are commonly used to describe these shifts, that you will see in the literature, and with which you should be familiar.

:'''Bathochromic''': a shift of a band to lower energy or longer wavelength (often called a red shift).

:'''Hypsochromic''': a shift of a band to higher energy or shorter wavelength (often called a blue shift).

:'''Hyperchromic''': an increase in the molar absorptivity.

:'''Hypochromic''': an decrease in the molar absorptivity.

=== Solvatochromism ===
[[Image:Solvatocrhomism.png|thumb|300px|Negative and positive solvatochromism]]

If you do something that moves to lower energy with a longer wavelength, it is referred to as a Bathochromic shift or (also called) red shift. The color will move more toward the red. Conversely, if you have something that moves to higher energy, it will be referred to as a hypsochromic shift. If there is an increase in the absorptivity or cause the spectrum to become more intense, it will be referred to as a hyperchromic shift. But a decrease is referred to as a hypochromic shift. There is a variety of factors that can cause these changes. One of the factors is found in a process known as solvatochromism. This explains why certain molecules can, in a profound way, look very different in terms of their color depending on whether the molecules are in a polar or non-polar solvent.

Solvatochromism is the property of a molecule changing its color as a function of the solvent polarity. But it is actually more complex than that. It can be related to the solvent polarizability as well. Basically it is the change in the color of a material, or change in the spectrum, as a function of the dielectric properties of the solvent. The dielectric properties of the solvent have polarizability and polarity built into them. Therefore, if molecules go from a less polar solvent to a more polar solvent and a red shift or a bathochromic shift occurs, then the substance is referred to as being positively solvatochromic. Conversely if you put molecules into a more polar solvent and a blue shift occurs, i.e. higher energy, the molecules are referred to as being negatively solvatochromic.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Introduction to Absorption|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Absorption and Emission of Light|Return to Absorption and Emission Menu]]</td>
<td style="text-align: right; width: 33%">[[Jablonksi Diagram| Next Topic]]</td>
</tr>
</table>

Electronic States vs Molecular Levels

2009-08-27T00:54:08Z

Nsylvain: /* 2Ag excited state */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Electrical Properties|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>

</tr>
</table>

The literature can be confusing in using the terminology of electronic states vs electrical levels.

=== Point Groups and symmetry ===
[[Image:C2h_pointgroup.jpg|thumb|300px|Hexatriene C2h point group]]
There is always an identity symmetry (E). The hexatriene molecule is symmetrical around the C2 axis. The C2 axis is perpendicular from the plane of the molecule. A rotation of 180 degrees brings the same structure. The σ h denotes symmetry above and below the plane, there must be the same electronic density above the plane as below. There is also an inversion symmetry (i) where the C2 axis meets the plane. These together have a point group called C2h.

Molecular level is a '''one electron''' wavefunction which is by definition a molecular orbital. Molecular levels corresponds to molecular orbitals (HOMO, LUMO etc).

Molecular states (which is what you can observe) is a wavefunction that results from '''all the electrons''' in the molecule.

[[Image:sixMO_config.JPG|thumb|300px|Electronic configuration and molecular orbitals]]
The electronic configuration is shown with six molecular orbitals, three of which on the π level are fully occupied. You can build many more configurations in which individual electrons move up to the π* level, these are called '''singly excited'''. You can also elevate two electrons from occupied levels into unoccupied levels, called '''doubly excited''' configurations. The way you put the electrons into the orbitals corresponds to electronic configurations.

The wavefunctions can be classified according their symmetry. The lowest level is the fully bonding an is unsymmetrical; it is labeled '''ungerade''' (u- german for odd). The next level up has a node in the middle and it is symmetral and is labeled '''gerade''' (g - german for even). In π conjugated systems molecular orbitals alternative between gerade and ungerade with each level.

One can generate other electronic configurations by promoting one or more electrons from occupied levels to unoccupied levels. One promotion is known as "singly excited configuration", and two or three electrons promoted are called doubly or triply excited.

=== The Ground State ===

An electronic state is described by a weighted combination of electronic configurations. In many instances however the ground state and the lowest excited states can be described by only one or very dominant configurations.

In polyenes the ground state with all π and σ levels doubly occupied is denoted '''Ag''' symmetry, or '''1Ag'''. This is also called state S0.

The lowest ground stat π-type excited states can be of Ag or Bu symmetry.

=== 1Bu Excited state ===
[[Image:1Bustate.png|thumb|200px|The 1 Buexcited state]]
The 1 Bu excited state is dominated by a (singly excited) configuration with promotion of one electron from HOMO to LUMO. Suppose you shine a light on a molecule and cause an electron to enter the excited state. In order for the transition to be allowed you need to have different symmetries for the ground state and for the excited state. If the ground state is gerade (even) the excited state must be ungerade (odd). This is because transition dipole operator is an odd function.

This can be noted using Dirac notation.

:<math>\langle g \vert \overrightarrow{r} \vert u \rangle\,\!</math>

The integral over all space of initial state gerade (g) the the dipole operator (r) and the wave function for the final state (u). In order for the integral to non zero the product for all three terms must be even for a level. Therefore the transition to the 1Bu excited state is possible.

:<math>B_u symmetry \rightarrow 1B_u state\,\!</math>

If there were a transition to a gerade excited state the product (g, r and g) would be odd and therefore the integral would be zero. The probability for an electron to go from the 1Ag to the 2Ag state is taken as the square of the transition dipole function, so if the function is zero, its square is zero and therefore the probability of the transition is zero. If an molecule has no symmetry it go from the ground state to any excited state. As soon as a molecule has symmetry there are selection rules that apply. Experimentally it is impossible to excite a polyene molecule to the 2Ag state. The transition dipole is zero, this is known as a forbidden transition. The fact that a transition dipole is non-zero does not tell about the probability in itself; it could be exceedingly small probability. However a non-zero transition dipole has at least some probability and therefore is not forbidden transition.

=== 2Ag excited state ===
[[Image:2Agstates.JPG|thumb|300px|Three possible electronic states for the 2Ag excited state]]

For a long time the one-photon allowed 1Bu was thought to be the lowest possible excited state for a polyene molecule also called S1 (the lowest singlet excted state). But the papers by Hudson and Kolher showed that the lowest excited state in a long polyene is actually the '''2Ag''' (the second state of Ag symmetry). There is not a single electronic state that dominates that wave function. One involves the promotion from HOMO to the LUMO plus 1 which is gerade. A second case promotes an electron from the HOMO-1 to the LUMO, which is also gerade. There is also a doubly excited situation in which two electrons are promoted from the HOMO to the LUMO resulting in a gerade state.

As long as they have the same symmetries they can mix. These three configurations have approximately the same energies so they can mix strongly. They are one photon forbidden (but two photon allowed). This results in a 2Ag state that is energetically lower than the 1Bu state, thus it should be considered the S1 state.
 

[[Image:2Ag to 1Bu excitation.JPG|thumb|300px|Transition from 1Bu to 2Ag results in a metastable state.]]
There can not be an optical transition from the 1Ag to the 2Ag state but there will be absorption indicating a transition to the 1Bu state which is higher in energy. From there it can vibrate and lower its energy through the process of internal conversion to the 2Ag state where it stays because the transition to the 1Ag state is forbidden. This results in a metastable state which is observed as quenching of the luminescence.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Electrical Properties|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>

</tr>
</table>

Electrical Properties

2009-08-27T00:53:46Z

Nsylvain: /* Temperature and mobility */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Bloch's Theorem|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronic States vs Molecular Levels| Next Topic]]</td>
</tr>
</table>

=== Bandgap and conductivity ===
[[Image:BandGap-Comparison-E.PNG|thumb|300px|Bandgap diagram for metals, semiconductors and insulators]]
In order to have a net electrical current electrons must jump from completely filled levels to empty levels across the bandgap. If the bandgap (Eg) is large, upon applying an external electric field at room temperature, there will be few electrons that have the necessary energy to jump from the valence band to the conduction band. Thermal energy can in some cases provide enough energy to elevate an electron into the conduction band.

Thermal Energy is :

:at 300K: kT ~ 0.025 eV (1/40 eV)
:~ 0.6 kcal/mol
:~ 2.5 kJ/mol

For '''Eg >=2eV is considered an insulator'''. Very few electrons can jump the bandgap.

:σRT is typically;
:<= 10-10 Ω -1 cm-1
:<= 10-10 S/ cm

The best insulators go down to 10-16 S/cm

'''0 < Eg ≤ 2eV: semiconductor'''

:10-10 ≤ σRT ≤ 102 S/cm

:polyacetylene Eg ~ 1.5 eV

:Si Eg ~ 1.1 eV

for '''Eg → 0: metal'''

:σRT ≥ 102 S/cm

Electrical conductivities vary over 25 orders of magnitude from the best insulator to the best conductor. A single crystal of copper at room temperature has a conductivity of 6 x 105 S/cm

:σRT Cu ~ 6 x 105 S/cm

:Ag, Au ~ 106 S/cm

=== Conductivity Defined ===
Electrical conductivity (σ) can be described with 3 terms.

:<math>\sigma = n \cdot \mu \cdot q\,\!</math>

'''Where'''
:'''n''' = density of charge carriers (cm-3)

:'''μ''' = mobility of charge carrier (cm 2/Vs)

:'''q'''= charge (Coulombs, Cb)

To have conduction you must have charge carriers. For example in transpolyacetylene at 0°K has valence band that is completely full and a conduction band that is completely empty. You have electrons but they can not participate in the charge transfer process. If an electron is elevated to the excited state it enters the conduction band (LUMO level) becomes a charge carrier. If you then apply an electric field the charges need to be able to move in order for there to be a current. It the bond between the ethylene subunits were much longer there would be no overlap between their atomic orbitals and there would be zero electronic coupling between the pi orbitals. The mobility of the charge would be zero.

In fact the bond is only 1.45 angstrom and there is a bonding and antibonding level that allows for electronic coupling between subunits. Energy splitting is of these levels is a measure of the strength of electronic coupling between the subunits. In transpolyacetyline the bandwidth of the valence and conduction bands is about 5eV, which corresponds to an energy splitting electronic coupling (t) of 2t. In organic semiconductors you may have an electronic coupling of t= 2.5eV, which is quite large.

Charge mobility is the average speed of diffusion of the charge carriers (cm/s) as a function of applied electric field (V/cm). Organic transistors or an electronic device made from organic material must have good mobility of the charge carriers. There are many papers in the literature which characterize the charge mobility of organic compounds. A good mobility is 1 or larger.

:<math>\mu = \frac {\frac {cm} {s}} {\frac {V} {cm}} = \frac {cm^2} {V \cdot s} \,\!</math>

We can use '''dimensional analysis''' to make sure that all the units reflect the components of the equations.

:<math>\sigma = \frac {S} {cm} = \frac {1} {\Omega cm} \equiv q \cdot \mu \cdot n\,\!</math>

:::<math>\equiv Cb \cdot \frac {cm^2} {V \cdot s} \cdot \frac {1} {cm^3}\,\!</math>

:<math>\frac {1} {\Omega cm } \equiv \frac{ Cb} {V \cdot s} \cdot \frac {1} {cm}\,\!</math>

:'''(R = V/i and Ω = V.s /Cb)'''

=== Charge Factor ===

Usually the charge (q) is simply the electronic charge of an electron. When an ionic compound such as salt is dissolved in water it forms ions.

Solid salt is white. This is an indication that it is an insulator because it means that does not absorb light in the visible spectrum. The visible spectrum is 1.5eV – 3eV so as white material absorbs beyond the visible spectrum. If the first optical transition is beyond the visible, bandgap is larger than 3eV it is an insulator. If the crystal is perfect such as diamond you could have an optical bandgap of 9eV. Salt looks white because of multiple scattering from smaller crystals. But when you put sodium chloride in water it dissociates into ions. The ions are charge carriers. In the case of calcium sulfate each ion will have a charge that is twice as large (+2 or -2 charge). Thus some ions charge carrier will have charge multiples greater than 1.

=== Temperature and mobility ===

At 0 ° Kelvin semiconductors and insulators there is a significant bandgap and no charge carriers. Increasing the temperature increases the chance for electrons to jump from the valence band to the conduction band. Charge carrier density rises exponentially. Charge mobility on the other hand goes down with higher temperature. However the increase in charge density with temperature is far less than the decrease in mobility with temperature so the there is a net increase in conductivity with increase temperature.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Bloch's Theorem|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronic States vs Molecular Levels| Next Topic]]</td>
</tr>
</table>

Bloch's Theorem

2009-08-27T00:53:25Z

Nsylvain: /* Band Gap */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[The Polyene Series|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Electrical Properties| Next Topic]]</td>
</tr>
</table>

=== Bloch’s Theorem ===
[[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 function 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. Note: Polyacetylene is made from acetylene but it does not contain triple bonds. 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>\downarrow\,\!</math>

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

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

=== Phase Factor ===
The '''phase factor''' takes into account that there has been a translation.
:<math>exp (i \overrightarrow{k}j\overrightarrow{a})\,\!</math> is the phase factor

The phase factor can be calculated using algebraic substitutions:

:<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>

:<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 this convention for phase factor:

:<math>exp(i 2 \pi \overrightarrow{k} j \overrightarrow{a})\,\!</math>

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

=== Electron Momentum and Wavelength ===

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

Any object has an associated wavelength that is given by:

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

:<math>\overrightarrow{\lambda} \equiv electron wavelength \equiv \frac {h} {m\overrightarrow{v}}\,\!</math>

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

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

'''Where'''

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

:'''m''' is mass

:'''v''' is velocity

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

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

'''Example: What is the human wavelength?'''

Mass times 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 Å.

But in the range of a the electron mass of 10-27 this leads to electron wavelengths in the range of Å which is measureable.

 

=== Band Structure of Polyacetylene ===
[[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.

The zone from –π/a to +π/a is called the first '''Brillouin zone''' of the polymer. The whole information about a periodic molecule is contained between 0 and π/a (the first half of the Brillouin Zone, since the <math>E(\overrightarrow{k})\,\!</math> curves are periodic (periodicity <math>2\pi/\overrightarrow{a})\,\!</math> and <math>E(\overrightarrow{k}) = E(-\overrightarrow{k})\,\!</math> ).
 
[[Image:brillouinzone.JPG|thumb|300px|This is first half of the Brillouin zone]]
The first half of the Brillouin zone is located (0 to π/a). In a three dimensional system the only difference is that the unit cell has a,b and c components, and in reciprocal space a corresponding set of reciprocal values. As we move from the center of the Brillouin zone (0) to the edge of the Brillouin zone (π/a) the π band goes up in energy while the π* band goes down.

 

=== π Band ===
[[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 .

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.
 
[[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.

 

=== π* Band ===
[[Image:pi*band.JPG|thumb|300px|π* band for various k values]]
Consider the '''π*''' band for the ethylene subunit.

At <math>\overrightarrow{k}=0\,\!</math> the phase factor is 1 everywhere. This is the fully antibonding case which has the maximum number of nodes, one node in the middle of each bond along the chain.

At <math>\overrightarrow{k}=\frac {\pi} {\overrightarrow{a}}\,\!</math> there is an antibonding situation within the unit cell but a bonding situation between unit cells. The LUMO bonding and antibonding is exactly opposite of that of the HOMO. So there is antibonding where there is a double-like bond factor, and bonding where there is a single-like ground state bond factor.
 

=== Band Gap ===

[[Image:piband_temperature.JPG|thumb|300px|]]
The π band goes up 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 bandgap ( here Eg~1.5eV) is said to be a direct bandgap at <math>\overrightarrow{k}=\frac {\pi} {\overrightarrow{a}}\,\!</math>

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

When :<math>\vec{k}=0\,\!</math> the wavelength is infinite

When :<math>\vec{k}=\pi/a\,\!</math> the wavelength is 2a

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.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[The Polyene Series|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Electrical Properties| Next Topic]]</td>
</tr>
</table>

The Polyene Series

2009-08-27T00:52:57Z

Nsylvain: /* External Links */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Electronic Structure of Hydrogen|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Bloch's Theorem| Next Topic]]</td>
</tr>
</table>

Now we will explore the electronic properties of the polyenes; methylene, ethylene, butadiene etc. Photonic properties of the polyene series can be generalized as we start with simple molecules and construct longer and longer chains.

=== Methyl radical ===
[[Image:methyl_radical.JPG|thumb|300px|The Methyl Radical]]
Its useful first to look at the methyl radical CH3·. It has a planar structure with the carbon and three of the four valence in the plane. The hydrogens are in σ orbitals which are symmetrical about the plane of the bond. The unpaired electron or radical is in a 2pz atomic orbital that is perpendicular to the plane of the molecule. The π AO for that radical has a lobe above the plane, and a negative lobe below the plane. The probability of the electron is always zero at the nucleus. The lobes in the diagram represent the surface within which there is an 80% probability of finding the electron.
 
[[Image:Sigmapi_compared.jpg|thumb|100px|Relative energy of π and σ orbitals]]
The energy of the π electronic orbital is much higher than the σ orbitals. All of the optical and electrical properties of the polyenes are due to highest occupied molecular orbitals which are the π orbitals.
 

=== Ethylene ===
[[Image:ethylene.JPG|thumb|300px|Ethylene molecular orbital]]
Ethylene is π- conjugated molecule. It can be thought of as two methyl radicals coming together and losing two hydrogens. It is coplanar molecule with a σ bond between the carbons. Each carbon has a π- electron orbital that overlaps with the other because the carbons are close. In a perfectly planar system the π- electrons will not interact with the σ bond. But as soon as the molecular becomes non coplanar there can be interaction.
 
[[Image:Ethylene_energy.jpg|thumb|300px|Homo / Lumo diagram for ethylene]]
If both π wavefunctions are the same sign you get a bonding molecular orbital with a higher density between the carbon nuclei (no nodes). If the sign are different you get an antibonding orbital with a node between the nuclei. From a methyl radical like π atomic orbital contribute two pi electrons in to the bonding molecular orbital which will be the highest occupied molecular orbital (HOMO). The antibonding π * (* denotes unoccupied) corresponds to the lowest unoccupied molecular orbital (LUMO). The energy difference between HOMO and LUMO is 7ev. This is the energy that would have to be supplied by a photon to bring promote an electron to the LUMO level. The visible part of the EM spectrum is between 1.5eV and 3eV. So the first optical transition occurs above the visible spectrum.

 

=== Butadiene ===
[[Image:butadiene energy.JPG|thumb|300px|Butadiene Energy Levels]]

From the point of view of the π level butadiene corresponds to the interaction between two ethylene sub-units. The π homo level of one ethylene subunit interacts with the π homo level of the other subunit. There can be a positive combination and a negative combination, one leads to a level that is more stable one that is less stable. The same thing happens for the π* levels. The π * LUMO has come down while the π HOMO has increased. As a result the energy difference between HOMO and LUMO (the first optical transition ) has decreased to 5.4 eV.
 
[[Image:butadienewave.JPG|thumb|300px|Butadience wavefunctions]]
Considering the wavefunction. The positive combination gives you all bonding for the lowest π bonding orbital. Then mixing two positive with two negative (++--) you fill the HOMO level resulting in a single node. In the π * orbitals the LUMO includes a negative combination (+--+) which results in bonding in the middle with 2 nodes. The fully antibonding molecular orbital has three nodes.
 

The bonding – antibonding character of the HOMO wavefunction translates to the double-bond / single-bond character of the geometry in the ground state. Higher electron density in the π MO corresponds to the double bond that allows the nuclei get closer (shorter bond length), lower density in single bond corresponds to the node between the antibonding orbitals.
[[Image:Butadiene_bonding.jpg|thumb|left|200px|Butadience Bonding and Antibonding- HOMO]]
[[Image:Butadiene_bonding_lumo.jpg|thumb|200px|Butadience Bonding and Antibonding- LUMO]]
 

=== Hexatriene ===
[[Image:hexatriene.JPG|thumb|left|300px|Hexatriene has three ethylene like sub units, 3 occupied levels and 3 unoccuppied pi * levels.]]
[[Image:hexatriene_wavefunctions.JPG|thumb|300px|Hexatriene wavefunctions. The wavefunctions for hexadiene can have 0, 1, or 2 nodes in the HOMO. The gap between the HOMO and LUMO is 4.7eV.]]

 
The energy of the π MO’s goes UP as a function of the # of nodes this is related to the kinetic term in the Schrödinger equation.

:<math>\hat{H}\Psi = E\Psi\,\!</math>

:<math>\hat{H}\rightarrow \hat{K} + \hat{V}\,\!</math>

:<math>
\hat{K} \rightarrow (\frac{ \partial} {\partial x^2}+ \frac{ \partial} {\partial y^2} +\frac{ \partial} {\partial z^2})\,\!</math>

[[Image:H2wavecombined.JPG|thumb|300px|The smoother the evolution of the wavefunction the smaller the kinetic energy term that is associated with it.]]

 

=== Geometry ===
[[Image:Hexatriene_nopi.png|thumb|300px|In the absence of pi electrons all of the C-C bond lengths would be nearly equal (about 1.51Angstrom for sp2 bonds).
]]
[[Image:hexatriene_geometry.JPG|thumb|300px|]]
When π electrons are added they will be distributed unevenly over the bonds. So there are shorted bonds in the double bonds (1.34 A). Even the single bond will be shorter (1.47 a) than a single bond because of the π conjugated system. The HOMO wavefunction reflects the single double bond alternation in the ground state. If we add an electron to hexatriene (reduce it) this electron will go to the LUMO causing an antibonding factor between the first carbons. This will cause the first bond to lengthen. So a rule of thumb is that when an electron is added to an antibonding wavefunction the bond will elongate. If it is added to a bonding factor then it will shorten.

As you go from the ground state geometry to the reduced state you will have lesser degree of bond length alternation because double bonds will have lengthened and the single bonds will have shortened. If you oxidize the molecule (remove an electron) the electron is lost from the HOMO, from a bonding factor weakens the bonding characteristic and that bond will elongate. But if you remove the electron from an antibonding wavefunction, that allows the bond to shorten. As a consequence when you put an electron on the LUMO (i.e. reduce the molecule) or remove an electron from HOMO (oxidize the molecule) the double bonds will elongate and the single bond will shorten. When you go the excited state (taking and electron from the HOMO to the LUMO) you should get longer double bonds and shorter single bonds compared to the ground state. This is because the π electron densities on the bonds changes the geometry of the molecule.

 

=== Chain size ===
[[Image:polyacetylene_band.JPG|thumb|300px|The electron band structure of polyacetylene.]]
The energy of the first optical transition (π to π* or HOMO to LUMO) goes down approximately as 1/n where n = number of atoms. The plot of energy vs n is linear.

The energy difference between consecutive π levels becomes smaller and smaller as the chain gets longer. At the limit of an infinite chain the change between levels is infinitely small. This results energy bands.

The π band is referred to as the Valence band and the π* is called the Conduction band. This a assumes a perfectly coplanar chain. The width of the Valence Band and the Conduction band is about 5eV. The energy gap or band gap Eg is about 1.5eV, this also known as the ionization potential. The bottom of the Conduction band is also known as the electron affinity. With trans polyacetylene and other conjugated polymers you have to refine the picture because of the connection between the geometries and electronic properties.

=== External Links ===

[http://138.253.125.24/~ng/external/orbitalsethene.htm MO orbital visualization of ethylene]

[http://www.jce.divched.org/JCEDLib/LivTexts/pChem/JCE2005p1880_2LTXT/QuantumStates/ Quantum States of Atoms and Molecules]

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Electronic Structure of Hydrogen|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Bloch's Theorm| Next Topic]]</td>
</tr>
</table>

Electronic Structure of Hydrogen

2009-08-27T00:52:39Z

Nsylvain: /* Additional Links */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Introduction|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[The Polyene Series| Next Topic]]</td>
</tr>
</table>

=== Electronc structure of hydrogen ===

[[Image:H2wavefunction.jpg|thumb|300px|Hydrogen wavefunction]]
The hydrogen atom has just a electron and a proton. When you bring them together the atom holds together because of the coulombic attraction between the electron and the nucleus. That attraction leads to stabilization of the system of -13.6 ev (one Rydberg). This is how much energy it take to separate the proton from the electron. This does not happen at room temperature. The electron has a 1s atomic orbital. The wavefunction decreases as you move away from the nucleus. The square of the wavefunction gives the probability of finding an electron. It will not actually touch the nucleus but it will stay very close.

When two hydrogen atoms approach one another: their ψ1s wavefunctions start overlapping and the 1s electrons start interacting. Molecular orbitals (MO) can be described using the atomic orbitals of the atoms forming the molecule. The hydrogen atom is the only element for which you can exactly solve Schrodingers equation. Even Helium can not be solved exactly. Hydrogen molecule can be described using '''Linear Combination of Atomic Orbitals'''(LCAO). For n atomic orbitals you get n molecular orbitals.
 

=== Adding wavefunctions ===

[[Image:h2wavecombined.JPG|thumb|300px|Hydrogen Molecular Orbital]]
The two wave functions 1SA and 1SB can be added constructively to produce a bonding orbital, or with opposite sign they add destructively to produce an antibonding molecular orbital. This describes the system quite well. The bonding orbital will have lower energy and create a more stable system. There has to be an increased density of electrons between the nuclei in order to form a molecule. The node in the antibonding MO decreases the density electrons and decreases the stability required to bring the two protons together.
 

=== Bonding and antibonding ===

[[Image:h2energelevels.JPG|thumb|300px|Energy levels for hydrogen electrons before and after bonding.]]
The energy of the bonding MO is lower than the energy of the two original atomic orbitals . If two electrons share the same wavefunction they have to opposite spins. When 2 hydrogen atoms form from electrons and protons the energy is 27.4 eV. The energy of the hydrogen bond 4eV; far smaller than the atomic bond energy.

=== Additional Links ===

See Wikipedia on [http://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_method LCAO]

See Wikipedia on [http://en.wikipedia.org/wiki/Huckel_method Huckel method]

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Introduction|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[The Polyene Series| Next Topic]]</td>
</tr>
</table>

Introduction to Band Structure

2009-08-27T00:52:03Z

Nsylvain:

<table id="toc" style="width: 100%">
<tr>

<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronic Structure of Hydrogen| Next Topic]]</td>
</tr>
</table>

[[Image:semiconductpolymer.JPG|thumb|300px|Semiconducting Polymers]]

Many of the electrical and optical properties of NLO materials are related to the electronic band structure of the compounds.

Semiconducting polymers combine the electrical and optical properties of metals and semiconductors with the mechanical properties of plastics. Highly electrically conductive polymers were first discovered in the 1970s. Now these organic materials will be used just as silicon is used today. The goal is not to replace silicon but rather to do new things that silicon is not well suited for. Silicon has to be extremely pure, it is rigid and brittle, it is difficult to process. Organic materials can be simply printed on surfaces and may remain flexible. Finally organic materials can be synthesized to have specific characteristics.

 
<table id="toc" style="width: 100%">
<tr>

<td style="text-align: center; width: 33%">[[Main_Page#Electronic Band Structure of Organic Materials|Return to Band Structure Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronic Structure of Hydrogen| Next Topic]]</td>
</tr>
</table>

Electronic Coupling Between Orbitals

2009-08-27T00:51:18Z

Nsylvain: /* Pentadienyl Radical */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Polarization and Polarizability|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Donors and Acceptors|Next Topic]]</td>

</tr>
</table>

[[Image:Beta-adjacent.png|thumb|200px|Adjacent p orbitals can interact]]

We will discuss the electronic structure of π-conjugated organic molecules at various levels of complexity. π-conjugated molecules have a sigma electron framework and π electron framework.

=== Hückel Molecular Orbital Theory===
Here we start at the simplest level that is basically derived from a Hückel Molecular Orbital approach in order to understand conjugated systems.

In this model, orbitals that are on atoms that are directly σ-bonded to one another and whose p-orbitals are in a plane can interact. This interaction is called the "electronic coupling" between the orbitals and has units of energy.

In the Hückel approximation this energy is the same for all carbon p-orbitals that are adjacent to on another and is called β.
 
[[Image:Beta-nonadjacent.png|thumb|200px|Non- Adjacent p orbitals do not interact]]
For atoms that are not adjacent to one another β is taken to be zero. Thus in the Hückel approximation there is no interaction between nonadjacent atoms. Even if you distort the system so that nonadjacent atoms are very close they will not be considered because there is not bonding.

Through this coupling of atomic orbitals, molecular orbitals can formed by a constructive and destructive combination of the atomic orbitals. If you begin to fill the orbitals from the lowest to the highest energy you fill the bonding orbital first. This is also known as the highes occupied molecular orbital (HOMO). The antibonding orbital is known as the lowest unoccupied molecular orbital (LUMO).

*The energy of a carbon orbital is taken to be zero, neither stabilized or destabilized. Fluorine would be defined as less than zero. As go to a more electronegative atom you will pull down the energy of the orbital.
*In ethylene the bonding orbital has no nodes and the antibonding orbital has a node that is in between the atoms.
*The bonding orbital is basically stabilized by amount roughly equal to β and the antibonding orbital is destabilized by a roughly equal amount. In reality, the coupling of the orbitals will be a function orbital overlap which itself depends of how diffuse the orbitals are, their relative orientation and their center to center distance. If bonds are longer the degree of overlap there will be less coupling which decreases the homo / lumo gap.
*An atom more electronegative than carbon will have an p-orbital that lies lower in energy than that of carbon and one more electropositive will have an energy that lies above that of carbon. This will have consequence in the molecular orbitals that are derived from atomic orbitals.

=== Perturbation theory ===
[[Image:Beta-energydiagram.png|thumb|300px|Bonding between atoms with equal electronegativity]]

If the orbitals that mix are the some energy the mixing can be treated using '''first order perturbation theory'''.

[[Image:Beta-energydiagram_unequal.png|thumb|300px|Bonding between atoms with unequal electronegativity, for example oxygen and carbon]]
If the orbitals are at different energy they are treated using '''second-order perturbation theory''' in which the energy gap between the orbital is taken into account and in general the mixing of orbitals that are degenerate is greater than those that are not; also the greater the energy difference between the orbital, in general the less the mixing. The amount the bonding orbital is stabilized and the anti-bonding orbital is destabilized decreases.

As a consequence of this the coefficient of the stabilized orbital is larger on the atom to which it is closer in energy and the same if true for the unfilled orbital. In this example there would be a higher electron density around the oxygen than around the carbon. This molecule would have a dipole moment. The antibonding orbital has a higher coefficient on the carbon. This a proton would tend to react with the oxygen because the highest occupied orbital has a higher electron density with the the oxygen. On the other hand the carbon will be the more electrophillic site. This allows us to make very simple predictions about reactivity.

If you promote an electron from the HOMO to the LUMO the dipole moment of the molecule would decrease.
 

=== Butadiene Orbitals ===
[[Image:Butadiene.png|thumb|300px|Orbital energy for diagram for butadiene]]
Butadiene starts with four p orbitals.
*The energy scale is in terms of β and the most stable orbital here has the most positive value of β, because β itself is take to have a negative value.

*For each orbital that has nodes, these will be situated between (not on) atoms.

*The gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is less than that in ethylene.

*The orbital energy are symmetrically disposed around zero.

*The LUMO in butadiene is lower than that in ethylene and the HOMO is higher than that in ethylene. Butadiene should be both easier to reduce than ethylene and easier to oxidize.

We can make molecules easier to oxidize and easier reduce as we increase the length and add more orbitals. From an electron chemical standpoint butadiene is a better electron donor.

===Allyl- Odd number of orbitals ===
[[Image:Allyl_energy.png|thumb|300px|Orbital energy diagram for allyl]]
Lets now consider a system with an odd number of atoms, the simplest being allyl.

Note that the are several significant similarities and differences between the orbital in allyl and those of butadiene or ethylene:

*The HOMO of allyl radical (anion) is at the some level of a carbon p-orbital (i.e.) it is neither stabilized nor destabilized relative to carbon.

*The HOMO has one node and that node lies on a carbon atom, not between atoms.

*Due to the node between carbon the central carbon (carbon 2) and the fact that in the Hückel approximation there is no interaction between carbons 1 and 3. This orbital neither bonding or antibonding and is thus called non-bonding.

*Allyl can exist as a allyl radical, allyl cation, allyl anion and each with different orbital filling.
The energy difference between the lowest and highest orbital of allyl and is less than the difference with ethylene ( 1.4β compared to 2β). Allyl has a small gap than ethylene.

What do we expect the result of changing the carbon for a more electronegative element to be?

A more electronegative element has its electrons held more tightly and closer to the nucleus and as a result as we noted earlier the energy level of a p-orbital on a more electronegative element should be lower than that of carbon. Therefore it should stabilize molecular orbitals that have significant coefficients on the atom that is more electronegative.
 
[[Image:Allyl nitrogen.png|thumb|300px|Relative energy levels of carbon chain with and without nitrogen.]]
For example if you made a molecule allyl cation that has only two electrons in the homo, and then made a corresponding molecule with a nitrogen in the middle. The homo /lumo gap will increase because the HOMO energy level is lowered while the LUMO stays the same. Thus, the most orbitals are stabilized by inclusion of the nitrogen
Note that the orbital that has a node on the central atom is not affected by this change.

This is how molecular orbital theory helps you predict how the properties of molecules change with additions of particular atoms.
 

=== Pentadienyl Radical ===
[[Image:Pentadienyl_energy.png|thumb|300px|Pentadiene]]

If you consider the pentadienyl anion, radical and cation, that each has nodes on either odd or even atoms in the HOMO or LUMO and that the nodes do not match up in position. The middle orbital has the same energy as carbon. The gap between levels decreases more rapidly than for even numbered orbital molecules.

This is the basis for making some simple predictions about the spectroscopy of so-called cyanine-like molecules upon substitution of atoms along the chain with either electronegative or electropositive elements. Likewise the addition of electron donating and withdrawing groups can have similar effects.
[[Image:polyene.png|thumb|200px|Polyenes]]
Another distinction between cyanine-like dyes and polyene like dyes is the in polyene there is a complete reversal of the bond order between the π-bonding HOMO and the π-antibonding LUMO.
 
[[Image:Cyanene.png|thumb|200px|Cyanines]]In contrast, in cyanines because typically either the HOMO or the LUMO is non-bonding, and because there are nodes at the atoms rather than between bonds in the frontier orbitals, there tend to be smaller changes in bond-length between the HOMO and LUMO. This will have spectroscopic implications. Anything that pushes electron density onto an atom increases the energy of the atomic orbital associated with that atom. Anything that takes electron density away will decrease the corresponding orbital energy.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Polarization and Polarizability|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Donors and Acceptors|Next Topic]]</td>

</tr>
</table>

Polarization and Polarizability

2009-08-27T00:50:56Z

Nsylvain: /* Applying an Oscillating Electric Field */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Sigma and pi Orbitals|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronic Coupling Between Orbitals|Next Topic]]</td>

</tr>
</table>

Polarization is an important concept in designing electro optical materials.

=== Mechanisms for polarization ===

If you have any mechanism to apply an external electrical field this may lead to polarization of the molecules.

In '''electronic polarization''' the field pulls electrons more than it repells the nucleus because electrons are far lighter than the protons and neutrons. The timescale for polarization of a atom due to a field is 10 -15 seconds. This is about as fast as you can do things.

If two atoms with unequal electronegavity are bonded the molecule will have a dipole moment. A vibration of the atoms results in a change in the position of the nuclei and the electrons follow the nuclei. This '''vibrational polarization''' happens on a timescale of picoseconds. Infrared leads to vibrations of molecules. The reciprocal of the frequency of IR has a time constant of 100 femptoseconds.

You can have molecules that already have a dipole moment and place them in an electronic field the molecules will align. This time constant for this '''rotational polarization''' is in the area of 10s of nanoseconds. Microwave radiation can be used to cause rotations in molecules.

If there are ions in the presence of an electric field there will be a bulk motion or '''ionic motion polarization'''.

If there is an oscillating electronic field it needs to be matched to the timescale of the polarization mechanism. Only those components that react as fast or faster than the frequency will make a contribution to the bulk polarization of the material. The following animation demonstrates these four types. Click the switches to apply an electric field.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/01 Polarization Mechanisms.swf</swf>

Here is an hydrogen atom in an oscillating electric field. The atom is is oscillating as the electrons are being displaced and the energy levels are changing.

[[Image:Hydrogen_polar.gif]]

In an atom with π orbitals there is more polarizability in the π orbitals than in the σ orbitals. Dyes are typically π conjugated compounds because π orbitals are more polarizable in the visible spectrum.

[[Image:PI orbit anim.gif]]

=== Linear polarizability ===
[[Image:Dipole_align.jpg|thumb|300px|The torque experienced by a dipole in an electric field depends on the orientation of the molecule at that moment.]]

Molecules that have a dipole moment will orient in an electric field. These are known as second order non-linear materials. The force that is on the system is a torque that is related to angle between dipole and the electric field, and upon the magnitude of the field and the dipole. If the molecule is aligned with the field it will not feel a torque. If it oriented at 90 degrees it will feel a substantial torque.

Dipole is vector quantity, it has a magnitude and a direction. An opposite field will induce a dipole in the opposite direction.

If you plot dipole moment against the electric field you get a straight line. The slope (α) of the line for the function is the linear polarizability. A higher α means a greater amount of polarization for the same amount of applied field.

[[Image:polar_plot.JPG|thumb|400px|In a linear material the induced dipole is directly proportional to the applied field]]

:<math>\mu = \alpha \Epsilon\,\!</math>

where

:<math>\mu\,\!</math> is the dipole moment

:<math>\alpha\,\!</math> is the linear polarizability, (slope of the line)

:<math>\Epsilon\,\!</math> is electric field

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/04 Polarizability.swf</swf>

 

=== Anisotropic Polarizability ===

[[Image:anisotropic_polar.JPG|thumb|300px|The degree of polarizability depends on the orientation of the molecule.]]

An organic molecule, 2-4 Hexadiene has two triple bonds and single bonds between two of the carbons. The triple bonds form a cylinder of electrons. If you apply and electic field along the long axis of the cyclinder you will get a larger induced dipole moment. If you assume that the polarizability at 90° to the axis is negible compare to the that of the along the long axis then electrons are going to move along the axis only. The interaction of the electric field with the electrons is greatest in the long direction, and becomes zero at 90 °. This can be described as a dot product (a cosine function). An oscillating field will cause electrons to move up and down along the long axis and creates an induced dipole. Molecules that are oriented at 45 ° angle will have some induced dipole due to the applied field. This induced field can be used with cross polarizers to allow or not allow light to pass through the material.

=== Tensor calculation ===

To describe the polarizability of this kind of molecule you have to use a tensor to describe all the components of polarizability. It is possible to apply an electric field along y but induce a dipole moment along y. Each entry of the tensor is a component of the polarizability.

:<math>\begin{pmatrix}
\mu_x\\
\mu_y\\
\mu_z
\end{pmatrix}
= \begin{pmatrix}
\alpha_{xx}& \alpha_{xy}& \alpha_{xz} & \\
\alpha_{yx}& \alpha_{yy}& \alpha_{yz} & \\
\alpha_{zx}& \alpha_{zy}& \alpha_{zz} &
\end{pmatrix}
=
\begin{pmatrix}
E_x \\
E_y \\
E_z
\end{pmatrix}
\,\!</math>

You can calculate the degree of polarization in any direction with the tensor product;

:<math>\mu_x = \alpha_{xx} E_x + \alpha_{xy} E_y + \alpha_{xz} E_z\,\!</math>

[[Image:Balloon_deformation.png|thumb|300px|An anisotropic material would be stretched by a force to varying degrees depend on its orientation.]]
Consider the electric field as a deforming force and the dipole moment as the deformation. The balloon can be stretched in one direction resulting in changes in the other dimensions. A balloon could be isotropic meaning it would be equally easy to stretch in all directions. Or it could have fibers built in that would limit the deformation along one axis. An electric field from any direction can induce a dipole in any direction. The components along the diagonal tend to be the strongest. It is possible to have more than one electric field applied (along x and y). To describe this situation you actually need a 3 x 3 x 3 (third rank) tensor, a matrix with 27 components.

You could even have 3 electric fields with 81 components.
<swf width="500" height="400">http://depts.washington.edu/cmditr/media/06 Tensorial Polarization.swf</swf>

=== Dielectric Constant ===
When use the dielectric constant we often are referring to a solvent. A refractive index refers to speed of light. For a bulk material we describe the polarization density in a material. Instead of looking at a single molecule, we can look at the polarizability of the entire bulk material induced by an elecric field.

In bulk materials, the linear polarization is given by:

:<math>P_i(\omega) = \sum_{ij}\chi_{ij}(\mu)E_j(\omega)\,\!</math> (4)

where

:<math>\chi_i( \omega )\,\!</math> is the polarization density in direction i.

:<math>\omega\,\!</math> is the frequency

:<math>\chi_{ij}(\omega)\,\!</math> is the linear susceptibility of an ensemble of molecules.

Note that the vectorial and tensor aspects of E and <math>\chi\,\!</math> have been ignored to simplify notation.

The total electric field (the "displaced" field, D) within the material becomes:

:<math>D = E + 4\pi P = (1 + 4\pi \chi)E \,\!</math> (5)

:<math>P = \chi E\,\!</math> (Equation (4)),
:<math>4 \pi \chi E\,\!</math> is the internal electric field created by the induced displacement (polarization) of charges.

=== The Index of Refraction===
The dielectric constant and the refractive index n(ω) are two bulk parameters that characterize the susceptibility of a material.

ε (ω) in a given direction is defined as the ratio of the displaced internal field to the applied field (ε = D/E) in that direction.

:<math>\varepsilon_{ij} (\omega) = 1 + 4\pi \chi_{ij} (\omega)\,\!</math> (6)

The dielectric constant of the material relates to the susceptiblity of the material, which relates to the polarizability of the molecules. Since χ is frequency dependent because of the different mechanisms that can induce polarization, then ε the dielectric constant is also frequency dependent.

The frequency dependence of the dielectric constant provides insight into the mechanism of charge polarization. with low frequency you can get contribution from electronic, vibronic, and rotational polarization. As you increase the frequency the dielectric constant drops until the point where the only contributing factor is the electronic polarization.
[[Image:Freq_dielectric.jpg|thumb|300px|The dielectric constant ε decreases as frequency increases.]]

The ratio of the speed of light in a vacuum, c, to the speed of light in a material, v, is called the index of refraction (n):

:<math>n = \frac {c}{v}\,\!</math> (7)

Index of refraction is typically measured at low frequencies. At optical frequencies the dielectric constant equals the square of the refractive index:

:<math>\epsilon_\infty(\omega) = n^2(\omega)\,\!</math> (8)

Consequently, we can relate the refractive index to the bulk linear (first-order) susceptibility:

:<math>n^2(\omega ) = 1+ 4\pi \chi(\omega)\,\!</math> (9)

Index of refraction depends therefore on chemical structure. For example iodine has a higher index of refraction than fluorine because iodine is has a higher electron affinity, and therefore has more polarizability, so as material it has as a material greater susceptibility, therefore the refractive index is higher. This is a simple and essential factor of optics. As an organic chemist you can manipulate the atoms of a molecule to change the polarizability and thereby control the optical properties of the material.

=== Birefringence ===
Isotropic molecules appear and behave the same regardless of their orientation. Anisotropic molecules have different properties depending on their angle.

Consider a material that is comprised of molecules that are dipolar and have polarizability anisotropy, then apply and electric field that couples with the dipole and induces a preferred orientation for the molecules:
[[Image:Birefringence.jpg|thumb|300px|Molecules aligned due to an electric field.]]

An electric field can induce birefrignece. The molecules will have a different susceptibility, therefore a different polarization, therefore a different dielectric constant therefore a different index of refraction. Light that passes through a birefringent materials with one polarization will have a slower speed than the light with a different polarization. Consequently the light waves emerge at different times. Examples of birefringent materials include quartzite, quartz feldspar and potassium dihydrogen phosphate (KDP). This property is significant for non-linear optics and liquid crystals.

 

=== Classical Polarization ===
[[Image:Distortion_coordinate.jpg|thumb|200px|Symmetrical function of electron density]]

For linear polarizability, the electrons are bound to the atoms in a harmonic potential, i.e., the restoring force for the electron is linearly proportional to its displacement from the nucleus:

:<math>F = -\Kappa x\,\!</math> (10)

The electrons see a potential energy surface:

:<math>V = 1/2 \Kappa x^2\,\!</math> (11)

The force is the derivative of the potential energy. This means that there is a symmetrical distribution of electron density around the atom with an equal ease of charge displacement in both the +x and -x directions. The extent of polarization if linearly relate to the applied field.

:<math>P_{ind} = \alpha E\,\!</math>

===Applying an Oscillating Electric Field ===
[[Image:Induced_sinefield.jpg|thumb|300px|Induced polarization has the same frequency as the exciting field.]]
Application of an oscillating electric field will induce an oscillating polarization in a material.
For linear polarizability, the polarization will have the same frequency as the applied electric field. If a sinusoidal wave field is applied the induced field will be sinusoidal. It may not have the same amplitude as the applied field but it will follow it. Light is an oscillating electromagnetic field. This induced polarization is itself a source of light which will propagate through the material in the same direction and with the same frequency as the light beam that created it. Because the electrons respond linearly to the applied field the induced wave is similar to the applied wave. However they the two waves may not have the same phase because it takes time for the electrons to move during induction process.

During the process there is no net transfer of energy to the material. However if the frequency of light approaches the natural resonance of the molecule it is possible to transfer to energy to the molecule giving it an excited state, or producing heat or other chemicial reactions.
 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Sigma and pi Orbitals|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronic Coupling Between Orbitals|Next Topic]]</td>

</tr>
</table>

Sigma and pi Orbitals

2009-08-27T00:50:29Z

Nsylvain: /* Covalent bonding */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Electronegativity and Bonding Between Atoms|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Polarization and Polarizability|Next Topic]]</td>

</tr>
</table>

[[Image:2sigma_orbitals.png|thumb|300px|Hydrogen 1 and Hydrogen 2 combine to form a new molecular orbital.]]

=== Molecular Orbital Theory ===

Molecular orbital theory was developed in the early part of the last century to help rationalize why bonds form and to explain the properties of molecules. In molecular orbital theory the atomic orbitals from each atom can overlap with those on other atoms. Since the atomic orbitals are wavefunctions and behave like waves it is possible for them to overlap in a constructive manner to form a bonding orbital.

:<math>\Psi_{new} = \Psi_1 + \Psi_2\,\!</math>

Or the two functions can combine in a destructive manner.

:<math>\Psi_{new} = \Psi_1 - \Psi_2\,\!</math>

This is sometimes referred to as Linear Combination of Atomic Orbitals (LCAO). When there is a destructive combination of the two waves there is a node between the two atoms where there is a zero probability of finding the electron. The first case is known as a bonding orbital and the latter case is known as an anti-bonding orbital.

[[Image:H2_antibonding_orbital.png|thumb|300px|Electrons from two atoms combine to form a σ bonding molecular orbital.]]
Note that the electron density is higher between the nuclei in the bonding molecular orbital and the that the orbital is stabilized, that is lower in energy relative to the two isolated hydrogren atoms. The sign of both wavefunctions is positive. There is a decreased electron density between the atoms in the antibonding molecular orbital and the orbital is destabilized, that is higher in energy than the two isolated hydrogen atoms. In very simple molecular orbital theory we treat the destabilization energy of the antibonding orbital as identical to the stabilization energy of the bonding orbital but in fact it is slightly more destabilized than the bond orbital is stabilized.
 

 

=== Orbital Overlap ===
[[Image:Sigma_bond.png|thumb|300px|The sign of wavefunctions is shown as light and dark lobes.]]

The amount of stabilization or destabilization is referred to a their orbital overlap. Mathematically the overlap is an integral of from negative infinity to positive infinity of a product of two wavefunctions over all space.

:<math>\int_{-\infty}^{\infty} \Psi_1^* \Psi_2 dxdydz\,\!</math> = Overlap

The area under the curve for these functions is the orbital overlap.
 
[[Image:Orbital_overlap.png|thumb|300px|Possible overlap of orbitals]]
If the s orbital is lined up exactly over the node of a p orbital you will end up with positive overlap on one side, and equal and opposite overlap on the other side. The orbital overlap when you integrate over all space will be identically zero. If you take two p orbitals that are oriented at 90 degrees the product is zero.

If you shift the s orbital a little bit one direction the overlap of the s orbital with one of the lobes will be non-zero. If you have two p orbitals at some angle you will have non-zero overlap. Or if a s orbital overlaps only on side of a the p orbital you would get some orbital overlap.

The two p orbitals that are aligned with the same sign will have the maxiumum orbital overlap and the maxium bonding interaction. If they are overlapping with opposite signs you have the maximum anti-bonding interaction. If you rotate them orthogonally they will be neither bonding or nor antibonding.

 

=== p-p σ Bonding ===

[[Image:P_orbital_overlap.png|thumb|300px|P atomic orbitals combine to form $sigma; molecular orbitals]]
Two p orbitals can combine contructively or destructively. In constructive situation you have a lower energy bonding orbital. If they align destructively you have the higher energy antibonding orbital.
In a σ bond, if you look down the axis between the atoms bonded to each other, the orbital will appear cylindrically symmetric. In such cases the atoms at the end of the bonds can rotate without breaking the bond. This is the case from the bonds between the s orbitals in hydrogen. P orbitals can overlap in two different ways. If they overlap head-on a σ bond is formed. 

 

=== p-p π Bonding ===
[[Image:Pi bond.png|thumb|300px|P atomic orbitals form π molecular orbitals.]]
If two p-orbitals overlap constructively then a π bond is formed. If you look down the axis of a π bond the orbital will not appear cylindrically symmetrical. The top has one sign the bottom has the opposite sign. There will be a node in the plane of the bond. The overlap of the orbital depends on critically on the angle between the orbitals. If the orbitals are not exactly lined up, the bond will begin to break and at 90° there be no πbonding. The stabilization of two p orbitals forming a σ bond is greater than two p orbitals forming a π bonds because p and σ bonds have more orbital overlap. As a consequence σ bonds are more stabilized and stonger than π bonds.

 
=== Effect of field on π electrons ===

[[Image:PI_orbit_anim.gif‎]]

[[Image:chargetransfer.JPG|thumb|300px|]]
σ electrons have electron density between the nuclei so they are tightly bound and do not change position as much when exposed to an external field. π electrons have the electron density above and below of the plane of the nuclei. When you apply a field you change the distribution and induce a dipole.

For a typical molecule you go from a neutral ground state to a charge separated state by application of an electric field. Quantum -mechanically the electric field causes a mixing of these two states.

=== Covalent bonding ===
[[Image:H2bondstrength.png|thumb|300px|Energy vs distance for bonding (blue) and excited state (red)]]

When two atoms with duets or octets that have not been satisfied get within close proximity, they can interact in such a way as to create a bond by "sharing" electrons. Sharing of electrons can achieve a stable electron configuration corresponding to a noble gas configuration. On the right side of the diagram the hydrogen atoms are far apart and do not interact. As they move closer the system is stabilized by the formation of a bond through overlap of their orbitals and the lowering of the kinetic energy of the system. If the atoms get too close there will be an extremely strong nuclear - nuclear repulsion that greatly destabilizes the system. The correct balance between these interactions is found at the equilibrium bond length.

How does the potential energy surface for a hydrogen-hydrogen σ bond in the ground state (where the electrons are in the lower orbital) vs the excited states (ie where one electron is in an antibonding orbital)? These two states have a different energy dependence as a function of distance. A σ bonding combination initially becomes increasingly stabilized as you move from infinity until the point where the internuclear interaction (repulsion between protons) causes the energy to increase greatly. This point for hydrogen occurs at a distance of about .74 Å. The bond strength is 104 Kcal/mol, which is a very strong bond. The equilibrium constant for hydrogen radicals vs bond hydrogen would be 99999999.... favoring the bound state.

In the excited state the electrons are increasingly destabilized as the atoms get closer, the surface is repulsive. If you have two atoms infinitely far apart the system is neither stabilized nor destabilized. If you bring atoms to a correct geometry the ground state is stabilized. If you bring them too close you get a destabilized state. The excited state is always increasingly destabilized with decreasing distance.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Electronegativity and Bonding Between Atoms|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Polarization and Polarizability|Next Topic]]</td>

</tr>
</table>

Electronegativity and Bonding Between Atoms

2009-08-27T00:49:50Z

Nsylvain:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Atomic Orbitals and Nodes|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[sigma and pi Orbitals|Next Topic]]</td>

</tr>
</table>

[[Image:Ionization_pot.jpg|thumb|300px|Energy level diagram showing the relation between electronegativity (χ), electron affinity (EA), ionization potential (IP) to the vacuum level and the valence band]]
Electronegativity describes how much an atom hold its electrons tightly. An atom that is less electronegative will be more likely to give up its electrons. The electronegativity of atoms determines the degree that electrons are transferred between molecules or shared between bonding atoms.

Mulliken presents a useful definition of electronegativity (χ) as the average of the ionization potential (IP) and the electron affinity (EA).

:<math>\chi = \frac{IP + EA}{ 2}\,\!</math>

Note that the ionization potential is the energy required to remove an electron from an atom (to vacuum) and the electron affinity is the energy released when an atom captures an electron (into the lowest unfilled orbital).

The more electronegative the atom is the lower the energy of the orbital within a given row. Moving to the right on the periodic table across a row the electronegativity increases because the number of protons increases which increases the pull on the electrons. When the difference in electronegativity is great, as in Na+ and Cl-, the electron is transferred from the electropositive species to the more electronegative species. These ionic compounds are held together by electrostatic forces and form a salt. In pure covalent bonds between like atoms each of the bonding electrons are shared equally. The electron density is highest between the atoms an there is no dipole moment. When there is slight difference in electronegativities for example in in C-F bond, the fluorine holds the electrons closer, the carbon less so. The whole bond has a polarity with unequal distribution of electrons..

Here is a summary of the different kinds of bonds.

{|border="1"
|-
! Ionic Bonding
! Polar Covalent Bonding
! Covalent Bonding
|-
| Forms salts that dissociated in water
| Forms bonds which have dipole
| Forms bonds with no dipole
|-
| Between atoms of the far left and far right of the periodic table
| Betwen atoms of different columns usually on the right side of the table
| Between two like atoms
|-
| Large differences in electronegativities
| Moderate differences in electronegativities
| No difference in electronegativities
|-
| Very unequal sharing, electron is transferred form one atom to another
| Unequal sharing of electrons
| Equal sharing of electrons
|-
| Na+Cl-
| C-F bond
| F-F bond

|}
 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Atomic Orbitals and Nodes|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[sigma and pi Orbitals|Next Topic]]</td>

</tr>
</table>

Atomic Orbitals and Nodes

2009-08-27T00:49:22Z

Nsylvain: /* Bridging the Language Barrier */

<table id="toc" style="width: 100%">
<tr>

<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronegativity and Bonding Between Atoms|Next Topic]]</td>

</tr>
</table>

[[Image:Wavefunction.jpg|thumb|300px|Electron Probability, peak density and electron density as a function of distance from the nucleus.]]

=== Atomic Orbitals ===

Orbitals are important because they determine the distribution of electrons in molecules, which in turn determines the electronic and optical properties of materials.
Atomic orbitals are wave functions that are solutions to the Schrödinger equation. This equation allows us to figure out the wave functions and associated energies in atomic orbitals.

The square of the wave function gives the probability of finding an electron at a certain point.

The integral of the wavefunction over a volume gives the enclosed electron density within that volume. The most likely position to find the 1s electron is at the nucleus.However the most likely radius is at some distance from the nucleus. The graph of wavefunction vs distance falls off exponentially as you move away from the nucleus. The electron density builds quadratically with distance from the nuclues. The peak electron density will be the product of the these two functions. This results in a curve with a density peak at a certain distance. Wavefunctions alone do not tell you the electron density.
 
[[Image:800px-S-p-Orbitals.svg.png|thumb|300px|Approximate shape of atomic orbitals viewed as a surface.]]

=== Orbital nodes ===

[[Image:HAtomOrbitals.png|thumb|300px|Visualization of electron density as cross sections for orbitals and shells.]]
The p orbitals have orientations along the x, y and z axes. A node is a place where there is zero probability of finding an electron. A radial node has a spherical surface with zero probability. P orbitals have an angular node along axes. We usually indicate the sign of the wave function in drawings by shading the orbital as black and white, or blue and green.

1s: no node
2s: one radial node, 2p one angular node
3s: two radial nodes, 3p one radial node one angular node, 3d two angular nodes

The more nodes the higher the energy of the orbitals.
The more the function varies spatially, the higher the energy.

=== Core and valence electrons ===
'''Core electrons''' are very tightly bound tot he nucleus and spend most of their time very close to the nucleus. They are largely unaffected by the presence of nearby atoms.

'''Valence electrons''' are less tightly bound to the nucleus and are in the outermost "shell". These electrons are easily affected by the presence of other atoms and the ones that are critical for bonding between atoms.

According to the '''Aufbau''' principle we start adding electrons to the 2s orbital and then to the three 2p orbitals, each of which can have up to two electrons. The orbitals are filled from lowest energy to the highest. Here are the electron configuration for row two of the periodic table. The 1s orbitals are core while the two 2s and 2p orbitals are valence electrons. Neon has a full octet so it is non-reactive, ie a noble gas.

{| border="1"
|-
! <math>element</math>
! <math>1s</math>
! <math>2s</math>
! <math>2p</math>

|-
! Li
| 1s2 || 2s1 ||
|-
! Be
| 1s2 || 2s2 ||
|-
! B
| 1s2 || 2s2 || 2p1
|-
! C
| 1s2 || 2s2 || 2p2
|-
! N
| 1s2 || 2s2 || 2p3
|-
! O
|1s2 || 2s2 || 2p4
|-
! F
|1s2|| 2s2 || 2p5
|-
! Ne
|1s2 || 2s2 || 2p6
|}

Here is the general rules for filling orbitals.

{| border="1"
|-
!
! <math>s</math>
! <math>p</math>
! <math>d</math>
! <math>f</math>
! <math>g</math>
|-
! 1
| 1 || || || ||
|-
! 2
| 2 || 3 || || ||
|-
! 3
| 4 || 5 || 7 || ||
|-
! 4
| 6 || 8 || 10 || 13 ||
|-
! 5
| 9 || 11 || 14 || 17 || 21
|-
! 6
|12 || 15 || 18 || 22 || 26
|-
! 7
|16 || 19 || 23 || 27 || 31
|-
! 8
|20 || 24 || 28 || 32 || 36
|}

=== Bridging the Language Barrier ===

In working in an interdisciplinary research its easy to get confused with different terms for the same concepts as used by physicists and organic chemists. This table might help:

{| class="wikitable" border ="1"
|-
! Physicist Speak
! Organic Chemist Speak

|-
| Bands
| Molecular Orbitals
|-
| Band Gap
| Excited State Energy
|-
| Excitons
| Excited States
|-
| High work function material
| Electron deficient - acceptor
|-
| Low work function material
| Electron rich -donor
|-
| !?&* stuff that messes up my vacuum chamber (fill in your favorite apparatus)
| Organic compound or polymer
|-
| Wavelength division mulitiplexer (fill you your favorite device system)
| Device thingy

|}

 
<table id="toc" style="width: 100%">
<tr>

<td style="text-align: center; width: 33%">[[Main_Page#Molecular Orbitals|Return to Molecular Orbitals Menu]]</td>
<td style="text-align: right; width: 33%">[[Electronegativity and Bonding Between Atoms|Next Topic]]</td>

</tr>
</table>

Lasers

2009-08-27T00:48:40Z

Nsylvain: /* Lasers */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Dispersion and Attenuation Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Optical Fibers, Waveguides, and Lasers|Return to Optical Fibers, Waveguides, and Lasers Menu]]</td>
</tr>
</table>
== Lasers ==

Laser is light amplification via stimulated emission of radiation. Stimulated emission is key to the ability of the process to amplify light. Mainman was the first person to develop the first visible laser using a ruby.

See Wikipedia on Lasers http://en.wikipedia.org/wiki/Laser

See Mainman 1960 <ref>T.H. Mainman, Nature 187, 493 (1960) DOI: [http://dx.doi.org/10.1038/187493a0 10.1038/187493a0]</ref>

See Cord Course on Lasers [http://cord.org/cm/leot1.htm Lasers]

== Emission and absorption of radiation ==

=== Two state system ===

Consider a simple two level system. There are two population of molecules (N1 and N2) at two possible energy levels (E1, E2). An atom on level E1 can be elevated to E2.
[[Image:Laser_init_final.png|thumb|400px|In stimulated emission the photon released is identical in energy, polarization and phase as the stimulating photon.]]
The absorption of a photon with the correct energy hν (the difference between E1 and E2) results in a stimulated absorption and the system moves into the excited state. This leads to the emission of photon in a lifetime τ21. In spontaneous emission the emitted photon can have differing energies, polarization and phase than the stimulating photon.

In stimulated emission the absorbed photon causes the excited photon to emit light with specific characteristics.

The stimulating and stimulated photon have the same energy, same polarization, and the same phase. Emitted energy is perfectly coherent and the amplitude of the electric field can interfere completely constructively.
 

== Einstein relations ==
[[Image:Laser_energypops.png|thumb|300px|]]

When the system is in thermal equilibrium the rate of upward and downward transitions are equal (Einstein 1917). This is a dynamic equilibrium, there is still emission and absorption but the relative rates are equal.

'''upward transition rate'''

The upward transition rate is
:<math>N_1 \rho_\nu B_{12}\,\!</math>

where

:<math>N_1\,\!</math> is the atoms per unit volume with energy E1
:<math>\rho_\nu\,\!</math> is the energy density at frequency nu corresponding to the difference in energy between E1 and E2 (hν)

:<math>\rho_\nu = N_{\nu} h\nu \,\!</math>

Where Nν is the number of photons per unit volume having frequency ν

The coefficient B12 gives the probability of upward transition in a stimulated process.

'''stimulated emission rate'''

The stimulated transition rate from level 2 to level 1 is given:

:<math>N_2\rho_{\nu}B_{21}\,\!</math>

Where

:<math>N_2\,\!</math> is the number of atoms per unit volume in the collection with energy E2.
:<math>B_{21}\,\!</math> is the probability of the photons stimulating the downward transition.

The total downward transition rate is the sum of the induced and spontaneous contributions

:<math>N_2\rho_{\nu}B_{21} + N_2 A_{21}\,\!</math>

and

:<math>A_{21} = 1 /\tau_{21}\,\!</math>

The spontaneous term is only dependent on the population of the upper level, it is not affected by the photons, and by the lifetime of the excitation on level 2 going to level 1.

The '''Einstein coefficients''' A21 B21 and B12 are dependent on the material. For a system in thermal equilibrium the upward and downward transition rates must be equal.

:<math>N_1 \rho_\nu B_{12} = N_2 \rho_\nu B_{21} + N_2 A_{21}\,\!</math>

Rearranging the equation in order to get the expression in terms of energy density.
:<math>
N_1 \rho_\nu B_{12} = N_2 \rho_\nu B_{21} + N_2 A_{21}\,\!</math>

We move :<math>\rho_\nu\,\!</math> to the left:

:<math>\rho\nu = \frac {N_2A_{21}} { N_1B_{12} – N_2B_{21}}\,\!</math>

Then divide the right side numerator and denominator by N2:

:<math>\rho_\nu = \frac {A_{21}/B_{21}} {[(B_{12}/B_{21}) (N_1 N_2)] -1}\,\!</math>

The ratio of the stimulated absorption emission coefficients times the ratio of the populations goes to one.

=== Bolztman Statistics ===
We can express N1 and N2 the ratio of the populations at thermal equilibrium using Botzmann statistics :

:<math>N_j = N_o \frac {g_j exp (-E_j/kT)} {\sum {g_i exp(-E_i/kT)}}\,\!</math>

:<math>N_j\,\!</math> is the population density of energy level Ej
:<math>N_o\,\!</math> is the total population density
:<math>g_j\,\!</math> is the degeneracy of the jth level (g terms go to 1 if there no degeneracy).
:<math>E_j\,\!</math> is the energy of the jth level

The denominator is the sum across all possible levels i. If we consider N1/N2 we only need the numerator because the term corresponding to the summation cancels out.

This gives:

:<math>\frac {N_1} {N_2} = \frac {g_1}{ g_2} exp [(E_2-E_1)/kt]= \frac {g_1} {g_2} exp(h\nu/kT)\,\!</math>

Then plug it into the previous expression

:<math>\rho_\nu = \frac {A_{21}/B_{21}} {[(g_1/g_2) (B_{12}/B_{21}) exp (h\nu /kT)] -1} \,\!</math>

Since the collection of atoms in the system is in thermal equilibrium the radiation is blackbody radiation.
Blackbody radiation or “cavity radiation” refers to an object or system which absorbs all radiation incident upon it and reradiates energy which is characteristic of the radiating system only, and is not dependent on the type of radiation that is incident on it. The sun can be considered a blackbody. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating.

See GSU hyperphysics on http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html

=== Classical versus quantum model ===
[[Image:Ultravioletcatastrofe.PNG|thumb|300px|The red line shows the classical model in which intensity increases with decreasing wavelength. The actual observation (green line) shows a peak and then decreasing intensity with the shortest wavelengths.]]
The cavity absorbs all radiation incident upon it leading to radiation. The more the number of modes the higher the energy.
In a classical description the amount of radiation should be proportional to the number of modes at a given frequency with the probability of modes per unit frequency per unit volume is equal. The average energy of emission is only dependent on the kT and thus the energy should increase indefinitely with temperature. This is true in the lower energy range but deviates significantly at shorter wavelengths. This was referred to the “ultraviolet catastrophe” because at higher energies there is a lower probability of finding the correct mode.

{| class="wikitable" border ="1"
|-
!
! #Modes /unit frequency, volume
! Probability of occupying modes
! Average energy per mode
|-
| '''Classical'''
| :<math>\frac {8\pi \nu^2 } {c^3}\,\!</math>
| Equal for all modes
| <math>kT\,\!</math>
|-
| '''Quantum'''
| <math>\frac {8\pi \nu^2 } {c^3}\,\!</math>
| Quantized, hν required, upper modes less probable
| <math>\frac {h\nu} {exp(h\nu/kT) -1}\,\!</math>
|}

Quantum mechanics was able to explain this. The number modes is the same in classical and quantum physics varies as the square of the frequency. But in quantum physics the average energy is quantized by hnu and the probability of reaching the exact required state for a given wavelength changes exponentially at higher energy levels.

The '''blackbody radiation density''' is

:<math>\rho_\nu = \frac {8\pi h\nu^3 n^3} {c^3} \left( \frac {1} {exp(h\nu/kT) -1} \right)\,\!</math>

Where
:<math>n\,\!</math> is the refractive index of the medium

The prefactor depends on the cube of the frequency, and the cube of the speed of light in a vacuum.
Now we can compare to expressions for :<math>\rho_\nu\,\!</math>

:<math>\rho_\nu = \frac {a_{21}/B_{21}} [(g_1/g_2) {(B_{12}/B_{21})exp(h\nu /kT)]-1}\,\!</math>

and

:<math>\rho_\nu = \frac {8\pi h\nu^3 n^3} {c^3} \left( \frac {1} {exp(h\nu/kT) -1} \right)\,\!</math>

'''Einstein relations''':

If we eliminate the degeneracy factor the oscillator strength for going from the ground state to the excited state is the same as that going from the excited state back down.

:<math>g_1B_{12} = g_2B_{21}\,\!</math>

Which leads to the ratio of the Einstein coefficients for spontaneous and stimulated emissions equal to:

:<math>\frac {A_{21}} {B_{21}} = \frac {8\pi h \nu^3 n^3} {c^3}\,\!</math>

The ratio of R of the rate of spontaneous absorption to the rate of stimulated emission for a given pair of energy levels.

:<math>R = \frac {A_{21}}{ \rho_\nu B_{21}}\,\!</math>

Spontaneous emission (A21) does not depend on energy density but stimulated emission does.

:<math>R = exp(h\nu/kT)-1\,\!</math>

Thus for emission in the middle of the visible spectrum (2.5 eV) at room temperature kT is relatively large, e100 -1 is very small. Thus a thermal equilibrium stimulated emission is not an important process.

The goal is to increase stimulated emission because this light is completely coherent allowing extremely high intensities.

For a given pair of energy levels:

Stimulated emission is described

:<math>N_2\rho_\nu B_{21}\,\!</math>

*We must increase both the radiation density (rhonu) and the population density N2 of the upper level in relation to the population density, and for a given material B21 is fixed.

*We must create a condition in which N2>(g2/g1) N1 even though E2>E1

This requires a '''population inversion''' in which the higher level is more populated than the lower level. Mainman did this with ruby laser but it was very difficult because the lower energy level was the ground state which is heavily populated. Other lasers have been developed where the energy transition is between two excited states. Thus it is much easier to have a inversion of population.

== Absorption of radiation ==
[[Image:Laser_irradiance.png|thumb|300px|]]

Consider a collimated beam of perfectly monochromatic radiation of unit cross-sectional area passing through an absorbing medium. Also assume for simplicity that it is a two level system with transition between E1 and E2.

As the beam with irradiance I(x) enters the medium at x and moves to position Delta x

:<math>\Delta I(x) = I (x + \Delta x) – I(x)\,\!</math>

( the sign of Δ I(x) is negative because it is an absorbing medium)

The '''beer lambert law ''' states that for a homogeneous medium:

:<math>\Delta I(x) = - \alpha I(x) \Delta X\,\!</math>

Where:

Alpha is the absorption coefficient which is positive

:<math>\frac {dI(x)} {dx} = -\alpha I(x)\,\!</math>

If we integrate:

:<math>I= I_o exp(-\alpha x)\,\!</math>

Where
:<math>I_0\,\!</math> is the incident irradiance
:<math>x\,\!</math> is the length of the path through the medium

The net rate of loss of photons per unit volume from the beam as it moves through a volume element of medium of thickness Delta X and unit cross-sectional area is given by:

:<math>- \frac {dN_\nu} {dt} = N_1 \rho_\nu B_{12} – N_2 \rho_\nu B_{21}\,\!</math>

Where

:<math>N_1 \rho_\nu B_{12}\,\!</math> is the rate of absorption

:<math>N_2 \rho_\nu B_{21}\,\!</math> is the rate of emission

Ignoring the degeneracy factors B21 equals B12. The rate of emission is dependent on the population of the two levels.

:<math>-\frac {dN_\nu} {dt} = (\frac {g_2}{g_1} N_1 - N_2) \rho_\nu B_{21}\,\!</math>

If we ignore photons created by spontaneous emission and scattering losses.

The irradiance is the energy cross a unit area per unit time, which is the energy density times the speed of light in the medium.

Using dimensional analysis:

Energy crossing an area in a unit time has units J/m2 t which is equivalent to energy density J /m3 times the speed of light in m/s

For photons of frequency nu:

:<math>I_\nu = \frac {\rho_\nu c}{n} = \frac {N_\nu h\nu c}{n} ; N_\nu = \frac {I_\nu n}{[h\nu c]}\,\!</math>

Where
:<math>I_\nu\,\!</math> is the irradiance at frequency nu
:<math>n\,\!</math> is the refractive index of the material
:<math>\nu\,\!</math> is frequency that can excite from the ground to the excited state
:<math>N_nu\,\!</math> is the energy density

The more that light is slowed down within the medium the more time there is for absorption.

The photon density decreases as it passes through a distance Δ x of f the volume element according to:

:<math>-dN_\nu(x)= [I_\nu (x) – I_\nu (x + \Delta x] \frac {n} {h\nu_{21}c}\,\!</math>

If Δx is sufficiently small you can integrate:

:<math>-dN_\nu(x) = - \frac {dI_\nu(x)} {dx} \cdot \frac {\Delta x n} {h\nu_{21}c}\,\!</math>

and

:<math>(\Delta x n/c = \Delta t)\,\!</math>

This implies that the rate of decay of photon density in a time interval dt:

:<math>\frac {dN_\nu} {dt} = \frac {dI_\nu(x)} {dx} \cdot \frac {1} {h\nu_{21}}\,\!</math>

:<math>\frac {dN_\nu} {dt} = -\alpha I_\nu (x) \cdot \frac {1} {h\nu_{21}} = -\alpha \rho_\nu \frac {c}{n} \cdot \frac {1} {h\nu_{21}}\,\!</math>

where

:<math>-\alpha I_\nu(x)\,\!</math> is the variation of the irradiance by unit volume

We can build a rate of disappearance of photon density as a function of absorption coefficient and the expression of photon density as a function of populations.

:<math>\alpha \rho_\nu \frac {c}{n} \frac {1}{h \nu_{21}} = \left (\frac {g_2}{g_1} N_1-N_2 \right) \rho_\nu B_{21}\,\!</math>

And finally put it all in terms of alpha:

:<math>\alpha = \left( \frac {g_2}{g_1} N_1 – N_2 \right) \frac {B_{21} h\nu_{21} n} {c}\,\!</math>

Thus the absorption coefficient alpha depends on the difference in the population difference between levels E1 and E2.

For a collection of atoms in thermal equilibrium since E2>E1 then:

:<math>(g_2/g_1) N_1 > N_2\,\!</math>

Hence α is positive and normally N2 is very small. But if we can create a situation for which:

:<math>(g_2/g_1) N_1 < N_2\,\!</math>

That is if there is an inversion of population, then α becomes negative.

If the beam is right energy hν and the excited state population is larger than the ground state, then the probability of a stimulated emission from the excited state is higher than the probability of a stimulated absorption from the ground state.

In this case the irradiance of the beam grows as it propagates through the medium.

Recalling that: I = I0exp(-αx) we obtain an exponential relationship:

:<math>I = I_0 exp(kx)\,\!</math>

:<math>k\,\!</math> is referred to the '''small signal gain coefficient''' or simply '''gain coefficient'''

:<math>k =(N_2- \frac {g_2}{g_1} N_1) B_{21} \frac {h\nu_{21}n} {c}\,\!</math>

== Population Inversion ==
[[Image:Laser_popinvert.png|thumb|300px|Normal and inverted populations]]
The plots show the energy of the two states (E2, E1 - y axis) and the population of each excited state (N2, N1 - x axis). The dotted line is the exp(-Eν/kt) thermal equilibrium distribution.

In the two level system at thermal equilibrium N2 is always smaller than N1. In order to reach a population inversion one has to pump the system into the excited state. Light or electrical current can be used to pump a system as in a diode laser. In the pumped condition the N1 population is far smaller than the boltzman distribution while the N2 population is far greater.

The ratio of the population in thermal equilibrium is given by boltzman statistics:

:<math>\frac {N_1} {N_2} = \frac {g_1} {g_2} exp [(E_2-E_1)/kt]= \frac {g_1}{g_2} exp(h\nu/kT)\,\!</math>

If hν ~ 2.5 eV (visible part of spectrum) and g2 =g1 at room temperature (kT = .025 eV)

:<math>N_2/N_1 = exp (-h\nu/kt)\,\!</math> ~ e-100 ~ 10-43 (10 ~ e2.3 so to make a conversion divide e exponent (that is 100) by 2.3)

This is a very small ratio.

'''Example'''
If we consider an octamer of phenylene vinylene which absorbes around 2.5 eV (= E2-E1). Assume the molar mass is ~ 1000g and the density is 1 g per cm3

What is roughly the length of the side of a cube containing oligophenylene vinylene in which on average one molecule will be be in the excited state at room temperature?

'''Answer'''
You would need 1043 molecules which weighs 1.6 x 1023 grams which take up 1023 cm3 or a cube that is 1017 m3 that is 108 km3 which is a cube 464 km on a side.

In order to get an population inversion we must supply a large amount of energy to excite atoms into the upper E2. Since B12 and B21 are equal (assuming g1 = g2). Once atoms are excited into the upper level the probabilities of further stimulated absorption or emission are equal. The best that can be achieved with a two level system is equality of the populations of the two levels. Thus you can not get to the state of inverted population. This is the reason to consider a three level system.

=== 3 Level System ===
[[Image:Laser_3level.png|thumb|300px|A three level system]]
The idea of three level system was proposed by Bloembergen in 1957 and received a Nobel prize in physics in 1981 for his work with nonlinear optics and lasers. He currently has a position at the University of Arizona optical science center.

In the three level system E2 is a second excited state. If atoms are intensely illuminated electrons can be pumped into E2 from the ground state E0. From E2 the electrons decay by non-radiative processes to level E1 so that a population inversion may be created between E1 and E0. Transition from level E2 to E1 should be very rapid thereby ensuring that there are always vacant states at E2. Transition from E1 to E0 should be very slow; E1 has to be metastable state so that you can accumulate a population.

Level E2 should preferably consist of a large number of closely spaced levels so that pumping uses as wide a part of the spectral range of the pumping radiation as possible.

Three level lasers such as ruby require a very high pump power because the terminal level of the laser transition is the ground state.

=== 4 level system ===
[[Image:Laser_4level.png|thumb|300px|Four level system]]
In a 4 level system there is population inversion between two excited state (E2 and E1). Pumping puts electron directly into a series of levels at E3. Internal conversion brings the electrons rapidly down to a metastable E2 where they accumulate and the population is inverted with respect to E1. E2 to E1 is a radiative transfer. From E1 top E0 there should be rapid decay so that the level is replenished for pumping. Even with pumping the population of E1, E2 and E3 is less than that of E0.

The frequency of light used for pumping must be correct for jumping between E0 and E3. The laser light produced reflects the transition between E2 and E1.

Most scientific lasers are solid state. They have a host materials (such as ruby or YAG) with dopants (niodinium) that have the lasing transitions.

In the Nd-YAG laser
*T32 ~ 10-8s
*T21~ .5 ms
*T10~ 30 ns

Much of laser research is the search for lasing host materials with energy transitions and time characteristics that are favorable to generating the right balance when pumped.

=== Optical Feedback ===
[[Image:Optical-cavity1.png|thumb|300px|Various configurations of an optical cavity]]
Positive feedback may be obtained by placing gain medium between a pair of mirrors which form a optical cavity. (a Fabry-Perot resonator) This allows the beam to make repeated passes through the medium gaining energy with each pass due to stimulated emission. A semitransparent mirror on one end finally allows the beam to escape when it has reached sufficient intensity. There are many systems that provide amplification of the signal and there are conjugated polymers that allow amplified stimulated emission. But that is not the same as laser. Three groups discovered lasing in conjugated polymers in the mid 1990s, Allan Heeger at Santa Barbara, Richard Friend at Cambridge, and Z.V. Vardeny at University of Utah. Since then there have been many efforts to create a diode laser from a conjugated polymer, but without success.

The problem is that you need to build a large population in the excited state. In an LED electrons are injected from ones side and holes are injected from the other and when they combine there must be large densities in the excited state. However the presence of injected charge (polarons) quenches the exciton. You don’t have that issue with optical pumping because you start with nothing, the atoms absorbs light and directly build up the population in the excited state. There are no polarons or charges floating around.

== Laser Modes ==

=== Axial Modes ===

The two mirrors of the laser form a resonant cavity and standing wave patterns are set up between the mirrors. Standing waves satisfy the condition:

:<math>p \frac {\lambda} {2} = L\,\!</math>

:<math>L\,\!</math> is the optical path between the mirrors (which is a product of the physical length and the index of refraction).
:<math>p\,\!</math> is an integer

The length of cavity needs to be and integral multiple of half the wavelength in order for each reflection to be coherent with the preceding waves.

Substituting for wavelength for c/ν :

:<math>\nu = \frac{ pc}{2L}\,\!</math>

There are many values of for p that will satisfy this equation.

[[Image:400px-Longitudinal_mode_v2.svg.png|thumb|300px|Modes of oscillation for p = 1 to p = 6]]

Example

If L = .5m and λ ~500 nm, then there are 2 x 106 possible p values.

Each value of p satisfying the above equations defines an axial (or longitudinal) mode of the cavity. The frequency separation (Δ p =1) is given by:

:<math>\Delta \nu = \frac {c}{2L}\,\!</math>

For L= .5m Δ ν = 300 Mhz, this corresponds to an photon energy difference of 10-6 eV(1eV ~ 2 x 1014 hz)
 
[[Image:Laser_irradiance_distribution.png|thumb|left|300px|Theoretical distribution of stimulated emission at each mode.]]
[[Image:Laser_axial_osc.png|thumb|300px|Oscilloscope of actual data plot shows a normal distribution]]

Thus the modes of oscillation of the laser cavity consist of a large number of frequencies separated by c/2L.
Specific frequencies are able to stimulate additional photons of the same wavelengths (when the beam goes back and forth in a resonating cavity). This results in a distribution of photons that are separated in energy by 300 mhz or a millionth of an eV.

The envelope corresponds to a number of photons with very precise energy.
 

=== Transverse modes ===
[[Image:Laser_transverse.png|thumb|300px|]]
Up to this point we have been discussing only the axial or longitudinal axis in which waves travel along a line joining the centers of the mirrors. For any real laser cavity there will be waves travelling just off the axis that are able to replicate themselves after covering a closed path. It is possible to have beams that are reflected at different angles. These transverse electromagnetic fields also give rise to resonant transverse electromagnetic modes (TEM) but the because they have components of their electromagnetic field that is transverse to the direction of propagation. The TEM subscripts indicate the number of modes along the x and y axis and you can see these patterns from lasers producing these modes.

== Classes of Lasers ==
===Doped insulator lasers ===

These are lasers whose active medium consists of a crystalline or amorphous (glassy) host material containing active ions, typically from the transition metal and rare earth groups in the periodic table.

=== Nd:YAG laser. ===
[[Image:Laser_ndyag.png|thumb|300px|3 step energy diagram for Nd:YAG laser]]
This is a common laser used in research. When you see the wavelength reported in research as 1.064 micron this indicates that a Nd:YAG laser was used. Similarly a wavelength 532 nm this means they used a Nd:YAG but cut the wavelength in half using a nonlinear process.

The active material is yttrium aluminum garnet (Y3Al5 O12) is a garnet in which the aluminum oxide in garnet is replaced by ytrium in an orderly fashion. It has the rare earth metal ion neodymium Nd3+ present as an impurity. Nd3+ ions are randomly distrubuted as substitutions on lattice sites normally occupied by Y3+ ions. The Nd3+ provide the energy levels for both the lasing transitions and pumping.

These are the transitions within the neodymium ion. There are three levels in E3 which can be pumped into and which all undergo fast nonradiative decay into the E2 state. The laser transition happens between E2 and E1 with a 1.05 Um wavelength (in the near IR). From N1 there is a rapid non-radiative decay to the groundstate N0. The lifetime of the lasing transition is long which is what allows it to build up the population in E2 for population inversion. It is important to be aware of the relative energy levels and the speed of various transition that makes each the laser system work.

Pumping is normally achieved using an intense flash of white light from a xenon flashtube. The presence of several possible pumping transition E3 levels contributes to the efficiency of the laser when using a pumping source with a broad spectral output. Each level is able to capture a slight different part of the pump light spectrum.

=== Ruby laser ===
[[Image:Ruby_energy.png|thumb|300px|Energy diagram for a ruby laser]]
Ruby was the first visible laser that was developed.
The active materials is a synthetically grown crystal of ruby, that is aluminum oxide with about 0.05% by weight of chromium as an impurity. Most precious gems have impurities in a very common material.
The ruby has transitions at the upper end of the visible spectrum in the green and violet (2-3 eV) which is what gives the ruby its characteristic deep red color.

=== Vibronic laser ===
[[Image:Laser_vibronic.png|thumb|300px|Vibronic laser energy diagram]]
Vibronic laser such as a alexandrite and Ti-sapphire are similar to other solid state lasers such as Nd:YAG in that light from a external pump source excites the impurity ions in a transparent host. The sapphire laser is similar to the ruby, it is aluminum oxide with impurities such as titanium.

They are fundamentally different in that laser gain is possible over a broad range of wavelengths so that they can produce either tunable outputs or ultrashort pulses.

Vibronic laser such as a alexandrite and Ti-sapphire are similar to other solid state lasers such as Nd:YAG in that light from a external pump source excites the impurity ions in a transparent host. The sapphire laser is similar to the ruby, it is aluminum oxide with impurities such as titanium.

They are fundamentally different in that laser gain is possible over a broad range of wavelengths so that they can produce either tunable outputs or ultrashort pulses.

Vibronic solid state lasers have four level systems. The pumping radiations excites the active ions to a vibronic band. The ions then lose vibrational energy and drop to the bottom of the band which is the upper laser level. The laser transition then occurs to a vibrationally excited sublevel of the n1 state; this is followed by the ions relaxing to the lowest sublevel of the ground state.

Alexandrite laser comprises Cr3+ ions in a beryllium aluminate (BeAl2O4) host.

The pumping leads from the lowest vibration level of the ground state to a high vibrational level of E3, followed by relaxation to the zeroth level of the excited state E3. Then the excited emission (laser) occurs between some level in E2 and some vibrational level of E1. This gives a wide range of combinations of energy transitions that make it possible to tune the laser by filtering what comes out of the laser. This gives a range of 701 to 826 nm. The broad energy range of emission means that the time of the emission can be very short. Ti-sapphire is the most used laser for femptosecond pulse studies.

The Heisenberg uncertainty principle tells us that if the precision on energy is high the precision on time will be low. Femptosecond pulses will have energy scattered over 1.5 eV. When doing spectroscopy where accuracy in energy domain (milli- eV precision) is important then it is better to use longer pulses in the picoseconds.

=== Semiconductor lasers ===

Semiconductor lasers such as the diode laser are made with p-n junctions. It users an LED with a very high density of emission.

=== Gas lasers ===
[[Image:He-ne_energy.png|thumb|300px|]]
Gas lasers are the most widely used type lasers. The helium laser is used for the red laser pointer, or the CO2 laser used in medicine.
Basically there are three classes of gas lasers according to whether the transitions are
Between electronic energy levels of atoms
or
Between electronic energy energy of ions
or
Between the vibrational/ rotational levels of molecules.

Atomic laser – the HE-NE laser
The active medium is a mixture of 10 parts helium to one part neon. Neon provides the energy levels for the laser transition. Helium provide an efficient excitation mechanism for the neon atoms.
Excitation usually takes place in a DC discharge created by applying a high voltage ( ~2-4 kV) across the gas. This creates high energy electrons in the gas mixture. The electrons hit the helium atoms (which are most common) bringing them to the excited state, and then transfer energy to the neon atoms when the energy of the excited states match. The neon can then relax to laser transitions levels such 3s level to 3p levels and then 3p to 2s and 2p levels.

:<math>E1 + He = He* +e2\,\!</math>

Resonant transfer of energy is proportional to exp (-ΔE/kt) where ΔE is the energy difference between the excited states of the two atoms involved.

:<math>He* + Ne = Ne* +He\,\!</math>

=== Liquid Dye lasers ===
[[Image:Laser_dye_Abs_Emis.png|thumb|300px|]]
Liquids have useful advantages over solids and gases. Solids are very difficult to prepare with the requisite degree of optical homogeneity and they suffer permanent damage if overheated. Gases do not have these difficulties but they have a much smaller density of active atoms. The dye laser has the advantage that they are simple and they can be tuned over a significant wavelength range.

The active medium is an organic dye dissolved in a solvent. It is a four level system that uses piconjugated chromophores. The pumping makes the dyes go from the electronic groundstate to an excited vibrational level of an excited state. The E3 level is the top vibronic level of the excited state curve, relaxation occurs down to the bottom of the E2 level. Laser emission can occur from all the vibronic levels.

[[Image:Potentialenergy surfacechang.png|thumb|300px|Potential energy surface diagram]]

The broad range of emission can be filtered for specific wavelengths.

Rhodamine is a common laser dye with a cyanine type structure. It is dissolved in methanol. It has as broad tuning range (570-660 nm) All dye lasers are optically pumped.

See Cord Module on Dye lasers [http://cord.org/cm/leot/course03_mod10/mod03_10.htm Dye lasers]

== References ==

<references/>

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Dispersion and Attenuation Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Optical Fibers, Waveguides, and Lasers|Return to Optical Fibers, Waveguides, and Lasers Menu]]</td>
</tr>
</table>

Lasers

2009-08-27T00:38:48Z

Nsylvain: /* References */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Dispersion and Attenuation Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Optical Fibers, Waveguides, and Lasers|Return to Optical Fibers, Waveguides, and Lasers Menu]]</td>
</tr>
</table>
== Lasers ==

Laser is light amplification via stimulated emission of radiation. Stimulated emission is key to the ability of the process to amplify light. Mainman was the first person to develop the first visible laser using a ruby.

See Wikipedia on Lasers http://en.wikipedia.org/wiki/Laser

See Mainman 1960 <ref>T.H. Mainman, Nature 187, 493 (1960)</ref>

See Cord Course on Lasers [http://cord.org/cm/leot1.htm Lasers]

== Emission and absorption of radiation ==

=== Two state system ===

Consider a simple two level system. There are two population of molecules (N1 and N2) at two possible energy levels (E1, E2). An atom on level E1 can be elevated to E2.
[[Image:Laser_init_final.png|thumb|400px|In stimulated emission the photon released is identical in energy, polarization and phase as the stimulating photon.]]
The absorption of a photon with the correct energy hν (the difference between E1 and E2) results in a stimulated absorption and the system moves into the excited state. This leads to the emission of photon in a lifetime τ21. In spontaneous emission the emitted photon can have differing energies, polarization and phase than the stimulating photon.

In stimulated emission the absorbed photon causes the excited photon to emit light with specific characteristics.

The stimulating and stimulated photon have the same energy, same polarization, and the same phase. Emitted energy is perfectly coherent and the amplitude of the electric field can interfere completely constructively.
 

== Einstein relations ==
[[Image:Laser_energypops.png|thumb|300px|]]

When the system is in thermal equilibrium the rate of upward and downward transitions are equal (Einstein 1917). This is a dynamic equilibrium, there is still emission and absorption but the relative rates are equal.

'''upward transition rate'''

The upward transition rate is
:<math>N_1 \rho_\nu B_{12}\,\!</math>

where

:<math>N_1\,\!</math> is the atoms per unit volume with energy E1
:<math>\rho_\nu\,\!</math> is the energy density at frequency nu corresponding to the difference in energy between E1 and E2 (hν)

:<math>\rho_\nu = N_{\nu} h\nu \,\!</math>

Where Nν is the number of photons per unit volume having frequency ν

The coefficient B12 gives the probability of upward transition in a stimulated process.

'''stimulated emission rate'''

The stimulated transition rate from level 2 to level 1 is given:

:<math>N_2\rho_{\nu}B_{21}\,\!</math>

Where

:<math>N_2\,\!</math> is the number of atoms per unit volume in the collection with energy E2.
:<math>B_{21}\,\!</math> is the probability of the photons stimulating the downward transition.

The total downward transition rate is the sum of the induced and spontaneous contributions

:<math>N_2\rho_{\nu}B_{21} + N_2 A_{21}\,\!</math>

and

:<math>A_{21} = 1 /\tau_{21}\,\!</math>

The spontaneous term is only dependent on the population of the upper level, it is not affected by the photons, and by the lifetime of the excitation on level 2 going to level 1.

The '''Einstein coefficients''' A21 B21 and B12 are dependent on the material. For a system in thermal equilibrium the upward and downward transition rates must be equal.

:<math>N_1 \rho_\nu B_{12} = N_2 \rho_\nu B_{21} + N_2 A_{21}\,\!</math>

Rearranging the equation in order to get the expression in terms of energy density.
:<math>
N_1 \rho_\nu B_{12} = N_2 \rho_\nu B_{21} + N_2 A_{21}\,\!</math>

We move :<math>\rho_\nu\,\!</math> to the left:

:<math>\rho\nu = \frac {N_2A_{21}} { N_1B_{12} – N_2B_{21}}\,\!</math>

Then divide the right side numerator and denominator by N2:

:<math>\rho_\nu = \frac {A_{21}/B_{21}} {[(B_{12}/B_{21}) (N_1 N_2)] -1}\,\!</math>

The ratio of the stimulated absorption emission coefficients times the ratio of the populations goes to one.

=== Bolztman Statistics ===
We can express N1 and N2 the ratio of the populations at thermal equilibrium using Botzmann statistics :

:<math>N_j = N_o \frac {g_j exp (-E_j/kT)} {\sum {g_i exp(-E_i/kT)}}\,\!</math>

:<math>N_j\,\!</math> is the population density of energy level Ej
:<math>N_o\,\!</math> is the total population density
:<math>g_j\,\!</math> is the degeneracy of the jth level (g terms go to 1 if there no degeneracy).
:<math>E_j\,\!</math> is the energy of the jth level

The denominator is the sum across all possible levels i. If we consider N1/N2 we only need the numerator because the term corresponding to the summation cancels out.

This gives:

:<math>\frac {N_1} {N_2} = \frac {g_1}{ g_2} exp [(E_2-E_1)/kt]= \frac {g_1} {g_2} exp(h\nu/kT)\,\!</math>

Then plug it into the previous expression

:<math>\rho_\nu = \frac {A_{21}/B_{21}} {[(g_1/g_2) (B_{12}/B_{21}) exp (h\nu /kT)] -1} \,\!</math>

Since the collection of atoms in the system is in thermal equilibrium the radiation is blackbody radiation.
Blackbody radiation or “cavity radiation” refers to an object or system which absorbs all radiation incident upon it and reradiates energy which is characteristic of the radiating system only, and is not dependent on the type of radiation that is incident on it. The sun can be considered a blackbody. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating.

See GSU hyperphysics on http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html

=== Classical versus quantum model ===
[[Image:Ultravioletcatastrofe.PNG|thumb|300px|The red line shows the classical model in which intensity increases with decreasing wavelength. The actual observation (green line) shows a peak and then decreasing intensity with the shortest wavelengths.]]
The cavity absorbs all radiation incident upon it leading to radiation. The more the number of modes the higher the energy.
In a classical description the amount of radiation should be proportional to the number of modes at a given frequency with the probability of modes per unit frequency per unit volume is equal. The average energy of emission is only dependent on the kT and thus the energy should increase indefinitely with temperature. This is true in the lower energy range but deviates significantly at shorter wavelengths. This was referred to the “ultraviolet catastrophe” because at higher energies there is a lower probability of finding the correct mode.

{| class="wikitable" border ="1"
|-
!
! #Modes /unit frequency, volume
! Probability of occupying modes
! Average energy per mode
|-
| '''Classical'''
| :<math>\frac {8\pi \nu^2 } {c^3}\,\!</math>
| Equal for all modes
| <math>kT\,\!</math>
|-
| '''Quantum'''
| <math>\frac {8\pi \nu^2 } {c^3}\,\!</math>
| Quantized, hν required, upper modes less probable
| <math>\frac {h\nu} {exp(h\nu/kT) -1}\,\!</math>
|}

Quantum mechanics was able to explain this. The number modes is the same in classical and quantum physics varies as the square of the frequency. But in quantum physics the average energy is quantized by hnu and the probability of reaching the exact required state for a given wavelength changes exponentially at higher energy levels.

The '''blackbody radiation density''' is

:<math>\rho_\nu = \frac {8\pi h\nu^3 n^3} {c^3} \left( \frac {1} {exp(h\nu/kT) -1} \right)\,\!</math>

Where
:<math>n\,\!</math> is the refractive index of the medium

The prefactor depends on the cube of the frequency, and the cube of the speed of light in a vacuum.
Now we can compare to expressions for :<math>\rho_\nu\,\!</math>

:<math>\rho_\nu = \frac {a_{21}/B_{21}} [(g_1/g_2) {(B_{12}/B_{21})exp(h\nu /kT)]-1}\,\!</math>

and

:<math>\rho_\nu = \frac {8\pi h\nu^3 n^3} {c^3} \left( \frac {1} {exp(h\nu/kT) -1} \right)\,\!</math>

'''Einstein relations''':

If we eliminate the degeneracy factor the oscillator strength for going from the ground state to the excited state is the same as that going from the excited state back down.

:<math>g_1B_{12} = g_2B_{21}\,\!</math>

Which leads to the ratio of the Einstein coefficients for spontaneous and stimulated emissions equal to:

:<math>\frac {A_{21}} {B_{21}} = \frac {8\pi h \nu^3 n^3} {c^3}\,\!</math>

The ratio of R of the rate of spontaneous absorption to the rate of stimulated emission for a given pair of energy levels.

:<math>R = \frac {A_{21}}{ \rho_\nu B_{21}}\,\!</math>

Spontaneous emission (A21) does not depend on energy density but stimulated emission does.

:<math>R = exp(h\nu/kT)-1\,\!</math>

Thus for emission in the middle of the visible spectrum (2.5 eV) at room temperature kT is relatively large, e100 -1 is very small. Thus a thermal equilibrium stimulated emission is not an important process.

The goal is to increase stimulated emission because this light is completely coherent allowing extremely high intensities.

For a given pair of energy levels:

Stimulated emission is described

:<math>N_2\rho_\nu B_{21}\,\!</math>

*We must increase both the radiation density (rhonu) and the population density N2 of the upper level in relation to the population density, and for a given material B21 is fixed.

*We must create a condition in which N2>(g2/g1) N1 even though E2>E1

This requires a '''population inversion''' in which the higher level is more populated than the lower level. Mainman did this with ruby laser but it was very difficult because the lower energy level was the ground state which is heavily populated. Other lasers have been developed where the energy transition is between two excited states. Thus it is much easier to have a inversion of population.

== Absorption of radiation ==
[[Image:Laser_irradiance.png|thumb|300px|]]

Consider a collimated beam of perfectly monochromatic radiation of unit cross-sectional area passing through an absorbing medium. Also assume for simplicity that it is a two level system with transition between E1 and E2.

As the beam with irradiance I(x) enters the medium at x and moves to position Delta x

:<math>\Delta I(x) = I (x + \Delta x) – I(x)\,\!</math>

( the sign of Δ I(x) is negative because it is an absorbing medium)

The '''beer lambert law ''' states that for a homogeneous medium:

:<math>\Delta I(x) = - \alpha I(x) \Delta X\,\!</math>

Where:

Alpha is the absorption coefficient which is positive

:<math>\frac {dI(x)} {dx} = -\alpha I(x)\,\!</math>

If we integrate:

:<math>I= I_o exp(-\alpha x)\,\!</math>

Where
:<math>I_0\,\!</math> is the incident irradiance
:<math>x\,\!</math> is the length of the path through the medium

The net rate of loss of photons per unit volume from the beam as it moves through a volume element of medium of thickness Delta X and unit cross-sectional area is given by:

:<math>- \frac {dN_\nu} {dt} = N_1 \rho_\nu B_{12} – N_2 \rho_\nu B_{21}\,\!</math>

Where

:<math>N_1 \rho_\nu B_{12}\,\!</math> is the rate of absorption

:<math>N_2 \rho_\nu B_{21}\,\!</math> is the rate of emission

Ignoring the degeneracy factors B21 equals B12. The rate of emission is dependent on the population of the two levels.

:<math>-\frac {dN_\nu} {dt} = (\frac {g_2}{g_1} N_1 - N_2) \rho_\nu B_{21}\,\!</math>

If we ignore photons created by spontaneous emission and scattering losses.

The irradiance is the energy cross a unit area per unit time, which is the energy density times the speed of light in the medium.

Using dimensional analysis:

Energy crossing an area in a unit time has units J/m2 t which is equivalent to energy density J /m3 times the speed of light in m/s

For photons of frequency nu:

:<math>I_\nu = \frac {\rho_\nu c}{n} = \frac {N_\nu h\nu c}{n} ; N_\nu = \frac {I_\nu n}{[h\nu c]}\,\!</math>

Where
:<math>I_\nu\,\!</math> is the irradiance at frequency nu
:<math>n\,\!</math> is the refractive index of the material
:<math>\nu\,\!</math> is frequency that can excite from the ground to the excited state
:<math>N_nu\,\!</math> is the energy density

The more that light is slowed down within the medium the more time there is for absorption.

The photon density decreases as it passes through a distance Δ x of f the volume element according to:

:<math>-dN_\nu(x)= [I_\nu (x) – I_\nu (x + \Delta x] \frac {n} {h\nu_{21}c}\,\!</math>

If Δx is sufficiently small you can integrate:

:<math>-dN_\nu(x) = - \frac {dI_\nu(x)} {dx} \cdot \frac {\Delta x n} {h\nu_{21}c}\,\!</math>

and

:<math>(\Delta x n/c = \Delta t)\,\!</math>

This implies that the rate of decay of photon density in a time interval dt:

:<math>\frac {dN_\nu} {dt} = \frac {dI_\nu(x)} {dx} \cdot \frac {1} {h\nu_{21}}\,\!</math>

:<math>\frac {dN_\nu} {dt} = -\alpha I_\nu (x) \cdot \frac {1} {h\nu_{21}} = -\alpha \rho_\nu \frac {c}{n} \cdot \frac {1} {h\nu_{21}}\,\!</math>

where

:<math>-\alpha I_\nu(x)\,\!</math> is the variation of the irradiance by unit volume

We can build a rate of disappearance of photon density as a function of absorption coefficient and the expression of photon density as a function of populations.

:<math>\alpha \rho_\nu \frac {c}{n} \frac {1}{h \nu_{21}} = \left (\frac {g_2}{g_1} N_1-N_2 \right) \rho_\nu B_{21}\,\!</math>

And finally put it all in terms of alpha:

:<math>\alpha = \left( \frac {g_2}{g_1} N_1 – N_2 \right) \frac {B_{21} h\nu_{21} n} {c}\,\!</math>

Thus the absorption coefficient alpha depends on the difference in the population difference between levels E1 and E2.

For a collection of atoms in thermal equilibrium since E2>E1 then:

:<math>(g_2/g_1) N_1 > N_2\,\!</math>

Hence α is positive and normally N2 is very small. But if we can create a situation for which:

:<math>(g_2/g_1) N_1 < N_2\,\!</math>

That is if there is an inversion of population, then α becomes negative.

If the beam is right energy hν and the excited state population is larger than the ground state, then the probability of a stimulated emission from the excited state is higher than the probability of a stimulated absorption from the ground state.

In this case the irradiance of the beam grows as it propagates through the medium.

Recalling that: I = I0exp(-αx) we obtain an exponential relationship:

:<math>I = I_0 exp(kx)\,\!</math>

:<math>k\,\!</math> is referred to the '''small signal gain coefficient''' or simply '''gain coefficient'''

:<math>k =(N_2- \frac {g_2}{g_1} N_1) B_{21} \frac {h\nu_{21}n} {c}\,\!</math>

== Population Inversion ==
[[Image:Laser_popinvert.png|thumb|300px|Normal and inverted populations]]
The plots show the energy of the two states (E2, E1 - y axis) and the population of each excited state (N2, N1 - x axis). The dotted line is the exp(-Eν/kt) thermal equilibrium distribution.

In the two level system at thermal equilibrium N2 is always smaller than N1. In order to reach a population inversion one has to pump the system into the excited state. Light or electrical current can be used to pump a system as in a diode laser. In the pumped condition the N1 population is far smaller than the boltzman distribution while the N2 population is far greater.

The ratio of the population in thermal equilibrium is given by boltzman statistics:

:<math>\frac {N_1} {N_2} = \frac {g_1} {g_2} exp [(E_2-E_1)/kt]= \frac {g_1}{g_2} exp(h\nu/kT)\,\!</math>

If hν ~ 2.5 eV (visible part of spectrum) and g2 =g1 at room temperature (kT = .025 eV)

:<math>N_2/N_1 = exp (-h\nu/kt)\,\!</math> ~ e-100 ~ 10-43 (10 ~ e2.3 so to make a conversion divide e exponent (that is 100) by 2.3)

This is a very small ratio.

'''Example'''
If we consider an octamer of phenylene vinylene which absorbes around 2.5 eV (= E2-E1). Assume the molar mass is ~ 1000g and the density is 1 g per cm3

What is roughly the length of the side of a cube containing oligophenylene vinylene in which on average one molecule will be be in the excited state at room temperature?

'''Answer'''
You would need 1043 molecules which weighs 1.6 x 1023 grams which take up 1023 cm3 or a cube that is 1017 m3 that is 108 km3 which is a cube 464 km on a side.

In order to get an population inversion we must supply a large amount of energy to excite atoms into the upper E2. Since B12 and B21 are equal (assuming g1 = g2). Once atoms are excited into the upper level the probabilities of further stimulated absorption or emission are equal. The best that can be achieved with a two level system is equality of the populations of the two levels. Thus you can not get to the state of inverted population. This is the reason to consider a three level system.

=== 3 Level System ===
[[Image:Laser_3level.png|thumb|300px|A three level system]]
The idea of three level system was proposed by Bloembergen in 1957 and received a Nobel prize in physics in 1981 for his work with nonlinear optics and lasers. He currently has a position at the University of Arizona optical science center.

In the three level system E2 is a second excited state. If atoms are intensely illuminated electrons can be pumped into E2 from the ground state E0. From E2 the electrons decay by non-radiative processes to level E1 so that a population inversion may be created between E1 and E0. Transition from level E2 to E1 should be very rapid thereby ensuring that there are always vacant states at E2. Transition from E1 to E0 should be very slow; E1 has to be metastable state so that you can accumulate a population.

Level E2 should preferably consist of a large number of closely spaced levels so that pumping uses as wide a part of the spectral range of the pumping radiation as possible.

Three level lasers such as ruby require a very high pump power because the terminal level of the laser transition is the ground state.

=== 4 level system ===
[[Image:Laser_4level.png|thumb|300px|Four level system]]
In a 4 level system there is population inversion between two excited state (E2 and E1). Pumping puts electron directly into a series of levels at E3. Internal conversion brings the electrons rapidly down to a metastable E2 where they accumulate and the population is inverted with respect to E1. E2 to E1 is a radiative transfer. From E1 top E0 there should be rapid decay so that the level is replenished for pumping. Even with pumping the population of E1, E2 and E3 is less than that of E0.

The frequency of light used for pumping must be correct for jumping between E0 and E3. The laser light produced reflects the transition between E2 and E1.

Most scientific lasers are solid state. They have a host materials (such as ruby or YAG) with dopants (niodinium) that have the lasing transitions.

In the Nd-YAG laser
*T32 ~ 10-8s
*T21~ .5 ms
*T10~ 30 ns

Much of laser research is the search for lasing host materials with energy transitions and time characteristics that are favorable to generating the right balance when pumped.

=== Optical Feedback ===
[[Image:Optical-cavity1.png|thumb|300px|Various configurations of an optical cavity]]
Positive feedback may be obtained by placing gain medium between a pair of mirrors which form a optical cavity. (a Fabry-Perot resonator) This allows the beam to make repeated passes through the medium gaining energy with each pass due to stimulated emission. A semitransparent mirror on one end finally allows the beam to escape when it has reached sufficient intensity. There are many systems that provide amplification of the signal and there are conjugated polymers that allow amplified stimulated emission. But that is not the same as laser. Three groups discovered lasing in conjugated polymers in the mid 1990s, Allan Heeger at Santa Barbara, Richard Friend at Cambridge, and Z.V. Vardeny at University of Utah. Since then there have been many efforts to create a diode laser from a conjugated polymer, but without success.

The problem is that you need to build a large population in the excited state. In an LED electrons are injected from ones side and holes are injected from the other and when they combine there must be large densities in the excited state. However the presence of injected charge (polarons) quenches the exciton. You don’t have that issue with optical pumping because you start with nothing, the atoms absorbs light and directly build up the population in the excited state. There are no polarons or charges floating around.

== Laser Modes ==

=== Axial Modes ===

The two mirrors of the laser form a resonant cavity and standing wave patterns are set up between the mirrors. Standing waves satisfy the condition:

:<math>p \frac {\lambda} {2} = L\,\!</math>

:<math>L\,\!</math> is the optical path between the mirrors (which is a product of the physical length and the index of refraction).
:<math>p\,\!</math> is an integer

The length of cavity needs to be and integral multiple of half the wavelength in order for each reflection to be coherent with the preceding waves.

Substituting for wavelength for c/ν :

:<math>\nu = \frac{ pc}{2L}\,\!</math>

There are many values of for p that will satisfy this equation.

[[Image:400px-Longitudinal_mode_v2.svg.png|thumb|300px|Modes of oscillation for p = 1 to p = 6]]

Example

If L = .5m and λ ~500 nm, then there are 2 x 106 possible p values.

Each value of p satisfying the above equations defines an axial (or longitudinal) mode of the cavity. The frequency separation (Δ p =1) is given by:

:<math>\Delta \nu = \frac {c}{2L}\,\!</math>

For L= .5m Δ ν = 300 Mhz, this corresponds to an photon energy difference of 10-6 eV(1eV ~ 2 x 1014 hz)
 
[[Image:Laser_irradiance_distribution.png|thumb|left|300px|Theoretical distribution of stimulated emission at each mode.]]
[[Image:Laser_axial_osc.png|thumb|300px|Oscilloscope of actual data plot shows a normal distribution]]

Thus the modes of oscillation of the laser cavity consist of a large number of frequencies separated by c/2L.
Specific frequencies are able to stimulate additional photons of the same wavelengths (when the beam goes back and forth in a resonating cavity). This results in a distribution of photons that are separated in energy by 300 mhz or a millionth of an eV.

The envelope corresponds to a number of photons with very precise energy.
 

=== Transverse modes ===
[[Image:Laser_transverse.png|thumb|300px|]]
Up to this point we have been discussing only the axial or longitudinal axis in which waves travel along a line joining the centers of the mirrors. For any real laser cavity there will be waves travelling just off the axis that are able to replicate themselves after covering a closed path. It is possible to have beams that are reflected at different angles. These transverse electromagnetic fields also give rise to resonant transverse electromagnetic modes (TEM) but the because they have components of their electromagnetic field that is transverse to the direction of propagation. The TEM subscripts indicate the number of modes along the x and y axis and you can see these patterns from lasers producing these modes.

== Classes of Lasers ==
===Doped insulator lasers ===

These are lasers whose active medium consists of a crystalline or amorphous (glassy) host material containing active ions, typically from the transition metal and rare earth groups in the periodic table.

=== Nd:YAG laser. ===
[[Image:Laser_ndyag.png|thumb|300px|3 step energy diagram for Nd:YAG laser]]
This is a common laser used in research. When you see the wavelength reported in research as 1.064 micron this indicates that a Nd:YAG laser was used. Similarly a wavelength 532 nm this means they used a Nd:YAG but cut the wavelength in half using a nonlinear process.

The active material is yttrium aluminum garnet (Y3Al5 O12) is a garnet in which the aluminum oxide in garnet is replaced by ytrium in an orderly fashion. It has the rare earth metal ion neodymium Nd3+ present as an impurity. Nd3+ ions are randomly distrubuted as substitutions on lattice sites normally occupied by Y3+ ions. The Nd3+ provide the energy levels for both the lasing transitions and pumping.

These are the transitions within the neodymium ion. There are three levels in E3 which can be pumped into and which all undergo fast nonradiative decay into the E2 state. The laser transition happens between E2 and E1 with a 1.05 Um wavelength (in the near IR). From N1 there is a rapid non-radiative decay to the groundstate N0. The lifetime of the lasing transition is long which is what allows it to build up the population in E2 for population inversion. It is important to be aware of the relative energy levels and the speed of various transition that makes each the laser system work.

Pumping is normally achieved using an intense flash of white light from a xenon flashtube. The presence of several possible pumping transition E3 levels contributes to the efficiency of the laser when using a pumping source with a broad spectral output. Each level is able to capture a slight different part of the pump light spectrum.

=== Ruby laser ===
[[Image:Ruby_energy.png|thumb|300px|Energy diagram for a ruby laser]]
Ruby was the first visible laser that was developed.
The active materials is a synthetically grown crystal of ruby, that is aluminum oxide with about 0.05% by weight of chromium as an impurity. Most precious gems have impurities in a very common material.
The ruby has transitions at the upper end of the visible spectrum in the green and violet (2-3 eV) which is what gives the ruby its characteristic deep red color.

=== Vibronic laser ===
[[Image:Laser_vibronic.png|thumb|300px|Vibronic laser energy diagram]]
Vibronic laser such as a alexandrite and Ti-sapphire are similar to other solid state lasers such as Nd:YAG in that light from a external pump source excites the impurity ions in a transparent host. The sapphire laser is similar to the ruby, it is aluminum oxide with impurities such as titanium.

They are fundamentally different in that laser gain is possible over a broad range of wavelengths so that they can produce either tunable outputs or ultrashort pulses.

Vibronic laser such as a alexandrite and Ti-sapphire are similar to other solid state lasers such as Nd:YAG in that light from a external pump source excites the impurity ions in a transparent host. The sapphire laser is similar to the ruby, it is aluminum oxide with impurities such as titanium.

They are fundamentally different in that laser gain is possible over a broad range of wavelengths so that they can produce either tunable outputs or ultrashort pulses.

Vibronic solid state lasers have four level systems. The pumping radiations excites the active ions to a vibronic band. The ions then lose vibrational energy and drop to the bottom of the band which is the upper laser level. The laser transition then occurs to a vibrationally excited sublevel of the n1 state; this is followed by the ions relaxing to the lowest sublevel of the ground state.

Alexandrite laser comprises Cr3+ ions in a beryllium aluminate (BeAl2O4) host.

The pumping leads from the lowest vibration level of the ground state to a high vibrational level of E3, followed by relaxation to the zeroth level of the excited state E3. Then the excited emission (laser) occurs between some level in E2 and some vibrational level of E1. This gives a wide range of combinations of energy transitions that make it possible to tune the laser by filtering what comes out of the laser. This gives a range of 701 to 826 nm. The broad energy range of emission means that the time of the emission can be very short. Ti-sapphire is the most used laser for femptosecond pulse studies.

The Heisenberg uncertainty principle tells us that if the precision on energy is high the precision on time will be low. Femptosecond pulses will have energy scattered over 1.5 eV. When doing spectroscopy where accuracy in energy domain (milli- eV precision) is important then it is better to use longer pulses in the picoseconds.

=== Semiconductor lasers ===

Semiconductor lasers such as the diode laser are made with p-n junctions. It users an LED with a very high density of emission.

=== Gas lasers ===
[[Image:He-ne_energy.png|thumb|300px|]]
Gas lasers are the most widely used type lasers. The helium laser is used for the red laser pointer, or the CO2 laser used in medicine.
Basically there are three classes of gas lasers according to whether the transitions are
Between electronic energy levels of atoms
or
Between electronic energy energy of ions
or
Between the vibrational/ rotational levels of molecules.

Atomic laser – the HE-NE laser
The active medium is a mixture of 10 parts helium to one part neon. Neon provides the energy levels for the laser transition. Helium provide an efficient excitation mechanism for the neon atoms.
Excitation usually takes place in a DC discharge created by applying a high voltage ( ~2-4 kV) across the gas. This creates high energy electrons in the gas mixture. The electrons hit the helium atoms (which are most common) bringing them to the excited state, and then transfer energy to the neon atoms when the energy of the excited states match. The neon can then relax to laser transitions levels such 3s level to 3p levels and then 3p to 2s and 2p levels.

:<math>E1 + He = He* +e2\,\!</math>

Resonant transfer of energy is proportional to exp (-ΔE/kt) where ΔE is the energy difference between the excited states of the two atoms involved.

:<math>He* + Ne = Ne* +He\,\!</math>

=== Liquid Dye lasers ===
[[Image:Laser_dye_Abs_Emis.png|thumb|300px|]]
Liquids have useful advantages over solids and gases. Solids are very difficult to prepare with the requisite degree of optical homogeneity and they suffer permanent damage if overheated. Gases do not have these difficulties but they have a much smaller density of active atoms. The dye laser has the advantage that they are simple and they can be tuned over a significant wavelength range.

The active medium is an organic dye dissolved in a solvent. It is a four level system that uses piconjugated chromophores. The pumping makes the dyes go from the electronic groundstate to an excited vibrational level of an excited state. The E3 level is the top vibronic level of the excited state curve, relaxation occurs down to the bottom of the E2 level. Laser emission can occur from all the vibronic levels.

[[Image:Potentialenergy surfacechang.png|thumb|300px|Potential energy surface diagram]]

The broad range of emission can be filtered for specific wavelengths.

Rhodamine is a common laser dye with a cyanine type structure. It is dissolved in methanol. It has as broad tuning range (570-660 nm) All dye lasers are optically pumped.

See Cord Module on Dye lasers [http://cord.org/cm/leot/course03_mod10/mod03_10.htm Dye lasers]

== References ==

<references/>

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Dispersion and Attenuation Phenomena|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Optical Fibers, Waveguides, and Lasers|Return to Optical Fibers, Waveguides, and Lasers Menu]]</td>
</tr>
</table>

Optical Fiber Waveguides

2009-08-27T00:38:01Z

Nsylvain:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Planar Dielectric Waveguides|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Optical Fibers, Waveguides, and Lasers|Return to Optical Fibers, Waveguides, and Lasers Menu]]</td>
<td style="text-align: right; width: 33%">[[Dispersion and Attenuation Phenomena|Next Topic]]</td>
</tr>
</table>
[[Image:Circular_dielect_wg.png|thumb|300px|Circular dielectric waveguide]]
A circular dielectric waveguide or fiber optic has an internal core that has a higher index of refraction than the cladding. At a certain diameter there is an angle that is less than the critical angle so there is total internal reflection.

[[Image:SI_fiber.png|thumb|300px|Step index fiber]]
These are referred to as step index fibers. The y axis refers to the index of refraction. The x axis is the distance from the center of the core. So the highest index of refraction (n1) is found in the core and there is a significant step in the index of refraction (n2) at the interface with the cladding.

In most practical waveguides the refractive indices of the core and cladding differ from each other by only a few percent.

The number of modes in a step-index circular waveguide is determined by the V parameter where

:<math>V= \frac {2\pi a} {\lambda_0} (n^2_1 - n^2_2)^{1/2}\,\!</math>

where
:<math>a\,\!</math> is the core radius

:<math>(n^2_1 - n^2_2)^{1/2}\,\!</math> is referred to as the numerical aperature, the larger this term the larger the number of modes.

The fiber can support only one mode when V < 2.405

For n1 = 1.48 and n2 = 1.46, the radius of the core a <2.7μm

 
[[Image:GI_fiber.png|thumb|300px|Graded index fiber]]

There is current research to develop a core with a graded index of refraction that gradually decreases to equal that of the cladding. This is referred to as a graded index fiber.

 
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Planar Dielectric Waveguides|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Optical Fibers, Waveguides, and Lasers|Return to Optical Fibers, Waveguides, and Lasers Menu]]</td>
<td style="text-align: right; width: 33%">[[Dispersion and Attenuation Phenomena|Next Topic]]</td>
</tr>
</table>

Talk:NSU PHD Qualifying Exam

2009-08-03T21:57:42Z

Nsylvain: /* Physics */

Topics and Sample Questions for Oral Exam for PhD Qualifiers – 1/26/09

=== Mathematics ===

1. Algebra, inversely and directly proportional, basic functions of one and two variables and their graphical representation.

2. Derivatives and integrals of common functions. Definition and application of differentials and derivatives.

3. Complex numbers: transformation between Cartesian and polar forms. Argand’s plane representation.

4. Basic Concepts of Linear Algebra: Vector and Matrix operations, determinants

=== Physics ===

1. Explain the law of conservation of energy; kinetic and potential energies; work and heat.

2. Know the ideal Gas Laws, and their assumptions. Absolute Temperature and change of temperature units.

3. Know the First and Second Laws of Thermodynamics. Heat, and how it relates to heat capacity, phase transformations, and change in temperature.

4. Know the definition and basic equations related to Electric field and Electric Potential, Coulomb’s law, Gauss's law, Dielectrics and polarization, Maxwell equations.

5. Explain and use in conceptual problems: polarization, interference, diffraction.([http://depts.washington.edu/cmditr/mediawiki/index.php?title=Main_Page#Basics_of_Light Basics of Light])

6. Electromagnetic spectrum: relationship between frequency, energy and wavelength; classification of different parts of the electromagnetic spectrum, and interaction of electromagnetic radiation with matter, depending on the radiation wavelength. ([[Electromagnetic Spectrum]])

=== Elementary Quantum Physics ===

Know the basic concepts, and basic equations related to:

1. Wave-particle dualism

2. Blackbody Radiation

3. Photoelectric effect

4. Plank, De Broglie, Einstein relations (dispersion relations for particles with the mass and photons)

5. Schrödinger equation

6. “Particle in the box”, harmonic oscillator

7. Uncertainty relations (momentum-position, time-energy)

8. Barriers, tunneling

9. Electronic structure of hydrogenic atom

10. Spin, angular momentum

11. Fermi’s golden rule

=== Chemistry and Polymers ===

1. Identify and define: materials, atoms and ions, sub-atomic particles, elements and isotopes.

2. Understand the periodic table and its arrangement: atomic numbers, atomic weight, and general chemical and physical properties of the elements, periodic properties. (Molecular Orbitals)

3. Differentiate between the structure and properties of metals, non-metals, inorganic materials, organic materials, polymers, ceramics, and composites; crystalline and amorphous materials.

4. Know the chemical structure, molecular formula, classification and how to calculate molecular weights of common organic and inorganic compounds (acids, based, oxides, salts)

5. Define oxidizers, reducers, and oxidation-reduction reactions

6. Key types of organic compounds and functional groups, properties, reactions, and applications

7. Basic polymer forming reaction methods, biological and inorganic polymers

8. Polymer molecular weights, morphology, structure-property relationships ([http://depts.washington.edu/cmditr/mediawiki/index.php?title=Main_Page#Electronic_Band_Structure_of_Organic_Materials Electronic Band Structure of Organic Materials])

9. Electronic and Optoelectronic Polymers and Applications: definition, explain structure-property relationship ([[Structure-Property Relationships]])

=== Materials Science, Electronic and Photonic Materials ===

1. Crystal Structure: Directions and planes

2. Lattice vibrations: Acoustical and optical branches; phonons

3. Thermal properties: Specific heats. Thermal expansion, Thermal conductivity

4. Diffusion, Fick’s Laws

5. Mechanical behavior. Plastic and elastic deformations. Young’s modulus

6. Phases, phase diagram. One component and binary systems. The Gibbs Phase rule.

7. Magnetic Properties. Basic concepts. Magnetic moment and permeability. Paramagnetic, ferromagnetic, antiferromagnetic materials

8. Electrical properties of metals and semiconductors. Hall effect. Intrinsic and extrinsic semiconductors; electrons and holes, electrical conductivity, statistics of electrons and holes, recombination and injection, life-time. (Electrical Properties)

9. Electric permittivity and susceptibility in dielectrics (Lorentz model) and in metals (Drude model).

10. Basics of semiconductor devices. Concept of p-n junction, Schottky junction and its electronic properties, MOS structure and basics of the field-effect, structure and operation principles of the bipolar transistors, basics of photonics devices (LED, photodetectors, solar cells, semiconductor lasers) (Electro Optical Components)

11. Basic methods and principles for materials characterization: Optical spectroscopy (X-ray, UV-visible, IR, Raman), Electron spectroscopy, Magnetic resonance spectroscopy (NMR, ESR).

12. Refraction, reflection, and transmission in loss-less dielectrics: index of refraction, Snell’s law, Brewster angle, total internal reflection, Frensel formulas for reflection and transmission. ([http://depts.washington.edu/cmditr/mediawiki/index.php?title=Main_Page#Basics_of_Light Basics of Light])

13. Electric permittivity and succeptibility in dielectrics (Lorentz model) and in metals (Drude model). Classical and quantum approaches to absorption in materials.

14. Basics of laser materials: principles of operation

History of Liquid Crystals

2009-07-27T23:27:13Z

Nsylvain: /* History of Liquid Crystals */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Double Refraction and Birefringence|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Liquid Crystals and Displays|Return to Liquid Crystal Menu]]</td>
<td style="text-align: right; width: 33%">[[Director – Degrees of Order in Liquid Crystals|Next Topic]]</td>
</tr>
</table>

=== What is a liquid crystal? ===

[[Image:LiquidCrystal-Ordering.jpg|thumb|300px|Crystalline materials have both orientational and positional order. Liquid crystals can have orientational order under the right conditions.]]
Liquid crystals are anisotropic materials. This is important for their function in LC displays.

In a crystal there is periodicity of a lattice in all three directions. In a liquid there is neither orientational or positional order.

A liquid crystal phase or mesomorphic phase is a phase comprising molecules having a higher degree of orientational order than is found in a liquid and less positional order than is found in a crystal. Under certain conditions they can flow.

There are many classes of liquid crystals having differing degrees of order. We need to ask if a molecule is capable of being liquid cyrstalline under certain conditions. Some molecules can be crystalline, liquid crystalline (and exist in different phases of liquid crystalline) and isotropic, depending on the conditions.
 
[[Image:MBBA.png|thumb|300px|Methoxybenzilidenebutylaniline (MBBA) has a liquid crystal phase between 21°C and 45°C]]
As shown in the diagram to the right there are discrete phase transitions between the crystal phase, the LC phase and the liquid phase. At each phase transition there is a corresponding heat of enthalpy for the transition. Much of the research in this area is german, as a result the letter K is often used to denote the cyrstalline phase (krystal in german). Nematic refers to the liquid cyrstalline phase (abbreviated N). Liquid phase may be abbreviated as l or I (for isotropic). Under the arrrow is temperature at which there is a phase transition. These transitions are usually quite sharp.

=== History of Liquid Crystals ===

One of the first papers about liquid cyrstals appeared more than 100 years ago.

'''1888''' - F Reinitzer (<ref>Montash. Chem. 1888, 9, 421</ref>) observed two melting points in cholesteryl benzoate. Today these are known as cholesteric liquid crystals.

'''1911''' - C. Mauguin (<ref>Bull. Soc. Fr. Min. 1911, 34, 71</ref>) performed and described the first electro-optical experiments involving a twisted nematic phase.

'''1922''' - M. G. Friedel (<ref>Ann. Phys. 1922, 18, 273</ref>) recognized three kinds of LC phases and introduced a new terminology.
*Smectic "soapy"

*Nematic "thread"

*Cholesteric (twisted nematic)

Shampoo in a bottle or soap in dish may appear irridescent. This is called a lyotropic liquid crystal in which phase transition is a function of concentration. Thermotropic liquid crystals have a phase change that is dependent on temperature. Lipid bilayers are phase transitions that are lyotropic.

'''1927''' - W. Friederickz et al. (<ref>Z. Phys. 1927, 42, 532</ref>) described the influence of electrical and magnetic fields on smectic, nematic and cholesteric LCs.
Marconi Wireless Telegraph Co was awarded a patent on a light valve in 1936.

1959 - W. Maier and S. Saupe (<ref>Z. Naturforsch. 1959, 13a, 564</ref>) published what is now known as the Maier-Saupe theory of liquid crystals which suggests the temperature at which material move from liquid crystalline to isotropic. It is summarized in the equation:

:<math>T_{is} = A/4.55kv^2\,\!</math>

Where:
A is a molecular parameter including the polarizability anisotropy,
k is Boltzmann's constant,
v is the molar volume.

'''1963''' - W. Richards and G. Heilmeier (RCA Corporation)(<ref>J. Chem. Phys. 1963, 39, 384</ref>) foresaw "TV on a wall".

Then there was a rediscovery of electro-optical effects and their applications in LC display:

'''1969''' - Dichroic dye; host guest (<ref>G. H. Heilmeier et al. Appl. Phys. Lett. 1969, 13, 46</ref>).

'''1969''' - Phase change displays (<ref>G. H. Heilmeier et al. Proc. IEEE. 1969, 57, 34</ref>).

'''1973''' - Twisted nematic field effect (<ref>J. L. Fergason US patent # 3,731,986, 1973</ref>).

'''1973''' - G.W. Gray et al. (<ref>Electron. Lett. 1973, 9, 130</ref>) developed LCD technology based upon cyanobiphenyls.

'''1977''' - S. Chandrasekhar et al. (<ref>Pramana 1977, 9, 471</ref>) discovered discotic liquid crystals.

'''1991''' - Pierre-Gilles de Gennes, Nobel Prize Physics 1991, studied how extremely complex forms of matter behave during the transition from order to disorder. He showed how electrically or mechanically induced phase changes transform liquid crystals from a transparent to an opaque state, the phenomenon exploited in liquid-crystal.

So there is a long research history in liquid crystals and much current work as well.

== References ==
<references/>

[[category:liquid crystals]]
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Double Refraction and Birefringence|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Liquid Crystals and Displays|Return to Liquid Crystal Menu]]</td>
<td style="text-align: right; width: 33%">[[Director – Degrees of Order in Liquid Crystals|Next Topic]]</td>
</tr>
</table>

History of Liquid Crystals

2009-07-27T23:26:46Z

Nsylvain: /* History of Liquid Crystals */

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Double Refraction and Birefringence|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Liquid Crystals and Displays|Return to Liquid Crystal Menu]]</td>
<td style="text-align: right; width: 33%">[[Director – Degrees of Order in Liquid Crystals|Next Topic]]</td>
</tr>
</table>

=== What is a liquid crystal? ===

[[Image:LiquidCrystal-Ordering.jpg|thumb|300px|Crystalline materials have both orientational and positional order. Liquid crystals can have orientational order under the right conditions.]]
Liquid crystals are anisotropic materials. This is important for their function in LC displays.

In a crystal there is periodicity of a lattice in all three directions. In a liquid there is neither orientational or positional order.

A liquid crystal phase or mesomorphic phase is a phase comprising molecules having a higher degree of orientational order than is found in a liquid and less positional order than is found in a crystal. Under certain conditions they can flow.

There are many classes of liquid crystals having differing degrees of order. We need to ask if a molecule is capable of being liquid cyrstalline under certain conditions. Some molecules can be crystalline, liquid crystalline (and exist in different phases of liquid crystalline) and isotropic, depending on the conditions.
 
[[Image:MBBA.png|thumb|300px|Methoxybenzilidenebutylaniline (MBBA) has a liquid crystal phase between 21°C and 45°C]]
As shown in the diagram to the right there are discrete phase transitions between the crystal phase, the LC phase and the liquid phase. At each phase transition there is a corresponding heat of enthalpy for the transition. Much of the research in this area is german, as a result the letter K is often used to denote the cyrstalline phase (krystal in german). Nematic refers to the liquid cyrstalline phase (abbreviated N). Liquid phase may be abbreviated as l or I (for isotropic). Under the arrrow is temperature at which there is a phase transition. These transitions are usually quite sharp.

=== History of Liquid Crystals ===

One of the first papers about liquid cyrstals appeared more than 100 years ago.

'''1888''' - F Reinitzer (<ref>Montash. Chem. 1888, 9, 421</ref>) observed two melting points in cholesteryl benzoate. Today these are known as cholesteric liquid crystals.

'''1911''' - C. Mauguin (<ref>Bull. Soc. Fr. Min. 1911, 34, 71</ref>) performed and described the first electro-optical experiments involving a twisted nematic phase.

'''1922''' - M. G. Friedel (<ref>Ann. Phys. 1922, 18, 273</ref>) recognized three kinds of LC phases and introduced a new terminology.
*Smectic "soapy"

*Nematic "thread"

*Cholesteric (twisted nematic)

Shampoo in a bottle or soap in dish may appear irridescent. This is called a lyotropic liquid crystal in which phase transition is a function of concentration. Thermotropic liquid crystals have a phase change that is dependent on temperature. Lipid bilayers are phase transitions that are lyotropic.

'''1927''' - W. Friederickz et al. (<ref>Z. Phys. 1927, 42, 532</ref>) described the influence of electrical and magnetic fields on smectic, nematic and cholesteric LCs.
Marconi Wireless Telegraph Co was awarded a patent on a light valve in 1936.

1959 - W. Maier and S. Saupe (<ref>Z. Naturforsch. 1959, 13a, 564</ref>) published what is now known as the Maier-Saupe theory of liquid crystals which suggests the temperature at which material move from liquid crystalline to isotropic. It is summarized in the equation:

:<math>T_{is} = A/4.55kv^2\,\!</math>

Where:
A is a molecular parameter including the polarizability anisotropy
k is Boltzmann's constant
v is the molar volume.

'''1963''' - W. Richards and G. Heilmeier (RCA Corporation)(<ref>J. Chem. Phys. 1963, 39, 384</ref>) foresaw "TV on a wall".

Then there was a rediscovery of electro-optical effects and their applications in LC display:

'''1969''' - Dichroic dye; host guest (<ref>G. H. Heilmeier et al. Appl. Phys. Lett. 1969, 13, 46</ref>).

'''1969''' - Phase change displays (<ref>G. H. Heilmeier et al. Proc. IEEE. 1969, 57, 34</ref>).

'''1973''' - Twisted nematic field effect (<ref>J. L. Fergason US patent # 3,731,986, 1973</ref>).

'''1973''' - G.W. Gray et al. (<ref>Electron. Lett. 1973, 9, 130</ref>) developed LCD technology based upon cyanobiphenyls.

'''1977''' - S. Chandrasekhar et al. (<ref>Pramana 1977, 9, 471</ref>) discovered discotic liquid crystals.

'''1991''' - Pierre-Gilles de Gennes, Nobel Prize Physics 1991, studied how extremely complex forms of matter behave during the transition from order to disorder. He showed how electrically or mechanically induced phase changes transform liquid crystals from a transparent to an opaque state, the phenomenon exploited in liquid-crystal.

So there is a long research history in liquid crystals and much current work as well.

== References ==
<references/>

[[category:liquid crystals]]
<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Double Refraction and Birefringence|Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Liquid Crystals and Displays|Return to Liquid Crystal Menu]]</td>
<td style="text-align: right; width: 33%">[[Director – Degrees of Order in Liquid Crystals|Next Topic]]</td>
</tr>
</table>