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Summary Optical diagnostics for combustion and fluid flow
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Chapter 1 – Molecular PhysicsInternal energy of a stable molecule consists of:
- Potential energy Te of electrons
- Vibrational energy Tvib of atomic cores
- Rotational energy Trot of the whole
The internal energy of a diatomic molecule can be calculated due to the Born-Oppenheimer
approximation:
E ( e , v , J ) =T e +T vib +T rot=T e +ω e v + ( 12 )+ B J ( J +1 )
e
Where:
- Te, Be, ωe are molecular constants
- v,J are quantum numbers
Optical diagnostics is based on absorption, emission and scattering of individual photons by
individual molecules. Photons are energy packages of light.
Molecules consist of at least two atoms. A molecule is a system with multiple degrees of freedom,
and there are 4 contributions :
- Rigid-body translation (of the center of mass)
- Rigid rotation (of the nuclear frame)
- Internal vibrations (of the nuclear frame)
- Electronic degrees of freedom
The first one is not dependent on the structure of the molecule, but the last three do. These 3
determine the internal energy of that molecule.
Molecules are stick together due to the electronic structure. This means the distribution of electrons
in the field of the positively charged nuclei. The elements are labelled by what is called the atomic
number Z; the number of protons in the nucleus of each atom of that element.
Nuclei contain ≥ Z neutrons, and the neutral atom also contains Z electrons. Typical sizes of nuclei are
Fluorescence=light emission High energy is low wavelength Temperature rise, emission to blue
Excitatie is een natuurkundige term voor het tijdelijk verhuizen van een elektron van een schil naar een andere schil, die hoger in het
energiespectrum ligt, binnen hetzelfde atoom
Broadband mode = it emits a continuous range of wavelengths
,on the order of 10-14-10-15m, size of an atom is on the order 10 -10m (1A). Nuclei can be considered as
point masses, and the size of an atom is determined by the distribution of the electrons around the
nucleus. An atom consist of i) an atom core made up of the nucleus plus all electrons in inner shells,
ii) the valence electrons perturbed and rearranged under the influence of both atomic cores.
An electronic state describes a distribution of electrons. The total energy associated with such a
distribution also depends on the distance between the nuclei. For diatomic molecules this
information is typically provided using potential energy curves; plots of the total electronic energy of
a given electron distribution as a function of internuclear distance r. A potential energy curve
associated with some particular electron distribution takes one of the two characteristics shapes
illustrated here:
The behaviour of these characteristic curves is the same at very small and very large r, but there is a
significant difference between:
- Potential well: There is a distinct minimum,
that defines the equilibrium internuclear
distance re, of the stable molecule for that
particular electronic state. If r -> 0, the energy
increases steeply because the positively
charged atomic cores push into each other,
and there is no room left for electrons in
between. If r goes to infinity the curve levels
off, because the molecule dissociates into two
free atoms.
- Gradual decay: The energy decreases
monotonically if the nuclei move away from
each other. This does not correspond to a
stable, bound state, but rather to what is
called a dissociative state.
Vibrational energy levels
In all models, the contribution of the electrons to the vibrational energy of a molecule is neglected,
because of their small mass relative to the nuclei. So, for the vibrational energy only the nuclear
framework needs to be considered. Since this is a bound system, the vibrational energies are
discrete. In the harmonic oscillator approximation the possible energies are given by a very simple
expression:
1
T vib=ωe (v + )
2
Where:
- ωe vibrational constant
- v vibrational quantum number
Usually the vibrational energy is given in units of wavenumbers cm -1.
In a diatomic molecule there is only one coordinate along which vibration can occur, which is the
internuclear axis.
Fluorescence=light emission High energy is low wavelength Temperature rise, emission to blue
Excitatie is een natuurkundige term voor het tijdelijk verhuizen van een elektron van een schil naar een andere schil, die hoger in het
energiespectrum ligt, binnen hetzelfde atoom
Broadband mode = it emits a continuous range of wavelengths
,Rotational energy levels
Like for the vibrational energy the contribution of the electrons to the rotational energy of a
molecule is neglected, because of their small mass relative to the nuclei. Thus, for the rotational
energy only the nuclear framework needs to be considered.
The rotational energy is expressed in terms of the moments of inertia. The expression for the
rotational energy depends on the molecular symmetry, and we will treat here only the case of linear
molecules (where the moment of inertia for rotation around the internuclear axis is zero). For this
case the toolbox expression is again very simple:
T rot =BJ ( J +1 )
Where:
- B rotational constant (molecule specific)
- J rotational quantum number
The rotational constant can be calculated due to:
ℏ2
B= ∨¿
2I
Where:
- ℏ reduced Planck constant, defined in terms of the Planck constant h as ℏ=h/(2∏)
- I moment of inertia relative to any axis perpendicular to the internuclear axis
Electronic energy levels
The electronic energy is independent of any vibrations or rotations of the nuclear framework. The
possible electron distributions are labelled by so-called molecular term symbols, which summarize a
number of symmetry properties of the molecule. Traditionally, the general notation of these term
symbols reads:
2 S +1 ±
name Λ u /g
Name: A single letter that serves a name. Capital X is reserved for the electronic ground state (the
lowest possible electronic energy), the rest runs alphabetically, approximately in the order of
increasing energy. Capital letters usually denote states with the same net electron spin as the ground
state, lower case letters denote states with other spin.
2S+1: S is the total spin of the electron configuration, which can be integral or half integral.
Terminology: When 2S+1 equals 1,2,3,4 or 5, we speak of singlet, doublet, triplet, quartet or quintet
states.
Λ : A Greek-letter-code for the component of electron orbital angular momentum along the
internuclear axis. This is an integer, and the code is ∑ ( for Λ=0 ) , Π ( for Λ=1 ) , Δ ( for Λ=2 )
u/g: Only for homonuclear diatomics, in which it specifies the parity, the symmetry of the
wavefunction under inversion of all coordinates.
Fluorescence=light emission High energy is low wavelength Temperature rise, emission to blue
Excitatie is een natuurkundige term voor het tijdelijk verhuizen van een elektron van een schil naar een andere schil, die hoger in het
energiespectrum ligt, binnen hetzelfde atoom
Broadband mode = it emits a continuous range of wavelengths
, ±: Specifies the symmetry of the wavefunction under reflection through any plane containing the
internuclear axis. Both symmetries occur in states with Λ>0, so this property is only indicated for Σ-
states (which can be either Σ+ as Σ-).
Boltzmann distribution
In the previous sections we have introduced expressions for the possible internal energies of a
diatomic molecule. Any individual molecule can be found in any of these levels, but not in between.
We may now ask whether we can predict in which of these possible energy states we would find a
molecule, if we did an appropriate measurement. The answer to that question must be that we
cannot predict this. In practice one doesn’t look at individual molecules, but at an ensemble, at the
net response of all molecules in a certain probe volume. This allows a statistical description of the
properties of interest. And, even when we cannot predict with certainty in which energy state any
particular molecule would be found, we can predict very accurately the average number of molecules
per available quantum state. This so-called Boltzmann statistics holds for ensembles in thermal
equilibrium only.
The average number of molecules N that will be in a particular energy state characterized by
quantum numbers (e,v,J) in a system, at temperature T is given by:
−E
gevJ∗g 1 exp ( ) −E
kB T , and where Z ( T )=∑ gevJ ∗g 1 exp ( ), which is the
N ( e , v , J )= N0 evJ kB T
Z (T )
partition function.
The fraction of the population of the level divided by the total number of molecules is called the
Boltzmann fraction: N/N0. The population difference between any two states can most conveniently
be expressed by the population ratio, because then the partition function Z cancels:
N' −Δ E
N } = {g'} over {g
exp
kB T ( )
, where Δ E=E ' −E' ' .
In the Born-Oppenheimer approximation, the electronic, vibrational and rotational energies are
completely independent. Therefore, if you want to calculate for instance only the vibrational states
separately, you get:
−E v
exp ( )
kBT −E J
N v =N 0 , where Z vib ( T )=∑ g J g I exp ( )
Z vib (T ) J k BT
The thermal energy available in gas phase collisions is k BT. At room temperature, this corresponds to
about 200cm-1, far too little to excite electronic states, and even too little to excite many vibrational
states. Thus, under conditions of room temperature, you can typically consider all molecules in a
gas to be in their ground electronic and vibronic state. Even in a flame of about 2200 K (1500cm -1),
electronic states are not populated, but vibronic states may carry significant population. In all cases,
many rotational states are typically populated. So:
Rotational Vibronic Electronic
Gas in room temperature Many states Ground state Ground state
Flame Many states Significant population Not populated
A particular state is said to be populated if a significant fraction of molecules is found in that
state.
Fluorescence=light emission High energy is low wavelength Temperature rise, emission to blue
Excitatie is een natuurkundige term voor het tijdelijk verhuizen van een elektron van een schil naar een andere schil, die hoger in het
energiespectrum ligt, binnen hetzelfde atoom
Broadband mode = it emits a continuous range of wavelengths
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