3. Absorption Spectroscopy and Data analysis
3.2. Absorption spectra and band structures
In the following, the physics leading to characteristic absorption spectra of different trace gases is introduced. For a detailed description, the reader is referred to the common specialized textbooks (e.g., Hellwege, 1990; Mayer-Kuckuck, 1997; Demtröder, 2010).
The absorption and emission spectra of molecules show characteristic band structures, which are a consequence of inner-molecular energy levelsEk,n,L (with the quantum numbers k, n, L denoting
the energy levels). Similar to atoms, the states of a molecule (and therefore the position of the respective energy levels) can be described by the time-independent Schrödinger equation:
Hˆ0ψ(r,R) =Ek,n,Lψ(r,R) (3.3)
whereψ(r,R) is the wavefunction of the molecule,rrepresents the position vectors of the electrons and R the nuclei position vectors.
Solving the Schrödinger equation for molecules is much more complex than for atoms and can be performed only in approximation (e.g. Born-Oppenheimer approximation or LCAO) as molecules consist of two or more nuclei and thus the electrical potential in the Hamiltonian ˆH is polycentric.
As a result of being not spherically symmetric, molecules have also degrees of freedom in rotation and vibration leading to further energy levels. In an approximation, the molecule’s wave function solving the Schrödinger equation can be separated in a component for the electron shell’s state, a component for rotations and a component for vibrations. The total energy is then the sum of all energies (eigen values of the Schrödinger equation):
Ek,n,L=Ekel+Envib+ELrot (3.4)
wherekcomprises all quantum numbers for the state of the electron shell,ndenotes the vibrational state and L the rotational state.
Interference with an external electromagnetic field (light) or quantum fluctuations can change the state of a molecule. As a result, energy is taken from or released to the external field in form of a photon whose wavelength corresponds to the difference in energies related to the initial and final state of the molecule (induced absorption and emission). Interference of an excited molecule with quantum fluctuations can only lead to the emission of a photon (spontaneous emission).
The transition from an initial to a final state cannot be described by a time-independent Schrödinger equation. Instead, the problem is described by means of pertubation theory. If ˆH0 is the Hamil-tonian of the independent Schrödinger equation, then the pertubation is described by a time-dependent term ˆH0(t) so that ˆH = ˆH0+ ˆH0(t) and the time-dependent Schrödinger equation has the form
i~∂ϕ(r,R, t)
∂t = [ ˆH0+ ˆH0(t)]ϕ(r,R, t) (3.5) If the pertubation is an electromagnetic field and the wavelength is much larger than the molecule’s diameter (which is the case for visible light), the calculation leads to transition dipole moments, also called transition matrix elements (coupling between the electromagnetic field and higher multipole moments is possible, but in practice only coupling with the dipole moment is relevant)
Mif = Z
u∗i p uf dτel·dτN (3.6)
whereiandf denote all quantum numbers of the initial and final state, respectively,ui, uf are the position-depending parts of ϕi, ϕf and the integration is over all electron and nuclei coordinates (denoted by eland N).
The symbol pis the dipole moment operator. For example, in case of a 2-atomic molecule (withj
electrons and two nuclei of atomic numbers Z1 and Z2 at positions R1 and R2, respectively), the dipole moment operator is
p=−e·X
j
rj+Z1eR1+Z2eR2 =pel+pN (3.7) The probability of a transition from state i to f is proportional to the squared transition dipole moment (|Mif|2).
In the Born-Oppenheimer approximation, the wavefunction of a molecule is broken into its elec-tronic and nuclear (vibrational, rotational) components:
utotal=uel·χN (3.8)
The transition matrix element is then Mif =
Z
u∗i,el·χ∗i,N (pel+pN) uf,el·χf,N dτel·dτN (3.9) Two cases can be distinguished:
1. The initial and final state belong to the same electron state (ui,el = uf,el). Then, only the nuclear wavefunction and the nuclear dipole momentpN contribute to the transition matrix element. This is a pure rotational-vibrational transition of the molecule.
2. A mixed electron vibrational-rotational transition, i.e. the final state has another electron quantum number than the initial one.
The physical meaning of transitions is that the electromagnetic field (light) interacts with the charge density distribution and therefore the molecule’s dipole moment. If the dipole moment does not change between states i and f, the transition is called forbidden. Otherwise, a photon of the wavelength corresponding to the transition (the energy difference betweeniandf) can be absorbed (or emitted) and therefore a peak occurs at the respective position in the absorption or emission spectrum. The strength of the respective line is propotional to the square of the dipole transition moment’s magnitude.
3.2.1. Electron transitions
The potential of a covalent bonded molecule is a function of the nuclei distance: If the distance between the nuclei is large, the electronic energy levels will match those of single (isolated) atoms.
If the distance is short, the Coulomb repulsion between the nuclei will dominate increasing the potential. Between these extrema is a local minimum in energy at the equilibrium distance R0. The shape of the resulting potential can be approximated by the Morse potential (Fig. 3.2)
V(r) =C 1−e−ar2 (3.10)
whereC is a molecule specific constant.
The depth of the minimum depends on the electron density between the nuclei, which constitutes the strength of the binding. In excited states, the electron wave functions expand decreasing the electron density between the nuclei. As a consequence, the potential flattens and the equilibrium
distance R0 increases.
As discussed above, the incident light beam can change the state of a molecule leading to absorption (or emission) of a photon. If the configuration of the electron shell changes, this is called an electronic transition. The energies related to electronic transitions are in the order of 1 eV to 10 eV. The corresponding wavelengths are according to
λ= hc
E (3.11)
in the range of ≈100 to 1000 nm, covering the UV, visible and near-infrared region.
3.2.2. Vibrational transitions
Atoms are often considered as ideal (point-like) particels. Consequently, they have no degrees of freedom in rotation as being rotationally symmetric. In contrast, molecules consists of two or more single atoms, thus having a three dimensional structure and therefore degrees of freedom in rotations as well as vibrations. In general, a molecule consisting ofN atoms has in total 3N degrees of freedom for translations, vibrations and rotations:
3N =ftrans+fvib+frot (3.12)
A point mass withN = 1 has only 3 degrees of freedom for translations, a moldecule ofN = 2 atoms has 3 degrees of freedom for translations, 1 for vibrations and 2 for rotations (the rotation around the molecule’s axis is no degree of freedom as no physical property would change). A molecule with N = 3 has 3 degrees of freedom for each, translations, vibrations and rotations (for illustration, see Fig. 7.5). As a result, every molecule has degrees of freedom leading to both, rotational and vibrational energy levels.
Considering a molecule consisting of two atoms, an approximation for describing vibrational states is the harmonic oscillator: If the distanceR between the nuclei changes from the stable distance in equilibrium R0, the consequene is a restoring force. Close to the equlibriumR0, the potential can be approximated by a parabola (see Fig. 3.2). The energy eigen values are then given by
Envib = (n+1
2)hν , n= 0,1,2, ... (3.13)
Thus, the vibrational energy levels are equally spaced. In general, wavelengths for vibrational transitions are in the infrared (e.g. ≈3µm for H2O).
3.2.3. Rotational transitions
Similar to Sect. 3.2.2, a two-atomic molecule is considered. This time, the approximation is that the molecule can rotate, but the distance between the nuclei cannot change (rigid rotator). According to eq. 3.12, this molecule has two degrees of freedom for rotations, meaning that it can rotate around two linear independent axes (as it is linear, rotation around the molecule’s axis does not change its properties). The energy eigen values for rotations are then
ELrot= L(L+ 1)~2
2I (3.14)
Figure 3.2: The Franck-Condon principle. The Morse potential for the ground state and an ex-cited electronic state are displayed in blue. Vibrational and rota-tional energy levels are displayed in black. In green, vibrational wave functions (approximation of an har-monic oscillator) are displayed for the ground state and a 2nd excited vibrational state.
with the quantum number for rotations (L) and the moment of inertia (I). The rotational energy levels are not equally spaced, but proportional toL:
∆Erot=EL−EL−1 = L~2
I (3.15)
For most molecules, the factor~2/I is small (10−4 eV to 10−3 eV), which is smaller than the kinetic energy of gas molecules at room temperature (2.5·10−2 eV). Thus, collisions of gas molecules can easily lead to excitation of rotational levels. The corresponding wavelengths for pure rotational transitions are in the far infrared to microwave region (typically 1 mm to 1 cm).
The selection rule for pure rotational transitions is ∆L=±1 (photons carry an angular momentum of 1). In the absorption spectrum, peaks corresponding to transitions of ∆L = −1 are referred to as P-branch and those of ∆L = +1 are referred to as R-branch. P and R-branches are only possible for molecules with no unpaired electron, as an unpaired electron’s spin could flip taking the photon’s angular momentum (as a consequence, ∆Lwould be 0 and the rotational energy according to eq. 3.14 would not change). In pure rotational spectra, a gap occurs between P and R-branch, calledQ-branch (corresponding to ∆L= 0). However, in spectroscopy often combined electron and vibrational-rotational transitions are observed. In that case, the photon’s angular momentum can change the electron angular momentum and a peak appears at the position of the Q-branch.
3.2.4. Band structures and Frank-Condon principle
As seen in Sect. 3.2 to 3.2.3, molecules can change their state under the influence of an external eletromagnetic field (light) and absorb a photon of the wavelength corresponding to the molecule’s transition. As a result, sharp peaks occur in the absorption and emission spectra at the corre-sponding wavelengths (eq. 3.11). These peaks are broadened according to natural, doppler and pressure broadening. The probabilty of absorption and therefore the strength of the absorption lines is given by the dipole transition moments (eq. 3.6). Under the approximation of a harmonic
Figure 3.3: Absorption cross-section of NO2 at 294 K between 400 and 500 nm as measured by Vandaele et al.
(1998). The literature cross-section has been convolved with the MAX-DOAS in-strument’s slit function from 14 October 2009 (during the TransBrom campaign, see Sect. 6.1).
potential and a rigid rotator, the vibrational and rotational energies can be calculated according to eq. 3.13 and eq. 3.14. It is
Eel> Evib> Erot (3.16) Thus, the typical energy structure of a molecule consists of energy levels of the electron shell (sum-marized in the symbolk) and a system of vibrational levels that is build on each of these electron energy levels. Further, on each vibrational level a system of rotational levels is build on. The resulting structure of energy levels is indicated in Fig. 3.2. Mostly, the absorption of a photon is a combined excitation of electronic, vibrational and rotational states.
As discussed in Sect. 3.2.1, the equilibrium distanceR0 of the nuclei inceases for excited electronic states. Typical time spans for electronic transitions are in the order of 10−15 s whereas typical periods for vibrations of the nuclei are in the order of 10−13 s. As a consequence, the distance between the nuclei cannot adjust to the new equilibrium distance during an electronic transition.
This leads to additional selection rules for combined electronic and vibrational (vibronic) transi-tions, which is illustrated in Fig. 3.2. The transition is indicated by a dashed arrow and the wave functions for a harmonic oscillator are displayed in green. The initial state is the ground state, both electronic and vibrational. Because the nuclei distance cannot change during the transition, the arrow starting at the point of highest wave function amplitude (largest probability density) is straight vertical. It hits the curve for the excited electronic state at a point where the wave function of the second vibrational mode has a large amplitude. Consequently, transitions between n = 0 in the electronic ground state and n= 2 in the electronic excited state are favoured. This effect is called the Franck-Condon principle. It is important that the wave functions of the initial and final state have large amplitudes (as in this example) since their overlap determines the dipole transition moment (eq. 3.6). Transitions between states with smaller overlap of wave functions can occur, but are less probable. The corresponding absorption lines would be less intense while absorption lines corresponding to states with higher overlap have more intensity.
The inner-molecular energy levels and transitions between them as well as mechanisms determining the probability of different transitions as explained in the previous sections result in characteristic absorption spectra (cross-sections) for each trace gas according to its physical properties. As an
Figure 3.4: Blue curve: An induced dipole momentum oscillates with the fre-quencyω0 of the incident light, which po-larizes the molecule. If the polarizabil-ity is a function of natural oscillations ωr
(green), the induced dipole momentum’s oscillation is modulated by this frequency.
example, Fig. 3.3 shows the cross section for nitrogen dioxide (NO2). In remote sensing techniques, these absorption spectra are used like afingerprint to identify the respective trace gases in recorded sunlight spectra, which are attenuated by absorption in the Earth’s atmosphere.