Line profiles

Doppler

This is a binned Doppler line profile which conserves the area at each grid point for any resolution used. It requires a mass of the molecule (Dalton) defined.

Example

temperature 2000.0
absorption
Doppler
mass 20.0

Doppler Sampling or Doppl0

temperature 2000.0
absorption
Doppler Sampling
mass 20.0

Doppler Sampling is used for the simple sampling method.

Gaussian

This is a binned Gaussian line profile which conserves the area at each grid point for any resolution used. It requires an HWHM value to be defined.

Example

temperature 2000.0
absorption
gaussian
hwhm .321

Gaussian Sampling or Gauss0

Gaussian Sampling is used for the simple sampling method.

Example

temperature 2000.0
absorption
gaussian Sampling
hwhm .321

Voigt (Sampling)

Example

temperature 2000.0
absorption
voigt
hwhm .321
mass 16.0313
cutoff 25 (cm-1)

Lorentzian

This profile uses a sampling method, where the cross sections at a given wavenumber points represents an aveerage over the wavenumber bin.

Example

temperature 2000.0
absorption
lorentzian
hwhm .321
cutoff 25 (cm-1)

A binned version can be invoked using the BINNING keyword or with the Lorentz0 (LORENTZIAN0') line profile, e.g.

temperature 2000.0
absorption
lorentz0
hwhm .321
cutoff 25 (cm-1)

or

temperature 2000.0 absorption lorentzian binning hwhm .5 cutoff 25 (cm-1)

For the binned profile method, the cross sections at a given wavenumber point represents an average over the wavenumber bin. The main feature of the binned Lorentzian is to preserve the total intensity. It is therefore normalised to 1 after the cutoff applied.

\[ \begin{align}\begin{aligned}\sigma_{\rm binned Lor}(\tilde\nu) = \sum_{i} \frac{\sigma_{i}}{\bar\sigma},\\where\end{aligned}\end{align} \]

\[\sigma_i = \int_{\tilde\nu_i-h}^{\tilde\nu_i+h} \sigma_{\rm Lor}(\tilde\nu) d\tilde\nu,\]

\[\sigma_{\rm Lor}(\tilde\nu) = \frac{\gamma}{\pi}\frac{1}{(\tilde\nu-\tilde\nu_{if})^+\gamma^2},\]

\[\bar\sigma = \int_{\tilde\nu_{if}-\delta\tilde\nu}^{\tilde\nu_{if}+\delta\tilde\nu} \sigma_{\rm Lor}(\tilde\nu) d\tilde\nu,\]

\(\sigma_{\rm Lor}(\tilde\nu)\) is the Lorentzian line profile, \(\tilde\nu_{if}\) is the line centre, \(\bar\sigma\) is the average of the line profile over the integration interval, \(\delta\tilde\nu\) is the line cutoff distance, \(\Delta \nu\) is the grid spacing, \(\tilde\nu_i\) is a wavenumber grid point.

Box

Is to compute cross sections using a rectangular line profile (normalised to 1) with the width the same as the wavenumber grid spacing (size of the wavenumber bin). That is, HWHM is assumed to be \(1/2\) of the grid spacing \(\Delta \nu\).

Example:

(ScH box spectrum)
Temperature 1500.0
Range 0.  16000.0

Npoints 16001

abundance 0.97

absorption
box
threshold 1e-29

output ScH_1500K_box
States       ScH.states
Transitions  ScH.trans

Here the width (2 \(\times\) HWHM) of the line is 1 cm_-1.

Rect

Is to compute cross sections using a rectangular line profile of length \(L = 2{\rm HWHM}\) (normalised to 1) with the HWHM defined in the input as given by

\[I(\nu) = \sum_i \frac{I_i(\nu)}{L}\]

For example:

(ScH box spectrum)
Temperature 1500.0
Range 0.  16000.0

Npoints 16001

abundance 0.97

absorption
rect
HWHM 0.5

output ScH_1500K_box
States       ScH.states
Transitions  ScH.trans

Here the HWHM of the line is 0.5 cm_-1. In order not to lose the density for small number of points, the frequency range rectangle is taken as .. math:

L = (\nu_{{\rm last}}-\nu_{{\rm first}})

where \(\nu_{\rm last}\) and \(\nu_{\rm first}\) are the last and fist grid points within the box, i.e. \(L \le \nu_{\rm last}- \nu_{\rm first}\).

To invoke the error cross section calculations, a free floating ERROR keyword is used.

Here, ExoCross uses the energies uncertainties to define uncertainties of the cross-sections in a form of absorption (emission) error cross-sections for different line profiles as given by

\[\left(\Delta \sigma(\tilde{\nu})\right)^2 = \sum_{i,j} \left(\frac{\partial \sigma(\tilde{\nu})}{\partial \tilde{\nu}_{ij}}\right)^2 \left[(\Delta \tilde{E}_i)^2 + (\Delta \tilde{E}_j)^2 \right],\]

where \(\Delta \tilde{E}_i\) and \(\Delta \tilde{E}_j\) are the uncertainties of the upper and lower states. Assuming a given line-profile \(f(\tilde\nu)\), the derivative wrt the energy is given by

\[\frac{\partial \sigma(\tilde{\nu})}{\partial \tilde{\nu}_{ij}} = I_{if} \frac{\partial f(\tilde{\nu})}{\partial \tilde{\nu}_{ij}}.\]

Here is an input example:

elorentz
error
hwhm 0.2

It is important that the energy uncertainties are provided in the column 5 of the States file. The error cross sections can be combined with the uncertainty filters:

filter
  unc 0.01
end

Error cross sections for the Lorentzian profile (Elorentz)

For the Lorentzian line profile centred at \(\tilde{\nu}_{ij}\) with HWHM \(\gamma\) given by

\[f(\tilde\nu,\tilde\nu_{ij},\gamma)_{\rm Lo} = \frac{\gamma}{\pi} \frac{1}{(\tilde{\nu}-\tilde{\nu}_{ij})^2+\gamma^2}\]

the corresponding derivative wrt \(\tilde{\nu}_{ij}\) is given by

\[\frac{\partial f(\tilde\nu)_{\rm Lo}}{\partial \tilde\nu_{ij}} = \frac{\gamma}{\pi} \frac{2(\tilde{\nu}_{ij}-\tilde{\nu})}{\left[(\tilde{\nu}-\tilde{\nu}_{ij})^2+\gamma^2\right]^2}.\]

Cross sections with energy uncertainties as line broadening (Voigt-unc)

If we assume that the energy uncertainties distributed according with the normal distribution with \(\sigma\) as the uncertainty of the corresponding line position \(\tilde\nu_{ij}\):

\[\sigma_{ij}^{\rm unc} = \sqrt{\Delta \tilde{E}_{i}^2+ \Delta \tilde{E}_{j}^2},\]

we can obtain the cross sections with the energy uncertainties included by convolving a Gaussian with \(\alpha_{ij}^{\rm G} = \sqrt{2\ln(2)} \sigma_{ij}\) with the Lorentzian (Voigt) of \(\gamma_{\rm V}\). The resulted line profile us Voigt. The Doppler broadening can be added to the uncertainty broadening as follows

\[\alpha_{ij}^{\rm G} = \sqrt{2\ln(2) (\sigma_{ij}^{\rm unc})^2 + (\alpha_{ij}^{\rm Doppl})^2}.\]

This feature is implemented as the Voigt-unc type and can be used as in the following example:

temperature 1000 Range  0 20000

Npoints 2000000

mass 64

absorption
Voigt-unc

pressure  1

species
 H2  gamma 0.0468  n 0.500 t0 296.0  ratio 0.85
 He  gamma 0.0468  n 0.5   t0 296.0  ratio 0.15
end


Output TiO_Voigt-unc_T1000K_P1atm

States 48Ti-16O__Toto.states

Transitions 48Ti-16O__Toto.trans

Pre-dissociative line profiles

In case of pre-dissociative effects, the lines van be broadened beyond the collisional or Doppler broadening. For these cases, ExoCross can use the lifetimes \(\tau\) from the States file (usually column 6 after the uncertainty column) to estimate the pre-dissociative line broadening (HWHM) via

\[\gamma_{\rm prediss} = \frac{1}{2\pi c \tau} \frac{1}{2}\]

where \(c\) is the speed of light in cm/s, \(\tau\) is the lifetime in s and the factor \(1/2\) is to convert to the HWHM \(\gamma\). ExoCross will apply the largest of the two line broadening values, \(\gamma_{\rm prediss}\) and \(\gamma_{\rm collis}\). This option is activated via a free floating keyword PREDISSOCIATION. By default, the lifetimes column is assumed to be column 6. Otherwise it is important to specify the number of the lifetime column as part of the QN section using the keyword lifetime

predissociation

QN
 lifetime 5
END

Note

The lifetime column specification can be combined with the non-LTE section, since QN and non-LTE are essentially aliases of each other.

Pseudo-Voigt

See wiki

The Pseudo-Voigt Profile (or Pseudo-Voigt Function) is an approximation of the Voigt Profile V(x), using a linear combination of a Gaussian curve G(x) and a Lorentzian curve L(x) instead of their convolution.

The Pseudo-Voigt Function is often used for calculations of experimental Spectral line shape profiles.

The mathematical definition of the normalized Pseudo-Voigt profile is given by:

\(V_p(x)= \eta \cdot L(x) + (1-\eta) \cdot G(x)\)

with \(0 < \eta < 1\)

There is several possible choices for the eta parameter. A simple formula, accurate to 1%, is: \(\eta = 1.36603\) (\(f_L/f\)) - 0.47719 \((f_L/f)^2 + 0.11116(f_L/f)^3\) where

\(f = [f_G^5 + 2.69269 f_G^4 f_L + 2.42843 f_G^3 f_L^2 + 4.47163 f_G^2 f_L^3 + 0.07842 f_G f_L^4 + f_L^5]^{1/5}\)

Example

absorption
pseudo
hwhm .321
mass 16.0313
cutoff 25 (cm-1)

Pseudo-Liu

(Liu_Lin_JOptSocAmB_2001)

Example

absorption
pseudo-Liu
hwhm .321
mass 16.0313
cutoff 25 (cm-1)

Pseudo-Rocco

(Rocco_Cruzado_ActaPhysPol_2012)

Example

absorption
pseudo-Rocco
hwhm .321
mass 16.0313
cutoff 25 (cm-1)

Voigt-parameters

Species or Broadener starts a section to define the Voigt-type broadening parameters

\(\gamma(Voigt) = \sum_i \gamma_i (T^0_i/T)^n P/P^0_i {\rm ratio}_i\)

The keywords are:

gamma or gamma0 is the reference HWHM (cm-1), n is the exponent n_i, T0 is the reference T (K),usually 298, P0 is the reference pressure in bar, usually 1, ratio is the mixing ratio of the species (unitless), for example the solar mixing ratio of H2 and He is 0.9 and 0.1.

The name of the species should be the first thing on the line.

The pressure value in bar must be specified (otherwise P=1 bar is assumed).

The effective molar mass of the molecule/atom mass be specified (1.0 is the default).

Example

mass 16.0
pressure 0.5
Temperature 1300.0
Species
  H2  gamma 0.05 n 0.4 t0 298.0 ratio 0.9
  He  gamma 0.04 n 1.0 t0 298.0 ratio 0.1
end

Voigt-Quad

Voigt-Quad is the Voigt obtained using the Guass-Hermite quadrature integrations. An analytical integration of the Lorentzian is used for the average contribution for each bin. The effect of the line truncation with cutoff parameter is folded back into the main part using the analytical expression. The line guarantees the area to conserve.

Example

Temperature   500  (K)
pressure 10. (bar)
absorption
Voigt-Quad
mass 16.0313
cutoff 25 (cm-1)
nquad   20   (N quadrature points)

Species
  H2  gamma 0.05 n 0.4 t0 298.0 ratio 0.9
  He  gamma 0.04 n 1.0 t0 298.0 ratio 0.1
end

ExoMol diet and broadening recipes

ExoCross is equipped to work with the ExoMol diet, which provides line broadening parameters in .broad files. The ExoMol diet allows for the line broadening parameters \(\gamma\) and \(n\) to depend, at least in principle, on any quantum numbers used in .states files. In practice, only simplest quantum number cases are currently in use, mainly for the rotational quantum number \(J\). Different quantum number cases (‘recipes’) are labeled with two character strings including a0 (\(\gamma\) and \(n\) depend on \(J''\) only), a1 (\(J''\) and \(J'\)), m0 and m1 (rotational index \(|m|\) and \(m\), respectively), v0 (\(v'\)), v1 (\(J''\) and \(v'\)), k1 (\(J''\) and \(k'\)). Currently, the vibrational \(v\) or rotational quantum \(k\) numbers can only be specified for the upper states.

A \(J''\)-dependent set of broadening parameters can be thus provided in an external file using the ExoMol Diet structure, recipe a0, e.g.

mass 16.0
pressure 0.5
Temperature 1300.0

species
  H2  gamma 0.0207 n 0.44 t0 298.0 file 1H2-16O__H2.broad  ratio 0.84
  He  gamma 0.043  n 0.02 t0 298.0 file  1H2-16O__He.broad ratio 0.16
end

where file is the filename with the parameters \(\gamma\) and \(n\). An example of the a0 recipe is as follows

a0   0.0145 0.500       0
a0   0.0156 0.417       1
a0   0.0164 0.350       2

where the 1st column describes the Diet model, the following two columns are the Voigt’s \(\gamma\) and \(n\), and the last one is \(J''\) . The values gamma and n in the species section are the default values in case of missing \(J\) in the broadening file. For \(J>J_{\rm max}\), the values \(\gamma\) and \(n\) the values of \(J=J_{\rm max}\) are assumed.

The a1 recipe allows providing both the upper and lower state \(J\), expecting :math`|J’-J’’|le 2`. An example of an a1 recipe broadening file has is as follows:

a1   0.0145 0.500       0       1
a1   0.0156 0.417       1       2
a1   0.0164 0.350       2       3

Here the last two columns list \(J''\) and :math`J’` (i.e. opposite to the conventional order).

Another example of .broad for m1:

m1   0.0156 0.417      -2
m1   0.0164 0.350      -1
m1   0.0145 0.500       0
m1   0.0156 0.417       1
m1   0.0164 0.350       2