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Models: Voigt profiles

Classes used:

The Voigt profile (after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution (with half-width at half-maximum gamma) and a Gaussian distribution (with standard deviation sigma). It is often used for analyzing spectroscopic data.

In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms: life-time broadening (Lorentzian part) and inhomogeneous broadening (Gaussian part).

Recipe

Listing 3 Concrete example of a recipe used to create Voigt profiles from the respective model aspecd.model.Voigtian. As the Voigt profile is a convolution of a Gaussian and Lorentzian line, setting the contribution of either component to zero should result in pure Lorentzian or Gaussian lines, respectively. This is demonstrated, and furthermore the variation of the parameters gamma and sigma for the line widths of the Lorentzian and Gaussian component, respectively. The latter reproduces the figure shown in the documentation of the underlying function scipy.special.voigt_profile().
  1format:
  2  type: ASpecD recipe
  3  version: '0.2'
  4
  5tasks:
  6  - kind: model
  7    type: Zeros
  8    properties:
  9      parameters:
 10        shape: 1001
 11        range: [-10, 10]
 12    result: dummy
 13
 14  - kind: model
 15    type: Voigtian
 16    from_dataset: dummy
 17    result: voigtian
 18    comment: >
 19      Create Voigt profile with default parameters
 20
 21  - kind: singleplot
 22    type: SinglePlotter1D
 23    properties:
 24      properties:
 25        axes:
 26          ylim: [-0.01, 0.22]
 27          ylabel: Null
 28      parameters:
 29        tight_layout: True
 30      filename: model-voigt-profile.pdf
 31    apply_to: voigtian
 32    comment: >
 33      Plot Voigt profile with default parameters
 34
 35  - kind: model
 36    type: NormalisedGaussian
 37    from_dataset: dummy
 38    result: gaussian
 39    comment: >
 40      Create normalised Gaussian to compare with Voigt with gamma = 0
 41
 42  - kind: model
 43    type: NormalisedLorentzian
 44    from_dataset: dummy
 45    result: lorentzian
 46    comment: >
 47      Create normalised Lorentzian to compare with Voigt with sigma = 0
 48
 49  - kind: model
 50    type: Voigtian
 51    from_dataset: dummy
 52    properties:
 53      parameters:
 54        gamma: 0
 55    result: voigtian_gamma_0
 56    comment: >
 57      Create Voigt profile with gamma = 0, i.e., purely Gaussian line shape
 58
 59  - kind: model
 60    type: Voigtian
 61    from_dataset: dummy
 62    properties:
 63      parameters:
 64        sigma: 0
 65    result: voigtian_sigma_0
 66    comment: >
 67      Create Voigt profile with sigma = 0, i.e., purely Lorentzian line shape
 68
 69  - kind: multiplot
 70    type: MultiPlotter1D
 71    properties:
 72      properties:
 73        axes:
 74          ylim: [-0.01, 0.33]
 75          ylabel: Null
 76        drawings:
 77          - label: Voigt
 78          - label: Lorentz
 79      parameters:
 80        tight_layout: True
 81        show_legend: True
 82      filename: model-voigt-compare-lorentz.pdf
 83    apply_to:
 84      - voigtian_sigma_0
 85      - lorentzian
 86    comment: >
 87      Compare Voigt profile with sigma = 0 to Lorentzian
 88
 89  - kind: multiplot
 90    type: MultiPlotter1D
 91    properties:
 92      properties:
 93        axes:
 94          ylim: [-0.01, 0.41]
 95          ylabel: Null
 96        drawings:
 97          - label: Voigt
 98          - label: Gauss
 99      parameters:
100        tight_layout: True
101        show_legend: True
102      filename: model-voigt-compare-gauss.pdf
103    apply_to:
104      - voigtian_gamma_0
105      - gaussian
106    comment: >
107      Compare Voigt profile with gamma = 0 to Gaussian
108
109  - kind: model
110    type: Voigtian
111    from_dataset: dummy
112    properties:
113      parameters:
114        sigma: 1.5
115        gamma: 0
116    result: voigtian_sigma_1_5_gamma_0
117    comment: >
118      Create Voigt profile with sigma = 1.5, gamma = 0
119
120  - kind: model
121    type: Voigtian
122    from_dataset: dummy
123    properties:
124      parameters:
125        sigma: 1.3
126        gamma: 0.5
127    result: voigtian_sigma_1_3_gamma_0_5
128    comment: >
129      Create Voigt profile with sigma = 1.3, gamma = 0.5
130
131  - kind: model
132    type: Voigtian
133    from_dataset: dummy
134    properties:
135      parameters:
136        sigma: 0
137        gamma: 1.8
138    result: voigtian_sigma_0_gamma_1.8
139    comment: >
140      Create Voigt profile with sigma = 0, gamma = 1.8
141
142  - kind: multiplot
143    type: MultiPlotter1D
144    properties:
145      properties:
146        axes:
147          ylim: [-0.01, 0.28]
148          ylabel: Null
149        drawings:
150          - label: $\sigma = 1.5, \gamma = 0.0$
151          - label: $\sigma = 1.3, \gamma = 0.5$
152          - label: $\sigma = 0.0, \gamma = 1.8$
153          - label: $\sigma = 1.0, \gamma = 1.0$
154      parameters:
155        tight_layout: True
156        show_legend: True
157      filename: model-voigt-variations.pdf
158    apply_to:
159      - voigtian_sigma_1_5_gamma_0
160      - voigtian_sigma_1_3_gamma_0_5
161      - voigtian_sigma_0_gamma_1.8
162      - voigtian
163    comment: >
164      Compare Voigt profiles as shown in scipy.special.voigt_profile example

Comments

  • The recipe shown here does not import any data, hence does not have the usual top-level block datasets, but directly starts with the tasks block.

  • To remove the y label, set it to Null in the recipe, the YAML analogon to None in Python.

Results

Examples for the figures created in the recipe are given below. While in the recipe, the output format has been set to PDF, for rendering them here they have been converted to PNG.

Note

The command line magic used to convert the PDF images to PNG images was:

for k in model-voigt*pdf; do echo ${k%.*}; convert -density 180 $k ${k%.*}.png; done

A Voigt profile with standard parameters, i.e. gamma = 1 and sigma = 1, results in a line shape with additive line widths of Lorentzian and Gaussian contributions. Hence, the line is clearly wider than the individual components, as shown below.

../_images/model-voigt-profile.png

Fig. 5 Voigt profile with standard parameters, i.e. gamma = 1 and sigma = 1.

As a Voigt profile is a convolution of Lorentzian and Gaussian contributions, setting the line width of either contribution to zero should result in a pure line shape, either Lorentzian or Gaussian. This is demonstrated below for both cases.

../_images/model-voigt-compare-lorentz.png

Fig. 6 Comparing a Voigt profile with purely Lorentzian contributions, i.e. gamma = 1 and sigma = 0, to a (normalised) Lorentzian. The lines completely overlap, as expected.

../_images/model-voigt-compare-gauss.png

Fig. 7 Comparing a Voigt profile with purely Gaussian contributions, i.e. gamma = 0 and sigma = 1, to a (normalised) Gaussian. The lines completely overlap, as expected.

Last but not least, a figure from the documentation of the scipy.special.voigt_profile() used to compute the Voigt profile is recreated and shown below. For didactic purposes, the line widths of the Lorentzian (gamma) and Gaussian (sigma) components are varied.

../_images/model-voigt-variations.png

Fig. 8 Voigt profile with varying parameters for the Lorentzian (gamma) and Gaussian (sigma) contributions.