.. _sphx_glr_auto_gallery_dispersion:

The dispersion models
---------------------

Each gas's fitted model of the refractive index against the data it rests on, and the
noble gases in the extreme ultraviolet.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="The refitted CO2 model (OEP 3) from 0.25 to 12 µm at 1 atm and 296 K. Top: n - 1. Bottom: the group-velocity dispersion&#x27;s driver, |D_2| = |d^2(\omega\alpha)/d\omega^2|, proportional to \beta_2 in the dilute limit, with its statistical one-sigma uncertainty and the representation bound of the infrared terms. The shaded regions are the gaps between the qualified intervals, where the gas absorbs or the bands&#x27; Gaussian rules do not meet the tolerance. The model is not to be used there, and warns.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_co2_windows_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_co2_windows`

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      <div class="sphx-glr-thumbnail-title">CO2 from the ultraviolet to its infrared windows</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Each molecule&#x27;s compact model (OEP 2) keeps its fit&#x27;s electronic terms and replaces its vibrational poles, thousands of them for CO2 and N2O, by three to five effective ones. Top: the reduction&#x27;s residuals, compact minus parent, in units of the parent&#x27;s statistical one-sigma, for the polarizability and the dispersion derivatives D_m = d^m(\omega\alpha)/d\omega^m (D_2 drives the group-velocity dispersion, D_3 the third-order dispersion), over each compact model&#x27;s interval. The tolerance is 0.1 (dashed). Bottom: the infrared terms&#x27; share of D_2, which the compact poles have to carry: below 0.1 % in the ultraviolet, 1-17 % at 800 nm, and at the long-wavelength end more than the electronic part, so that \beta_2 changes sign inside the interval of every molecule but CH4 (CO at 1969 nm, CO2 at 1315 nm, N2O at 1465 nm, SF6 at 1271 nm), where the share diverges.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_compact_models_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_compact_models`

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      <div class="sphx-glr-thumbnail-title">The compact models against their parents</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The complex refractive index of helium at 1 bar and 20 °C from 20 to 120 nm, from its constrained oscillator-strength distribution (``euv/He``): Samson et al.&#x27;s measured photoionisation cross section, Henke&#x27;s scattering factors above 120 eV and the NIST Rydberg lines. Above the ionisation threshold (50.4 nm) \mathrm{Re}\,n - 1 is negative and the absorption index \mathrm{Im}\,n appears; between the first line (58.4 nm) and the threshold the Rydberg lines make the index singular. For comparison: the index from Henke&#x27;s f_1 and f_2 (dotted, unreliable below about 30 eV by Henke et al.&#x27;s own account), Liu&#x27;s ab initio absorption (squares) and the ab initio fit fit/He below the first resonance (dashed).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_euv_he_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_euv_he`

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      <div class="sphx-glr-thumbnail-title">Helium in the extreme ultraviolet</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The complex refractive index at 1 bar and 20 °C, from 20 nm to beyond the first resonance, from each gas&#x27;s constrained oscillator-strength distribution (``euv/Ne``, euv/Ar, euv/Kr, euv/Xe): Chan et al.&#x27;s oscillator strengths, Henke&#x27;s scattering factors at high energy. Shaded: the region of the discrete lines, where the index is singular and the model carries no line widths. Dashed: the fitted model below the first resonance.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_euv_heavy_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_euv_heavy`

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      <div class="sphx-glr-thumbnail-title">Neon, argon, krypton and xenon in the extreme ultraviolet</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The argon model (``fit/Ar``): the 104.8 nm and 106.7 nm resonance lines at their NIST energies and two effective oscillators, with the magnetic susceptibility \chi_0 = -2.30\times10^{-4} a.u. that makes the electric polarizability the refractive one. The absolute level comes from Lesiuk &amp; Jeziorski&#x27;s ab initio Cauchy moments, the dielectric static polarizability (Gaiser &amp; Fellmuth, via Lesiuk) and Egan&#x27;s refractometry at 633 nm, which agree to a few 1e-5; the shape from 468 nm to 2.06 um from Peck &amp; Fisher, whose level is 3.5e-4 high and is fitted free; the vacuum ultraviolet from Bideau-Mehu and Gordon. Squares are validators, not fitted: Zhang&#x27;s comb value at 800 nm (+2.6e-4), Birch at 633 nm, Borzsonyi, Rusch&#x27;s infrared point, and the Lyman-alpha values of Gill and Heddle.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_ar_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_ar`

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      <div class="sphx-glr-thumbnail-title">Argon: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The methane model (``fit/CH4``): two effective electronic oscillators fitted to the Cuthbertsons&#x27; (1920) and Watson &amp; Ramaswamy&#x27;s visible dispersion (level free), Achtermann&#x27;s 633 nm and Burns et al.&#x27;s 458-633 nm values, Zhao et al.&#x27;s 1566 nm value and Shelton&#x27;s N2/CH4 ratios, with HITRAN&#x27;s vibrational bands below 4500 cm^-1 fixed (reduced chi-squared 0.68; TC-797). Shown from 240 nm to 1.4 um, across the absorption gaps; the qualified intervals (240 to 1123 nm and the infrared windows) are listed in /gas_models (TC-797).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_ch4_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_ch4`

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      <div class="sphx-glr-thumbnail-title">Methane: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The CO model (``fit/CO``): two effective electronic oscillators fitted to Smith et al.&#x27;s 167-288 nm interferometry, the Cuthbertsons&#x27; 1920 visible dispersion (level free), Watson &amp; Ramaswamy&#x27;s green line and Zhao et al.&#x27;s 2026 refractometry at 1566 nm, with HITRAN&#x27;s fundamental and first overtone fixed (reduced chi-squared 0.37). Shown over the optical range. The main qualified interval extends from 167 to 2288 nm, with two further infrared windows; see /gas_models (TC-797).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_co_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_co`

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      <div class="sphx-glr-thumbnail-title">Carbon monoxide: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The CO2 model (``fit/CO2``): two effective electronic oscillators fitted to Old, Gentili &amp; Peck (480-1600 nm, level free), the visible data, the 633 and 1542 nm metrology, Hohm&#x27;s 325-633 nm dispersion, Shelton&#x27;s N2/CO2 ratios with Marchetti&#x27;s CO2-laser values as validators, with HITRAN&#x27;s vibrational bands below 7000 cm^-1 fixed (reduced chi-squared 4.01; TC-797). Shown over the model&#x27;s main range, 250 nm to 1.4 um; seven additional qualified infrared intervals extend to 8.941 um, with gaps listed in /gas_models.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_co2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_co2`

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      <div class="sphx-glr-thumbnail-title">Carbon dioxide: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The deuterium model (``fit/D2``) uses Raj et al.&#x27;s (2018) ab initio average for normal deuterium and Shelton&#x27;s D2/H2 and H2/D2 ratios. Published refractometry is shown as independent evidence where identified as a validator. See /gas_models for the current data selection and /accuracy for limitations.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_d2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_d2`

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      <div class="sphx-glr-thumbnail-title">Deuterium: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The hydrogen model (``fit/H2``) is based on Raj et al.&#x27;s (2018) ab initio average for normal hydrogen, with a 0.1 % allowance, the dispersion shapes of Peck and Hung and of Smith et al., and Shelton&#x27;s He/H2 phase-matching ratio. The current model&#x27;s statistical uncertainty and the remaining absolute-scale evidence are summarised in /gas_models and /accuracy.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_h2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_h2`

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      <div class="sphx-glr-thumbnail-title">Hydrogen: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The helium model (``fit/He``): the 1s2p (58.4 nm) and 1s3p (53.7 nm) lines at their NIST energies and four effective oscillators, with the magnetic susceptibility \chi_0 = -2.12\times10^{-5} a.u. that makes the electric polarizability the refractive one. The backbone is Puchalski et al.&#x27;s ab initio dispersion coefficients of 4He, with every relativistic and QED correction (the static term to 1e-7), and Schmidt et al.&#x27;s microwave static polarizability: the model reproduces both to their uncertainties, so its one-sigma band is below 1 ppm from the infrared to 110 nm and 7 ppm at 76 nm.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_he_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_he`

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      <div class="sphx-glr-thumbnail-title">Helium: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The krypton model (``fit/Kr``): the 116.5 nm and 123.6 nm resonance lines at their NIST energies and two effective oscillators. The only ab initio dispersion in the library (Pawlowski et al.&#x27;s CC3 moments) is not converged, so the fit rests on data: the dielectric static polarizabilities of Orcutt &amp; Cole and of Huot &amp; Bose (via Kumar &amp; Thakkar; 3.6 sigma from the others), Birch&#x27;s and Hohm &amp; Kerl&#x27;s 633 nm values, Burns et al.&#x27;s 458-633 nm molar refractivities, Smith, Huber &amp; Parkinson&#x27;s 28 ultraviolet points (0.06 %), Bideau-Mehu&#x27;s revised (1976) and vacuum-ultraviolet (1981) values and the Cuthbertsons&#x27; visible shape (level free). Leonard&#x27;s absolute values lie 0.2 % low and his recommended one 0.04 % high. The magnetic term (about -25 ppm) is neglected. Chopra &amp; Heddle&#x27;s Lyman-alpha value lies between the two resonance lines, where the sign of n - 1 is not measured.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_kr_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_kr`

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      <div class="sphx-glr-thumbnail-title">Krypton: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The nitrogen model (``fit/N2``): effective oscillators fitted to Griesmann &amp; Burnett&#x27;s vacuum ultraviolet (level free), Peck &amp; Khanna&#x27;s visible and near infrared, and the 633 nm, microwave, dielectric and 10.57 um values agree to their uncertainties (reduced chi-squared 1.1).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_n2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_n2`

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      <div class="sphx-glr-thumbnail-title">Nitrogen: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The N2O model (``fit/N2O``): two effective electronic oscillators fitted to Hohm&#x27;s 325-633 nm dispersion (1e-5 relative) and the Cuthbertsons&#x27; (1914) visible dispersion (both level free) and the 633 nm metrology of Birch and Egan et al., with HITRAN&#x27;s vibrational bands below 4500 cm^-1 fixed and the Thomas-Reiche-Kuhn sum as a prior (reduced chi-squared 3.01: the two dispersions differ in shape by about 1e-4; TC-797). The main qualified interval is 320-1516 nm, with three additional infrared windows; see /gas_models.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_n2o_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_n2o`

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      <div class="sphx-glr-thumbnail-title">Nitrous oxide: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The neon model (``fit/Ne``): the 73.6 nm and 74.4 nm resonance lines at their NIST energies and two effective oscillators, with the magnetic susceptibility \chi_0 = -8.48\times10^{-5} a.u. The level comes from Lesiuk, Przybytek &amp; Jeziorski&#x27;s ab initio Cauchy moments and Gaiser &amp; Fellmuth&#x27;s 2018 dielectric static polarizability (via Lesiuk; their 2010 value is 16 ppm higher); the shape from the Cuthbertsons&#x27; visible data (level free: their standard state is ambiguous, and it comes out 0.3 % high) and Bideau-Mehu&#x27;s vacuum ultraviolet.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_ne_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_ne`

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      <div class="sphx-glr-thumbnail-title">Neon: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The oxygen model (``fit/O2``): effective oscillators fitted to Smith, Huber &amp; Parkinson&#x27;s ultraviolet above the Schumann-Runge bands, Egan et al.&#x27;s 2026 values at 633 and 1542 nm, Birch at 633 nm, Hohm&#x27;s 325-633 nm dispersion, Swift and Marchetti; Zhang&#x27;s 800 nm value (A band) and Kumar &amp; Thakkar&#x27;s sums are validators.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_o2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_o2`

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      <div class="sphx-glr-thumbnail-title">Oxygen: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The O3 model (``fit/O3``) absorbs. Its imaginary part is the measured absorption spectrum (Serdyuchenko et al. 2014 at 293 K from 213 nm, Yoshino et al. 1993, Mason et al. 1996 and Ogawa &amp; Cook 1958 to 12 eV), smoothed with a 50 cm^-1 Gaussian. Its real part is that spectrum&#x27;s Kramers-Kronig transform, plus:">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_o3_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_o3`

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      <div class="sphx-glr-thumbnail-title">Ozone: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The SF6 model (``fit/SF6``): two effective electronic oscillators fitted to Swift&#x27;s, Vukovic&#x27;s, Achtermann&#x27;s and Burns et al.&#x27;s values and Shelton&#x27;s N2/SF6 ratios, with HITRAN&#x27;s measured infrared cross-sections as fixed bands (reduced chi-squared 4.1; the two Shelton sets prefer the bands times 1.036, TC-146). The bands give the gas a zero-dispersion wavelength near 1.2 um. Shown from 230 nm to 5 um.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_sf6_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_sf6`

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      <div class="sphx-glr-thumbnail-title">Sulphur hexafluoride: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The xenon model (``fit/Xe``): the 146.96 nm and 129.56 nm resonance lines at their NIST energies and two effective oscillators, one of which settles at 121.6 nm, between the strong 119.2 nm and the 125.0 nm lines. The level comes from Huot &amp; Bose&#x27;s dielectric static polarizability (via Kumar &amp; Thakkar), Burns et al.&#x27;s 458-633 nm molar refractivities and Hohm &amp; Kerl&#x27;s 633 nm value; the shape from Abjean et al.&#x27;s and Bideau-Mehu et al.&#x27;s ultraviolet and vacuum ultraviolet, which crosses the 146.96 nm line (140.4 nm is between the two lines), and the Cuthbertsons&#x27; visible data (level free). The fitted strength of the 146.96 nm line, 0.2688, is Bideau-Mehu&#x27;s measured 0.268(8). Leonard&#x27;s recommended value is 0.13 % high. The magnetic term (about -25 ppm) is neglected.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_xe_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_xe`

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      <div class="sphx-glr-thumbnail-title">Xenon: the fitted model</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /auto_gallery/dispersion/plot_co2_windows
   /auto_gallery/dispersion/plot_compact_models
   /auto_gallery/dispersion/plot_euv_he
   /auto_gallery/dispersion/plot_euv_heavy
   /auto_gallery/dispersion/plot_fit_ar
   /auto_gallery/dispersion/plot_fit_ch4
   /auto_gallery/dispersion/plot_fit_co
   /auto_gallery/dispersion/plot_fit_co2
   /auto_gallery/dispersion/plot_fit_d2
   /auto_gallery/dispersion/plot_fit_h2
   /auto_gallery/dispersion/plot_fit_he
   /auto_gallery/dispersion/plot_fit_kr
   /auto_gallery/dispersion/plot_fit_n2
   /auto_gallery/dispersion/plot_fit_n2o
   /auto_gallery/dispersion/plot_fit_ne
   /auto_gallery/dispersion/plot_fit_o2
   /auto_gallery/dispersion/plot_fit_o3
   /auto_gallery/dispersion/plot_fit_sf6
   /auto_gallery/dispersion/plot_fit_xe