:orphan:

Examples
========

This gallery walks through every public feature of ``chemistrykit.spectro``:
the Beer-Lambert absorbance law and its stray-light deviation from
linearity; rigid-rotor rotational spectra with isotope shifts; harmonic
vs. Morse-potential vibrational band positions, plus a genuine triatomic
normal-mode calculation; Franck-Condon vibronic progressions for
electronic spectra; NMR from Larmor frequencies and chemical shifts to
J-coupling multiplets, the Karplus relation, and Fourier-transform NMR;
atomic line spectra (Fraunhofer, Kirchhoff-Bunsen, Balmer-Rydberg); and
spectral lineshapes.

Each script in this gallery is self-contained and can be run directly with
``python examples/spectro/<section>/<script>.py``. Every script also
carries an RST module docstring as its title/description and uses ``# %%``
markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source of
truth for what you see, not a copy of it.

Sections
--------

- **beer_lambert** -- the Beer-Lambert absorbance law and its stray-light
  deviation from linearity at high concentration.
- **atomic** -- Fraunhofer's dark solar lines, Kirchhoff and Bunsen's
  flame emission/absorption spectra, and the hydrogen series from the
  Balmer-Rydberg formula.
- **rotational** -- rigid-rotor rotational spectra, relative line
  intensities, and microwave isotope shifts.
- **vibrational** -- harmonic vs. Morse IR band positions and
  anharmonicity constants, Wilson GF-matrix normal modes for CO2 and H2O,
  Raman/IR mutual exclusion, and a linear molecule's degenerate bend.
- **electronic** -- Franck-Condon vibronic progressions in a UV-Vis
  absorption band.
- **nmr** -- Larmor frequencies and chemical shifts, first-order
  J-coupling multiplets, the Karplus relation, and Fourier-transform NMR.
- **lineshapes** -- Gaussian, Lorentzian, and Voigt line profiles.


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Atomic line spectra
--------------------

Discrete atomic lines as chemical fingerprints: Fraunhofer's dark solar
lines, Kirchhoff and Bunsen's flame emission and absorption spectra, and
the Balmer-Rydberg formula for the hydrogen series.


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    <div class="sphx-glr-thumbcontainer" tooltip="Fraunhofer (1814) found that sunlight is not a smooth rainbow but a continuum crossed by hundreds of sharp dark lines at fixed, reproducible wavelengths, and he labelled the strongest ones A to K. This example builds a sun-like continuum from Planck&#x27;s law at 5800 K, removes light at the tabulated positions of Fraunhofer&#x27;s lettered lines using narrow Lorentzian absorption profiles, and marks each letter above its line.">

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  .. image:: /api/gallery/spectro/atomic/images/thumb/sphx_glr_plot_01_fraunhofer_dark_lines_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/atomic/plot_01_fraunhofer_dark_lines`

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      <div class="sphx-glr-thumbnail-title">Fraunhofer's dark lines: sharp absorption lines at fixed wavelengths in the solar spectrum</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Bunsen and Kirchhoff (1859-1861) vaporized salts in a nearly colourless flame and found that each element gives bright lines at its own fixed wavelengths. The top panel shows the strongest visible and near-visible resonance lines of the alkali metals they studied, including caesium&#x27;s blue doublet and rubidium&#x27;s red and near-infrared lines, the lines that revealed those two new elements. The bottom panel illustrates Kirchhoff&#x27;s law: a cooler sodium vapour placed in front of a continuous source absorbs at exactly the wavelengths where hot sodium emits. This is what explains Fraunhofer&#x27;s D lines.">

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  .. image:: /api/gallery/spectro/atomic/images/thumb/sphx_glr_plot_02_kirchhoff_bunsen_flame_spectra_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/atomic/plot_02_kirchhoff_bunsen_flame_spectra`

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      <div class="sphx-glr-thumbnail-title">Kirchhoff and Bunsen's flame spectra: bright emission lines and matching dark absorption lines</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Balmer (1885) fitted the four visible hydrogen lines with \lambda=B\,n^2/(n^2-4). Rydberg (1890) rewrote this in wavenumbers as \tilde\nu=R(1/n_1^2-1/n_2^2), and Bohr (1913) derived it from quantized electron energies E_n=-hcR/n^2. This example computes the Lyman, Balmer, and Paschen series with rydberg_wavenumber, checks the result against Balmer&#x27;s original formula, and shows each series bunching toward its series limit.">

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  .. image:: /api/gallery/spectro/atomic/images/thumb/sphx_glr_plot_03_balmer_rydberg_hydrogen_series_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/atomic/plot_03_balmer_rydberg_hydrogen_series`

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      <div class="sphx-glr-thumbnail-title">The hydrogen spectrum: Balmer's formula, the Rydberg series, and Bohr's energy levels</div>
    </div>


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    </div>

Beer-Lambert law
-----------------

The Beer-Lambert absorbance law, and its stray-light deviation from
linearity at high concentration.


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    <div class="sphx-glr-thumbcontainer" tooltip="Absorbance is exactly linear in concentration by the ideal Beer-Lambert law. A small amount of instrumental stray light (light that reaches the detector without having passed through the fully absorbing sample path) causes the apparent absorbance to fall increasingly below the true value as concentration -- and hence true absorbance -- increases, the classic &quot;rolling over&quot; deviation seen in real UV-Vis instruments.">

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  .. image:: /api/gallery/spectro/beer_lambert/images/thumb/sphx_glr_plot_01_beer_lambert_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/beer_lambert/plot_01_beer_lambert`

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      <div class="sphx-glr-thumbnail-title">Beer-Lambert absorbance, and its deviation from linearity at high concentration</div>
    </div>


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    </div>

Electronic spectra
--------------------

Franck-Condon vibronic progressions for a UV-Vis electronic absorption
band.


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    <div class="sphx-glr-thumbcontainer" tooltip="Builds a vibronic progression from the Huang-Rhys displacement parameter, verifies the Franck-Condon factors&#x27; normalization (they must sum to 1 -- the required correctness check for any Franck-Condon calculation), and shows how a larger excited-state displacement shifts the most-intense vibronic peak to higher v&#x27;.">

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  .. image:: /api/gallery/spectro/electronic/images/thumb/sphx_glr_plot_01_franck_condon_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/electronic/plot_01_franck_condon`

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      <div class="sphx-glr-thumbnail-title">Franck-Condon vibronic progressions in a UV-Vis absorption band</div>
    </div>


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Spectral lineshapes
---------------------

The Gaussian, Lorentzian, and Voigt lineshapes, and the two physically
distinct broadening mechanisms (Doppler/inhomogeneous vs.
lifetime-collisional/homogeneous) that produce them.


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares the three standard spectral lineshapes at matched FWHM: a Lorentzian (the shape a classically radiating damped oscillator, or a finite-lifetime quantum transition, produces -- &quot;homogeneous&quot; broadening, identical for every molecule in the sample), a Gaussian (the shape a Maxwell-Boltzmann spread of line-of-sight velocities produces via the Doppler effect -- &quot;inhomogeneous&quot; broadening, different molecules contributing different, momentarily Doppler-shifted, sub-lines), and their convolution, the Voigt profile, which is what a real spectrum typically shows once both mechanisms contribute together. The Lorentzian has visibly heavier tails than the Gaussian at the same FWHM -- the qualitative signature used to diagnose which mechanism dominates a real measured line.">

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  .. image:: /api/gallery/spectro/lineshapes/images/thumb/sphx_glr_plot_01_lineshapes_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/lineshapes/plot_01_lineshapes`

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      <div class="sphx-glr-thumbnail-title">Gaussian, Lorentzian, and Voigt lineshapes: two broadening mechanisms and their convolution</div>
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NMR spectroscopy
-----------------

Nuclear magnetic resonance: Larmor frequencies and the chemical shift,
first-order spin-spin coupling multiplets (the n+1 rule and a doublet of
triplets), the Karplus dihedral-angle dependence of vicinal couplings,
and Fourier-transform NMR from a free-induction decay.


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    <div class="sphx-glr-thumbcontainer" tooltip="Indirect spin-spin coupling carried through the bonding electrons (Ramsey and Purcell, 1952) splits an NMR resonance into a multiplet. The splitting is J Hz regardless of field strength. In the first-order limit, n equivalent spin-1/2 neighbours give n+1 lines with Pascal&#x27;s-triangle intensities. This example simulates ethanol&#x27;s -CH3 triplet and -CH2- quartet, then a doublet of triplets from two inequivalent coupling partners with different J values.">

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  .. image:: /api/gallery/spectro/nmr/images/thumb/sphx_glr_plot_01_nmr_multiplets_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/nmr/plot_01_nmr_multiplets`

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      <div class="sphx-glr-thumbnail-title">Spin-spin (J) coupling and the n+1 rule: ethanol's triplet/quartet and a doublet of triplets</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Bloch and Purcell (1946) showed that nuclei in a static field B_0 absorb radio-frequency energy sharply at the Larmor frequency \nu_0=\gamma B_0/(2\pi). The effect became a chemical tool when it was found that a nucleus&#x27;s surrounding electrons shield it slightly, so the exact resonance frequency depends on its chemical environment. This example computes ^1\mathrm{H} and ^{13}\mathrm{C} Larmor frequencies against field strength. It then places three proton environments of methyl acetate-like shielding on the frequency axis at two field strengths and shows that converting to ppm with chemical_shift_ppm makes the spectrum field-independent.">

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  .. image:: /api/gallery/spectro/nmr/images/thumb/sphx_glr_plot_02_nmr_resonance_chemical_shift_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/nmr/plot_02_nmr_resonance_chemical_shift`

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      <div class="sphx-glr-thumbnail-title">Nuclear magnetic resonance: Larmor frequencies and the field-independent chemical shift</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Karplus (1959) used valence-bond theory to show that the three-bond (vicinal) H-C-C-H coupling constant depends on the dihedral angle \phi. It is large for eclipsed (0 degrees) and anti (180 degrees) protons and nearly zero near 90 degrees. This lets a measured ^3J report molecular conformation. This example plots Karplus&#x27;s 1959 curve from karplus_coupling, reads off the axial-axial and axial-equatorial couplings of a cyclohexane chair, and shows the resulting doublet splittings.">

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  .. image:: /api/gallery/spectro/nmr/images/thumb/sphx_glr_plot_03_karplus_vicinal_coupling_thumb.png
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  :doc:`/api/gallery/spectro/nmr/plot_03_karplus_vicinal_coupling`

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      <div class="sphx-glr-thumbnail-title">The Karplus relation: vicinal 3J(H,H) coupling as a function of dihedral angle</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Ernst and Anderson (1966) replaced the slow frequency sweep of continuous-wave NMR with a single short radio-frequency pulse. The pulse excites all resonances at once. The resulting free-induction decay (FID) is a sum of decaying oscillations, one per resonance, and its Fourier transform gives the whole spectrum. Because each FID takes about as long as a single scan and many can be averaged, the method is far more sensitive. This example simulates a noisy FID for three resonances with free_induction_decay, transforms it with fid_to_spectrum, and shows signal averaging: signal-to-noise grows as \sqrt{N} over N co-added scans.">

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  .. image:: /api/gallery/spectro/nmr/images/thumb/sphx_glr_plot_04_fourier_transform_nmr_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/nmr/plot_04_fourier_transform_nmr`

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      <div class="sphx-glr-thumbnail-title">Fourier-transform NMR: from a free-induction decay to a spectrum</div>
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Rotational spectra
-------------------

Rigid-rotor rotational spectra built on ``chemistrykit.quantum``'s rigid
rotor: quantized levels, evenly spaced transition wavenumbers and
Boltzmann line intensities, and microwave isotope shifts of CO
isotopologues.


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    <div class="sphx-glr-thumbcontainer" tooltip="Dennison (1926) gave the quantum-mechanical rotational energy levels E_J=\hbar^2J(J+1)/(2I). With the \Delta J=+1 selection rule, these levels produce absorption lines at 2B(J+1), which are evenly spaced by 2B. This example builds an HCl-like rigid rotor from chemistrykit.quantum, prints its level energies and line positions, checks the constant 2B spacing, and adds the Boltzmann-population intensity pattern seen in a real far-infrared or microwave spectrum.">

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  .. image:: /api/gallery/spectro/rotational/images/thumb/sphx_glr_plot_01_rotational_spectrum_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/rotational/plot_01_rotational_spectrum`

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      <div class="sphx-glr-thumbnail-title">Dennison's quantized rigid rotor: HCl rotational lines spaced by 2B, with Boltzmann intensities</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Rotational lines of light molecules lie in the microwave region, which became routinely accessible after Cleeton and Williams (1934) and postwar microwave spectroscopy by Townes and others. Microwave frequencies can be measured very precisely, so isotope shifts are easy to see. Isotopic substitution leaves the bond length unchanged but changes the reduced mass, so every line of the heavier isotopologue moves down by \mu_a/\mu_b, the ratio isotope_shift_ratio predicts. This example computes the J=1\leftarrow0 lines of ^{12}\mathrm{C}^{16}\mathrm{O}, ^{13}\mathrm{C}^{16}\mathrm{O} and ^{12}\mathrm{C}^{18}\mathrm{O} in GHz and compares them with the measured values (115.271, 110.201 and 109.782 GHz).">

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  .. image:: /api/gallery/spectro/rotational/images/thumb/sphx_glr_plot_02_microwave_isotope_shift_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/rotational/plot_02_microwave_isotope_shift`

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      <div class="sphx-glr-thumbnail-title">Microwave spectroscopy and the isotope shift: CO isotopologues' J=1-0 lines in GHz</div>
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Vibrational spectra
---------------------

Harmonic vs. Morse-potential IR band positions and anharmonicity
constants; Wilson GF-matrix normal modes for CO2 and H2O; the Raman/IR
mutual exclusion rule for CO2; and the doubly degenerate bend of a
linear triatomic.


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    <div class="sphx-glr-thumbcontainer" tooltip="Predicts HCl&#x27;s fundamental IR absorption from a harmonic oscillator, compares it against the exact (anharmonic) Morse-potential fundamental and first overtone, then inverts the observed 0\to1/0\to2 band positions to recover the spectroscopic constants \omega_e/\omega_ex_e -- exactly the way real anharmonicity constants are measured from an observed overtone spectrum.">

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  .. image:: /api/gallery/spectro/vibrational/images/thumb/sphx_glr_plot_01_harmonic_vs_morse_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/vibrational/plot_01_harmonic_vs_morse`

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      <div class="sphx-glr-thumbnail-title">Harmonic vs. Morse IR band positions, and recovering anharmonicity from overtones</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="This example uses Wilson&#x27;s GF-matrix method to calculate normal modes for a linear triatomic (CO2) and a bent one (H2O) with TriatomicNormalModes. The molecule&#x27;s geometry and masses give the kinetic-energy matrix G=BM^{-1}B^T, where B is the Wilson B-matrix. A diagonal valence force field gives the potential-energy matrix F. Solving the eigenvalue problem GFL=L\Lambda, with \lambda_k=(2\pi c\tilde\nu_k)^2, gives the normal-mode wavenumbers. Changing only the bend entry of F shifts the bend frequency, while the stretches barely move.">

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  .. image:: /api/gallery/spectro/vibrational/images/thumb/sphx_glr_plot_02_triatomic_normal_modes_thumb.png
    :alt:

  :doc:`/api/gallery/spectro/vibrational/plot_02_triatomic_normal_modes`

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      <div class="sphx-glr-thumbnail-title">Wilson's GF-matrix method: CO2 and H2O normal-mode frequencies from the G and F matrices</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Raman and Krishnan (1928) found that a small fraction of scattered light is shifted in frequency by a molecule&#x27;s vibrational wavenumbers. A mode is Raman-active if it changes the molecule&#x27;s polarizability. It is infrared-active if it changes the dipole moment. For a centrosymmetric molecule such as CO2, no mode is active in both spectra.">

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  .. image:: /api/gallery/spectro/vibrational/images/thumb/sphx_glr_plot_03_raman_ir_mutual_exclusion_thumb.png
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  :doc:`/api/gallery/spectro/vibrational/plot_03_raman_ir_mutual_exclusion`

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      <div class="sphx-glr-thumbnail-title">Raman scattering vs. infrared absorption: the mutual exclusion rule for CO2</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Herzberg&#x27;s Infrared and Raman Spectra of Polyatomic Molecules (1945) set out the rules for counting and labelling vibrations of polyatomic molecules. A nonlinear molecule of N atoms has 3N-6 vibrations. A linear molecule has 3N-5, because rotation about its axis does not count as a degree of freedom. For CO2 this means four vibrations: the symmetric stretch, the antisymmetric stretch, and a doubly degenerate bend (in-plane and out-of-plane bending have the same frequency by cylindrical symmetry). This example compares linear CO2 with bent H2O using TriatomicNormalModes, whose is_linear flag marks the bend as a degenerate pair.">

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  .. image:: /api/gallery/spectro/vibrational/images/thumb/sphx_glr_plot_04_linear_bend_degeneracy_thumb.png
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  :doc:`/api/gallery/spectro/vibrational/plot_04_linear_bend_degeneracy`

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      <div class="sphx-glr-thumbnail-title">Herzberg's linear-molecule rules: CO2's doubly degenerate bend and 3N-5 vibrations</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/spectro/atomic/index.rst
   /api/gallery/spectro/beer_lambert/index.rst
   /api/gallery/spectro/electronic/index.rst
   /api/gallery/spectro/lineshapes/index.rst
   /api/gallery/spectro/nmr/index.rst
   /api/gallery/spectro/rotational/index.rst
   /api/gallery/spectro/vibrational/index.rst



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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_