.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/photochem/fluorescence/plot_02_perrin_anisotropy.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_photochem_fluorescence_plot_02_perrin_anisotropy.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_photochem_fluorescence_plot_02_perrin_anisotropy.py:


Perrin equation: fluorescence anisotropy and molecular rotation
=================================================================

Perrin (1926) related the polarization of fluorescence to how far a
molecule rotates during its excited-state lifetime :math:`\tau`:
:math:`r=r_0/(1+\tau/\theta)`, with rotational correlation time
:math:`\theta=\eta V/k_BT`
(:func:`~chemistrykit.photochem.perrin_anisotropy`,
:func:`~chemistrykit.photochem.rotational_correlation_time`). A small dye
depolarizes almost completely in water but stays polarized in viscous
glycerol, and a "Perrin plot" of :math:`1/r` against :math:`T/\eta` is a
straight line whose intercept gives :math:`r_0` and whose slope gives the
molecular volume.

.. GENERATED FROM PYTHON SOURCE LINES 18-39

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import K_B
    from chemistrykit.photochem import perrin_anisotropy, rotational_correlation_time

    r0, tau = 0.4, 4.0e-9  # fundamental anisotropy, lifetime (s)
    V = 1.5e-27  # hydrodynamic volume, m^3 (~1.5 nm^3)
    T = 298.15
    eta = np.logspace(-3.5, 0.5, 200)  # Pa*s, from below water to glycerol-like

    r = perrin_anisotropy(r0, tau, rotational_correlation_time(eta, V, T))
    fig, ax = plt.subplots()
    ax.semilogx(eta * 1e3, r)
    ax.axhline(r0, color="k", ls=":", label=r"$r_0$ (no rotation)")
    ax.set_xlabel("Solvent viscosity (mPa s)")
    ax.set_ylabel("Steady-state anisotropy $r$")
    ax.set_title("Perrin equation: anisotropy rises as rotation slows")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/photochem/fluorescence/images/sphx_glr_plot_02_perrin_anisotropy_001.png
   :alt: Perrin equation: anisotropy rises as rotation slows
   :srcset: /api/gallery/photochem/fluorescence/images/sphx_glr_plot_02_perrin_anisotropy_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 40-41

Perrin plot from synthetic measurements at several temperatures.

.. GENERATED FROM PYTHON SOURCE LINES 41-61

.. code-block:: Python

    rng = np.random.default_rng(1)
    T_meas = np.linspace(283.0, 333.0, 6)
    eta_meas = 5.0e-3
    r_meas = perrin_anisotropy(r0, tau, rotational_correlation_time(eta_meas, V, T_meas)) * (1 + rng.normal(0, 0.005, 6))
    x = T_meas / eta_meas
    # 1/r = 1/r0 + (tau k_B / (r0 V)) T/eta, so intercept = 1/r0 and V = intercept tau k_B / slope.
    slope, intercept = np.polyfit(x, 1.0 / r_meas, 1)
    V_fit = intercept * tau * K_B / slope
    print(f"Fitted r0 = {1 / intercept:.3f} (true {r0}), fitted V = {V_fit * 1e27:.2f} nm^3 (true 1.50)")

    fig2, ax2 = plt.subplots()
    ax2.plot(x, 1.0 / r_meas, "o", label="measured")
    ax2.plot(x, intercept + slope * x, "k--", label="linear fit")
    ax2.set_xlabel(r"$T/\eta$ (K Pa$^{-1}$ s$^{-1}$)")
    ax2.set_ylabel("$1/r$")
    ax2.set_title("Perrin plot")
    ax2.legend()
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/photochem/fluorescence/images/sphx_glr_plot_02_perrin_anisotropy_002.png
   :alt: Perrin plot
   :srcset: /api/gallery/photochem/fluorescence/images/sphx_glr_plot_02_perrin_anisotropy_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Fitted r0 = 0.405 (true 0.4), fitted V = 1.48 nm^3 (true 1.50)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.089 seconds)


.. _sphx_glr_download_api_gallery_photochem_fluorescence_plot_02_perrin_anisotropy.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_perrin_anisotropy.ipynb <plot_02_perrin_anisotropy.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_perrin_anisotropy.py <plot_02_perrin_anisotropy.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_perrin_anisotropy.zip <plot_02_perrin_anisotropy.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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