.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/photochem/energy_transfer/plot_01_forster_fret_efficiency.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_energy_transfer_plot_01_forster_fret_efficiency.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_photochem_energy_transfer_plot_01_forster_fret_efficiency.py:


Förster resonance energy transfer: the inverse-sixth-power distance law
=========================================================================

Förster (1948) showed that an excited donor can hand its energy to a
nearby acceptor through dipole-dipole coupling at a rate
:math:`k_T=\tau_D^{-1}(R_0/r)^6`, so the transfer efficiency
:math:`E=1/(1+(r/R_0)^6)` falls from nearly 1 to nearly 0 over a narrow
range around the Förster distance :math:`R_0`
(:func:`~chemistrykit.photochem.forster_radius`,
:func:`~chemistrykit.photochem.forster_efficiency`). That steepness is
why FRET works as a "spectroscopic ruler" for distances of 1-10 nm.

.. GENERATED FROM PYTHON SOURCE LINES 16-37

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.photochem import forster_efficiency, forster_radius, forster_rate

    R0 = forster_radius(kappa2=2 / 3, n=1.4, quantum_yield_donor=0.5, overlap_J=2.0e15) / 10.0  # nm
    tau_D = 4.0e-9
    print(f"Forster distance R0 = {R0:.2f} nm")

    r = np.linspace(1.0, 12.0, 400)
    E = forster_efficiency(r, R0)
    fig, ax = plt.subplots()
    ax.plot(r, E)
    ax.axvline(R0, color="k", ls=":", label=rf"$R_0$ = {R0:.1f} nm, $E$ = 0.5")
    ax.axvspan(0.5 * R0, 1.5 * R0, color="0.92", label=r"useful ruler range $0.5$-$1.5\,R_0$")
    ax.set_xlabel("Donor-acceptor distance $r$ (nm)")
    ax.set_ylabel("Transfer efficiency $E$")
    ax.set_title(r"FRET efficiency $1/(1+(r/R_0)^6)$")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/photochem/energy_transfer/images/sphx_glr_plot_01_forster_fret_efficiency_001.png
   :alt: FRET efficiency $1/(1+(r/R_0)^6)$
   :srcset: /api/gallery/photochem/energy_transfer/images/sphx_glr_plot_01_forster_fret_efficiency_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Forster distance R0 = 4.98 nm




.. GENERATED FROM PYTHON SOURCE LINES 38-40

The donor lifetime shortens with transfer: tau_DA = 1/(1/tau_D + k_T),
and 1 - tau_DA/tau_D reproduces E exactly.

.. GENERATED FROM PYTHON SOURCE LINES 40-45

.. code-block:: Python

    for d in (0.5 * R0, R0, 1.5 * R0):
        tau_DA = 1.0 / (1.0 / tau_D + forster_rate(d, R0, tau_D))
        print(f"r = {d:4.2f} nm: E = {forster_efficiency(d, R0):.3f}, 1 - tau_DA/tau_D = {1 - tau_DA / tau_D:.3f}")

    plt.show()




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

 .. code-block:: none

    r = 2.49 nm: E = 0.985, 1 - tau_DA/tau_D = 0.985
    r = 4.98 nm: E = 0.500, 1 - tau_DA/tau_D = 0.500
    r = 7.47 nm: E = 0.081, 1 - tau_DA/tau_D = 0.081





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

   **Total running time of the script:** (0 minutes 0.041 seconds)


.. _sphx_glr_download_api_gallery_photochem_energy_transfer_plot_01_forster_fret_efficiency.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_forster_fret_efficiency.ipynb <plot_01_forster_fret_efficiency.ipynb>`

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

      :download:`Download Python source code: plot_01_forster_fret_efficiency.py <plot_01_forster_fret_efficiency.py>`

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

      :download:`Download zipped: plot_01_forster_fret_efficiency.zip <plot_01_forster_fret_efficiency.zip>`


.. only:: html

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

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