.. _sphx_glr_api_gallery_crystal_madelung:

Madelung constant
-------------------

Madelung's electrostatic lattice sum for NaCl, shell by shell, and
Evjen's genuinely converging summation of it, contrasted with a naive
truncated sum that does not converge.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="The NaCl Madelung-constant lattice sum is only conditionally convergent: a naive truncated sum over a growing cube of ions does not settle down as the cutoff grows, it oscillates. madelung_constant_nacl uses Evjen&#x27;s method (fractional boundary-charge weighting) to build a genuinely converging summation instead, and this example demonstrates both behaviors side by side against the literature value.">

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  .. image:: /api/gallery/crystal/madelung/images/thumb/sphx_glr_plot_01_madelung_convergence_thumb.png
    :alt:

  :doc:`/api/gallery/crystal/madelung/plot_01_madelung_convergence`

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      <div class="sphx-glr-thumbnail-title">Evjen's method: converging the NaCl Madelung constant</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Madelung (1918) asked for the electrostatic energy of one ion in the field of every other ion of an infinite crystal. For rock salt it is U=-Me^2/(4\pi\varepsilon_0r_0) per ion pair, with the Madelung constant M=\sum_j \pm r_0/r_j summed shell by shell: 6 opposite charges at r_0, 12 like charges at \sqrt2r_0, 8 opposite charges at \sqrt3r_0, and so on. madelung_constant_nacl returns the converged value.">

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  .. image:: /api/gallery/crystal/madelung/images/thumb/sphx_glr_plot_02_madelung_lattice_sum_thumb.png
    :alt:

  :doc:`/api/gallery/crystal/madelung/plot_02_madelung_lattice_sum`

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      <div class="sphx-glr-thumbnail-title">Madelung's lattice sum: the electrostatic energy of rock salt</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/crystal/madelung/plot_01_madelung_convergence
   /api/gallery/crystal/madelung/plot_02_madelung_lattice_sum