.. _sphx_glr_api_gallery_condensed_phonons:

Phonons
-------

Lattice vibrations: the phonon dispersion of 1D chains and a 2D square
lattice with its acoustic sound speeds, and the heat capacity of the
phonon gas, from Debye's :math:`T^3` law to the Dulong-Petit limit.


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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="Born and von Karman (1912) treated a crystal as masses joined by springs with periodic boundary conditions, so that its vibrations are plane waves with a dispersion relation \omega(\mathbf k). This example shows the monatomic chain (~physicskit.condensed.phonons.monatomic_chain_dispersion), the diatomic chain with its acoustic and optical branches and zone-boundary gap (~physicskit.condensed.phonons.diatomic_chain_dispersion), and a 2D square lattice (~physicskit.condensed.phonons.square_lattice_phonon_dispersion) whose acoustic branches start linearly with the sound speeds c_L = a\sqrt{(K_1+K_2)/m} and c_T = a\sqrt{K_2/m}.">

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  .. image:: /api/gallery/condensed/phonons/images/thumb/sphx_glr_plot_born_von_karman_phonon_dispersion_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/phonons/plot_born_von_karman_phonon_dispersion`

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      <div class="sphx-glr-thumbnail-title">Born and von Karman: phonon dispersion of crystal lattices</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Einstein&#x27;s 1907 model, with every atom vibrating at one frequency, explained why heat capacities drop below the classical Dulong-Petit value 3Nk_B at low temperature, but it predicted an exponential drop. Measurements showed a power law. Debye (1912) replaced the single frequency by sound waves with \omega = ck up to a cutoff \Theta_D and found C \approx \frac{12\pi^4}{5}Nk_B(T/\Theta_D)^3 at low T. debye_heat_capacity is compared here with the Einstein model and with the heat capacity of an actual lattice dispersion (~physicskit.condensed.phonons.lattice_heat_capacity).">

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  .. image:: /api/gallery/condensed/phonons/images/thumb/sphx_glr_plot_debye_heat_capacity_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/phonons/plot_debye_heat_capacity`

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      <div class="sphx-glr-thumbnail-title">Debye's heat capacity and the T³ law</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/condensed/phonons/plot_born_von_karman_phonon_dispersion
   /api/gallery/condensed/phonons/plot_debye_heat_capacity