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Image for plasma implantation approximation
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KulaginVladimir committed Dec 12, 2023
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11 changes: 9 additions & 2 deletions docs/source/theory.rst
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Expand Up @@ -190,14 +190,21 @@ Henry’s condition (see Equations :eq:`eq_DirichletBC_Sievert` and
c_\mathrm{m} = K_H P~\text{on}~\delta\Omega
Plasma implantation approximattion
Plasma implantation approximation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Dirichlet’s boundary conditions can also be used to approximate plasma implantation
in near surface regions to be more computationally efficient [:ref:`5 <Delaporte-Mathurin et al. 2>`].
Let us consider a volumetric source term of hydrogen :math:`\Gamma=\varphi_{\mathrm{imp}}f(x)`,
where :math:`f(x)` is a narrow Gaussian distribution. The concentration profile of mobile
species can be approximated by a triangular shape [:ref:`6 <Schmid>`] with maximum at :math:`x=R_p`.
species can be approximated by a triangular shape [:ref:`6 <Schmid>`] with maximum at :math:`x=R_p`
(see the figure below).

.. figure:: docs/source/images/recomb_sketch.png
:align: center
:alt: Concentration profile with recombination flux and volumetric source term at :math:`x=R_p`. Dashed lines correspond to the time evolution

Concentration profie with recombination flux and volumetric source term at :math:`x=R_p`. Dashed lines correspond to the time evolution

The expression of maximum concentration value :math:`c_{\mathrm{m}}` can be obtained by expressing
the flux balance at equilibrium:
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