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fixed theory
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simongravelle committed Jan 4, 2024
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2 changes: 1 addition & 1 deletion docs/source/index.rst
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Expand Up @@ -49,7 +49,7 @@ DNMR from MD
:hidden:

theory/context
theory/theory
theory/dipolar_relaxation
theory/best-practice

.. toctree::
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Theory
======

Dipolar relaxation
------------------
==================

.. container:: justify

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.. math::
G^{(m)} (\tau) = \dfrac{\alpha_m^2}{N}
G^{(\text{m)} (\tau) = \dfrac{\alpha_m^2}{N}
\sum_i \sum_{j \ne i} \dfrac{Y_2^{(m)} [\Omega_{ij} (0)]}{r_{ij}^3 (0)} \dfrac{Y_2^{*(m)} [\Omega_{ij} (\tau)]}{r_{ij}^3 (\tau)},
.. container:: justify
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where :math:`j \in M_i` and :math:`j \notin M_i` refer to summation on spin from the
same molecule as :math:`i`, and from different molecules as :math:`i`, respectively.

.. container:: justify

Intra-molecular relaxation is usually attributed to the rotational motion of the
molecule, and inter-molecular relaxation to the translational motion. This is only an
molecules, and inter-molecular relaxation to their translational motion. This is only an
assumption that simplify the interpretation of results, and can lead to
error :cite:`hubbardTheoryNuclearMagnetic1963`.

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.. container:: justify

For isotropic system, the correlation functions are proportional to each others,
For isotropic systems, the correlation functions are proportional to each others,
and only :math:`G^{(0)} (t)` needs to be calculated.

In that case, :math:`G^{(0)} = 6 G^{(1)}`, and :math:`G^{(0)} = 6 / 4 G^{(2)}`
:cite:`becherMolecularDynamicsSimulations2021`, which can easily be checked, for instance
for a bulk water system:

.. image:: ../figures/best-practices/proportionality-dark.png
:class: only-dark
:alt: NMR results obtained from the LAMMPS simulation of water

.. image:: ../figures/best-practices/proportionality-light.png
:class: only-light
:alt: NMR results obtained from the LAMMPS simulation of water

.. container:: justify

Therefore, the spectrums can be calculated as:
In that case, :math:`G^{(0)} = 6 G^{(1)}`, and :math:`G^{(0)} = 6 / 4 G^{(2)}`
:cite:`becherMolecularDynamicsSimulations2021`,
and the spectrums can be calculated as:

.. math::
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