.. _nmr.pake:

Wideline spectra
^^^^^^^^^^^^^^^^

Calculation of spectra
----------------------

In general, time signals are calculated by integration of all orientations (see also :ref:`list.orienations`):

.. math::
   g(t) = \int f[\omega_\text{int}(\theta, \phi)t]\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi

with :math:`f(\theta, \phi, t) = \cos[\omega_\text{int}(\theta, \phi) t]` or :math:`\exp[i\omega_\text{int}(\theta) t]`
and fourier transform for a spectrum.
However, summation over :math:`\theta`, :math:`\phi`, and calculating :math:`f(\theta, \phi, t)` for each orientation is time consuming.

Alternatively, if the orientations are equidistant in :math:`\cos\theta`, one can get to the spectrum directly by creating a histogram of :math:`\omega_\text{int}(\theta, \phi)`, thus circumventing a lot of calculations.


De-Paked spectra
----------------

A superposition of different Pake spectra complicates the evaluation of relaxation times or similar.
The idea is to deconvolute these broad spectra into one line corresponding to relative orientation :math:`\theta = 0` [mccabe97]_.

For :math:`\omega_\text{int}(\theta) \propto (3\cos^2\theta -1)/2 = P_2(\cos\theta)`, the property :math:`\omega_\text{int}(\theta) = \omega_\text{int}(0) \omega_\text{int}(\theta)` is used to write

.. math::
   g(t) = \int_0^{1} f[0, \omega_\text{int}(\theta)t]\,\mathrm{d}\cos\theta.

This way, the integration is not over orientations at one time :math:`t`, but over times at one orientation 0.
After some integrations, rearrangenments, and substitutions, a spectrum can be calculated by

.. math::
  F(-2\omega) = \sqrt{\frac{3|\omega |}{2\pi}}(1\pm i) \text{FT}[g(t)\sqrt{t}]

with :math:`1+i` for :math:`\omega > 0` and :math:`1-i` for :math:`\omega > 0`.

.. figure:: depake.png
   :scale: 50 %


.. [mccabe97] M.A. McCabe, S.R. Wassail: Rapid deconvolution of NMR powder spectra by weighted fast Fourier transformation, Solid State Nuclear Magnetic Resonance (1997). https://doi.org/10.1016/S0926-2040(97)00024-6