44 lines
2.0 KiB
ReStructuredText
44 lines
2.0 KiB
ReStructuredText
.. _nmr.pake:
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Wideline spectra
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^^^^^^^^^^^^^^^^
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Calculation of spectra
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----------------------
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In general, time signals are calculated by integration of all orientations (see also :ref:`list.orienations`):
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.. math::
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g(t) = \int f[\omega_\text{int}(\theta, \phi)t]\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi
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with :math:`f(\theta, \phi, t) = \cos[\omega_\text{int}(\theta, \phi) t]` or :math:`\exp[i\omega_\text{int}(\theta) t]`
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and fourier transform for a spectrum.
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However, summation over :math:`\theta`, :math:`\phi`, and calculating :math:`f(\theta, \phi, t)` for each orientation is time consuming.
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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.
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De-Paked spectra
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----------------
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A superposition of different Pake spectra complicates the evaluation of relaxation times or similar.
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The idea is to deconvolute these broad spectra into one line corresponding to relative orientation :math:`\theta = 0` [mccabe97]_.
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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
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.. math::
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g(t) = \int_0^{1} f[0, \omega_\text{int}(\theta)t]\,\mathrm{d}\cos\theta.
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This way, the integration is not over orientations at one time :math:`t`, but over times at one orientation 0.
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After some integrations, rearrangenments, and substitutions, a spectrum can be calculated by
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.. math::
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F(-2\omega) = \sqrt{\frac{3|\omega |}{2\pi}}(1\pm i) \text{FT}[g(t)\sqrt{t}]
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with :math:`1+i` for :math:`\omega > 0` and :math:`1-i` for :math:`\omega > 0`.
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.. figure:: depake.png
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:scale: 50 %
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.. [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 |