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update of the docu. Added the beta-NMR docu written by Jonas Krieger.

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doc/html/_sources/user-libs.txt
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@ -407,3 +407,341 @@ Nonlocal superconductivity related Meissner screening functions (AS libs)
-------------------------------------------------------------------------
To be written yet ...
.. index:: BNMR-libs
.. _BNMR-libs:
Functions to analyze |bgr|-NMR data (BNMR libs)
-------------------------------------------------------------------------
This is a collection of ``C++`` classes using the ``musrfit`` :ref:`user-functions <user-functions>`
interface in order to facilitate the usage in conjunction with ``musrfit``. It consists of two libraries:
* ``libBNMR`` contains functions to fit spin lattice relaxation (SLR) data.
* ``libLineProfile`` contains functions to fit resonance lineshapes.
.. note::
Currently it is recommended to read in the data in ASCII format as a non-|mgr|\SR fit :ref:`(fit type 8) <non-musr-fit>`.
.. index:: libBNMR
libBNMR
++++++++++
In |bgr|-NMR the SLR is usually measured by implanting a pulse of :math:`^8`\ Li with a length :math:`t_0` into the sample.
The asymmetry is measured both during the pulse and afterwards. For a a general spin relaxation function :math:`f(t)` the time evolution of the asymmetry is then given by [`Z. Salman, et al., PRL 96, 147601 (2006) <http://dx.doi.org/10.1103/PhysRevLett.96.147601>`_]:
.. index:: SLR
.. _SLR:
.. math::
P(t) = \left\{\begin{matrix}
\frac{\int_0^t e^{-(t-t')/\tau_{\mathrm{Li}}}f(t-t')dt'}{\int_0^t e^{-t'/\tau_{\mathrm{Li}}}dt' } & t\leq t_0\\[6pt]
\frac{\int_0^{t_0}e^{-(t_0-t')/\tau_{\mathrm{Li}}}f(t-t')dt'}{\int_0^{t_0}e^{-t'/\tau_{\mathrm{Li}}}dt'} & t> t_0,
\end{matrix}\right.
where :math:`\tau_{\mathrm{Li}}=1.21`\ s is the :math:`^8`\ Li lifetime.
Functions
^^^^^^^^^^^^
The ``libLineProfile`` library currently contains the following functions:
.. index:: ExpRlx
**Exponential relaxation**
::
userFcn libBNMR ExpRlx 1 2
The parameters are:
#. pulse length :math:`t_0` (ms)
#. relaxation rate :math:`\sigma` (ms\ :math:`^{-1}`\ )
This function implements :math:`f(t)=e^{-\sigma t}`.
.. index:: SExpRlx
**Stretched exponential relaxation**
::
userFcn libBNMR SExpRlx 1 2 3
The parameters are:
#. pulse length :math:`t_0` (ms)
#. relaxation rate :math:`\sigma` (ms\ :math:`^{-1}`\ )
#. stretching exponent :math:`\beta`
This function implements :math:`f(t)=e^{-(\sigma t)^{\beta}}`.
.. index:: libLineProfile
libLineProfile
+++++++++++++++++
In addition to some simple line shapes ``libLineProfile`` contains functions to fit chemical shift anisotropies in the powder average.
Their functional form can be found in `M. Mehring, Principles of High Resolution NMR in Solids (Springer 1983) <http://dx.doi.org/10.1007/978-3-642-68756-3_2>`_.
For an axially symmetric interaction it is given by:
.. index:: Iax
.. _Iax:
.. math::
I_{\mathrm ax}(f)=\left\{\begin{matrix} \frac{1}{2\sqrt{(f_\parallel-f_\perp)(f-f_\perp)}}& f\in(f_\perp,f_\parallel)\cup(f_\parallel,f_\perp)\\[6pt] 0 & \text{otherwise}\end{matrix} \right.
where :math:`f_\parallel` and :math:`f_\perp` are the frequencies that would be observed if the field is oriented paralell or perpendicular to the symmetry axis, respectively.
| In case of a completely anisotropic interaction, the powder average can be described by the frequencies along the three principle axis :math:`f_1,f_2,f_3`.
| Assume without loss of generality that :math:`f_1<f_2<f_3`, then
.. index:: Ianiso
.. _Ianiso:
.. math::
I(f)&=\left\{\begin{matrix}
\frac{K(m)}{\pi\sqrt{(f-f_1)(f_3-f_2)}},& f_3\geq f>f_2 \\[9pt]
\frac{K(m)}{\pi\sqrt{(f_3-f)(f_2-f_1)}},& f_2>f\geq f_1\\[9pt]
0 & \text{otherwise}
\end{matrix} \right. \\
\\
m&=\left\{\begin{matrix}
\frac{(f_2-f_1)(f_3-f)}{(f_3-f_2)(f-f_1)},& f_3\geq f>f_2 \\[6pt]
\frac{(f-f_1)(f_3-f_2)}{(f_3-f)(f_2-f_1)},& f_2>f\geq f_1\\[6pt]
\end{matrix} \right. \\
\\
K(m)&=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},
:math:`K(m)` is the complete elliptic integral of the first kind.
Functions
^^^^^^^^^^^^
The ``libLineProfile`` library currently contains the following functions:
.. index:: LineGauss
**Gaussian**
::
userFcn libLineProfile LineGauss 1 2
The parameters are:
#. center of the line :math:`f_0`
#. FWHM of the line :math:`\sigma`
| The height of the peak is 1.
| The functional form is given by
.. math::
A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}}
.. index:: LineLorentzian
**Lorentzian**
::
userFcn libLineProfile LineLorentzian 1 2
The parameters are:
#. center of the line :math:`f_0`
#. FWHM of the line :math:`w`
| The height of the peak is 1.
| The functional form is given by
.. math::
A(f)= \frac{w^2}{4(f-f_0)^2+w^2}
.. index:: LineLaplace
**Laplacian**
::
userFcn libLineProfile LineLaplace 1 2
The parameters are:
#. center of the line :math:`f_0`
#. FWHM of the line :math:`w`
| The height of the peak is 1.
| The functional form is given by
.. math::
A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{w}\right|}
.. index:: LineSkewLorentzian
**Skewed Lorentzian**
::
userFcn libLineProfile LineSkewLorentzian 1 2 3
The parameters are:
#. center of the line :math:`f_0`
#. width of the line :math:`w`
#. skewness parameter :math:`a`
| The height of the peak is 1.
| The functional form is given by
.. math::
A(f)= \frac{w w_a}{4(f-f_0)^2+w_a^2}, \quad w_a=\frac{2w}{1+e^{a(f-f_0)}}
.. index:: LineSkewLorentzian2
**Skewed Lorentzian 2**
::
userFcn libLineProfile LineSkewLorentzian2 1 2 3
The parameters are:
#. center of the line :math:`f_0`
#. width left of the center :math:`w_1`
#. width right of the center :math:`w_2`
| The height of the peak is 1.
| The functional form is given by
.. math::
A(f)= \left\{\begin{matrix}\frac{{w_1}^2}{4{(f-f_0)}^2+{w_1}^2},&f\leq f_0\\[9pt] \frac{{w_2}^2}{4{(f-f_0)}^2+{w_2}^2},&f>f_0\end{matrix}\right.
.. index:: PowderLineAxialLor
**Powder average of an axially symmetric interaction convoluted with a Lorentzian**
::
userFcn libLineProfile PowderLineAxialLor 1 2 3
The parameters are:
#. frequency for the field oriented paralell to the symmetry axis :math:`f_\parallel`
#. frequency for the field oriented perpendicular to the symmetry axis :math:`f_\parallel`
#. FWHM of the Lorentzian :math:`w`
| The height of the peak is :math:`\sim`\ 1.
| The functional form is given by
.. math::
A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)
with :math:`I_{\mathrm ax}(f)` defined :ref:`above <Iax>`.
.. index:: PowderLineAxialGss
**Powder average of an axially symmetric interaction convoluted with a Gaussian**
::
userFcn libLineProfile PowderLineAxialGss 1 2 3
The parameters are:
#. frequency for the field oriented paralell to the symmetry axis :math:`f_\parallel`
#. frequency for the field oriented perpendicular to the symmetry axis :math:`f_\parallel`
#. FWHM of the Gaussian :math:`\sigma`
| The height of the peak is :math:`\sim`\ 1.
| The functional form is given by
.. math::
A(f)= I_{\mathrm ax}(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)
with :math:`I_{\mathrm ax}(f)` defined :ref:`above <Iax>`.
.. index:: PowderLineAsymLor
**Powder average of an anisotropic interaction convoluted with a Lorentzian**
::
userFcn libLineProfile PowderLineAsymLor 1 2 3 4
The parameters are:
#. :math:`f_1`
#. :math:`f_1`
#. :math:`f_3` frequencies along the principal axes
#. FWHM of the Lorentzian :math:`w`
| The height of the peak is :math:`\sim`\ 1.
| The functional form is given by
.. math::
A(f)= I(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)
with :math:`I(f)` defined :ref:`above <Ianiso>`. Note that :math:`f_1<f_2<f_3` is not required by the code.
.. index:: PowderLineAsymGss
**Powder average of an anisotropic interaction convoluted with a Gaussian**
::
userFcn libLineProfile PowderLineAsymGss 1 2 3 4
The parameters are:
#. :math:`f_1`
#. :math:`f_1`
#. :math:`f_3` frequencies along the principal axes
#. FWHM of the Gaussian :math:`\sigma`
| The height of the peak is :math:`\sim`\ 1.
| The functional form is given by
.. math::
A(f)= I(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)
with :math:`I(f)` defined :ref:`above <Ianiso>`. Note that :math:`f_1<f_2<f_3` is not required by the code.