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update of the docu including the SECTOR cmd.

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:tocdepth: 3
.. include:: <isogrk1.txt>
.. index:: user-libs
.. _user-libs:
Documentation of user libs (user functions)
===========================================
.. index:: BMW-libs
.. _BMW-libs:
Meissner-Profiles / Vortex-Lattice related functions (BMW libs)
---------------------------------------------------------------
.. index:: libFitPofB
libFitPofB
++++++++++
Introduction
^^^^^^^^^^^^
``libFitPofB`` 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``. The classes contained in this
library generally implement calculations of one-dimensional static magnetic field distributions
:math:`p(B)` which lead to the muon-spin depolarization functions
.. math::
{\cal P}(t) = \int p(B) \cos(\gamma_\mu B t + \varphi) dB,
where :math:`\gamma_\mu = 2 \pi \times 135.54` MHz/T is the gyromagnetic ratio of the muon and :math:`\varphi`
is the initial phase of the muon spins with respect to the positron detector. At the moment the only available
implementations deal with field distributions measured in local isotropic superconductors, either by means of
low-energy |mgr|\SR (see `<https://www.psi.ch/smus/lem>`_) in the Meissner state or by bulk |mgr|\SR in the mixed state.
In the following the basic usage of the library in ``musrfit`` is explained—the calculations by themselves are only
outlined. For further information please refer to the original literature and/or the source code of the implementation.
.. note::
In order to supply certain information needed for the calculations but not suited to be stored in the ``musrfit``
msr files an ``XML`` configuration file in the working directory is used. For details, see below.
.. note::
The implementations in this library heavily rely on `FFTW3 <http://fftw.org/>`_. In principle, it always checks what
is the best way to do efficient Fourier transforms for a given machine before the transforms are actually done. If
repeatedly Fourier transforms of the same (sizable) length should be done, it might be worth storing the once
obtained information in an external file and just load it the next time this information is needed
(`wisdom handling <http://fftw.org/fftw3_doc/Wisdom.html>`_). In case this feature shall be used, a valid wisdom
file has to be specified in the ``XML`` file.
.. note::
The model functions described in the following do generally *not behave nicely* in conjunction with ``MINUIT``
function minimizations (or maximizations). The analysis process at the moment in most cases involves some
tedious trial-and-error procedure, where the displayed MINUIT information as always deserves attention.
This is especially true if small effects should be analyzed (*e.g.* small diamagnetic shifts in superconductors).
The parameter uncertainty in many cases has to be estimated independently. Due to these limitations, also
the use of the fit option of ``msr2data`` *cannot* be advised.
.. note::
If these classes still prove useful and results obtained through them are part of scientific publications,
an acknowledgment of the use of the library is appreciated.
LE-|mgr|\SR
^^^^^^^^^^^
.. index:: 1D-London-Meissner
One-dimensional London model for the Meissner state of isotropic superconductors
""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
The models for analyzing LE-|mgr|\SR data assume the magnetic induction :math:`B(z)` to vary only in the
dimension parallel to the momentum of the incident muons. In such a case the magnetic field distribution is given by
.. math::
p(B) = n(z) \left| \frac{dB(z)}{dz} \right|^{-1}
where :math:`n(z)` is the muon implantation profile simulated by ``TRIM.SP``.
Assuming an array of *N* isotropic local superconductors with a total thickness *d* in the Meissner state
the magnetic induction is given by solving the 1D London equation
.. math::
\frac{\partial^2}{\partial z^2}B_i(z) = \frac{1}{\lambda_i^2}B_i(z)
for each layer *i* taking into account the boundary conditions (F. London, Superfluids: Macroscopic Theory of Superconductivity, Dover (1961), p. 34)
.. math::
B_1(0) = B_N(d) = \mu_0H
B_i(d_i) = B_{i+1}(d_i)
\lambda_i^2B_i'(z)\Big\vert_{z=d_i} = \lambda_{i+1}^2B_{i+1}'(z)\Big\vert_{z=d_i},
where the :math:`d_i` specify the interfaces between two adjacent layers and :math:`\lambda_i` is
the magnetic field penetration depth in the constituent :math:`i`.
The calculation of the field distribution has been set up for a superconducting half-space as well
as superconducting thin films with up to three superconducting layers with different penetration depths.
The muon-spin depolarization functions are calculated using the following lines in the ``THEORY`` block
of a ``musrfit`` msr file:
.. index:: TLondon1DHS
**Superconducting half-space**
::
userFcn libFitPofB TLondon1DHS 1 2 3 4 5
The parameters are:
#. phase (deg)
#. muon implantation energy as specified in the :ref:`XML startup <BMWlibs-XML>` file (keV)
#. applied field (G)
#. thickness of the dead layer (nm)
#. magnetic field penetration depth (nm)
.. index:: TLondon1D1L
**Superconducting thin film (one layer)**
::
userFcn libFitPofB TLondon1D1L 1 2 3 4 5 6 [a b]
The mandatory parameters are:
#. phase (deg)
#. muon implantation energy as specified in the :ref:`XML startup <BMWlibs-XML>` file (keV)
#. applied field (G)
#. thickness of the dead layer (nm)
#. thickness of the actually superconducting layer (nm)
#. magnetic field penetration depth (nm)
The optional parameters are:
a. fraction f\ :sub:`1` of muons in the thin film contributing to the signal (0 ≤ f\ :sub:`1` ≤ 1)
b. fraction f\ :sub:`s` of muons in the substrate contributing to the signal (0 ≤ f\ :sub:`s` ≤ 1)
.. index:: TLondon1D2L
**Superconducting thin-film bilayer heterostructure**
::
userFcn libFitPofB TLondon1D2L 1 2 3 4 5 6 7 8 [a b c]
The mandatory parameters are:
#. phase (deg)
#. muon implantation energy as specified in the :ref:`XML startup <BMWlibs-XML>` file (keV)
#. applied field (G)
#. thickness of the dead layer (nm)
#. thickness of the actually superconducting first layer (nm)
#. thickness of the actually superconducting second layer (nm)
#. magnetic field penetration depth of the first layer (nm)
#. magnetic field penetration depth of the second layer (nm)
The optional parameters are:
a. fraction f\ :sub:`1` of muons in the dead and first layer contributing to the signal (0 ≤ f\ :sub:`1` ≤ 1)
b. fraction f\ :sub:`2` of muons in the second layer contributing to the signal (0 ≤ f\ :sub:`2` ≤ 1)
c. fraction f\ :sub:`s` of muons in the substrate contributing to the signal (0 ≤ f\ :sub:`s` ≤ 1)
.. index:: TLondon1D3L
**Superconducting thin-film trilayer heterostructure**
::
userFcn libFitPofB TLondon1D3L 1 2 3 4 5 6 7 8 9 10 [a b c d]
The mandatory parameters are:
#. phase (deg)
#. muon implantation energy as specified in the :ref:`XML startup <BMWlibs-XML>` file (keV)
#. applied field (G)
#. thickness of the dead layer (nm)
#. thickness of the actually superconducting first layer (nm)
#. thickness of the actually superconducting second layer (nm)
#. thickness of the actually superconducting third layer (nm)
#. magnetic field penetration depth of the first layer (nm)
#. magnetic field penetration depth of the second layer (nm)
#. magnetic field penetration depth of the third layer (nm)
The optional parameters are:
a. fraction f\ :sub:`1` of muons in the dead and first layer contributing to the signal (0 ≤ f\ :sub:`1` ≤ 1)
b. fraction f\ :sub:`2` of muons in the second layer contributing to the signal (0 ≤ f\ :sub:`2` ≤ 1)
c. fraction f\ :sub:`3` of muons in the third layer contributing to the signal (0 ≤ f\ :sub:`3` ≤ 1)
d. fraction f\ :sub:`s` of muons in the substrate contributing to the signal (0 ≤ f\ :sub:`s` ≤ 1)
Bulk |mgr|\SR
^^^^^^^^^^^^^
.. index:: Vortex-State-Isotropic
Field distributions in the mixed state of isotropic superconductors
"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
When investigating superconductors in the mixed state by means of conventional |mgr|\SR a
two-dimensional flux-line lattice is probed randomly by the muons. The spatial field
distributions within such an ordered lattice are modeled using the Fourier series
.. math::
B(\mathbf{r}) = \langle B \rangle \sum\limits_{\mathbf{K}}B_{\mathbf{K}}\exp(-\imath\mathbf{K}\mathbf{r}),
where :math:`\mathbf{r}=(x,y)`, **K** are the reciprocal lattice vectors of a two-dimensional
vortex lattice and the :math:`B_{\mathbf{K}}` are the Fourier coefficients depending on the
magnetic penetration depth :math:`\lambda` and the superconducting coherence length :math:`\xi`.
The :math:`B_{\mathbf{K}}` for some specific models are as follows:
**London model with Gaussian cutoff** (E.H. Brandt, `J. Low Temp. Phys. 73, 355 (1988) <http://dx.doi.org/10.1007/BF00683568>`_.)
.. math::
B_{\mathbf{K}} = \frac{\exp\left({-K^2\xi^2/2}\right)}{1 + K^2\lambda^2}
**Modified London model** (T.M. Riseman *et al.*, `Phys. Rev. B 52, 10569 (1995) <http://dx.doi.org/10.1103/PhysRevB.52.10569>`_.)
.. math::
B_{\mathbf{K}} = \frac{\exp\left({-K^2\xi^2/2(1-b)}\right)}{1 + K^2\lambda^2/(1-b)},
where :math:`b = \langle B \rangle / (\mu_0 H_{\rm c2})`.
**Analytical Ginzburg-Landau model** ( A. Yaouanc, P. Dalmas de Réotier and E.H. Brandt, `Phys. Rev. B 55, 11107 (1997) <http://dx.doi.org/10.1103/PhysRevB.55.11107>`_)
.. math::
B_{\mathbf{K}} = \frac{f_{\infty}K_1\left(\frac{\xi_v}{\lambda}\sqrt{f_{\infty}^2+\lambda^2K^2}\right)}{K_1\left(\frac{\xi_v}{\lambda}f_{\infty}\right)\sqrt{f_{\infty}^2+\lambda^2K^2}},
where :math:`f_{\infty} = 1 - b^4,~\xi_v = \xi\left(\sqrt{2}-{3\xi}/\left({4\lambda}\right)\right)\sqrt{(1+b^4)(1-2b(1-b)^2)}` and
:math:`K_1` is a modified Bessel function.
Apart from the mentioned analytic models the **numerical Ginzburg-Landau model** (`E.H. Brandt, Phys. Rev. B 68, 054506 (2003). <http://dx.doi.org/10.1103/PhysRevB.68.054506>`_) is available. In this case :math:`B(\mathbf{r})` is obtained by an iterative minimization of the free energy of the vortex lattice.
**Concerning the applicability (e.g. field regions) of each of the mentioned models please refer to the original publications!**
At the moment, the calculation of the field distribution has been implemented for *triangular* flux-line lattices.
The number of grid lines in which the inter-vortex distance is divided for the calculations to be specified through
the :ref:`XML startup <BMWlibs-XML>`.
The muon-spin depolarization functions finally are calculated using the following lines in the THEORY block of a ``musrfit`` msr file:
.. index:: Vortex-Gaussian-CutOff
**2D triangular vortex lattice, London model with Gaussian cutoff**
::
userFcn libFitPofB TBulkTriVortexLondon 1 2 3 4
The parameters are:
#. phase (deg)
#. mean magnetic induction (G)
#. magnetic penetration depth (nm)
#. Ginzburg-Landau coherence length (nm)
.. index:: Vortex-London-modified
**2D triangular vortex lattice, modified London model**
::
userFcn libFitPofB TBulkTriVortexML 1 2 3 4
The parameters are:
#. phase (deg)
#. mean magnetic induction (G)
#. magnetic penetration depth (nm)
#. Ginzburg-Landau coherence length (nm)
.. index:: Vortex-Analytic-GL
**2D triangular vortex lattice, analytic Ginzburg-Landau model**
::
userFcn libFitPofB TBulkTriVortexAGL 1 2 3 4
The parameters are:
#. phase (deg)
#. mean magnetic induction (G)
#. magnetic penetration depth (nm)
#. Ginzburg-Landau coherence length (nm)
.. index:: Vortex-Numeric-GL
**2D triangular vortex lattice, numerical Ginzburg-Landau model**
::
userFcn libFitPofB TBulkTriVortexNGL 1 2 3 4
The parameters are:
#. phase (deg)
#. mean magnetic induction (G)
#. magnetic penetration depth (nm)
#. Ginzburg-Landau coherence length (nm)
.. note::
In order to improve the convergence of ``MIGRAD`` it has proven useful to use the log-likelihood
maximization instead of the :math:`\chi^2` minimization routines and to choose sufficiently large
initial steps for the parameters. Calling ``MINOS`` in conjunction with these functions is futile.
Therefore, the :ref:`COMMANDS block <msr-commands-block>` of the msr file could look like:
::
COMMANDS
STRATEGY 2
MAX_LIKELIHOOD
MIGRAD
HESSE
SAVE
.. index:: BMWlibs-XML
.. _BMWlibs-XML:
The XML startup file
^^^^^^^^^^^^^^^^^^^^
``BMW_startup.xml`` is a configuration file located in the working directory. In this file some settings
like the time and field resolution of the calculations as well as the present muon implantation profiles
for a LE-|mgr|\SR analysis have to be defined. The following XML tags are allowed to define settings:
**<debug>ONE_OR_ZERO</debug>**
activate the debugging output of the settings read from the XML file by setting 1, deactivate it with 0.
**<wisdom>PATH_TO_FILE</wisdom>**
specify the ``PATH_TO_FILE`` to an `FFTW3 wisdom file <http://fftw.org/fftw3_doc/Wisdom.html#Wisdom>`_
that should be used; if the ``PATH_TO_FILE`` is invalid, no ``FFTW3`` wisdom will be used.
**<delta_t>ResT</delta_t>**
set the time resolution ``ResT`` for the calculated depolarization function in microseconds.
**<delta_B>ResB</delta_B>**
set the field resolution ``ResB`` for the calculated field distribution in Gauss.
**<VortexLattice></VortexLattice>**
set the parameters used for the calculation of the spatial field distribution of a vortex lattice.
**<N_VortexGrid>N</N_VortexGrid>**
specify the number of points **N** (in each of the two dimensions) for which the fields within the
vortex lattice are calculated (inside a **<VortexLattice>** environment)
**<LEM></LEM>**
set the parameters used for the calculation of LE-|mgr|\SR field distributions
**<data_path>DATA_PATH_PREFIX</data_path>**
specify the ``DATA_PATH_PREFIX`` to the ``TRIM.SP`` implantation profiles (inside a **<LEM>** environment)
**<N_theory>N_THEORY</N_theory>**
specify the number of points **N_THEORY** for which *B(z)* is calculated (inside a **<LEM>** environment)
The specification of this number is not needed if the calculation of the inverse of *B(z)* is implemented!
**<energy_list></energy_list>**
set the energies for which ``TRIM.SP`` implantation profiles are available (inside a **<LEM>** environment)
**<energy_label>LABEL</energy_label>**
specify the **LABEL** within the file name of a available ``TRIM.SP`` ``RGE`` file (inside a **<energy_list>** environment)
The expected name of the ``RGE`` file will be: ``DATA_PATH_PREFIX + LABEL + .rge``
**<energy>E</energy>**
specify the muon energy *E* (in keV) belonging to the ``TRIM.SP`` ``RGE`` file given above (inside a **<energy_list>** environment)
An example XML file looks as follows:
.. code-block:: xml
<?xml version="1.0" encoding="UTF-8"?>
<BMW>
<debug>0</debug>
<wisdom>/home/user/WordsOfWisdom.dat</wisdom>
<delta_t>0.01</delta_t>
<delta_B>0.5</delta_B>
<VortexLattice>
<N_VortexGrid>1024</N_VortexGrid>
</VortexLattice>
<LEM>
<data_path>/home/user/TrimSP/some-sample-</data_path>
<N_theory>5000</N_theory>
<energy_list>
<energy_label>02_0</energy_label>
<energy>2.0</energy>
<energy_label>03_0</energy_label>
<energy>3.0</energy>
<energy_label>03_6</energy_label>
<energy>3.6</energy>
<energy_label>05_0</energy_label>
<energy>5.0</energy>
<energy_label>05_3</energy_label>
<energy>5.3</energy>
</energy_list>
</LEM>
</BMW>
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 ``libBNMR`` library currently contains the following functions:
.. index:: ExpRlx
**Exponential relaxation**
::
userFcn libBNMR ExpRlx 1 2
The parameters are:
#. pulse length :math:`t_0` (s)
#. relaxation rate :math:`\lambda` (s\ :math:`^{-1}`\ )
This function implements :math:`f(t)=e^{-\lambda t}`.
.. index:: SExpRlx
**Stretched exponential relaxation**
::
userFcn libBNMR SExpRlx 1 2 3
The parameters are:
#. pulse length :math:`t_0` (s)
#. relaxation rate :math:`\lambda` (s\ :math:`^{-1}`\ )
#. stretching exponent :math:`\beta`
This function implements :math:`f(t)=e^{-(\lambda 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.