Documentation of user libs (user functions)

Meissner-Profiles / Vortex-Lattice related functions (BMW libs)

libFitPofB

Introduction

libFitPofB is a collection of C++ classes using the musrfit 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 \(p(B)\) which lead to the muon-spin depolarization functions

\[{\cal P}(t) = \int p(B) \cos(\gamma_\mu B t + \varphi) dB,\]

where \(\gamma_\mu = 2 \pi \times 135.54\) MHz/T is the gyromagnetic ratio of the muon and \(\varphi\) 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 μSR (see https://www.psi.ch/smus/lem) in the Meissner state or by bulk μSR 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. 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). 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-μSR

One-dimensional London model for the Meissner state of isotropic superconductors

The models for analyzing LE-μSR data assume the magnetic induction \(B(z)\) 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

\[p(B) = n(z) \left| \frac{dB(z)}{dz} \right|^{-1}\]

where \(n(z)\) 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

\[\frac{\partial^2}{\partial z^2}B_i(z) = \frac{1}{\lambda_i^2}B_i(z)\]

for each layer i taking into account the boundary conditions (F. London, Superfluids: Macroscopic Theory of Superconductivity, Dover (1961), p. 34)

\[ \begin{align}\begin{aligned}B_1(0) = B_N(d) = \mu_0H\\B_i(d_i) = B_{i+1}(d_i)\\\lambda_i^2B_i'(z)\Big\vert_{z=d_i} = \lambda_{i+1}^2B_{i+1}'(z)\Big\vert_{z=d_i},\end{aligned}\end{align} \]

where the \(d_i\) specify the interfaces between two adjacent layers and \(\lambda_i\) is the magnetic field penetration depth in the constituent \(i\).

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:

Superconducting half-space

userFcn  libFitPofB TLondon1DHS 1 2 3 4 5

The parameters are:

1. phase (deg)
2. muon implantation energy as specified in the XML startup file (keV)
3. applied field (G)
4. thickness of the dead layer (nm)
5. magnetic field penetration depth (nm)

Superconducting thin film (one layer)

userFcn  libFitPofB TLondon1D1L 1 2 3 4 5 6 [a b]

The mandatory parameters are:

1. phase (deg)
2. muon implantation energy as specified in the XML startup file (keV)
3. applied field (G)
4. thickness of the dead layer (nm)
5. thickness of the actually superconducting layer (nm)
6. magnetic field penetration depth (nm)

The optional parameters are:

1. fraction f₁ of muons in the thin film contributing to the signal (0 ≤ f₁ ≤ 1)
2. fraction f_s of muons in the substrate contributing to the signal (0 ≤ f_s ≤ 1)

Superconducting thin-film bilayer heterostructure

userFcn  libFitPofB TLondon1D2L 1 2 3 4 5 6 7 8 [a b c]

The mandatory parameters are:

1. phase (deg)
2. muon implantation energy as specified in the XML startup file (keV)
3. applied field (G)
4. thickness of the dead layer (nm)
5. thickness of the actually superconducting first layer (nm)
6. thickness of the actually superconducting second layer (nm)
7. magnetic field penetration depth of the first layer (nm)
8. magnetic field penetration depth of the second layer (nm)

The optional parameters are:

1. fraction f₁ of muons in the dead and first layer contributing to the signal (0 ≤ f₁ ≤ 1)
2. fraction f₂ of muons in the second layer contributing to the signal (0 ≤ f₂ ≤ 1)
3. fraction f_s of muons in the substrate contributing to the signal (0 ≤ f_s ≤ 1)

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:

1. phase (deg)
2. muon implantation energy as specified in the XML startup file (keV)
3. applied field (G)
4. thickness of the dead layer (nm)
5. thickness of the actually superconducting first layer (nm)
6. thickness of the actually superconducting second layer (nm)
7. thickness of the actually superconducting third layer (nm)
8. magnetic field penetration depth of the first layer (nm)
9. magnetic field penetration depth of the second layer (nm)
10. magnetic field penetration depth of the third layer (nm)

The optional parameters are:

1. fraction f₁ of muons in the dead and first layer contributing to the signal (0 ≤ f₁ ≤ 1)
2. fraction f₂ of muons in the second layer contributing to the signal (0 ≤ f₂ ≤ 1)
3. fraction f₃ of muons in the third layer contributing to the signal (0 ≤ f₃ ≤ 1)
4. fraction f_s of muons in the substrate contributing to the signal (0 ≤ f_s ≤ 1)

Bulk μSR

Field distributions in the mixed state of isotropic superconductors

When investigating superconductors in the mixed state by means of conventional μSR 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

\[B(\mathbf{r}) = \langle B \rangle \sum\limits_{\mathbf{K}}B_{\mathbf{K}}\exp(-\imath\mathbf{K}\mathbf{r}),\]

where \(\mathbf{r}=(x,y)\), K are the reciprocal lattice vectors of a two-dimensional vortex lattice and the \(B_{\mathbf{K}}\) are the Fourier coefficients depending on the magnetic penetration depth \(\lambda\) and the superconducting coherence length \(\xi\). The \(B_{\mathbf{K}}\) for some specific models are as follows:

London model with Gaussian cutoff (E.H. Brandt, J. Low Temp. Phys. 73, 355 (1988).)

\[B_{\mathbf{K}} = \frac{\exp\left({-K^2\xi^2/2}\right)}{1 + K^2\lambda^2}\]

Modified London model (T.M. Riseman et al., Phys. Rev. B 52, 10569 (1995).)

\[B_{\mathbf{K}} = \frac{\exp\left({-K^2\xi^2/2(1-b)}\right)}{1 + K^2\lambda^2/(1-b)},\]

where \(b = \langle B \rangle / (\mu_0 H_{\rm c2})\).

Analytical Ginzburg-Landau model ( A. Yaouanc, P. Dalmas de Réotier and E.H. Brandt, Phys. Rev. B 55, 11107 (1997))

\[B_{\mathbf{K}} = \frac{f_{\infty}K_1\left(\frac{\xi_v}{\lambda}\sqrt{f_{\infty}^2+\lambda^2K^2}\right)}{K_1\left(\frac{\xi_v}{\lambda}f_{\infty}\right)\sqrt{f_{\infty}^2+\lambda^2K^2}},\]

where \(f_{\infty} = 1 - b^4,~\xi_v = \xi\left(\sqrt{2}-{3\xi}/\left({4\lambda}\right)\right)\sqrt{(1+b^4)(1-2b(1-b)^2)}\) and \(K_1\) 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).) is available. In this case \(B(\mathbf{r})\) 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 XML startup. The muon-spin depolarization functions finally are calculated using the following lines in the THEORY block of a musrfit msr file:

2D triangular vortex lattice, London model with Gaussian cutoff

userFcn  libFitPofB TBulkTriVortexLondon 1 2 3 4

The parameters are:

1. phase (deg)
2. mean magnetic induction (G)
3. magnetic penetration depth (nm)
4. Ginzburg-Landau coherence length (nm)

2D triangular vortex lattice, modified London model

userFcn  libFitPofB TBulkTriVortexML 1 2 3 4

The parameters are:

1. phase (deg)
2. mean magnetic induction (G)
3. magnetic penetration depth (nm)
4. Ginzburg-Landau coherence length (nm)

2D triangular vortex lattice, analytic Ginzburg-Landau model

userFcn  libFitPofB TBulkTriVortexAGL 1 2 3 4

The parameters are:

1. phase (deg)
2. mean magnetic induction (G)
3. magnetic penetration depth (nm)
4. Ginzburg-Landau coherence length (nm)

2D triangular vortex lattice, numerical Ginzburg-Landau model

userFcn  libFitPofB TBulkTriVortexNGL 1 2 3 4

The parameters are:

1. phase (deg)
2. mean magnetic induction (G)
3. magnetic penetration depth (nm)
4. 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 \(\chi^2\) minimization routines and to choose sufficiently large initial steps for the parameters. Calling MINOS in conjunction with these functions is futile.

Therefore, the COMMANDS block of the msr file could look like:

COMMANDS
STRATEGY 2
MAX_LIKELIHOOD
MIGRAD
HESSE
SAVE

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-μSR 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 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-μSR 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:

<?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>

Supeconducting Gap-Integrals to calculate \(1/\lambda^2\) vs \(T\)

The details about the various superconducting gap-integrals are found in the pdf-file GapIntegrals.pdf which can be found in the musrfit source under <musrfit-dir>/src/external/libGapIntegrals/.

Nonlocal superconductivity related Meissner screening functions (AS libs)

This library allows to calculate the magnetic field profile \(B(z)\) for nonlocal superconductors. For details see A. Suter, et al., PRB 72, 024506 (2005), and references therein.

The provided function calculates the muon spin polarization

\[P(t, E) = \int n(z, E)\, \cos(\gamma_\mu B(z) t + \phi) \, dz,\]

where \(B(z)\) is calculated in the limit of specular reflection. The corresponding user function is called as

userFcn  libPNL_PippardFitter PNL_PippardFitter 1 2 3 4 5 6 7 8 9

with the parameters

1. implantation energy in (keV).
2. reduced temperature \(t=T/T_c\).
3. thickness in (nm).
4. electron mean path, \(\ell\) in (nm).
5. superconducting coherence length, \(\xi\) in (nm).
6. London penetration length, \(\lambda_{\rm L}\) in (nm).
7. external magnetic field strength in (G).
8. the effective detector phase, \(\varphi\) in \((^\circ)\).
9. a “dead layer” thickness in (nm).

Typically this function needs to be multiplied by a Gaussian in order to take into account: nuclear dipole broadening, partial trapped flux, etc.

In order to find the muon stopping profile, \(n(z,E)\), needed for the calculation, the library needs to find the corresponding trimsp rge-files (muon stoppping profiles). For this the library reads at start-up the following xml-file (example):

<?xml version="1.0" encoding="UTF-8"?>
<nonlocal xmlns="http://nemu.web.psi.ch/musrfit/nonlocal">
    <comment>
        nonlocal_startup.xml
    </comment>
    <nonlocal_par>
        <fourier_points>262144</fourier_points>
    </nonlocal_par>
    <trim_sp>
        <data_path>./profiles/</data_path>
        <rge_fln_pre>Sn_E</rge_fln_pre>
        <energy_list>
            <energy>1000</energy>
            <energy>2000</energy>
            <energy>4000</energy>
            <energy>6000</energy>
            <energy>8000</energy>
            <energy>10000</energy>
            <energy>12000</energy>
            <energy>14100</energy>
            <energy>18000</energy>
            <energy>22000</energy>
            <energy>25000</energy>
            <energy>27300</energy>
        </energy_list>
    </trim_sp>
</nonlocal>

Here the number of Fourier points needed in the calculation can be defined (fourier_points). The trim_sp section contains all the information needed to load the proper muon stopping profiles. data_path is the path to the needed rge-files. rge_fln_pre is the rge-file prefix, and energy are all the energy tags. E.g. ./profile/Sn_E1000.rge would be the first muon stopping profile for an energy of \(E=1000\) (eV).

The name of the xml-file has to be nonlocal_startup.xml and needs to be placed in the directory where the analysis takes place, i.e. in the directory of all the msr-files.

Depth resolved information (AS libs)

A method to extract depth-resolved information from the implantation energy dependence of the experimental parameters in a low-energy muon spin spectroscopy experiment. For details see A. F. A. Simões, et al. Review of Scientific Instruments. 2020; 91(2): 023906 (7 pp.).

If you have a layered material (e.g. \(N\) layers), properties like the asymmetry might depend on the layer in which the muons are stopped. For instance there might be different probabilities for muonium formation depending on the material, charge transfer layers, etc.

Since the muon stopping distribution, \(n(z)\), has some finite range, these properties will be smeared out. For the following we define the stopping probability, \(p_{i}(E)\) for a finite slice \(i\), ranging from \(z \in [a, b]\) as

\[p_{i}(E) = \int_a^b n(z,E) \, dz\]

Furthermore it is assumes that there is a sharp transition between the layers of the property of interest, e.g. the diamagnetic fraction, \(f_i\). Hence the measured property as function of energy is

\[f(E) = \sum_{i=1}^N p_{i}(E) \cdot f_i.\]

Note

Currently it is recommended to read in the data in ASCII or DAT format as a non-μSR fit (fit type 8).

The user library for the depth profile analysis looks for a 3 layer material, assuming one is looking for the diamagnetic fraction, like

###############################################################
FITPARAMETER
#      Nr.   Name        Value     Step      Pos_Error  Boundaries
        1    f1          0.54540   0.00072     none        0       1
        2    f2          0.23957   0.00048     none
        3    f3          0.05615   0.00047     none        0       1
        4    x1          63.5000   0.0014      none
        5    x2          101.5001  0.0044      none

###############################################################
THEORY
userFcn  libPDepthProfile PDepthProfile 1 2 3 4 5

Here, \(f1\) is the diamagnetic fraction of the first layer, etc. \(x1\) is the thickness of the first layer, etc.

In order to find the muon stopping profile, \(n(z,E)\), needed for the calculation, the library needs to find the corresponding trimsp rge-files (muon stoppping profiles). For this the library reads at start-up the following xml-file (example):

<?xml version="1.0" encoding="UTF-8"?>
<depthProf xmlns="http://nemu.web.psi.ch/musrfit/depthProf">
    <comment>
        TrimSp information
    </comment>
    <trim_sp>
        <data_path>./TRIMSP/</data_path>
        <rge_fln_pre>SiO2_70nm2.0_30nm2.2_SiC_E</rge_fln_pre>
        <energy_vect start="1000" stop="22000" step="1000"/>
    </trim_sp>
</depthProf>

Functions to analyze β-NMR data (BNMR libs)

This is a collection of C++ classes using the musrfit 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-μSR fit (fit type 8).

libBNMR

In β-NMR the SLR is usually measured by implanting a pulse of \(^8\)Li with a length \(t_0\) into the sample. The asymmetry is measured both during the pulse and afterwards. For a a general spin relaxation function \(f(t)\) the time evolution of the asymmetry is then given by [Z. Salman, et al., PRL 96, 147601 (2006)]:

\[\begin{split}P(t) = \left\{\begin{matrix} \frac{\int_0^t e^{-(t-t')/\tau_{\mathrm{Li}}}f(t-t')dt'}{\int_0^t e^{-t'/\tau_{\mathrm{Li}}}dt' } & t\leq t_0\\[6pt] \frac{\int_0^{t_0}e^{-(t_0-t')/\tau_{\mathrm{Li}}}f(t-t')dt'}{\int_0^{t_0}e^{-t'/\tau_{\mathrm{Li}}}dt'} & t> t_0, \end{matrix}\right.\end{split}\]

where \(\tau_{\mathrm{Li}}=1.21\)s is the \(^8\)Li lifetime.

Functions

The libBNMR library currently contains the following functions:

Exponential relaxation

userFcn libBNMR ExpRlx 1 2

The parameters are:

1. pulse length \(t_0\) (s)
2. relaxation rate \(\lambda\) (s\(^{-1}\))

This function implements \(f(t)=e^{-\lambda t}\).

Stretched exponential relaxation

userFcn libBNMR SExpRlx 1 2 3

The parameters are:

1. pulse length \(t_0\) (s)
2. relaxation rate \(\lambda\) (s\(^{-1}\))
3. stretching exponent \(\beta\)

This function implements \(f(t)=e^{-(\lambda t)^{\beta}}\).

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).

For an axially symmetric interaction it is given by:

\[\begin{split}I_{\mathrm ax}(f)=\left\{\begin{matrix} \frac{1}{2\sqrt{(f_\parallel-f_\perp)(f-f_\perp)}}& f\in(f_\perp,f_\parallel)\cup(f_\parallel,f_\perp)\\[6pt] 0 & \text{otherwise}\end{matrix} \right.\end{split}\]

where \(f_\parallel\) and \(f_\perp\) 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 \(f_1,f_2,f_3\).

Assume without loss of generality that \(f_1<f_2<f_3\), then

\[\begin{split}I(f)&=\left\{\begin{matrix} \frac{K(m)}{\pi\sqrt{(f-f_1)(f_3-f_2)}},& f_3\geq f>f_2 \\[9pt] \frac{K(m)}{\pi\sqrt{(f_3-f)(f_2-f_1)}},& f_2>f\geq f_1\\[9pt] 0 & \text{otherwise} \end{matrix} \right. \\ \\ m&=\left\{\begin{matrix} \frac{(f_2-f_1)(f_3-f)}{(f_3-f_2)(f-f_1)},& f_3\geq f>f_2 \\[6pt] \frac{(f-f_1)(f_3-f_2)}{(f_3-f)(f_2-f_1)},& f_2>f\geq f_1\\[6pt] \end{matrix} \right. \\ \\ K(m)&=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\end{split}\]

\(K(m)\) is the complete elliptic integral of the first kind.

Functions

The libLineProfile library currently contains the following functions:

Gaussian

userFcn  libLineProfile LineGauss 1 2

The parameters are:

1. center of the line \(f_0\)
2. FWHM of the line \(\sigma\)

The height of the peak is 1.

The functional form is given by

\[A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}}\]

Lorentzian

userFcn  libLineProfile LineLorentzian 1 2

The parameters are:

1. center of the line \(f_0\)
2. FWHM of the line \(w\)

The height of the peak is 1.

The functional form is given by

\[A(f)= \frac{w^2}{4(f-f_0)^2+w^2}\]

Laplacian

userFcn  libLineProfile LineLaplace 1 2

The parameters are:

1. center of the line \(f_0\)
2. FWHM of the line \(w\)

The height of the peak is 1.

The functional form is given by

\[A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{w}\right|}\]

Skewed Lorentzian

userFcn  libLineProfile LineSkewLorentzian 1 2 3

The parameters are:

1. center of the line \(f_0\)
2. width of the line \(w\)
3. skewness parameter \(a\)

The height of the peak is 1.

The functional form is given by

\[A(f)= \frac{w w_a}{4(f-f_0)^2+w_a^2}, \quad w_a=\frac{2w}{1+e^{a(f-f_0)}}\]

Skewed Lorentzian 2

userFcn  libLineProfile LineSkewLorentzian2 1 2 3

The parameters are:

1. center of the line \(f_0\)
2. width left of the center \(w_1\)
3. width right of the center \(w_2\)

The height of the peak is 1.

The functional form is given by

\[\begin{split}A(f)= \left\{\begin{matrix}\frac{{w_1}^2}{4{(f-f_0)}^2+{w_1}^2},&f\leq f_0\\[9pt] \frac{{w_2}^2}{4{(f-f_0)}^2+{w_2}^2},&f>f_0\end{matrix}\right.\end{split}\]

Powder average of an axially symmetric interaction convoluted with a Lorentzian

userFcn  libLineProfile PowderLineAxialLor 1 2 3

The parameters are:

1. frequency for the field oriented paralell to the symmetry axis \(f_\parallel\)
2. frequency for the field oriented perpendicular to the symmetry axis \(f_\parallel\)
3. FWHM of the Lorentzian \(w\)

The height of the peak is \(\sim\)1.

The functional form is given by

\[A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)\]

with \(I_{\mathrm ax}(f)\) defined above.

Powder average of an axially symmetric interaction convoluted with a Gaussian

userFcn  libLineProfile PowderLineAxialGss 1 2 3

The parameters are:

1. frequency for the field oriented paralell to the symmetry axis \(f_\parallel\)
2. frequency for the field oriented perpendicular to the symmetry axis \(f_\parallel\)
3. FWHM of the Gaussian \(\sigma\)

The height of the peak is \(\sim\)1.

The functional form is given by

\[A(f)= I_{\mathrm ax}(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)\]

with \(I_{\mathrm ax}(f)\) defined above.

Powder average of an anisotropic interaction convoluted with a Lorentzian

userFcn  libLineProfile PowderLineAsymLor 1 2 3 4

The parameters are:

1. \(f_1\)
2. \(f_1\)
3. \(f_3\) frequencies along the principal axes
4. FWHM of the Lorentzian \(w\)

The height of the peak is \(\sim\)1.

The functional form is given by

\[A(f)= I(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)\]

with \(I(f)\) defined above. Note that \(f_1<f_2<f_3\) is not required by the code.

Powder average of an anisotropic interaction convoluted with a Gaussian

userFcn  libLineProfile PowderLineAsymGss 1 2 3 4

The parameters are:

1. \(f_1\)
2. \(f_1\)
3. \(f_3\) frequencies along the principal axes
4. FWHM of the Gaussian \(\sigma\)

The height of the peak is \(\sim\)1.

The functional form is given by

\[A(f)= I(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)\]

with \(I(f)\) defined above. Note that \(f_1<f_2<f_3\) is not required by the code.