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