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<div id="patternMainContents"><div class="patternTop"><span class="patternHomePath foswikiLeft"><a href="WebHome.html" class="foswikiCurrentWebHomeLink">MUSR</a><span class='natSep'> » </span><a href="MusrFit.html">MusrFit</a><span class='natSep'> » </span><a href="BmwLibs.html">BmwLibs</a><span class='natSep'> » </span><a href="LibZFRelaxation.html" class="foswikiCurrentTopicLink">LibZFRelaxation</a></span><span class="patternToolBar foswikiRight"><span><a href='LibZFRelaxationc99a.html?cover=print;cover=print' rel='nofollow' title='Printable version of this topic' accesskey='p'><span class='foswikiAccessKey'>P</span>rint version</a></span><a href="https://intranet.psi.ch/MUSR/LibZFRelaxation?restrictions=on" id="restriction_link"><img src='../pub/System/PsiSkin/lock_open.png' class='lock' height='16' width='23' title='open in PSI - click for more information'/></a></span><div class="foswikiClear"></div></div><div class="foswikiContentHeader">
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<a name="TopIc"></a>
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<h1><a name="libZFRelaxation"></a> libZFRelaxation </h1>
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<a name="foswikiTOC"></a><div class="foswikiToc"> <ul>
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<li> <a href="LibZFRelaxation.html#libZFRelaxation"> libZFRelaxation </a>
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</li> <li> <a href="LibZFRelaxation.html#A_1_Introduction"> 1 Introduction </a>
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</li> <li> <a href="LibZFRelaxation.html#A_2_Isotropic_static_Gaussian_broadened_randomly_oriented_internal_fields"> 2 Isotropic static Gaussian broadened randomly oriented internal fields </a>
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</li> <li> <a href="LibZFRelaxation.html#A_3_Isotropic_static_Lorentzian_broadened_randomly_oriented_internal_fields"> 3 Isotropic static Lorentzian broadened randomly oriented internal fields </a>
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</li> <li> <a href="LibZFRelaxation.html#A_4_Static_Gaussian_distributed_fields_with_uniaxial_anisotropy"> 4 Static Gaussian distributed fields with uniaxial anisotropy </a>
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</li></ul>
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<a name="IntroDuction"></a>
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<h1><a name="A_1_Introduction"></a> 1 Introduction </h1>
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<p></p>
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<code>libZFRelaxation</code> is a collection of <code>C++</code> classes using the <code><a href="MusrFit.html">musrfit</a></code> <code><a href="MusrFit.html#UserFunctions1">user-function interface</a></code> in order to facilitate the usage in conjunction with <code>musrfit</code>. The classes contained in this library implement the calculation of some additional zero-field μSR relaxation functions which are not built-in into <code>musrfit</code>.
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<p></p>
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In the following the use of the implemented functions is shortly introduced—for details on the applicability and derivation of these functions please refer to the original publications.
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<p></p>
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If these classes prove useful and results obtained through them are part of scientific publications,
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an acknowledgement of the use of the library is appreciated.
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<p></p>
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<h1><a name="A_2_Isotropic_static_Gaussian_broadened_randomly_oriented_internal_fields"></a> 2 Isotropic static Gaussian broadened randomly oriented internal fields </h1>
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<p></p>
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E. I. Kornilov and V. Yu. Pomjakushin, Phys. Lett. A <b>153</b>, 364–367 (1991), doi: <a href="http://dx.doi.org/10.1016/0375-9601(91)90959-C" target="_top">10.1016/0375-9601(91)90959-C</a>
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<p></p>
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<p style="text-align:center">
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<img alt="\mathcal{P}(t) = \left(1-\alpha\right)\exp\left(-\frac{\sigma_{\mathrm{L}}^2t^2}{2}\right)+\alpha\left(\cos\left(2\pi\nu t\right)-\frac{\sigma_{\mathrm{T}}^2t}{2\pi\nu}\sin\left(2\pi\nu t\right)\right)\exp\left(-\frac{\sigma_{\mathrm{T}}^2t^2}{2}\right)" class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_fca91646b118061038b53ada5563c17c.png" title="ZFMagGss" />
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</p>
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<p></p>
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<code>musrfit</code> THEORY-block line: <pre>
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userFcn libZFRelaxation ZFMagGss 1 2 3 4
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</pre>
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The parameters are: <ol>
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<li> the precessing fraction α (= 2/3 in the article),
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</li> <li> the "internal frequency" ν (MHz),
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</li> <li> the "transverse depolarization rate" σ<sub>T</sub> (μs<sup>-1</sup>),
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</li> <li> the "longitudinal depolarization rate" σ<sub>L</sub> (μs<sup>-1</sup>) (= 0 in the article).
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</li></ol>
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<p></p>
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<h1><a name="A_3_Isotropic_static_Lorentzian_broadened_randomly_oriented_internal_fields"></a> 3 Isotropic static Lorentzian broadened randomly oriented internal fields </h1>
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<p></p>
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M. I. Larkin, Y. Fudamoto, I. M. Gat, A. Kinkhabwala, K. M. Kojima, G. M. Luke, J. Merrin, B. Nachumi, Y. J. Uemura, M. Azuma, T. Saito, and M. Takano, Physica B <b>289–290</b>, 153–156 (2000), doi: <a href="http://dx.doi.org/10.1016/S0921-4526(00)00337-9"" target="_top">10.1016/S0921-4526(00)00337-9</a>
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<p></p>
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<p style="text-align:center">
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<img alt="\mathcal{P}(t) = \left(1-\alpha\right)\exp\left(-a_{\mathrm{L}}t\right) + \alpha\left(\cos\left(2\pi\nu t\right)-\frac{a_{\mathrm{T}}}{2\pi\nu}\sin\left(2\pi\nu t\right)\right)\exp\left(-a_{\mathrm{T}}t\right)" class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_975f948b57199037b2b8bfb5e986b6a9.png" title="ZFMagExp" />
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</p>
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<p></p>
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<code>musrfit</code> THEORY-block line: <pre>
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userFcn libZFRelaxation ZFMagExp 1 2 3 4
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</pre>
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The parameters are: <ol>
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<li> the precessing fraction α (= 2/3 in the article),
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</li> <li> the "internal frequency" ν (MHz),
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</li> <li> the "transverse depolarization rate" a<sub>T</sub> (μs<sup>-1</sup>),
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</li> <li> the "longitudinal depolarization rate" a<sub>L</sub> (μs<sup>-1</sup>) (= 0 in the article).
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</li></ol>
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<p></p>
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<h1><a name="A_4_Static_Gaussian_distributed_fields_with_uniaxial_anisotropy"></a> 4 Static Gaussian distributed fields with uniaxial anisotropy </h1>
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<p></p>
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G. Solt, Hyperfine Interactions <b>96</b>, 167–175 (1995), doi: <a href="http://dx.doi.org/10.1007/BF02066280" target="_top">10.1007/BF02066280</a>
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<p></p>
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<p style="text-align:center">
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<img alt="\mathcal{P}(t) = \frac{1}{2}\sin^2\Theta + \left(\frac{3}{2}\cos^2\Theta-\frac{1}{2}\right)f_1\left(\epsilon\right) + \sin^2\Theta f_2(\epsilon, \sigma_1, t) + \left(\frac{3}{2}\cos^2\Theta-\frac{1}{2}\right)f_3\left(\epsilon, \sigma_1, t\right)," class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_4611c56f1556464dd8a6eaabbe070c05.png" title="UniaxialStatGssKT" />
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</p>
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where
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<p style="text-align:center">
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<img alt="\epsilon = \left(\frac{\sigma_1}{\sigma_3}\right)^2-1,\quad p(x) = 1+\epsilon x^2," class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_2b4a75902dcb0b8de5a662656d3b9066.png" title="epsilon_p(x)" />
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</p>
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<p style="text-align:center">
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<img alt="f_1\left(\epsilon\right) = 1 - \frac{\sqrt{1+\epsilon}}{\epsilon}\left(\sqrt{1+\epsilon}-\left\lbrace \begin{matrix} \left(\mathrm{asinh}\sqrt{\epsilon}\right)/\sqrt{\epsilon} \\ \left(\arcsin\sqrt{-\epsilon}\right)/\sqrt{-\epsilon} \end{matrix} \right\rbrace\right),\:\mathrm{for}\,\bigg\lbrace\begin{matrix}\epsilon > 0 \\ \epsilon < 0\end{matrix}\,," class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_5100b245762fc5c6dee8db3b05032a18.png" title="f1" />
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</p>
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<p style="text-align:center">
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<img alt="f_2\left(\epsilon, \sigma_1, t\right) = \sqrt{1+\epsilon}\int_0^1\mathrm{d}x\left(\frac{1}{p(x)^{3/2}}-\frac{\gamma_{\mu}^2\sigma_1^2t^2}{p(x)^{5/2}}\right)\exp\left(-\frac{\gamma_{\mu}^2\sigma_1^2t^2}{2\,p(x)}\right)," class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_0a70ac778f3ffbf2c93d46b23559d371.png" title="f2" />
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</p>
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and
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<p style="text-align:center">
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<img alt="f_3\left(\epsilon, \sigma_1, t\right) = \sqrt{1+\epsilon}\int_0^1\mathrm{d}x\left(1-x^2\right)\left(\frac{1}{p(x)^{3/2}}-\frac{\gamma_{\mu}^2\sigma_1^2t^2}{p(x)^{5/2}}\right)\exp\left(-\frac{\gamma_{\mu}^2\sigma_1^2t^2}{2\,p(x)}\right)." class="mmpImage" src="../pub/MUSR/LibZFRelaxation/_MathModePlugin_52c5f826fb399ae721887c5fb4e31de0.png" title="f3" />
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</p>
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<p></p>
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<code>musrfit</code> THEORY-block line: <pre>
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userFcn libZFRelaxation UniaxialStatGssKT 1 2 3
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</pre>
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The parameters are: <ol>
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<li> σ<sub>1</sub> (μs<sup>-1</sup>),
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</li> <li> σ<sub>3</sub> (μs<sup>-1</sup>),
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</li> <li> Θ (°).
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</li></ol>
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<p></p>
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-- <a href="http://www.fsf.org/register_form?referrer=8369" target="_top">BMW</a> & <a href="http://lmu.web.psi.ch/lem/group.html" target="_top">AS</a> </div>
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