.. _func-IkedaCarpenterPV: ================ IkedaCarpenterPV ================ .. index:: IkedaCarpenterPV Description ----------- This peakshape function is designed to be used to fit time-of-flight peaks. In particular this function is the convolution of the Ikeda-Carpenter function (Ref [1]), which aims to model the neutron pulse shape from a moderator, and a pseudo-Voigt that models any broadening to the peak due to sample properties etc. The convolution of the Ikeda-Carpenter function with psuedo-Voigt is (Ref [3]) .. math:: N \left[ (1-\eta)\Omega_G + \eta \Omega_L \right] where :math:`\Omega_G` and :math:`\Omega_L` are the Gaussian and Lorentzian parts of the function, respectively: .. math:: \Omega_G = N_u e^u erfc(y_u) + N_v e^v erfc(y_v) + N_s e^s erfc(y_s) + N_r e^r erfc(y_r) .. math:: \Omega_L = -\frac{2}{\pi} \left[ N_u Im[e^{z_u}E_1(z_u)] + N_v Im[e^{z_v}E_1(z_v)] + N_s Im[e^{z_s}E_1(z_s)] + N_r Im[e^{z_r}E_1(z_r)] \right] :math:`erfc` is the complementary error function, :math:`E_1` is the complex exponential integral, and :math:`Im` is the imaginary component. The parameters used above are defined as (from Panel 13 of Ref [3]): +------------------------------------+------------------------------------------------------------------------+--------------------------------------------------------------+------------------------------------------------------+---------------------------------------------------------------------+ | .. math:: k = 0.05 | .. math:: z_s = -\alpha dt + i\frac{1}{2} \alpha \gamma | .. math:: u = \frac{1}{2} \alpha^- (\alpha^- \sigma^2 - 2dt) | .. math:: N = \frac{1}{4} \alpha \frac{(1-k^2)}{k^2} | .. math:: y_u = \frac{ (\alpha^- \sigma^2 - dt) }{\sqrt{2\sigma^2}} | | .. math:: \alpha^- = \alpha(1 - k) | .. math:: z_u = -\alpha^- dt + i\frac{1}{2} \alpha^- \gamma = (1-k)z_s | .. math:: v = \frac{1}{2} \alpha^+ (\alpha^+ \sigma^2 - 2dt) | .. math:: N_u = 1 - R \frac{\alpha^-}{x} | .. math:: y_v = \frac{ (\alpha^+ \sigma^2 - dt) }{\sqrt{2\sigma^2}} | | .. math:: \alpha^+ = \alpha(1 + k) | .. math:: z_v = -\alpha^+ dt + i\frac{1}{2} \alpha^+ \gamma = (1+k)z_s | .. math:: s = \frac{1}{2} \alpha (\alpha \sigma^2 - 2dt) | .. math:: N_v = 1 - R \frac{\alpha^+}{z} | .. math:: y_s = \frac{ (\alpha \sigma^2 - dt) }{\sqrt{2\sigma^2}} | | .. math:: x = \alpha^- - \beta | .. math:: z_r = -\beta dt + i\frac{1}{2} \beta \gamma | .. math:: r = \frac{1}{2} \beta (\beta \sigma^2 - 2dt) | .. math:: N_s = -2(1 - R\frac{\alpha}{y}) | .. math:: y_r = \frac{ (\beta \sigma^2 - dt) }{\sqrt{2\sigma^2}} | | .. math:: y = \alpha - \beta | | | .. math:: N_r = 2R\alpha^2\beta \frac{k^2}{xyz} | | | .. math:: z = \alpha^+ - \beta | | | | | +------------------------------------+------------------------------------------------------------------------+--------------------------------------------------------------+------------------------------------------------------+---------------------------------------------------------------------+ where :math:`dt = T_i - T_h` is the shift in microseconds with respect to the Bragg position, :math:`\alpha` and :math:`\beta` are the fast and slow neutron decay constants respectively, and :math:`R` is a maxing coefficient that relates to the moderator temperature. :math:`\alpha` and :math:`R` are further modelled to depend on wavelength and using the notation in the Fullprof manual (Ref [2]). The refineable Ikeda-Carpenter parameters are Alpha0, Alpha1, Beta0 and Kappa and these are defined as .. math:: \alpha=1/(\mbox{Alpha0}+\lambda*\mbox{Alpha1}) .. math:: \beta = 1/\mbox{Beta0} .. math:: R = \exp (-81.799/(\mbox{Kappa}*\lambda^2)) , where :math:`\lambda` is the neutron wavelength. *In general when fitting a single peak it is not recommended to refine both Alpha0 and Alpha1 at the same time since these two parameters will effectively be 100% correlated because the wavelength over a single peak is likely effectively constant*. All parameters are constrained to be non-negative. The pseudo-Voigt function is defined as a linear combination of a Lorentzian and Gaussian and is a computational efficient way of calculation a Voigt function. The Voigt parameters are related to the pseudo-Voigt parameters through a relation (see Fullprof manual eq. (3.16) which in revision July2001 is missing a power 1/5). It is the two Voigt parameters which you can refine with this peakshape function: SigmaSquared (for the Gaussian part) and Gamma (for the Lorentzian part). Notice the Voigt Gaussian FWHM=SigmaSquared\*8\*ln(2) and the Voigt Lorentzian FWHM=Gamma. For information about how to create instrument specific values for the parameters of this fitting function see :ref:`CreateIkedaCarpenterParameters `. The implementation of the IkedaCarpenterPV peakshape function here follows the analytical expression for this function as presented in Panels 13-17 of Ref[3]. References: #. S. Ikeda and J. M. Carpenter, `Nuclear Inst. and Meth. in Phys. Res. A239, 536 (1985) `_ #. Fullprof manual, see http://www.ill.eu/sites/fullprof/ #. J. Rodriguez-Carvajal, `Using FullProf to analyze Time of Flight Neutron Powder Diffraction data `_ The figure below illustrate this peakshape function fitted to a TOF peak: .. figure:: /images/IkedaCarpenterPVwithBackground.png :alt: IkedaCarpenterPVwithBackground.png .. attributes:: .. properties:: .. categories:: .. sourcelink::