Changeset ccbbc3b in sasmodels for doc/guide/sesans/sans_to_sesans.rst
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 Mar 28, 2019 12:41:49 PM (5 months ago)
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doc/guide/sesans/sans_to_sesans.rst
rd7af1c6 rccbbc3b 31 31 32 32 in which :math:`t` is the thickness of the sample and :math:`\lambda` is the wavelength of the neutrons. 33 34 Log Spaced SESANS35 36 37 For computational efficiency, the integral in the Hankel transform is38 converted into a Reimann sum39 40 41 .. math:: G(\delta) \approx42 2 \pi43 \sum_{Q=q_{min}}^{q_{max}} J_0(Q \delta)44 \frac{d \Sigma}{d \Omega} (Q)45 Q \Delta Q \!46 47 However, this model approximates more than is strictly necessary.48 Specifically, it is approximating the entire integral, when it is only49 the scattering function that cannot be handled analytically. A better50 approximation might be51 52 .. math:: G(\delta) \approx53 \sum_{n=0} 2 \pi \frac{d \Sigma}{d \Omega} (q_n)54 \int_{q_{n1}}^{q_n} J_0(Q \delta) Q dQ55 =56 \sum_{n=0} \frac{2 \pi}{\delta} \frac{d \Sigma}{d \Omega} (q_n)57 (q_n J_1(q_n \delta)  q_{n1}J_1(q_{n1} \delta))\!,58 59 Assume that vectors :math:`q_n` and :math:`I_n` represent the q points60 and corresponding intensity data, respectively. Further assume that61 :math:`\delta_m` and :math:`G_m` are the spin echo lengths and62 corresponding Hankel transform value.63 64 .. math:: G_m = H_{nm} I_n65 66 where67 68 .. math:: H_{nm} = \frac{2 \pi}{\delta_m}69 (q_n J_1(q_n \delta_m)  q_{n1} J_1(q_{n1} \delta_m))70 71 Also not that, for the limit as :math:`\delta_m` approaches zero,72 73 .. math:: G(0)74 =75 \sum_{n=0} \pi \frac{d \Sigma}{d \Omega} (q_n) (q_n^2  q_{n1}^2)
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