Changeset e46dde6 in sasmodels


Ignore:
Timestamp:
Jun 12, 2018 5:39:34 AM (4 years ago)
Author:
Adam Washington <adam.washington@…>
Branches:
master
Children:
02e3c54
Parents:
38935ec
Message:

Start writing up log_sesans theory

File:
1 edited

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  • doc/guide/sesans/sans_to_sesans.rst

    rf0fc507 re46dde6  
    3131 
    3232in which :math:`t` is the thickness of the sample and :math:`\lambda` is the wavelength of the neutrons. 
     33 
     34Log Spaced SESANS 
     35----------------- 
     36 
     37For computational efficiency, the integral in the Hankel transform is 
     38converted into a Reimann sum 
     39 
     40 
     41.. math:: G(\delta) \approx 
     42          2 \pi 
     43          \sum_{Q=q_{min}}^{q_{max}} J_0(Q \delta) 
     44          \frac{d \Sigma}{d \Omega} (Q) 
     45          Q \Delta Q \! 
     46 
     47However, this model approximates more than is strictly necessary. 
     48Specifically, it is approximating the entire integral, when it is only 
     49the scattering function that cannot be handled analytically.  A better 
     50approximation might be 
     51 
     52.. math:: G(\delta) \approx 
     53          \sum_{n=0} 2 \pi \frac{d \Sigma}{d \Omega} (q_n) 
     54          \int_{q_n}^{q_{n+1}} J_0(Q \delta) Q dQ 
     55          = 
     56          \sum_{n=0} \frac{2 \pi}{\delta} \frac{d \Sigma}{d \Omega} (q_n) 
     57          (q_{n+1}J_1(q_{n+1} \delta) - q_{n}J_1(q_{n} \delta))\!, 
     58 
     59Assume that vectors :math:`q_n` and :math:`I_n` represent the q points 
     60and corresponding intensity data, respectively.  Further assume that 
     61:math:`\delta_m` and :math:`G_m` are the spin echo lengths and 
     62corresponding Hankel transform value. 
     63 
     64.. math:: G_m = H_{nm} I_n 
     65 
     66where 
     67 
     68.. math:: H_{nm} = \frac{2 \pi}{\delta_m} 
     69          (q_{n+1} J_1(q_{n+1} \delta_m) - q_n J_1(q_n \delta_m)) 
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