| 1 | .. _lamellarPC: |
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| 2 | |
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| 3 | Lamellarpc |
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| 4 | ======================================================= |
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| 5 | |
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| 6 | Random lamellar sheet with paracrystal structure factor |
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| 7 | |
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| 8 | ================ ================================= ============ ============= |
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| 9 | Parameter Description Units Default value |
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| 10 | ================ ================================= ============ ============= |
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| 11 | scale Source intensity None 1 |
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| 12 | background Source background |cm^-1| 0 |
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| 13 | thickness sheet thickness |Ang| 33 |
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| 14 | Nlayers Number of layers None 20 |
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| 15 | spacing d-spacing of paracrystal stack |Ang| 250 |
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| 16 | spacing_polydisp d-spacing of paracrystal stack |Ang| 0 |
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| 17 | sld layer scattering length density |1e-6Ang^-2| 1 |
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| 18 | solvent_sld Solvent scattering length density |1e-6Ang^-2| 6.34 |
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| 19 | ================ ================================= ============ ============= |
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| 20 | |
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| 21 | The returned value is scaled to units of |cm^-1|. |
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| 22 | |
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| 23 | |
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| 24 | This model calculates the scattering from a stack of repeating lamellar |
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| 25 | structures. The stacks of lamellae (infinite in lateral dimension) are |
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| 26 | treated as a paracrystal to account for the repeating spacing. The repeat |
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| 27 | distance is further characterized by a Gaussian polydispersity. **This model |
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| 28 | can be used for large multilamellar vesicles.** |
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| 29 | |
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| 30 | *2.1.33.1. Definition* |
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| 31 | |
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| 32 | The scattering intensity *I(q)* is calculated as |
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| 33 | |
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| 34 | .. image:: img/image145.jpg |
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| 35 | |
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| 36 | The form factor of the bilayer is approximated as the cross section of an |
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| 37 | infinite, planar bilayer of thickness *t* |
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| 38 | |
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| 39 | .. image:: img/image146.jpg |
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| 40 | |
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| 41 | Here, the scale factor is used instead of the mass per area of the |
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| 42 | bilayer (*G*). The scale factor is the volume fraction of the material in |
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| 43 | the bilayer, *not* the total excluded volume of the paracrystal. |
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| 44 | *Z*\ :sub:`N`\ *(q)* describes the interference effects for aggregates |
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| 45 | consisting of more than one bilayer. The equations used are (3-5) |
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| 46 | from the Bergstrom reference below. |
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| 47 | |
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| 48 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 49 | the lower and higher values |
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| 50 | |
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| 51 | .. image:: img/image147.jpg |
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| 52 | |
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| 53 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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| 54 | of the *q* vector which is defined as |
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| 55 | |
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| 56 | .. math:: |
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| 57 | |
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| 58 | Q = \sqrt{Q_x^2 + Q_y^2} |
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| 59 | |
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| 60 | The parameters of the model are *Nlayers* = no. of layers, and |
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| 61 | *pd_spacing* = polydispersity of spacing. |
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| 62 | |
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| 63 | ============== ======== ============= |
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| 64 | Parameter name Units Default value |
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| 65 | ============== ======== ============= |
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| 66 | background |cm^-1| 0 |
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| 67 | scale None 1 |
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| 68 | Nlayers None 20 |
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| 69 | pd_spacing None 0.2 |
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| 70 | sld_layer |Ang^-2| 1e-6 |
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| 71 | sld_solvent |Ang^-2| 6.34e-6 |
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| 72 | spacing |Ang| 250 |
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| 73 | thickness |Ang| 33 |
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| 74 | ============== ======== ============= |
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| 75 | |
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| 76 | .. image:: img/image148.jpg |
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| 77 | |
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| 78 | *Figure. 1D plot using the default values above (w/20000 data point).* |
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| 79 | |
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| 80 | Our model uses the form factor calculations implemented in a C library |
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| 81 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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| 82 | |
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| 83 | REFERENCE |
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| 84 | |
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| 85 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, |
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| 86 | *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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| 87 | |
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| 88 | |
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