1 | #mono_gauss_coil model |
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2 | #conversion of DebyeModel.py |
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3 | #converted by Steve King, Mar 2016 |
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4 | r""" |
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5 | This Debye Gaussian coil model strictly describes the scattering from |
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6 | *monodisperse* polymer chains in theta solvents or polymer melts, conditions |
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7 | under which the distances between segments follow a Gaussian distribution. |
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8 | Provided the number of segments is large (ie, high molecular weight polymers) |
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9 | the single-chain form factor P(Q) is that described by Debye (1947). |
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10 | |
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11 | To describe the scattering from *polydisperse* polymer chains see the |
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12 | :ref:`poly-gauss-coil` model. |
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13 | |
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14 | Definition |
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15 | ---------- |
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16 | |
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17 | .. math:: |
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18 | |
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19 | I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background} |
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20 | |
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21 | where |
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22 | |
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23 | .. math:: |
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24 | |
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25 | I_0 &= \phi_\text{poly} \cdot V |
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26 | \cdot (\rho_\text{poly} - \rho_\text{solv})^2 \\ |
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27 | P(q) &= 2 [\exp(-Z) + Z - 1] / Z^2 \\ |
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28 | Z &= (q R_g)^2 \\ |
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29 | V &= M / (N_A \delta) |
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30 | |
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31 | Here, $\phi_\text{poly}$ is the volume fraction of polymer, $V$ is the |
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32 | volume of a polymer coil, *M* is the molecular weight of the polymer, |
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33 | $N_A$ is Avogadro's Number, $\delta$ is the bulk density of the polymer, |
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34 | $\rho_\text{poly}$ is the sld of the polymer, $\rho\text{solv}$ is the |
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35 | sld of the solvent, and $R_g$ is the radius of gyration of the polymer coil. |
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36 | |
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37 | The 2D scattering intensity is calculated in the same way as the 1D, |
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38 | but where the *q* vector is redefined as |
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39 | |
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40 | .. math:: |
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41 | |
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42 | q = \sqrt{q_x^2 + q_y^2} |
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43 | |
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44 | References |
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45 | ---------- |
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46 | |
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47 | P Debye, *J. Phys. Colloid. Chem.*, 51 (1947) 18. |
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48 | |
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49 | R J Roe, *Methods of X-Ray and Neutron Scattering in Polymer Science*, |
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50 | Oxford University Press, New York (2000). |
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51 | |
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52 | http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf |
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53 | """ |
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54 | |
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55 | import numpy as np |
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56 | from numpy import inf |
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57 | |
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58 | name = "mono_gauss_coil" |
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59 | title = "Scattering from monodisperse polymer coils" |
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60 | |
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61 | description = """ |
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62 | Evaluates the scattering from |
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63 | monodisperse polymer chains. |
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64 | """ |
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65 | category = "shape-independent" |
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66 | |
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67 | # pylint: disable=bad-whitespace, line-too-long |
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68 | # ["name", "units", default, [lower, upper], "type", "description"], |
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69 | parameters = [ |
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70 | ["i_zero", "1/cm", 70.0, [0.0, inf], "", "Intensity at q=0"], |
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71 | ["rg", "Ang", 75.0, [0.0, inf], "volume", "Radius of gyration"], |
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72 | ] |
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73 | # pylint: enable=bad-whitespace, line-too-long |
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74 | |
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75 | source = ["mono_gauss_coil.c"] |
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76 | have_Fq = False |
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77 | effective_radius_type = ["R_g", "2R_g", "3R_g", "sqrt(5/3)*R_g"] |
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78 | |
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79 | |
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80 | def random(): |
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81 | rg = 10**np.random.uniform(0, 4) |
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82 | #rg = 1e3 |
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83 | pars = dict( |
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84 | #scale=1, background=0, |
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85 | i_zero=1e7, # i_zero is a simple scale |
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86 | rg=rg, |
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87 | ) |
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88 | return pars |
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89 | |
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90 | demo = dict(scale=1.0, i_zero=70.0, rg=75.0, background=0.0) |
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91 | |
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92 | # these unit test values taken from SasView 3.1.2 |
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93 | tests = [ |
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94 | [{'scale': 1.0, 'i_zero': 70.0, 'rg': 75.0, 'background': 0.0}, |
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95 | [0.0106939, 0.469418], [57.1241, 0.112859]], |
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96 | ] |
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