1 | #correlation length model |
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2 | # Note: model title and parameter table are inserted automatically |
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3 | r""" |
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4 | Definition |
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5 | ---------- |
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6 | |
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7 | The scattering intensity I(q) is calculated as |
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8 | |
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9 | .. math:: |
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10 | I(Q) = \frac{A}{Q^n} + \frac{C}{1 + (Q\xi)^m} + \text{background} |
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11 | |
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12 | The first term describes Porod scattering from clusters (exponent = $n$) and |
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13 | the second term is a Lorentzian function describing scattering from |
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14 | polymer chains (exponent = $m$). This second term characterizes the |
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15 | polymer/solvent interactions and therefore the thermodynamics. The two |
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16 | multiplicative factors $A$ and $C$, and the two exponents $n$ and $m$ are |
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17 | used as fitting parameters. (Respectively *porod_scale*, *lorentz_scale*, |
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18 | *porod_exp* and *lorentz_exp* in the parameter list.) The remaining |
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19 | parameter $\xi$ (*cor_length* in the parameter list) is a correlation |
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20 | length for the polymer chains. Note that when $m=2$ this functional form |
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21 | becomes the familiar Lorentzian function. Some interpretation of the |
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22 | values of $A$ and $C$ may be possible depending on the values of $m$ and $n$. |
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23 | |
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24 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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25 | where the q vector is defined as |
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26 | |
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27 | .. math:: q = \sqrt{q_x^2 + q_y^2} |
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28 | |
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29 | References |
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30 | ---------- |
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31 | |
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32 | B Hammouda, D L Ho and S R Kline, Insight into Clustering in |
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33 | Poly(ethylene oxide) Solutions, Macromolecules, 37 (2004) 6932-6937 |
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34 | """ |
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35 | |
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36 | from numpy import inf, errstate |
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37 | |
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38 | name = "correlation_length" |
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39 | title = """Calculates an empirical functional form for SAS data characterized |
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40 | by a low-Q signal and a high-Q signal.""" |
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41 | description = """ |
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42 | """ |
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43 | category = "shape-independent" |
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44 | # pylint: disable=bad-continuation, line-too-long |
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45 | # ["name", "units", default, [lower, upper], "type","description"], |
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46 | parameters = [ |
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47 | ["lorentz_scale", "", 10.0, [0, inf], "", "Lorentzian Scaling Factor"], |
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48 | ["porod_scale", "", 1e-06, [0, inf], "", "Porod Scaling Factor"], |
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49 | ["cor_length", "Ang", 50.0, [0, inf], "", "Correlation length, xi, in Lorentzian"], |
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50 | ["porod_exp", "", 3.0, [0, inf], "", "Porod Exponent, n, in q^-n"], |
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51 | ["lorentz_exp", "1/Ang^2", 2.0, [0, inf], "", "Lorentzian Exponent, m, in 1/( 1 + (q.xi)^m)"], |
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52 | ] |
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53 | # pylint: enable=bad-continuation, line-too-long |
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54 | |
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55 | def Iq(q, lorentz_scale, porod_scale, cor_length, porod_exp, lorentz_exp): |
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56 | """ |
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57 | 1D calculation of the Correlation length model |
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58 | """ |
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59 | with errstate(divide='ignore'): |
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60 | porod = porod_scale / q**porod_exp |
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61 | lorentz = lorentz_scale / (1.0 + (q * cor_length)**lorentz_exp) |
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62 | inten = porod + lorentz |
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63 | return inten |
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64 | Iq.vectorized = True |
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65 | |
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66 | # parameters for demo |
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67 | demo = dict(lorentz_scale=10.0, porod_scale=1.0e-06, cor_length=50.0, |
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68 | porod_exp=3.0, lorentz_exp=2.0, background=0.1, |
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69 | ) |
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70 | |
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71 | tests = [[{}, 0.001, 1009.98], |
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72 | [{}, 0.150141, 0.175645], |
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73 | [{}, 0.442528, 0.0213957]] |
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