[5054e80] | 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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[40a87fa] | 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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[a807206] | 18 | *porod_exp* and *lorentz_exp* in the parameter list.) The remaining |
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[40a87fa] | 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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[5054e80] | 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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[40a87fa] | 27 | .. math:: q = \sqrt{q_x^2 + q_y^2} |
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[5054e80] | 28 | |
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[aa2edb2] | 29 | References |
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| 30 | ---------- |
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[5054e80] | 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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[2c74c11] | 36 | from numpy import inf, errstate |
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[5054e80] | 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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[326281f] | 49 | ["cor_length", "Ang", 50.0, [0, inf], "", "Correlation length, xi, in Lorentzian"], |
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[a807206] | 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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[5054e80] | 52 | ] |
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| 53 | # pylint: enable=bad-continuation, line-too-long |
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| 54 | |
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[a807206] | 55 | def Iq(q, lorentz_scale, porod_scale, cor_length, porod_exp, lorentz_exp): |
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[5054e80] | 56 | """ |
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| 57 | 1D calculation of the Correlation length model |
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| 58 | """ |
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[2c74c11] | 59 | with errstate(divide='ignore'): |
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[a807206] | 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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[5054e80] | 62 | inten = porod + lorentz |
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| 63 | return inten |
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[2c74c11] | 64 | Iq.vectorized = True |
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[5054e80] | 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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[a807206] | 68 | porod_exp=3.0, lorentz_exp=2.0, background=0.1, |
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[5054e80] | 69 | ) |
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| 70 | |
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[3e6c5c1] | 71 | tests = [[{}, 0.001, 1009.98], |
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[6dd90c1] | 72 | [{}, 0.150141, 0.175645], |
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| 73 | [{}, 0.442528, 0.0213957]] |
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