| 1 | r""" |
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| 2 | This model calculates an empirical functional form for SAS data characterized |
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| 3 | by a broad scattering peak. Many SAS spectra are characterized by a broad peak |
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| 4 | even though they are from amorphous soft materials. For example, soft systems |
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| 5 | that show a SAS peak include copolymers, polyelectrolytes, multiphase systems, |
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| 6 | layered structures, etc. |
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| 7 | |
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| 8 | The d-spacing corresponding to the broad peak is a characteristic distance |
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| 9 | between the scattering inhomogeneities (such as in lamellar, cylindrical, or |
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| 10 | spherical morphologies, or for bicontinuous structures). |
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| 11 | |
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| 12 | Definition |
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| 13 | ---------- |
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| 14 | |
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| 15 | The scattering intensity $I(q)$ is calculated as |
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| 16 | |
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| 17 | .. math:: |
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| 18 | |
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| 19 | I(q) = \frac{A}{q^n} + \frac{C}{1 + (q\xi)^m} + B |
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| 20 | |
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| 21 | Here the peak position is related to the d-spacing as $q_o = 2\pi / d_o$. |
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| 22 | |
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| 23 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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| 24 | where the $q$ vector is defined as |
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| 25 | |
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| 26 | .. math:: |
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| 27 | |
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| 28 | q = \sqrt{q_x^2 + q_y^2} |
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| 29 | |
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| 30 | |
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| 31 | .. figure:: img/broad_peak_1d.jpg |
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| 32 | |
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| 33 | 1D plot using the default values (w/200 data point). |
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| 34 | |
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| 35 | References |
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| 36 | ---------- |
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| 37 | |
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| 38 | None. |
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| 39 | |
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| 40 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
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| 41 | |
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| 42 | """ |
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| 43 | |
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| 44 | from numpy import inf, sqrt |
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| 45 | |
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| 46 | name = "broad_peak" |
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| 47 | title = "Broad Lorentzian type peak on top of a power law decay" |
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| 48 | description = """\ |
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| 49 | I(q) = scale_p/pow(q,exponent)+scale_l/ |
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| 50 | (1.0 + pow((fabs(q-q_peak)*length_l),exponent_l) )+ background |
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| 51 | |
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| 52 | List of default parameters: |
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| 53 | porod_scale = Porod term scaling |
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| 54 | porod_exp = Porod exponent |
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| 55 | lorentz_scale = Lorentzian term scaling |
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| 56 | lorentz_length = Lorentzian screening length [A] |
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| 57 | peak_pos = peak location [1/A] |
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| 58 | lorentz_exp = Lorentzian exponent |
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| 59 | background = Incoherent background""" |
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| 60 | category = "shape-independent" |
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| 61 | |
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| 62 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 63 | parameters = [["porod_scale", "", 1.0e-05, [-inf, inf], "", "Power law scale factor"], |
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| 64 | ["porod_exp", "", 3.0, [-inf, inf], "", "Exponent of power law"], |
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| 65 | ["lorentz_scale", "", 10.0, [-inf, inf], "", "Scale factor for broad Lorentzian peak"], |
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| 66 | ["lorentz_length", "Ang", 50.0, [-inf, inf], "", "Lorentzian screening length"], |
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| 67 | ["peak_pos", "1/Ang", 0.1, [-inf, inf], "", "Peak postion in q"], |
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| 68 | ["lorentz_exp", "", 2.0, [-inf, inf], "", "exponent of Lorentz function"], |
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| 69 | ] |
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| 70 | |
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| 71 | def Iq(q, porod_scale, porod_exp, lorentz_scale, lorentz_length, peak_pos, lorentz_exp): |
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| 72 | inten = (porod_scale / q ** porod_exp + lorentz_scale |
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| 73 | / (1.0 + (abs(q - peak_pos) * lorentz_length) ** lorentz_exp)) |
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| 74 | return inten |
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| 75 | Iq.vectorized = True # Iq accepts an array of q values |
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| 76 | |
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| 77 | def Iqxy(qx, qy, *args): |
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| 78 | return Iq(sqrt(qx ** 2 + qy ** 2), *args) |
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| 79 | Iqxy.vectorized = True # Iqxy accepts an array of qx, qy values |
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| 80 | |
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| 81 | |
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| 82 | demo = dict(scale=1, background=0, |
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| 83 | porod_scale=1.0e-05, porod_exp=3, |
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| 84 | lorentz_scale=10, lorentz_length=50, peak_pos=0.1, lorentz_exp=2) |
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| 85 | |
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| 86 | oldname = "BroadPeakModel" |
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| 87 | oldpars = dict(porod_scale='scale_p', porod_exp='exponent_p', |
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| 88 | lorentz_scale='scale_l', lorentz_length='length_l', peak_pos='q_peak', |
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| 89 | lorentz_exp='exponent_l', scale=None) |
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