[19b6d28] | 1 | r""" |
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| 2 | This model fits the Porod function |
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| 3 | |
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[40a87fa] | 4 | .. math:: I(q) = C/q^4 |
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[19b6d28] | 5 | |
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| 6 | to the data directly without any need for linearisation (cf. Log I(q) vs Log q). |
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| 7 | |
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[40a87fa] | 8 | Here $C = 2\pi (\Delta\rho)^2 S_v$ is the scale factor where $S_v$ is |
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| 9 | the specific surface area (ie, surface area / volume) of the sample, and |
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| 10 | $\Delta\rho$ is the contrast factor. |
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[19b6d28] | 11 | |
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[cc3fac6] | 12 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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[16bb433] | 13 | where the q vector is defined as |
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[19b6d28] | 14 | |
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[40a87fa] | 15 | .. math:: q = \sqrt{q_x^2+q_y^2} |
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[19b6d28] | 16 | |
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[2f63032] | 17 | References |
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| 18 | ---------- |
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| 19 | |
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| 20 | G Porod. *Kolloid Zeit*. 124 (1951) 83. |
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[40a87fa] | 21 | |
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| 22 | L A Feigin, D I Svergun, G W Taylor. *Structure Analysis by Small-Angle |
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| 23 | X-ray and Neutron Scattering*. Springer. (1987) |
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[19b6d28] | 24 | """ |
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| 25 | |
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[2c74c11] | 26 | from numpy import power, inf, errstate |
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[19b6d28] | 27 | |
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| 28 | name = "porod" |
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| 29 | title = "Porod function" |
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| 30 | description = """\ |
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| 31 | I(q) = scale/q^4 + background |
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| 32 | """ |
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| 33 | |
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| 34 | category = "shape-independent" |
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| 35 | |
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| 36 | parameters = [] |
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| 37 | |
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| 38 | def Iq(q): |
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| 39 | """ |
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| 40 | @param q: Input q-value |
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| 41 | """ |
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[82923a6] | 42 | with errstate(divide='ignore'): |
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[4962519] | 43 | return q**-4 |
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[19b6d28] | 44 | |
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| 45 | Iq.vectorized = True # Iq accepts an array of q values |
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| 46 | |
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[232bb12] | 47 | def random(): |
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| 48 | import numpy as np |
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| 49 | sld, solvent = np.random.uniform(-0.5, 12, size=2) |
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| 50 | radius = 10**np.random.uniform(1, 4.7) |
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| 51 | Vf = 10**np.random.uniform(-3, -1) |
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| 52 | scale = 1e-4 * Vf * 2*np.pi*(sld-solvent)**2/(3*radius) |
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| 53 | pars = dict( |
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| 54 | scale=scale, |
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| 55 | ) |
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| 56 | return pars |
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| 57 | |
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[19b6d28] | 58 | demo = dict(scale=1.5, background=0.5) |
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| 59 | |
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[82923a6] | 60 | tests = [ |
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| 61 | [{'scale': 0.00001, 'background':0.01}, 0.04, 3.916250], |
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| 62 | [{}, 0.0, inf], |
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| 63 | ] |
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