[80768fc] | 1 | r""" |
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[f224873] | 2 | Calculates the scattering from a randomly distributed, two-phase system based on |
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| 3 | the Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system |
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| 4 | is characterized by a single length scale, the correlation length, which is a |
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| 5 | measure of the average spacing between regions of phase 1 and phase 2. **The |
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| 6 | model also assumes smooth interfaces between the phases** and hence exhibits |
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[eb69cce] | 7 | Porod behavior $(I \sim q^{-4})$ at large $q$, $(qL \gg 1)$. |
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[80768fc] | 8 | |
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| 9 | The DAB model is ostensibly a development of the earlier Debye-Bueche model. |
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| 10 | |
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[eb69cce] | 11 | Definition |
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| 12 | ---------- |
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| 13 | |
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| 14 | .. math:: |
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[80768fc] | 15 | |
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[eb69cce] | 16 | I(q) = \text{scale}\cdot\frac{L^3}{(1 + (q\cdot L)^2)^2} + \text{background} |
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[80768fc] | 17 | |
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| 18 | where scale is |
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| 19 | |
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[eb69cce] | 20 | .. math:: \text{scale} = 8 \pi \phi (1-\phi) \Delta\rho^2 |
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[80768fc] | 21 | |
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[eb69cce] | 22 | and the parameter $L$ is the correlation length. |
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[f224873] | 23 | |
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[eb69cce] | 24 | For 2D data the 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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[f224873] | 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 | |
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[eb69cce] | 30 | References |
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| 31 | ---------- |
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[80768fc] | 32 | |
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[f224873] | 33 | P Debye, H R Anderson, H Brumberger, *Scattering by an Inhomogeneous Solid. II. |
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| 34 | The Correlation Function and its Application*, *J. Appl. Phys.*, 28(6) (1957) 679 |
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[80768fc] | 35 | |
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[f224873] | 36 | P Debye, A M Bueche, *Scattering by an Inhomogeneous Solid*, *J. Appl. Phys.*, |
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| 37 | 20 (1949) 518 |
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[80768fc] | 38 | |
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| 39 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
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| 40 | """ |
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| 41 | |
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[2d81cfe] | 42 | import numpy as np |
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[80768fc] | 43 | from numpy import inf |
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| 44 | |
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| 45 | name = "dab" |
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| 46 | title = "DAB (Debye Anderson Brumberger) Model" |
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| 47 | description = """\ |
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| 48 | |
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[f224873] | 49 | F(q)= scale * L^3/(1 + (q*L)^2)^2 |
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[80768fc] | 50 | |
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| 51 | L: the correlation length |
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| 52 | |
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| 53 | """ |
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[a5d0d00] | 54 | category = "shape-independent" |
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[80768fc] | 55 | |
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[3e428ec] | 56 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[a807206] | 57 | parameters = [["cor_length", "Ang", 50.0, [0, inf], "", "correlation length"], |
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[3e428ec] | 58 | ] |
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[80768fc] | 59 | |
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| 60 | Iq = """ |
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[4962519] | 61 | double numerator = cube(cor_length); |
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| 62 | double denominator = square(1 + square(q*cor_length)); |
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[404ebbd] | 63 | |
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[80768fc] | 64 | return numerator / denominator ; |
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| 65 | """ |
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| 66 | |
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[404ebbd] | 67 | def random(): |
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| 68 | pars = dict( |
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| 69 | scale=10**np.random.uniform(1, 4), |
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| 70 | cor_length=10**np.random.uniform(0.3, 3), |
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| 71 | #background = 0, |
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| 72 | ) |
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| 73 | pars['scale'] /= pars['cor_length']**3 |
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| 74 | return pars |
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| 75 | |
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[80768fc] | 76 | # ER defaults to 1.0 |
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| 77 | |
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| 78 | # VR defaults to 1.0 |
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| 79 | |
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[a807206] | 80 | demo = dict(scale=1, background=0, cor_length=50) |
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