[80768fc] | 1 | r""" |
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| 2 | |
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[f224873] | 3 | Calculates the scattering from a randomly distributed, two-phase system based on |
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| 4 | the Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system |
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| 5 | is characterized by a single length scale, the correlation length, which is a |
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| 6 | measure of the average spacing between regions of phase 1 and phase 2. **The |
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| 7 | model also assumes smooth interfaces between the phases** and hence exhibits |
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[eb69cce] | 8 | Porod behavior $(I \sim q^{-4})$ at large $q$, $(qL \gg 1)$. |
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[80768fc] | 9 | |
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| 10 | The DAB model is ostensibly a development of the earlier Debye-Bueche model. |
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| 11 | |
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[eb69cce] | 12 | Definition |
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| 13 | ---------- |
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| 14 | |
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| 15 | .. math:: |
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[80768fc] | 16 | |
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[eb69cce] | 17 | I(q) = \text{scale}\cdot\frac{L^3}{(1 + (q\cdot L)^2)^2} + \text{background} |
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[80768fc] | 18 | |
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| 19 | where scale is |
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| 20 | |
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[eb69cce] | 21 | .. math:: \text{scale} = 8 \pi \phi (1-\phi) \Delta\rho^2 |
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[80768fc] | 22 | |
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[eb69cce] | 23 | and the parameter $L$ is the correlation length. |
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[f224873] | 24 | |
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[eb69cce] | 25 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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| 26 | where the $q$ vector is defined as |
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[f224873] | 27 | |
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| 28 | .. math:: q = \sqrt{q_x^2 + q_y^2} |
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| 29 | |
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| 30 | .. figure:: img/dab_1d.jpg |
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| 31 | |
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| 32 | 1D plot using the default values (w/200 data point). |
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| 33 | |
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| 34 | |
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[eb69cce] | 35 | References |
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| 36 | ---------- |
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[80768fc] | 37 | |
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[f224873] | 38 | P Debye, H R Anderson, H Brumberger, *Scattering by an Inhomogeneous Solid. II. |
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| 39 | The Correlation Function and its Application*, *J. Appl. Phys.*, 28(6) (1957) 679 |
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[80768fc] | 40 | |
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[f224873] | 41 | P Debye, A M Bueche, *Scattering by an Inhomogeneous Solid*, *J. Appl. Phys.*, |
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| 42 | 20 (1949) 518 |
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[80768fc] | 43 | |
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| 44 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
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| 45 | |
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| 46 | """ |
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| 47 | |
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| 48 | from numpy import inf |
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| 49 | |
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| 50 | name = "dab" |
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| 51 | title = "DAB (Debye Anderson Brumberger) Model" |
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| 52 | description = """\ |
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| 53 | |
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[f224873] | 54 | F(q)= scale * L^3/(1 + (q*L)^2)^2 |
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[80768fc] | 55 | |
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| 56 | L: the correlation length |
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| 57 | |
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| 58 | """ |
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[a5d0d00] | 59 | category = "shape-independent" |
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[80768fc] | 60 | |
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[3e428ec] | 61 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 62 | parameters = [["length", "Ang", 50.0, [0, inf], "", "correlation length"], |
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| 63 | ] |
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[80768fc] | 64 | |
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| 65 | Iq = """ |
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| 66 | double numerator = pow(length, 3); |
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| 67 | double denominator = pow(1 + pow(q*length,2), 2); |
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| 68 | |
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| 69 | return numerator / denominator ; |
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| 70 | """ |
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| 71 | |
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| 72 | Iqxy = """ |
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| 73 | // never called since no orientation or magnetic parameters. |
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| 74 | //return -1.0; |
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| 75 | return Iq(sqrt(qx*qx + qy*qy), length); |
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| 76 | """ |
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| 77 | |
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| 78 | # ER defaults to 1.0 |
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| 79 | |
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| 80 | # VR defaults to 1.0 |
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| 81 | |
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[3e428ec] | 82 | demo = dict(scale=1, background=0, length=50) |
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[80768fc] | 83 | oldname = "DABModel" |
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| 84 | oldpars = dict(length='length') |
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