1 | r""" |
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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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7 | Porod behavior $(I \sim q^{-4})$ at large $q$, $(qL \gg 1)$. |
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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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11 | Definition |
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12 | ---------- |
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13 | |
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14 | .. math:: |
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15 | |
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16 | I(q) = \text{scale}\cdot\frac{L^3}{(1 + (q\cdot L)^2)^2} + \text{background} |
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17 | |
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18 | where scale is |
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19 | |
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20 | .. math:: \text{scale} = 8 \pi \phi (1-\phi) \Delta\rho^2 |
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21 | |
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22 | and the parameter $L$ is the correlation length. |
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23 | |
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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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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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30 | References |
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31 | ---------- |
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32 | |
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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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35 | |
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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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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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42 | import numpy as np |
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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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49 | F(q)= scale * L^3/(1 + (q*L)^2)^2 |
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50 | |
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51 | L: the correlation length |
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52 | |
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53 | """ |
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54 | category = "shape-independent" |
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55 | |
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56 | # ["name", "units", default, [lower, upper], "type", "description"], |
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57 | parameters = [["cor_length", "Ang", 50.0, [0, inf], "", "correlation length"], |
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58 | ] |
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59 | |
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60 | Iq = """ |
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61 | double numerator = cube(cor_length); |
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62 | double denominator = square(1 + square(q*cor_length)); |
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63 | |
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64 | return numerator / denominator ; |
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65 | """ |
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66 | |
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67 | def random(): |
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68 | """Return a random parameter set for the model.""" |
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69 | pars = dict( |
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70 | scale=10**np.random.uniform(1, 4), |
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71 | cor_length=10**np.random.uniform(0.3, 3), |
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72 | #background = 0, |
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73 | ) |
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74 | pars['scale'] /= pars['cor_length']**3 |
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75 | return pars |
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76 | |
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77 | demo = dict(scale=1, background=0, cor_length=50) |
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