1 | r""" |
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2 | |
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3 | Scattering model class for the DAB (Debye Anderson Brumberger) Model |
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4 | |
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5 | Definition |
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6 | ---------- |
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7 | |
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8 | Calculates the scattering from a randomly distributed, two-phase system based on |
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9 | the Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system |
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10 | is characterized by a single length scale, the correlation length, which is a |
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11 | measure of the average spacing between regions of phase 1 and phase 2. **The |
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12 | model also assumes smooth interfaces between the phases** and hence exhibits |
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13 | Porod behavior (I ~ *q*\ :sup:`-4`) at large *q* (*QL* >> 1). |
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14 | |
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15 | The DAB model is ostensibly a development of the earlier Debye-Bueche model. |
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16 | |
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17 | *Definition* |
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18 | |
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19 | .. math:: I(q) = \text{scale}\cdot\frac{L^3}{(1 + (q \cdot L)^2)^2} + \text{background} |
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20 | |
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21 | where scale is |
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22 | |
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23 | .. math:: \text{scale} = 8 \pi \phi (1-\phi)(\Delta \rho)^2 |
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24 | |
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25 | and the parameter *L* is the correlation length. |
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26 | |
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27 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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28 | where the *q* vector is defined as |
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29 | |
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30 | .. math:: q = \sqrt{q_x^2 + q_y^2} |
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31 | |
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32 | ============== ======== ============= |
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33 | Parameter name Units Default value |
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34 | ============== ======== ============= |
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35 | scale None 1.0 |
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36 | corr length L |Ang| 50.0 |
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37 | background |cm^-1| 0.0 |
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38 | ============== ======== ============= |
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39 | |
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40 | |
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41 | .. figure:: img/dab_1d.jpg |
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42 | |
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43 | 1D plot using the default values (w/200 data point). |
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44 | |
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45 | |
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46 | Reference |
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47 | --------- |
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48 | |
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49 | P Debye, H R Anderson, H Brumberger, *Scattering by an Inhomogeneous Solid. II. |
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50 | The Correlation Function and its Application*, *J. Appl. Phys.*, 28(6) (1957) 679 |
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51 | |
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52 | P Debye, A M Bueche, *Scattering by an Inhomogeneous Solid*, *J. Appl. Phys.*, |
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53 | 20 (1949) 518 |
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54 | |
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55 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
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56 | |
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57 | """ |
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58 | |
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59 | from numpy import inf |
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60 | |
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61 | name = "dab" |
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62 | title = "DAB (Debye Anderson Brumberger) Model" |
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63 | description = """\ |
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64 | |
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65 | F(q)= scale * L^3/(1 + (q*L)^2)^2 |
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66 | |
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67 | L: the correlation length |
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68 | |
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69 | """ |
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70 | category = "shape-independent" |
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71 | |
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72 | # ["name", "units", default, [lower, upper], "type", "description"], |
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73 | parameters = [["length", "Ang", 50.0, [0, inf], "", "correlation length"], |
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74 | ] |
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75 | |
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76 | Iq = """ |
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77 | double numerator = pow(length, 3); |
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78 | double denominator = pow(1 + pow(q*length,2), 2); |
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79 | |
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80 | return numerator / denominator ; |
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81 | """ |
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82 | |
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83 | Iqxy = """ |
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84 | // never called since no orientation or magnetic parameters. |
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85 | //return -1.0; |
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86 | return Iq(sqrt(qx*qx + qy*qy), length); |
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87 | """ |
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88 | |
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89 | # ER defaults to 1.0 |
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90 | |
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91 | # VR defaults to 1.0 |
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92 | |
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93 | demo = dict(scale=1, background=0, length=50) |
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94 | oldname = "DABModel" |
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95 | oldpars = dict(length='length') |
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