1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | This model fits the Guinier function |
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6 | |
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7 | .. math:: |
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8 | |
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9 | I(q) = \text{scale} \cdot \exp{\left[ \frac{-Q^2 R_g^2 }{3} \right]} |
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10 | + \text{background} |
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11 | |
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12 | to the data directly without any need for linearisation (*cf*. the usual |
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13 | plot of $\ln I(q)$ vs $q^2$\ ). Note that you may have to restrict the data |
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14 | range to include small q only, where the Guinier approximation actually |
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15 | applies. See also the guinier_porod model. |
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16 | |
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17 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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18 | where the $q$ vector is defined as |
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19 | |
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20 | .. math:: q = \sqrt{q_x^2 + q_y^2} |
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21 | |
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22 | In scattering, the radius of gyration $R_g$ quantifies the objects's |
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23 | distribution of SLD (not mass density, as in mechanics) from the objects's |
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24 | SLD centre of mass. It is defined by |
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25 | |
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26 | .. math:: R_g^2 = \frac{\sum_i\rho_i\left(r_i-r_0\right)^2}{\sum_i\rho_i} |
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27 | |
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28 | where $r_0$ denotes the object's SLD centre of mass and $\rho_i$ is the SLD at |
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29 | a point $i$. |
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30 | |
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31 | Notice that $R_g^2$ may be negative (since SLD can be negative), which happens |
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32 | when a form factor $P(Q)$ is increasing with $Q$ rather than decreasing. This |
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33 | can occur for core/shell particles, hollow particles, or for composite |
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34 | particles with domains of different SLDs in a solvent with an SLD close to the |
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35 | average match point. (Alternatively, this might be regarded as there being an |
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36 | internal inter-domain "structure factor" within a single particle which gives |
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37 | rise to a peak in the scattering). |
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38 | |
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39 | To specify a negative value of $R_g^2$ in SasView, simply give $R_g$ a negative |
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40 | value ($R_g^2$ will be evaluated as $R_g |R_g|$). Note that the physical radius |
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41 | of gyration, of the exterior of the particle, will still be large and positive. |
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42 | It is only the apparent size from the small $Q$ data that will give a small or |
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43 | negative value of $R_g^2$. |
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44 | |
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45 | References |
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46 | ---------- |
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47 | |
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48 | .. [#] A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley & Sons, New York (1955) |
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49 | |
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50 | Source |
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51 | ------ |
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52 | |
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53 | `guinier.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/guinier.py>`_ |
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54 | |
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55 | Authorship and Verification |
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56 | ---------------------------- |
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57 | |
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58 | * **Author:** |
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59 | * **Last Modified by:** |
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60 | * **Last Reviewed by:** |
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61 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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62 | """ |
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63 | |
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64 | import numpy as np |
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65 | from numpy import inf |
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66 | |
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67 | name = "guinier" |
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68 | title = "" |
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69 | description = """ |
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70 | I(q) = scale.exp ( - rg^2 q^2 / 3.0 ) |
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71 | |
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72 | List of default parameters: |
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73 | scale = scale |
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74 | rg = Radius of gyration |
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75 | """ |
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76 | category = "shape-independent" |
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77 | |
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78 | # ["name", "units", default, [lower, upper], "type","description"], |
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79 | parameters = [["rg", "Ang", 60.0, [-inf, inf], "", "Radius of Gyration"]] |
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80 | |
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81 | Iq = """ |
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82 | double exponent = fabs(rg)*rg*q*q/3.0; |
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83 | double value = exp(-exponent); |
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84 | return value; |
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85 | """ |
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86 | |
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87 | def random(): |
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88 | """Return a random parameter set for the model.""" |
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89 | scale = 10**np.random.uniform(1, 4) |
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90 | # Note: compare.py has Rg cutoff of 1e-30 at q=1 for guinier, so use that |
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91 | # log_10 Ae^(-(q Rg)^2/3) = log_10(A) - (q Rg)^2/ (3 ln 10) > -30 |
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92 | # => log_10(A) > Rg^2/(3 ln 10) - 30 |
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93 | q_max = 1.0 |
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94 | rg_max = np.sqrt(90*np.log(10) + 3*np.log(scale))/q_max |
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95 | rg = 10**np.random.uniform(0, np.log10(rg_max)) |
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96 | pars = dict( |
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97 | #background=0, |
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98 | scale=scale, |
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99 | rg=rg, |
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100 | ) |
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101 | return pars |
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102 | |
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103 | # parameters for demo |
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104 | demo = dict(scale=1.0, background=0.001, rg=60.0) |
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105 | |
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106 | # parameters for unit tests |
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107 | tests = [[{'rg' : 31.5}, 0.005, 0.992756]] |
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