| 1 | <body> |
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| 2 | <h4>Generic Scattering Calculator:</h4> |
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| 3 | Polarization and Magnetic Scattering |
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| 4 | <br> |
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| 5 | <br> |
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| 6 | <ul> |
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| 7 | <li><a href="#theory">Theory:</a></li> |
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| 8 | <li><a href="#gui">GUI</a></li> |
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| 9 | <li><a href="#pdb">PDB Data</a></li> |
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| 10 | </ul> |
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| 11 | <br> |
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| 12 | <br> |
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| 13 | <b>1. <a name="theory">Theory</a> </b> |
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| 14 | <br> |
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| 15 | In general, a particle with a volume V can be described by an ensemble containing N |
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| 16 | 3-dimensional rectangular pixels where each pixels are much smaller than V. |
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| 17 | Assuming that |
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| 18 | all the pixel sizes are same, the elastic scattering intensity |
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| 19 | by the particle |
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| 20 | <p> |
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| 21 | <img src="gen_i.png"/> |
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| 22 | </p> |
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| 23 | <br> |
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| 24 | where β<sub>j</sub> and r<sub>j</sub> are the scattering |
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| 25 | length density and the position |
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| 26 | of the j'th pixel respectively. And the total volume |
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| 27 | <p> |
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| 28 | <img src="v_j.png"/> |
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| 29 | </p> |
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| 30 | <br> |
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| 31 | for β<sub>j</sub> ≠ 0 where v<sub>j</sub> is the volume of the j'th pixel |
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| 32 | (or the j'th natural atomic volume (= atomic mass/natural molar density/Avogadro number) |
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| 33 | for the atomic structures). The total volume V can be corrected by users. |
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| 34 | This correction is useful especially for an atomic structure (taken from a pdb file) to get the right |
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| 35 | normalization. Note that the β<sub>j</sub> displayed in GUI may be incorrect but will not |
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| 36 | affect the scattering computation if the correction of the total volume is made. |
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| 37 | <br> |
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| 38 | The scattering length density (SLD) of each pixel where the SLD is uniform, is a combination of the nuclear and magnetic SLDs |
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| 39 | and depends on the spin states of the neutrons as follows: |
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| 40 | <br> |
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| 41 | <br> |
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| 42 | For magnetic scattering, only the magnetization component, <b>M</b><sub>perp</sub>, |
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| 43 | perpendicular to the scattering vector <b>Q</b> contributes to the the magnetic |
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| 44 | scattering length. (Figure below). |
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| 45 | <p> |
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| 46 | <img src="mag_vector.png"/> |
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| 47 | </p> |
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| 48 | <br> |
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| 49 | The magnetic scattering length density is then |
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| 50 | <p> |
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| 51 | <img src="dm_eq.png"/> |
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| 52 | </p> |
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| 53 | <br> |
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| 54 | where γ = -1.913 the gyromagnetic ratio, μ<sub>B</sub> is the Bohr magneton, |
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| 55 | r<sub>0</sub> is the classical radius of electron, |
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| 56 | and <b>σ</b> is the Pauli spin. |
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| 57 | <br> |
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| 58 | For polarized neutron, the magnetic scattering is depending on the spin states. |
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| 59 | Let's consider that the incident neutrons are polarized parallel (+)/anti-parallel |
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| 60 | (–) to the x' axis (See both Figures above). |
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| 61 | The possible out-coming states then are + and - states for both incident states. |
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| 62 | <br> |
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| 63 | - Non-spin-flips: (+ +) and (- -) |
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| 64 | <br> |
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| 65 | - Spin-flips: (+ -) and (- +) |
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| 66 | <br> |
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| 67 | <p> |
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| 68 | <img src="gen_mag_pic.png"/> |
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| 69 | </p> |
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| 70 | <br> |
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| 71 | <br> |
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| 72 | Now, let's assume that the angles of the <b>Q</b> vector and the spin-axis (x') from x-axis |
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| 73 | are φ and θ<sub>up</sub>, respectively (See Figure above). |
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| 74 | Then, depending upon the polarization (spin) state of neutrons, the scattering length |
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| 75 | densities , including the nuclear scattering length density (β <sub>N</sub>) are given as, for non-spin-flips, |
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| 76 | <p> |
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| 77 | <img src="sld1.png"/> |
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| 78 | </p> |
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| 79 | <br> |
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| 80 | <br> |
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| 81 | for spin-flips, |
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| 82 | <p> |
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| 83 | <img src="sld2.png"/> |
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| 84 | </p> |
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| 85 | <br> |
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| 86 | <br> |
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| 87 | where |
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| 88 | <p> |
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| 89 | <img src="mxp.png"/> |
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| 90 | </p> |
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| 91 | <p> |
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| 92 | <img src="myp.png"/> |
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| 93 | </p> |
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| 94 | <p> |
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| 95 | <img src="mzp.png"/> |
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| 96 | </p> |
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| 97 | <p> |
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| 98 | <img src="mqx.png"/> |
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| 99 | </p> |
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| 100 | <p> |
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| 101 | <img src="mqy.png"/> |
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| 102 | </p> |
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| 103 | <br> |
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| 104 | <br> |
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| 105 | Here, the M<sub>0x</sub>, M<sub>0y</sub> and M<sub>0z</sub> are the x, y and z |
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| 106 | components of the magnetization vector given in the xyz lab frame. |
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| 107 | <br> |
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| 108 | <br> |
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| 109 | <b>2. <a name="gui">GUI</a> </b> |
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| 110 | <br> |
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| 111 | <p> |
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| 112 | <img src="gen_gui_help.png"/> |
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| 113 | </p> |
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| 114 | <br> |
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| 115 | <p> |
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| 116 | After the computation, the result will be listed in the 'Theory' box |
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| 117 | in the data explorer panel on the main window. |
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| 118 | <br> |
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| 119 | The 'Up_frac_in' and 'Up_frac_out' are the ratio, (spin up) /(spin up + spin down) neutrons |
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| 120 | before the sample and at the analyzer, respectively. |
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| 121 | </p> |
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| 122 | <br> |
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| 123 | *Note I: The values of 'Up_frac_in' and 'Up_frac_out' must be in the range between 0 and 1. |
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| 124 | For example, both values are 0.5 for unpolarized neutrons. |
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| 125 | <br> |
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| 126 | *Note II: This computation is totally based on the pixel (or atomic) data fixed |
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| 127 | in the xyz coordinates. Thus no angular orientational averaging is considered. |
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| 128 | <br> |
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| 129 | *Note III: For the nuclear scattering length density, only the real component is taken account. |
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| 130 | <br> |
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| 131 | <br> |
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| 132 | <b>3. <a name="pdb">PDB Data</a> </b> |
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| 133 | <br> |
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| 134 | This Generic scattering calculator also supports some pdb files without considering polarized/magnetic scattering |
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| 135 | so that the related parameters such as Up_*** will be ignored (see the Picture below). The calculation for fixed orientation uses (the first) Equation above resulting |
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| 136 | in a 2D output, whileas the scattering calculation averaged over all the orientations uses the Debye equation providing a 1D output: |
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| 137 | <p> |
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| 138 | <img src="gen_debye_eq.png"/> |
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| 139 | </p> |
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| 140 | <br> |
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| 141 | where v<sub>j</sub>β<sub>j</sub> ≡ b<sub>j</sub> is the scattering length of the j'th atom. |
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| 142 | The resultant outputs will be displayed in the DataExporer for further uses. |
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| 143 | <br> |
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| 144 | <p> |
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| 145 | <img src="pdb_combo.jpg"/> |
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| 146 | </p> |
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| 147 | </body> |
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