| 1 | <body> |
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| 2 | <br> |
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| 3 | <p><a name="magnetic"></a><strong><span style="font-size: 14pt;"> Polarization and Magnetic Scattering</span></strong></p> |
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| 4 | <br> |
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| 5 | <br> |
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| 6 | The magnetic scattering is implemented in five (2D) models, SphereModel, CoreShellModel, |
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| 7 | CoreMultiShellModel, CylinderModel, and ParallelepipedModel. |
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| 8 | In general, the scattering length density (SLD) in each regions where the SLD (=β) |
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| 9 | is uniform, is a combination of the nuclear and magnetic SLDs |
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| 10 | and depends on the spin states of the neutrons as follows: |
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| 11 | <br> |
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| 12 | <br> |
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| 13 | For magnetic scattering, only the magnetization component, <b>M</b><sub>perp</sub>, |
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| 14 | perpendicular to the scattering vector <b>Q</b> contributes to the the magnetic |
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| 15 | scattering length. (Figure below). |
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| 16 | <p> |
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| 17 | <img src="img/mag_vector.bmp"/> |
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| 18 | </p> |
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| 19 | <br> |
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| 20 | The magnetic scattering length density is then |
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| 21 | <p> |
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| 22 | <img src="img/dm_eq.gif"/> |
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| 23 | </p> |
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| 24 | <br> |
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| 25 | where γ = -1.913 the gyromagnetic ratio, μ<sub>B</sub> is the Bohr magneton, |
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| 26 | r<sub>0</sub> is the classical radius of electron, |
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| 27 | and <b>σ</b> is the Pauli spin. |
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| 28 | <br> |
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| 29 | For polarized neutron, the magnetic scattering is depending on the spin states. |
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| 30 | Let's consider that the incident neutrons are polarized parallel (+)/anti-parallel |
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| 31 | (–) to the x' axis (See both Figures above). |
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| 32 | The possible out-coming states then are + and - states for both incident states. |
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| 33 | <br> |
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| 34 | - Non-spin-flips: (+ +) and (- -) |
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| 35 | <br> |
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| 36 | - Spin-flips: (+ -) and (- +) |
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| 37 | <br> |
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| 38 | <p> |
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| 39 | <img src="img/M_angles_pic.bmp"/> |
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| 40 | </p> |
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| 41 | <br> |
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| 42 | <br> |
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| 43 | Now, let's assume that the angles of the <b>Q</b> vector and the spin-axis (x') against x-axis |
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| 44 | are φ and θ<sub>up</sub>, respectively (See Figure above). |
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| 45 | Then, depending upon the polarization (spin) state of neutrons, the scattering length |
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| 46 | densities , including the nuclear scattering length density (β <sub>N</sub>) are given as, for non-spin-flips, |
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| 47 | <p> |
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| 48 | <img src="img/sld1.gif"/> |
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| 49 | </p> |
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| 50 | <br> |
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| 51 | <br> |
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| 52 | for spin-flips, |
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| 53 | <p> |
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| 54 | <img src="img/sld2.gif"/> |
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| 55 | </p> |
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| 56 | <br> |
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| 57 | <br> |
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| 58 | where |
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| 59 | <p> |
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| 60 | <img src="img/mxp.gif"/> |
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| 61 | </p> |
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| 62 | <p> |
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| 63 | <img src="img/myp.gif"/> |
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| 64 | </p> |
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| 65 | <p> |
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| 66 | <img src="img/mzp.gif"/> |
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| 67 | </p> |
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| 68 | <p> |
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| 69 | <img src="img/mqx.gif"/> |
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| 70 | </p> |
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| 71 | <p> |
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| 72 | <img src="img/mqy.gif"/> |
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| 73 | </p> |
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| 74 | <br> |
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| 75 | <br> |
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| 76 | 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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| 77 | components of the magnetization vector given in the xyz lab frame. |
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| 78 | The angles of the magnetization, θ<sub>M</sub> and φ<sub>M</sub> as defined in the |
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| 79 | Figure (above), |
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| 80 | <p> |
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| 81 | <img src="img/m0x_eq.gif"/> |
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| 82 | </p> |
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| 83 | <p> |
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| 84 | <img src="img/m0y_eq.gif"/> |
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| 85 | </p> |
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| 86 | <p> |
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| 87 | <img src="img/m0z_eq.gif"/> |
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| 88 | </p> |
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| 89 | <br> |
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| 90 | <p> |
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| 91 | The user input parameters are M0_sld = D<sub>M</sub>M<sub>0</sub>, Up_theta = θ<sub>up</sub>, |
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| 92 | M_theta = θ<sub>M</sub>, and M_phi = φ<sub>M</sub>. |
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| 93 | The 'Up_frac_i' and 'Up_frac_f' are the ratio, (spin up) /(spin up + spin down) neutrons |
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| 94 | before the sample and at the analyzer, respectively. |
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| 95 | </p> |
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| 96 | <br> |
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| 97 | *Note: The values of the 'Up_frac_i' and 'Up_frac_f' must be in the range between 0 and 1. |
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| 98 | </body> |
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| 99 | |
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| 100 | |
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| 101 | |
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| 102 | |
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