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1<body>
2<br>
3<p><a name="magnetic"></a><strong><span style="font-size: 14pt;"> Polarization and Magnetic Scattering</span></strong></p>
4<br>
5<br>
6The magnetic scattering is implemented in five (2D) models, SphereModel, CoreShellModel,
7CoreMultiShellModel, CylinderModel, and ParallelepipedModel.
8In general, the scattering length density (SLD) in each regions where the SLD (=&#946;)
9 is uniform, is a combination of the nuclear and magnetic SLDs
10and depends on the spin states of the neutrons as follows:
11<br>
12<br>
13For magnetic scattering, only the magnetization component, <b>M</b><sub>perp</sub>,
14perpendicular to the scattering vector <b>Q</b> contributes to the the magnetic
15scattering length. (Figure below).
16<p>
17<img src="img/mag_vector.bmp"/>
18</p>
19<br>
20The magnetic scattering length density is then
21<p>
22<img src="img/dm_eq.gif"/>
23</p>
24<br>
25where &#947; = -1.913 the gyromagnetic ratio,   &#956;<sub>B</sub> is the Bohr magneton,
26r<sub>0</sub> is the classical radius of electron,
27and <b>&#963;</b> is the Pauli spin.
28<br>
29For polarized neutron, the magnetic scattering is depending on the spin states.
30Let's consider that the incident neutrons are polarized parallel (+)/anti-parallel
31(&#8211;) to the x' axis (See both Figures above).
32The possible out-coming states then are + and - states for both incident states.
33<br>
34 - Non-spin-flips:      (+ +) and       (- -)
35<br>
36 - Spin-flips:          (+ -) and       (- +)
37<br>
38<p>
39<img src="img/M_angles_pic.bmp"/>
40</p>
41<br>
42<br>
43Now, let's assume that the angles of the <b>Q</b> vector and the spin-axis (x') against x-axis
44are &#966; and  &#952;<sub>up</sub>, respectively (See Figure above).
45Then, depending upon the polarization (spin) state of neutrons, the scattering length
46densities , including the nuclear scattering length density (&#946; <sub>N</sub>) are given as, for non-spin-flips,
47<p>
48<img src="img/sld1.gif"/>
49</p>
50<br>
51<br>
52for spin-flips,
53<p>
54<img src="img/sld2.gif"/>
55</p>
56<br>
57<br>
58where
59<p>
60<img src="img/mxp.gif"/>
61</p>
62<p>
63<img src="img/myp.gif"/>
64</p>
65<p>
66<img src="img/mzp.gif"/>
67</p>
68<p>
69<img src="img/mqx.gif"/>
70</p>
71<p>
72<img src="img/mqy.gif"/>
73</p>
74<br>
75<br>
76Here, the M<sub>0x</sub>, M<sub>0y</sub> and M<sub>0z</sub> are the x, y and z
77components of the magnetization vector given in the xyz lab frame.
78The angles of the magnetization, &#952;<sub>M</sub> and &#966;<sub>M</sub> as defined in the
79Figure (above),
80<p>
81<img src="img/m0x_eq.gif"/>
82</p>
83<p>
84<img src="img/m0y_eq.gif"/>
85</p>
86<p>
87<img src="img/m0z_eq.gif"/>
88</p>
89<br>
90<p>
91The user input parameters are M0_sld = D<sub>M</sub>M<sub>0</sub>, Up_theta = &#952;<sub>up</sub>,
92M_theta = &#952;<sub>M</sub>, and M_phi = &#966;<sub>M</sub>.
93The 'Up_frac_i' and 'Up_frac_f' are the ratio, (spin up) /(spin up + spin down) neutrons
94before the sample and at the analyzer, respectively.
95</p>
96<br>
97*Note: The values of the 'Up_frac_i' and 'Up_frac_f' must be in the range between 0 and 1.
98</body>
99
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