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1<body>
2<h4>Generic Scattering Calculator:</h4>
3Polarization and Magnetic Scattering
4<br>
5<br>
6<ul>
7<li><a href="#theory">Theory:</a></li>
8<li><a href="#gui">GUI</a></li>
9<li><a href="#pdb">PDB Data</a></li>
10</ul>
11<br>
12<br>
13<b>1. <a name="theory">Theory</a> </b>
14<br>
15In general, a particle with a volume V can be described by an ensemble containing N
163-dimensional rectangular pixels where each pixels are much smaller than V.
17Assuming that
18all the pixel sizes are same, the elastic scattering intensity
19by the particle
20<p>
21<img src="gen_i.gif"/>
22</p>
23<br>
24where &#946;<sub>j</sub> and r<sub>j</sub> are the scattering
25length density and the position
26of the j'th pixel respectively. And the total volume
27<p>
28<img src="v_j.gif"/>
29</p>
30<br>
31for &#946;<sub>j</sub> &#8800; 0 where v<sub>j</sub> is the volume of the j'th pixel
32(or the j'th natural atomic volume (= atomic mass/natural molar density/Avogadro number)
33 for the atomic structures). The total volume V can be corrected by users.
34This correction is useful especially for an atomic structure (taken from a pdb file) to get the right
35normalization. Note that the  &#946;<sub>j</sub> displayed in GUI may be incorrect but will not
36affect the scattering computation if the correction of the total volume is made.
37<br>
38The scattering length density (SLD) of each pixel where the SLD is uniform, is a combination of the nuclear and magnetic SLDs
39and depends on the spin states of the neutrons as follows:
40<br>
41<br>
42For magnetic scattering, only the magnetization component, <b>M</b><sub>perp</sub>,
43perpendicular to the scattering vector <b>Q</b> contributes to the the magnetic
44scattering length. (Figure below).
45<p>
46<img src="mag_vector.bmp"/>
47</p>
48<br>
49The magnetic scattering length density is then
50<p>
51<img src="dm_eq.gif"/>
52</p>
53<br>
54where &#947; = -1.913 the gyromagnetic ratio,   &#956;<sub>B</sub> is the Bohr magneton,
55r<sub>0</sub> is the classical radius of electron,
56and <b>&#963;</b> is the Pauli spin.
57<br>
58For polarized neutron, the magnetic scattering is depending on the spin states.
59Let's consider that the incident neutrons are polarized parallel (+)/anti-parallel
60(&#8211;) to the x' axis (See both Figures above).
61The possible out-coming states then are + and - states for both incident states.
62<br>
63 - Non-spin-flips:      (+ +) and       (- -)
64<br>
65 - Spin-flips:          (+ -) and       (- +)
66<br>
67<p>
68<img src="gen_mag_pic.bmp"/>
69</p>
70<br>
71<br>
72Now, let's assume that the angles of the <b>Q</b> vector and the spin-axis (x') from x-axis
73are &#966; and  &#952;<sub>up</sub>, respectively (See Figure above).
74Then, depending upon the polarization (spin) state of neutrons, the scattering length
75densities , including the nuclear scattering length density (&#946; <sub>N</sub>) are given as, for non-spin-flips,
76<p>
77<img src="sld1.gif"/>
78</p>
79<br>
80<br>
81for spin-flips,
82<p>
83<img src="sld2.gif"/>
84</p>
85<br>
86<br>
87where
88<p>
89<img src="mxp.gif"/>
90</p>
91<p>
92<img src="myp.gif"/>
93</p>
94<p>
95<img src="mzp.gif"/>
96</p>
97<p>
98<img src="mqx.gif"/>
99</p>
100<p>
101<img src="mqy.gif"/>
102</p>
103<br>
104<br>
105Here, the M<sub>0x</sub>, M<sub>0y</sub> and M<sub>0z</sub> are the x, y and z
106components of the magnetization vector given in the xyz lab frame.
107<br>
108<br>
109<b>2. <a name="gui">GUI</a> </b>
110<br>
111<p>
112<img src="gen_gui_help.bmp"/>
113</p>
114<br>
115<p>
116After the computation, the result will be listed in the 'Theory' box
117in the data explorer panel on the main window.
118<br>
119The 'Up_frac_in' and 'Up_frac_out' are the ratio, (spin up) /(spin up + spin down) neutrons
120before the sample and at the analyzer, respectively.
121</p>
122<br>
123*Note I: The values of 'Up_frac_in' and 'Up_frac_out' must be in the range between 0 and 1.
124For example, both values are 0.5 for unpolarized neutrons.
125<br>
126*Note II: This computation is totally based on the pixel (or atomic) data fixed
127in the xyz coordinates. Thus no angular orientational averaging is considered.
128<br>
129*Note III: For the nuclear scattering length density, only the real component is taken account.
130<br>
131<br>
132<b>3. <a name="pdb">PDB Data</a> </b>
133<br>
134This Generic scattering calculator also supports some pdb files without considering polarized/magnetic scattering
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
136in a 2D output, whileas the scattering calculation averaged over all the orientations uses the Debye equation providing a 1D output:
137 <p>
138<img src="gen_debye_eq.gif"/>
139</p>
140<br>
141where v<sub>j</sub>&#946;<sub>j</sub> &#8801; b<sub>j</sub> is the scattering length of the j'th atom.
142The resultant outputs will be displayed in the DataExporer for further uses.
143<br>
144 <p>
145<img src="pdb_combo.jpg"/>
146</p>
147</body>
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