Ignore:
Timestamp:
Dec 18, 2012 10:55:24 AM (12 years ago)
Author:
Jae Cho <jhjcho@…>
Branches:
master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, costrafo411, magnetic_scatt, release-4.1.1, release-4.1.2, release-4.2.2, release_4.0.1, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
Children:
6550b64
Parents:
0203ade
Message:

Added polarization and magnetic stuffs

File:
1 edited

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  • sansmodels/src/sans/models/media/smear_computation.html

    r17574ae r318b5bbb  
    1 <html> 
    21 
    32<head> 
    … …  
    3029smeared scattering intensity for SANS is defined by</span></p> 
    3130 
    32 <p class=MsoNormal><img width=349 height=49 
     31<p class=MsoNormal><img  
    3332src="./img/sm_image002.gif" align=left hspace=12></p> 
    3433 
    3534<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>            
    36                                  1)</span><br clear=all> 
     35                                 ---- 1)</span><br clear=all> 
    3736<span style='font-family:"Times New Roman","serif"'>where Norm = <span 
    38 style='position:relative;top:15.0pt'><img width=137 height=49 
     37style='position:relative;top:15.0pt'><img  
    3938src="./img/sm_image003.gif"></span>.</span></p> 
    40  
     39<br> 
    4140<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The 
    42 functions <span style='position:relative;top:6.0pt'><img width=43 height=25 
    43 src="./img/sm_image004.gif"></span>and <span style='position: 
    44 relative;top:6.0pt'><img width=43 height=25 
    45 src="./img/sm_image005.gif"></span>refer to the slit width weighting 
     41functions <span style='position:relative;top:6.0pt'><img  
     42src="./img/sm_image004.gif"></span> and <span style='position: 
     43relative;top:6.0pt'><img  
     44src="./img/sm_image005.gif"></span> refer to the slit width weighting 
    4645function and the slit height weighting determined at the q point, respectively. 
    4746 Here, we assumes that the weighting function is described by a rectangular 
    4847function, i.e.,</span></p> 
    4948 
    50 <p class=MsoNormal><span style='position:relative;top:7.0pt'><img width=134 
    51 height=26 src="./img/sm_image006.gif">                                                                                                        
     49<p class=MsoNormal><span style='position:relative;top:7.0pt'><img  
     50 src="./img/sm_image006.gif">                                                                                                        
    5251  </span><span style='font-family:"Times New Roman","serif";position:relative; 
    53 top:7.0pt'>2)</span></p> 
     52top:7.0pt'> ---- 2)</span></p> 
    5453 
    5554<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>and </span></p> 
    5655 
    57 <p class=MsoNormal><span style='position:relative;top:7.0pt'><img width=136 
    58 height=26 src="./img/sm_image007.gif"></span>,                                                                               
    59                          <span style='font-family:"Times New Roman","serif"'>3)</span></p> 
    60  
    61 <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>so that </span><span 
    62 style='position:relative;top:6.0pt'><img width=58 height=23 
    63 src="./img/sm_image008.gif"></span> <span style='position:relative; 
    64 top:16.0pt'><img width=76 height=51 src="./img/sm_image009.gif"></span> <span 
    65 style='font-family:"Times New Roman","serif"'>for</span>  <span 
    66 style='position:relative;top:3.0pt'><img width=40 height=15 
    67 src="./img/sm_image010.gif"></span> <span style='font-family: 
    68 "Times New Roman","serif"'>and <i>u</i>. The </span><span style='position:relative; 
    69 top:6.0pt'><img width=28 height=24 src="./img/sm_image011.gif"></span> <span 
    70 style='font-family:"Times New Roman","serif"'>and </span><span 
    71 style='position:relative;top:6.0pt'><img width=28 height=24 
    72 src="./img/sm_image012.gif"> </span><span style='font-family: 
    73 "Times New Roman","serif"'>stand for the slit height (FWHM/2) and the slit 
     56<p class=MsoNormal><span style='position:relative;top:7.0pt'><img  
     57 src="./img/sm_image007.gif"></span>,                                                                               
     58                         <span style='font-family:"Times New Roman","serif"'> ---- 3)</span></p> 
     59 
     60<p>so that <img  
     61src="./img/sm_image008.gif"> <img src="./img/sm_image009.gif">  for <img  
     62src="./img/sm_image010.gif"> and <i>u</i>. The <img src="./img/sm_image011.gif">  
     63 and <img src="./img/sm_image012.gif">  stand for the slit height (FWHM/2) and the slit 
    7464width (FWHM/2) in the q space. Now the integral of Eq. (1) is simplified to</span></p> 
    7565 
    76 <p class=MsoNormal><img width=283 height=52 
     66<p class=MsoNormal><img  
    7767src="./img/sm_image013.gif" align=left hspace=12><span 
    7868style='font-family:"Times New Roman","serif"'>                                                   
    79          4)</span></p> 
     69          ---- 4)</span></p> 
    8070 
    8171<p class=MsoNormal><span style='font-family:"Times New Roman","serif"; 
    … …  
    8979style='font-family:"Times New Roman","serif"'>1)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    9080</span></span><span style='font-family:"Times New Roman","serif"'>For </span><span 
    91 style='position:relative;top:6.0pt'><img width=28 height=24 
    92 src="./img/sm_image014.gif"></span>= 0  <span style='font-family: 
     81style='position:relative;top:6.0pt'><img  
     82src="./img/sm_image012.gif"></span>= 0  <span style='font-family: 
    9383"Times New Roman","serif"'>and </span><span style='position:relative; 
    94 top:6.0pt'><img width=28 height=24 src="./img/sm_image015.gif"></span> = 
     84top:6.0pt'><img src="./img/sm_image011.gif"></span> = 
    9585<span style='font-family:"Times New Roman","serif"'>constant:</span></p> 
    9686 
    97 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> 
     87<p> 
    9888<img src="./img/sm_image016.gif"></p> 
    9989 
    100 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    101 style='font-family:"Times New Roman","serif"'>For discrete q values, at the q 
     90<p> For discrete q values, at the q 
    10291values from the data points and at the q values extended up to  q<sub>N</sub>= 
    103 q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img width=28 
    104 height=24 src="./img/sm_image011.gif"></span><span 
    105 style='font-family:"Times New Roman","serif"'>, the smeared intensity can be 
    106 calculated approximately,</span></p> 
    107  
    108 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><img  
     92q<sub>i</sub> +  <img src="./img/sm_image011.gif"> , the smeared intensity can be 
     93calculated approximately, </p> 
     94 
     95<p><img  
    10996src="./img/sm_image017.gif">.                                                            
    110 <span style='font-family:"Times New Roman","serif"'>5)</span></p> 
    111  
    112 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    113 style='position:relative;top:7.0pt'><img width=23 height=25 
     97 ---- 5)</p> 
     98 
     99<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
     100style='position:relative;top:7.0pt'><img  
    114101src="./img/sm_image018.gif"></span> <span style='font-family: 
    115102"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span 
    … …  
    123110style='font-family:"Times New Roman","serif"'>2)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    124111</span></span><span style='font-family:"Times New Roman","serif"'>For  </span><span 
    125 style='position:relative;top:6.0pt'><img width=28 height=24 
    126 src="./img/sm_image014.gif"></span>= <span style='font-family: 
     112style='position:relative;top:6.0pt'><img  
     113src="./img/sm_image012.gif"></span>= <span style='font-family: 
    127114"Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and 
    128 </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
    129 src="./img/sm_image015.gif"></span> = <span style='font-family: 
     115</span><span style='position:relative;top:6.0pt'><img  
     116src="./img/sm_image011.gif"></span> = <span style='font-family: 
    130117"Times New Roman","serif"'>0:</span></p> 
    131118 
    … …  
    135122<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> 
    136123<img src="./img/sm_image019.gif">                                                                                         
    137 <span style='font-family:"Times New Roman","serif"'>6)</span></p> 
     124<span style='font-family:"Times New Roman","serif"'> ---- 6)</span></p> 
    138125 
    139126<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    140127style='font-family:"Times New Roman","serif"'>for  q<sub>p</sub> = q<sub>i</sub> 
    141 - </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
     128- </span><span style='position:relative;top:6.0pt'><img  
    142129src="./img/sm_image012.gif"></span><span style='font-family: 
    143130"Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub> 
    144131= q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img 
    145 width=28 height=24 src="./img/sm_image012.gif"></span>.  <span 
    146 style='position:relative;top:7.0pt'><img width=23 height=25 
     132 src="./img/sm_image012.gif"></span>.  <span 
     133style='position:relative;top:7.0pt'><img  
    147134src="./img/sm_image018.gif"></span> <span style='font-family: 
    148135"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span 
    … …  
    155142style='font-family:"Times New Roman","serif"'>3)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    156143</span></span><span style='font-family:"Times New Roman","serif"'>For  </span><span 
    157 style='position:relative;top:6.0pt'><img width=28 height=24 
    158 src="./img/sm_image014.gif"></span>= <span style='font-family: 
     144style='position:relative;top:6.0pt'><img  
     145src="./img/sm_image011.gif"></span>= <span style='font-family: 
    159146"Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and 
    160 </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
    161 src="./img/sm_image015.gif"></span> = <span style='font-family: 
     147</span><span style='position:relative;top:6.0pt'><img  
     148src="./img/sm_image011.gif"></span> = <span style='font-family: 
    162149"Times New Roman","serif"'>constant:</span></p> 
    163150 
    … …  
    177164<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> 
    178165<img src="./img/sm_image020.gif">    <span style='font-family: 
    179 "Times New Roman","serif"'>(7)</span></p> 
     166"Times New Roman","serif"'> ---- (7)</span></p> 
    180167 
    181168<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    182169style='font-family:"Times New Roman","serif"'>for  q<sub>p</sub> = q<sub>i</sub> 
    183 - </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
     170- </span><span style='position:relative;top:6.0pt'><img  
    184171src="./img/sm_image012.gif"></span><span style='font-family: 
    185172"Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub> 
    186173= q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img 
    187 width=28 height=24 src="./img/sm_image012.gif"></span>.  <span 
    188 style='position:relative;top:7.0pt'><img width=23 height=25 
     174 src="./img/sm_image012.gif"></span>.  <span 
     175style='position:relative;top:7.0pt'><img  
    189176src="./img/sm_image018.gif"></span> <span style='font-family: 
    190177"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span 
    … …  
    207194 
    208195<p class=MsoNormal><img src="./img/sm_image021.gif"><span 
    209 style='font-family:"Times New Roman","serif"'>                                                         (8)</span></p> 
     196style='font-family:"Times New Roman","serif"'>   ---- (8)</span></p> 
    210197 
    211198<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>For all 
    … …  
    228215 
    229216<p class=MsoNormal><img src="./img/sm_image022.gif"><span 
    230 style='font-family:"Times New Roman","serif"'>                   (9)</span></p> 
     217style='font-family:"Times New Roman","serif"'>     ---- (9)</span></p> 
    231218 
    232219<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>In Eq 
    233 (9), x<sub>0</sub> = qcos</span><span style='font-family:Symbol'>(theta)</span><span 
    234 style='font-family:"Times New Roman","serif"'> and y<sub>0</sub>=qsin</span><span 
    235 style='font-family:Symbol'>(theta)</span><span style='font-family:"Times New Roman","serif"'> 
    236 , and the primed axes are in the coordinate rotated by an angle </span><span 
    237 style='font-family:Symbol'>theta</span><span style='font-family:"Times New Roman","serif"'> 
    238 around z-axis (below) so that x’<sub>0</sub> =  x<sub>0</sub>cos</span><span 
    239 style='font-family:Symbol'>(theta) + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
    240 sin</span><span style='font-family:Symbol'>(theta)  </span><span style='font-family: 
    241 "Times New Roman","serif"'>and y’<sub>0</sub> =  -x<sub>0</sub>sin</span><span 
    242 style='font-family:Symbol'>(theta) + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
    243 cos</span><span style='font-family:Symbol'>(theta) .</span><span style='font-family: 
     220(9), x<sub>0</sub> = qcos&#952;</span><span 
     221style='font-family:"Times New Roman","serif"'> and y<sub>0</sub> = qsin&#952;</span><span style='font-family:"Times New Roman","serif"'> 
     222, and the primed axes are in the coordinate rotated by an angle &#952;</span><span style='font-family:"Times New Roman","serif"'> 
     223around z-axis (below) so that x’<sub>0</sub> =  x<sub>0</sub>cos&#952; + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
     224sin&#952;  </span><span style='font-family: 
     225"Times New Roman","serif"'>and y’<sub>0</sub> =  -x<sub>0</sub>sin&#952; + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
     226cos&#952;.</span><span style='font-family: 
    244227"Times New Roman","serif"'> Note that the rotation angle is zero for x-y 
    245 symmetric elliptical Gaussian distribution</span><span style='font-family:Symbol'>. 
    246 </span><span style='font-family:"Times New Roman","serif"'>The  A is a 
     228symmetric elliptical Gaussian distribution</span>. 
     229<span style='font-family:"Times New Roman","serif"'>The  A is a 
    247230normalization factor.</span></p> 
    248231 
    249232<p class=MsoNormal align=center style='text-align:center'><span 
    250 style='font-family:"Times New Roman","serif"'><img width=439 height=376 
     233style='font-family:"Times New Roman","serif"'><img  
    251234id="Object 1" src="./img/sm_image023.gif"></span></p> 
    252235 
    … …  
    254237 
    255238<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Now we 
    256 consider a numerical integration where each bins in </span><span 
    257 style='font-family:Symbol'>THETA</span><span style='font-family:"Times New Roman","serif"'> 
     239consider a numerical integration where each bins in </span> &#920; </span><span style='font-family:"Times New Roman","serif"'> 
    258240and R are <b>evenly </b>(this is to simplify the equation below) distributed by 
    259 </span><span style='font-family:Symbol'>Delta_THETA </span><span style='font-family: 
    260 "Times New Roman","serif"'>and </span><span style='font-family:Symbol'>Delta</span><span 
     241</span>&#916;&#920; </span><span style='font-family: 
     242"Times New Roman","serif"'>and </span> &#916;</span><span 
    261243style='font-family:"Times New Roman","serif"'>R, respectively, and it is 
    262244assumed that I(x’, y’) is constant within the bins which in turn becomes</span></p> 
    … …  
    265247 
    266248<p class=MsoNormal>                                                                                                                                                                                <span 
    267 style='font-family:"Times New Roman","serif"'>(10)</span></p> 
     249style='font-family:"Times New Roman","serif"'> ---- (10)</span></p> 
    268250 
    269251<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Since we 
    270252have found the weighting factor on each bin points, it is convenient to 
    271 transform x’-y’ back to x-y coordinate (rotating it by -</span><span 
    272 style='font-family:Symbol'>(theta)</span><span style='font-family:"Times New Roman","serif"'> 
     253transform x’-y’ back to x-y coordinate (rotating it by -&#952;</span><span style='font-family:"Times New Roman","serif"'> 
    273254around z axis).  Then, for the polar symmetric smear,</span></p> 
    274255 
    275256<p class=MsoNormal><img src="./img/sm_image025.gif"><span 
    276 style='position:relative;top:35.0pt'>                                                         </span>(11)</p> 
     257style='position:relative;top:35.0pt'>      </span> ---- (11)</p> 
    277258 
    278259<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p> 
    279260 
    280 <p class=MsoNormal><img src="./img/sm_image026.gif"></p> 
     261<p class=MsoNormal><img src="./img/sm_image026.gif">,</p> 
    281262 
    282263<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>while 
    … …  
    284265 
    285266<p class=MsoNormal><img src="./img/sm_image027.gif"><span 
    286 style='font-family:"Times New Roman","serif"'>                                                                                          (12)</span></p> 
     267style='font-family:"Times New Roman","serif"'>  ---- (12)</span></p> 
    287268 
    288269<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p> 
    … …  
    301282</body> 
    302283 
    303 </html> 
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