Changeset 93b6fcc in sasview for src/sans/models/media
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src/sans/models/media/model_functions.rst (modified) (93 diffs)
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src/sans/models/media/model_functions.rst
r58eccf6 r93b6fcc 34 34 .. |psi| unicode:: U+03C8 35 35 .. |omega| unicode:: U+03C9 36 .. |biggamma| unicode:: U+0393 36 37 .. |bigdelta| unicode:: U+0394 37 .. |big gamma| unicode:: U+039338 .. |bigzeta| unicode:: U+039E 38 39 .. |bigpsi| unicode:: U+03A8 39 40 .. |drho| replace:: |bigdelta|\ |rho| … … 62 63 .. Actual document starts here... 63 64 65 This is xi, |xi| 66 67 This is zeta, |zeta| 68 64 69 SasView Model Functions 65 70 ======================= … … 211 216 212 217 - AbsolutePower_Law_ 213 - BEPolyelectrolyte 214 - BroadPeakModel 215 - CorrLength 216 - DABModel 217 - Debye 218 - FractalModel 219 - FractalCoreShell 220 - GaussLorentzGel 221 - Guinier 222 - GuinierPorod 223 - Lorentz 224 - MassFractalModel 225 - MassSurfaceFractal 218 - BEPolyelectrolyte_ 219 - BroadPeakModel_ 220 - CorrLength_ 221 - DABModel_ 222 - Debye_ 223 - FractalModel_ 224 - FractalCoreShell_ 225 - GaussLorentzGel_ 226 - Guinier_ 227 - GuinierPorod_ 228 - Lorentz_ 229 - MassFractalModel_ 230 - MassSurfaceFractal_ 226 231 - PeakGaussModel 227 232 - PeakLorentzModel … … 231 236 - RPA10Model 232 237 - StarPolymer 233 - SurfaceFractalModel 234 - Teubner Strey238 - SurfaceFractalModel_ 239 - TeubnerStrey_ 235 240 - TwoLorentzian 236 241 - TwoPowerLaw … … 275 280 276 281 *Small-Angle Scattering of X-Rays* 277 A . Guinier and G.Fournet282 A Guinier and G Fournet 278 283 John Wiley & Sons, New York (1955) 279 284 280 P . Stckel, R. May, I. Strell, Z. Cejka, W. Hoppe, H. Heumann, W. Zillig and H.Crespi285 P Stckel, R May, I Strell, Z Cejka, W Hoppe, H Heumann, W Zillig and H Crespi 281 286 *Eur. J. Biochem.*, 112, (1980), 411-417 282 287 283 G .Porod288 G Porod 284 289 in *Small Angle X-ray Scattering* 285 (editors) O . Glatter and O.Kratky290 (editors) O Glatter and O Kratky 286 291 Academic Press (1982) 287 292 288 293 *Structure Analysis by Small-Angle X-Ray and Neutron Scattering* 289 L.A . Feigin and D. I.Svergun294 L.A Feigin and D I Svergun 290 295 Plenum Press, New York (1987) 291 296 292 S .Hansen297 S Hansen 293 298 *J. Appl. Cryst.* 23, (1990), 344-346 294 299 295 S .J.Henderson300 S J Henderson 296 301 *Biophys. J.* 70, (1996), 1618-1627 297 302 298 B .C. McAlister and B.P. Grady, B.P303 B C McAlister and B P Grady 299 304 *J. Appl. Cryst.* 31, (1998), 594-599 300 305 301 S .R.Kline306 S R Kline 302 307 *J Appl. Cryst.* 39(6), (2006), 895 303 308 … … 356 361 REFERENCE 357 362 358 A .Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955)363 A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 359 364 360 365 *2.1.1.2. Validation of the SphereModel* … … 368 373 The parameters were set to: Scale=1.0, Radius=60 |Ang|, Contrast=1e-6 |Ang^-2|, and Background=0.01 |cm^-1|. 369 374 370 *2013/09/09 and 2014/01/06 - Description reviewed by S . King and P.Parker.*375 *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* 371 376 372 377 … … 421 426 REFERENCE 422 427 423 N . W. Ashcroft and D. C.Langreth, *Physical Review*, 156 (1967) 685-692428 N W Ashcroft and D C Langreth, *Physical Review*, 156 (1967) 685-692 424 429 [Errata found in *Phys. Rev.* 166 (1968) 934] 425 430 … … 484 489 REFERENCE 485 490 486 M . Stieger, J. S. Pedersen, P. Lindner, W.Richtering, *Langmuir*, 20 (2004) 7283-7292491 M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*, 20 (2004) 7283-7292 487 492 488 493 … … 541 546 REFERENCE 542 547 543 K . Larson-Smith, A. Jackson, and D.C.Pozzo, *Small angle scattering model for Pickering emulsions and raspberry*548 K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for Pickering emulsions and raspberry* 544 549 *particles*, *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41 545 550 … … 592 597 REFERENCE 593 598 594 A . Guinier and G.Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955)599 A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 595 600 596 601 *2.1.5.2. Validation of the core-shell sphere model* … … 726 731 REFERENCE 727 732 728 S . King, P. Griffiths, J. Hone, and T.Cosgrove, *SANS from Adsorbed Polymer Layers*,733 S King, P Griffiths, J. Hone, and T Cosgrove, *SANS from Adsorbed Polymer Layers*, 729 734 *Macromol. Symp.*, 190 (2002) 33-42 730 735 … … 774 779 REFERENCE 775 780 776 B .Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions: New Methods of Investigation*, Ch.2,777 Surfactant Science Series Vol. 22, Ed. R . Zana and M.Dekker, New York, (1987).781 B Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions: New Methods of Investigation*, Ch.2, 782 Surfactant Science Series Vol. 22, Ed. R Zana and M Dekker, New York, (1987). 778 783 779 784 … … 893 898 REFERENCE 894 899 895 L . A. Feigin and D. I.Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*,900 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, 896 901 Plenum Press, New York, (1987). 897 902 … … 953 958 REFERENCE 954 959 955 A .Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955)960 A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 956 961 957 962 … … 1082 1087 REFERENCE 1083 1088 1084 L . A. Feigin and D. I.Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*,1089 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, 1085 1090 Plenum Press, New York, (1987) 1086 1091 … … 1128 1133 REFERENCE 1129 1134 1130 A . V. Dobrynin, M. Rubinstein and S. P.Obukhov, *Macromol.*, 29 (1996) 2974-29791135 A V Dobrynin, M Rubinstein and S P Obukhov, *Macromol.*, 29 (1996) 2974-2979 1131 1136 1132 1137 … … 1193 1198 REFERENCE 1194 1199 1195 R . Schweins and K.Huber, *Particle Scattering Factor of Pearl Necklace Chains*, *Macromol. Symp.* 211 (2004) 25-42 20041200 R Schweins and K Huber, *Particle Scattering Factor of Pearl Necklace Chains*, *Macromol. Symp.* 211 (2004) 25-42 2004 1196 1201 1197 1202 … … 1342 1347 REFERENCE 1343 1348 1344 L . A. Feigin and D. I.Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press,1349 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, 1345 1350 New York, (1987) 1346 1351 … … 1427 1432 REFERENCE 1428 1433 1429 H .Kaya, *J. Appl. Cryst.*, 37 (2004) 223-2301430 1431 H .Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata)1434 H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230 1435 1436 H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) 1432 1437 1433 1438 … … 1612 1617 REFERENCE 1613 1618 1614 L . A. Feigin and D. I.Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum,1619 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, 1615 1620 New York, (1987) 1616 1621 … … 1672 1677 REFERENCE 1673 1678 1674 J . S. Pedersen and P.Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume*1679 J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume* 1675 1680 *effects*. *Macromolecules*, 29 (1996) 7602-7612 1676 1681 1677 1682 Correction of the formula can be found in 1678 1683 1679 W -R Chen, P. D. Butler and L. J.Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from*1684 W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from* 1680 1685 *Cationic Wormlike Micelles*. *Langmuir*, 22(15) 2006 6539ââ¬â6548 1681 1686 … … 1760 1765 REFERENCE 1761 1766 1762 J . S. Pedersen and P.Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume*1767 J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume* 1763 1768 *effects*. *Macromolecules*, 29 (1996) 7602-7612 1764 1769 1765 1770 Correction of the formula can be found in 1766 1771 1767 W -R Chen, P. D. Butler and L. J.Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from*1772 W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from* 1768 1773 *Cationic Wormlike Micelles*. *Langmuir*, 22(15) 2006 6539ââ¬â6548 1769 1774 … … 1823 1828 REFERENCE 1824 1829 1825 L . A. Feigin and D. I.Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press,1830 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, 1826 1831 New York, (1987) 1827 1832 … … 1911 1916 REFERENCE 1912 1917 1913 H .Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-2301914 1915 H . Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata)1918 H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 1919 1920 H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) 1916 1921 1917 1922 … … 1998 2003 REFERENCE 1999 2004 2000 A . Guinier and G.Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, 19552001 2002 O . Kratky and G.Porod, *J. Colloid Science*, 4, (1949) 352003 2004 J . S. Higgins and H. C.Benoit, *Polymers and Neutron Scattering*, Clarendon, Oxford, 19942005 A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, 1955 2006 2007 O Kratky and G Porod, *J. Colloid Science*, 4, (1949) 35 2008 2009 J S Higgins and H C Benoit, *Polymers and Neutron Scattering*, Clarendon, Oxford, 1994 2005 2010 2006 2011 … … 2047 2052 REFERENCE 2048 2053 2049 S .Alexandru Rautu, Private Communication.2054 S Alexandru Rautu, Private Communication. 2050 2055 2051 2056 … … 2143 2148 REFERENCE 2144 2149 2145 L . A. Feigin and D. I.Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum,2150 L A Feigin and D I Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, 2146 2151 New York, 1987. 2147 2152 … … 2207 2212 REFERENCE 2208 2213 2209 M . Kotlarchyk, S.-H.Chen, *J. Chem. Phys.*, 79 (1983) 24612210 2211 S . J.Berr, *Phys. Chem.*, 91 (1987) 47602214 M Kotlarchyk, S H Chen, *J. Chem. Phys.*, 79 (1983) 2461 2215 2216 S J Berr, *Phys. Chem.*, 91 (1987) 4760 2212 2217 2213 2218 … … 2272 2277 REFERENCE 2273 2278 2274 R . K.Heenan, Private communication2279 R K Heenan, Private communication 2275 2280 2276 2281 … … 2347 2352 REFERENCE 2348 2353 2349 L . A. Feigin and D. I.Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum,2354 L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, 2350 2355 New York, 1987. 2351 2356 … … 2397 2402 REFERENCE 2398 2403 2399 F . Nallet, R. Laversanne, and D.Roux, J. Phys. II France, 3, (1993) 487-5022404 F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 2400 2405 2401 2406 also in J. Phys. Chem. B, 105, (2001) 11081-11088 … … 2451 2456 REFERENCE 2452 2457 2453 F . Nallet, R. Laversanne, and D.Roux, J. Phys. II France, 3, (1993) 487-5022458 F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 2454 2459 2455 2460 also in J. Phys. Chem. B, 105, (2001) 11081-11088 2456 2461 2457 *2014/04/17 - Description reviewed by S . King and P.Butler.*2462 *2014/04/17 - Description reviewed by S King and P Butler.* 2458 2463 2459 2464 … … 2518 2523 REFERENCE 2519 2524 2520 F . Nallet, R. Laversanne, and D.Roux, J. Phys. II France, 3, (1993) 487-5022525 F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 2521 2526 2522 2527 also in J. Phys. Chem. B, 105, (2001) 11081-11088 … … 2590 2595 REFERENCE 2591 2596 2592 F . Nallet, R. Laversanne, and D.Roux, J. Phys. II France, 3, (1993) 487-5022597 F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 2593 2598 2594 2599 also in J. Phys. Chem. B, 105, (2001) 11081-11088 … … 2651 2656 REFERENCE 2652 2657 2653 M . Bergstrom, J. S. Pedersen, P. Schurtenberger, S. U.Egelhaaf, *J. Phys. Chem. B*, 103 (1999) 9888-98972658 M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, *J. Phys. Chem. B*, 103 (1999) 9888-9897 2654 2659 2655 2660 … … 3004 3009 REFERENCE 3005 3010 3006 P . Mittelbach and G.Porod, *Acta Physica Austriaca*, 14 (1961) 185-2113011 P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 3007 3012 Equations (1), (13-14). (in German) 3008 3013 … … 3112 3117 REFERENCE 3113 3118 3114 P . Mittelbach and G.Porod, *Acta Physica Austriaca*, 14 (1961) 185-2113119 P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 3115 3120 Equations (1), (13-14). (in German) 3116 3121 … … 3126 3131 **2.2.1. Debye (Gaussian Coil Model)** 3127 3132 3128 The Debye model is a form factor for a linear polymer chain. In addition 3129 to the radius of gyration, Rg, a scale factor "scale", and a constant 3130 background term are included in the calculation. 3133 The Debye model is a form factor for a linear polymer chain. In addition to the radius of gyration, *Rg*, a scale factor 3134 *scale*, and a constant background term are included in the calculation. **NB: No size polydispersity is included in** 3135 **this model, use the** Poly_GaussCoil_ **Model instead** 3131 3136 3132 3137 .. image:: img/image172.PNG 3133 3138 3134 For 2D plot, the wave transfer is defined as3139 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3135 3140 3136 3141 .. image:: img/image040.GIF … … 3150 3155 REFERENCE 3151 3156 3152 Roe, R.-J., "Methods of X-Ray and Neutron Scattering in 3153 Polymer Science", Oxford University Press, New York (2000). 3157 R J Roe, *Methods of X-Ray and Neutron Scattering in Polymer Science*, Oxford University Press, New York (2000) 3154 3158 3155 3159 … … 3159 3163 **2.2.2. BroadPeakModel** 3160 3164 3161 Calculate an empirical functional form for SANS data characterized by a 3162 broad scattering peak. Many SANS spectra are characterized by a broad 3163 peak even though they are from amorphous soft materials. The d-spacing 3164 corresponding to the broad peak is a characteristic distance between the 3165 scattering inhomogeneities (such as in lamellar, cylindrical, or 3166 spherical morphologiesor for bicontinuous structures).3165 This model calculates an empirical functional form for SANS data characterized by a broad scattering peak. Many SANS 3166 spectra are characterized by a broad peak even though they are from amorphous soft materials. For example, soft systems 3167 that show a SANS peak include copolymers, polyelectrolytes, multiphase systems, layered structures, etc. 3168 3169 The d-spacing corresponding to the broad peak is a characteristic distance between the scattering inhomogeneities (such 3170 as in lamellar, cylindrical, or spherical morphologies, or for bicontinuous structures). 3167 3171 3168 3172 The returned value is scaled to units of |cm^-1|, absolute scale. 3169 3173 3170 The scattering intensity *I(q)* is calculated by: 3174 *2.2.2.1. Definition* 3175 3176 The scattering intensity *I(q)* is calculated as 3171 3177 3172 3178 .. image:: img/image174.JPG 3173 3179 3174 Here the peak position is related to the d-spacing as Q0 = 2pi/d0. Soft 3175 systems that show a SANS peak include copolymers, polyelectrolytes, 3176 multiphase systems, layered structures, etc. 3177 3178 For 2D plot, the wave transfer is defined as 3180 Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*. 3181 3182 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3179 3183 3180 3184 .. image:: img/image040.GIF 3181 3185 3182 =============== ======== =============3183 Parameter name Units Default value3184 =============== ======== =============3185 scale_l (=C) None 103186 scale_p (=A) None 1e-053187 length_l (=x) |Ang| 503188 q_peak (=Q0) |Ang^-1| 0.13189 exponent_p (=n) None 23190 exponent_l (=m) None 33191 Background (=B) |cm^-1| 0.13192 =============== ======== =============3186 ================== ======== ============= 3187 Parameter name Units Default value 3188 ================== ======== ============= 3189 scale_l (=C) None 10 3190 scale_p (=A) None 1e-05 3191 length_l (= |xi| ) |Ang| 50 3192 q_peak (=Q0) |Ang^-1| 0.1 3193 exponent_p (=n) None 2 3194 exponent_l (=m) None 3 3195 Background (=B) |cm^-1| 0.1 3196 ================== ======== ============= 3193 3197 3194 3198 .. image:: img/image175.JPG … … 3200 3204 None. 3201 3205 3202 *2013/09/09 - Description reviewed by King, S .and Parker, P.*3206 *2013/09/09 - Description reviewed by King, S and Parker, P.* 3203 3207 3204 3208 … … 3208 3212 **2.2.3. CorrLength (Correlation Length Model)** 3209 3213 3210 Calculate an empirical functional form for SANS data characterized by a 3211 low-Q signal and a high-Q signal 3214 Calculates an empirical functional form for SANS data characterized by a low-Q signal and a high-Q signal. 3212 3215 3213 3216 The returned value is scaled to units of |cm^-1|, absolute scale. 3214 3217 3215 The scattering intensity *I(q)* is calculated by: 3218 *2.2.3. Definition* 3219 3220 The scattering intensity *I(q)* is calculated as 3216 3221 3217 3222 .. image:: img/image176.JPG 3218 3223 3219 The first term describes Porod scattering from clusters (exponent = n) 3220 and the second term is a Lorentzian function describing scattering from 3221 polymer chains (exponent = m). This second term characterizes the 3222 polymer/solvent interactions and therefore the thermodynamics. The two 3223 multiplicative factors A and C, the incoherent background B and the two 3224 exponents n and m are used as fitting parameters. The final parameter 3225 (xi) is a correlation length for the polymer chains. Note that when m = 3226 2, this functional form becomes the familiar Lorentzian function. 3227 3228 For 2D plot, the wave transfer is defined as 3224 The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function 3225 describing scattering from polymer chains (exponent = *m*). This second term characterizes the polymer/solvent 3226 interactions and therefore the thermodynamics. The two multiplicative factors *A* and *C*, the incoherent 3227 background *B* and the two exponents *n* and *m* are used as fitting parameters. The final parameter |xi| is a 3228 correlation length for the polymer chains. Note that when *m*\ =2 this functional form becomes the familiar Lorentzian 3229 function. 3230 3231 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3229 3232 3230 3233 .. image:: img/image040.GIF 3231 3234 3232 =============== ======== =============3233 Parameter name Units Default value3234 =============== ======== =============3235 scale_l (=C) None  103236 scale_p (=A) None  1e-063237 length_l (= x) |Ang| 503238 exponent_p (=n) None  23239 exponent_l (=m) None 33240 Background (=B) |cm^-1| 0.13241 =============== ======== =============3235 ==================== ======== ============= 3236 Parameter name Units Default value 3237 ==================== ======== ============= 3238 scale_l (=C) None  10 3239 scale_p (=A) None  1e-06 3240 length_l (= |xi| ) |Ang| 50 3241 exponent_p (=n) None  2 3242 exponent_l (=m) None 3 3243 Background (=B) |cm^-1| 0.1 3244 ==================== ======== ============= 3242 3245 3243 3246 .. image:: img/image177.JPG … … 3247 3250 REFERENCE 3248 3251 3249 B . Hammouda, D.L. Ho and S.R. Kline, Insight into Clustering in3250 Poly(ethylene oxide) Solutions, Macromolecules 37, 6932-6937 (2004). 3251 3252 *2013/09/09 - Description reviewed by King, S .and Parker, P.*3252 B Hammouda, D L Ho and S R Kline, *Insight into Clustering in Poly(ethylene oxide) Solutions*, *Macromolecules*, 37 3253 (2004) 6932-6937 3254 3255 *2013/09/09 - Description reviewed by King, S and Parker, P.* 3253 3256 3254 3257 … … 3258 3261 **2.2.4. Lorentz (Ornstein-Zernicke Model)** 3259 3262 3260 The Ornstein-Zernicke model is defined by: 3263 *2.2.4.1. Definition* 3264 3265 The Ornstein-Zernicke model is defined by 3261 3266 3262 3267 .. image:: img/image178.PNG 3263 3268 3264 The parameter L is referred to as the screening length.3265 3266 For 2D plot, the wave transfer is defined as3269 The parameter *L* is the screening length. 3270 3271 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3267 3272 3268 3273 .. image:: img/image040.GIF … … 3278 3283 .. image:: img/image179.JPG 3279 3284 3280 ** Figure. 1D plot using the default values (w/200 data point).** 3285 * Figure. 1D plot using the default values (w/200 data point).* 3286 3287 REFERENCE 3288 3289 None. 3281 3290 3282 3291 … … 3286 3295 **2.2.5. DABModel (Debye-Anderson-Brumberger Model)** 3287 3296 3288 Calculates the scattering from a randomly distributed, two-phase system 3289 based on the Debye-Anderson-Brumberger (DAB) model for such systems. The 3290 two-phase system is characterized by a single length scale, the 3291 correlation length, which is a measure of the average spacing between 3292 regions of phase 1 and phase 2. The model also assumes smooth interfaces 3293 between the phases and hence exhibits Porod behavior (I ~ Q-4) at large 3294 Q (Q\*correlation length >> 1). 3297 Calculates the scattering from a randomly distributed, two-phase system based on the Debye-Anderson-Brumberger (DAB) 3298 model for such systems. The two-phase system is characterized by a single length scale, the correlation length, which 3299 is a measure of the average spacing between regions of phase 1 and phase 2. **The model also assumes smooth interfaces** 3300 **between the phases** and hence exhibits Porod behavior (I ~ *q*\ :sup:`-4`) at large *q* (*QL* >> 1). 3301 3302 The DAB model is ostensibly a development of the earlier Debye-Bueche model. 3303 3304 *2.2.5.1. Definition* 3295 3305 3296 3306 .. image:: img/image180.PNG 3297 3307 3298 The parameter L is referred to as the correlation length.3299 3300 For 2D plot, the wave transfer is defined as3308 The parameter *L* is the correlation length. 3309 3310 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3301 3311 3302 3312 .. image:: img/image040.GIF … … 3312 3322 .. image:: img/image181.JPG 3313 3323 3314 ** Figure. 1D plot using the default values (w/200 data point).** 3315 3316 REFERENCE 3317 3318 Debye, Anderson, Brumberger, "Scattering by an Inhomogeneous Solid. II. 3319 The Correlation Function and its Application", J. Appl. Phys. 28 (6), 3320 679 (1957). 3321 3322 Debye, Bueche, "Scattering by an Inhomogeneous Solid", J. Appl. Phys. 3323 20, 518 (1949). 3324 3325 *2013/09/09 - Description reviewed by King, S. and Parker, P.* 3324 * Figure. 1D plot using the default values (w/200 data point).* 3325 3326 REFERENCE 3327 3328 P Debye, H R Anderson, H Brumberger, *Scattering by an Inhomogeneous Solid. II. The Correlation Function* 3329 *and its Application*, *J. Appl. Phys.*, 28(6) (1957) 679 3330 3331 P Debye, A M Bueche, *Scattering by an Inhomogeneous Solid*, *J. Appl. Phys.*, 20 (1949) 518 3332 3333 *2013/09/09 - Description reviewed by King, S and Parker, P.* 3326 3334 3327 3335 … … 3331 3339 **2.2.6. AbsolutePower_Law** 3332 3340 3333 This model describes a power law with background.3341 This model describes a simple power law with background. 3334 3342 3335 3343 .. image:: img/image182.PNG 3336 3344 3337 Note the minus sign in front of the exponent. 3345 Note the minus sign in front of the exponent. The parameter *m* should therefore be entered as a **positive** number. 3338 3346 3339 3347 ============== ======== ============= … … 3349 3357 *Figure. 1D plot using the default values (w/200 data point).* 3350 3358 3351 3352 3353 .. _Teubner Strey: 3354 3355 **2.2.7. Teubner Strey (Model)** 3356 3357 This function calculates the scattered intensity of a two-component 3358 system using the Teubner-Strey model. 3359 REFERENCE 3360 3361 None. 3362 3363 3364 3365 .. _TeubnerStrey: 3366 3367 **2.2.7. TeubnerStrey (Model)** 3368 3369 This function calculates the scattered intensity of a two-component system using the Teubner-Strey model. Unlike the 3370 DABModel_ this function generates a peak. 3371 3372 *2.2.7.1. Definition* 3359 3373 3360 3374 .. image:: img/image184.PNG 3361 3375 3362 For 2D plot, the wave transfer is defined as3376 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3363 3377 3364 3378 .. image:: img/image040.GIF … … 3379 3393 REFERENCE 3380 3394 3381 Teubner, M; Strey, R. J. Chem. Phys., 87, 3195 (1987). 3382 3383 Schubert, K-V., Strey, R., Kline, S. R. and E. W. Kaler, J. Chem. Phys., 3384 101, 5343 (1994). 3395 M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195 3396 3397 K V Schubert, R Strey, S R Kline and E W Kaler, *J. Chem. Phys.*, 101 (1994) 5343 3385 3398 3386 3399 … … 3390 3403 **2.2.8. FractalModel** 3391 3404 3392 Calculates the scattering from fractal-like aggregates built from 3393 spherical building blocks following the Texiera reference. The value 3394 returned is in cm-1. 3405 Calculates the scattering from fractal-like aggregates built from spherical building blocks following the Texiera 3406 reference. 3407 3408 The value returned is in |cm^-1|\ . 3409 3410 *2.2.8.1. Definition* 3395 3411 3396 3412 .. image:: img/image186.PNG 3397 3413 3398 The scale parameter is the volume fraction of the building blocks, R0 is 3399 the radius of the building block, Df is the fractal dimension, ß is the 3400 correlation length, *Ãï¿œsolvent* is the scattering length density of the 3401 solvent, and *Ãï¿œblock* is the scattering length density of the building 3402 blocks. 3403 3404 **The polydispersion in radius is provided.** 3405 3406 For 2D plot, the wave transfer is defined as 3414 The *scale* parameter is the volume fraction of the building blocks, *R0* is the radius of the building block, *Df* is 3415 the fractal dimension, |xi| is the correlation length, |rho|\ *solvent* is the scattering length density of the 3416 solvent, and |rho|\ *block* is the scattering length density of the building blocks. 3417 3418 **Polydispersity on the radius is provided for.** 3419 3420 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3407 3421 3408 3422 .. image:: img/image040.GIF … … 3426 3440 REFERENCE 3427 3441 3428 J . Teixeira, (1988) J. Appl. Cryst., vol. 21, p781-7853442 J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785 3429 3443 3430 3444 … … 3434 3448 **2.2.9. MassFractalModel** 3435 3449 3436 Calculates the scattering from fractal-like aggregates based on the 3437 Mildner reference (below). 3450 Calculates the scattering from fractal-like aggregates based on the Mildner reference. 3451 3452 *2.2.9.1. Definition* 3438 3453 3439 3454 .. image:: img/mass_fractal_eq1.JPG 3440 3455 3441 The R is the radius of the building block, Dm is the mass fractal 3442 dimension, ß is the correlation (or cutt-off) length, *Ãï¿œsolvent* is the 3443 scattering length density of the solvent, and *Ãï¿œparticle* is the 3444 scattering length density of particles. 3445 3446 Note:  The mass fractal dimension is valid for 1<mass_dim<6.3456 where *R* is the radius of the building block, *Dm* is the **mass** fractal dimension, |zeta| is the cut-off length, 3457 |rho|\ *solvent* is the scattering length density of the solvent, and |rho|\ *particle* is the scattering length 3458 density of particles. 3459 3460 Note:  The mass fractal dimension *Dm* is only valid if 1 < mass_dim < 6. It is also only valid over a limited 3461 *q* range (see the reference for details). 3447 3462 3448 3463 ============== ======== ============= … … 3458 3473 .. image:: img/mass_fractal_fig1.JPG 3459 3474 3460 *Figure. 1D plot *3461 3462 REFERENCE 3463 3464 D . Mildner, and P. Hall, J. Phys. D.: Appl. Phys., 19, 1535-1545Â3465 (1986), Equation(9). 3466 3467 *2013/09/09 - Description reviewed by King, S .and Parker, P.*3475 *Figure. 1D plot using default values.* 3476 3477 REFERENCE 3478 3479 D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 3480 Equation(9) 3481 3482 *2013/09/09 - Description reviewed by King, S and Parker, P.* 3468 3483 3469 3484 … … 3473 3488 **2.2.10. SurfaceFractalModel** 3474 3489 3475 Calculates the scattering based on the Mildner reference (below). 3490 Calculates the scattering from fractal-like aggregates based on the Mildner reference. 3491 3492 *2.2.10.1. Definition* 3476 3493 3477 3494 .. image:: img/surface_fractal_eq1.GIF 3478 3495 3479 The R is the radius of the building block, Ds is the surface fractal 3480 dimension, ß is the correlation (or cutt-off) length, *Ãï¿œsolvent* is the 3481 scattering length density of the solvent, and *Ãï¿œparticle* is the 3482 scattering length density of particles. 3483 3484  Note:  The surface fractal dimension is valid for 1<surface_dim<3. 3485  Also it is valid in a limited q range (see the reference for details). 3496 where *R* is the radius of the building block, *Ds* is the **surface** fractal dimension, |zeta| is the cut-off length, 3497 |rho|\ *solvent* is the scattering length density of the solvent, and |rho|\ *particle* is the scattering length 3498 density of particles. 3499 3500 Note:  The surface fractal dimension *Ds* is only valid if 1 < surface_dim < 3. It is also only valid over a limited 3501 *q* range (see the reference for details). 3486 3502 3487 3503 ============== ======== ============= … … 3497 3513 .. image:: img/surface_fractal_fig1.JPG 3498 3514 3499 *Figure. 1D plot *3500 3501 REFERENCE 3502 3503 D . Mildner, and P. Hall, J. Phys. D.: Appl. Phys., 19, 1535-1545Â3504 (1986), Equation(13). 3515 *Figure. 1D plot using default values.* 3516 3517 REFERENCE 3518 3519 D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 3520 Equation(13) 3505 3521 3506 3522 … … 3510 3526 **2.2.11. MassSurfaceFractal (Model)** 3511 3527 3512 A number of natural and commercial processes form high-surface area 3513 materials as a result of the vapour-phase aggregation of primary 3514 particles. Examples of such materials include soots, aerosols, and 3515 fume or pyrogenic silicas. These are all characterised by cluster mass 3516 distributions (sometimes also cluster size distributions) and internal 3517 surfaces that are fractal in nature.  The scattering from such 3518 materials displays two distinct breaks in log-log representation, 3519 corresponding to the radius-of-gyration of the primary particles, rg, 3520 and the radius-of-gyration of the clusters (aggregates), Rg. Between 3521 these boundaries the scattering follows a power law related to the mass 3522 fractal dimension, Dm, whilst above the high-Q boundary the scattering 3523 follows a power law related to the surface fractal dimension of the 3524 primary particles, Ds. 3525 3526 The scattered intensity *I(q)* is then calculated using a modified 3527 Ornstein-Zernicke equation: 3528 A number of natural and commercial processes form high-surface area materials as a result of the vapour-phase 3529 aggregation of primary particles. Examples of such materials include soots, aerosols, and fume or pyrogenic silicas. 3530 These are all characterised by cluster mass distributions (sometimes also cluster size distributions) and internal 3531 surfaces that are fractal in nature. The scattering from such materials displays two distinct breaks in log-log 3532 representation, corresponding to the radius-of-gyration of the primary particles, *rg*, and the radius-of-gyration of 3533 the clusters (aggregates), *Rg*. Between these boundaries the scattering follows a power law related to the mass 3534 fractal dimension, *Dm*, whilst above the high-Q boundary the scattering follows a power law related to the surface 3535 fractal dimension of the primary particles, *Ds*. 3536 3537 *2.2.11.1. Definition* 3538 3539 The scattered intensity *I(q)* is calculated using a modified Ornstein-Zernicke equation 3528 3540 3529 3541 .. image:: img/masssurface_fractal_eq1.JPG 3530 3542 3531 The Rg is for the cluster, rg is for the primary, Ds is the surface 3532 fractal dimension, Dm is the mass fractal dimension, *Ãï¿œsolvent* is the 3533 scattering length density of the solvent, and *Ãï¿œp* is the scattering 3534 length density of particles. 3535 3536  Note:  The surface and mass fractal dimensions are valid for 3537 0<surface_dim<6, 0<mass_dim<6, and (surface_mass+mass_dim)<6. 3543 where *Rg* is the size of the cluster, *rg* is the size of the primary particle, *Ds* is the surface fractal dimension, 3544 *Dm* is the mass fractal dimension, |rho|\ *solvent* is the scattering length density of the solvent, and |rho|\ *p* is 3545 the scattering length density of particles. 3546 3547 Note:  The surface (*Ds*) and mass (*Dm*) fractal dimensions are only valid if 0 < *surface_dim* < 6, 3548 0 < *mass_dim* < 6, and (*surface_dim*+*mass_dim*) < 6. 3538 3549 3539 3550 ============== ======== ============= … … 3550 3561 .. image:: img/masssurface_fractal_fig1.JPG 3551 3562 3552 *Figure. 1D plot* 3553 3554 REFERENCE 3555 3556 P. Schmidt, J Appl. Cryst., 24, 414-435 (1991), Equation(19). 3557 3558 Hurd, Schaefer, Martin, Phys. Rev. A, 35, 2361-2364 (1987), Equation(2). 3563 *Figure. 1D plot using default values.* 3564 3565 REFERENCE 3566 3567 P Schmidt, *J Appl. Cryst.*, 24 (1991) 414-435 3568 Equation(19) 3569 3570 A J Hurd, D W Schaefer, J E Martin, *Phys. Rev. A*, 35 (1987) 2361-2364 3571 Equation(2) 3559 3572 3560 3573 … … 3564 3577 **2.2.12. FractalCoreShell (Model)** 3565 3578 3566 Calculates the scattering from a fractal structure with a primary 3567 building block of core-shell spheres. 3579 Calculates the scattering from a fractal structure with a primary building block of core-shell spheres, as opposed to 3580 just homogeneous spheres in the FractalModel_. This model could find use for aggregates of coated particles, or 3581 aggregates of vesicles. 3582 3583 The returned value is scaled to units of |cm^-1|, absolute scale. 3584 3585 *2.2.12.1. Definition* 3568 3586 3569 3587 .. image:: img/fractcore_eq1.GIF 3570 3588 3571 The formfactor P(q) is `CoreShellModel <#CoreShellModel>`__ with bkg 3572 = 0, 3589 The form factor *P(q)* is that from CoreShellModel_ with *bkg* = 0 3573 3590 3574 3591 .. image:: img/image013.PNG 3575 3592 3576 while the fractal structure factor S(q) ;3593 while the fractal structure factor S(q) is 3577 3594 3578 3595 .. image:: img/fractcore_eq3.gif 3579 3596 3580 where Df = frac_dim, ß = cor_length, rc = (core) radius, and scale 3581 = volfraction. 3582 3583 The fractal structure is as documented in the fractal model. This model 3584 could find use for aggregates of coated particles, or aggregates of 3585 vesicles. The polydispersity computation of radius and thickness is 3586 provided. 3587 3588 The returned value is scaled to units of |cm^-1|, absolute scale. 3589 3590 See each of these individual models for full documentation. 3591 3592 For 2D plot, the wave transfer is defined as 3597 where *Df* = frac_dim, |xi| = cor_length, *rc* = (core) radius, and *scale* = volume fraction. 3598 3599 The fractal structure is as documented in the FractalModel_. Polydispersity of radius and thickness is provided for. 3600 3601 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3593 3602 3594 3603 .. image:: img/image040.GIF … … 3614 3623 REFERENCE 3615 3624 3616 See the PolyCore and Fractal documentation.\3625 See the CoreShellModel_ and FractalModel_ descriptions. 3617 3626 3618 3627 … … 3622 3631 **2.2.13. GaussLorentzGel(Model)** 3623 3632 3624 Calculates the scattering from a gel structure, typically a physical 3625 network. It is modeled as a sum of a low-q exponential decay plus a 3626 lorentzian at higher q-values. It is generally applicable to gel 3627 structures. 3633 Calculates the scattering from a gel structure, but typically a physical rather than chemical network. It is modeled as 3634 a sum of a low-*q* exponential decay plus a lorentzian at higher *q*-values. 3628 3635 3629 3636 The returned value is scaled to units of |cm^-1|, absolute scale. 3630 3637 3631 The scattering intensity *I(q)* is calculated as (eqn 5 from the 3632 reference): 3638 *2.2.13.1. Definition* 3639 3640 The scattering intensity *I(q)* is calculated as (eqn 5 from the reference) 3633 3641 3634 3642 .. image:: img/image189.JPG 3635 3643 3636 Uppercase Zeta is the static correlations in the gel, which can be 3637 attributed to the "frozen-in" crosslinks of some gels. Lowercase zeta is 3638 the dynamic correlation length, which can be attributed to the 3639 fluctuating polymer chain between crosslinks. IG(0) and IL(0) are the 3640 scaling factors for each of these structures. Your physical system may 3641 be different, so think about the interpretation of these parameters. 3642 3643 Note that the peaked structure at higher q values (from Figure 2 of the 3644 reference below) is not reproduced by the model. Peaks can be introduced 3645 into the model by summing this model with the PeakGauss Model function. 3646 3647 For 2D plot, the wave transfer is defined as 3644 |bigzeta| is the length scale of the static correlations in the gel, which can be attributed to the "frozen-in" 3645 crosslinks. |xi| is the dynamic correlation length, which can be attributed to the fluctuating polymer chains between 3646 crosslinks. *I*\ :sub:`G`\ *(0)* and *I*\ :sub:`L`\ *(0)* are the scaling factors for each of these structures. **Think carefully about how** 3647 **these map to your particular system!** 3648 3649 NB: The peaked structure at higher *q* values (Figure 2 from the reference) is not reproduced by the model. Peaks can 3650 be introduced into the model by summing this model with the PeakGaussModel_ function. 3651 3652 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3648 3653 3649 3654 .. image:: img/image040.GIF … … 3665 3670 REFERENCE 3666 3671 3667 G. Evmenenko, E. Theunissen, K. Mortensen, H. Reynaers, Polymer 42 3668 (2001) 2907-2913. 3672 G Evmenenko, E Theunissen, K Mortensen, H Reynaers, *Polymer*, 42 (2001) 2907-2913 3669 3673 3670 3674 … … 3674 3678 **2.2.14. BEPolyelectrolyte (Model)** 3675 3679 3676 Calculates the structure factor of a polyelectrolyte solution with the 3677 RPA expression derived by Borue and Erukhimovich. The value returned is 3678 in cm-1. 3680 Calculates the structure factor of a polyelectrolyte solution with the RPA expression derived by Borue and Erukhimovich. 3681 3682 The value returned is in |cm^-1|. 3683 3684 *2.2.14.1. Definition* 3679 3685 3680 3686 .. image:: img/image191.PNG 3681 3687 3682 K is a contrast factor of the polymer, Lb is the Bjerrum length, h is 3683 the virial parameter, b is the monomer length, Cs is the concentration 3684 of monovalent salt, ñ is the ionization degree, Ca is the polymer molar 3685 concentration, and background is the incoherent background. 3686 3687 For 2D plot, the wave transfer is defined as 3688 where *K* is the contrast factor for the polymer, *Lb* is the Bjerrum length, *h* is the virial parameter, *b* is the 3689 monomer length, *Cs* is the concentration of monovalent salt, |alpha| is the ionization degree, *Ca* is the polymer 3690 molar concentration, and *background* is the incoherent background. 3691 3692 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3688 3693 3689 3694 .. image:: img/image040.GIF … … 3706 3711 REFERENCE 3707 3712 3708 Borue, V. Y., Erukhimovich, I. Y. Macromolecules 21, 3240 (1988). 3709 3710 Joanny, J.-F., Leibler, L. Journal de Physique 51, 545 (1990). 3711 3712 Moussaid, A., Schosseler, F., Munch, J.-P., Candau, S. J. Journal de 3713 Physique II France 3, 573 (1993). 3714 3715 Raphaël, E., Joanny, J.-F., Europhysics Letters 11, 179 (1990). 3713 V Y Borue, I Y Erukhimovich, *Macromolecules*, 21 (1988) 3240 3714 3715 J F Joanny, L Leibler, *Journal de Physique*, 51 (1990) 545 3716 3717 A Moussaid, F Schosseler, J P Munch, S Candau, *J. Journal de Physique II France*, 3 (1993) 573 3718 3719 E Raphael, J F Joanny, *Europhysics Letters*, 11 (1990) 179 3716 3720 3717 3721 … … 3721 3725 **2.2.15. Guinier (Model)** 3722 3726 3723 A Guinier analysis is done by linearizing the data at low q by plotting 3724 it as log(I) versus Q2. The Guinier radius Rg can be obtained by fitting 3725 the following model: 3727 This model fits the Guinier function 3726 3728 3727 3729 .. image:: img/image192.PNG 3728 3730 3729 For 2D plot, the wave transfer is defined as 3731 to the data directly without any need for linearisation (*cf*. Ln *I(q)* vs *q*\ :sup:`2`). 3732 3733 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3730 3734 3731 3735 .. image:: img/image040.GIF … … 3738 3742 ============== ======== ============= 3739 3743 3744 REFERENCE 3745 3746 A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley & Sons, New York (1955) 3747 3740 3748 3741 3749 … … 3744 3752 **2.2.16. GuinierPorod (Model)** 3745 3753 3746 Calculates the scattering for a generalized Guinier/power law object. 3747 This is an empirical model that can be used to determine the size and 3748 dimensionality of scattering objects. 3749 3750 The returned value is P(Q) as written in equation (1), plus the 3751 incoherent background term. The result is in the units of |cm^-1|, 3752 absolute scale. 3753 3754 A Guinier-Porod empirical model can be used to fit SAS data from 3755 asymmetric objects such as rods or platelets. It also applies to 3756 intermediate shapes between spheres and rod or between rods and 3757 platelets. The following functional form is used: 3758 3759 .. image:: img/image193.JPG  (1) 3760 3761 This is based on the generalized Guinier law for such elongated objects 3762 [2]. For 3D globular objects (such as spheres), s = 0 and one recovers 3763 the standard Guinier formula. For 2D symmetry (such as for rods) s = 1 3764 and for 1D symmetry (such as for lamellae or platelets) s = 2. A 3765 dimensionality parameter 3-s is defined, and is 3 for spherical objects, 3766 2 for rods, and 1 for plates. 3767 3768 Enforcing the continuity of the Guinier and Porod functions and their 3769 derivatives yields: 3754 Calculates the scattering for a generalized Guinier/power law object. This is an empirical model that can be used to 3755 determine the size and dimensionality of scattering objects, including asymmetric objects such as rods or platelets, and 3756 shapes intermediate between spheres and rods or between rods and platelets. 3757 3758 The result is in the units of |cm^-1|, absolute scale. 3759 3760 *2.2.16.1 Definition* 3761 3762 The following functional form is used 3763 3764 .. image:: img/image193.JPG 3765 3766 This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D 3767 globular objects (such as spheres), *s* = 0 and one recovers the standard Guinier_ formula. For 2D symmetry (such as 3768 for rods) *s* = 1, and for 1D symmetry (such as for lamellae or platelets) *s* = 2. A dimensionality parameter (3-*s*) 3769 is thus defined, and is 3 for spherical objects, 2 for rods, and 1 for plates. 3770 3771 Enforcing the continuity of the Guinier and Porod functions and their derivatives yields 3770 3772 3771 3773 .. image:: img/image194.JPG … … 3775 3777 .. image:: img/image195.JPG 3776 3778 3777 Note that the radius of gyration for a sphere of radius R is given by Rg 3778 = R sqrt(3/5) , 3779 3780  that for the cross section of an randomly oriented cylinder of radius R 3781 is given by Rg = R / sqrt(2). 3782 3783 The cross section of a randomly oriented lamella of thickness T is given 3784 by Rg = T / sqrt(12). 3785 3786 The intensity given by Eq. 1 is the calculated result, and is plotted 3787 below for the default parameter values. 3788 3789 REFERENCE 3790 3791 [1] Guinier, A.; Fournet, G. "Small-Angle Scattering of X-Rays", John 3792 Wiley and Sons, New York, (1955). 3793 3794 [2] Glatter, O.; Kratky, O., Small-Angle X-Ray Scattering, Academic 3795 Press (1982). Check out Chapter 4 on Data Treatment, pages 155-156. 3796 3797 For 2D plot, the wave transfer is defined as 3779 Note that 3780 3781 the radius of gyration for a sphere of radius *R* is given by *Rg* = *R* sqrt(3/5) 3782 3783  the cross-sectional radius of gyration for a randomly oriented cylinder of radius *R* is given by *Rg* = *R* / sqrt(2) 3784 3785 the cross-sectional radius of gyration of a randomly oriented lamella of thickness *T* is given by *Rg* = *T* / sqrt(12) 3786 3787 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3798 3788 3799 3789 .. image:: img/image008.PNG … … 3813 3803 *Figure. 1D plot using the default values (w/500 data points).* 3814 3804 3805 REFERENCE 3806 3807 A Guinier, G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 3808 3809 O Glatter, O Kratky, *Small-Angle X-Ray Scattering*, Academic Press (1982) 3810 Check out Chapter 4 on Data Treatment, pages 155-156. 3811 3815 3812 3816 3813 … … 3830 3827 The background term is added for data analysis. 3831 3828 3832 For 2D plot, the wave transfer is defined as3829 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3833 3830 3834 3831 .. image:: img/image040.GIF … … 3860 3857 None 3861 3858 3862 For 2D plot, the wave transfer is defined as3859 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3863 3860 3864 3861 .. image:: img/image040.GIF … … 3896 3893 None 3897 3894 3898 For 2D plot, the wave transfer is defined as3895 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3899 3896 3900 3897 .. image:: img/image040.GIF … … 3940 3937 The polydispersion in rg is provided. 3941 3938 3942 For 2D plot, the wave transfer is defined as3939 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3943 3940 3944 3941 .. image:: img/image040.GIF … … 3946 3943 TEST DATASET 3947 3944 3948 This example dataset is produced by running the Poly_GaussCoil, using 200 data points, *qmin* = 0.001 |Ang^-1| \,3945 This example dataset is produced by running the Poly_GaussCoil, using 200 data points, *qmin* = 0.001 |Ang^-1| , 3949 3946 qmax = 0.7 |Ang^-1| and the default values below. 3950 3947 … … 3964 3961 REFERENCE 3965 3962 3966 Glatter & Kratky - pg.404. 3967 3968 J.S. Higgins, and H.C. Benoit, Polymers and Neutron Scattering, Oxford 3969 Science Publications (1996). 3963 Glatter & Kratky - p404 3964 3965 J S Higgins, and H C Benoit, Polymers and Neutron Scattering, Oxford Science Publications (1996) 3970 3966 3971 3967 … … 4038 4034 REFERENCE 4039 4035 4040 Benoit, H., Comptes Rendus (1957). 245, 2244-2247. 4041 4042 Hammouda, B., SANS from Homogeneous Polymer Mixtures  A Unified 4043 Overview, Advances in Polym. Sci. (1993), 106, 87-133. 4044 4045 For 2D plot, the wave transfer is defined as 4036 H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247 4037 4038 B Hammouda, *SANS from Homogeneous Polymer Mixtures  A Unified Overview*, *Advances in Polym. Sci.*, 106 (1993) 87-133 4039 4040 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4046 4041 4047 4042 .. image:: img/image040.GIF … … 4049 4044 TEST DATASET 4050 4045 4051 This example dataset is produced, using 200 data points, qmin = 0.001 |Ang^-1| \, qmax = 0.2 |Ang^-1|  and the4046 This example dataset is produced, using 200 data points, qmin = 0.001 |Ang^-1| , qmax = 0.2 |Ang^-1|  and the 4052 4047 default values below. 4053 4048 … … 4122 4117 REFERENCE 4123 4118 4124 A .Z. Akcasu, R. Klein and B. Hammouda, Macromolecules 26, 4136 (1993)4119 A Z Akcasu, R Klein and B Hammouda, *Macromolecules*, 26 (1993) 4136 4125 4120 4126 4121 Fitting parameters for Case0 Model … … 4180 4175 The background term is added for data analysis. 4181 4176 4182 For 2D plot, the wave transfer is defined as4177 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4183 4178 4184 4179 .. image:: img/image040.GIF … … 4226 4221 Be sure to enter the power law exponents as positive values. 4227 4222 4228 For 2D plot, the wave transfer is defined as4223 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4229 4224 4230 4225 .. image:: img/image040.GIF … … 4258 4253 4259 4254 Otherwise, program incorporates the empirical multiple level unified 4260 Exponential/Power-law fit method developed by G .Beaucage. Four4255 Exponential/Power-law fit method developed by G Beaucage. Four 4261 4256 functions are included so that One, Two, Three, or Four levels can be 4262 4257 used. … … 4277 4272 parameters. 4278 4273 4279 For 2D plot, the wave transfer is defined as4274 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4280 4275 4281 4276 .. image:: img/image040.GIF … … 4304 4299 REFERENCE 4305 4300 4306 G .Beaucage (1995). J. Appl. Cryst., vol. 28, p717-728.4307 4308 G .Beaucage (1996). J. Appl. Cryst., vol. 29, p134-146.4301 G Beaucage (1995). J. Appl. Cryst., vol. 28, p717-728. 4302 4303 G Beaucage (1996). J. Appl. Cryst., vol. 29, p134-146. 4309 4304 4310 4305 … … 4434 4429 REFERENCE 4435 4430 4436 Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C. Han, J. Chem. Phys. 4437 1992, 97 (9), 6829-6841. 4438 4439 Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R. Rennie, Erik 4440 Geissler, Macromolecules 1991, 24, 543-548. 4431 Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C Han, J. Chem. Phys. 1992, 97 (9), 6829-6841 4432 4433 Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie, Erik Geissler, Macromolecules 1991, 24, 543-548 4441 4434 4442 4435 … … 4458 4451 REFERENCE 4459 4452 4460 H .Benoit,  J. Polymer Science., 11, 596-599 (1953)4453 H Benoit,  J. Polymer Science., 11, 596-599 (1953) 4461 4454 4462 4455 … … 4497 4490 REFERENCE 4498 4491 4499 J . K. Percus, J.Yevick, *J. Phys. Rev.*, 110, (1958) 14492 J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1 4500 4493 4501 4494 … … 4521 4514 where *r* is the distance from the center of the sphere of a radius *R*. 4522 4515 4523 For 2D plot, the wave transfer is defined as4516 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4524 4517 4525 4518 .. image:: img/image040.GIF … … 4540 4533 REFERENCE 4541 4534 4542 R . V. Sharma, K. C.Sharma, *Physica*, 89A (1977) 2134535 R V Sharma, K C Sharma, *Physica*, 89A (1977) 213 4543 4536 4544 4537 … … 4560 4553 multivalent salts. The counterions are also assumed to be monovalent. 4561 4554 4562 For 2D plot, the wave transfer is defined as4555 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4563 4556 4564 4557 .. image:: img/image040.gif … … 4581 4574 REFERENCE 4582 4575 4583 J . B. Hayter and J.Penfold, *Molecular Physics*, 42 (1981) 109-1184584 4585 J . P. Hansen and J. B.Hayter, *Molecular Physics*, 46 (1982) 651-6564576 J B Hayter and J Penfold, *Molecular Physics*, 42 (1981) 109-118 4577 4578 J P Hansen and J B Hayter, *Molecular Physics*, 46 (1982) 651-656 4586 4579 4587 4580 … … 4619 4612 until the optimization does not hit the constraints. 4620 4613 4621 For 2D plot, the wave transfer is defined as4614 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4622 4615 4623 4616 .. image:: img/image040.GIF … … 4638 4631 REFERENCE 4639 4632 4640 S . V. G. Menon, C. Manohar, and K. S.Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-91904633 S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 4641 4634 4642 4635
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