Changeset eacf6a8c in sasview
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src/sans/models/media/model_functions.rst (modified) (137 diffs)
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src/sans/models/media/model_functions.rst
r2005bb5 reacf6a8c 25 25 .. |pi| unicode:: U+03C0 26 26 .. |rho| unicode:: U+03C1 27 .. |sigma| unicode:: U+03C 227 .. |sigma| unicode:: U+03C3 28 28 .. |tau| unicode:: U+03C4 29 29 .. |upsilon| unicode:: U+03C5 … … 32 32 .. |psi| unicode:: U+03C8 33 33 .. |omega| unicode:: U+03C9 34 35 .. |bigdelta| unicode:: U+0394 36 .. |biggamma| unicode:: U+0393 37 38 .. |drho| replace:: |bigdelta|\ |rho| 34 39 35 40 .. |Ang| unicode:: U+212B … … 43 48 .. |cm^3| replace:: cm\ :sup:`3` 44 49 .. |cm^-3| replace:: cm\ :sup:`-3` 50 .. |sr^-1| replace:: sr\ :sup:`-1` 45 51 46 52 .. |P0| replace:: P\ :sub:`0`\ 53 .. |A2| replace:: A\ :sub:`2`\ 47 54 48 55 … … 140 147 - SphereModel_ (including magnetic 2D version) 141 148 - BinaryHSModel_ 142 - FuzzySphereModel 143 - RaspBerryModel 144 - CoreShellModel (including magnetic 2D version)145 - CoreMultiShellModel (including magnetic 2D version)146 - Core2ndMomentModel 147 - MultiShellModel 148 - OnionExpShellModel 149 - VesicleModel 150 - SphericalSLDModel 151 - LinearPearlsModel 152 - PearlNecklaceModel 149 - FuzzySphereModel_ 150 - RaspBerryModel_ 151 - CoreShellModel_ (including magnetic 2D version) 152 - CoreMultiShellModel_ (including magnetic 2D version) 153 - Core2ndMomentModel_ 154 - MultiShellModel_ 155 - OnionExpShellModel_ 156 - VesicleModel_ 157 - SphericalSLDModel_ 158 - LinearPearlsModel_ 159 - PearlNecklaceModel_ 153 160 154 161 Cylinder-based … … 247 254 ------------------------ 248 255 249 - testmodel 250 - testmodel_2 251 - sum_p1_p2 252 - sum_Ap1_1_Ap2 253 - polynomial5 254 - sph_bessel_jn 256 - testmodel_ 257 - testmodel_2_ 258 - sum_p1_p2_ 259 - sum_Ap1_1_Ap2_ 260 - polynomial5_ 261 - sph_bessel_jn_ 255 262 256 263 … … 329 336 The 2D scattering intensity is the same as above, regardless of the orientation of the q vector. 330 337 331 The returned value is scaled to units of |cm^-1| and the parameters of the sphere model are the following:338 The returned value is scaled to units of |cm^-1| and the parameters of the SphereModel are the following: 332 339 333 340 ============== ======== ============= … … 344 351 Research (Kline, 2006). 345 352 353 REFERENCE 354 A. Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 355 346 356 *2.1.1.2. Validation of the SphereModel* 347 357 … … 380 390 .. image:: img/image008.PNG 381 391 382 The parameters of the binary hard sphereare the following (in the names, *l* (or *ls*\ ) stands for larger spheres392 The parameters of the BinaryHSModel are the following (in the names, *l* (or *ls*\ ) stands for larger spheres 383 393 while *s* (or *ss*\ ) for the smaller spheres). 384 394 … … 392 402 solvent_sld |Ang^-2| 6e-6 393 403 s_radius |Ang| 25.0 394 vol_frac_ls 395 vol_frac_ss 404 vol_frac_ls None 0.1 405 vol_frac_ss None 0.2 396 406 ============== ======== ============= 397 407 … … 406 416 407 417 REFERENCE 408 N. W. Ashcroft and D. C. Langreth, Physical Review, v. 156 (1967) 685-692 409 410 [Errata found in Phys. Rev. 166 (1968) 934.] 418 N. W. Ashcroft and D. C. Langreth, *Physical Review*, 156 (1967) 685-692 419 [Errata found in *Phys. Rev.* 166 (1968) 934] 411 420 412 421 … … 416 425 **2.1.3. FuzzySphereModel** 417 426 418 **This model is to calculate the scattering from spherical particles 419 with a "fuzzy" interface.** 427 This model is to calculate the scattering from spherical particles with a "fuzzy" interface. 420 428 421 429 *2.1.3.1. Definition* 422 430 431 The scattering intensity *I(q)* is calculated as: 432 433 .. image:: img/image010.PNG 434 435 where the amplitude *A(q)* is given as the typical sphere scattering convoluted with a Gaussian to get a gradual 436 drop-off in the scattering length density 437 438 .. image:: img/image011.PNG 439 440 Here |A2|\ *(q)* is the form factor, *P(q)*. The scale is equivalent to the volume fraction of spheres, each of 441 volume, *V*\. Contrast (|drho|) is the difference of scattering length densities of the sphere and the surrounding 442 solvent. 443 444 Poly-dispersion in radius and in fuzziness is provided for. 445 446 The returned value is scaled to units of |cm^-1|\ |sr^-1|; ie, absolute scale. 447 448 From the reference 449 450 The "fuzziness" of the interface is defined by the parameter |sigma| :sub:`fuzzy`\ . The particle radius *R* 451 represents the radius of the particle where the scattering length density profile decreased to 1/2 of the core 452 density. The |sigma| :sub:`fuzzy`\ is the width of the smeared particle surface; i.e., the standard deviation 453 from the average height of the fuzzy interface. The inner regions of the microgel that display a higher density 454 are described by the radial box profile extending to a radius of approximately *Rbox* ~ *R* - 2\ |sigma|\ . The 455 profile approaches zero as *Rsans* ~ *R* + 2\ |sigma|\ . 456 457 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 458 459 .. image:: img/image008.PNG 460 461 This example dataset is produced by running the FuzzySphereModel, using 200 data points, *qmin* = 0.001 -1, 462 *qmax* = 0.7 |Ang^-1| and the default values 463 464 ============== ======== ============= 465 Parameter name Units Default value 466 ============== ======== ============= 467 scale None 1.0 468 radius |Ang| 60 469 fuzziness |Ang| 10 470 sldSolv |Ang^-2| 3e-6 471 sldSph |Ang^-2| 1e-6 472 background |cm^-1| 0.001 473 ============== ======== ============= 474 475 .. image:: img/image012.JPG 476 477 *Figure. 1D plot using the default values (w/200 data point).* 478 479 REFERENCE 480 M. Stieger, J. S. Pedersen, P. Lindner, W. Richtering, *Langmuir*, 20 (2004) 7283-7292 481 482 483 484 .. _RaspBerryModel: 485 486 **2.1.4. RaspBerryModel** 487 488 Calculates the form factor, *P(q)*, for a "Raspberry-like" structure where there are smaller spheres at the surface 489 of a larger sphere, such as the structure of a Pickering emulsion. 490 491 *2.1.4.1. Definition* 492 493 The structure is: 494 495 .. image:: img/raspberry_pic.JPG 496 497 where *Ro* = the radius of the large sphere, *Rp* = the radius of the smaller sphere on the surface, |delta| = the 498 fractional penetration depth, and surface coverage = fractional coverage of the large sphere surface (0.9 max). 499 500 The large and small spheres have their own SLD, as well as the solvent. The surface coverage term is a fractional 501 coverage (maximum of approximately 0.9 for hexagonally-packed spheres on a surface). Since not all of the small 502 spheres are necessarily attached to the surface, the excess free (small) spheres scattering is also included in the 503 calculation. The function calculated follows equations (8)-(12) of the reference below, and the equations are not 504 reproduced here. 505 506 The returned value is scaled to units of |cm^-1|. No inter-particle scattering is included in this model. 507 508 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 509 510 .. image:: img/image008.PNG 511 512 This example dataset is produced by running the RaspBerryModel, using 2000 data points, *qmin* = 0.0001 |Ang^-1|, 513 *qmax* = 0.2 |Ang^-1| and the default values below, where *Ssph/Lsph* stands for smaller or larger sphere, respectively, 514 and *surfrac_Ssph* is the surface fraction of the smaller spheres. 515 516 ============== ======== ============= 517 Parameter name Units Default value 518 ============== ======== ============= 519 delta_Ssph None 0 520 radius_Lsph |Ang| 5000 521 radius_Ssph |Ang| 100 522 sld_Lsph |Ang^-2| -4e-07 523 sld_Ssph |Ang^-2| 3.5e-6 524 sld_solv |Ang^-2| 6.3e-6 525 surfrac_Ssph None 0.4 526 volf_Lsph None 0.05 527 volf_Lsph None 0.005 528 background |cm^-1| 0 529 ============== ======== ============= 530 531 .. image:: img/raspberry_plot.JPG 532 533 *Figure. 1D plot using the values of /2000 data points.* 534 535 REFERENCE 536 K. Larson-Smith, A. Jackson, and D.C. Pozzo, *Small angle scattering model for Pickering emulsions and raspberry* 537 *particles*, *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41 538 539 540 541 .. _CoreShellModel: 542 543 **2.1.5. CoreShellModel** 544 545 This model provides the form factor, *P(q)*, for a spherical particle with a core-shell structure. The form factor is 546 normalized by the particle volume. 547 548 For information about polarised and magnetic scattering, click here_. 549 550 *2.1.5.1. Definition* 551 552 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 553 554 .. image:: img/image013.PNG 555 556 where *scale* is a scale factor, *Vs* is the volume of the outer shell, *Vc* is the volume of the core, *rs* is the 557 radius of the shell, *rc* is the radius of the core, *c* is the scattering length density of the core, *s* is the 558 scattering length density of the shell, *solv* is the scattering length density of the solvent, and *bkg* is the 559 background level. 560 561 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector. 562 563 NB: The outer most radius (ie, = *radius* + *thickness*) is used as the effective radius for *S(Q)* when 564 *P(Q)* \* *S(Q)* is applied. 565 566 The returned value is scaled to units of |cm^-1| and the parameters of the CoreShellModel are the following 567 568 ============== ======== ============= 569 Parameter name Units Default value 570 ============== ======== ============= 571 scale None 1.0 572 (core) radius |Ang| 60 573 thickness |Ang| 10 574 core_sld |Ang^-2| 1e-6 575 shell_sld |Ang^-2| 2e-6 576 solvent_sld |Ang^-2| 3e-6 577 background |cm^-1| 0.001 578 ============== ======== ============= 579 580 Here, *radius* = the radius of the core and *thickness* = the thickness of the shell. 581 582 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron 583 Research (Kline, 2006). 584 585 REFERENCE 586 A. Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 587 588 *2.1.5.2. Validation of the core-shell sphere model* 589 590 Validation of our code was done by comparing the output of the 1D model to the output of the software provided by 591 NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 592 593 .. image:: img/image014.JPG 594 595 Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with the output of the NIST SANS 596 analysis software. The parameters were set to: *Scale* = 1.0, *Radius* = 60 , *Contrast* = 1e-6 |Ang^-2|, and 597 *Background* = 0.001 |cm^-1|. 598 599 600 601 .. _CoreMultiShellModel: 602 603 **2.1.6. CoreMultiShellModel** 604 605 This model provides the scattering from a spherical core with 1 to 4 concentric shell structures. The SLDs of the core 606 and each shell are individually specified. 607 608 For information about polarised and magnetic scattering, click here_. 609 610 *2.1.6.1. Definition* 611 612 This model is a trivial extension of the CoreShell function to a larger number of shells. See the CoreShell function 613 for a diagram and documentation. 614 615 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 616 617 Be careful! The SLDs and scale can be highly correlated. Hold as many of these parameters fixed as possible. 618 619 The 2D scattering intensity is the same as P(q) of 1D, regardless of the orientation of the q vector. 620 621 NB: The outer most radius (ie, = *radius* + 4 *thicknesses*) is used as the effective radius for *S(Q)* when 622 *P(Q)* \* *S(Q)* is applied. 623 624 The returned value is scaled to units of |cm^-1| and the parameters of the CoreMultiShell model are the following 625 626 ============== ======== ============= 627 Parameter name Units Default value 628 ============== ======== ============= 629 scale None 1.0 630 rad_core |Ang| 60 631 sld_core |Ang^-2| 6.4e-6 632 sld_shell1 |Ang^-2| 1e-6 633 sld_shell2 |Ang^-2| 2e-6 634 sld_shell3 |Ang^-2| 3e-6 635 sld_shell4 |Ang^-2| 4e-6 636 sld_solv |Ang^-2| 6.4e-6 637 thick_shell1 |Ang| 10 638 thick_shell2 |Ang| 10 639 thick_shell3 |Ang| 10 640 thick_shell4 |Ang| 10 641 background |cm^-1| 0.001 642 ============== ======== ============= 643 644 NB: Here, *rad_core* = the radius of the core, *thick_shelli* = the thickness of the shell *i* and 645 *sld_shelli* = the SLD of the shell *i*. *sld_core* and the *sld_solv* are the SLD of the core and the solvent, 646 respectively. 647 648 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron 649 Research (Kline, 2006). 650 651 This example dataset is produced by running the CoreMultiShellModel using 200 data points, *qmin* = 0.001 -1, 652 *qmax* = 0.7 -1 and the above default values. 653 654 .. image:: img/image015.JPG 655 656 *Figure: 1D plot using the default values (w/200 data point).* 657 658 The scattering length density profile for the default sld values (w/ 4 shells). 659 660 .. image:: img/image016.JPG 661 662 *Figure: SLD profile against the radius of the sphere for default SLDs.* 663 664 REFERENCE 665 See the CoreShell documentation. 666 667 668 669 .. _Core2ndMomentModel: 670 671 **2.1.7. Core2ndMomentModel** 672 673 This model describes the scattering from a layer of surfactant or polymer adsorbed on spherical particles under the 674 conditions that (i) the particles (cores) are contrast-matched to the dispersion medium, (ii) *S(Q)* ~ 1 (ie, the 675 particle volume fraction is dilute), (iii) the particle radius is >> layer thickness (ie, the interface is locally 676 flat), and (iv) scattering from excess unadsorbed adsorbate in the bulk medium is absent or has been corrected for. 677 678 Unlike a core-shell model, this model does not assume any form for the density distribution of the adsorbed species 679 normal to the interface (cf, a core-shell model which assumes the density distribution to be a homogeneous 680 step-function). For comparison, if the thickness of a (core-shell like) step function distribution is *t*, the second 681 moment, |sigma| = sqrt((*t* :sup:`2` )/12). The |sigma| is the second moment about the mean of the density distribution 682 (ie, the distance of the centre-of-mass of the distribution from the interface). 683 684 *2.1.7.1. Definition* 685 686 The *I* :sub:`0` is calculated in the following way (King, 2002) 687 688 .. image:: img/secondmeq1.JPG 689 690 where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer, *solv* is the sld of the 691 solvent/medium and cores, |phi|\ :sub:`cores` is the volume fraction of the core paraticles, and |biggamma| and 692 |delta| are the adsorbed amount and the bulk density of the polymers respectively. The |sigma| is the second moment 693 of the thickness distribution. 694 695 Note that all parameters except the |sigma| are correlated for fitting so that fitting those with more than one 696 parameter will generally fail. Also note that unlike other shape models, no volume normalization is applied to this 697 model (the calculation is exact). 698 699 The returned value is scaled to units of |cm^-1| and the parameters are the following 700 701 ============== ======== ============= 702 Parameter name Units Default value 703 ============== ======== ============= 704 scale None 1.0 705 density_poly g/cm2 0.7 706 radius_core |Ang| 500 707 ads_amount mg/m 2 1.9 708 second_moment |Ang| 23.0 709 volf_cores None 0.14 710 sld_poly |Ang^-2| 1.5e-6 711 sld_solv |Ang^-2| 6.3e-6 712 background |cm^-1| 0.0 713 ============== ======== ============= 714 715 .. image:: img/secongm_fig1.JPG 716 717 REFERENCE 718 S. King, P. Griffiths, J. Hone, and T. Cosgrove, *SANS from Adsorbed Polymer Layers*, 719 *Macromol. Symp.*, 190 (2002) 33-42 720 721 722 723 .. _MultiShellModel: 724 725 **2.1.8. MultiShellModel** 726 727 This model provides the form factor, *P(q)*, for a multi-lamellar vesicle with *N* shells where the core is filled with 728 solvent and the shells are interleaved with layers of solvent. For *N* = 1, this returns the VesicleModel (above). 729 730 .. image:: img/image020.JPG 731 732 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 733 734 .. image:: img/image008.PNG 735 736 NB: The outer most radius (= *core_radius* + *n_pairs* \* *s_thickness* + (*n_pairs* - 1) \* *w_thickness*) is used 737 as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 738 739 The returned value is scaled to units of |cm^-1| and the parameters of the MultiShellModel are the following 740 741 ============== ======== ============= 742 Parameter name Units Default value 743 ============== ======== ============= 744 scale None 1.0 745 core_radius |Ang| 60.0 746 n_pairs None 2.0 747 core_sld |Ang^-2| 6.3e-6 748 shell_sld |Ang^-2| 0.0 749 background |cm^-1| 0.0 750 s_thickness |Ang| 10 751 w_thickness |Ang| 10 752 ============== ======== ============= 753 754 NB: *s_thickness* is the shell thickness while the *w_thickness* is the solvent thickness, and *n_pair* 755 is the number of shells. 756 757 .. image:: img/image021.JPG 758 759 *Figure. 1D plot using the default values (w/200 data point).* 760 761 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron 762 Research (Kline, 2006). 763 764 REFERENCE 765 B. Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions: New Methods of Investigation*, Ch.2, 766 Surfactant Science Series Vol. 22, Ed. R. Zana and M. Dekker, New York, (1987). 767 768 769 770 .. _OnionExpShellModel: 771 772 **2.1.9. OnionExpShellModel** 773 774 This model provides the form factor, *P(q)*, for a multi-shell sphere where the scattering length density (SLD) of the 775 each shell is described by an exponential (linear, or flat-top) function. The form factor is normalized by the volume 776 of the sphere where the SLD is not identical to the SLD of the solvent. We currently provide up to 9 shells with this 777 model. 778 779 *2.1.9.1. Definition* 780 423 781 The 1D scattering intensity is calculated in the following way 424 (Guinier, 1955): 425 426 The returned value is scaled to units of [cm-1 sr-1], absolute scale. 427 428 The scattering intensity I(q) is calculated as: 429 430 431 432 where the amplitude A(q) is given as the typical sphere scattering 433 convoluted with a Gaussian to get a gradual drop-off in the scattering 434 length density: 435 436 437 438 Here A2(q) is the form factor, P(q). The scale is equivalent to the 439 volume fraction of spheres, each of volume, V. Contrast ( * ) is the 440 difference of scattering length densities of the sphere and the 441 surrounding solvent. 442 443 The poly-dispersion in radius and in fuzziness is provided. 444 445 (direct from the reference) 446 447 The "fuzziness" of the interface is defined by the parameter 448 (sigma)fuzzy. The particle radius R represents the radius of the 449 particle where the scattering length density profile decreased to 1/2 450 of the core density. The (sigma)fuzzy is the width of the smeared 451 particle surface: i.e., the standard deviation from the average height 452 of the fuzzy interface. The inner regions of the microgel that display 453 a higher density are described by the radial box profile extending to 454 a radius of approximately Rbox ~ R - 2(sigma). the profile approaches 455 zero as Rsans ~ R + 2(sigma). 456 457 For 2D data: The 2D scattering intensity is calculated in the same way 458 as 1D, where the *q* vector is defined as . 782 783 .. image:: img/image022.GIF 784 785 .. image:: img/image023.GIF 786 787 where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 788 789 .. image:: img/image024.GIF 790 791 so that 792 793 .. image:: img/image025.GIF 794 795 .. image:: img/image026.GIF 796 797 .. image:: img/image027.GIF 798 799 Here we assumed that the SLDs of the core and solvent are constant against *r*. 800 801 Now lets consider the SLD of a shell, *r*\ :sub:`shelli`, defined by 802 803 .. image:: img/image028.GIF 804 805 An example of a possible SLD profile is shown below where *sld_in_shelli* (|rho|\ :sub:`in`\ ) and 806 *thick_shelli* (|bigdelta|\ *t* :sub:`shelli`\ ) stand for the SLD of the inner side of the *i*\ th shell and the 807 thickness of the *i*\ th shell in the equation above, respectively. 808 809 For \| *A* \| > 0, 810 811 .. image:: img/image029.GIF 812 813 For *A* ~ 0 (eg., *A* = -0.0001), this function converges to that of the linear SLD profile (ie, 814 |rho|\ :sub:`shelli`\ *(r)* = *A*\ :sup:`'` ( *r* - *r*\ :sub:`shelli` - 1) / |bigdelta|\ *t* :sub:`shelli`) + *B*\ :sup:`'`), 815 so this case is equivalent to 816 817 .. image:: img/image030.GIF 818 819 .. image:: img/image031.GIF 820 821 .. image:: img/image032.GIF 822 823 .. image:: img/image033.GIF 824 825 For *A* = 0, the exponential function has no dependence on the radius (so that *sld_out_shell* (|rho|\ :sub:`out`) is 826 ignored this case) and becomes flat. We set the constant to |rho|\ :sub:`in` for convenience, and thus the form 827 factor contributed by the shells is 828 829 .. image:: img/image034.GIF 830 831 .. image:: img/image035.GIF 832 833 In the equation 834 835 .. image:: img/image036.GIF 836 837 Finally, the form factor can be calculated by 838 839 .. image:: img/image037.GIF 840 841 where 842 843 .. image:: img/image038.GIF 844 845 and 846 847 .. image:: img/image039.GIF 848 849 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 850 defined as 851 852 .. image:: img/image040.GIF 853 854 NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 855 856 The returned value is scaled to units of |cm^-1| and the parameters of this model (for only one shell) are the following 857 858 ============== ======== ============= 859 Parameter name Units Default value 860 ============== ======== ============= 861 A_shell1 None 1 862 scale None 1.0 863 rad_core |Ang| 200 864 thick_shell1 |Ang| 50 865 sld_core |Ang^-2| 1.0e-06 866 sld_in_shell1 |Ang^-2| 1.7e-06 867 sld_out_shell1 |Ang^-2| 2.0e-06 868 sld_solv |Ang^-2| 6.4e-06 869 background |cm^-1| 0.0 870 ============== ======== ============= 871 872 NB: *rad_core* represents the core radius (*R1*) and *thick_shell1* (*R2* - *R1*) is the thickness of the shell1, etc. 873 874 .. image:: img/image041.JPG 875 876 *Figure. 1D plot using the default values (w/400 point).* 877 878 .. image:: img/image042.JPG 879 880 *Figure. SLD profile from the default values.* 459 881 460 882 REFERENCE 461 462 M. Stieger, J. S. Pedersen, P. Lindner, W. Richtering, Langmuir 20 463 (2004) 7283-7292. 464 465 *2.1.3.2. Validation of the fuzzy sphere model* 466 467 This example dataset is produced by running the FuzzySphereModel, 468 using 200 data points, qmin = 0.001 -1, qmax = 0.7 A-1 and the default 469 values: 470 471 Parameter name 472 473 Units 474 475 Default value 476 477 scale 478 479 None 480 481 1.0 482 483 radius 484 485 486 487 60 488 489 fuzziness 490 491 492 493 10 494 495 sldSolv 496 497 -2 498 499 3e-6 500 501 sldSph 502 503 -2 504 505 1e-6 506 507 background 508 509 cm-1 510 511 0.001 512 513 883 L. A. Feigin and D. I. Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, 884 Plenum Press, New York, (1987). 885 886 887 888 .. _VesicleModel: 889 890 **2.1.10. VesicleModel** 891 892 This model provides the form factor, *P(q)*, for an unilamellar vesicle. The form factor is normalized by the volume 893 of the shell. 894 895 *2.1.10.1. Definition* 896 897 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 898 899 .. image:: img/image017.PNG 900 901 where *scale* is a scale factor, *Vshell* is the volume of the shell, *V1* is the volume of the core, *V2* is the total 902 volume, *R1* is the radius of the core, *R2* is the outer radius of the shell, |rho|\ :sub:`1` is the scattering 903 length density of the core and the solvent, |rho|\ :sub:`2` is the scattering length density of the shell, *bkg* is 904 the background level, and *J1* = (sin\ *x*- *x* cos\ *x*)/ *x* :sup:`2`\ . The functional form is identical to a 905 "typical" core-shell structure, except that the scattering is normalized by the volume that is contributing to the 906 scattering, namely the volume of the shell alone. Also, the vesicle is best defined in terms of a core radius (= *R1*) 907 and a shell thickness, *t*. 908 909 .. image:: img/image018.JPG 910 911 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 912 defined as 913 914 .. image:: img/image008.PNG 915 916 For P*S: toward S(Q) when P(Q)*S(Q) is applied. 917 918 NB: The outer most radius (= *radius* + *thickness*) is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* 919 is applied. 920 921 The returned value is scaled to units of |cm^-1| and the parameters of the VesicleModel are the following 922 923 ============== ======== ============= 924 Parameter name Units Default value 925 ============== ======== ============= 926 scale None 1.0 927 radius |Ang| 100 928 thickness |Ang| 30 929 core_sld |Ang^-2| 6.3e-6 930 shell_sld |Ang^-2| 0 931 background |cm^-1| 0.0 932 ============== ======== ============= 933 934 NB: *radius* represents the core radius (*R1*) and the *thickness* (*R2* - *R1*) is the shell thickness. 935 936 .. image:: img/image019.JPG 514 937 515 938 *Figure. 1D plot using the default values (w/200 data point).* 516 517 518 519 .. _RaspBerryModel:520 521 **2.1.4. RaspBerryModel**522 523 Calculates the form factor, P(q), for a "Raspberry-like" structure524 where there are smaller spheres at the surface of a larger sphere,525 such as the structure of a Pickering emulsion.526 527 *2.1.4.1. Definition*528 529 The structure is:530 531 532 533 Ro = the radius of thelarge sphere534 Rp = the radius of the smaller sphere on the surface535 delta = the fractional penetration depth536 surface coverage = fractional coverage of the large sphere surface537 (0.9 max)538 539 540 The large and small spheres have their own SLD, as well as the541 solvent. The surface coverage term is a fractional coverage (maximum542 of approximately 0.9 for hexagonally packed spheres on a surface).543 Since not all of the small spheres are necessarily attached to the544 surface, the excess free (small) spheres scattering is also included545 in the calculation. The function calculated follows equations (8)-(12)546 of the reference below, and the equations are not reproduced here.547 548 The returned value is scaled to units of [cm-1]. No interparticle549 scattering is included in this model.550 551 For 2D data: The 2D scattering intensity is calculated in the same way552 as 1D, where the *q* vector is defined as .553 554 REFERENCE555 Kjersta Larson-Smith, Andrew Jackson, and Danilo C Pozzo, "Small angle556 scattering model for Pickering emulsions and raspberry particles."557 Journal of Colloid and Interface Science (2010) vol. 343 (1) pp.558 36-41.559 560 *2.1.4.2. Validation of the RaspBerry Model*561 562 This example dataset is produced by running the RaspBerryModel, using563 2000 data points, qmin = 0.0001 -1, qmax = 0.2 A-1 and the default564 values, where Ssph/Lsph stands for Smaller/Large sphere565 andsurfrac_Ssph for the surface fraction of the smaller spheres.566 567 Parameter name568 569 Units570 571 Default value572 delta_Ssph 0 radius_Lsph 5000 radius_Ssph 100 sld_Lsph -2 -4e-07573 sld_Ssph574 575 -2576 577 3.5e-6578 579 sld_solv580 581 -2582 583 6.3e-6584 585 surfrac_Ssph586 587 588 589 0.4590 591 volf_Lsph592 593 0.05594 595 volf_Lsph596 597 598 599 0.005600 601 background602 603 cm-1604 605 0606 607 608 609 *Figure. 1D plot using the values of /2000 data points.*610 611 612 613 .. _CoreShellModel:614 615 **2.1.5. CoreShellModel**616 617 This model provides the form factor, P( *q*), for a spherical particle618 with a core-shell structure. The form factor is normalized by the619 particle volume.620 621 For information about polarised and magnetic scattering, click here_.622 623 *2.1.5.1. Definition*624 625 The 1D scattering intensity is calculated in the following way626 (Guinier, 1955):627 628 629 630 631 632 where *scale* is a scale factor, *Vs* is the volume of the outer633 shell, *Vc* is the volume of the core, *rs* is the radius of the634 shell, *rc* is the radius of the core, *c* is the scattering length635 density of the core, *s* is the scattering length density of the636 shell, solv is the scattering length density of the solvent, and *bkg*637 is the background level.638 639 The 2D scattering intensity is the same as P(q) above, regardless of640 the orientation of the q vector.641 642 For P*S: The outer most radius (= radius + thickness) is used as the643 effective radius toward S(Q) when P(Q)*S(Q) is applied.644 645 The returned value is scaled to units of [cm-1] and the parameters of646 the core-shell sphere model are the following:647 648 Here, radius = the radius of the core and thickness = the thickness of649 the shell.650 651 Parameter name652 653 Units654 655 Default value656 657 scale658 659 None660 661 1.0662 663 (core) radius664 665 666 667 60668 669 thickness670 671 672 673 10674 675 core_sld676 677 -2678 679 1e-6680 681 shell_sld682 683 -2684 685 2e-6686 687 solvent_sld688 689 -2690 691 3e-6692 693 background694 695 cm-1696 697 0.001698 939 699 940 Our model uses the form factor calculations implemented in a c-library 700 941 provided by the NIST Center for Neutron Research (Kline, 2006). 701 942 702 703 704 943 REFERENCE 705 706 Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John 707 Wiley and Sons, New York, (1955). 708 709 *2.1.5.2. Validation of the core-shell sphere model* 710 711 Validation of our code was done by comparing the output of the 1D 712 model to the output of the software provided by the NIST (Kline, 713 2006). Figure 1 shows a comparison of the output of our model and the 714 output of the NIST software. 715 716 717 718 Figure 7: Comparison of the DANSE scattering intensity for a core- 719 shell sphere with the output of the NIST SANS analysis software. The 720 parameters were set to: Scale=1.0, Radius=60 , Contrast=1e-6 -2, and 721 Background=0.001 cm -1. 722 723 724 725 .. _CoreMultiShellModel: 726 727 **2.1.6. CoreMultiShellModel** 728 729 This model provides the scattering from spherical core with from 1 up 730 to 4 shell structures. Ithas a core of a specified radius, with four 731 shells. The SLDs of the core and each shell are individually 732 specified. 733 734 For information about polarised and magnetic scattering, click here_. 735 736 *1.1. Definition* 737 738 The returned value is scaled to units of [cm-1sr-1], absolute scale. 739 740 This model is a trivial extension of the CoreShell function to a 741 larger number of shells. See the CoreShell function for a diagram and 742 documentation. 743 744 Be careful that the SLDs and scale can be highly correlated. Hold as 745 many of these fixed as possible. 746 747 The 2D scattering intensity is the same as P(q) of 1D, regardless of 748 the orientation of the q vector. 749 750 For P*S: The outer most radius (= radius + 4 thicknesses) is used as 751 the effective radius toward S(Q) if P(Q)*S(Q) is applied. 752 753 The returned value is scaled to units of [cm-1] and the parameters of 754 the CoreFourshell sphere model are the following: 755 756 Here, rad_core = the radius of the core, thick_shelli = the thickness 757 of the shell i and sld_shelli = the SLD of the shell i. 758 759 And the sld_core and the sld_solv are the SLD of the core and the 760 solvent, respectively. 761 762 Parameter name 763 764 Units 765 766 Default value 767 768 scale 769 770 None 771 772 1.0 773 774 rad_core 775 776 777 778 60 779 780 sld_core 781 782 -2 783 784 6.4e-6 785 786 sld_shell1 787 788 -2 789 790 1e-6 791 792 sld_shell2 793 794 -2 795 796 2e-6 797 798 sld_shell3 799 800 -2 801 802 3e-6 803 804 sld_shell4 805 806 -2 807 808 4e-6 809 810 sld_solv 811 812 -2 813 814 6.4e-6 815 816 thick_shell1 817 818 819 820 10 821 822 thick_shell2 823 824 825 826 10 827 828 thick_shell3 829 830 831 832 10 833 834 thick_shell4 835 836 837 838 10 839 840 background 841 842 cm-1 843 844 0.001 845 846 Our model uses the form factor calculations implemented in a c-library 847 provided by the NIST Center for Neutron Research (Kline, 2006). 848 849 944 A. Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) 945 946 947 948 .. _SphericalSLDModel: 949 950 **2.1.11. SphericalSLDModel** 951 952 Similarly to the OnionExpShellModel, this model provides the form factor, *P(q)*, for a multi-shell sphere, where the 953 interface between the each neighboring shells can be described by one of a number of functions including error, 954 power-law, and exponential functions. This model is to calculate the scattering intensity by building a continuous 955 custom SLD profile against the radius of the particle. The SLD profile is composed of a flat core, a flat solvent, 956 a number (up to 9 ) flat shells, and the interfacial layers between the adjacent flat shells (or core, and solvent) 957 (see below). Unlike the OnionExpShellModel (using an analytical integration), the interfacial layers here are 958 sub-divided and numerically integrated assuming each of the sub-layers are described by a line function. The number 959 of the sub-layer can be given by users by setting the integer values of *npts_inter* in the GUI. The form factor is 960 normalized by the total volume of the sphere. 961 962 *2.1.11.1. Definition* 963 964 The 1D scattering intensity is calculated in the following way: 965 966 .. image:: img/image022.GIF 967 968 .. image:: img/image043.GIF 969 970 where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 971 972 .. image:: img/image024.GIF 973 974 so that 975 976 .. image:: img/image044.GIF 977 978 .. image:: img/image045.GIF 979 980 .. image:: img/image046.GIF 981 982 .. image:: img/image047.GIF 983 984 .. image:: img/image048.GIF 985 986 .. image:: img/image027.GIF 987 988 Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between 989 shells, |rho|\ :sub:`inter_i`, is calculated with a function chosen by an user, where the functions are 990 991 1) Exp 992 993 .. image:: img/image049.GIF 994 995 2) Power-Law 996 997 .. image:: img/image050.GIF 998 999 3) Erf 1000 1001 .. image:: img/image051.GIF 1002 1003 The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is 1004 continuous at the boundaries of the interface as well as each sub-layers. Thus *B* and *C* are determined. 1005 1006 Once |rho|\ :sub:`rinter_i` is found at the boundary of the sub-layer of the interface, we can find its contribution 1007 to the form factor *P(q)* 1008 1009 .. image:: img/image052.GIF 1010 1011 .. image:: img/image053.GIF 1012 1013 .. image:: img/image054.GIF 1014 1015 where we assume that |rho|\ :sub:`inter_i`\ *(r)* can be approximately linear within a sub-layer *j*. 1016 1017 In the equation 1018 1019 .. image:: img/image055.GIF 1020 1021 Finally, the form factor can be calculated by 1022 1023 .. image:: img/image037.GIF 1024 1025 where 1026 1027 .. image:: img/image038.GIF 1028 1029 and 1030 1031 .. image:: img/image056.GIF 1032 1033 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 1034 defined as 1035 1036 .. image:: img/image040.GIF 1037 1038 NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 1039 1040 The returned value is scaled to units of |cm^-1| and the parameters of this model (for just one shell) are the following 1041 1042 ============== ======== ============= 1043 Parameter name Units Default value 1044 ============== ======== ============= 1045 background |cm^-1| 0.0 1046 npts_inter None 35 1047 scale None 1 1048 sld_solv |Ang^-2| 1e-006 1049 func_inter1 None Erf 1050 nu_inter None 2.5 1051 thick_inter1 |Ang| 50 1052 sld_flat1 |Ang^-2| 4e-006 1053 thick_flat1 |Ang| 100 1054 func_inter0 None Erf 1055 nu_inter0 None 2.5 1056 rad_core0 |Ang| 50 1057 sld_core0 |Ang^-2| 2.07e-06 1058 thick_core0 |Ang| 50 1059 ============== ======== ============= 1060 1061 NB: *rad_core0* represents the core radius (*R1*). 1062 1063 .. image:: img/image057.JPG 1064 1065 *Figure. 1D plot using the default values (w/400 point).* 1066 1067 .. image:: img/image058.JPG 1068 1069 *Figure. SLD profile from the default values.* 850 1070 851 1071 REFERENCE 852 853 See the CoreShell documentation. 854 855 TEST DATASET 856 857 This example dataset is produced by running the CoreMultiShellModel 858 using 200 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the above 859 default values. 860 861 862 863 *Figure: 1D plot using the default values (w/200 data point).* 864 865 The scattering length density profile for the default sld values (w/ 4 866 shells). 867 868 869 870 *Figure: SLD profile against the radius of the sphere for default 871 SLDs.* 872 873 874 875 .. _Core2ndMomentModel: 876 877 **2.1.7. Core2ndMomentModel** 878 879 This model describes the scattering from a layer of surfactant or 880 polymer adsorbed on spherical particles under the conditions that (i) 881 theparticles (cores) are contrast-matched to the dispersion medium, 882 (ii) S(Q)~1 (ie, the particle volume fraction is dilute), (iii) the 883 particle radius is >> layer thickness (ie, the interface is locally 884 flat), and (iv) scattering from excess unadsorbed adsorbate in the 885 bulk medium is absent or has been corrected for. 886 887 Unlike a core-shell model, this model does not assume any form for the 888 density distribution of the adsorbed species normal to the interface 889 (cf, a core-shell model which assumes the density distribution to be a 890 homogeneous step-function). For comparison, if the thickness of a 891 (core-shell like) step function distribution is t, the second moment, 892 sigma = sqrt((t^2)/12). Thesigma is the second moment about the mean 893 of the density distribution (ie, the distance of the centre-of-mass of 894 the distribution from the interface). 895 896 *1.1. Definition* 897 898 The I0 is calculated in the following way (King, 2002): 899 900 901 902 903 904 where *scale* is a scale factor, *poly* is the sld of the polymer (or 905 surfactant) layer,solv is the sld of the solvent/medium and cores, 906 phi_cores is the volume fraction of the core paraticles, and Gamma and 907 delta arethe adsorbed amount and the bulk density of the polymers 908 respectively. The sigma is the second moment of the thickness 909 distribution. 910 911 912 913 Note that all parameters except the 'sigma' are correlated for fitting 914 so that fittingthose with more than one parameters will be generally 915 failed. And note that unlike other shape models, no volume 916 normalization was applied to this model. 917 918 The returned value is scaled to units of [cm-1] and the parameters are 919 the following: 920 921 Parameter name 922 923 Units 924 925 Default value 926 927 scale 928 929 None 930 931 1.0 932 933 density_poly 934 935 g/cm2 936 937 0.7 938 939 radius_core 940 941 942 943 500 944 945 ads_amount 946 947 mg/m2 948 949 1.9 950 second_moment 23.0 volf_cores None 0.14 951 sld_poly 952 953 -2 954 955 1.5e-6 956 957 sld_solv 958 959 -2 960 961 6.3e-6 962 963 background 964 965 cm-1 966 967 0.0 968 969 1072 L. A. Feigin and D. I. Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, 1073 Plenum Press, New York, (1987) 1074 1075 1076 1077 .. _LinearPearlsModel: 1078 1079 **2.1.12. LinearPearlsModel** 1080 1081 This model provides the form factor for *N* spherical pearls of radius *R* linearly joined by short strings (or segment 1082 length or edge separation) *l* (= *A* - 2\ *R*)). *A* is the center-to-center pearl separation distance. The thickness 1083 of each string is assumed to be negligible. 1084 1085 .. image:: img/linearpearls.jpg 1086 1087 *2.1.12.1. Definition* 1088 1089 The output of the scattering intensity function for the LinearPearlsModel is given by (Dobrynin, 1996) 1090 1091 .. image:: img/linearpearl_eq1.gif 1092 1093 where the mass *m*\ :sub:`p` is (SLD\ :sub:`pearl` - SLD\ :sub:`solvent`) \* (volume of *N* pearls). V is the total 1094 volume. 1095 1096 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector. 1097 1098 The returned value is scaled to units of |cm^-1| and the parameters of the LinearPearlsModel are the following 1099 1100 =============== ======== ============= 1101 Parameter name Units Default value 1102 =============== ======== ============= 1103 scale None 1.0 1104 radius |Ang| 80.0 1105 edge_separation |Ang| 350.0 1106 num_pearls None 3 1107 sld_pearl |Ang^-2| 1e-6 1108 sld_solv |Ang^-2| 6.3e-6 1109 background |cm^-1| 0.0 1110 =============== ======== ============= 1111 1112 NB: *num_pearls* must be an integer. 1113 1114 .. image:: img/linearpearl_plot.jpg 970 1115 971 1116 REFERENCE 972 973 S. King, P. Griffiths, J. Hone, and T. Cosgrove, "SANS from Adsorbed 974 Polymer Lyaers", Macromol. Symp. 190, 33-42 (2002). 975 976 977 978 .. _MultiShellModel: 979 980 **2.1.8. MultiShellModel** 981 982 This model provides the form factor, P( *q*), for a multi-lamellar 983 vesicle with N shells where the core is filled with solvent and the 984 shells are interleaved with layers of solvent. For N = 1, this return 985 to the vesicle model (above). 986 987 988 989 The 2D scattering intensity is the same as 1D, regardless of the 990 orientation of the *q* vector which is defined as . 991 992 For P*S: The outer most radius (= core_radius + n_pairs * s_thickness 993 + (n_pairs -1) * w_thickness) is used as the effective radius toward 994 S(Q) when P(Q)*S(Q) is applied. 995 996 The returned value is scaled to units of [cm-1] and the parameters of 997 the multi-shell model are the following: 998 999 In the parameters, the s_thickness is the shell thickness while the 1000 w_thickness is the solvent thickness, and the n_pair is the number of 1001 shells. 1002 1003 Parameter name 1004 1005 Units 1006 1007 Default value 1008 1009 scale 1010 1011 None 1012 1013 1.0 1014 1015 core_radius 1016 1017 1018 1019 60.0 1020 1021 n_pairs 1022 1023 None 1024 1025 2.0 1026 1027 core_sld 1028 1029 -2 1030 1031 6.3e-6 1032 1033 shell_sld 1034 1035 -2 1036 1037 0.0 1038 1039 background 1040 1041 cm-1 1042 1043 0.0 1044 1045 s_thickness 1046 1047 1048 1049 10 1050 1051 w_thickness 1052 1053 1054 1055 10 1056 1057 1058 1059 *Figure. 1D plot using the default values (w/200 data point).* 1060 1061 Our model uses the form factor calculations implemented in a c-library 1062 provided by the NIST Center for Neutron Research (Kline, 2006). 1117 A. V. Dobrynin, M. Rubinstein and S. P. Obukhov, *Macromol.*, 29 (1996) 2974-2979 1118 1119 1120 1121 .. _PearlNecklaceModel: 1122 1123 **2.1.13. PearlNecklaceModel** 1124 1125 This model provides the form factor for a pearl necklace composed of two elements: *N* pearls (homogeneous spheres 1126 of radius *R*) freely jointed by *M* rods (like strings - with a total mass *Mw* = *M* \* *m*\ :sub:`r` + *N* \* *m*\ :sub:`s`, 1127 and the string segment length (or edge separation) *l* (= *A* - 2\ *R*)). *A* is the center-to-center pearl separation 1128 distance. 1129 1130 .. image:: img/pearl_fig.jpg 1131 1132 *2.1.13.1. Definition* 1133 1134 The output of the scattering intensity function for the PearlNecklaceModel is given by (Schweins, 2004) 1135 1136 .. image:: img/pearl_eq1.gif 1137 1138 where 1139 1140 .. image:: img/pearl_eq2.gif 1141 1142 .. image:: img/pearl_eq3.gif 1143 1144 .. image:: img/pearl_eq4.gif 1145 1146 .. image:: img/pearl_eq5.gif 1147 1148 .. image:: img/pearl_eq6.gif 1149 1150 and 1151 1152 .. image:: img/pearl_eq7.gif 1153 1154 where the mass *m*\ :sub:`i` is (SLD\ :sub:`i` - SLD\ :sub:`solvent`) \* (volume of the *N* pearls/rods). *V* is the 1155 total volume of the necklace. 1156 1157 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector. 1158 1159 The returned value is scaled to units of |cm^-1| and the parameters of the PearlNecklaceModel are the following 1160 1161 =============== ======== ============= 1162 Parameter name Units Default value 1163 =============== ======== ============= 1164 scale None 1.0 1165 radius |Ang| 80.0 1166 edge_separation |Ang| 350.0 1167 num_pearls None 3 1168 sld_pearl |Ang^-2| 1e-6 1169 sld_solv |Ang^-2| 6.3e-6 1170 sld_string |Ang^-2| 1e-6 1171 thick_string 1172 (=rod diameter) |Ang| 2.5 1173 background |cm^-1| 0.0 1174 =============== ======== ============= 1175 1176 NB: *num_pearls* must be an integer. 1177 1178 .. image:: img/pearl_plot.jpg 1063 1179 1064 1180 REFERENCE 1065 1066 Cabane, B., Small Angle Scattering Methods, Surfactant Solutions: New 1067 Methods of Investigation, Ch.2, Surfactant Science Series Vol. 22, Ed. 1068 R. Zana, M. Dekker, New York, 1987. 1069 1070 1071 1072 .. _OnionExpShellModel: 1073 1074 **2.1.9. OnionExpShellModel** 1075 1076 This model provides the form factor, *P*( *q*), for a multi-shell 1077 sphere where the scattering length density (SLD) of the each shell is 1078 described by an exponential (linear, or flat-top) function. The form 1079 factor is normalized by the volume of the sphere where the SLD is not 1080 identical to the SLD of the solvent. We currently provide up to 9 1081 shells with this model. 1082 1083 The 1D scattering intensity is calculated in the following way: 1084 1085 1086 1087 1088 1089 where, for a spherically symmetric particle with a particle density 1090 *r*( *r*) [L.A.Feigin and D.I.Svergun, Structure Analysis by Small- 1091 Angle X-Ray and Neutron Scattering, Plenum Press, New York, 1987], 1092 1093 1094 1095 so that 1096 1097 1098 1099 1100 1101 1102 1103 1104 Here we assumed that the SLDs of the core and solvent are constant 1105 against *r*. Now lets consider the SLD of a shell, *rshelli*, 1106 defineded by 1107 1108 1109 1110 An example of a possible SLD profile is shown below where 1111 sld_in_shelli ( *rin* ) and thick_shelli ( *Dtshelli* ) stand for the 1112 SLD of the inner side of the ith shell and the thickness of the ith 1113 shell in the equation above, respectively. 1114 1115 For \|A\|>0, 1116 1117 1118 1119 For A *~ *0 (eg., A = - 0.0001), this function converges to that of 1120 the linear SLD profile (ie, *rshelli*( *r*) = *A \*( *r* - 1121 *rshelli-1*) / *Dtshelli*) + *B \*), so this case it is equivalent 1122 to 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 For A = 0, the exponential function has no dependence on the radius 1133 (so that sld_out_shell# ( *rout*) is ignored this case) and becomes 1134 flat. We set the constant to *rin* for convenience, and thus the form 1135 factor contributed by the shells is 1136 1137 1138 1139 1140 1141 In the equation, 1142 1143 1144 1145 Finally, the form factor can be calculated by 1146 1147 1148 1149 where 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 The 2D scattering intensity is the same as *P*( *q*) above, regardless 1162 of the orientation of the *q* vector which is defined as . 1163 1164 For P*S: The outer most radius is used as the effective radius toward 1165 S(Q) when P(Q)*S(Q) is applied. 1166 1167 The returned value is scaled to units of [cm-1] and the parameters of 1168 this model are the following: 1169 1170 In the parameters, the rad_core represents the core radius (R1) and 1171 the thick_shell1 (R2 R1) is the thickness of the shell1, etc. 1172 1173 Note: Only No. of shells = 1 is given below. 1174 1175 Parameter name 1176 1177 Units 1178 1179 Default value 1180 1181 A_shell1 1182 1183 None 1184 1185 1 1186 1187 scale 1188 1189 None 1190 1191 1.0 1192 1193 rad_core 1194 1195 1196 1197 200 1198 1199 thick_shell1 1200 1201 1202 1203 50 1204 1205 sld_core 1206 1207 -2 1208 1209 1.0e-06 1210 1211 sld_in_shell1 1212 1213 -2 1214 1215 1.7e-06 1216 1217 sld_out_shell1 1218 1219 -2 1220 1221 2.0e-06 1222 1223 sld_solv 1224 1225 -2 1226 1227 6.4e-06 1228 1229 background 1230 1231 cm-1 1232 1233 0.0 1234 1235 1236 1237 *Figure. 1D plot using the default values (w/400 point).* 1238 1239 1240 1241 *Figure. SLD profile from the default values.* 1242 1243 REFERENCE 1244 1245 L.A.Feigin and D.I.Svergun, Structure Analysis by Small-Angle X-Ray 1246 and Neutron Scattering, Plenum Press, New York, 1987 1247 1248 1249 1250 .. _VesicleModel: 1251 1252 **2.1.10. VesicleModel** 1253 1254 This model provides the form factor, P( *q*), for an unilamellar 1255 vesicle. The form factor is normalized by the volume of the shell. 1256 1257 The 1D scattering intensity is calculated in the following way 1258 (Guinier, 1955): 1259 1260 1261 1262 1263 1264 where *scale* is a scale factor, *Vshell* is the volume of the shell, *V1* is the volume of the core, *V2* is the total 1265 volume, *R1* is the radius of the core, *r2* is the outer radius of the shell, *1* is the scattering length density of 1266 the core and the solvent, *2* is the scattering length density of the shell, and *bkg* is the background level. And 1267 *J1* = (sin *x *- *x*cos *x*)/ *x*2. The functional form is identical to a "typical" core-shell structure, except that 1268 the scattering is normalized by the volume that is contributing to the scattering, namely the volume of the shell alone. 1269 Also, the vesicle is best defined in terms of a core radius (= R1) and a shell thickness, t. 1270 1271 1272 1273 The 2D scattering intensity is the same as *P*( *q*) above, regardless 1274 of the orientation of the *q* vector which is defined as . 1275 1276 For P*S: The outer most radius (= radius + thickness) is used as the 1277 effective radius toward S(Q) when P(Q)*S(Q) is applied. 1278 1279 The returned value is scaled to units of [cm-1] and the parameters of 1280 the vesicle model are the following: 1281 1282 In the parameters, the radius represents the core radius (R1) and the 1283 thickness (R2 R1) is the shell thickness. 1284 1285 Parameter name 1286 1287 Units 1288 1289 Default value 1290 1291 scale 1292 1293 None 1294 1295 1.0 1296 1297 radius 1298 1299 1300 1301 100 1302 1303 thickness 1304 1305 1306 1307 30 1308 1309 core_sld 1310 1311 -2 1312 1313 6.3e-6 1314 1315 shell_sld 1316 1317 -2 1318 1319 0 1320 1321 background 1322 1323 cm-1 1324 1325 0.0 1326 1327 1328 1329 *Figure. 1D plot using the default values (w/200 data point).* 1330 1331 Our model uses the form factor calculations implemented in a c-library 1332 provided by the NIST Center for Neutron Research (Kline, 2006). 1333 1334 REFERENCE 1335 1336 Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John 1337 Wiley and Sons, New York, (1955). 1338 1339 1340 1341 .. _SphericalSLDModel: 1342 1343 **2.1.11. SphericalSLDModel** 1344 1345 Similarly to the OnionExpShellModel, this model provides the form 1346 factor, *P*( *q*), for a multi-shell sphere, where the interface 1347 between the each neighboring shells can be described by one of the 1348 functions including error, power-law, and exponential functions. This 1349 model is to calculate the scattering intensity by building a 1350 continuous custom SLD profile against the radius of the particle. The 1351 SLD profile is composed of a flat core, a flat solvent, a number (up 1352 to 9 shells) of flat shells, and the interfacial layers between the 1353 adjacent flat shells (or core, and solvent) (See below). Unlike 1354 OnionExpShellModel (using an analytical integration), the interfacial 1355 layers are sub-divided and numerically integrated assuming each sub- 1356 layers are described by a line function. The number of the sub-layer 1357 can be given by users by setting the integer values of npts_inter# in 1358 GUI. The form factor is normalized by the total volume of the sphere. 1359 1360 The 1D scattering intensity is calculated in the following way: 1361 1362 1363 1364 1365 1366 where, for a spherically symmetric particle with a particle density 1367 *r*( *r*) [L.A.Feigin and D.I.Svergun, Structure Analysis by Small- 1368 Angle X-Ray and Neutron Scattering, Plenum Press, New York, 1987], 1369 1370 1371 1372 so that 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 Here we assumed that the SLDs of the core and solvent are constant 1387 against *r*. The SLD at the interface between shells, *rinter_i*, is 1388 calculated with a function chosen by an user, where the functions are: 1389 1390 1) Exp; 1391 1392 1393 1394 2) Power-Law; 1395 1396 1397 1398 1399 1400 3) Erf; 1401 1402 1403 1404 1405 1406 1407 1408 Then the functions are normalized so that it varies between 0 and 1 1409 and they are constrained such that the SLD is continuous at the 1410 boundaries of the interface as well as each sub-layers and thus the B 1411 and C are determined. 1412 1413 Once the *rinter_i* is found at the boundary of the sub-layer of the 1414 interface, we can find its contribution to the form factor P(q); 1415 1416 1417 1418 1419 1420 1421 1422 where we assume that rho_inter_i (r) can be approximately linear 1423 within a sub-layer j. 1424 1425 In the equation, 1426 1427 1428 1429 Finally, the form factor can be calculated by 1430 1431 1432 1433 where 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 The 2D scattering intensity is the same as *P*( *q*) above, regardless 1446 of the orientation of the *q* vector which is defined as . 1447 1448 For P*S: The outer most radius is used as the effective radius toward 1449 S(Q) when P(Q)*S(Q) is applied. 1450 1451 The returned value is scaled to units of [cm-1] and the parameters of 1452 this model are the following: 1453 1454 In the parameters, the rad_core0 represents the core radius (R1). 1455 1456 Note: Only No. of shells = 1 is given below. 1457 1458 Parameter name 1459 1460 Units 1461 1462 Default value 1463 1464 background 1465 1466 cm-1 1467 1468 0.0 1469 1470 npts_inter 1471 1472 35 1473 1474 scale 1475 1476 1 1477 1478 sld_solv 1479 1480 -2 1481 1482 1e-006 1483 1484 func_inter1 1485 1486 Erf 1487 1488 nu_inter 1489 1490 2.5 1491 1492 thick_inter1 1493 1494 1495 1496 50 1497 1498 sld_flat1 1499 1500 -2 1501 1502 4e-006 1503 1504 thick_flat1 1505 1506 1507 1508 100 1509 1510 func_inter0 1511 1512 Erf 1513 1514 nu_inter0 1515 1516 2.5 1517 1518 rad_core0 1519 1520 1521 1522 50 1523 1524 sld_core0 1525 1526 -2 1527 1528 2.07e-06 1529 1530 thick_core0 1531 1532 1533 1534 50 1535 1536 1537 1538 *Figure. 1D plot using the default values (w/400 point).* 1539 1540 1541 1542 *Figure. SLD profile from the default values.* 1543 1544 REFERENCE 1545 1546 L.A.Feigin and D.I.Svergun, Structure Analysis by Small-Angle X-Ray 1547 and Neutron Scattering, Plenum Press, New York, 1987 1548 1549 1550 1551 .. _LinearPearlsModel: 1552 1553 **2.1.12. LinearPearlsModel** 1554 1555 This model provides the form factor for pearls linearly joined by 1556 short strings: N pearls (homogeneous spheres), the radius R and the 1557 string segment length (or edge separation) l (= A- 2R)). The A is the 1558 center to center pearl separation distance. The thickness of each 1559 string is assumed to be negligable. 1560 1561 1562 1563 1564 1565 *1.1. Definition* 1566 1567 1568 1569 The output of the scattering intensity function for the linearpearls 1570 model is given by (Dobrynin, 1996): 1571 1572 1573 1574 where the mass mp is (sld(of a pearl) sld(of solvent)) * (volume of 1575 the N pearls), V is the total volume. 1576 1577 The 2D scattering intensity is the same as P(q) above, regardless of 1578 the orientation of the q vector. 1579 1580 The returned value is scaled to units of [cm-1] and the parameters are 1581 the following: 1582 1583 Parameter name 1584 1585 Units 1586 1587 Default value 1588 1589 scale 1590 1591 None 1592 1593 1.0 1594 1595 radius 1596 1597 1598 1599 80.0 1600 1601 edge_separation 1602 1603 1604 1605 350.0 1606 1607 num_pearls 1608 1609 (integer) 1610 1611 3 1612 1613 sld_pearl 1614 1615 -2 1616 1617 1e-6 1618 1619 sld_solv 1620 1621 -2 1622 1623 6.3e-6 1624 1625 background 1626 1627 cm-1 1628 1629 0.0 1630 1631 1632 1633 1634 1635 REFERENCE 1636 1637 A. V. Dobrynin, M. Rubinstein and S. P. Obukhov, Macromol. 29, 1638 2974-2979, 1996. 1639 1640 1641 1642 .. _PearlNecklaceModel: 1643 1644 **2.1.13. PearlNecklaceModel** 1645 1646 This model provides the form factor for a pearl necklace composed of 1647 two elements: N pearls (homogeneous spheres) freely jointed by M rods 1648 (like strings) (with a total mass Mw = M *mr + N * ms, the radius R 1649 and the string segment length (or edge separation) l (= A- 2R)). The A 1650 is the center to center pearl separation distance. 1651 1652 1653 1654 1655 1656 *1.1. Definition* 1657 1658 The output of the scattering intensity function for the pearlnecklace 1659 model is given by (Schweins, 2004): 1660 1661 1662 1663 where 1664 1665 , 1666 1667 , 1668 1669 , 1670 1671 , 1672 1673 , 1674 1675 and 1676 1677 . 1678 1679 1680 1681 where the mass mi is (sld(of i) sld(of solvent)) * (volume of the N 1682 pearls/rods), V is the total volume of the necklace. 1683 1684 The 2D scattering intensity is the same as P(q) above, regardless of 1685 the orientation of the q vector. 1686 1687 The returned value is scaled to units of [cm-1] and the parameters are 1688 the following: 1689 1690 Parameter name 1691 1692 Units 1693 1694 Default value 1695 1696 scale 1697 1698 None 1699 1700 1.0 1701 1702 radius 1703 1704 1705 1706 80.0 1707 1708 edge_separation 1709 1710 1711 1712 350.0 1713 1714 num_pearls 1715 1716 (integer) 1717 1718 3 1719 1720 sld_pearl 1721 1722 -2 1723 1724 1e-6 1725 1726 sld_solv 1727 1728 -2 1729 1730 6.3e-6 1731 1732 sld_string 1733 1734 -2 1735 1736 1e-6 1737 1738 thick_string 1739 1740 (=rod diameter) 1741 1742 1743 1744 2.5 1745 1746 background 1747 1748 cm-1 1749 1750 0.0 1751 1752 1753 1754 1755 1756 REFERENCE 1757 1758 R. Schweins and K. Huber, Particle Scattering Factor of Pearl Necklace 1759 Chains, Macromol. Symp., 211, 25-42, 2004. 1181 R. Schweins and K. Huber, *Particle Scattering Factor of Pearl Necklace Chains*, *Macromol. Symp.* 211 (2004) 25-42 2004 1760 1182 1761 1183 … … 1774 1196 1775 1197 The output of the 2D scattering intensity function for oriented 1776 cylinders is given by (Guinier, 1955) :1198 cylinders is given by (Guinier, 1955) 1777 1199 1778 1200 … … 1803 1225 toward S(Q) when P(Q)*S(Q) is applied. 1804 1226 1805 The returned value is scaled to units of [cm-1]and the parameters of1227 The returned value is scaled to units of |cm^-1| and the parameters of 1806 1228 the cylinder model are the following: 1807 1229 … … 1838 1260 background 1839 1261 1840 cm-1 1262 |cm^-1| 1841 1263 1842 1264 0.0 … … 1886 1308 1887 1309 1888 Figure 3: Comparison of the DANSEscattering intensity for a cylinder1310 Figure 3: Comparison of the SasView scattering intensity for a cylinder 1889 1311 with the output of the NIST SANS analysis software. The parameters 1890 1312 were set to: Scale=1.0, Radius=20 , Length=400 , Contrast=3e-6 -2, and … … 1916 1338 hollow cylinder have the same SLD. 1917 1339 1918 The 1D scattering intensity is calculated in the following way 1919 (Guinier, 1955): 1920 1921 1922 1923 1924 1925 where *scale* is a scale factor, *J1* is the 1st order Bessel 1926 function, *J1* (x)= (sin *x *- *x*cos *x*)/ *x*2. 1340 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 1341 1342 1343 1344 1345 1346 where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1(x)* = (sin *x - *x* cos *x*)/ *x*\ :sup:`2`. 1927 1347 1928 1348 … … 1986 1406 background 1987 1407 1988 cm-1 1408 |cm^-1| 1989 1409 1990 1410 0.01 … … 2028 1448 *1.1. Definition* 2029 1449 2030 The returned value is scaled to units of [ cm-1sr-1], absolute scale.1450 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 2031 1451 2032 1452 The Capped Cylinder geometry is defined as: … … 2077 1497 2078 1498 This example dataset is produced by running the Macro 2079 CappedCylinder(), using 200 data points, qmin = 0.001 -1, qmax= 0.71499 CappedCylinder(), using 200 data points, *qmin* = 0.001 -1, *qmax* = 0.7 2080 1500 -1 and the above default values. 2081 1501 … … 2190 1610 used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. 2191 1611 2192 The returned value is scaled to units of [cm-1]and the parameters of1612 The returned value is scaled to units of |cm^-1| and the parameters of 2193 1613 the core-shell cylinder model are the following: 2194 1614 … … 2243 1663 background 2244 1664 2245 cm-1 1665 |cm^-1| 2246 1666 2247 1667 0.0 … … 2284 1704 2285 1705 2286 Figure 8: Comparison of the DANSEscattering intensity for a core-1706 Figure 8: Comparison of the SasView scattering intensity for a core- 2287 1707 shell cylinder with the output of the NIST SANS analysis software. The 2288 1708 parameters were set to: Scale=1.0, Radius=20 , Thickness=10 , … … 2354 1774 normalized by the particle volume: P(q) = scale*<f^2>/V . 2355 1775 2356 The returned value is scaled to units of [cm-1].1776 The returned value is scaled to units of |cm^-1|. 2357 1777 2358 1778 To provide easy access to the orientation of the elliptical, we define … … 2452 1872 2453 1873 2454 *Figure. The intensities averaged from 2D over different number * 2455 2456 *of points of binning of angles.* 1874 *Figure. The intensities averaged from 2D over different numbers of bins and angles.* 2457 1875 2458 1876 REFERENCE … … 2481 1899 flexible cylinder can be considered a rigid rod. The Kuhn length (b = 2482 1900 2*lp) is also used to describe the stiffness of a chain. The returned 2483 value is in units of [cm-1], on absolute scale. In the parameters, the1901 value is in units of |cm^-1|, on absolute scale. In the parameters, the 2484 1902 sldCyl and sldSolv represent SLD (chain/cylinder) and SLD (solvent) 2485 1903 respectively. … … 2527 1945 background 2528 1946 2529 cm-1 1947 |cm^-1| 2530 1948 2531 1949 0.01 … … 2568 1986 **2.1.20 FlexCylEllipXModel** 2569 1987 2570 *Flexible Cylinder with Elliptical Cross-Section: *Calculates the1988 *Flexible Cylinder with Elliptical Cross-Section:* Calculates the 2571 1989 form factor for a flexible cylinder with an elliptical cross section 2572 1990 and a uniform scattering length density. The non-negligible diameter 2573 1991 of the cylinder is included by accounting for excluded volume 2574 1992 interactions within the walk of a single cylinder. The form factor is 2575 normalized by the particle volume such that P(q) = scale *<f^2>/Vol +1993 normalized by the particle volume such that P(q) = scale\*<f^2>/Vol + 2576 1994 bkg, where < > is an average over all possible orientations of the 2577 1995 flexible cylinder. … … 2586 2004 for the details. 2587 2005 2588 N OTE: there are several typos in the original reference that have been2006 NB: there are several typos in the original reference that have been 2589 2007 corrected by WRC. Details of the corrections are in the reference 2590 2008 below. … … 2612 2030 curve fitting to maintain this inequality. 2613 2031 2614 The returned value is in units of [cm-1], on absolute scale.2032 The returned value is in units of |cm^-1|, on absolute scale. 2615 2033 2616 2034 The sldCyl = SLD (chain), sldSolv = SLD (solvent). The scale, and the … … 2644 2062 2645 2063 This example dataset is produced by running the Macro 2646 FlexCylEllipXModel, using 200 data points, qmin = 0.001 -1, qmax= 0.72064 FlexCylEllipXModel, using 200 data points, *qmin* = 0.001 -1, *qmax* = 0.7 2647 2065 -1 and the default values below. 2648 2066 … … 2659 2077 background 2660 2078 2661 cm-1 2079 |cm^-1| 2662 2080 2663 2081 0.0001 … … 2717 2135 2718 2136 2719 The returned value is scaled to units of [cm-1]and the parameters of2137 The returned value is scaled to units of |cm^-1| and the parameters of 2720 2138 the core-shell cylinder model are the following: 2721 2139 … … 2770 2188 background 2771 2189 2772 cm-1 2190 |cm^-1| 2773 2191 2774 2192 0.0 … … 2829 2247 *1.1. Definition* 2830 2248 2831 The returned value is scaled to units of [ cm-1sr-1], absolute scale.2832 2833 The barbell geometry is defined as :2249 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 2250 2251 The barbell geometry is defined as 2834 2252 2835 2253 … … 2837 2255 r is the radius of the cylinder. All other parameters are as defined 2838 2256 in the diagram. Since the end cap radius R >= r and by definition for 2839 this geometry h > 0, h is then defined by r and R as :2257 this geometry h > 0, h is then defined by r and R as 2840 2258 2841 2259 h = sqrt(R^2 - r^2). … … 2880 2298 2881 2299 This example dataset is produced by running the Macro PlotBarbell(), 2882 using 200 data points, qmin = 0.001 -1, qmax= 0.7 -1 and the above2300 using 200 data points, *qmin* = 0.001 -1, *qmax* = 0.7 -1 and the above 2883 2301 default values. 2884 2302 … … 2975 2393 2976 2394 2977 The returned value is in units of [ cm-1 sr-1], on absolute scale.2395 The returned value is in units of [|cm^-1| |sr^-1|, on absolute scale. 2978 2396 2979 2397 The scattering intensity I(q) is: … … 3022 2440 background 3023 2441 3024 cm-1 2442 |cm^-1| 3025 2443 3026 2444 0.001 … … 3112 2530 3113 2531 3114 The returned value is in units of [cm-1], on absolute scale.2532 The returned value is in units of |cm^-1|, on absolute scale. 3115 2533 3116 2534 The form factor calculated is: … … 3135 2553 background 3136 2554 3137 cm-1 2555 |cm^-1| 3138 2556 3139 2557 0.0 … … 3223 2641 effective radius toward S(Q) when P(Q)*S(Q) is applied. 3224 2642 3225 The returned value is scaled to units of [cm-1]and the parameters of2643 The returned value is scaled to units of |cm^-1| and the parameters of 3226 2644 the ellipsoid model are the following: 3227 2645 … … 3264 2682 background 3265 2683 3266 cm-1 2684 |cm^-1| 3267 2685 3268 2686 0.0 … … 3314 2732 The NIST software performs that integration with a 76-point Gaussian 3315 2733 quadrature rule, which will become imprecise at high q where the 3316 amplitude varies quickly as a function of q. The DANSEresult shown2734 amplitude varies quickly as a function of q. The SasView result shown 3317 2735 has been obtained by summing over 501 equidistant points in . Our 3318 2736 result was found to be stable over the range of q shown for a number … … 3321 2739 * * 3322 2740 3323 Figure 5: Comparison of the DANSEscattering intensity for an2741 Figure 5: Comparison of the SasView scattering intensity for an 3324 2742 ellipsoid with the output of the NIST SANS analysis software. The 3325 2743 parameters were set to: Scale=1.0, Radius_a=20 , Radius_b=400 , … … 3351 2769 3352 2770 3353 The returned value is in units of [cm-1], on absolute scale.2771 The returned value is in units of |cm^-1|, on absolute scale. 3354 2772 3355 2773 The form factor calculated is: … … 3386 2804 background 3387 2805 3388 cm-1 2806 |cm^-1| 3389 2807 3390 2808 0.001 … … 3472 2890 3473 2891 3474 The returned value is in units of [cm-1], on absolute scale.2892 The returned value is in units of |cm^-1|, on absolute scale. 3475 2893 3476 2894 The form factor calculated is: … … 3508 2926 background 3509 2927 3510 cm-1 2928 |cm^-1| 3511 2929 3512 2930 0.0 … … 3601 3019 3602 3020 3603 The returned value is in units of [cm-1], on absolute scale. In the3021 The returned value is in units of |cm^-1|, on absolute scale. In the 3604 3022 parameters, sld_bi = SLD of the bilayer, sld_sol = SLD of the solvent, 3605 3023 and bi_thick = the thickness of the bilayer. … … 3615 3033 background 3616 3034 3617 cm-1 3035 |cm^-1| 3618 3036 3619 3037 0.0 … … 3682 3100 3683 3101 3684 The returned value is in units of [cm-1], on absolute scale. In the3102 The returned value is in units of |cm^-1|, on absolute scale. In the 3685 3103 parameters, sld_tail = SLD of the tail group, and sld_head = SLD of 3686 3104 the head group. … … 3696 3114 background 3697 3115 3698 cm-1 3116 |cm^-1| 3699 3117 3700 3118 0.0 … … 3781 3199 B=compression modulus, and N = number of lamellar plates (n_plates). 3782 3200 3783 N ote: When the Caille parameter is greater than approximately 0.8 to3201 NB: When the Caille parameter is greater than approximately 0.8 to 3784 3202 1.0, the assumptions of the model are incorrect. And due to the 3785 3203 complication of the model function, users are responsible to make sure … … 3790 3208 the *q* vector is defined as . 3791 3209 3792 The returned value is in units of [cm-1], on absolute scale.3210 The returned value is in units of |cm^-1|, on absolute scale. 3793 3211 3794 3212 … … 3802 3220 background 3803 3221 3804 cm-1 3222 |cm^-1| 3805 3223 3806 3224 0.0 … … 3888 3306 (n_plates). 3889 3307 3890 N ote: When the Caille parameter is greater than approximately 0.8 to3308 NB: When the Caille parameter is greater than approximately 0.8 to 3891 3309 1.0, the assumptions of the model are incorrect. And due to the 3892 3310 complication of the model function, users are responsible to make sure … … 3899 3317 3900 3318 3901 The returned value is in units of [cm-1], on absolute scale. In the3319 The returned value is in units of |cm^-1|, on absolute scale. In the 3902 3320 parameters, sld_tail = SLD of the tail group, sld_head = SLD of the 3903 3321 head group, and sld_solvent = SLD of the solvent. … … 3913 3331 background 3914 3332 3915 cm-1 3333 |cm^-1| 3916 3334 3917 3335 0.001 … … 4028 3446 background 4029 3447 4030 cm-1 3448 |cm^-1| 4031 3449 4032 3450 0 … … 4095 3513 characterized by a Gaussian distribution. 4096 3514 4097 The returned value is scaled to units of [ cm-1sr-1], absolute scale.4098 4099 The scattering intensity I(q) is calculated as :3515 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 3516 3517 The scattering intensity I(q) is calculated as 4100 3518 4101 3519 … … 4135 3553 4136 3554 4137 N OTE: The calculation of Z(q) is a double numerical integral that must3555 NB: The calculation of Z(q) is a double numerical integral that must 4138 3556 be carried out with a high density of points to properly capture the 4139 3557 sharp peaks of the paracrystalline scattering. So be warned that the … … 4160 3578 background 4161 3579 4162 cm-1 3580 |cm^-1| 4163 3581 4164 3582 0 … … 4198 3616 TEST DATASET 4199 3617 4200 This example dataset is produced using 200 data points, qmin= 0.014201 -1, qmax= 0.1 -1 and the above default values.3618 This example dataset is produced using 200 data points, *qmin* = 0.01 3619 -1, *qmax* = 0.1 -1 and the above default values. 4202 3620 4203 3621 … … 4237 3655 characterized by a Gaussian distribution. 4238 3656 4239 The returned value is scaled to units of [ cm-1sr-1], absolute scale.3657 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 4240 3658 4241 3659 The scattering intensity I(q) is calculated as: … … 4279 3697 4280 3698 4281 N OTE: The calculation of Z(q) is a double numerical integral that must3699 NB: The calculation of Z(q) is a double numerical integral that must 4282 3700 be carried out with a high density of points to properly capture the 4283 3701 sharp peaks of the paracrystalline scattering. So be warned that the … … 4306 3724 background 4307 3725 4308 cm-1 3726 |cm^-1| 4309 3727 4310 3728 0 … … 4344 3762 TEST DATASET 4345 3763 4346 This example dataset is produced using 200 data points, qmin= 0.014347 -1, qmax= 0.1 -1 and the above default values.3764 This example dataset is produced using 200 data points, *qmin* = 0.01 3765 -1, *qmax* = 0.1 -1 and the above default values. 4348 3766 4349 3767 … … 4371 3789 Paracrystalline distortion is assumed to be isotropic and 4372 3790 characterized by a Gaussian distribution.The returned value is scaled 4373 to units of [ cm-1sr-1], absolute scale.3791 to units of [|cm^-1|\ |sr^-1|, absolute scale. 4374 3792 4375 3793 The scattering intensity I(q) is calculated as: … … 4413 3831 4414 3832 4415 N OTE: The calculation of Z(q) is a double numerical integral that must3833 NB: The calculation of Z(q) is a double numerical integral that must 4416 3834 be carried out with a high density of points to properly capture the 4417 3835 sharp peaks of the paracrystalline scattering. So be warned that the … … 4440 3858 background 4441 3859 4442 cm-1 3860 |cm^-1| 4443 3861 4444 3862 0 … … 4478 3896 TEST DATASET 4479 3897 4480 This example dataset is produced using 200 data points, qmin= 0.0014481 -1, qmax= 0.1 -1 and the above default values.3898 This example dataset is produced using 200 data points, *qmin* = 0.001 3899 -1, *qmax* = 0.1 -1 and the above default values. 4482 3900 4483 3901 … … 4535 3953 4536 3954 The scattering intensity per unit volume is returned in the unit of 4537 [cm-1]; I(q) = fP(q).3955 |cm^-1|; I(q) = fP(q). 4538 3956 4539 3957 For P*S: The 2nd virial coefficient of the solid cylinder is calculate … … 4566 3984 background 4567 3985 4568 cm-1 3986 |cm^-1| 4569 3987 4570 3988 0.0 … … 4681 4099 an error, but the results will not be correct. 4682 4100 4683 The returned value is in units of [cm-1], on absolute scale.4101 The returned value is in units of |cm^-1|, on absolute scale. 4684 4102 4685 4103 For P*S: The 2nd virial coefficient of this CSPP is calculate based on … … 4708 4126 4709 4127 This example dataset is produced by running the Macro 4710 Plot_CSParallelepiped(), using 100 data points, qmin = 0.001 -1, qmax4128 Plot_CSParallelepiped(), using 100 data points, *qmin* = 0.001 -1, *qmax* 4711 4129 = 0.7 -1 and the below default values. 4712 4130 … … 4719 4137 background 4720 4138 4721 cm-1 4139 |cm^-1| 4722 4140 4723 4141 0.06 … … 4856 4274 background 4857 4275 4858 cm-1 4276 |cm^-1| 4859 4277 4860 4278 0.0 … … 4879 4297 structures). 4880 4298 4881 The returned value is scaled to units of [ cm-1sr-1], absolute scale.4299 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 4882 4300 4883 4301 The scattering intensity I(q) is calculated by: … … 4933 4351 Background (=B) 4934 4352 4935 cm-1 4353 |cm^-1| 4936 4354 4937 4355 0.1 … … 4952 4370 a low-Q signal and a high-Q signal 4953 4371 4954 The returned value is scaled to units of [ cm-1sr-1], absolute scale.4372 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 4955 4373 4956 4374 The scattering intensity I(q) is calculated by: … … 5004 4422 Background (=B) 5005 4423 5006 cm-1 4424 |cm^-1| 5007 4425 5008 4426 0.1 … … 5061 4479 background 5062 4480 5063 cm-1 4481 |cm^-1| 5064 4482 5065 4483 0.0 … … 5113 4531 background 5114 4532 5115 cm-1 4533 |cm^-1| 5116 4534 5117 4535 0.0 … … 5134 4552 2013/09/09 - Description reviewed by King, S. and Parker, P. 5135 4553 5136 *3.6. Absolute Power_Law * 4554 4555 4556 **3.6. AbsolutePowerLaw** 5137 4557 5138 4558 This model describes a power law with background. … … 5166 4586 Background 5167 4587 5168 cm-1 4588 |cm^-1| 5169 4589 5170 4590 0.0 … … 5217 4637 background 5218 4638 5219 cm-1 4639 |cm^-1| 5220 4640 5221 4641 0.0 … … 5238 4658 Calculates the scattering from fractal-like aggregates built from 5239 4659 spherical building blocks following the Texiera reference. The value 5240 returned is in cm-1.4660 returned is in |cm^-1|. 5241 4661 5242 4662 … … 5302 4722 background 5303 4723 5304 cm-1 4724 |cm^-1| 5305 4725 5306 4726 0.0 … … 5339 4759 scattering length density of particles. 5340 4760 5341 N ote: The mass fractal dimension is valid for 1<mass_dim<6.4761 NB: The mass fractal dimension is valid for 1<mass_dim<6. 5342 4762 5343 4763 … … 5416 4836 scattering length density of particles. 5417 4837 5418 N ote: The surface fractal dimension is valid for 1<surface_dim<3. Also4838 NB: The surface fractal dimension is valid for 1<surface_dim<3. Also 5419 4839 it is valid in a limited q range (see the reference for details). 5420 4840 … … 5507 4927 length density of particles. 5508 4928 5509 N ote: The surface and mass fractal dimensions are valid for4929 NB: The surface and mass fractal dimensions are valid for 5510 4930 0<surface_dim<6, 0<mass_dim<6, and (surface_mass+mass_dim)<6. 5511 4931 … … 5583 5003 provided. 5584 5004 5585 The returned value is scaled to units of [ cm-1sr-1], absolute scale.5005 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 5586 5006 5587 5007 See each of these individual models for full documentation. … … 5641 5061 background 5642 5062 5643 cm-1 5063 |cm^-1| 5644 5064 5645 5065 0.0 … … 5664 5084 structures. 5665 5085 5666 The returned value is scaled to units of [ cm-1sr-1], absolute scale.5086 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 5667 5087 5668 5088 The scattering intensity I(q) is calculated as (eqn 5 from the … … 5717 5137 background 5718 5138 5719 cm-1 5139 |cm^-1| 5720 5140 5721 5141 0.0 … … 5738 5158 Calculates the structure factor of a polyelectrolyte solution with the 5739 5159 RPA expression derived by Borue and Erukhimovich. The value returned 5740 is in cm-1.5160 is in |cm^-1|. 5741 5161 5742 5162 … … 5803 5223 background 5804 5224 5805 cm-1 5225 |cm^-1| 5806 5226 5807 5227 0.0 … … 5844 5264 scale 5845 5265 5846 cm-1 5266 |cm^-1| 5847 5267 5848 5268 1.0 … … 5863 5283 5864 5284 The returned value is P(Q) as written in equation (1), plus the 5865 incoherent background term. The result is in the units of [ cm-1sr-1],5285 incoherent background term. The result is in the units of [|cm^-1|\ |sr^-1|, 5866 5286 absolute scale. 5867 5287 … … 5925 5345 Scale(=Guinier scale, G) 5926 5346 5927 cm-1 5347 |cm^-1| 5928 5348 5929 5349 1.0 … … 5990 5410 background 5991 5411 5992 cm-1 5412 |cm^-1| 5993 5413 5994 5414 0 … … 6023 5443 scale 6024 5444 6025 cm-1 5445 |cm^-1| 6026 5446 6027 5447 100 … … 6080 5500 scale 6081 5501 6082 cm-1 5502 |cm^-1| 6083 5503 6084 5504 100 … … 6112 5532 molecular weight distribution. 6113 5533 6114 The returned value is scaled to units of [ cm-1sr-1], absolute scale.5534 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 6115 5535 6116 5536 … … 6135 5555 6136 5556 This example dataset is produced by running the Poly_GaussCoil, using 6137 200 data points, qmin = 0.001 -1, qmax= 0.7 -1 and the default values5557 200 data points, *qmin* = 0.001 -1, *qmax* = 0.7 -1 and the default values 6138 5558 below. 6139 5559 … … 6164 5584 background 6165 5585 6166 cm-1 5586 |cm^-1| 6167 5587 6168 5588 0.001 … … 6189 5609 Calculates the scattering from polymers with excluded volume effects. 6190 5610 6191 The returned value is scaled to units of [ cm-1sr-1], absolute scale.5611 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 6192 5612 6193 5613 The returned value is P(Q) as written in equation (2), plus the 6194 incoherent background term. The result is in the units of [ cm-1sr-1],5614 incoherent background term. The result is in the units of [|cm^-1|\ |sr^-1|, 6195 5615 absolute scale. 6196 5616 … … 6260 5680 TEST DATASET 6261 5681 6262 This example dataset is produced, using 200 data points, qmin= 0.0016263 -1, qmax= 0.2 -1 and the default values below.5682 This example dataset is produced, using 200 data points, *qmin* = 0.001 5683 -1, *qmax* = 0.2 -1 and the default values below. 6264 5684 6265 5685 Parameter name … … 6287 5707 background 6288 5708 6289 cm-1 5709 |cm^-1| 6290 5710 6291 5711 0.0 … … 6325 5745 Case 9: A-B-C-D Four-block copolymer 6326 5746 6327 N ote: the case numbers are different from the IGOR/NIST SANS package.5747 NB: the case numbers are different from the IGOR/NIST SANS package. 6328 5748 6329 5749 … … 6332 5752 will overwrite the original parameter waves. 6333 5753 6334 The returned value is scaled to units of [cm-1].5754 The returned value is scaled to units of |cm^-1|. 6335 5755 6336 5756 Component D is assumed to be the "background" component (all contrasts … … 6368 5788 background 6369 5789 6370 cm-1 5790 |cm^-1| 6371 5791 6372 5792 0.0 … … 6447 5867 a two Lorentzian functions. 6448 5868 6449 The returned value is scaled to units of [ cm-1sr-1], absolute scale.5869 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 6450 5870 6451 5871 The scattering intensity I(q) is calculated by: … … 6504 5924 Background(=B) 6505 5925 6506 cm-1 5926 |cm^-1| 6507 5927 6508 5928 0.1 … … 6525 5945 two power laws. 6526 5946 6527 The returned value is scaled to units of [ cm-1sr-1], absolute scale.5947 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 6528 5948 6529 5949 … … 6570 5990 background 6571 5991 6572 cm-1 5992 |cm^-1| 6573 5993 6574 5994 0.0 … … 6586 6006 *3.25. UnifiedPower(Law and)Rg(Model)* 6587 6007 6588 The returned value is scaled to units of [ cm-1sr-1], absolute scale.6008 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 6589 6009 6590 6010 Note that the level 0 is an extra function that is the inverse … … 6643 6063 G2 6644 6064 6645 cm-1sr-1 6065 |cm^-1|\ |sr^-1| 6646 6066 6647 6067 3 … … 6649 6069 B2 6650 6070 6651 cm-1sr-1 6071 |cm^-1|\ |sr^-1| 6652 6072 6653 6073 0.0006 … … 6665 6085 G1 6666 6086 6667 cm-1sr-1 6087 |cm^-1|\ |sr^-1| 6668 6088 6669 6089 400 … … 6671 6091 B1 6672 6092 6673 cm-1sr-1 6093 |cm^-1|\ |sr^-1| 6674 6094 6675 6095 4.5e-006 … … 6677 6097 background 6678 6098 6679 cm-1 6099 |cm^-1| 6680 6100 6681 6101 0.0 … … 6708 6128 terms. 6709 6129 6710 *N ote:* For 2D plot, I(q) = I(qx)*I(qy) which is defined differently6130 *NB:* For 2D plot, I(q) = I(qx)*I(qy) which is defined differently 6711 6131 from other shape independent models. 6712 6132 … … 6719 6139 A 6720 6140 6721 cm-1 6141 |cm^-1| 6722 6142 6723 6143 1.0 … … 6743 6163 sigma=roughness). 6744 6164 6745 N ote: This model was contributed by an interested user.6165 NB: This model was contributed by an interested user. 6746 6166 6747 6167 … … 6782 6202 6783 6203 6784 N ote: This model was implemented by an interested user.6204 NB: This model was implemented by an interested user. 6785 6205 6786 6206 *3.29. GelFitModel* … … 6812 6232 2.8. 6813 6233 6814 N ote: This model was implemented by an interested user.6234 NB: This model was implemented by an interested user. 6815 6235 6816 6236 *Default input parameter values* … … 6824 6244 Background 6825 6245 6826 cm-1 6246 |cm^-1| 6827 6247 6828 6248 0.01 … … 6830 6250 Guinier scale 6831 6251 6832 cm-1 6252 |cm^-1| 6833 6253 6834 6254 1.7 … … 6836 6256 Lorentzian scale 6837 6257 6838 cm-1 6258 |cm^-1| 6839 6259 6840 6260 3.5 … … 6863 6283 6864 6284 *Figure. 1D plot using the default values (w/300 data points, 6865 qmin=0.001, and qmax=0.3).*6285 *qmin*=0.001, and *qmax*=0.3).* 6866 6286 6867 6287 … … 6877 6297 6878 6298 6879 * 3.30. Star Polymer with Gaussian Statistics*6299 **3.30. Star Polymer with Gaussian Statistics** 6880 6300 6881 6301 For a star with *f* arms: … … 6903 6323 ------------------------------ 6904 6324 6905 The information in this section is originated from NIST SANS IgorPro 6906 package. 6907 6908 *5.1. HardSphere Structure * 6325 The information in this section is originated from NIST SANS IgorPro package. 6326 6327 **2.3.1. HardSphereStructure Factor** 6909 6328 6910 6329 This calculates the interparticle structure factor for monodisperse spherical particles interacting through hard sphere (excluded volume) interactions. The calculation uses the Percus-Yevick closure where the interparticle potential is: … … 6942 6361 Percus, J. K.; Yevick, J. Phys. Rev. 110, 1. (1958). 6943 6362 6944 *5.2. SquareWell Structure * 6363 6364 6365 **2.3.2. SquareWellStructure Factor** 6945 6366 6946 6367 This calculates the interparticle structure factor for a square well fluid spherical particles The mean spherical … … 7002 6423 7003 6424 7004 * 5.3. HayterMSA Structure*6425 **2.3.3. HayterMSAStructure Factor** 7005 6426 7006 6427 This calculates the Structure factor (the Fourier transform of the pair correlation function g(r)) for a system of … … 7062 6483 JB Hayter and J Penfold, Molecular Physics 42, 109-118 (1981). 7063 6484 7064 *5.4. StickyHS Structure * 6485 6486 6487 **2.3.4. StickyHSStructure Factor** 7065 6488 7066 6489 This calculates the interparticle structure factor for a hard sphere … … 7149 6572 7150 6573 7151 7152 7153 6574 2.4 Customised Functions 7154 6575 ------------------------------ 7155 6576 7156 6577 7157 Customized model functions can be redefined or added by users (See 7158 SansView tutorial for details). 7159 7160 *4.1. testmodel* 7161 7162 6578 Customized model functions can be redefined or added to by users (See SansView tutorial for details). 6579 6580 .. _testmodel: 6581 6582 **2.4.1. testmodel** 7163 6583 7164 6584 This function, as an example of a user defined function, calculates 7165 the intensity = A + Bcos(2q) + Csin(2q). 7166 7167 *4.2. testmodel_2 * 6585 6586 *I(q)* = *A* + *B* cos(2\ *q*\ ) + *C* sin(2\ *q*\ ) 6587 6588 6589 6590 .. _testmodel_2: 6591 6592 **4.2. testmodel_2** 7168 6593 7169 6594 This function, as an example of a user defined function, calculates 7170 the intensity = scale * sin(f)/f, where f = A + Bq + Cq2 + Dq3 + Eq4 + 7171 Fq5. 7172 7173 *4.3. sum_p1_p2 * 6595 6596 *I(q)* = *scale* * sin(*f*\ )/*f* 6597 6598 where 6599 6600 *f* = *A* + *Bq* + *Cq*\ :sup:`2` + *Dq*\ :sup:`3` + *Eq*\ :sup:`4` + *Fq*\ :sup:`5` 6601 6602 6603 6604 .. _sum_p1_p2: 6605 6606 **4.3. sum_p1_p2** 7174 6607 7175 6608 This function, as an example of a user defined function, calculates 7176 the intensity = scale_factor * (CylinderModel + PolymerExclVolume 7177 model). To make your own sum(P1+P2) model, select 'Easy Custom Sum' 7178 from the Fitting menu, or modify and compile the file named 7179 'sum_p1_p2.py' from 'Edit Custom Model' in the 'Fitting' menu. It 7180 works only for single functional models. 7181 7182 *4.4. sum_Ap1_1_Ap2 * 6609 6610 *I(q)* = *scale_factor* \* (CylinderModel + PolymerExclVolumeModel) 6611 6612 To make your own (*p1 + p2*) model, select 'Easy Custom Sum' from the Fitting menu, or modify and compile the file 6613 named 'sum_p1_p2.py' from 'Edit Custom Model' in the 'Fitting' menu. 6614 6615 NB: Summing models only works only for single functional models (ie, single shell models, two-component RPA models, etc). 6616 6617 6618 6619 .. _sum_Ap1_1_Ap2: 6620 6621 **4.4. sum_Ap1_1_Ap2** 7183 6622 7184 6623 This function, as an example of a user defined function, calculates 7185 the intensity = (scale_factor * CylinderModel + (1-scale_factor) * 7186 PolymerExclVolume model). To make your own A*p1+(1-A)*p2 model, modify 7187 and compile the file named 'sum_Ap1_1_Ap2.py' from 'Edit Custom Model' 7188 in the 'Fitting' menu. It works only for single functional models. 7189 7190 *4.5. polynomial5 * 6624 6625 *I(q)* = (*scale_factor* \* CylinderModel + (1 - *scale_factor*\ ) \* PolymerExclVolume model) 6626 6627 To make your own (*A*\ * *p1* + (1-*A*) \* *p2*) model, modify and compile the file named 'sum_Ap1_1_Ap2.py' from 6628 'Edit Custom Model' in the 'Fitting' menu. 6629 6630 NB: Summing models only works only for single functional models (ie, single shell models, two-component RPA models, etc). 6631 6632 6633 6634 .. _polynomial5: 6635 6636 **4.5. polynomial5** 7191 6637 7192 6638 This function, as an example of a user defined function, calculates 7193 the intensity = A + Bq + Cq2 + Dq3 + Eq4 + Fq5. This model can be 7194 modified and compiled from 'Edit Custom Model' in the 'Fitting' menu. 7195 7196 *4.6. sph_bessel_jn * 6639 6640 *I(q)* = *A* + *Bq* + *Cq*\ :sup:`2` + *Dq*\ :sup:`3` + *Eq*\ :sup:`4` + *Fq*\ :sup:`5` 6641 6642 This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu. 6643 6644 6645 6646 .. _sph_bessel_jn: 6647 6648 **4.6. sph_bessel_jn** 7197 6649 7198 6650 This function, as an example of a user defined function, calculates 7199 the intensity = C*sph_jn(Ax+B)+D where the sph_jn is spherical Bessel 7200 function of the order n. This model can be modified and compiled from 7201 'Edit Custom Model' in the 'Fitting' menu. 6651 6652 *I(q)* = *C* \* *sph_jn(Ax+B)+D* 6653 6654 where *sph_jn* is a spherical Bessel function of order *n*. 6655 6656 This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu.
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