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src/sans/models/media/model_functions.rst
rbf8c07b r1127c32 186 186 -------- 187 187 188 - LamellarModel 189 - LamellarFFHGModel 190 - LamellarPSModel 191 - LamellarPSHGModel 188 - LamellarModel_ 189 - LamellarFFHGModel_ 190 - LamellarPSModel_ 191 - LamellarPSHGModel_ 192 192 193 193 Paracrystals 194 194 ------------ 195 195 196 - LamellarPCrystalModel 196 - LamellarPCrystalModel_ 197 197 - SCCrystalModel 198 198 - FCCrystalModel … … 2356 2356 **2.1.29. LamellarModel** 2357 2357 2358 This model provides the scattering intensity, I( *q*), for a lyotropic 2359 lamellar phase where a uniform SLD and random distribution in solution 2360 are assumed. The ploydispersion in the bilayer thickness can be 2361 applied from the GUI. 2362 2363 The scattering intensity I(q) is: 2364 2365 2366 2367 The form factor is, 2368 2369 2370 2371 where d = bilayer thickness. 2372 2373 The 2D scattering intensity is calculated in the same way as 1D, where 2374 the *q* vector is defined as . 2375 2376 2377 2378 The returned value is in units of |cm^-1|, on absolute scale. In the 2379 parameters, sld_bi = SLD of the bilayer, sld_sol = SLD of the solvent, 2380 and bi_thick = the thickness of the bilayer. 2358 This model provides the scattering intensity, *I(q)*, for a lyotropic lamellar phase where a uniform SLD and random 2359 distribution in solution are assumed. Polydispersity in the bilayer thickness can be applied from the GUI. 2360 2361 *2.1.29.1. Definition* 2362 2363 The scattering intensity *I(q)* is 2364 2365 .. image:: img/image133.PNG 2366 2367 The form factor is 2368 2369 .. image:: img/image134.PNG 2370 2371 where |delta| = bilayer thickness. 2372 2373 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2374 2375 .. image:: img/image040.GIF 2376 2377 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_bi* = SLD of the bilayer, 2378 *sld_sol* = SLD of the solvent, and *bi_thick* = thickness of the bilayer. 2381 2379 2382 2380 ============== ======== ============= … … 2390 2388 ============== ======== ============= 2391 2389 2392 2390 .. image:: img/image135.JPG 2393 2391 2394 2392 *Figure. 1D plot using the default values (w/1000 data point).* 2395 2393 2396 Our model uses the form factor calculations implemented in a c-library 2397 provided by the NIST Center for Neutron Research (Kline, 2006): 2394 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 2395 (Kline, 2006). 2398 2396 2399 2397 REFERENCE … … 2409 2407 **2.1.30. LamellarFFHGModel** 2410 2408 2411 This model provides the scattering intensity, I( *q*), for a lyotropic 2412 lamellar phase where a random distribution in solution are assumed. 2413 The SLD of the head region is taken to be different from the SLD of 2414 the tail region. 2415 2416 The scattering intensity I(q) is: 2417 2418 2419 2420 The form factor is, 2421 2422 2423 2424 where dT = tail length (or t_length), dH = heasd thickness (or 2425 h_thickness) , DrH = SLD (headgroup) - SLD(solvent), and DrT = SLD 2426 (tail) - SLD(headgroup). 2427 2428 The 2D scattering intensity is calculated in the same way as 1D, where 2429 the *q* vector is defined as . 2430 2431 2432 2433 The returned value is in units of |cm^-1|, on absolute scale. In the 2434 parameters, sld_tail = SLD of the tail group, and sld_head = SLD of 2435 the head group. 2409 This model provides the scattering intensity, *I(q)*, for a lyotropic lamellar phase where a random distribution in 2410 solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region. 2411 2412 *2.1.31.1. Definition* 2413 2414 The scattering intensity *I(q)* is 2415 2416 .. image:: img/image136.PNG 2417 2418 The form factor is 2419 2420 .. image:: img/image137.JPG 2421 2422 where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), 2423 |bigdelta|\ |rho|\ H = SLD(headgroup) - SLD(solvent), and |bigdelta|\ |rho|\ T = SLD(tail) - SLD(headgroup). 2424 2425 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2426 2427 .. image:: img/image040.GIF 2428 2429 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, 2430 and *sld_head* = SLD of the head group. 2436 2431 2437 2432 ============== ======== ============= … … 2447 2442 ============== ======== ============= 2448 2443 2449 2444 .. image:: img/image138.JPG 2450 2445 2451 2446 *Figure. 1D plot using the default values (w/1000 data point).* 2452 2447 2453 Our model uses the form factor calculations implemented in a c-library 2454 provided by the NIST Center for Neutron Research (Kline, 2006): 2448 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 2449 (Kline, 2006). 2455 2450 2456 2451 REFERENCE … … 2466 2461 **2.1.31. LamellarPSModel** 2467 2462 2468 This model provides the scattering intensity ( *form factor* \* 2469 *structure factor*), I( *q*), for a lyotropic lamellar phase where a 2470 random distribution in solution are assumed. 2471 2472 The scattering intensity I(q) is: 2473 2474 2463 This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lyotropic lamellar phase where a random 2464 distribution in solution are assumed. 2465 2466 *2.1.31.1. Definition* 2467 2468 The scattering intensity *I(q)* is 2469 2470 .. image:: img/image139.PNG 2475 2471 2476 2472 The form factor is 2477 2473 2478 2479 2480 and the structure is2481 2482 2474 .. image:: img/image134.PNG 2475 2476 and the structure factor is 2477 2478 .. image:: img/image140.PNG 2483 2479 2484 2480 where 2485 2481 2486 2487 2488 2489 2490 2491 2492 Here d= (repeat) spacing, d = bilayer thickness, the contrast Dr = SLD 2493 (headgroup) - SLD(solvent), K=smectic bending elasticity, 2494 B=compression modulus, and N = number of lamellar plates (n_plates). 2495 2496 NB: When the Caille parameter is greater than approximately 0.8 to 2497 1.0, the assumptions of the model are incorrect. And due to the 2498 complication of the model function, users are responsible to make sure 2499 that all the assumptions are handled accurately: see the original 2500 reference (below) for more details. 2501 2502 The 2D scattering intensity is calculated in the same way as 1D, where 2503 the *q* vector is defined as . 2482 .. image:: img/image141.PNG 2483 2484 Here *d* = (repeat) spacing, |delta| = bilayer thickness, the contrast |bigdelta|\ |rho| = SLD(headgroup) - SLD(solvent), 2485 K = smectic bending elasticity, B = compression modulus, and N = number of lamellar plates (*n_plates*). 2486 2487 NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** 2488 And due to a complication of the model function, users are responsible for making sure that all the assumptions are 2489 handled accurately (see the original reference below for more details). 2490 2491 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2492 2493 .. image:: img/image040.GIF 2504 2494 2505 2495 The returned value is in units of |cm^-1|, on absolute scale. … … 2517 2507 ============== ======== ============= 2518 2508 2519 2509 .. image:: img/image142.JPG 2520 2510 2521 2511 *Figure. 1D plot using the default values (w/6000 data point).* 2522 2512 2523 Our model uses the form factor calculations implemented in a c-library 2524 provided by the NIST Center for Neutron Research (Kline, 2006): 2513 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 2514 (Kline, 2006). 2525 2515 2526 2516 REFERENCE … … 2536 2526 **2.1.32. LamellarPSHGModel** 2537 2527 2538 This model provides the scattering intensity ( *form factor* \* 2539 *structure factor*), I( *q*), for a lyotropic lamellar phase where a 2540 random distribution in solution are assumed. The SLD of the head 2541 region is taken to be different from the SLD of the tail region. 2542 2543 The scattering intensity I(q) is: 2544 2545 2546 2547 The form factor is, 2548 2549 2528 This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lyotropic lamellar phase where a random 2529 distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail 2530 region. 2531 2532 *2.1.32.1. Definition* 2533 2534 The scattering intensity *I(q)* is 2535 2536 .. image:: img/image139.PNG 2537 2538 The form factor is 2539 2540 .. image:: img/image143.PNG 2550 2541 2551 2542 The structure factor is 2552 2543 2553 2544 .. image:: img/image140.PNG 2554 2545 2555 2546 where 2556 2547 2557 2558 2559 2560 2561 2562 2563 where dT = tail length (or t_length), dH = heasd thickness (or 2564 h_thickness) , DrH = SLD (headgroup) - SLD(solvent), and DrT = SLD 2565 (tail) - SLD(headgroup). Here d= (repeat) spacing, K=smectic bending 2566 elasticity, B=compression modulus, and N = number of lamellar plates 2567 (n_plates). 2568 2569 NB: When the Caille parameter is greater than approximately 0.8 to 2570 1.0, the assumptions of the model are incorrect. And due to the 2571 complication of the model function, users are responsible to make sure 2572 that all the assumptions are handled accurately: see the original 2573 reference (below) for more details. 2574 2575 The 2D scattering intensity is calculated in the same way as 1D, where 2576 the *q* vector is defined as . 2577 2578 2579 2580 The returned value is in units of |cm^-1|, on absolute scale. In the 2581 parameters, sld_tail = SLD of the tail group, sld_head = SLD of the 2582 head group, and sld_solvent = SLD of the solvent. 2548 .. image:: img/image141.PNG 2549 2550 where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), 2551 |bigdelta|\ |rho|\ H = SLD(headgroup) - SLD(solvent), and |bigdelta|\ |rho|\ T = SLD(tail) - SLD(headgroup). 2552 Here *d* = (repeat) spacing, *K* = smectic bending elasticity, *B* = compression modulus, and N = number of lamellar 2553 plates (*n_plates*). 2554 2555 NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** 2556 And due to a complication of the model function, users are responsible for making sure that all the assumptions are 2557 handled accurately (see the original reference below for more details). 2558 2559 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2560 2561 .. image:: img/image040.GIF 2562 2563 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, 2564 *sld_head* = SLD of the head group, and *sld_solvent* = SLD of the solvent. 2583 2565 2584 2566 ============== ======== ============= … … 2597 2579 ============== ======== ============= 2598 2580 2599 2600 2581 .. image:: img/image144.JPG 2601 2582 2602 2583 *Figure. 1D plot using the default values (w/6000 data point).* 2603 2584 2604 Our model uses the form factor calculations implemented in a c-library 2605 provided by the NIST Center for Neutron Research (Kline, 2006): 2585 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 2586 (Kline, 2006). 2606 2587 2607 2588 REFERENCE … … 2617 2598 **2.1.33. LamellarPCrystalModel** 2618 2599 2619 Lamella ParaCrystal Model: Calculates the scattering from a stack of 2620 repeating lamellar structures. The stacks of lamellae (infinite in 2621 lateral dimension) are treated as a paracrystal to account for the 2622 repeating spacing. The repeat distance is further characterized by a 2623 Gaussian polydispersity. This model can be used for large 2624 multilamellar vesicles. 2625 2626 The scattering intensity I(q) is calculated as: 2627 2628 2629 2630 The form factor of the bilayer is approximated as the cross section of 2631 an infinite, planar bilayer of thickness t. 2632 2633 2634 2635 Here, the scale factor is used instead of the mass per area of the 2636 bilayer (G). The scale factor is the volume fraction of the material 2637 in the bilayer, not the total excluded volume of the paracrystal. 2638 ZN(q) describes the interference effects for aggregates consisting of 2639 more than one bilayer. The equations used are (3-5) from the Bergstrom 2640 reference below. 2641 2642 Non-integer numbers of stacks are calculated as a linear combination 2643 of the lower and higher values: 2644 2645 2646 2647 The 2D scattering intensity is the same as 1D, regardless of the 2648 orientation of the *q* vector which is defined as . 2649 2650 The parameters of the model are the following (Nlayers= no. of layers, 2651 pd_spacing= polydispersity of spacing): 2600 This model calculates the scattering from a stack of repeating lamellar structures. The stacks of lamellae (infinite 2601 in lateral dimension) are treated as a paracrystal to account for the repeating spacing. The repeat distance is further 2602 characterized by a Gaussian polydispersity. **This model can be used for large multilamellar vesicles.** 2603 2604 *2.1.33.1. Definition* 2605 2606 The scattering intensity *I(q)* is calculated as 2607 2608 .. image:: img/image145.JPG 2609 2610 The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* 2611 2612 .. image:: img/image146.JPG 2613 2614 Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume 2615 fraction of the material in the bilayer, *not* the total excluded volume of the paracrystal. *ZN(q)* describes the 2616 interference effects for aggregates consisting of more than one bilayer. The equations used are (3-5) from the 2617 Bergstrom reference below. 2618 2619 Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values 2620 2621 .. image:: img/image147.JPG 2622 2623 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 2624 2625 .. image:: img/image040.GIF 2626 2627 The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. 2652 2628 2653 2629 ============== ======== ============= … … 2664 2640 ============== ======== ============= 2665 2641 2666 2667 2668 *Figure. 1D plot using the default values above (w/20000 data 2669 point).* 2670 2671 Our model uses the form factor calculations implemented in a c-library 2672 provided by the NIST Center for Neutron Research (Kline, 2006). 2673 2674 See the reference for details. 2642 .. image:: img/image148.JPG 2643 2644 *Figure. 1D plot using the default values above (w/20000 data point).* 2645 2646 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 2647 (Kline, 2006). 2675 2648 2676 2649 REFERENCE
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