6. formfactor (ff)¶
Particle form factors
If possible the scattering intensity (Iq) of a single particle with real scattering length densities is calculated. It is the normalized particle form factor (Fq) if the scattering length is not unique defined as e.g. for beaucage model.
The scattering per particle is I(q)= I_0 F(q) with particle form factor F(q)=<F_a(q)F^*_a(q)>=<|F_a(q)|^2>. <> indicates the ensemble average.
The particle scattering amplitude
F_a(q)= \int_V b(r) e^{iqr} \mathrm{d}r / \int_V b(r) \mathrm{d}r = \sum_N b_i e^{iqr} / \sum_N b_i
The forward scattering per particle is I_0=V_p^2(SLD-solventSLD)^2 with particle volume V_p.
In this module units for I(q) and I_0 are nm^2=10^{-14} cm^2 per particle.
The scattering of particles with concentration c in mol/liter in units of \frac{1}{cm} is I_{[1/cm]}(q)=N_A \frac{c}{1000} 10^{-14} I_{[nm^2]}(q).
The scattering of arbitrary shaped particles can be calculated by cloudScattering()
as a cloud of points representing the desired shape as a kind of volume integration.
In the same way distributions of particles as e.g. clusters of particles or nanocrystals can be calculated.
Oriented scattering of e.g. magnetic nanoclusters in a magnetic field can be calculated by
orientedCloudScattering()
.
Methods to build clouds of scatterers e.g. a cube decorated with spheres at the corners can be found in Lattice with examples. The advantage here is that there is no double counted overlap.
- Some scattering length densities as guide to choose realistic values for SLD and solventSLD:
- neutron scattering unit nm^-2:
- protonated polyethylene glycol = 0.640e-6 A^-2 = 0.640e-4 nm^-2
- SiO2 = 4.186e-6 A^-2 = 4.186e-4 nm^-2
- D2O = 6.335e-6 A^-2 = 6.335e-4 nm^-2
- H2O =-0.560e-6 A^-2 =-0.560e-4 nm^-2
- gold = 4.400e-6 A^-2 = 4.400e-4 nm^-2
- Xray scattering unit nm^-2:
- polyethylene glycol = 1.09e-3 nm^-2 = 387 e/nm**3
- SiO2 = 2.20e-3 nm^-2 = 781 e/nm**3
- D2O = 0.94e-3 nm^-2 = 332 e/nm**3
- H2O = 0.94e-3 nm^-2 = 333 e/nm**3
- protein = 1.20e-3 nm^-2 = 433 e/nm**3
- gold = 12.9e-3 nm^-2 =4589 e/nm**3
Return values are dataArrays were useful. To get only Y values use .Y
6.1. Form Factors¶
beaucage (q[, Rg, G, d]) |
Beaucage introduced a model based on the polymer fractal model. |
genGuinier (q[, Rg, A, alpha]) |
Generalized Guinier approximation for low wavevector q scattering q*Rg< 1-1.3 |
guinier (q[, Rg, A]) |
Classical Guinier |
gaussianChain (q, Rg[, nu]) |
General formfactor of a gaussian polymer chain with excluded volume parameter. |
ringPolymer (q, Rg) |
General formfactor of a polymer ring in theta solvent. |
sphere (q, radius[, contrast]) |
Scattering of a single homogeneous sphere. |
sphereCoreShell (q, Rc, Rs, bc, bs[, solventSLD]) |
Scattering of a spherical core shell particle. |
ellipsoid (q, Ra, Rb[, SLD, solventSLD, …]) |
Form factor for a simple ellipsoid (ellipsoid of revolution). |
multiShellSphere (q, shellthickness, shellSLD) |
Scattering of spherical multi shell particle including linear contrast variation in subshells. |
multiShellEllipsoid (q, poleshells, …[, …]) |
Scattering of multi shell ellipsoidal particle with variing shell thickness at pole and equator. |
multiShellCylinder (q, L, shellthickness, …) |
Multi shell cylinder with caps in solvent averaged over axis orientations. |
multilamellarVesicles (Q, R, N, phi[, …]) |
Scattering intensity of a multilamellar vesicle with random displacements of the inner vesicles [1]. |
cuboid (q, a[, b, c, SLD, solventSLD, NN]) |
Formfactor of cuboid. |
sphereFuzzySurface (q, R, sigmasurf, contrast) |
Scattering of a sphere with a fuzzy interface. |
sphereGaussianCorona (q, R, Rg, Ncoil, coilequR) |
Scattering of a sphere surrounded by gaussian coils as model for grafted polymers on particle e.g. |
pearlNecklace (Q, Rc, l, N[, A1, A2, A3, ms, mr]) |
Formfactor of a pearlnecklace (freely jointed chain of pearls connected by rods) |
wormlikeChain (q, N, a[, R, SLD, solventSLD, …]) |
Scattering of a wormlike chain, which correctly reproduces the rigid-rod and random-coil limits. |
teubnerStrey (q, xi, d[, eta2]) |
Phenomenological model for the scattering intensity of a two-component system using the Teubner-Strey model [1]. |
ellipsoidFilledCylinder ([q, R, L, Ra, Rb, …]) |
Scattering of a cylinder filled with ellipsoidal particles. |
superball (q, R, p[, SLD, solventSLD, nGrid, …]) |
A superball is a general mathematical shape that can be used to describe rounded cubes, sphere and octahedra. |
6.2. Addons¶
cloudScattering (q, cloud[, relError, V, …]) |
Scattering of a cloud of scatterers with variable scattering length. |
orientedCloudScattering (qxz, cloud[, rms, …]) |
2D scattering of an oriented cloud of scatterers with equal or variable scattering length. |
scatteringFromSizeDistribution (q, …[, …]) |
Scattering of a size distribution of objects with form factor fffunction |
Particle form factors
If possible the scattering intensity (Iq) of a single particle with real scattering length densities is calculated. It is the normalized particle form factor (Fq) if the scattering length is not unique defined as e.g. for beaucage model.
The scattering per particle is I(q)= I_0 F(q) with particle form factor F(q)=<F_a(q)F^*_a(q)>=<|F_a(q)|^2>. <> indicates the ensemble average.
The particle scattering amplitude
F_a(q)= \int_V b(r) e^{iqr} \mathrm{d}r / \int_V b(r) \mathrm{d}r = \sum_N b_i e^{iqr} / \sum_N b_i
The forward scattering per particle is I_0=V_p^2(SLD-solventSLD)^2 with particle volume V_p.
In this module units for I(q) and I_0 are nm^2=10^{-14} cm^2 per particle.
The scattering of particles with concentration c in mol/liter in units of \frac{1}{cm} is I_{[1/cm]}(q)=N_A \frac{c}{1000} 10^{-14} I_{[nm^2]}(q).
The scattering of arbitrary shaped particles can be calculated by cloudScattering()
as a cloud of points representing the desired shape as a kind of volume integration.
In the same way distributions of particles as e.g. clusters of particles or nanocrystals can be calculated.
Oriented scattering of e.g. magnetic nanoclusters in a magnetic field can be calculated by
orientedCloudScattering()
.
Methods to build clouds of scatterers e.g. a cube decorated with spheres at the corners can be found in Lattice with examples. The advantage here is that there is no double counted overlap.
- Some scattering length densities as guide to choose realistic values for SLD and solventSLD:
- neutron scattering unit nm^-2:
- protonated polyethylene glycol = 0.640e-6 A^-2 = 0.640e-4 nm^-2
- SiO2 = 4.186e-6 A^-2 = 4.186e-4 nm^-2
- D2O = 6.335e-6 A^-2 = 6.335e-4 nm^-2
- H2O =-0.560e-6 A^-2 =-0.560e-4 nm^-2
- gold = 4.400e-6 A^-2 = 4.400e-4 nm^-2
- Xray scattering unit nm^-2:
- polyethylene glycol = 1.09e-3 nm^-2 = 387 e/nm**3
- SiO2 = 2.20e-3 nm^-2 = 781 e/nm**3
- D2O = 0.94e-3 nm^-2 = 332 e/nm**3
- H2O = 0.94e-3 nm^-2 = 333 e/nm**3
- protein = 1.20e-3 nm^-2 = 433 e/nm**3
- gold = 12.9e-3 nm^-2 =4589 e/nm**3
Return values are dataArrays were useful. To get only Y values use .Y
-
jscatter.formfactor.
beaucage
(q, Rg=1, G=1, d=3)[source]¶ Beaucage introduced a model based on the polymer fractal model.
Beaucage used the numerical integration form (Benoit, 1957) although the analytical integral form was available [1]. This is an artificial connection of Guinier and Porod Regime . Better use the polymer fractal model [1] used in gaussiaChain.
Parameters: - q : array
Wavevector
- Rg : float
Radius of gyration in 1/q units
- G : float
Guinier scaling factor, transition between Guinier and Porod
- d : float
Porod exponent for large wavevectors
Returns: - dataArray [q,Fq]
Notes
Polymer fractals:
d = 5/3 fully swollen chains,d = 2 ideal Gaussian chains andd = 3 collapsed chains. (volume scattering)d = 4 surface scattering at a sharp interface/surfaced = 6-dim rough surface area with a dimensionality dim between 2-3 (rough surface)d = 3 Volume scatteringd < r mass fractals (eg gaussian chain)The Beaucage model is used to analyze small-angle scattering (SAS) data from fractal and particulate systems. It models the Guinier and Porod regions with a smooth transition between them and yields a radius of gyration and a Porod exponent. This model is an approximate form of an earlier polymer fractal model that has been generalized to cover a wider scope. The practice of allowing both the Guinier and the Porod scale factors to vary independently during nonlinear least-squares fits introduces undesired artefacts in the fitting of SAS data to this model.
[1] (1, 2) Analysis of the Beaucage model Boualem Hammouda J. Appl. Cryst. (2010). 43, 1474–1478 http://dx.doi.org/10.1107/S0021889810033856
-
jscatter.formfactor.
cloudScattering
(q, cloud, relError=50, V=0, formfactor=None, rms=0, ffpolydispersity=0, ncpu=0)[source]¶ Scattering of a cloud of scatterers with variable scattering length. Using multiprocessing.
Cloud can represent any object described by a cloud of scatterers with scattering amplitudes as constant, sphere scattering amplitude, Gaussian scattering amplitude or explicitly given one. The result is normalized by sum(scattering length )**2 to equal one for q=0 (except for polydispersity). Rememeber that the atomic bond length are on the order 0.1-0.2 nm. Methods to build clouds of scatterers e.g. a cube decorated with spheres at the corners can be found in Lattice with examples.
Parameters: - q : array, ndim= Nx1
wavevectors in 1/nm
- cloud : array Nx3 or Nx4
- Center of mass positions (in nm) of the N scatterers in the cloud.
- If given cloud[3] is the scattering length b at positions cloud[:3], otherwise b=1.
- relError : float
- relError>0 Explicit calculation of spherical average with Fibonacci lattice of 2*relError+1 points.
- 0<relError<1 Monte Carlo integration until relative error is smaller than relError.
- (Monte carlo integratio with pseudo random numbers, see sphereAverage)
- relError=0 The Debye equation is used.
- (no asymmetry factor beta, no multiprocessing, no rms, no ffpolydispersity).
- rms : float, default=0
Root mean square displacement =<u**2>**0.5 of the positions in cloud as random (Gaussian) displacements in units nm. Dispacement u is random for each orientation in sphere scattering. rms can be used to simulate a Debye-Waller factor.
- V : float, default=0
Volume of the scatterers for scattering amplitude (see formfactor).
- formfactor : None,’gauss’,’sphere’,’cube’
- Gridpoint scattering amplitudes F(q) are described by:
- None : const scattering amplitude.
- ‘sphere’: Sphere scattering amplitude according to [3].
- The sphere radius is R=(\frac{3V}{4\pi})^{1/3}
- ‘gauss’ : Gaussian function b_i(q)=bVexp(- \frac{V^{2/3.}}{4\pi}q^2) according to [2].
- ‘cube’ : Cube method not yet implemented.
- Explicit isotropic form factor ff as array with [q,ff] e.g. from multishell. The normalized scattering amplitude fa for each gridpoint is calculated as fa=ff**0.5/fa(0). Missing values are linear intepolated (np.interp), q values outside interval are mapped to qmin or qmax.
- ffpolydispersity : float
Polydispersity of the gridpoints in relative units for sphere, gauss, explicit. Assuming F(q*R) for each gridpoint F is scaled as F(q*f*R) with f as normal distribution around 1 and standard deviation ffpolydispersity. The scattering length b is scaled according to the respective volume change by f**3. (f<0 is set to zero . ) This results in a change of the forward scattering because of the stronger weight of larger objects.
- ncpu : int, default 0
- Number of cpus used in the pool for multiprocessing.
- not given or 0 : all cpus are used
- int>0 : min(ncpu, mp.cpu_count)
- int<0 : ncpu not to use
- 1 : single core usage for testing or comparing speed to Debye
Returns: - dataArray with columns [q, Pq, beta]
- .sumblength : Sum of blength with I(q=0)=sumblength**2
- .formfactoramplitude_q : formfactoramplitude of cloudpoints according to type for all q values.
Notes
We calculate the scattering amplitude F(q) for N atoms in a volume V with scattering length density b(r)
F(q)= \int_V b(r) e^{iqr} \mathrm{d}r / \int_V b(r) \mathrm{d}r = \sum_N b_i e^{iqr} / \sum_N b_i
with the form factor P(Q) after orientational average <>
P(Q)=< F(q) \cdot F^*(q) >=< |F(q)|^2 >
The scattering intensity of a single object represented by the cloud is I(Q)=P(Q) \cdot V^2 \cdot (\int_V b(r) \mathrm{d}r)^2.
beta is the asymmetry factor [1] beta =|< F(q) >|^2 / < |F(q)|^2 >
One has to expect a peak at q=2\pi/d_{NN} with d_{NN} as the distance between scatterers.
On the other side the cloud scattering can represent the scattering of a cluster of particles with polydispersity and position distortion according to root mean square displacements (rms). Polydispersity and rms displacements are randomly changed within the orientational average to represent an ensemble average (opposite to the time average of a single cluster). See
latticeStructureFactor()
for nanocubes and example A nano cube build of different lattices .References
[1] (1, 2) - Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).1
[2] (1, 2) An improved method for calculating the contribution of solvent to the X-ray diffraction pattern of biological molecules Fraser R MacRae T Suzuki E IUCr Journal of Applied Crystallography 1978 vol: 11 (6) pp: 693-694 [3] (1, 2) X-ray diffuse scattering by proteins in solution. Consideration of solvent influence B. A. Fedorov, O. B. Ptitsyn and L. A. Voronin J. Appl. Cryst. (1974). 7, 181-186 doi: 10.1107/S0021889874009137 Examples
The example compares to the analytic solution for an ellipsoid. For other shapes the grid may be better rotated away from the object symmetry or a random grid should be used. The example shows a good approximation with NN=20. Because of the grid peak at q=2\pi/d_{NN} the grid scatterer distance d_{NN} should be d_{NN} < \frac{1}{3} 2\pi/q_{max} .
Inspecting A nano cube build of different lattices shows other posibilities building a grid. Also a pseudorandom grid can be used
pseudoRandomLattice()
.import jscatter as js import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D # cubic grid points R=3;NN=20;relError=50 grid= np.mgrid[-R:R:1j*NN, -R:R:1j*NN,-2*R:2*R:2j*NN].reshape(3,-1).T # points inside of sphere with radius R p=1;p2=1*2 # p defines a superball with 1->sphere p=inf cuboid .... inside=lambda xyz,R1,R2,R3:(np.abs(xyz[:,0])/R1)**p2+(np.abs(xyz[:,1])/R2)**p2+(np.abs(xyz[:,2])/R3)**p2<=1 insidegrid=grid[inside(grid,R,R,2*R)] q=np.r_[0:5:0.1] p=js.grace() p.title('compare formfactors of an ellipsoid') ffe=js.ff.cloudScattering(q,insidegrid,relError=relError) p.plot(ffe,legend='cloud ff explicit') ffa=js.ff.ellipsoid(q,2*R,R) p.plot(ffa.X,ffa.Y/ffa.I0,li=1,sy=0,legend='analytic formula') p.legend() # fig = plt.figure() ax = fig.add_subplot(111, projection='3d') # show only each 20th point pxyz=insidegrid[np.random.randint(len(insidegrid),size=int(relError**3/20))] # ax.scatter(pxyz[:,0],pxyz[:,1],pxyz[:,2],color="k",s=20) ax.set_xlim([-5,5]) ax.set_ylim([-5,5]) ax.set_zlim([-5,5]) ax.set_aspect("equal") plt.tight_layout() plt.show(block=False)
An objekt with explicit given formfactor for each gridpoint.
# 5 coreshell particles in line with polydispersity rod0 = np.zeros([5, 3]) rod0[:, 1] = np.r_[0, 1, 2, 3, 4] * 4 q = js.loglist(0.01, 7, 100) cs = js.ff.sphereCoreShell(q=q, Rc=1, Rs=2, bc=0.1, bs=1, solventSLD=0) ffe = js.ff.cloudScattering(q, rod0, formfactor=cs,relError=100,ffpolydispersity=0.1) p=js.grace() p.plot(ffe)
Using cloudscattering as fit model.
We have to define a model that parametrizes the building of the cloud that we get fit parameters. As example we use two overlapping spheres. The model can be used to fit some data. The build of the model is important as it describes how the overlapp is treated e.g. as average.
It is important that the model is continious in its parameters to avoid steps as any fit algorithm cannot handle this. Therfore avoid to use the same gridpoint distance with changing number of points e.g. inside of your object. Instead change the gridpoint distance with same number of points as e.g. in the following examples.
#: test if distance from point on X axis isInside=lambda x,A,R:((x-np.r_[A,0,0])**2).sum(axis=1)**0.5<R #: model def dumbbell(q,A,R1,R2,b1,b2,bgr=0,dx=0.3,relError=100): # D sphere distance # R1, R2 radii # b1,b2 scattering length # bgr background # dx grid distance not a fit parameter!! mR=max(R1,R2) # xyz coordinates grid=np.mgrid[-A/2-mR:A/2+mR:dx,-mR:mR:dx,-mR:mR:dx].reshape(3,-1).T insidegrid=grid[isInside(grid,-A/2.,R1) | isInside(grid,A/2.,R2)] # add blength column insidegrid=np.c_[insidegrid,insidegrid[:,0]*0] # set the corresponding blength; the order is important as here b2 overwrites b1 insidegrid[isInside(insidegrid[:,:3],-A/2.,R1),3]=b1 insidegrid[isInside(insidegrid[:,:3],A/2.,R2),3]=b2 # and maybe a mix ; this depends on your model insidegrid[isInside(insidegrid[:,:3],-A/2.,R1) & isInside(insidegrid[:,:3],A/2.,R2),3]=(b2+b1)/2. # calc the scattering result=js.ff.cloudScattering(q,insidegrid,relError=relError) result.Y=result.Y+bgr # add attributes for later usage result.A=A result.R1=R1 result.R2=R2 result.dx=dx result.insidegrid=insidegrid return result # # test it q=np.r_[0.01:10:0.02] data=dumbbell(q,4,2,2,0.5,1.5) # show result configuration js.mpl.scatter3d(data.insidegrid[:,0],data.insidegrid[:,1],data.insidegrid[:,2]) # # Fit your data like this (I know that b1 abd b2 are wrong). # It may be a good idea to use not the highest resolution in the beginning because of speed. # If you have a good set of starting parameters you can decrease dx. data2=data.prune(number=100) data2.makeErrPlot(yscale='l') data2.fit(model=dumbbell, freepar={'A':3}, fixpar={'R1':2,'R2':2,'dx':0.3,'b1':1,'b2':2,'bgr':0}, mapNames={'q':'X'})
Fit a sphere formfactor. The quality of the grid approximation (number of gridpoints) may improve the correct description of higher order minima.
import numpy as np import jscatter as js # a function to discriminate what is inside of the sphere # basically a superball p2=2 is a sphere inside=lambda xyz,R1,p2:(np.abs(xyz[:,0]))**p2+(np.abs(xyz[:,1]))**p2+(np.abs(xyz[:,2]))**p2<=R1**2 def test(q,R,b,p2=2,relError=20): # make cubic grid with right size (increase NN for betetr approximation) NN=20 grid= np.mgrid[-R:R:1j*NN, -R:R:1j*NN,-R:R:1j*NN].reshape(3,-1).T # cut the edges to get a sphere insidegrid=grid[inside(grid,R,p2)] # add scattering length for points # the average scattering length density is sum(b)/sphereVolume insidegrid=np.c_[insidegrid,insidegrid[:,0]*0] insidegrid[:,3]=b # calc formfactor (normalised) for a single sphere ffs=js.ff.cloudScattering(q,insidegrid,relError=relError) # the total scattering is sumblength**2 ffs.Y*=ffs.sumblength**2 # save radius and the grid for later ffs.R=R ffs.insidegrid=insidegrid return ffs ####main q=np.r_[0:3:0.01] sp=js.formfactor.sphere(q,3,1) sp.makeErrPlot(yscale='l') # show intermediate results sp.setlimit(R=[0.3,10]) # set some reasonable limits for R sp.fit(model=test, freepar={'b':6,'R':2.1}, fixpar={}, mapNames={'q':'X'}) # show the resulting sphere grid resultgrid=sp.lastfit.insidegrid js.mpl.scatter3d(resultgrid[:,0],resultgrid[:,1],resultgrid[:,2])
Here we compare explicit calculation with the Debye equation as the later gets quite slow for larger numbers.
import jscatter as js import numpy as np R=3;NN=20 grid= np.mgrid[-R:R:1j*NN, -R:R:1j*NN,-R:R:1j*NN].reshape(3,-1).T p=1;p2=1*2 # p defines a superball with 1->sphere p=inf cuboid .... inside=lambda xyz,R:(np.abs(xyz[:,0])/R)**p2+(np.abs(xyz[:,1])/R)**p2+(np.abs(xyz[:,2])/R)**p2<=1 insidegrid=grid[inside(grid,R)] q=np.r_[0:5:0.1] ffe=js.ff.cloudScattering(q,insidegrid,relError=50) # takes about 1.9 s on single core ffd=js.ff.cloudScattering(q,insidegrid,relError=0) # takes about 47 s on single core
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jscatter.formfactor.
cuboid
(q, a, b=None, c=None, SLD=1, solventSLD=0, NN=300)[source]¶ Formfactor of cuboid.
Parameters: - q : array
Wavevector in 1/nm
- a,b,c : float, None
Edge length, for a=b=c its a cube, Units in nm. If b=None b=a. If c=None c=b.
- SLD : float, default =1
Scattering length density of cuboid.unit nm^-2 e.g. SiO2 = 4.186*1e-6 A^-2 = 4.186*1e-4 nm^-2 for neutrons
- solventSLD : float, default =0
Scattering length density of solvent. unit nm^-2 e.g. D2O = 6.335*1e-6 A^-2 = 6.335*1e-4 nm^-2 for neutrons
- NN : int
Number of gridpoints on Fibonacci lattice is 2*NN+1 for spherical average.
Returns: - dataArray [q,Iq]
- .edges
- .contrast
- .gridpoints
Notes
The orientational average is calculated on Fibonacci lattice.
References
[1] Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Pedersen, Jan Skov Advances in Colloid and Interface Science 70, 171 (1997) http://dx.doi.org/10.1016/S0001-8686(97)00312-6
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jscatter.formfactor.
cylinder
(q, L, radius, SLD, solventSLD=0, alpha=[0, 1.5707963267948966])[source]¶ Cylinder form factor (open cap).
See multiShellCylinder
-
jscatter.formfactor.
ellipsoid
(q, Ra, Rb, SLD=1, solventSLD=0, alpha=[0, 90], tol=1e-06, beta=False)[source]¶ Form factor for a simple ellipsoid (ellipsoid of revolution).
Parameters: - q : float
scattering vector unit e.g. 1/A or 1/nm 1/Ra
- Ra : float
radius rotation axis units in 1/unit(q)
- Rb : float
radius rotated axis units in 1/unit(q)
- SLD : float, default =1
Scattering length density of unit nm^-2 e.g. SiO2 = 4.186*1e-6 A^-2 = 4.186*1e-4 nm^-2 for neutrons
- solventSLD : float, default =0
Scattering length density of solvent. unit nm^-2 e.g. D2O = 6.335*1e-6 A^-2 = 6.335*1e-4 nm^-2 for neutrons
- alpha : [float,float] , default [0,90]
alpha is angle between rotation axis Ra and scattering vector q in unit grad between these angles orientation is averaged alpha=0 axis and q are parallel, other orientation is averaged
- beta : True,default False
beta is asymmetry factor according to [3]. beta = |<F(Q)>|²/<|F(Q)|²> with scattering amplitude F(Q) and form factor P(Q)=<|F(Q)|²>
- tol : float
relative tolerance for integration between alpha
Returns: - dataArray with columns [q; Iq; beta ] # if beta=True
- .RotationAxisRadius
- .RotatedAxisRadius
- .EllipsoidVolume
- .I0 forward scattering q=0
References
[1] Structure Analysis by Small-Angle X-Ray and Neutron Scattering Feigin, L. A, and D. I. Svergun, Plenum Press, New York, (1987). [2] http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html [3] (1, 2) - Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).
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jscatter.formfactor.
ellipsoidFilledCylinder
(q=1, R=10, L=0, Ra=1, Rb=2, eta=0.1, SLDcylinder=0.1, SLDellipsoid=1, SLDmatrix=0, alpha=90, epsilon=[0, 90], fPY=1, dim=3)[source]¶ Scattering of a cylinder filled with ellipsoidal particles.
Ellipsoids of revolution with a fluid like distribution and hard core interaction leading to Percus-Yevick structure factor between ellipsoids. Ellipsoids can be oriented along cylinder axis. If cylinders are in a lattice, the .Y component is seen in the diffusive scattering and the dominating cylinder contributes only to the bragg peaks.
Parameters: - q : array
wavevectors in units 1/nm
- R : float
Cylinder radius in nm
- L : float
Length of the cylinder in nm If zero infinite length is assumed, but absolute intensity is not valid, only relative intensity.
- Ra : float
radius rotation axis units in nm
- Rb : float
radius rotated axis units in nm
- eta : float
Volume fraction of ellipsoids in cylinder for use in PercusYevick structure factor. Radius in PY corresponds to sphere with same Volume as the ellipsoid.
- SLDcylinder : float,default 1
Scattering length density cylinder material in nm**-2
- SLDellipsoid : float,default 1
Scattering length density of ellipsoids in cylinder in nm**-2
- SLDmatrix : float
Scattering length desnity of the matrix outside the cylinder in nm**-2
- alpha : float, default 90
orientation of the cylinder axis to wavevector in degrees
- epsilon : [float,float], default [0,90]
min,max orientations of ellipsoids rotation axis relative to cylinder axis in degrees
- fPY : float
Factor between radius of ellipsoids Rv (equivalent volume) and radius used in structure factor Rpy Rpy=fPY*(Ra*Rb*Rb)**(1/3)
- dim : 3,1, default 3
Dimensionality of the PercusYevick structure factor 1 is one dimensional stricture factor, anything else is 3 dimesional (normal PY)
Returns: - dataArray [q,n*conv ellipsoids + cylinder, n *conv ellipsoids, cylinder, n * ellipsoids , structure factor, beta ellipsoids ]
Each contributing formfactor is given with its absolute contribution V**2*contrast**2 (NOT normalized to 1) beta ellipsoids is the asymmetry factor of Chen and Kotlarchyk.
References
to be published
Examples
import jscatter as js p=js.grace() q=js.loglist(0.01,5,800) ff=js.ff.ellipsoidFilledCylinder(q,L=100,R=5,Ra=2,Rb=1.5,eta=0.4,alpha=90,epsilon=[0,90]) p.plot(ff.X,ff[2],legend='convolution cylinder x ellipsoids') p.plot(ff.X,ff[3],legend='cylinder only') p.plot(ff.X,ff[4],legend='ellipsoid only') p.plot(ff.X,ff[5],legend='structure factor ellipsoids') p.plot(ff.X,ff.Y,legend='conv. ellipsoid + filled cylinder') p.legend() p.yaxis(scale='l',label='I(q)') p.xaxis(scale='l',label='q / nm\S-1') # an angular averaged formfactor def averageEFC(q,R,L,Ra,Rb,eta,alpha=[alpha0,alpha1],fPY=fPY) res=js.dL() alphas=np.deg2rad(np.r_[alpha0:alpha1:13j]) for alpha in alphas: ffe=js.ff.ellipsoidFilledCylinder(q,R=R,L=L,Ra=Ra,Rb=Rb,eta=ata,alpha=alpha,) res.append(ffe) result=res[0].copy() result.Y=scipy.integrate.simps(res.Y,alphas)/(alpha1-alpha0) return result
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jscatter.formfactor.
gaussianChain
(q, Rg, nu=0.5)[source]¶ General formfactor of a gaussian polymer chain with excluded volume parameter.
For nu=0.5 this is the Debye model for Gaussian chain in theta solvent. nu>0.5 for good solvents,nu<0.5 for bad solvents.
Parameters: - q : array
Scattering vector, unit eg 1/A or 1/nm
- Rg : float
Radius of gyration, units in 1/unit(q)
- nu : float, default=0.5
ν is the excluded volume parameter, which is related to the Porod exponent d as ν = 1/d and [5/3 <= d <= 3].
Returns: - dataArray [q,Fq]
- .radiusOfGyration
- .nu excluded volume parameter
Notes
Rg^2=l^2 N^{2\nu} with monomer length l and monomer number N.
calcs
P(Q) = \frac{1}{\nu U^{\frac{1}{2\nu}}} \gamma_{inc}(\frac{1}{2\nu}, U) - \frac{1}{\nu U^{\frac{1}{\nu}}} \gamma_{inc}(\frac{1}{\nu}, U)
with U=(qR_g)^2 and \gamma_{inc} as lower incomplete gamma function.
From [1]: “Note that this model describing polymer chains with excluded volume applies only in the mass fractal range ([5/3 <= d <= 3]) and does not apply to surface fractals ([3 < d < 4]). It does not reproduce the rigid-rod limit (d = 1) because it assumes chain flexibility from the outset, nor does it describe semi-flexible chains ([1 < d < 5/3]). “
References
[1] (1, 2) Analysis of the Beaucage model Boualem Hammouda J. Appl. Cryst. (2010). 43, 1474–1478 http://dx.doi.org/10.1107/S0021889810033856 [2] SANS from homogeneous polymer mixtures: A unified overview. Hammouda, B. in Polymer Characteristics 87–133 (Springer-Verlag, 1993). doi:10.1007/BFb0025862 Examples
import jscatter as js import numpy as np q=js.loglist(0.1,8,100) p=js.grace() for nu in np.r_[0.3:0.61:0.05]: iq=js.ff.gaussianChain(q,2,nu) p.plot(iq,le='nu= $nu') p.yaxis(scale='l') p.xaxis(scale='l') p.legend(x=0.2,y=0.5)
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jscatter.formfactor.
genGuinier
(q, Rg=1, A=1, alpha=0)[source]¶ Generalized Guinier approximation for low wavevector q scattering q*Rg< 1-1.3
Parameters: - q : array of float
Wavevector
- Rg : float
Radius of gyration in units=1/q
- alpha : float
Shape [α = 0] spheroid, [α = 1] rod-like [α = 2] plane
- A : float
Amplitudes
Returns: - dataArray [q,Fq]
Notes
Quantitative analysis of particle size and shape starts with the Guinier approximations.For three-dimensional objects the Guinier approximation is given byI(q) = exp( − Rg**2*q**2 / 3)This approximation can be extended also to rod-like and plane objects byI(q) =(1 if α=0 or (α*pi*q^-α) if α=(1 or 2) ) * A*exp( − Rg^2q^2 / (3-α))If the particle has one dimension of length L, that is, much larger than the others (i.e., elongated, rod-like, or worm-like), then there is a q range such that qR_c < 1 << qL, where α = 1.
for more details see http://sasfit.ingobressler.net/manual/Generalized_Guinier_Approximation
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jscatter.formfactor.
guinier
(q, Rg=1, A=1)[source]¶ Classical Guinier
see genGuinier with alpha=0
Parameters: - q :array
- A : float
- Rg : float
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jscatter.formfactor.
multiShellCylinder
(q, L, shellthickness, shellSLD, solventSLD=0, alpha=[0, 1.5707963267948966], h=None, nalpha=30, ncap=31)[source]¶ Multi shell cylinder with caps in solvent averaged over axis orientations.
Each shell has a constant SLD and may have a cap with same SLD sequence. Caps may be globular (barbell) or small (like lenses). For zero length L a lens shaped disc or a double sphere like shape is recovered.
Parameters: - q : array
Wavevectors, units 1/nm
- L : float
Length of cylinder, units nm L=0 infinite cylinder if h=None.
- shellthickness : list of float or float, all >0
Thickness of shells in sequence, units nm. radii r=cumulativeSum(shellthickness)
- shellSLD : list of float/list
Scattering length density of shells in nm^-2. A shell can be divided in sub shells if instead of a single float a list of floats is given. These list values are used as scattering length of equal thickness subshells. E.g. [1,2,[3,2,1]] results in the last shell with 3 subshell of euqal thickness. The sum of subshell thickness is the thickness given in shellthickness. See second example. SiO2 = 4.186*1e-6 A^-2 = 4.186*1e-4 nm^-2
- solventSLD : float
Scattering length density of surrounding solvent in nm^-2. D2O = 6.335*1e-6 A^-2 = 6.335*1e-4 nm^-2
- h : float, default=None
Geometry of the caps with cap radii R=(r**2+h**2)**0.5 h is distance of cap center with radius R from the flat cylinder cap and r as radii of the cylinder shells.
- None No caps, flat ends as default.
- 0 cap radii equal cylinder radii (same shellthickness as cylinder shells)
- >0 cap radius larger cylinder radii as barbell
- <0 cap radius smaller cylinder radii as lens caps
- alpha : float, [float,float] , unit rad
Orientation, angle between the cylinder axis and the scattering vector q. 0 means parallel, pi/2 is perpendicular If alpha =[start,end] is integrated between start,end start > 0, end < pi/2
- nalpha : int, default 30
Number of points in Gauss integration along alpha.
- ncap : int, default=31
Number of points in Gauss integration for cap.
Returns: - dataArray [q ,Iq ]
- .outerCylinderVolume
- .Radius
- .cylindeLength
- .alpha
- .shellthickness
- .shellSLD
- .solventSLD
- .modelname
- .contrastprofile
- .capRadii
Notes
- Multishell of types:
- flat cap cylinder L>0, radii>0, h=None
- lens cap cylinder L>0, radii>0, h<0
- globular cap cylinder L>0, radii>0, h>0
- lens L=0, radii>0, h<0
- barbell no cylinder L=0, radii>0, h>0
- infinite flat disc L=0. h=None
References
Single cylinder
[1] Guinier, A. and G. Fournet, “Small-Angle Scattering of X-Rays”, John Wiley and Sons, New York, (1955) [2] http://www.ncnr.nist.gov/resources/sansmodels/Cylinder.html Double cylinder
[3] Use of viscous shear alignment to study anisotropic micellar structure by small-angle neutron scattering, J. B. Hayter and J. Penfold J. Phys. Chem., 88:4589–4593, 1984 [4] http://www.ncnr.nist.gov/resources/sansmodels/CoreShellCylinder.html Barbell, cylinder with small end-caps, circular lens
[5] Scattering from cylinders with globular end-caps Kaya (2004). J. Appl. Cryst. 37, 223-230] DOI: 10.1107/S0021889804000020 Scattering from capped cylinders. Addendum H. Kaya and Nicolas-RaphaeÈl de Souza J. Appl. Cryst. (2004). 37, 508-509 DOI: 10.1107/S0021889804005709 Examples
Alternating shells with different thickness 0.3 nm h2o and 0.2 nm d2o in vacuum:
import jscatter as js import numpy as np x=np.r_[0.0:10:0.01] ashell=js.ff.multiShellCylinder(x,20,[0.4,0.6]*5,[-0.56e-4,6.39e-4]*5) #plot it p=js.grace() p.multi(2,1) p[0].plot(ashell) p[1].plot(ashell.contrastprofile,li=1) # a contour of the SLDs
Double shell with exponential decreasing exterior shell to solvent scattering:
import jscatter as js import numpy as np x=np.r_[0.0:10:0.01] def doubleexpshells(q,L,d1,d2,e3,sd1,sd2,sol): # The third layer will have 9 subshells with combined thickness of e3. # The scattering length decays to e**(-3) in last subshell. return js.ff.multiShellCylinder(q,L,[d1,d2,e3],[sd1,sd2,((sd2-sol)*np.exp(-np.r_[0:3:9j])+sol)],solventSLD=sol) dde=doubleexpshells(x,10,0.5,0.5,3,1e-4,2e-4,0) #plot it p=js.grace() p.multi(2,1) p[0].plot(dde) p[1].plot(dde.contrastprofile,li=1) # a countour of the SLDs
Cylinder with cap:
x=np.r_[0.1:10:0.01] p=js.grace() p.title('Comaprison of dumbell cylinder with simple models') p.subtitle('thin lines correspnd to simple models as sphere and dshell sphere') p.plot(js.ff.multiShellCylinder(x,0,[10],[1],h=0),sy=[1,0.5,2],le='simple sphere') p.plot(js.ff.sphere(x,10),sy=0,li=1) p.plot(js.ff.multiShellCylinder(x,0,[2,1],[1,2],h=0),sy=[1,0.5,3],le='double shell sphere') p.plot(js.ff.multiShellSphere(x,[2,1],[1,2]),sy=0,li=1) p.plot(js.ff.multiShellCylinder(x,10,[3],[20],h=-5),sy=[1,0.5,4],le='thin lens cap cylinder=flat cap cylinder') p.plot(js.ff.multiShellCylinder(x,10,[3],[20],h=None),sy=0,li=[1,2,1],le='flat cap cylinder') p.plot(js.ff.multiShellCylinder(x,10,[3],[20],h=-0.5),sy=0,li=[3,2,6],le='thick lens cap cylinder') p.yaxis(scale='l') p.xaxis(scale='l') p.legend(x=0.15,y=0.01)
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jscatter.formfactor.
multiShellEllipsoid
(q, poleshells, equatorshells, shellSLD, solventSLD=0, alpha=[0, 90], tol=1e-06)[source]¶ Scattering of multi shell ellipsoidal particle with variing shell thickness at pole and equator.
Shell thicknesses add up to form complex particles with any combination of axial ratios and shell thickness. A const axial ratio means different shell thickness at equator and pole.
Parameters: - q : array
Wavevectors, unit 1/nm
- equatorshells : list of float
Thickness of shells starting from inner most for rotated axis Re making the equator. unit nm. The absolute values are used.
- poleshells : list of float
Thickness of shells starting from inner most for rotating axis Rp pointing to pole. unit nm. The absolute values are used.
- shellSLD : list of float
List of scattering length densities of the shells in sequence corresponding to shellthickness. unit nm^-2.
- solventSLD : float, default=0
Scattering length density of the surrounding solvent. unit nm^-2
Returns: - dataArray [q, Iq]
- Iq, scattering cross section in units nm**2
- .contrastprofile as radius and contrast values at edge points of equatorshells
- .equatorshellthicknes consecutive shell thickness
- .poleshellthickness
- .shellcontrast contrast of the shells to the solvent
- .equatorshellradii outer radius of the shells
- .poleshellradii
- .outerVolume Volume of complete sphere
- .I0 forward scattering for Q=0
References
[1] Structure Analysis by Small-Angle X-Ray and Neutron Scattering Feigin, L. A, and D. I. Svergun, Plenum Press, New York, (1987). [2] http://www.ncnr.nist.gov/resources/sansmodels/Ellipsoid.html [3] - Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).
Examples
Simple ellipsoid in vacuum:
x=np.r_[0.0:10:0.01] Rp=2. Re=1. ashell=js.ff.multiShellEllipsoid(x,Rp,Re,1) #plot it p=js.grace() p.multi(2,1) p[0].plot(ashell) p[1].plot(ashell.contrastprofile,li=1) # a contour of the SLDs
Alternating shells with thickness 0.3 nm h2o and 0.2 nm d2o in vacuum:
x=np.r_[0.0:10:0.01] shell=np.r_[[0.3,0.2]*3] sld=[-0.56e-4,6.39e-4]*3 # constant axial ratio for all shells but nonconstant shell thickness axialratio=2 ashell=js.ff.multiShellEllipsoid(x,axialratio*shell,shell,sld) # shell with constant shellthickness of one component and other const axialratio pshell=shell[:] pshell[0]=shell[0]*axialratio pshell[2]=shell[2]*axialratio pshell[4]=shell[4]*axialratio bshell=js.ff.multiShellEllipsoid(x,pshell,shell,sld) #plot it p=js.grace() p.multi(2,1) p[0].plot(ashell,le='const. axial ratio') p[1].plot(ashell.contrastprofile,li=2) # a contour of the SLDs p[0].plot(bshell,le='const shell thickness') p[1].plot(bshell.contrastprofile,li=2) # a contour of the SLDs p[0].legend()
double shell with exponential decreasing exterior shell to solvent scattering:
x=np.r_[0.0:10:0.01] def doubleexpshells(q,d1,ax,d2,e3,sd1,sd2,sol): shells =[d1 ,d2]+[e3]*9 shellsp=[d1*ax,d2]+[e3]*9 sld=[sd1,sd2]+list(((sd2-sol)*np.exp(-np.r_[0:3:9j]))) return js.ff.multiShellEllipsoid(q,shellsp,shells,sld,solventSLD=sol) dde=doubleexpshells(x,0.5,1,0.5,1,1e-4,2e-4,0) #plot it p=js.grace() p.multi(2,1) p[0].plot(dde) p[1].plot(dde.contrastprofile,li=1) # a countour of the SLDs
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jscatter.formfactor.
multiShellSphere
(q, shellthickness, shellSLD, solventSLD=0)[source]¶ Scattering of spherical multi shell particle including linear contrast variation in subshells.
The results needs to be multiplied with the concentration to get the measured scattering.
Parameters: - q : array
Wavevectors to calculate form factor, unit e.g. 1/nm.
- shellthickness : list of float
Thickness of shells starting from inner most, unit in 1/[q units].
- shellSLD : list of float or list
- List of scattering length densities of the shells in sequence corresponding to shellthickness. unit in nm**-2
- Innermost shell needs to be constant shell.
- If an element of the list is itself a list of SLD values it is interpreted as equal thick subshells with linear progress between SLD values in sum giving shellthickness.
- If subshell list has only one float e.g. [1e.4] the second value is the SLD of the following shell.
- If empty list is given as [] the SLD of the previous and following shells are used as smooth transition.
- solventSLD : float, default=0
Scattering length density of the surrounding solvent. If equal to zero (default) then in profile the contrast is given. Unit in [q unit]**2 e.g. 1/nm**2
Returns: - dataArray [wavevector, Iq]
- Iq scattering cross section in units nm**2
- .contrastprofile as radius and contrast values at edge points
- .shellthickness consecutive shell thickness
- .shellcontrast contrast of the shells to the solvent
- .shellradii outer radius of the shells
- .slopes slope of linear increase of each shell
- .outerVolume Volume of complete sphere
- .I0 forward scattering for Q=0
Notes
The solution is unstable (digital resolution) for really low QR values, which are set to the I0 scattering.
Examples
Alternating shells with 5 alternating thickness 0.4 nm and 0.6 nm with h2o, d2o scattering contrast in vacuum:
x=np.r_[0.0:10:0.01] ashell=js.ff.multiShellSphere(x,[0.4,0.6]*5,[-0.56e-4,6.39e-4]*5) #plot it p=js.grace() p.multi(2,1) p[0].plot(ashell) p[1].plot(ashell.contrastprofile,li=1) # a contour of the SLDs
Double shell with exponential decreasing exterior shell to solvent scattering:
x=np.r_[0.0:10:0.01] def doubleexpshells(q,d1,d2,e3,sd1,sd2,sol): return js.ff.multiShellSphere(q,[d1,d2,e3*3],[sd1,sd2,((sd2-sol)*np.exp(-np.r_[0:3:9j]))],solventSLD=sol) dde=doubleexpshells(x,0.5,0.5,1,1e-4,2e-4,0) #plot it p=js.grace() p.multi(2,1) p[0].plot(dde) p[1].plot(dde.contrastprofile,li=1) # a countour of the SLDs
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jscatter.formfactor.
multilamellarVesicles
(Q, R, N, phi, displace=0, dR=0, dN=0, shellthickness=0, ds=0, SLD=1, solventSLD=0, nGauss=100)[source]¶ Scattering intensity of a multilamellar vesicle with random displacements of the inner vesicles [1].
The result contains the full scattering, the structure factor of the shells and a multilayer formfactor of the lamellar layer structure. Other layer structures as mentioned in [2].
Parameters: - Q : float
Wavevector in 1/nm.
- R : float
Outer radius of the Vesicle in units nm.
- dR : float
Width of outer radius distribution in units nm.
- displace : float
Displacements of the vesicles centers in units nm. This describes the displacement steps in a random walk of the centers. displace=0 it is concentric, all have same center. displace< R/N.
- N : int
Number of layers.
- dN : int, default=0
Width of distribution for number of layers. (dN< 0.4 is single N) A zero truncated normal distribution is used with N>0 and N<R/displace. Check .Ndistribution and .Nweight = Nweigth for the used distribtion.
- shellthickness: float,list of float,default=0
Thickness of shells in symetric layer in units nm. Zero assumes infinit thin layer with constant formfactor. List gives consecutive layer thickness from center to outside. [4,1] result in a [1,4,1] symmetric layer.
- ds : float, not working , set to zero
Thickness fluctuation of the innermost layer in shellthickness. unit is nm. A Gaussian is used which is cut at 0.1*shellthickness[0]. ds should be significant smaller than shellthickness[0].
- phi : float
Volume fraction \phi of layers inside of vesicle.
- SLD : float
Scattering length density of shells in nm^-2.
- solventSLD
Solvent scattering length density in nm^-2.
Returns: - dataArray with [q,I(q),S(q),F(q)]
- .columnname=’q;Iq;Sq;Fq’
- .outerShellVolume
- .Ndistribution
- .Nweight
- .displace
- .phi
- .shellthickness
- .SLD
- .solventSLD
- .shellfluctuations=ds
- .preFactor=phi*Voutershell**2
Notes
The left shows a concentric lamellar structure. The right shows the random path of the consecutive centers of the spheres. See Multilamellar Vesicles for resulting scattering curves.
The function returns I(Q) as (see [1] equ. 17 )
I(Q)=\phi V_{outershell} S(Q) F(Q)
F(Q)= ( \sum_i d \rho_i sinc( Q d_i) )^2
with d\rho as scattering length density difference to next layer with thickness d_i and the shell structure factor S(Q) as described in equ. A2 in [1].
The amphiphile concentration phi is roughly given by phi = d/a, with d being the bilayer thickness and a being the spacing of the shells. The spacing of the shells is given by the scattering vector of the first correlation peak, i.e., a = 2pi/Q. Once the MLVs leave considerable space between each other then phi < d/a holds. This condition coincides with the assumption of dilution of the Guinier law. (from [1])
Structure factor part is normalized that S(0)=\sum_{j=1}^N (j/N)^2
To use a different shell form factor the structure factor is given explicitly.
Comparing a unilamellar vesicle (N=1) with multiShellSphere shows that R is located in the center of the shell:
Q=js.loglist(0.0001,5,1000)#np.r_[0.01:5:0.01] ffmV=js.ff.multilamellarVesicles p=js.grace() # comparison double sphere mV=ffmV(Q=Q, R=100., displace=0, dR=0,N=1,dN=0, phi=1,shellthickness=6, SLD=1e-4,nGauss=20) p.plot(mV) p.plot(js.ff.multiShellSphere(Q,[97,6],[0,1e-4]),li=1) p.yaxis(label='S(Q)',scale='l',min=1e-10,max=1e6,ticklabel=['power',0]) p.xaxis(label='Q / nm\S-1',scale='l',min=1e-3,max=5,ticklabel=['power',0])
References
[1] (1, 2, 3, 4, 5, 6) Small-angle scattering model for multilamellar vesicles H. Frielinghaus Physical Review E 76, 051603 (2007) [2] Small-Angle Scattering from Homogenous and Heterogeneous Lipid Bilayers N. Kučerka Advances in Planar Lipid Bilayers and Liposomes 12, 201-235 (2010) Examples
import jscatter as js import numpy as np ffmV=js.ff.multilamellarVesicles Q=js.loglist(0.01,5,500) dd=1.5 dR=5 nG=100 ds=0 R=50 N=5 st=[3.5,(6.5-3.5)/2] p=js.grace(1,1) p.title('Lipid bilayer in SAXS/SANS') # SAXS saxm=ffmV(Q=Q, R=R, displace=dd, dR=dR,N=N,dN=0, phi=0.2,shellthickness=st,ds=ds, SLD=[0.6e-3,0.07e-3],solventSLD=0.94e-3,nGauss=nG) p.plot(saxm,sy=[1,0.3,1],le='SAXS multilamellar') saxu=ffmV(Q=Q, R=R, displace=0, dR=dR,N=1,dN=0, phi=0.2,shellthickness=st,ds=ds,SLD=[0.6e-3,0.07e-3],solventSLD=0.94e-3,nGauss=100) p.plot(saxu,sy=0,li=[3,2,1],le='SAXS unilamellar') # SANS sanm=ffmV(Q=Q, R=R, displace=dd, dR=dR,N=N,dN=0, phi=0.2,shellthickness=st,ds=ds, SLD=[1.5e-4,0.3e-4],solventSLD=6.335e-4,nGauss=nG) p.plot( sanm,sy=[1,0.3,2],le='SANS multilamellar') sanu=ffmV(Q=Q, R=R, displace=0, dR=dR,N=1,dN=0, phi=0.2,shellthickness=st,ds=ds,SLD=[1.5e-4,0.3e-4],solventSLD=6.335e-4,nGauss=100) p.plot(sanu,sy=0,li=[3,2,2],le='SANS unilamellar') # p.legend(x=0.015,y=1e-1) p.subtitle('R=50 nm, N=5, shellthickness=[1.5,3.5,1.5] nm, dR=5, ds=0.') p.yaxis(label='S(Q)',scale='l',min=1e-6,max=1e4,ticklabel=['power',0]) p.xaxis(label='Q / nm\S-1',scale='l',min=1e-2,max=5,ticklabel=['power',0])
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jscatter.formfactor.
orientedCloudScattering
(qxz, cloud, rms=0, coneangle=10, nCone=50, V=0, formfactor=None, ncpu=0)[source]¶ 2D scattering of an oriented cloud of scatterers with equal or variable scattering length. Using multiprocessing.
Cloud can represent an object described by a cloud of isotropic scatterers with orientation averaged in a cone. Scattering amplitudes may be constant, sphere scattering amplitude, Gaussian scattering amplitude or explicitly given form factor. Rememeber that the atomic bond length are on the order 0.1-0.2 nm and one expects Bragg peaks.
Parameters: - qxz : array, ndim= Nx3
wavevectors in 1/nm
- cloud : array Nx3 or Nx4
- Center of mass positions (in nm) of the N scatterers in the cloud.
- If given cloud[3] is the scattering length b at positions cloud[:3], otherwise b=1.
- coneangle : float
Coneangle in units degrees.
- rms : float, default=0
Root mean square displacement =<u**2>**0.5 of the positions in cloud as random (Gaussian) displacements in units nm. Dispacement u is random for each orientation nCone. rms can be used to simulate a Debye-Waller factor. Larger nCone is advised to smooth data.
- nCone : int
Cone average as average over nCone Fibonacci lattice points in cone.
- V : float, default=0
Volume of the scatterers for formfactor ‘gauss’ and ‘sphere’.
- formfactor : ‘gauss’,’sphere’,array 2xN,default=None
- Gridpoint scattering amplitudes are described by:
- None : const scattering amplitude, point like particle.
- ‘sphere’: Sphere scattering amplitude according to [3]. The sphere radius is R=(\\frac{3V}{4\\pi})^{1/3}
- ‘gauss’ : Gaussian function b_i(q)=bVexp(- \\frac{V^{2/3.}}{4\pi}q^2) according to [2].
- explicit isotropic form factor ff as array with [q,ff] e.g. from multishell. The normalized scattering amplitude fa for each gridpoint is calculated as fa=ff**0.5/fa(0). Missing values are linear intepolated (np.interp), q values outside interval are mapped to qmin or qmax.
- ncpu : int, default 0
- Number of cpus used in the pool for multiprocessing.
- not given or 0 : all cpus are used
- int>0 : min(ncpu, mp.cpu_count)
- int<0 : ncpu not to use
- 1 : single core usage for testing or comparing speed to Debye
Returns: - dataArray [qx,qz, Pq]
- The forward scattering is Pq(q=0)= sumblength**2
- .sumblength : Sum of blength with sumblength**2
- .formfactoramplitude : formfactoramplitude of cloudpoints according to type for all q values.
- .formfactoramplitude_q :corresponding q values.
References
[1] - Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).1
[2] (1, 2) An improved method for calculating the contribution of solvent to the X-ray diffraction pattern of biological molecules Fraser R MacRae T Suzuki E IUCr Journal of Applied Crystallography 1978 vol: 11 (6) pp: 693-694 [3] (1, 2) X-ray diffuse scattering by proteins in solution. Consideration of solvent influence B. A. Fedorov, O. B. Ptitsyn and L. A. Voronin J. Appl. Cryst. (1974). 7, 181-186 doi: 10.1107/S0021889874009137 Examples
How to use orientedCloudScattering for fitting see last Example in cloudScattering.
import jscatter as js import numpy as np # two points along y result in pattern independent of x but cos**2 for z # with larger coneangle Ix becomes qx dependent rod0=np.zeros([2,3]) rod0[:,1]=np.r_[0,np.pi] qxz=np.mgrid[-6:6:50j, -6:6:50j].reshape(2,-1).T ffe=js.ff.orientedCloudScattering(qxz,rod0,coneangle=5,nCone=10,rms=0) fig=js.mpl.surface(ffe.X,ffe.Z,ffe.Y) fig.axes[0].set_title(r'cos**2 for Z and slow decay for X due to 5 degree cone') fig.show() # noise in positions ffe=js.ff.orientedCloudScattering(qxz,rod0,coneangle=5,nCone=100,rms=0.1) fig=js.mpl.surface(ffe.X,ffe.Z,ffe.Y) fig.axes[0].set_title('cos**2 for Y and slow decay for X with position noise') fig.show() # # two points along z result in symmetric pattern around zero # asymetry reflects fibonacci lattice -> increase nCone rod0=np.zeros([2,3]) rod0[:,2]=np.r_[0,np.pi] ffe=js.ff.orientedCloudScattering(qxz,rod0,coneangle=45,nCone=10,rms=0.005) fig2=js.mpl.surface(ffe.X,ffe.Z,ffe.Y) fig2.axes[0].set_title('symmetric because of orientation along z; \n nCone needs to be larger for large cones') fig2.show() # # 5 spheres in line with small position distortion rod0 = np.zeros([5, 3]) rod0[:, 1] = np.r_[0, 1, 2, 3, 4] * 3 qxz = np.mgrid[-6:6:50j, -6:6:50j].reshape(2, -1).T ffe = js.ff.orientedCloudScattering(qxz, rod0, formfactor='sphere', V=4/3.*np.pi*2**3, coneangle=20, nCone=30, rms=0.02) fig4 = js.mpl.surface(ffe.X, ffe.Z, np.log10(ffe.Y), colorMap='gnuplot') fig4.axes[0].set_title('5 spheres with R=2 along Z with noise (rms=0.02)') fig4.show() # # 5 core shell particles in line with small position distortion (Gaussian) rod0 = np.zeros([5, 3]) rod0[:, 1] = np.r_[0, 1, 2, 3, 4] * 3 qxz = np.mgrid[-6:6:50j, -6:6:50j].reshape(2, -1).T # only as demo : extract q from qxz qxzy = np.c_[qxz, np.zeros_like(qxz[:, 0])] qrpt = js.formel.xyz2rphitheta(qxzy) q = np.unique(sorted(qrpt[:, 0])) # or use interpolation q = js.loglist(0.01, 7, 100) cs = js.ff.sphereCoreShell(q=q, Rc=1, Rs=2, bc=0.1, bs=1, solventSLD=0) ffe = js.ff.orientedCloudScattering(qxz, rod0, formfactor=cs, coneangle=20, nCone=100, rms=0.05) fig4 = js.mpl.surface(ffe.X, ffe.Z, np.log10(ffe.Y), colorMap='gnuplot') fig4.axes[0].set_title('5 core shell particles with R=2 along Z with noise (rms=0.05)') fig4.show()
Make a slice for an angular region but with higher resolution to see the additional peaks due to allignement
rod0 = np.zeros([5, 3]) rod0[:, 1] = np.r_[0, 1, 2, 3, 4] * 3 qxz = np.mgrid[-4:4:150j, -4:4:150j].reshape(2, -1).T # only as demo : extract q from qxz qxzy = np.c_[qxz, np.zeros_like(qxz[:, 0])] qrpt = js.formel.xyz2rphitheta(qxzy) q = np.unique(sorted(qrpt[:, 0])) # or use interpolation q = js.loglist(0.01, 7, 100) cs = js.ff.sphereCoreShell(q=q, Rc=1, Rs=2, bc=0.1, bs=1, solventSLD=0) ffe = js.ff.orientedCloudScattering(qxz, rod0, formfactor=cs, coneangle=20, nCone=100, rms=0.05) fig4 = js.mpl.surface(ffe.X, ffe.Z, np.log10(ffe.Y), colorMap='gnuplot') fig4.axes[0].set_title('5 core shell particles with R=2 along Z with noise (rms=0.05)') fig4.show() # # transform X,Z to spherical coordinates qphi=js.formel.xyz2rphitheta([ffe.X,ffe.Z,abs(ffe.X*0)],transpose=True )[:,:2] # add qphi or use later rp[1] for selection ffb=ffe.addColumn(2,qphi.T) # select a portion of the phi angles phi=np.pi/2 dphi=0.2 ffn=ffb[:,(ffb[-1]<phi+dphi)&(ffb[-1]>phi-dphi)] ffn.isort(-2) # sort along radial q p=js.grace() p.plot(ffn[-2],ffn.Y,le='oriented spheres form factor') # compare to coreshell formfactor scaled p.plot(cs.X,cs.Y/cs.Y[0]*25,li=1,le='coreshell form factor') p.yaxis(label='F(Q,phi=90°+-11°)', scale='log') p.title('5 alligned core shell particle with additional interferences',size=1.) p.subtitle(' due to sphere allignement dependent on observation angle') # 2: direct way with 2D q in xz plane rod0 = np.zeros([5, 3]) rod0[:, 1] = np.r_[0, 1, 2, 3, 4] * 3 x=np.r_[0.0:6:0.05] qxzy = np.c_[x, x*0,x*0] for alpha in np.r_[0:91:30]: R=js.formel.rotationMatrix(np.r_[0,0,1],np.deg2rad(alpha)) # rotate around Z axis qa=np.dot(R,qxzy.T).T[:,:2] ffe = js.ff.orientedCloudScattering(qa, rod0, formfactor=cs, coneangle=20, nCone=100, rms=0.05) p.plot(x,ffe.Y,li=[1,2,-1],sy=0,le='alpha=%g' %alpha) p.xaxis(label=r'Q / nm\S-1') p.legend()
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jscatter.formfactor.
pearlNecklace
(Q, Rc, l, N, A1=None, A2=None, A3=None, ms=None, mr=None)[source]¶ Formfactor of a pearlnecklace (freely jointed chain of pearls connected by rods)
The formfactor is normalized that the pearls contribution equals 1.
Parameters: - Q : array
wavevector in nm
- Rc : float
pearl radius in nm
- N : float
number of pearls (homogeneous spheres)
- l : float
physical length of the rods
- A1, A2, A3 : float
Amplitudes of pearl-pearl, rod-rod and pearl-rod scattering. Can be calculated with the number of chemical monomers in a pearl ms and rod mr (see below for further information) If ms and mr are given A1,A2,A3 are calculated from these.
- ms : float, default None
number of chemical monomers in each pearl
- mr : float, default None
number of chemical monomers in rod like strings
Returns: - dataArray [q, Iq]
- .pearlRadius
- .A1
- .A2
- .A3
- .numberPearls
- .mr
- .ms
- .stringLength
Notes
- M : number of rod like strings (M=N-1)
- A1 = ms²/(M*mr+N*ms)²
- A2 = mr²/(M*mr+N*ms)²
- A3 = (mr*ms)/(M*mr+N*ms)²
References
[1] - Schweins, K. Huber, Macromol. Symp., 211, 25-42, 2004.
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jscatter.formfactor.
ringPolymer
(q, Rg)[source]¶ General formfactor of a polymer ring in theta solvent.
Parameters: - q : array
Scattering vector, unit eg 1/A or 1/nm
- Rg : float
Radius of gyration, units in 1/unit(q)
Returns: - dataArray [q,Fq]
- .radiusOfGyration
References
[1] SANS from homogeneous polymer mixtures: A unified overview. Hammouda, B. in Polymer Characteristics 87–133 (Springer-Verlag, 1993). doi:10.1007/BFb0025862 Examples
import jscatter as js import numpy as np q=js.loglist(0.1,8,100) p=js.grace() iq=js.ff.ringPolymer(q,5) p.plot(iq.X,iq.Y*iq.X**2) p.yaxis(scale='l') p.xaxis(scale='l') p.legend(x=0.2,y=0.5)
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jscatter.formfactor.
scatteringFromSizeDistribution
(q, sizedistribution, fffunction=<function beaucage>, fffkwargs={'G': 1, 'd': 3}, scatteringlength=<function <lambda>>)[source]¶ Scattering of a size distribution of objects with form factor fffunction
scattering of a single size:I(q) = n * drho^2 * V^2 * P(q)*S(q)n number densitydrho difference in scattering length density (contrast)V scattering Volume of particle ~r³ ~ massP(q) formfactor with P(0)=1; here fffunctionS(q) structure factor with S(inf)=>1 for simplicitysize distributionI(q)= sum_over_sizes[ scatteringlength(size) * probability(size) * fffunction(q,size,**fffkwargs).Y ] with S(q)=1Parameters: - q : list/array of float; unit = 1/unit(size distribution)
Wavevectors to calculate scattering
- sizedistribution : dataArray or array
Distribution.T[i] should give pairs [size,probability,…] (X or dimension 0) -> list of sizes; (Y or dimension 1) list of probability eg. from DLS measurement
- fffunction : lambda or function
Function that describes the form factor like a beaucage (default) first arguments (q,R,… should return dataArray with .Y as result
- fffkwargs : args
Any additional keyword arguments for fffunction as dictionary eg. {‘d’:3}
- scatteringlength : function
Scattering_length(r)**2= drho**2 * V**2 Volume objects dimension 3 with scattering length density constant Number distribution: scatteringlength per particle ~R**6 Volume distribution: scatteringlength per particle ~R**2 because V=r**3 Intensity distribution: scatteringlength per particle const because V=r**3
Returns: - dataArray [q,Iq or Fq]
Notes
To look at the contribution of aggregates of size 70 to 12 nm particles:
# equal concentration p.plot(s.formel.scatteringFromSizeDistribution(s.loglist(0.01,6,100),[[12,70],[1,1]]),legend='with aggregates') # 2:1 concentration p.plot(s.formel.scatteringFromSizeDistribution(s.loglist(0.01,6,100),[[12,70],[2,1]]),legend='no aggregates')
To look at a part of the distribution use sizedistribution.prune(min,max)
If size,probability is not in column 0,1 use as for the Malvern frequencydistribution: sizedistribution=frequencydistribution[np.r_[4,1]] if probability is column 1 and soze column 4
It is instructive to plot the result together with q*R~const p.plot(1/sizedistribution[0],sizedistribution[1])
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jscatter.formfactor.
sphere
(q, radius, contrast=1)[source]¶ Scattering of a single homogeneous sphere.
Parameters: - q : float
Wavevector in units of 1/nm
- radius : float
Radius in units nm
- contrast : float, default=1
Difference in scattering length to the solvent = contrast
Returns: - dataArray [q,Iq]
- Iq scattering intensity
- .I0 forward scattering
Notes
The first minimum of the form factor is at q*R=4.493
www.ncnr.nist.gov/resources/sansmodels/Sphere.html
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jscatter.formfactor.
sphereCoreShell
(q, Rc, Rs, bc, bs, solventSLD=0)[source]¶ Scattering of a spherical core shell particle.
Parameters: - q : float
Wavevector in units of 1/(R units)
- Rc,Rs : float
Radius core and radius of shell Rs>Rc
- bc,bs : float
Contrast to solvent scattering length density of core and shell.
- solventSLD : float, default =0
Scattering length density of the surrounding solvent. If equal to zero (default) then in profile the contrast is given.
Returns: - dataArray [wavevector ,Iq ]
Notes
Calls multiShellSphere.
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jscatter.formfactor.
sphereFuzzySurface
(q, R, sigmasurf, contrast)[source]¶ Scattering of a sphere with a fuzzy interface.
Parameters: - q : float
Wavevector in units of 1/(R units)
- R : float
The particle radius R represents the radius of the particle where the scattering length density profile decreased to 1/2 of the core density.
- sigmasurf : float
Sigmasurf is the width of the smeared particle surface.
- contrast : float
Difference in scattering length to the solvent = contrast
Returns: - dataArray [q, Iq]
- Iq scattering intensity related to sphere volume.
- .I0 forward scattering
Notes
The “fuzziness” of the interface is defined by the parameter sigmasurf. The particle radius R represents the radius of the particle where the scattering length density profile decreased to 1/2 of the core density. sigmasurf is the width of the smeared particle surface. The inner regions of the microgel that display a higher density are described by the radial box profile extending to a radius of approximately Rbox ~ R - 2(sigma). In dilute solution, the profile approaches zero as Rsans ~ R + 2(sigma).
References
[1] - Stieger, J. S. Pedersen, P. Lindner, W. Richtering, Langmuir 20 (2004) 7283-7292
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jscatter.formfactor.
sphereGaussianCorona
(q, R, Rg, Ncoil, coilequR, coilSLD=6.4e-05, sphereSLD=0.0004186, solventSLD=0.0006335, d=1)[source]¶ Scattering of a sphere surrounded by gaussian coils as model for grafted polymers on particle e.g. a micelle.
The additional scattering is uniformly distributed at the surface, which might fail for lower aggregation numbers as 1, 2, 3. Instead of aggregation number in [1] we use sphere volume and a equivalent volume of the gaussian coils.
Parameters: - q: array of float
wavevectors in units 1/nm
- R : float
sphere radius unit nm
- Rg : float
radius of gyration of coils unit nm
- d : float, default 1
Coils centre loacated d*Rg away from the sphere surface
- Ncoil : float
number of coils at the surface (aggregation number)
- coilequR : float
Equivalent radius to calc volume of the coil mass if densely packed as a sphere. Needed to calculate absolute scattering of the coil.
- coilSLD : float
Scattering length density of coil. unit nm^-2. default hPEG = 0.64*1e-6 A^-2 = 0.64*1e-4 nm^-2
- sphereSLD : float
Scattering length density of sphere.unit nm^-2. default SiO2 = 4.186*1e-6 A^-2 = 4.186*1e-4 nm^-2
- solventSLD : float
Scattering length density of solvent. unit nm^-2. default D2O = 6.335*1e-6 A^-2 = 6.335*1e-4 nm^-2
Returns: - dataArray [q,Iq]
- .coilRg
- .sphereRadius
- .numberOfCoils
- .coildistancefactor
- .coilequVolume
- .coilSLD
- .sphereSLD
- .solventSLD
Notes
- The defaults result in a silica sphere with hPEG grafted at the surface in D2O.
- Rg=N**0.5*b with N monomers of length b
- Vcoilsphere=4/3.*np.pi*coilequR**3
- Vcoilsphere=N*monomerVolume
- coilequR=(N*monomerVolume/(4/3.*np.pi))**(1/3.)
References
[1] (1, 2) Form factors of block copolymer micelles with spherical, ellipsoidal and cylindrical cores Pedersen J Journal of Applied Crystallography 2000 vol: 33 (3) pp: 637-640 [2] Hammouda, B. (1992).J. Polymer Science B: Polymer Physics30 , 1387–1390
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jscatter.formfactor.
superball
(q, R, p, SLD=1, solventSLD=0, nGrid=15, returngrid=False)[source]¶ A superball is a general mathematical shape that can be used to describe rounded cubes, sphere and octahedra.
The numerical integration is done by a grid of scatterers, refined at the surface with an insidegrid and an outsidegrid averaged.
Parameters: - q : array
Wavevector in 1/nm
- R : float, None
2R = edge length
- p : float, 0<p<100
Parameter that describes shape | p=0 empty space | p<0.5 concave octahedra | p=0.5 octahedra | 0.5<p<1 convex octahedra | p=1 spheres | p>1 rounded cubes | p->inf cubes
- SLD : float, default =1
Scattering length density of cuboid.unit nm^-2
- solventSLD : float, default =0
Scattering length density of solvent. unit nm^-2
- nGrid : int
Number of gridpoints in start lattice is nGrid**3. This is refined at the surface.til griddistance/3. relError=nGrid*4 is used for Fibonacci lattice with 2*relError+1 orientations in spherical average.
Returns: - dataArray [q,Iq, beta]
Notes
For large Q we observe a peak corresponding to the grid that is used for the integration. This is a sign that the density of the grid is not high enough as we see something like internal grid structure. The same happens in standard integration routines if you integrate for the Q values with a fixed resolution. In these cases please choose a smaller grid and wait. :-) Use the time to think about the advantages of an analytical solution if it is availible and atomic resolution.
References
[1] Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems Yager, K.G.; Zhang, Y.; Lu, F.; Gang, O. Journal of Applied Crystallography 2014, 47, 118–129. doi: 10.1107/S160057671302832X [2] http://gisaxs.com/index.php/Form_Factor:Superball Examples
import jscatter as js import numpy as np # q=np.r_[0:5:0.02] R=6; p=js.grace() p.multi(2,1) p[0].yaxis(scale='l') ss=js.ff.superball(q,R,p=1,) p[0].plot(ss,legend='superball p=1') p[0].plot(js.ff.sphere(q,R),li=1,sy=0,legend='sphere ff') p[0].legend(x=1,y=1e-1) # p[1].yaxis(scale='l') cc=js.ff.superball(q,R,p=100) p[1].plot(cc,sy=[1,0.3,4],legend='superball p=100') p[1].plot(js.ff.cuboid(q,2*R),li=4,sy=0,legend='cuboid, but never reached') p[1].legend(x=1,y=1e6) p[1].text('internal structure of grid',x=3.5,y=4e4) # # visualisation import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D # cubic grid points fig = plt.figure() ax = fig.add_subplot(111, projection='3d') q=np.r_[0:5:0.1] R=3 pxyz=js.ff.superball(q,R,p=200,nGrid=10,returngrid=True).grid pxyz=pxyz[pxyz[:,0]>0] ax.scatter(pxyz[:,0],pxyz[:,1],pxyz[:,2],color="k",s=20) ax.set_xlim([-3,3]) ax.set_ylim([-3,3]) ax.set_zlim([-3,3]) ax.set_aspect("equal") plt.tight_layout() plt.show(block=False)
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jscatter.formfactor.
teubnerStrey
(q, xi, d, eta2=1)[source]¶ Phenomenological model for the scattering intensity of a two-component system using the Teubner-Strey model [1].
Often used for bi-continuous micro-emulsions.
Parameters: - q : array
wavevectors
- xi : float
correlation length
- d : float
characteristic domain size, periodicity
- eta2 : float, default=1
squared scattering length density contrast
Returns: - dataArray [q, Iq]
Notes
- q_{max}=((2\pi/d)^2-\xi^{-2})^{1/2}
References
[1] (1, 2, 3) M. Teubner and R. Strey, Origin of the scattering peak in microemulsions, J. Chem. Phys., 87:3195, 1987 [2] K. V. Schubert, R. Strey, S. R. Kline, and E. W. Kaler, Small angle neutron scattering near lifshitz lines: Transition from weakly structured mixtures to microemulsions, J. Chem. Phys., 101:5343, 1994 Examples
Fit Teubner-Strey with background and a power law for low Q
#import jscatter as js #import numpy as np def tbpower(q,B,xi,dd,A,beta,bgr): # Model Teubner Strey + power law and background tb=js.ff.teubnerStrey(q=q,xi=xi,d=dd) # add power law and background tb.Y=B*tb.Y+A*q**beta+bgr tb.A=A tb.bgr=bgr tb.beta=beta return tb # simulate some data q=js.loglist(0.01,5,600) data=tbpower(q,1,10,20,0.002,-3,0.1) # or read them # data=js.dA('filename.chi') # plot data p=js.grace() p.plot(data,legend='simulated data') p.xaxis(scale='l',label=r'Q / nm\S-1') p.yaxis(scale='l',label='I(Q) / a.u.') p.title('TeubnerStrey model with power and background')
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jscatter.formfactor.
wormlikeChain
(q, N, a, R=None, SLD=1, solventSLD=0, rtol=0.02)[source]¶ Scattering of a wormlike chain, which correctly reproduces the rigid-rod and random-coil limits.
The forward scattering is :math:´I0=V^2(SLD-solventSLD)^2´ volume :math:´V=piR^2N´.
Parameters: - q : array
wavevectors in 1/nm
- N : float
Chain length, units of 1/q
- a : float
Persistence length with l=2a l=Kuhn length (segment length), units of nm.
- R : float
Radius in units of nm.
- SLD : float
Scattering length density segments.
- solventSLD :
Solvent scattering length density.
- rtol : float
Maximum relative tolerance in integration.
Returns: - dataArray [q, Iq]
- .chainRadius
- .chainLength
- .persistenceLength
- .Rg
- .volume
- .contrast
Notes
- From [1] :
- The Kratky plot (Figure 4 ) is not the most convenient way to determine a as was pointed out in ref 20. Figure 5 provides an alternative way of measuring a by plotting the experimentally measurable combination Nk2S(k) versus a for fixed wavelength k. As Figure 5 indicates, this plot is rather insensitive to the chain length N and therefore is universal. The numerical analysis of eq 17 shows that this remains true for as long as k is not too small. Taking into account that the excluded-volume effects leave S(k) practically unchanged (e.g., see Figures 2 and 4 of ref 231, the plot of Figure 5 can serve as a useful alternative to the Kratky plot which, in addition, does not suffer from the polydispersity effects
References
[1] (1, 2) Analytical calculation of the scattering function for polymers of arbitrary flexibility using the dirac propagator, A. L. Kholodenko, Macromolecules, 26:4179–4183, 1993 [2] (1, 2) The structure factor of a wormlike chain and the random-phase-approximation solution for the spinodal line of a diblock copolymer melt Zhang X et. al. Soft Matter 10, 5405 (2014) [3] (1, 2) Models of Polymer Chains Teraoka I. in Polymer Solutions: An Introduction to Physical Properties pp: 1-67, New York, John Wiley & Sons, Inc. Examples
import jscatter as js p=js.grace() p.multi(2,1) p.title('figure 3 (2 scaled) of ref Kholodenko Macromolecules 26, 4179 (1993)',size=1) q=js.loglist(0.01,10,100) for a in [1,2.5,5,20,50,1000]: ff=js.ff.wormlikeChain(q,200,a) p[0].plot(ff.X,200*ff.Y*ff.X**2,legend='a=%.4g' %ff.persistenceLength) p[1].plot(ff.X,ff.Y,legend='a=%.4g' %ff.persistenceLength) p[0].legend() p[0].yaxis(label='Nk\S2\NS(k)') p[1].xaxis(label='k',scale='l') p[1].yaxis(label='S(k)',scale='l') # p=js.grace() p.multi(2,1) p.title('figure 4 of ref Kholodenko Macromolecules 26, 4179 (1993)',size=1) # fig 4 seems to be wrong scale in [Re57b872e77e7-1]_ as for large N with a=1 fig 2 and 4 should have same plateau. a=1 q=js.loglist(0.01,4./a,100) for NN in [1,20,50,150,500]: ff=js.ff.wormlikeChain(q,NN,a) p[0].plot(ff.X*a,NN*a*ff.Y*ff.X**2,legend='N=%.4g' %ff.chainLength) p[1].plot(ff.X,ff.Y,legend='a=%.4g' %ff.persistenceLength) p[0].legend() p[0].yaxis(label='(N/a)(ka)\S2\NS(k)') p[0].xaxis(label='ka') p[1].xaxis(label='k',scale='l') p[1].yaxis(label='S(k)',scale='l')