13. Examples

These examples show how to use Jscatter. Use showExampleList to get a full list. Maybe first study the Beginners Guide / Help

Examples are mainly based on XmGrace for plotting as this is more convenient for interactive inspection of data and used for the shown plots.

Matplotlib can be used by setting usempl=True in runExample and showExample (automatically set if Grace not present). With matplotlib the plots are not optimized but still show the possibilities.

Image quality is for HTML. File formats in XmGrace can jpg, png eps,pdf…. in high resolution in publication quality.

For publication use .eps or .pdf with image width 8.6 cm and 600 dpi (see .save of a plot).

The shown examples can be run by copy and pasting to a shell or by using js.example.runExample(17)

showExampleList
showExample
runExample
runAll

Try Jscatter Demo in a Jupyter Notebook at .

13.1. In a hurry and short

Daily use example to show how short it can be.

Comments are shown in next examples.

# example in fast hurry without fine tuning of plots

import jscatter as js
import numpy as np

# read data with 16 intermediate scattering functions from NSE measurement of protein diffusion
i5 = js.dL(js.examples.datapath + '/iqt_1hho.dat')
# manipulate data
for dat in i5:
    dat.X = dat.X /1.  # conversion from ps to ns
    dat.q *= 1  # conversion to 1/nm
# define model as simple diffusion with elastic background
diffusion = lambda A, D, t, elastic, wavevector=0: A * np.exp(-wavevector ** 2 * D * t) + elastic
# make ErrPlot to see progress of intermediate steps with residuals (updated all 2 seconds)
i5.makeErrPlot(title='diffusion model residual plot')
# fit it
i5.fit(model=diffusion,  # the fit function
       freepar={'D': [0.2, 0.25], 'A': 1},  # freepar with start values; [..] indicate independent fit parameter
       fixpar={'elastic': 0.0},  # fixed parameters, single values indicates common fit parameter
       mapNames={'t': 'X', 'wavevector': 'q'},  # map names of the model to names of data attributes
       condition=lambda a: (a.X > 0.01) & (a.Y > 0.01))  # a condition to include only specific values
#
i5.lastfit.savetxt('iqt_proteindiffusion_fit.dat')  # save fit result with errors and covariance matrix
# plot it together with lastfit result
p = js.grace()
p.plot(i5, symbol=[-1, 0.4, -1], legend='Q=$q')  # plot as alternating symbols and colors with size 0.4
p.plot(i5.lastfit, symbol=0, line=[1, 1, -1])  # plot a line with alternating colors

# plot result with error bars
p1 = js.grace(2, 2)  # plot with a defined size
p1.plot(i5.lastfit.wavevector, i5.lastfit.D, i5.lastfit.D_err, symbol=[2, 1, 1, ''], legend='average effective D')
p1.save('Diffusioncoefficients.agr')  # save as XmGrace plot

13.2. How to build simple models

# How to build simple models
# which are actually not so simple....

import numpy as np
import jscatter as js

# Build models in one line using lambda
# directly calc the values and return only Y values
diffusion = lambda A, D, t, wavevector, elastic=0: A * np.exp(-wavevector ** 2 * D * t) + elastic

# use a model from the libraries
# here Teubner-Strey adding background and power law
# this returns as above only Y values
tbpower = lambda q, B, xi, dd, A, beta, bgr: js.ff.teubnerStrey(q=q, xi=xi, d=dd).Y * B + A * q ** beta + bgr


# The same as above in a function definition
def diffusion2(A, D, t, elastic, wavevector=0):
    Y = A * np.exp(-wavevector ** 2 * D * t) + elastic
    return Y


# returning dataArray allows additional attributes to be included in the result
# this returns a dataArray with X, Y values and attributes
def diffusion3(A, D, t, wavevector, elastic=0):
    Y = A * np.exp(-wavevector ** 2 * D * t) + elastic
    result = js.dA(np.c_[t, Y].T)
    result.diffusioncoefficient = D
    result.wavevector = wavevector
    result.columnname = 'time;Iqt'
    return result


def tbpower2(q, B, xi, dd, A, beta, bgr):
    """Model Teubner Strey  + power law and background"""
    # save different contributions for later analysis
    tb = js.ff.teubnerStrey(q=q, xi=xi, d=dd)
    pl = A * q ** beta  # power law
    tb = tb.addZeroColumns(2)
    tb[-2] = pl  # save power law in new last column
    tb[-1] = tb.Y  # save Teubner-Strey in last column
    tb.Y = B * tb.Y + pl + bgr  # put full model to Y values (usually tb[1])
    # save the additional parameters ; xi and d already included in teubnerStrey
    tb.A = A
    tb.bgr = bgr
    tb.beta = beta
    tb.columnname = 'q;Iq,IqTb,Iqpower'
    return tb

# How to add a numpy like docstring see in the example "How to build a complex model".

13.3. How to build a more complex model

# How to build a complex model

import jscatter as js


# build a complex model of different components

def particlesWithInteraction(q, Ra, Rb, molarity, bgr, contrast, collimation=None, beta=True):
    """
    Particles with interaction and ellipsoid form factor as a model for e.g. dense protein solutions.

    Document your model if needed for later use that you know what you did and why.
    Or make it short without all the nasty documentation for testing.
    The example neglects the protein exact shape and non constant scattering length density.
    Proteins are more potato shaped  and nearly never like a ellipsoid or sphere.
    So this model is only valid at low Q as an approximation.

    Parameters
    ----------
    q : float
        Wavevector
    Ra,Rb : float
        Radius
    molarity : float
        Concentration in mol/l
    contrast : float
        Contrast between ellipsoid and solvent.
    bgr : float
        Background e.g. incoherent scattering
    collimation : float
        Collimation length for SANS. For SAXS use None.
    beta : bool
        True include asymmetry factor beta of
        M. Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).

    Returns
    -------
        dataArray

    Notes
    -----
    Explicitly:
    **The return value can be a dataArray OR only Y values**. Both is working for fitting.

    """
    # We need to multiply form factor and structure factor and add an additional background.
    # formfactor of ellipsoid returns dataArray with beta at last column.
    ff = js.ff.ellipsoid(q, Ra, Rb, SLD=contrast)
    V = ff.EllipsoidVolume
    # the structure factor returns also dataArray
    # we need to supply a radius calculated from Ra Rb, this is an assumption of effective radius for the interaction.
    R = (Ra * Rb * Rb) ** (1 / 3.)
    # the volume fraction is concentration * volume
    # the units have to be converted as V is usually nm**3 and concentration is mol/l
    sf = js.sf.PercusYevick(q, R, molarity=molarity)
    if beta:
        # beta is asymmetry factor according to M. Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).
        # correction to apply for the structure factor
        # noinspection PyProtectedMember
        sf.Y = 1 + ff._beta * (sf.Y - 1)
    #
    # molarity (mol/l) with conversion to number/nm**3 result is in cm**-1
    ff.Y = molarity * 6.023e23 / (1000 * 1e7 ** 2) * ff.Y * sf.Y + bgr
    # add parameters for later use; ellipsoid parameters are already included in ff
    # if data are saved these are included in the file as documentation
    # or can be used for further calculations e.g. if volume fraction is needed (V*molarity)
    ff.R = R
    ff.bgr = bgr
    ff.Volume = V
    ff.molarity = molarity
    # for small angle neutron scattering we may need instrument resolution smearing
    if collimation is not None:
        # For SAX we set collimation=None and this is ignored
        # For SANS we set some reasonable values as 2000 (mm) or 20000 (mm)
        # as attribute in the data we want to fit.
        result = js.sas.resFunct(ff, collimation, 10, collimation, 10, 0.7, 0.15)
    else:
        result = ff
    # we return the complex model for fitting
    return result


p = js.grace()
q = js.loglist(0.1, 5, 300)
for m in [0.01, 0.1, 1, 2, 3, 4]:
    data = particlesWithInteraction(q, Ra=3, Rb=4, molarity=m * 1e-3, bgr=0, contrast=1)
    p.plot(data, sy=[-1, 0.3, -1], le='c= $molarity M')

p.legend()
p.yaxis(min=100, max=1e9, scale='l', label='I(Q) / cm\S-1', tick=[10, 9])
p.xaxis(scale='n', label='Q / nm\S-1')
p.title('Ellipsoidal particles with interaction')
p.subtitle('Ra=3, Rb=4 and PercusYevick structure factor')
# Hint for fitting SANS data or other parameter dependent fit:
# For a combined fit of several collimation distances each dataset should contain an attribute data.collimation.
# This is automatically used in the fit, if there is not explicit fit parameter with this name.

if 0:
    p.save('interactingParticles.agr')
    p.save('interactingParticles.jpeg')

13.4. 1D fits with attributes

Some Sinusoidal fits with different kinds to use data attributes

# Basic fit examples with synthetic data. Usually data are loaded from a file.

import jscatter as js
import numpy as np

sinus = lambda x, A, a, B: A * np.sin(a * x) + B  # define model

# Fit sine to simulated data
x = np.r_[0:10:0.1]
data = js.dA(np.c_[x, np.sin(x) + 0.2 * np.random.randn(len(x)), x * 0 + 0.2].T)  # simulate data with error
data.fit(sinus, {'A': 1.2, 'a': 1.2, 'B': 0}, {}, {'x': 'X'})  # fit data
data.showlastErrPlot()  # show fit
data.errPlotTitle('Fit Sine')

# Fit sine to simulated data using an attribute in data with same name
data = js.dA(np.c_[x, 1.234 * np.sin(x) + 0.1 * np.random.randn(len(x)), x * 0 + 0.1].T)  # create data
data.A = 1.234  # add attribute
data.makeErrPlot()  # makes errorPlot prior to fit
data.fit(sinus, {'a': 1.2, 'B': 0}, {}, {'x': 'X'})  # fit using .A
data.errPlotTitle('Fit Sine with attribute')

# Fit sine to simulated data using an attribute in data with different name and fixed B
data = js.dA(np.c_[x, 1.234 * np.sin(x) + 0.1 * np.random.randn(len(x)), x * 0 + 0.1].T)  # create data
data.dd = 1.234  # add attribute
data.fit(sinus, {'a': 1.2, }, {'B': 0}, {'x': 'X', 'A': 'dd'})  # fit data
data.showlastErrPlot()  # show fit
data.errPlotTitle('Fit Sine with attribut and fixed B')

A 2D fit using an attribute B stored in the data as second dimension. This might be extended to several attributes allowing multidimensional fitting.

# first one common parameter then as parameter list
# create data
data = js.dL()
ef = 0.1  # increase this to increase error bars of final result
for ff in [0.001, 0.4, 0.8, 1.2, 1.6]:
    data.append(js.dA(np.c_[x, (1.234 + ff) * np.sin(x + ff) + ef * ff * np.random.randn(len(x)), x * 0 + ef * ff].T))
    data[-1].B = 0.2 * ff / 2  # add attributes

# fit with a single parameter for all data, obviously wrong result
data.fit(lambda x, A, a, B, p: A * np.sin(a * x + p) + B, {'a': 1.2, 'p': 0, 'A': 1.2}, {}, {'x': 'X'})
data.showlastErrPlot()  # show fit
data.errPlotTitle('Fit Sine with attribut and common fit parameter')

# now allowing multiple p,A,B as indicated by the list starting value
data.fit(lambda x, A, a, B, p: A * np.sin(a * x + p) + B, {'a': 1.2, 'p': [0], 'B': [0, 0.1], 'A': [1]}, {}, {'x': 'X'})
data.errPlotTitle('Fit Sine with attribut and non common fit parameter')

# plot p against A , just as demonstration
p = js.grace()
p.plot(data.A, data.p, data.p_err, sy=[1, 0.3, 1])
p.xaxis(label='Amplitude')
p.yaxis(label='phase')

13.5. 2D fitting

Unlike the previous we use here data with two dimensions in X,Z coordinates. Additional one could use again attributes.

Another example is shown in Fitting the 2D scattering of a lattice.

import jscatter as js
import numpy as np

# 2D fit data with an X,Z grid data and Y values (For 3D we would use X,Z,W )
# For 2D fit we calc Y values from X,Z coordinates (only for scalar Y data).
# For fitting we need data with X,Z,Y columns indicated .

#  We create synthetic 2D data with X,Z axes and Y values as Y=f(X,Z)
# This is ONE way to make a grid. For fitting it can be unordered, non-gridded X,Z data
x, z = np.mgrid[-5:5:0.25, -5:5:0.25]
xyz = js.dA(np.c_[x.flatten(), z.flatten(), 0.3 * np.sin(x * z / np.pi).flatten() + 0.01 * np.random.randn(
    len(x.flatten())), 0.01 * np.ones_like(x).flatten()].T)
# set columns where to find X,Z coordinates and Y values and eY errors )
xyz.setColumnIndex(ix=0, iz=1, iy=2, iey=3)


# define model
def ff(x, z, a, b):
    return a * np.sin(b * x * z)


xyz.fit(model=ff, freepar={'a': 1, 'b': 1 / 3.}, fixpar={}, mapNames={'x': 'X', 'z': 'Z'})

# show in 2D plot
import matplotlib.pyplot as plt
from matplotlib import cm

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.set_title('2D Sinusoidal fit')
# plot data as points
ax.scatter(xyz.X, xyz.Z, xyz.Y)
# plot fit as contour lines
ax.tricontour(xyz.lastfit.X, xyz.lastfit.Z, xyz.lastfit.Y, cmap=cm.coolwarm, antialiased=False)
plt.show(block=False)

13.6. Simple diffusion fit of not so simple diffusion case

Here the long part with description from the first example.

This is the diffusion of a protein in solution. This is NOT constant as for Einstein diffusion.

These simulated data are similar to data measured by Neutron Spinecho Spectroscopy, which measures on the length scale of the protein and therefore also rotational diffusion contributes to the signal. At low wavevectors additional the influence of the structure factor leads to an upturn, which is neglected in the simulated data. To include the correction D_T(q)=D_{T0} H(q)/S(q) look at hydrodynamicFunct().

For details see this tutorial review Biehl et al. Softmatter 7,1299 (2011)

# import jscatter and numpy
import numpy as np
import jscatter as js

# read the data (16 sets) with attributes as q, Dtrans .... into dataList
i5 = js.dL(js.examples.datapath + '/iqt_1hho.dat')

# define a model for the fit
diffusion = lambda A, D, t, elastic, wavevector=0: A * np.exp(-wavevector ** 2 * D * t) + elastic
# do the fit
i5.fit(model=diffusion,  # the fit function
       freepar={'D': [0.08], 'A': 0.98},  # start parameters, "[]" -> independent fit
       fixpar={'elastic': 0.0},  # fixed parameters
       mapNames={'t': 'X', 'wavevector': 'q'})  # map names from the model to names from the data
# single valued start parameters are the same for all dataArrays
# list start parameters indicate independent fitting for datasets
# the command line shows progress and the final result, which is found in .lastfit
i5.showlastErrPlot(yscale='l')  # opens plot with residuals

# open a plot with fixed size and plot
p = js.grace(1.2, 0.8)
# plot the data with Q values in legend as symbols
p.plot(i5, symbol=[-1, 0.4, -1], legend='Q=$q')
# plot fit results in lastfit as lines without symbol or legend
p.plot(i5.lastfit, symbol=0, line=[1, 1, -1])

# pretty up if needed
p.yaxis(min=0.02, max=1.1, scale='log', charsize=1.5, label='I(Q,t)/I(Q,0)')
p.xaxis(min=0, max=130, charsize=1.5, label='t / ns')
p.legend(x=110, y=0.9, charsize=1)
p.title('I(Q,t) as measured by Neutron Spinecho Spectroscopy', size=1.3)
p.text('for diffusion a single exp. decay', x=60, y=0.35, rot=360 - 20, color=4)
p.text(r'f(t)=A*e\S-Q\S2\N\SDt', x=100, y=0.025, rot=0, charsize=1.5)

if 0:  # optional; save in different formats
    p.save('DiffusionFit.agr')
    p.save('DiffusionFit.jpg')

# This is the basis of the simulated data above
D = js.dA('./exampleData/1hho.Dq')

# Plot the result in an additional plot
p1 = js.grace(1, 1)  # plot with a defined size
p1.plot(i5.q, i5.D, i5.D_err, symbol=[2, 1, 1, ''], legend='average effective D')
p1.plot(D.X, D.Y * 1000., sy=0, li=1, legend='diffusion coefficient 1hho')
# pretty up if needed
p1.title('diffusion constant of a dilute protein in solution', size=1.5)
p1.subtitle('the increase is due to rotational diffusion on top of translational diffusion at Q=0', size=1)
p1.xaxis(min=0, max=3, charsize=1.5, label='Q / nm\S-1')  # xaxis numbers in size 1.5
p1.yaxis(min=0.05, max=0.15, charsize=1.5, label=r'D\seff\N / nm\S2\N/ns')
p1.legend(x=1, y=0.14)

if 0:
    p1.save('effectiveDiffusion.agr')
    p1.save('effectiveDiffusion.jpg')

13.7. How to smooth Xray data and make an inset in the plot

import jscatter as js
import numpy as np
from scipy.interpolate import LSQUnivariateSpline

# load data
ao50 = js.dA(js.examples.datapath + '/a0_336.dat')
ao50.conc = 50
ao10 = js.dA(js.examples.datapath + '/a0_338.dat')
ao10.conc = 2

p = js.grace(1.5, 1)
p.clear()

p.plot(ao50.X, ao50.Y, legend='50 mg/ml')
p.plot(ao10.X, ao10.Y, line=0, symbol=[1, 0.05, 2], legend='2mg/ml')
p.xaxis(0, 6, label='Q / nm\S-1')
p.yaxis(0.05, 200, scale='logarithmic', label='I(Q) / a.u.')
p.title('smoothed X-ray data')
p.subtitle('inset is the extracted structure factor at low Q')

# smoothing with a spline
# determine the knots of the spline 
# less points than data points
t = np.r_[ao10.X[1]:ao10.X[-2]:30j]
# calculate the spline
f = LSQUnivariateSpline(ao10.X, ao10.Y, t)
# calculate the new y values of the spline at the x points
ys = f(ao10.X)
p.plot(ao10.X, ys, symbol=[1, 0.2, 5, 5], legend='2 mg/ml spline ')
p.plot(t, f(t), line=0, symbol=[1, 0.2, 2, 1], legend='knot of spline')

# other idea: use lower number of points with averages in intervals
# this makes 100 intervals with average X and Y values and errors if wanted. Check prune how to use it!
# this is the best solution and additionally creates good error estimate!!!
p.plot(ao10.prune(number=100), line=0, symbol=[1, 0.3, 4], legend='2mg/ml prune to 100 points')
p.legend(x=1, y=100, charsize=0.7)
p.xaxis(0, 6)
p.yaxis(0.05, 200, scale='logarithmic')

# make a smaller plot inside for the structure factor
p.new_graph()
p[1].SetView(0.6, 0.4, 0.9, 0.8)
p[1].plot(ao50.X, ao50.Y / ao10.Y, symbol=[1, 0.2, 1, ''], legend='structure factor')
p[1].yaxis(0.5, 1.3, label='S(Q)')
p[1].xaxis(0, 2, label='Q / nm\S-1')
p[1].legend(x=0.5, y=0.8)

13.8. Analyse SAS data

In small angle scattering a typical situation is that you want to measure a formfactor (particle shape) or structure factor (particle interaction). For this a concentration series is measured and we need to extrapolate to zero concentration to get the formfactor. Afterwards we can divide the measurement by the formfactor to get the structure factor. So we have three key parts :

• Correction for transmission, dark and empty cell scattering to get instrument independent datasets.
• Extrapolating concentration scaled data to get the formfactor.
• Divide by formfactor to get structure factor

Correction

Brulet at al [1] describe the data correction for SANS, which is in principle also valid for SAXS, if incoherent contributions are neglected.

The difference is, that SAXS has typical transmission around ~0.3 for 1mm water sample in quartz cell due to absorption, while in SANS typical values are around ~0.9 for D2O. Larger volume fractions in the sample play a more important rule for SANS as hydrogenated ingredients reduce the transmission significantly, while in SAXS still the water and the cell (quartz) dominate.

One finds for a sample inside of a container with thicknesses (z) for sample, buffer (solvent), empty cell and empty beam measurement (omitting the overall q dependence):

I_s = \frac{1}{z_S}\big((\frac{I_S-I_{dark}}{T_S}-I_{b}T_S\big) -\big(\frac{I_{EC}-I_{dark}}{T_{EC}}-I_{b}T_{EC})\big) - \frac{1}{z_B}\big((\frac{I_B-I_{dark}}{T_B}-I_{b}T_B\big) -\big(\frac{I_{EC}-I_{dark}}{T_{EC}}-I_{b}T_{EC})\big)

where

• I_s is the interesting species
• I_S is the sample of species in solvent (buffer)
• I_B is the pure solvent (describing a constant background)
• I_{dark} is the dark current measurement
• I_b is the empty beam measurement
• I_{EC} is the empty cell measurement
• z_x corresponding sample thickness
• T_x corresponding transmission

The recurring pattern \big((\frac{I-I_{dark}}{T}-I_{b}T\big) shows that the the beam tail (border of primary beam not absorbed by the beam stop) is attenuated by the corresponding sample.

For equal sample thickness z the empty beam is included in subtraction of I_B :

I_s = \frac{1}{z} \big((\frac{I_S-I_{dark}}{T_S}-I_{b}T_S) - (\frac{I_B-I_{dark}}{T_B}-I_{b}T_B)\big)

• The simple case

  If the transmissions are nearly equal as for e.g. protein samples with low concentration (T_S \approx T_B) we only need to subtract the transmission and dark current corrected buffer measurement from the sample.

  I_s = \frac{1}{z} \big((\frac{I_S-I_{dark}}{T_S}) - (\frac{I_B-I_{dark}}{T_B}\big)

• Higher accuracy for large volume fractions

  For larger volume fractions \Phi the transmission might be different and we have to take into account that only 1-\Phi of solvent contributes to I_S. We may incorporate this in the sense of an optical density changing the effective thickness \frac{1}{z_B}\rightarrow\frac{1-\Phi}{z_B} resulting in different thicknesses z_S \neq z_B

[1]	Improvement of data treatment in small-angle neutron scattering Brûlet et al Journal of Applied Crystallography 40, 165-177 (2007)

Extrapolation and dividing

We assume that the above correction was correctly applied and we have a transmission corrected sample and buffer (background) measurement. This is what you typically get from SANS instruments as e.g KWS1-3 from MLZ Garching or D11-D33 at ILL, Grenoble.

The key part is dataff=datas.extrapolate(molarity=0)[0] to extrapolate to zero molarity.

# Analyse SAS data by extrapolating the form factor followed by structure factor determination

import jscatter as js
import numpy as np

# generate some synthetic data just for this demonstration
# import the model described in example_buildComplexModel
# it describes ellipsoids with PercusYevick structure factor
from js import particlesWithInteraction as PWI

NN = 100
q = js.loglist(0.1, 5, NN)
data = js.dL()
bgr = js.dA(np.c_[q, 0.2 + np.random.randn(NN) * 1e-3, [1e-3] * NN].T)  # background
for m in [0.02, 0.05, 0.2, 0.6]:
    pwi = PWI(q, Ra=3, Rb=4, molarity=m * 1e-3, bgr=0, contrast=6e-4)
    pwi = pwi.addColumn(1, np.random.randn(NN) * 1e-3)
    pwi.Y = pwi.Y + bgr.Y
    pwi.setColumnIndex(iey=-1)
    data.append(pwi)

# With measured data the above is just reading data and background
# with an attribut molarity or concentration.
# This might look like this
if 0:
    data = js.dL()
    bgr = js.dA('backgroundmeasurement.dat')
    for name in ['data_conc01.dat', 'data_conc02.dat', 'data_conc05.dat', 'data_conc08.dat']:
        data.append(name)
        data[-1].molarity = float(name.split('.')[0][-2:])

p = js.grace(2, 0.8)
p.multi(1, 4)
p[0].plot(data, sy=[-1, 0.3, -1], le='c= $molarity M')
p[0].yaxis(min=1e-4, max=1e2, scale='l', label='I(Q) / cm\S-1', tick=[10, 9], charsize=1, ticklabel=['power', 0, 1])
p[0].xaxis(scale='l', label='Q / nm\S-1', min=1e-1, max=1e1, charsize=1)
p[0].text(r'original data\nlike from measurement', y=50, x=1)
p[0].legend(x=0.12, y=0.003)

# Using the synthetic data we extract again the form factor and structure factor
# subtract background and scale data by concentration or volume fraction
datas = data.copy()
for dat in datas:
    dat.Y = (dat.Y - bgr.Y) / dat.molarity
    dat.eY = (dat.eY + bgr.eY) / dat.molarity  # errors increase
p[1].plot(datas, sy=[-1, 0.3, -1], le='c= $molarity M')
p[1].yaxis(min=1, max=1e5, scale='l', tick=[10, 9], charsize=1, ticklabel=['power', 0])
p[1].xaxis(scale='l', label='Q / nm\S-1', min=1e-1, max=1e1, charsize=1)
p[1].text(r'bgr subtracted and\n conc. scaled', y=5e4, x=0.8)

# extrapolate to zero concentration to get the  form factor
# dataff=datas.extrapolate(molarity=0,func=lambda y:-1/y,invfunc=lambda y:-1/y)
dataff = datas.extrapolate(molarity=0)[0]
# as error *estimate* we may use the mean of the errors which is not absolutely correct
dataff = dataff.addColumn(1, datas.eY.array.mean(axis=0))
dataff.setColumnIndex(iey=2)
p[2].plot(datas[0], li=[1, 2, 4], sy=0, le='low molarity')
p[2].plot(dataff, sy=[1, 0.5, 1, 1], le='extrapolated')
p[2].yaxis(min=1, max=1e5, scale='l', tick=[10, 9], charsize=1, ticklabel=['power', 0])
p[2].xaxis(scale='l', label='Q / nm\S-1', min=1e-1, max=1e1, charsize=1)
p[2].legend(x=0.13, y=200)
p[2].text(r'extrapolated formfactor \ncompared to lowest conc.', y=5e4, x=0.7)

# calc the structure factor by dividing by the form factor
sf = datas.copy()
for dat in sf:
    dat.Y = dat.Y / dataff.Y
    dat.eY = (dat.eY ** 2 / dataff.Y ** 2 + dataff.eY ** 2 * (dat.Y / dataff.Y ** 2) ** 2) ** 0.5
    dat.volfrac = dat.Volume * dat.molarity
p[3].plot(sf, sy=[-1, 0.3, -1], le=r'\xF\f{}= $volfrac')
p[3].yaxis(min=0, max=2, scale='n', label=['S(Q)', 1, 'opposite'], charsize=1, ticklabel=['General', 0, 1, 'opposite'])
p[3].xaxis(scale='n', label='Q / nm\S-1', min=1e-1, max=2, charsize=1)
p[3].text('structure factor', y=1.5, x=1)
p[3].legend(x=0.8, y=0.5)

# remember so safe the form factor and structurefactor
# sf.save('uniquenamestructurefactor.dat')
# dataff.save('uniquenameformfactor.dat')

# save the figures
# p.save('SAS_sf_extraction.agr')
# p.save('SAS_sf_extraction.png',size=(2500,1000),dpi=300)

13.9. How to fit SANS data including the resolution for different detector distances

First this example shows the influence of smearing, then how to do a fit including

smearing a la Pedersen in 2 versions.

import jscatter as js
import numpy as np

# prepare profiles SANS for typical 2m and 8m measurement
# smear calls resFunc with the respective parameters; smear also works with line collimation SAXS if needed
resol2m = js.sas.prepareBeamProfile('SANS', detDist=2000, collDist=2000., wavelength=0.4, wavespread=0.15)
resol8m = js.sas.prepareBeamProfile('SANS', detDist=8000, collDist=8000., wavelength=0.4, wavespread=0.15)

# demonstration smearing effects

# generate some data, or load them from measurement
a, b = 2, 3
obj = js.dL()
temp = js.ff.ellipsoid(np.r_[0.01:1:0.01], a, b)
temp.Y += 2  # add background
obj.append(js.sas.smear(temp, resol8m))
temp = js.ff.ellipsoid(np.r_[0.5:5:0.05], a, b)
temp.Y += 2
obj.append(js.sas.smear(temp, resol2m))

# here we compare the difference between the 2 profiles using for both the full q range
obj2 = js.dL()
temp = js.ff.ellipsoid(np.r_[0.01:5:0.01], a, b)
temp.Y += 2
obj2.append(js.sas.smear(temp, resol8m))
temp = js.ff.ellipsoid(np.r_[0.01:5:0.01], a, b)
temp.Y += 2
obj2.append(js.sas.smear(temp, resol2m))

# plot it
p = js.grace()
ellip = js.ff.ellipsoid(np.r_[0.01:5:0.01], a, b)
ellip.Y += 2
p.plot(ellip, sy=[1, 0.3, 1], legend='unsmeared ellipsoid')
p.yaxis(label='Intensity / a.u.', scale='l', min=1, max=1e4)
p.xaxis(label='Q / nm\S-1', scale='n')
p.plot(obj, legend='smeared $rf_detDist')
p.plot(obj2[0], li=[1, 1, 4], sy=0, legend='8m smeared full range')
p.plot(obj2[1], li=[3, 1, 4], sy=0, legend='2m smeared full range')
p.legend(x=2.5, y=8000)
p.title('SANS smearing of ellipsoid')


# p.save('SANSsmearing.jpg')

# now we use the simulated data to fit this to a model

# first possibility
# obj has the information about the used settings in rf_detDist (set in resFunct )
# for experimental data this needs to be added to the loaded data
# another possibility is to use resFunc directly and have the detDist attribute in the data as shown below
def smearedellipsoid(q, A, a, b, rf_detDist, bgr):
    ff = js.ff.ellipsoid(q, a, b)  # calc model
    ff.Y = ff.Y * A + bgr  # multiply amplitude factor and add bgr
    # smear
    if rf_detDist == 2000:
        ffs = js.sas.smear(ff, resol2m)
    elif rf_detDist == 8000:
        ffs = js.sas.smear(ff, resol8m)
    elif rf_detDist == 0:
        # this shows unsmeared model
        ffs = ff
    return ffs


# fit it , here no errors
obj.makeErrPlot(yscale='l', fitlinecolor=[1, 2, 5])
obj.fit(smearedellipsoid, {'A': 1, 'a': 2.5, 'b': 3.5, 'bgr': 0}, {}, {'q': 'X'})
# show the unsmeared model
obj.errPlot(obj.modelValues(rf_detDist=0), li=[3, 2, 4], sy=0, legend='unsmeared fit')

if 0:
    # second possibility for model: use resFunc directly
    obj[0].detDist = 8000  # set detDist for your data
    obj[1].detDist = 2000


    def smearedellipsoid2(q, A, a, b, detDist, bgr):
        ff = js.ff.ellipsoid(q, a, b)  # calc model
        ff.Y = ff.Y * A + bgr  # multiply amplitude factor and add bgr
        # smear
        if detDist > 0:
            ffs = js.sas.resFunct(ff, detDist=detDist, collDist=detDist, wavelength=0.4, wavespread=0.15)
        elif detDist == 0:
            # this shows unsmeared model
            ffs = ff
        return ffs


    # fit it , here no errors
    obj.makeErrPlot(yscale='l', fitlinecolor=5)
    obj.fit(smearedellipsoid2,
            {'A': 1, 'a': 2.5, 'b': 2.5, 'bgr': 0}, {}, {'q': 'X'})

13.10. Smearing and desmearing of SAX and SANS data

import jscatter as js
import numpy as np

# Here we examine the effect of instrumental smearing for SAX (Kratky Camera, line! ) and SANS
# and how we can use the Lake algorithm for desmearing.

# some data
q = np.r_[0.01:7:0.01]
# obj=js.ff.sphere(q,5)
obj = js.ff.ellipsoid(q, 2, 3)
# add background
obj.Y += 2
# load data for beam width profile
empty = js.dA(js.examples.datapath + '/buffer_averaged_corrected_despiked.pdh', usecols=[0, 1],
              lines2parameter=[2, 3, 4])
# beam length profile measurement for a slit (Kratky Camera)
beam = js.dA(js.examples.datapath + '/BeamProfile.pdh', usecols=[0, 1], lines2parameter=[2, 3, 4])

# fit beam width
bwidth = js.sas.getBeamWidth(empty, 'auto')
# prepare measured beamprofile
mbeam = js.sas.prepareBeamProfile(beam, a=2, b=1, bxw=bwidth, dIW=1.)
# prepare profile with trapezoidal shape
tbeam = js.sas.prepareBeamProfile('trapz', a=mbeam.a, b=mbeam.b, bxw=bwidth, dIW=1)
# prepare profile SANS a la Pedersen
Sbeam = js.sas.prepareBeamProfile('SANS', detDist=2000, wavelength=0.4, wavespread=0.15)

if 0:
    p = js.sas.plotBeamProfile(mbeam)
    p = js.sas.plotBeamProfile(mbeam, p)

# smear
datasm = js.sas.smear(obj, mbeam)
datast = js.sas.smear(obj, tbeam)
datasS = js.sas.smear(obj, Sbeam)
# add noise
datasm.Y += np.random.normal(0, 0.5, len(datasm.X))
datast.Y += np.random.normal(0, 0.5, len(datast.X))
datasS.Y += np.random.normal(0, 0.5, len(datasS.X))

# desmear
ws = 11
NI = -15
dsm = js.sas.desmear(datasm, mbeam, NIterations=NI, windowsize=ws)
dst = js.sas.desmear(datast, tbeam, NIterations=NI, windowsize=ws)
dsS = js.sas.desmear(datasS, Sbeam, NIterations=NI, windowsize=ws)

p = js.grace(2, 1.4)
p.plot(obj, sy=[1, 0.3, 3], le='original ellipsoid')
p.plot(dst, sy=0, li=[1, 2, 1], le='desmeared SAX line collimation')
p.plot(dsS, sy=0, li=[1, 2, 2], le='desmeared SANS')
p.plot(datasm, li=[3, 2, 1], sy=0, le='smeared SAX line collimation')
p.plot(datasS, li=[3, 2, 4], sy=0, le='smeared SANS')
p.yaxis(max=1e4, min=0.1, scale='l', label='Intensity / a.u.', size=1.7)
p.xaxis(label='q / nm\S-1', size=1.7)
p.legend(x=3, y=5500, charsize=1.7)
p.title('Smeared ellipsoid and desmearing by Lake algorithm')

# The conclusion is to better fit smeared models than to desmear and fit unsmeared models.
# p.save('SASdesmearing.png')

13.11. A long example for diffusion and how to analyze step by step

This is a long example to show possibilities.

A main feature of the fit is that we can change from a constant fit parameters to a parameter dependent one by simply changing A to [A].

from __future__ import print_function
import jscatter as js
import numpy as np

# load example data and show them in a nice plot as
i5 = js.dL(js.examples.datapath + '/iqt_1hho.dat')

# make a fixed size plot with the data
p = js.grace(1.5, 1)
p.plot(i5, symbol=[-1, 0.4, -1], legend='Q=$q')
p.legend(charsize=1.)
p.yaxis(0.01, 1.1, scale='log', charsize=1.5, label='I(Q,t)/I(Q,0)')
p.title('Intermediate scattering function', size=2)
p.xaxis(charsize=1.5, label='t / ns')

# defining model to use in fit
# simple diffusion
diffusion = lambda A, D, t, wavevector, elastic=0: A * np.exp(-wavevector ** 2 * D * t) + elastic
# or if you want to include in a library with description
# see examples in formel and formfactor

# in the data we have X as coordinate for time so we have to map the name
# same for the wavevector which is usually 'q' in these data
# the wavevector is available in the data for all i as i5[i].q
# or as a list as i5.q
# so test these

# analyzing the data
# to see the results we open an errorplot with Y-log scale
i5.makeErrPlot(yscale='l')
# '----------------------------------------'
# ' a first try model which is bad because of fixed high elastic fraction'
i5.fit(model=diffusion,
       freepar={'D': 0.1, 'A': 1},
       fixpar={'elastic': 0.5},
       mapNames={'t': 'X', 'wavevector': 'q'})
# '--------------------------------------'
# ' Now we try it with constant D and a worse A as starting parameters'
i5.fit(model=diffusion,
       freepar={'D': 0.1, 'A': 18},
       fixpar={'elastic': 0.0},
       mapNames={'t': 'X', 'wavevector': 'q'})
print(i5.lastfit.D, i5.lastfit.D_err)
print(i5.lastfit.A, i5.lastfit.A_err)
# A is close to 1 (as it should be here) but the fits dont look good
# '--------------------------------------'
# ' A free amplitude dependent on wavevector might improve '
i5.fit(model=diffusion,
       freepar={'D': 0.1, 'A': [1]},
       fixpar={'elastic': 0.0},
       mapNames={'t': 'X', 'wavevector': 'q'})
# and a second plot to see the results of A
pr = js.grace()
pr.plot(i5.lastfit.wavevector, i5.lastfit.A, i5.lastfit.A_err, legend='A')
# The fit is ok only the chi^2 is to high in this case of simulated data
# '--------------------------------------'
# ' now with free diffusion coefficient dependent on wavevector; is this the best solution?'
i5.fit(model=diffusion,
       freepar={'D': [0.1], 'A': [1]},
       fixpar={'elastic': 0.0},
       mapNames={'t': 'X', 'wavevector': 'q'})

pr.clear()  # removes the old stuff
pr.plot(i5.lastfit.wavevector, i5.lastfit.D, i5.lastfit.D_err, legend='D')
pr.plot(i5.lastfit.wavevector, i5.lastfit.A, i5.lastfit.A_err, legend='A')
# Ahh
# Now the amplitude is nearly constant and the diffusion is changing
# the fit is ok
# '--------------------------------------'
# ' now with changing diffusion and constant amplitude '
i5.fit(model=diffusion,
       freepar={'D': [0.1], 'A': 1},
       fixpar={'elastic': 0.0},
       mapNames={'t': 'X', 'wavevector': 'q'})
pr.clear()  # removes the old stuff
pr.plot(i5.lastfit.wavevector, i5.lastfit.D, i5.lastfit.D_err, legend='D')

# Booth fits are very good, but the last has less parameter.
# From simulation i know it should be equal to 1 for all amplitudes :-))))).

13.12. Sedimentation of two particle sizes and resulting scattering: a Simulation

# I had the question how long do i need to centrifuge to get rid of
# the larger aggregates and not just guess somewhat.

import numpy as np
import jscatter as js

t1 = np.r_[100:2e3:11j]  # time in seconds

# open plot()
p = js.grace(1.5, 1.5)
p[0].SetView(0.15, 0.12, 0.9, 0.85)
# calculate sedimentation profile for two different particles
# data correspond to Fresco 21 with dual rotor
# default is solvent='h2o',temp=293
Rh1 = 2  # nm
Rh2 = 40  # nm
g = 21000.  # g # RZB number
omega = g * 246 / 21000
profiles1 = js.formel.sedimentationProfile(t=t1, Rh=Rh1, c0=0.05, omega=omega, rm=48, rb=85)
profiles2 = js.formel.sedimentationProfile(t=t1, Rh=Rh2, c0=0.05, omega=omega, rm=48, rb=85)

# plot it
p.plot(profiles1, li=-1, sy=0, legend='%s nm -> t=$time s' % Rh1)
p.plot(profiles2, li=[2, 2, -1], sy=0, legend='%s nm-> t=$time s' % Rh2)

# label the plot with some explanations
p.title(r'sedimentation of %s nm species and %s nm species \nafter t seconds centrifugation ' % (Rh1, Rh2), size=1)
p.subtitle(r'rotor speed %s rps=%sg, r\smeniscus\N=48mm, r\sbottom\N=85mm' % (omega, g))
p.yaxis(max=0.2, min=0, label='concentration')
p.xaxis(label='position in cell / mm')
p.legend(x=40, y=0.2, charsize=0.5)
p.text(r'%s nm particles \nnearly not sedimented \nin sedimentation time of %s nm' % (Rh1, Rh2), 44, 0.07)
p.text(r'%snm sediment\nquite fast' % Rh2, 73, 0.105)
p[0].line(80, 0.1, 84, 0.08, 5, arrow=2)

# corresponding small angle scattering for the above
# centrifugation is done to remove the large fraction
# how long do you need to centrifuge and how does it look without centrifugation?
# scattering for a 2 nm particle and 40 nm particle with same intensity in DLS
# with DLS you can see easily the aggregates

p = js.grace()
# equal intensity  of both species in DLS as in SANS
p.plot(js.ff.scatteringFromSizeDistribution(js.loglist(1e-2, 4, 100), [[Rh1, Rh2], [1, 1]]))
# larger has 10% of smaller species intensity
p.plot(js.ff.scatteringFromSizeDistribution(js.loglist(1e-2, 4, 100), [[Rh1, Rh2], [1, 0.1]]))
# larger species particles
p.plot(js.ff.scatteringFromSizeDistribution(js.loglist(1e-2, 4, 100), [[Rh1, Rh2], [1, 0]]))

p.xaxis(min=0.01, max=5, scale='l', label=r'Q / nm\S-1')
p.yaxis(min=0, max=2, label='I(Q)')
p.title('How does %.2g nm aggregates influence SANS scattering' % Rh2, size=1)
p.subtitle('Beaucage form factor with d=3 for spherical particles')
p.text('Here Rg is determined', x=0.2, y=1)
p.text(r'10%% intensity as\n %.2g nm in DLS' % Rh1, x=0.011, y=1.2)
p.text(r'same intensity as\n %.2g nm in DLS' % Rh1, x=0.02, y=1.9)
p.text('no big aggregates', x=0.011, y=0.9)

13.13. Create a stacked chart of some curves

import numpy as np
import jscatter as js

# create a stacked chart of 10 plots

# create some data
mean = 5
x = np.r_[mean - 3 * 3:mean + 3 * 3:200j]
data = js.dL()  # empty dataList
for sigma in np.r_[3:0.3:10j]:
    temp = js.dA(np.c_[x, np.exp(-0.5 * (x - mean) ** 2 / sigma ** 2) / sigma / np.sqrt(2 * np.pi)].T)
    temp.sigma = sigma
    data.append(temp)

p = js.grace()
# each shifted by hshift,vshift
# the yaxis is switched off for all except the first
p.stacked(10, hshift=0.02, vshift=0.01, yaxis='off')
# plot some Gaussians in each graph
for i, dat in enumerate(data):
    p[i].plot(dat, li=[1, 2, i + 1], sy=0, legend='sigma=$sigma')
# choose the same yscale for the data but no ticks for the later plots
# adjusting the scale and the size of the xaxis ticks

p[0].yaxis(min=0, max=1, tick=[0.2, 5, 0.3, 0.1])
p[0].xaxis(min=min(x), max=max(x), tick=[1, 1, 0.3, 0.1])
for pp in p.g[1:]:
    pp.yaxis(min=0, max=1, tick=False)
    pp.xaxis(min=min(x), max=max(x), tick=[1, 1, 0.3, 0.1])

# This plot is shown below; no fine tuning
if 0:
    p.save('stackedGaussians.agr')
    p.save('stackedGaussians', format='jpeg')

# change the stacking to improve plot (or not)
p.stacked(10, hshift=-0.015, vshift=-0.01, yaxis='off')
p.title('stacked')
# create a plot with exponential decaying function but shifted consecutively by factors of 2
x = js.loglist(0.01, 5, 100)
p = js.grace()
for i in np.r_[1:10]:
    p.plot(x, np.exp(-i ** 2 * x ** 2))

p.shiftbyfactor(xfactors=2 * np.r_[10:1:-1], yfactors=2 * np.r_[10:1:-1])
p.yaxis(scale='l')
p.xaxis(scale='l')

13.14. A comparison of different dynamic models in frequency domain

# Comparison inelastic neutron scattering models
# Compare different kinds of diffusion in restricted geometry by the HWHM from the spectra.

import jscatter as js
import numpy as np

# make a plot of the spectrum
w = np.r_[-100:100:1]
ql = np.r_[1:15:.5]
p = js.grace()
p.title('Inelastic neutron scattering ')
p.subtitle('diffusion in a sphere')
iqw = js.dL([js.dynamic.diffusionInSphere_w(w=w, q=q, D=0.14, R=0.2) for q in ql])
p.plot(iqw)
p.yaxis(scale='l', label='I(q,w) / a.u.', min=1e-6, max=1, )
p.xaxis(scale='n', label='w / ns\S-1', min=-100, max=100, )

# Parameters
ql = np.r_[0.5:15.:0.2]
D = 0.1
R = 0.5  # diffusion coefficient and radius
w = np.r_[-100:100:0.1]
u0 = R / 4.33 ** 0.5
t0 = R ** 2 / 4.33 / D  # corresponding values for Gaussian restriction, see Volino et al.

# In the following we calc the spectra and then extract the FWHM to plot it

# calc spectra
iqwD = js.dL([js.dynamic.transDiff_w(w=w, q=q, D=D) for q in ql[5:]])
iqwDD = js.dL([js.dynamic.time2frequencyFF(js.dynamic.simpleDiffusion, 'elastic', w=w, q=q, D=D) for q in ql])
iqwS = js.dL([js.dynamic.diffusionInSphere_w(w=w, q=q, D=D, R=R) for q in ql])
iqwG3 = js.dL([js.dynamic.diffusionHarmonicPotential_w(w=w, q=q, rmsd=u0, tau=t0, ndim=3) for q in ql])
iqwG2 = js.dL([js.dynamic.diffusionHarmonicPotential_w(w=w, q=q, rmsd=u0, tau=t0, ndim=2) for q in ql])
iqwG11 = js.dL([js.dynamic.t2fFF(js.dynamic.diffusionHarmonicPotential, 'elastic', w=np.r_[-100:100:0.01], q=q, rmsd=u0,
                                           tau=t0, ndim=1) for q in ql])
iqwG22 = js.dL([js.dynamic.t2fFF(js.dynamic.diffusionHarmonicPotential, 'elastic', w=np.r_[-100:100:0.01], q=q, rmsd=u0,
                                           tau=t0, ndim=2) for q in ql])
iqwG33 = js.dL([js.dynamic.t2fFF(js.dynamic.diffusionHarmonicPotential, 'elastic', w=np.r_[-100:100:0.01], q=q, rmsd=u0,
                                           tau=t0, ndim=3) for q in ql])
# iqwCH3=js.dL([js.dynamic.t2fFF(js.dynamic.methylRotation,'elastic',w=np.r_[-100:100:0.1],q=q ) for q in ql])

# plot HWHM  in a scaled plot
p1 = js.grace(1.5, 1.5)
p1.title('Inelastic neutron scattering models')
p1.subtitle('Comparison of HWHM for different types of diffusion')
p1.plot([0.1, 60], [4.33296] * 2, li=[1, 1, 1], sy=0)
p1.plot((R * iqwD.wavevector.array) ** 2, [js.dynamic.getHWHM(dat)[0] / (D / R ** 2) for dat in iqwD], sy=[1, 0.5, 1],
        le='free')
p1.plot((R * iqwS.wavevector.array) ** 2, [js.dynamic.getHWHM(dat)[0] / (D / R ** 2) for dat in iqwS], sy=[2, 0.5, 3],
        le='in sphere')
p1.plot((R * iqwG3.wavevector.array) ** 2, [js.dynamic.getHWHM(dat)[0] / (D / R ** 2) for dat in iqwG3], sy=[3, 0.5, 4],
        le='harmonic 3D')
p1.plot((R * iqwG2.wavevector.array) ** 2, [js.dynamic.getHWHM(dat)[0] / (D / R ** 2) for dat in iqwG2], sy=[4, 0.5, 7],
        le='harmonic 2D')
p1.plot((R * iqwDD.wavevector.array) ** 2, [js.dynamic.getHWHM(dat)[0] / (D / R ** 2) for dat in iqwDD], sy=0,
        li=[1, 2, 7], le='free fft')
p1.plot((R * iqwG11.wavevector.array) ** 2, [js.dynamic.getHWHM(dat, gap=0.04)[0] / (D / R ** 2) for dat in iqwG11],
        sy=0, li=[1, 2, 1], le='harmonic 1D fft')
p1.plot((R * iqwG22.wavevector.array) ** 2, [js.dynamic.getHWHM(dat, gap=0.04)[0] / (D / R ** 2) for dat in iqwG22],
        sy=0, li=[2, 2, 1], le='harmonic 2D fft')
p1.plot((R * iqwG33.wavevector.array) ** 2, [js.dynamic.getHWHM(dat, gap=0.04)[0] / (D / R ** 2) for dat in iqwG33],
        sy=0, li=[3, 2, 1], le='harmonic 3D fft')
# 1DGauss as given in the reference (see help for this function) seems to have a mistake
# Use the t2fFF from time domain
# p1.plot((0.12*iqwCH3.wavevector.array)**2,[js.dynamic.getHWHM(dat,gap=0.04)[0]/(D/0.12**2) for dat in iqwCH3],
#                                                                             sy=0,li=[1,2,4], le='1D harmonic')


# jump diffusion
r0 = .5
t0 = r0 ** 2 / 2 / D
w = np.r_[-100:100:0.02]
iqwJ = js.dL([js.dynamic.jumpDiff_w(w=w, q=q, r0=r0, t0=t0) for q in ql])
ii = 54
p1.plot((r0 * iqwJ.wavevector.array[:ii]) ** 2, [js.dynamic.getHWHM(dat)[0] / (D / r0 ** 2) for dat in iqwJ[:ii]])
p1.text(r'jump diffusion', x=8, y=1.4, rot=0, charsize=1.25)
p1.plot((R * iqwG33.wavevector.array) ** 2, (R * iqwG33.wavevector.array) ** 2, sy=0, li=[3, 2, 1])
p1.yaxis(min=0.3, max=100, scale='l', label='HWHM/(D/R**2)')
p1.xaxis(min=0.3, max=140, scale='l', label='(q*R)\S2')
p1.legend(x=0.35, y=80, charsize=1.25)
# The finite Q resolution results in  js.dynamic.getHWHM (linear interpolation to find [max/2])
# in an offset for very narrow spectra
p1.text(r'free diffusion', x=8, y=10, rot=45, color=2, charsize=1.25)
p1.text(r'free ', x=70, y=65, rot=0, color=1, charsize=1.25)
p1.text(r'3D ', x=70, y=45, rot=0, color=4, charsize=1.25)
p1.text(r'sphere', x=70, y=35, rot=0, color=3, charsize=1.25)
p1.text(r'2D ', x=70, y=15, rot=0, color=7, charsize=1.25)
p1.text(r'1D ', x=70, y=7, rot=0, color=2, charsize=1.25)
# p1.save('DynamicModels.png',size=(900,900),dpi=150)

13.15. Protein incoherent scattering in frequency domain

# A model for protein incoherent scattering
# neglecting coherent and maybe some other contributions

import jscatter as js
import numpy as np
import urllib

# get pdb structure file with 3rn3 atom coordinates and filter for CA positions in A
# this results in C-alpha representation of protein Ribonuclease A.
url = 'https://files.rcsb.org/download/3RN3.pdb'
try:
    pdbtxt = urllib.urlopen(url).read()
except AttributeError:
    # python3
    pdbtxt = urllib.request.urlopen(url).read().decode('utf8')
CA = [line[31:55].split() for line in pdbtxt.split('\n') if line[13:16].rstrip() == 'CA' and line[:4] == 'ATOM']
# conversion to nm and center of mass
ca = np.array(CA, float) / 10.
cloud = ca - ca.mean(axis=0)
Dtrans = 0.086  # nm**2/ns  for 3rn3 in D2O at 20°C
Drot = 0.0163  # 1/ns

# A list of wavevectors and frequencies and a resolution.
# If bgr is zero we have to add it later after the convolution with resolution as constant instrument background.
# As resolution a single q independent (unrealistic) Gaussian is used.
ql = np.r_[0.1, 0.5, 1, 2:15.:2, 20, 30, 40]
w = np.r_[-100:100:0.1]
start = {'s0': 0.5, 'm0': 0, 'a0': 1, 'bgr': 0.00}
resolution = lambda w: js.dynamic.resolution_w(w=w, **start)
res = resolution(w)

# translational diffusion
p = js.grace(3, 0.75)
p.multi(1, 4)
p[0].title(' ' * 50 + 'Inelastic neutron scattering for protein Ribonuclease A', size=2)
iqwRtr = js.dL([js.dynamic.convolve(js.dynamic.transDiff_w(w, q=q, D=Dtrans), res, normB=True) for q in ql])
p[0].plot(iqwRtr, le='$wavevector')
iqwRt = js.dL([js.dynamic.transDiff_w(w, q=q, D=Dtrans) for q in ql])
p[0].plot(iqwRt, li=1, sy=0)
p[0].yaxis(min=1e-8, max=1e3, scale='l', ticklabel=['power', 0, 1], label=[r'I(Q,\xw\f{})', 1.5])
p[0].xaxis(min=-149, max=99, label=r'\xw\f{} / 1/ns', charsize=1)
p[0].legend(charsize=1, x=-140, y=1)
p[0].text(r'translational diffusion', y=80, x=-130, charsize=1.5)
p[0].text(r'Q / nm\S-1', y=2, x=-140, charsize=1.5)

# rotational diffusion
iqwRrr = js.dL(
    [js.dynamic.convolve(js.dynamic.rotDiffusion_w(w, q=q, cloud=cloud, Dr=Drot), res, normB=True) for q in ql])
p[1].plot(iqwRrr, le='$wavevector')
iqwRr = js.dL([js.dynamic.rotDiffusion_w(w, q=q, cloud=cloud, Dr=Drot) for q in ql])
p[1].plot(iqwRr, li=1, sy=0)
p[1].yaxis(min=1e-8, max=1e3, scale='l', ticklabel=['power', 0, 1, 'normal'])
p[1].xaxis(min=-149, max=99, label=r'\xw\f{} / 1/ns', charsize=1)
# p[1].legend()
p[1].text(r'rotational diffusion', y=50, x=-130, charsize=1.5)

# restricted diffusion
iqwRgr = js.dL(
    [js.dynamic.convolve(js.dynamic.diffusionHarmonicPotential_w(w, q=q, tau=0.15, rmsd=0.3), res, normB=True) for q in
     ql])
p[2].plot(iqwRgr, le='$wavevector')
iqwRg = js.dL([js.dynamic.diffusionHarmonicPotential_w(w, q=q, tau=0.15, rmsd=0.3) for q in ql])
p[2].plot(iqwRg, li=1, sy=0)
p[2].yaxis(min=1e-8, max=1e3, scale='l', ticklabel=['power', 0, 1], label='')
p[2].xaxis(min=-149, max=99, label=r'\xw\f{} / 1/ns', charsize=1)
# p[2].legend()
p[2].text(r'restricted diffusion \n(harmonic)', y=50, x=-130, charsize=1.5)

# amplitudes at w=0 and w=10
p[3].title('amplitudes w=[0, 10]')
p[3].subtitle(r'\xw\f{}>10 restricted diffusion > translational diffusion')
ww = 10
wi = np.abs(w - ww).argmin()
p[3].plot(iqwRtr.wavevector, iqwRtr.Y.array.max(axis=1), sy=[1, 0.3, 1], li=[1, 2, 1], le='trans + res')
p[3].plot(iqwRt.wavevector, iqwRt.Y.array.max(axis=1), sy=[1, 0.3, 1], li=[2, 2, 1], le='trans ')
p[3].plot(iqwRt.wavevector, iqwRt.Y.array[:, wi], sy=[2, 0.3, 1], li=[3, 3, 1], le='trans w=%.2g' % ww)
p[3].plot(iqwRrr.wavevector, iqwRrr.Y.array.max(axis=1), sy=[1, 0.3, 2], li=[1, 2, 2], le='rot + res')
p[3].plot(iqwRr.wavevector, iqwRr.Y.array.max(axis=1), sy=[1, 0.3, 2], li=[2, 2, 2], le='rot')
p[3].plot(iqwRr.wavevector, iqwRr.Y.array[:, wi], sy=[2, 0.3, 2], li=[3, 3, 2], le='rot w=%.2g' % ww)
p[3].plot(iqwRgr.wavevector, iqwRgr.Y.array.max(axis=1), sy=[1, 0.3, 3], li=[1, 2, 3], le='rest + res')
p[3].plot(iqwRg.wavevector, iqwRg.Y.array.max(axis=1), sy=[8, 0.3, 1], li=[2, 2, 3], le='rest')
p[3].plot(iqwRg.wavevector, iqwRg.Y.array[:, wi], sy=[3, 0.3, 3], li=[3, 3, 3], le='rest w=%.2g' % ww)

p[3].yaxis(min=1e-6, max=1e3, scale='l', ticklabel=['power', 0, 1, 'opposite'],
           label=[r'I(Q,\xw\f{}=[0,10])', 1, 'opposite'])
p[3].xaxis(min=0.1, max=50, scale='l', label='Q / nm\S-1', charsize=1)
p[3].legend(charsize=1., x=3, y=1.5e-3)

p[3].text(r'translation', y=3e-2, x=15, rot=330, charsize=1.5, color=1)
p[3].text(r'rotation', y=6e1, x=15, rot=330, charsize=1.5, color=2)
p[3].text(r'harmonic', y=6e-4, x=15, rot=330, charsize=1.5, color=3)
p[3].text(r'\xw\f{}=10', y=1e-3, x=0.2, rot=30, charsize=1.5, color=1)
p[3].text(r'resolution', y=0.23, x=0.2, rot=0, charsize=1.5, color=1)
p[3].line(0.17, 5e-3, 0.17, 5e-4, 5, arrow=2)
p[3].line(0.17, 0.8, 0.17, 0.1, 5, arrow=2)
# p.save('inelasticNeutronScattering.png',size=(2400,600),dpi=200)

# all together in a combined model  ----------------------------
conv = js.dynamic.convolve
start = {'s0': 0.5, 'm0': 0, 'a0': 1, 'bgr': 0.00}
resolution = lambda w: js.dynamic.resolution_w(w=w, **start)


def transrotsurfModel(w, q, Dt, Dr, exR, tau, rmsd):
    """
    A model for trans/rot diffusion with a partial local restricted diffusion at the protein surface.

    See Fast internal dynamics in alcohol dehydrogenase The Journal of Chemical Physics 143, 075101 (2015);
    https://doi.org/10.1063/1.4928512

    Parameters
    ----------
    w   frequencies
    q   wavevector
    Dt  translational diffusion
    Dr  rotational diffusion
    exR outside this radius additional restricted diffusion with t0 u0
    tau  correlation time
    rmsd  Root mean square displacement Ds=u0**2/t0

    Returns
    -------

    """
    natoms = cloud.shape[0]
    trans = js.dynamic.transDiff_w(w, q, Dt)
    rot = js.dynamic.rotDiffusion_w(w, q, cloud, Dr)
    fsurf = ((cloud ** 2).sum(axis=1) ** 0.5 > exR).sum() * 1. / natoms  # fraction close to surface
    loc = js.dynamic.diffusionHarmonicPotential_w(w, q, tau, rmsd)
    # only a fraction at surface contributes to local restricted diffusion
    loc.Y = js.dynamic.elastic_w(w).Y * (1 - fsurf) + fsurf * loc.Y
    final = conv(trans, rot)
    final = conv(final, loc)
    final.setattr(rot, 'rot_')
    final.setattr(loc, 'loc_')
    res = resolution(w)
    finalres = conv(final, res, normB=True)
    # finalres.Y+=0.0073
    finalres.q = q
    finalres.fsurf = fsurf
    return finalres


ql = np.r_[0.1, 0.5, 1, 2:15.:4, 20]
p = js.grace(1, 1)
p.title('Protein incoherent scattering')
p.subtitle('Ribonuclease A')
iqwR = js.dL([transrotsurfModel(w, q=q, Dt=Dtrans, Dr=Drot, exR=0, tau=0.15, rmsd=0.3) for q in ql])
p.plot(iqwR[0], sy=0, li=[1, 2, 1], le='exR=0')
p.plot(iqwR[1:], sy=0, li=[1, 2, 1])
iqwR = js.dL([transrotsurfModel(w, q=q, Dt=Dtrans, Dr=Drot, exR=1, tau=0.15, rmsd=0.3) for q in ql])
p.plot(iqwR[0], sy=0, li=[1, 2, 2], le='exR=1')
p.plot(iqwR[1:], sy=0, li=[1, 2, 2])
iqwR = js.dL([transrotsurfModel(w, q=q, Dt=Dtrans, Dr=Drot, exR=1.4, tau=0.15, rmsd=0.3) for q in ql])
p.plot(iqwR[0], sy=0, li=[3, 3, 3], le='exR=1.4')
p.plot(iqwR[1:], sy=0, li=[3, 3, 3])
p.yaxis(min=1e-4, max=10, label=r'I(q,\xw\f{})', scale='l', ticklabel=['power', 0, 1])
p.xaxis(min=0.1, max=100, label=r'\xw\f{} / ns\S-1', scale='l')
p.legend(x=0.5, y=0.001)
p[0].line(1, 5e-2, 1, 5e-3, 2, arrow=1)
p.text('resolution', x=0.9, y=7e-3, rot=90)
p.text('q=0.1 nm\S-1', x=0.2, y=6, rot=0)
p.text('q=20 nm\S-1', x=0.2, y=0.04, rot=0)
p[0].line(0.17, 6, 0.17, 0.04, 2, arrow=2)
p.text(r'Variation \nfixed/harmonic protons', x=0.5, y=2e-3, rot=0)
p[0].line(1.4, 2.3e-3, 3, 2.3e-3, arrow=2)

# p.save('Ribonuclease_inelasticNeutronScattering.png',size=(1200,1200),dpi=150)

13.16. Fitting a multiShellcylinder in various ways

import jscatter as js
import numpy as np

# Can we trust a fit?

# simulate some data with noise
# in reality read some data
x = js.loglist(0.1, 7, 1000)
R1 = 2
R2 = 2
L = 20
# this is a three shell cylinder with the outer as a kind of "hydration layer"
simcyl = js.ff.multiShellCylinder(x, L, [R1, R2, 0.5], [4e-4, 2e-4, 6.5e-4], solventSLD=6e-4)

p = js.grace()
p.plot(simcyl, li=1, sy=0)
# noinspection PyArgumentList
simcyl.Y += np.random.randn(len(simcyl.Y)) * simcyl.Y[simcyl.X > 4].mean()
simcyl = simcyl.addColumn(1, simcyl.Y[simcyl.X > 4].mean())
simcyl.setColumnIndex(iey=2)
p.plot(simcyl, li=0, sy=1)
p.yaxis(min=2e-7, max=0.1, scale='l')


# create a model to fit
# We use the model of a double cylinder with background (The intention is to use a wrong but close model).

def dcylinder(q, L, R1, R2, b1, b2, bgr):
    # assume D2O for the solvent
    result = js.ff.multiShellCylinder(q, L, [R1, R2], [b1 * 1e-4, b2 * 1e-4], solventSLD=6.335e-4)
    result.Y += bgr
    return result


simcyl.makeErrPlot(yscale='l')
simcyl.fit(dcylinder,
           freepar={'L': 20, 'R1': 1, 'R2': 2, 'b1': 2, 'b2': 3},
           fixpar={'bgr': 0},
           mapNames={'q': 'X'})


# There are still systematic deviations in the residuals due to the missing layer
# but the result is very promising
# So can we trust such a fit :-)
# The outer 0.5 nm layer modifies the layer thicknesses and scattering length density.
# Here prior knowledge about the system might help.

def dcylinder3(q, L, R1, R2, R3, b1, b2, b3, bgr):
    # assume D2O for the solvent
    result = js.ff.multiShellCylinder(q, L, [R1, R2, R3], [b1 * 1e-4, b2 * 1e-4, b3 * 1e-4], solventSLD=6.335e-4)
    result.Y += bgr
    return result


simcyl.makeErrPlot(yscale='l')
# noinspection PyBroadException
try:
    # The fit will need quite long and fails as it runs in a wrong direction.
    simcyl.fit(dcylinder3,
               freepar={'L': 20, 'R1': 1, 'R2': 2, 'R3': 2, 'b1': 2, 'b2': 3, 'b3': 0},
               fixpar={'bgr': 0},
               mapNames={'q': 'X'}, maxfev=3000)
except:
    # this try : except is only to make the script run as it is
    pass

# Try the fit with a better guess for the starting parameters.
# Prior knowledge by a good idea what is fitted helps to get a good result and
# prevents from running in a wrong minimum of the fit.

simcyl.fit(dcylinder3,
           freepar={'L': 20, 'R1': 2, 'R2': 2, 'R3': 0.5, 'b1': 4, 'b2': 2, 'b3': 6.5},
           fixpar={'bgr': 0}, ftol=1e-5,
           mapNames={'q': 'X'}, condition=lambda a: a.X < 4)

# Finally look at the errors.
# Was the first the better model with less parameters as we cannot get all back due to the noise in the "measurement"?

13.17. Hydrodynamic function

import numpy as np
import jscatter as js

# Hydrodynamic function and structure factor of hard spheres
# The hydrodynamic function describes how diffusion of spheres in concentrated suspension
# is changed due to hydrodynamic interactions.
# The diffusion is changed according to D(Q)=D0*H(Q)/S(Q)
# with the hydrodynamic function H(Q), structure factor S(Q)
# and Einstein diffusion of sphere D0

# some  wavevectors
q = np.r_[0:5:0.03]

p = js.grace(2, 1.5)
p.multi(1, 2)
p[0].title('Hydrodynamic function H(Q)')
p[0].subtitle('concentration dependence')
Rh = 2.2
for ii, mol in enumerate(np.r_[0.1:20:5]):
    H = js.sf.hydrodynamicFunct(q, Rh, molarity=0.001 * mol, )
    p[0].plot(H, sy=[1, 0.3, ii + 1], legend='H(Q) c=%.2g mM' % mol)
    p[0].plot(H.X, H[3], sy=0, li=[1, 2, ii + 1], legend='structure factor')

p[0].legend(x=2, y=2.4)
p[0].yaxis(min=0., max=2.5, label='S(Q); H(Q)')
p[0].xaxis(min=0., max=5., label='Q / nm\S-1')

# hydrodynamic function and structure factor of charged spheres
p[1].title('Hydrodynamic function H(Q)')
p[1].subtitle('screening length dependence')
for ii, scl in enumerate(np.r_[0.1:30:6]):
    H = js.sf.hydrodynamicFunct(q, Rh, molarity=0.0005, structureFactor=js.sf.RMSA,
                                structureFactorArgs={'R': Rh * 2, 'scl': scl, 'gamma': 5}, )
    p[1].plot(H, sy=[1, 0.3, ii + 1], legend='H(Q) scl=%.2g nm' % scl)
    p[1].plot(H.X, H[3], sy=0, li=[1, 2, ii + 1], legend='structure factor')

p[1].legend(x=2, y=2.4)
p[1].yaxis(min=0., max=2.5, label='S(Q); H(Q)')
p[1].xaxis(min=0., max=5., label='Q / nm\S-1')

p[0].text(r'high Q shows \nreduction in D\sself', x=3, y=0.22)
p[1].text(r'low Q shows reduction \ndue to stronger interaction', x=0.5, y=0.25)

# p.save('HydrodynamicFunction.png')

13.18. Multilamellar Vesicles

import jscatter as js

Q = js.loglist(0.001, 5, 500)  # np.r_[0.01:5:0.01]

ffmV = js.ff.multilamellarVesicles
save = 0

# correlation peak sharpness depends on disorder
dR = 20
nG = 200
p = js.grace(1, 1)
for dd in [0.1, 6, 10]:
    p.plot(ffmV(Q=Q, R=100, displace=dd, dR=dR, N=10, dN=0, phi=0.2, layers=0, SLD=1e-4, nGauss=nG),
           le='displace= %g ' % dd)

p.legend(x=0.3, y=10)
p.title('Scattering of multilamellar vesicle')
p.subtitle('lamella N=10, layers 0 nm, dR=20, R=100')
p.yaxis(label='S(Q)', scale='l', min=1e-7, max=1e2, ticklabel=['power', 0])
p.xaxis(label='Q / nm\S-1', scale='l', min=1e-3, max=5, ticklabel=['power', 0])
p.text('Guinier range', x=0.005, y=10)
p.text(r'Correlation peaks\nsharpness decreases with disorder', x=0.02, y=0.00001)
if save: p.save('multilamellar1.png')

# Correlation peak position depends on average layer distance
dd = 0
dR = 20
nG = 200
p = js.grace(1, 1)
for N in [1, 3, 10, 30, 100]:
    p.plot(ffmV(Q=Q, R=100, displace=dd, dR=dR, N=N, dN=0, phi=0.2, layers=0, SLD=1e-4, nGauss=nG), le='N= %g ' % N)

p.legend(x=0.3, y=10)
p.title('Scattering of multilamellar vesicle')
p.subtitle('shellnumber N, layers 0 nm, dR=20, R=100')
p.yaxis(label='S(Q)', scale='l', min=1e-7, max=1e2, ticklabel=['power', 0])
p.xaxis(label='Q / nm\S-1', scale='l', min=1e-3, max=5, ticklabel=['power', 0])

p.text('Guinier range', x=0.005, y=40)
p.text(r'Correlation peaks\n at 2\xp\f{}N/R', x=0.2, y=0.01)
if save: p.save('multilamellar2.png')

# including the shell formfactor with fluctuations of layer thickness
dd = 2
dR = 20
nG = 200
p = js.grace(1, 1)
# multi lamellar structure factor
mV = ffmV(Q=Q, R=100, displace=dd, dR=dR, N=10, dN=0, phi=0.2, layers=6, SLD=1e-4, nGauss=nG)
for i, ds in enumerate([0.001, 0.1, 0.6, 1.2], 1):
    # calc layer fomfactor
    lf = js.formel.pDA(js.ff.multilayer, ds, 'layerd', q=Q, layerd=6, SLD=1e-4)
    p.plot(mV.X, mV._Sq * lf.Y / lf.Y[0], sy=[i, 0.3, i], le='ds= %g ' % ds)
    p.plot(mV.X, lf.Y, sy=0, li=[3, 3, i])
    p.plot(mV.X, mV._Sq, sy=0, li=[2, 3, i])

p.legend(x=0.003, y=1)
p.title('Scattering of multilamellar vesicle')
p.subtitle('shellnumber N=10, layers 6 nm, dR=20, R=100')
p.yaxis(label='S(Q)', scale='l', min=1e-12, max=1e2, ticklabel=['power', 0])
p.xaxis(label='Q / nm\S-1', scale='l', min=2e-3, max=5, ticklabel=['power', 0])

p.text('Guinier range', x=0.005, y=10)
p.text(r'Correlation peak\n at 2\xp\f{}N/R', x=0.4, y=5e-3)
p.text('Shell form factor', x=0.03, y=1e-6)
p.text(r'Shell structure factor', x=0.02, y=2e-5)
p[0].line(0.08, 1e-5, 2, 1e-5, 2, arrow=2)
if save: p.save('multilamellar3.png')

# Comparing multilamellar and unilamellar vesicle
dd = 2
dR = 5
nG = 100
ds = 0.2
p = js.grace(1, 1)
for i, R in enumerate([40, 50, 60], 1):
    mV = ffmV(Q=Q, R=R, displace=dd, dR=dR, N=4, dN=0, phi=0.2, layers=6, ds=ds, SLD=1e-4, nGauss=nG)
    p.plot(mV, sy=[i, 0.3, i], le='R= %g ' % R)
    p.plot(mV.X, mV[-1], sy=0, li=[3, 3, i])
    p.plot(mV.X, mV[-2] * 0.01, sy=0, li=[2, 3, i])

# comparison double sphere
mV = ffmV(Q=Q, R=50., displace=0, dR=5, N=1, dN=0, phi=1, layers=6, ds=ds, SLD=1e-4, nGauss=100)
p.plot(mV, sy=0, li=[1, 2, 4], le='unilamellar R=50 nm')
mV = ffmV(Q=Q, R=60., displace=0, dR=5, N=1, dN=0, phi=1, layers=6, ds=ds, SLD=1e-4, nGauss=100)
p.plot(mV, sy=0, li=[3, 2, 4], le='unilamellar R=60 nm')

p.legend(x=0.3, y=2e3)
p.title('Comparing multilamellar and unilamellar vesicle')
p.subtitle('R=%.2g nm, N=%.1g, layers=%s nm, dR=%.1g, ds=%.2g' % (R, N, 6, dR, ds))
p.yaxis(label='S(Q)', scale='l', min=1e-10, max=1e4, ticklabel=['power', 0])
p.xaxis(label='Q / nm\S-1', scale='l', min=1e-2, max=5, ticklabel=['power', 0])

p.text('Guinier range', x=0.02, y=1000)
p.text(r'Correlation peaks\n at 2\xp\f{}N/R', x=0.8, y=0.1)
p[0].line(0.8, 4e-2, 0.6, 4e-2, 2, arrow=2)
p.text('Shell form factor', x=1, y=1e-2, rot=335)
# p[0].line(0.2,4e-5,0.8,4e-5,2,arrow=2)
p.text(r'Shell structure factor\n x0.01', x=0.011, y=0.1, rot=0)
p.text('Shell form factor ', x=0.02, y=2e-6, rot=0)
if save: p.save('multilamellar4.png')

# Lipid bilayer in SAXS/SANS
# Values for layer thickness can be found in
# Structure of lipid bilayers
# John F. Nagle et al Biochim Biophys Acta. 1469, 159–195. (2000)
Q = js.loglist(0.01, 5, 500)
dd = 1.5
dR = 5
nG = 100
ds = 0.15  # variation of hydrocarbon layer thickness
R = 50
sd = [1.5, 3.5, 1.5]
N = 2

p = js.grace()
p.title('Multilamellar/unilamellar vesicle for SAXS/SANS')
# SAXS
sld = [0.07e-3, 0.6e-3, 0.07e-3]
saxm = ffmV(Q=Q, R=R, displace=dd, dR=dR, N=N, dN=0, phi=0.2, layers=sd, ds=ds, SLD=sld, solventSLD=0.94e-3, nGauss=nG)
p.plot(saxm, sy=0, li=[1, 1, 1], le='SAXS multilamellar')
saxu = ffmV(Q=Q, R=R, displace=0, dR=dR, N=1, dN=0, phi=0.2, layers=sd, ds=ds, SLD=sld, solventSLD=0.94e-3, nGauss=100)
p.plot(saxu, sy=0, li=[3, 2, 1], le='SAXS unilamellar')
saxu = ffmV(Q=Q, R=R, displace=0, dR=dR, N=1, dN=0, phi=0.2, layers=sd, ds=0, SLD=sld, solventSLD=0.94e-3, nGauss=100)
p.plot(saxu, sy=0, li=[2, 0.3, 1], le='SAXS unilamellar ds=0')

# SANS
sld = [0.3e-4, 1.5e-4, 0.3e-4]
sanm = ffmV(Q=Q, R=R, displace=dd, dR=dR, N=N, dN=0, phi=0.2, layers=sd, ds=ds, SLD=sld, solventSLD=6.335e-4, nGauss=nG)
p.plot(sanm, sy=0, li=[1, 1, 2], le='SANS multilamellar')
sanu = ffmV(Q=Q, R=R, displace=0, dR=dR, N=1, dN=0, phi=0.2, layers=sd, ds=ds, SLD=sld, solventSLD=6.335e-4, nGauss=100)
p.plot(sanu, sy=0, li=[3, 2, 2], le='SANS unilamellar')
sanu = ffmV(Q=Q, R=R, displace=0, dR=dR, N=1, dN=0, phi=0.2, layers=sd, ds=0, SLD=sld, solventSLD=6.335e-4, nGauss=100)
p.plot(sanu, sy=0, li=[2, 0.3, 2], le='SANS unilamellar ds=0')
sanu2 = ffmV(Q=Q, R=R, displace=0, dR=dR, N=1, dN=0, phi=0.2, layers=sd, ds=0, SLD=[0.09e-3, 0.6e-3, 0.07e-3],
             solventSLD=6.335e-4, nGauss=100)
p.plot(sanu2, sy=0, li=[1, 0.5, 4], le='SANS unilamellar asymmetric layer ds=0')

p.legend(x=0.013, y=3e-1, boxcolor=0, boxfillpattern=0)
p.title('Comparing multilamellar and unilamellar vesicle')
p.subtitle('R=%.2g nm, N=%.1g, layers=%s nm, dR=%.1g, ds=%.2g' % (R, N, sd, dR, ds))
p.yaxis(label='S(Q)', scale='l', min=1e-5, max=5e3, ticklabel=['power', 0])
p.xaxis(label='Q / nm\S-1', scale='l', min=1e-2, max=5, ticklabel=['power', 0])

p.text('Guinier range', x=0.03, y=2000)
p.text(r'Correlation peaks\n at 2\xp\f{}N/R', x=0.3, y=20)
p.text(r'For multilayers \n shoulder and peaky', x=0.8, y=0.8)
p[0].line(1.5, 0.1, 3.2, 0.25, 2, arrow=1)
p[0].line(0.7, 0.2, 1, 0.25, 2, arrow=1)
p.text('Shell form factor ds=0', x=0.1, y=0.5e-4)
p[0].line(0.2, 4e-5, 0.6, 4e-5, 2, arrow=2)
if save: p.save('multilamellar5.png')

13.19. 2D oriented scattering

Formfactors of oriented particles or particle complexes

import jscatter as js
import numpy as np

# Examples for scattering of 2D scattering of some spheres oriented in space relative to incoming beam
# incoming beam along Y
# detector plane X,Z
# For latter possibility to fit 2D data we have Y=f(X,Z)

# two points
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=10, nCone=10, rms=0)
fig = js.mpl.surface(ffe.X, ffe.Z, ffe.Y)
fig.axes[0].set_title('cos**2 for Z and slow decay for X')
fig.show()
# noise in positions
ffe = js.ff.orientedCloudScattering(qxz, rod0, coneangle=10, 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
# asymmetry is due to small nCone and reflects the used Fibonacci lattice
rod0 = np.zeros([2, 3])
rod0[:, 2] = np.r_[0, np.pi]
ffe = js.ff.orientedCloudScattering(qxz, rod0, coneangle=45, nCone=10, rms=0.05)
fig2 = js.mpl.surface(ffe.X, ffe.Z, ffe.Y)
fig2.axes[0].set_title('symmetric around zero')
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)
# explicitly given isotropic form factor
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()

# Extracting 1D data
# 1. average angular region (similar to experimental detector data)
# 2. direct calculation
# Here with higher resolution to see the additional peaks due to alignment.
#
# 1:
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 aligned core shell particle with additional interferences', size=1.)
p.subtitle(' due to sphere alignment 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()

Oriented crystal structure factors in 2D

# Comparison of different domain sizes dependent on direction of scattering ::

import jscatter as js
import numpy as np

# make xy grid in q space
R = 8  # maximum
N = 50  # number of points
qxy = np.mgrid[-R:R:N * 1j, -R:R:N * 1j].reshape(2, -1).T
# add z=0 component
qxyz = np.c_[qxy, np.zeros(N ** 2)]
# create sc lattice which includes reciprocal lattice vectors and methods to get peak positions
sclattice = js.lattice.scLattice(2.1, 5)
sclattice.rotatehkl2Vector([1, 0, 0], [0, 0, 1])
# define crystal size in directions
ds = [[20, 1, 0, 0], [5, 0, 1, 0], [5, 0, 0, 1]]
# We orient to 100 direction perpendicular to center of qxy plane
ffs = js.sf.orientedLatticeStructureFactor(qxyz, sclattice, domainsize=ds, rmsd=0.1, hklmax=2)
fig = js.mpl.surface(qxyz[:, 0], qxyz[:, 1], ffs[3].array)
fig.axes[0].set_title('symmetric peaks: thinner direction perpendicular to scattering plane')
fig.show()
# We orient to 010 direction perpendicular to center of qxy plane
sclattice.rotatehkl2Vector([0, 1, 0], [0, 0, 1])
ffs = js.sf.orientedLatticeStructureFactor(qxyz, sclattice, domainsize=ds, rmsd=0.1, hklmax=2)
fig2 = js.mpl.surface(qxyz[:, 0], qxyz[:, 1], ffs[3].array)
fig2.axes[0].set_title('asymmetric peaks: thin direction is parallel to scattering plane')
fig2.show()

# rhombic lattice simple and body centered

import jscatter as js
import numpy as np

# make xy grid in q space
R = 8  # maximum
N = 50  # number of points
qxy = np.mgrid[-R:R:N * 1j, -R:R:N * 1j].reshape(2, -1).T
# add z=0 component
qxyz = np.c_[qxy, np.zeros(N ** 2)]
# create rhombic  bc lattice which includes reciprocal lattice vectors and methods to get peak positions
rblattice = js.lattice.rhombicLattice([[2, 0, 0], [0, 3, 0], [0, 0, 1]], size=[5, 5, 5],
                                      unitCellAtoms=[[0, 0, 0], [0.5, 0.5, 0.5]])
# We orient to 100 direction perpendicular to xy plane
rblattice.rotatehkl2Vector([1, 0, 0], [0, 0, 1])

# define crystal size in directions
ds = [[20, 1, 0, 0], [5, 0, 1, 0], [5, 0, 0, 1]]
ffs = js.sf.orientedLatticeStructureFactor(qxyz, rblattice, domainsize=ds, rmsd=0.1, hklmax=2)
fig = js.mpl.surface(ffs.X, ffs.Z, ffs[3].array)
fig.axes[0].set_title('rhombic body centered lattice')
fig.show()
# same without body centered atom
tlattice = js.lattice.rhombicLattice([[2, 0, 0], [0, 3, 0], [0, 0, 1]], size=[5, 5, 5])
tlattice.rotatehkl2Vector([1, 0, 0], [0, 0, 1])
ffs = js.sf.orientedLatticeStructureFactor(qxyz, tlattice, domainsize=ds, rmsd=0.1, hklmax=2)
fig2 = js.mpl.surface(ffs.X, ffs.Z, ffs[3].array)
fig2.axes[0].set_title('rhombic lattice')
fig2.show()

# Rotation of 10 degrees along [1,1,1] axis. It looks spiky because of low number of points in xy plane ::

import jscatter as js
import numpy as np

# make xy grid in q space
R = 8  # maximum
N = 800  # number of points
qxy = np.mgrid[-R:R:N * 1j, -R:R:N * 1j].reshape(2, -1).T
# add z=0 component
qxyz = np.c_[qxy, np.zeros(N ** 2)]  # as position vectors
# create sc lattice which includes reciprocal lattice vectors and methods to get peak positions
sclattice = js.lattice.scLattice(2.1, 5)
# We orient to 111 direction perpendicular to xy plane
sclattice.rotatehkl2Vector([1, 1, 1], [0, 0, 1])
# this needs crystal rotation by 15 degrees to be aligned to xy plane after rotation to 111 direction
# The crystals rotates by 10 degrees around 111 to broaden peaks.
ds = 15
fpi = np.pi / 180.
ffs = js.sf.orientedLatticeStructureFactor(qxyz, sclattice, rotation=[1, 1, 1, 10 * fpi],
                                           domainsize=ds, rmsd=0.1, hklmax=2, nGauss=23)
fig = js.mpl.surface(ffs.X, ffs.Z, ffs[3].array)
fig.axes[0].set_title('10 degree rotation around 111 direction')
fig.show()

Ewald Sphere of simple cubic lattice

import jscatter as js
import numpy as np

import matplotlib.pyplot as pyplot
from matplotlib import cm

phi, theta = np.meshgrid(np.r_[0:np.pi:45j], np.r_[0:2 * np.pi:90j])

# The Cartesian coordinates of the sphere
q = 3
x = q * (np.sin(phi) * np.cos(theta) + 1)
y = q * np.sin(phi) * np.sin(theta)
z = q * np.cos(phi)
qxzy = np.asarray([x, y, z]).reshape(3, -1).T

sclattice = js.lattice.scLattice(2.1, 5)
ffe = js.ff.orientedCloudScattering(qxzy, sclattice.XYZ, coneangle=5, nCone=20, rms=0.02)

# log scale for colors
ffey = np.log(ffe.Y)
fmax, fmin = ffey.max(), ffey.min()
ffeY = (np.reshape(ffey, x.shape) - fmin) / (fmax - fmin)

# Set the aspect ratio to 1 so our sphere looks spherical
fig = pyplot.figure(figsize=pyplot.figaspect(1.))
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x, y, z, rstride=1, cstride=1, facecolors=cm.seismic(ffeY))

# Turn off the axis planes
ax.set_axis_off()
pyplot.show(block=False)

# ------------------------------------

phi, theta = np.meshgrid(np.r_[0:np.pi:90j], np.r_[0:1 * np.pi:90j])
# The Cartesian coordinates of the unit sphere
q = 8
x = q * (np.sin(phi) * np.cos(theta) + 1)
y = q * np.sin(phi) * np.sin(theta)
z = q * np.cos(phi)
qxzy = np.asarray([x, y, z]).reshape(3, -1).T

# Set the aspect ratio to 1 so our sphere looks spherical
fig = pyplot.figure(figsize=pyplot.figaspect(1.))
ax = fig.add_subplot(111, projection='3d')

# create lattice and show it scaled
sclattice = js.lattice.scLattice(2.1, 1)
sclattice.rotatehkl2Vector([1, 1, 1], [0, 0, 1])
gg = ax.scatter(sclattice.X / 3 + q, sclattice.Y / 3, sclattice.Z / 3, c='k', alpha=0.9)
gg.set_visible(True)

ds = 15
fpi = np.pi / 180.
ffs = js.sf.orientedLatticeStructureFactor(qxzy, sclattice, rotation=None, domainsize=ds, rmsd=0.1, hklmax=2, nGauss=23)
# show scattering in log scale on Ewald sphere
ffsy = np.log(ffs.Y)
fmax, fmin = ffsy.max(), ffsy.min()
ffsY = (np.reshape(ffsy, x.shape) - fmin) / (fmax - fmin)

ax.plot_surface(x, y, z, rstride=1, cstride=1, facecolors=cm.hsv(ffsY), alpha=0.4)
ax.plot([q, 2 * q], [0, 0], [0, 0], color='k')
pyplot.show(block=False)

#  ad flat detector xy plane
xzw = np.mgrid[-8:8:80j, -8:8:80j]
qxzw = np.stack([np.zeros_like(xzw[0]), xzw[0], xzw[1]], axis=0)

ff2 = js.sf.orientedLatticeStructureFactor(qxzw.reshape(3, -1).T, sclattice, rotation=None, domainsize=ds, rmsd=0.1,
                                           hklmax=2, nGauss=23)
ffs2 = np.log(ff2.Y)
fmax, fmin = ffs2.max(), ffs2.min()
ff2Y = (np.reshape(ffs2, xzw[0].shape) - fmin) / (fmax - fmin)
ax.plot_surface(qxzw[0], qxzw[1], qxzw[2], rstride=1, cstride=1, facecolors=cm.gist_ncar(ff2Y), alpha=0.3)
ax.set_xlabel('x axis')
ax.set_ylabel('y axis')
ax.set_zlabel('z axis')

fig.suptitle('Scattering plane and Ewald sphere of lattice \nmatplotlib is limited in 3D')
pyplot.show(block=False)
# matplotlib is not real 3D rendering
# maybe use later something else

13.20. Fitting the 2D scattering of a lattice

This example shows how to fit a 2D scattering pattern as e.g. measured by small angle scattering.

Because of the topology of the problem with a large high plateau and small minima most fit algorithm (using a gradient) will fail. One has to take care that the first estimate of the starting parameters is already close to a very good guess or one may use a method to find a global minimum as differential_evolution or brute force scanning.

In the following we limit the model to a few parameters. One needs basically to include more as e.g. the orientation of the crystal in 3D and more parameters influencing the measurement.

Another possibility is to normalize the data e.g. to peak maximum=1 with high q data also fixed.

import jscatter as js
import numpy as np

# load a image with hexagonal peaks (synthetic data)
image = js.sas.sasImage(js.examples.datapath + '/2Dhex.tiff')
# transform to dataarray with X,Z as wavevectors
# The fit algorithm works also with masked areas
hexa = image.asdataArray(0)


def latticemodel(qx, qz, R, ds, rmsd, hklmax, bgr, I0):
    # a hexagonal grid with background, domain size and Debye Waller-factor (rmsd)
    # 3D wavevector
    qxyz = np.c_[qx, qz, np.zeros_like(qx)]
    # define lattice
    grid = js.sf.hcpLattice(R, [3, 3, 3])
    # here one may rotate the crystal

    # calc scattering
    lattice = js.sf.orientedLatticeStructureFactor(qxyz, grid, domainsize=ds, rmsd=rmsd, hklmax=hklmax)
    # add I0 and background
    lattice.Y = I0 * lattice.Y + bgr
    return lattice


# Because of the high plateau in the chi2 landscape
# we first need to use a algorithm finding a global minimum
# this needs around 2300 evaluations and some time
if 0:
    # Please do this if you want to wait
    hexa.fit(latticemodel, {'R': 2, 'ds': 10, 'rmsd': 0.1, 'bgr': 1, 'I0': 10},
             {'hklmax': 4, }, {'qx': 'X', 'qz': 'Z'}, method='differential_evolution')
    #
    fig = js.mpl.surface(hexa.X, hexa.Z, hexa.Y)
    fig = js.mpl.surface(hexa.lastfit.X, hexa.lastfit.Z, hexa.lastfit.Y)

# We use as starting parameters the result of the previous fit
# Now we use LevenbergMarquardt algorithm to polish the result
hexa.fit(latticemodel, {'R': 3.02, 'ds': 8.7, 'rmsd': 0.193, 'bgr': 3, 'I0': 31},
         {'hklmax': 4, }, {'qx': 'X', 'qz': 'Z'})

fig = js.mpl.surface(hexa.X, hexa.Z, hexa.Y)
fig.suptitle('original')
fig.show()
fig2 = js.mpl.surface(hexa.lastfit.X, hexa.lastfit.Z, hexa.lastfit.Y)
fig2.suptitle('fit result')
fig2.show()

13.21. A nano cube build of different lattices

Here we build a nano cube from different crystal structures.

We can observe the cube formfactor and at larger Q the crystal peaks broadened by the small size. The scattering is explicitly calculated from the grids by ff.cloudScattering. Smoothing with a Gaussian considers experimental broadening (and smooth the result:-).

relError is set to higher value to speed it up. For precise calculation decrease it. For the pictures it was 0.02, this takes about 26 s on 6 core CPU for last bcc example with about 10000 atoms and 900 Q values.

Model with explicit atom positions in small grid

The peaks at large Q are Bragg peaks. Due to the small size extinction rules are not fulfilled completely, which is best visible at first peak positions where we still observe forbidden peaks for bcc and fcc. All Bragg peaks are shown in the second example. The formfactor shows reduced amplitudes due to Debye-Waller factor (rms) and the number of grid atoms. The analytical solution has infinite high density of atoms and a sharp interface.

import jscatter as js
import numpy as np

q = np.r_[js.loglist(0.01, 3, 200), 3:40:800j]
unitcelllength = 1.5
N = 8
rms = 0.05
relError = 20  # 0.02 for picture below

# make grids and calc scattering
scgrid = js.sf.scLattice(unitcelllength, N)
sc = js.ff.cloudScattering(q, scgrid.points, relError=relError, rms=rms)
bccgrid = js.sf.bccLattice(unitcelllength, N)
bcc = js.ff.cloudScattering(q, bccgrid.points, relError=relError, rms=rms)
fccgrid = js.sf.fccLattice(unitcelllength, N)
fcc = js.ff.cloudScattering(q, fccgrid.points, relError=relError, rms=rms)
#
# plot  data
p = js.grace(1.5, 1)
# smooth with Gaussian to include instrument resolution
p.plot(sc.X, js.formel.smooth(sc, 10, window='gaussian'), legend='sc')
p.plot(bcc.X, js.formel.smooth(bcc, 10, window='gaussian'), legend='bcc')
p.plot(fcc.X, js.formel.smooth(fcc, 10, window='gaussian'), legend='fcc')
#
# diffusive scattering
# while cloudScattering is normalized to one (without normalization ~ N**2),
# diffusive is proportional to scattering of single atoms (incoherent ~ N)
q = q = js.loglist(1, 35, 100)
p.plot(q, (1 - np.exp(-q * q * rms ** 2)) / scgrid.numberOfAtoms(), li=[3, 2, 1], sy=0, le='sc diffusive')
p.plot(q, (1 - np.exp(-q * q * rms ** 2)) / bccgrid.numberOfAtoms(), li=[3, 2, 2], sy=0, le='bcc diffusive')
p.plot(q, (1 - np.exp(-q * q * rms ** 2)) / fccgrid.numberOfAtoms(), li=[3, 2, 3], sy=0, le='fcc diffusive')
#
# cuboid formfactor for small Q
q = js.loglist(0.01, 2, 300)
cube = js.ff.cuboid(q, unitcelllength * (2 * N + 1))
p.plot(cube.X, js.formel.smooth(cube, 10, window='gaussian') / cube.Y[0], sy=0, li=1, le='cube form factor')
#
p.title('Comparison sc, bcc, fcc lattice for a nano cube')
p.yaxis(scale='l', label='I(Q)', max=1, min=5e-7)
p.xaxis(scale='l', label='Q / A\S-1', max=50, min=0.01)
p.legend(x=0.02, y=0.001, charsize=1.5)
p.text('cube formfactor', x=0.02, y=0.05, charsize=1.4)
p.text('Bragg peaks', x=4, y=0.05, charsize=1.4)
p.text('diffusive scattering', x=4, y=1e-6, charsize=1.4)
# p.save('jscatter/examples/LatticeComparison.png')

Analytical model assuming crystal lattice with limited size

This shows the Bragg peaks of crystal structures with broadening due to limited size.

The low Q scattering from the chrystal shape is not covered well as only the asymptotic behaviour is governed.

import jscatter as js
import numpy as np

q = np.r_[js.loglist(0.5, 3, 50), 3:80:1200j]
unitcelllength = 1.5
N = 8
a = 1.5  # unit cell length
domainsize = a * (2 * N + 1)
rms = 0.05
p = js.grace(1.5, 1)
p.title('structure factor for sc, bcc and fcc 3D lattice')
p.subtitle('with diffusive scattering,asymmetry factor beta=1')

scgrid = js.sf.scLattice(unitcelllength, N)
sc = js.sf.latticeStructureFactor(q, lattice=scgrid, rmsd=rms, domainsize=domainsize)
p.plot(sc, li=[1, 3, 1], sy=0, le='sc')
p.plot(sc.X, 1 - sc[-3], li=[3, 2, 1], sy=0)  # diffusive scattering

bccgrid = js.sf.bccLattice(unitcelllength, N)
bcc = js.sf.latticeStructureFactor(q, lattice=bccgrid, rmsd=rms, domainsize=domainsize)
p.plot(bcc, li=[1, 3, 2], sy=0, le='bcc')
p.plot(bcc.X, 1 - bcc[-3], li=[3, 2, 2], sy=0)

fccgrid = js.sf.fccLattice(unitcelllength, N)
fcc = js.sf.latticeStructureFactor(q, lattice=fccgrid, rmsd=rms, domainsize=domainsize)
p.plot(fcc, li=[1, 3, 3], sy=0, le='fcc')
p.plot(fcc.X, 1 - fcc[-3], li=[3, 2, 3], sy=0)

p.text(r"broken lines \nshow diffusive scattering", x=10, y=0.1)
p.yaxis(label='S(q)', scale='l', max=50, min=0.05)
p.xaxis(label='q / A\S-1', scale='l', max=50, min=0.5)
p.legend(x=1, y=30, charsize=1.5)
# p.save('jscatter/examples/LatticeComparison2.png')

A direct comparison between both models for bcc cube

Differences are due to incomplete extinction of peaks and due to explicit dependence on the edges (incomplete elementary cells)

relError=0.02 samples until the error is smaller 0.02 for a q point with pseudorandom numbers. The same can be done with relError=400 on a fixed Fibonacci lattice. Both need longer for the computation.

#end3
"""

import jscatter as js
import numpy as np

q = np.r_[js.loglist(0.1, 3, 100), 3:40:800j]
unitcelllength = 1.5
N = 8
rms = 0.05
relError = 20  # 0.02 for the picture below
domainsize = unitcelllength * (2 * N + 1)

bccgrid = js.sf.bccLattice(unitcelllength, N)
bcc = js.ff.cloudScattering(q, bccgrid.points, relError=relError, rms=rms)
p = js.grace(1.5, 1)
p.plot(bcc.X, js.formel.smooth(bcc, 10, window='gaussian') * bccgrid.numberOfAtoms(), legend='bcc explicit')

q = np.r_[js.loglist(0.1, 3, 200), 3:40:1600j]
sc = js.sf.latticeStructureFactor(q, lattice=bccgrid, rmsd=rms, domainsize=domainsize)
p.plot(sc, li=[1, 3, 4], sy=0, le='bcc analytic')
p.yaxis(scale='l', label='I(Q)', max=20000, min=0.05)
p.xaxis(scale='l', label='Q / A\S-1', max=50, min=0.1)
p.legend(x=0.5, y=1000, charsize=1.5)
p.title('Comparison explicit and implicit model for a crystal cube')
p.text('cube formfactor', x=0.11, y=1, charsize=1.4)
p.text('bcc Bragg peaks', x=4, y=100, charsize=1.4)
p.text('diffusive scattering', x=10, y=0.1, charsize=1.4)
# p.save('jscatter/examples/LatticeComparison3.png')

13.22. Using cloudscattering as fit model

At the end a complex shaped object: A cube decorated with spheres of different scattering length.

import numpy as np

# Using cloudScattering as fit model.
# We have to define a model that parametrize the building of the cloud that we get a fit parameter.
# As example we use two overlapping spheres. The model can be used to fit some data.


#: 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):
    # A 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 += bgr
    # add attributes for later usage
    result.A = A
    result.R1 = R1
    result.R2 = R2
    result.dx = dx
    result.bgr = bgr
    result.b1 = b1
    result.b2 = b2
    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)
#
# 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=200)
data2.makeErrPlot(yscale='l')
data2.setlimit(A=[0, None, 0])
data2.fit(model=dumbbell,
          freepar={'A': 3},
          fixpar={'R1': 2, 'R2': 2, 'dx': 0.3, 'b1': 0.5, 'b2': 1.5, 'bgr': 0},
          mapNames={'q': 'X'})

# An example to demonstrate how to build a complex shaped object with a simple cubic lattice
# Methods are defined in lattice objects.
# A cube with the corners decorated by spheres

import jscatter as js
import numpy as np

grid = js.sf.scLattice(0.2, 2 * 15, b=[0])
v1 = np.r_[4, 0, 0]
v2 = np.r_[0, 4, 0]
v3 = np.r_[0, 0, 4]
grid.inParallelepiped(v1, v2, v3, b=1)
grid.inSphere(1, center=[0, 0, 0], b=2)
grid.inSphere(1, center=v1, b=3)
grid.inSphere(1, center=v2, b=4)
grid.inSphere(1, center=v3, b=5)
grid.inSphere(1, center=v1 + v2, b=6)
grid.inSphere(1, center=v2 + v3, b=7)
grid.inSphere(1, center=v3 + v1, b=8)
grid.inSphere(1, center=v3 + v2 + v1, b=9)
grid.show()