8. dynamic¶
Models describing dynamic processes mainly for inelastic neutron scattering.
Models in the time domain have a parameter t for time. -> intermediate scattering function
Models in the frequency domain have a parameter w for frequency and _w appended. -> dynamic structure factor
Models in time domain can be transformed to frequency domain by time2frequencyFF()
implementing the Fourier transform .
In time domain the combination of processes is done by multiplication,
including instrument resolution
:
.
# multiplying and creating new dataArray
I(t,q) = js.dA( np.c[t, I1(t,q,..).Y*I2(t,q,..).Y*R(t,q,..).Y ].T)
In frequency domain it is a convolution, including the instrument resolution.
.
conv=js.formel.convolve
S(w,q)=conv(conv(S1(w,q,..),S2(w,q,..)),res(w,q,..),normB=True) # normB normalizes resolution
FFT from time domain by time2frequencyFF()
may include the resolution where it acts like a
window function to reduce spectral leakage with vanishing values at .
If not used
needs to be large (see tfactor) to reduce spectral leakage.
The last step is to shift the model spectrum to the symmetry point of the instrument
as found in the resolution measurement and optional binning over frequency channels.
Both is done by shiftAndBinning()
.
Example
Let us describe the diffusion of a particle inside a diffusing invisible sphere by mixing time domain and frequency domain.
resolutionparameter={'s0':5,'m0':0,'a0':1,'bgr':0.00}
w=np.r_[-100:100:0.5]
resolution=js.dynamic.resolution_w(w,**resolutionparameter)
# model
def diffindiffSphere(w,q,R,Dp,Ds,w0,bgr):
# time domain with transform to frequency domain
diff_w=js.dynamic.time2frequencyFF(js.dynamic.simpleDiffusion,resolution,q=q,D=Ds)
# last convolution in frequency domain, resolution is already included in time domain.
Sx=js.formel.convolve(js.dynamic.diffusionInSphere_w(w=w,q=q,D=Dp,R=R),diff_w)
Sxsb=js.dynamic.shiftAndBinning(Sx,w=w,w0=w0)
Sxsb.Y+=bgr # add background
return Sxsb
#
Iqw=diffindiffSphere(w=w,q=5.5,R=0.5,Dp=1,Ds=0.035,w0=1,bgr=1e-4)
For more complex systems with different scattering length or changing contributions the fraction of contributing atoms (with scattering length) has to be included.
Accordingly, if desired, the mixture of coherent and incoherent scattering needs to be accounted for by corresponding scattering length. This additionally is dependent on the used instrument e.g. for spin echo only 1/3 of the incoherent scattering contributes to the signal. An example model for protein dynamics is given in Protein incoherent scattering in frequency domain.
A comparison of different dynamic models in frequency domain is given in examples. A comparison of different dynamic models in frequency domain.
For conversion to energy use E = ℏ*w = js.dynamic.hbar*w with h/2π = 4.13566/2π [µeV*ns] = 0.6582 [µeV*ns]
Return values are dataArrays were useful. To get only Y values use .Y
8.1. Transform between domains¶
|
Shift spectrum and average (binning) in intervals. |
|
Fast Fourier transform from time domain to frequency domain for inelastic neutron scattering. |
8.2. Time domain¶
|
Resolution in time domain as multiple Gaussians for inelastic measurement as back scattering or time of flight instrument. |
|
Intermediate scattering function for diffusing particles. |
|
Two exponential decaying functions describing diffusion. |
|
Cumulant of order ki with cumulants as diffusion coefficients. |
|
Cumulant of order ki. |
|
Cumulant analysis for dynamic light scattering assuming Gaussian size distribution. |
|
Rouse dynamics of a finite chain with N beads of bonds length l and internal friction. |
|
Zimm dynamics of a finite chain with N beads with internal friction and hydrodynamic interactions. |
|
Conformational dynamics of an ideal chain with hydrodynamic interaction, coherent scattering. |
|
Stretched exponential function. |
|
Incoherent intermediate scattering function of translational jump diffusion in the time domain. |
|
Incoherent intermediate scattering function of CH3 methyl rotation in the time domain. |
|
ISF corresponding to the standard OU process for diffusion in harmonic potential for dimension 1,2,3. |
|
Fractional diffusion of a particle in a periodic potential. |
|
Translational + rotational diffusion of an object (dummy atoms); dynamic structure factor in time domain. |
|
Dynamics of bicontinuous micro emulsion phases. |
|
Dynamics of lamellar microemulsion phases. |
8.3. Frequency domain¶
Planck constant in µeV*ns |
|
h/2π reduced Planck constant in µeV*ns |
|
|
Find half width at half maximum of a distribution around zero. |
|
Convolve A and B with proper tracking of the output X axis. |
|
Transform from S(w,q) to the imaginary part of the dynamic susceptibility. |
|
Resolution as multiple Gaussians for inelastic measurement as backscattering or time of flight instrument in w domain. |
|
Elastic line; dynamic structure factor in w domain. |
|
Translational diffusion; dynamic structure factor in w domain. |
|
Jump diffusion; dynamic structure factor in w domain. |
|
Diffusion in a harmonic potential for dimension 1,2,3 (isotropic averaged), dynamic structure factor in w domain. |
|
Diffusion inside of a sphere; dynamic structure factor in w domain. |
|
Rotational diffusion of an object (dummy atoms); dynamic structure factor in w domain. |
|
Random walk among N equidistant sites (isotropic averaged); dynamic structure factor in w domain. |
Models describing dynamic processes mainly for inelastic neutron scattering.
Models in the time domain have a parameter t for time. -> intermediate scattering function
Models in the frequency domain have a parameter w for frequency and _w appended. -> dynamic structure factor
Models in time domain can be transformed to frequency domain by time2frequencyFF()
implementing the Fourier transform .
In time domain the combination of processes is done by multiplication,
including instrument resolution
:
.
# multiplying and creating new dataArray
I(t,q) = js.dA( np.c[t, I1(t,q,..).Y*I2(t,q,..).Y*R(t,q,..).Y ].T)
In frequency domain it is a convolution, including the instrument resolution.
.
conv=js.formel.convolve
S(w,q)=conv(conv(S1(w,q,..),S2(w,q,..)),res(w,q,..),normB=True) # normB normalizes resolution
FFT from time domain by time2frequencyFF()
may include the resolution where it acts like a
window function to reduce spectral leakage with vanishing values at .
If not used
needs to be large (see tfactor) to reduce spectral leakage.
The last step is to shift the model spectrum to the symmetry point of the instrument
as found in the resolution measurement and optional binning over frequency channels.
Both is done by shiftAndBinning()
.
Example
Let us describe the diffusion of a particle inside a diffusing invisible sphere by mixing time domain and frequency domain.
resolutionparameter={'s0':5,'m0':0,'a0':1,'bgr':0.00}
w=np.r_[-100:100:0.5]
resolution=js.dynamic.resolution_w(w,**resolutionparameter)
# model
def diffindiffSphere(w,q,R,Dp,Ds,w0,bgr):
# time domain with transform to frequency domain
diff_w=js.dynamic.time2frequencyFF(js.dynamic.simpleDiffusion,resolution,q=q,D=Ds)
# last convolution in frequency domain, resolution is already included in time domain.
Sx=js.formel.convolve(js.dynamic.diffusionInSphere_w(w=w,q=q,D=Dp,R=R),diff_w)
Sxsb=js.dynamic.shiftAndBinning(Sx,w=w,w0=w0)
Sxsb.Y+=bgr # add background
return Sxsb
#
Iqw=diffindiffSphere(w=w,q=5.5,R=0.5,Dp=1,Ds=0.035,w0=1,bgr=1e-4)
For more complex systems with different scattering length or changing contributions the fraction of contributing atoms (with scattering length) has to be included.
Accordingly, if desired, the mixture of coherent and incoherent scattering needs to be accounted for by corresponding scattering length. This additionally is dependent on the used instrument e.g. for spin echo only 1/3 of the incoherent scattering contributes to the signal. An example model for protein dynamics is given in Protein incoherent scattering in frequency domain.
A comparison of different dynamic models in frequency domain is given in examples. A comparison of different dynamic models in frequency domain.
For conversion to energy use E = ℏ*w = js.dynamic.hbar*w with h/2π = 4.13566/2π [µeV*ns] = 0.6582 [µeV*ns]
Return values are dataArrays were useful. To get only Y values use .Y
-
jscatter.dynamic.
cumulant
(x, k0=0, k1=0, k2=0, k3=0, k4=0, k5=0)[source]¶ Cumulant of order ki.
- Parameters
- xfloat
Wavevector
- k0,k1, k2,k3,k4,k5float
- Cumulant coefficients; units 1/x
k0 amplitude
k1 expected value
k2 variance with
relative standard deviation
higher order see Wikipedia
- Returns
- dataArray
-
jscatter.dynamic.
cumulantDLS
(t, A, G, sigma, skewness=0, bgr=0.0)[source]¶ Cumulant analysis for dynamic light scattering assuming Gaussian size distribution.
- Parameters
- tarray
Time
- Afloat
Amplitude at t=0; Intercept
- Gfloat
Mean relaxation time as 1/decay rate in units of t.
- sigmafloat
relative standard deviation if a gaussian distribution is assumed
should be smaller 1 or the Taylor expansion is not valid
k2=variance=sigma**2/G**2
- skewnessfloat,default 0
Relative skewness k3=skewness**3/G**3
- bgrfloat; default 0
Constant background
- Returns
- dataArray
References
- 1
Revisiting the method of cumulants for the analysis of dynamic light-scattering data Barbara J. Frisken APPLIED OPTICS 40, 4087 (2001)
-
jscatter.dynamic.
cumulantDiff
(t, q, k0=0, k1=0, k2=0, k3=0, k4=0, k5=0)[source]¶ Cumulant of order ki with cumulants as diffusion coefficients.
- Parameters
- tarray
Time
- qfloat
Wavevector
- k0float
Amplitude
- k1float
Diffusion coefficient in units of 1/([q]*[t])
- k2,k3,k4,k5float
Higher coefficients in same units as k1
- Returns
- dataArray :
-
jscatter.dynamic.
diffusionHarmonicPotential
(t, q, rmsd, tau, beta=0, ndim=3)[source]¶ ISF corresponding to the standard OU process for diffusion in harmonic potential for dimension 1,2,3.
The intermediate scattering function corresponding to the standard OU process for diffusion in an harmonic potential [1]. It is used for localized translational motion in incoherent neutron scattering [2] as improvement for the diffusion in a sphere model. Atomic motion may be restricted to ndim=1,2,3 dimensions and are isotropic averaged. The correlation is assumed to be exponential decaying.
- Parameters
- tarray
Time values in units ns
- qfloat
Wavevector in unit 1/nm
- rmsdfloat
Root mean square displacement <u**2>**0.5 in potential in units nm. <u**2>**0.5 is the width of the potential According to [2] 5*u**2=R**2 compared to the diffusion in a sphere.
- taufloat
Correlation time
in units ns. Diffusion constant in sphere Ds=u**2/tau
- betafloat, default 0
- Exponent in correlation function
.
- Exponent in correlation function
- ndim1,2,3, default=3
Dimensionality of the diffusion potential.
- Returns
- dataArray
Notes
We use equ. 18-20 from [2] and correlation time
with equal amplitudes
in the dimensions as
3 dim case:
2 dim case:
1 dim case:
with erf as the error function and erfi is the imaginary error function erf(iz)/i
References
- 1
Quasielastic neutron scattering and relaxation processes in proteins: analytical and simulation-based models G. R. Kneller Phys. ChemChemPhys. ,2005, 7,2641–2655
- 2(1,2,3,4,5,6,7)
Gaussian model for localized translational motion: Application to incoherent neutron scattering F. Volino, J.-C. Perrin and S. Lyonnard, J. Phys. Chem. B 110, 11217–11223 (2006)
Examples
import numpy as np import jscatter as js t=np.r_[0.1:6:0.1] p=js.grace() p.plot(js.dynamic.diffusionHarmonicPotential(t,1,2,1,1),le='1D ') p.plot(js.dynamic.diffusionHarmonicPotential(t,1,2,1,2),le='2D ') p.plot(js.dynamic.diffusionHarmonicPotential(t,1,2,1,3),le='3D ') p.legend() p.yaxis(label='I(Q,t)') p.xaxis(label='Q / ns') p.subtitle('Figure 2 of ref Volino J. Phys. Chem. B 110, 11217')
-
jscatter.dynamic.
diffusionHarmonicPotential_w
(w, q, tau, rmsd, ndim=3, nmax='auto')[source]¶ Diffusion in a harmonic potential for dimension 1,2,3 (isotropic averaged), dynamic structure factor in w domain.
An approach worked out by Volino et al [1] assuming Gaussian confinement and leads to a more efficient formulation by replacing the expression for diffusion in a sphere with a simpler expression pertaining to a soft confinement in harmonic potential. Ds = ⟨u**2⟩/t0
- Parameters
- warray
Frequencies in 1/ns
- qfloat
Wavevector in nm**-1
- taufloat
Mean correlation time time. In units ns.
- rmsdfloat
Root mean square displacement (width) of the Gaussian in units nm.
- ndim1,2,3, default=3
Dimensionality of the potential.
- nmaxint,’auto’
Order of expansion. ‘auto’ -> nmax = min(max(int(6*q * q * u2),30),1000)
- Returns
- dataArray
Notes
Volino et al [1] compared the behaviour of this approach to the well known expression for diffusion in a sphere. Even if the details differ, the salient features of both models match if the radius R**2 ≃ 5*u0**2 and the diffusion constant inside the sphere relates to the relaxation time of particle correlation t0= ⟨u**2⟩/Ds towards the Gaussian with width u0=⟨u**2⟩**0.5.
- ndim=3
Here we use the Fourier transform of equ 23 with equ. 27a+b in [1]. For order n>30 the Stirling approximation for n! in equ 27b of [1] is used.
- ndim=2
Here we use the Fourier transform of equ 23 with equ. 28a+b in [1].
with
Kummer confluent hypergeometric function, Gamma function
and erfi is the imaginary error function erf(iz)/i
- ndim=1
The equation given by Volino (29a+b in [1]) seems to be wrong as a comparison with the Fourier transform and the other dimensions shows. Use the model from time domain and use FFT as shown in the example.
For experts: To test this remove a flag in the source code and compare.
References
- 1(1,2,3,4,5,6)
Gaussian model for localized translational motion: Application to incoherent neutron scattering. Volino, F., Perrin, J. C. & Lyonnard, S. J. Phys. Chem. B 110, 11217–11223 (2006).
Examples
import jscatter as js import numpy as np t2f=js.dynamic.time2frequencyFF dHP=js.dynamic.diffusionHarmonicPotential w=np.r_[-100:100] ql=np.r_[1:14.1:6j] iqt3=js.dL([js.dynamic.diffusionHarmonicPotential_w(w=w,q=q,tau=0.14,rmsd=0.34,ndim=3) for q in ql]) iqt2=js.dL([js.dynamic.diffusionHarmonicPotential_w(w=w,q=q,tau=0.14,rmsd=0.34,ndim=2) for q in ql]) # as ndim=1 is a wrong solution use this instead # To move spectral leakage out of our window we increase w and interpolate. # The needed factor (here 23) depends on the quality of your data and background contribution. # You may test it using ndim=2 in this example. iqt1=js.dL([t2f(dHP,'elastic',w=w*23,q=q, rmsd=0.34, tau=0.14 ,ndim=1).interpolate(w) for q in ql]) p=js.grace() p.multi(2,3) p[1].title('diffusionHarmonicPotential for ndim= 1,2,3') for i,(i3,i2,i1) in enumerate(zip(iqt3,iqt2,iqt1)): p[i].plot(i3,li=1,sy=0,le='q=$wavevector nm\S-1') p[i].plot(i2,li=2,sy=0) p[i].plot(i1,li=4,sy=0) p[i].yaxis(scale='log') if i in [1,2,4,5]:p[i].yaxis(ticklabel=0) p[i].legend(x=5,y=1, charsize=0.7)
-
jscatter.dynamic.
diffusionInSphere_w
(w, q, D, R)[source]¶ Diffusion inside of a sphere; dynamic structure factor in w domain.
- Parameters
- warray
Frequencies in 1/ns
- qfloat
Wavevector in nm**-1
- Dfloat
Diffusion coefficient in units nm**2/ns
- Rfloat
Radius of the sphere in units nm.
- Returns
- dataArray
Notes
Here we use equ. 33 in [1]
with
as the first 99 solutions of equ 27 a+b as given in [1] and
This is valid for qR<20 with accuracy of ~0.001 as given in [1]. If we look at a comparison with free diffusion the valid range seems to be smaller.
A comparison of diffusion in different restricted geometry is show in example A comparison of different dynamic models in frequency domain.
References
- 1(1,2,3)
Neutron incoherent scattering law for diffusion in a potential of spherical symmetry: general formalism and application to diffusion inside a sphere. Volino, F. & Dianoux, A. J., Mol. Phys. 41, 271–279 (1980). https://doi.org/10.1080/00268978000102761
Examples
import jscatter as js import numpy as np w=np.r_[-100:100] ql=np.r_[1:14.1:1.3] p=js.grace() 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')
-
jscatter.dynamic.
diffusionPeriodicPotential
(t, q, u, rt, Dg, gamma=1)[source]¶ Fractional diffusion of a particle in a periodic potential.
The diffusion describes a fast dynamics inside of the potential trap with a mean square displacement before a jump and a fractional long time diffusion. For fractional coefficient gamma=1 normal diffusion is recovered.
- Parameters
- tarray
Time points, units ns.
- qfloat
Wavevector, units 1/nm
- ufloat
Root mean square displacement in the trap, units nm.
- rtfloat
Relaxation time of fast dynamics in the trap; units ns ( = 1/lambda in [1] )
- gammafloat
Fractional exponent gamma=1 is normal diffusion
- Dgfloat
Long time fractional diffusion coefficient; units nm**2/ns.
- Returns
- dataArray :
[t, Iqt , Iqt_diff, Iqt_trap]
Notes
We use equ. 4 of [1] for fractional diffusion coefficient
with fraction
as
with the Mittag Leffler function
and Gamma function
and
.
The first term in msd describes the long time fractional diffusion while the second describes the additional mean-square displacement inside the trap
.
For
simplifying the equation to normal diffusion with traps.
References
- 1(1,2,3)
Gupta, S.; Biehl, R.; Sill, C.; Allgaier, J.; Sharp, M.; Ohl, M.; Richter, D. Macromolecules 2016, 49 (5), 1941.
Examples
Example similar to protein diffusion in a mesh of high molecular weight PEG as found in [1].
import jscatter as js import numpy as np t=js.loglist(0.1,50,100) p=js.grace() for i,q in enumerate(np.r_[0.1:2:0.3],1): iq=js.dynamic.diffusionPeriodicPotential(t,q,0.5,5,0.036) p.plot(iq,symbol=[1,0.3,i],legend='q=$wavevector') p.plot(iq.X,iq._Iqt_diff,sy=0,li=[1,0.5,i]) p.title('Diffusion in periodic potential traps') p.subtitle('lines show long time diffusion contribution') p.yaxis(max=1,min=1e-2,scale='log',label='I(Q,t)/I(Q,0)') p.xaxis(min=0,max=50,label='t / ns') p.legend(x=110,y=0.8) # p.save(js.examples.imagepath+'/fractalDiff.jpg')
-
jscatter.dynamic.
doubleDiffusion
(q, t, A0, D0, w0=0, A1=0, D1=0, w1=0)[source]¶ Two exponential decaying functions describing diffusion.
- Parameters
- qfloat, array
Wavevector
- tfloat, array
Time list
- A0,A1float
Prefactor
- D0,D1float
Diffusion coefficient in units [ [q]**-2/[t] ]
- w0,w1float
Width of diffusion coefficient distributions in D units.
- Returns
- dataArray
-
jscatter.dynamic.
dynamicSusceptibility
(data, Temp)[source]¶ Transform from S(w,q) to the imaginary part of the dynamic susceptibility.
with Bose distribution for integer spin particles
- Parameters
- datadataArray
Data to transform with w units in 1/ns
- Tempfloat
Measurement temperature in K.
- Returns
- dataArray
Notes
“Whereas relaxation processes on different time scales are usually hard to identify in S(w,q), they appear as distinct peaks in dynamic susceptibility with associated relaxation times :math:´1/2piw´ [1].”
References
- 1
Roh et al. ,Biophys. J. 91, 2573 (2006)
Examples
start={'s0':5,'m0':0,'a0':1,'bgr':0.00} w=np.r_[-100:100:0.5] resolution=js.dynamic.resolution_w(w,**start) # model def diffindiffSphere(w,q,R,Dp,Ds,w0,bgr): diff_w=js.dynamic.transDiff_w(w,q,Ds) rot_w=js.dynamic.diffusionInSphere_w(w=w,q=q,D=Dp,R=R) Sx=js.formel.convolve(rot_w,diff_w) Sxsb=js.dynamic.shiftAndBinning(Sx,w=w,w0=w0) Sxsb.Y+=bgr # add background return Sxsb # q=5.5;R=0.5;Dp=1;Ds=0.035;w0=1;bgr=1e-4 Iqw=diffindiffSphere(w,q,R,Dp,Ds,w0,bgr) IqwR=js.dynamic.diffusionInSphere_w(w,q,Dp,R) IqwT=js.dynamic.transDiff_w(w,q,Ds) Xqw=js.dynamic.dynamicSusceptibility(Iqw,293) XqwR=js.dynamic.dynamicSusceptibility(IqwR,293) XqwT=js.dynamic.dynamicSusceptibility(IqwT,293) p=js.grace() p.plot(Xqw) p.plot(XqwR) p.plot(XqwT) p.yaxis(scale='l',label='X(w,q) / a.u.') p.xaxis(scale='l',label='w / ns\S-1')
-
jscatter.dynamic.
elastic_w
(w)[source]¶ Elastic line; dynamic structure factor in w domain.
- Parameters
- warray
Frequencies in 1/ns
- Returns
- dataArray
-
jscatter.dynamic.
finiteRouse
(t, q, NN=None, pmax=None, l=None, frict=None, Dcm=None, Wl4=None, Dcmfkt=None, tintern=0.0, Temp=293, ftype=None, specm=None, specb=None, rk=None)[source]¶ Rouse dynamics of a finite chain with N beads of bonds length l and internal friction.
The Rouse model describes the conformational dynamics of an ideal chain. The single chain diffusion is represented by Brownian motion of beads connected by harmonic springs. No excluded volume, random thermal force, drag force with solvent, no hydrodynamic interaction and optional internal friction. Coherent + incoherent scattering.
- Parameters
- tarray
Time in units nanoseconds
- qfloat, list
Scattering vector, units nm^-1 For a list a dataList is returned otherwise a dataArray is returned
- NNinteger
Number of chain beads.
- lfloat, default 1
Bond length between beads; unit nm.
- pmaxinteger, list of floats
integer => maximum mode number (
)
list =>
list of amplitudes>0 for individual modes to allow weighing; not given modes have weight zero
- frictfloat
Friction of a single bead/monomer
, units Pas*m=kg/s=1e-6 g/ns.
A sphere with R=0.1 nm in H2O(20°C) (1 mPas) => 1.89e-12 Pas*m
- Wl4float
Characteristic value to calc friction and Dcm.
and characteristic Rouse variable
- Dcmfloat
- Center of mass diffusion in nm^2/ns.
with
= friction of single bead in solvent
- Dcmfktarray 2xN, function
Function f(q) or array with [qi, f(qi) ] as correction for Dcm like Diff = Dcm*f(q). e.g. for inclusion of structure factor or hydrodynamic function with f(q)=H(Q)/S(q). Missing values are interpolated. Only array input can be pickled to speedup by using formel.memoize .
- tinternfloat>0
Relaxation time due to internal friction between neighbouring beads in ns.
- ftype‘rni’, ‘rap’,’nonspec’ default = ‘rif’
- Type of internal friction. See [7] for a description and respective references.
‘rif’ Internal friction between neighboring beads in chain.
- ‘rni’ Rouse model with non-local interactions (RNI).
Additional friction between random close approaching beads.
- ‘rap’ Rouse model with anharmonic potentials due to stiffness of the chain
- ‘specrif’ Specific interactions of strength
between beads separated by m bonds. See [7] .
- ‘specrif’ Specific interactions of strength
- ‘crif’ Bead confining potential with internal friction. The beads are confined in an additional
harmonic potential with
leading to a more compact configuration.
describes the relative strength compared to the force between beads
.
- Tempfloat
Temperature Kelvin = 273+T[°C]
- specm,specb: float
Parameters m, b used in internal friction models ‘spec’ and ‘specrif’.
- rkNone , float
describes the relative force constant for ftype ‘crif’.
- Returns
- dataArrayfor single q
- dataListmultiple q
[time; Sqt; Sqt_inf; Sqtinc]
time units ns
Sqt is coherent scattering with diffusion and mode contributions
Sqt_inf is coherent scattering with ONLY diffusion
Sqtinc is incoherent scattering with diffusion and mode contributions (no separate diffusion)
.q wavevector
.Wl4
.Re end to end distance
- .trouse rotational correlation time or rouse time
.tintern relaxation time due to internal friction
.tr_p characteristic times
.beadfriction
.ftype type of internal friction
…. use .attr to see all attributes
Notes
The Rouse model for the coherent intermediate scattering function
is [1] [2] :
and for incoherent intermediate scattering function the same with
in the first sum.
- with
mode amplitude (usual
)
mode relaxation time with Rouse time
center of mass diffusion
force constant k between beads.
single bead friction
in solvent (e.g. surrounding melt)
additional relaxation time due to internal friction
Modifications (ftype) for internal friction and additional interaction (see [7] and [9]):
- RNIRouse model with non-local interactions as additional friction between spatial close beads [5] .
- RAPRouse model with anharmonic potentials in bonds describing the stiffness of the chain [6].
- SPECRIFSpecific interactions of relative strength
between beads separated by m bonds.
Internal friction between neighboring beads as in RIF is added.
relative strength of both interactions.
The interaction is between all pairs separated by m.
- SPECRIFSpecific interactions of relative strength
- CRIFCompacted Rouse with internal friction [9].
The beads are confined in an additional harmonic potential with
leading to a more compact configuration.
describes the relative strength compared to the force between beads
. Typically
.
The mode amplitude prefactor changes from Rouse type to modified confined amplitudes
The mode relaxation time changes from Rouse type to modified confined
allows to determine
from small angle scattering data
We assume here that the additional potential is
with
as the polymer center of mass. As the Langevin equation only depends on relative distances the internal motions are not affected. The center of mass diffusion math:D_{cm} is not affected as the mode dependent friction coefficients dont change [9].
With
the original Rouse is recovered for amplitudes, relaxation and
.
A combination of different effects is possible [7] (but not implemented).
The amplitude
allows for additional suppression of specific modes e.g. by topological constraints (see [8]).
- From above the triple Dcm,l,NN are fixed.
If 2 are given 3rd is calculated
If all 3 are given the given values are used
For an example see example_Zimm. To speedup see example
memoize()
References
- 1
Doi Edwards Theory of Polymer dynamics in the appendix the equation is found
- 2
Nonflexible Coils in Solution: A Neutron Spin-Echo Investigation of Alkyl-Substituted Polynorbonenes in Tetrahydrofuran Michael Monkenbusch et al Macromolecules 2006, 39, 9473-9479 The exponential is missing a “t” http://dx.doi.org/10.1021/ma0618979
about internal friction
- 3
Exploring the role of internal friction in the dynamics of unfolded proteins using simple polymer models Cheng et al JOURNAL OF CHEMICAL PHYSICS 138, 074112 (2013) http://dx.doi.org/10.1063/1.4792206
- 4
Rouse Model with Internal Friction: A Coarse Grained Framework for Single Biopolymer Dynamics Khatri, McLeish| Macromolecules 2007, 40, 6770-6777 http://dx.doi.org/10.1021/ma071175x
- 5
Origin of internal viscosities in dilute polymer solutions P. G. de Gennes J. Chem. Phys. 66, 5825 (1977); https://doi.org/10.1063/1.433861
- 6
Microscopic theory of polymer internal viscosity: Mode coupling approximation for the Rouse model. Adelman, S. A., & Freed, K. F. (1977). The Journal of Chemical Physics, 67(4), 1380–1393. https://doi.org/10.1063/1.435011
- 7(1,2,3,4)
Internal friction in an intrinsically disordered protein - Comparing Rouse-like models with experiments A. Soranno, F. Zosel, H. Hofmann J. Chem. Phys. 148, 123326 (2018) http://aip.scitation.org/doi/10.1063/1.5009286
- 8
Onset of topological constraints in polymer melts: A mode analysis by neutron spin echo spectroscopy D. Richter, L. Willner, A. Zirkel, B. Farago, L. J. Fetters, and J. S. Huang Phys. Rev. Lett. 71, 4158 https://doi.org/10.1103/PhysRevLett.71.4158
- 9(1,2,3)
Looping dynamics of a flexible chain with internal friction at different degrees of compactness. Samanta, N., & Chakrabarti, R. (2015). Physica A: Statistical Mechanics and Its Applications, 436, 377–386. https://doi.org/10.1016/j.physa.2015.05.042
Examples
Coherent and incoherent contributions to Rouse dynamics. To mix the individual q dependent contributions have to be weighted with the according formfactor respectivly incoherent scattering length and instrument specific measurement technique.
import jscatter as js import numpy as np t = js.loglist(0.02, 100, 40) q=np.r_[0.1:2:0.2] l=0.38 # nm , bond length amino acids rr = js.dynamic.finiteRouse(x, qq, 124, 7, l=0.38, Dcm=0.37, tintern=0., Temp=273 + 60) p=js.grace() p.multi(2,1) p[0].xaxis(scale='log') p[0].yaxis(label='I(q,t)\scoherent') p[1].xaxis(label=r't / ns',scale='log') p[1].yaxis(label=r'I(q,t)\sincoherent') p[0].title('Rouse dynamics in a solvent') for i, z in enumerate(rr1, 1): p[0].plot(z.X, z.Y, line=[1, 1, i], symbol=0, legend='q=%g' % z.q) p[0].plot(z.X, z._Sqt_inf, line=[3, 2, i], symbol=0, legend='q=%g diff' % z.q) p[1].plot(z.X, z._Sqtinc, line=[1, 2, i], symbol=0, legend='q=%g diff' % z.q) #p.save(js.examples.imagepath+'/Rousecohinc.jpg')
-
jscatter.dynamic.
finiteZimm
(t, q, NN=None, pmax=None, l=None, Dcm=None, Dcmfkt=None, tintern=0.0, mu=0.5, viscosity=1.0, ftype=None, rk=None, Temp=293)[source]¶ Zimm dynamics of a finite chain with N beads with internal friction and hydrodynamic interactions.
The Zimm model describes the conformational dynamics of an ideal chain with hydrodynamic interaction between beads. The single chain diffusion is represented by Brownian motion of beads connected by harmonic springs. Coherent + incoherent scattering.
- Parameters
- tarray
Time in units nanoseconds.
- q: float, array
Scattering vector in units nm^-1. If q is list a dataList is returned otherwise a dataArray is returned.
- NNinteger
Number of chain beads.
- lfloat, default 1
Bond length between beads; units nm.
- pmaxinteger, list of float, default is NN
integer => maximum mode number taken into account.
list => list of amplitudes
for individual modes to allow weighing. Not given modes have weight zero.
- Dcmfloat
- Center of mass diffusion in nm^2/ns
for theta solvent with
for good solvent with
- Dcmfktarray 2xN, function
Function f(q) or array with [qi, f(qi)] as correction for Dcm like Diff = Dcm*f(q). e.g. for inclusion of structure factor and hydrodynamic function with f(q)=H(Q)/S(q). Missing values are interpolated. Only array input can be pickled to speedup by using formel.memoize .
- tinternfloat>0
Additional relaxation time due to internal friction between neighbouring beads in units ns.
- mufloat in range [0.1,0.9]
describes solvent quality.
<0.5 collapsed
=0.5 theta solvent 0.5 (gaussian chain)
=0.6 good solvent
>0.6 swollen chain
- viscosityfloat
in units cPoise=mPa*s e.g. water
- Tempfloat, default 273+20
Temperature in Kelvin.
- ftype‘czif’, default = ‘zif’
- Type of internal friction and interaction modification.
Default Zimm is used with
- ‘zif’ Internal friction between neighboring beads in chain [3].
- ‘czif’ Bead confining harmonic potential with internal friction, only for
[6] .
The beads are confined in an additional harmonic potential with
leading to a more compact configuration.
describes the relative strength compared to the force between beads
.
- ‘czif’ Bead confining harmonic potential with internal friction, only for
- rkNone , float
describes the relative force constant for ftype ‘czif’.
- Returns
- dataArrayfor single q
- dataListfor multiple q
[time; Sqt; Sqt_inf; Sqtinc]
time units ns
Sqt is coherent scattering with diffusion and mode contributions
Sqt_inf is coherent scattering with ONLY diffusion
Sqtinc is incoherent scattering with diffusion and mode contributions (no separate diffusion)
.q wavevector
.modecontribution :math:`a_p`of coherent modes i in sequence as in PRL 71, 4158 equ (3)
.Re
.tzimm => Zimm time or rotational correlation time
.t_p characteristic times
…. use .attr for all attributes
Notes
The Zimm model describes beads connected by harmonic springs with hydrodynamic interaction. The
parameter scales between theta solvent
and good solvent
(excluded volume or swollen chain). The coherent intermediate scattering function
is
and for incoherent intermediate scattering function the same with
in the first sum.
- with
mode amplitude (usual
)
mode relaxation time
Zimm mode relaxation time
end to end distance
force constant between beads
single bead friction in solvent with viscosity
additional amplitude for suppression of specific modes e.g. by topological constraints (see [5]).
Modifications (ftype) for internal friction and additional interaction:
- CZIFCompacted Zimm with internal friction [6].
Restricted to
, a combination with excluded volume is not valid. In [9]_ the beads are confined in an additional harmonic potential around the origin with
leading to a more compact configuration.
describes the relative strength compared to the force between beads
. Typically
.
The mode amplitude prefactor changes from Zimm type to modified confined amplitudes
The mode relaxation time changes from Zimm type to modified confined with
allows to determine
from small angle scattering data
For a free diffusing chain we assume here (not given in [9]_ ) that the additional potential is
with
as the polymer center of mass. As the Langevin equation only depends on position distances the internal motions are not affected. The center of mass diffusion
can be calculated similar to the Zimm
in [1] assuming a Gaussian configuration with width
. We find
With
the original Zimm is recovered for amplitudes, relaxation and
.
- From above the triple Dcm,l,N are fixed.
If 2 are given 3rd is calculated.
If all 3 are given the given values are used.
For an example see example_Zimm. To speedup see example
memoize()
References
- 1
Doi Edwards Theory of Polymer dynamics in appendix the equation is found
- 2
Nonflexible Coils in Solution: A Neutron Spin-Echo Investigation of Alkyl-Substituted Polynorbonenes in Tetrahydrofuran Michael Monkenbusch et al Macromolecules 2006, 39, 9473-9479 The exponential is missing a “t” http://dx.doi.org/10.1021/ma0618979
about internal friction
- 3(1,2)
Exploring the role of internal friction in the dynamics of unfolded proteins using simple polymer models Cheng et al JOURNAL OF CHEMICAL PHYSICS 138, 074112 (2013) http://dx.doi.org/10.1063/1.4792206
- 4
Rouse Model with Internal Friction: A Coarse Grained Framework for Single Biopolymer Dynamics Khatri, McLeish| Macromolecules 2007, 40, 6770-6777 http://dx.doi.org/10.1021/ma071175x
mode contribution factors from
- 5
Onset of Topological Constraints in Polymer Melts: A Mode Analysis by Neutron Spin Echo Spectroscopy D. Richter et al PRL 71,4158-4161 (1993)
- 6(1,2)
Looping dynamics of a flexible chain with internal friction at different degrees of compactness. Samanta, N., & Chakrabarti, R. (2015). Physica A: Statistical Mechanics and Its Applications, 436, 377–386. https://doi.org/10.1016/j.physa.2015.05.042
Examples
Coherent and incoherent contributions to Rouse dynamics. To mix the individual q dependent contributions have to be weighted with the according formfactor respectivly incoherent scattering length and instrument specific measurement technique.
import jscatter as js import numpy as np t = js.loglist(0.02, 100, 40) q=np.r_[0.1:2:0.2] l=0.38 # nm , bond length amino acids zz = js.dynamic.finiteZimm(x, qq, 124, 7, l=0.38, Dcm=0.37, tintern=0., Temp=273 + 60) p=js.grace() p.multi(2,1) p[0].xaxis(scale='log') p[0].yaxis(label='I(q,t)\scoherent') p[1].xaxis(label=r't / ns',scale='log') p[1].yaxis(label=r'I(q,t)\sincoherent') p[0].title('Zimm dynamics in a solvent') for i, z in enumerate(zz, 1): p[0].plot(z.X, z.Y, line=[1, 1, i], symbol=0, legend='q=%g' % z.q) p[0].plot(z.X, z._Sqt_inf, line=[3, 2, i], symbol=0, legend='q=%g diff' % z.q) p[1].plot(z.X, z._Sqtinc, line=[1, 2, i], symbol=0, legend='q=%g diff' % z.q) #p.save(js.examples.imagepath+'/Zimmcohinc.jpg')
-
jscatter.dynamic.
getHWHM
(data, center=0, gap=0)[source]¶ Find half width at half maximum of a distribution around zero.
The hwhm is determined from cubic spline between Y values to find Y.max/2. Requirement Y.max/2>Y.min and increasing X values. If nothing is found an empty list is returned
- Parameters
- datadataArray
Distribution
- center: float, default=0
Center (symmetry point) of data. If None the position of the maximum is used.
- gapfloat, default 0
Exclude values around center as it may contain a singularity. Excludes values within X<= abs(center-gap).
- Returns
- list of float with hwhm X>0 , X<0 if existing
-
jscatter.dynamic.
h
= 4.135667696923859¶ Planck constant in µeV*ns
-
jscatter.dynamic.
hbar
= 0.6582119569509066¶ h/2π reduced Planck constant in µeV*ns
-
jscatter.dynamic.
integralZimm
(t, q, Temp=293, viscosity=0.001, amp=1, rtol=0.02, tol=0.02, limit=50)[source]¶ Conformational dynamics of an ideal chain with hydrodynamic interaction, coherent scattering.
Integral version Zimm dynamics.
- Parameters
- tarray
Time points in ns
- qfloat
Wavevector in 1/nm
- Tempfloat
Temperature in K
- viscosityfloat
Viscosity in cP=mPa*s
- ampfloat
Amplitude
- rtol,tolfloat
Relative and absolute tolerance in scipy.integrate.quad
- limitint
Limit in scipy.integrate.quad.
- Returns
- dataArray
Notes
The Zimm model describes the conformational dynamics of an ideal chain with hydrodynamic interaction between beads. We use equ 85 and 86 from [1] as
with
See [1] for details.
References
- 1(1,2)
Neutron Spin Echo Investigations on the Segmental Dynamics of Polymers in Melts, Networks and Solutions in Neutron Spin Echo Spectroscopy Viscoelasticity Rheology Volume 134 of the series Advances in Polymer Science pp 1-129 DOI 10.1007/3-540-68449-2_1
Examples
import jscatter as js import numpy as np t=np.r_[0:10:0.2] p=js.grace() for q in np.r_[0.26,0.40,0.53,0.79,1.06]: iqt=js.dynamic.integralZimm(t=t,q=q,viscosity=0.2e-3) p.plot(iqt) #p.plot((iqt.X*iqt.q**3)**(2/3.),iqt.Y)
-
jscatter.dynamic.
jumpDiff_w
(w, q, t0, r0)[source]¶ Jump diffusion; dynamic structure factor in w domain.
Jump diffusion as a Markovian random walk. Jump length distribution is a Gaussian with width r0 and jump rate distribution with width G (Poisson). Diffusion coefficient D=r0**2/2t0.
- Parameters
- warray
Frequencies in 1/ns
- qfloat
Wavevector in nm**-1
- t0float
Mean residence time in a Poisson distribution of jump times. In units ns. G = 1/tg = Mean jump rate
- r0float
Root mean square jump length in 3 dimensions <r**2> = 3*r_0**2
- Returns
- dataArray
Notes
Equ 6 + 8 in [1] :
References
- 1
Incoherent neutron scattering functions for random jump diffusion in bounded and infinite media. Hall, P. L. & Ross, D. K. Mol. Phys. 42, 637–682 (1981).
-
jscatter.dynamic.
jumpDiffusion
(t, Q, t0, l0)[source]¶ Incoherent intermediate scattering function of translational jump diffusion in the time domain.
- Parameters
- tarray
Times, units ns
- Qfloat
Wavevector, units nm
- t0float
Residence time, units ns
- l0float
Mean square jump length, units nm
- Returns
- dataArray
Notes
We use equ. 3-5 from [1] for random jump diffusion as
with residence time
and mean jump length
and diffusion constant
in
References
- 1
Experimental determination of the nature of diffusive motions of water molecules at low temperatures J. Teixeira, M.-C. Bellissent-Funel, S. H. Chen, and A. J. Dianoux Phys. Rev. A 31, 1913 – Published 1 March 1985
-
jscatter.dynamic.
methylRotation
(t, q, t0=0.001, fraction=1, rhh=0.12, beta=0.8)[source]¶ Incoherent intermediate scattering function of CH3 methyl rotation in the time domain.
- Parameters
- tarray
List of times, units ns
- qfloat
Wavevector, units nm
- t0float, default 0.001
Residence time, units ns
- fractionfloat, default 1
Fraction of protons contributing.
- rhhfloat, default=0.12
Mean square jump length, units nm
- betafloat, default 0.8
exponent
- Returns
- dataArray
Notes
According to [1]:
with
residence time,
proton jump distance.
References
- 1
Bée, Quasielastic Neutron Scattering (Adam Hilger, 1988).
- 2
Monkenbusch et al. J. Chem. Phys. 143, 075101 (2015)
Examples
import jscatter as js import numpy as np # make a plot of the spectrum w=np.r_[-100:100] ql=np.r_[1:15:1] iqwCH3=js.dL([js.dynamic.time2frequencyFF(js.dynamic.methylRotation,'elastic',w=np.r_[-100:100:0.1],q=q ) for q in ql]) p=js.grace() p.plot(iqwCH3,le='CH3') p.yaxis(min=1e-5,max=10,scale='l')
-
jscatter.dynamic.
nSiteJumpDiffusion_w
(w, q, N, t0, r0)[source]¶ Random walk among N equidistant sites (isotropic averaged); dynamic structure factor in w domain.
E.g. for CH3 group rotational jump diffusion over 3 sites.
- Parameters
- warray
Frequencies in 1/ns
- q: float
Wavevector in units 1/nm
- Nint
Number of jump sites, jump angle 2pi/N
- r0float
Distance of sites from center of rotation. For CH3 eg 0.12 nm.
- t0float
Rotational correlation time.
- Returns
- dataArray
Notes
Equ. 24 [1] :
with
,
References
- 1
Incoherent scattering law for neutron quasi-elastic scattering in liquid crystals. Dianoux, A., Volino, F. & Hervet, H., Mol. Phys. 30, 37–41 (1975). https://doi.org/10.1080/00268977500102721
Examples
import jscatter as js import numpy as np w=np.r_[-100:100:0.1] ql=np.r_[1:14.1:1.3] p=js.grace() iqw=js.dL([js.dynamic.nSiteJumpDiffusion_w(w=w,q=q,N=3,t0=0.01,r0=0.12) for q in ql]) p.plot(iqw) p.yaxis(scale='l')
-
jscatter.dynamic.
resolution
(t, s0=1, m0=0, s1=None, m1=None, s2=None, m2=None, s3=None, m3=None, s4=None, m4=None, s5=None, m5=None, a0=1, a1=1, a2=1, a3=1, a4=1, a5=1, bgr=0, resolution=None)[source]¶ Resolution in time domain as multiple Gaussians for inelastic measurement as back scattering or time of flight instrument.
Multiple Gaussians define the function to describe a resolution measurement. Use resolution_w to fit with the appropriate normalized Gaussians. See Notes
- Parameters
- tarray
Times
- s0,s1,…float
Width of Gaussian functions representing a resolution measurement. The number of si not None determines the number of Gaussians.
- m0, m1,….float, None
Means of the Gaussian functions representing a resolution measurement.
- a0, a1,….float, None
Amplitudes of the Gaussian functions representing a resolution measurement.
- bgrfloat, default=0
Background
- resolutiondataArray
- Resolution with attributes sigmas, amps which are used instead of si, ai.
If from w domain this represents the Fourier transform from w to t domain. The means are NOT used from as these result only in a phase shift, instead m0..m5 are used.
If from t domain the resolution is recalculated.
- Returns
- dataArray
Notes
In a typical inelastic experiment the resolution is measured by e.g. a vanadium measurement (elastic scatterer). This is described in w domain by a multi Gaussian function as in resw=resolution_w(w,…) with amplitudes ai_w, width si_w and common mean m_w. resolution(t,resolution_w=resw) defines the Fourier transform of resolution_w using the same coefficients. mi_t are set by default to zero as mi_w lead only to a phase shift. It is easiest to shift w values in w domain as it corresponds to a shift of the elastic line.
The used Gaussians are normalized that they are a pair of Fourier transforms:
under the Fourier transform defined as
Examples
Using the result of a fit in w domain to represent the resolution in time domain :
import jscatter as js # resw is a resolution in w domain maybe as a result from a fit to vanadium data # resw contains all parameters w=np.r_[-100:100:0.5] resw=js.dynamic.resolution_w(w, s0=12, m0=0, a0=2) # representing the Fourier transform of resw as a gaussian transforms to time domain t=np.r_[0:1:0.01] rest=js.dynamic.resolution(t,resolution=resw) t2=np.r_[0:0.5:0.005] rest2=js.dynamic.resolution(t2,resolution=rest)
-
jscatter.dynamic.
resolution_w
(w, s0=1, m0=0, s1=None, m1=None, s2=None, m2=None, s3=None, m3=None, s4=None, m4=None, s5=None, m5=None, a0=1, a1=1, a2=1, a3=1, a4=1, a5=1, bgr=0, resolution=None)[source]¶ Resolution as multiple Gaussians for inelastic measurement as backscattering or time of flight instrument in w domain.
Multiple Gaussians define the function to describe a resolution measurement. Use only a common mi to account for a shift. See resolution for transform to time domain.
- Parameters
- warray
Frequencies
- s0,s1,…float
Sigmas of several Gaussian functions representing a resolution measurement. The number of si not none determines the number of Gaussians.
- m0, m1,….float, None
Means of the Gaussian functions representing a resolution measurement.
- a0, a1,….float, None
Amplitudes of the Gaussian functions representing a resolution measurement.
- bgrfloat, default=0
Background
- resolutiondataArray
- Resolution with attributes sigmas, amps which are used instead of si, ai.
If from t domain this represents the Fourier transform from w to t domain. The means are NOT used from as these result only in a phase shift, instead m0..m5 are used.
If from w domain the resolution is recalculated.
- Returns
- dataArray
.means .amps .sigmas
Notes
In a typical inelastic experiment the resolution is measured by e.g. a vanadium measurement (elastic scatterer). This is described in w domain by a multi Gaussian function as in resw=resolution_w(w,…) with amplitudes ai_w, width si_w and common mean m_w. resolution(t,resolution_w=resw) defines the Fourier transform of resolution_w using the same coefficients. mi_t are set by default to 0 as mi_w lead only to a phase shift. It is easiest to shift w values in w domain as it corresponds to a shift of the elastic line.
The used Gaussians are normalized that they are a pair of Fourier transforms:
under the Fourier transform defined as
Examples
Transform from and to time domain
import jscatter as js # resw is a resolution in w domain maybe as a result from a fit to vanadium data # resw contains all parameters w=np.r_[-100:100:0.5] resw=js.dynamic.resolution_w(w, s0=12, m0=0, a0=2) w2=np.r_[0:50:0.2] rest2=js.dynamic.resolution_w(w2,resolution=resw) # representing the Fourier transform of to time domain t=np.r_[0:1:0.01] rest=js.dynamic.resolution(t,resolution=resw)
Sequential fit in w domain to a measurement with realistic data. The data file is from the SPHERE instrument at MLZ Garching (usually not gziped). The file needs to be split to be easily read.
import jscatter as js import numpy as np import gzip with gzip.open(js.examples.datapath +'/Vana.inx.gz','rt') as f: lines = f.readlines() vana = js.dL() for j in np.r_[0:int(len(lines)//(563))]: vana.append(js.dA(lines[j*563:(j+1)*563],lines2parameter=[0,2,3],usecols=[1,2,3])) vana[-1].q=float(vana[-1].line_2[0]) # extract q values start={'s0':0.5,'m0':0,'a0':1,'s1':1,'m1':0,'a1':1,'s2':10,'m2':0,'a2':1,'bgr':0.0073} dm=5 for van in vana: van.setlimit(m0=[-dm,dm],m1=[-dm,dm],m2=[-dm,dm],m3=[-dm,dm],m4=[-dm,dm],m5=[-dm,dm]) van.fit(js.dynamic.resolution_w,start,{},{'w':'X'}) van.showlastErrPlot(yscale='l', fitlinecolor=11) # vana[7].savelastErrPlot(js.examples.imagepath+'/resolutionfit.jpg')
-
jscatter.dynamic.
rotDiffusion_w
(w, q, cloud, Dr, lmax='auto')[source]¶ Rotational diffusion of an object (dummy atoms); dynamic structure factor in w domain.
A cloud of dummy atoms can be used for coarse graining of a non-spherical object e.g. for amino acids in proteins. On the other hand its just a way to integrate over an object e.g. a sphere or ellipsoid. We use [2] for an objekt of arbitrary shape modified for incoherent scattering.
- Parameters
- warray
Frequencies in 1/ns
- qfloat
Wavevector in units 1/nm
- cloudarray Nx3, Nx4 or Nx5 or float
A cloud of N dummy atoms with positions cloud[:3] that describe an object.
If given, cloud[3] is the incoherent scattering length
otherwise its equal 1.
If given, cloud[4] is the coherent scattering length otherwise its equal 1.
If cloud is single float the value is used as radius of a sphere with 10x10x10 grid.
- Drfloat
Rotational diffusion constant in units 1/ns.
- lmaxint
Maximum order of spherical bessel function. ‘auto’ -> lmax > 2pi*r.max()*q/6.
- Returns
- dataArray
Columns [w; Iqwinc; Iqwcoh] Input parameters as attributes.
Notes
See
transRotDiffusion()
for more details. The Fourier transform of the exp function is a Lorentzian so the exp should be exchange.References
- 1
Incoherent scattering law for neutron quasi-elastic scattering in liquid crystals. Dianoux, A., Volino, F. & Hervet, H. Mol. Phys. 30, 37–41 (1975).
- 2
Effect of rotational diffusion on quasielastic light scattering from fractal colloid aggregates. Lindsay, H., Klein, R., Weitz, D., Lin, M. & Meakin, P. Phys. Rev. A 38, 2614–2626 (1988).
Examples
import jscatter as js import numpy as np R=2;NN=10 Drot=js.formel.Drot(R) ql=np.r_[0.5:15.:2] w=np.r_[-100:100:0.1] grid=js.ff.superball(ql,R,p=1,nGrid=NN,returngrid=True) p=js.grace() iqwR1=js.dL([js.dynamic.rotDiffusion_w(w,q,grid.XYZ,Drot) for q in ql]) p.plot(iqwR1,le=f'NN={NN:.0f} q=$wavevector nm\S-1') p.yaxis(scale='l') p.legend()
-
jscatter.dynamic.
shiftAndBinning
(data, w=None, dw=None, w0=0)[source]¶ Shift spectrum and average (binning) in intervals.
The intention is to shift spectra and average over intervals. It should be used after convolution with the instrument resolution, when singular values at zero are smeared by resolution.
- Parameters
- datadataArray
Data (from model) to be shifted and averaged in intervals to meet experimental data.
- warray
New X values (e.g. from experiment). If w is None data.X values are used.
- w0float
Shift by w0 that wnew=wold+w0
- dwfloat, default
Average over intervals between [w[i]-dw,w[i]+dw] to average over a detector pixel width. If None dw is half the interval to neighbouring points. If 0 the value is only linear interpolated to w values and not averaged (about 10 times faster).
- Returns
- dataArray
Notes
For averaging over intervals scipy.interpolate.CubicSpline is used with integration in the intervals.
Examples
import jscatter as js import numpy as np w=np.r_[-100:100:0.5] start={'s0':6,'m0':0,'a0':1,'s1':None,'m1':0,'a1':1,'bgr':0.00} resolution=js.dynamic.resolution_w(w,**start) p=js.grace() p.plot(resolution) p.plot(js.dynamic.shiftAndBinning(resolution,w0=5,dw=0))
-
jscatter.dynamic.
simpleDiffusion
(q, t, D, w=0, amplitude=1)[source]¶ Intermediate scattering function for diffusing particles.
- Parameters
- qfloat, array
Wavevector
- tfloat, array
Times
- amplitudefloat
Prefactor
- Dfloat
Diffusion coefficient in units [ [q]**-2/[t] ]
- wfloat
Width of diffusion coefficient distribution in D units.
- Returns
- dataArray
-
jscatter.dynamic.
stretchedExp
(t, gamma, beta, amp=1)[source]¶ Stretched exponential function.
- Parameters
- tarray
Times
- gammafloat
Relaxation rate in units 1/[unit t]
- betafloat
Stretched exponent
- ampfloat default 1
Amplitude
- Returns
- dataArray
-
jscatter.dynamic.
t2fFF
(timemodel, resolution, w=None, tfactor=7, **kwargs)¶ Fast Fourier transform from time domain to frequency domain for inelastic neutron scattering.
Shortcut t2fFF calls this function.
- Parameters
- timemodelfunction, None
Model for I(t,q) in time domain. t in units of ns. The values for t are determined from w as
with
and
.
is the minimal width of the Gaussians given in resolution. If None a constant function (elastic scattering) is used.
- resolutiondataArray, float, string
dataArray that describes the resolution function as multiple Gaussians (use resolution_w). A nonzero bgr in resolution is ignored and needs to be added afterwards.
- floatvalue is used as width of a single Gaussian in units 1/ns (w is needed below).
Resolution width is in the range of 6 1/ns (IN5 TOF) to 1 1/ns (Spheres BS).
string : no resolution (‘elastic’)
- warray
Frequencies for the result, e.g. from experimental data. If w is None the frequencies resolution.X are used. This allows to use the fit of a resolution to be used with same w values.
- kwargskeyword args
Additional keyword arguments that are passed to timemodel.
- tfactorfloat, default 7
Factor to determine max time for timemodel to minimize spectral leakage. tmax=1/(min(resolution_width)*tfactor) determines the resolution to decay as
. The time step is dt=1/max(|w|). A minimum of len(w) steps is used (which might increase tmax). Increase tfactor if artifacts (wobbling) from the limited time window are visible as the limited time interval acts like a window function (box) for the Fourier transform.
- Returns
- dataArrayA symmetric spectrum of the Fourier transform is returned.
.Sq
Integration is done by a cubic spline in w domain on the ‘raw’ fourier transform of timemodel.
- .Iqt timemodel(t,kwargs) dataArray as returned from timemodel.
Implicitly this is the Fourier transform to time domain after a successful fit in w domain. Using a heuristic model in time domain as multiple Gaussians or stretched exponential allows a convenient transform to time domain of experimental data.
Notes
We use Fourier transform with real signals. The transform is defined as
The resolution function is defined as (see resolution_w)
using the resolution in time domain with same coefficients
The Fourier transform of a timemodel I(q,t) is
The integral is calculated by Fast Fourier transform as
. Due to the cutoff at
a wobbling might appear indicating spectral leakage. Spectral leakage results from the cutoff, which can be described as multiplication with a box function. The corresponding Fourier Transform of the box is a sinc function visible in the frequency spectrum as wobbling. If the resolution is included in time domain, it acts like a window function to reduce spectral leakage with vanishing values at
. The second possibility (default) is to increase
(increase tfactor) to make the sinc sharp and with low wobbling amplitude.
Mixed domain models
Associativity and Convolution theorem allow to mix models from frequency domain and time domain. After transformation to frequency domain the w domain models have to be convoluted with the FFT transformed model.
Examples
Other usage example with a comparison of w domain and transformed from time domain can be found in A comparison of different dynamic models in frequency domain or in the example of
diffusionHarmonicPotential_w()
.Compare transDiffusion transform from time domain with direct convolution in w domain.
import jscatter as js import numpy as np w=np.r_[-100:100:0.5] start={'s0':6,'m0':0,'a0':1,'s1':None,'m1':0,'a1':1,'bgr':0.00} resolution=js.dynamic.resolution_w(w,**start) p=js.grace() D=0.035;qq=3 # diffusion coefficient of protein alcohol dehydrogenase (140 kDa) is 0.035 nm**2/ns p.title('Inelastic spectrum IN5 like') p.subtitle(r'resolution width about 6 ns\S-1\N, Q=%.2g nm\S-1\N' %(qq)) # compare diffusion with convolution and transform from time domain diff_ffw=js.dynamic.time2frequencyFF(js.dynamic.simpleDiffusion,resolution,q=qq,D=D) diff_w=js.dynamic.transDiff_w(w, q=qq, D=D) p.plot(diff_w,sy=0,li=[1,3,3],le=r'original diffusion D=%.3g nm\S2\N/ns' %(D)) p.plot(diff_ffw,sy=[2,0.3,2],le='transform from time domain') p.plot(diff_ffw.X,diff_ffw.Y+diff_ffw.Y.max()*1e-3,sy=[2,0.3,7],le=r'transform from time domain with 10\S-3\N bgr') # resolution has to be normalized in convolve diff_cw=js.dynamic.convolve(diff_w,resolution,normB=1) p.plot(diff_cw,sy=0,li=[1,3,4],le='after convolution in w domain') p.plot(resolution.X,resolution.Y/resolution.integral,sy=0,li=[1,1,1],le='resolution') p.yaxis(min=1e-6,max=5,scale='l',label='S(Q,w)') p.xaxis(min=-100,max=100,label='w / ns\S-1') p.legend() p.text(string=r'convolution edge ==>\nmake broader and cut',x=10,y=8e-6)
Compare the resolutions direct and from transform from time domain.
p=js.grace() fwres=js.dynamic.time2frequencyFF(None,resolution) p.plot(fwres,le='fft only resolution') p.plot(resolution,sy=0,li=2,le='original resolution')
Compare diffusionHarmonicPotential to show simple usage
import jscatter as js import numpy as np t2f=js.dynamic.time2frequencyFF dHP=js.dynamic.diffusionHarmonicPotential w=np.r_[-100:100] ql=np.r_[1:14.1:6j] iqw=js.dL([js.dynamic.diffusionHarmonicPotential_w(w=w,q=q,tau=0.14,rmsd=0.34,ndim=3) for q in ql]) # To move spectral leakage out of our window we increase w and interpolate. # The needed factor (here 23) depends on the quality of your data and background contribution. iqt=js.dL([t2f(dHP,'elastic',w=w*13,q=q, rmsd=0.34, tau=0.14 ,ndim=3,tfactor=14).interpolate(w) for q in ql]) p=js.grace() p.multi(2,3) p[1].title('Comparison direct and FFT for ndim= 3') for i,(iw,it) in enumerate(zip(iqw,iqt)): p[i].plot(iw,li=1,sy=0,le='q=$wavevector nm\S-1') p[i].plot(it,li=2,sy=0) p[i].yaxis(min=1e-5,max=2,scale='log') if i in [1,2,4,5]:p[i].yaxis(ticklabel=0) p[i].legend(x=5,y=1, charsize=0.7)
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jscatter.dynamic.
time2frequencyFF
(timemodel, resolution, w=None, tfactor=7, **kwargs)[source]¶ Fast Fourier transform from time domain to frequency domain for inelastic neutron scattering.
Shortcut t2fFF calls this function.
- Parameters
- timemodelfunction, None
Model for I(t,q) in time domain. t in units of ns. The values for t are determined from w as
with
and
.
is the minimal width of the Gaussians given in resolution. If None a constant function (elastic scattering) is used.
- resolutiondataArray, float, string
dataArray that describes the resolution function as multiple Gaussians (use resolution_w). A nonzero bgr in resolution is ignored and needs to be added afterwards.
- floatvalue is used as width of a single Gaussian in units 1/ns (w is needed below).
Resolution width is in the range of 6 1/ns (IN5 TOF) to 1 1/ns (Spheres BS).
string : no resolution (‘elastic’)
- warray
Frequencies for the result, e.g. from experimental data. If w is None the frequencies resolution.X are used. This allows to use the fit of a resolution to be used with same w values.
- kwargskeyword args
Additional keyword arguments that are passed to timemodel.
- tfactorfloat, default 7
Factor to determine max time for timemodel to minimize spectral leakage. tmax=1/(min(resolution_width)*tfactor) determines the resolution to decay as
. The time step is dt=1/max(|w|). A minimum of len(w) steps is used (which might increase tmax). Increase tfactor if artifacts (wobbling) from the limited time window are visible as the limited time interval acts like a window function (box) for the Fourier transform.
- Returns
- dataArrayA symmetric spectrum of the Fourier transform is returned.
.Sq
Integration is done by a cubic spline in w domain on the ‘raw’ fourier transform of timemodel.
- .Iqt timemodel(t,kwargs) dataArray as returned from timemodel.
Implicitly this is the Fourier transform to time domain after a successful fit in w domain. Using a heuristic model in time domain as multiple Gaussians or stretched exponential allows a convenient transform to time domain of experimental data.
Notes
We use Fourier transform with real signals. The transform is defined as
The resolution function is defined as (see resolution_w)
using the resolution in time domain with same coefficients
The Fourier transform of a timemodel I(q,t) is
The integral is calculated by Fast Fourier transform as
. Due to the cutoff at
a wobbling might appear indicating spectral leakage. Spectral leakage results from the cutoff, which can be described as multiplication with a box function. The corresponding Fourier Transform of the box is a sinc function visible in the frequency spectrum as wobbling. If the resolution is included in time domain, it acts like a window function to reduce spectral leakage with vanishing values at
. The second possibility (default) is to increase
(increase tfactor) to make the sinc sharp and with low wobbling amplitude.
Mixed domain models
Associativity and Convolution theorem allow to mix models from frequency domain and time domain. After transformation to frequency domain the w domain models have to be convoluted with the FFT transformed model.
Examples
Other usage example with a comparison of w domain and transformed from time domain can be found in A comparison of different dynamic models in frequency domain or in the example of
diffusionHarmonicPotential_w()
.Compare transDiffusion transform from time domain with direct convolution in w domain.
import jscatter as js import numpy as np w=np.r_[-100:100:0.5] start={'s0':6,'m0':0,'a0':1,'s1':None,'m1':0,'a1':1,'bgr':0.00} resolution=js.dynamic.resolution_w(w,**start) p=js.grace() D=0.035;qq=3 # diffusion coefficient of protein alcohol dehydrogenase (140 kDa) is 0.035 nm**2/ns p.title('Inelastic spectrum IN5 like') p.subtitle(r'resolution width about 6 ns\S-1\N, Q=%.2g nm\S-1\N' %(qq)) # compare diffusion with convolution and transform from time domain diff_ffw=js.dynamic.time2frequencyFF(js.dynamic.simpleDiffusion,resolution,q=qq,D=D) diff_w=js.dynamic.transDiff_w(w, q=qq, D=D) p.plot(diff_w,sy=0,li=[1,3,3],le=r'original diffusion D=%.3g nm\S2\N/ns' %(D)) p.plot(diff_ffw,sy=[2,0.3,2],le='transform from time domain') p.plot(diff_ffw.X,diff_ffw.Y+diff_ffw.Y.max()*1e-3,sy=[2,0.3,7],le=r'transform from time domain with 10\S-3\N bgr') # resolution has to be normalized in convolve diff_cw=js.dynamic.convolve(diff_w,resolution,normB=1) p.plot(diff_cw,sy=0,li=[1,3,4],le='after convolution in w domain') p.plot(resolution.X,resolution.Y/resolution.integral,sy=0,li=[1,1,1],le='resolution') p.yaxis(min=1e-6,max=5,scale='l',label='S(Q,w)') p.xaxis(min=-100,max=100,label='w / ns\S-1') p.legend() p.text(string=r'convolution edge ==>\nmake broader and cut',x=10,y=8e-6)
Compare the resolutions direct and from transform from time domain.
p=js.grace() fwres=js.dynamic.time2frequencyFF(None,resolution) p.plot(fwres,le='fft only resolution') p.plot(resolution,sy=0,li=2,le='original resolution')
Compare diffusionHarmonicPotential to show simple usage
import jscatter as js import numpy as np t2f=js.dynamic.time2frequencyFF dHP=js.dynamic.diffusionHarmonicPotential w=np.r_[-100:100] ql=np.r_[1:14.1:6j] iqw=js.dL([js.dynamic.diffusionHarmonicPotential_w(w=w,q=q,tau=0.14,rmsd=0.34,ndim=3) for q in ql]) # To move spectral leakage out of our window we increase w and interpolate. # The needed factor (here 23) depends on the quality of your data and background contribution. iqt=js.dL([t2f(dHP,'elastic',w=w*13,q=q, rmsd=0.34, tau=0.14 ,ndim=3,tfactor=14).interpolate(w) for q in ql]) p=js.grace() p.multi(2,3) p[1].title('Comparison direct and FFT for ndim= 3') for i,(iw,it) in enumerate(zip(iqw,iqt)): p[i].plot(iw,li=1,sy=0,le='q=$wavevector nm\S-1') p[i].plot(it,li=2,sy=0) p[i].yaxis(min=1e-5,max=2,scale='log') if i in [1,2,4,5]:p[i].yaxis(ticklabel=0) p[i].legend(x=5,y=1, charsize=0.7)
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jscatter.dynamic.
transDiff_w
(w, q, D)[source]¶ Translational diffusion; dynamic structure factor in w domain.
- Parameters
- warray
Frequencies in 1/ns
- qfloat
Wavevector in nm**-1
- Dfloat
Diffusion constant in nm**2/ns
- Returns
- dataArray
Notes
Equ 33 in [1]_
References
- 0
Scattering of Slow Neutrons by a Liquid Vineyard G Physical Review 1958 vol: 110 (5) pp: 999-1010
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jscatter.dynamic.
transRotDiffusion
(t, q, cloud, Dr, Dt=0, lmax='auto')[source]¶ Translational + rotational diffusion of an object (dummy atoms); dynamic structure factor in time domain.
A cloud of dummy atoms can be used for coarse graining of a non-spherical object e.g. for amino acids in proteins. On the other hand its just a way to integrate over an object e.g. a sphere or ellipsoid (see example). We use [2] for an objekt of arbitrary shape modified for incoherent scattering.
- Parameters
- tarray
Times in ns.
- qfloat
Wavevector in units 1/nm
- cloudarray Nx3, Nx4 or Nx5 or float
A cloud of N dummy atoms with positions cloud[:3] in units nm that describe an object .
If given, cloud[3] is the incoherent scattering length
otherwise its equal 1.
If given, cloud[4] is the coherent scattering length
otherwise its equal 1.
If cloud is single float the value is used as radius of a sphere with 10x10x10 grid points.
- Drfloat
Rotational diffusion constant (scalar) in units 1/ns.
- Dtfloat, default=0
Translational diffusion constant (scalar) in units nm²/ns.
- lmaxint
Maximum order of spherical bessel function. ‘auto’ -> lmax > 2π r.max()*q/6.
- Returns
- dataArray :
- Columns [t; Iqtinc; Iqtcoh; Iqttrans]
.radiusOfGyration
.Iq_coh coherent scattering (formfactor)
.Iq_inc incoherent scattering
.wavevector
.rotDiffusion
.transDiffusion
.lmax
Notes
We calculate the field autocorrelation function given in equ 24 in [2] for an arbitrary rigid object without additional internal dynamic as
where
is the rotational diffusion contribution and
and coh/inc scattering length
, position vector
and orientation of atoms
, spherical Bessel function
, spherical harmonics
.
The incoherent intermediate scattering function is res.Y/res.Iq_inc or res._Iqtinc/res.Iq_inc
The coherent intermediate scattering function is res._Iqtcoh/res.Iq_coh
For real scattering data as backscattering or spinecho coherent and incoherent have to be mixed according to the polarisation conditions of the experiment accounting also for spin flip probability of coherent and incoherent scattering. For the simple case of non-polarised measurement we get
References
- 1
Incoherent scattering law for neutron quasi-elastic scattering in liquid crystals. Dianoux, A., Volino, F. & Hervet, H. Mol. Phys. 30, 37–41 (1975).
- 2(1,2)
Effect of rotational diffusion on quasielastic light scattering from fractal colloid aggregates. Lindsay, H., Klein, R., Weitz, D., Lin, M. & Meakin, P. Phys. Rev. A 38, 2614–2626 (1988).
Examples
A bit artificial look at only rotational diffusion of a superball build from dummy atoms. (rotational diffusion should only show if also translational diffusion is seen) Change p to change from spherical shape (p=1) to cube (p>10) or star like (p<0.5) (use grid.show() to take a look at the shape) The coherent contribution is suppressed for low q if the particle is spherical .
import jscatter as js import numpy as np R=2;NN=10 ql=np.r_[0.4:2.:0.3,2.1:15:2] t=js.loglist(0.001,50,50) # get superball p2=1 grid=js.ff.superball(ql,R,p=p2,nGrid=NN,returngrid=True) Drot=js.formel.Drot(R) Dtrans=js.formel.Dtrans(R) p=js.grace(1.5,1) p.new_graph(xmin=0.23,xmax=0.43,ymin=0.25,ymax=0.55) iqt=js.dL([js.dynamic.transRotDiffusion(t,q,grid.XYZ,Drot,lmax=30) for q in ql]) for i,iiqt in enumerate(iqt,1): p[0].plot(iiqt.X,iiqt.Y/iiqt.Iq_inc,li=[1,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1') p[0].plot(iiqt.X,iiqt._Iqtcoh/iiqt.Iq_coh,li=[3,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1') p[1].plot(iqt.wavevector,iqt.Iq_coh.array/grid.numberOfAtoms(),li=1) p[1].plot(iqt.wavevector,iqt.Iq_inc.array/grid.numberOfAtoms(),li=1) p[0].xaxis(scale='l',label='t / ns',max=200,min=0.001) p[0].yaxis(scale='n',label='I(q,t)/I(q,0)') p[1].xaxis(scale='n',label='q / nm\S-1') p[1].yaxis(scale='l',label='I(q,t=0)') p[0].legend(x=60,y=1.1,charsize=0.7) p[0].title(f'rotational diffusion of superball with p={p2:.2f}') p[0].subtitle(f'coh relevant only at high q for sphere') p[1].subtitle('coh + inc scattering') p[0].text(x=0.0015,y=0.8,string=r'lines inc\ndashed coh',charsize=1.5) #p.save(js.examples.imagepath+'/rotDiffusion.jpg') # Second example # non-polarized experiment p=js.grace(1.5,1) grid=js.ff.superball(ql,R,p=1.,nGrid=10,returngrid=True) iqt=js.dL([js.dynamic.transRotDiffusion(t,q,grid.XYZ,Drot,Dtrans,lmax=30) for q in ql]) for i,iiqt in enumerate(iqt,1): p.plot(iiqt.X,(iiqt._Iqtinc+iiqt._Iqtcoh)/(iiqt.Iq_inc+iiqt.Iq_coh),li=[1,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1') p.plot(iiqt.X,iiqt._Iqtcoh/iiqt.Iq_coh,li=[3,3,i],sy=0,le=f'q={iiqt.wavevector:.1f} nm\S-1') p.xaxis(scale='l',label='t / ns',max=200,min=0.001) p.yaxis(scale='n',label='I(q,t)/I(q,0)') p[0].legend(x=60,y=1.1,charsize=0.7) p[0].title(f'translational/rotational diffusion of superball with p={p2:.2f}') p[0].text(x=0.0015,y=0.5,string=r'lines coh+inc\ndashed only coh',charsize=1.5) #p.save(js.examples.imagepath+'/transrotDiffusion.jpg')
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jscatter.dynamic.
zilmanGranekBicontinious
(t, q, xi, kappa, eta, mt=1, amp=1, eps=1, nGauss=60)[source]¶ Dynamics of bicontinuous micro emulsion phases. Zilman-Granek model as equ B10 in [1]. Coherent scattering.
On very local scales (however larger than the molecular size) Zilman and Granek represent the amphiphile layer in the bicontinuous network as consisting of an ensemble of independent patches at random orientation of size equal to the correlation length xi. Uses Gauss integration and multiprocessing.
- Parameters
- tarray
Time values in ns
- qfloat
Scattering vector in 1/A
- xifloat
Correlation length related to the size of patches which are locally planar and determine the width of the peak in static data. unit A A result of the teubnerStrey model to e.g. SANS data. Determines kmin=eps*pi/xi .
- kappafloat
Apparent single membrane bending modulus, unit kT
- etafloat
Solvent viscosity, unit kT*A^3/ns=100/(1.38065*T)*eta[unit Pa*s] Water about 0.001 Pa*s = 0.000243 kT*A^3/ns
- ampfloat, default = 1
Amplitude scaling factor
- epsfloat, default=1
Scaling factor in range [1..1.3] for kmin=eps*pi/xi and rmax=xi/eps. See [1].
- mtfloat, default 0.1
Membrane thickness in unit A as approximated from molecular size of material. Determines kmax=pi/mt. About 12 Angstrom for tenside C10E4.
- nGaussint, default 60
Number of points in Gauss integration
- Returns
- dataList
Notes
See equ B10 in [1] :
with
,
as Bessel function of order 0
For technical reasons, in order to avoid numerical difficulties, the real space upper (rmax integration) cutoff was realized by multiplying the integrand with a Gaussian having a width of eps*xi and integrating over [0,3*eps*xi].
References
- 1(1,2,3)
Dynamics of bicontinuous microemulsion phases with and without amphiphilic block-copolymers M. Mihailescu, M. Monkenbusch et al J. Chem. Phys. 115, 9563 (2001); http://dx.doi.org/10.1063/1.1413509
Examples
import jscatter as js import numpy as np t=js.loglist(0.1,30,20) p=js.grace() iqt=js.dynamic.zilmanGranekBicontinious(t=t,q=np.r_[0.03:0.2:0.04],xi=110,kappa=1.,eta=0.24e-3,nGauss=60) p.plot(iqt) # to use the multiprocessing in a fit of data use memoize data=iqt # this represent your measured data tt=list(set(data.X.flatten)) # a list of all time values tt.sort() # use correct values from data for q -> interpolation is exact for q and tt zGBmem=js.formel.memoize(q=data.q,t=tt)(js.dynamic.zilmanGranekBicontinious) def mfitfunc(t, q, xi, kappa, eta, amp): # this will calculate in each fit step for for Q (but calc all) and then take from memoized values res= zGBmem(t=t, q=q, xi=xi, kappa=kappa, eta=eta, amp=amp) return res.interpolate(q=q,X=t)[0] # use mfitfunc for fitting with multiprocessing
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jscatter.dynamic.
zilmanGranekLamellar
(t, q, df, kappa, eta, mu=0.001, eps=1, amp=1, mt=0.1, nGauss=40)[source]¶ Dynamics of lamellar microemulsion phases. Zilman-Granek model as Equ 16 in [1]. Coherent scattering.
Oriented lamellar phases at the length scale of the inter membrane distance and beyond are performed using small-angle neutrons scattering and neutron spin-echo spectroscopy.
- Parameters
- tarray
Time in ns
- qfloat
Scattering vector
- dffloat
film-film distance. unit A
This represents half the periodicity of the structure, generally denoted by d=0.5df which determines the peak position and determines kmin=eps*pi/df
- kappafloat
Apparent single membrane bending modulus, unit kT
- mufloat, default 0.001
Angle between q and surface normal in unit rad. For lamellar oriented system this is close to zero in NSE.
- etafloat
Solvent viscosity, unit kT*A^3/ns = 100/(1.38065*T)*eta[unit Pa*s] Water about 0.001 Pa*s = 0.000243 kT*A^3/ns
- epsfloat, default=1
Scaling factor in range [1..1.3] for kmin=eps*pi/xi and rmax=xi/eps
- ampfloat, default 1
Amplitude scaling factor
- mtfloat, default 0.1
Membrane thickness in unit A as approximated from molecular size of material. Determines kmax=pi/mt About 12 Angstrom for tenside C10E4.
- nGaussint, default 40
Number of points in Gauss integration
- Returns
- dataList
Notes
See equ 16 in [1] :
- with
,
,
as Bessel function of order 0. For details see [1].
The integrations are done by nGauss point Gauss quadrature, except for the kmax-kmin integration which is done by adaptive Gauss integration with rtol=0.1/nGauss k< kmax/8 and rtol=1./nGauss k> kmax/8.
References
- 1(1,2,3)
Neutron scattering study on the structure and dynamics of oriented lamellar phase microemulsions M. Mihailescu, M. Monkenbusch, J. Allgaier, H. Frielinghaus, D. Richter, B. Jakobs, and T. Sottmann Phys. Rev. E 66, 041504 (2002)
Examples
import jscatter as js import numpy as np t=js.loglist(0.1,30,20) ql=np.r_[0.08:0.261:0.03] p=js.grace() iqt=js.dynamic.zilmanGranekLamellar(t=t,q=ql,df=100,kappa=1,eta=2*0.24e-3) p.plot(iqt)