Source code for jscatter.dynamic

# -*- coding: utf-8 -*-
# written by Ralf Biehl at the Forschungszentrum Jülich ,
# Jülich Center for Neutron Science 1 and Institute of Complex Systems 1
#    Jscatter is a program to read, analyse and plot data
#    Copyright (C) 2015-2019  Ralf Biehl
#
#    This program is free software: you can redistribute it and/or modify
#    it under the terms of the GNU General Public License as published by
#    the Free Software Foundation, either version 3 of the License, or
#    (at your option) any later version.
#
#    This program is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#    GNU General Public License for more details.
#
#    You should have received a copy of the GNU General Public License
#    along with this program.  If not, see <http://www.gnu.org/licenses/>.
#

r"""
Models describing dynamic processes mainly for inelastic neutron scattering.

- Models in the time domain have a parameter t for time. -> intermediate scattering function :math:`I(t,q)`
- Models in the frequency domain have a parameter w for frequency and _w appended. ->
  dynamic structure factor :math:`S(w,q)`

Models in time domain can be transformed to frequency domain by :py:func:`~.dynamic.time2frequencyFF`
implementing the Fourier transform :math:`S(w,q)=F(I(t,q))`.

In time domain the combination of processes :math:`I_i(t,q)` is done by multiplication,
including instrument resolution :math:`R(t,q)`:

:math:`I(t,q)=I_1(t,q)I_2(t,q)R(t,q)`.
::

 # 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.

:math:`S(w,q) = S_1(w,q) \otimes S_2(w,q) \otimes R(w,q)`.
::

 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 :py:func:`time2frequencyFF` may include the resolution where it acts like a
window function to reduce spectral leakage with vanishing values at :math:`t_{max}`.
If not used :math:`t_{max}` 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 :py:func:`~.dynamic.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 :ref:`Protein incoherent scattering in frequency domain`.

A comparison of different dynamic models in frequency domain is given in examples.
:ref:`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

"""

import inspect
import math
import os
import sys
import numbers

import numpy as np
from numpy import linalg as la
import scipy
import scipy.integrate
import scipy.interpolate
import scipy.constants
import scipy.special as special

from . import dataArray as dA
from . import dataList as dL
from . import formel
from . import parallel
from .formel import convolve

try:
    from . import fscatter

    useFortran = True
except ImportError:
    useFortran = False

pi = np.pi
_path_ = os.path.realpath(os.path.dirname(__file__))

#: Planck constant in µeV*ns
h = scipy.constants.Planck / scipy.constants.e * 1E15  # µeV*ns

#: h/2π  reduced Planck constant in µeV*ns
hbar = h/2/pi  # µeV*ns

try:
    # change in scipy 18
    spjn = special.spherical_jn
except AttributeError:
    spjn = lambda n, z: special.jv(n + 1 / 2, z) * np.sqrt(pi / 2) / (np.sqrt(z))


[docs]def simpleDiffusion(q, t, D, w=0, amplitude=1): """ Intermediate scattering function for diffusing particles. .. math:: I(q,t)=Ae^{-q^2 (Dt + 0.5w^2t^2)} Parameters ---------- q : float, array Wavevector t : float, array Times amplitude : float Prefactor D : float Diffusion coefficient in units [ [q]**-2/[t] ] w : float Width of diffusion coefficient distribution in D units. Returns ------- dataArray """ result = dA(np.c_[t, amplitude * np.exp(-q ** 2 * (D * t + 1 / 2. * abs(w) * w * t * t))].T) result.amplitude = amplitude result.Diffusioncoefficient = D result.wavevector = q result.columnname = 't;Iqt' result.setColumnIndex(iey=None) result.modelname = inspect.currentframe().f_code.co_name return result
[docs]def doubleDiffusion(q, t, A0, D0, w0=0, A1=0, D1=0, w1=0): """ Two exponential decaying functions describing diffusion. .. math:: I(q,t)=A_1e^{-q^2 (D_1t + 0.5w_1^2t^2)} + A_2e^{-q^2 (D_2t + 0.5w_2^2t^2)} Parameters ---------- q : float, array Wavevector t : float, array Time list A0,A1 : float Prefactor D0,D1 : float Diffusion coefficient in units [ [q]**-2/[t] ] w0,w1 : float Width of diffusion coefficient distributions in D units. Returns ------- dataArray """ result = dA(np.c_[t, A0 * np.exp(-q ** 2 * (D0 * t + 1 / 2. * abs(w0) * w0 * t * t)) + A1 * np.exp(-q ** 2 * (D1 * t + 1 / 2. * abs(w1) * w1 * t * t))].T) result.amplitude0 = amplitude0 result.D0 = D0 result.wavevector = q result.amplitude1 = amplitude1 result.D1 = D1 result.modelname = inspect.currentframe().f_code.co_name result.columnname = 't;Iqt' result.setColumnIndex(iey=None) return result
[docs]def cumulantDiff(t, q, k0=0, k1=0, k2=0, k3=0, k4=0, k5=0): """ Cumulant of order ki with cumulants as diffusion coefficients. .. math:: I(t,q)=k0 exp(-q^2(k_1x+1/2(k_2x)^2+1/6(k_3x)^3+1/24(k_4x)^4+1/120(k_5x)^5)) Parameters ---------- t : array Time q : float Wavevector k0 : float Amplitude k1 : float Diffusion coefficient in units of 1/([q]*[t]) k2,k3,k4,k5 : float Higher coefficients in same units as k1 Returns ------- dataArray : """ t = np.atleast_1d(t) res = k0 * ( np.exp(-q ** 2. * (k1 * t + 1 / 2. * abs(k2) * k2 * t * t + 1. / 6 * k3 * k3 * k3 * t * t * t + 1. / 24 * abs(k4) * k4 * k4 * k4 * t * t * t * t + 1. / 120 * (k5 * t) ** 5))) result = dA(np.c_[t, res].T) result.k0tok5 = [k0, k1, k2, k3, k4, k5] result.wavevector = q result.modelname = inspect.currentframe().f_code.co_name result.columnname = 't;Iqt' result.setColumnIndex(iey=None) return result
[docs]def cumulant(x, k0=0, k1=0, k2=0, k3=0, k4=0, k5=0): r""" Cumulant of order ki. .. math:: I(x) = k_0 exp(-k_1x+1/2k_2x^2-1/6 k_3x^3+1/24k_4x^4-1/120k_5x^5) Parameters ---------- x : float Wavevector k0,k1, k2,k3,k4,k5 : float Cumulant coefficients; units 1/x - k0 amplitude - k1 expected value - k2 variance with :math:`\sqrt(k2/k1) =` relative standard deviation - higher order see Wikipedia Returns ------- dataArray """ x = np.atleast_1d(x) res = k0 * np.exp(- k1 * x + 1 / 2. * k2 * x ** 2 - 1 / 6. * k3 * x ** 3 + 1 / 24 * k4 * x ** 4 - 1 / 120 * k5 * x ** 5) result = dA(np.c_[x, res].T) result.k0tok5 = [k0, k1, k2, k3, k4, k5] result.modelname = inspect.currentframe().f_code.co_name result.columnname = 't;Iqt' result.setColumnIndex(iey=None) return result
[docs]def cumulantDLS(t, A, G, sigma, skewness=0, bgr=0.): r""" Cumulant analysis for dynamic light scattering assuming Gaussian size distribution. .. math:: I(t,q) = A exp(-t/G) \big( 1+(sigma/G t)^2/2. - (skewness/G t)^3/6. \big) + bgr Parameters ---------- t : array Time A : float Amplitude at t=0; Intercept G : float Mean relaxation time as 1/decay rate in units of t. sigma : float - 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 skewness : float,default 0 Relative skewness k3=skewness**3/G**3 bgr : float; 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) """ t = np.atleast_1d(t) if skewness == 0: res = A * np.exp(-t / G) * (1 + (sigma / G * t) ** 2 / 2.) + bgr else: res = A * np.exp(-t / G) * (1 + (sigma / G * t) ** 2 / 2. - (skewness / G * t) ** 3 / 6.) + bgr result = dA(np.c_[t, res].T) result.A = A result.relaxationtime = G result.sigma = sigma result.skewness = skewness result.elastic = bgr result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't;Iqt' return result
[docs]def stretchedExp(t, gamma, beta, amp=1): r""" Stretched exponential function. .. math:: I(t) = amp\, e^{-(t\gamma)^\beta} Parameters ---------- t : array Times gamma : float Relaxation rate in units 1/[unit t] beta : float Stretched exponent amp : float default 1 Amplitude Returns ------- dataArray """ t = np.atleast_1d(t) res = amp * np.exp(-(t * gamma) ** beta) result = dA(np.c_[t, res].T) result.amp = amp result.gamma = gamma result.beta = beta result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't;Iqt' return result
[docs]def jumpDiffusion(t, Q, t0, l0): r""" Incoherent intermediate scattering function of translational jump diffusion in the time domain. Parameters ---------- t : array Times, units ns Q : float Wavevector, units nm t0 : float Residence time, units ns l0 : float Mean square jump length, units nm Returns ------- dataArray Notes ----- We use equ. 3-5 from [1]_ for random jump diffusion as .. math:: T(Q,t) = exp(-\Gamma(Q)t) with residence time :math:`\tau_0` and mean jump length :math:`<l^2>^{1/2}_{av}` and diffusion constant :math:`D` in .. math:: \Gamma(Q) = \frac{DQ^2}{1+DQ^2\tau_0} .. math:: D=\frac{ <l^2>_{av}}{6\tau_0} 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 """ t = np.atleast_1d(t) D = l0 ** 2 / 6. / t0 gamma = D * Q * Q / (1 + D * Q * Q * t0) tdif = np.exp(-gamma * t) result = dA(np.c_[t, tdif].T) result.residencetime = t0 result.jumplength = l0 result.diffusioncoefficient = D result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't;Iqt' return result
[docs]def methylRotation(t, q, t0=0.001, fraction=1, rhh=0.12, beta=0.8): r""" Incoherent intermediate scattering function of CH3 methyl rotation in the time domain. Parameters ---------- t : array List of times, units ns q : float Wavevector, units nm t0 : float, default 0.001 Residence time, units ns fraction : float, default 1 Fraction of protons contributing. rhh : float, default=0.12 Mean square jump length, units nm beta : float, default 0.8 exponent Returns ------- dataArray Notes ----- According to [1]_: .. math:: I(q,t) = (EISF + (1-EISF) e^{-(\frac{t}{t_0})^{\beta}} ) .. math:: EISF=\frac{1}{3}+\frac{2}{3}\frac{sin(qr_{HH})}{qr_{HH}} with :math:`t_0` residence time, :math:`r_{HH}` proton jump distance. 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') References ---------- .. [1] M. Bée, Quasielastic Neutron Scattering (Adam Hilger, 1988). .. [2] Monkenbusch et al. J. Chem. Phys. 143, 075101 (2015) """ t = np.atleast_1d(t) EISF = (1 + 2 * np.sinc(q * rhh / np.pi)) / 3. Iqt = (1 - fraction) + fraction * (EISF + (1 - EISF) * np.exp(-(t / t0) ** beta)) result = dA(np.c_[t, Iqt].T) result.wavevector = q result.residencetime = t0 result.rhh = rhh result.beta = beta result.EISF = EISF result.methylfraction = fraction result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't;Iqt' return result
[docs]def diffusionHarmonicPotential(t, q, rmsd, tau, beta=0, ndim=3): r""" 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 ---------- t : array Time values in units ns q : float Wavevector in unit 1/nm rmsd : float 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. tau : float Correlation time :math:`\tau_0` in units ns. Diffusion constant in sphere Ds=u**2/tau beta : float, default 0 Exponent in correlation function :math:`\rho(t)`. - beta=0 : :math:`\rho(t) = exp(-t/\tau_0)` normal liquids where memory effects are presumably weak or negligible [2]_. - 0<beta,inf : :math:`\rho(t,beta) = (1+\frac{t}{\beta\tau_0})^{-\beta}`. See [2]_ equ. 21a. supercooled liquids or polymers, where memory effects may be important correlation functions with slower decay rates should be introduced [2]_. See [2]_ equ. 21b. ndim : 1,2,3, default=3 Dimensionality of the diffusion potential. Returns ------- dataArray Notes ----- We use equ. 18-20 from [2]_ and correlation time :math:`\tau_0` with equal amplitudes :math:`u` in the dimensions as 3 dim case: .. math:: I_s(Q,t) = e^{-Q^2\langle u^2_x \rangle (1-\rho(t))} 2 dim case: .. math:: I_s(Q,t) = \frac{\pi^{0.5}}{2} e^{-g^2(t)} \frac{erfi(g(t))}{g(t)} \ with \ g(t) = \sqrt{Q^2\langle u^2_x \rangle (1-\rho(t))} 1 dim case: .. math:: I_s(Q,t) = \frac{\pi^{0.5}}{2} \frac{erf(g(t))}{g(t)} \ with \ g(t) = \sqrt{Q^2\langle u^2_x \rangle (1-\rho(t))} with *erf* as the error function and *erfi* is the imaginary error function *erf(iz)/i* 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') 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] 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) """ erf = special.erf erfi = special.erfi q2u2 = q ** 2 * rmsd ** 2 if beta <=0: ft = (1 - np.exp(-t / tau)) else: ft = (1 - (1+t/tau/beta)**(-beta)) ft[t == 0] = 1e-8 # avoid zero to prevent zero division and overwrite later with EISF if ndim == 3: Iqt = np.exp(-q2u2 * ft) EISF = np.exp(-q2u2) Iqt[t == 0] = EISF elif ndim == 2: q2u2exp = q2u2 * ft Iqt = 0.5 * pi ** 0.5 * np.exp(-q2u2exp) * erfi(q2u2exp ** 0.5) / q2u2exp ** 0.5 EISF = 0.5 * pi ** 0.5 * np.exp(-q2u2) * erfi(q2u2 ** 0.5) / q2u2 ** 0.5 Iqt[t == 0] = EISF elif ndim == 1: q2u2exp = q2u2 * ft Iqt = 0.5 * pi ** 0.5 * erf(q2u2exp ** 0.5) / q2u2exp ** 0.5 EISF = 0.5 * pi ** 0.5 * erf(q2u2 ** 0.5) / q2u2 ** 0.5 Iqt[t == 0] = EISF else: raise Exception('ndim should be one of 1,2,3 ') result = dA(np.c_[t, Iqt].T) result.tau = tau result.Ds = rmsd ** 2 / tau result.rmsd = rmsd result.EISF = EISF result.wavevector = q result.dimension = ndim result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't;Iqt' return result
[docs]def finiteZimm(t, q, NN=None, pmax=None, l=None, Dcm=None, Dcmfkt=None, tintern=0., mu=0.5, viscosity=1., ftype=None, rk=None, Temp=293): r""" 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 ---------- t : array 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. NN : integer Number of chain beads. l : float, default 1 Bond length between beads; units nm. pmax : integer, list of float, default is NN - integer => maximum mode number taken into account. - list => list of amplitudes :math:`a_p > 0` for individual modes to allow weighing. Not given modes have weight zero. Dcm : float Center of mass diffusion in nm^2/ns - :math:`=0.196 k_bT/(R_e visc)` for theta solvent with :math:`\mu=0.5` - :math:`=0.203 k_bT/(R_e visc)` for good solvent with :math:`\mu=0.6` Dcmfkt : array 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 . tintern : float>0 Additional relaxation time due to internal friction between neighbouring beads in units ns. mu : float in range [0.1,0.9] :math:`\mu` describes solvent quality. - <0.5 collapsed - =0.5 theta solvent 0.5 (gaussian chain) - =0.6 good solvent - >0.6 swollen chain viscosity : float :math:`\eta` in units cPoise=mPa*s e.g. water :math:`visc(T=293 K) =1 mPas` Temp : float, default 273+20 Temperature in Kelvin. ftype : 'czif', default = 'zif' Type of internal friction and interaction modification. - Default Zimm is used with :math:`t_{intern}=0` - 'zif' Internal friction between neighboring beads in chain [3]_. :math:`t_{zp}=t_z p^{-3\mu}+t_{intern}` - 'czif' Bead confining harmonic potential with internal friction, only for :math:`\mu=0.5` [6]_ . The beads are confined in an additional harmonic potential with :math:`k_c/2(r_n-0)^2` leading to a more compact configuration. :math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`. rk : None , float :math:`rk= k_c/k` describes the relative force constant for *ftype* 'czif'. Returns ------- dataArray : for single q dataList : for 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 :math:`\mu` parameter scales between theta solvent :math:`\mu=0.5` and good solvent :math:`\mu=0.6` (excluded volume or swollen chain). The coherent intermediate scattering function :math:`S(q,t)/S(q,0)` is .. math:: \frac{S(q,t)}{S(q,0)} = \frac{1}{N} e^{-q^2D_{cm}t}\sum_{n,m}^N e^{-\frac{1}{6}q^2B(n,m,t)} .. math:: B(n,m,t)=|n-m|^{2\mu}l^2 + \sum_{p=1}^{N-1} A_p cos(\pi pn/N)cos(\pi pm/N) (1-e^{-t/t_{zp}}) and for incoherent intermediate scattering function the same with :math:`n=m` in the first sum. with - :math:`A_p = a_p\frac{4R_e^2}{\pi^2}\frac{1}{p^{2\mu+1}}` mode amplitude (usual :math:`a_p=1`) - :math:`t_{zp} = t_z p^{-3\mu}` mode relaxation time - :math:`t_z = \eta R_e^3/(\sqrt(3\pi) k_bT)` Zimm mode relaxation time - :math:`R_e=l N^{\mu}` end to end distance - :math:`k=3kT/l^2` force constant between beads - :math:`\xi=6\pi\eta l` single bead friction in solvent with viscosity :math:`\eta` - :math:`a_p` additional amplitude for suppression of specific modes e.g. by topological constraints (see [5]_). - :math:`D_{cm} = \frac{8}{3(6\pi^3)^{1/2}} \frac{k_bT}{\eta R_e} = 0.196 \frac{k_bT}{\eta R_e}` Modifications (*ftype*) for internal friction and additional interaction: - ZIF : Zimm with internal friction between neighboring beads in chain [3]_ [4]_. - :math:`t_{zp}=t_z p^{{-3\mu}}+t_{intern}` - :math:`\xi_i=t_{intern}k=t_{intern}3k_bT/l^2` internal friction per bead - CZIF : Compacted Zimm with internal friction [6]_. Restricted to :math:`\mu=0.5` , a combination with excluded volume is not valid. In [9]_ the beads are confined in an additional harmonic potential around the origin with :math:`k_c/2(r_n-0)^2` leading to a more compact configuration. :math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`. Typically :math:`rk << 1` . - The mode amplitude prefactor changes from Zimm type to modified confined amplitudes .. math:: A_p =\frac{4Nl^2}{\pi^2}\frac{1}{p^2}\Rightarrow A_p^c = \frac{4Nl^2}{\pi^2}\frac{1}{\frac{N^2k_c}{\pi^2k}+p^2} - The mode relaxation time changes from Zimm type to modified confined with :math:`t_{z} = \frac{\eta N^{3/2} l^3}{\sqrt(3\pi) k_bT}` .. math:: t_{zp} = t_z \frac{1}{p^{3/2}} \Rightarrow t_{zp}^c = t_z \frac{p^{1/2}}{\frac{N^2k_c}{\pi^2k} + p^2} - :math:`R_e^c` allows to determine :math:`k_c/k` from small angle scattering data .. math:: (R_e^c)^2 = \frac{2l^2}{\sqrt{k_c/k}}tanh(\frac{N}{2}\sqrt{k_c/k}) - For a free diffusing chain we assume here (not given in [9]_ ) that the additional potential is :math:`k_c/2(r_n-r_0)^2` with :math:`r_0` 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 :math:`D_{cm}` can be calculated similar to the Zimm :math:`D_{cm}` in [1]_ assuming a Gaussian configuration with width :math:`R_e`. We find .. math:: D_{cm} = \frac{kT}{\xi_{p=0}} = \frac{8}{3(6\pi^3)^{1/2}} \frac{kT}{\eta R_e} - With :math:`rk=k_c/k \rightarrow 0` the original Zimm is recovered for amplitudes, relaxation and :math:`R_e` . 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 :py:func:`~.formel.memoize` 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') .. image:: ../../examples/images/Zimmcohinc.jpg :align: center :width: 50 % :alt: Zimm 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] 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] 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 """ kb = 1.3806505e-23 # in SI units # convert to Pa*s viscosity *= 1e-3 q = np.atleast_1d(q) # check mu between 0.1 and 0.9 mu = max(mu, 0.1) mu = min(mu, 0.9) # avoid l=0 from stupid users if l == 0: l = None # and linear interpolate prefactor ffact = 8 / (3 * 6 ** 0.5 * np.pi ** (3 / 2)) fact = ffact + (mu - 0.5) / (0.6 - 0.5) * (0.203 - 0.196) NN = int(NN) if pmax is None: pmax = NN # if a list pmax of modes is given these are amplitudes for the modes # pmax is length of list if isinstance(pmax, numbers.Number): pmax = min(int(pmax), NN) modeamplist = np.ones(pmax) elif isinstance(pmax, list): modeamplist = np.abs(pmax) else: raise TypeError('pmax should be integer or list of amplitudes') # create correction for diffusion if Dcmfkt is not None: if formel._getFuncCode(Dcmfkt): # is already an interpolation function Dcmfunktion = Dcmfkt elif np.shape(Dcmfkt)[0] == 2: Dcmfunktion = lambda qq: dA(Dcmfkt).interp(qq) else: raise TypeError('Shape of Dcmfkt is not 2xN!') else: # by default no correction Dcmfunktion = lambda qq: 1. if ftype == 'czif': # compacted zimm with internal friction if mu != 0.5: raise ValueError('For ftype "czif" only mu=0.5 is allowed. ') if Dcm is None and l is not None and NN is not None: Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5) # end to end distance Dcm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9 # diffusion constant in nm^2/ns elif Dcm is not None and l is None and NN is not None: Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9 # end to end distance l = Re * (rk ** 0.5 / 2 / np.tanh(NN / 2 * rk ** 0.5)) ** 0.5 elif Dcm is not None and l is not None and NN is None: Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9 # end to end distance NN = 2 / rk ** 0.5 * np.arctanh(Re ** 2 / 2 / l ** 2) elif Dcm is not None and l is not None and NN is not None: Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5) else: raise TypeError('finiteZimm takes at least 2 arguments from Dcm,NN,l') # determine mode relaxation times # slowest zimm time tz1 = viscosity * NN ** (3 / 2) * (l * 1e-9) ** 3 / (np.sqrt(3 * pi) * kb * Temp) * 1e9 # mode amplitudes p = np.r_[1:len(modeamplist) + 1] modeamplist = 4 * NN * l ** 2 / pi ** 2 * modeamplist tzp = tz1 * p ** 0.5 / (NN ** 2 / np.pi ** 2 * rk + p ** 2) + abs(tintern) modeamplist = modeamplist / (NN ** 2 / np.pi ** 2 * rk + p ** (2 * mu + 1)) else: # ZIF with constant internal friction time added as default if Dcm is None and l is not None and NN is not None: Re = l * NN ** mu # end to end distance Dcm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9 # diffusion constant in nm^2/ns elif Dcm is not None and l is None and NN is not None: Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9 # end to end distance l = Re / NN ** mu # bond length elif Dcm is not None and l is not None and NN is None: Re = fact * kb * Temp / (Dcm * 1e-9 * viscosity) * 1e9 # end to end distance NN = int((Re / l) ** (1. / mu)) elif Dcm is not None and l is not None and NN is not None: Re = l * NN ** mu else: raise TypeError('finiteZimm takes at least 2 arguments from Dcm,NN,l') # determine mode relaxation times # slowest zimm time tz1 = viscosity * (Re * 1e-9) ** 3 / (np.sqrt(3 * pi) * kb * Temp) * 1e9 # mode amplitudes p = np.r_[1:len(modeamplist) + 1] modeamplist = 4 * Re ** 2 / pi ** 2 * modeamplist # characteristic Zimm time of mode p adding internal friction ti tzp = tz1 * p ** (-3 * mu) + abs(tintern) modeamplist = modeamplist / (p ** (2 * mu + 1)) ftype = 'zif' # prepend 0 and append infinite time t = np.r_[0, np.atleast_1d(t)] # calc array of mode contributions including first constant element as list # do the calculation as an array of bnm=[n*m , len(t)] elements # sum up contributions for modes: all, diff+ mode1, only diffusion, t=0 amplitude for normalisation if useFortran: BNM = fscatter.dynamic.bnmt(t, NN, l, mu, modeamplist, tzp) BNMmodes = BNM[:, -len(modeamplist):] BNMi = BNM[:, len(t):2*len(t)] BNM = BNM[:, :len(t)] else: raise ImportError('finiteZimm only with working Fortran.') result = dL() for qq in q: # diffusion for all t Sqt = np.exp(-qq ** 2 * Dcm * Dcmfunktion(qq) * t[1:]) # only diffusion contribution # amplitude at t=0 expB0 = np.sum(np.exp(-qq ** 2 / 6. * BNM[:, 0])) # is S(qq,t=0)/Sqt coherent expB0i = np.sum(np.exp(-qq ** 2 / 6. * BNMi[:, 0])) # is S(qq,t=0)/Sqt incoherent # diffusion for infinite times in modes expBinf = np.sum(np.exp(-qq ** 2 / 6. * np.sum(BNMmodes, axis=1))) # is S(qq,t=inf)/Sqt # contribution all modes expB = np.sum(np.exp(-qq ** 2 / 6. * BNM[:, 1:]), axis=0) # coherent expBi = np.sum(np.exp(-qq ** 2 / 6. * BNMi[:, 1:]), axis=0) # incoherent # contribution only first modes result.append(np.c_[t[1:], Sqt * expB / expB0, Sqt * expBinf / expB0, Sqt * expBi / expB0i].T) result[-1].modecontribution = (np.sum(np.exp(-qq ** 2 / 6. * BNMmodes), axis=0) / expB0).flatten() result[-1].q = qq result[-1].Re = Re result[-1].ll = l result[-1].pmax = pmax result[-1].Dcm = Dcm result[-1].effectiveDCM = Dcm * Dcmfunktion(qq) DZimm = fact * kb * Temp / (Re * 1e-9 * viscosity) * 1e9 result[-1].DZimm = DZimm result[-1].mu = mu result[-1].viscosity = viscosity result[-1].Temperature = Temp result[-1].tzimm = tz1 result[-1].moderelaxationtimes = tzp result[-1].tintern = tintern result[-1].modeAmplist = modeamplist result[-1].Drot = 1. / 6. / tz1 result[-1].N = NN result[-1].columnname = ' time; Sqt; Sqt_inf; Sqtinc' result[-1].ftype = ftype result[-1].rk = rk if len(result) == 1: return result[0] result.setColumnIndex(iey=None) result.modelname = inspect.currentframe().f_code.co_name return result
[docs]def finiteRouse(t, q, NN=None, pmax=None, l=None, frict=None, Dcm=None, Wl4=None, Dcmfkt=None, tintern=0., Temp=293, ftype=None, specm=None, specb=None, rk=None): r""" 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 ---------- t : array Time in units nanoseconds q : float, list Scattering vector, units nm^-1 For a list a dataList is returned otherwise a dataArray is returned NN : integer Number of chain beads. l : float, default 1 Bond length between beads; unit nm. pmax : integer, list of floats - integer => maximum mode number (:math:`a_p=1`) - list => :math:`a_p` list of amplitudes>0 for individual modes to allow weighing; not given modes have weight zero frict : float Friction of a single bead/monomer :math:`\xi = 6\pi\eta l`, 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 Wl4 : float :math:`W_l^4` Characteristic value to calc friction and Dcm. :math:`D_{cm}=\frac{W_l^4}{3R_e^2}` and characteristic Rouse variable :math:`\Omega_Rt=(q^2/6)^2 W_l^4 t` Dcm : float Center of mass diffusion in nm^2/ns. - :math:`D_{cm}=k_bT/(N\xi)` with :math:`\xi` = friction of single bead in solvent - :math:`D_{cm}=W_l^4/(3Nl^2)=W_l^4/(3Re^2)` Dcmfkt : array 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 . tintern : float>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. :math:`t_{rp}=t_r p^{-2}+t_{intern}` - 'rni' Rouse model with non-local interactions (RNI). Additional friction between random close approaching beads. :math:`t_{rp}=t_r p^{-2}+N/p t_{intern}` - 'rap' Rouse model with anharmonic potentials due to stiffness of the chain :math:`t_{rp}=t_r p^{-2}+t_{intern}ln(N/p\pi)` - 'specrif' Specific interactions of strength :math:`b` between beads separated by *m* bonds. See [7]_ . :math:`t_{rp}=t_r p^{-2} (1+bm^2)^{-1} + (1+m^2/(1+bm^2))t_{intern}` - 'crif' Bead confining potential with internal friction. The beads are confined in an additional harmonic potential with :math:`k_c/2(r_n-0)^2` leading to a more compact configuration. :math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`. Temp : float Temperature Kelvin = 273+T[°C] specm,specb: float Parameters *m, b* used in internal friction models 'spec' and 'specrif'. rk : None , float :math:`rk= k_c/k` describes the relative force constant for *ftype* 'crif'. Returns ------- dataArray : for single q dataList : 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 - .Wl4 - .Re end to end distance :math:`R_e^2=l^2N` - .trouse rotational correlation time or rouse time :math:`tr_1 = \xi N^2 l^2/(3 \pi^2 k_bT)= <R_e^2>/(3\pi D_{cm}) = N^2\xi/(pi^2k)` - .tintern relaxation time due to internal friction - .tr_p characteristic times :math:`tr_p=tr_1 p^{-2}+t_{intern}` - .beadfriction - .ftype type of internal friction - .... use .attr to see all attributes Notes ----- The Rouse model for the coherent intermediate scattering function :math:`S(q,t)/S(q,0)` is [1]_ [2]_ : .. math:: \frac{S(q,t)}{S(q,0)} = \frac{1}{N} e^{-q^2D_{cm}t} \sum_{n,m}^N e^{-\frac{1}{6}q^2B(n,m,t)} .. math:: B(n,m,t)=|n-m|^{2\mu}l^2 + \sum_{p=1}^{N-1} A_p cos(\pi pn/N)cos(\pi pm/N) (1-e^{-t/t_{rp}}) and for incoherent intermediate scattering function the same with :math:`n=m` in the first sum. with - :math:`A_p = a_p\frac{4R_e^2}{\pi^2}\frac{1}{p^2}` mode amplitude (usual :math:`a_p=1`) - :math:`t_{rp} = \frac{t_r}{p^2}` mode relaxation time with Rouse time :math:`t_r =\frac{\xi N R_e^2 }{3\pi^2 k_bT} = \frac{R_e^2}{3\pi^2 D_{cm}} = \frac{N^2 \xi}{\pi^2 k}` - :math:`D_{cm}=kT/{N\xi}` center of mass diffusion - :math:`k=3k_bT/l^2` force constant k between beads. - :math:`\xi=6\pi visc R` single bead friction :math:`\xi` in solvent (e.g. surrounding melt) - :math:`t_{intern}=\xi_i/k` additional relaxation time due to internal friction :math:`\xi_i` Modifications (*ftype*) for internal friction and additional interaction (see [7]_ and [9]_): - RIF : Rouse with internal friction between neighboring beads in chain [3]_ [4]_. - :math:`t_{rp}=t_r p^{-2}+t_{intern}` - :math:`\xi_i=t_{intern}k=t_{intern}3k_bT/l^2` internal friction per bead - RNI : Rouse model with non-local interactions as additional friction between spatial close beads [5]_ . - :math:`t_{rp}=t_r p^{-2}+Nt_{intern}/p` - RAP : Rouse model with anharmonic potentials in bonds describing the stiffness of the chain [6]_. - :math:`t_{rp}=t_r p^{-2}+t_{intern}ln(N/p\pi)` - SPECRIF : Specific interactions of relative strength :math:`b` between beads separated by *m* bonds. Internal friction between neighboring beads as in RIF is added. - :math:`t_{rp}=t_r p^{-2} (1+bm^2)^{-1} + (1+\frac{m^2}{1+bm^2})t_{intern}` - :math:`b=k_{specific}/k_{neighbor}` relative strength of both interactions. - The interaction is between **all** pairs separated by m. - CRIF : Compacted Rouse with internal friction [9]_. The beads are confined in an additional harmonic potential with :math:`k_c/2(r_n-0)^2` leading to a more compact configuration. :math:`rk= k_c/k` describes the relative strength compared to the force between beads :math:`k`. Typically :math:`rk << 1` . - The mode amplitude prefactor changes from Rouse type to modified confined amplitudes .. math:: A_p =\frac{4R_e^2}{\pi^2}\frac{1}{p^2}\Rightarrow A_p^c = \frac{4R_e^2}{\pi^2}\frac{1}{\frac{N^2k_c}{\pi^2k}+p^2} - The mode relaxation time changes from Rouse type to modified confined .. math:: t_{rp} = \frac{t_r}{p^2} \Rightarrow t_{rp}^c = \frac{t_r}{\frac{N^2k_c}{\pi^2k} + p^2} - :math:`R_e` allows to determine :math:`k_c/k` from small angle scattering data .. math:: R_e^2 = \frac{2l^2}{\sqrt{k_c/k}}tanh(\frac{N}{2}\sqrt{k_c/k}) - We assume here that the additional potential is :math:`k_c/2(r_n-r_0)^2` with :math:`r_0` 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 :math:`rk=k_c/k \rightarrow 0` the original Rouse is recovered for amplitudes, relaxation and :math:`R_e` . A combination of different effects is possible [7]_ (but not implemented). The amplitude :math:`A_p` 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 :py:func:`~.formel.memoize` 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') .. image:: ../../examples/images/Rousecohinc.jpg :align: center :width: 50 % :alt: Rouse 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] 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] 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 """ kb = 1.3806505e-23 # in SI units # assure flatt arrays t = np.atleast_1d(t) q = np.atleast_1d(q) # avoid l=0 if l == 0: l = None NN = int(NN) if pmax is None: pmax = NN # if a list pmax of modes is given these are amplitudes for the modes # pmax is length of list if isinstance(pmax, numbers.Number): pmax = min(int(pmax), NN) modeamplist = np.ones(pmax) elif isinstance(pmax, list): modeamplist = np.abs(pmax) else: raise TypeError('pmax should be integer or list of amplitudes') # create correction for diffusion if Dcmfkt is not None: if formel._getFuncCode(Dcmfkt): # is already an interpolation function Dcmfunktion = Dcmfkt elif np.shape(Dcmfkt)[0] == 2: Dcmfunktion = lambda qq: dA(Dcmfkt).interp(qq) else: raise TypeError('Shape of Dcmfkt is not 2xN!') else: # by default no correction Dcmfunktion = lambda qq: 1. # calc the cases of not given parameters for Dcm,NN,l # kB*Temp is in SI so convert all to SI then back to ns if rk is not None: # [9]_ equ 17 for rk->0 this goes to l*NN**0.5 Re = 2 * l ** 2 / rk ** 0.5 * np.tanh(NN / 2 * rk ** 0.5) else: # end to end distance Re = l * np.sqrt(NN) # friction or Dcm must be given # Dcm is independent of rk as no HI in Rouse if Dcm is not None and frict is not None: pass elif Dcm is not None and frict is None: frict = kb * Temp / NN / (Dcm * 1e-9) # diffusion constant in nm^2/ns elif Dcm is None and frict is not None: Dcm = kb * Temp / NN / frict * 1e9 # diffusion constant in nm^2/ns elif Dcm is None and frict is None and Wl4 is not None: Dcm = Wl4 / (3 * Re ** 2) frict = kb * Temp / NN / (Dcm * 1e-9) else: raise TypeError('fqtfiniteRouse takes at least 1 arguments from Dcm, frict, Wl4') # slowest relaxation time is rouse time tr1 = frict * NN ** 2 * l ** 2 / (3 * pi ** 2 * kb * Temp) * 1e-9 # different models for internal friction p = np.r_[1:len(modeamplist) + 1] modeamplist = 4 * Re ** 2 / pi ** 2 * modeamplist if ftype == 'rni': # rouse with non-local interactions # frict = f_s + p *f_i trp = tr1 / p ** 2 + NN * abs(tintern) / p modeamplist = modeamplist / p ** 2 elif ftype == 'rap': # rouse model with anharmonic potentials trp = tr1 / p ** 2 + abs(tintern) * np.log(NN / p * np.pi) modeamplist = modeamplist / p ** 2 elif ftype == 'specrif': # rouse model with specific interactions between bead separated by specm of relative strength specb # + rif trp = tr1 / p ** 2 / (1 + specb * specm ** 2) + (1 + specm ** 2 / (1 + specb * specm ** 2)) * abs(tintern) modeamplist = modeamplist / p ** 2 elif ftype == 'crif': # compacted rouse with internal friction trp = tr1 / (NN ** 2 / np.pi ** 2 * rk + p ** 2) + abs(tintern) modeamplist = modeamplist / (NN ** 2 / np.pi ** 2 * rk + p ** 2) else: # RIF with constant internal friction time added as default trp = tr1 / p ** 2 + abs(tintern) modeamplist = modeamplist / p ** 2 ftype = 'rif' # prepend 0 t = np.r_[0, np.atleast_1d(t)] # do the calculation as an array of bnm=[n*m , len(t)] elements # sum up contributions for modes: all, diff+ mode1, only diffusion, t=0 amplitude for normalisation if useFortran: RNM = fscatter.dynamic.bnmt(t, NN, l, 0.5, modeamplist, trp) RNMmodes = RNM[:, -len(modeamplist):] RNMi = RNM[:, len(t):(2*len(t))] # incoherent RNM = RNM[:, :len(t)] # coherent else: raise ImportError('finiteRouse only with working Fortran.') result = dL() for qq in q: # diffusion for all t Sqt = np.exp(-qq ** 2 * Dcm * Dcmfunktion(qq) * t[1:]) # only diffusion contribution # amplitude at t=0 expB0 = np.sum(np.exp(-qq ** 2 / 6. * RNM[:, 0])) # is S(qq,t=0)/Sqt # coherent expB0i = np.sum(np.exp(-qq ** 2 / 6. * RNMi[:, 0])) # is S(qq,t=0)/Sqt incoherent # diffusion for infinite times in modes expBinf = np.sum(np.exp(-qq ** 2 / 6. * np.sum(RNMmodes, axis=1))) # is S(qq,t=inf)/Sqt # contribution all modes expB = np.sum(np.exp(-qq ** 2 / 6. * RNM[:, 1:]), axis=0) # coherent expBi = np.sum(np.exp(-qq ** 2 / 6. * RNMi[:, 1:]), axis=0) # incoherent # contribution only first modes result.append(dA(np.c_[t[1:], Sqt * expB / expB0, Sqt * expBinf / expB0, Sqt * expBi / expB0i].T)) result[-1].setColumnIndex(iey=None) result[-1].modecontribution = (np.sum(np.exp(-qq ** 2 / 6. * RNMmodes), axis=0) / expB0).flatten() result[-1].q = qq result[-1].Re = Re result[-1].ll = l result[-1].pmax = pmax result[-1].Dcm = Dcm result[-1].effectiveDCM = Dcm * Dcmfunktion(qq) result[-1].Dcmrouse = kb * Temp / NN / frict * 1e9 result[-1].Temperature = Temp result[-1].trouse = tr1 result[-1].tintern = tintern result[-1].moderelaxationtimes = trp result[-1].modeamplitudes = modeamplist result[-1].beadfriction = frict result[-1].Drot = 1. / 6. / tr1 result[-1].N = NN result[-1].internalfriction_g_ns = (tintern * 1e-9) * 3 * kb * Temp / (l * 1e-9) ** 2 * 1e-6 result[-1].columnname = 'time; Sqt; Sqt_inf; Sqtinc' result[-1].ftype = ftype result[-1].rk = rk if specm is not None: result[-1].specm = specm result[-1].specb = specb if len(result) == 1: return result[0] result.setColumnIndex(iey=None) result.modelname = inspect.currentframe().f_code.co_name return result
@formel.memoize() def _msd_trap(t, u, rt, gamma=1): # defined here to memoize it # msd in trap ; equ 4 right part res = np.zeros_like(t) + u ** 2 res[t < rt * 30] = 6 * u ** 2 * (1 - formel.Ea(-(t[t < rt * 30] / rt) ** gamma, gamma)) return res
[docs]def diffusionPeriodicPotential(t, q, u, rt, Dg, gamma=1): r""" 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 ---------- t : array Time points, units ns. q : float Wavevector, units 1/nm u : float Root mean square displacement in the trap, units nm. rt : float Relaxation time of fast dynamics in the trap; units ns ( = 1/lambda in [1]_ ) gamma : float Fractional exponent gamma=1 is normal diffusion Dg : float 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 :math:`D_{\gamma}` with fraction :math:`\gamma` as .. math:: I(Q,t) = exp(-\frac{1}{6}Q^2 msd(t)) .. math:: msd(t) = \langle (x(t)-x(0))^2 \rangle = 6\Gamma^{-1}(\gamma+1)D_{\gamma}t^{\gamma} + 6\langle u^2 \rangle (1-E_{\gamma}(-(\lambda t)^{\gamma})) with the Mittag Leffler function :math:`E_{\gamma}(-at^{\gamma})` and Gamma function :math:`\Gamma` and :math:`\lambda =1/r_t`. The first term in *msd* describes the long time fractional diffusion while the second describes the additional mean-square displacement inside the trap :math:`\langle u^2 \rangle`. For :math:`\gamma=1 \to E_{\gamma}(-at^{\gamma}) \to exp(-at)` simplifying the equation to normal diffusion with traps. 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') .. image:: ../../examples/images/fractalDiff.jpg :align: center :height: 300px :alt: fractalDiff References ---------- .. [1] Gupta, S.; Biehl, R.; Sill, C.; Allgaier, J.; Sharp, M.; Ohl, M.; Richter, D. Macromolecules 2016, 49 (5), 1941. """ # q=np.atleast_1d(q) # mean square displacement for diffusion in periodic potential no trap; equ 4 left part msd = lambda t, Dg, u, rt, gamma=1: 6 * Dg * t ** gamma / scipy.special.gamma(gamma + 1) # Trap contribution in _msd_trap. This is memoized as it is independent of the wavevector # but for fitting with several Q it is needed multiple times. Cache size is 128 entries. # the above but extrapolation to t=0 without trap as contribution of long time diffusion at short times msd_0 = lambda t, Dg, u, rt, gamma=1: 6 * Dg * t ** gamma / scipy.special.gamma(gamma + 1) + 6 * u ** 2 # intermediate scattering function of diffusion in periodic... sqt = lambda q, t, Dg, u, rt, gamma=1: np.exp(-q ** 2 / 6 * (msd(t, Dg, u, rt, gamma))) sqttrap = lambda q, t, Dg, u, rt, gamma=1: np.exp(-q ** 2 / 6 * (_msd_trap(t, u, rt, gamma))) sqt_0 = lambda q, t, Dg, u, rt, gamma=1: np.exp(-q ** 2 / 6 * msd_0(t, Dg, u, rt, gamma)) result = dA(np.c_[t, sqt(q, t, Dg, u, rt, gamma) * sqttrap(q, t, Dg, u, rt, gamma), sqt_0(q, t, Dg, u, rt, gamma), sqttrap(q, t, Dg,u, rt, gamma)].T) result.wavevector = q result.fractionalDiffusionCoefficient = Dg result.displacement_u = u result.relaxationtime = rt result.fractionalCoefficient_gamma = gamma result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't;Iqt;Iqt_diff;Iqt_trap' return result
[docs]def zilmanGranekBicontinious(t, q, xi, kappa, eta, mt=1, amp=1, eps=1, nGauss=60): r""" 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 ---------- t : array Time values in ns q : float Scattering vector in 1/A xi : float 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 . kappa : float Apparent single membrane bending modulus, unit kT eta : float 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 amp : float, default = 1 Amplitude scaling factor eps : float, default=1 Scaling factor in range [1..1.3] for kmin=eps*pi/xi and rmax=xi/eps. See [1]_. mt : float, 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. nGauss : int, default 60 Number of points in Gauss integration Returns ------- dataList Notes ----- See equ B10 in [1]_ : .. math:: S(q,t) = \frac{2\pi\xi^2}{a^4} \int_0^1 d\mu \int_0^{r_{max}} dr rJ_0(qr\sqrt{1-\mu^2}) e^{-kT/(2\pi\kappa)q^2\mu^2 \int_{k_{min}}^{k_{max}} dk[1-J_0(kr)e^{w(k)t}]/k^3} with :math:`\mu = cos(\sphericalangle(q,surface normal))` , :math:`J_0` 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]. 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 References ---------- .. [1] 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 """ tt = np.r_[0., t] qq = np.r_[q] result = dL() nres = parallel.doForList(_zgbicintegral, looplist=qq, loopover='q', t=tt, xi=xi, kappa=kappa, eta=eta, mt=mt, eps=eps, nGauss=nGauss) for qi, res in zip(qq, nres): S0 = res[0] result.append(dA(np.c_[t, res[1:]].T)) result[-1].setColumnIndex(iey=None) result[-1].Y *= amp / S0 result[-1].q = qi result[-1].xi = xi result[-1].kappa = kappa result[-1].eta = eta result[-1].eps = eps result[-1].mt = mt result[-1].amp = amp result[-1].setColumnIndex(iey=None) result[-1].columnname = 't;Iqt' return result
def _zgbicintegral(t, q, xi, kappa, eta, eps, mt, nGauss): """integration of gl. B10 in Mihailescu, JCP 2001""" quad = formel.parQuadratureFixedGauss aquad = formel.parQuadratureAdaptiveGauss def _zgintegrand_k(k, r, t, kappa, eta): """kmin-kmax integrand of gl. B10 in Mihailescu, JCP 2001""" tmp = -kappa / 4. / eta * k ** 3 * t res = (1. - special.j0(k * r) * np.exp(tmp)) / k ** 3 return res def _zgintegral_k(r, t, xi, kappa, eta): """kmin-kmax integration of gl. B10 in Mihailescu, JCP 2001 integration is done in 2 intervals to weight the lower stronger. """ kmax = pi / mt # use higher accuracy at lower k res0 = aquad(_zgintegrand_k, eps * pi / xi, kmax / 8., 'k', r=r, t=t[None, :], kappa=kappa, eta=eta, rtol=0.1 / nGauss, maxiter=250) res1 = aquad(_zgintegrand_k, kmax / 8., kmax, 'k', r=r, t=t[None, :], kappa=kappa, eta=eta, rtol=1. / nGauss, maxiter=250) return res0 + res1 def _zgintegrand_mu_r(r, mu, q, t, xi, kappa, eta): """Mu-r integration of gl. B10 in Mihailescu, JCP 2001 aus numerischen Gruenden Multiplikation mit Gaussfunktion mit Breite xi""" tmp = (-1 / (2 * pi * kappa) * q * q * mu * mu * _zgintegral_k(r, t, xi, kappa, eta)[0] - r * r / ( 2 * (eps * xi) ** 2)) tmp[tmp < -500] = -500 # otherwise overflow error in np.exp y = r * special.j0(q * r * np.sqrt(1 - mu ** 2)) * np.exp(tmp - r ** 2 / (2 * (eps * xi) ** 2)) return y def _gaussBorder(mu, q, t, xi, kappa, eta): # For technical reasons, in order to avoid numerical difficulties, the real # space upper cutoff was realized by multiplying the integrand with a # Gaussian having a width of eps*xi. y = quad(_zgintegrand_mu_r, 0, eps * 3 * xi, 'r', mu=mu, q=q, t=t, xi=xi, kappa=kappa, eta=eta, n=nGauss) return y y = quad(_gaussBorder, 0., 1., 'mu', q=q, t=t, xi=xi, kappa=kappa, eta=eta, n=nGauss) return y
[docs]def zilmanGranekLamellar(t, q, df, kappa, eta, mu=0.001, eps=1, amp=1, mt=0.1, nGauss=40): r""" 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 ---------- t : array Time in ns q : float Scattering vector df : float - 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 kappa : float Apparent single membrane bending modulus, unit kT mu : float, default 0.001 Angle between q and surface normal in unit rad. For lamellar oriented system this is close to zero in NSE. eta : float 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 eps : float, default=1 Scaling factor in range [1..1.3] for kmin=eps*pi/xi and rmax=xi/eps amp : float, default 1 Amplitude scaling factor mt : float, 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. nGauss : int, default 40 Number of points in Gauss integration Returns ------- dataList 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) Notes ----- See equ 16 in [1]_ : .. math:: S(q,t) \propto \int_0^{r_{max}} dr r J_0(q_{\perp}r) exp \Big( -\frac{kT}{2\pi\kappa} q^2\mu^2 \int_{k_{min}}^{k_{max}} \frac{dk}{k^3} [1-J_0(kr) e^{w^\infty(k)t}] \Big) with :math:`w^{\infty(k) = k^3\kappa/4\overline{\eta}}`, :math:`\mu = cos(\sphericalangle(q,surface normal))` , :math:`J_0` 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] 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) """ tt = np.r_[0., t] qq = np.atleast_1d(q) result = dL() nres = parallel.doForList(_zglamintegral, looplist=qq, loopover='q', t=tt, kappa=kappa, eta=eta, df=df, mu=mu, mt=mt, eps=eps, nGauss=nGauss) for qi, res in zip(qq, nres): S0 = res[0] result.append(dA(np.c_[t, res[1:]].T)) result[-1].setColumnIndex(iey=None) result[-1].Y *= amp / S0 result[-1].q = qi result[-1].df = df result[-1].kappa = kappa result[-1].eta = eta result[-1].eps = eps result[-1].mt = mt result[-1].amp = amp result[-1].setColumnIndex(iey=None) result[-1].columnname = 't;Iqt' return result
def _zglamintegral(t, q, df, kappa, eta, eps, mu, mt, nGauss): """integration of gl. 16""" # quad=scipy.integrate.quad quad = formel.parQuadratureFixedGauss aquad = formel.parQuadratureAdaptiveGauss def _zgintegrand_k(k, r, t, kappa, eta): """kmin-kmax integrand o""" tmp = -kappa / 4. / eta * k ** 3 * t res = (1. - special.j0(k * r) * np.exp(tmp)) / k ** 3 return res def _zgintegral_k(r, t, df, kappa, eta): """ kmin-kmax integration of gl. B10 in Mihailescu, JCP 2001 """ kmax = pi / mt # use higher accuracy at lower k res0 = aquad(_zgintegrand_k, eps * pi / df, kmax / 8., 'k', r=r, t=t[None, :], kappa=kappa, eta=eta, rtol=0.1 / nGauss, maxiter=250) res1 = aquad(_zgintegrand_k, kmax / 8., kmax, 'k', r=r, t=t[None, :], kappa=kappa, eta=eta, rtol=1. / nGauss, maxiter=250) return res0 + res1 def _zgintegrand_r(r, mu, q, t, df, kappa, eta): """Mu-r integration """ smu = np.sin(mu) tmp = (-1 / (2 * pi * kappa) * q * q * (1 - smu ** 2) * _zgintegral_k(r, t, df, kappa, eta)[0]) tmp[tmp < -500] = -500 # otherwise overflow error in np.exp y = r * special.j0(q * r * smu) * np.exp(tmp) return y y = quad(_zgintegrand_r, 0, df / eps, 'r', mu=mu, q=q, t=t, df=df, kappa=kappa, eta=eta, n=nGauss) return y
[docs]def integralZimm(t, q, Temp=293, viscosity=1.0e-3, amp=1, rtol=0.02, tol=0.02, limit=50): r""" Conformational dynamics of an ideal chain with hydrodynamic interaction, coherent scattering. Integral version Zimm dynamics. Parameters ---------- t : array Time points in ns q : float Wavevector in 1/nm Temp : float Temperature in K viscosity : float Viscosity in cP=mPa*s amp : float Amplitude rtol,tol : float Relative and absolute tolerance in scipy.integrate.quad limit : int 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 .. math:: S(Q,t) = \frac{12}{Q^2l^2} \int_0^{\infty} e^{-u-(\Omega_Z t)^{2/3} g(u(\Omega_Z t)^{2/3})} du with .. math:: g(y) = \frac{2}{\pi} \int_0^{\infty} x^{-2}cos(xy)(1-e^{-2^{-0.5}x^{2/3}}) dx .. math:: \Omega_z = \frac{kTQ^3}{6\pi\eta_s} See [1]_ for details. 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) References ---------- .. [1] 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 """ quad = scipy.integrate.quad kb = 1.3806503e-23 tt = np.r_[t] * 1e-9 tt[t == 0] = 1e-20 # avoid zero # Zimm diffusion coefficient OmegaZ = (q * 1e9) ** 3 * kb * Temp / (6 * pi * viscosity) _g_integrand = lambda x, y: math.cos(y * x) / x / x * (1 - math.exp(-x ** (3. / 2.) / math.sqrt(2))) _g = lambda y: 2. / pi * quad(_g_integrand, 0, np.inf, args=(y,), epsrel=rtol, epsabs=tol, limit=limit)[0] _z_integrand = lambda u, t: math.exp(-u - (OmegaZ * t) ** (2. / 3.) * _g(u * (OmegaZ * t) ** (2. / 3.))) y1 = [quad(_z_integrand, 0, np.inf, args=(ttt,), epsrel=rtol, epsabs=tol, limit=limit)[0] for ttt in tt] result = dA(np.c_[t, amp * np.r_[y1]].T) result.setColumnIndex(iey=None) result.columnname = 't;Iqt' result.q = q result.OmegaZimm = OmegaZ result.Temperature = Temp result.viscosity = viscosity result.amplitude = amp return result
[docs]def transRotDiffusion(t, q, cloud, Dr, Dt=0, lmax='auto'): r""" 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 ---------- t : array Times in ns. q : float Wavevector in units 1/nm cloud : array 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 :math:`b_{inc}` otherwise its equal 1. - If given, cloud[4] is the coherent scattering length :math:`b_{coh}` otherwise its equal 1. - If cloud is single float the value is used as radius of a sphere with 10x10x10 grid points. Dr : float Rotational diffusion constant (scalar) in units 1/ns. Dt : float, default=0 Translational diffusion constant (scalar) in units nm²/ns. lmax : int 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 .. math:: I(q,t) = e^{-q^2D_tt} I_{rot}(q,t) = e^{-q^2D_tt} \sum_l S_{l,i/c}(q)e^{-l(l+1)D_rt} where :math:`I_{rot}(q,t)` is the rotational diffusion contribution and .. math:: S_{l,c}(q) &= 4\pi \sum_m |\sum_i b_{i,coh} j_l(qr_i) Y_{l,m}(\Omega_i)|^2 & coherent scattering \\ S_{l,i}(q) &= \sum_m \sum_i (2l+1) b_{i,inc}^2 |j_l(qr_i)|^2 & incoherent scattering\\ and coh/inc scattering length :math:`b_{i,coh/inc}`, position vector :math:`r_i` and orientation of atoms :math:`\Omega_i`, spherical Bessel function :math:`j_l(x)`, spherical harmonics :math:`Y_{l,m}(\Omega_i)`. - 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 .. math:: I(q,t)/I(q,0) = \frac{I_{coh}(q,t)+I_{inc}(q,t)}{I_{coh}(q,0)+I_{inc}(q,0)} 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') .. image:: ../../examples/images/rotDiffusion.jpg :width: 50 % :align: center :alt: rotDiffusion .. image:: ../../examples/images/transrotDiffusion.jpg :width: 50 % :align: center :alt: transrotDiffusion 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). """ Ylm = special.sph_harm #: Lorentzian expo = lambda t, ll1D: np.exp(-ll1D * t) if isinstance(cloud, numbers.Number): R = cloud NN = 10 grid = np.mgrid[-R:R:1j * NN, -R:R:1j * NN, -R:R:1j * NN].reshape(3, -1).T inside = lambda xyz, R: la.norm(grid, axis=1) < R cloud = grid[inside(grid, R)] if cloud.shape[1] == 5: # last columns are incoherent and coherent scattering length blinc = cloud[:, 3] blcoh = cloud[:, 4] cloud = cloud[:, :3] elif cloud.shape[1] == 4: # last column is scattering length blinc = cloud[:, 3] blcoh = np.ones(cloud.shape[0]) cloud = cloud[:, :3] else: blinc = np.ones(cloud.shape[0]) blcoh = blinc t = np.array(t, float) bi2 = blinc ** 2 r, p, th = formel.xyz2rphitheta(cloud).T pp = p[:, None] tt = th[:, None] qr = q * r if not isinstance(lmax, numbers.Integral): # lmax = pi * r.max() * q / 6. # a la Cryson lmax = min(max(2 * int(pi * qr.max() / 6.), 6), 100) # We calc here the field autocorrelation function as in equ 24 # incoherent with i=j -> Sum_m(Ylm) leads to (2l+1)/4pi bjlylminc = [(bi2 * spjn(l, qr) ** 2 * (2 * l + 1)).sum() for l in np.r_[:lmax + 1]] # add time dependence Iqtinc = np.c_[[bjlylminc[l].real * expo(t, l * (l + 1) * Dr) for l in np.r_[:lmax + 1]]].sum(axis=0) Iq_inc = np.sum(bjlylminc).real # coh is sum over i then (abs)squared and sum over m see Lindsay equ 19 or 20 bjlylmcoh = [4 * np.pi * np.sum(np.abs((blcoh * spjn(l, qr) * Ylm(np.r_[-l:l + 1], l, pp, tt).T).sum(axis=1)) ** 2) for l in np.r_[:lmax + 1]] Iqtcoh = np.c_[[bjlylmcoh[l].real * expo(t, l * (l + 1) * Dr) for l in np.r_[:lmax + 1]]].sum(axis=0) Iq_coh = np.sum(bjlylmcoh).real Iq_trans = np.exp(-q ** 2 * Dt * t) result = dA(np.c_[t, Iq_trans * Iqtinc, Iq_trans * Iqtcoh].T) result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 't; Iqtinc; Iqtcoh; Iqttrans' result.radiusOfGyration = np.sum(r ** 2) ** 0.5 result.Iq_coh = Iq_coh result.Iq_inc = Iq_inc result.wavevector = q result.rotDiffusion = Dr result.transDiffusion = Dt result.lmax = lmax return result
# noinspection PyIncorrectDocstring
[docs]def 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): r""" 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 ---------- t : array 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. bgr : float, default=0 Background resolution : dataArray 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: .. math:: R_t(t,m_i,s_i,a_i)=\sum_i a_i s_i e^{-\frac{1}{2}s_i^2 t^2} \Leftrightarrow R_w(w,m_i,s_i,a_i)= \sum_i a_i e^{-\frac{1}{2}(\frac{w-m_i}{s_i})^2} under the Fourier transform defined as .. math:: F(f(t)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt .. math:: F(f(w)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(\omega) e^{i\omega t} d\omega 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) """ def gauss(x, mean, sigma): return np.exp(-0.5 * (x - mean) ** 2 / sigma ** 2) / np.sqrt(2 * pi) / sigma if resolution is None: means = [m0, m1, m2, m3, m4, m5] sigmas = [s0, s1, s2, s3, s4, s5] amps = [a0, a1, a2, a3, a4, a5] else: if resolution.modelname[-1] == 'w': means = [m0, m1, m2, m3, m4, m5] sigmas = [1. / s if s is not None else s for s in resolution.sigmas] amps = resolution.amps else: means = resolution.means sigmas = resolution.sigmas amps = resolution.amps t = np.atleast_1d(t) Y = np.r_[[a * gauss(t, m, s) for s, m, a in zip(sigmas, means, amps) if (s is not None) & (m is not None)]].sum( axis=0) result = dA(np.c_[t, Y + bgr].T) result.setColumnIndex(iey=None) result.modelname = inspect.currentframe().f_code.co_name result.columnname = 't; Rqt' result.means = means result.sigmas = sigmas result.amps = amps return result
################################################################## # frequency domain # ################################################################## # noinspection PyBroadException
[docs]def getHWHM(data, center=0, gap=0): """ 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 ---------- data : dataArray Distribution center: float, default=0 Center (symmetry point) of data. If None the position of the maximum is used. gap : float, 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 """ gap = abs(gap) if center is None: # determine center center = data.X[data.Y.argmax()] data1 = data[:, data.X >= center + gap] data2 = data[:, data.X <= center - gap] data1.X = data1.X - center data2.X = data2.X - center res = [] try: max = data1.Y.max() min = data1.Y.min() if min < max / 2. and np.all(np.diff(data1.X) > 0): hwhm1 = np.interp((max - min) / 2., data1.Y.astype(float)[::-1], data1.X.astype(float)[::-1]) res.append(np.abs(hwhm1)) except: res.append(None) try: max = data2.Y.max() min = data2.Y.min() if min < max / 2. and np.all(np.diff(data2.X) > 0): hwhm2 = np.interp((max - min) / 2., data2.Y.astype(float), data2.X.astype(float)) res.append(np.abs(hwhm2)) except: res.append(None) return res
[docs]def elastic_w(w): """ Elastic line; dynamic structure factor in w domain. Parameters ---------- w : array Frequencies in 1/ns Returns ------- dataArray """ Iqw = np.zeros_like(w) Iqw[np.abs(w) < 1e-8] = 1. result = dA(np.c_[w, Iqw].T) result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' result.modelname = inspect.currentframe().f_code.co_name return result
[docs]def transDiff_w(w, q, D): r""" Translational diffusion; dynamic structure factor in w domain. Parameters ---------- w : array Frequencies in 1/ns q : float Wavevector in nm**-1 D : float Diffusion constant in nm**2/ns Returns ------- dataArray Notes ----- Equ 33 in [1]_ .. math:: I(\omega,q) = \frac{1}{\pi} \frac{Dq^2}{(Dq^2)^2 + \omega^2} References ---------- .. [0] Scattering of Slow Neutrons by a Liquid Vineyard G Physical Review 1958 vol: 110 (5) pp: 999-1010 """ dw = q * q * D res = 1 / pi * dw / (dw * dw + w * w) result = dA(np.c_[w, res].T) result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' result.modelname = inspect.currentframe().f_code.co_name result.wavevector = q result.D = D return result
[docs]def jumpDiff_w(w, q, t0, r0): r""" 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 ---------- w : array Frequencies in 1/ns q : float Wavevector in nm**-1 t0 : float Mean residence time in a Poisson distribution of jump times. In units ns. G = 1/tg = Mean jump rate r0 : float Root mean square jump length in 3 dimensions <r**2> = 3*r_0**2 Returns ------- dataArray Notes ----- Equ 6 + 8 in [1]_ : .. math:: S_{inc}(q,\omega) = \frac{1}{\pi} \frac{\Delta\omega}{\Delta\omega^2 + \omega^2} with \; \Delta\omega = \frac{1-e^{-q^2 r_0^2/2}}{t_0} 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). """ Ln = lambda w, dw: dw / (dw * dw + w * w) / pi dw = 1. / t0 * (1 - np.exp(-q ** 2 * r0 ** 2 / 2.)) result = dA(np.c_[w, Ln(w, dw)].T) result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' result.wavevector = q result.meanresidencetime = t0 result.meanjumplength = r0 return result
_erfi = special.erfi _G = special.gamma _h1f1 = special.hyp1f1 _erf = special.erf _Gi = special.gammainc
[docs]def diffusionHarmonicPotential_w(w, q, tau, rmsd, ndim=3, nmax='auto'): r""" 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 ---------- w : array Frequencies in 1/ns q : float Wavevector in nm**-1 tau : float Mean correlation time time. In units ns. rmsd : float Root mean square displacement (width) of the Gaussian in units nm. ndim : 1,2,3, default=3 Dimensionality of the potential. nmax : int,'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. .. math:: I_s(Q_x,\omega) = A_0(Q) + \sum_n^{\infty} A_n(Q) L_n(\omega) \; with \; L_n(\omega) = \frac{\tau_0 n}{\pi (n^2+ \omega^2\tau_0^2)} 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. .. math:: A_0(Q) = e^{-Q^2\langle u^2_x \rangle} .. math:: A_n(Q,\omega) = e^{-Q^2\langle u^2_x \rangle} \frac{(Q^2\langle u^2_x \rangle)^n}{n!} ndim=2 Here we use the Fourier transform of equ 23 with equ. 28a+b in [1]_. .. math:: A_0(Q) = \frac{\sqrt{\pi} e^{-Q^2\langle u^2_x \rangle}}{2} \frac{erfi(\sqrt{Q^2\langle u^2_x \rangle})}{\sqrt{Q^2\langle u^2_x \rangle}} .. math:: A_n(Q,\omega) = \frac{\sqrt{\pi} (Q^2\langle u^2_x \rangle)^n}{2} \frac{F_{1,1}(1+n;3/2+n;-Q^2\langle u^2_x \rangle)}{\Gamma(3/2+n)} with :math:`F_{1,1}(a,b,z)` Kummer confluent hypergeometric function, Gamma function :math:`\Gamma` 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. 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) References ---------- .. [1] 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). """ w = np.array(w, float) u2 = rmsd ** 2 if not isinstance(nmax, numbers.Integral): nmax = min(max(int(6 * q * q * u2), 30), 1000) Ln = lambda w, t0, n: t0 / pi * n / (n * n + w * w * t0 * t0) # equ 25a if ndim == 3: # 3D case A0 = lambda q: np.exp(-q * q * u2) # EISF equ 27a def An(q, n): s = (n < 30) # select not to large n and use for the other the Stirling equation An = np.r_[ (q * q * u2) ** n[s] / special.factorial(n[s]), (q * q * u2 / n[~s] * np.e) ** n[~s] / ( 2 * pi * n[~s]) ** 0.5] An *= np.exp(-q * q * u2) return An n = np.r_[:nmax] + 1 an = An(q, n) sel = np.isfinite(an) # remove An with inf or nan Iqw = (an[sel, None] * Ln(w, tau, n[sel, None])).sum(axis=0) # equ 23 after ft Iqw[np.abs(w) < 1e-8] += A0(q) elif ndim == 2: # 2D case A0 = lambda q: pi ** 0.5 / 2. * np.exp(-q * q * u2) * _erfi((q * q * u2) ** 0.5) / ( q * q * u2) ** 0.5 # EISF equ 28a An = lambda q, n: pi ** 0.5 / 2. * (q * q * u2) ** n * _h1f1(1 + n, 1.5 + n, -q * q * u2) / _G( 1.5 + n) # equ 28b n = np.r_[:nmax] + 1 Iqw = (An(q, n)[:, None] * Ln(w, tau, n[:, None])).sum(axis=0) # equ 23 after ft Iqw[np.abs(w) < 1e-8] += A0(q) elif ndim == 1 and False: print(' THis seems to be wrong as given in the paper') # 1D case A0 = lambda q: pi ** 0.5 / 2. * _erf((q * q * u2) ** 0.5) / (q * q * u2) ** 0.5 # EISF equ 29a An = lambda q, n: (_G(0.5 + n) - _Gi(0.5 + n, q * q * u2)) / (2 * (q * q * u2) ** 0.5 * _G(1 + n)) # equ 29b n = np.r_[:nmax] + 1 an = An(q, n) sel = np.isfinite(an) # remove An with inf or nan Iqw = (an[sel, None] * Ln(w, tau, n[sel, None])).sum(axis=0) # equ 23 after ft Iqw[np.abs(w) < 1e-8] += A0(q) else: raise Exception('ndim should be one of 2 or 3; for 1 use fourier tranform from time domain, see doc.') result = dA(np.c_[w, Iqw].T) result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' result.u0 = rmsd result.dimension = ndim result.wavevector = q result.meancorrelationtime = tau result.gaussWidth = rmsd result.nmax = nmax result.Ds = rmsd ** 2 / tau return result
#: First 99 coefficients from Volino for diffusionInSphere_w # VolinoCoefficient=np.loadtxt(os.path.join(_path_,'data','VolinoCoefficients.dat')) # numpy cannot load because of utf8 with open(os.path.join(_path_, 'data', 'VolinoCoefficients.dat')) as f: VolinoC = f.readlines() VolinoCoefficient = np.array([line.strip().split() for line in VolinoC if line[0] != '#'], dtype=float)
[docs]def diffusionInSphere_w(w, q, D, R): r""" Diffusion inside of a sphere; dynamic structure factor in w domain. Parameters ---------- w : array Frequencies in 1/ns q : float Wavevector in nm**-1 D : float Diffusion coefficient in units nm**2/ns R : float Radius of the sphere in units nm. Returns ------- dataArray Notes ----- Here we use equ. 33 in [1]_ .. math:: S(q,\omega) = A_0^0(q) \delta(\omega) + \frac{1}{\pi} \sum_{l,n\ne 0,0}(2l+1)A_n^l(q) \frac{(x_n^l)^2D/a^2}{[(x_n^l)^2D/a^2]^2 + \omega^2} with :math:`x_n^l` as the first 99 solutions of equ 27 a+b as given in [1]_ and .. math:: A_0^0(q) = \big[ \frac{3j_1(qa)}{qa} \big]^2 , \; (l,n) = (0,0) .. math:: A_n^l(q) &= \frac{6(x_n^l)^2}{(x_n^l)^2-l(l+1)} \big[\frac{qaj_{l+1}(qa)-lj_l(qa)}{(qa)^2-(x_n^l)^2}\big]^2 \; for \; qa\ne x_n^l &= \frac{3}{2}j_l^2(x_n^l) \frac{(x_n^l)^2-l(l+1)}{(x_n^l)^2} \; for \; qa = x_n^l 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 :ref:`A comparison of different dynamic models in frequency domain`. 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') References ---------- .. [1] 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 """ nmax = 99 qR = q * R x = VolinoCoefficient[1:nmax, 0] # x_n_l x2 = x ** 2 l = VolinoCoefficient[1:nmax, 1].astype(int) # n = VolinoCoefficient[1:50, 2].astype(int) w = np.array(w, float) Ln = lambda w, g: g / (g * g + w * w) A0 = lambda qa: (3 * spjn(1, qa) / qa) ** 2 def Anl(qa): # equ 31 a+b in [1]_ res = np.zeros_like(x) s = (x == qa) if np.any(s): res[s] = 1.5 * spjn(l[s], x[s]) ** 2 * (x2[s] - l[s] * (l[s] + 1)) / x2[s] if np.any(~s): s = ~s # not s res[s] = 6 * x2[s] / (x2[s] - l[s] * (l[s] + 1)) * ( (qa * spjn(l[s] + 1, qa) - l[s] * spjn(l[s], qa)) / (qa ** 2 - x2[s])) ** 2 return res Iqw = 1 / pi * (((2 * l + 1) * Anl(qR))[:, None] * Ln(w, x2[:, None] * D / R ** 2)).sum(axis=0) # equ 33 Iqw[np.abs(w) < 1e-8] += A0(q) result = dA(np.c_[w, Iqw].T) result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' result.radius = R result.wavevector = q result.diffusion = D return result
[docs]def rotDiffusion_w(w, q, cloud, Dr, lmax='auto'): r""" 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 ---------- w : array Frequencies in 1/ns q : float Wavevector in units 1/nm cloud : array 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 :math:`b_{inc}` 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. Dr : float Rotational diffusion constant in units 1/ns. lmax : int 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 :py:func:`~.dynamic.transRotDiffusion` for more details. The Fourier transform of the *exp* function is a Lorentzian so the *exp* should be exchange. 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() 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). """ Ylm = special.sph_harm #: Lorentzian Ln = lambda w, g: g / (g * g + w * w) / pi if isinstance(cloud, numbers.Number): R = cloud NN = 10 grid = np.mgrid[-R:R:1j * NN, -R:R:1j * NN, -R:R:1j * NN].reshape(3, -1).T inside = lambda xyz, R: la.norm(grid, axis=1) < R cloud = grid[inside(grid, R)] if cloud.shape[1] == 5: # last columns are incoherent and coherent scattering length blinc = cloud[:, 3] blcoh = cloud[:, 4] cloud = cloud[:, :3] elif cloud.shape[1] == 4: # last column is scattering length blinc = cloud[:, 3] blcoh = np.ones(cloud.shape[0]) cloud = cloud[:, :3] else: blinc = np.ones(cloud.shape[0]) blcoh = blinc w = np.array(w, float) bi2 = blinc ** 2 r, p, t = formel.xyz2rphitheta(cloud).T pp = p[:, None] tt = t[:, None] qr = q * r if not isinstance(lmax, numbers.Integral): # lmax = pi * r.max() * q / 6. # a la CRYSON (SANS/SAXS) # we need a factor of 2 more compared to CRYSON for Q>10 nm**-1 lmax = min(max(2 * int(pi * qr.max() / 6. * 2), 7), 100) # We calc here the field autocorrelation function as in equ 24 # Fourier transform of the exp result in lorentz function # incoherent with i=j -> Sum_m(Ylm) leads to (2l+1)/4pi bjlylminc = [(bi2 * spjn(l, qr) ** 2 * (2 * l + 1)).sum() for l in np.r_[:lmax + 1]] # add Lorentzian Iqwinc = np.c_[[bjlylminc[l].real * Ln(w, l * (l + 1) * Dr) for l in np.r_[:lmax + 1]]].sum(axis=0) Iq_inc = np.sum(bjlylminc).real # coh is sum over i then squared and sum over m see Lindsay equ 19 bjlylmcoh = [4 * np.pi * np.sum(np.abs((blcoh * spjn(l, qr) * Ylm(np.r_[-l:l + 1], l, pp, tt).T).sum(axis=1)) ** 2) for l in np.r_[:lmax + 1]] Iqwcoh = np.c_[[bjlylmcoh[l].real * Ln(w, l * (l + 1) * Dr) for l in np.r_[:lmax + 1]]].sum(axis=0) Iq_coh = np.sum(bjlylmcoh).real result = dA(np.c_[w, Iqwinc, Iqwcoh].T) result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 'w; Iqwinc; Iqwcoh' result.radiusOfGyration = np.sum(r ** 2) ** 0.5 result.Iq_coh = Iq_coh result.Iq_inc = Iq_inc result.wavevector = q result.rotDiffusion = Dr result.lmax = lmax return result
[docs]def nSiteJumpDiffusion_w(w, q, N, t0, r0): r""" 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 ---------- w : array Frequencies in 1/ns q: float Wavevector in units 1/nm N : int Number of jump sites, jump angle 2pi/N r0 : float Distance of sites from center of rotation. For CH3 eg 0.12 nm. t0 : float Rotational correlation time. Returns ------- dataArray Notes ----- Equ. 24 [1]_ : .. math:: S_{inc}^{rot}(Q,\omega) = B_0(Qa)\delta(\omega) + \frac{1}{\pi} \sum_{n=1}^{N-1} B_n(Qa) \frac{\tau_n}{1+(\omega\tau_n)^2} with :math:`\tau_1=\frac{\tau}{1-cos(2\pi/N)}` , :math:`\tau_n=\tau_1\frac{sin^2(\pi/N)}{sin^2(n\pi/N)}` .. math:: B_n(Qa) = \frac{1}{N} \sum_{p=1}^{N} j_0 \Big( 2Qa sin(\frac{\pi p}{N}) \Big) cos(n\frac{2\pi p}{N}) 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') 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 """ w = np.array(w, float) #: Lorentzian Ln = lambda w, tn: tn / (1 + (w * tn) ** 2) / pi def Bn(qa, n): return np.sum([spjn(0, 2 * qa * np.sin(pi * p / N)) * np.cos(n * 2 * pi * p / N) for p in np.r_[:N] + 1]) / N B0 = np.sum([spjn(0, 2 * q * r0 * np.sin(pi * p / N)) for p in np.r_[:N] + 1]) / N t1 = t0 / (1 - np.cos(2 * pi / N)) tn = lambda n: t1 * np.sin(pi / N) ** 2 / np.sin(n * pi / N) ** 2 Iqw = np.c_[[Bn(q * r0, n) * Ln(w, tn(n)) for n in np.r_[1:N]]].sum(axis=0) Iqw[np.abs(w) < 1e-8] += B0 result = dA(np.c_[w, Iqw].T) result.modelname = inspect.currentframe().f_code.co_name result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' result.r0 = r0 result.wavevector = q result.t0 = t0 result.N = N return result
# noinspection PyIncorrectDocstring
[docs]def 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): r""" 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 ---------- w : array 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. bgr : float, default=0 Background resolution : dataArray 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: .. math:: R_t(t,m_i,s_i,a_i)=\sum_i a_i s_i e^{-\frac{1}{2}s_i^2 t^2} \Leftrightarrow R_w(w,m_i,s_i,a_i)=\sum_i a_i e^{-\frac{1}{2}(\frac{w-m_i}{s_i})^2} under the Fourier transform defined as .. math:: F(f(t)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt .. math:: F(f(w)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(\omega) e^{i\omega t} d\omega 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') .. image:: ../../examples/images/resolutionfit.jpg :align: center :width: 50 % :alt: worm """ def gauss(x, mean, sigma): return np.exp(-0.5 * ((x - mean) / sigma) ** 2) if resolution is None: means = [m0, m1, m2, m3, m4, m5] sigmas = [s0, s1, s2, s3, s4, s5] amps = [a0, a1, a2, a3, a4, a5] else: if resolution.modelname[-1] == 'w': # resolution from w domain means = resolution.means sigmas = resolution.sigmas amps = resolution.amps else: means = [m0, m1, m2, m3, m4, m5] sigmas = [1. / s if s is not None else s for s in resolution.sigmas] amps = resolution.amps w = np.atleast_1d(w) if isinstance(resolution, str): # elastic Y = np.zeros_like(w) Y[np.abs(w - m0) < 1e-8] = 1. integral = 1 else: Y = np.r_[ [a * gauss(w, m, s) for s, m, a in zip(sigmas, means, amps) if (s is not None) & (m is not None)]].sum( axis=0) integral = np.trapz(Y, w) result = dA(np.c_[w, Y + bgr].T) result.setColumnIndex(iey=None) result.modelname = inspect.currentframe().f_code.co_name result.columnname = 'w;Rw' result.means = means result.sigmas = sigmas result.amps = amps result.integral = integral return result
[docs]def time2frequencyFF(timemodel, resolution, w=None, tfactor=7, **kwargs): r""" Fast Fourier transform from time domain to frequency domain for inelastic neutron scattering. Shortcut t2fFF calls this function. Parameters ---------- timemodel : function, None Model for I(t,q) in time domain. t in units of ns. The values for t are determined from w as :math:`t=[0..n_{max}]\Delta t` with :math:`\Delta t=1/max(|w|)` and :math:`n_{max}=w_{max}/\sigma_{min} tfactor`. :math:`\sigma_{min}` is the minimal width of the Gaussians given in resolution. If None a constant function (elastic scattering) is used. resolution : dataArray, 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. - float : value 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') w : array 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. kwargs : keyword args Additional keyword arguments that are passed to timemodel. tfactor : float, 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 :math:`e^{-tfactor^2/2}`. 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 ------- dataArray : A symmetric spectrum of the Fourier transform is returned. .Sq :math:`\rightarrow S(q)=\int_{-\omega_{min}}^{\omega_{max}} S(Q,\omega)d\omega \approx \int_{-\infty}^{\infty} S(Q,\omega)d\omega = I(q,t=0)` 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 .. math:: F(f(t)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt .. math:: F(f(w)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(\omega) e^{i\omega t} d\omega The resolution function is defined as (see resolution_w) .. math:: R_w(w,m_i,s_i,a_i)&= \sum_i a_i e^{-\frac{1}{2}(\frac{w-m_i}{s_i})^2} = F(R_t(t)) \\ &=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \sum_i{a_i s_i e^{-\frac{1}{2}s_i^2t^2}} e^{-i\omega t} dt using the resolution in time domain with same coefficients :math:`R_t(t,m_i,s_i,a_i)=\sum_i a_i s_i e^{-\frac{1}{2}s_i^2 t^2}` The Fourier transform of a timemodel I(q,t) is .. math:: I(q,w) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} I(q,t) e^{-i\omega t} dt The integral is calculated by Fast Fourier transform as .. math:: I(q,m\Delta w) = \frac{1}{\sqrt{2\pi}} \Delta t \sum_{n=-N}^{N} I(q,n\Delta t) e^{-i mn/N} :math:`t_{max}=tfactor/min(s_i)`. Due to the cutoff at :math:`t_{max}` 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 :math:`t_{max}=N\Delta t`. The second possibility (default) is to increase :math:`t_{max}` (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 :ref:`A comparison of different dynamic models in frequency domain` or in the example of :py:func:`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) """ if w is None: w = resolution.X if timemodel is None: timemodel = lambda t, **kwargs: dA(np.c_[t, np.ones_like(t)].T) gauss = lambda t, si: si * np.exp(-0.5 * (si * t) ** 2) if isinstance(resolution, numbers.Number): si = np.r_[resolution] ai = np.r_[1] # mi = np.r_[0] elif isinstance(resolution, str): si = np.r_[0.5] # just a dummy ai = np.r_[1] # mi = np.r_[0] else: # filter for given values (remove None) and drop bgr in resolution sma = np.r_[[[si, mi, ai] for si, mi, ai in zip(resolution.sigmas, resolution.means, resolution.amps) if (si is not None) & (mi is not None)]] si = sma[:, 0, None] # mi = sma[:, 1, None] # ignored ai = sma[:, 2, None] # determine the times and differences dt dt = 1. / np.max(np.abs(w)) nn = int(np.max(w) / si.min() * tfactor) nn = max(nn, len(w)) tt = np.r_[0:nn] * dt # calc values if isinstance(resolution, str): timeresol = np.ones_like(tt) else: timeresol = ai * gauss(tt, si) # resolution normalized to timeresol(w=0)=1 if timeresol.ndim > 1: timeresol = np.sum(timeresol, axis=0) timeresol = timeresol / (timeresol[0]) # That S(Q)= integral[-w_min,w_max] S(Q,w)= = I(Q, t=0) kwargs.update(t=tt) tm = timemodel(**kwargs) RY = timeresol * tm.Y # resolution * timemodel # make it symmetric zero only once RY = np.r_[RY[:0:-1], RY] # do rfft from -N to N # using spectrum from -N,N the shift theorem says we get a # exp[-j*2*pi*f*N/2] phase leading to alternating sign => use the absolute value wn = 2 * pi * np.fft.rfftfreq(2 * nn - 1, dt) # frequencies wY = dt * np.abs(np.fft.rfft(RY).real) / (2 * pi) # fft # now try to average or interpolate for needed w values wn = np.r_[-wn[:0:-1], wn] wY = np.r_[wY[:0:-1], wY] integral = scipy.integrate.simps(wY, wn) result = dA(np.c_[wn, wY].T) result.setattr(tm) try: result.modelname += '_t2w' except AttributeError: result.modelname = '_t2w' result.Sq = integral result.Iqt = tm result.timeresol = timeresol result.setColumnIndex(iey=None) result.columnname = 'w;Iqw' return result
t2fFF = time2frequencyFF
[docs]def shiftAndBinning(data, w=None, dw=None, w0=0): """ 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 ---------- data : dataArray Data (from model) to be shifted and averaged in intervals to meet experimental data. w : array New X values (e.g. from experiment). If w is None data.X values are used. w0 : float Shift by w0 that wnew=wold+w0 dw : float, 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). Notes ----- For averaging over intervals scipy.interpolate.CubicSpline is used with integration in the intervals. Returns ------- dataArray 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)) """ if w is None: w = data.X.copy() data.X += w0 if dw == 0: iwY = data.interp(w) else: if dw is None: dw = np.diff(w) else: dw = np.zeros(len(w) - 1) * dw csp = scipy.interpolate.CubicSpline(data.X, data.Y) iwY = [csp.integrate(wi - dwl, wi + dwr) / (dwl + dwr) for wi, dwl, dwr in zip(w, np.r_[0, dw], np.r_[dw, 0])] result = dA(np.c_[w, iwY].T) result.setattr(data) result.setColumnIndex(data) return result
[docs]def dynamicSusceptibility(data, Temp): r""" Transform from S(w,q) to the imaginary part of the dynamic susceptibility. .. math:: \chi (w,q) &= \frac{S(w,q)}{n(w)} (gain side) &= \frac{S(w,q)}{n(w)+1} (loss side) with Bose distribution for integer spin particles .. math:: with \ n(w)=\frac{1}{e^{hw/kT}-1} Parameters ---------- data : dataArray Data to transform with w units in 1/ns Temp : float 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:´1/2\piw´ [1]_." References ---------- .. [1] H. 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') """ ds = data.copy() ds.Y[ds.X > 0] = ds.Y[ds.X > 0] / formel.boseDistribution(ds.X[ds.X > 0], Temp).Y ds.Y[ds.X < 0] = ds.Y[ds.X < 0] / (formel.boseDistribution(-ds.X[ds.X < 0], Temp).Y + 1) ds.Y[ds.X == 0] = 0 ds.modelname = data.modelname + '_Susceptibility' return ds