# -*- 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 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 inealstic neutron scattering.
- Models in the time domain have a parameter t for time. -> intermediate scattering function I(q,t)
- Models in the frequency domain have a parameter w for frequency and _w in the name. -> dynamic structure factor S(q,w)
Models in time domain can be transformed to frequncy domain by :py:func:`~.dynamic.time2frequencyFF`.
In time domain the combination of processes is done by multiplication, including instrument resolution.
:math:`I_x(t,q)=I_1(t,q)I_2(t,q)R(t,q)`.
::
# multiplying and creating new dataArray
Ix(q,t) = 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_x(t,q) = S_1(t,q) \otimes S_2(t,q) \otimes R(w,q)`.
::
conv=js.formel.convolve
Sx(q,w)=conv(conv(S1(w,q,..),S2(w,q,..)),res(w,q,..),normB=True) # normB normalizes resolution
:py:func:`time2frequencyFF` allows mixing between timedomain and frequncy domain models.
This FFT from timedomain needs the resolution included in the timedomain as it acts like a
window function to reduce spectral leakage with vanishing values at :math:`t_{max}`.
The last step is to shift the model spectrum to the symmetry as found in the resolution measurement
and do the binning over frequency channels by :py:func:`~.dynamic.shiftAndBinning`.
Let us describe the diffusion of a particle inside a diffusing invisible sphere by mixing time domain and frequency domain.
::
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):
# 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 was 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.
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=js.dynamic.h*w with h=4.13566 [µeV*ns]
Return values are dataArrays were useful.
To get only Y values use .Y
"""
from __future__ import division
import inspect
import numpy as np
import os
import scipy
import scipy.integrate
import scipy.special as special
from scipy.misc import factorial
import scipy.interpolate
import math
import sys
from . import dataArray as dA
from . import dataList as dL
from . import formel
from . import parallel
from .formel import convolve
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*nsec
try:
# change in scipy 18
spjn=special.spherical_jn
except:
spjn = lambda n, z: special.jv(n + 1 / 2, z) * np.sqrt(pi / 2) / (np.sqrt(z))
# normalized Gaussian
_gauss=lambda x,mean,sigma:np.exp(-0.5*(x-mean)**2/sigma**2)/np.sqrt(2*pi)/sigma
[docs]def simpleDiffusion(q,t,D,amplitude=1):
"""
Intermediate scattering function for diffusing particles.
Parameters
----------
q : float, array
wavevector
t : float, array
times
amplitude : float
prefactor
D : float
diffusion coefficient in units [ [q]**-2/[t] ]
Returns
-------
dataArray
Notes
-----
.. math:: I(q,t)=Ae^{-q^2Dt}
"""
result=dA(np.c_[t,amplitude*np.exp(-q**2*D*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
# relaxation with 2 diffusion processes
[docs]def doubleDiffusion(q,t,amplitude0,D0,amplitude1=0,D1=0):
"""
Two exponential decaying functions.
Parameters
----------
q : float, array
wavevector
t : float, array
timelist
amplitude0,amplitude1 : float
prefactor
D0,D1 : float
diffusion coefficient in units [ [q]**-2/[t] ]
Returns
-------
dataArray
"""
result=dA(np.c_[t,amplitude0*np.exp(-q**2*D0*t)+amplitude1*np.exp(-q**2*D1*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.
means gamma_1 =q^2*D_1 in the linear term
k0*(exp(-q**2.*(k1*x+1/2*(k2*x)**2+1/6*(k3*x)**3+1/24*(k4*x)**4+1/120*(k5*x)**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):
"""
Cumulant of order ki
k0*(exp(-k1*x+1/2*(k2*x)**2-1/6*(k3*x)**3+1/24*(k4*x)**4-1/120*(k5*x)**5))
Parameters
----------
x : float
wavevector
k0,k1, k2,k3,k4,k5 : float
coefficients all in units 1/x
k2/k1 = relative standard deviation if a gaussian distribution is assumed
k3/k1 = relative skewness k3=skewness**3/G**3
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.):
"""
Cumulant analysis for dynamic light scattering
A*np.exp(-t/G)*(1+(sigma/G*t)**2/2.-(skewness/G*t)**3/6.)+elastic
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
a 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):
"""
Stretched exponential function.
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):
"""
Incoherent intermediate scattering function of translational jump diffusion in the time domain.
Parameters
----------
t : array
list of times, units ns
Q : float
wavevector, units nm
t0 : float
residence time, units ns
l0 : float
mean square jump length, units nm
Returns
-------
dataArray
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, 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
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],dw=0,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).
"""
t=np.atleast_1d(t)
EISF=(1+2*np.sinc(q*rhh))/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,ndim=3):
"""
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 potenital [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 resticted to ndim=1,2,3 dimensions and are isotropical averaged.
The correlation is assumed to be exponential decaying.
Parameters
----------
t : array
timevalues in units ns
q : float
wavevector in unit 1/nm
rmsd : float
Root mean square displacement <u**2>**0.5 in potetial 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 in units ns.
Diffusion constant in sphere Ds=u**2/tau
ndim : 1,2,3, default=3
Dimensionality of the diffusion potential.
Returns
-------
dataArray
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
ft=(1 - np.exp(-t / tau))
ft[t==0]=1e-8 # avoid zero to prevent zero divison 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,ll=None,Dcm=None,Dcmfkt=lambda q:1.,tintern=0.,mu=0.5,viscosity=1.,Temp=293):
"""
Zimm dynamics with internal friction of a finite chain with N beads of bonds length l. Coherent scattering.
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.
no excluded volume, random thermal force, drag force with solvent, hydrodynamics between beads
and optional internal friction.
Parameters
----------
t : array
Time in nanoseconds
q: float, array
Scattering vector in nm^-1
If q is list a dataList is returned otherwise a dataArray is returned
NN : integer
Number of chain beads
ll : float, default 1
Bond length between beads; units nm
pmax : integer, default is NN
- integer => maximum mode number
- list => list of amplitudes>0 for individual modes
to allow weighing; not given modes have weigth zero
Dcm : float
Center of mass diffusion in nm^2/ns
- 0.196 kb T/(Re*viscosity) theta solvent with mu=0.6
- 0.203 kb T/(Re*viscosity) good solvent with mu=0.5
Dcmfkt : function returning array
Function to modify Dcm as Dcm(q)=Dcm*Dcmfkt(q) e.g. for inclusion of structure factor Dcmfkt=lambda q:1/S(q)
tintern : float>0
Additional relaxation time due to internal friction
(if a tuple as (0,1,1,1,) the mode p will get tintern[p])
Automatically extended to length pmax
mu : float in range [0.5,0.6]
varies between good solvent 0.6 and theta solvent 0.5 (gaussian chain)
viscosity : float
cPoise=mPa*s as water 20+273.15 K =1 mPa*s
Temp : float
temperatur Kelvin = 273+20
Returns
-------
dataArray : for single q
- [wavevector q , Iqt_diff+modes, Iqt_diffusion]
- dataArray.modecontribution of modes i in sequence
dataList : multiple q
- datalist with dataArrays as for single q as above
Notes
-----
Additional attributes defined:
- Re end to end distance Re^2=l^2*N^2mu
- tz1 rotational corrleation time tz1 = visc*Re^3/(sqrt(3 pi)kb*T)
- t_p characteristic times t_p=tz1*p^-3mu+tintern
- modecontribution is modecontribution as in PRL 71, 4158 equ (3)
From above the triple Dcm,ll,NN are fixed.
- If 2 are given 3rd is calculated
- If all 3 are given the given values are used
Remind:
- k=3kT/ll**2 forsce constant between beads.
- f=6pi*eta*R single bead friction in solvent
- tintern=fi/k additional relaxation time due to internal friction fi
- fi=tintern*k=tintern*3kT/ll**2 internal friction per bead
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)
"""
kb=1.3806505e-23 # in SI units
# convert to Pa*s
viscosity*=1e-3
# assure flatt arrays
t=np.atleast_1d(t)
q=np.atleast_1d(q)
# check mu between 0.5 and 0.6
mu=max(mu,0.5)
mu=min(mu,0.6)
# avoid ll=0 from stupid users
if ll==0: ll=None
# and linear interpolate prefactor
fact=0.196+(mu-0.5)/(0.6-0.5)*(0.203-0.196)
NN=int(NN)
if pmax==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,(int,float)):
pmax=min(int(pmax),NN)
ps=range(1,pmax+1)
modeamplist=np.ones_like(ps)
elif isinstance(pmax,list):
ps=range(1,len(pmax)+1)
modeamplist=np.abs(pmax)
else:
raise TypeError('pmax should be integer or list of amplitudes')
# calc the cases of not given parameters for Dcm,NN,ll
if Dcm is None and ll is not None and NN is not None:
Re=np.sqrt(ll**2*NN**(2*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 ll is None and NN is not None:
Re=fact*kb*Temp/(Dcm*1e-9*viscosity)*1e9 # end to end distance
ll=Re/NN**mu # bond length
elif Dcm is not None and ll is not None and NN is None:
Re=fact*kb*Temp/(Dcm*1e-9*viscosity)*1e9 # end to end distance
NN=int((Re/ll)**(1./mu))
elif Dcm is not None and ll is not None and NN is not None:
Re=np.sqrt(ll**2*NN**(2*mu))
else:
raise TypeError('fqtfiniteZimm takes at least 2 arguments from Dcm,NN,ll')
# slowest zimm time
tz1=viscosity*(Re*1e-9)**3/(np.sqrt(3*pi)*kb*Temp)*1e9
# characteristic Zimm time of mode p with internal friction ti
if isinstance(tintern,tuple): # allow different tintern
tintern=(tintern+(tintern[-1],)*len(modeamplist))[:len(modeamplist)]
# remember p starts at 1
tzp=lambda p,ti=1:tz1*p**(-3*mu)+abs(tintern[p-1])*ti
elif isinstance(tintern,(float,int)): # for a common tintern
tzp=lambda p,ti=1:tz1*p**(-3*mu)+abs(tintern)*ti
else:
raise TypeError('tintern should be float or tuple as (1,2,3) of float')
# define functions
spmax=lambda t,NN,n,m,mu,moAmplist,ps:[
4*Re**2/pi**2*pamplitude*(1./(ip**(2*mu+1))*
np.cos(pi*ip*n/NN)*np.cos(pi*ip*m/NN)*
(1-np.exp(-t/(tzp(ip)))))
for ip,pamplitude in zip(ps,moAmplist)]
# calc array of mode contributions including first constant element as list
Bmn=lambda t,NN,l,mu:np.array([[np.array([abs(n-m)**(2*mu)*l**2]*len(t))]+spmax(t,NN,n,m,mu,modeamplist,ps)
for n in range(1,NN+1) for m in range(1,NN+1)])
# do the calculation as an array of bnm=[n*m ,pmax, len(t)] elements
bnm=Bmn(t,NN,ll,mu)
# sum up contributions for modes: all, diff+ mode1, only diffusion, t=0 amplitude for normalisation
BNM=np.sum(bnm[:,:,:],axis=1) # summation over pmax axis
BNM0=bnm[:,:1,0] # only 0. element for t=0
bmninf=Bmn(np.r_[tzp(1,0)*1e6],NN,ll,mu) # relaxed after long time for t=inf
BNMinf=np.sum(bmninf[:,:],axis=1) # summation over pmax axis again
BNMmcontrib=bmninf[:,1:]+bmninf[:,0:1,:] # t=0 contrib + single modes contrib
result=dL()
for qq in q:
# diffusion for all t
Sqt=np.exp(-qq**2*Dcm*Dcmfkt(qq)*t) # only diffusion contribution
# amplitude at t=0
expB0=np.sum(np.exp(-qq**2/6.*BNM0)) # is S(qq,t=0)/Sqt
# diffusion for infinite times in modes
expBinf=np.sum(np.exp(-qq**2/6.*BNMinf)) # is S(qq,t=inf)/Sqt
# contribution all modes
expB=np.sum(np.exp(-qq**2/6.*BNM),axis=0)
# contribution only first modes
result.append(dA(np.r_[[t,Sqt*expB/expB0,
Sqt*expBinf/expB0,
]]))
result[-1].modecontribution=(np.sum(np.exp(-qq**2/6.*BNMmcontrib),axis=0)/expB0).flatten()
result[-1].q=qq
result[-1].Re=Re
result[-1].ll=ll
result[-1].pmax=pmax
result[-1].Dcm=Dcm
result[-1].effectiveDCM=Dcm*Dcmfkt(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].tintern=tintern
result[-1].modeAmplist=modeamplist
result[-1].Drot=1./6./tz1
result[-1].N=NN
result[-1].columnname=' time; Sqt; Sqt_inf'
if len(result)==1:
return result[0]
result.setColumnIndex(iey=None)
result.modelname=sys._getframe().f_code.co_name
# update parameter
return result
[docs]def finiteRouse(t,q,NN=None,pmax=None,ll=None,frict=None,Dcm=None,Wl4=None,Dcmfkt=lambda q:1.,tintern=0.,Temp=293):
"""
Rouse dynamics of a finite chain with N beads of bonds length l and internal friction. Coherent scattering.
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 and optional internal friction.
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.
ll : float, default 1
Bond length between beads; unit nm.
pmax : integer
Maximum mode number, default is NN.
As list =>list of amplitudes>0 for individual modes to allow weighting; not given modes have weigth zero.
frict : float
Friction of a single bead, units Pas*m=kg/s=1e-6 g/ns.
A sphere with R=1 nm in water = 1.88e-11 kg/s=1.88e-17 g/ns
Wl4 : float
needed to calc friction and Dcm
Dcm : float
Center of mass diffusion in nm^2/ns
- =kT/(NN*f) with f = friction of single bead in solvent
- =Wl^4/(3*N*l^2)=Wl^4/(3* Re^2)
Dcmfkt : function returning array
function to modify Dcm as Dcm(q)=Dcm*Dcmfkt(q)
eg for inclusion of structure factor Dcmfkt=lambda q:1/S(q)
tintern : float>0
relaxation time due to internal friction in ns
Temp : float
temperature Kelvin = 273+T[°C]
Returns
-------
dataArray
Notes
-----
Additional Attributes
- Wl4
- Re end to end distance Re^2=l^2*N
- tr1 is rotational corrleation time or rouse time
tr1 = f*NN^2*ll^2/(3 pi^2*kb*T)= <Re^2>/(3*pi*Dcm) = N**2*f/(pi**2*k)
- tintern relaxation time due to internal friction
- t_p characteristic times t_p=tr1*p^2+tintern
From above the triple Dcm,ll,NN are fixed.
- If 2 are given 3rd is calculated
- If all 3 are given the given values are used
Remind:
- k=3kT/ll**2 force constant k between beads.
- f=6pi*eta*R single bead friction f in solvent (e.g. surrounding melt)
- tintern=fi/k additional relaxation time due to internal friction fi
- fi=tintern*k=tintern*3kT/ll**2 internal friction per bead
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
"""
kb=1.3806505e-23 # in SI units
# assure flatt arrays
t=np.atleast_1d(t)
q=np.atleast_1d(q)
# avoid ll=0
if ll==0: ll=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,(int,float)):
pmax=min(int(pmax),NN)
ps=range(1,pmax+1)
modeamplist=np.ones_like(ps)
elif isinstance(pmax,list):
ps=range(1,len(pmax)+1)
modeamplist=pmax
else:
raise TypeError('pmax should be integer or list of amplitudes')
# calc the cases of not given parameters for Dcm,NN,ll
Re=ll*np.sqrt(NN) # end to end distance
if Dcm is not None and frict is None:
frict=kb*Temp/NN/(Dcm*1e9) # 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*1e9)
else:
raise TypeError('fqtfiniteRouse takes at least 1 arguments from Dcm,frict')
# slowest rouse time
tr1=Re**2/(3*pi**2*Dcm)
# characteristic rouse time of mode p with internal friction ti
trp=lambda p,ti=tintern:tr1/p**(-2)+abs(ti)
# define functions
spmax=lambda t,NN,n,m,moAmplist,ps:[
4*Re**2/pi**2*pamplitude*(1./(ip**2)*
np.cos(pi*ip*n/NN)*np.cos(pi*ip*m/NN)*
(1-np.exp(-t/(trp(ip)))))
for ip,pamplitude in zip(ps,modeamplist)]
# B=lambda t,NN,l,n,m:abs(n-m)*l**2+sumpmax(t,NN,n,m)
# Bmn=lambda t,NN,l:np.array([B(t,NN,l,n,m) for n in range(1,NN+1) for m in range(1,NN+1)])
Bmn=lambda t,NN,l:np.array([[np.array([abs(n-m)*l**2]*len(t))]+spmax(t,NN,n,m,modeamplist,ps)
for n in range(1,NN+1) for m in range(1,NN+1)])
# do the calculation as an array of bnm=[n*m ,pmax, len(t)] elements
bnm=Bmn(t,NN,ll)
BNM=np.sum(bnm[:,:,:],axis=1) # summation over pmax axis
BNM0=bnm[:,:1,0] # only 0. element for t=0
bmninf=Bmn(np.r_[trp(1,0)*1e6],NN,ll) # relaxed after long time for t=inf
BNMinf=np.sum(bmninf[:,:],axis=1) # summation over pmax axis again
BNMmcontrib=bmninf[:,1:]+bmninf[:,0:1,:] # t=0 contrib + single modes contrib
result=dL()
for qq in q:
# diffusion for all t
Sqt=np.exp(-qq**2*Dcm*Dcmfkt(qq)*t) # only diffusion contribution
# amplitude at t=0
expB0=np.sum(np.exp(-qq**2/6.*BNM0))# /float(NN) # is S(qq,t=0)/Sqt
# diffusion for infinite times in modes
expBinf=np.sum(np.exp(-qq**2/6.*BNMinf))# /float(NN) # is S(qq,t=inf)/Sqt
# contribution all modes
expB=np.sum(np.exp(-qq**2/6.*BNM),axis=0)# /float(NN)
# contribution only first modes
result.append(dA(np.r_[[t,Sqt*expB/expB0,
Sqt*expBinf/expB0,
]]))
result[-1].setColumnIndex(iey=None)
result[-1].modecontribution=(np.sum(np.exp(-qq**2/6.*BNMmcontrib),axis=0)/expB0).flatten()
result[-1].q=qq
result[-1].Re=Re
result[-1].ll=ll
result[-1].pmax=pmax
result[-1].Dcm=Dcm
result[-1].Dcmrouse=kb*Temp/NN/frict/1e9
result[-1].Temperature=Temp
result[-1].trouse=tr1
result[-1].tintern=tintern
result[-1].friction=frict
result[-1].Drot=1./6./tr1
result[-1].N=NN
result[-1].internalfriction_g_ns=(tintern*1e-9)*3*kb*Temp/(ll*1e-9)**2*1e-6
result[-1].columnname='time[ns]; Sqt; Sqt_inf'
if len(result)==1:
return result[0]
result.setColumnIndex(iey=None)
result.modelname=inspect.currentframe().f_code.co_name
# update parameter
return result
[docs]def diffusionPeriodicPotential(t,q,u,rt,Dg,gamma=1,NN=100):
"""
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
q : float
Wavevector
u : float
Mean displacement in the trap
rt : float
Relaxation time of fast dynamics in the trap (1/lambda in [1]_ )
gamma : float
Fractional exponent gamma=1 is normal diffusion
Dg : float
Long time fractional diffusion coefficient
NN : int, default 100
Order for approximating Mittag Leffler function
sum([x**k/scipy.special.gamma(a*k+1) for k in range(NN)])
Test this for your needed time range. See Examples
Returns
-------
dataArray : .
[times, intermediate scattering function , intermediate scattering function only diffusional part]
References
----------
.. [1] Gupta, S.; Biehl, R.; Sill, C.; Allgaier, J.; Sharp, M.; Ohl, M.; Richter, D.
Macromolecules 2016, 49 (5), 1941.
Examples
--------
::
t=js.loglist(1,10000,1000)
q=0.5
p=js.grace()
f100=js.dynamic.diffusionPeriodicPotential(t,q,0.5,15,0.036,NN=100)
f30= js.dynamic.diffusionPeriodicPotential(t,q,0.5,15,0.036,NN=30)
p.plot(f100,legend='NN=100')
p.plot(f100,legend='NN=30')
"""
Ea=formel.Ea
# q=np.atleast_1d(q)
# mean square displacement for diffusion in periodic potential
msd6=lambda t,Dg,u,rt,gamma=1:Dg*t**gamma/scipy.special.gamma(gamma+1)
def msd6trap(t,Dg,u,rt,gamma=1):
res=t*0+u**2
res[t<rt*30]= u**2*(1-Ea(-(t[t<rt*30]/rt)**gamma, gamma))
return res
msd6_0=lambda t,Dg,u,rt,gamma=1:Dg*t**gamma/scipy.special.gamma(gamma+1)+u**2
# intermediate scattering function of diffusion in periodic...
sqt=lambda q,t,Dg,u,rt,gamma=1:np.exp(-q**2*(msd6(t,Dg,u,rt,gamma)))
sqttrap=lambda q,t,Dg,u,rt,gamma=1:np.exp(-q**2*(msd6trap(t,Dg,u,rt,gamma)))
sqt_0=lambda q,t,Dg,u,rt,gamma=1:np.exp(-q**2*msd6_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;Sqt_inf'
return result
[docs]def zilmanGranekBicontinious(t, q, xi, kappa, eta, mt=1, amp=1, eps=1 ,nGauss=60):
"""
Dynamics of bicontinuous microemulsion 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 Angstroem for tenside C10E4.
nGauss : int, default 60
Number of points in Gauss integration
Returns
-------
dataList
Notes
-----
- 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 forst 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. Mihailescu1, 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 doen in 2 intervalls to weigth 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 Grnden 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 owerflow 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):
"""
Dynamics of lamellar microemulsion phases. Zilman-Granek model as Equ B10 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 lamelar 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 Angstroem 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
-----
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 owerflow 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):
"""
Conformational dynamics of an ideal chain with hydrodynamic interaction Integral version Zimm dynamics. Coherent scattering.
The Zimm model describes the conformational dynamics of an ideal chain with hydrodynamic
interaction between beads. See [1]_.
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
Returns
-------
dataArray
Examples
--------
::
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 rotDiffusion(t,q,cloud,Dr,lmax='auto'):
"""
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 nonspherical 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
----------
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] that describe an object.
- If given, cloud[3] is the incoherent scattering length :math:`b_{inc}`.
- If given, cloud[4] is the coherent scattering length :math:`b_{coh}`
- If cloud[3] not given :math:`b_{inc}=b_{coh}=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 > pi*r.max()*q/6.
Returns
-------
dataArray with [t;Iqtinc;Iqtcoh]
.radiusOfGyration
.Iq_coh coherent formfactor
.Iq_inc
.wavevector
.rotDiffusion
.lmax
Notes
-----
- The incoherent intermediate scattering function is res.Y/res.Iq_inc
- The coherent intermediate scattering function is res[2]/res.Iq_coh
Examples
--------
::
import jscatter as js
import numpy as np
R=2;NN=5
grid= np.mgrid[-R:R:1j*NN, -R:R:1j*NN,-R:R:1j*NN].reshape(3,-1).T
# points inside of sphere with radius R
p2=1*2*0.5 # p defines a superball with 1->sphere p=inf cuboid ....
inside=lambda xyz,R:(np.abs(xyz[:,0])/R)**p2+(np.abs(xyz[:,1])/R)**p2+(np.abs(xyz[:,2])/R)**p2<=1
insidegrid=grid[inside(grid,R)]
Drot=js.formel.Drot(R)
ql=np.r_[0.5:15.:1]
t=js.loglist(1,200,100)
p=js.grace()
p.new_graph(xmin=0.25,xmax=0.55,ymin=0.2,ymax=0.5)
iqt=js.dL([js.dynamic.rotDiffusion(t,q,insidegrid,Drot) for q in ql])
for iiqt in iqt:
#p[0].plot(iiqt.X,iiqt.Y/iiqt.Iq_inc,le='q=%.3g nm\S-1' %(iiqt.wavevector))
p[0].plot(iiqt.X,iiqt[2]/iiqt.Iq_coh,le='q=%.3g nm\S-1' %(iiqt.wavevector))
p[1].plot(iqt.wavevector,iqt.Iq_coh,li=1)
p[0].xaxis(min=1,max=100,scale='l')
p[0].yaxis(min=0.8,max=1.03,scale='n')
p[0].legend()
# Dependent on the contributing spherical harmonics for a given q value positive correlation
# in the intermediate scattering function may appear.
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,(float,int)):
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:(np.abs(xyz[:,0])/R)**2+(np.abs(xyz[:,1])/R)**2+(np.abs(xyz[:,2])/R)**2<=1
cloud=grid[inside(grid,R)]
if cloud.shape[1]==4:
# last column is scattering length
blinc=cloud[:,3]
blcoh=None
cloud=cloud[:,:3]
elif cloud.shape[1]==5:
# last columns are incoherent and coherent scattering length
blinc=cloud[:,3]
blcoh=cloud[:,4]
cloud=cloud[:,:3]
else:
blinc=np.ones(cloud.shape[0])
blcoh=blinc
t = np.array(t, float)
bi2 = blinc ** 2
r,p,th=np.r_[[formel.xyz2rphitheta(r) for r in cloud]].T
pp=p[:,None]
tt=th[:,None]
qr=q*r
if not isinstance(lmax,int):
# lmax = pi * r.max() * q / 6. # a la Cryson
lmax = min(max(int(pi*qr.max()/6.),7),100)
# 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
if blcoh is not None:
# coh is sum over i then squared and sum over m see Lindsay equ 19
bjlylmcoh = [ np.sum((blcoh*spjn(l,qr) *Ylm(np.r_[-l:l+1],l,pp,tt).T).sum(axis=0)**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
else:
Iqtcoh=np.zeros_like(Iqtinc)
result=dA(np.c_[t,Iqtinc,Iqtcoh].T)
result.modelname=sys._getframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname='t;Iqtinc;Iqtcoh'
result.radiusOfGyration=np.sum(r**2)**0.5
if blcoh is not None:
result.Iq_coh=Iq_coh
result.Iq_inc=Iq_inc
result.wavevector=q
result.rotDiffusion=Dr
result.lmax=lmax
return result
[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_w=None):
r"""
Resolution in time domain as multiple Gaussians for inelastic measurement as backscattering or time of flight instruement.
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_w : dataArray
Resolution in w domain with attributes sigmas, amps which are used instead of si, ai.
This represents the Fourier transform of multi gauss resolution from w to t domain.
The m0..m5 are NOT used as these result only in a phase shift.
Returns
-------
dataArray
Notes
-----
In a typical inelastic experiment the resolution is measured by e.g. a vanadium meausrement (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
--------
::
import jscatter as js
resw=js.dynamic.resolution_w(w, s0=12, m0=0, a0=2) # resolution in w domain
# representing the fouriertransform of resw as a gaussian transfoms to ag gaussian
rest=js.dynamic.resolution(t,resolution=resw)
"""
gauss=lambda x,mean,sigma:sigma*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:
means = [m0,m1,m2,m3,m4,m5]
sigmas = 1./np.array(resolution.sigmas)
amps = np.array(resolution.amps)
w=np.atleast_1d(w)
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.columnname = 't;Rqt'
result.means = means
result.sigmas= sigmas
result.amps = amps
return result
##################################################################
# frequenxy domain #
##################################################################
[docs]def getHWHM(data,center=0,gap=0):
"""
Find half width at half maximum of a distribution around zero.
The hwhm is determined from cubicspline 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=scipy.interpolate.interp1d(data1.Y.astype(float)[::-1],data1.X.astype(float)[::-1], kind=2)((max-min) / 2.)
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=scipy.interpolate.interp1d(data1.Y.astype(float),data1.X.astype(float), kind=2)((max-min)/ 2.)
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
w0 : float
Position of elastic line :math:`\delta(w=w0)=1`
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=sys._getframe().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
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=sys._getframe().f_code.co_name
result.wavevector = q
result.D = D
return result
[docs]def jumpDiff_w(w,q,t0,r0):
"""
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
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=sys._getframe().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'):
"""
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.
ndim=3
Here we use the Fourier transform of equ 23 with equ. 29a+b in [1]_.
For order n larger 30 the Stirling approximation for n! in equ 27b of [1]_ is used.
ndim=2
Here we use the Fourier transform of equ 23 with equ. 28a+b in [1]_.
ndim=1
The equation given by Violino seems to be wrong !!!!
Dont use this !!!!!!!!
Use the model from time domain and use FFT as in example given
Here we use the Fourier transform of equ 23 with equ. 29a+b in [1]_.
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()
iqt3=js.dL([js.dynamic.gaussDiffusion3D_w(w=w,q=q,t0=0.14,u0=0.34,ndim=3) for q in ql])
iqt2=js.dL([js.dynamic.gaussDiffusion3D_w(w=w,q=q,t0=0.14,u0=0.34,ndim=2) for q in ql])
# as ndim=1 is a wrong solution use this instead
iqt1=js.dL([js.dynamic.time2frequencyFF(js.dynamic.diffusionHarmonicPotential,'elastic',w=np.r_[-100:100:0.01],dw=0,q=q, rmsd=u0, tau=t0 ,ndim=1) for q in ql])
p.plot(iqt2)
p.plot(iqt3)
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,int):
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]/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:
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 1,2,3 ')
result=dA(np.c_[w, Iqw].T)
result.modelname=sys._getframe().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(_path_+'/VolinoCoefficients.dat') # numpy cannot load because of utf8
with open(_path_+'/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):
"""
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]_ with the first 99 solutions of equ 27 a+b as given in [1]_.
This is valid for q*R<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.
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')
Compare different kinds of diffusion in restricted geometry.
::
import jscatter as js
import numpy as np
# compare the HWHM
ql=np.r_[0.5:15.:0.2]
D=0.1;R=0.5
w=np.r_[-js.loglist(0.01,100,100)[::-1],0,js.loglist(0.01,100,100)]
iqwS=js.dL([js.dynamic.diffusionInSphere_w(w=w,q=q,D=D,R=R) for q in ql])
iqwD=js.dL([js.dynamic.transDiff_w(w=w,q=q,D=D) for q in ql[:]])
u0=R/4.33**0.5;t0=R**2/4.33/D
iqwG3=js.dL([js.dynamic.gaussDiffusion3D_w(w=w,q=q,u0=u0,t0=t0) for q in ql])
iqwG2=js.dL([js.dynamic.gaussDiffusion2D_w(w=w,q=q,u0=u0,t0=t0) for q in ql])
p1=js.grace()
p1.subtitle('Comparison of HWHM for different types of diffusion')
p1.plot((R*iqwD.wavevector.array)**2,[js.dynamic.getHWHM(dat)[0]/(D/R**2) for dat in iqwD], le='free diffusion')
p1.plot((R*iqwS.wavevector.array)**2,[js.dynamic.getHWHM(dat)[0]/(D/R**2) for dat in iqwS], le='diffusion in sphere')
p1.plot([0.1,60],[4.33296]*2,li=[1,1,1])
p1.plot((R*iqwG3.wavevector.array)**2,[js.dynamic.getHWHM(dat)[0]/(D/R**2) for dat in iqwG3], le='diffusion 3D Gauss')
p1.plot((R*iqwG2.wavevector.array)**2,[js.dynamic.getHWHM(dat)[0]/(D/R**2) for dat in iqwG2], le='diffusion 2D Gauss')
r0=.5;t0=r0**2/2./D
iqwJ=js.dL([js.dynamic.jumpDiff_w(w=w,q=q,r0=r0,t0=t0) for q in ql])
ii=54;p1.plot((r0*iqwJ.wavevector.array[:ii])**2,[js.dynamic.getHWHM(dat)[0]/(D/r0**2) for dat in iqwJ[:ii]], le='jump diffusion')
p1.yaxis(min=0.1,max=100,scale='l',label='HWHM/(D/R**2)')
p1.xaxis(min=0.1,max=100,scale='l',label='(Q*R)\S2')
p1.legend(x=0.2,y=50)
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).
"""
qR=q*R
x=VolinoCoefficient[1:50,0] # x_n_l
x2=x**2
l=VolinoCoefficient[1:50,1].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=sys._getframe().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'):
"""
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 nonspherical 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}`.
- If given, cloud[4] is the coherent scattering length
- If cloud[3] not given :math:`b_{inc}=b_{coh}=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 > pi*r.max()*q/6.
Returns
-------
dataArray with [w;Iqwinc;Iqwcoh]
Examples
--------
::
import jscatter as js
import numpy as np
R=2;NN=5
grid= np.mgrid[-R:R:1j*NN, -R:R:1j*NN,-R:R:1j*NN].reshape(3,-1).T
# points inside of sphere with radius R
p2=1*2 # p defines a superball with 1->sphere p=inf cuboid ....
inside=lambda xyz,R:(np.abs(xyz[:,0])/R)**p2+(np.abs(xyz[:,1])/R)**p2+(np.abs(xyz[:,2])/R)**p2<=1
insidegrid=grid[inside(grid,R)]
Drot=js.formel.Drot(R)
ql=np.r_[0.5:15.:2]
w=np.r_[-100:100:0.1]
p=js.grace()
iqwR=js.dL([js.dynamic.rotDiffusion_w(w,q,insidegrid,Drot) for q in ql])
p.plot(iqwR,le='NN=%.1g q=$wavevector nm\S-1' %(NN))
iqwR=js.dL([js.dynamic.rotDiffusion_w(w,q,2,Drot) for q in ql])
p.plot(iqwR,li=1,sy=0,le='NN=10 $wavevector nm\S-1')
p.yaxis(min=-0.001,max=0.001,scale='n')
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,(float,int)):
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:(np.abs(xyz[:,0])/R)**2+(np.abs(xyz[:,1])/R)**2+(np.abs(xyz[:,2])/R)**2<=1
cloud=grid[inside(grid,R)]
if cloud.shape[1]==4:
# last column is scattering length
blinc=cloud[:,3]
blcoh=None
cloud=cloud[:,:3]
elif cloud.shape[1]==5:
# last columns are incoherent and coherent scattering length
blinc=cloud[:,3]
blcoh=cloud[:,4]
cloud=cloud[:,:3]
else:
blinc=np.ones(cloud.shape[0])
blcoh=blinc
w = np.array(w, float)
bi2 = blinc ** 2
r,p,t=np.r_[[formel.xyz2rphitheta(r) for r in cloud]].T
pp=p[:,None]
tt=t[:,None]
qr=q*r
if not isinstance(lmax,int):
# lmax = pi * r.max() * q / 6. # a la Cryson
lmax = min(max(int(pi*qr.max()/6.),7),100)
# 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
if blcoh is not None:
# coh is sum over i then squared and sum over m see Lindsay equ 19
bjlylmcoh = [ np.sum((blcoh*spjn(l,qr) *Ylm(np.r_[-l:l+1],l,pp,tt).T).sum(axis=0)**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
else:
Iqwcoh=np.zeros_like(Iqwinc)
result=dA(np.c_[w,Iqwinc,Iqwcoh].T)
result.modelname=sys._getframe().f_code.co_name
result.setColumnIndex(iey=None)
result.columnname='w;Iqwinc;Iqwcoh'
result.radiusOfGyration=np.sum(r**2)**0.5
if blcoh is not None:
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):
"""
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
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.nSiteJumpDiffusion_w(w=w,q=q,N=3,t0=0.001,r0=0.12) for q in ql])
p.plot(iqw)
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).
"""
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=sys._getframe().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
[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 instruement 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 in t space with attributes means, sigmas, amps which are used instead of si, mi, ai.
This represents the fourier transform of multi gauss resolution from t to w space.
The mi are used as mi from resolution_w result in a phase shift.
Returns
-------
dataArray
.means
.amps
.sigmas
Notes
-----
In a typical inelastic experiment the resolution is measured by e.g. a vanadium meausrement (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
--------
::
import jscatter as js
# read data
vana=js.dL('vana_0p2mm.scat') # use your files here
start={'s0':0.5,'m0':0,'a0':1,'bgr':0.0073}
dm=5
vana[0].setlimit(m0=[-dm,dm],m1=[-dm,dm],m2=[-dm,dm],m3=[-dm,dm],m4=[-dm,dm],m5=[-dm,dm])
vana[0].fit(js.dynamic.resolution_w,start,{},{'w':'X'})
"""
gauss=lambda x,mean,sigma: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:
means = [m0,m1,m2,m3,m4,m5]
sigmas = 1./np.array(resolution.sigmas)
amps = np.array(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.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(q,t) in time domain. t in units of ns.
If None a constant function equal one is used like elastic scattering to fit resolution measurement.
resolution : dataArray
dataArray that describes the resolution function from a fit with resolution_w or parametrized.
A nonzero bgr in resolution is ignored and needs to be added afterwards.
Resolution width are in the range of 6 1/ns (IN5 TOF) or 1 1/ns (Spheres BS).
w : array
Frequencies for the result, e.g. from experimental data.
If w is None the frequencies w of the resolution 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.
tmax=1/(min(resolution_width)*tfactor) determines the resolution to decay as :math:`e^{-tfactor^2/2}`.
THe timestep is dt=1/max(|w|). A minimum of len(w) steps is used (which might increase tmax).
Increase tfactor if artefacts (wobbling) from the limited timewindow are visible as the limited timeintervall
acts like a window function for the Fourier transform.
Returns
-------
dataArray : A symmetric spectrum 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 transfom of timemodel.
.Iqt timemodel(t,kwargs) dataarray returned from timemodel.
Implicitly this is the Fourier transform to timedomain after a succesfull fit in w domain.
Using a heuristic model in timedomain as multiple Gaussians or stretched exponentials allows a convenient
transform to timedomain 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} \\
&=\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 timedomain :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 tranform of the timemodel I(q,t) is
.. math:: I(q,w) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} R_t(t,m_i,s_i,a_i) 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} R_t(n\Delta t,...) I(q,n\Delta t) e^{-i mn/N}
with :math:`t_{max}=tfactor/min(s_i)` large enough that the resolution decayed to be negligible.
Actually the resolution acts like a window function to reduce spectral leakage with vanishing values at :math:`t_{max}` .
Nevertheless, due to the cutoff at :math:`t_{max}` a wobbling migth appear indicating that :math:`t_{max}` needs to be larger.
**Mixed domain models**
Associativity and Convolution theorem allow to mix models from frequency domain and time domain.
Fourier transform needs the resolution included in time domain as it acts like a window function.
After tranformation to frequency domain the w domain models have to be convoluted with FFT transformed model.
Resolution is already taken into account in this way.
Examples
--------
Other usage example with a comparison of w domain and transfomed from time domain can be found in
:ref:`A comparison of different dynamic models in frequency domain` .
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('resolution width about 6 ns\S-1\N, Q=%.2g nm\S-1\N' %(qq))
# compare diffusion with convolution and transform from timedomain
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='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='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')
"""
# prerequisites
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 )
# filter for given values (remove None) and drop bgr in resolution
if isinstance(resolution,str):
si=np.r_[0.5]
else:
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)).sum(axis=0) # resolution normalized to timeresol(w=0)=1
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=result.modelname+'_t2w'
except:
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 intervalls.
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 None
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=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