# -*- 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/>.
#
"""
Structure factors and directly related functions for scattering
related to interaction potentials between particles.
Return values are dataArrays were useful.
To get only Y values use .Y
For RMSA an improved algorithm is used based on the original idea (see Notes in RMSA).
"""
from __future__ import print_function
from __future__ import division
import sys
import os
import inspect
import math,cmath
import time
import numpy as np
from numpy import linalg as la
import scipy
import scipy.integrate
import scipy.fftpack
import scipy.constants as constants
import scipy.special as special
from .dataarray import dataArray as dA
from .dataarray import dataList as dL
from . import parallel
from .graceplot import GracePlot as grace
from . import formel
from .lattice import lattice,rhombicLattice, bravaisLattice, scLattice,bccLattice,\
fccLattice,diamondLattice,hexLattice,hcpLattice,pseudoRandomLattice
try:
from . import fscatter
useFortran=True
except:
useFortran = False
_path_=os.path.realpath(os.path.dirname(__file__))
# variable to allow printout for debugging as if debug:print 'message'
# set it to integer value above debuglevel
debug=False
def _sqcoefOriginalHP(ir,eta,gek,ak,a=0.,b=0.,c=0.,f=0.,u=0.,v=0.,gamk=0.,seta=0.,sgek=0.,sak=0.,scal=0.,g1=0.):
"""
CALCULATES RESCALED VOLUME FRACTION AND CORRESPONDING COEFFICIENTS
This is only for documenting the difference to the old algorithm.
This is the iterative part to find rescaling parameter to get G(1+)>0 (Gillian condition) if G(1+)>0
Returns:
ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
seta IS THE RESCALED VOLUME FRACTION.
sgek IS THE RESCALED CONTACT POTENTIAL.
sak IS THE RESCALED SCREENING CONSTANT.
a,b,c,f,u,v ARE THE MSA COEFFICIENTS.
g1=G(1+) IS THE CONTACT VALUE OF G(R/SIG);
FOR THE GILLAN CONDITION, THE DIFFERENCE FROM
ZERO INDICATES THE COMPUTATIONAL ACCURACY.
IR > 0: NORMAL EXIT, IR IS THE NUMBER OF ITERATIONS.
< 0: FAILED TO CONVERGE.
This is equivalent to the original HP Fortran code.
The different conditions might have saved computing time in 1981.
For some parameter conditons the rescaling is needed but not done.
Also for some parameter contributions the wrong root for Fwww is used.
"""
# set to zero to get debug messages; debuglevel>10 no messages
debuglevel=1
itm=40 # original 40
acc=5.e-6
if debug>debuglevel: print('-- ')
if ak>=(1+8.*eta):
# for large screening (scl is small and ak is large)
# ix=1 SOLVE FOR LARGE K, RETURN G(1+)
ix,ir,g1,eta,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(1,ir,g1,eta,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1,useHP=True)
if debug>debuglevel: print('large screening ', ir, g1, ak, gamk, 'abcfuv', a, b, c, f, u, v)
if ir<0 or g1>=0: # error or already a good solution is returned
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
else:
# we have to rescale the solution in the later as here g+<0
pass
seta=min(eta,0.2)
if ak>=(1+8.*eta) or gamk>=0.15:
# find a rescaled eta with g+>=0 for strong coupling or low volume fraction
j=0.
f1=0.
f2=0.
while True: # loop for Newton iteration to find g+=0
j=j+1
if (j>itm):
return -1,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
if (seta<=0.0): seta=eta/j # g+<0 -> rescale eta
if (seta>0.6): seta=0.35/j # rescaled eta>0.6 rescale to smaller value
e1=seta # e1 first eta
# ix=2 RETURN FUNCTION TO SOLVE FOR ETA(GILLAN)
ix,ir,f1,e1,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(2,ir,f1,e1,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1,useHP=True)
e2=seta*1.01 # increase scaled eta
ix,ir,f2,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(2,ir,f2,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1,useHP=True)
e2=e1-(e2-e1)*f1/(f2-f1) # new approximation for scaled eta
seta=e2 # save for next iteration or as result
delta=abs((e2-e1)/e1) # relative change
if delta<acc: break # if changes are small enough then break
if debug>debuglevel: print('rescaling with %i iterations leads to scaling by %.3g' % (j, seta / eta))
# ix=4 RETURN G(1+) FOR ETA=ETA(GILLAN).
ix,ir,g1,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g11= \
_sqfun(4,ir,g1,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1,useHP=True)
ir=j
###---------------end of Newton loop
if debug>debuglevel: print( 'rescaled ',ir,g1,ak,gamk,'abcfuv',a,b,c,f,u,v,'ak>,seta>eta ',ak>=(1+8.*eta), seta>=eta)
if ak>=(1+8.*eta): # in this case return anyway
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
else:
if seta>=eta: # seta>eta indicates succesfull rescaling with g1 as zero
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
ix,ir,g1,eta,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(3,ir,g1,eta,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1,useHP=True)
if debug>debuglevel: print('after scaling ',ir,g1,ak,gamk,'abcfuv',a,b,c,f,u,v)
if ir>=0 :
if g1<0.: ir=-3 # rescaling not successful
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
def _sqcoef(ir,eta,gek,ak,a=0.,b=0.,c=0.,f=0.,u=0.,v=0.,gamk=0.,seta=0.,sgek=0.,sak=0.,scal=0.,g1=0.):
"""
CALCULATES RESCALED VOLUME FRACTION AND CORRESPONDING COEFFICIENTS
This is the iterative part to find rescaling parameter to get G(1+)>0 (Gillian condition) if G(1+)>0
Returns:
ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
seta IS THE RESCALED VOLUME FRACTION.
sgek IS THE RESCALED CONTACT POTENTIAL.
sak IS THE RESCALED SCREENING CONSTANT.
a,b,c,f,u,v ARE THE MSA COEFFICIENTS.
g1=G(1+) IS THE CONTACT VALUE OF G(R/SIG);
FOR THE GILLAN CONDITION, THE DIFFERENCE FROM
ZERO INDICATES THE COMPUTATIONAL ACCURACY.
IR > 0: NORMAL EXIT, IR IS THE NUMBER OF ITERATIONS.
< 0: FAILED TO CONVERGE.
This is a shorter version of sqcoef which is easier to understand and allows
no bypassing between the conditions in original code which leads to errors for harmless parameter settings.
The idea is the original idea (see [2]_) to calculate the MSA and to rescale if g+<0 .
"""
# set to zero to get debug messages; debuglevel>10 no messages
debuglevel=1
itm=80 # original 40
acc=5.e-6
fix=0.5
if debug>debuglevel: print('-- ')
# just try to solve
ix,ir,g1,eta,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(1,ir,g1,eta,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1)
if debug>debuglevel: print('first try ',ir,g1,ak,gamk,'abcfuv',a,b,c,f,u,v)
if (ir == -2):
# FAILED TO CONVERGE in Newton algorith to find zero, only in classical HP solution,
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
elif (ir == -4):
# no root found in first try
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
elif (g1<0):
# we have to rescale the solution in the later as here g+<0
pass
elif (g1>=0): # already a good solution is returned
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
seta=min(eta,0.2)
# find a rescaled eta with g+>=0 for strong coupling or low volume fraction
j=0.
f1=0.
f2=0.
while True: # loop for Newton iteration to find g+=0
j=j+1
if (j>itm):
return -1,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
if (seta<=0.0): seta=eta/j # g+<0 -> rescale eta
if (seta>0.6): seta=0.35/j # rescaled eta>0.6 rescale to smaller value
e1=seta # e1 first eta
# ix=2 RETURN FUNCTION TO SOLVE FOR ETA(GILLAN)
ix,ir,f1,e1,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(2,ir,f1,e1,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1)
e2=seta*1.01 # increase scaled eta
ix,ir,f2,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1= \
_sqfun(2,ir,f2,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1)
e2=e1-(e2-e1)*f1/(f2-f1) # new approximation for scaled eta
seta=e2 # save for next iteration or as result
delta=abs((e2-e1)/e1) # relative change
if delta<acc: break # changes are small enough then break
if debug>debuglevel: print('rescaling with %i iterations leads to scaling by %.3g'%(j,seta/eta))
# ix=4 RETURN G(1+) FOR ETA=ETA(GILLAN) with all parameters.
ix,ir,g1,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g11= \
_sqfun(4,ir,g1,e2,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1)
if (seta>0.64) or (seta<eta):
ir=-3 # rescaling not successful
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
ir=j
###---------------end of Newton loop
if debug>debuglevel: print('rescaled ',ir,g1,ak,gamk,'abcfuv',a,b,c,f,u,v,'ak>,seta,eta ',ak>=(fix+8.*eta), seta,eta)
return ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
def _sqfun(ix,ir,fval,evar,reta,rgek,rak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1,useHP=False):
"""
CALCULATES VARIOUS COEFFICIENTS AND FUNCTION VALUES FOR _sqcoef
this is the NOT rescaled solution! == MSA
Options
ix =1: SOLVE FOR LARGE K, RETURN G(1+).
2: RETURN FUNCTION TO SOLVE FOR ETA(GILLAN).
3: ASSUME NEAR GILLAN, SOLVE, RETURN G(1+).
4: RETURN G(1+) FOR ETA=ETA(GILLAN).
SETA IS THE RESCALED VOLUME FRACTION.
SGEK IS THE RESCALED CONTACT POTENTIAL.
SAK IS THE RESCALED SCREENING CONSTANT.
A,B,C,F,U,V ARE THE MSA COEFFICIENTS.
G1=G(1+) IS THE CONTACT VALUE OF G(R/SIG);
FOR THE GILLAN CONDITION, THE DIFFERENCE FROM
ZERO INDICATES THE COMPUTATIONAL ACCURACY.
IR > 0: NORMAL EXIT, IR IS THE NUMBER OF ITERATIONS.
< 0: FAILED TO CONVERGE.
The root of the quartic F = w4*fa**4+w3*fa**3+w2*fa**2+w1*fa+w0 needs to be found.
in this code we have two choices in the source code.
One for documentation and the second as the correct solution:
1. to use the original HayterPenfold algorithm from the Fortran code as also eg used in SASVIEW and SASFIT
with an estimate for the root of Fwww which is refined by Newton algorithm
which results under specific conditions in the wrong root
test with e.g.
for scl in np.r_[1:10]:p.plot(js.sf.RMSA(q=x,R=3.1,scl=scl, gamma=1.1, eta=0.5),legend='%.3g' %scl)
the correct branch can be verified by using the PercusYevivk as limit
2. original idea from Hayter paper [1]_ as *default solution*
find all roots (by numpy.roots) and take the physical root with g(r/diameter<1)=0
in this code there is no difference between ix=1 and 3
with structurefactor.debug=11 you get output for g(r) and the zeros of Fwww (see source code)
"""
# set to zero to get debug messages; debuglevel>10 no messages
debuglevel=1
acc=1e-6 # stop criterion for Newton
itm=40 # max number of iterations
# needed parameters with changes for iteration
eta=evar # volume fraction
scal=(reta/evar)**(1/3.) # scaling factor
sak=rak/scal # scaled dimensionless screening constant
val=rgek if abs(rgek)>1e-9 else 1e-9 # prevent zero and just take small value
sgek=val*scal*math.exp(rak-sak) # scaled contact potential
gek=sgek
ak=sak
# -----------------reproduce original fortran code
# using these variables is important to reduce the dependency on accuracy of float64
# and maybe it makes it a bit faster
eta2=eta**2
eta3=eta2*eta
e12=12.*eta
e24=e12+e12
ak2=ak**2
ak1=1+ak
dak2=1.0/ak2
dak4=dak2*dak2
d=1-eta
d2=d*d
dak=d/ak
dd2=1.0/d2
dd4=dd2*dd2
dd45=dd4*2.0e-1
eta3d=3.*eta
eta6d=eta3d+eta3d
eta32=eta3+eta3
eta2d=eta+2.0
eta2d2=eta2d*eta2d
eta21=2.0*eta+1.0
eta22=eta21*eta21
# all coefficients from appendix in the paper [1]
al1=-eta21*dak
al2=(14*eta2-4*eta-1)*dak2
al3=36*eta2*dak4
b1=-(eta2+7.*eta+1.)*dak
b2=9.*eta*(eta2+4.*eta-2.)*dak2
b3=12.*eta*(2*eta2+8.*eta-1.)*dak4
n1=-(eta3+3.*eta2+45.*eta+5.)*dak
n2=(eta32+3.*eta2+42.*eta-20.)*dak2
n3=(eta32+30.*eta-5.)*dak4
n4=n1+24.*eta*ak*n3
n5=eta6d*(n2+4.*n3)
f1=eta6d/ak
f2=d-12.*eta*dak2
ff1=f1*f1
ff2=f2*f2
ff=ff1+ff2
f1f2=2.*f1*f2
t1=(eta+5.)/(5.*ak)
t2=eta2d*dak2
t3=-12.*eta*gek*(t1+t2)
t4=eta3d*ak2*(t1*t1-t2*t2)
t5=eta3d*(eta+8.)*0.1-2.*eta22*dak2
# ------------
a1=(e24*gek*(al1+al2+ak1*al3)-eta22)*dd4
bb1=(1.5*eta*eta2d2-12.*eta*gek*(b1+b2+ak1*b3))*dd4
v1=(eta21*(eta2-2.*eta+10.)*0.25-gek*(n4+n5))*dd45
p1=(gek*(ff1+ff2-f1f2)-0.5*eta2d)*dd2
T1=t3+t4*a1+t5*bb1
if (sak>15) and (ix==1):
if debug>debuglevel: print('(sak>15) and (ix==1)',ak)
# this corresponds to ibig=1 in original Hayter-Penfold code for large screening
# large screening means the screenig length 1/kappa is small compared to 2R and we are in the hard sphere limit
# if ak is big -> cosh = sinh and a lot simplifies in asymptotic solution
# but at same time cosh(ak) may exceeds numerical limits for really large ak
a3=e24*(eta22*dak2-0.5*d2-al3)*dd4
bb3=e12*(0.5*d2*eta2d-eta3d*eta2d2*dak2+b3)*dd4
v3=((eta3-6.*eta2+5.)*d-eta6d*(2.*eta3-3.*eta2+18.*eta+10.)*dak2+e24*n3)*dd45
p3=(ff1-ff2)*dd2
T3=t4*a3+t5*bb3+e12*t2-0.4*eta*(eta+10.)-1.
M6=T3*a3-e12*v3*v3
M5=T1*a3+a1*T3-e24*v1*v3
M4=T1*a1-e12*v1*v1
L6=e12*p3*p3
L5=e24*p1*p3-2.*bb3-ak2
L4=e12*p1*p1-2.*bb1
W56=M5*L6-L5*L6
W46=M4*L6-L4*M6
fa=-W46/W56
ca=-fa
f=fa
c=ca
b=bb1+bb3*fa
a=a1+a3*fa
v=v1+v3*fa
g1=-(p1+p3*fa)
fval=g1 if g1>1e-3 else 0.
seta=evar
# g24 = e24*gek*math.exp(ak) # prevent math range error in exp for large ak (-> small scl)
# u = (ak2*ak*ca-g24)/(ak2*g24) # so we rewrite this to have exp(-ak)
u=ak*ca/e24/gek*math.exp(-ak)-1/ak2 # same as above two lines but this prevents math range error
return ix,ir,fval,evar,reta,rgek,rak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
# small sak for the remaining
sk=math.sinh(ak)
ck=math.cosh(ak)
ckma=ck-1.-ak*sk
skma=sk-ak*ck
a2=e24*(al3*skma+al2*sk-al1*ck)*dd4
a3=e24*(eta22*dak2-0.5*d2+al3*ckma-al1*sk+al2*ck)*dd4
bb2=e12*(-b3*skma-b2*sk+b1*ck)*dd4
bb3=e12*(0.5*d2*eta2d-eta3d*eta2d2*dak2-b3*ckma+b1*sk-b2*ck)*dd4
v2=(n4*ck-n5*sk)*dd45
v3=((eta3-6.*eta2+5.)*d-eta6d*(2.*eta3-3.*eta2+18.*eta+10.)*dak2+e24*n3+n4*sk-n5*ck)*dd45
# define...
p2=(ff*sk+f1f2*ck)*dd2
p3=(ff*ck+f1f2*sk+ff1-ff2)*dd2
T2=t4*a2+t5*bb2+e12*(t1*ck-t2*sk)
T3=t4*a3+t5*bb3+e12*(t1*sk-t2*(ck-1.))-0.4*eta*(eta+10.)-1.
M1=T2*a2-e12*v2*v2
M2=T1*a2+T2*a1-e24*v1*v2
M3=T2*a3+T3*a2-e24*v2*v3
M4=T1*a1-e12*v1*v1
M5=T1*a3+T3*a1-e24*v1*v3
M6=T3*a3-e12*v3*v3
# ix is defined from the _sqcoef
# large k or close to GILLAN CONDITION g1==0 as explained in [1]
if ix==1 or ix==3:
# YES - G(X=1+) = 0
# COEFFICIENTS AND FUNCTION VALUE
L1=e12*p2*p2
L2=e24*p1*p2-2.*bb2
L3=e24*p2*p3
L4=e12*p1*p1-2.*bb1
L5=e24*p1*p3-2.*bb3-ak2
L6=e12*p3*p3
W16=M1*L6-L1*M6
W15=M1*L5-L1*M5
W14=M1*L4-L1*M4
W13=M1*L3-L1*M3
W12=M1*L2-L1*M2
W26=M2*L6-L2*M6
W25=M2*L5-L2*M5
W24=M2*L4-L2*M4
W36=M3*L6-L3*M6
W35=M3*L5-L3*M5
W34=M3*L4-L3*M4
W32=M3*L2-L3*M2
W46=M4*L6-L4*M6
W56=M5*L6-L5*M6
#####QUARTIC COEFFICIENTS W(I)
# these are used in
# fun = w0+(w1+(w2+(w3+w4*fa)*fa)*fa)*fa =w4*fa**4+w3*fa**3+w2*fa**2+w1*fa+w0
w4=W16*W16-W13*W36
w3=2.*W16*W15-W13*(W35+W26)-W12*W36
w2=W15*W15+2.*W16*W14-W13*(W34+W25)-W12*(W35+W26)
w1=2.*W15*W14-W13*W24-W12*(W34+W25)
w0=W14*W14-W12*W24
# now find root of fun
if useHP:
# this documents the original HayterPenfold algorithm as found in original fortran code
# to find the correct root an estimate is used and refined by Newton method
# fails eg for R=3.1 gam=1.1 eta=0.5 when scl 6.1999 -> 6,2 as sak changes over 1
# or scl=1.37382379588 R=2.5 gam=5.1 eta=0.6 as the found root results in g(r<1)>0
# reason: in Newton refining an arbitrary root is found
if (ix==1): # large screening
# LARGE K estimate for the zero of Fwwww
fap=(W14-W34-W46)/(W12-W15+W35-W26+W56-W32)
else: # ix=3 no large screening
# ASSUME NOT TOO FAR FROM GILLAN CONDITION.
# IF BOTH RGEK AND RAK ARE SMALL, USE P-W ESTIMATE.of the zero of Fwwww
g1=0.5*eta2d*dd2*math.exp(-gek)
pg=p1+g1
ca=ak2*pg+2.*(bb3*pg-bb1*p3)+e12*g1*g1*p3
ca=-ca/(ak2*p2+2.*(bb3*p2-bb2*p3))
fap2=-(pg+p2*ca)/p3
if ((gek>0) and (sgek<=2.0) and (sak<=1.0)):
# gek>0 as this is only for positive contact potentials
#this was introduced in the SASFIT conversion (C code)
e24g=e24*gek*math.exp(ak)
pwk=math.sqrt(e24g)
qpw=(1.-math.sqrt(1.+2.*d2*d*pwk/eta22))*eta21/d
g1=-qpw*qpw/e24+0.5*(eta2d)*dd2
pg=p1+g1
ca=ak2*pg+2.*(bb3*pg-bb1*p3)+e12*g1*g1*p3
ca=-ca/(ak2*p2+2.*(bb3*p2-bb2*p3))
fap=-(pg+p2*ca)/p3
# print('PWEstimate',fap,fap2,( sgek<=2.0) and ( sak<=1.0))
# now find a better estimate of the zero by Newton iteration
# RB: this algorithm finds different roots dependent on sgek and sak
# the roots are somehow arbitrary in the 4 possible ones,
# the main time it is one of the two centered which make no big jumps but the outer ones make large jumps in the result
ii=0
while True:
ii=ii+1
if (ii>itm): # FAILED TO CONVERGE IN ITM ITERATIONS
ir=-2
return ix,ir,fval,evar,reta,rgek,rak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
fa=fap # estimated zero pole of fun
fun=w0+(w1+(w2+(w3+w4*fa)*fa)*fa)*fa # function to minimize
fund=w1+(2.*w2+(3.*w3+4.*w4*fa)*fa)*fa # derivative of fun
fap=fa-fun/fund # new value as next estimate
if fa==0: continue # fa is 0 if gek is zero
delta=abs((fap-fa)/fa) # difference
if delta<acc: break
# found one and use this zero
ir=ir+ii
fa=fap
ca=-(W16*fa*fa+W15*fa+W14)/(W13*fa+W12)
g1=-(p1+p2*ca+p3*fa)
else:
# original idea from Hayter paper [1]_
# take all roots and use the physical root with g(r/diameter<1)=0
# in this code there is no difference between ix=1 or 3
# The algorithm relies on computing the eigenvalues of the companion matrix
x0=np.roots([w4,w3,w2,w1,w0]) # 114µs slower than direct calculation, but this is not the bottle neck
if np.all((x0.imag/x0.real)<1e-3):
# if the imaginary part of complex roots is small use also these
# in some cases this is the correct solution in gr
fa=x0.real
else:
fa=x0[np.isreal(x0)].real # 6.5µs
fa.sort() # we have up to 4 real roots and each of the following has up to 4 values
ca=-(W16*fa*fa+W15*fa+W14)/(W13*fa+W12)
g1=-(p1+p2*ca+p3*fa)
b=bb1+bb2*ca+bb3*fa
a=a1+a2*ca+a3*fa
# choose the correct root by calculating g(r) (sin transform) and using the one with g(r<1)=0
# here i choose explicitly 1-delta
delta=0.05
nn=(2**13+0) # n number of points to get reliable fft
dqr2=np.r_[0,delta:nn*delta:delta] # points to calculate S(dqr2)
kk=1//delta # index of last point smaller 1
# calc the valueof g(x) with x=1-delta=kk*delta in equ.12 of[1]_
gr1=[delta*np.sum((_SQMSA(dqr2,scal,eta,ak,gek,aa,bb,cca,ffa)-1)*dqr2*np.sin(kk*delta*dqr2))
for aa,bb,cca,ffa in zip(a,b,ca,fa)]
grval=[1+ggr/(12*np.pi*eta*kk*delta) for ggr in gr1]
if len(fa)==0 or np.min(grval)>0.1:
# no real root found or not grval close to zero
ir = -4
return ix,ir,fval,evar,reta,rgek,rak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1.max() if np.size(g1) else g1
# chose the one with grval close to zero
chooseone=np.argmin(np.abs(grval))
if debug>debuglevel:
# this writes the calculated g(r) to a file for checking of g(r)
rrr=2*np.pi*np.fft.rfftfreq(len(dqr2),d=delta) # points in r domain from rfft r/diameter
# doing sin transform with rfft results in minus in front of imag part compared to equation 12 in HP paper [1]
# delta* is to get correct integral
#gr=[delta*np.fft.rfft((_SQMSA(dqr2,scal,eta,ak,gek,aa,bb,cca,ffa)-1)*dqr2).imag
# for aa,bb,cca,ffa in zip(a,b,ca,fa)]
gr=[delta*scipy.fftpack.dst(_SQMSA(dqr2,scal,eta,ak,gek,aa,bb,cca,ffa)-1)*dqr2
for aa,bb,cca,ffa in zip(a,b,ca,fa)]
# [1:] to avoid rrr=zero
gr=[1-ggr[1:]/(12*np.pi*eta*rrr[1:]) for ggr in gr]
# choose one with minimum mean value g(r) for rrr<1 which should be zero
# above we use only one value and choose smallest grval
# here we choose the smallest mean value which is often not correct but here it is only for demo
grval=[grr[rrr[1:]<0.9][1:].mean() for grr in gr]
print('grval ',grval)
temp=dL()
for i,grr in enumerate(gr):
temp.append(np.c_[rrr[1:],grr].T)
temp[-1].choosen=chooseone
temp[-1].zero=fa[i]
temp[-1].g1=g1[i]
temp[-1].legend='g(r<1)= %.3g' %(grval[i])
temp.savetxt('testgr.dat')
print('zeros,g1,choosen zero',fa,g1,chooseone)
fa=fa[chooseone]
ca=ca[chooseone]
g1=-(p1+p2*ca+p3*fa)
# end searching the root- recalculating final result------------------------
fval=(g1 if abs(g1)>1e-3 else 0.)
seta=evar
f=fa
c=ca
b=bb1+bb2*ca+bb3*fa
a=a1+a2*ca+a3*fa
v=(v1+v2*ca+v3*fa)/a
else:
# -> ix==2 or ix==4
ca=ak2*p1+2.*(bb3*p1-bb1*p3)
ca=-ca/(ak2*p2+2.0*(bb3*p2-bb2*p3))
fa=-(p1+p2*ca)/p3
# fval will contain g1 for Newton iteration ix=2,4
if ix==2: fval=M1*ca*ca+(M2+M3*fa)*ca+M4+M5*fa+M6*fa*fa
if ix==4: fval=-(p1+p2*ca+p3*fa)
f=fa
c=ca
b=bb1+bb2*ca+bb3*fa
a=a1+a2*ca+a3*fa
v=(v1+v2*ca+v3*fa)/a
g24=e24*gek*math.exp(ak)
u=(ak2*ak*ca-g24)/(ak2*g24)
return ix,ir,fval,evar,reta,rgek,rak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1
def _SQMSA(qR2,scal,eta,ak,gek,a,b,c,f):
"""
equation 14 Hayter-Penfold paper [1] in sfRMSA to calculate the final structure factor
"""
K=np.where(qR2==0,1e-15,qR2/scal) # catch zero
if ak>25: # c==-f and
# avoid to large ak to prevent math range error
# ak>15 has f=-c from sqfun so the sinh and cosh terms cancel for large ak
sinhsk=0.
coshsk=0.
else:
sinhsk=math.sinh(ak)
coshsk=math.cosh(ak)
sink=np.sin(K)
cosk=np.cos(K)
K2=K*K
K3=K2*K
K4=K3*K
KK2ak2=1/K/(K2+ak**2)
a_K=a*(sink-K*cosk)/K3 \
+b*((2./K**2-1)*K*cosk+2*sink-2./K)/K3 \
+a*eta*(24./K3+4.*(1-6./K2)*sink-(1-12./K2+24./K4)*K*cosk)/2./K3 \
+c*(ak*coshsk*sink-K*sinhsk*cosk)*KK2ak2 \
+f*(ak*sinhsk*sink-K*(coshsk*cosk-1))*KK2ak2 \
+f*(cosk-1)/K2 \
-gek*(ak*sink+K*cosk)*KK2ak2
msa=1/(1-24.*eta*a_K)
MSA=np.where(qR2==0,-1/a,msa) # -1/a is correct solution for qR2==zero
return MSA
[docs]def RMSA(q, R, scl, gamma, molarity=None, eta=None,useHP=False):
"""
Structure factor for a screened coulomb interaction in rescaled mean spherical approximation (RMSA).
Structure factor according to Hayter-Penfold [1]_ [2]_ .
Consider a scattering system consisting of macro ions, counter ions and solvent.
Here an improved algorithm is used based on the original idea described in [1]_ (see Notes).
Parameters
----------
q : array; N dim
Scattering vector; units 1/nm
R : float
Radius of the object; units nm
molarity : float
Number density n in units mol/l. Overides eta, if both given.
scl : float>0
Screening length; units nm; negative values evaluate to scl=0.
gamma : float
| contact potential; units kT
| gamma=Zm/(pi*e0*e*R*(2+k*R))
| Zm = Z* effective surface charge
| e0,e free permittivity and dielectric constant
| k=1/scl inverse screening length of Debye-Hückel potential
eta : float
Volume fraction as eta=4/3*pi*R**3*n with number density n.
useHP : True, default False
To use the original Hayter/Penfold algorithm. This gives wrong results for some parameter conditions.
It should ONLY be used for testing.
See example examples/test_newRMSAAlgorithm.py for a direct comparison.
Returns
-------
dataArray : .
- .volumeFraction = eta
- .rescaledVolumeFraction
- .screeningLength
- .gamma=gamma
- .contactpotential
- .S0 structure factor at q=0
- .scalingfactor factor for rescaling to get g+1=0; if =1 nothing was scaled and it is MSA
Notes
-----
Improved algorithm
The Python code is deduced from the original Hayter-Penfold Fortran code (1981, ILL Grenoble).
This is also used in other common SAS programs as SASfit or SASview (translated to C).
The original algorithm determines the root of a quartic F(w1,w2,w3,w4) by an estimate (named PW estimate),
refining it by a Newton algorithm. As the PW estimate is somtimes not good enough this results in an
arbitrary root of the quartic in the Newton algorithm. The solution therefore jumps between different
posibilities by small changes of the parameters.
We use here the original idea from [1]_ to calculate G(r<0) for all four roots of F(w1,w2,w3,w4) and use
the physical solution with G(r<R)=0.
See examples/test_newRMSAAlgorithm.py for a direct comparison.
Validity
The calculation of charge at the surface or screening length from a solute ion concentration is explicitly dedicate
to the user. The Debye-Hückel theory for a macro ion in screened solution is a far field theory as a linearization
of the Poisson-Boltzmann (PB) theory and from limited validity (far field or low charge -> linearization).
Things like reverting charge layer, ion condensation at the surface, pH changes at the surface or other things
might appear. Before calculating please take these things into account. Close to the surface the PB has to be solved.
The DH theory can still be used if the charge is thus an effective charge named Z*, which might be different from the
real surface charge.
See Ref [3]_ for details.
References
----------
.. [1] J. B. Hayter and J. Penfold, Mol. Phys. 42, 109 (1981).
.. [2] J.-P. Hansen and J. B. Hayter, Mol. Phys. 46, 651 (2006).
.. [3] L. Belloni, J. Phys. Condens. Matter 12, R549 (2000).
"""
"""
Original Doc of the Hayter Penfold Fortran routine::
seta is the rescaled volume fraction.
sgek is the rescaled contact potential.
sak is the rescaled screening constant.
a,b,c,f,u,v are the msa coefficients.
g1=g(1+) is the contact value of g(r/sig);
for the Gillan condition, the difference from
zero indicates the computational accuracy.
ROUTINE TO CALCULATE S(Q*SIG) FOR A SCREENED COULOMB
POTENTIAL BETWEEN FINITE PARTICLES OF DIAMETER 'SIG'
AT ANY VOLUME FRACTION. THIS ROUTINE IS MUCH MORE POWER-
FUL THAN "SQHP" AND SHOULD BE USED TO REPLACE THE LATTER
IN EXISTING PROGRAMS. NOTE THAT THE COMMON AREA IS
CHANGED; IN PARTICULAR, THE POTENTIAL IS PASSED
DIRECTLY AS 'GEK' = GAMMA*EXP(-K) IN THE PRESENT ROUTINE.
JOHN B.HAYTER (I.L.L.) 19-AUG-81
CALLING SEQUENCE:
CALL SQHPA(QQ,SQ,NPT,IERR)
QQ: ARRAY OF DIMENSION NPT CONTAINING THE VALUES OF Q*SIG AT WHICH S(Q*SIG) WILL BE CALCULATED.
SQ: ARRAY OF DIMENSION NPT INTO WHICH VALUES OF S(Q*SIG) WILL BE RETURNED.
NPT: NUMBER OF VALUES OF Q*SIG.
IERR > 0: NORMAL EXIT; IERR=NUMBER OF ITERATIONS.
-1: NEWTON ITERATION NON-CONVERGENT IN "SQCOEF"
-2: NEWTON ITERATION NON-CONVERGENT IN "SQFUN".
-3: CANNOT RESCALE TO G(1+) > 0.
ON ENTRY:
ETA: VOLUME FRACTION
GEK: THE CONTACT POTENTIAL GAMMA*EXP(-K)
AK: THE DIMENSIONLESS SCREENING CONSTANT
AK = KAPPA*SIG WHERE KAPPA IS THE INVERSE SCREENING
LENGTH AND SIG IS THE PARTICLE DIAMETER.
ON EXIT:
GAMK IS THE COUPLING: 2*GAMMA*S*EXP(-K/S), S=ETA**(1/3).
SETA, SGEK AND SAK ARE THE RESCALED INPUT PARAMETERS.
SCAL IS THE RESCALING FACTOR: (ETA/SETA)**(1/3).
G1=G(1+), THE CONTACT VALUE OF G(R/SIG).
A,B,C,F,U,V ARE THE CONSTANTS APPEARING IN THE ANALYTIC
SOLUTION OF THE MSA (HAYTER-PENFOLD; MOL.PHYS. 42: 109 (1981))
NOTES:
(A) AFTER THE FIRST CALL TO SQHPA, S(Q*SIG) MAY BE EVALUATED
AT OTHER Q*SIG VALUES BY REDEFINING THE ARRAY QQ AND CALLING
"SQHCAL" DIRECTLY FROM THE MAIN PROGRAM.
(B) THE RESULTING S(Q*SIG) MAY BE TRANSFORMED TO G(R/SIG)
USING THE ROUTINE "TROGS".
(C) NO ERROR CHECKING OF INPUT PARAMETERS IS PERFORMED;
IT IS THE RESPONSIBILITY OF THE CALLING PROGRAM TO VERIFY
VALIDITY.
SUBROUTINES REQUIRED BY SQHPA:
(1) SQCOEF RESCALES THE PROBLEM AND CALULATES THE
APPROPRIATE COEFFICIENTS FOR "SQHCAL".
(2) SQFUN CALCULATES VARIOUS VALUES FOR "SQCOEF".
(3) SQHCAL CALCULATES H-P S(Q*SIG) GIVEN A,B,C,F.
"""
R=abs(R)
error={-1: 'NEWTON ITERATION NON-CONVERGENT IN _sqcoef',
-2: 'NEWTON ITERATION NON-CONVERGENT IN _sqfun',
-3: 'CANNOT RESCALE TO G(1+) > 0.',
-4: 'no physical root with G(r<1) < 0.1 in _sqfun found'} # added for new algorithm
# get volume fraction eta from number density and radius R
if isinstance(molarity,(float,int)):
molarity=abs(molarity)
numdens=constants.N_A*molarity*1e-24 # from mol/l to particles/nm**3
eta=4/3.*np.pi*R**3*numdens
elif isinstance(eta,(float,int)):
numdens=eta/(4/3.*np.pi*R**3)
molarity=numdens/(constants.N_A*1e-24)
else:
raise Exception('one of molarity/eta needs to be given.') # dimensionless screening constant ak
if eta <= 0.: eta = 1e-10
#if eta>1: raise Exception('eta needs to be smaller 1.')
if scl<=0:
ak=1e20
else:
ak=2*R/scl
# to large ak make math error in exp , anyway then we have a hard sphere
if ak>200:ak=200
# the contact potential in kT
gek=gamma*math.exp(-ak)
# coupling
gamk=2.*eta**(1/3.)*gek*math.exp(ak-ak/eta**(1/3.))
# ----------do the rescaling in _sqcoef--------------------
# _sqcoef does the rescaling to satisfy the Gillian condition with g1==0 according to [2]
# therein _sqfun calculates the NOT rescaled solution described in [1]
if useHP:
ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1=_sqcoefOriginalHP(ir=0,eta=eta,gek=gek,ak=ak,gamk=gamk)
else:
ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1=_sqcoef(ir=0,eta=eta,gek=gek,ak=ak,gamk=gamk)
# catch error
if ir<0:
print(ir,error[ir],'g+ =',g1,'ak=',ak)
return ir
# dimensionless q scale
q=np.atleast_1d(q)
qR2=2*R*q
# calc values by _SQMSA
sq=_SQMSA(qR2,scal,seta,sak,sgek,a,b,c,f)
result=dA(np.r_[[q,sq]])
result.setColumnIndex(iey=None)
# add important parameters
result.volumeFraction=eta
result.rescaledVolumeFraction=seta
result.molarity=molarity
result.screeningLength=scl
result.gamma=gamma
result.contactpotential=gek
result.S0=-1/a
result.scalingfactor=scal
result.gplus1=[g1,ir]
result.modelname=inspect.currentframe().f_code.co_name
result._coefficients={key: value for (value,key ) in zip([ir,eta,gek,ak,a,b,c,f,u,v,gamk,seta,sgek,sak,scal,g1],
['ir','eta','gek','ak','a','b','c','f','u','v','gamk','seta','sgek','sak','scal','g1'])}
return result
[docs]def sq2gr(Sq,R,interpolatefactor=2):
r"""
Radial distribution function g(r) from structure factor Sq.
The result strongly depends on quality of S(Q) (number of data points, Q range, smoothness).
Read [2]_ for details of this inversion problem and why it may fail.
Parameters
----------
Sq : dataArray
Structure factor e.g. in units as [Q]=1/nm
- .X wavevector
- .Y structure factor
R : float
Estimate for the radius of the particles.
interpolatefactor : int
Number of points between points in interpolation for rfft.
2 doubles the points
Returns
-------
dataArray
.n0 approximated from :math:`2\pi^2 n_0=\int_0^{Q_{max}} [S(Q) -1]Q^2 dQ`
Notes
-----
One finds that
.. math:: g(r)-1=(2\pi^2 n_0 r)^{-1} \int_0^\infty [S(Q) -1]Qsin(qr)dQ
with :math:`2\pi^2 n_0=\int_0^\infty [S(Q) -1]Q^2 dQ` defining :math:`n_0`.
As we have only a limited Q range (:math:`0 < Q < \infty` ), limited accuracy and number of Q values
we require that :math:`mean(g(R/2<r<R3/4))=0`.
Examples
--------
::
import jscatter as js
import numpy as np
p=js.grace()
p.multi(2,1)
q=js.loglist(0.01,100,2**13)
p[0].clear();p[1].clear()
R=2.5
eta=0.3;scl=5
n=eta/(4/3.*np.pi*R**3) # unit 1/nm**3
sf=js.sf.RMSA(q=q,R=R,scl=scl, gamma=50, eta=eta)
gr=js.sf.sq2gr(sf,R,interpolatefactor=1)
sfcut=js.sf.RMSA(js.loglist(0.01,10,2**10),R=R,scl=scl, gamma=50, eta=eta)
grcut=js.sf.sq2gr(sfcut,R,interpolatefactor=5)
p[0].plot(sf.X*2*R,sf.Y,le=r'\xG=50')
p[1].plot(gr.X/2/R,gr[1],le=r'\xG=50')
p[1].plot(grcut.X/2/R,grcut[1],le=r'\xG=50 \f{}Q\smax\N=10')
sfh=js.sf.RMSA(q=q,R=R,scl=scl, gamma=0.01, eta=eta)
grh=js.sf.sq2gr(sfh,R,interpolatefactor=1)
p[0].plot(sfh.X*2*R,sfh.Y,le=r'\xG=0.01')
p[1].plot(grh.X/2/R,grh[1],le=r'\xG=0.01')
p[0].xaxis(max=20,label='2RQ')
p[1].xaxis(max=4*R,label='r/(2R)')
p[0].yaxis(max=2,min=0,label='S(Q)')
p[1].yaxis(max=2.5,min=0,label='g(r)')
p[0].legend(x=10,y=1.8)
p[1].legend(x=4,y=1.8)
p[0].title('Comparison RMSA')
p[0].subtitle('R=%.2g, eta=%.2g, scl=%.2g' %(R,eta,scl))
References
----------
.. [1] Yarnell, J. L., Katz, M. J., Wenzel, R. G., & Koenig, S. H. (1973).
Structure factor and radial distribution function for liquid argon at 85 K.
Physical Review A, 7(6), 2130.
.. [2] On the determination of the pair correlation function from liquid structure factor measurements
A.K. Soper Chemical Physics 107, 61-74, (1986)
"""
nn=interpolatefactor*Sq.X.shape[0]
delta=Sq.X.max()/nn
Q=np.r_[0:Sq.X.max():1j*nn]
rrr=2*np.pi*scipy.fftpack.fftfreq(nn,delta)[1:nn//2]
# interpolation for more or smoother points
Yminus1=np.interp(Q,Sq.X,Sq.Y-1)
#Yminus1=scipy.interpolate.interp1d(Sq.X,Sq.Y-1,kind=2)(Q)
# doing sine transform to solve the sin integral
Sqdst=scipy.fftpack.dst(Yminus1*Q)
gr=1/(2*np.pi**2*rrr)*Sqdst[2::2]/(2*np.pi)
#grminus=1/(2*np.pi**2*rrr)*Sqdst[3::2]/(2*np.pi)
n0=-1/(2*np.pi**2)*scipy.integrate.simps(Sq.X**2*(Sq.Y-1),Sq.X)
factor=abs(gr[abs(rrr-R)<R/2].mean())
gr=1+gr/factor
#grminus=1+grminus/factor
result=dA(np.c_[rrr,gr].T)
result.n0=n0
result.modelname=inspect.currentframe().f_code.co_name
return result
[docs]def PercusYevick(q,R,molarity=None,eta=None):
"""
The Percus-Yevick structure factor.
Structure factor for the potential U(r)= (inf for 0<r<R) and (0 for R<r).
Parameters
----------
q : array; N dim
scattering vector; units 1/(R[unit])
R : float
Radius of the object
eta : float
volume fraction as eta=4/3*pi*R**3*n with number density n in units or R
molarity : float
number density in mol/l and defines q and R units to 1/nm and nm to be correct
preferred over eta if both given
Returns
-------
dataArray
structure factor for given q
Notes
-----
Problem is given in [1]_; solution in [2]_ and best description of the solution is in [3]_.
References
----------
.. [1] J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958).
.. [2] M. S. Wertheim, Phys. Rev. Lett. 10, 321 (1963).
.. [3] D. J. Kinning and E. L. Thomas, Macromolecules 17, 1712 (1984).
"""
q=np.atleast_1d(q)
R=abs(R)
# get volume fraction eta from number density and radius R
if isinstance(molarity,(float,int)):
molarity=abs(molarity)
numdens=constants.N_A*molarity*1e-24 # from mol/l to particles/nm**3
eta=4/3.*np.pi*R**3*numdens
elif isinstance(eta,(float,int)):
eta=abs(eta)
numdens=eta/(4/3.*np.pi*R**3)
molarity=numdens/(constants.N_A*1e-24)
else:
raise Exception('one of molarity/eta needs to be given.')
if R==0 or eta==0:
Sq=np.ones_like(q)
a=1.
else:
u=q*R*2
u=np.where(u>=0.01,u,np.ones_like(u)*0.01) # problems with number limits for to small u and avoid zero
a=(1+2*eta)**2/(1-eta)**4
b=-3/2*eta*(eta+2)**2/(1-eta)**4
UU=(a*(np.sin(u)-u*np.cos(u))+
b*((2/u**2-1)*u*np.cos(u)+2*np.sin(u)-2/u)+
eta*a/2*(24/u**3+4*(1-6/u**2)*np.sin(u)-
(1-12/u**2+24/u**4)*u*np.cos(u)))
_Sq=1/(1+24*eta/u**3*UU)
Sq=np.where(u>0.02,_Sq,np.ones_like(u)/a) # for low u we use the S(q=0) = 1/a
result=dA(np.r_[[q,Sq]])
result.setColumnIndex(iey=None)
result.modelname=inspect.currentframe().f_code.co_name
result.eta=eta
result.molarity=molarity
result.radius=R
result.Sq0=1/a
return result
[docs]def PercusYevick2D(q,R=1,eta=0.1,a=None):
"""
The PercusYevick structure factor in 2D
Structure factor for the potential U(r)= (inf for 0<r<R) and (0 for R<r).
Parameters
----------
q : array; N dim
scattering vector; units 1/(R[unit])
R : float, default 1
Radius of the object
eta : float, default 0.1
packing fraction as eta=pi*R**2*n with number density n
maximum hexagonal closed = (np.pi*R**2)/(3/2.*3**0.5*a**2)
Rmax=a*3**0.5/2. with max packing of 0.9069
a : float, default None
hexagonal lattice constant
if not None the packing fraction in hexagonal lattice
as eta=(np.pi*R**2)/(3/2.*3**0.5*a**2) is used
Returns
-------
dataArary
References
----------
.. [1] Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions:
Structure factors for the hard-disk (D=2) mixtures in simple explicit form
Yaakov Rosenfeld Phys. Rev. A 42, 5978
"""
if a!=None:
eta=(np.pi*R**2)/(3/2.*3**0.5*a**2)
q=np.atleast_1d(q)
if R==0 or eta==0:
Sq=np.ones_like(q)
else:
qR=lambda q: q*R
u=np.piecewise(q,[q==0],[1e-8,qR]) # exchange q=zero with small Q as limit
Xi=(1+eta)/(1-eta)**3
G=(1-eta)**-1
Z=(1-eta)**-2
A=(1+(2*eta-1)*Xi+2*eta*G)/eta
B=((1-eta)*Xi-1-3*eta*G)/eta
UU=4*eta*(A*(special.j1(u)/u)**2+B*special.j0(u)*special.j1(u)/u+G*special.j1(2*u)/u)
Sq=1/(1+UU)
result=dA(np.r_[[q,Sq]])
result.setColumnIndex(iey=None)
result.packingfraction=eta
result.R=R
result.a=(np.pi*R**2/(eta*3/2.*3**0.5))**0.5
result.modelname=inspect.currentframe().f_code.co_name
return result
[docs]def PercusYevick1D(q,R=1,eta=0.1):
"""
The PercusYevick structure factor in 1D
Structure factor for the potential U(r)= (inf for 0<r<R) and (0 for R<r).
Parameters
----------
q : array; N dim
scattering vector; units 1/(R[unit])
R : float
Radius of the object
eta : float
packing fraction as eta=2*R*n with number density n
Returns
-------
dataArray
[q,structure factor]
Notes
-----
.. [1] Exact solution of the Percus-Yevick equation for a hard-core fluid in odd dimensions
Leutheusser E Physica A 1984 vol: 127 (3) pp: 667-676
.. [2] On the equivalence of the Ornstein–Zernike relation and Baxter’s relations for a one-dimensional simple fluid
Chen M Journal of Mathematical Physics 1975 vol: 16 (5) pp: 1150
"""
q=np.atleast_1d(q)
D=2.*R
nn=eta/D
if R==0 or eta==0:
Sq=np.ones_like(q)
else:
# exchange q=zero with small Q as limit
Q=np.piecewise(q,[q==0],[1e-8,lambda q: q])
xi=(1-D*nn)
cQ=-2*(1./Q/xi*np.sin(Q*D) + nn/Q**2/xi**2*(1-np.cos(Q*D)))
Sq=(1-cQ*nn)**-1 # =1/A eq 6 and 8b of [1]_
result=dA(np.r_[[q,Sq]])
result.setColumnIndex(iey=None)
result.packingfraction=eta
result.R=R
result.nkTkappa=xi**2
result.modelname=inspect.currentframe().f_code.co_name
return result
[docs]def stickyHardSphere(q,R,width,depth,molarity=None,phi=None):
"""
Structure factor of a square well potential with depth and width (sticky hard spheres).
Sticky hard sphere model is derived using a perturbative solution of the factorized
form of the Ornstein-Zernike equation and the Percus-Yevick closure relation.
The perturbation parameter is width/(width+2R)
Parameters
----------
q : array; N dim
Scattering vector; units 1/(R[unit])
R : float
Radius of the hard sphere
phi : float
Volume fraction of the hard core particles
molarity : float
Number density in mol/l and defines q and R units to 1/nm and nm to be correct
Preferred over phi if both given.
depth : float
Potential well depth in kT
depth >0 (U<0); positive potential allowed (repulsive) see [1]_.
width : float
Width of the square well
Notes
-----
| The potential U(r) is defined as
| r<2R U(r)=infinity
| 2R<r<2R+width U(r)=-depth in [kT]
| r >2R+width U(r)=0
eps=width/(2*R+width)
stickyness=exp(-depth)/12./eps
References
----------
.. [1] S.V. G. Menon, C. Manohar, and K. S. Rao, J. Chem. Phys. 95, 9186 (1991)
.. [2] M. Sztucki, T. Narayanan, G. Belina, A. Moussaïd, F. Pignon, and H. Hoekstra, Phys. Rev. E 74, 051504 (2006)
.. [3] W.-R. Chen, S.-H. Chen, and F. Mallamace, Phys. Rev. E 66, 021403 (2002)
.. [4] G. Foffi, E. Zaccarelli, F. Sciortino, P. Tartaglia, and K. A. Dawson, J. Stat. Phys. 100, 363 (2000)
"""
# get volume fraction eta from number density and radius R
if isinstance(molarity,(float,int)):
numdens=constants.N_A*molarity*1e-24 # from mol/l to particles/nm**3
phi=4/3.*np.pi*R**3*numdens
elif isinstance(phi,(float,int)):
numdens=phi/(4/3.*np.pi*R**3)
molarity=numdens/(constants.N_A*1e-24)
else:
raise Exception('one of molarity/eta needs to be given.')
# to prevent math range error in fits
if depth<-200:depth=-200
q=np.atleast_1d(q)
Q=np.piecewise(q,[q==0],[1e-8,lambda q: q]) # avoid zero
eps=width/(2*R+width) # pertubation parameter
tau=math.exp(-depth)/12./eps # stickyness
eta=phi*(1-eps)**3
lam=(1+0.5*eta)/(1-eta)**2/(eta**2/(1-eta)-eta/12.+tau)
mu=lam*eta*(1-eta)
al=(1+2*eta-mu)/(1-eta)**2
be=(-3*eta+mu)/2./(1-eta)**2
k=Q*(2*R+width)
sink=np.sin(k)
cosk=np.cos(k)
Ak=1+12*eta*(al*((sink-k*cosk)/k**3)+be*(1-cosk)/k**2-lam/12.*sink/k)
Bk=12*eta*(al*(0.5/k-sink/k**2+(1-cosk)/k**3)+be*(1/k-sink/k**2)-lam/12.*((1-cosk)/k))
Sk=1./(Ak*Ak+Bk*Bk)
result=dA(np.r_[[q,Sk]])
result.welldepth=depth
result.weelwidth=width
result.stickiness=tau
result.volumefraction=phi
result.eta=eta
result.molarity=molarity
result.modelname=inspect.currentframe().f_code.co_name
return result
[docs]def adhesiveHardSphere(q,R,tau,delta,molarity=None,eta=None):
"""
structure factor of a adhesive hard sphere potential (a square well potential)
Parameters
----------
q : array; N dim
scattering vector; units 1/(R[unit])
R : float
radius of the hard core
eta : float
volume fraction of the hard core particles
molarity : float
number density in mol/l and defines q and R units to 1/nm and nm to be correct
preferred over eta if both given
tau : float
stickiness
delta : float
width of the square well
Notes
-----
The potential U(r) is defined as
r<2R U(r)=infinity
2R<r<2R+delta U(r)=-depth=ln(12*tau*delta/(2R+delta))
r >2R+delta U(r)=0
References
----------
.. [1] C. Regnaut and J. C. Ravey, J. Chem. Phys. 91, 1211 (1989).
.. [2] C. Regnaut and J. C. Ravey, J. Chem. Phys. 92 (5) (1990), 3250 Erratum
"""
# get volume fraction eta from number density and radius R
if isinstance(molarity,(float,int)):
numdens=constants.N_A*molarity*1e-24 # from mol/l to particles/nm**3
eta=4/3.*np.pi*R**3*numdens
elif isinstance(eta,(float,int)):
numdens=eta/(4/3.*np.pi*R**3)
molarity=numdens/(constants.N_A*1e-24)
else:
raise Exception('one of molarity/eta needs to be given.')
q=np.atleast_1d(q)
sigma=2.*R+delta
k=np.piecewise(q,[q==0],[1e-8,lambda q: q*sigma])
phi=eta*(sigma/(2*R))**3
lam=6.*(tau/phi+1.0/(1.-phi))
try:
lam1=lam+math.sqrt(lam**2-12./phi*(1.+0.5*phi)/(1-phi)**2)
lam2=lam-math.sqrt(lam**2-12./phi*(1.+0.5*phi)/(1-phi)**2)
lambd=lam1 if abs(lam1)<abs(lam2) else lam2
except ValueError: # complex root
return -1
mu=lambd*phi*(1.-phi)
A=0.5*(1.+2.*phi-mu)/(1.-phi)**2
B=0.5*sigma*(mu-3.*phi)/(1.-phi)**2
C=-A*sigma**2-B*sigma+lambd*sigma**2/12.
sink=np.sin(k)
cosk=np.cos(k)
I0=sink/k
I1=(cosk+k*sink-1.0)/k**2
I2=(k**2*sink-2.0*sink+2.0*k*cosk)/k**3
J0=(1-cosk)/k
J1=(sink-k*cosk)/k**2
J2=(2.*sink*k+2.*cosk-k**2*cosk-2.)/k**3
alpha=1.0-12.0*phi*(C/sigma**2*I0+B/sigma*I1+A*I2)
beta=12.0*phi*(C/sigma**2*J0+B/sigma*J1+A*J2)
SQ=1./(alpha*alpha+beta*beta)
result=dA(np.r_[[q,SQ]])
try:
result.welldepth=math.log(12*tau*delta/sigma)
except:
result.welldepth=-np.inf
result.wellwidth=delta
result.stickiness=tau
result.welldepth=math.log(12*tau*delta/sigma) if 12*tau*delta/sigma>0 else np.inf
result.HSvolumefraction=eta
result.phi=phi
result.modelname=inspect.currentframe().f_code.co_name
return result
[docs]def criticalSystem(q,corrlength,isothermalcompress):
"""
Structure factor of a critical system should take the Ornstein-Zernike form.
Parameters
----------
q : array; N dim
scattering vector; units 1/(R[unit])
corrlength : float
correlation length
isothermalcompress : float
isothermal compressibility of the system
Notes
-----
The peaking of the structure factor near Q=0 region is due to attractive interaction.
Away from it the structure factor should be close to the hard sphere structure factor.
Near the critical point we should find
S(Q)=S_PY(Q)+S_OZ(Q)
S_PY Percus Yevik structure factor
S_OZ this function
References
----------
.. [1] Analysis of Critical Scattering Data from AOT/D2O/n-Decane Microemulsions
S. H. Chen, T. L. Lin, M. Kotlarchyk
Surfactants in Solution pp 1315-1330
"""
Q=np.atleast_1d(q)
result=dA(np.r_[[Q,isothermalcompress/(1+Q**2*corrlength**2)]])
result.corrlength=corrlength
result.isothermalcompress=isothermalcompress
result.modelname=inspect.currentframe().f_code.co_name
return result
[docs]def latticeStructureFactor(q, lattice=None, domainsize=1000, asym=0, lg=1, rmsd=0.02, beta=None, hklmax=7, c=1.):
r"""
Radial structure factor S(q) of a crystal lattice with particle asymmetry, DebyeWaller factor, diffusive scattering and domain size.
The peak shape is a Voigt function.
To get the full scattering the formfactor needs to be included (See Notes and Examples).
1-3 dimensional lattice structures with basis containing multiple atoms.
Parameters
----------
q : float
Wavevector in inverse units of lattice constant, units 1/A or 1/nm
domainsize : float
Domainsize of the crystal, units as lattice constant of lattice.
According to Debye-Scherrer equation :math:`fwhm=2\pi/domainsize` the peak width is determined [2]_.
lattice : lattice object
The crystal structure as defined in a lattice object. The size of the lattice is ignored.
One of rhombicLattice, bravaisLattice, scLattice, bccLattice, fccLattice, diamondLattice, hexLattice, hcpLattice.
See respective definitions.
lg : float, default = 1
Lorenzian/gaussian fraction describes the contributions of gaussian and lorenzian shape in peak shape.
- lorenzian/gaussian >> 1 lorenzian,
- lorenzian/gaussian ~ 1 central part gaussian, outside lorenzian wings
- lorenzian/gaussian << 1. gaussian
asym : float, default=0
Asymmetry factor in sigmoidal as :math:`2*fwhm/(1+np.exp(asym*(x-center)))`
For a=0 the Voigt is symmetric with fwhm. See formel.voigt .
rmsd : float, default=0.02
Root mean square displacement :math:`<u^2>^{0.5}` determining the Debye Waller factor.
Units as a.
beta : float, None, dataArray
Asymmetry factor of the formfactor or reduction due to polydispersity.
- None beta=1, No beta assumed (spherical symmetric formfactor, no polydispersity)
- dataArray explicitly given as dataArray with beta in .Y column.
Missing values are interpolated.
- An approximation for polydisperse beta can be found in [1]_ equ.17.
This can be realized by beta=js.dA(np.vstack(q,np.exp(-(q*sr*R)**2)))
with sr as relative standard deviation of gaussian distribution of the size R.
- See .formfactor for different formfactors which explicit calculation of beta.
hklmax : int
Maximum order of the Bragg peaks to include.
c : float, default=1
Porod constant. See 3.8 in [1]_.
Returns
-------
dataArray with columns [q,Sq,DW,beta,Z0q]
- q wavevector
- Sq = S(q) = 1+beta(q)*(Z0(q)-1)*DW(q) structure factor
- DW(q) Debye-Waller factor with (1-DW)=diffusive scattering.
- beta(q) asymmetry factor of the formfactor.
- Z0q lattice factor Z0(q)
Attributes
- .q_hkl peak positions
- .fhkl symmetry factor
- .mhkl multiplicity
Notes
-----
- The scattering intensity of a crystal domain is
.. math:: I(q)={\Delta\rho}^2 n P(q) S(q)
with
- :math:`\Delta\rho` scattering length difference between matrix and particles
- :math:`n` number density (of elementary cells)
- :math:`P(q)` form factor
- :math:`S(q)` structure factor :math:`S(q)`
For inhomogenious particles we can incorporate :math:`\Delta\rho(r)` in the formfactor :math:`P(q)`
if this includes the integrated scattering length differences.
- The structure factor is [1]_ :
.. math:: S(q)=1+ \beta(q)(Z_0(q)-1)*DW(Q)
with
- :math:`\beta(q)=<F(q)>^2/<F(q)^2>` as asymmetry factor [3]_ dependent on the
scattering amplitude :math:`F(q)` and particle polydiispersity
- :math:`DW(q)` Debye Waller factor
- The lattice factor is [1]_ :
.. math :: Z_0(q) = \frac{(2\pi)^{d-1}c}{nv_dq^{d-1}} \sum\limits_{hkl}m_{hkl}f_{hkl}^2L_{hkl}(q)
with
- :math:`n` number of particles per unit cell
- :math:`f_{hkl}` unit cell structure factor that takes into account symmetry-related extinction rules
- :math:`v_d` volume of the d-dimensional unit cell
- :math:`hkl` reflections
- :math:`m_{hkl}` peak multiplicity
- :math:`c` Porod constant :math:`\simeq 1`
- We use a Voigt function for the peak shape :math:`L_{hkl}(q)` (see formel.voigt).
- DW is a Debye Waller like factor as :math:`DW(q)=e^{-q^2<u^2>}` leading to a reduction
of scattered intensity and diffusive scattering.
It has contributions from thermal lattice disorder ( DW factor with 1/3 factor in 3D),
surface roughness and size polydispersity.
- For the limitting behaviour q->0 see the discussion in [1]_ in 3.9. :
"... The zero-order peak is not explicitely considered because of the q^(1-dim) singularity and
because its intensity depends also on the scattering length difference between the lattice inside and outside......
Due to the singularity and since structural features on length scales d > a,
such as packing defects, grain boundaries or fluctuations decaying on larger length scales
are only indirectly considered via the domain size D, eq 30 is not expected to give good agreement with
experimentally determined scattering curves in the range of scattering vectors q < 2π/a.
However, for q > 2π/a, this approach describes remarkably well experimentally measured
high-resolution scattering curves...."
A good description of the real scattering for low Q is shown in
example :ref:`A nano cube build of different lattices`.
Examples
--------
::
import jscatter as js
import numpy as np
q = np.r_[0.001:1:800j]
a = 50.
R=15
sr=0.1
p = js.grace()
beta=js.dA(np.vstack([q,np.exp(-(q*sr*R)**2)]))
p.title('structure factor for hexagonal 2D lattice')
p.subtitle('with diffusive scattering and asymmetry factor beta')
for i,rmsd in enumerate([1., 3., 10., 30.],1):
grid=js.sf.hexLattice(50,5)
hex = js.sf.latticeStructureFactor(q, rmsd=rmsd, domainsize=500., beta=beta,lattice=grid)
p.plot(hex, li=[1, 2, i], sy=0, le='rmsd=$rmsd')
p.plot(hex.X,1-hex[-3], li=[3, 2, i], sy=0)
p.plot(hex.X, hex[-2], li=[2, 2, i], sy=0, le='beta')
p.text(r'broken lines \nshow diffusive scattering',x=0.4,y=6)
p.yaxis(label='S(q)')
p.xaxis(label='q / nm')
p.legend(x=0.6,y=4)
import jscatter as js
import numpy as np
q=np.r_[js.loglist(0.1,3,200),3:40:800j]
unitcelllength=1.5
N=2
R=0.5
sr=0.1
beta=js.dA(np.vstack([q,np.exp(-(q*sr*R)**2)]))
rmsd=0.02
#
scgrid= js.lattice.scLattice(unitcelllength,N)
sc=js.sf.latticeStructureFactor(q, rmsd=rmsd, domainsize=50., beta=beta,lattice=scgrid)
bccgrid= js.lattice.bccLattice(unitcelllength,N)
bcc=js.sf.latticeStructureFactor(q, rmsd=rmsd, domainsize=50., beta=beta,lattice=bccgrid)
fccgrid= js.lattice.fccLattice(unitcelllength,N)
fcc=js.sf.latticeStructureFactor(q, rmsd=rmsd, domainsize=50., beta=beta,lattice=fccgrid)
#
p=js.grace()
p.plot(sc,legend='sc')
p.plot(bcc,legend='bcc')
p.plot(fcc,legend='fcc')
p.text(r'broken lines \nshow diffusive scattering',x=10,y=0.1)
p.yaxis(label='S(q)',scale='l',max=50,min=0.05)
p.xaxis(label='q / nm',scale='l',max=50,min=0.5)
p.legend(x=1,y=30,charsize=1.5)
References
----------
.. [1] Scattering curves of ordered mesoscopic materials.
Förster, S. et al. J. Phys. Chem. B 109, 1347–1360 (2005).
.. [2] Patterson, A.
The Scherrer Formula for X-Ray Particle Size Determination
Phys. Rev. 56 (10): 978–982 (1939)
doi:10.1103/PhysRev.56.978.
.. [3] M. Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).1
"""
# Förster et al, Journal of Applied Crystallography 43 (2010), no. 3, 639–646.
qq=q.copy()
qq[q==0]=min(q[q>0])*1e-4 # avoid zero
n = len(lattice.unitCellAtoms)
vd = lattice.unitCellVolume
dim = len(lattice.latticeVectors)
qhkl,f2hkl,mhkl = lattice.getRadialReciprocalLattice(hklmax)
# Bragg peak shape
_Lhkl=lambda q,center,lg,domainsize,asym: formel.voigt(q,center=center,lg=lg,fwhm=2*np.pi/domainsize,asym=asym).Y
# Debye Waller factor
DW=np.exp(-q ** 2 * rmsd ** 2 )
# lattice factor
Z0q = np.c_[[ m * f2 * _Lhkl(q,qr,lg,domainsize,asym) for qr,f2,m in zip(qhkl, f2hkl, mhkl) ]].sum(axis=0)
Z0q *= (2 * np.pi) ** (dim - 1) * c / n / vd / q ** (dim - 1)
if beta is None:
beta=np.ones_like(q)
elif hasattr(beta,'_isdataArray'):
beta=beta.interp(q)
# structure factor
Sq=1+beta*(Z0q-1)*DW
# prepare result
result=dA(np.vstack([q ,Sq ,DW ,beta ,Z0q ]))
result.setColumnIndex(iey=None)
result.q_hkl= qhkl
result.fhkl = f2hkl
result.mhkl = mhkl
result.latticeconstants=la.norm(lattice.latticeVectors,axis=1)
result.peakFWHM=2*np.pi/domainsize
result.peaksigma=(result.peakFWHM/ (2*np.sqrt(2*np.log(2))))
result.peakAsymmetry=asym
result.domainsize=domainsize
result.rmsd=rmsd
result.lorenzianOverGaussian=lg
result.modelname=sys._getframe().f_code.co_name
return result
def _Z0q(qhkl,qpeaks,f2peaks,peakWidthSigma,rotvector,angle=0):
# rotate qhkl
if rotvector is not None and angle!=0:
R=formel.rotationMatrix(rotvector,angle)
qqhkl=np.einsum('ij,kj->ki',R,qhkl)
else:
qqhkl=qhkl.copy()
# calc Z0q
if useFortran:
lhkl=fscatter.cloud.lhkl
Z0q = np.c_[[f2 * lhkl(qqhkl, q, peakWidthSigma) for q, f2 in zip(qpeaks, f2peaks) if la.norm(q)>0]].sum(axis=0)
else:
# Bragg peak shape
def _Lhkl(q, center, pWsigma):
Lhkl = np.multiply.reduce(np.exp(-0.5*((q-center)/pWsigma)**2)/pWsigma/np.sqrt(2*np.pi),axis=1)
return Lhkl
Z0q = np.c_[[f2 * _Lhkl(qqhkl, q, peakWidthSigma) for q, f2 in zip(qpeaks, f2peaks) if la.norm(q)>0]].sum(axis=0)
return Z0q
[docs]def latticeStructureFactor2D(qxyz,lattice,orientation=None,rotation=None,domainsize=1000,
rmsd=0.02,beta=None,hklmax=3,nGauss=13):
r"""
2D structure factor S(q) of a crystal lattice including particle asymmetry, DebyeWaller factor,
diffusive scattering, rotation and domain size.
To get the full scattering the formfactor needs to be included (See Notes and Examples).
1-3 dimensional lattice structures with basis containing multiple atoms.
Parameters
----------
qxyz : array 3xN
Wavevector array representing a surface in q space (3D), units 1/A or 1/nm.
This can describe a detector plane, section of the Ewald sphere or a line in reciprocal space.
lattice : lattice object
Lattice object with arbitrary atoms particles in unit cell,
or predefined lattice from rhombicLattice, bravaisLattice, scLattice,bccLattice,
fccLattice, diamondLattice, hexLattice, hcpLattice with scattering length of unit cell atoms.
See lattices for examples.
orientation : 3 float as [h,k,l], None
Orientation of the crystal relative to qxyz surface.
- [h,k,l] rotates the crystal that the lattice hkl direction points to the center of qxyz (mean position).
e.g. points to plane center.
- If None the orientation is not changed.
rotation : 5 float as [mean,sigma,h,k,l], None
Rotation of the crystal around axis hkl with Gaussian distribution around mean and width sigma (units rad).
domainsize : float,list, list of directions
Domainsize of the crystal, units as lattice constant of lattice.
According to Debye-Scherrer equation :math:`fwhm=2\pi/domainsize` the peak width is determined [2]_.
- float : assume same domainsize in all directions.
- list 3 float : domainsize in directions of latticeVectors.
- list 4 x 3 : 3 times domainsize in hkl direction as [[size,h,k,l] ,[..],[..] ]
[[3,1,1,1],[100,1,-1,0],[100,1,1,-2]] is thin in 111 direction and others are thick
The user should take care that the directions are nearly orthogonal.
rmsd : float, default=0.02
Root mean square displacement :math:`<u^2>^{0.5}` determining the Debye Waller factor.
Units as lattice constant.
beta : float, None, dataArray
Asymmetry factor of the formfactor or reduction due to polydispersity.
- None beta=1, No beta assumed (spherical symmetric formfactor, no polydispersity)
- dataArray beta explicitly given as dataArray with beta in .Y column.
Missing values are interpolated.
- An approximation for polydisperse beta can be found in [1]_ equ.17.
This can be realized by beta=js.dA(np.vstack(q,np.exp(-(q*sr*R)**2)))
with sr as relative standard deviation of gaussian distribution of the size R.
- See .formfactor for different formfactors which explicit calculation of beta.
hklmax : int
Maximum order of the Bragg peaks.
nGauss : int, default 13
Number of points in integration over Gaussian for rotation width sigma.
Returns
-------
dataArray with columns [qx,qy,qz,Sq,DW,beta,Z0q]
- q wavevector
- Sq = S(q) = 1+beta(q)*(Z0(q)-1)*DW(q) structure factor
- DW(q) Debye-Waller factor with (1-DW)=diffusive scattering.
- beta(q) asymmetry factor of the formfactor.
- Z0q lattice factor Z0(q)
Attributes (+ input parameters)
- .q_hkl peak positions
- .hkl Miller indices
- .peakFWHM full width half maximum
Notes
-----
- The scattering intensity of a crystal domain is
.. math:: I(q)={\Delta\rho}^2 n P(q) S(q)
with
- :math:`\Delta\rho` scattering length difference between matrix and particles
- :math:`n` number density (of elementary cells)
- :math:`P(q)` form factor
- :math:`S(q)` structure factor :math:`S(q)`
For inhomogenious particles we can incorporate :math:`\Delta\rho(r)` in the formfactor :math:`P(q)`
if this includes the integrated scattering length differences.
- The structure factor is [1]_ :
.. math:: S(q)=1+ \beta(q)(Z_0(q)-1)*DW(Q)
with
- :math:`\beta(q)=<F(q)>^2/<F(q)^2>` as asymmetry factor [3]_ dependent on the
scattering amplitude :math:`F(q)` and particle polydispersity
- :math:`DW(q)` Debye Waller factor
- The lattice factor is [1]_ :
.. math :: Z_0(q) = \frac{(2\pi)^3}{mv} \sum\limits_{hkl}f_{hkl}^2L_{hkl}(q,g_{hkl})
with
- :math:`g_{hkl}` peak positions
- :math:`m` number of particles per unit cell
- :math:`f_{hkl}` unit cell structure factor that takes into account symmetry-related extinction rules
- :math:`v` volume of the unit cell
- :math:`hkl` reflections
- The peak shape function is
.. math :: L_{hkl}(q,g_{hkl}) = \frac{1}{ \sqrt{2\pi} \sigma} e^{-\frac{(q-g_{hkl})^2}{2\sigma^2}}
with :math:`\sigma=fwhm/2\sqrt{2log(2)}` related to the domainsize.
Correspondingly :math:`\sigma` is a vector describing the peak shapes in all directions.
- Distributions of domain orientation are included by the parameter rotation that describes
gaussian distributions with mean and sigma around an axis defined by the corresponding hkl indices.
- DW is a Debye Waller like factor as :math:`DW(q)=e^{-q^2<u^2>}` leading to a reduction
of scattered intensity and diffusive scattering.
It has contributions from thermal lattice disorder
( DW factor with 1/3 factor in 3D).
Examples
--------
Comparison fcc and sc to demonstrate selection rules ::
import jscatter as js
import numpy as np
R=8
N=50
ds=10
fcclattice= js.lattice.fccLattice(3.1, 5)
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
qxyz=np.c_[qxy,np.zeros(N**2)]
ffe=js.sf.latticeStructureFactor2D(qxyz,fcclattice,orientation=[1,1,1],domainsize=ds,rmsd=0.1,hklmax=4)
fig=js.mpl.surface(qxyz[:,0],qxyz[:,1],ffe[3].array)
sclattice= js.lattice.scLattice(3.1, 5)
ffs=js.sf.latticeStructureFactor2D(qxyz,sclattice,orientation=[1,1,1],domainsize=ds,rmsd=0.1,hklmax=4)
fig=js.mpl.surface(qxyz[:,0],qxyz[:,1],ffs[3].array)
Comparison of different domainsizes dependent on direction of scattering ::
import jscatter as js
import numpy as np
R=8
N=50
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
qxyz=np.c_[qxy,np.zeros(N**2)]
sclattice= js.lattice.scLattice(2.1, 5)
ds=[[20,1,0,0],[5,0,1,0],[5,0,0,1]]
ffs=js.sf.latticeStructureFactor2D(qxyz,sclattice,orientation=[1,0,0],domainsize=ds,rmsd=0.1,hklmax=2)
fig=js.mpl.surface(qxyz[:,0],qxyz[:,1],ffs[3].array)
fig.axes[0].set_title('symmetric peaks: thiner direction perpendicular to scattering plane')
fig.show()
ffs=js.sf.latticeStructureFactor2D(qxyz,sclattice,orientation=[0,1,0],domainsize=ds,rmsd=0.1,hklmax=2)
fig2=js.mpl.surface(qxyz[:,0],qxyz[:,1],ffs[3].array)
fig2.axes[0].set_title('asymmetric peaks: thin direction is parallel to scattering plane')
fig2.show()
Rotation along [1,1,1] axis. It looks spiky because of low number of points in xy plane ::
import jscatter as js
import numpy as np
# make xy grid in q space
R=8 # maximum
N=800 # number of points
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
# add z=0 component
qxyz=np.c_[qxy,np.zeros(N**2)] # as position vectors
# create sc lattice which includes reciprocal lattice vectors and methods to get peak positions
sclattice= js.lattice.scLattice(2.1, 5)
# We orient to 111 direction perpendicular to center of qxy plane
# this needs crystal rotation by 15 degrees to be alligned to xy plane after rotation to 111 direction
# The crystals rotates by 10 degrees around 111 to broaden peaks.
ds=15; fpi=np.pi/180.
ffs=js.sf.latticeStructureFactor2D(qxyz,sclattice,orientation=[1,1,1],
rotation=[(30+15)*fpi,10*fpi,1,1,1],
domainsize=ds,rmsd=0.1,hklmax=2,nGauss=23)
fig=js.mpl.surface(qxyz[:,0],qxyz[:,1],ffs[3].array)
fig.axes[0].set_title('10 degree rotation around 111 direction')
fig.show()
References
----------
.. [1] Order causes secondary Bragg peaks in soft materials
Förster et al Nature Materials doi: 10.1038/nmat1995
.. [2] Patterson, A.
The Scherrer Formula for X-Ray Particle Size Determination
Phys. Rev. 56 (10): 978–982 (1939)
doi:10.1103/PhysRev.56.978.
.. [3] M. Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 (1983).1
"""
vd=lattice.unitCellVolume
n=len(lattice.unitCellAtoms)
dim=3 # dimensionality
# peakWidthSigma describes Bragg peak width as 3D vector relative to lattice
if isinstance(domainsize,(float,int)):
domainsize=np.array([domainsize]*3)
fwhm = 2 * np.pi / np.abs(domainsize)
peakWidthSigma = fwhm / (2 * np.sqrt(2 * np.log(2)))
elif isinstance(domainsize,list):
if np.ndim(domainsize)==1:
# use latticevector direction
domainsize=np.atleast_1d(domainsize)
# broadening due to domainsize in direction of latticeVectors
fwhm = 2 * np.pi / np.abs(domainsize)
sigma = fwhm / (2 * np.sqrt(2 * np.log(2)))
peakWidthSigma=np.sum([ s*lV/la.norm(lV) for lV,s in zip(lattice.latticeVectors,sigma)],axis=1)
else:
# we assume that width with Miller indices is given
ds=np.array(domainsize)
sigma=2 * np.pi /np.abs(ds[:,0])/ (2 * np.sqrt(2 * np.log(2))) # width as sigma
# transform hkl to real directions by using the latticeVectors
hkldirection=np.einsum('ij,lj',lattice.latticeVectors,ds[:,1:4])
peakWidthSigma = np.sum([s * lV / la.norm(lV) for lV, s in zip(hkldirection, sigma)], axis=1)
else:
raise TypeError('domainsize cannot be interpreted.')
if orientation is not None:
qhkl=lattice.rotate2hkl(qxyz,orientation[0],orientation[1],orientation[2])
if rotation is not None:
# rotation direction
rotvector=rotation[2]*lattice.latticeVectors[0]+\
rotation[3]*lattice.latticeVectors[1]+\
rotation[4]*lattice.latticeVectors[2]
R=formel.rotationMatrix(rotvector,rotation[0])
qhkl=np.einsum('ij,kj->ki',R,qhkl)
else:
rotvector=None
# Debye Waller factor
qr=(qxyz**2).sum(axis=1)**0.5
DW=np.exp(-qr ** 2 * rmsd ** 2 )
# reciprocal lattice
peaks=lattice.getReciprocalLattice(hklmax)
qpeaks=peaks[:,:3] # positions
f2peaks=peaks[:,3] # scattering intensity
if rotation is not None and abs(rotation[1])>0:
# gauss distribution
Z0q=formel.parDistributedAverage(_Z0q, abs(rotation[1]), parname='angle', nGauss=nGauss,
qhkl=qhkl,qpeaks=qpeaks,f2peaks=f2peaks,
peakWidthSigma=peakWidthSigma,rotvector=rotvector,angle=0,ncpu=0 )
else:
# single orientation
Z0q=_Z0q(qhkl,qpeaks,f2peaks,peakWidthSigma,rotvector,0)
Z0q *= (2 * np.pi) ** dim / n / vd
if beta is None:
beta=np.ones_like(qr)
elif hasattr(beta,'_isdataArray'):
beta=beta.interp(qr)
# structure factor
Sq=1+beta*(Z0q-1)*DW
# prepare result
result=dA(np.vstack([qxyz.T ,Sq ,DW ,beta ,Z0q ]))
result.setColumnIndex(iey=None,ix=0,iy=1,iz=2)
result.q_hkl=qpeaks
result.peaksigma=peakWidthSigma
result.domainsize=domainsize
result.rmsd=rmsd
result.rotation=rotation
result.orientation = orientation
result.modelname=sys._getframe().f_code.co_name
return result
#------------------------------------------------------------------
# hydrodynamic function
#see Beenakker ref 2 Table 1, given is phi*gamma0^m/n
_tablegamma0='0.0 0.0 0.0 0.0 0.0 \
0.05 0.0553 0.0542 0.0533 0.0525 0.10 0.1228 0.1177 0.1135 0.1104 0.15 0.2048 0.1918 0.1813 0.1738 \
0.20 0.3038 0.2777 0.2574 0.2432 0.25 0.4224 0.3766 0.3423 0.3186 0.30 0.5627 0.4895 0.4364 0.4005 \
0.35 0.7267 0.6172 0.5402 0.4888 0.40 0.9157 0.7601 0.6538 0.5839 0.45 1.1310 0.9183 0.7776 0.6856'
_gamma0=np.fromstring(_tablegamma0,sep=' ').reshape(-1,5).T
#interpolate polynom order 3
_gamma0poly=np.polyfit(_gamma0[0],_gamma0[1:].T,4) #about 200µs
#calc values as np.polyval(_gamma0poly,xx)
def _Sg(xx,mm1):
"""
from Genz [1] equ 6 with gamma0 from [2]_ Table 1
Sg=C(x) + .......
this is for all ak (see Beenakker ref 2 ) and accurate in (volume fraction)**2
returns array
"""
x=np.where(xx==0,np.ones_like(xx)*1e-5,xx) # avoid zero
x2=2*x
x3=x*x*x
x4=x3*x
cxx=9/2.*(special.sici(x2)[0]/x+ 0.5*np.cos(x2)/x/x+ 0.25*np.sin(x2)/x3-
np.sin(x)**2/x4- 4/x3/x3*(np.sin(x)-x*np.cos(x))**2)
Cx=np.where(xx==0,np.ones_like(xx)*2.5,cxx) # zero is equal 2.5
func=( Cx + 9./4 * np.pi *5/9.* mm1[0] * 9./x3 * special.jn(1.5, x) ** 2 +
9./4 * np.pi * 1. * mm1[1] *25./x3 * special.jn(2.5, x) ** 2 +
9./4 * np.pi * 1. * mm1[2] *49./x3 * special.jn(3.5, x) ** 2 +
9./4 * np.pi * 1. * mm1[3] *81./x3 * special.jn(4.5, x) ** 2 )
return func
def _HINTEGRAL(Q,Rh,molarity,sffunc,sfargs={},numberOfPoints=50):
"""
calculation of hydrodynamic function for one Q
see hydrodynamicFunct
"""
# set to zero to get debug messages; debuglevel>10 no messages
debuglevel=10
phi=4/3*np.pi*Rh**3*constants.N_A*molarity*1e-24
if phi>0.5 :
print('to large volume fraction %.3g in H' %(phi))
return -1
# coefficients for the gamma0^m/n
mm1=np.polyval(_gamma0poly,phi)/phi-1
def Sq(q):
"""structure factor; infinite S(Q=inf)=1 """
# ravel q
sf=sffunc(q.ravel(),**sfargs)
# reshape sf to q shape
if sf._isdataArray: return sf.Y.reshape(q.shape)
return sf.reshape(q.shape)
ak=np.r_[0:np.pi*3:numberOfPoints*3j,np.pi*2:np.pi*53:numberOfPoints*4j]
k=ak/Rh
x= np.cos(np.r_[np.pi:0:-numberOfPoints*2j]) # x is cos(angle(k,k`))
Qmk=np.sqrt(Q**2+k**2-2*Q*k*x[:,None])
#(Sq(Qmk)-1) is correct as compared with [2]_ equ 5.7 and 5.9
integrand=np.sinc(ak/np.pi)**2/(1+phi*_Sg(ak,mm1))*(1-x[:,None]**2)*(Sq(Qmk)-1)
integrandak=np.trapz(integrand,x=x,axis=0)
integral=np.trapz(integrandak,x=ak,axis=0)
return np.r_[Q,3./2./np.pi*integral]
def _HINTEGRALDs(Rh,molarity,numberOfPoints=50):
"""
calculation of hydrodynamic function for the self diffusion Ds
see hydrodynamicFunct
number of points is number of points in integration in a pi interval
"""
# set to zero to get debug messages; debuglevel>10 no messages
debuglevel=10
phi=4./3*np.pi*Rh**3*constants.N_A*molarity*1e-24
if phi>0.5 :
print('to large volume fraction %.3g in Ds' %(phi))
return -1
# coefficients for the gamma0^m/n
mm1=np.polyval(_gamma0poly,phi)/phi-1
ak=np.r_[0:np.pi*3:numberOfPoints*3j,np.pi*3:np.pi*153:numberOfPoints*50j]
integrandDs=np.sinc(ak/np.pi)**2/(1+phi*_Sg(ak,mm1))
integralDs=np.trapz(integrandDs,x=ak)
return 2/np.pi*integralDs
[docs]def hydrodynamicFunct(wavevector,Rh,molarity,intrinsicVisc=None,DsoverD0=None,
structureFactor=None,structureFactorArgs={},
numberOfPoints=50,ncpu=-1):
"""
Hydrodynamic function H(Q) from hydrodynamic pair interaction of spheres in suspension.
We use the theory from Beenakker and Mazur [2]_ as given by Genz [1]_.
The :math:`\delta\gamma`-expansion of Beenakker expresses many body hydrodynamic
interaction within the renormalization approach dependent on the structure factor S(q).
Parameters
----------
wavevector : array
scattering vector q in units 1/nm
Rh : float
effective hydrodynamic radius of particles in nm.
molarity : float
| molarity in mol/l
| This overides a parameter 'molarity' in the structureFactorArgs.
| Rh and molarity define the hydrodynamic interaction, the volume fraction and Ds/D0 for H(Q).
| The structure factor may have a radius different from Rh e.g. for attenuated hydrodynamic interactions.
DsoverD0 : float
| The high Q limit of the hydrodynamic function is for low volume fractions
| Ds/D0= 1/(1+intrinsicVisc * volumeFraction ) with self diffusion Ds.
| Ds is calculated from molarity and Rh.
| This explicit value overrides intrinsic viscosity and calculated Ds/D0.
structureFactor : function, None
| Structure factor S(q) with S(q=inf)=1.0 recommended.
| 1: If structurefactor is None a Percus-Yevik is assumed with molarity and R=Rh.
| 2: A function S(q,...) is given as structure factor, which might be an
| empirical function (e.g. polynominal fit of a measurement)
| First parameter needs to be wavevector q .
| If "molarity" parameter is present it is overwriten by molarity above.
structureFactorArgs : dictionary
Any extra arguments to structureFactor e.g. structFactorArgs={'x':0.123,R=3,....}
intrinsicVisc : float
| Defines the high q limit for the hydrodynamic function.
| effective_viscosity= eta_solvent * (1-intrinsicVisc*Volumefraction )
| intrinsicVisc = 2.5 Einstein hard sphere density 1 g/cm**3
| For proteins instead of volume fraction the protein concentration in g/ml with typical
| protein density 1.37 g/cm^3 is often used.
| Intrinsic Viscosity depends on protein shape (see HYDROPRO).
| Typical real values for intrinsicVisc in practical units cm^3/g
| sphere 1.76 cm^3/g= 2.5 sphere with protein density
| ADH 3.9 = 5.5 a tetrameric protein
| PGK 4.0 = 5.68 two domains with hinge-> elongated
| Rnase 3.2 = 4.54 one domain
| eta_solvent/effective_viscosity = (1-intrinsicVisc * Volumefraction )=Dself/D0
numberOfPoints : integer, default 50
Determines number of integration points in equ 5 of ref [1]_ and therfore accuracy of integration.
The typical accuracy of this function is <1e-4 for (H(q) -highQLimit) and <1e-3 for Ds/D0.
Returns
-------
dataArray
| [0] q values
| [1] hydrodynamic function
| [2] hydrodynamic function only Q dependent part = H(q) -highQLimit
| [3] structure factor for H(q) calculation
| .selfdiffusion Ds
Notes
-----
Ds is calculated according to equ 11 in [1]_ which is valid for volume fractions up to 0.5.
With this assumption the deviation of self diffusion Ds from
Ds=Do*[1-1.73*phi+0.88*phi**2+ O(phi**3)] is smaller 5% for phi<0.2 (10% for phi<0.3)
References
----------
.. [1] U. Genz and R. Klein, Phys. A Stat. Mech. Its Appl. 171, 26 (1991).
.. [2] C. W. J. Beenakker and P. Mazur, Phys. A Stat. Mech. Its Appl. 126, 349 (1984).
.. [3] C. W. J. Beenakker and P. Mazur, Phys. A Stat. Mech. Its Appl. 120, 388 (1983).
"""
# set to zero to get debug messages; debuglevel>10 no messages
debuglevel=0
if structureFactor==None:
# we use PercusYevic Structure factor --> hard spheres
structureFactor=PercusYevick
if 'R' not in structureFactorArgs:
structureFactorArgs={'R':Rh}
sfcode=formel._getFuncCode(structureFactor)
if 'molarity' in structureFactorArgs or \
'molarity' in sfcode.co_varnames[:sfcode.co_argcount] :
# the last examines the function
# overwrite or append 'molarity'
structureFactorArgs=dict(structureFactorArgs, **{'molarity':molarity})
if debug>debuglevel:
p=grace()
XX=np.r_[min(wavevector)/10.:max(wavevector)*2:100j]
p.plot(structureFactor(XX,**structureFactorArgs),line=1,symbol=0)
p.plot(structureFactor(wavevector,**structureFactorArgs))
# Volume fraction
phi=lambda mol,R: 4/3.*np.pi*R**3*constants.N_A*mol/(10e7)**3
if phi(molarity,Rh)>0.5:
raise ValueError('Volume fraction %.3g to high; Chose appropriate Rh or molarity for Volume fraction <0.5' %phi(molarity,Rh))
res=[]
qqq=np.atleast_1d(wavevector)
columnname=['q','HydDynFun','HydDynfun-DsoverD0','structureFactor']
if debug>debuglevel:
print(columnname[:-1])
def cb(res): # for intermediate results
print(res[0],1+res[1],(0+res[1]))
else:
cb=None
Ds=_HINTEGRALDs(Rh=Rh,molarity=molarity,numberOfPoints=numberOfPoints)
if DsoverD0!=None:
Hinf=DsoverD0
elif intrinsicVisc!=None:
DsintrVisc=1/(1+intrinsicVisc*phi(molarity,Rh))
Hinf=DsintrVisc
DsoverD0=DsintrVisc
else:
Hinf=Ds
DsoverD0=Ds
# in parallel for production run
# if debug!= None it will be single thread
res=parallel.doForQlist(_HINTEGRAL,
qqq,
Rh=Rh,
molarity=molarity,
sffunc=structureFactor,
sfargs=structureFactorArgs,
numberOfPoints=numberOfPoints,
ncpu=ncpu,
cb=cb,)
# and calc final result from this
result=dA(np.c_[qqq,
Hinf+np.array(res)[:,1],
np.array(res)[:,1],
structureFactor(wavevector,**structureFactorArgs).Y].T)
if debug>debuglevel:
p.plot(result)
result.Sq=structureFactor
result.SqArgs=str(structureFactorArgs)
result.Rh=Rh
result.molarity=molarity
result.intrinsicVisc=intrinsicVisc
result.phi_Rh=phi(molarity,Rh)
result.DsoverD0=DsoverD0
result.numberOfPoints=numberOfPoints
result.columnname=columnname
result.setColumnIndex(iey=None)
return result