#/usr/bin/env python
# -*- coding: utf-8 -*-
'''
*GSASII small angle calculation module*
=======================================
'''
########### SVN repository information ###################
# $Date: 2019-08-16 15:58:08 +0000 (Fri, 16 Aug 2019) $
# $Author: vondreele $
# $Revision: 4095 $
# $URL: trunk/GSASIIsasd.py $
# $Id: GSASIIsasd.py 4095 2019-08-16 15:58:08Z vondreele $
########### SVN repository information ###################
from __future__ import division, print_function
import os
import math
import numpy as np
import scipy.special as scsp
import scipy.optimize as so
#import pdb
import GSASIIpath
GSASIIpath.SetVersionNumber("$Revision: 4095 $")
import GSASIIpwd as G2pwd
# trig functions in degrees
sind = lambda x: math.sin(x*math.pi/180.)
asind = lambda x: 180.*math.asin(x)/math.pi
tand = lambda x: math.tan(x*math.pi/180.)
atand = lambda x: 180.*math.atan(x)/math.pi
atan2d = lambda y,x: 180.*math.atan2(y,x)/math.pi
cosd = lambda x: math.cos(x*math.pi/180.)
acosd = lambda x: 180.*math.acos(x)/math.pi
rdsq2d = lambda x,p: round(1.0/math.sqrt(x),p)
#numpy versions
npsind = lambda x: np.sin(x*np.pi/180.)
npasind = lambda x: 180.*np.arcsin(x)/math.pi
npcosd = lambda x: np.cos(x*math.pi/180.)
npacosd = lambda x: 180.*np.arccos(x)/math.pi
nptand = lambda x: np.tan(x*math.pi/180.)
npatand = lambda x: 180.*np.arctan(x)/np.pi
npatan2d = lambda y,x: 180.*np.arctan2(y,x)/np.pi
npT2stl = lambda tth, wave: 2.0*npsind(tth/2.0)/wave
npT2q = lambda tth,wave: 2.0*np.pi*npT2stl(tth,wave)
###############################################################################
#### Particle form factors
###############################################################################
def SphereFF(Q,R,args=()):
''' Compute hard sphere form factor - can use numpy arrays
:param float Q: Q value array (usually in A-1)
:param float R: sphere radius (Usually in A - must match Q-1 units)
:param array args: ignored
:returns: form factors as array as needed (float)
'''
QR = Q[:,np.newaxis]*R
return (3./(QR**3))*(np.sin(QR)-(QR*np.cos(QR)))
def SphericalShellFF(Q,R,args=()):
''' Compute spherical shell form factor - can use numpy arrays
:param float Q: Q value array (usually in A-1)
:param float R: sphere radius (Usually in A - must match Q-1 units)
:param array args: [float r]: controls the shell thickness: R_inner = min(r*R,R), R_outer = max(r*R,R)
:returns float: form factors as array as needed
Contributed by: L.A. Avakyan, Southern Federal University, Russia
'''
r = args[0]
if r < 0: # truncate to positive value
r = 0
if r < 1: # r controls inner sphere radius
return SphereFF(Q,R) - SphereFF(Q,R*r)
else: # r controls outer sphere radius
return SphereFF(Q,R*r) - SphereFF(Q,R)
def SpheroidFF(Q,R,args):
''' Compute form factor of cylindrically symmetric ellipsoid (spheroid)
- can use numpy arrays for R & AR; will return corresponding numpy array
param float Q : Q value array (usually in A-1)
param float R: radius along 2 axes of spheroid
param array args: [float AR]: aspect ratio so 3rd axis = R*AR
returns float: form factors as array as needed
'''
NP = 50
AR = args[0]
if 0.99 < AR < 1.01:
return SphereFF(Q,R,0)
else:
cth = np.linspace(0,1.,NP)
try:
Rct = R[:,np.newaxis]*np.sqrt(1.+(AR**2-1.)*cth**2)
except:
Rct = R*np.sqrt(1.+(AR**2-1.)*cth**2)
return np.sqrt(np.sum(SphereFF(Q[:,np.newaxis],Rct,0)**2,axis=2)/NP)
def CylinderFF(Q,R,args):
''' Compute form factor for cylinders - can use numpy arrays
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: [float L]: cylinder length (A)
returns float: form factor
'''
L = args[0]
NP = 200
alp = np.linspace(0,np.pi/2.,NP)
QL = Q[:,np.newaxis]*L
LBessArg = 0.5*QL[:,:,np.newaxis]**np.cos(alp)
LBess = np.where(LBessArg<1.e-6,1.,np.sin(LBessArg)/LBessArg)
QR = Q[:,np.newaxis]*R
SBessArg = QR[:,:,np.newaxis]*np.sin(alp)
SBess = np.where(SBessArg<1.e-6,0.5,scsp.jv(1,SBessArg)/SBessArg)
LSBess = LBess*SBess
return np.sqrt(2.*np.pi*np.sum(np.sin(alp)*LSBess**2,axis=2)/NP)
def CylinderDFF(Q,L,args):
''' Compute form factor for cylinders - can use numpy arrays
param float Q: Q value array (A-1)
param float L: cylinder half length (A)
param array args: [float R]: cylinder radius (A)
returns float: form factor
'''
R = args[0]
return CylinderFF(Q,R,args=[2.*L])
def CylinderARFF(Q,R,args):
''' Compute form factor for cylinders - can use numpy arrays
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: [float AR]: cylinder aspect ratio = L/D = L/2R
returns float: form factor
'''
AR = args[0]
return CylinderFF(Q,R,args=[2.*R*AR])
def UniSphereFF(Q,R,args=0):
''' Compute form factor for unified sphere - can use numpy arrays
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: ignored
returns float: form factor
'''
Rg = np.sqrt(3./5.)*R
B = np.pi*1.62/(Rg**4) #are we missing *np.pi? 1.62 = 6*(3/5)**2/(4/3) sense?
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*Rg/np.sqrt(6)))**3
return np.sqrt(np.exp((-Q[:,np.newaxis]**2*Rg**2)/3.)+(B/QstV**4))
def UniRodFF(Q,R,args):
''' Compute form factor for unified rod - can use numpy arrays
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: [float R]: cylinder radius (A)
returns float: form factor
'''
L = args[0]
Rg2 = np.sqrt(R**2/2+L**2/12)
B2 = np.pi/L
Rg1 = np.sqrt(3.)*R/2.
G1 = (2./3.)*R/L
B1 = 4.*(L+R)/(R**3*L**2)
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*Rg2/np.sqrt(6)))**3
FF = np.exp(-Q[:,np.newaxis]**2*Rg2**2/3.)+(B2/QstV)*np.exp(-Rg1**2*Q[:,np.newaxis]**2/3.)
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*Rg1/np.sqrt(6)))**3
FF += G1*np.exp(-Q[:,np.newaxis]**2*Rg1**2/3.)+(B1/QstV**4)
return np.sqrt(FF)
def UniRodARFF(Q,R,args):
''' Compute form factor for unified rod of fixed aspect ratio - can use numpy arrays
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: [float AR]: cylinder aspect ratio = L/D = L/2R
returns float: form factor
'''
AR = args[0]
return UniRodFF(Q,R,args=[2.*AR*R,])
def UniDiskFF(Q,R,args):
''' Compute form factor for unified disk - can use numpy arrays
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: [float T]: disk thickness (A)
returns float: form factor
'''
T = args[0]
Rg2 = np.sqrt(R**2/2.+T**2/12.)
B2 = 2./R**2
Rg1 = np.sqrt(3.)*T/2.
RgC2 = 1.1*Rg1
G1 = (2./3.)*(T/R)**2
B1 = 4.*(T+R)/(R**3*T**2)
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*Rg2/np.sqrt(6)))**3
FF = np.exp(-Q[:,np.newaxis]**2*Rg2**2/3.)+(B2/QstV**2)*np.exp(-RgC2**2*Q[:,np.newaxis]**2/3.)
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*Rg1/np.sqrt(6)))**3
FF += G1*np.exp(-Q[:,np.newaxis]**2*Rg1**2/3.)+(B1/QstV**4)
return np.sqrt(FF)
def UniTubeFF(Q,R,args):
''' Compute form factor for unified tube - can use numpy arrays
assumes that core of tube is same as the matrix/solvent so contrast
is from tube wall vs matrix
param float Q: Q value array (A-1)
param float R: cylinder radius (A)
param array args: [float L,T]: tube length & wall thickness(A)
returns float: form factor
'''
L,T = args[:2]
Ri = R-T
DR2 = R**2-Ri**2
Vt = np.pi*DR2*L
Rg3 = np.sqrt(DR2/2.+L**2/12.)
B1 = 4.*np.pi**2*(DR2+L*(R+Ri))/Vt**2
B2 = np.pi**2*T/Vt
B3 = np.pi/L
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*Rg3/np.sqrt(6)))**3
FF = np.exp(-Q[:,np.newaxis]**2*Rg3**2/3.)+(B3/QstV)*np.exp(-Q[:,np.newaxis]**2*R**2/3.)
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*R/np.sqrt(6)))**3
FF += (B2/QstV**2)*np.exp(-Q[:,np.newaxis]**2*T**2/3.)
QstV = Q[:,np.newaxis]/(scsp.erf(Q[:,np.newaxis]*T/np.sqrt(6)))**3
FF += B1/QstV**4
return np.sqrt(FF)
###############################################################################
#### Particle volumes
###############################################################################
def SphereVol(R,args=()):
''' Compute volume of sphere
- numpy array friendly
param float R: sphere radius
param array args: ignored
returns float: volume
'''
return (4./3.)*np.pi*R**3
def SphericalShellVol(R,args):
''' Compute volume of spherical shell
- numpy array friendly
param float R: sphere radius
param array args: [float r]: controls shell thickness, see SphericalShellFF description
returns float: volume
'''
r = args[0]
if r < 0:
r = 0
if r < 1:
return SphereVol(R) - SphereVol(R*r)
else:
return SphereVol(R*r) - SphereVol(R)
def SpheroidVol(R,args):
''' Compute volume of cylindrically symmetric ellipsoid (spheroid)
- numpy array friendly
param float R: radius along 2 axes of spheroid
param array args: [float AR]: aspect ratio so radius of 3rd axis = R*AR
returns float: volume
'''
AR = args[0]
return AR*SphereVol(R)
def CylinderVol(R,args):
''' Compute cylinder volume for radius & length
- numpy array friendly
param float R: diameter (A)
param array args: [float L]: length (A)
returns float:volume (A^3)
'''
L = args[0]
return np.pi*L*R**2
def CylinderDVol(L,args):
''' Compute cylinder volume for length & diameter
- numpy array friendly
param float: L half length (A)
param array args: [float D]: diameter (A)
returns float:volume (A^3)
'''
D = args[0]
return CylinderVol(D/2.,[2.*L,])
def CylinderARVol(R,args):
''' Compute cylinder volume for radius & aspect ratio = L/D
- numpy array friendly
param float: R radius (A)
param array args: [float AR]: =L/D=L/2R aspect ratio
returns float:volume
'''
AR = args[0]
return CylinderVol(R,[2.*R*AR,])
def UniSphereVol(R,args=()):
''' Compute volume of sphere
- numpy array friendly
param float R: sphere radius
param array args: ignored
returns float: volume
'''
return SphereVol(R)
def UniRodVol(R,args):
''' Compute cylinder volume for radius & length
- numpy array friendly
param float R: diameter (A)
param array args: [float L]: length (A)
returns float:volume (A^3)
'''
L = args[0]
return CylinderVol(R,[L,])
def UniRodARVol(R,args):
''' Compute rod volume for radius & aspect ratio
- numpy array friendly
param float R: diameter (A)
param array args: [float AR]: =L/D=L/2R aspect ratio
returns float:volume (A^3)
'''
AR = args[0]
return CylinderARVol(R,[AR,])
def UniDiskVol(R,args):
''' Compute disk volume for radius & thickness
- numpy array friendly
param float R: diameter (A)
param array args: [float T]: thickness
returns float:volume (A^3)
'''
T = args[0]
return CylinderVol(R,[T,])
def UniTubeVol(R,args):
''' Compute tube volume for radius, length & wall thickness
- numpy array friendly
param float R: diameter (A)
param array args: [float L,T]: tube length & wall thickness(A)
returns float: volume (A^3) of tube wall
'''
L,T = args[:2]
return CylinderVol(R,[L,])-CylinderVol(R-T,[L,])
################################################################################
#### Distribution functions & their cumulative fxns
################################################################################
def LogNormalDist(x,pos,args):
''' Standard LogNormal distribution - numpy friendly on x axis
ref: http://www.itl.nist.gov/div898/handbook/index.htm 1.3.6.6.9
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: width of distribution (m)
param float shape: shape - (sigma of log(LogNormal))
returns float: LogNormal distribution
'''
scale,shape = args
return np.exp(-np.log((x-pos)/scale)**2/(2.*shape**2))/(np.sqrt(2.*np.pi)*(x-pos)*shape)
def GaussDist(x,pos,args):
''' Standard Normal distribution - numpy friendly on x axis
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: width of distribution (sigma)
param float shape: not used
returns float: Normal distribution
'''
scale = args[0]
return (1./(scale*np.sqrt(2.*np.pi)))*np.exp(-(x-pos)**2/(2.*scale**2))
def LSWDist(x,pos,args=[]):
''' Lifshitz-Slyozov-Wagner Ostwald ripening distribution - numpy friendly on x axis
ref:
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: not used
param float shape: not used
returns float: LSW distribution
'''
redX = x/pos
result = (81.*2**(-5/3.))*(redX**2*np.exp(-redX/(1.5-redX)))/((1.5-redX)**(11/3.)*(3.-redX)**(7/3.))
return np.nan_to_num(result/pos)
def SchulzZimmDist(x,pos,args):
''' Schulz-Zimm macromolecule distribution - numpy friendly on x axis
ref: http://goldbook.iupac.org/S05502.html
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: width of distribution (sigma)
param float shape: not used
returns float: Schulz-Zimm distribution
'''
scale = args[0]
b = (2.*pos/scale)**2
a = b/pos
if b<70: #why bother?
return (a**(b+1.))*x**b*np.exp(-a*x)/scsp.gamma(b+1.)
else:
return np.exp((b+1.)*np.log(a)-scsp.gammaln(b+1.)+b*np.log(x)-(a*x))
def LogNormalCume(x,pos,args):
''' Standard LogNormal cumulative distribution - numpy friendly on x axis
ref: http://www.itl.nist.gov/div898/handbook/index.htm 1.3.6.6.9
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: width of distribution (sigma)
param float shape: shape parameter
returns float: LogNormal cumulative distribution
'''
scale,shape = args
lredX = np.log((x-pos)/scale)
return (scsp.erf((lredX/shape)/np.sqrt(2.))+1.)/2.
def GaussCume(x,pos,args):
''' Standard Normal cumulative distribution - numpy friendly on x axis
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: width of distribution (sigma)
param float shape: not used
returns float: Normal cumulative distribution
'''
scale = args[0]
redX = (x-pos)/scale
return (scsp.erf(redX/np.sqrt(2.))+1.)/2.
def LSWCume(x,pos,args=[]):
''' Lifshitz-Slyozov-Wagner Ostwald ripening cumulative distribution - numpy friendly on x axis
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: not used
param float shape: not used
returns float: LSW cumulative distribution
'''
nP = 500
xMin,xMax = [0.,2.*pos]
X = np.linspace(xMin,xMax,nP,True)
fxn = LSWDist(X,pos)
mat = np.outer(np.ones(nP),fxn)
cume = np.sum(np.tril(mat),axis=1)/np.sum(fxn)
return np.interp(x,X,cume,0,1)
def SchulzZimmCume(x,pos,args):
''' Schulz-Zimm cumulative distribution - numpy friendly on x axis
param float x: independent axis (can be numpy array)
param float pos: location of distribution
param float scale: width of distribution (sigma)
param float shape: not used
returns float: Normal distribution
'''
scale = args[0]
nP = 500
xMin = np.fmax([0.,pos-4.*scale])
xMax = np.fmin([pos+4.*scale,1.e5])
X = np.linspace(xMin,xMax,nP,True)
fxn = LSWDist(X,pos)
mat = np.outer(np.ones(nP),fxn)
cume = np.sum(np.tril(mat),axis=1)/np.sum(fxn)
return np.interp(x,X,cume,0,1)
return []
################################################################################
#### Structure factors for condensed systems
################################################################################
def DiluteSF(Q,args=[]):
''' Default: no structure factor correction for dilute system
'''
return np.ones_like(Q) #or return 1.0
def HardSpheresSF(Q,args):
''' Computes structure factor for not dilute monodisperse hard spheres
Refs.: PERCUS,YEVICK PHYS. REV. 110 1 (1958),THIELE J. CHEM PHYS. 39 474 (1968),
WERTHEIM PHYS. REV. LETT. 47 1462 (1981)
param float Q: Q value array (A-1)
param array args: [float R, float VolFrac]: interparticle distance & volume fraction
returns numpy array S(Q)
'''
R,VolFr = args
denom = (1.-VolFr)**4
num = (1.+2.*VolFr)**2
alp = num/denom
bet = -6.*VolFr*(1.+VolFr/2.)**2/denom
gamm = 0.5*VolFr*num/denom
A = 2.*Q*R
A2 = A**2
A3 = A**3
A4 = A**4
Rca = np.cos(A)
Rsa = np.sin(A)
calp = alp*(Rsa/A2-Rca/A)
cbet = bet*(2.*Rsa/A2-(A2-2.)*Rca/A3-2./A3)
cgam = gamm*(-Rca/A+(4./A)*((3.*A2-6.)*Rca/A4+(A2-6.)*Rsa/A3+6./A4))
pfac = -24.*VolFr/A
C = pfac*(calp+cbet+cgam)
return 1./(1.-C)
def SquareWellSF(Q,args):
'''Computes structure factor for not dilute monodisperse hard sphere with a
square well potential interaction.
Refs.: SHARMA,SHARMA, PHYSICA 89A,(1977),213-
:param float Q: Q value array (A-1)
:param array args: [float R, float VolFrac, float depth, float width]:
interparticle distance, volume fraction (<0.08), well depth (e/kT<1.5kT),
well width
:returns: numpy array S(Q)
well depth > 0 attractive & values outside above limits nonphysical cf.
Monte Carlo simulations
'''
R,VolFr,Depth,Width = args
eta = VolFr
eta2 = eta*eta
eta3 = eta*eta2
eta4 = eta*eta3
etam1 = 1. - eta
etam14 = etam1**4
alp = ( ( (1. + 2.*eta)**2 ) + eta3*( eta-4.0 ) )/etam14
bet = -(eta/3.0) * ( 18. + 20.*eta - 12.*eta2 + eta4 )/etam14
gam = 0.5*eta*( (1. + 2.*eta)**2 + eta3*(eta-4.) )/etam14
SK = 2.*Q*R
SK2 = SK*SK
SK3 = SK*SK2
SK4 = SK3*SK
T1 = alp * SK3 * ( np.sin(SK) - SK * np.cos(SK) )
T2 = bet * SK2 * ( 2.*SK*np.sin(SK) - (SK2-2.)*np.cos(SK) - 2.0 )
T3 = ( 4.0*SK3 - 24.*SK ) * np.sin(SK)
T3 = T3 - ( SK4 - 12.0*SK2 + 24.0 )*np.cos(SK) + 24.0
T3 = gam*T3
T4 = -Depth*SK3*(np.sin(Width*SK) - Width*SK*np.cos(Width*SK)+ SK*np.cos(SK) - np.sin(SK) )
CK = -24.0*eta*( T1 + T2 + T3 + T4 )/SK3/SK3
return 1./(1.-CK)
def StickyHardSpheresSF(Q,args):
''' Computes structure factor for not dilute monodisperse hard spheres
Refs.: PERCUS,YEVICK PHYS. REV. 110 1 (1958),THIELE J. CHEM PHYS. 39 474 (1968),
WERTHEIM PHYS. REV. LETT. 47 1462 (1981)
param float Q: Q value array (A-1)
param array args: [float R, float VolFrac]: sphere radius & volume fraction
returns numpy array S(Q)
'''
R,VolFr,epis,sticky = args
eta = VolFr/(1.0-epis)/(1.0-epis)/(1.0-epis)
sig = 2.0 * R
aa = sig/(1.0 - epis)
etam1 = 1.0 - eta
# SOLVE QUADRATIC FOR LAMBDA
qa = eta/12.0
qb = -1.0*(sticky + eta/(etam1))
qc = (1.0 + eta/2.0)/(etam1*etam1)
radic = qb*qb - 4.0*qa*qc
# KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL
lam1 = (-1.0*qb-np.sqrt(radic))/(2.0*qa)
lam2 = (-1.0*qb+np.sqrt(radic))/(2.0*qa)
lam = min(lam1,lam2)
mu = lam*eta*etam1
alp = (1.0 + 2.0*eta - mu)/(etam1*etam1)
bet = (mu - 3.0*eta)/(2.0*etam1*etam1)
# CALCULATE THE STRUCTURE FACTOR
kk = Q*aa
k2 = kk*kk
k3 = kk*k2
ds = np.sin(kk)
dc = np.cos(kk)
aq1 = ((ds - kk*dc)*alp)/k3
aq2 = (bet*(1.0-dc))/k2
aq3 = (lam*ds)/(12.0*kk)
aq = 1.0 + 12.0*eta*(aq1+aq2-aq3)
bq1 = alp*(0.5/kk - ds/k2 + (1.0 - dc)/k3)
bq2 = bet*(1.0/kk - ds/k2)
bq3 = (lam/12.0)*((1.0 - dc)/kk)
bq = 12.0*eta*(bq1+bq2-bq3)
sq = 1.0/(aq*aq +bq*bq)
return sq
#def HayterPenfoldSF(Q,args): #big & ugly function - do later (if ever)
# pass
def InterPrecipitateSF(Q,args):
''' Computes structure factor for precipitates in a matrix
Refs.: E-Wen Huang, Peter K. Liaw, Lionel Porcar, Yun Liu, Yee-Lang Liu,
Ji-Jung Kai, and Wei-Ren Chen,APPLIED PHYSICS LETTERS 93, 161904 (2008)
R. Giordano, A. Grasso, and J. Teixeira, Phys. Rev. A 43, 6894 1991
param float Q: Q value array (A-1)
param array args: [float R, float VolFr]: "radius" & volume fraction
returns numpy array S(Q)
'''
R,VolFr = args
QV2 = Q**2*VolFr**2
top = 1 - np.exp(-QV2/4)*np.cos(2.*Q*R)
bot = 1-2*np.exp(-QV2/4)*np.cos(2.*Q*R) + np.exp(-QV2/2)
return 2*(top/bot) - 1
################################################################################
##### SB-MaxEnt
################################################################################
def G_matrix(q,r,contrast,FFfxn,Volfxn,args=()):
'''Calculates the response matrix :math:`G(Q,r)`
:param float q: :math:`Q`
:param float r: :math:`r`
:param float contrast: :math:`|\\Delta\\rho|^2`, the scattering contrast
:param function FFfxn: form factor function FF(q,r,args)
:param function Volfxn: volume function Vol(r,args)
:returns float: G(Q,r)
'''
FF = FFfxn(q,r,args)
Vol = Volfxn(r,args)
return 1.e-4*(contrast*Vol*FF**2).T #10^-20 vs 10^-24
'''
sbmaxent
Entropy maximization routine as described in the article
J Skilling and RK Bryan; MNRAS 211 (1984) 111 - 124.
("MNRAS": "Monthly Notices of the Royal Astronomical Society")
:license: Copyright (c) 2013, UChicago Argonne, LLC
:license: This file is distributed subject to a Software License Agreement found
in the file LICENSE that is included with this distribution.
References:
1. J Skilling and RK Bryan; MON NOT R ASTR SOC 211 (1984) 111 - 124.
2. JA Potton, GJ Daniell, and BD Rainford; Proc. Workshop
Neutron Scattering Data Analysis, Rutherford
Appleton Laboratory, UK, 1986; ed. MW Johnson,
IOP Conference Series 81 (1986) 81 - 86, Institute
of Physics, Bristol, UK.
3. ID Culverwell and GP Clarke; Ibid. 87 - 96.
4. JA Potton, GK Daniell, & BD Rainford,
J APPL CRYST 21 (1988) 663 - 668.
5. JA Potton, GJ Daniell, & BD Rainford,
J APPL CRYST 21 (1988) 891 - 897.
'''
class MaxEntException(Exception):
'''Any exception from this module'''
pass
def MaxEnt_SB(datum, sigma, G, base, IterMax, image_to_data=None, data_to_image=None, report=False):
'''
do the complete Maximum Entropy algorithm of Skilling and Bryan
:param float datum[]:
:param float sigma[]:
:param float[][] G: transformation matrix
:param float base[]:
:param int IterMax:
:param obj image_to_data: opus function (defaults to opus)
:param obj data_to_image: tropus function (defaults to tropus)
:returns float[]: :math:`f(r) dr`
'''
TEST_LIMIT = 0.05 # for convergence
CHI_SQR_LIMIT = 0.01 # maximum difference in ChiSqr for a solution
SEARCH_DIRECTIONS = 3 # <10. This code requires value = 3
RESET_STRAYS = 1 # was 0.001, correction of stray negative values
DISTANCE_LIMIT_FACTOR = 0.1 # limitation on df to constrain runaways
MAX_MOVE_LOOPS = 5000 # for no solution in routine: move,
MOVE_PASSES = 0.001 # convergence test in routine: move
def tropus (data, G):
'''
tropus: transform data-space -> solution-space: [G] * data
default definition, caller can use this definition or provide an alternative
:param float[M] data: observations, ndarray of shape (M)
:param float[M][N] G: transformation matrix, ndarray of shape (M,N)
:returns float[N]: calculated image, ndarray of shape (N)
'''
return G.dot(data)
def opus (image, G):
'''
opus: transform solution-space -> data-space: [G]^tr * image
default definition, caller can use this definition or provide an alternative
:param float[N] image: solution, ndarray of shape (N)
:param float[M][N] G: transformation matrix, ndarray of shape (M,N)
:returns float[M]: calculated data, ndarray of shape (M)
'''
return np.dot(G.T,image) #G.transpose().dot(image)
def Dist(s2, beta):
'''measure the distance of this possible solution'''
w = 0
n = beta.shape[0]
for k in range(n):
z = -sum(s2[k] * beta)
w += beta[k] * z
return w
def ChiNow(ax, c1, c2, s1, s2):
'''
ChiNow
:returns tuple: (ChiNow computation of ``w``, beta)
'''
bx = 1 - ax
a = bx * c2 - ax * s2
b = -(bx * c1 - ax * s1)
beta = ChoSol(a, b)
w = 1.0
for k in range(SEARCH_DIRECTIONS):
w += beta[k] * (c1[k] + 0.5*sum(c2[k] * beta))
return w, beta
def ChoSol(a, b):
'''
ChoSol: ? chop the solution vectors ?
:returns: new vector beta
'''
n = b.shape[0]
fl = np.zeros((n,n))
bl = np.zeros_like(b)
#print_arr("ChoSol: a", a)
#print_vec("ChoSol: b", b)
if (a[0][0] <= 0):
msg = "ChoSol: a[0][0] = "
msg += str(a[0][0])
msg += ' Value must be positive'
raise MaxEntException(msg)
# first, compute fl from a
# note fl is a lower triangular matrix
fl[0][0] = math.sqrt (a[0][0])
for i in (1, 2):
fl[i][0] = a[i][0] / fl[0][0]
for j in range(1, i+1):
z = 0.0
for k in range(j):
z += fl[i][k] * fl[j][k]
#print "ChoSol: %d %d %d z = %lg" % ( i, j, k, z)
z = a[i][j] - z
if j == i:
y = math.sqrt(max(0.,z))
else:
y = z / fl[j][j]
fl[i][j] = y
#print_arr("ChoSol: fl", fl)
# next, compute bl from fl and b
bl[0] = b[0] / fl[0][0]
for i in (1, 2):
z = 0.0
for k in range(i):
z += fl[i][k] * bl[k]
#print "\t", i, k, z
bl[i] = (b[i] - z) / fl[i][i]
#print_vec("ChoSol: bl", bl)
# last, compute beta from bl and fl
beta = np.empty((n))
beta[-1] = bl[-1] / fl[-1][-1]
for i in (1, 0):
z = 0.0
for k in range(i+1, n):
z += fl[k][i] * beta[k]
#print "\t\t", i, k, 'z=', z
beta[i] = (bl[i] - z) / fl[i][i]
#print_vec("ChoSol: beta", beta)
return beta
def MaxEntMove(fSum, blank, chisq, chizer, c1, c2, s1, s2):
'''
move beta one step closer towards the solution
'''
a_lower, a_upper = 0., 1. # bracket "a"
cmin, beta = ChiNow (a_lower, c1, c2, s1, s2)
#print "MaxEntMove: cmin = %g" % cmin
if cmin*chisq > chizer:
ctarg = (1.0 + cmin)/2
else:
ctarg = chizer/chisq
f_lower = cmin - ctarg
c_upper, beta = ChiNow (a_upper, c1, c2, s1, s2)
f_upper = c_upper - ctarg
fx = 2*MOVE_PASSES # just to start off
loop = 1
while abs(fx) >= MOVE_PASSES and loop <= MAX_MOVE_LOOPS:
a_new = (a_lower + a_upper) * 0.5 # search by bisection
c_new, beta = ChiNow (a_new, c1, c2, s1, s2)
fx = c_new - ctarg
# tighten the search range for the next pass
if f_lower*fx > 0:
a_lower, f_lower = a_new, fx
if f_upper*fx > 0:
a_upper, f_upper = a_new, fx
loop += 1
if abs(fx) >= MOVE_PASSES or loop > MAX_MOVE_LOOPS:
msg = "MaxEntMove: Loop counter = "
msg += str(MAX_MOVE_LOOPS)
msg += ' No convergence in alpha chop'
raise MaxEntException(msg)
w = Dist (s2, beta);
m = SEARCH_DIRECTIONS
if (w > DISTANCE_LIMIT_FACTOR*fSum/blank): # invoke the distance penalty, SB eq. 17
for k in range(m):
beta[k] *= math.sqrt (fSum/(blank*w))
chtarg = ctarg * chisq
return w, chtarg, loop, a_new, fx, beta
#MaxEnt_SB starts here
if image_to_data == None:
image_to_data = opus
if data_to_image == None:
data_to_image = tropus
n = len(base)
npt = len(datum)
# Note that the order of subscripts for
# "xi" and "eta" has been reversed from
# the convention used in the FORTRAN version
# to enable parts of them to be passed as
# as vectors to "image_to_data" and "data_to_image".
xi = np.zeros((SEARCH_DIRECTIONS, n))
eta = np.zeros((SEARCH_DIRECTIONS, npt))
beta = np.zeros((SEARCH_DIRECTIONS))
# s1 = np.zeros((SEARCH_DIRECTIONS))
# c1 = np.zeros((SEARCH_DIRECTIONS))
s2 = np.zeros((SEARCH_DIRECTIONS, SEARCH_DIRECTIONS))
c2 = np.zeros((SEARCH_DIRECTIONS, SEARCH_DIRECTIONS))
# TODO: replace blank (scalar) with base (vector)
blank = sum(base) / len(base) # use the average value of base
chizer, chtarg = npt*1.0, npt*1.0
f = base * 1.0 # starting distribution is base
fSum = sum(f) # find the sum of the f-vector
z = (datum - image_to_data (f, G)) / sigma # standardized residuals, SB eq. 3
chisq = sum(z*z) # Chi^2, SB eq. 4
for iter in range(IterMax):
ox = -2 * z / sigma # gradient of Chi^2
cgrad = data_to_image (ox, G) # cgrad[i] = del(C)/del(f[i]), SB eq. 8
sgrad = -np.log(f/base) / (blank*math.exp (1.0)) # sgrad[i] = del(S)/del(f[i])
snorm = math.sqrt(sum(f * sgrad*sgrad)) # entropy term, SB eq. 22
cnorm = math.sqrt(sum(f * cgrad*cgrad)) # ChiSqr term, SB eq. 22
tnorm = sum(f * sgrad * cgrad) # norm for gradient term TEST
a = 1.0
b = 1.0 / cnorm
if iter == 0:
test = 0.0 # mismatch between entropy and ChiSquared gradients
else:
test = math.sqrt( ( 1.0 - tnorm/(snorm*cnorm) )/2 ) # SB eq. 37?
a = 0.5 / (snorm * test)
b *= 0.5 / test
xi[0] = f * cgrad / cnorm
xi[1] = f * (a * sgrad - b * cgrad)
eta[0] = image_to_data (xi[0], G); # image --> data
eta[1] = image_to_data (xi[1], G); # image --> data
ox = eta[1] / (sigma * sigma)
xi[2] = data_to_image (ox, G); # data --> image
a = 1.0 / math.sqrt(sum(f * xi[2]*xi[2]))
xi[2] = f * xi[2] * a
eta[2] = image_to_data (xi[2], G) # image --> data
# print_arr("MaxEnt: eta.transpose()", eta.transpose())
# print_arr("MaxEnt: xi.transpose()", xi.transpose())
# prepare the search directions for the conjugate gradient technique
c1 = xi.dot(cgrad) / chisq # C_mu, SB eq. 24
s1 = xi.dot(sgrad) # S_mu, SB eq. 24
# print_vec("MaxEnt: c1", c1)
# print_vec("MaxEnt: s1", s1)
for k in range(SEARCH_DIRECTIONS):
for l in range(k+1):
c2[k][l] = 2 * sum(eta[k] * eta[l] / sigma/sigma) / chisq
s2[k][l] = -sum(xi[k] * xi[l] / f) / blank
# print_arr("MaxEnt: c2", c2)
# print_arr("MaxEnt: s2", s2)
# reflect across the body diagonal
for k, l in ((0,1), (0,2), (1,2)):
c2[k][l] = c2[l][k] # M_(mu,nu)
s2[k][l] = s2[l][k] # g_(mu,nu)
beta[0] = -0.5 * c1[0] / c2[0][0]
beta[1] = 0.0
beta[2] = 0.0
if (iter > 0):
w, chtarg, loop, a_new, fx, beta = MaxEntMove(fSum, blank, chisq, chizer, c1, c2, s1, s2)
f_old = f.copy() # preserve the last image
f += xi.transpose().dot(beta) # move the image towards the solution, SB eq. 25
# As mentioned at the top of p.119,
# need to protect against stray negative values.
# In this case, set them to RESET_STRAYS * base[i]
#f = f.clip(RESET_STRAYS * blank, f.max())
for i in range(n):
if f[i] <= 0.0:
f[i] = RESET_STRAYS * base[i]
df = f - f_old
fSum = sum(f)
fChange = sum(df)
# calculate the normalized entropy
S = sum((f/fSum) * np.log(f/fSum)) # normalized entropy, S&B eq. 1
z = (datum - image_to_data (f, G)) / sigma # standardized residuals
chisq = sum(z*z) # report this ChiSq
if report:
print (" MaxEnt trial/max: %3d/%3d" % ((iter+1), IterMax))
print (" Residual: %5.2lf%% Entropy: %8lg" % (100*test, S))
print (" Function sum: %.6lg Change from last: %.2lf%%\n" % (fSum,100*fChange/fSum))
# See if we have finished our task.
# do the hardest test first
if (abs(chisq/chizer-1.0) < CHI_SQR_LIMIT) and (test < TEST_LIMIT):
print (' Convergence achieved.')
return chisq,f,image_to_data(f, G) # solution FOUND returns here
print (' No convergence! Try increasing Error multiplier.')
return chisq,f,image_to_data(f, G) # no solution after IterMax iterations
###############################################################################
#### IPG/TNNLS Routines
###############################################################################
def IPG(datum,sigma,G,Bins,Dbins,IterMax,Qvec=[],approach=0.8,Power=-1,report=False):
''' An implementation of the Interior-Point Gradient method of
Michael Merritt & Yin Zhang, Technical Report TR04-08, Dept. of Comp. and
Appl. Math., Rice Univ., Houston, Texas 77005, U.S.A. found on the web at
http://www.caam.rice.edu/caam/trs/2004/TR04-08.pdf
Problem addressed: Total Non-Negative Least Squares (TNNLS)
:param float datum[]:
:param float sigma[]:
:param float[][] G: transformation matrix
:param int IterMax:
:param float Qvec: data positions for Power = 0-4
:param float approach: 0.8 default fitting parameter
:param int Power: 0-4 for Q^Power weighting, -1 to use input sigma
'''
if Power < 0:
GmatE = G/sigma[:np.newaxis]
IntE = datum/sigma
pwr = 0
QvecP = np.ones_like(datum)
else:
GmatE = G[:]
IntE = datum[:]
pwr = Power
QvecP = Qvec**pwr
Amat = GmatE*QvecP[:np.newaxis]
AAmat = np.inner(Amat,Amat)
Bvec = IntE*QvecP
Xw = np.ones_like(Bins)*1.e-32
calc = np.dot(G.T,Xw)
nIter = 0
err = 10.
while (nIter 1.):
#Step 1 in M&Z paper:
Qk = np.inner(AAmat,Xw)-np.inner(Amat,Bvec)
Dk = Xw/np.inner(AAmat,Xw)
Pk = -Dk*Qk
#Step 2 in M&Z paper:
alpSt = -np.inner(Pk,Qk)/np.inner(Pk,np.inner(AAmat,Pk))
alpWv = -Xw/Pk
AkSt = [np.where(Pk[i]<0,np.min((approach*alpWv[i],alpSt)),alpSt) for i in range(Pk.shape[0])]
#Step 3 in M&Z paper:
shift = AkSt*Pk
Xw += shift
#done IPG; now results
nIter += 1
calc = np.dot(G.T,Xw)
chisq = np.sum(((datum-calc)/sigma)**2)
err = chisq/len(datum)
if report:
print (' Iteration: %d, chisq: %.3g, sum(shift^2): %.3g'%(nIter,chisq,np.sum(shift**2)))
return chisq,Xw,calc
###############################################################################
#### SASD Utilities
###############################################################################
def SetScale(Data,refData):
Profile,Limits,Sample = Data
refProfile,refLimits,refSample = refData
x,y = Profile[:2]
rx,ry = refProfile[:2]
Beg = np.fmax([rx[0],x[0],Limits[1][0],refLimits[1][0]])
Fin = np.fmin([rx[-1],x[-1],Limits[1][1],refLimits[1][1]])
iBeg = np.searchsorted(x,Beg)
iFin = np.searchsorted(x,Fin)+1 #include last point
sum = np.sum(y[iBeg:iFin])
refsum = np.sum(np.interp(x[iBeg:iFin],rx,ry,0,0))
Sample['Scale'][0] = refSample['Scale'][0]*refsum/sum
def Bestimate(G,Rg,P):
return (G*P/Rg**P)*np.exp(scsp.gammaln(P/2))
def SmearData(Ic,Q,slitLen,Back):
Np = Q.shape[0]
Qtemp = np.concatenate([Q,Q[-1]+20*Q])
Ictemp = np.concatenate([Ic,Ic[-1]*(1-(Qtemp[Np:]-Qtemp[Np])/(20*Qtemp[Np-1]))])
Icsm = np.zeros_like(Q)
Qsm = 2*slitLen*(np.interp(np.arange(2*Np)/2.,np.arange(Np),Q)-Q[0])/(Q[-1]-Q[0])
Sp = np.searchsorted(Qsm,slitLen)
DQsm = np.diff(Qsm)[:Sp]
Ism = np.interp(np.sqrt(Q[:,np.newaxis]**2+Qsm**2),Qtemp,Ictemp)
Icsm = np.sum((Ism[:,:Sp]*DQsm),axis=1)
Icsm /= slitLen
return Icsm
###############################################################################
#### Size distribution
###############################################################################
def SizeDistribution(Profile,ProfDict,Limits,Sample,data):
shapes = {'Spheroid':[SpheroidFF,SpheroidVol],'Cylinder':[CylinderDFF,CylinderDVol],
'Cylinder AR':[CylinderARFF,CylinderARVol],'Unified sphere':[UniSphereFF,UniSphereVol],
'Unified rod':[UniRodFF,UniRodVol],'Unified rod AR':[UniRodARFF,UniRodARVol],
'Unified disk':[UniDiskFF,UniDiskVol],'Sphere':[SphereFF,SphereVol],
'Cylinder diam':[CylinderDFF,CylinderDVol],
'Spherical shell': [SphericalShellFF, SphericalShellVol]}
Shape = data['Size']['Shape'][0]
Parms = data['Size']['Shape'][1:]
if data['Size']['logBins']:
Bins = np.logspace(np.log10(data['Size']['MinDiam']),np.log10(data['Size']['MaxDiam']),
data['Size']['Nbins']+1,True)/2. #make radii
else:
Bins = np.linspace(data['Size']['MinDiam'],data['Size']['MaxDiam'],
data['Size']['Nbins']+1,True)/2. #make radii
Dbins = np.diff(Bins)
Bins = Bins[:-1]+Dbins/2.
Contrast = Sample['Contrast'][1]
Scale = Sample['Scale'][0]
Sky = 10**data['Size']['MaxEnt']['Sky']
BinsBack = np.ones_like(Bins)*Sky*Scale/Contrast
Back = data['Back']
Q,Io,wt,Ic,Ib = Profile[:5]
Qmin = Limits[1][0]
Qmax = Limits[1][1]
wtFactor = ProfDict['wtFactor']
Ibeg = np.searchsorted(Q,Qmin)
Ifin = np.searchsorted(Q,Qmax)+1 #include last point
BinMag = np.zeros_like(Bins)
Ic[:] = 0.
Gmat = G_matrix(Q[Ibeg:Ifin],Bins,Contrast,shapes[Shape][0],shapes[Shape][1],args=Parms)
if 'MaxEnt' == data['Size']['Method']:
chisq,BinMag,Ic[Ibeg:Ifin] = MaxEnt_SB(Scale*Io[Ibeg:Ifin]-Back[0],
Scale/np.sqrt(wtFactor*wt[Ibeg:Ifin]),Gmat,BinsBack,
data['Size']['MaxEnt']['Niter'],report=True)
elif 'IPG' == data['Size']['Method']:
chisq,BinMag,Ic[Ibeg:Ifin] = IPG(Scale*Io[Ibeg:Ifin]-Back[0],Scale/np.sqrt(wtFactor*wt[Ibeg:Ifin]),
Gmat,Bins,Dbins,data['Size']['IPG']['Niter'],Q[Ibeg:Ifin],approach=0.8,
Power=data['Size']['IPG']['Power'],report=True)
Ib[:] = Back[0]
Ic[Ibeg:Ifin] += Back[0]
print (' Final chi^2: %.3f'%(chisq))
data['Size']['Distribution'] = [Bins,Dbins,BinMag/(2.*Dbins)]
################################################################################
#### Modelling
################################################################################
def PairDistFxn(Profile,ProfDict,Limits,Sample,data):
def CalcMoore():
def MoorePOR(cw,r,dmax): #lines 1417-1428
n = 0
nmax = len(cw)
POR = np.zeros(len(r))
while n < nmax:
POR += 4.*r*cw[n]*np.sin((n+1.)*np.pi*r/dmax)
n += 1
return POR
def MooreIOREFF(cw,q,dmax): #lines 1437-1448
n = 0
nmax = len(cw)
POR = np.zeros(len(q))
dq = dmax*q
fpd = 8.*(np.pi**2)*dmax*np.sin(dq)/q
while n < nmax:
POR += cw[n]*(n+1.)*((-1)**n)*fpd/(((n+1.)*np.pi)**2-dq**2)
n += 1
return POR
def calcSASD(values,Q,Io,wt,Ifb,dmax,ifBack):
if ifBack:
M = np.sqrt(wt)*(MooreIOREFF(values[:-1],Q,dmax)+Ifb+values[-1]-Io)
else:
M = np.sqrt(wt)*(MooreIOREFF(values,Q,dmax)+Ifb-Io)
return M
Q,Io,wt,Ic,Ib,Ifb = Profile[:6]
Qmin = Limits[1][0]
Qmax = Limits[1][1]
wtFactor = ProfDict['wtFactor']
Back,ifBack = data['Back']
Ibeg = np.searchsorted(Q,Qmin)
Ifin = np.searchsorted(Q,Qmax)+1 #include last point
Ic[Ibeg:Ifin] = 0
Bins = np.linspace(0.,pairData['MaxRadius'],pairData['NBins']+1,True)
Dbins = np.diff(Bins)
Bins = Bins[:-1]+Dbins/2.
N = pairData['Moore']
if ifBack:
N += 1
MPV = np.ones(N)*1.e-5
dmax = pairData['MaxRadius']
if 'User' in pairData['Errors']:
Wt = wt[Ibeg:Ifin]
elif 'Sqrt' in pairData['Errors']:
Wt = 1./Io[Ibeg:Ifin]
elif 'Percent' in pairData['Errors']:
Wt = 1./(pairData['Percent error']*Io[Ibeg:Ifin])
result = so.leastsq(calcSASD,MPV,full_output=True,epsfcn=1.e-8, #ftol=Ftol,
args=(Q[Ibeg:Ifin],Io[Ibeg:Ifin],wtFactor*Wt,Ifb[Ibeg:Ifin],dmax,ifBack))
if ifBack:
MPVR = result[0][:-1]
data['Back'][0] = result[0][-1]
Back = data['Back'][0]
else:
MPVR = result[0]
Back = 0.
chisq = np.sum(result[2]['fvec']**2)
covM = result[1]
Ic[Ibeg:Ifin] = MooreIOREFF(MPVR,Q[Ibeg:Ifin],dmax)+Ifb[Ibeg:Ifin]+Back
ncalc = result[2]['nfev']
GOF = chisq/(Ifin-Ibeg-N)
Rwp = np.sqrt(chisq/np.sum(wt[Ibeg:Ifin]*Io[Ibeg:Ifin]**2))*100. #to %
print (' Results of small angle data modelling fit of P(R):')
print ('Number of function calls: %d Number of observations: %d Number of parameters: %d'%(ncalc,Ifin-Ibeg,N))
print ('Rwp = %7.2f%%, chi**2 = %12.6g, reduced chi**2 = %6.2f'%(Rwp,chisq,GOF))
if len(covM):
sig = np.sqrt(np.diag(covM)*GOF)
for val,esd in zip(result[0],sig):
print(' parameter: %.4g esd: %.4g'%(val,esd))
BinMag = MoorePOR(MPVR,Bins,dmax)/2.
return Bins,Dbins,BinMag
pairData = data['Pair']
if pairData['Method'] == 'Regularization': #not used
print('Regularization calc; dummy Gaussian - TBD')
pairData['Method'] = 'Moore'
elif pairData['Method'] == 'Moore':
Bins,Dbins,BinMag = CalcMoore()
BinSum = np.sum(BinMag*Dbins)
BinMag /= BinSum
data['Pair']['Distribution'] = [Bins,Dbins,BinMag] #/(2.*Dbins)]
if 'Pair Calc' in data['Pair']:
del data['Pair']['Pair Calc']
################################################################################
#### Modelling
################################################################################
def ModelFit(Profile,ProfDict,Limits,Sample,Model):
shapes = {'Spheroid':[SpheroidFF,SpheroidVol],'Cylinder':[CylinderDFF,CylinderDVol],
'Cylinder AR':[CylinderARFF,CylinderARVol],'Unified sphere':[UniSphereFF,UniSphereVol],
'Unified rod':[UniRodFF,UniRodVol],'Unified rod AR':[UniRodARFF,UniRodARVol],
'Unified disk':[UniDiskFF,UniDiskVol],'Sphere':[SphereFF,SphereVol],
'Unified tube':[UniTubeFF,UniTubeVol],'Cylinder diam':[CylinderDFF,CylinderDVol],
'Spherical shell':[SphericalShellFF,SphericalShellVol]}
sfxns = {'Dilute':DiluteSF,'Hard sphere':HardSpheresSF,'Square well':SquareWellSF,
'Sticky hard sphere':StickyHardSpheresSF,'InterPrecipitate':InterPrecipitateSF,}
parmOrder = ['Volume','Radius','Mean','StdDev','MinSize','G','Rg','B','P','Cutoff',
'PkInt','PkPos','PkSig','PkGam',]
FFparmOrder = ['Aspect ratio','Length','Diameter','Thickness','Shell thickness']
SFparmOrder = ['Dist','VolFr','epis','Sticky','Depth','Width']
def GetModelParms():
parmDict = {'Scale':Sample['Scale'][0],'SlitLen':Sample.get('SlitLen',0.0),}
varyList = []
values = []
levelTypes = []
Back = Model['Back']
if Back[1]:
varyList += ['Back',]
values.append(Back[0])
parmDict['Back'] = Back[0]
partData = Model['Particle']
for i,level in enumerate(partData['Levels']):
cid = str(i)+';'
controls = level['Controls']
Type = controls['DistType']
levelTypes.append(Type)
if Type in ['LogNormal','Gaussian','LSW','Schulz-Zimm','Monodisperse']:
if Type not in ['Monodosperse',]:
parmDict[cid+'NumPoints'] = controls['NumPoints']
parmDict[cid+'Cutoff'] = controls['Cutoff']
parmDict[cid+'FormFact'] = shapes[controls['FormFact']][0]
parmDict[cid+'FFVolume'] = shapes[controls['FormFact']][1]
parmDict[cid+'StrFact'] = sfxns[controls['StrFact']]
parmDict[cid+'Contrast'] = controls['Contrast']
for item in FFparmOrder:
if item in controls['FFargs']:
parmDict[cid+item] = controls['FFargs'][item][0]
if controls['FFargs'][item][1]:
varyList.append(cid+item)
values.append(controls['FFargs'][item][0])
for item in SFparmOrder:
if item in controls.get('SFargs',{}):
parmDict[cid+item] = controls['SFargs'][item][0]
if controls['SFargs'][item][1]:
varyList.append(cid+item)
values.append(controls['SFargs'][item][0])
distDict = controls['DistType']
for item in parmOrder:
if item in level[distDict]:
parmDict[cid+item] = level[distDict][item][0]
if level[distDict][item][1]:
values.append(level[distDict][item][0])
varyList.append(cid+item)
return levelTypes,parmDict,varyList,values
def SetModelParms():
print (' Refined parameters: Histogram scale: %.4g'%(parmDict['Scale']))
if 'Back' in varyList:
Model['Back'][0] = parmDict['Back']
print (' %15s %15.4f esd: %15.4g'%('Background:',parmDict['Back'],sigDict['Back']))
partData = Model['Particle']
for i,level in enumerate(partData['Levels']):
controls = level['Controls']
Type = controls['DistType']
if Type in ['LogNormal','Gaussian','LSW','Schulz-Zimm','Monodisperse']:
print (' Component %d: Type: %s: Structure Factor: %s Contrast: %12.3f' \
%(i,Type,controls['StrFact'],controls['Contrast']))
else:
print (' Component %d: Type: %s: '%(i,Type,))
cid = str(i)+';'
if Type in ['LogNormal','Gaussian','LSW','Schulz-Zimm','Monodisperse']:
for item in FFparmOrder:
if cid+item in varyList:
controls['FFargs'][item][0] = parmDict[cid+item]
print (' %15s: %15.4g esd: %15.4g'%(cid+item,parmDict[cid+item],sigDict[cid+item]))
for item in SFparmOrder:
if cid+item in varyList:
controls['SFargs'][item][0] = parmDict[cid+item]
print (' %15s: %15.4g esd: %15.4g'%(cid+item,parmDict[cid+item],sigDict[cid+item]))
distDict = controls['DistType']
for item in level[distDict]:
if cid+item in varyList:
level[distDict][item][0] = parmDict[cid+item]
print (' %15s: %15.4g esd: %15.4g'%(cid+item,parmDict[cid+item],sigDict[cid+item]))
def calcSASD(values,Q,Io,wt,Ifb,levelTypes,parmDict,varyList):
parmDict.update(zip(varyList,values))
M = np.sqrt(wt)*(getSASD(Q,levelTypes,parmDict)+Ifb-parmDict['Scale']*Io)
return M
def getSASD(Q,levelTypes,parmDict):
Ic = np.zeros_like(Q)
for i,Type in enumerate(levelTypes):
cid = str(i)+';'
if Type in ['LogNormal','Gaussian','LSW','Schulz-Zimm']:
FFfxn = parmDict[cid+'FormFact']
Volfxn = parmDict[cid+'FFVolume']
SFfxn = parmDict[cid+'StrFact']
FFargs = []
SFargs = []
for item in [cid+'Aspect ratio',cid+'Length',cid+'Thickness',cid+'Diameter',cid+'Shell thickness']:
if item in parmDict:
FFargs.append(parmDict[item])
for item in [cid+'Dist',cid+'VolFr',cid+'epis',cid+'Sticky',cid+'Depth',cid+'Width']:
if item in parmDict:
SFargs.append(parmDict[item])
distDict = {}
for item in [cid+'Volume',cid+'Mean',cid+'StdDev',cid+'MinSize',]:
if item in parmDict:
distDict[item.split(';')[1]] = parmDict[item]
contrast = parmDict[cid+'Contrast']
rBins,dBins,dist = MakeDiamDist(Type,parmDict[cid+'NumPoints'],parmDict[cid+'Cutoff'],distDict)
Gmat = G_matrix(Q,rBins,contrast,FFfxn,Volfxn,FFargs).T
dist *= parmDict[cid+'Volume']
Ic += np.dot(Gmat,dist)*SFfxn(Q,args=SFargs)
elif 'Unified' in Type:
Rg,G,B,P,Rgco = parmDict[cid+'Rg'],parmDict[cid+'G'],parmDict[cid+'B'], \
parmDict[cid+'P'],parmDict[cid+'Cutoff']
Qstar = Q/(scsp.erf(Q*Rg/np.sqrt(6)))**3
Guin = G*np.exp(-(Q*Rg)**2/3)
Porod = (B/Qstar**P)*np.exp(-(Q*Rgco)**2/3)
Ic += Guin+Porod
elif 'Porod' in Type:
B,P,Rgco = parmDict[cid+'B'],parmDict[cid+'P'],parmDict[cid+'Cutoff']
Porod = (B/Q**P)*np.exp(-(Q*Rgco)**2/3)
Ic += Porod
elif 'Mono' in Type:
FFfxn = parmDict[cid+'FormFact']
Volfxn = parmDict[cid+'FFVolume']
SFfxn = parmDict[cid+'StrFact']
FFargs = []
SFargs = []
for item in [cid+'Aspect ratio',cid+'Length',cid+'Thickness',cid+'Diameter',cid+'Shell thickness']:
if item in parmDict:
FFargs.append(parmDict[item])
for item in [cid+'Dist',cid+'VolFr',cid+'epis',cid+'Sticky',cid+'Depth',cid+'Width']:
if item in parmDict:
SFargs.append(parmDict[item])
contrast = parmDict[cid+'Contrast']
R = parmDict[cid+'Radius']
Gmat = G_matrix(Q,R,contrast,FFfxn,Volfxn,FFargs)
Ic += Gmat[0]*parmDict[cid+'Volume']*SFfxn(Q,args=SFargs)
elif 'Bragg' in Type:
Ic += parmDict[cid+'PkInt']*G2pwd.getPsVoigt(parmDict[cid+'PkPos'],
parmDict[cid+'PkSig'],parmDict[cid+'PkGam'],Q)
Ic += parmDict['Back'] #/parmDict['Scale']
slitLen = Sample['SlitLen']
if slitLen:
Ic = SmearData(Ic,Q,slitLen,parmDict['Back'])
return Ic
Q,Io,wt,Ic,Ib,Ifb = Profile[:6]
Qmin = Limits[1][0]
Qmax = Limits[1][1]
wtFactor = ProfDict['wtFactor']
Ibeg = np.searchsorted(Q,Qmin)
Ifin = np.searchsorted(Q,Qmax)+1 #include last point
Ic[:] = 0
levelTypes,parmDict,varyList,values = GetModelParms()
if varyList:
result = so.leastsq(calcSASD,values,full_output=True,epsfcn=1.e-8, #ftol=Ftol,
args=(Q[Ibeg:Ifin],Io[Ibeg:Ifin],wtFactor*wt[Ibeg:Ifin],Ifb[Ibeg:Ifin],levelTypes,parmDict,varyList))
parmDict.update(zip(varyList,result[0]))
chisq = np.sum(result[2]['fvec']**2)
ncalc = result[2]['nfev']
covM = result[1]
else: #nothing varied
M = calcSASD(values,Q[Ibeg:Ifin],Io[Ibeg:Ifin],wtFactor*wt[Ibeg:Ifin],Ifb[Ibeg:Ifin],levelTypes,parmDict,varyList)
chisq = np.sum(M**2)
ncalc = 0
covM = []
sig = []
sigDict = {}
result = []
Rvals = {}
Rvals['Rwp'] = np.sqrt(chisq/np.sum(wt[Ibeg:Ifin]*Io[Ibeg:Ifin]**2))*100. #to %
Rvals['GOF'] = chisq/(Ifin-Ibeg-len(varyList)) #reduced chi^2
Ic[Ibeg:Ifin] = getSASD(Q[Ibeg:Ifin],levelTypes,parmDict)
Msg = 'Failed to converge'
try:
Nans = np.isnan(result[0])
if np.any(Nans):
name = varyList[Nans.nonzero(True)[0]]
Msg = 'Nan result for '+name+'!'
raise ValueError
Negs = np.less_equal(result[0],0.)
if np.any(Negs):
name = varyList[Negs.nonzero(True)[0]]
Msg = 'negative coefficient for '+name+'!'
raise ValueError
if len(covM):
sig = np.sqrt(np.diag(covM)*Rvals['GOF'])
sigDict = dict(zip(varyList,sig))
print (' Results of small angle data modelling fit:')
print ('Number of function calls: %d Number of observations: %d Number of parameters: %d'%(ncalc,Ifin-Ibeg,len(varyList)))
print ('Rwp = %7.2f%%, chi**2 = %12.6g, reduced chi**2 = %6.2f'%(Rvals['Rwp'],chisq,Rvals['GOF']))
SetModelParms()
covMatrix = covM*Rvals['GOF']
return True,result,varyList,sig,Rvals,covMatrix,parmDict,''
except (ValueError,TypeError): #when bad LS refinement; covM missing or with nans
return False,0,0,0,0,0,0,Msg
def getSASDRg(Q,parmDict):
Ic = np.zeros_like(Q)
Rg,G,B = parmDict['Rg'],parmDict['G'],parmDict['B']
Qstar = Q/(scsp.erf(Q*Rg/np.sqrt(6)))**3
Guin = G*np.exp(-(Q*Rg)**2/3)
Porod = (B/Qstar**4) #*np.exp(-(Q*B)**2/3)
Ic += Guin+Porod+parmDict['Back']
return Ic
def RgFit(Profile,ProfDict,Limits,Sample,Model):
print('unified fit single Rg to data to estimate a size')
def GetModelParms():
parmDict = {}
varyList = []
values = []
Back = Model['Back']
if Back[1]:
varyList += ['Back',]
values.append(Back[0])
parmDict['Back'] = Back[0]
pairData = Model['Pair']
parmDict['G'] = pairData.get('Dist G',Io[Ibeg])
parmDict['Rg'] = pairData['MaxRadius']/2.5
parmDict['B'] = pairData.get('Dist B',Io[Ifin-1]*Q[Ifin-1]**4)
varyList += ['G','Rg','B']
values += [parmDict['G'],parmDict['Rg'],parmDict['B']]
return parmDict,varyList,values
def calcSASD(values,Q,Io,wt,Ifb,parmDict,varyList):
parmDict.update(zip(varyList,values))
M = np.sqrt(wt)*(getSASDRg(Q,parmDict)-Io)
return M
def SetModelParms():
print (' Refined parameters: ')
if 'Back' in varyList:
Model['Back'][0] = parmDict['Back']
print (' %15s %15.4f esd: %15.4g'%('Background:',parmDict['Back'],sigDict['Back']))
pairData = Model['Pair']
pairData['Dist G'] = parmDict['G']
pairData['MaxRadius'] = parmDict['Rg']*2.5
pairData['Dist B'] = parmDict['B']
for item in varyList:
print (' %15s: %15.4g esd: %15.4g'%(item,parmDict[item],sigDict[item]))
Q,Io,wt,Ic,Ib,Ifb = Profile[:6]
Qmin = Limits[1][0]
Qmax = Limits[1][1]
wtFactor = ProfDict['wtFactor']
Ibeg = np.searchsorted(Q,Qmin)
Ifin = np.searchsorted(Q,Qmax)+1 #include last point
Ic[:] = 0
pairData = Model['Pair']
if 'User' in pairData['Errors']:
Wt = wt[Ibeg:Ifin]
elif 'Sqrt' in pairData['Errors']:
Wt = 1./Io[Ibeg:Ifin]
elif 'Percent' in pairData['Errors']:
Wt = 1./(pairData['Percent error']*Io[Ibeg:Ifin])
parmDict,varyList,values = GetModelParms()
result = so.leastsq(calcSASD,values,full_output=True,epsfcn=1.e-12,factor=0.1, #ftol=Ftol,
args=(Q[Ibeg:Ifin],Io[Ibeg:Ifin],wtFactor*Wt,Ifb[Ibeg:Ifin],parmDict,varyList))
parmDict.update(dict(zip(varyList,result[0])))
chisq = np.sum(result[2]['fvec']**2)
ncalc = result[2]['nfev']
covM = result[1]
Rvals = {}
Rvals['Rwp'] = np.sqrt(chisq/np.sum(wt[Ibeg:Ifin]*Io[Ibeg:Ifin]**2))*100. #to %
Rvals['GOF'] = chisq/(Ifin-Ibeg-len(varyList)) #reduced chi^2
Ic[Ibeg:Ifin] = getSASDRg(Q[Ibeg:Ifin],parmDict)
Msg = 'Failed to converge'
try:
Nans = np.isnan(result[0])
if np.any(Nans):
name = varyList[Nans.nonzero(True)[0]]
Msg = 'Nan result for '+name+'!'
raise ValueError
Negs = np.less_equal(result[0],0.)
if np.any(Negs):
name = varyList[Negs.nonzero(True)[0]]
Msg = 'negative coefficient for '+name+'!'
raise ValueError
if len(covM):
sig = np.sqrt(np.diag(covM)*Rvals['GOF'])
sigDict = dict(zip(varyList,sig))
print (' Results of Rg fit:')
print ('Number of function calls: %d Number of observations: %d Number of parameters: %d'%(ncalc,Ifin-Ibeg,len(varyList)))
print ('Rwp = %7.2f%%, chi**2 = %12.6g, reduced chi**2 = %6.2f'%(Rvals['Rwp'],chisq,Rvals['GOF']))
SetModelParms()
covMatrix = covM*Rvals['GOF']
return True,result,varyList,sig,Rvals,covMatrix,parmDict,''
except (ValueError,TypeError): #when bad LS refinement; covM missing or with nans
return False,0,0,0,0,0,0,Msg
return [None,]
def ModelFxn(Profile,ProfDict,Limits,Sample,sasdData):
shapes = {'Spheroid':[SpheroidFF,SpheroidVol],'Cylinder':[CylinderDFF,CylinderDVol],
'Cylinder AR':[CylinderARFF,CylinderARVol],'Unified sphere':[UniSphereFF,UniSphereVol],
'Unified rod':[UniRodFF,UniRodVol],'Unified rod AR':[UniRodARFF,UniRodARVol],
'Unified disk':[UniDiskFF,UniDiskVol],'Sphere':[SphereFF,SphereVol],
'Unified tube':[UniTubeFF,UniTubeVol],'Cylinder diam':[CylinderDFF,CylinderDVol],
'Spherical shell':[SphericalShellFF,SphericalShellVol]}
sfxns = {'Dilute':DiluteSF,'Hard sphere':HardSpheresSF,'Square well':SquareWellSF,
'Sticky hard sphere':StickyHardSpheresSF,'InterPrecipitate':InterPrecipitateSF,}
# pdb.set_trace()
partData = sasdData['Particle']
Back = sasdData['Back']
Q,Io,wt,Ic,Ib,Ifb = Profile[:6]
Qmin = Limits[1][0]
Qmax = Limits[1][1]
Ibeg = np.searchsorted(Q,Qmin)
Ifin = np.searchsorted(Q,Qmax)+1 #include last point
Ib[:] = Back[0]
Ic[:] = 0
Rbins = []
Dist = []
for level in partData['Levels']:
controls = level['Controls']
distFxn = controls['DistType']
if distFxn in ['LogNormal','Gaussian','LSW','Schulz-Zimm']:
parmDict = level[controls['DistType']]
FFfxn = shapes[controls['FormFact']][0]
Volfxn = shapes[controls['FormFact']][1]
SFfxn = sfxns[controls['StrFact']]
FFargs = []
SFargs = []
for item in ['Dist','VolFr','epis','Sticky','Depth','Width',]:
if item in controls.get('SFargs',{}):
SFargs.append(controls['SFargs'][item][0])
for item in ['Aspect ratio','Length','Thickness','Diameter','Shell thickness']:
if item in controls['FFargs']:
FFargs.append(controls['FFargs'][item][0])
contrast = controls['Contrast']
distDict = {}
for item in parmDict:
distDict[item] = parmDict[item][0]
rBins,dBins,dist = MakeDiamDist(controls['DistType'],controls['NumPoints'],controls['Cutoff'],distDict)
Gmat = G_matrix(Q[Ibeg:Ifin],rBins,contrast,FFfxn,Volfxn,FFargs).T
dist *= level[distFxn]['Volume'][0]
Ic[Ibeg:Ifin] += np.dot(Gmat,dist)*SFfxn(Q[Ibeg:Ifin],args=SFargs)
Rbins.append(rBins)
Dist.append(dist/(4.*dBins))
elif 'Unified' in distFxn:
parmDict = level[controls['DistType']]
Rg,G,B,P,Rgco = parmDict['Rg'][0],parmDict['G'][0],parmDict['B'][0], \
parmDict['P'][0],parmDict['Cutoff'][0]
Qstar = Q[Ibeg:Ifin]/(scsp.erf(Q[Ibeg:Ifin]*Rg/np.sqrt(6)))**3
Guin = G*np.exp(-(Q[Ibeg:Ifin]*Rg)**2/3)
Porod = (B/Qstar**P)*np.exp(-(Q[Ibeg:Ifin]*Rgco)**2/3)
Ic[Ibeg:Ifin] += Guin+Porod
Rbins.append([])
Dist.append([])
elif 'Porod' in distFxn:
parmDict = level[controls['DistType']]
B,P,Rgco = parmDict['B'][0],parmDict['P'][0],parmDict['Cutoff'][0]
Porod = (B/Q[Ibeg:Ifin]**P)*np.exp(-(Q[Ibeg:Ifin]*Rgco)**2/3)
Ic[Ibeg:Ifin] += Porod
Rbins.append([])
Dist.append([])
elif 'Mono' in distFxn:
parmDict = level[controls['DistType']]
R = level[controls['DistType']]['Radius'][0]
FFfxn = shapes[controls['FormFact']][0]
Volfxn = shapes[controls['FormFact']][1]
SFfxn = sfxns[controls['StrFact']]
FFargs = []
SFargs = []
for item in ['Dist','VolFr','epis','Sticky','Depth','Width',]:
if item in controls.get('SFargs',{}):
SFargs.append(controls['SFargs'][item][0])
for item in ['Aspect ratio','Length','Thickness','Diameter','Shell thickness']:
if item in controls['FFargs']:
FFargs.append(controls['FFargs'][item][0])
contrast = controls['Contrast']
Gmat = G_matrix(Q[Ibeg:Ifin],R,contrast,FFfxn,Volfxn,FFargs)
Ic[Ibeg:Ifin] += Gmat[0]*level[distFxn]['Volume'][0]*SFfxn(Q[Ibeg:Ifin],args=SFargs)
Rbins.append([])
Dist.append([])
elif 'Bragg' in distFxn:
parmDict = level[controls['DistType']]
Ic[Ibeg:Ifin] += parmDict['PkInt'][0]*G2pwd.getPsVoigt(parmDict['PkPos'][0],
parmDict['PkSig'][0],parmDict['PkGam'][0],Q[Ibeg:Ifin])
Rbins.append([])
Dist.append([])
Ic[Ibeg:Ifin] += Back[0]
slitLen = Sample.get('SlitLen',0.)
if slitLen:
Ic[Ibeg:Ifin] = SmearData(Ic,Q,slitLen,Back[0])[Ibeg:Ifin]
sasdData['Size Calc'] = [Rbins,Dist]
def MakeDiamDist(DistName,nPoints,cutoff,distDict):
if 'LogNormal' in DistName:
distFxn = 'LogNormalDist'
cumeFxn = 'LogNormalCume'
pos = distDict['MinSize']
args = [distDict['Mean'],distDict['StdDev']]
step = 4.*np.sqrt(np.exp(distDict['StdDev']**2)*(np.exp(distDict['StdDev']**2)-1.))
mode = distDict['MinSize']+distDict['Mean']/np.exp(distDict['StdDev']**2)
minX = 1. #pos
maxX = 1.e4 #np.min([mode+nPoints*step*40,1.e5])
elif 'Gauss' in DistName:
distFxn = 'GaussDist'
cumeFxn = 'GaussCume'
pos = distDict['Mean']
args = [distDict['StdDev']]
mode = distDict['Mean']
minX = np.max([mode-4.*distDict['StdDev'],1.])
maxX = np.min([mode+4.*distDict['StdDev'],1.e5])
elif 'LSW' in DistName:
distFxn = 'LSWDist'
cumeFxn = 'LSWCume'
pos = distDict['Mean']
args = []
minX,maxX = [0.,2.*pos]
elif 'Schulz' in DistName:
distFxn = 'SchulzZimmDist'
cumeFxn = 'SchulzZimmCume'
pos = distDict['Mean']
args = [distDict['StdDev']]
minX = np.max([1.,pos-4.*distDict['StdDev']])
maxX = np.min([pos+4.*distDict['StdDev'],1.e5])
nP = 500
Diam = np.logspace(0.,5.,nP,True)
TCW = eval(cumeFxn+'(Diam,pos,args)')
CumeTgts = np.linspace(cutoff,(1.-cutoff),nPoints+1,True)
rBins = np.interp(CumeTgts,TCW,Diam,0,0)
dBins = np.diff(rBins)
rBins = rBins[:-1]+dBins/2.
return rBins,dBins,eval(distFxn+'(rBins,pos,args)')
################################################################################
#### MaxEnt testing stuff
################################################################################
def print_vec(text, a):
'''print the contents of a vector to the console'''
n = a.shape[0]
print ("%s[ = (" % text,end='')
for i in range(n):
s = " %g, " % a[i]
print (s,end='')
print (")")
def print_arr(text, a):
'''print the contents of an array to the console'''
n, m = a.shape
print ("%s[][] = (" % text)
for i in range(n):
print (" (",end='')
for j in range(m):
print (" %g, " % a[i][j],end='')
print ("),")
print (")")
def test_MaxEnt_SB(report=True):
def readTextData(filename):
'''return q, I, dI from a 3-column text file'''
if not os.path.exists(filename):
raise Exception("file not found: " + filename)
buf = [line.split() for line in open(filename, 'r').readlines()]
buf = zip(*buf) # transpose rows and columns
q = np.array(buf[0], dtype=np.float64)
I = np.array(buf[1], dtype=np.float64)
dI = np.array(buf[2], dtype=np.float64)
return q, I, dI
print ("MaxEnt_SB: ")
test_data_file = os.path.join( 'testinp', 'test.sas')
rhosq = 100 # scattering contrast, 10^20 1/cm^-4
bkg = 0.1 # I = I - bkg
dMin, dMax, nRadii = 25, 9000, 40
defaultDistLevel = 1.0e-6
IterMax = 40
errFac = 1.05
r = np.logspace(math.log10(dMin), math.log10(dMax), nRadii)/2
dr = r * (r[1]/r[0] - 1) # step size
f_dr = np.ndarray((nRadii)) * 0 # volume fraction histogram
b = np.ndarray((nRadii)) * 0 + defaultDistLevel # MaxEnt "sky background"
qVec, I, dI = readTextData(test_data_file)
G = G_matrix(qVec,r,rhosq,SphereFF,SphereVol,args=())
chisq,f_dr,Ic = MaxEnt_SB(I - bkg, dI*errFac, b, IterMax, G, report=report)
if f_dr is None:
print ("no solution")
return
print ("solution reached")
for a,b,c in zip(r.tolist(), dr.tolist(), f_dr.tolist()):
print ('%10.4f %10.4f %12.4g'%(a,b,c))
def tests():
test_MaxEnt_SB(report=True)
if __name__ == '__main__':
tests()