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TensorScatteringClass.py
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#github comments
#default wyckoff to 'a'
#all_letters now attribute
#now have glide_screw atribute and message to report presence of glide or screw operators
import sys, pprint
from copy import deepcopy
from numpy.linalg import inv
from numpy.random import rand
from scipy.special import factorial
import numpy as np
from numpy.linalg import det, norm
import matplotlib.pyplot as plt
#sys.path.append('/home/spc93/python/PyCifRW-3.1.2')
#sys.path.append('/media/DCF0769CF0767D18/python/PyCifRW-3.1.2')
class TensorScatteringClass():
'''
Python class for resonant tensor scattering.
To run test with supplied CIF file in ipython (sometimes need to repeat for plot): %run TensorScatteringClass
While this currently has limited capability for magnetic systems, magnetic symmetry operators are used throughout
If no Site keyword arg supplied then available sites will be displayed before exiting
Useful methods:
latt2b compute reciprocal or real-space B matrix from lattice
equiv_sites compute symmetry-equivalent sites for selected site
invert inverts current spacegroup operators and sites
isGroup(sg) Returns True if sg forms a group or False and shows message if not (self.isGroup(self.sglist) should return True)
TensorCalc Calculate tensor properties for crystal and reflection; save tensors as attributes; print tensor information
print_tensors() Display crystal/atomic/structure factor spherical/Cartesian tensors
CalculateIntensityInPolarizationChannels calculate four intensity channels vs psi
PlotIntensityInPolarizationChannels plot sigma or pi intensity vs azimuthal angle
Useful parameters:
lattice
B
sglist
pglist
crystalpglist
Ts_crystal, Ts_atom, Fs (spherical tensors)
Tc_crystal, Tc_atom, Fc (spherical tensors)
'''
def __init__(self, CIFfile=None, Site=None, spacegroup_number = None, wyckoff_letter = 'a', lattice = None, TimeEven=False, verbose = True):
self.tensortypes = ['E1E1','E1E2','E2E2'] # supported tensor types
self.processes = self.tensortypes+['E1E1mag','NonResMag'] # processes list includes magnetic scattering (not treated in tensor framework)
self.Fs = None #placeholder for spherical scattering tensor (if defined)
self.verbose = verbose
self.fmt='\n%28s: '
if not CIFfile == None:
(self.symxyz, self.sitevec, self.Site, self.lattice) = self.symInfoFromCifFile(CIFfile, Site)
elif not spacegroup_number == None:
(self.symxyz, self.sitevec, self.Site) = self.symInfoFromSpacegroupAndWyckoff(spacegroup_number, wyckoff_letter)
else:
raise ValueError('=== Must specify either CIF file and site label or Spacegroup number and Wyckoff letter')
if not lattice == None:
self.lattice = lattice # use lattice parameters if supplied
elif CIFfile == None: # if not supplied and no CIF file then use default lattice
self.lattice = [0.5, 0.5, 0.5, 90, 90, 90]
self.sglist=self.spacegroup_list_from_genpos_list(self.symxyz)
if TimeEven==True: #add time-reversal symmetry operator - doubles size of spacegroup
__sgnew=deepcopy(self.sglist)
for __sym in __sgnew:
__sym[2]=-__sym[2]
self.sglist+=__sgnew
#calculate B matrix
self.B=self.latt2b(self.lattice)
self.pglist= self.site_sym(self.sglist, self.sitevec) #point group of site
self.crystalpglist = self.crystal_point_sym(self.sglist)
if CIFfile == None:
self.CIFfile = 'Spacegroup # %i' % spacegroup_number
self.all_labels = ''
else:
self.CIFfile = CIFfile
if self.verbose:
print(self.__repr__())
def symInfoFromSpacegroupAndWyckoff(self, sg_num, wyck_letter, site = None):
'''
Get symxyz (general positions as strings from spacegroup and Wyckoff letter
returns site after substituting either given x, y, z values or random numbers
Requires CCTBX package
'''
import cctbx.sgtbx as sg
sgi = sg.space_group_info(sg_num)
sg = sgi.group()
w = sgi.wyckoff_table()
self.all_letters = list([w.position(i).letter() for i in range(w.size())])
if wyck_letter in self.all_letters:
opxyz = str(w.position(wyck_letter).special_op()) # generic xyz string for site
else:
print('=== Invalid Wyckoff letter for spacegroup. Valid letters:', self.all_letters)
raise(ValueError)
opxyz = opxyz.replace('/','./') # fix for Python 2.x int divide - not needed for Python 3
if not site == None: #use random values for x,y,z if not specified
x, y, z = site
else:
x, y, z = rand(), rand(), rand()
sitevec = np.array(eval(opxyz)) # numerical array for site
sitevec = self.firstCell(sitevec) # put in first unit cell
symxyz = [str(s.as_xyz()) for s in sg.all_ops()] # all sg ops as xyz list
sitestr = '%s %s' % (str(w.position(wyck_letter).multiplicity()), wyck_letter)
return (symxyz, sitevec, sitestr)
def symInfoFromCifFile(self, CIFfile, Site):
try:
import CifFile
except:
print("=== You need to install the PyCifRW module and add it to the Python path.\n=== Use pip install PyCifRW or visit https://pypi.org/project/PyCifRW/")
self.CIFfile = CIFfile
self.Site = Site
self.cif_obj = CifFile.CifFile(self.CIFfile)
firstkey = self.cif_obj.keys()[0]; cb = self.cifblock=self.cif_obj[firstkey]
lattice = [float(cb['_cell_length_a'].partition('(')[0]), float(cb['_cell_length_b'].partition('(')[0]), float(cb['_cell_length_c'].partition('(')[0]), float(cb['_cell_angle_alpha'].partition('(')[0]), float(cb['_cell_angle_beta'].partition('(')[0]), float(cb['_cell_angle_gamma'].partition('(')[0])]
self.all_labels=', '.join(cb['_atom_site_label'])
try:
self.atom_index = cb['_atom_site_label'].index(Site)
except:
print("=== Error: site keyword string must be in the atomic site list: " + self.all_labels)
return
sitevec = np.array([float(cb['_atom_site_fract_x'][self.atom_index].split('(')[0]), float(cb['_atom_site_fract_y'][self.atom_index].split('(')[0]), float(cb['_atom_site_fract_z'][self.atom_index].split('(')[0]) ])
try:
symxyz=cb['_symmetry_equiv_pos_as_xyz']
except:
symxyz=cb['_space_group_symop_operation_xyz'] #assume this is full group, not just generators
return (symxyz, sitevec, Site, lattice)
def __repr__(self):
if self.Site==None:
return "=== Atomic site labels: \n" + self.all_labels + "\n=== Use Site keyword to specific a site, e.g. Site = 'Fe1'"
#self.fmt='\n%28s: ' ########## delete
return '\nCrystal properties\n' \
+ (self.fmt+'%s') % ('CIF file',self.CIFfile) \
+ (self.fmt+'%.3f %.3f %.3f %.2f %.2f %.2f') % ('Lattice',self.lattice[0], self.lattice[1], self.lattice[2], self.lattice[3], self.lattice[4], self.lattice[5]) \
+ (self.fmt+'%s') % ('All sites', self.all_labels) \
+ (self.fmt+'%s') % ('Site selected', self.Site) \
+ (self.fmt+'%.3f %.3f %.3f') % ('Site vector', self.sitevec[0], self.sitevec[1], self.sitevec[2]) \
+ (self.fmt+'%i') % ('No. of spacegroup ops', len(self.sglist)) \
+ (self.fmt+'%i') % ('No. of sym ops at site', len(self.pglist)) \
+ (self.fmt+'%i') % ('No. of equiv. sites in cell', int(len(self.sglist)/len(self.pglist))) \
+ (self.fmt+'%i') % ('No. of pg ops for crystal', len(self.crystalpglist))
def TensorCalc(self, hkl=np.array([0,0,0]), K=None, Parity=+1, Time=+1):
'''
hkl, hkln: hkl values for reflection and azimuthal reference
'''
self.K = K
self.hkl = hkl
self.Parity = Parity
self.Time = Time
txtoe=['Even', 'Odd', 'Either', 'Either'];
outstr = '\nTensor properties\n'\
+ (self.fmt+'%s') % ('Required parity', self.msg(self.Parity, txtoe)) \
+ (self.fmt+'%s') % ('Required time sym.', self.msg(self.Time, txtoe)) \
outstr += self.SF_symmetry(self.sitevec, self.hkl, self.sglist)
outstr += (self.fmt+'%r') % ('Glide or screw', self.glide_screw)
#populate tensor with random complex numbers that satisfy the requirements for a Hermitian tensor
self.Ts=list(np.zeros(2*self.K+1))
self.Ts[self.K]=rand()
for Qp in range(1,self.K+1):
Qn=-Qp;
rndr=rand(); rndi=rand();
self.Ts[K-Qp]=rndr+rndi*1.J;
self.Ts[K+Qp]=(-1)**Qp*(rndr-rndi*1.J);
self.Tc1=self.spherical_to_cart_tensor(self.Ts) #convert to cartesian tensor of same rank
self.Tc_atom=self.apply_sym(self.Tc1, self.pglist, self.B, P=Parity, T=Time); #apply site symmetry using site point group ops and B matrix
self.Tc_crystal=self.apply_sym(self.Tc1, self.crystalpglist, self.B, P=Parity, T=Time); #apply site symmetry using crystal point group ops and B matrix
self.Fc=self.norm_array(self.calc_SF(self.Tc1, self.sitevec, self.hkl, self.sglist, self.B, P=Parity, T=Time)); #calc SF Crt tensor using crystal space group and B matrix
self.Ts_atom=self.norm_array(self.cart_to_spherical_tensor(self.Tc_atom)); #atomic spherical tensor
self.Ts_crystal=self.norm_array(self.cart_to_spherical_tensor(self.Tc_crystal)); #crystal spherical tensor
self.Fs=self.norm_array(self.cart_to_spherical_tensor(self.Fc)); #SF spherical tensor
if self.verbose:
print(outstr)
return
def calcXrayVectors(self, lam, psi, hkl, hkln):
'''
calculate relevant Cartesian vector in sample reference frame
return(h, q0, q1, esig, e0pi, e1pi)
'''
self.theta = theta = np.arcsin(lam*norm(np.dot(self.B, hkl))/2)
h=x=np.array([1,0,0])
q0=np.array([-np.sin(theta), np.cos(theta),0])
q1=np.array([np.sin(theta),np.cos(theta),0])
esig=z=np.array([0,0,1])
epi0=np.array([np.cos(theta),np.sin(theta),0])
epi1=np.array([np.cos(theta),-np.sin(theta),0])
y=np.array([0,1,0])
Uctheta=self.theta_to_cartesian(hkl,hkln,psi,self.B)
h_c=x_c=np.dot(Uctheta, h)
q0_c=np.dot(Uctheta,q0)
q1_c=np.dot(Uctheta,q1)
esig_c=z_c=np.dot(Uctheta,esig)
e0pi_c=np.dot(Uctheta,epi0)
e1pi_c=np.dot(Uctheta,epi1)
y_c=np.dot(Uctheta,y)
return(h_c, q0_c, q1_c, esig_c, e0pi_c, e1pi_c)
def Xtensor(self, process, rank, time, parity, e0, e1, q0, q1):
'''
Calculates resonant scattering tensor as per Lovesey
This version is a dump of the output of a Mathematica implementation (hence messy!)
The Sympy version of this calculation carries out the same tensor calculation
'''
[e0x, e0y, e0z]=e0
[e1x, e1y, e1z]=e1
[q0x, q0y, q0z]=q0
[q1x, q1y, q1z]=q1
Complex=self.Complex #make it easier to convert Mathematica Fortran output
Sqrt=self.Sqrt
if process=='E1E1' and rank==2 and parity==1:
X2=np.array([
((e0x-Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y))/2.,
(e0z*(e1x-Complex(0,1)*e1y)+(e0x-Complex(0,1)*e0y)*e1z)/2.,
-((e0x*e1x+e0y*e1y-2*e0z*e1z)/Sqrt(6)),
(-(e0z*e1x)-Complex(0,1)*e0z*e1y-e0x*e1z-Complex(0,1)*e0y*e1z)/2.,
((e0x+Complex(0,1)*e0y)*(e1x+Complex(0,1)*e1y))/2.
])
return X2
if process=='E1E2' and rank==3 and parity==-1:
#E1E2 rank 3
n3t=np.array([
((e0x-Complex(0,1)*e0y)*(Complex(0,1)*e1x+e1y)*
(q0x-Complex(0,1)*q0y))/(2.*Sqrt(10)),
(e0z*(Complex(0,1)*e1x+e1y)*(q0x-Complex(0,1)*q0y)+
(Complex(0,1)*e0x+e0y)*
(e1z*(q0x-Complex(0,1)*q0y)+(e1x-Complex(0,1)*e1y)*q0z))/
(2.*Sqrt(15)),(Complex(0,0.1)*
(-((e0x-Complex(0,1)*e0y)*(e1x+Complex(0,1)*e1y)*
(q0x-Complex(0,1)*q0y))+
4*e1z*(e0z*(q0x-Complex(0,1)*q0y)+
(e0x-Complex(0,1)*e0y)*q0z)-
2*(e1x-Complex(0,1)*e1y)*(e0x*q0x+e0y*q0y-2*e0z*q0z)))/Sqrt(6),
(Complex(0,-0.2)*(e0x*e1z*q0x+e0y*e1z*q0y+e0x*e1x*q0z+
e0y*e1y*q0z+e0z*(e1x*q0x+e1y*q0y-2*e1z*q0z)))/Sqrt(2),
(Complex(0,0.1)*((e0x+Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*
(q0x+Complex(0,1)*q0y)-
4*e1z*(e0z*(q0x+Complex(0,1)*q0y)+
(e0x+Complex(0,1)*e0y)*q0z)+
2*(e1x+Complex(0,1)*e1y)*(e0x*q0x+e0y*q0y-2*e0z*q0z)))/Sqrt(6),
(Complex(0,0.5)*(e0z*(e1x+Complex(0,1)*e1y)*(q0x+Complex(0,1)*q0y)+
(e0x+Complex(0,1)*e0y)*
(e1z*(q0x+Complex(0,1)*q0y)+(e1x+Complex(0,1)*e1y)*q0z)))/
Sqrt(15),((e0x+Complex(0,1)*e0y)*(Complex(0,-1)*e1x+e1y)*
(q0x+Complex(0,1)*q0y))/(2.*Sqrt(10))
])
n3=np.array([
((e0x-Complex(0,1)*e0y)*(Complex(0,1)*e1x+e1y)*
(q1x-Complex(0,1)*q1y))/(2.*Sqrt(10)),
(e0z*(Complex(0,1)*e1x+e1y)*(q1x-Complex(0,1)*q1y)+
(Complex(0,1)*e0x+e0y)*
(e1z*(q1x-Complex(0,1)*q1y)+(e1x-Complex(0,1)*e1y)*q1z))/
(2.*Sqrt(15)),(Complex(0,0.1)*
(-((e0x+Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*
(q1x-Complex(0,1)*q1y))+
4*e0z*(e1z*(q1x-Complex(0,1)*q1y)+
(e1x-Complex(0,1)*e1y)*q1z)-
2*(e0x-Complex(0,1)*e0y)*(e1x*q1x+e1y*q1y-2*e1z*q1z)))/Sqrt(6),
(Complex(0,-0.2)*(e0x*e1z*q1x+e0y*e1z*q1y+e0x*e1x*q1z+
e0y*e1y*q1z+e0z*(e1x*q1x+e1y*q1y-2*e1z*q1z)))/Sqrt(2),
(Complex(0,0.1)*((e0x-Complex(0,1)*e0y)*(e1x+Complex(0,1)*e1y)*
(q1x+Complex(0,1)*q1y)-
4*e0z*(e1z*(q1x+Complex(0,1)*q1y)+
(e1x+Complex(0,1)*e1y)*q1z)+
2*(e0x+Complex(0,1)*e0y)*(e1x*q1x+e1y*q1y-2*e1z*q1z)))/Sqrt(6),
(Complex(0,0.5)*(e0z*(e1x+Complex(0,1)*e1y)*(q1x+Complex(0,1)*q1y)+
(e0x+Complex(0,1)*e0y)*
(e1z*(q1x+Complex(0,1)*q1y)+(e1x+Complex(0,1)*e1y)*q1z)))/
Sqrt(15),((e0x+Complex(0,1)*e0y)*(Complex(0,-1)*e1x+e1y)*
(q1x+Complex(0,1)*q1y))/(2.*Sqrt(10))
])
if time==1:
return n3t-n3
elif time==-1:
return n3t+n3
if process=='E1E2' and rank==2 and parity==-1:
#E1E2 rank 2
n2t=np.array([(e0z*(Complex(0,1)*e1x+e1y)*(q0x-Complex(0,1)*q0y)+(e0x-Complex(0,1)*e0y)*(Complex(0,-2)*e1z*q0x-2*e1z*q0y+Complex(0,1)*e1x*q0z+e1y*q0z))/(2.*Sqrt(30)),(e0z*(Complex(0,-1)*e1z*q0x-e1z*q0y+Complex(0,2)*e1x*q0z+2*e1y*q0z)+e0y*(Complex(0,1)*e1y*q0x+e1x*(q0x-Complex(0,2)*q0y)-e1z*q0z)+e0x*(-2*e1y*q0x+e1x*q0y+Complex(0,1)*e1y*q0y-Complex(0,1)*e1z*q0z))/(2.*Sqrt(30)),(-(e0z*e1y*q0x)+e0z*e1x*q0y+e0y*e1x*q0z-e0x*e1y*q0z)/(2.*Sqrt(5)),(e0z*(Complex(0,-1)*e1z*q0x+e1z*q0y+Complex(0,2)*e1x*q0z-2*e1y*q0z)+e0x*(2*e1y*q0x-e1x*q0y+Complex(0,1)*e1y*q0y-Complex(0,1)*e1z*q0z)+e0y*(Complex(0,1)*e1y*q0x-e1x*(q0x+Complex(0,2)*q0y)+e1z*q0z))/(2.*Sqrt(30)),(e0z*(Complex(0,-1)*e1x+e1y)*(q0x+Complex(0,1)*q0y)+(e0x+Complex(0,1)*e0y)*(Complex(0,2)*e1z*q0x-2*e1z*q0y-Complex(0,1)*e1x*q0z+e1y*q0z))/(2.*Sqrt(30))])
n2=np.array([(Complex(0,-2)*e0z*(e1x-Complex(0,1)*e1y)*(q1x-Complex(0,1)*q1y)+(Complex(0,1)*e0x+e0y)*(e1z*(q1x-Complex(0,1)*q1y)+(e1x-Complex(0,1)*e1y)*q1z))/(2.*Sqrt(30)),(Complex(0,-1)*e0z*(e1z*q1x-Complex(0,1)*e1z*q1y+e1x*q1z-Complex(0,1)*e1y*q1z)+e0x*(e1y*q1x+e1x*q1y-Complex(0,2)*e1y*q1y+Complex(0,2)*e1z*q1z)+e0y*(-2*e1x*q1x+Complex(0,1)*e1y*q1x+Complex(0,1)*e1x*q1y+2*e1z*q1z))/(2.*Sqrt(30)),(-(e0y*(e1z*q1x+e1x*q1z))+e0x*(e1z*q1y+e1y*q1z))/(2.*Sqrt(5)),(e0z*(Complex(0,-1)*e1z*q1x+e1z*q1y-Complex(0,1)*e1x*q1z+e1y*q1z)+e0y*(2*e1x*q1x+Complex(0,1)*e1y*q1x+Complex(0,1)*e1x*q1y-2*e1z*q1z)-e0x*(e1y*q1x+e1x*q1y+Complex(0,2)*e1y*q1y-Complex(0,2)*e1z*q1z))/(2.*Sqrt(30)),(Complex(0,2)*e0z*(e1x+Complex(0,1)*e1y)*(q1x+Complex(0,1)*q1y)+(Complex(0,-1)*e0x+e0y)*(e1z*(q1x+Complex(0,1)*q1y)+(e1x+Complex(0,1)*e1y)*q1z))/(2.*Sqrt(30))])
if time==1:
return n2t-n2
elif time==-1:
return n2t+n2
if process=='E1E2' and rank==1 and parity==-1:
#E1E2 rank 1
print("xxxxx E1E2 rank 1 not tested")
n1t=np.array([(Complex(0,0.1)*(-3*(e0x-Complex(0,1)*e0y)*(e1x+Complex(0,1)*e1y)*(q0x-Complex(0,1)*q0y)-3*e1z*(e0z*(q0x-Complex(0,1)*q0y)+(e0x-Complex(0,1)*e0y)*q0z)-(e1x-Complex(0,1)*e1y)*(e0x*q0x+e0y*q0y-2*e0z*q0z)))/Sqrt(6),(Complex(0,-0.1)*(-2*e0x*e1z*q0x-2*e0y*e1z*q0y+3*e0x*e1x*q0z+3*e0y*e1y*q0z+e0z*(3*e1x*q0x+3*e1y*q0y+4*e1z*q0z)))/Sqrt(3),(Complex(0,0.1)*(3*(e0x+Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*(q0x+Complex(0,1)*q0y)+3*e1z*(e0z*(q0x+Complex(0,1)*q0y)+(e0x+Complex(0,1)*e0y)*q0z)+(e1x+Complex(0,1)*e1y)*(e0x*q0x+e0y*q0y-2*e0z*q0z)))/Sqrt(6)])
n1=np.array([(Complex(0,0.1)*(-3*(e0x+Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*(q1x-Complex(0,1)*q1y)-3*e0z*(e1z*(q1x-Complex(0,1)*q1y)+(e1x-Complex(0,1)*e1y)*q1z)-(e0x-Complex(0,1)*e0y)*(e1x*q1x+e1y*q1y-2*e1z*q1z)))/Sqrt(6),(Complex(0,0.1)*(-3*(e0x*e1z*q1x+e0y*e1z*q1y+e0x*e1x*q1z+e0y*e1y*q1z)+2*e0z*(e1x*q1x+e1y*q1y-2*e1z*q1z)))/Sqrt(3),(Complex(0,0.1)*(3*(e0x-Complex(0,1)*e0y)*(e1x+Complex(0,1)*e1y)*(q1x+Complex(0,1)*q1y)+3*e0z*(e1z*(q1x+Complex(0,1)*q1y)+(e1x+Complex(0,1)*e1y)*q1z)+(e0x+Complex(0,1)*e0y)*(e1x*q1x+e1y*q1y-2*e1z*q1z)))/Sqrt(6)])
if time==1:
return n1t-n1
elif time==-1:
return n1t+n1
if process=='E2E2' and rank==0 and parity==1 and time==1:
XQQ0=np.array([(e0y*(3*e1y*q0x*q1x-2*e1x*q0y*q1x+3*e1x*q0x*q1y+4*e1y*q0y*q1y+
3*e1z*q0z*q1y-2*e1z*q0y*q1z+3*e1y*q0z*q1z)+
e0z*(3*e1z*q0x*q1x-2*e1x*q0z*q1x+3*e1z*q0y*q1y-2*e1y*q0z*q1y+
3*e1x*q0x*q1z+3*e1y*q0y*q1z+4*e1z*q0z*q1z)+
e0x*(3*e1y*q0y*q1x+3*e1z*q0z*q1x-2*e1y*q0x*q1y-2*e1z*q0x*q1z+
e1x*(4*q0x*q1x+3*q0y*q1y+3*q0z*q1z)))/(6.*Sqrt(5))])
return XQQ0
if process=='E2E2' and rank==1 and parity==1 and time==-1:
XQQ1=np.array([(e0x*(-2*e1x*q0z*q1x+Complex(0,1)*e1y*q0z*q1x+
Complex(0,1)*e1x*q0z*q1y+2*e1x*q0x*q1z-
Complex(0,1)*e1x*q0y*q1z+e1y*q0y*q1z+
e1z*(2*q0x*q1x-Complex(0,1)*q0y*q1x+q0y*q1y+2*q0z*q1z))-
Complex(0,1)*e0y*(Complex(0,-1)*e1y*q0z*q1x-
Complex(0,1)*e1x*q0z*q1y-2*e1y*q0z*q1y+e1x*q0x*q1z+
Complex(0,1)*e1y*q0x*q1z+2*e1y*q0y*q1z+
e1z*(q0x*q1x+Complex(0,1)*q0x*q1y+2*q0y*q1y+2*q0z*q1z))-
e0z*(e1x*(2*q0x*q1x-Complex(0,1)*q0x*q1y+q0y*q1y+2*q0z*q1z)-
Complex(0,1)*(2*e1z*(Complex(0,1)*q0z*q1x+q0z*q1y-
Complex(0,1)*q0x*q1z-q0y*q1z)+
e1y*(q0x*q1x+Complex(0,1)*q0y*q1x+2*q0y*q1y+2*q0z*q1z))))/
(4.*Sqrt(5)),(Complex(0,0.5)*
(e0z*(e1z*q0y*q1x-e1z*q0x*q1y-e1y*q0x*q1z+e1x*q0y*q1z)-
e0x*(2*e1y*q0x*q1x-2*e1x*q0y*q1x+2*e1x*q0x*q1y+
2*e1y*q0y*q1y+e1z*q0z*q1y+e1y*q0z*q1z)+
e0y*(2*e1y*q0y*q1x+e1z*q0z*q1x-2*e1y*q0x*q1y+
e1x*(2*q0x*q1x+2*q0y*q1y+q0z*q1z))))/Sqrt(10),
(e0x*(e1x*(-2*q0z*q1x-Complex(0,1)*q0z*q1y+2*q0x*q1z+
Complex(0,1)*q0y*q1z)+e1y*(Complex(0,-1)*q0z*q1x+q0y*q1z)+
e1z*(2*q0x*q1x+Complex(0,1)*q0y*q1x+q0y*q1y+2*q0z*q1z))-
e0z*(2*e1z*(q0z*(q1x+Complex(0,1)*q1y)-
(q0x+Complex(0,1)*q0y)*q1z)+
e1y*(Complex(0,1)*q0x*q1x+q0y*q1x+Complex(0,2)*q0y*q1y+
Complex(0,2)*q0z*q1z)+
e1x*(2*q0x*q1x+Complex(0,1)*q0x*q1y+q0y*q1y+2*q0z*q1z))+
e0y*(-(e1x*q0z*q1y)+Complex(0,1)*e1x*q0x*q1z+
e1y*(-(q0z*(q1x+Complex(0,2)*q1y))+
(q0x+Complex(0,2)*q0y)*q1z)+
e1z*(q0x*(Complex(0,1)*q1x+q1y)+
Complex(0,2)*(q0y*q1y+q0z*q1z))))/(4.*Sqrt(5))])
return XQQ1
if process=='E2E2' and rank==2 and parity==1 and time==1:
XQQ2=np.array([(e0z*(-3*e1z*(q0x-Complex(0,1)*q0y)*(q1x-Complex(0,1)*q1y)+
(e1x-Complex(0,1)*e1y)*
(4*q0z*q1x-Complex(0,4)*q0z*q1y-3*q0x*q1z+
Complex(0,3)*q0y*q1z))+
e0y*(e1z*(Complex(0,3)*q0z*q1x+3*q0z*q1y-Complex(0,4)*q0x*q1z-
4*q0y*q1z)+Complex(0,1)*e1x*
(2*q0x*q1x+2*q0y*q1y+3*q0z*q1z)+
e1y*(Complex(0,2)*q0y*q1x+Complex(0,2)*q0x*q1y+4*q0y*q1y+
3*q0z*q1z))+e0x*(e1x*
(-4*q0x*q1x+Complex(0,2)*q0y*q1x+Complex(0,2)*q0x*q1y-
3*q0z*q1z)+Complex(0,1)*
(e1z*(Complex(0,3)*q0z*q1x+3*q0z*q1y-Complex(0,4)*q0x*q1z-
4*q0y*q1z)+e1y*(2*q0x*q1x+2*q0y*q1y+3*q0z*q1z))))/
(4.*Sqrt(21)),(e0y*(e1y*(-3*q0z*q1x+Complex(0,2)*q0z*q1y-
3*q0x*q1z+Complex(0,2)*q0y*q1z)+
e1x*(Complex(0,-4)*q0z*q1x-3*q0z*q1y+Complex(0,3)*q0x*q1z+
4*q0y*q1z)+e1z*(Complex(0,3)*q0x*q1x+4*q0y*q1x-
3*q0x*q1y+Complex(0,2)*q0y*q1y+Complex(0,2)*q0z*q1z))-
e0x*(e1x*(2*q0z*q1x-Complex(0,3)*q0z*q1y+2*q0x*q1z-
Complex(0,3)*q0y*q1z)+
e1y*(Complex(0,-3)*q0z*q1x-4*q0z*q1y+Complex(0,4)*q0x*q1z+
3*q0y*q1z)+e1z*(Complex(0,-3)*q0y*q1x+
2*q0x*(q1x+Complex(0,2)*q1y)+3*q0y*q1y+2*q0z*q1z))+
e0z*(-2*e1z*(q0z*q1x-Complex(0,1)*q0z*q1y+q0x*q1z-
Complex(0,1)*q0y*q1z)+
e1y*(Complex(0,3)*q0x*q1x-3*q0y*q1x+4*q0x*q1y+
Complex(0,2)*q0y*q1y+Complex(0,2)*q0z*q1z)-
e1x*(2*q0x*q1x+Complex(0,4)*q0y*q1x-Complex(0,3)*q0x*q1y+
3*q0y*q1y+2*q0z*q1z)))/(4.*Sqrt(21)),
(e0y*(6*e1y*q0x*q1x-8*e1x*q0y*q1x+6*e1x*q0x*q1y+4*e1y*q0y*q1y-
3*e1z*q0z*q1y+4*e1z*q0y*q1z-3*e1y*q0z*q1z)-
e0z*(3*e1z*q0x*q1x-4*e1x*q0z*q1x+3*e1z*q0y*q1y-4*e1y*q0z*q1y+
3*e1x*q0x*q1z+3*e1y*q0y*q1z+8*e1z*q0z*q1z)+
e0x*(6*e1y*q0y*q1x-3*e1z*q0z*q1x-8*e1y*q0x*q1y+4*e1z*q0x*q1z+
e1x*(4*q0x*q1x+6*q0y*q1y-3*q0z*q1z)))/(6.*Sqrt(14)),
(e0y*(e1x*(Complex(0,-4)*q0z*q1x+3*q0z*q1y+Complex(0,3)*q0x*q1z-
4*q0y*q1z)+e1y*(3*q0z*q1x+Complex(0,2)*q0z*q1y+
3*q0x*q1z+Complex(0,2)*q0y*q1z)+
e1z*(-4*q0y*q1x+Complex(0,2)*q0y*q1y+
3*q0x*(Complex(0,1)*q1x+q1y)+Complex(0,2)*q0z*q1z))+
e0x*(e1x*(2*q0z*q1x+Complex(0,3)*q0z*q1y+2*q0x*q1z+
Complex(0,3)*q0y*q1z)+
e1y*(Complex(0,3)*q0z*q1x-4*q0z*q1y-Complex(0,4)*q0x*q1z+
3*q0y*q1z)+e1z*(Complex(0,3)*q0y*q1x+
2*q0x*(q1x-Complex(0,2)*q1y)+3*q0y*q1y+2*q0z*q1z))+
e0z*(2*e1z*(q0z*q1x+Complex(0,1)*q0z*q1y+q0x*q1z+
Complex(0,1)*q0y*q1z)+
e1y*(Complex(0,3)*q0x*q1x+3*q0y*q1x-4*q0x*q1y+
Complex(0,2)*q0y*q1y+Complex(0,2)*q0z*q1z)+
e1x*(2*q0x*q1x-Complex(0,4)*q0y*q1x+Complex(0,3)*q0x*q1y+
3*q0y*q1y+2*q0z*q1z)))/(4.*Sqrt(21)),
(Complex(0,-1)*(e0z*(3*e1z*(Complex(0,-1)*q0x+q0y)*
(q1x+Complex(0,1)*q1y)+
(e1x+Complex(0,1)*e1y)*
(Complex(0,4)*q0z*q1x-4*q0z*q1y-Complex(0,3)*q0x*q1z+
3*q0y*q1z))+e0y*
(e1z*(3*q0z*(q1x+Complex(0,1)*q1y)-
4*(q0x+Complex(0,1)*q0y)*q1z)+
e1y*(2*q0y*(q1x+Complex(0,2)*q1y)+2*q0x*q1y+
Complex(0,3)*q0z*q1z)+
e1x*(2*q0x*q1x+2*q0y*q1y+3*q0z*q1z)))-
e0x*(e1x*(4*q0x*q1x+Complex(0,2)*q0y*q1x+Complex(0,2)*q0x*q1y+
3*q0z*q1z)+Complex(0,1)*
(e1z*(Complex(0,-3)*q0z*q1x+3*q0z*q1y+Complex(0,4)*q0x*q1z-
4*q0y*q1z)+e1y*(2*q0x*q1x+2*q0y*q1y+3*q0z*q1z))))/
(4.*Sqrt(21))])
return XQQ2
if process=='E2E2' and rank==3 and parity==1 and time==-1:
XQQ3=np.array([(e0z*(e1x-Complex(0,1)*e1y)*(q0x-Complex(0,1)*q0y)*
(q1x-Complex(0,1)*q1y)-
(e0x-Complex(0,1)*e0y)*e1z*(q0x-Complex(0,1)*q0y)*
(q1x-Complex(0,1)*q1y)+
(e0x-Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*q0z*
(q1x-Complex(0,1)*q1y)-
(e0x-Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*
(q0x-Complex(0,1)*q0y)*q1z)/(4.*Sqrt(2)),
(Complex(0,0.25)*(Complex(0,-2)*e0z*(e1x-Complex(0,1)*e1y)*q0z*
(q1x-Complex(0,1)*q1y)+
e0y*(e1y*q0y*q1x-e1y*q0x*q1y+
e1x*(-(q0x*q1x)+Complex(0,2)*q0y*q1x+q0y*q1y)+
2*e1z*q0x*q1z-Complex(0,2)*e1z*q0y*q1z)+
e0x*(e1y*q0x*q1x-e1x*q0y*q1x+e1x*q0x*q1y-
Complex(0,2)*e1y*q0x*q1y-e1y*q0y*q1y+
Complex(0,2)*e1z*q0x*q1z+2*e1z*q0y*q1z)))/Sqrt(3),
(Complex(0,-1)*e0y*(e1x*(5*q0z*q1x-Complex(0,3)*q0z*q1y+
3*q0x*q1z-Complex(0,5)*q0y*q1z)+
e1y*(Complex(0,-3)*q0z*q1x-q0z*q1y+Complex(0,3)*q0x*q1z+
q0y*q1z)+e1z*(Complex(0,-5)*q0y*q1x+
3*q0x*(q1x+Complex(0,1)*q1y)+q0y*q1y-4*q0z*q1z))+
e0x*(e1x*(-(q0z*q1x)+Complex(0,3)*q0z*q1y+q0x*q1z-
Complex(0,3)*q0y*q1z)+
e1y*(Complex(0,3)*q0z*q1x+5*q0z*q1y+Complex(0,5)*q0x*q1z+
3*q0y*q1z)+e1z*(Complex(0,-3)*q0y*q1x+
q0x*(q1x+Complex(0,5)*q1y)+3*q0y*q1y-4*q0z*q1z))+
e0z*(4*e1z*(q0z*q1x-Complex(0,1)*q0z*q1y-q0x*q1z+
Complex(0,1)*q0y*q1z)-
e1x*(Complex(0,5)*q0y*q1x+q0x*(q1x-Complex(0,3)*q1y)+
3*q0y*q1y-4*q0z*q1z)+
e1y*(Complex(0,3)*q0x*q1x-3*q0y*q1x+5*q0x*q1y+
Complex(0,1)*q0y*q1y-Complex(0,4)*q0z*q1z)))/(4.*Sqrt(30)),
(Complex(0,0.5)*(2*e0z*(-(e1z*q0y*q1x)+e1z*q0x*q1y+e1y*q0x*q1z-
e1x*q0y*q1z)+e0y*(e1y*q0y*q1x-2*e1z*q0z*q1x-e1y*q0x*q1y+
e1x*(q0x*q1x+q0y*q1y-2*q0z*q1z))+
e0x*(e1x*q0y*q1x-e1x*q0x*q1y+2*e1z*q0z*q1y-
e1y*(q0x*q1x+q0y*q1y-2*q0z*q1z))))/Sqrt(10),
(Complex(0,1)*e0y*(e1x*(5*q0z*q1x+Complex(0,3)*q0z*q1y+3*q0x*q1z+
Complex(0,5)*q0y*q1z)+
e1y*(Complex(0,3)*q0z*q1x-q0z*q1y-Complex(0,3)*q0x*q1z+
q0y*q1z)+e1z*(Complex(0,5)*q0y*q1x+
3*q0x*(q1x-Complex(0,1)*q1y)+q0y*q1y-4*q0z*q1z))+
e0x*(e1x*(-(q0z*(q1x+Complex(0,3)*q1y))+
(q0x+Complex(0,3)*q0y)*q1z)+
e1y*(Complex(0,-3)*q0z*q1x+5*q0z*q1y-Complex(0,5)*q0x*q1z+
3*q0y*q1z)+e1z*(Complex(0,3)*q0y*q1x+
q0x*(q1x-Complex(0,5)*q1y)+3*q0y*q1y-4*q0z*q1z))-
e0z*(4*e1z*(-(q0z*(q1x+Complex(0,1)*q1y))+
(q0x+Complex(0,1)*q0y)*q1z)+
e1x*(Complex(0,-5)*q0y*q1x+q0x*(q1x+Complex(0,3)*q1y)+
3*q0y*q1y-4*q0z*q1z)+
e1y*(Complex(0,3)*q0x*q1x+3*q0y*q1x-5*q0x*q1y+
Complex(0,1)*q0y*q1y-Complex(0,4)*q0z*q1z)))/(4.*Sqrt(30)),
(Complex(0,0.25)*(Complex(0,2)*e0z*(e1x+Complex(0,1)*e1y)*q0z*
(q1x+Complex(0,1)*q1y)+
e0y*(e1y*q0y*q1x-e1y*q0x*q1y+
e1x*(-(q0x*q1x)-Complex(0,2)*q0y*q1x+q0y*q1y)+
2*e1z*q0x*q1z+Complex(0,2)*e1z*q0y*q1z)+
e0x*(e1y*q0x*q1x-e1x*q0y*q1x+e1x*q0x*q1y+
Complex(0,2)*e1y*q0x*q1y-e1y*q0y*q1y-
Complex(0,2)*e1z*q0x*q1z+2*e1z*q0y*q1z)))/Sqrt(3),
(e0z*(e1x+Complex(0,1)*e1y)*(q0x+Complex(0,1)*q0y)*
(q1x+Complex(0,1)*q1y)-
(e0x+Complex(0,1)*e0y)*(e1z*(q0x+Complex(0,1)*q0y)*
(q1x+Complex(0,1)*q1y)-
(e1x+Complex(0,1)*e1y)*
(q0z*(q1x+Complex(0,1)*q1y)-(q0x+Complex(0,1)*q0y)*q1z)))/
(4.*Sqrt(2))])
return XQQ3
if process=='E2E2' and rank==4 and parity==1 and time==1:
#nedit replace \n \r and ' ' with nothing.
#List((( -> array([(( and add square bracket after first round, also at end
XQQ4=np.array([((e0x-Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*
(q0x-Complex(0,1)*q0y)*(q1x-Complex(0,1)*q1y))/4.,
(e0z*(e1x-Complex(0,1)*e1y)*(q0x-Complex(0,1)*q0y)*
(q1x-Complex(0,1)*q1y)+
(e0x-Complex(0,1)*e0y)*e1z*(q0x-Complex(0,1)*q0y)*
(q1x-Complex(0,1)*q1y)+
(e0x-Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*q0z*
(q1x-Complex(0,1)*q1y)+
(e0x-Complex(0,1)*e0y)*(e1x-Complex(0,1)*e1y)*
(q0x-Complex(0,1)*q0y)*q1z)/(4.*Sqrt(2)),
(e0x*(e1x*(-2*q0x*q1x+Complex(0,1)*q0y*q1x+Complex(0,1)*q0x*q1y+
2*q0z*q1z)+Complex(0,1)*
(Complex(0,-2)*e1z*(q0z*q1x-Complex(0,1)*q0z*q1y+q0x*q1z-
Complex(0,1)*q0y*q1z)+e1y*(q0x*q1x+q0y*q1y-2*q0z*q1z))\
)+Complex(0,1)*(Complex(0,-2)*e0z*
(e1z*(q0x-Complex(0,1)*q0y)*(q1x-Complex(0,1)*q1y)+
(e1x-Complex(0,1)*e1y)*
(q0z*(q1x-Complex(0,1)*q1y)+(q0x-Complex(0,1)*q0y)*q1z))\
+e0y*(-2*e1z*(q0z*q1x-Complex(0,1)*q0z*q1y+q0x*q1z-
Complex(0,1)*q0y*q1z)+
e1x*(q0x*q1x+q0y*q1y-2*q0z*q1z)+
e1y*(q0y*q1x+q0x*q1y-Complex(0,2)*q0y*q1y+
Complex(0,2)*q0z*q1z))))/(4.*Sqrt(7)),
(-(e0x*(e1x*(3*q0z*q1x-Complex(0,1)*q0z*q1y+3*q0x*q1z-
Complex(0,1)*q0y*q1z)+
e1y*(Complex(0,-1)*q0z*q1x+q0z*q1y-Complex(0,1)*q0x*q1z+
q0y*q1z)+e1z*(3*q0x*q1x-Complex(0,1)*q0y*q1x-
Complex(0,1)*q0x*q1y+q0y*q1y-4*q0z*q1z)))-
e0z*(-4*e1z*(q0z*q1x-Complex(0,1)*q0z*q1y+q0x*q1z-
Complex(0,1)*q0y*q1z)+
e1x*(3*q0x*q1x-Complex(0,1)*q0y*q1x-Complex(0,1)*q0x*q1y+
q0y*q1y-4*q0z*q1z)+
e1y*(Complex(0,-1)*q0x*q1x+q0y*q1x+q0x*q1y-
Complex(0,3)*q0y*q1y+Complex(0,4)*q0z*q1z))-
e0y*(e1y*(q0z*q1x-Complex(0,3)*q0z*q1y+q0x*q1z-
Complex(0,3)*q0y*q1z)+
e1x*(Complex(0,-1)*q0z*q1x+q0z*q1y-Complex(0,1)*q0x*q1z+
q0y*q1z)+e1z*(q0y*q1x-Complex(0,3)*q0y*q1y+
q0x*(Complex(0,-1)*q1x+q1y)+Complex(0,4)*q0z*q1z)))/
(4.*Sqrt(14)),(e0y*(e1y*q0x*q1x+e1x*q0y*q1x+e1x*q0x*q1y+
3*e1y*q0y*q1y-4*e1z*q0z*q1y-4*e1z*q0y*q1z-4*e1y*q0z*q1z)-
4*e0z*(e1z*q0x*q1x+e1x*q0z*q1x+e1z*q0y*q1y+e1y*q0z*q1y+
e1x*q0x*q1z+e1y*q0y*q1z-2*e1z*q0z*q1z)+
e0x*(e1y*q0y*q1x-4*e1z*q0z*q1x+e1y*q0x*q1y-4*e1z*q0x*q1z+
e1x*(3*q0x*q1x+q0y*q1y-4*q0z*q1z)))/(2.*Sqrt(70)),
(e0x*(e1x*(3*q0z*q1x+Complex(0,1)*q0z*q1y+3*q0x*q1z+
Complex(0,1)*q0y*q1z)+
e1y*(Complex(0,1)*q0z*q1x+q0z*q1y+Complex(0,1)*q0x*q1z+
q0y*q1z)+e1z*(3*q0x*q1x+Complex(0,1)*q0y*q1x+
Complex(0,1)*q0x*q1y+q0y*q1y-4*q0z*q1z))+
e0z*(-4*e1z*(q0z*q1x+Complex(0,1)*q0z*q1y+q0x*q1z+
Complex(0,1)*q0y*q1z)+
e1x*(3*q0x*q1x+Complex(0,1)*q0y*q1x+Complex(0,1)*q0x*q1y+
q0y*q1y-4*q0z*q1z)+
e1y*(Complex(0,1)*q0x*q1x+q0y*q1x+q0x*q1y+
Complex(0,3)*q0y*q1y-Complex(0,4)*q0z*q1z))+
e0y*(e1y*(q0z*q1x+Complex(0,3)*q0z*q1y+q0x*q1z+
Complex(0,3)*q0y*q1z)+
e1x*(Complex(0,1)*q0z*q1x+q0z*q1y+Complex(0,1)*q0x*q1z+
q0y*q1z)+e1z*(q0y*q1x+Complex(0,3)*q0y*q1y+
q0x*(Complex(0,1)*q1x+q1y)-Complex(0,4)*q0z*q1z)))/
(4.*Sqrt(14)),(2*e0z*(e1z*(q0x+Complex(0,1)*q0y)*
(q1x+Complex(0,1)*q1y)+
(e1x+Complex(0,1)*e1y)*
(q0z*(q1x+Complex(0,1)*q1y)+(q0x+Complex(0,1)*q0y)*q1z))-
Complex(0,1)*e0y*(-2*e1z*
(q0z*q1x+Complex(0,1)*q0z*q1y+q0x*q1z+
Complex(0,1)*q0y*q1z)+e1x*(q0x*q1x+q0y*q1y-2*q0z*q1z)+
e1y*(q0y*q1x+q0x*q1y+Complex(0,2)*q0y*q1y-
Complex(0,2)*q0z*q1z))+
e0x*(-(e1x*(2*q0x*q1x+Complex(0,1)*q0y*q1x+
Complex(0,1)*q0x*q1y-2*q0z*q1z))-
Complex(0,1)*(Complex(0,2)*e1z*
(q0z*q1x+Complex(0,1)*q0z*q1y+q0x*q1z+
Complex(0,1)*q0y*q1z)+e1y*(q0x*q1x+q0y*q1y-2*q0z*q1z)))\
)/(4.*Sqrt(7)),(-(e0z*(e1x+Complex(0,1)*e1y)*(q0x+Complex(0,1)*q0y)*
(q1x+Complex(0,1)*q1y))-
(e0x+Complex(0,1)*e0y)*
(e1z*(q0x+Complex(0,1)*q0y)*(q1x+Complex(0,1)*q1y)+
(e1x+Complex(0,1)*e1y)*
(q0z*(q1x+Complex(0,1)*q1y)+(q0x+Complex(0,1)*q0y)*q1z)))/
(4.*Sqrt(2)),((e0x+Complex(0,1)*e0y)*(e1x+Complex(0,1)*e1y)*
(q0x+Complex(0,1)*q0y)*(q1x+Complex(0,1)*q1y))/4.])
return XQQ4
else:
raise ValueError('Unknown tensor type')
def Complex(self, a, b): #allow FortranForm Mathematica output with only minor mods need to delete spaces, List( ... )->array([...])
return a+b*1j
def Sqrt(self, a):
return np.sqrt(a)
def TensorScatteringMatrix(self, mpol, Fs, time, parity, esig_c, e0pi_c, e1pi_c, q0_c, q1_c):
'''
Calculate 2x2 scattering amplitude matrix for tensor scattering
'''
K=(len(Fs)-1)/2 #get rank K from tensor size
X_ss=self.Xtensor(mpol, K, time, parity, esig_c, esig_c, q0_c, q1_c)
X_sp=self.Xtensor(mpol, K, time, parity, esig_c, e1pi_c, q0_c, q1_c)
X_ps=self.Xtensor(mpol, K, time, parity, e0pi_c, esig_c, q0_c, q1_c)
X_pp=self.Xtensor(mpol, K, time, parity, e0pi_c, e1pi_c, q0_c, q1_c)
f_ss=self.scalar_contract(X_ss, Fs)
f_sp=self.scalar_contract(X_sp, Fs)
f_ps=self.scalar_contract(X_ps, Fs)
f_pp=self.scalar_contract(X_pp, Fs)
self.G=np.array([[f_ss, f_ps], [f_sp, f_pp]]) #scattering matrix
return self.G
def NonResonantMagneticScatteringMatrix(self, sk, lk, esig_c, e0pi_c, e1pi_c, q0_c, q1_c):
'''
Calculate 2x2 scattering amplitude matrix for non-resonant magnetic scattering
spin and orbital components (complex) for reflection are sk, lk
BB and AA are B (spin) and A (orbit) coupling vectors from SWL, Blume etc
'''
e0=esig_c; e1=esig_c;
BB=np.cross(e1,e0)+np.cross(q1_c,e1)*np.dot(q1_c,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)-np.cross(np.cross(q1_c,e1),np.cross(q0_c,e0))
AA=2*(1-np.dot(q0_c,q1_c))*np.cross(e1,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)+np.cross(q1_c,e1)*np.dot(q1_c,e0)
f_ss=1j*(np.dot(sk,BB)+np.dot(lk,AA));
e0=esig_c; e1=e1pi_c;
BB=np.cross(e1,e0)+np.cross(q1_c,e1)*np.dot(q1_c,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)-np.cross(np.cross(q1_c,e1),np.cross(q0_c,e0))
AA=2*(1-np.dot(q0_c,q1_c))*np.cross(e1,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)+np.cross(q1_c,e1)*np.dot(q1_c,e0)
f_sp=1j*(np.dot(sk,BB)+np.dot(lk,AA));
e0=e0pi_c; e1=esig_c;
BB=np.cross(e1,e0)+np.cross(q1_c,e1)*np.dot(q1_c,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)-np.cross(np.cross(q1_c,e1),np.cross(q0_c,e0))
AA=2*(1-np.dot(q0_c,q1_c))*np.cross(e1,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)+np.cross(q1_c,e1)*np.dot(q1_c,e0)
f_ps=1j*(np.dot(sk,BB)+np.dot(lk,AA));
e0=e0pi_c; e1=e1pi_c;
BB=np.cross(e1,e0)+np.cross(q1_c,e1)*np.dot(q1_c,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)-np.cross(np.cross(q1_c,e1),np.cross(q0_c,e0))
AA=2*(1-np.dot(q0_c,q1_c))*np.cross(e1,e0)-np.cross(q0_c,e0)*np.dot(q0_c,e1)+np.cross(q1_c,e1)*np.dot(q1_c,e0)
f_pp=1j*(np.dot(sk,BB)+np.dot(lk,AA));
self.G=np.array([[f_ss, f_ps], [f_sp, f_pp]]) #scattering matrix
return self.G
def E1E1ResonantMagneticScatteringMatrix(self, mk, esig_c, e0pi_c, e1pi_c, q0_c, q1_c):
'''
Calculate 2x2 scattering amplitude matrix for E1E1 resonant magnetic scattering
'''
e0=esig_c; e1=esig_c;
E1E1=np.cross(e1,e0)
f_ss=1j*(np.dot(mk,E1E1));
e0=esig_c; e1=e1pi_c;
E1E1=np.cross(e1,e0)
f_sp=1j*(np.dot(mk,E1E1));
e0=e0pi_c; e1=esig_c;
E1E1=np.cross(e1,e0)
f_ps=1j*(np.dot(mk,E1E1));
e0=e0pi_c; e1=e1pi_c;
E1E1=np.cross(e1,e0)
f_pp=1j*(np.dot(mk,E1E1));
self.G=np.array([[f_ss, f_ps], [f_sp, f_pp]]) #scattering matrix
return self.G
# def CalculateIntensityInPolarizationChannels(self, process, lam, hkl, hkln, psideg, K=None, Time=None, Parity=None, mk=None, lk=None, sk=None):
# '''
# Calculate intensity in four linear polarization channels
# psi can be a scalar or array/list
# '''
#
# tensortypes=['E1E1','E1E2','E2E2']
# processes=tensortypes+['E1E1mag','NonResMag']
# assert process in processes, '=== First argument (process) should be in ' + str(processes)
#
# assert process not in tensortypes or (K != None and Time != None and Parity != None), '=== Need keywords K, Time, Parity for tensor processes'
# assert process != 'E1E1mag' or mk is not None, '=== Need keyword mk for E1E1 resonant magnetic scattering'
# assert process != 'NonResMag' or (sk is not None and lk is not None), '=== Need keywords sk, lk (arrays) for non-resonant magnetic scattering'
#
# if process in tensortypes:
# self.TensorCalc(K=K, hkl=hkl, Parity=Parity, Time=Time)
#
# try:
# psi=[float(psideg)*np.pi/180]
# except:
# psi=np.array(psideg*np.pi/180)
#
# Iss, Isp,Ips, Ipp = [], [], [],[]
# for psival in psi:
#
# (h, q0, q1, esig, e0pi, e1pi) = self.calcXrayVectors(lam, psival, hkl, hkln)
#
# if process in tensortypes:
# self.Xtensor(process, K, Time, Parity, esig, e1pi, q0, q1)
# G=self.TensorScatteringMatrix(process, self.Fs, Time, Parity, esig, e0pi, e1pi, q0, q1)
#
# elif process == 'E1E1mag':
# G=self.E1E1ResonantMagneticScatteringMatrix(mk, esig, e0pi, e1pi, q0, q1)
#
# elif process == 'NonResMag':
# G=self.NonResonantMagneticScatteringMatrix(sk, lk, esig, e0pi, e1pi, q0, q1)
#
# else:
# raise ValueError("== Don't know what to do with process type %s: " % process)
#
# Iss+=[abs(G[0,0])**2]; Isp+=[abs(G[1,0])**2]; Ips+=[abs(G[0,1])**2]; Ipp+=[abs(G[1,1])**2];
# if len(Iss)>1:
# return (np.array(Iss), np.array(Isp), np.array(Ips), np.array(Ipp))
# else:
# return (Iss[0], Isp[0], Ips[0], Ipp[0])
def CalculateIntensityInPolarizationChannels(self, process, lam, hkl, hkln, psideg, K=None, Time=None, Parity=None, mk=None, lk=None, sk=None):
'''
Calculate intensity in four linear polarization channels
psi can be a scalar or array/list
'''
assert process in self.processes, '=== First argument (process) should be in ' + str(self.processes)
assert process not in self.tensortypes or (K != None and Time != None and Parity != None), '=== Need keywords K, Time, Parity for tensor processes'
assert process != 'E1E1mag' or mk is not None, '=== Need keyword mk for E1E1 resonant magnetic scattering'
assert process != 'NonResMag' or (sk is not None and lk is not None), '=== Need keywords sk, lk (arrays) for non-resonant magnetic scattering'
if process in self.tensortypes:
self.TensorCalc(K=K, hkl=hkl, Parity=Parity, Time=Time)
try:
psi=[float(psideg)*np.pi/180]
except:
psi=np.array(psideg)*np.pi/180
Iss, Isp,Ips, Ipp = [], [], [],[]
for psival in psi:
G = self.CalculateScatteringMatrixG(process, lam, psival, hkl, hkln, Fs = self.Fs, K = K, Time = Time, Parity = Parity, mk = mk, sk = sk, lk = lk)
Iss+=[abs(G[0,0])**2]; Isp+=[abs(G[1,0])**2]; Ips+=[abs(G[0,1])**2]; Ipp+=[abs(G[1,1])**2];
if len(Iss)>1:
return (np.array(Iss), np.array(Isp), np.array(Ips), np.array(Ipp))
else:
return (Iss[0], Isp[0], Ips[0], Ipp[0])
def CalculateIntensityFromPolarizationAnalyser(self, process, lam, hkl, hkln, psideg, pol_eta_deg, pol_th_deg = 45, stokesvec_swl = [0, 0, 1], K = None, Time = None, Parity = None, mk = None, lk = None, sk = None):
'''
Calculate intensity from polarization analyser vs pol_eta (analyser rotation)
pol_eta_deg can be a scalar or array/list
pol_th_deg is polarizer theta angle (deg) (default 45)
stokesvec_swl is Stokes as per SWL papers (P3 = horizontal linear, default [0 ,0, 1])
'''
assert process in self.processes+['Scalar'], '=== First argument (process) should be in ' + str(self.processes+['Scalar'])
assert process not in self.tensortypes or (K != None and Time != None and Parity != None), '=== Need keywords K, Time, Parity for tensor processes'
assert process != 'E1E1mag' or mk is not None, '=== Need keyword mk for E1E1 resonant magnetic scattering'
assert process != 'NonResMag' or (sk is not None and lk is not None), '=== Need keywords sk, lk (arrays) for non-resonant magnetic scattering'
if process in self.tensortypes:
self.TensorCalc(K=K, hkl=hkl, Parity=Parity, Time=Time)
try:
pol_eta = [float(pol_eta_deg) * np.pi/180]
except:
pol_eta = np.array(pol_eta_deg) * np.pi/180
pol_theta = pol_th_deg * np.pi/180
if process=='Scalar': #special case - scalar scattering not considered in the rest of the class
self.calcXrayVectors(lam, 0, hkl, hkln) # needed to calculate theta
G = np.array([[1, 0], [0, np.cos(2*self.theta)]])
else:
G = self.CalculateScatteringMatrixG(process, lam, psideg*np.pi/180, hkl, hkln, Fs = self.Fs, K = K, Time = Time, Parity = Parity, mk = mk, sk = sk, lk = lk)
[P1, P2, P3] = stokesvec_swl
mu = 1./2*np.array([[1.+P3, P1-1.J*P2], [ P1+1.J*P2, 1.-P3]])
I_pol=[]
for eta in pol_eta:
A = np.array([[np.cos(eta), np.sin(eta)], [-np.cos(2*pol_theta)*np.sin(eta), np.cos(2*pol_theta)*np.cos(eta) ]])
I_pol += [np.dot(A, np.dot(G, np.dot(mu,np.dot(np.conjugate(G.T), np.conjugate(A.T))))).trace()]
if len(I_pol)>1:
return np.array(I_pol)
else:
return I_pol[0]
def CalculateScatteringMatrixG(self, process, lam, psival, hkl, hkln, Fs = None, K=None, Time = None, Parity = None, mk = None, sk = None, lk = None):
'''
Calculate G for specified scattering process; require lam, psival, hkl, hkln
Fs (structure factor spherical tensor), Time & Parity symmetry, mk, sk, lk are required for specific processes only
2 x 2 G matrix defined in SWL papers
'''
(h, q0, q1, esig, e0pi, e1pi) = self.calcXrayVectors(lam, psival, hkl, hkln)
(h, q0, q1, esig, e0pi, e1pi) = self.calcXrayVectors(lam, psival, hkl, hkln)
if process in self.tensortypes:
assert Fs is not None and K is not None
self.Xtensor(process, K, Time, Parity, esig, e1pi, q0, q1)
G=self.TensorScatteringMatrix(process, Fs, Time, Parity, esig, e0pi, e1pi, q0, q1)
elif process == 'E1E1mag':
G=self.E1E1ResonantMagneticScatteringMatrix(mk, esig, e0pi, e1pi, q0, q1)
elif process == 'NonResMag':
G=self.NonResonantMagneticScatteringMatrix(sk, lk, esig, e0pi, e1pi, q0, q1)
else:
raise ValueError("== Don't know what to do with process type %s: " % process)
return G
def PlotIntensityInPolarizationChannels(self, process, lam, hkl, hkln, psideg=None, K=None, Time=None, Parity=None, mk=None, lk=None, sk=None , sigmapi=None, savefile=None):
'''
Plot azimuthal dependence of sigma or pi intensity and save figure if savefile keyword string (fine name root) given
'''
assert sigmapi in ('sigma', 'pi'), "=== sigmapi keyword must be 'sigma' or 'pi'"
if psideg == None:
psideg = np.array(range(361))
Iss, Isp,Ips, Ipp = self.CalculateIntensityInPolarizationChannels(process, lam, hkl, hkln, psideg=psideg, K=K, Time=Time, Parity=Parity, mk=mk, lk=lk, sk=sk)
#sig-sig, sig-pi, sig-total
titlestr = process+' hkl=[%.1f, %.1f, %.1f] $\psi_0$=[%.1f, %.1f, %.1f]' % (tuple(hkl)+tuple(hkln))
if sigmapi == 'sigma':
Ixs, Ixp, Itot, polchar = Iss, Isp, Iss + Isp, '\sigma'
elif sigmapi == 'pi':
Ixs, Ixp, Itot, polchar = Ips, Ipp, Ips + Ipp, '\pi'
plt.figure();
plt.plot(psideg, Itot, 'k',label='$\sigma$ Total',linewidth=2.0);
plt.plot(psideg, Ixs, 'r',label='$'+polchar+'\sigma$',linewidth=2.0);
plt.plot(psideg, Ixp, 'b',label='$'+polchar+'\pi$',linewidth=2.0);
plt.legend(loc='best'); plt.ylabel('Intensity (aribtrary units)'); plt.axis('tight'); plt.xlabel('$\psi$ (degrees)'); plt.title(titlestr); plt.grid(1)
if max(abs(Itot)) < 1e-20:
plt.ylim([0,1])
plt.show()
if savefile != None:
plt.savefig(('%s '+sigmapi+'.pdf') % savefile)
def PlotIntensityVsPolarizationAnalyserRotation(self, process, lam, hkl, hkln, psideg, pol_eta_deg, pol_th_deg = 45, stokesvec_swl = [0, 0, 1], K = None, Time = None, Parity = None, mk = None, lk = None, sk = None, savefile=None):
'''
Plot intensity vs PA rotation and save figure if savefile keyword string (fine name root) given
'''
I_pol = self.CalculateIntensityFromPolarizationAnalyser(process, lam, hkl, hkln, psideg, pol_eta_deg, pol_th_deg = pol_th_deg, stokesvec_swl = stokesvec_swl, K = K, Time = Time, Parity = Parity, mk = mk, lk = lk, sk = sk)
titlestr = process+' hkl=[%.1f, %.1f, %.1f] $\psi_0$=[%.1f, %.1f, %.1f]' % (tuple(hkl)+tuple(hkln))
plt.figure();
plt.plot(pol_eta_deg, I_pol, label='Polarization\nanalyser\nintensity',linewidth=2.0);
plt.legend(loc='best'); plt.ylabel('Intensity (aribtrary units)'); plt.axis('tight'); plt.xlabel('$\eta$ (degrees)'); plt.title(titlestr); plt.grid(1)
if max(abs(I_pol)) < 1e-20:
plt.ylim([0,1])
plt.show()
if savefile != None:
plt.savefig('%s '+etascan+'.pdf' % savefile)
def theta_to_cartesian(self, hkl, hkln, psi, B):
'''
Unitary matrix for transformation from theta to cartesian coordinate system
'''
Rx=np.array([
[1, 0, 0],
[0, np.cos(psi), np.sin(psi)],
[0, -np.sin(psi), np.cos(psi)]
])
xp=np.dot(B, hkl)/norm(np.dot(B, hkl))
cp=np.cross(np.dot(B, hkl),np.dot(B, hkln));
zp=cp/norm(cp)
yp=np.cross(zp, xp);
Ucpsi=np.array([xp, yp, zp]).T;
Uctheta=np.dot(Ucpsi, Rx)
return Uctheta
def scalar_contract(self, X, T):
if len(X)!=len(T):
raise ValueError("Can't form scalar contraction of tensors with different rank")
K=int(len(X)/2)
scalar=0
for kk in range(len(X)):
q=kk-K
scalar+=(-1)**q * X[2*K-kk] * T[kk]
return scalar
def print_tensors(self):
np.set_printoptions(precision=3, suppress=True)
print('\nTensor components\n',\
self.fmt % 'Crystal (spherical)', self.Ts_crystal,\
self.fmt % 'Atom (spherical)', self.Ts_atom,\
self.fmt % 'Struct. factor (spherical)', self.Fs,\
'\n\nCrystal (Cartesian):\n\n', self.Tc_crystal,\
'\n\nAtom (Cartesian):\n\n', self.Tc_atom,\
'\n\nStruct. factor (Cartesian):\n\n', self.Fc, '\n')
def cart_to_spherical_tensor(self, Tc):
K=len(Tc.shape); #Cartesian tensor rank
Cconj=self.StoneSphericalToCartConversionCoefs(K)
Ts=np.zeros(2*K+1, dtype=complex)
for kk in range(-K, K+1):
Ts[kk+K]=np.sum(Cconj[kk+K]*Tc)
return Ts
def apply_sym(self, Tensor, symop_list, Bmat=np.array([[1, 0, 0],[0,1,0],[0,0,1]]), P=None, T=+1):
#apply point sym ops in symop_list to tensor of rank K
#Optional Bmat is used to transform arrays to Cartesian from crystal basis
#Default time (T) sym +1; no default for parity (P)
Tnew=Tensor*0.0
for sym in symop_list:
tsign=T**int((1-sym[1])/2) #sign change of time-odd tensor under time inversion
Tnew+=self.transform_cart(Tensor, self.crystal_to_cart_operator(sym[0], Bmat),P)*tsign
return Tnew
def norm_array(self, Array, Minval=0.001):
#Normalise array by largest abs value if >Minval (avoids trying to renormalise zero array)
greatest=0.0
flatarray=Array.flat
for i in range(len(flatarray)):
if abs(flatarray[i])>abs(greatest):
greatest=flatarray[i]
if abs(greatest)>Minval:
newarray=Array/greatest
else:
newarray=Array
return newarray
def calc_SF(self, Tensor, R, hkl, spacegroup_list, Bmat=np.array([[1, 0, 0],[0,1,0],[0,0,1]]), P=None, T=+1):
#calc structure factor tensor for symop_list to tensot T of rank K at position R hkl=Q
#Optional Bmat is used to transform arrays to Cartesian from crystal basis
Tnew=Tensor*0.0
for sym in spacegroup_list:
mat=sym[0]
vec=sym[1]
time=sym[2]
tsign=T**int((1-time)/2) #sign change of time-odd tensor under time inversion
newR=np.dot(mat, R)+vec
phase=np.exp(np.pi*2.j * np.dot(hkl, newR))
newbit=self.transform_cart(Tensor, self.crystal_to_cart_operator(mat, Bmat),P)*phase*tsign
Tnew=Tnew+newbit
return Tnew
def transform_cart(self, T, S, P=0):
#transform Cart tensor rank K using symmetry operator S
#If optional parameter P (parity) is given then a correction is made to account for the otherwise incorrect
#tranformation of Cartesian tensors derived from spherical pseudotensors (see Mittelwihr paper)
d=3
k=len(T.shape); #rank of T
if P==0:
Sfac=1
elif P==1 or P==-1:
Sfac=det(S)**int((3+P*(-1)**k)/2)
else:
raise ValueError('Parity should be +1 (even), -1 (odd) or 0 (ignored)')