DAMASK_EICMD/python/damask/_crystal.py

877 lines
38 KiB
Python
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

import numpy as np
from . import util
from . import Rotation
lattice_symmetries = {
'aP': 'triclinic',
'mP': 'monoclinic',
'mS': 'monoclinic',
'oP': 'orthorhombic',
'oS': 'orthorhombic',
'oI': 'orthorhombic',
'oF': 'orthorhombic',
'tP': 'tetragonal',
'tI': 'tetragonal',
'hP': 'hexagonal',
'cP': 'cubic',
'cI': 'cubic',
'cF': 'cubic',
}
class Crystal():
"""Crystal lattice."""
def __init__(self,*,
family = None,
lattice = None,
a = None,b = None,c = None,
alpha = None,beta = None,gamma = None,
degrees = False):
"""
Representation of crystal in terms of crystal family or Bravais lattice.
Parameters
----------
family : {'triclinic', 'monoclinic', 'orthorhombic', 'tetragonal', 'hexagonal', 'cubic'}, optional.
Name of the crystal family.
Will be inferred if 'lattice' is given.
lattice : {'aP', 'mP', 'mS', 'oP', 'oS', 'oI', 'oF', 'tP', 'tI', 'hP', 'cP', 'cI', 'cF'}, optional.
Name of the Bravais lattice in Pearson notation.
a : float, optional
Length of lattice parameter 'a'.
b : float, optional
Length of lattice parameter 'b'.
c : float, optional
Length of lattice parameter 'c'.
alpha : float, optional
Angle between b and c lattice basis.
beta : float, optional
Angle between c and a lattice basis.
gamma : float, optional
Angle between a and b lattice basis.
degrees : bool, optional
Angles are given in degrees. Defaults to False.
"""
if family not in [None] + list(lattice_symmetries.values()):
raise KeyError(f'invalid crystal family "{family}"')
if lattice is not None and family is not None and family != lattice_symmetries[lattice]:
raise KeyError(f'incompatible family "{family}" for lattice "{lattice}"')
self.family = lattice_symmetries[lattice] if family is None else family
self.lattice = lattice
if self.lattice is not None:
self.a = 1 if a is None else a
self.b = b
self.c = c
self.a = float(self.a) if self.a is not None else \
(self.b / self.ratio['b'] if self.b is not None and self.ratio['b'] is not None else
self.c / self.ratio['c'] if self.c is not None and self.ratio['c'] is not None else None)
self.b = float(self.b) if self.b is not None else \
(self.a * self.ratio['b'] if self.a is not None and self.ratio['b'] is not None else
self.c / self.ratio['c'] * self.ratio['b']
if self.c is not None and self.ratio['b'] is not None and self.ratio['c'] is not None else None)
self.c = float(self.c) if self.c is not None else \
(self.a * self.ratio['c'] if self.a is not None and self.ratio['c'] is not None else
self.b / self.ratio['b'] * self.ratio['c']
if self.c is not None and self.ratio['b'] is not None and self.ratio['c'] is not None else None)
self.alpha = np.radians(alpha) if degrees and alpha is not None else alpha
self.beta = np.radians(beta) if degrees and beta is not None else beta
self.gamma = np.radians(gamma) if degrees and gamma is not None else gamma
if self.alpha is None and 'alpha' in self.immutable: self.alpha = self.immutable['alpha']
if self.beta is None and 'beta' in self.immutable: self.beta = self.immutable['beta']
if self.gamma is None and 'gamma' in self.immutable: self.gamma = self.immutable['gamma']
if \
(self.a is None) \
or (self.b is None or ('b' in self.immutable and self.b != self.immutable['b'] * self.a)) \
or (self.c is None or ('c' in self.immutable and self.c != self.immutable['c'] * self.b)) \
or (self.alpha is None or ('alpha' in self.immutable and self.alpha != self.immutable['alpha'])) \
or (self.beta is None or ('beta' in self.immutable and self.beta != self.immutable['beta'])) \
or (self.gamma is None or ('gamma' in self.immutable and self.gamma != self.immutable['gamma'])):
raise ValueError (f'incompatible parameters {self.parameters} for crystal family {self.family}')
if np.any(np.array([self.alpha,self.beta,self.gamma]) <= 0):
raise ValueError ('lattice angles must be positive')
if np.any([np.roll([self.alpha,self.beta,self.gamma],r)[0]
>= np.sum(np.roll([self.alpha,self.beta,self.gamma],r)[1:]) for r in range(3)]):
raise ValueError ('each lattice angle must be less than sum of others')
else:
self.a = self.b = self.c = None
self.alpha = self.beta = self.gamma = None
def __repr__(self):
"""Represent."""
return '\n'.join([f'Crystal family {self.family}']
+ ([] if self.lattice is None else [f'Bravais lattice {self.lattice}']+
list(map(lambda x:f'{x[0]}:{x[1]:.5g}',
zip(['a','b','c','alpha','beta','gamma',],
self.parameters))))
)
def __eq__(self,other):
"""
Equal to other.
Parameters
----------
other : Crystal
Crystal to check for equality.
"""
return self.lattice == other.lattice and \
self.parameters == other.parameters and \
self.family == other.family
@property
def parameters(self):
"""Return lattice parameters a, b, c, alpha, beta, gamma."""
return (self.a,self.b,self.c,self.alpha,self.beta,self.gamma)
@property
def immutable(self):
"""Return immutable lattice parameters."""
_immutable = {
'cubic': {
'b': 1.0,
'c': 1.0,
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': np.pi/2.,
},
'hexagonal': {
'b': 1.0,
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': 2.*np.pi/3.,
},
'tetragonal': {
'b': 1.0,
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': np.pi/2.,
},
'orthorhombic': {
'alpha': np.pi/2.,
'beta': np.pi/2.,
'gamma': np.pi/2.,
},
'monoclinic': {
'alpha': np.pi/2.,
'gamma': np.pi/2.,
},
'triclinic': {}
}
return _immutable[self.family]
@property
def standard_triangle(self):
"""
Corners of the standard triangle.
Notes
-----
Not yet defined for monoclinic.
References
----------
Bases are computed from
>>> basis = {
... 'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,1.]/np.sqrt(2.), # green
... [1.,1.,1.]/np.sqrt(3.)]).T), # blue
... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # blue
... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [1.,1.,0.]/np.sqrt(2.)]).T), # blue
... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red
... [1.,0.,0.], # green
... [0.,1.,0.]]).T), # blue
... }
"""
_basis = {
'cubic': {'improper':np.array([ [-1. , 0. , 1. ],
[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
[ 0. , np.sqrt(3.) , 0. ] ]),
'proper':np.array([ [ 0. , -1. , 1. ],
[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
[ np.sqrt(3.) , 0. , 0. ] ]),
},
'hexagonal':
{'improper':np.array([ [ 0. , 0. , 1. ],
[ 1. , -np.sqrt(3.) , 0. ],
[ 0. , 2. , 0. ] ]),
'proper':np.array([ [ 0. , 0. , 1. ],
[-1. , np.sqrt(3.) , 0. ],
[ np.sqrt(3.) , -1. , 0. ] ]),
},
'tetragonal':
{'improper':np.array([ [ 0. , 0. , 1. ],
[ 1. , -1. , 0. ],
[ 0. , np.sqrt(2.) , 0. ] ]),
'proper':np.array([ [ 0. , 0. , 1. ],
[-1. , 1. , 0. ],
[ np.sqrt(2.) , 0. , 0. ] ]),
},
'orthorhombic':
{'improper':np.array([ [ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.] ]),
'proper':np.array([ [ 0., 0., 1.],
[-1., 0., 0.],
[ 0., 1., 0.] ]),
}}
return _basis.get(self.family,None)
@property
def ratio(self):
"""Return axes ratios of own lattice."""
_ratio = { 'hexagonal': {'c': np.sqrt(8./3.)}}
return dict(b = self.immutable['b']
if 'b' in self.immutable else
_ratio[self.family]['b'] if self.family in _ratio and 'b' in _ratio[self.family] else None,
c = self.immutable['c']
if 'c' in self.immutable else
_ratio[self.family]['c'] if self.family in _ratio and 'c' in _ratio[self.family] else None,
)
@property
def basis_real(self):
"""
Return orthogonal real space crystal basis.
References
----------
C.T. Young and J.L. Lytton, Journal of Applied Physics 43:14081417, 1972
https://doi.org/10.1063/1.1661333
"""
if None in self.parameters:
raise KeyError('missing crystal lattice parameters')
return np.array([
[1,0,0],
[np.cos(self.gamma),np.sin(self.gamma),0],
[np.cos(self.beta),
(np.cos(self.alpha)-np.cos(self.beta)*np.cos(self.gamma)) /np.sin(self.gamma),
np.sqrt(1 - np.cos(self.alpha)**2 - np.cos(self.beta)**2 - np.cos(self.gamma)**2
+ 2 * np.cos(self.alpha) * np.cos(self.beta) * np.cos(self.gamma))/np.sin(self.gamma)],
],dtype=float).T \
* np.array([self.a,self.b,self.c])
@property
def basis_reciprocal(self):
"""Return reciprocal (dual) crystal basis."""
return np.linalg.inv(self.basis_real.T)
@property
def lattice_points(self):
"""Return lattice points."""
_lattice_points = {
'P': [
],
'S': [
[0.5,0.5,0],
],
'I': [
[0.5,0.5,0.5],
],
'F': [
[0.0,0.5,0.5],
[0.5,0.0,0.5],
[0.5,0.5,0.0],
],
'hP': [
[2./3.,1./3.,0.5],
],
}
if self.lattice is None: raise KeyError('no lattice type specified')
return np.array([[0,0,0]]
+ _lattice_points.get(self.lattice if self.lattice == 'hP' else \
self.lattice[-1],None),dtype=float)
def to_lattice(self,*,direction=None,plane=None):
"""
Calculate lattice vector corresponding to crystal frame direction or plane normal.
Parameters
----------
direction|plane : numpy.ndarray of shape (...,3)
Vector along direction or plane normal.
Returns
-------
Miller : numpy.ndarray of shape (...,3)
Lattice vector of direction or plane.
Use util.scale_to_coprime to convert to (integer) Miller indices.
"""
if (direction is not None) ^ (plane is None):
raise KeyError('specify either "direction" or "plane"')
axis,basis = (np.array(direction),self.basis_reciprocal.T) \
if plane is None else \
(np.array(plane),self.basis_real.T)
return np.einsum('il,...l',basis,axis)
def to_frame(self,*,uvw=None,hkl=None):
"""
Calculate crystal frame vector along lattice direction [uvw] or plane normal (hkl).
Parameters
----------
uvw|hkl : numpy.ndarray of shape (...,3)
Miller indices of crystallographic direction or plane normal.
Returns
-------
vector : numpy.ndarray of shape (...,3)
Crystal frame vector along [uvw] direction or (hkl) plane normal.
"""
if (uvw is not None) ^ (hkl is None):
raise KeyError('specify either "uvw" or "hkl"')
axis,basis = (np.array(uvw),self.basis_real) \
if hkl is None else \
(np.array(hkl),self.basis_reciprocal)
return np.einsum('il,...l',basis,axis)
def kinematics(self,mode):
"""
Return crystal kinematics systems.
Parameters
----------
mode : {'slip','twin'}
Deformation mode.
Returns
-------
direction_plane : dictionary
Directions and planes of deformation mode families.
"""
_kinematics = {
'cF': {
'slip' :[np.array([
[+0,+1,-1, +1,+1,+1],
[-1,+0,+1, +1,+1,+1],
[+1,-1,+0, +1,+1,+1],
[+0,-1,-1, -1,-1,+1],
[+1,+0,+1, -1,-1,+1],
[-1,+1,+0, -1,-1,+1],
[+0,-1,+1, +1,-1,-1],
[-1,+0,-1, +1,-1,-1],
[+1,+1,+0, +1,-1,-1],
[+0,+1,+1, -1,+1,-1],
[+1,+0,-1, -1,+1,-1],
[-1,-1,+0, -1,+1,-1]]),
np.array([
[+1,+1,+0, +1,-1,+0],
[+1,-1,+0, +1,+1,+0],
[+1,+0,+1, +1,+0,-1],
[+1,+0,-1, +1,+0,+1],
[+0,+1,+1, +0,+1,-1],
[+0,+1,-1, +0,+1,+1]])],
'twin' :[np.array([
[-2, 1, 1, 1, 1, 1],
[ 1,-2, 1, 1, 1, 1],
[ 1, 1,-2, 1, 1, 1],
[ 2,-1, 1, -1,-1, 1],
[-1, 2, 1, -1,-1, 1],
[-1,-1,-2, -1,-1, 1],
[-2,-1,-1, 1,-1,-1],
[ 1, 2,-1, 1,-1,-1],
[ 1,-1, 2, 1,-1,-1],
[ 2, 1,-1, -1, 1,-1],
[-1,-2,-1, -1, 1,-1],
[-1, 1, 2, -1, 1,-1]])]
},
'cI': {
'slip' :[np.array([
[+1,-1,+1, +0,+1,+1],
[-1,-1,+1, +0,+1,+1],
[+1,+1,+1, +0,-1,+1],
[-1,+1,+1, +0,-1,+1],
[-1,+1,+1, +1,+0,+1],
[-1,-1,+1, +1,+0,+1],
[+1,+1,+1, -1,+0,+1],
[+1,-1,+1, -1,+0,+1],
[-1,+1,+1, +1,+1,+0],
[-1,+1,-1, +1,+1,+0],
[+1,+1,+1, -1,+1,+0],
[+1,+1,-1, -1,+1,+0]]),
np.array([
[-1,+1,+1, +2,+1,+1],
[+1,+1,+1, -2,+1,+1],
[+1,+1,-1, +2,-1,+1],
[+1,-1,+1, +2,+1,-1],
[+1,-1,+1, +1,+2,+1],
[+1,+1,-1, -1,+2,+1],
[+1,+1,+1, +1,-2,+1],
[-1,+1,+1, +1,+2,-1],
[+1,+1,-1, +1,+1,+2],
[+1,-1,+1, -1,+1,+2],
[-1,+1,+1, +1,-1,+2],
[+1,+1,+1, +1,+1,-2]]),
np.array([
[+1,+1,-1, +1,+2,+3],
[+1,-1,+1, -1,+2,+3],
[-1,+1,+1, +1,-2,+3],
[+1,+1,+1, +1,+2,-3],
[+1,-1,+1, +1,+3,+2],
[+1,+1,-1, -1,+3,+2],
[+1,+1,+1, +1,-3,+2],
[-1,+1,+1, +1,+3,-2],
[+1,+1,-1, +2,+1,+3],
[+1,-1,+1, -2,+1,+3],
[-1,+1,+1, +2,-1,+3],
[+1,+1,+1, +2,+1,-3],
[+1,-1,+1, +2,+3,+1],
[+1,+1,-1, -2,+3,+1],
[+1,+1,+1, +2,-3,+1],
[-1,+1,+1, +2,+3,-1],
[-1,+1,+1, +3,+1,+2],
[+1,+1,+1, -3,+1,+2],
[+1,+1,-1, +3,-1,+2],
[+1,-1,+1, +3,+1,-2],
[-1,+1,+1, +3,+2,+1],
[+1,+1,+1, -3,+2,+1],
[+1,+1,-1, +3,-2,+1],
[+1,-1,+1, +3,+2,-1]])],
'twin' :[np.array([
[-1, 1, 1, 2, 1, 1],
[ 1, 1, 1, -2, 1, 1],
[ 1, 1,-1, 2,-1, 1],
[ 1,-1, 1, 2, 1,-1],
[ 1,-1, 1, 1, 2, 1],
[ 1, 1,-1, -1, 2, 1],
[ 1, 1, 1, 1,-2, 1],
[-1, 1, 1, 1, 2,-1],
[ 1, 1,-1, 1, 1, 2],
[ 1,-1, 1, -1, 1, 2],
[-1, 1, 1, 1,-1, 2],
[ 1, 1, 1, 1, 1,-2]])]
},
'hP': {
'slip' :[np.array([
[+2,-1,-1,+0, +0,+0,+0,+1],
[-1,+2,-1,+0, +0,+0,+0,+1],
[-1,-1,+2,+0, +0,+0,+0,+1]]),
np.array([
[+2,-1,-1,+0, +0,+1,-1,+0],
[-1,+2,-1,+0, -1,+0,+1,+0],
[-1,-1,+2,+0, +1,-1,+0,+0]]),
np.array([
[-1,+1,+0,+0, +1,+1,-2,+0],
[+0,-1,+1,+0, -2,+1,+1,+0],
[+1,+0,-1,+0, +1,-2,+1,+0]]),
np.array([
[-1,+2,-1,+0, +1,+0,-1,+1],
[-2,+1,+1,+0, +0,+1,-1,+1],
[-1,-1,+2,+0, -1,+1,+0,+1],
[+1,-2,+1,+0, -1,+0,+1,+1],
[+2,-1,-1,+0, +0,-1,+1,+1],
[+1,+1,-2,+0, +1,-1,+0,+1]]),
np.array([
[-2,+1,+1,+3, +1,+0,-1,+1],
[-1,-1,+2,+3, +1,+0,-1,+1],
[-1,-1,+2,+3, +0,+1,-1,+1],
[+1,-2,+1,+3, +0,+1,-1,+1],
[+1,-2,+1,+3, -1,+1,+0,+1],
[+2,-1,-1,+3, -1,+1,+0,+1],
[+2,-1,-1,+3, -1,+0,+1,+1],
[+1,+1,-2,+3, -1,+0,+1,+1],
[+1,+1,-2,+3, +0,-1,+1,+1],
[-1,+2,-1,+3, +0,-1,+1,+1],
[-1,+2,-1,+3, +1,-1,+0,+1],
[-2,+1,+1,+3, +1,-1,+0,+1]]),
np.array([
[-1,-1,+2,+3, +1,+1,-2,+2],
[+1,-2,+1,+3, -1,+2,-1,+2],
[+2,-1,-1,+3, -2,+1,+1,+2],
[+1,+1,-2,+3, -1,-1,+2,+2],
[-1,+2,-1,+3, +1,-2,+1,+2],
[-2,+1,+1,+3, +2,-1,-1,+2]])],
'twin' :[np.array([
[-1, 0, 1, 1, 1, 0,-1, 2], # shear = (3-(c/a)^2)/(sqrt(3) c/a) <-10.1>{10.2}
[ 0,-1, 1, 1, 0, 1,-1, 2],
[ 1,-1, 0, 1, -1, 1, 0, 2],
[ 1, 0,-1, 1, -1, 0, 1, 2],
[ 0, 1,-1, 1, 0,-1, 1, 2],
[-1, 1, 0, 1, 1,-1, 0, 2]]),
np.array([
[-1,-1, 2, 6, 1, 1,-2, 1], # shear = 1/(c/a) <11.6>{-1-1.1}
[ 1,-2, 1, 6, -1, 2,-1, 1],
[ 2,-1,-1, 6, -2, 1, 1, 1],
[ 1, 1,-2, 6, -1,-1, 2, 1],
[-1, 2,-1, 6, 1,-2, 1, 1],
[-2, 1, 1, 6, 2,-1,-1, 1]]),
np.array([
[ 1, 0,-1,-2, 1, 0,-1, 1], # shear = (4(c/a)^2-9)/(4 sqrt(3) c/a) <10.-2>{10.1}
[ 0, 1,-1,-2, 0, 1,-1, 1],
[-1, 1, 0,-2, -1, 1, 0, 1],
[-1, 0, 1,-2, -1, 0, 1, 1],
[ 0,-1, 1,-2, 0,-1, 1, 1],
[ 1,-1, 0,-2, 1,-1, 0, 1]]),
np.array([
[ 1, 1,-2,-3, 1, 1,-2, 2], # shear = 2((c/a)^2-2)/(3 c/a) <11.-3>{11.2}
[-1, 2,-1,-3, -1, 2,-1, 2],
[-2, 1, 1,-3, -2, 1, 1, 2],
[-1,-1, 2,-3, -1,-1, 2, 2],
[ 1,-2, 1,-3, 1,-2, 1, 2],
[ 2,-1,-1,-3, 2,-1,-1, 2]])]
},
}
master = _kinematics[self.lattice][mode]
if self.lattice == 'hP':
return {'direction':[util.Bravais_to_Miller(uvtw=m[:,0:4]) for m in master],
'plane': [util.Bravais_to_Miller(hkil=m[:,4:8]) for m in master]}
else:
return {'direction':[m[:,0:3] for m in master],
'plane': [m[:,3:6] for m in master]}
def relation_operations(self,model):
"""
Crystallographic orientation relationships for phase transformations.
Parameters
----------
model : str
Name of orientation relationship.
Returns
-------
operations : (string, damask.Rotation)
Resulting lattice and rotations characterizing the orientation relationship.
References
----------
S. Morito et al., Journal of Alloys and Compounds 577:s587-s592, 2013
https://doi.org/10.1016/j.jallcom.2012.02.004
K. Kitahara et al., Acta Materialia 54(5):1279-1288, 2006
https://doi.org/10.1016/j.actamat.2005.11.001
Y. He et al., Journal of Applied Crystallography 39:72-81, 2006
https://doi.org/10.1107/S0021889805038276
H. Kitahara et al., Materials Characterization 54(4-5):378-386, 2005
https://doi.org/10.1016/j.matchar.2004.12.015
Y. He et al., Acta Materialia 53(4):1179-1190, 2005
https://doi.org/10.1016/j.actamat.2004.11.021
"""
_orientation_relationships = {
'KS': {
'cF' : np.array([
[[-1, 0, 1],[ 1, 1, 1]],
[[-1, 0, 1],[ 1, 1, 1]],
[[ 0, 1,-1],[ 1, 1, 1]],
[[ 0, 1,-1],[ 1, 1, 1]],
[[ 1,-1, 0],[ 1, 1, 1]],
[[ 1,-1, 0],[ 1, 1, 1]],
[[ 1, 0,-1],[ 1,-1, 1]],
[[ 1, 0,-1],[ 1,-1, 1]],
[[-1,-1, 0],[ 1,-1, 1]],
[[-1,-1, 0],[ 1,-1, 1]],
[[ 0, 1, 1],[ 1,-1, 1]],
[[ 0, 1, 1],[ 1,-1, 1]],
[[ 0,-1, 1],[-1, 1, 1]],
[[ 0,-1, 1],[-1, 1, 1]],
[[-1, 0,-1],[-1, 1, 1]],
[[-1, 0,-1],[-1, 1, 1]],
[[ 1, 1, 0],[-1, 1, 1]],
[[ 1, 1, 0],[-1, 1, 1]],
[[-1, 1, 0],[ 1, 1,-1]],
[[-1, 1, 0],[ 1, 1,-1]],
[[ 0,-1,-1],[ 1, 1,-1]],
[[ 0,-1,-1],[ 1, 1,-1]],
[[ 1, 0, 1],[ 1, 1,-1]],
[[ 1, 0, 1],[ 1, 1,-1]],
],dtype=float),
'cI' : np.array([
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
],dtype=float),
},
'GT': {
'cF' : np.array([
[[ -5,-12, 17],[ 1, 1, 1]],
[[ 17, -5,-12],[ 1, 1, 1]],
[[-12, 17, -5],[ 1, 1, 1]],
[[ 5, 12, 17],[ -1, -1, 1]],
[[-17, 5,-12],[ -1, -1, 1]],
[[ 12,-17, -5],[ -1, -1, 1]],
[[ -5, 12,-17],[ -1, 1, 1]],
[[ 17, 5, 12],[ -1, 1, 1]],
[[-12,-17, 5],[ -1, 1, 1]],
[[ 5,-12,-17],[ 1, -1, 1]],
[[-17, -5, 12],[ 1, -1, 1]],
[[ 12, 17, 5],[ 1, -1, 1]],
[[ -5, 17,-12],[ 1, 1, 1]],
[[-12, -5, 17],[ 1, 1, 1]],
[[ 17,-12, -5],[ 1, 1, 1]],
[[ 5,-17,-12],[ -1, -1, 1]],
[[ 12, 5, 17],[ -1, -1, 1]],
[[-17, 12, -5],[ -1, -1, 1]],
[[ -5,-17, 12],[ -1, 1, 1]],
[[-12, 5,-17],[ -1, 1, 1]],
[[ 17, 12, 5],[ -1, 1, 1]],
[[ 5, 17, 12],[ 1, -1, 1]],
[[ 12, -5,-17],[ 1, -1, 1]],
[[-17,-12, 5],[ 1, -1, 1]],
],dtype=float),
'cI' : np.array([
[[-17, -7, 17],[ 1, 0, 1]],
[[ 17,-17, -7],[ 1, 1, 0]],
[[ -7, 17,-17],[ 0, 1, 1]],
[[ 17, 7, 17],[ -1, 0, 1]],
[[-17, 17, -7],[ -1, -1, 0]],
[[ 7,-17,-17],[ 0, -1, 1]],
[[-17, 7,-17],[ -1, 0, 1]],
[[ 17, 17, 7],[ -1, 1, 0]],
[[ -7,-17, 17],[ 0, 1, 1]],
[[ 17, -7,-17],[ 1, 0, 1]],
[[-17,-17, 7],[ 1, -1, 0]],
[[ 7, 17, 17],[ 0, -1, 1]],
[[-17, 17, -7],[ 1, 1, 0]],
[[ -7,-17, 17],[ 0, 1, 1]],
[[ 17, -7,-17],[ 1, 0, 1]],
[[ 17,-17, -7],[ -1, -1, 0]],
[[ 7, 17, 17],[ 0, -1, 1]],
[[-17, 7,-17],[ -1, 0, 1]],
[[-17,-17, 7],[ -1, 1, 0]],
[[ -7, 17,-17],[ 0, 1, 1]],
[[ 17, 7, 17],[ -1, 0, 1]],
[[ 17, 17, 7],[ 1, -1, 0]],
[[ 7,-17,-17],[ 0, -1, 1]],
[[-17, -7, 17],[ 1, 0, 1]],
],dtype=float),
},
'GT_prime': {
'cF' : np.array([
[[ 0, 1, -1],[ 7, 17, 17]],
[[ -1, 0, 1],[ 17, 7, 17]],
[[ 1, -1, 0],[ 17, 17, 7]],
[[ 0, -1, -1],[ -7,-17, 17]],
[[ 1, 0, 1],[-17, -7, 17]],
[[ 1, -1, 0],[-17,-17, 7]],
[[ 0, 1, -1],[ 7,-17,-17]],
[[ 1, 0, 1],[ 17, -7,-17]],
[[ -1, -1, 0],[ 17,-17, -7]],
[[ 0, -1, -1],[ -7, 17,-17]],
[[ -1, 0, 1],[-17, 7,-17]],
[[ -1, -1, 0],[-17, 17, -7]],
[[ 0, -1, 1],[ 7, 17, 17]],
[[ 1, 0, -1],[ 17, 7, 17]],
[[ -1, 1, 0],[ 17, 17, 7]],
[[ 0, 1, 1],[ -7,-17, 17]],
[[ -1, 0, -1],[-17, -7, 17]],
[[ -1, 1, 0],[-17,-17, 7]],
[[ 0, -1, 1],[ 7,-17,-17]],
[[ -1, 0, -1],[ 17, -7,-17]],
[[ 1, 1, 0],[ 17,-17, -7]],
[[ 0, 1, 1],[ -7, 17,-17]],
[[ 1, 0, -1],[-17, 7,-17]],
[[ 1, 1, 0],[-17, 17, -7]],
],dtype=float),
'cI' : np.array([
[[ 1, 1, -1],[ 12, 5, 17]],
[[ -1, 1, 1],[ 17, 12, 5]],
[[ 1, -1, 1],[ 5, 17, 12]],
[[ -1, -1, -1],[-12, -5, 17]],
[[ 1, -1, 1],[-17,-12, 5]],
[[ 1, -1, -1],[ -5,-17, 12]],
[[ -1, 1, -1],[ 12, -5,-17]],
[[ 1, 1, 1],[ 17,-12, -5]],
[[ -1, -1, 1],[ 5,-17,-12]],
[[ 1, -1, -1],[-12, 5,-17]],
[[ -1, -1, 1],[-17, 12, -5]],
[[ -1, -1, -1],[ -5, 17,-12]],
[[ 1, -1, 1],[ 12, 17, 5]],
[[ 1, 1, -1],[ 5, 12, 17]],
[[ -1, 1, 1],[ 17, 5, 12]],
[[ -1, 1, 1],[-12,-17, 5]],
[[ -1, -1, -1],[ -5,-12, 17]],
[[ -1, 1, -1],[-17, -5, 12]],
[[ -1, -1, 1],[ 12,-17, -5]],
[[ -1, 1, -1],[ 5,-12,-17]],
[[ 1, 1, 1],[ 17, -5,-12]],
[[ 1, 1, 1],[-12, 17, -5]],
[[ 1, -1, -1],[ -5, 12,-17]],
[[ 1, 1, -1],[-17, 5,-12]],
],dtype=float),
},
'NW': {
'cF' : np.array([
[[ 2, -1, -1],[ 1, 1, 1]],
[[ -1, 2, -1],[ 1, 1, 1]],
[[ -1, -1, 2],[ 1, 1, 1]],
[[ -2, -1, -1],[ -1, 1, 1]],
[[ 1, 2, -1],[ -1, 1, 1]],
[[ 1, -1, 2],[ -1, 1, 1]],
[[ 2, 1, -1],[ 1, -1, 1]],
[[ -1, -2, -1],[ 1, -1, 1]],
[[ -1, 1, 2],[ 1, -1, 1]],
[[ 2, -1, 1],[ -1, -1, 1]],
[[ -1, 2, 1],[ -1, -1, 1]],
[[ -1, -1, -2],[ -1, -1, 1]],
],dtype=float),
'cI' : np.array([
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
[[ 0, -1, 1],[ 0, 1, 1]],
],dtype=float),
},
'Pitsch': {
'cF' : np.array([
[[ 1, 0, 1],[ 0, 1, 0]],
[[ 1, 1, 0],[ 0, 0, 1]],
[[ 0, 1, 1],[ 1, 0, 0]],
[[ 0, 1, -1],[ 1, 0, 0]],
[[ -1, 0, 1],[ 0, 1, 0]],
[[ 1, -1, 0],[ 0, 0, 1]],
[[ 1, 0, -1],[ 0, 1, 0]],
[[ -1, 1, 0],[ 0, 0, 1]],
[[ 0, -1, 1],[ 1, 0, 0]],
[[ 0, 1, 1],[ 1, 0, 0]],
[[ 1, 0, 1],[ 0, 1, 0]],
[[ 1, 1, 0],[ 0, 0, 1]],
],dtype=float),
'cI' : np.array([
[[ 1, -1, 1],[ -1, 0, 1]],
[[ 1, 1, -1],[ 1, -1, 0]],
[[ -1, 1, 1],[ 0, 1, -1]],
[[ -1, 1, -1],[ 0, -1, -1]],
[[ -1, -1, 1],[ -1, 0, -1]],
[[ 1, -1, -1],[ -1, -1, 0]],
[[ 1, -1, -1],[ -1, 0, -1]],
[[ -1, 1, -1],[ -1, -1, 0]],
[[ -1, -1, 1],[ 0, -1, -1]],
[[ -1, 1, 1],[ 0, -1, 1]],
[[ 1, -1, 1],[ 1, 0, -1]],
[[ 1, 1, -1],[ -1, 1, 0]],
],dtype=float),
},
'Bain': {
'cF' : np.array([
[[ 0, 1, 0],[ 1, 0, 0]],
[[ 0, 0, 1],[ 0, 1, 0]],
[[ 1, 0, 0],[ 0, 0, 1]],
],dtype=float),
'cI' : np.array([
[[ 0, 1, 1],[ 1, 0, 0]],
[[ 1, 0, 1],[ 0, 1, 0]],
[[ 1, 1, 0],[ 0, 0, 1]],
],dtype=float),
},
'Burgers' : {
'cI' : np.array([
[[ -1, 1, 1],[ 1, 1, 0]],
[[ -1, 1, -1],[ 1, 1, 0]],
[[ 1, 1, 1],[ 1, -1, 0]],
[[ 1, 1, -1],[ 1, -1, 0]],
[[ 1, 1, -1],[ 1, 0, 1]],
[[ -1, 1, 1],[ 1, 0, 1]],
[[ 1, 1, 1],[ -1, 0, 1]],
[[ 1, -1, 1],[ -1, 0, 1]],
[[ -1, 1, -1],[ 0, 1, 1]],
[[ 1, 1, -1],[ 0, 1, 1]],
[[ -1, 1, 1],[ 0, -1, 1]],
[[ 1, 1, 1],[ 0, -1, 1]],
],dtype=float),
'hP' : np.array([
[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
],dtype=float),
},
}
orientation_relationships = {k:v for k,v in _orientation_relationships.items() if self.lattice in v}
if model not in orientation_relationships:
raise KeyError(f'unknown orientation relationship "{model}"')
r = orientation_relationships[model]
sl = self.lattice
ol = (set(r)-{sl}).pop()
m = r[sl]
o = r[ol]
p_,_p = np.zeros(m.shape[:-1]+(3,)),np.zeros(o.shape[:-1]+(3,))
p_[...,0,:] = m[...,0,:] if m.shape[-1] == 3 else util.Bravais_to_Miller(uvtw=m[...,0,0:4])
p_[...,1,:] = m[...,1,:] if m.shape[-1] == 3 else util.Bravais_to_Miller(hkil=m[...,1,0:4])
_p[...,0,:] = o[...,0,:] if o.shape[-1] == 3 else util.Bravais_to_Miller(uvtw=o[...,0,0:4])
_p[...,1,:] = o[...,1,:] if o.shape[-1] == 3 else util.Bravais_to_Miller(hkil=o[...,1,0:4])
return (ol,Rotation.from_parallel(p_,_p))