877 lines
38 KiB
Python
877 lines
38 KiB
Python
import numpy as np
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from . import util
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from . import Rotation
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lattice_symmetries = {
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'aP': 'triclinic',
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'mP': 'monoclinic',
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'mS': 'monoclinic',
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'oP': 'orthorhombic',
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'oS': 'orthorhombic',
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'oI': 'orthorhombic',
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'oF': 'orthorhombic',
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'tP': 'tetragonal',
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'tI': 'tetragonal',
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'hP': 'hexagonal',
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'cP': 'cubic',
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'cI': 'cubic',
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'cF': 'cubic',
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}
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class Crystal():
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"""Crystal lattice."""
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def __init__(self,*,
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family = None,
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lattice = None,
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a = None,b = None,c = None,
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alpha = None,beta = None,gamma = None,
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degrees = False):
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"""
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Representation of crystal in terms of crystal family or Bravais lattice.
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Parameters
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----------
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family : {'triclinic', 'monoclinic', 'orthorhombic', 'tetragonal', 'hexagonal', 'cubic'}, optional.
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Name of the crystal family.
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Will be inferred if 'lattice' is given.
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lattice : {'aP', 'mP', 'mS', 'oP', 'oS', 'oI', 'oF', 'tP', 'tI', 'hP', 'cP', 'cI', 'cF'}, optional.
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Name of the Bravais lattice in Pearson notation.
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a : float, optional
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Length of lattice parameter 'a'.
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b : float, optional
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Length of lattice parameter 'b'.
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c : float, optional
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Length of lattice parameter 'c'.
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alpha : float, optional
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Angle between b and c lattice basis.
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beta : float, optional
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Angle between c and a lattice basis.
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gamma : float, optional
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Angle between a and b lattice basis.
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degrees : bool, optional
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Angles are given in degrees. Defaults to False.
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"""
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if family not in [None] + list(lattice_symmetries.values()):
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raise KeyError(f'invalid crystal family "{family}"')
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if lattice is not None and family is not None and family != lattice_symmetries[lattice]:
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raise KeyError(f'incompatible family "{family}" for lattice "{lattice}"')
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self.family = lattice_symmetries[lattice] if family is None else family
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self.lattice = lattice
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if self.lattice is not None:
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self.a = 1 if a is None else a
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self.b = b
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self.c = c
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self.a = float(self.a) if self.a is not None else \
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(self.b / self.ratio['b'] if self.b is not None and self.ratio['b'] is not None else
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self.c / self.ratio['c'] if self.c is not None and self.ratio['c'] is not None else None)
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self.b = float(self.b) if self.b is not None else \
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(self.a * self.ratio['b'] if self.a is not None and self.ratio['b'] is not None else
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self.c / self.ratio['c'] * self.ratio['b']
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if self.c is not None and self.ratio['b'] is not None and self.ratio['c'] is not None else None)
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self.c = float(self.c) if self.c is not None else \
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(self.a * self.ratio['c'] if self.a is not None and self.ratio['c'] is not None else
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self.b / self.ratio['b'] * self.ratio['c']
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if self.c is not None and self.ratio['b'] is not None and self.ratio['c'] is not None else None)
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self.alpha = np.radians(alpha) if degrees and alpha is not None else alpha
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self.beta = np.radians(beta) if degrees and beta is not None else beta
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self.gamma = np.radians(gamma) if degrees and gamma is not None else gamma
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if self.alpha is None and 'alpha' in self.immutable: self.alpha = self.immutable['alpha']
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if self.beta is None and 'beta' in self.immutable: self.beta = self.immutable['beta']
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if self.gamma is None and 'gamma' in self.immutable: self.gamma = self.immutable['gamma']
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if \
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(self.a is None) \
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or (self.b is None or ('b' in self.immutable and self.b != self.immutable['b'] * self.a)) \
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or (self.c is None or ('c' in self.immutable and self.c != self.immutable['c'] * self.b)) \
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or (self.alpha is None or ('alpha' in self.immutable and self.alpha != self.immutable['alpha'])) \
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or (self.beta is None or ('beta' in self.immutable and self.beta != self.immutable['beta'])) \
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or (self.gamma is None or ('gamma' in self.immutable and self.gamma != self.immutable['gamma'])):
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raise ValueError (f'incompatible parameters {self.parameters} for crystal family {self.family}')
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if np.any(np.array([self.alpha,self.beta,self.gamma]) <= 0):
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raise ValueError ('lattice angles must be positive')
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if np.any([np.roll([self.alpha,self.beta,self.gamma],r)[0]
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>= np.sum(np.roll([self.alpha,self.beta,self.gamma],r)[1:]) for r in range(3)]):
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raise ValueError ('each lattice angle must be less than sum of others')
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else:
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self.a = self.b = self.c = None
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self.alpha = self.beta = self.gamma = None
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def __repr__(self):
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"""Represent."""
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return '\n'.join([f'Crystal family {self.family}']
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+ ([] if self.lattice is None else [f'Bravais lattice {self.lattice}']+
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list(map(lambda x:f'{x[0]}: {x[1]:.5g}',
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zip(['a','b','c','α','β','γ',],
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self.parameters))))
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)
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def __eq__(self,other):
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"""
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Equal to other.
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Parameters
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----------
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other : Crystal
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Crystal to check for equality.
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"""
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return self.lattice == other.lattice and \
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self.parameters == other.parameters and \
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self.family == other.family
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@property
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def parameters(self):
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"""Return lattice parameters a, b, c, alpha, beta, gamma."""
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return (self.a,self.b,self.c,self.alpha,self.beta,self.gamma)
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@property
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def immutable(self):
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"""Return immutable lattice parameters."""
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_immutable = {
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'cubic': {
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'b': 1.0,
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'c': 1.0,
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'alpha': np.pi/2.,
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'beta': np.pi/2.,
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'gamma': np.pi/2.,
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},
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'hexagonal': {
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'b': 1.0,
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'alpha': np.pi/2.,
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'beta': np.pi/2.,
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'gamma': 2.*np.pi/3.,
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},
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'tetragonal': {
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'b': 1.0,
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'alpha': np.pi/2.,
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'beta': np.pi/2.,
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'gamma': np.pi/2.,
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},
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'orthorhombic': {
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'alpha': np.pi/2.,
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'beta': np.pi/2.,
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'gamma': np.pi/2.,
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},
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'monoclinic': {
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'alpha': np.pi/2.,
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'gamma': np.pi/2.,
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},
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'triclinic': {}
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}
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return _immutable[self.family]
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@property
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def standard_triangle(self):
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"""
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Corners of the standard triangle.
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Notes
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-----
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Not yet defined for monoclinic.
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References
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----------
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Bases are computed from
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>>> basis = {
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... 'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,1.]/np.sqrt(2.), # green
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... [1.,1.,1.]/np.sqrt(3.)]).T), # blue
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... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,0.], # green
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... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # blue
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... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,0.], # green
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... [1.,1.,0.]/np.sqrt(2.)]).T), # blue
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... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,0.], # green
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... [0.,1.,0.]]).T), # blue
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... }
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"""
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_basis = {
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'cubic': {'improper':np.array([ [-1. , 0. , 1. ],
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[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
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[ 0. , np.sqrt(3.) , 0. ] ]),
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'proper':np.array([ [ 0. , -1. , 1. ],
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[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
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[ np.sqrt(3.) , 0. , 0. ] ]),
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},
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'hexagonal':
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{'improper':np.array([ [ 0. , 0. , 1. ],
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[ 1. , -np.sqrt(3.) , 0. ],
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[ 0. , 2. , 0. ] ]),
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'proper':np.array([ [ 0. , 0. , 1. ],
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[-1. , np.sqrt(3.) , 0. ],
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[ np.sqrt(3.) , -1. , 0. ] ]),
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},
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'tetragonal':
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{'improper':np.array([ [ 0. , 0. , 1. ],
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[ 1. , -1. , 0. ],
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[ 0. , np.sqrt(2.) , 0. ] ]),
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'proper':np.array([ [ 0. , 0. , 1. ],
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[-1. , 1. , 0. ],
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[ np.sqrt(2.) , 0. , 0. ] ]),
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},
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'orthorhombic':
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{'improper':np.array([ [ 0., 0., 1.],
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[ 1., 0., 0.],
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[ 0., 1., 0.] ]),
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'proper':np.array([ [ 0., 0., 1.],
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[-1., 0., 0.],
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[ 0., 1., 0.] ]),
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}}
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return _basis.get(self.family,None)
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@property
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def ratio(self):
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"""Return axes ratios of own lattice."""
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_ratio = { 'hexagonal': {'c': np.sqrt(8./3.)}}
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return dict(b = self.immutable['b']
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if 'b' in self.immutable else
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_ratio[self.family]['b'] if self.family in _ratio and 'b' in _ratio[self.family] else None,
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c = self.immutable['c']
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if 'c' in self.immutable else
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_ratio[self.family]['c'] if self.family in _ratio and 'c' in _ratio[self.family] else None,
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)
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@property
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def basis_real(self):
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"""
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Return orthogonal real space crystal basis.
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References
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----------
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C.T. Young and J.L. Lytton, Journal of Applied Physics 43:1408–1417, 1972
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https://doi.org/10.1063/1.1661333
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"""
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if None in self.parameters:
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raise KeyError('missing crystal lattice parameters')
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return np.array([
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[1,0,0],
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[np.cos(self.gamma),np.sin(self.gamma),0],
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[np.cos(self.beta),
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(np.cos(self.alpha)-np.cos(self.beta)*np.cos(self.gamma)) /np.sin(self.gamma),
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np.sqrt(1 - np.cos(self.alpha)**2 - np.cos(self.beta)**2 - np.cos(self.gamma)**2
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+ 2 * np.cos(self.alpha) * np.cos(self.beta) * np.cos(self.gamma))/np.sin(self.gamma)],
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],dtype=float).T \
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* np.array([self.a,self.b,self.c])
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@property
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def basis_reciprocal(self):
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"""Return reciprocal (dual) crystal basis."""
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return np.linalg.inv(self.basis_real.T)
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@property
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def lattice_points(self):
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"""Return lattice points."""
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_lattice_points = {
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'P': [
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],
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'S': [
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[0.5,0.5,0],
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],
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'I': [
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[0.5,0.5,0.5],
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],
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'F': [
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[0.0,0.5,0.5],
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[0.5,0.0,0.5],
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[0.5,0.5,0.0],
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],
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'hP': [
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[2./3.,1./3.,0.5],
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],
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}
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if self.lattice is None: raise KeyError('no lattice type specified')
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return np.array([[0,0,0]]
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+ _lattice_points.get(self.lattice if self.lattice == 'hP' else \
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self.lattice[-1],None),dtype=float)
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def to_lattice(self,*,direction=None,plane=None):
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"""
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Calculate lattice vector corresponding to crystal frame direction or plane normal.
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Parameters
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----------
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direction|plane : numpy.ndarray of shape (...,3)
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Vector along direction or plane normal.
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Returns
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-------
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Miller : numpy.ndarray of shape (...,3)
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Lattice vector of direction or plane.
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Use util.scale_to_coprime to convert to (integer) Miller indices.
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"""
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if (direction is not None) ^ (plane is None):
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raise KeyError('specify either "direction" or "plane"')
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axis,basis = (np.array(direction),self.basis_reciprocal.T) \
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if plane is None else \
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(np.array(plane),self.basis_real.T)
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return np.einsum('il,...l',basis,axis)
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def to_frame(self,*,uvw=None,hkl=None):
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"""
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Calculate crystal frame vector along lattice direction [uvw] or plane normal (hkl).
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Parameters
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----------
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uvw|hkl : numpy.ndarray of shape (...,3)
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Miller indices of crystallographic direction or plane normal.
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Returns
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-------
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vector : numpy.ndarray of shape (...,3)
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Crystal frame vector along [uvw] direction or (hkl) plane normal.
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"""
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if (uvw is not None) ^ (hkl is None):
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raise KeyError('specify either "uvw" or "hkl"')
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axis,basis = (np.array(uvw),self.basis_real) \
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if hkl is None else \
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(np.array(hkl),self.basis_reciprocal)
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return np.einsum('il,...l',basis,axis)
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def kinematics(self,mode):
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"""
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Return crystal kinematics systems.
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Parameters
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----------
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mode : {'slip','twin'}
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Deformation mode.
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Returns
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-------
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direction_plane : dictionary
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Directions and planes of deformation mode families.
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"""
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_kinematics = {
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'cF': {
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'slip' :[np.array([
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[+0,+1,-1, +1,+1,+1],
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[-1,+0,+1, +1,+1,+1],
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[+1,-1,+0, +1,+1,+1],
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[+0,-1,-1, -1,-1,+1],
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[+1,+0,+1, -1,-1,+1],
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[-1,+1,+0, -1,-1,+1],
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[+0,-1,+1, +1,-1,-1],
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[-1,+0,-1, +1,-1,-1],
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[+1,+1,+0, +1,-1,-1],
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[+0,+1,+1, -1,+1,-1],
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[+1,+0,-1, -1,+1,-1],
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[-1,-1,+0, -1,+1,-1]]),
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np.array([
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[+1,+1,+0, +1,-1,+0],
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[+1,-1,+0, +1,+1,+0],
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[+1,+0,+1, +1,+0,-1],
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[+1,+0,-1, +1,+0,+1],
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[+0,+1,+1, +0,+1,-1],
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[+0,+1,-1, +0,+1,+1]])],
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'twin' :[np.array([
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[-2, 1, 1, 1, 1, 1],
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[ 1,-2, 1, 1, 1, 1],
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[ 1, 1,-2, 1, 1, 1],
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[ 2,-1, 1, -1,-1, 1],
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[-1, 2, 1, -1,-1, 1],
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[-1,-1,-2, -1,-1, 1],
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[-2,-1,-1, 1,-1,-1],
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[ 1, 2,-1, 1,-1,-1],
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[ 1,-1, 2, 1,-1,-1],
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[ 2, 1,-1, -1, 1,-1],
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[-1,-2,-1, -1, 1,-1],
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[-1, 1, 2, -1, 1,-1]])]
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},
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'cI': {
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'slip' :[np.array([
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[+1,-1,+1, +0,+1,+1],
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[-1,-1,+1, +0,+1,+1],
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[+1,+1,+1, +0,-1,+1],
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[-1,+1,+1, +0,-1,+1],
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[-1,+1,+1, +1,+0,+1],
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[-1,-1,+1, +1,+0,+1],
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[+1,+1,+1, -1,+0,+1],
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[+1,-1,+1, -1,+0,+1],
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[-1,+1,+1, +1,+1,+0],
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[-1,+1,-1, +1,+1,+0],
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[+1,+1,+1, -1,+1,+0],
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[+1,+1,-1, -1,+1,+0]]),
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np.array([
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[-1,+1,+1, +2,+1,+1],
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[+1,+1,+1, -2,+1,+1],
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[+1,+1,-1, +2,-1,+1],
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[+1,-1,+1, +2,+1,-1],
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[+1,-1,+1, +1,+2,+1],
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[+1,+1,-1, -1,+2,+1],
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[+1,+1,+1, +1,-2,+1],
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[-1,+1,+1, +1,+2,-1],
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[+1,+1,-1, +1,+1,+2],
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[+1,-1,+1, -1,+1,+2],
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[-1,+1,+1, +1,-1,+2],
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[+1,+1,+1, +1,+1,-2]]),
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np.array([
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[+1,+1,-1, +1,+2,+3],
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[+1,-1,+1, -1,+2,+3],
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[-1,+1,+1, +1,-2,+3],
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[+1,+1,+1, +1,+2,-3],
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[+1,-1,+1, +1,+3,+2],
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[+1,+1,-1, -1,+3,+2],
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[+1,+1,+1, +1,-3,+2],
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[-1,+1,+1, +1,+3,-2],
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[+1,+1,-1, +2,+1,+3],
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[+1,-1,+1, -2,+1,+3],
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[-1,+1,+1, +2,-1,+3],
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[+1,+1,+1, +2,+1,-3],
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[+1,-1,+1, +2,+3,+1],
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[+1,+1,-1, -2,+3,+1],
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[+1,+1,+1, +2,-3,+1],
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[-1,+1,+1, +2,+3,-1],
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[-1,+1,+1, +3,+1,+2],
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[+1,+1,+1, -3,+1,+2],
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[+1,+1,-1, +3,-1,+2],
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[+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))
|