from typing import Union, Dict, List, Tuple import numpy as np from ._typehints import FloatSequence, CrystalFamily, CrystalLattice, CrystalKinematics from . import util from . import Rotation lattice_symmetries: Dict[CrystalLattice, CrystalFamily] = { '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(): """ Representation of a crystal as (general) crystal family or (more specific) as a scaled Bravais lattice. Examples -------- Cubic crystal family: >>> import damask >>> cubic = damask.Crystal(family='cubic') >>> cubic Crystal family: cubic Body-centered cubic Bravais lattice with parameters of iron: >>> import damask >>> Fe = damask.Crystal(lattice='cI', a=287e-12) >>> Fe Crystal family: cubic Bravais lattice: cI a=2.87e-10 m, b=2.87e-10 m, c=2.87e-10 m α=90°, β=90°, γ=90° """ def __init__(self, *, family: CrystalFamily = None, lattice: CrystalLattice = None, a: float = None, b: float = None, c: float = None, alpha: float = None, beta: float = None, gamma: float = None, degrees: bool = False): """ New representation of a crystal. 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 is not None and family not in 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') def __repr__(self): """Give short human-readable summary.""" family = f'Crystal family: {self.family}' return family if self.lattice is None else \ util.srepr([family, f'Bravais lattice: {self.lattice}', 'a={:.5g} m, b={:.5g} m, c={:.5g} m'.format(*self.parameters[:3]), 'α={:.5g}°, β={:.5g}°, γ={:.5g}°'.format(*np.degrees(self.parameters[3:]))]) def __eq__(self, other: object) -> bool: """ Equal to other. Parameters ---------- other : Crystal Crystal to check for equality. """ return NotImplemented if not isinstance(other, Crystal) else \ 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.""" if hasattr(self,'a'): 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) -> Union[Dict[str, np.ndarray], None]: """ 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: Dict[CrystalFamily, Dict[str, np.ndarray]] = { '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 symmetry_operations(self) -> Rotation: """Symmetry operations as Rotations.""" _symmetry_operations: Dict[CrystalFamily, List] = { 'cubic': [ [ 1.0, 0.0, 0.0, 0.0 ], [ 0.0, 1.0, 0.0, 0.0 ], [ 0.0, 0.0, 1.0, 0.0 ], [ 0.0, 0.0, 0.0, 1.0 ], [ 0.0, 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2) ], [ 0.0, 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2) ], [ 0.0, 0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2) ], [ 0.0, 0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2) ], [ 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ], [ 0.0, -0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ], [ 0.5, 0.5, 0.5, 0.5 ], [-0.5, 0.5, 0.5, 0.5 ], [-0.5, 0.5, 0.5, -0.5 ], [-0.5, 0.5, -0.5, 0.5 ], [-0.5, -0.5, 0.5, 0.5 ], [-0.5, -0.5, 0.5, -0.5 ], [-0.5, -0.5, -0.5, 0.5 ], [-0.5, 0.5, -0.5, -0.5 ], [-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], [ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], [-0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2), 0.0 ], [-0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2), 0.0 ], [-0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0, 0.0 ], [-0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0, 0.0 ], ], 'hexagonal': [ [ 1.0, 0.0, 0.0, 0.0 ], [-0.5*np.sqrt(3), 0.0, 0.0, -0.5 ], [ 0.5, 0.0, 0.0, 0.5*np.sqrt(3) ], [ 0.0, 0.0, 0.0, 1.0 ], [-0.5, 0.0, 0.0, 0.5*np.sqrt(3) ], [-0.5*np.sqrt(3), 0.0, 0.0, 0.5 ], [ 0.0, 1.0, 0.0, 0.0 ], [ 0.0, -0.5*np.sqrt(3), 0.5, 0.0 ], [ 0.0, 0.5, -0.5*np.sqrt(3), 0.0 ], [ 0.0, 0.0, 1.0, 0.0 ], [ 0.0, -0.5, -0.5*np.sqrt(3), 0.0 ], [ 0.0, 0.5*np.sqrt(3), 0.5, 0.0 ], ], 'tetragonal': [ [ 1.0, 0.0, 0.0, 0.0 ], [ 0.0, 1.0, 0.0, 0.0 ], [ 0.0, 0.0, 1.0, 0.0 ], [ 0.0, 0.0, 0.0, 1.0 ], [ 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ], [ 0.0, -0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ], [ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], [-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], ], 'orthorhombic': [ [ 1.0,0.0,0.0,0.0 ], [ 0.0,1.0,0.0,0.0 ], [ 0.0,0.0,1.0,0.0 ], [ 0.0,0.0,0.0,1.0 ], ], 'monoclinic': [ [ 1.0,0.0,0.0,0.0 ], [ 0.0,0.0,1.0,0.0 ], ], 'triclinic': [ [ 1.0,0.0,0.0,0.0 ], ]} return Rotation.from_quaternion(_symmetry_operations[self.family],accept_homomorph=True) @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) -> np.ndarray: """ Return orthogonal real space crystal basis. References ---------- C.T. Young and J.L. Lytton, Journal of Applied Physics 43:1408–1417, 1972 https://doi.org/10.1063/1.1661333 """ if self.parameters is None: 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) -> np.ndarray: """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: FloatSequence = None, plane: FloatSequence = None) -> np.ndarray: """ Calculate lattice vector corresponding to crystal frame direction or plane normal. Parameters ---------- direction|plane : numpy.ndarray, shape (...,3) Real space vector along direction or reciprocal space vector along plane normal. Returns ------- Miller : numpy.ndarray, 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"') basis,axis = (self.basis_reciprocal,np.array(direction)) \ if plane is None else \ (self.basis_real,np.array(plane)) return np.einsum('li,...l',basis,axis) def to_frame(self, *, uvw: FloatSequence = None, hkl: FloatSequence = None) -> np.ndarray: """ Calculate crystal frame vector corresponding to lattice direction [uvw] or plane normal (hkl). Parameters ---------- uvw|hkl : numpy.ndarray, shape (...,3) Miller indices of crystallographic direction or plane normal. Returns ------- vector : numpy.ndarray, shape (...,3) Crystal frame vector in real space along [uvw] direction or in reciprocal space along (hkl) plane normal. Examples -------- Crystal frame vector (real space) of Magnesium corresponding to [1,1,0] direction: >>> import damask >>> Mg = damask.Crystal(lattice='hP', a=321e-12, c=521e-12) >>> Mg.to_frame(uvw=[1, 1, 0]) array([1.60500000e-10, 2.77994155e-10, 0.00000000e+00]) Crystal frame vector (reciprocal space) of Titanium along (1,0,0) plane normal: >>> import damask >>> Ti = damask.Crystal(lattice='hP', a=0.295e-9, c=0.469e-9) >>> Ti.to_frame(hkl=(1, 0, 0)) array([ 3.38983051e+09, 1.95711956e+09, -4.15134508e-07]) """ if (uvw is not None) ^ (hkl is None): raise KeyError('specify either "uvw" or "hkl"') basis,axis = (self.basis_real,np.array(uvw)) \ if hkl is None else \ (self.basis_reciprocal,np.array(hkl)) return np.einsum('il,...l',basis,axis) def kinematics(self, mode: CrystalKinematics) -> Dict[str, List[np.ndarray]]: """ Return crystal kinematics systems. Parameters ---------- mode : {'slip','twin'} Deformation mode. Returns ------- direction_plane : dictionary Directions and planes of deformation mode families. """ _kinematics: Dict[CrystalLattice, Dict[CrystalKinematics, List[np.ndarray]]] = { '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,+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]])] }, 'tI': { 'slip': [np.array([ [+0,+0,+1, +1,+0,+0], [+0,+0,+1, +0,+1,+0]]), np.array([ [+0,+0,+1, +1,+1,+0], [+0,+0,+1, -1,+1,+0]]), np.array([ [+0,+1,+0, +1,+0,+0], [+1,+0,+0, +0,+1,+0]]), 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]]), np.array([ [+1,-1,+0, +1,+1,+0], [+1,+1,+0, +1,-1,+0]]), np.array([ [+0,+1,+1, +1,+0,+0], [+0,-1,+1, +1,+0,+0], [-1,+0,+1, +0,+1,+0], [+1,+0,+1, +0,+1,+0]]), np.array([ [+0,+1,+0, +0,+0,+1], [+1,+0,+0, +0,+0,+1]]), np.array([ [+1,+1,+0, +0,+0,+1], [-1,+1,+0, +0,+0,+1]]), np.array([ [+0,+1,-1, +0,+1,+1], [+0,-1,-1, +0,-1,+1], [-1,+0,-1, -1,+0,+1], [+1,+0,-1, +1,+0,+1]]), 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]]), np.array([ [+1,+0,+0, +0,+1,+1], [+1,+0,+0, +0,+1,-1], [+0,+1,+0, +1,+0,+1], [+0,+1,+0, +1,+0,-1]]), np.array([ [+0,+1,-1, +2,+1,+1], [+0,-1,-1, +2,-1,+1], [+1,+0,-1, +1,+2,+1], [-1,+0,-1, -1,+2,+1], [+0,+1,-1, -2,+1,+1], [+0,-1,-1, -2,-1,+1], [-1,+0,-1, -1,-2,+1], [+1,+0,-1, +1,-2,+1]]), np.array([ [-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, -2,+1,+1], [+1,-1,+1, -2,-1,+1], [-1,+1,+1, -1,-2,+1], [+1,+1,+1, +1,-2,+1]])] } } 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: str) -> Tuple[CrystalLattice, Rotation]: """ 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: Dict[str, Dict[CrystalLattice,np.ndarray]] = { '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))