from typing import Optional, Union, Dict, List, Tuple
import numpy as np
from ._typehints import FloatSequence, CrystalFamily, BravaisLattice, CrystalKinematics
from . import util
from . import Rotation
lattice_symmetries: Dict[Optional[BravaisLattice], 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',
}
orientation_relationships: Dict[str, Dict[str,List[np.ndarray]]] = {
'KS': { # https://doi.org/10.1016/j.jallcom.2012.02.004
'cF-->cI' : [
np.repeat(np.array([
[[-1, 0, 1],[ 1, 1, 1]],
[[ 0, 1,-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],[-1, 1, 1]],
[[-1, 1, 0],[ 1, 1,-1]],
[[ 0,-1,-1],[ 1, 1,-1]],
[[ 1, 0, 1],[ 1, 1,-1]],
],dtype=float),
2,axis=0),
np.tile(np.array([[[-1,-1, 1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]]],dtype=float),
(12,1,1)),
],
'cI-->cF' : [
np.repeat(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]],
],dtype=float),
2,axis=0),
np.tile(np.array([[[ 0, 1,-1],[ 1, 1, 1]],
[[ 0,-1, 1],[ 1, 1, 1]]],dtype=float),
(12,1,1)),
],
},
'GT': { # https://doi.org/10.1107/S0021889805038276
'cF-->cI' : [
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),
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),
],
'cI-->cF' : [
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),
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),
],
},
'GT_prime': { # https://doi.org/10.1107/S0021889805038276
'cF-->cI' : [
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),
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),
],
'cI-->cF' : [
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),
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),
],
},
'NW': { # https://doi.org/10.1016/j.matchar.2004.12.015
'cF-->cI' : [
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),
np.broadcast_to(np.array([[ 0,-1, 1],[ 0, 1, 1]],dtype=float),
(12,2,3)),
],
'cI-->cF' : [
np.repeat(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]],
[[ 1,-1, 0],[ 1, 1, 0]],
[[ 1, 1, 0],[ 1,-1, 0]],
],dtype=float),
2,axis=0),
np.tile(np.array([
[[ 2,-1,-1],[ 1, 1, 1]],
[[-2, 1, 1],[ 1, 1, 1]],
],dtype=float),
(6,1,1)),
],
},
'Pitsch': { # https://doi.org/10.1080/14786435908238253
'cF-->cI' : [
np.repeat(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]],
[[ 1, 1, 0],[ 0, 0, 1]],
[[ 1,-1, 0],[ 0, 0, 1]],
],dtype=float),
2,axis=0),
np.tile(np.array([
[[ 1, 1,-1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
],dtype=float),
(6,1,1)),
],
'cI-->cF' : [
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]],
],dtype=float),
np.broadcast_to(np.array([[ 1, 1, 0],[ 0, 0, 1]],dtype=float),
(12,2,3)),
],
},
'Bain': { # https://doi.org/10.1107/S0021889805038276
'cF-->cI' : [
np.array([
[[ 0, 1, 0],[ 1, 0, 0]],
[[ 0, 0, 1],[ 0, 1, 0]],
[[ 1, 0, 0],[ 0, 0, 1]],
],dtype=float),
np.broadcast_to(np.array([[ 1, 1, 0],[ 0, 0, 1]],dtype=float),
(3,2,3)),
],
'cI-->cF' : [
np.array([
[[ 0, 1, 1],[ 1, 0, 0]],
[[ 1, 0, 1],[ 0, 1, 0]],
[[ 1, 1, 0],[ 0, 0, 1]],
],dtype=float),
np.broadcast_to(np.array([[ 1, 0, 0],[ 0, 0, 1]],dtype=float),
(3,2,3)),
]
},
'Burgers' : { # https://doi.org/10.1016/S0031-8914(34)80244-3
'cI-->hP' : [
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]],
],dtype=float),
np.broadcast_to(np.array([[ 2,-1,-1, 0],[ 0, 0, 0, 1]],dtype=float),
(12,2,4)),
],
'hP-->cI' : [
np.repeat(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]],
],dtype=float),
2,axis=0),
np.tile(np.array([
[[ 1, 1,-1],[ 0, 1, 1]],
[[-1, 1,-1],[ 0, 1, 1]],
],dtype=float),
(3,1,1)),
]
},
}
[docs]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'))
Crystal family: cubic
Body-centered cubic Bravais lattice with parameters of iron:
>>> import damask
>>> (Fe := damask.Crystal(lattice='cI', a=287e-12))
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: Optional[CrystalFamily] = None,
lattice: Optional[BravaisLattice] = None,
a: Optional[float] = None, b: Optional[float] = None, c: Optional[float] = None,
alpha: Optional[float] = None, beta: Optional[float] = None, gamma: Optional[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):
"""
Return 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:
"""
Return self==other.
Test equality of 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) # type: ignore
@property
def parameters(self) -> Optional[Tuple]:
"""Return lattice parameters a, b, c, alpha, beta, gamma."""
return (self.a,self.b,self.c,self.alpha,self.beta,self.gamma) if hasattr(self,'a') else None
@property
def immutable(self) -> Dict[str, float]:
"""Return immutable lattice parameters."""
_immutable: Dict[CrystalFamily, Dict[str,float]] = {
'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 orientation_relationships(self) -> List[str]:
"""Return labels of orientation relationships."""
return [k for k,v in orientation_relationships.items() if np.any([m.startswith(str(self.lattice)) for m in v])]
@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:
"""
Return symmetry operations.
References
----------
U.F. Kocks et al.,
Texture and Anisotropy:
Preferred Orientations in Polycrystals and their Effect on Materials Properties.
Cambridge University Press 1998. Table II
"""
_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) -> np.ndarray:
"""Return lattice points."""
_lattice_points: Dict[str, List] = {
'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],[]),dtype=float)
[docs] def to_lattice(self, *,
direction: Optional[FloatSequence] = None,
plane: Optional[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)
[docs] def to_frame(self, *,
uvw: Optional[FloatSequence] = None,
hkl: Optional[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=295e-12, c=469e-12)
>>> 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)
[docs] 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[Optional[BravaisLattice], 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]
return {'direction':[util.Bravais_to_Miller(uvtw=m[:,0:4]) if self.lattice == 'hP'
else m[:,0:3] for m in master],
'plane': [util.Bravais_to_Miller(hkil=m[:,4:8]) if self.lattice == 'hP'
else m[:,3:6] for m in master]}
[docs] def relation_operations(self,
model: str,
target = None) -> Tuple[BravaisLattice, Rotation]:
"""
Crystallographic orientation relationships for phase transformations.
Parameters
----------
model : str
Name of orientation relationship.
target : Crystal, optional
Crystal to transform to.
Providing this parameter allows specification of non-standard lattice parameters.
Default is inferred from selected model and uses standard lattice parameters.
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
"""
m_l: BravaisLattice
o_l: BravaisLattice
if model not in self.orientation_relationships:
raise KeyError(f'unknown orientation relationship "{model}"')
sep = '-->'
search = self.lattice+sep+('' if target is None else target.lattice) # type: ignore
transform = [t for t in orientation_relationships[model].keys() if t.startswith(search)] # type: ignore
if len(transform) != 1:
raise ValueError(f'invalid target lattice "{search.split(sep)[1]}"')
m_l,o_l = transform[0].split(sep) # type: ignore
m_p,o_p = orientation_relationships[model][m_l+sep+o_l]
other = Crystal(lattice=o_l) if target is None else target
m_p = np.stack((self.to_frame(uvw=m_p[:,0] if len(m_p[0,0])==3 else util.Bravais_to_Miller(uvtw=m_p[:,0])),
self.to_frame(hkl=m_p[:,1] if len(m_p[0,1])==3 else util.Bravais_to_Miller(hkil=m_p[:,1]))),
axis=1)
o_p = np.stack((other.to_frame(uvw=o_p[:,0] if len(o_p[0,0])==3 else util.Bravais_to_Miller(uvtw=o_p[:,0])),
other.to_frame(hkl=o_p[:,1] if len(o_p[0,1])==3 else util.Bravais_to_Miller(hkil=o_p[:,1]))),
axis=1)
return (o_l,Rotation.from_parallel(a=m_p,b=o_p))