1077 lines
46 KiB
Python
1077 lines
46 KiB
Python
from typing import Optional, Union, Dict, List, Tuple
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import numpy as np
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from ._typehints import FloatSequence, CrystalFamily, BravaisLattice, CrystalKinematics
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from . import util
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from . import Rotation
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lattice_symmetries: Dict[BravaisLattice, CrystalFamily] = {
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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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orientation_relationships: Dict[str, Dict[BravaisLattice,np.ndarray]] = {
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'KS': {
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'cF': np.array([
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[[-1, 0, 1],[ 1, 1, 1]],
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[[-1, 0, 1],[ 1, 1, 1]],
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[[ 0, 1,-1],[ 1, 1, 1]],
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[[ 0, 1,-1],[ 1, 1, 1]],
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[[ 1,-1, 0],[ 1, 1, 1]],
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[[ 1,-1, 0],[ 1, 1, 1]],
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[[ 1, 0,-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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[[-1,-1, 0],[ 1,-1, 1]],
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[[ 0, 1, 1],[ 1,-1, 1]],
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[[ 0, 1, 1],[ 1,-1, 1]],
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[[ 0,-1, 1],[-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, 0,-1],[-1, 1, 1]],
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[[ 1, 1, 0],[-1, 1, 1]],
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[[ 1, 1, 0],[-1, 1, 1]],
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[[-1, 1, 0],[ 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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[[ 0,-1,-1],[ 1, 1,-1]],
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[[ 1, 0, 1],[ 1, 1,-1]],
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[[ 1, 0, 1],[ 1, 1,-1]],
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],dtype=float),
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'cI': 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],[ 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],[ 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],[ 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],[ 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],[ 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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],dtype=float),
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},
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'GT': {
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'cF': np.array([
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[[ -5,-12, 17],[ 1, 1, 1]],
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[[ 17, -5,-12],[ 1, 1, 1]],
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[[-12, 17, -5],[ 1, 1, 1]],
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[[ 5, 12, 17],[ -1, -1, 1]],
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[[-17, 5,-12],[ -1, -1, 1]],
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[[ 12,-17, -5],[ -1, -1, 1]],
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[[ -5, 12,-17],[ -1, 1, 1]],
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[[ 17, 5, 12],[ -1, 1, 1]],
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[[-12,-17, 5],[ -1, 1, 1]],
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[[ 5,-12,-17],[ 1, -1, 1]],
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[[-17, -5, 12],[ 1, -1, 1]],
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[[ 12, 17, 5],[ 1, -1, 1]],
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[[ -5, 17,-12],[ 1, 1, 1]],
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[[-12, -5, 17],[ 1, 1, 1]],
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[[ 17,-12, -5],[ 1, 1, 1]],
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[[ 5,-17,-12],[ -1, -1, 1]],
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[[ 12, 5, 17],[ -1, -1, 1]],
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[[-17, 12, -5],[ -1, -1, 1]],
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[[ -5,-17, 12],[ -1, 1, 1]],
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[[-12, 5,-17],[ -1, 1, 1]],
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[[ 17, 12, 5],[ -1, 1, 1]],
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[[ 5, 17, 12],[ 1, -1, 1]],
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[[ 12, -5,-17],[ 1, -1, 1]],
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[[-17,-12, 5],[ 1, -1, 1]],
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],dtype=float),
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'cI': np.array([
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[[-17, -7, 17],[ 1, 0, 1]],
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[[ 17,-17, -7],[ 1, 1, 0]],
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[[ -7, 17,-17],[ 0, 1, 1]],
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[[ 17, 7, 17],[ -1, 0, 1]],
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[[-17, 17, -7],[ -1, -1, 0]],
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[[ 7,-17,-17],[ 0, -1, 1]],
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[[-17, 7,-17],[ -1, 0, 1]],
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[[ 17, 17, 7],[ -1, 1, 0]],
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[[ -7,-17, 17],[ 0, 1, 1]],
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[[ 17, -7,-17],[ 1, 0, 1]],
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[[-17,-17, 7],[ 1, -1, 0]],
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[[ 7, 17, 17],[ 0, -1, 1]],
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[[-17, 17, -7],[ 1, 1, 0]],
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[[ -7,-17, 17],[ 0, 1, 1]],
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[[ 17, -7,-17],[ 1, 0, 1]],
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[[ 17,-17, -7],[ -1, -1, 0]],
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[[ 7, 17, 17],[ 0, -1, 1]],
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[[-17, 7,-17],[ -1, 0, 1]],
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[[-17,-17, 7],[ -1, 1, 0]],
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[[ -7, 17,-17],[ 0, 1, 1]],
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[[ 17, 7, 17],[ -1, 0, 1]],
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[[ 17, 17, 7],[ 1, -1, 0]],
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[[ 7,-17,-17],[ 0, -1, 1]],
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[[-17, -7, 17],[ 1, 0, 1]],
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],dtype=float),
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},
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'GT_prime': {
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'cF' : np.array([
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[[ 0, 1, -1],[ 7, 17, 17]],
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[[ -1, 0, 1],[ 17, 7, 17]],
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[[ 1, -1, 0],[ 17, 17, 7]],
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[[ 0, -1, -1],[ -7,-17, 17]],
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[[ 1, 0, 1],[-17, -7, 17]],
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[[ 1, -1, 0],[-17,-17, 7]],
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[[ 0, 1, -1],[ 7,-17,-17]],
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[[ 1, 0, 1],[ 17, -7,-17]],
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[[ -1, -1, 0],[ 17,-17, -7]],
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[[ 0, -1, -1],[ -7, 17,-17]],
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[[ -1, 0, 1],[-17, 7,-17]],
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[[ -1, -1, 0],[-17, 17, -7]],
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[[ 0, -1, 1],[ 7, 17, 17]],
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[[ 1, 0, -1],[ 17, 7, 17]],
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[[ -1, 1, 0],[ 17, 17, 7]],
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[[ 0, 1, 1],[ -7,-17, 17]],
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[[ -1, 0, -1],[-17, -7, 17]],
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[[ -1, 1, 0],[-17,-17, 7]],
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[[ 0, -1, 1],[ 7,-17,-17]],
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[[ -1, 0, -1],[ 17, -7,-17]],
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[[ 1, 1, 0],[ 17,-17, -7]],
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[[ 0, 1, 1],[ -7, 17,-17]],
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[[ 1, 0, -1],[-17, 7,-17]],
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[[ 1, 1, 0],[-17, 17, -7]],
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],dtype=float),
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'cI' : np.array([
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[[ 1, 1, -1],[ 12, 5, 17]],
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[[ -1, 1, 1],[ 17, 12, 5]],
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[[ 1, -1, 1],[ 5, 17, 12]],
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[[ -1, -1, -1],[-12, -5, 17]],
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[[ 1, -1, 1],[-17,-12, 5]],
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[[ 1, -1, -1],[ -5,-17, 12]],
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[[ -1, 1, -1],[ 12, -5,-17]],
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[[ 1, 1, 1],[ 17,-12, -5]],
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[[ -1, -1, 1],[ 5,-17,-12]],
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[[ 1, -1, -1],[-12, 5,-17]],
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[[ -1, -1, 1],[-17, 12, -5]],
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[[ -1, -1, -1],[ -5, 17,-12]],
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[[ 1, -1, 1],[ 12, 17, 5]],
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[[ 1, 1, -1],[ 5, 12, 17]],
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[[ -1, 1, 1],[ 17, 5, 12]],
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[[ -1, 1, 1],[-12,-17, 5]],
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[[ -1, -1, -1],[ -5,-12, 17]],
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[[ -1, 1, -1],[-17, -5, 12]],
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[[ -1, -1, 1],[ 12,-17, -5]],
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[[ -1, 1, -1],[ 5,-12,-17]],
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[[ 1, 1, 1],[ 17, -5,-12]],
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[[ 1, 1, 1],[-12, 17, -5]],
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[[ 1, -1, -1],[ -5, 12,-17]],
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[[ 1, 1, -1],[-17, 5,-12]],
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],dtype=float),
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},
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'NW': {
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'cF' : 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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],dtype=float),
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'cI' : np.array([
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[[ 0, -1, 1],[ 0, 1, 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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[[ 0, -1, 1],[ 0, 1, 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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[[ 0, -1, 1],[ 0, 1, 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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[[ 0, -1, 1],[ 0, 1, 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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],dtype=float),
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},
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'Pitsch': {
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'cF' : np.array([
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[[ 1, 0, 1],[ 0, 1, 0]],
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[[ 1, 1, 0],[ 0, 0, 1]],
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[[ 0, 1, 1],[ 1, 0, 0]],
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[[ 0, 1, -1],[ 1, 0, 0]],
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[[ -1, 0, 1],[ 0, 1, 0]],
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[[ 1, -1, 0],[ 0, 0, 1]],
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[[ 1, 0, -1],[ 0, 1, 0]],
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[[ -1, 1, 0],[ 0, 0, 1]],
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[[ 0, -1, 1],[ 1, 0, 0]],
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[[ 0, 1, 1],[ 1, 0, 0]],
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[[ 1, 0, 1],[ 0, 1, 0]],
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[[ 1, 1, 0],[ 0, 0, 1]],
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],dtype=float),
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'cI' : np.array([
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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],[ 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, -1, 0]],
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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],[ 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, 1, 0]],
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],dtype=float),
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},
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'Bain': {
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'cF' : np.array([
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[[ 0, 1, 0],[ 1, 0, 0]],
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[[ 0, 0, 1],[ 0, 1, 0]],
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[[ 1, 0, 0],[ 0, 0, 1]],
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],dtype=float),
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'cI' : np.array([
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[[ 0, 1, 1],[ 1, 0, 0]],
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[[ 1, 0, 1],[ 0, 1, 0]],
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[[ 1, 1, 0],[ 0, 0, 1]],
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],dtype=float),
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},
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'Burgers' : {
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'cI' : np.array([
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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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[[ 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],[ 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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],dtype=float),
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'hP' : np.array([
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[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
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[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
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[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
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[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
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[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
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[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
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[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
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[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
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[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
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[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
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[[ -1, 2, -1, 0],[ 0, 0, 0, 1]],
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[[ -1, -1, 2, 0],[ 0, 0, 0, 1]],
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],dtype=float),
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},
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}
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class Crystal():
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"""
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Representation of a crystal as (general) crystal family or (more specific) as a scaled Bravais lattice.
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Examples
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--------
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Cubic crystal family:
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>>> import damask
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>>> (cubic := damask.Crystal(family='cubic'))
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Crystal family: cubic
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Body-centered cubic Bravais lattice with parameters of iron:
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>>> import damask
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>>> (Fe := damask.Crystal(lattice='cI', a=287e-12))
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Crystal family: cubic
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Bravais lattice: cI
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a=2.87e-10 m, b=2.87e-10 m, c=2.87e-10 m
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α=90°, β=90°, γ=90°
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"""
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def __init__(self, *,
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family: Optional[CrystalFamily] = None,
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lattice: Optional[BravaisLattice] = None,
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a: Optional[float] = None, b: Optional[float] = None, c: Optional[float] = None,
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alpha: Optional[float] = None, beta: Optional[float] = None, gamma: Optional[float] = None,
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degrees: bool = False):
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"""
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New representation of a crystal.
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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 is not None and family not in 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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|
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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):
|
||
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
|
||
|
||
@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 orientation_relationships(self):
|
||
"""Return labels of orientation relationships."""
|
||
return [k for k,v in orientation_relationships.items() if self.lattice 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):
|
||
"""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: 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)
|
||
|
||
|
||
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=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[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]}
|
||
|
||
|
||
def relation_operations(self,
|
||
model: str) -> Tuple[BravaisLattice, 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
|
||
|
||
"""
|
||
my_relationships = {k:v for k,v in orientation_relationships.items() if self.lattice in v}
|
||
if model not in my_relationships:
|
||
raise KeyError(f'unknown orientation relationship "{model}"')
|
||
r = my_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))
|