import numpy as np from . import Rotation class Symmetry: """ Symmetry operations for lattice systems. References ---------- https://en.wikipedia.org/wiki/Crystal_system """ lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic'] def __init__(self, symmetry = None): """ Symmetry Definition. Parameters ---------- symmetry : str, optional label of the crystal system """ if symmetry is not None and symmetry.lower() not in Symmetry.lattices: raise KeyError('Symmetry/crystal system "{}" is unknown'.format(symmetry)) self.lattice = symmetry.lower() if isinstance(symmetry,str) else symmetry def __copy__(self): """Copy.""" return self.__class__(self.lattice) copy = __copy__ def __repr__(self): """Readable string.""" return '{}'.format(self.lattice) def __eq__(self, other): """ Equal to other. Parameters ---------- other : Symmetry Symmetry to check for equality. """ return self.lattice == other.lattice def __neq__(self, other): """ Not Equal to other. Parameters ---------- other : Symmetry Symmetry to check for inequality. """ return not self.__eq__(other) def __cmp__(self,other): """ Linear ordering. Parameters ---------- other : Symmetry Symmetry to check for for order. """ myOrder = Symmetry.lattices.index(self.lattice) otherOrder = Symmetry.lattices.index(other.lattice) return (myOrder > otherOrder) - (myOrder < otherOrder) def symmetryOperations(self,members=[]): """List (or single element) of symmetry operations as rotations.""" if self.lattice == 'cubic': symQuats = [ [ 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 ], ] elif self.lattice == 'hexagonal': symQuats = [ [ 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 ], ] elif self.lattice == 'tetragonal': symQuats = [ [ 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) ], ] elif self.lattice == 'orthorhombic': symQuats = [ [ 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 ], ] else: symQuats = [ [ 1.0,0.0,0.0,0.0 ], ] symOps = list(map(Rotation, np.array(symQuats)[np.atleast_1d(members) if members != [] else range(len(symQuats))])) try: iter(members) # asking for (even empty) list of members? except TypeError: return symOps[0] # no, return rotation object else: return symOps # yes, return list of rotations @property def symmetry_operations(self): """Symmetry operations as Rotations.""" if self.lattice == 'cubic': symQuats = [ [ 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 ], ] elif self.lattice == 'hexagonal': symQuats = [ [ 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 ], ] elif self.lattice == 'tetragonal': symQuats = [ [ 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) ], ] elif self.lattice == 'orthorhombic': symQuats = [ [ 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 ], ] else: symQuats = [ [ 1.0,0.0,0.0,0.0 ], ] return np.array(symQuats) def inFZ(self,rodrigues): """ Check whether given Rodrigues-Frank vector falls into fundamental zone of own symmetry. Fundamental zone in Rodrigues space is point symmetric around origin. """ if (len(rodrigues) != 3): raise ValueError('Input is not a Rodrigues-Frank vector.\n') if np.any(rodrigues == np.inf): return False # ToDo: MD: not sure if needed Rabs = abs(rodrigues) if self.lattice == 'cubic': return np.sqrt(2.0)-1.0 >= Rabs[0] \ and np.sqrt(2.0)-1.0 >= Rabs[1] \ and np.sqrt(2.0)-1.0 >= Rabs[2] \ and 1.0 >= Rabs[0] + Rabs[1] + Rabs[2] elif self.lattice == 'hexagonal': return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[2] \ and 2.0 >= np.sqrt(3)*Rabs[0] + Rabs[1] \ and 2.0 >= np.sqrt(3)*Rabs[1] + Rabs[0] \ and 2.0 >= np.sqrt(3) + Rabs[2] elif self.lattice == 'tetragonal': return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] \ and np.sqrt(2.0) >= Rabs[0] + Rabs[1] \ and np.sqrt(2.0) >= Rabs[2] + 1.0 elif self.lattice == 'orthorhombic': return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[2] else: return True def inDisorientationSST(self,rodrigues): """ Check whether given Rodrigues-Frank vector (of misorientation) falls into standard stereographic triangle of own symmetry. References ---------- A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991 https://doi.org/10.1107/S0108767391006864 """ if (len(rodrigues) != 3): raise ValueError('Input is not a Rodrigues-Frank vector.\n') R = rodrigues epsilon = 0.0 if self.lattice == 'cubic': return R[0] >= R[1]+epsilon and R[1] >= R[2]+epsilon and R[2] >= epsilon elif self.lattice == 'hexagonal': return R[0] >= np.sqrt(3)*(R[1]-epsilon) and R[1] >= epsilon and R[2] >= epsilon elif self.lattice == 'tetragonal': return R[0] >= R[1]-epsilon and R[1] >= epsilon and R[2] >= epsilon elif self.lattice == 'orthorhombic': return R[0] >= epsilon and R[1] >= epsilon and R[2] >= epsilon else: return True def inSST(self, vector, proper = False, color = False): """ Check whether given vector falls into standard stereographic triangle of own symmetry. proper considers only vectors with z >= 0, hence uses two neighboring SSTs. Return inverse pole figure color if requested. Bases are computed from >>> basis = {'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,1.]/np.sqrt(2.), # direction of green ... [1.,1.,1.]/np.sqrt(3.)]).T), # direction of blue ... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,0.], # direction of green ... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # direction of blue ... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,0.], # direction of green ... [1.,1.,0.]/np.sqrt(2.)]).T), # direction of blue ... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,0.], # direction of green ... [0.,1.,0.]]).T), # direction of blue ... } """ if self.lattice == 'cubic': basis = {'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. ] ]), } elif self.lattice == 'hexagonal': basis = {'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. ] ]), } elif self.lattice == 'tetragonal': basis = {'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. ] ]), } elif self.lattice == 'orthorhombic': basis = {'improper':np.array([ [ 0., 0., 1.], [ 1., 0., 0.], [ 0., 1., 0.] ]), 'proper':np.array([ [ 0., 0., 1.], [-1., 0., 0.], [ 0., 1., 0.] ]), } else: # direct exit for unspecified symmetry if color: return (True,np.zeros(3,'d')) else: return True v = np.array(vector,dtype=float) if proper: # check both improper ... theComponents = np.around(np.dot(basis['improper'],v),12) inSST = np.all(theComponents >= 0.0) if not inSST: # ... and proper SST theComponents = np.around(np.dot(basis['proper'],v),12) inSST = np.all(theComponents >= 0.0) else: v[2] = abs(v[2]) # z component projects identical theComponents = np.around(np.dot(basis['improper'],v),12) # for positive and negative values inSST = np.all(theComponents >= 0.0) if color: # have to return color array if inSST: rgb = np.power(theComponents/np.linalg.norm(theComponents),0.5) # smoothen color ramps rgb = np.minimum(np.ones(3,dtype=float),rgb) # limit to maximum intensity rgb /= max(rgb) # normalize to (HS)V = 1 else: rgb = np.zeros(3,dtype=float) return (inSST,rgb) else: return inSST def in_SST(self, vector, proper = False, color = False): """ Check whether given vector falls into standard stereographic triangle of own symmetry. proper considers only vectors with z >= 0, hence uses two neighboring SSTs. Return inverse pole figure color if requested. Bases are computed from >>> basis = {'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,1.]/np.sqrt(2.), # direction of green ... [1.,1.,1.]/np.sqrt(3.)]).T), # direction of blue ... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,0.], # direction of green ... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # direction of blue ... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,0.], # direction of green ... [1.,1.,0.]/np.sqrt(2.)]).T), # direction of blue ... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red ... [1.,0.,0.], # direction of green ... [0.,1.,0.]]).T), # direction of blue ... } """ if self.lattice == 'cubic': basis = {'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. ] ]), } elif self.lattice == 'hexagonal': basis = {'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. ] ]), } elif self.lattice == 'tetragonal': basis = {'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. ] ]), } elif self.lattice == 'orthorhombic': basis = {'improper':np.array([ [ 0., 0., 1.], [ 1., 0., 0.], [ 0., 1., 0.] ]), 'proper':np.array([ [ 0., 0., 1.], [-1., 0., 0.], [ 0., 1., 0.] ]), } else: # direct exit for unspecified symmetry if color: return (np.ones_like(vector[...,0],bool),np.zeros_like(vector)) else: return np.ones_like(vector[...,0],bool) b_p = np.broadcast_to(basis['proper'], vector.shape+(3,)) if proper: b_i = np.broadcast_to(basis['improper'],vector.shape+(3,)) improper = np.all(np.around(np.einsum('...ji,...i',b_i,vector),12)>=0.0,axis=-1,keepdims=True) theComponents = np.where(np.broadcast_to(improper,vector.shape), np.around(np.einsum('...ji,...i',b_i,vector),12), np.around(np.einsum('...ji,...i',b_p,vector),12)) else: vector_ = np.block([vector[...,0:2],np.abs(vector[...,2:3])]) # z component projects identical theComponents = np.around(np.einsum('...ji,...i',b_p,vector_),12) in_SST = np.all(theComponents >= 0.0,axis=-1) if color: # have to return color array with np.errstate(invalid='ignore',divide='ignore'): rgb = (theComponents/np.linalg.norm(theComponents,axis=-1,keepdims=True))**0.5 # smoothen color ramps rgb = np.minimum(1.,rgb) # limit to maximum intensity rgb /= np.max(rgb,axis=-1,keepdims=True) # normalize to (HS)V = 1 rgb[np.broadcast_to(~in_SST.reshape(vector[...,0].shape+(1,)),vector.shape)] = 0.0 return (in_SST,rgb) else: return in_SST # ****************************************************************************************** class Lattice: # ToDo: Make a subclass of Symmetry! """ Lattice system. Currently, this contains only a mapping from Bravais lattice to symmetry and orientation relationships. It could include twin and slip systems. References ---------- https://en.wikipedia.org/wiki/Bravais_lattice """ lattices = { 'triclinic':{'symmetry':None}, 'bct':{'symmetry':'tetragonal'}, 'hex':{'symmetry':'hexagonal'}, 'fcc':{'symmetry':'cubic','c/a':1.0}, 'bcc':{'symmetry':'cubic','c/a':1.0}, } def __init__(self, lattice): """ New lattice of given type. Parameters ---------- lattice : str Bravais lattice. """ self.lattice = lattice self.symmetry = Symmetry(self.lattices[lattice]['symmetry']) def __repr__(self): """Report basic lattice information.""" return 'Bravais lattice {} ({} symmetry)'.format(self.lattice,self.symmetry) # Kurdjomov--Sachs orientation relationship for fcc <-> bcc transformation # from S. Morito et al., Journal of Alloys and Compounds 577:s587-s592, 2013 # also see K. Kitahara et al., Acta Materialia 54:1279-1288, 2006 KS = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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'), 'directions': 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')} # Greninger--Troiano orientation relationship for fcc <-> bcc transformation # from Y. He et al., Journal of Applied Crystallography 39:72-81, 2006 GT = {'mapping':{'fcc':0,'bcc':1}, 'planes': np.array([ [[ 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],[ 0, -1, 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],[ 0, -1, 1]], [[ 1, -1, 1],[ 1, 0, 1]]],dtype='float'), 'directions': np.array([ [[ -5,-12, 17],[-17, -7, 17]], [[ 17, -5,-12],[ 17,-17, -7]], [[-12, 17, -5],[ -7, 17,-17]], [[ 5, 12, 17],[ 17, 7, 17]], [[-17, 5,-12],[-17, 17, -7]], [[ 12,-17, -5],[ 7,-17,-17]], [[ -5, 12,-17],[-17, 7,-17]], [[ 17, 5, 12],[ 17, 17, 7]], [[-12,-17, 5],[ -7,-17, 17]], [[ 5,-12,-17],[ 17, -7,-17]], [[-17, -5, 12],[-17,-17, 7]], [[ 12, 17, 5],[ 7, 17, 17]], [[ -5, 17,-12],[-17, 17, -7]], [[-12, -5, 17],[ -7,-17, 17]], [[ 17,-12, -5],[ 17, -7,-17]], [[ 5,-17,-12],[ 17,-17, -7]], [[ 12, 5, 17],[ 7, 17, 17]], [[-17, 12, -5],[-17, 7,-17]], [[ -5,-17, 12],[-17,-17, 7]], [[-12, 5,-17],[ -7, 17,-17]], [[ 17, 12, 5],[ 17, 7, 17]], [[ 5, 17, 12],[ 17, 17, 7]], [[ 12, -5,-17],[ 7,-17,-17]], [[-17,-12, 5],[-17,-7, 17]]],dtype='float')} # Greninger--Troiano' orientation relationship for fcc <-> bcc transformation # from Y. He et al., Journal of Applied Crystallography 39:72-81, 2006 GTprime = {'mapping':{'fcc':0,'bcc':1}, 'planes': np.array([ [[ 7, 17, 17],[ 12, 5, 17]], [[ 17, 7, 17],[ 17, 12, 5]], [[ 17, 17, 7],[ 5, 17, 12]], [[ -7,-17, 17],[-12, -5, 17]], [[-17, -7, 17],[-17,-12, 5]], [[-17,-17, 7],[ -5,-17, 12]], [[ 7,-17,-17],[ 12, -5,-17]], [[ 17, -7,-17],[ 17,-12, -5]], [[ 17,-17, -7],[ 5,-17,-12]], [[ -7, 17,-17],[-12, 5,-17]], [[-17, 7,-17],[-17, 12, -5]], [[-17, 17, -7],[ -5, 17,-12]], [[ 7, 17, 17],[ 12, 17, 5]], [[ 17, 7, 17],[ 5, 12, 17]], [[ 17, 17, 7],[ 17, 5, 12]], [[ -7,-17, 17],[-12,-17, 5]], [[-17, -7, 17],[ -5,-12, 17]], [[-17,-17, 7],[-17, -5, 12]], [[ 7,-17,-17],[ 12,-17, -5]], [[ 17, -7,-17],[ 5, -12,-17]], [[ 17,-17, -7],[ 17, -5,-12]], [[ -7, 17,-17],[-12, 17, -5]], [[-17, 7,-17],[ -5, 12,-17]], [[-17, 17, -7],[-17, 5,-12]]],dtype='float'), 'directions': 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]], [[ 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]]],dtype='float')} # Nishiyama--Wassermann orientation relationship for fcc <-> bcc transformation # from H. Kitahara et al., Materials Characterization 54:378-386, 2005 NW = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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]]],dtype='float'), 'directions': np.array([ [[ 2, -1, -1],[ 0, -1, 1]], [[ -1, 2, -1],[ 0, -1, 1]], [[ -1, -1, 2],[ 0, -1, 1]], [[ -2, -1, -1],[ 0, -1, 1]], [[ 1, 2, -1],[ 0, -1, 1]], [[ 1, -1, 2],[ 0, -1, 1]], [[ 2, 1, -1],[ 0, -1, 1]], [[ -1, -2, -1],[ 0, -1, 1]], [[ -1, 1, 2],[ 0, -1, 1]], [[ 2, -1, 1],[ 0, -1, 1]], #It is wrong in the paper, but matrix is correct [[ -1, 2, 1],[ 0, -1, 1]], [[ -1, -1, -2],[ 0, -1, 1]]],dtype='float')} # Pitsch orientation relationship for fcc <-> bcc transformation # from Y. He et al., Acta Materialia 53:1179-1190, 2005 Pitsch = {'mapping':{'fcc':0,'bcc':1}, 'planes': np.array([ [[ 0, 1, 0],[ -1, 0, 1]], [[ 0, 0, 1],[ 1, -1, 0]], [[ 1, 0, 0],[ 0, 1, -1]], [[ 1, 0, 0],[ 0, -1, -1]], [[ 0, 1, 0],[ -1, 0, -1]], [[ 0, 0, 1],[ -1, -1, 0]], [[ 0, 1, 0],[ -1, 0, -1]], [[ 0, 0, 1],[ -1, -1, 0]], [[ 1, 0, 0],[ 0, -1, -1]], [[ 1, 0, 0],[ 0, -1, 1]], [[ 0, 1, 0],[ 1, 0, -1]], [[ 0, 0, 1],[ -1, 1, 0]]],dtype='float'), 'directions': np.array([ [[ 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],[ 1, 1, -1]]],dtype='float')} # Bain orientation relationship for fcc <-> bcc transformation # from Y. He et al., Journal of Applied Crystallography 39:72-81, 2006 Bain = {'mapping':{'fcc':0,'bcc':1}, 'planes': np.array([ [[ 1, 0, 0],[ 1, 0, 0]], [[ 0, 1, 0],[ 0, 1, 0]], [[ 0, 0, 1],[ 0, 0, 1]]],dtype='float'), 'directions': np.array([ [[ 0, 1, 0],[ 0, 1, 1]], [[ 0, 0, 1],[ 1, 0, 1]], [[ 1, 0, 0],[ 1, 1, 0]]],dtype='float')} def relationOperations(self,model): """ Crystallographic orientation relationships for phase transformations. 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 """ models={'KS':self.KS, 'GT':self.GT, 'GT_prime':self.GTprime, 'NW':self.NW, 'Pitsch': self.Pitsch, 'Bain':self.Bain} try: relationship = models[model] except KeyError : raise KeyError('Orientation relationship "{}" is unknown'.format(model)) if self.lattice not in relationship['mapping']: raise ValueError('Relationship "{}" not supported for lattice "{}"'.format(model,self.lattice)) r = {'lattice':Lattice((set(relationship['mapping'])-{self.lattice}).pop()), # target lattice 'rotations':[] } myPlane_id = relationship['mapping'][self.lattice] otherPlane_id = (myPlane_id+1)%2 myDir_id = myPlane_id +2 otherDir_id = otherPlane_id +2 for miller in np.hstack((relationship['planes'],relationship['directions'])): myPlane = miller[myPlane_id]/ np.linalg.norm(miller[myPlane_id]) myDir = miller[myDir_id]/ np.linalg.norm(miller[myDir_id]) myMatrix = np.array([myDir,np.cross(myPlane,myDir),myPlane]) otherPlane = miller[otherPlane_id]/ np.linalg.norm(miller[otherPlane_id]) otherDir = miller[otherDir_id]/ np.linalg.norm(miller[otherDir_id]) otherMatrix = np.array([otherDir,np.cross(otherPlane,otherDir),otherPlane]) r['rotations'].append(Rotation.from_matrix(np.dot(otherMatrix.T,myMatrix))) return r