2020-02-21 03:59:12 +05:30
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import numpy as np
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class Symmetry:
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"""
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2020-06-30 16:25:08 +05:30
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Symmetry-related operations for crystal systems.
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2020-02-21 03:59:12 +05:30
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References
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----------
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https://en.wikipedia.org/wiki/Crystal_system
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"""
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2020-06-30 16:25:08 +05:30
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crystal_systems = [None,'orthorhombic','tetragonal','hexagonal','cubic']
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2020-02-21 03:59:12 +05:30
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2020-06-30 16:25:08 +05:30
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def __init__(self, system = None):
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2020-02-21 03:59:12 +05:30
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"""
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Symmetry Definition.
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Parameters
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----------
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2020-06-30 16:25:08 +05:30
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system : {None,'orthorhombic','tetragonal','hexagonal','cubic'}, optional
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Name of the crystal system. Defaults to 'None'.
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2020-02-21 03:59:12 +05:30
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"""
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2020-06-30 16:25:08 +05:30
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if system is not None and system.lower() not in self.crystal_systems:
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raise KeyError(f'Crystal system "{system}" is unknown')
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self.system = system.lower() if isinstance(system,str) else system
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def __copy__(self):
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"""Copy."""
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2020-06-30 16:25:08 +05:30
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return self.__class__(self.system)
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2020-02-21 03:59:12 +05:30
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copy = __copy__
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def __repr__(self):
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"""Readable string."""
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return f'{self.system}'
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2020-02-21 03:59:12 +05:30
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def __eq__(self, other):
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"""
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Equal to other.
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Parameters
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----------
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other : Symmetry
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Symmetry to check for equality.
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"""
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2020-06-30 16:25:08 +05:30
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return self.system == other.system
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2020-02-21 03:59:12 +05:30
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def __neq__(self, other):
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"""
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Not Equal to other.
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Parameters
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----------
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other : Symmetry
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Symmetry to check for inequality.
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"""
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return not self.__eq__(other)
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def __cmp__(self,other):
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"""
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Linear ordering.
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Parameters
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----------
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other : Symmetry
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Symmetry to check for for order.
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"""
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2020-06-30 16:25:08 +05:30
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myOrder = self.crystal_systems.index(self.system)
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otherOrder = self.crystal_systems.index(other.system)
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2020-02-21 03:59:12 +05:30
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return (myOrder > otherOrder) - (myOrder < otherOrder)
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2020-06-19 01:59:28 +05:30
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@property
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def symmetry_operations(self):
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"""Symmetry operations as quaternions."""
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if self.system == 'cubic':
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sym_quats = [
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[ 1.0, 0.0, 0.0, 0.0 ],
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[ 0.0, 1.0, 0.0, 0.0 ],
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[ 0.0, 0.0, 1.0, 0.0 ],
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[ 0.0, 0.0, 0.0, 1.0 ],
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[ 0.0, 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2) ],
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[ 0.0, 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2) ],
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[ 0.0, 0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2) ],
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[ 0.0, 0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2) ],
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[ 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ],
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[ 0.0, -0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ],
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[ 0.5, 0.5, 0.5, 0.5 ],
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[-0.5, 0.5, 0.5, 0.5 ],
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[-0.5, 0.5, 0.5, -0.5 ],
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[-0.5, 0.5, -0.5, 0.5 ],
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[-0.5, -0.5, 0.5, 0.5 ],
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[-0.5, -0.5, 0.5, -0.5 ],
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[-0.5, -0.5, -0.5, 0.5 ],
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[-0.5, 0.5, -0.5, -0.5 ],
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[-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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[ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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[-0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2), 0.0 ],
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[-0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2), 0.0 ],
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[-0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0, 0.0 ],
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[-0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0, 0.0 ],
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]
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elif self.system == 'hexagonal':
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sym_quats = [
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[ 1.0, 0.0, 0.0, 0.0 ],
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[-0.5*np.sqrt(3), 0.0, 0.0, -0.5 ],
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[ 0.5, 0.0, 0.0, 0.5*np.sqrt(3) ],
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[ 0.0, 0.0, 0.0, 1.0 ],
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[-0.5, 0.0, 0.0, 0.5*np.sqrt(3) ],
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[-0.5*np.sqrt(3), 0.0, 0.0, 0.5 ],
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[ 0.0, 1.0, 0.0, 0.0 ],
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[ 0.0, -0.5*np.sqrt(3), 0.5, 0.0 ],
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[ 0.0, 0.5, -0.5*np.sqrt(3), 0.0 ],
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[ 0.0, 0.0, 1.0, 0.0 ],
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[ 0.0, -0.5, -0.5*np.sqrt(3), 0.0 ],
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[ 0.0, 0.5*np.sqrt(3), 0.5, 0.0 ],
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]
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2020-06-30 16:25:08 +05:30
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elif self.system == 'tetragonal':
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sym_quats = [
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[ 1.0, 0.0, 0.0, 0.0 ],
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[ 0.0, 1.0, 0.0, 0.0 ],
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[ 0.0, 0.0, 1.0, 0.0 ],
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[ 0.0, 0.0, 0.0, 1.0 ],
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[ 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ],
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[ 0.0, -0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ],
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[ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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[-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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]
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elif self.system == 'orthorhombic':
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sym_quats = [
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[ 1.0,0.0,0.0,0.0 ],
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[ 0.0,1.0,0.0,0.0 ],
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[ 0.0,0.0,1.0,0.0 ],
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[ 0.0,0.0,0.0,1.0 ],
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]
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else:
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sym_quats = [
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[ 1.0,0.0,0.0,0.0 ],
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]
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2020-07-01 16:41:51 +05:30
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return np.array(sym_quats)
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2020-02-21 03:59:12 +05:30
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2020-07-01 01:13:57 +05:30
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def in_FZ(self,rho):
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2020-02-21 03:59:12 +05:30
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"""
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2020-07-01 01:13:57 +05:30
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Check whether given Rodrigues-Frank vector falls into fundamental zone.
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2020-02-21 03:59:12 +05:30
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Fundamental zone in Rodrigues space is point symmetric around origin.
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"""
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if(rho.shape[-1] != 3):
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raise ValueError('Input is not a Rodrigues-Frank vector field.')
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rho_abs = np.abs(rho)
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with np.errstate(invalid='ignore'):
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# using '*'/prod for 'and'
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if self.system == 'cubic':
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return np.where(np.prod(np.sqrt(2)-1. >= rho_abs,axis=-1) * \
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(1. >= np.sum(rho_abs,axis=-1)),True,False)
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elif self.system == 'hexagonal':
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return np.where(np.prod(1. >= rho_abs,axis=-1) * \
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(2. >= np.sqrt(3)*rho_abs[...,0] + rho_abs[...,1]) * \
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(2. >= np.sqrt(3)*rho_abs[...,1] + rho_abs[...,0]) * \
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(2. >= np.sqrt(3) + rho_abs[...,2]),True,False)
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elif self.system == 'tetragonal':
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return np.where(np.prod(1. >= rho_abs[...,:2],axis=-1) * \
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(np.sqrt(2) >= rho_abs[...,0] + rho_abs[...,1]) * \
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(np.sqrt(2) >= rho_abs[...,2] + 1.),True,False)
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elif self.system == 'orthorhombic':
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return np.where(np.prod(1. >= rho_abs,axis=-1),True,False)
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else:
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return np.where(np.all(np.isfinite(rho_abs),axis=-1),True,False)
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2020-02-21 03:59:12 +05:30
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2020-07-01 01:13:57 +05:30
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def in_disorientation_SST(self,rho):
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"""
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2020-07-01 01:13:57 +05:30
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Check whether given Rodrigues-Frank vector (of misorientation) falls into standard stereographic triangle.
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2020-02-21 03:59:12 +05:30
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References
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----------
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A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
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https://doi.org/10.1107/S0108767391006864
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"""
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if(rho.shape[-1] != 3):
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raise ValueError('Input is not a Rodrigues-Frank vector field.')
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2020-07-01 01:13:57 +05:30
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with np.errstate(invalid='ignore'):
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# using '*' for 'and'
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if self.system == 'cubic':
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return np.where((rho[...,0] >= rho[...,1]) * \
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(rho[...,1] >= rho[...,2]) * \
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(rho[...,2] >= 0),True,False)
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elif self.system == 'hexagonal':
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return np.where((rho[...,0] >= rho[...,1]*np.sqrt(3)) * \
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(rho[...,1] >= 0) * \
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(rho[...,2] >= 0),True,False)
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elif self.system == 'tetragonal':
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return np.where((rho[...,0] >= rho[...,1]) * \
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(rho[...,1] >= 0) * \
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(rho[...,2] >= 0),True,False)
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elif self.system == 'orthorhombic':
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return np.where((rho[...,0] >= 0) * \
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(rho[...,1] >= 0) * \
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(rho[...,2] >= 0),True,False)
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else:
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return np.ones_like(rho[...,0],dtype=bool)
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2020-06-19 14:24:13 +05:30
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2020-07-01 12:18:30 +05:30
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#ToDo: IPF color in separate function
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def in_SST(self,vector,proper=False,color=False):
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"""
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Check whether given vector falls into standard stereographic triangle of own symmetry.
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proper considers only vectors with z >= 0, hence uses two neighboring SSTs.
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Return inverse pole figure color if requested.
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Bases are computed from
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>>> basis = {'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,1.]/np.sqrt(2.), # direction of green
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... [1.,1.,1.]/np.sqrt(3.)]).T), # direction of blue
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... 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,0.], # direction of green
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... [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # direction of blue
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... 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,0.], # direction of green
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... [1.,1.,0.]/np.sqrt(2.)]).T), # direction of blue
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... 'orthorhombic': np.linalg.inv(np.array([[0.,0.,1.], # direction of red
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... [1.,0.,0.], # direction of green
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... [0.,1.,0.]]).T), # direction of blue
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... }
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"""
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2020-07-01 02:03:58 +05:30
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if(vector.shape[-1] != 3):
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raise ValueError('Input is not a 3D vector field.')
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if self.system == 'cubic':
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basis = {'improper':np.array([ [-1. , 0. , 1. ],
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[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
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[ 0. , np.sqrt(3.) , 0. ] ]),
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'proper':np.array([ [ 0. , -1. , 1. ],
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[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
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[ np.sqrt(3.) , 0. , 0. ] ]),
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}
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elif self.system == 'hexagonal':
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basis = {'improper':np.array([ [ 0. , 0. , 1. ],
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[ 1. , -np.sqrt(3.) , 0. ],
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[ 0. , 2. , 0. ] ]),
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'proper':np.array([ [ 0. , 0. , 1. ],
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[-1. , np.sqrt(3.) , 0. ],
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[ np.sqrt(3.) , -1. , 0. ] ]),
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}
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elif self.system == 'tetragonal':
|
2020-06-19 14:24:13 +05:30
|
|
|
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. ] ]),
|
|
|
|
|
}
|
2020-06-30 16:25:08 +05:30
|
|
|
elif self.system == 'orthorhombic':
|
2020-06-19 14:24:13 +05:30
|
|
|
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)
|
|
|
|
|
|
2020-07-01 02:03:58 +05:30
|
|
|
|
|
|
|
|
b_i = np.broadcast_to(basis['improper'],vector.shape+(3,))
|
2020-06-19 14:24:13 +05:30
|
|
|
if proper:
|
2020-07-01 02:03:58 +05:30
|
|
|
b_p = np.broadcast_to(basis['proper'], vector.shape+(3,))
|
2020-06-19 14:24:13 +05:30
|
|
|
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
|
2020-07-01 02:03:58 +05:30
|
|
|
theComponents = np.around(np.einsum('...ji,...i',b_i,vector_),12)
|
2020-06-19 14:24:13 +05:30
|
|
|
|
|
|
|
|
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
|
2020-06-21 14:07:09 +05:30
|
|
|
rgb[np.broadcast_to(~in_SST.reshape(vector[...,0].shape+(1,)),vector.shape)] = 0.0
|
2020-06-19 14:24:13 +05:30
|
|
|
return (in_SST,rgb)
|
|
|
|
|
else:
|
|
|
|
|
return in_SST
|
2020-02-21 03:59:12 +05:30
|
|
|
|
|
|
|
|
|
|
|
|
|
# ******************************************************************************************
|
2020-06-20 20:04:19 +05:30
|
|
|
class Lattice: # ToDo: Make a subclass of Symmetry!
|
2020-02-22 04:49:27 +05:30
|
|
|
"""
|
2020-07-01 12:18:30 +05:30
|
|
|
Bravais lattice.
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-07-01 12:18:30 +05:30
|
|
|
This contains only a mapping from Bravais lattice to symmetry
|
2020-02-22 04:49:27 +05:30
|
|
|
and orientation relationships. It could include twin and slip systems.
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
References
|
|
|
|
|
----------
|
|
|
|
|
https://en.wikipedia.org/wiki/Bravais_lattice
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
"""
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
lattices = {
|
2020-06-30 17:25:09 +05:30
|
|
|
'triclinic':{'system':None},
|
|
|
|
|
'bct': {'system':'tetragonal'},
|
|
|
|
|
'hex': {'system':'hexagonal'},
|
|
|
|
|
'fcc': {'system':'cubic','c/a':1.0},
|
|
|
|
|
'bcc': {'system':'cubic','c/a':1.0},
|
2020-02-22 04:49:27 +05:30
|
|
|
}
|
2020-02-21 03:59:12 +05:30
|
|
|
|
|
|
|
|
|
2020-07-01 12:18:30 +05:30
|
|
|
def __init__(self,lattice,c_over_a=None):
|
2020-02-22 04:49:27 +05:30
|
|
|
"""
|
|
|
|
|
New lattice of given type.
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
Parameters
|
|
|
|
|
----------
|
|
|
|
|
lattice : str
|
|
|
|
|
Bravais lattice.
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
"""
|
|
|
|
|
self.lattice = lattice
|
2020-06-30 17:25:09 +05:30
|
|
|
self.symmetry = Symmetry(self.lattices[lattice]['system'])
|
|
|
|
|
|
|
|
|
|
# transition to subclass
|
2020-07-01 12:18:30 +05:30
|
|
|
self.system = self.symmetry.system
|
|
|
|
|
self.in_SST = self.symmetry.in_SST
|
|
|
|
|
self.in_FZ = self.symmetry.in_FZ
|
|
|
|
|
self.in_disorientation_SST = self.symmetry.in_disorientation_SST
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
def __repr__(self):
|
|
|
|
|
"""Report basic lattice information."""
|
2020-07-01 16:41:51 +05:30
|
|
|
return f'Bravais lattice {self.lattice} ({self.symmetry} crystal system)'
|
2020-02-22 04:49:27 +05:30
|
|
|
|
|
|
|
|
|
|
|
|
|
# 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
|
2020-07-01 12:18:30 +05:30
|
|
|
_KS = {'mapping':{'fcc':0,'bcc':1},
|
2020-02-22 04:49:27 +05:30
|
|
|
'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
|
2020-07-01 12:18:30 +05:30
|
|
|
_GT = {'mapping':{'fcc':0,'bcc':1},
|
2020-02-22 04:49:27 +05:30
|
|
|
'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
|
2020-07-01 12:18:30 +05:30
|
|
|
_GTprime = {'mapping':{'fcc':0,'bcc':1},
|
2020-02-22 04:49:27 +05:30
|
|
|
'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]],
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[[ 1, 1, 0],[ 1, 1, -1]]],dtype='float')}
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# Nishiyama--Wassermann orientation relationship for fcc <-> bcc transformation
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# from H. Kitahara et al., Materials Characterization 54:378-386, 2005
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2020-07-01 12:18:30 +05:30
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_NW = {'mapping':{'fcc':0,'bcc':1},
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2020-02-22 04:49:27 +05:30
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'planes': 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]]],dtype='float'),
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'directions': np.array([
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[[ 2, -1, -1],[ 0, -1, 1]],
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[[ -1, 2, -1],[ 0, -1, 1]],
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[[ -1, -1, 2],[ 0, -1, 1]],
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[[ -2, -1, -1],[ 0, -1, 1]],
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[[ 1, 2, -1],[ 0, -1, 1]],
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[[ 1, -1, 2],[ 0, -1, 1]],
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[[ 2, 1, -1],[ 0, -1, 1]],
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[[ -1, -2, -1],[ 0, -1, 1]],
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[[ -1, 1, 2],[ 0, -1, 1]],
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[[ 2, -1, 1],[ 0, -1, 1]], #It is wrong in the paper, but matrix is correct
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[[ -1, 2, 1],[ 0, -1, 1]],
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[[ -1, -1, -2],[ 0, -1, 1]]],dtype='float')}
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# Pitsch orientation relationship for fcc <-> bcc transformation
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# from Y. He et al., Acta Materialia 53:1179-1190, 2005
|
2020-07-01 12:18:30 +05:30
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|
_Pitsch = {'mapping':{'fcc':0,'bcc':1},
|
2020-02-22 04:49:27 +05:30
|
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|
'planes': np.array([
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|
[[ 0, 1, 0],[ -1, 0, 1]],
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[[ 0, 0, 1],[ 1, -1, 0]],
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[[ 1, 0, 0],[ 0, 1, -1]],
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[[ 1, 0, 0],[ 0, -1, -1]],
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[[ 0, 1, 0],[ -1, 0, -1]],
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[[ 0, 0, 1],[ -1, -1, 0]],
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[[ 0, 1, 0],[ -1, 0, -1]],
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|
[[ 0, 0, 1],[ -1, -1, 0]],
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|
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|
[[ 1, 0, 0],[ 0, -1, -1]],
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|
|
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|
[[ 1, 0, 0],[ 0, -1, 1]],
|
|
|
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|
[[ 0, 1, 0],[ 1, 0, -1]],
|
|
|
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|
[[ 0, 0, 1],[ -1, 1, 0]]],dtype='float'),
|
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|
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|
'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
|
2020-07-01 12:18:30 +05:30
|
|
|
_Bain = {'mapping':{'fcc':0,'bcc':1},
|
2020-02-22 04:49:27 +05:30
|
|
|
'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')}
|
|
|
|
|
|
2020-07-01 12:18:30 +05:30
|
|
|
|
|
|
|
|
def relation_operations(self,model):
|
2020-02-22 04:49:27 +05:30
|
|
|
"""
|
|
|
|
|
Crystallographic orientation relationships for phase transformations.
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
References
|
|
|
|
|
----------
|
|
|
|
|
S. Morito et al., Journal of Alloys and Compounds 577:s587-s592, 2013
|
|
|
|
|
https://doi.org/10.1016/j.jallcom.2012.02.004
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
K. Kitahara et al., Acta Materialia 54(5):1279-1288, 2006
|
|
|
|
|
https://doi.org/10.1016/j.actamat.2005.11.001
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
Y. He et al., Journal of Applied Crystallography 39:72-81, 2006
|
|
|
|
|
https://doi.org/10.1107/S0021889805038276
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
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
|
|
|
|
|
|
|
|
|
|
"""
|
2020-07-01 12:18:30 +05:30
|
|
|
models={'KS':self._KS, 'GT':self._GT, 'GT_prime':self._GTprime,
|
|
|
|
|
'NW':self._NW, 'Pitsch': self._Pitsch, 'Bain':self._Bain}
|
2020-02-22 04:49:27 +05:30
|
|
|
try:
|
|
|
|
|
relationship = models[model]
|
|
|
|
|
except KeyError :
|
2020-06-25 01:04:51 +05:30
|
|
|
raise KeyError(f'Orientation relationship "{model}" is unknown')
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
if self.lattice not in relationship['mapping']:
|
2020-06-25 01:04:51 +05:30
|
|
|
raise ValueError(f'Relationship "{model}" not supported for lattice "{self.lattice}"')
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
r = {'lattice':Lattice((set(relationship['mapping'])-{self.lattice}).pop()), # target lattice
|
|
|
|
|
'rotations':[] }
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
myPlane_id = relationship['mapping'][self.lattice]
|
|
|
|
|
otherPlane_id = (myPlane_id+1)%2
|
2020-06-25 01:04:51 +05:30
|
|
|
myDir_id = myPlane_id +2
|
2020-02-22 04:49:27 +05:30
|
|
|
otherDir_id = otherPlane_id +2
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
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])
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
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])
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-07-01 12:18:30 +05:30
|
|
|
r['rotations'].append(np.dot(otherMatrix.T,myMatrix))
|
|
|
|
|
|
|
|
|
|
r['rotations'] = np.array(r['rotations'])
|
2020-02-21 03:59:12 +05:30
|
|
|
|
2020-02-22 04:49:27 +05:30
|
|
|
return r
|
