def three_perp_eigs(A):
testvecs = []
testindices = []
svecs = zeros((3, 3), dtype=float)
for k in range(A.nops):
# print; print k
op = array(A.symops[:, :, k])
[vals, vecs] = eig(op)
vecs = array(vecs)
#find operations with nondegenerate real eigenvalues
# print k, nonDegen(vals)
for i in nonDegen(vals):
if not matchDirection(transpose(vecs[:, i]),
testvecs): #keep only unique directions
testvecs.append(real(vecs[:, i]))
testindices.append([k, i])
if len(testvecs) >= 3:
testvecstrials = [list(x) for x in combinations(testvecs, 3)]
# print testvecstrials
for trial in testvecstrials:
#find superlattice
for i in range(3):
# print; print 'trial u',trial[i]
svecs[i, :] = lattvec_u(A.vecs, trial[i])
# print svecs[i,:]
# print 'lattice m for lattice vector', dot(inv(A.vecs),transpose(svecs[i,:]))
# print 'cosvecs', cosvecs(svecs[0,:],svecs[1,:]) , cosvecs(svecs[1,:],svecs[2,:]) , cosvecs(svecs[2,:],svecs[0,:])
# print arenormal(svecs[0,:],svecs[1,:]) , arenormal(svecs[1,:],svecs[2,:]) , arenormal(svecs[2,:],svecs[0,:])
if arenormal(svecs[0, :], svecs[1, :]) and arenormal(
svecs[1, :], svecs[2, :]) and arenormal(
svecs[2, :], svecs[0, :]):
S = transpose(array([svecs[0, :], svecs[1, :], svecs[2, :]]))
# print 'found 3 perp'; print unique_anorms(S)
# print S; print norm(S[:,0]); print norm(S[:,1]); print norm(S[:,2])
# print unique_anorms(S).count(True); print A.lattype
if A.lattype == 'Cubic' and unique_anorms(S).count(True) == 0:
return trimSmall(S)
if A.lattype == 'Tetragonal' and unique_anorms(S).count(
True) <= 1:
return trimSmall(S)
if A.lattype == 'Orthorhombic': #and unique_anorms(S).count(True) == 3:
return trimSmall(S)
sys.exit('error in three_perp_eigs')
def three_perp_eigs(A):
testvecs = []; testindices = []
svecs = zeros((3,3),dtype = float)
for k in range(A.nops):
# print; print k
op = array(A.symops[:,:,k])
[vals,vecs]=eig(op); vecs = array(vecs)
#find operations with nondegenerate real eigenvalues
# print k, nonDegen(vals)
for i in nonDegen(vals):
if not matchDirection(transpose(vecs[:,i]),testvecs): #keep only unique directions
testvecs.append(real(vecs[:,i]))
testindices.append([k,i])
if len(testvecs) >= 3:
testvecstrials = [list(x) for x in combinations(testvecs,3)]
# print testvecstrials
for trial in testvecstrials:
#find superlattice
for i in range(3):
# print; print 'trial u',trial[i]
svecs[i,:] = lattvec_u(A.vecs,trial[i])
# print svecs[i,:]
# print 'lattice m for lattice vector', dot(inv(A.vecs),transpose(svecs[i,:]))
# print 'cosvecs', cosvecs(svecs[0,:],svecs[1,:]) , cosvecs(svecs[1,:],svecs[2,:]) , cosvecs(svecs[2,:],svecs[0,:])
# print arenormal(svecs[0,:],svecs[1,:]) , arenormal(svecs[1,:],svecs[2,:]) , arenormal(svecs[2,:],svecs[0,:])
if arenormal(svecs[0,:],svecs[1,:]) and arenormal(svecs[1,:],svecs[2,:]) and arenormal(svecs[2,:],svecs[0,:]):
S = transpose(array([svecs[0,:],svecs[1,:],svecs[2,:]]))
# print 'found 3 perp'; print unique_anorms(S)
# print S; print norm(S[:,0]); print norm(S[:,1]); print norm(S[:,2])
# print unique_anorms(S).count(True); print A.lattype
if A.lattype == 'Cubic' and unique_anorms(S).count(True) == 0:
return trimSmall(S)
if A.lattype == 'Tetragonal' and unique_anorms(S).count(True) <= 1:
return trimSmall(S)
if A.lattype == 'Orthorhombic': #and unique_anorms(S).count(True) == 3:
return trimSmall(S)
sys.exit('error in three_perp_eigs')
def bestmeshEigen(Blatt,Nmesh):
'''The kmesh can be related to the reciprocal lattice B by B = KM, where M is an integer 3x3 matrix
So K = B Inv(M) . Work in the inverse space of this problem, where we can work with M instead of Inv(M).
T(InvK) = T(InvB)T(M).
Define S = T(InvK), and the real lattice A = T(InvB). So S = A T(M) is a superlattice on the real lattice.
Minimization scheme'''
##############################################################
########################## Script ############################
M = zeros((3,3),dtype = int)
S = zeros((3,3),dtype = fprec)
B = lattice()
A = lattice()
K = lattice()
B.vecs = Blatt/2/pi #Don't use 2pi constants in RL here.
#############End BCT lattice
eps = 1.0e-6
B.det = det(B.vecs)
B.Nmesh = Nmesh
print 'B vectors';print B.vecs #
#print 'B transpose'; print transpose(B.vecs)
print 'Det of B', B.det
print 'Orth Defect of B', orthdef(B.vecs)
print 'Surf/vol of B', surfvol(B.vecs)
[B.symops,B.nops] = getGroup(B.vecs)
print 'Number of symmetry operations', B.nops
eigendirs = zeros([3,3,B.nops],dtype = int)
#print 'symmetry operations of B\n'
#for j in range(nopsB):
# print j
# op = array(symopsB[:,:,j])
# print op
#find real lattice
A.vecs = transpose(inv(B.vecs))
A.det = det(A.vecs)
A.Nmesh = Nmesh
print;print 'A vectors';print A.vecs
print 'Det of A', A.det
print 'Orth Defect of A', orthdef(A.vecs)
print 'Surf/vol of A', surfvol(A.vecs)
[A.symops,A.nops] = getGroup(A.vecs)
if A.nops != B.nops:
sys.exit('Number of operations different for A and B; stop')
testvecs = [];testindices = []
# print 'symmetry operations R of A\n'
for k in range(A.nops):
print
print k
op = array(A.symops[:,:,k])
print'symop R of A'; print trimSmall(op)
m = trimSmall(dot(dot(inv(A.vecs), A.symops[:,:,k]),A.vecs))
print'symop m'; print m
# print 'det(m)', det(m)
'Take eigenvectors in cartesian space'
[vals,vecs]=eig(op)
print 'eigen of m',vals
print 'eigenvecs are calculated in cartesian space'; print vecs
#transform to m space
for i in range(3): vecs[:,i] = dot(inv(A.vecs),vecs[:,i])
print 'eigenvecs in m space'; print vecs
print 'scaled to integers'
for i in range(3): vecs[:,i] = vecs[:,i]/abs(vecs[:,i])[nonzero(vecs[:,i])].min()
vecs = rint(vecs)
print vecs
eigendirs[:,:,k]= vecs
#find operations with nondegenerate real eigenvalues
print 'nonDegen', nonDegen(vals)
for i in nonDegen(vals):
if not matchDirection(transpose(vecs[:,i]),testvecs): #keep only unique directions
testvecs.append(vecs[:,i].real/abs(vecs[:,i])[nonzero(vecs[:,i])].min())
testindices.append([k,i])
#print; print oplist;
print 'testvecs'; print testvecs
#print testindices
MT = zeros((3,3),dtype = fprec)
if len(testvecs) == 0:
print 'No eigen directions'
[M,K.vecs] = unconstrainedSVsearch(B)
if det(K.vecs)==0:
sys.exit('Det(K) is zero after unconstrained search! Stop')
if not checksymmetry(K.vecs,B):
sys.exit('Symmetry missing in mesh! Stop')
# MT = unconstrainedmin(B.vecs)
if len(testvecs) == 1:
print 'Only 1 eigen direction'
#Choose this one and the other two in the plane perpendicular to this.
MT[:,0] = testvecs[0]
# print 'testindices',testindices
kop = testindices[0][0] #useful operator
ieigen = testindices[0][1] #index of the only eigendirection
op = array(A.symops[:,:,kop])
# print trimSmall(op)
# find one other direction in the plane perp to the eigendireation; either degenerate eigenvalue will do.
otherindices = nonzero(array([0,1,2])-ieigen)
print eigendirs[:,:,otherindices[0][0]]
MT[:,1] = eigendirs[:,:,kop][:,otherindices[0][0]]
#Make 3rd vector perp as possible to the other two
ur0 = dot(A.vecs,MT[:,0])/norm(dot(A.vecs,MT[:,0])) #unit vectors in real space
ur1 = dot(A.vecs,MT[:,1])/norm(dot(A.vecs,MT[:,1]))
ur2 = cross(ur0,ur1)
print ur0
print ur1
print ur2
print 'ur2 transformed to m space'; print dot(inv(A.vecs),ur2)
mvec = dot(inv(A.vecs),ur2) #transformed to M space, but real
mvec = rint(mvec/abs(mvec[nonzero(mvec)]).min()) #Scale so smallest comp is 1
MT[:,2] = mvec
print 'MT from single operator';print MT
print 'starting superlattice'; print dot(A.vecs,MT)
# Q2 = MT2mesh_three_ns(MT,B)
Q2 = MT2mesh(MT,B,A)
if checksymmetry(Q2,B):
SV = surfvol(Q2)
# print round(surfvol(Q2),4),round(orthdef(Q2),4),'SV of Q2,','OD'
K.vecs = Q2
else:
print'Q from single operator fails symmetry'
if len(testvecs) == 2:
print 'Only 2 eigen directions'
MT[:,0] = testvecs[0]
MT[:,1] = testvecs[1]
#Make 3rd vector perp as possible to the other two
ur0 = dot(A.vecs,MT[:,0])/norm(dot(A.vecs,MT[:,0])) #unit vector in real space
ur1 = dot(A.vecs,MT[:,1])/norm(dot(A.vecs,MT[:,1]))
ur2 = cross(ur0,ur1)
MT[:,2] = rint(dot(inv(A.vecs),ur2))
print 'MT from two eigen directions';print MT
# Q2 = MT2mesh_three_ns(MT,B)
Q2 = MT2mesh(MT,B)
if checksymmetry(Q2,B):
SV = surfvol(Q2)
print round(surfvol(Q2),4),round(orthdef(Q2),4),'SV of Q2,','OD'
K.vecs = Q2
else:
print'Q fails symmetry'
if len(testvecs) >= 3:
print 'MT from three eigen directions'
testvecstrials = [list(x) for x in combinations(testvecs,3)]
print testvecstrials
bestindex = -1
bestcost = 1000
for i,vecs in enumerate(testvecstrials):
print; print 'trial',i
print vecs
MT[:,0] = vecs[0]
MT[:,1] = vecs[1]
MT[:,2] = vecs[2]
print 'MT'; print MT
print 'det MT', det(MT)
if not areEqual(det(MT),0):
Q2 = MT2mesh(MT,B)
if checksymmetry(Q2,B):
Nscale =1*.8; Ncost = Nscale * abs((B.det/det(Q2))-B.Nmesh)/B.Nmesh
cost = surfvol(Q2)*(1+Ncost)
print cost
if cost<bestcost:
bestcost = cost;
bestindex = i;
K.vecs = Q2
print round(surfvol(Q2),4),round(orthdef(Q2),4),'SV of Q2,','OD'
else:
print'Q from trial %i fails symmetry' % i
print '___________ Best mesh ___________'
print 'trial', bestindex
if checksymmetry(K.vecs,B):
print K.vecs
K.det = abs(det(K.vecs))
print 'N of mesh', B.det/K.det
SV = surfvol(K.vecs)
print round(surfvol(K.vecs),4),round(orthdef(K.vecs),4),'SV of Q2,','OD'
else:
print'K mesh fails symmetry'
def bestmeshEigen(Blatt, Nmesh):
'''The kmesh can be related to the reciprocal lattice B by B = KM, where M is an integer 3x3 matrix
So K = B Inv(M) . Work in the inverse space of this problem, where we can work with M instead of Inv(M).
T(InvK) = T(InvB)T(M).
Define S = T(InvK), and the real lattice A = T(InvB). So S = A T(M) is a superlattice on the real lattice.
Minimization scheme
Nothing calls this routine'''
##############################################################
########################## Script ############################
M = zeros((3, 3), dtype=int)
S = zeros((3, 3), dtype=fprec)
B = lattice()
A = lattice()
K = lattice()
B.vecs = Blatt / 2 / pi #Don't use 2pi constants in RL here.
#############End BCT lattice
eps = 1.0e-6
B.det = det(B.vecs)
B.Nmesh = Nmesh
print 'B vectors'
print B.vecs #
#print 'B transpose'; print transpose(B.vecs)
print 'Det of B', B.det
print 'Orth Defect of B', orthdef(B.vecs)
print 'Surf/vol of B', surfvol(B.vecs)
[B.symops, B.nops] = getGroup(B.vecs)
print 'Number of symmetry operations', B.nops
eigendirs = zeros([3, 3, B.nops], dtype=int)
#print 'symmetry operations of B\n'
#for j in range(nopsB):
# print j
# op = array(symopsB[:,:,j])
# print op
#find real lattice
A.vecs = transpose(inv(B.vecs))
A.det = det(A.vecs)
A.Nmesh = Nmesh
print
print 'A vectors'
print A.vecs
print 'Det of A', A.det
print 'Orth Defect of A', orthdef(A.vecs)
print 'Surf/vol of A', surfvol(A.vecs)
[A.symops, A.nops] = getGroup(A.vecs)
if A.nops != B.nops:
sys.exit('Number of operations different for A and B; stop')
testvecs = []
testindices = []
# print 'symmetry operations R of A\n'
for k in range(A.nops):
print
print k
op = array(A.symops[:, :, k])
print 'symop R of A'
print trimSmall(op)
m = trimSmall(dot(dot(inv(A.vecs), A.symops[:, :, k]), A.vecs))
print 'symop m'
print m
# print 'det(m)', det(m)
'Take eigenvectors in cartesian space'
[vals, vecs] = eig(op)
print 'eigen of m', vals
print 'eigenvecs are calculated in cartesian space'
print vecs
#transform to m space
for i in range(3):
vecs[:, i] = dot(inv(A.vecs), vecs[:, i])
print 'eigenvecs in m space'
print vecs
print 'scaled to integers'
for i in range(3):
vecs[:,
i] = vecs[:, i] / abs(vecs[:, i])[nonzero(vecs[:, i])].min()
vecs = rint(vecs)
print vecs
eigendirs[:, :, k] = vecs
#find operations with nondegenerate real eigenvalues
print 'nonDegen', nonDegen(vals)
for i in nonDegen(vals):
if not matchDirection(transpose(vecs[:, i]),
testvecs): #keep only unique directions
testvecs.append(vecs[:, i].real /
abs(vecs[:, i])[nonzero(vecs[:, i])].min())
testindices.append([k, i])
#print; print oplist;
print 'testvecs'
print testvecs
#print testindices
MT = zeros((3, 3), dtype=fprec)
if len(testvecs) == 0:
print 'No eigen directions'
[M, K.vecs] = unconstrainedSVsearch(B)
if det(K.vecs) == 0:
sys.exit('Det(K) is zero after unconstrained search! Stop')
if not checksymmetry(K.vecs, B):
sys.exit('Symmetry missing in mesh! Stop')
# MT = unconstrainedmin(B.vecs)
if len(testvecs) == 1:
print 'Only 1 eigen direction'
#Choose this one and the other two in the plane perpendicular to this.
MT[:, 0] = testvecs[0]
# print 'testindices',testindices
kop = testindices[0][0] #useful operator
ieigen = testindices[0][1] #index of the only eigendirection
op = array(A.symops[:, :, kop])
# print trimSmall(op)
# find one other direction in the plane perp to the eigendireation; either degenerate eigenvalue will do.
otherindices = nonzero(array([0, 1, 2]) - ieigen)
print eigendirs[:, :, otherindices[0][0]]
MT[:, 1] = eigendirs[:, :, kop][:, otherindices[0][0]]
#Make 3rd vector perp as possible to the other two
ur0 = dot(A.vecs, MT[:, 0]) / norm(dot(
A.vecs, MT[:, 0])) #unit vectors in real space
ur1 = dot(A.vecs, MT[:, 1]) / norm(dot(A.vecs, MT[:, 1]))
ur2 = cross(ur0, ur1)
print ur0
print ur1
print ur2
print 'ur2 transformed to m space'
print dot(inv(A.vecs), ur2)
mvec = dot(inv(A.vecs), ur2) #transformed to M space, but real
mvec = rint(
mvec /
abs(mvec[nonzero(mvec)]).min()) #Scale so smallest comp is 1
MT[:, 2] = mvec
print 'MT from single operator'
print MT
print 'starting superlattice'
print dot(A.vecs, MT)
# Q2 = MT2mesh_three_ns(MT,B)
Q2 = MT2mesh(MT, B, A)
if checksymmetry(Q2, B):
SV = surfvol(Q2)
# print round(surfvol(Q2),4),round(orthdef(Q2),4),'SV of Q2,','OD'
K.vecs = Q2
else:
print 'Q from single operator fails symmetry'
if len(testvecs) == 2:
print 'Only 2 eigen directions'
MT[:, 0] = testvecs[0]
MT[:, 1] = testvecs[1]
#Make 3rd vector perp as possible to the other two
ur0 = dot(A.vecs, MT[:, 0]) / norm(dot(
A.vecs, MT[:, 0])) #unit vector in real space
ur1 = dot(A.vecs, MT[:, 1]) / norm(dot(A.vecs, MT[:, 1]))
ur2 = cross(ur0, ur1)
MT[:, 2] = rint(dot(inv(A.vecs), ur2))
print 'MT from two eigen directions'
print MT
# Q2 = MT2mesh_three_ns(MT,B)
Q2 = MT2mesh(MT, B)
if checksymmetry(Q2, B):
SV = surfvol(Q2)
print round(surfvol(Q2), 4), round(orthdef(Q2),
4), 'SV of Q2,', 'OD'
K.vecs = Q2
else:
print 'Q fails symmetry'
if len(testvecs) >= 3:
print 'MT from three eigen directions'
testvecstrials = [list(x) for x in combinations(testvecs, 3)]
print testvecstrials
bestindex = -1
bestcost = 1000
for i, vecs in enumerate(testvecstrials):
print
print 'trial', i
print vecs
MT[:, 0] = vecs[0]
MT[:, 1] = vecs[1]
MT[:, 2] = vecs[2]
print 'MT'
print MT
print 'det MT', det(MT)
if not areEqual(det(MT), 0):
Q2 = MT2mesh(MT, B)
if checksymmetry(Q2, B):
Nscale = 1 * .8
Ncost = Nscale * abs((B.det / det(Q2)) - B.Nmesh) / B.Nmesh
cost = surfvol(Q2) * (1 + Ncost)
print cost
if cost < bestcost:
bestcost = cost
bestindex = i
K.vecs = Q2
print round(surfvol(Q2), 4), round(orthdef(Q2),
4), 'SV of Q2,', 'OD'
else:
print 'Q from trial %i fails symmetry' % i
print '___________ Best mesh ___________'
print 'trial', bestindex
if checksymmetry(K.vecs, B):
print K.vecs
K.det = abs(det(K.vecs))
print 'N of mesh', B.det / K.det
SV = surfvol(K.vecs)
print round(surfvol(K.vecs), 4), round(orthdef(K.vecs),
4), 'SV of Q2,', 'OD'
else:
print 'K mesh fails symmetry'
print trimSmall(op)
m = trimSmall(dot(dot(inv(A.vecs[:,:]), A.symops[:,:,k]),A.vecs[:,:]) )
[vals,vecs]=eig(m); vecs = array(vecs)
print 'symop m'; print m
print 'det(m)', det(m)
print 'eigen of m',vals
print vecs
print 'as rows, scaled to integers'
print vecs[:,0]/abs(vecs[:,0])[nonzero(vecs[:,0])].min()
print vecs[:,1]/abs(vecs[:,1])[nonzero(vecs[:,1])].min()
print vecs[:,2]/abs(vecs[:,2])[nonzero(vecs[:,2])].min()
#find operations with nondegenerate real eigenvalues
# print 'nonDegen', nonDegen(vals)
for i in nonDegen(vals):
if not matchDirection(transpose(vecs[:,i]),testvecs): #keep only unique directions
testvecs.append(vecs[:,i].real/abs(vecs[:,i])[nonzero(vecs[:,i])].min())
testindices.append([k,i])
#print; print oplist;
#print testvecs
#print testindices
MT = zeros((3,3),dtype = int)
if len(testvecs) == 0:
print 'No eigen directions'
K.vecs = unconstrainedSVsearch(B)
if det(K.vecs)==0:
sys.exit('Det(K) is zero after unconstrained search! Stop')
if not checksymmetry(K.vecs,B):
sys.exit('Symmetry missing in mesh! Stop')