Grid Quality Notebook



In [2]:

    
import numpy as np
from BZI.symmetry import make_ptvecs, make_rptvecs
from BZI.sampling import sphere_pts



In [5]:

    
# These lattice constants were calculated in Mathematica and 
# are such that the volumes are the same.
a_fcc = 1.
a_bcc = 0.793701
a_sc = 0.629961

fcc_consts = [a_fcc]*3
bcc_consts = [a_bcc]*3
sc_consts = [a_sc]*3

lat_angles = [np.pi/2]*3

fcc_lat_vecs = make_ptvecs("face", fcc_consts, lat_angles)
fcc_rlat_vecs = make_rptvecs(fcc_lat_vecs)
bcc_lat_vecs = make_ptvecs("body", bcc_consts, lat_angles)
bcc_rlat_vecs = make_rptvecs(bcc_lat_vecs)
sc_lat_vecs = make_ptvecs("prim", sc_consts, lat_angles)
sc_rlat_vecs = make_rptvecs(sc_lat_vecs)

Verify the volumes are the same.



In [6]:

    
vol_fcc = np.linalg.det(fcc_rlat_vecs)
vol_bcc = np.linalg.det(bcc_rlat_vecs)
vol_sc = np.linalg.det(sc_rlat_vecs)

print("Lattice Vectors","\nfcc:\n", fcc_rlat_vecs, "\nbcc:\n", bcc_rlat_vecs, "\nsc:\n", sc_rlat_vecs)
print("Volumes","\nfcc: ", vol_fcc, "\nbcc: ", vol_bcc, "\nsc: ", vol_sc)









    



Lattice Vectors 
fcc:
 [[-6.28318531  6.28318531  6.28318531]
 [ 6.28318531 -6.28318531  6.28318531]
 [ 6.28318531  6.28318531 -6.28318531]] 
bcc:
 [[ 0.         7.9163127  7.9163127]
 [ 7.9163127  0.         7.9163127]
 [ 7.9163127  7.9163127  0.       ]] 
sc:
 [[  9.97392744e+00   0.00000000e+00   0.00000000e+00]
 [ -6.10726916e-16   9.97392744e+00   0.00000000e+00]
 [ -6.10726916e-16  -6.10726916e-16   9.97392744e+00]]
Volumes 
fcc:  992.20085377 
bcc:  992.199076077 
sc:  992.19860912

Let's see what the volumes are in real space.



In [8]:

    
realvol_fcc = np.linalg.det(fcc_lat_vecs)
realvol_bcc = np.linalg.det(bcc_lat_vecs)
realvol_sc = np.linalg.det(sc_lat_vecs)

print("Reciprical Lattice Vectors","\nfcc:\n", fcc_lat_vecs, "\nbcc:\n", bcc_lat_vecs, "\nsc:\n", sc_lat_vecs)
print("Reciprical Space Volumes","\nfcc: ", realvol_fcc, "\nbcc: ", realvol_bcc, "\nsc: ", realvol_sc)









    



Reciprical Lattice Vectors 
fcc:
 [[  6.12323400e-17   5.00000000e-01   5.00000000e-01]
 [  5.00000000e-01   3.06161700e-17   5.00000000e-01]
 [  5.00000000e-01   5.00000000e-01   0.00000000e+00]] 
bcc:
 [[-0.3968505  0.3968505  0.3968505]
 [ 0.3968505 -0.3968505  0.3968505]
 [ 0.3968505  0.3968505 -0.3968505]] 
sc:
 [[  6.29961000e-01   3.85739861e-17   3.85739861e-17]
 [  0.00000000e+00   6.29961000e-01   3.85739861e-17]
 [  0.00000000e+00   0.00000000e+00   6.29961000e-01]]
Reciprical Space Volumes 
fcc:  0.25 
bcc:  0.250000447917 
sc:  0.250000565575

Apparently the volume of the supercell in real space is the same for all three. Next let's check the length of the vectors in real space.



In [9]:

    
fcc_rmin = np.linalg.norm(fcc_lat_vecs[:,0])
bcc_rmin = np.linalg.norm(bcc_lat_vecs[:,0])
sc_rmin = np.linalg.norm(sc_lat_vecs[:,0])

print("Cutoffs", "\nfcc: ", fcc_rmin, "\nbcc: ", bcc_rmin, "\nsc: ", sc_rmin)









    



Cutoffs 
fcc:  0.707106781187 
bcc:  0.687365229009 
sc:  0.629961

Face-centered cubic has the longest length cutoff or includes the most frequencies.



In [10]:

    
print("fcc/bcc: ", fcc_rmin/bcc_rmin*100 - 100)
print("fcc/sc: ", fcc_rmin/sc_rmin*100 - 100)
print("bcc/sc: ", bcc_rmin/sc_rmin*100 - 100)









    



fcc/bcc:  2.87206151028
fcc/sc:  12.2461201863
bcc/sc:  9.11234648004



In [11]:

    
from BZI.sampling import sphere_pts
from BZI.plots import PlotSphereMesh



In [12]:

    
fourier_sc = sphere_pts(sc_lat_vecs, sc_rmin, [0,0,0])
fourier_bcc = sphere_pts(bcc_lat_vecs, bcc_rmin, [0,0,0])
fourier_fcc = sphere_pts(fcc_lat_vecs, fcc_rmin, [0,0,0])

print("Number of Fourier terms sc: ", len(fourier_sc))
print("Number of Fourier terms bcc: ", len(fourier_bcc))
print("Number of Fourier terms fcc: ", len(fourier_fcc))









    



Number of Fourier terms sc:  7
Number of Fourier terms bcc:  15
Number of Fourier terms fcc:  13

An fcc grid would have the most Fourier terms for a set volume.

Let's calculate the average distance of these points. I think it would be more important to have more of the closer terms.



In [13]:

    
sc_average_length = np.sum([np.linalg.norm(pt) for pt in fourier_sc])/len(fourier_sc)
fcc_average_length = np.sum([np.linalg.norm(pt) for pt in fourier_fcc])/len(fourier_fcc)
bcc_average_length = np.sum([np.linalg.norm(pt) for pt in fourier_bcc])/len(fourier_bcc)
print("sc ", sc_average_length)
print("fcc ", fcc_average_length)
print("bcc ", bcc_average_length)









    



sc  0.539966571429
fcc  0.652713951865
bcc  0.684075188805