-    WILLEMITE     -    Zn2SiO4

The crystal structure is fully relaxed (both unit cell parameters and atomic positions under symmetry constraints) starting from an experimental structure similar to the one reported in AMCSD

Crystal Structure 


Because of the translational symmetry all the calculations are performed in the primitive unit cell and not in the conventional unit cell. The following information regarding the structure is given with respect to this primitive unit cell, which sometimes can take an unintuitive shape.

Symmetry (experimental): 

Space group:  148  R-3 
Lattice parameters (Å):  4.7104  10.2718  8.7476 
Angles (°):  90  90  120 

Symmetry (theoretical): 

Space group:  148  R-3 
Lattice parameters (Å):  8.5641  8.5641  8.5641 
Angles (°):  107.88  107.88  107.88 

Cell contents: 

Number of atoms:  42 
Number of atom types: 
Chemical composition: 

Atomic positions (theoretical):

Zn:  0.3974  0.6252  0.2236 
Zn:  0.0585  0.2962  0.8895 
Si:  0.7333  0.9616  0.5540 
O:  0.8598  0.8563  0.5348 
O:  0.7476  0.0731  0.4331 
O:  0.8098  0.1014  0.7671 
O:  0.5262  0.8101  0.4754 
Zn:  0.6252  0.2236  0.3974 
Zn:  0.2962  0.8895  0.0585 
Si:  0.9616  0.5540  0.7333 
O:  0.8563  0.5348  0.8598 
O:  0.0731  0.4331  0.7476 
O:  0.1014  0.7671  0.8098 
O:  0.8101  0.4754  0.5262 
Zn:  0.2236  0.3974  0.6252 
Zn:  0.8895  0.0585  0.2962 
Si:  0.5540  0.7333  0.9616 
O:  0.5348  0.8598  0.8563 
O:  0.4331  0.7476  0.0731 
O:  0.7671  0.8098  0.1014 
O:  0.4754  0.5262  0.8101 
Zn:  0.6026  0.3748  0.7764 
Zn:  0.9415  0.7038  0.1105 
Si:  0.2667  0.0384  0.4460 
O:  0.1402  0.1437  0.4652 
O:  0.2524  0.9269  0.5669 
O:  0.1902  0.8986  0.2329 
O:  0.4738  0.1899  0.5246 
Zn:  0.3748  0.7764  0.6026 
Zn:  0.7038  0.1105  0.9415 
Si:  0.0384  0.4460  0.2667 
O:  0.1437  0.4652  0.1402 
O:  0.9269  0.5669  0.2524 
O:  0.8986  0.2329  0.1902 
O:  0.1899  0.5246  0.4738 
Zn:  0.7764  0.6026  0.3748 
Zn:  0.1105  0.9415  0.7038 
Si:  0.4460  0.2667  0.0384 
O:  0.4652  0.1402  0.1437 
O:  0.5669  0.2524  0.9269 
O:  0.2329  0.1902  0.8986 
O:  0.5246  0.4738  0.1899 
Atom type 

We have listed here the reduced coordinates of all the atoms in the primitive unit cell.
It is enough to know only the position of the atoms from the assymetrical unit cell and then use the symmetry to build the whole crystal structure.

Visualization of the crystal structure: 

Size:

Nx:  Ny:  Nz: 
You can define the size of the supercell to be displayed in the jmol panel as integer translations along the three crys­tallo­gra­phic axis.
Please note that the structure is represented using the pri­mi­tive cell, and not the conventional one.
   

Dielectric Properties 


We define:

  • The Born effective charges, also called dynamical charges, are tensors that correspond to the energy derivative with respect to atomic displacements and electric fields or, equivalently, to the change in atomic force due to an electric field: The sum of the Born effective charges of all nuclei in one cell must vanish, element by element, along each of the three directions of the space.
  • The dielectric tensors are the energy derivative with respect to two electric fields. They also relate the induced polarization to the external electric field.

Born effective charges (Z): 

Zn: 2.0804 0.0684 -0.0446 
0.1044 2.0518 0.0542 
-0.0173 0.0522 2.1845 
Eig. Value: 2.1515 1.9620 2.2032 
Zn: 2.1962 -0.0354 0.0586 
-0.0004 1.9717 -0.0351 
0.0250 -0.0345 2.1693 
Eig. Value: 2.2315 1.9653 2.1406 
Si: 3.2459 0.0530 -0.0133 
-0.0377 3.0390 -0.0050 
-0.0181 -0.0165 3.2336 
Eig. Value: 3.2573 3.0382 3.2229 
O: -2.6708 -0.0726 -0.0111 
-0.0771 -1.1248 -0.0312 
0.0059 -0.0196 -1.7886 
Eig. Value: -2.6745 -1.1203 -1.7895 
O: -1.9672 0.2225 0.0106 
0.2270 -2.3252 -0.0075 
0.0334 0.0179 -1.1828 
Eig. Value: -1.8596 -2.4336 -1.1821 
O: -1.3297 -0.4344 0.1956 
-0.4458 -1.9310 0.2892 
0.1739 0.3005 -2.3064 
Eig. Value: -1.0967 -1.8809 -2.5894 
O: -1.3640 -0.4643 -0.1507 
-0.4333 -1.8722 -0.2767 
-0.1404 -0.2938 -2.3097 
Eig. Value: -1.1020 -1.8903 -2.5535 
Zn: 1.9841 -0.0488 0.0692 
-0.0128 2.1481 0.0115 
0.0538 -0.0111 2.1845 
Eig. Value: 1.9620 2.1515 2.2032 
Zn: 2.0433 0.0886 -0.0597 
0.1237 2.1246 -0.0332 
-0.0424 -0.0044 2.1693 
Eig. Value: 1.9653 2.2315 2.1406 
Si: 3.0841 0.1311 0.0024 
0.0404 3.2008 0.0140 
-0.0052 0.0239 3.2336 
Eig. Value: 3.0382 3.2573 3.2229 
O: -1.4465 -0.6298 -0.0215 
-0.6343 -2.3491 0.0252 
-0.0199 0.0047 -1.7886 
Eig. Value: -1.1203 -2.6745 -1.7895 
O: -2.4304 0.0404 -0.0118 
0.0449 -1.8621 -0.0055 
-0.0012 -0.0379 -1.1828 
Eig. Value: -2.4336 -1.8596 -1.1821 
O: -1.3995 0.4861 0.1526 
0.4748 -1.8612 -0.3140 
0.1733 -0.3008 -2.3064 
Eig. Value: -1.0967 -1.8809 -2.5894 
O: -1.3565 0.4289 -0.1643 
0.4600 -1.8797 0.2689 
-0.1842 0.2685 -2.3097 
Eig. Value: -1.1020 -1.8903 -2.5535 
Zn: 2.1337 -0.0736 -0.0246 
-0.0376 1.9985 -0.0657 
-0.0365 -0.0411 2.1845 
Eig. Value: 2.1515 1.9620 2.2032 
Zn: 2.0123 -0.1058 0.0011 
-0.0707 2.1556 0.0683 
0.0174 0.0389 2.1693 
Eig. Value: 1.9653 2.2315 2.1406 
Si: 3.0973 -0.0480 0.0110 
-0.1388 3.1875 -0.0090 
0.0233 -0.0075 3.2336 
Eig. Value: 3.0382 3.2573 3.2229 
O: -1.5761 0.7091 0.0326 
0.7046 -2.2195 0.0060 
0.0140 0.0149 -1.7886 
Eig. Value: -1.1203 -2.6745 -1.7895 
O: -2.0411 -0.2696 0.0012 
-0.2651 -2.2514 0.0129 
-0.0322 0.0200 -1.1828 
Eig. Value: -1.8596 -2.4336 -1.1821 
O: -2.1618 -0.0346 -0.3482 
-0.0460 -1.0989 0.0248 
-0.3472 0.0003 -2.3064 
Eig. Value: -1.8809 -1.0967 -2.5894 
O: -2.1338 -0.0112 0.3150 
0.0199 -1.1023 0.0078 
0.3247 0.0253 -2.3097 
Eig. Value: -1.8903 -1.1020 -2.5535 
Zn: 2.0804 0.0684 -0.0446 
0.1044 2.0518 0.0542 
-0.0173 0.0522 2.1845 
Eig. Value: 2.1515 1.9620 2.2032 
Zn: 2.1962 -0.0354 0.0586 
-0.0004 1.9717 -0.0351 
0.0250 -0.0345 2.1693 
Eig. Value: 2.2315 1.9653 2.1406 
Si: 3.2459 0.0530 -0.0133 
-0.0377 3.0390 -0.0050 
-0.0181 -0.0165 3.2336 
Eig. Value: 3.2573 3.0382 3.2229 
O: -2.6708 -0.0726 -0.0111 
-0.0771 -1.1248 -0.0312 
0.0059 -0.0196 -1.7886 
Eig. Value: -2.6745 -1.1203 -1.7895 
O: -1.9672 0.2225 0.0106 
0.2270 -2.3252 -0.0075 
0.0334 0.0179 -1.1828 
Eig. Value: -1.8596 -2.4336 -1.1821 
O: -1.3297 -0.4344 0.1956 
-0.4458 -1.9310 0.2892 
0.1739 0.3005 -2.3064 
Eig. Value: -1.0967 -1.8809 -2.5894 
O: -1.3640 -0.4643 -0.1507 
-0.4333 -1.8722 -0.2767 
-0.1404 -0.2938 -2.3097 
Eig. Value: -1.1020 -1.8903 -2.5535 
Zn: 1.9841 -0.0488 0.0692 
-0.0128 2.1481 0.0115 
0.0538 -0.0111 2.1845 
Eig. Value: 1.9620 2.1515 2.2032 
Zn: 2.0433 0.0886 -0.0597 
0.1237 2.1246 -0.0332 
-0.0424 -0.0044 2.1693 
Eig. Value: 1.9653 2.2315 2.1406 
Si: 3.0841 0.1311 0.0024 
0.0404 3.2008 0.0140 
-0.0052 0.0239 3.2336 
Eig. Value: 3.0382 3.2573 3.2229 
O: -1.4465 -0.6298 -0.0215 
-0.6343 -2.3491 0.0252 
-0.0199 0.0047 -1.7886 
Eig. Value: -1.1203 -2.6745 -1.7895 
O: -2.4304 0.0404 -0.0118 
0.0449 -1.8621 -0.0055 
-0.0012 -0.0379 -1.1828 
Eig. Value: -2.4336 -1.8596 -1.1821 
O: -1.3995 0.4861 0.1526 
0.4748 -1.8612 -0.3140 
0.1733 -0.3008 -2.3064 
Eig. Value: -1.0967 -1.8809 -2.5894 
O: -1.3565 0.4289 -0.1643 
0.4600 -1.8797 0.2689 
-0.1842 0.2685 -2.3097 
Eig. Value: -1.1020 -1.8903 -2.5535 
Zn: 2.1337 -0.0736 -0.0246 
-0.0376 1.9985 -0.0657 
-0.0365 -0.0411 2.1845 
Eig. Value: 2.1515 1.9620 2.2032 
Zn: 2.0123 -0.1058 0.0011 
-0.0707 2.1556 0.0683 
0.0174 0.0389 2.1693 
Eig. Value: 1.9653 2.2315 2.1406 
Si: 3.0973 -0.0480 0.0110 
-0.1388 3.1875 -0.0090 
0.0233 -0.0075 3.2336 
Eig. Value: 3.0382 3.2573 3.2229 
O: -1.5761 0.7091 0.0326 
0.7046 -2.2195 0.0060 
0.0140 0.0149 -1.7886 
Eig. Value: -1.1203 -2.6745 -1.7895 
O: -2.0411 -0.2696 0.0012 
-0.2651 -2.2514 0.0129 
-0.0322 0.0200 -1.1828 
Eig. Value: -1.8596 -2.4336 -1.1821 
O: -2.1618 -0.0346 -0.3482 
-0.0460 -1.0989 0.0248 
-0.3472 0.0003 -2.3064 
Eig. Value: -1.8809 -1.0967 -2.5894 
O: -2.1338 -0.0112 0.3150 
0.0199 -1.1023 0.0078 
0.3247 0.0253 -2.3097 
Eig. Value: -1.8903 -1.1020 -2.5535 
Atom type 

Dielectric tensors: 

 
Ɛ3.1652 0.0000 0.0000 
0.0000 3.1652 0.0000 
0.0000 0.0000 3.2866 
Eig. Value: 3.1652 3.1652 3.2866 
Refractive index (N): 1.7791 0.0000 0.0000 
0.0000 1.7791 0.0000 
0.0000 0.0000 1.8129 
Eig. Value: 1.7791 1.7791 1.8129 
Ɛ00.0000 0.0000 0.0000 
0.0000 0.0000 0.0000 
0.0000 0.0000 0.0000 
Eig. Value: 0.0000 0.0000 0.0000