-    FARRINGTONITE     -    Mg3PO4

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:  14  P2_1/n 
Lattice parameters (Å):  5.0800  5.0800  11.7000 
Angles (°):  90  86.48  90 

Symmetry (theoretical): 

Space group:  14  P2_1/n 
Lattice parameters (Å):  8.2073  7.6488  5.0534 
Angles (°):  136.35  136.35  63.43 

Cell contents: 

Number of atoms:  26 
Number of atom types:  3 
Chemical composition:  0 

Atomic positions (theoretical):

Mg:  0.6422  0.0998  0.1082 
Mg:  0.0000  0.0000  0.0000 
P:  0.3031  0.3009  0.0257 
O:  0.1311  0.2438  0.9571 
O:  0.3079  0.3750  0.3063 
O:  0.3487  0.4404  0.8162 
O:  0.4213  0.1448  0.9952 
Mg:  0.8578  0.5998  0.3918 
Mg:  0.5000  0.5000  0.5000 
P:  0.1969  0.8009  0.4743 
O:  0.3689  0.7438  0.5429 
O:  0.1921  0.8750  0.1937 
O:  0.1513  0.9404  0.6838 
O:  0.0787  0.6448  0.5048 
Mg:  0.3578  0.9002  0.8918 
P:  0.6969  0.6991  0.9743 
O:  0.8689  0.7562  0.0429 
O:  0.6921  0.6250  0.6937 
O:  0.6513  0.5596  0.1838 
O:  0.5787  0.8552  0.0048 
Mg:  0.1422  0.4002  0.6082 
P:  0.8031  0.1991  0.5257 
O:  0.6311  0.2562  0.4571 
O:  0.8079  0.1250  0.8063 
O:  0.8487  0.0596  0.3162 
O:  0.9213  0.3552  0.4952 
Atom type  X  Y  Z 

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): 

Mg: 1.9994 -0.0203 0.1767 
0.0214 1.9408 0.1404 
0.0551 0.1371 2.0515 
Eig. Value: 1.9766 1.8193 2.1958 
Mg: 2.2709 0.2165 -0.0960 
-0.2165 2.1443 0.0505 
0.0472 -0.1018 2.1579 
Eig. Value: 2.2762 2.1230 2.1740 
P: 3.1073 -0.0913 0.0823 
0.0013 3.2183 0.0620 
0.1336 -0.0283 3.4225 
Eig. Value: 3.0594 3.2328 3.4559 
O: -1.8988 -0.3489 -0.3015 
-0.3606 -1.3708 0.0088 
-0.2847 0.0310 -1.3328 
Eig. Value: -2.1569 -1.0766 -1.3689 
O: -1.2677 -0.0177 0.0354 
-0.0687 -1.2707 -0.5129 
-0.0023 -0.5094 -2.2987 
Eig. Value: -1.2775 -1.0500 -2.5095 
O: -1.2948 -0.1942 0.1478 
-0.2588 -1.5821 0.5744 
0.1352 0.5412 -1.8288 
Eig. Value: -1.2829 -1.0848 -2.3380 
O: -1.7808 0.5461 -0.0927 
0.5367 -2.0077 -0.1012 
-0.0606 -0.0394 -1.0927 
Eig. Value: -1.3788 -2.4474 -1.0550 
Mg: 1.9994 0.0203 0.1767 
-0.0214 1.9408 -0.1404 
0.0551 -0.1371 2.0515 
Eig. Value: 1.9766 1.8193 2.1958 
Mg: 2.2709 -0.2165 -0.0960 
0.2165 2.1443 -0.0505 
0.0472 0.1018 2.1579 
Eig. Value: 2.2762 2.1230 2.1740 
P: 3.1073 0.0913 0.0823 
-0.0013 3.2183 -0.0620 
0.1336 0.0283 3.4225 
Eig. Value: 3.0594 3.2328 3.4559 
O: -1.8988 0.3489 -0.3015 
0.3606 -1.3708 -0.0088 
-0.2847 -0.0310 -1.3328 
Eig. Value: -2.1569 -1.0766 -1.3689 
O: -1.2677 0.0177 0.0354 
0.0687 -1.2707 0.5129 
-0.0023 0.5094 -2.2987 
Eig. Value: -1.2775 -1.0500 -2.5095 
O: -1.2948 0.1942 0.1478 
0.2588 -1.5821 -0.5744 
0.1352 -0.5412 -1.8288 
Eig. Value: -1.2829 -1.0848 -2.3380 
O: -1.7808 -0.5461 -0.0927 
-0.5367 -2.0077 0.1012 
-0.0606 0.0394 -1.0927 
Eig. Value: -1.3788 -2.4474 -1.0550 
Mg: 1.9994 -0.0203 0.1767 
0.0214 1.9408 0.1404 
0.0551 0.1371 2.0515 
Eig. Value: 1.9766 1.8193 2.1958 
P: 3.1073 -0.0913 0.0823 
0.0013 3.2183 0.0620 
0.1336 -0.0283 3.4225 
Eig. Value: 3.0594 3.2328 3.4559 
O: -1.8988 -0.3489 -0.3015 
-0.3606 -1.3708 0.0088 
-0.2847 0.0310 -1.3328 
Eig. Value: -2.1569 -1.0766 -1.3689 
O: -1.2677 -0.0177 0.0354 
-0.0687 -1.2707 -0.5129 
-0.0023 -0.5094 -2.2987 
Eig. Value: -1.2775 -1.0500 -2.5095 
O: -1.2948 -0.1942 0.1478 
-0.2588 -1.5821 0.5744 
0.1352 0.5412 -1.8288 
Eig. Value: -1.2829 -1.0848 -2.3380 
O: -1.7808 0.5461 -0.0927 
0.5367 -2.0077 -0.1012 
-0.0606 -0.0394 -1.0927 
Eig. Value: -1.3788 -2.4474 -1.0550 
Mg: 1.9994 0.0203 0.1767 
-0.0214 1.9408 -0.1404 
0.0551 -0.1371 2.0515 
Eig. Value: 1.9766 1.8193 2.1958 
P: 3.1073 0.0913 0.0823 
-0.0013 3.2183 -0.0620 
0.1336 0.0283 3.4225 
Eig. Value: 3.0594 3.2328 3.4559 
O: -1.8988 0.3489 -0.3015 
0.3606 -1.3708 -0.0088 
-0.2847 -0.0310 -1.3328 
Eig. Value: -2.1569 -1.0766 -1.3689 
O: -1.2677 0.0177 0.0354 
0.0687 -1.2707 0.5129 
-0.0023 0.5094 -2.2987 
Eig. Value: -1.2775 -1.0500 -2.5095 
O: -1.2948 0.1942 0.1478 
0.2588 -1.5821 -0.5744 
0.1352 -0.5412 -1.8288 
Eig. Value: -1.2829 -1.0848 -2.3380 
O: -1.7808 -0.5461 -0.0927 
-0.5367 -2.0077 0.1012 
-0.0606 0.0394 -1.0927 
Eig. Value: -1.3788 -2.4474 -1.0550 
Atom type  X  Y  Z 

Dielectric tensors: 

 X Y Z 
Ɛ∞: 2.4589 0.0000 0.0350 
0.0000 2.4719 0.0000 
0.0350 0.0000 2.4988 
Eig. Value: 2.4386 2.4719 2.5191 
Refractive index (N): 1.5681 0.0000 0.1870 
0.0000 1.5722 0.0000 
0.1870 0.0000 1.5808 
Eig. Value: 1.5616 1.5722 1.5872 
Ɛ0: 0.0000 0.0000 0.0000 
0.0000 0.0000 0.0000 
0.0000 0.0000 0.0000 
Eig. Value: 0.0000 0.0000 0.0000