- APATITE-OH (ferroelectric) - Ca₅(PO₄)₃OH

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:	Exp 176, Theo 173		Exp P6_3/m, Theo P6_3
Lattice parameters (Å):	9.3940	9.3940		6.8967
Angles (°):	90	90		120

Symmetry (theoretical):

Space group:	Exp 176, Theo 173		Exp P6_3/m, Theo P6_3
Lattice parameters (Å):	9.3921	9.3921		6.8995
Angles (°):	90	90		120

Cell contents:

Number of atoms:	44
Number of atom types:	4
Chemical composition:	0

Atomic positions (theoretical):

Ca:	0.6667	0.3333	0.9957
Ca:	0.9927	0.2474	0.2461
P:	0.3659	0.3982	0.2472
O:	0.4807	0.3258	0.2487
O:	0.4642	0.5874	0.2417
O:	0.2527	0.3387	0.0695
O:	0.0000	0.0000	0.2841
Ca:	0.3333	0.6667	0.4957
Ca:	0.7453	0.9927	0.7461
P:	0.9677	0.3659	0.7472
O:	0.1549	0.4807	0.7487
O:	0.8768	0.4642	0.7417
O:	0.9140	0.2527	0.5695
O:	0.0000	0.0000	0.7841
Ca:	0.7526	0.7453	0.2461
P:	0.6018	0.9677	0.2472
O:	0.6742	0.1549	0.2487
O:	0.4126	0.8768	0.2417
O:	0.6613	0.9140	0.0695
Ca:	0.0073	0.7526	0.7461
P:	0.6341	0.6018	0.7472
O:	0.5193	0.6742	0.7487
O:	0.5358	0.4126	0.7417
O:	0.7473	0.6613	0.5695
Ca:	0.2547	0.0073	0.2461
P:	0.0323	0.6341	0.2472
O:	0.8451	0.5193	0.2487
O:	0.1232	0.5358	0.2417
O:	0.0860	0.7473	0.0695
Ca:	0.2474	0.2547	0.7461
P:	0.3982	0.0323	0.7472
O:	0.3258	0.8451	0.7487
O:	0.5874	0.1232	0.7417
O:	0.3387	0.0860	0.5695
Ca:	0.3333	0.6667	0.9950
O:	0.7427	0.6506	0.9284
Ca:	0.6667	0.3333	0.4950
O:	0.0921	0.7427	0.4284
O:	0.3494	0.0921	0.9284
O:	0.2573	0.3494	0.4284
O:	0.9079	0.2573	0.9284
O:	0.6506	0.9079	0.4284
H:	0.0000	0.0000	0.4254
H:	0.0000	0.0000	0.9254

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:

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):

Ca:	2.3064	0.0719	0.0000
	-0.0719	2.3064	-0.0000
	0.0000	0.0000	2.2898
Eig. Value:	2.3064	2.3064	2.2898
Ca:	2.3246	0.1021	0.0144
	0.1524	2.3241	-0.0296
	0.0167	-0.0265	2.2835
Eig. Value:	2.4521	2.1872	2.2930
P:	3.3544	-0.1913	-0.0072
	-0.1118	3.0828	0.0171
	-0.0037	0.0161	3.1700
Eig. Value:	3.4227	3.0140	3.1706
O:	-2.3912	0.4073	-0.0129
	0.4569	-1.4291	0.0022
	-0.0154	0.0009	-1.3551
Eig. Value:	-2.5569	-1.2634	-1.3551
O:	-1.3762	-0.0100	0.0033
	0.0118	-2.3497	0.0207
	-0.0089	0.0220	-1.2493
Eig. Value:	-1.3763	-2.3501	-1.2488
O:	-1.5512	-0.0931	-0.3198
	-0.1566	-1.3608	-0.3063
	-0.3497	-0.2757	-2.0578
Eig. Value:	-1.3869	-1.2553	-2.3277
O:	-1.6022	-0.0274	-0.0000
	0.0274	-1.6022	0.0000
	-0.0000	-0.0000	-1.0499
Eig. Value:	-1.6022	-1.6022	-1.0499
Ca:	2.3064	0.0719	-0.0000
	-0.0719	2.3064	0.0000
	0.0000	0.0000	2.2898
Eig. Value:	2.3064	2.3064	2.2898
Ca:	2.2140	-0.0886	0.0328
	-0.0383	2.4347	-0.0023
	0.0313	0.0012	2.2835
Eig. Value:	2.1872	2.4521	2.2930
P:	3.2819	0.1537	-0.0184
	0.2332	3.1553	0.0023
	-0.0158	0.0048	3.1700
Eig. Value:	3.4227	3.0140	3.1706
O:	-2.0439	-0.6575	-0.0083
	-0.6078	-1.7764	-0.0101
	-0.0085	-0.0129	-1.3551
Eig. Value:	-2.5569	-1.2634	-1.3551
O:	-2.1071	0.4102	-0.0162
	0.4320	-1.6188	0.0132
	-0.0235	0.0033	-1.2493
Eig. Value:	-2.3501	-1.3763	-1.2488
O:	-1.3003	0.0117	0.1053
	-0.0517	-1.6117	-0.4301
	0.0639	-0.4407	-2.0578
Eig. Value:	-1.2553	-1.3869	-2.3277
O:	-1.6022	-0.0274	0.0000
	0.0274	-1.6022	-0.0000
	-0.0000	-0.0000	-1.0499
Eig. Value:	-1.6022	-1.6022	-1.0499
Ca:	2.4344	-0.0890	0.0184
	-0.0386	2.2143	0.0273
	0.0146	0.0277	2.2835
Eig. Value:	2.4521	2.1872	2.2930
P:	3.0194	-0.0816	-0.0112
	-0.0021	3.4178	-0.0148
	-0.0121	-0.0113	3.1700
Eig. Value:	3.0140	3.4227	3.1706
O:	-1.2954	0.1757	0.0046
	0.2253	-2.5249	-0.0123
	0.0070	-0.0138	-1.3551
Eig. Value:	-1.2634	-2.5569	-1.3551
O:	-2.1055	-0.4329	-0.0196
	-0.4111	-1.6204	-0.0075
	-0.0146	-0.0187	-1.2493
Eig. Value:	-2.3501	-1.3763	-1.2488
O:	-1.5165	0.1766	0.4251
	0.1131	-1.3955	-0.1238
	0.4136	-0.1650	-2.0578
Eig. Value:	-1.2553	-1.3869	-2.3277
Ca:	2.3246	0.1021	-0.0144
	0.1524	2.3241	0.0296
	-0.0167	0.0265	2.2835
Eig. Value:	2.4521	2.1872	2.2930
P:	3.3544	-0.1913	0.0072
	-0.1118	3.0828	-0.0171
	0.0037	-0.0161	3.1700
Eig. Value:	3.4227	3.0140	3.1706
O:	-2.3912	0.4073	0.0129
	0.4569	-1.4291	-0.0022
	0.0154	-0.0009	-1.3551
Eig. Value:	-2.5569	-1.2634	-1.3551
O:	-1.3762	-0.0100	-0.0033
	0.0118	-2.3497	-0.0207
	0.0089	-0.0220	-1.2493
Eig. Value:	-1.3763	-2.3501	-1.2488
O:	-1.5512	-0.0931	0.3198
	-0.1566	-1.3608	0.3063
	0.3497	0.2757	-2.0578
Eig. Value:	-1.3869	-1.2553	-2.3277
Ca:	2.2140	-0.0886	-0.0328
	-0.0383	2.4347	0.0023
	-0.0313	-0.0012	2.2835
Eig. Value:	2.1872	2.4521	2.2930
P:	3.2819	0.1537	0.0184
	0.2332	3.1553	-0.0023
	0.0158	-0.0048	3.1700
Eig. Value:	3.4227	3.0140	3.1706
O:	-2.0439	-0.6575	0.0083
	-0.6078	-1.7764	0.0101
	0.0085	0.0129	-1.3551
Eig. Value:	-2.5569	-1.2634	-1.3551
O:	-2.1071	0.4102	0.0162
	0.4320	-1.6188	-0.0132
	0.0235	-0.0033	-1.2493
Eig. Value:	-2.3501	-1.3763	-1.2488
O:	-1.3003	0.0117	-0.1053
	-0.0517	-1.6117	0.4301
	-0.0639	0.4407	-2.0578
Eig. Value:	-1.2553	-1.3869	-2.3277
Ca:	2.4344	-0.0890	-0.0184
	-0.0386	2.2143	-0.0273
	-0.0146	-0.0277	2.2835
Eig. Value:	2.4521	2.1872	2.2930
P:	3.0194	-0.0816	0.0112
	-0.0021	3.4178	0.0148
	0.0121	0.0113	3.1700
Eig. Value:	3.0140	3.4227	3.1706
O:	-1.2954	0.1757	-0.0046
	0.2253	-2.5249	0.0123
	-0.0070	0.0138	-1.3551
Eig. Value:	-1.2634	-2.5569	-1.3551
O:	-2.1055	-0.4329	0.0196
	-0.4111	-1.6204	0.0075
	0.0146	0.0187	-1.2493
Eig. Value:	-2.3501	-1.3763	-1.2488
O:	-1.5165	0.1766	-0.4251
	0.1131	-1.3955	0.1238
	-0.4136	0.1650	-2.0578
Eig. Value:	-1.2553	-1.3869	-2.3277
Ca:	2.3031	0.0787	0.0000
	-0.0787	2.3031	-0.0000
	-0.0000	0.0000	2.3140
Eig. Value:	2.3031	2.3031	2.3140
O:	-1.5430	-0.0723	-0.3313
	-0.1230	-1.3494	-0.2588
	-0.3639	-0.2260	-2.0626
Eig. Value:	-1.3732	-1.2742	-2.3076
Ca:	2.3031	0.0787	-0.0000
	-0.0787	2.3031	0.0000
	0.0000	-0.0000	2.3140
Eig. Value:	2.3031	2.3031	2.3140
O:	-1.3133	-0.0097	0.0585
	-0.0603	-1.5791	-0.4163
	0.0138	-0.4282	-2.0626
Eig. Value:	-1.2742	-1.3732	-2.3076
O:	-1.4824	0.1579	0.3898
	0.1073	-1.4100	-0.1575
	0.3777	-0.2022	-2.0626
Eig. Value:	-1.2742	-1.3732	-2.3076
O:	-1.5430	-0.0723	0.3313
	-0.1230	-1.3494	0.2588
	0.3639	0.2260	-2.0626
Eig. Value:	-1.3732	-1.2742	-2.3076
O:	-1.3133	-0.0097	-0.0585
	-0.0603	-1.5791	0.4163
	-0.0138	0.4282	-2.0626
Eig. Value:	-1.2742	-1.3732	-2.3076
O:	-1.4824	0.1579	-0.3898
	0.1073	-1.4100	0.1575
	-0.3777	0.2022	-2.0626
Eig. Value:	-1.2742	-1.3732	-2.3076
H:	0.3899	0.0075	0.0000
	-0.0075	0.3899	-0.0000
	-0.0000	-0.0000	0.2599
Eig. Value:	0.3899	0.3899	0.2599
H:	0.3899	0.0075	-0.0000
	-0.0075	0.3899	0.0000
	-0.0000	-0.0000	0.2599
Eig. Value:	0.3899	0.3899	0.2599