- LANGBEINITE - K₂Mg₂(SO₄)₃

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:	198		P2_13
Lattice parameters (Å):	9.9190	9.9190		9.9190
Angles (°):	90	90		90

Symmetry (theoretical):

Space group:	198		P2_13
Lattice parameters (Å):	9.6450	9.6450		9.6450
Angles (°):	90	90		90

Cell contents:

Number of atoms:	76
Number of atom types:	4
Chemical composition:	0

Atomic positions (theoretical):

K:	0.0704	0.0704	0.0704
K:	0.3014	0.3014	0.3014
Mg:	0.5847	0.5847	0.5847
Mg:	0.8461	0.8461	0.8461
S:	0.6275	0.4693	0.2684
O:	0.6536	0.4947	0.4148
O:	0.7583	0.4921	0.1940
O:	0.5761	0.3267	0.2486
O:	0.5270	0.5670	0.2094
S:	0.2684	0.6275	0.4693
O:	0.4148	0.6536	0.4947
O:	0.1940	0.7583	0.4921
O:	0.2486	0.5761	0.3267
O:	0.2094	0.5270	0.5670
K:	0.4296	0.9296	0.5704
K:	0.1986	0.6986	0.8014
Mg:	0.9153	0.4153	0.0847
Mg:	0.6539	0.1539	0.3461
S:	0.8725	0.5307	0.7684
O:	0.8464	0.5053	0.9148
O:	0.7417	0.5079	0.6940
O:	0.9239	0.6733	0.7486
O:	0.9730	0.4330	0.7094
K:	0.9296	0.5704	0.4296
K:	0.6986	0.8014	0.1986
Mg:	0.4153	0.0847	0.9153
Mg:	0.1539	0.3461	0.6539
S:	0.3725	0.9693	0.2316
O:	0.3464	0.9947	0.0852
O:	0.2417	0.9921	0.3060
O:	0.4239	0.8267	0.2514
O:	0.4730	0.0670	0.2906
K:	0.5704	0.4296	0.9296
K:	0.8014	0.1986	0.6986
Mg:	0.0847	0.9153	0.4153
Mg:	0.3461	0.6539	0.1539
S:	0.1275	0.0307	0.7316
O:	0.1536	0.0053	0.5852
O:	0.2583	0.0079	0.8060
O:	0.0761	0.1733	0.7514
O:	0.0270	0.9330	0.7906
S:	0.7684	0.8725	0.5307
O:	0.9148	0.8464	0.5053
O:	0.6940	0.7417	0.5079
O:	0.7486	0.9239	0.6733
O:	0.7094	0.9730	0.4330
S:	0.2316	0.3725	0.9693
O:	0.0852	0.3464	0.9947
O:	0.3060	0.2417	0.9921
O:	0.2514	0.4239	0.8267
O:	0.2906	0.4730	0.0670
S:	0.7316	0.1275	0.0307
O:	0.5852	0.1536	0.0053
O:	0.8060	0.2583	0.0079
O:	0.7514	0.0761	0.1733
O:	0.7906	0.0270	0.9330
S:	0.4693	0.2684	0.6275
O:	0.4947	0.4148	0.6536
O:	0.4921	0.1940	0.7583
O:	0.3267	0.2486	0.5761
O:	0.5670	0.2094	0.5270
S:	0.5307	0.7684	0.8725
O:	0.5053	0.9148	0.8464
O:	0.5079	0.6940	0.7417
O:	0.6733	0.7486	0.9239
O:	0.4330	0.7094	0.9730
S:	0.9693	0.2316	0.3725
O:	0.9947	0.0852	0.3464
O:	0.9921	0.3060	0.2417
O:	0.8267	0.2514	0.4239
O:	0.0670	0.2906	0.4730
S:	0.0307	0.7316	0.1275
O:	0.0053	0.5852	0.1536
O:	0.0079	0.8060	0.2583
O:	0.1733	0.7514	0.0761
O:	0.9330	0.7906	0.0270

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):

K:	1.1277	-0.0109	-0.0139
	-0.0139	1.1277	-0.0109
	-0.0109	-0.0139	1.1277
Eig. Value:	1.1401	1.1030	1.1401
K:	1.1249	-0.0093	0.0189
	0.0189	1.1249	-0.0093
	-0.0093	0.0189	1.1249
Eig. Value:	1.1346	1.1201	1.1201
Mg:	2.2876	0.0079	0.2124
	0.2124	2.2876	0.0079
	0.0079	0.2124	2.2876
Eig. Value:	2.5079	2.1774	2.1774
Mg:	2.2786	0.0966	0.0073
	0.0073	2.2786	0.0966
	0.0966	0.0073	2.2786
Eig. Value:	2.3825	2.2266	2.2266
S:	3.4732	-0.3690	0.0003
	-0.1959	3.4327	0.0457
	-0.0135	-0.1018	3.4117
Eig. Value:	3.7368	3.1672	3.4136
O:	-1.0009	0.0394	-0.0642
	-0.0215	-1.0555	-0.3504
	-0.1016	-0.3447	-2.2139
Eig. Value:	-1.0136	-0.9420	-2.3147
O:	-1.9579	-0.0472	0.5792
	-0.0390	-1.0352	-0.0019
	0.6192	0.0567	-1.2902
Eig. Value:	-2.3120	-1.0332	-0.9380
O:	-1.0805	-0.2057	-0.0136
	-0.2297	-2.1270	-0.0368
	-0.0193	-0.0444	-0.9948
Eig. Value:	-1.0385	-2.1721	-0.9918
O:	-1.7235	0.7107	-0.4051
	0.6713	-1.4994	0.3539
	-0.3722	0.3367	-1.1574
Eig. Value:	-2.5111	-0.9092	-0.9601
S:	3.4117	-0.0135	-0.1018
	0.0003	3.4732	-0.3690
	0.0457	-0.1959	3.4327
Eig. Value:	3.4136	3.7368	3.1672
O:	-2.2139	-0.1016	-0.3447
	-0.0642	-1.0009	0.0394
	-0.3504	-0.0215	-1.0555
Eig. Value:	-2.3147	-1.0136	-0.9420
O:	-1.2902	0.6192	0.0567
	0.5792	-1.9579	-0.0472
	-0.0019	-0.0390	-1.0352
Eig. Value:	-0.9380	-2.3120	-1.0332
O:	-0.9948	-0.0193	-0.0444
	-0.0136	-1.0805	-0.2057
	-0.0368	-0.2297	-2.1270
Eig. Value:	-0.9918	-1.0385	-2.1721
O:	-1.1574	-0.3722	0.3367
	-0.4051	-1.7235	0.7107
	0.3539	0.6713	-1.4994
Eig. Value:	-0.9601	-2.5111	-0.9092
K:	1.1277	-0.0109	0.0139
	-0.0139	1.1277	0.0109
	0.0109	0.0139	1.1277
Eig. Value:	1.1401	1.1030	1.1401
K:	1.1249	-0.0093	-0.0189
	0.0189	1.1249	0.0093
	0.0093	-0.0189	1.1249
Eig. Value:	1.1346	1.1201	1.1201
Mg:	2.2876	0.0079	-0.2124
	0.2124	2.2876	-0.0079
	-0.0079	-0.2124	2.2876
Eig. Value:	2.5079	2.1774	2.1774
Mg:	2.2786	0.0966	-0.0073
	0.0073	2.2786	-0.0966
	-0.0966	-0.0073	2.2786
Eig. Value:	2.3825	2.2266	2.2266
S:	3.4732	-0.3690	-0.0003
	-0.1959	3.4327	-0.0457
	0.0135	0.1018	3.4117
Eig. Value:	3.7368	3.1672	3.4136
O:	-1.0009	0.0394	0.0642
	-0.0215	-1.0555	0.3504
	0.1016	0.3447	-2.2139
Eig. Value:	-1.0136	-0.9420	-2.3147
O:	-1.9579	-0.0472	-0.5792
	-0.0390	-1.0352	0.0019
	-0.6192	-0.0567	-1.2902
Eig. Value:	-2.3120	-1.0332	-0.9380
O:	-1.0805	-0.2057	0.0136
	-0.2297	-2.1270	0.0368
	0.0193	0.0444	-0.9948
Eig. Value:	-1.0385	-2.1721	-0.9918
O:	-1.7235	0.7107	0.4051
	0.6713	-1.4994	-0.3539
	0.3722	-0.3367	-1.1574
Eig. Value:	-2.5111	-0.9092	-0.9601
K:	1.1277	0.0109	-0.0139
	0.0139	1.1277	0.0109
	-0.0109	0.0139	1.1277
Eig. Value:	1.1401	1.1030	1.1401
K:	1.1249	0.0093	0.0189
	-0.0189	1.1249	0.0093
	-0.0093	-0.0189	1.1249
Eig. Value:	1.1346	1.1201	1.1201
Mg:	2.2876	-0.0079	0.2124
	-0.2124	2.2876	-0.0079
	0.0079	-0.2124	2.2876
Eig. Value:	2.5079	2.1774	2.1774
Mg:	2.2786	-0.0966	0.0073
	-0.0073	2.2786	-0.0966
	0.0966	-0.0073	2.2786
Eig. Value:	2.3825	2.2266	2.2266
S:	3.4732	0.3690	0.0003
	0.1959	3.4327	-0.0457
	-0.0135	0.1018	3.4117
Eig. Value:	3.7368	3.1672	3.4136
O:	-1.0009	-0.0394	-0.0642
	0.0215	-1.0555	0.3504
	-0.1016	0.3447	-2.2139
Eig. Value:	-1.0136	-0.9420	-2.3147
O:	-1.9579	0.0472	0.5792
	0.0390	-1.0352	0.0019
	0.6192	-0.0567	-1.2902
Eig. Value:	-2.3120	-1.0332	-0.9380
O:	-1.0805	0.2057	-0.0136
	0.2297	-2.1270	0.0368
	-0.0193	0.0444	-0.9948
Eig. Value:	-1.0385	-2.1721	-0.9918
O:	-1.7235	-0.7107	-0.4051
	-0.6713	-1.4994	-0.3539
	-0.3722	-0.3367	-1.1574
Eig. Value:	-2.5111	-0.9092	-0.9601
K:	1.1277	0.0109	0.0139
	0.0139	1.1277	-0.0109
	0.0109	-0.0139	1.1277
Eig. Value:	1.1401	1.1030	1.1401
K:	1.1249	0.0093	-0.0189
	-0.0189	1.1249	-0.0093
	0.0093	0.0189	1.1249
Eig. Value:	1.1346	1.1201	1.1201
Mg:	2.2876	-0.0079	-0.2124
	-0.2124	2.2876	0.0079
	-0.0079	0.2124	2.2876
Eig. Value:	2.5079	2.1774	2.1774
Mg:	2.2786	-0.0966	-0.0073
	-0.0073	2.2786	0.0966
	-0.0966	0.0073	2.2786
Eig. Value:	2.3825	2.2266	2.2266
S:	3.4732	0.3690	-0.0003
	0.1959	3.4327	0.0457
	0.0135	-0.1018	3.4117
Eig. Value:	3.7368	3.1672	3.4136
O:	-1.0009	-0.0394	0.0642
	0.0215	-1.0555	-0.3504
	0.1016	-0.3447	-2.2139
Eig. Value:	-1.0136	-0.9420	-2.3147
O:	-1.9579	0.0472	-0.5792
	0.0390	-1.0352	-0.0019
	-0.6192	0.0567	-1.2902
Eig. Value:	-2.3120	-1.0332	-0.9380
O:	-1.0805	0.2057	0.0136
	0.2297	-2.1270	-0.0368
	0.0193	-0.0444	-0.9948
Eig. Value:	-1.0385	-2.1721	-0.9918
O:	-1.7235	-0.7107	0.4051
	-0.6713	-1.4994	0.3539
	0.3722	0.3367	-1.1574
Eig. Value:	-2.5111	-0.9092	-0.9601
S:	3.4117	0.0135	0.1018
	-0.0003	3.4732	-0.3690
	-0.0457	-0.1959	3.4327
Eig. Value:	3.4136	3.7368	3.1672
O:	-2.2139	0.1016	0.3447
	0.0642	-1.0009	0.0394
	0.3504	-0.0215	-1.0555
Eig. Value:	-2.3147	-1.0136	-0.9420
O:	-1.2902	-0.6192	-0.0567
	-0.5792	-1.9579	-0.0472
	0.0019	-0.0390	-1.0352
Eig. Value:	-0.9380	-2.3120	-1.0332
O:	-0.9948	0.0193	0.0444
	0.0136	-1.0805	-0.2057
	0.0368	-0.2297	-2.1270
Eig. Value:	-0.9918	-1.0385	-2.1721
O:	-1.1574	0.3722	-0.3367
	0.4051	-1.7235	0.7107
	-0.3539	0.6713	-1.4994
Eig. Value:	-0.9601	-2.5111	-0.9092
S:	3.4117	-0.0135	0.1018
	0.0003	3.4732	0.3690
	-0.0457	0.1959	3.4327
Eig. Value:	3.4136	3.7368	3.1672
O:	-2.2139	-0.1016	0.3447
	-0.0642	-1.0009	-0.0394
	0.3504	0.0215	-1.0555
Eig. Value:	-2.3147	-1.0136	-0.9420
O:	-1.2902	0.6192	-0.0567
	0.5792	-1.9579	0.0472
	0.0019	0.0390	-1.0352
Eig. Value:	-0.9380	-2.3120	-1.0332
O:	-0.9948	-0.0193	0.0444
	-0.0136	-1.0805	0.2057
	0.0368	0.2297	-2.1270
Eig. Value:	-0.9918	-1.0385	-2.1721
O:	-1.1574	-0.3722	-0.3367
	-0.4051	-1.7235	-0.7107
	-0.3539	-0.6713	-1.4994
Eig. Value:	-0.9601	-2.5111	-0.9092
S:	3.4117	0.0135	-0.1018
	-0.0003	3.4732	0.3690
	0.0457	0.1959	3.4327
Eig. Value:	3.4136	3.7368	3.1672
O:	-2.2139	0.1016	-0.3447
	0.0642	-1.0009	-0.0394
	-0.3504	0.0215	-1.0555
Eig. Value:	-2.3147	-1.0136	-0.9420
O:	-1.2902	-0.6192	0.0567
	-0.5792	-1.9579	0.0472
	-0.0019	0.0390	-1.0352
Eig. Value:	-0.9380	-2.3120	-1.0332
O:	-0.9948	0.0193	-0.0444
	0.0136	-1.0805	0.2057
	-0.0368	0.2297	-2.1270
Eig. Value:	-0.9918	-1.0385	-2.1721
O:	-1.1574	0.3722	0.3367
	0.4051	-1.7235	-0.7107
	0.3539	-0.6713	-1.4994
Eig. Value:	-0.9601	-2.5111	-0.9092
S:	3.4327	0.0457	-0.1959
	-0.1018	3.4117	-0.0135
	-0.3690	0.0003	3.4732
Eig. Value:	3.1672	3.4136	3.7368
O:	-1.0555	-0.3504	-0.0215
	-0.3447	-2.2139	-0.1016
	0.0394	-0.0642	-1.0009
Eig. Value:	-0.9420	-2.3147	-1.0136
O:	-1.0352	-0.0019	-0.0390
	0.0567	-1.2902	0.6192
	-0.0472	0.5792	-1.9579
Eig. Value:	-1.0332	-0.9380	-2.3120
O:	-2.1270	-0.0368	-0.2297
	-0.0444	-0.9948	-0.0193
	-0.2057	-0.0136	-1.0805
Eig. Value:	-2.1721	-0.9918	-1.0385
O:	-1.4994	0.3539	0.6713
	0.3367	-1.1574	-0.3722
	0.7107	-0.4051	-1.7235
Eig. Value:	-0.9092	-0.9601	-2.5111
S:	3.4327	-0.0457	-0.1959
	0.1018	3.4117	0.0135
	-0.3690	-0.0003	3.4732
Eig. Value:	3.1672	3.4136	3.7368
O:	-1.0555	0.3504	-0.0215
	0.3447	-2.2139	0.1016
	0.0394	0.0642	-1.0009
Eig. Value:	-0.9420	-2.3147	-1.0136
O:	-1.0352	0.0019	-0.0390
	-0.0567	-1.2902	-0.6192
	-0.0472	-0.5792	-1.9579
Eig. Value:	-1.0332	-0.9380	-2.3120
O:	-2.1270	0.0368	-0.2297
	0.0444	-0.9948	0.0193
	-0.2057	0.0136	-1.0805
Eig. Value:	-2.1721	-0.9918	-1.0385
O:	-1.4994	-0.3539	0.6713
	-0.3367	-1.1574	0.3722
	0.7107	0.4051	-1.7235
Eig. Value:	-0.9092	-0.9601	-2.5111
S:	3.4327	-0.0457	0.1959
	0.1018	3.4117	-0.0135
	0.3690	0.0003	3.4732
Eig. Value:	3.1672	3.4136	3.7368
O:	-1.0555	0.3504	0.0215
	0.3447	-2.2139	-0.1016
	-0.0394	-0.0642	-1.0009
Eig. Value:	-0.9420	-2.3147	-1.0136
O:	-1.0352	0.0019	0.0390
	-0.0567	-1.2902	0.6192
	0.0472	0.5792	-1.9579
Eig. Value:	-1.0332	-0.9380	-2.3120
O:	-2.1270	0.0368	0.2297
	0.0444	-0.9948	-0.0193
	0.2057	-0.0136	-1.0805
Eig. Value:	-2.1721	-0.9918	-1.0385
O:	-1.4994	-0.3539	-0.6713
	-0.3367	-1.1574	-0.3722
	-0.7107	-0.4051	-1.7235
Eig. Value:	-0.9092	-0.9601	-2.5111
S:	3.4327	0.0457	0.1959
	-0.1018	3.4117	0.0135
	0.3690	-0.0003	3.4732
Eig. Value:	3.1672	3.4136	3.7368
O:	-1.0555	-0.3504	0.0215
	-0.3447	-2.2139	0.1016
	-0.0394	0.0642	-1.0009
Eig. Value:	-0.9420	-2.3147	-1.0136
O:	-1.0352	-0.0019	0.0390
	0.0567	-1.2902	-0.6192
	0.0472	-0.5792	-1.9579
Eig. Value:	-1.0332	-0.9380	-2.3120
O:	-2.1270	-0.0368	0.2297
	-0.0444	-0.9948	0.0193
	0.2057	0.0136	-1.0805
Eig. Value:	-2.1721	-0.9918	-1.0385
O:	-1.4994	0.3539	-0.6713
	0.3367	-1.1574	0.3722
	-0.7107	0.4051	-1.7235
Eig. Value:	-0.9092	-0.9601	-2.5111