- POTASSIUM CARBONATE - K₂CO₃

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:	12		C2/m
Lattice parameters (Å):	4.7123	2.7713		3.1989
Angles (°):	90	101.32		90

Symmetry (theoretical):

Space group:	12		C2/m
Lattice parameters (Å):	5.3119	5.3119		6.1685
Angles (°):	7.79946214E+01	1.02005379E+02		1.17440928E+02

Cell contents:

Number of atoms:	12
Number of atom types:	3
Chemical composition:	0

Atomic positions (theoretical):

K:	0.0000	0.0000	0.0000
K:	0.0000	0.0000	0.5000
K:	0.6661	0.3339	0.7272
C:	0.6650	0.3350	0.2521
O:	0.4082	0.1922	0.2991
O:	0.7784	0.2216	0.1635
K:	0.3339	0.6661	0.2728
C:	0.3350	0.6650	0.7479
O:	0.1922	0.4082	0.7009
O:	0.2216	0.7784	0.8365
O:	0.5918	0.8078	0.7009
O:	0.8078	0.5918	0.2991

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.

Parameters of the Calculation

All the calculations have been done using the ABINIT software. This is a list of the most representative parameteres used during the Raman calculation.

Number of electronic bands:	26
k-points
grid:	6 6 6
number of shifts:	1
shifts:	0.5 0.5 0.5
Kinetic energy cut-off:	40 Ha [=1088.464 eV ]
eXchange-Correlation functional:	LDA pw90

Pseudopotentials:
K:	potassium, fhi98PP : Trouiller-Martins-type, LDA Ceperley/Alder Perdew/Wang (1992), l= 0 local
C:	carbon, fhi98PP : Trouiller-Martins-type, LDA Ceperley/Alder Perdew/Wang (1992), l= 2 local
O:	oxygen, fhi98PP : Trouiller-Martins-type, LDA Ceperley/Alder Perdew/Wang (1992), l= 2 local

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:	1.1360	-0.0000	0.0034
	-0.0000	1.0945	-0.0000
	-0.0664	-0.0000	0.9918
Eig. Value:	1.1425	1.0945	0.9852
K:	1.1453	-0.0000	-0.0040
	0.0000	1.1384	-0.0000
	-0.0200	0.0000	0.9802
Eig. Value:	1.1462	1.1384	0.9793
K:	0.9605	-0.0000	-0.0420
	-0.0000	0.9635	-0.0000
	-0.0097	-0.0000	1.0669
Eig. Value:	0.9546	0.9635	1.0729
C:	2.2876	0.0000	-1.0524
	-0.0000	2.7187	-0.0000
	-1.0536	-0.0000	0.7454
Eig. Value:	2.8217	2.7187	0.2114
O:	-1.2249	-0.4710	0.2409
	-0.4858	-1.8959	0.3016
	0.2481	0.2221	-0.8817
Eig. Value:	-1.0148	-2.2333	-0.7543
O:	-1.9391	0.0000	0.6128
	0.0000	-1.0068	0.0000
	0.6103	0.0000	-1.0350
Eig. Value:	-2.2475	-1.0068	-0.7266
K:	0.9605	-0.0000	-0.0420
	-0.0000	0.9635	0.0000
	-0.0097	-0.0000	1.0669
Eig. Value:	0.9546	0.9635	1.0729
C:	2.2876	0.0000	-1.0524
	-0.0000	2.7187	-0.0000
	-1.0536	-0.0000	0.7454
Eig. Value:	2.8217	2.7187	0.2114
O:	-1.2249	0.4710	0.2409
	0.4858	-1.8959	-0.3016
	0.2481	-0.2221	-0.8817
Eig. Value:	-1.0148	-2.2333	-0.7543
O:	-1.9391	0.0000	0.6128
	0.0000	-1.0068	0.0000
	0.6103	0.0000	-1.0350
Eig. Value:	-2.2475	-1.0068	-0.7266
O:	-1.2249	-0.4710	0.2409
	-0.4858	-1.8959	0.3016
	0.2481	0.2221	-0.8817
Eig. Value:	-1.0148	-2.2333	-0.7543
O:	-1.2249	0.4710	0.2409
	0.4858	-1.8959	-0.3016
	0.2481	-0.2221	-0.8817
Eig. Value:	-1.0148	-2.2333	-0.7543

Atom type

Dielectric tensors:

	X	Y	Z
Ɛ_∞:	2.1971	0.0000	-0.1608
	0.0000	2.2684	0.0000
	-0.1608	0.0000	1.9607
Eig. Value:	2.2784	2.2684	1.8793
Refractive index (N):	1.4823	-0.0000	-0.4010
	-0.0000	1.5061	-0.0000
	-0.4010	-0.0000	1.4002
Eig. Value:	1.5094	1.5061	1.3709
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

Powder Raman

Powder Raman spectrum

The intensity of the Raman peaks is computed within the density-functional perturbation theory. The intensity depends on the temperature (for now fixed at 300K), frequency of the input laser (for now fixed at 21834 cm^-1, frequency of the phonon mode and the Raman tensor. The Raman tensor represents the derivative of the dielectric tensor during the atomic displacement that corresponds to the phonon vibration. The Raman tensor is related to the polarizability of a specific phonon mode.

Choose the polarization of the lasers.

I ∥

I ⊥

I Total

Horizontal:

Xmin:
Xmax:

Vertical:

Ymin:
Ymax:

Data about the phonon modes

Frequency of the transverse (TO) and longitudinal (LO) phonon modes in the zone-center. The longitudinal modes are computed along the three cartesian directions. You can visualize the atomic displacement pattern corresponding to each phonon by clicking on the appropriate cell in the table below.

1	ac	0	0	0	0
2	ac	0	0	0	0
3	ac	0	0	0	0
4	Au	90	90	92	90
5	Bg	93	93	93	93	7.092e+37 0.1	8.033e+37 0.1	1.513e+38 0.3
6	Au	96	96	114	96
7	Ag	131	131	131	131	2.079e+38 0.4	1.144e+38 0.2	3.222e+38 0.6
8	Bu	133	133	133	141
9	Ag	145	145	145	145	7.318e+38 1.3	8.065e+38 1.4	1.538e+39 2.7
10	Bu	146	146	146	157
11	Bg	157	157	157	157	6.407e+38 1.1	7.728e+38 1.4	1.414e+39 2.5
12	Au	171	171	179	171
13	Bu	181	185	181	181
14	Bg	196	196	196	196	2.658e+39 4.7	2.999e+39 5.3	5.657e+39 9.9
15	Ag	220	220	220	220	7.732e+38 1.4	7.062e+38 1.2	1.479e+39 2.6
16	Bg	226	226	226	226	6.694e+39 11.7	8.028e+39 14.1	1.472e+40 25.8
17	Bu	229	238	229	229
18	Ag	239	239	239	239	2.497e+39 4.4	2.337e+39 4.1	4.834e+39 8.5
19	Bu	251	255	251	255
20	Ag	255	259	255	257	4.487e+39 7.9	4.573e+39 8.0	9.059e+39 15.9
21	Au	259	262	259	259
22	Au	277	277	278	277
23	Bu	278	278	290	285
24	Bu	290	329	321	334
25	Bg	692	692	692	692	1.415e+39 2.5	2.372e+39 4.2	3.786e+39 6.6
26	Au	693	693	694	693
27	Bu	706	706	706	706
28	Bu	706	707	706	707	1.397e+39 2.4	1.136e+39 2.0	2.533e+39 4.4
29	Bu	866	867	866	872
30	Ag	872	872	872	873	1.419e+38 0.2	1.377e+38 0.2	2.795e+38 0.5
31	Ag	1101	1101	1101	1101	5.435e+40 95.2	2.737e+39 4.8	5.709e+40 100.0
32	Bu	1102	1102	1102	1102
33	Au	1444	1444	1446	1444
34	Bg	1446	1446	1449	1446	2.083e+39 3.6	3.496e+39 6.1	5.579e+39 9.8
35	Bu	1449	1468	1468	1468
36	Ag	1468	1543	1551	1477	1.564e+39 2.7	1.569e+39 2.7	3.133e+39 5.5

No.

Char.

ω TO

ω LOx

ω LOy

ω LOz

I ∥

I ⊥

I Total

You can define the size of the supercell for the visualization of the vibration.

Normalized

Raw

Options for intensity.

Single Crystal Raman spectra

Single crystal Raman spectrum

The Raman measurements performed on single crystals employ polarized lasers and allow for the selection of specific elements of the individual Raman tensors of the Raman-active modes.

By convention, in the following we assume a measurement as X(XZ)Z, i.e. incident laser polarized along the X axis, emergent light polarized along the Z axis. If the crystal is aligned with the xyz reference frame, we sample the α_xz element. As you rotate the crystal you can sample other entries of the Raman tensor or various linear combineations.