Comparison of crystal structures of the same symmetry P31m ( No. 157 )Structure #1157
8.7092 8.7092 5.2772 90 90 120
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Rb 1 3c 0.612170 0.000000 0.940070
V 1 2b 0.333333 0.666670 0.416870
V 2 3c 0.769550 0.769550 0.432700
O 1 2b 0.333333 0.666670 0.725700
O 2 3c 0.747400 0.747400 0.734000
O 3 3c 0.167400 0.167400 0.372700
O 4 6d 0.461000 0.169800 0.305400
Structure #2157
8.6970 8.6970 4.9434 90 90 120
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K 1 3c 0.393630 0.000000 0.454500
V 1 2b 0.333333 0.666670 0.920900
V 2 3c 0.229750 0.229750 0.932200
O 1 2b 0.333333 0.666670 0.248000
O 2 3c 0.248400 0.248400 0.255300
O 3 3c 0.831900 0.831900 0.862500
O 4 6d 0.461400 0.292200 0.801800
Description of Structure #2 in the most similar configuration to Structure #1157
8.697000 8.697000 4.943400 90.000000 90.000000 120.000000
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K 1 3c 0.606370 0.000000 0.949680
V 1 2b 0.666667 0.333330 0.416080
V 2 3c 0.770250 0.770250 0.427380
O 1 2b 0.666667 0.333330 0.743180
O 2 3c 0.751600 0.751600 0.750480
O 3 3c 0.168100 0.168100 0.357680
O 4 6d 0.538600 0.707800 0.296980
Transformation matrix ( P, p): -a,-b,c ; 0,0,0.50482 - Matrix form:
(P, p) = | | -1 0 0 0
0 -1 0 0
0 0 1 0.50482 | |
|
Atom pairings and distances
Atom Mappings |
---|
WP | Atom | Coordinates in S1 | Atom | Coordinates in S2 |
3c | (x,0,z) | V2 | (0.769550,0.769550,0.432700) | V2 | (0.770250,0.770250,0.427375) |
3c | (x,0,z) | O3 | (0.167400,0.167400,0.372700) | O3 | (0.168100,0.168100,0.357675) |
3c | (x,0,z) | Rb1 | (0.612170,0.000000,0.940070) | K1 | (0.606370,0.000000,0.949675) |
3c | (x,0,z) | O2 | (0.747400,0.747400,0.734000) | O2 | (0.751600,0.751600,0.750475) |
6d | (x,y,z) | O4 | (0.461000,0.169800,0.305400) | O4 | (0.461400,0.169200,0.296975) |
2b | (1/3,2/3,z) | V1 | (0.333333,0.666670,0.416870) | V1 | (0.333333,0.666667,0.416075) |
2b | (1/3,2/3,z) | O1 | (0.333333,0.666670,0.725700) | O1 | (0.333333,0.666667,0.743175) |
WP | Atom Structure1 | Atom Structure2 | Atomic Displacements |
---|
ux | uy | uz | |u| |
---|
3c | (x,0,z) | V2 | V2 | 0.0007 | 0.0007 | -0.0053 | 0.0288 | 3c | (x,0,z) | O3 | O3 | 0.0007 | 0.0007 | -0.0150 | 0.0795 | 3c | (x,0,z) | Rb1 | K1 | -0.0058 | 0.0000 | 0.0096 | 0.0716 | 3c | (x,0,z) | O2 | O2 | 0.0042 | 0.0042 | 0.0165 | 0.0943 | 6d | (x,y,z) | O4 | O4 | 0.0004 | -0.0006 | -0.0084 | 0.0451 | 2b | (1/3,2/3,z) | V1 | V1 | 0.0000 | -0.0000 | -0.0008 | 0.0042 | 2b | (1/3,2/3,z) | O1 | O1 | 0.0000 | -0.0000 | 0.0175 | 0.0922 |
NOTE: u x, u y and u z are given in relative units. |u| is the absolute distance given in +
Evaluation of the structure similarityS | dmax. (+) | dav. (+) | Δ | 0.0233 | 0.0943 | 0.0585 | 0.082 |
- Lattice and atomic position criteria:
- The degree of lattice distortion (S)
is the spontaneous strain (sum of the squared eigenvalues of the strain
tensor divided by 3). For the given two structures, the degree of lattice distortion (S) is 0.0233.
- The maximum distance (dmax.) shows the maximal displacement between the atomic positions of the paired atoms. The maximum distance (dmax.) in this case is: 0.0943 +
- The arithmetic mean (dav) of the distance. In this case, the arithmetic mean (dav) is 0.0585 +
. - The measure of similarity (Δ) (Bergerhoff et al.,
1998) is a function of the differences in atomic positions (weighted by
the multiplicities of the sites) and the ratios of the corresponding
lattice parameters of the structures. The measure of similarity (Δ) calculated for this case is 0.082.
The transformation of the Structure 2 to the most similar configurationThe transformation of the Structure 2 to the most similar configuration to Structure 1 is done in two steps:- Step 1: Transformation of structure 2 by the matrix (P, p)1
Transformation matrix (P, p)1: a,b,c ; 0,0,0- Matrix form:
(P, p)1 = | | 1 0 0 0
0 1 0 0
0 0 1 0 | |
|
Structure 2 transformed by (P, p)1
157
8.697000 8.697000 4.943400 90.000000 90.000000 120.000000
7
K 1 3c 0.393630 0.000000 0.454500
V 1 2b 0.333333 0.666670 0.920900
V 2 3c 0.229750 0.229750 0.932200
O 1 2b 0.333333 0.666670 0.248000
O 2 3c 0.248400 0.248400 0.255300
O 3 3c 0.831900 0.831900 0.862500
O 4 6d 0.461400 0.292200 0.801800
The transformation matrix (P, p)1 belongs to the affine normalizer of P31m (No. 157) and it is determined by the best fit of the lattice parameters of the Structure 2 to those of Structure 1.
Step 2: Transformation of the modified Structure 2 (step 1) by the transformation matrix (P, p)2Transformation matrix (P, p)2: -a,-b,c ; 0,0,0.50482- Matrix form:
(P, p)2 = | | -1 0 0 0
0 -1 0 0
0 0 1 0.50482 | |
|
Modified Structure 2 (step 1) transformed by (P, p)2157
8.697000 8.697000 4.943400 90.000000 90.000000 120.000000
7
K 1 3c 0.606370 0.000000 0.954500
V 2 3c 0.770250 0.770250 0.432200
V 3 3c 0.751600 0.751600 0.755300
O 4 3c 0.168100 0.168100 0.362500
O 1 2b 0.666667 0.333330 0.420900
O 2 2b 0.666667 0.333330 0.748000
O 1 6d 0.538600 0.707800 0.301800
The transformation (P, p)2 belongs to the Euclidean normalizer of P31m (No. 157) and it is determined by the best pairing of the atomic position of the two structures.
The (overall) transformation matrix (P, p) used for the transformation of Structure 2 to the most similar description to Structure 1 is equal to:
(P, p) = (P, p)1 (P, p)2
| -1 0 0 0
0 -1 0 0
0 0 1 0.50482 | |
| = | | 1 0 0 0
0 1 0 0
0 0 1 0 | |
| | -1 0 0 0
0 -1 0 0
0 0 1 0.50482 | |
|
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