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Comparison of crystal structures of the same symmetry P31m ( No. 157 )

Structure #1

157
8.7092 8.7092 5.2772 90 90 120
7
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 #2

157
8.6970 8.6970 4.9434 90 90 120
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

Description of Structure #2 in the most similar configuration to Structure #1

157
8.697000 8.697000 4.943400 90.000000 90.000000 120.000000 
7
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
WPAtomCoordinates in S1AtomCoordinates 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)


WPAtom
Structure1
Atom
Structure2
Atomic Displacements
uxuyuz|u|
3c(x,0,z)V2V2 0.0007 0.0007 -0.0053 0.0288
3c(x,0,z)O3O3 0.0007 0.0007 -0.0150 0.0795
3c(x,0,z)Rb1K1 -0.0058 0.0000 0.0096 0.0716
3c(x,0,z)O2O2 0.0042 0.0042 0.0165 0.0943
6d(x,y,z)O4O4 0.0004 -0.0006 -0.0084 0.0451
2b(1/3,2/3,z)V1V1 0.0000 -0.0000 -0.0008 0.0042
2b(1/3,2/3,z)O1O1 0.0000 -0.0000 0.0175 0.0922

NOTE: ux, uy and uz are given in relative units. |u| is the absolute distance given in +

Evaluation of the structure similarity

Sdmax. (+)dav. (+)Δ
0.02330.09430.05850.082

  • 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 configuration

The 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)2
  • Transformation 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)2
    157
    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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