Comparison of crystal structures of the same symmetry C2/c ( No. 15 ) [ unique axis b ]

Structure #1

15
13.800 5.691 9.420 90.0 102.3 90.0
7
Pb  1 4e  0.000000  0.291000  0.250000
Pb  2 8f  0.317000  0.309000  0.352000
P 1 8f  0.599000  0.241000  0.447000
O 1 8f  0.643000  0.030000  0.392000
O 2 8f  0.634000  0.464000  0.374000
O 3 8f  0.642000  0.280000  0.612000
O 4 8f  0.491000  0.222000  0.420000

Structure #2

15
13.967 5.560 40.778 90.0 166.713 90.0
7
Pb  1 4e  0.000000  0.000000  0.750000
Pb  2 8f  0.000000  0.000000  0.856300
P 1 8f  0.000000  0.000000  0.951100
O 1 8f  0.000000  0.000000  0.914500
O 2 8f  0.271500  0.728500  0.888500
O 3 8f  0.957000  0.500000  0.117000
O 4 8f  0.728500  0.271500  0.611500

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

015
13.967000 5.560000 9.630055 90.000000 103.295059 90.000000 
7
Pb  1 4e  0.500000  0.250000  0.750000
Pb  2 8f  0.818900  0.250000  0.856300
P 1 8f  0.103300  0.250000  0.951100
O 1 8f  0.993500  0.250000  0.914500
O 2 8f  0.644000  0.521500  0.888500
O 3 8f  0.644000  0.750000  0.117000
O 4 8f  0.356000  0.978500  0.611500

Transformation matrix (P, p): -a,-b,3a+c ; 1/4,1/4,0

Matrix form:

(P, p) =

-1 0 3 1/4 0 -1 0 1/4 0 0 1 0

Atom pairings and distances

Atom Mappings
WP		Atom	Coordinates in S1	Atom	Coordinates in S2
4e	(0,y,1/4)	Pb1	(0.000000,0.291000,0.250000)	Pb1	(0.000000,0.250000,0.250000)
8f	(x,y,z)	Pb2	(0.317000,0.309000,0.352000)	Pb2	(0.318900,0.250000,0.356300)
8f	(x,y,z)	P1	(0.599000,0.241000,0.447000)	P1	(0.603300,0.250000,0.451100)
8f	(x,y,z)	O1	(0.643000,0.030000,0.392000)	O4	(0.644000,0.021500,0.388500)
8f	(x,y,z)	O2	(0.634000,0.464000,0.374000)	O2	(0.644000,0.478500,0.388500)
8f	(x,y,z)	O3	(0.642000,0.280000,0.612000)	O3	(0.644000,0.250000,0.617000)
8f	(x,y,z)	O4	(0.491000,0.222000,0.420000)	O1	(0.493500,0.250000,0.414500)

WP		Atom	Atomic Displacements
			ux	uy	uz	|u|
4e	(0,y,1/4)	Pb1	0.0000	-0.0410	0.0000	0.2333
8f	(x,y,z)	Pb2	0.0019	-0.0590	0.0043	0.3386
8f	(x,y,z)	P1	0.0043	0.0090	0.0041	0.0816
8f	(x,y,z)	O1	0.0010	-0.0085	-0.0035	0.0617
8f	(x,y,z)	O2	0.0100	0.0145	0.0145	0.1910
8f	(x,y,z)	O3	0.0020	-0.0300	0.0050	0.1777
8f	(x,y,z)	O4	0.0025	0.0280	-0.0055	0.1733

NOTE: u_x, u_y and u_z are given in relative units. |u| is the absolute distance given in Å

Evaluation of the structure similarity

S	dmax. (Å)	dav. (Å)	Δ
0.0116	0.3386	0.1755	0.066

• 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 0.0116.
    
    
  • The maximum distance (d_max.) shows the maximal displacement between the atomic positions of the paired atoms. The maximum distance (d_max.) in this case is: 0.3386 Å
    
    
• The arithmetic mean (d_av) of the distance. In this case, the arithmetic mean (d_av) is 0.1755 Å
  .
• 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 0.066.
  

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,3a+c ; 1/4,1/4,0

Matrix form:

(P, p)1 =

-1 0 3 1/4 0 -1 0 1/4 0 0 1 0


Structure 2 transformed by (P, p)₁

015
13.967000 5.560000 9.630055 90.000000 103.295059 90.000000 
7
Pb  1 4e  0.500000  0.250000  0.750000
Pb  2 8f  0.818900  0.250000  0.856300
P 1 8f  0.103300  0.250000  0.951100
O 1 8f  0.993500  0.250000  0.914500
O 2 8f  0.644000  0.521500  0.888500
O 3 8f  0.644000  0.750000  0.117000
O 4 8f  0.356000  0.978500  0.611500

The transformation matrix (P, p)₁ belongs to the affine normalizer of C2/c (No. 15) 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

Matrix form:

(P, p)2 =

1 0 0 0 0 1 0 0 0 0 1 0


Modified Structure 2 (step 1) transformed by (P, p)₂

015
13.967000 5.560000 9.630055 90.000000 103.295059 90.000000 
7
Pb  1 4e  0.500000  0.250000  0.750000
Pb  2 8f  0.818900  0.250000  0.856300
P 1 8f  0.103300  0.250000  0.951100
O 1 8f  0.993500  0.250000  0.914500
O 2 8f  0.644000  0.521500  0.888500
O 3 8f  0.644000  0.750000  0.117000
O 4 8f  0.356000  0.978500  0.611500

The transformation (P, p)₂ belongs to the Euclidean normalizer of C2/c (No. 15) 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 3 1/4 0 -1 0 1/4 0 0 1 0

=

-1 0 3 1/4 0 -1 0 1/4 0 0 1 0

1 0 0 0 0 1 0 0 0 0 1 0

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