PTM reference templates

This page lists the ideal reference templates used by the Polyhedral template matching modifier. The local lattice orientations computed by the modifier are rotations relative to these templates, and the elastic deformation gradients describe the affine transformation of these templates that best fits the actual neighbor vectors.

Each template consists of the central atom (index 0), located at the origin, followed by its nearest neighbors (1st shell) and, for some structure types, the neighbors of neighbors (2nd shell). The neighbor positions are given in Cartesian coordinates. Each template is scaled such that the mean distance of the neighbors from the central atom is 1. This is a slight departure from the templates in the published PTM paper, in that the central atom is not included in this calculation. Coordinate values below are rounded to six decimal places.

For the HCP, cubic diamond, hexagonal diamond, and graphene structures, atoms occupying different sites of the crystal lattice have local environments that are rotated or inverted with respect to each other. The PTM therefore uses several template variants for these structures, one per type of lattice site. The orientation computed for an atom always refers to the first variant, i.e., all atoms of a perfect single crystal receive the same orientation regardless of the lattice site they occupy.

FCC

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.707107	0.707107	0.000000
2	1st	0.000000	0.707107	0.707107
3	1st	0.707107	0.000000	0.707107
4	1st	-0.707107	-0.707107	0.000000
5	1st	0.000000	-0.707107	-0.707107
6	1st	-0.707107	0.000000	-0.707107
7	1st	-0.707107	0.707107	0.000000
8	1st	0.000000	-0.707107	0.707107
9	1st	-0.707107	0.000000	0.707107
10	1st	0.707107	-0.707107	0.000000
11	1st	0.000000	0.707107	-0.707107
12	1st	0.707107	0.000000	-0.707107

HCP

Variant 1:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.500000	-0.866025	0.000000
2	1st	-1.000000	0.000000	0.000000
3	1st	-0.500000	0.288675	-0.816497
4	1st	0.500000	0.288675	-0.816497
5	1st	0.000000	-0.577350	-0.816497
6	1st	-0.500000	0.866025	0.000000
7	1st	0.500000	0.866025	0.000000
8	1st	1.000000	0.000000	0.000000
9	1st	-0.500000	-0.866025	0.000000
10	1st	0.000000	-0.577350	0.816497
11	1st	0.500000	0.288675	0.816497
12	1st	-0.500000	0.288675	0.816497

Variant 2:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	1.000000	0.000000	0.000000
2	1st	-0.500000	-0.866025	0.000000
3	1st	-0.500000	-0.288675	-0.816497
4	1st	0.000000	0.577350	-0.816497
5	1st	0.500000	-0.288675	-0.816497
6	1st	-1.000000	0.000000	0.000000
7	1st	-0.500000	0.866025	0.000000
8	1st	0.500000	0.866025	0.000000
9	1st	0.500000	-0.866025	0.000000
10	1st	0.500000	-0.288675	0.816497
11	1st	0.000000	0.577350	0.816497
12	1st	-0.500000	-0.288675	0.816497

BCC

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.541452	0.541452	0.541452
2	1st	-0.541452	0.541452	0.541452
3	1st	0.541452	0.541452	-0.541452
4	1st	-0.541452	-0.541452	0.541452
5	1st	0.541452	-0.541452	0.541452
6	1st	-0.541452	0.541452	-0.541452
7	1st	-0.541452	-0.541452	-0.541452
8	1st	0.541452	-0.541452	-0.541452
9	2nd	1.082904	0.000000	0.000000
10	2nd	-1.082904	0.000000	0.000000
11	2nd	0.000000	1.082904	0.000000
12	2nd	0.000000	-1.082904	0.000000
13	2nd	0.000000	0.000000	1.082904
14	2nd	0.000000	0.000000	-1.082904

Icosahedral (ICO)

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.000000	0.000000	1.000000
2	1st	0.000000	0.000000	-1.000000
3	1st	-0.525731	0.723607	-0.447214
4	1st	0.525731	-0.723607	0.447214
5	1st	0.000000	-0.894427	-0.447214
6	1st	0.000000	0.894427	0.447214
7	1st	0.850651	-0.276393	-0.447214
8	1st	-0.850651	0.276393	0.447214
9	1st	-0.850651	-0.276393	-0.447214
10	1st	0.850651	0.276393	0.447214
11	1st	0.525731	0.723607	-0.447214
12	1st	-0.525731	-0.723607	0.447214

Simple cubic (SC)

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.000000	0.000000	-1.000000
2	1st	0.000000	0.000000	1.000000
3	1st	0.000000	-1.000000	0.000000
4	1st	0.000000	1.000000	0.000000
5	1st	-1.000000	0.000000	0.000000
6	1st	1.000000	0.000000	0.000000

Cubic diamond

Variant 1:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.391492	0.391492	0.391492
2	1st	0.391492	-0.391492	-0.391492
3	1st	-0.391492	-0.391492	0.391492
4	1st	-0.391492	0.391492	-0.391492
5	2nd	0.782983	0.782983	0.000000
6	2nd	0.000000	0.782983	0.782983
7	2nd	0.782983	0.000000	0.782983
8	2nd	0.000000	-0.782983	-0.782983
9	2nd	0.782983	-0.782983	0.000000
10	2nd	0.782983	0.000000	-0.782983
11	2nd	-0.782983	-0.782983	0.000000
12	2nd	0.000000	-0.782983	0.782983
13	2nd	-0.782983	0.000000	0.782983
14	2nd	-0.782983	0.000000	-0.782983
15	2nd	-0.782983	0.782983	0.000000
16	2nd	0.000000	0.782983	-0.782983

Variant 2:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.391492	-0.391492	0.391492
2	1st	0.391492	0.391492	-0.391492
3	1st	-0.391492	-0.391492	-0.391492
4	1st	-0.391492	0.391492	0.391492
5	2nd	0.782983	0.000000	0.782983
6	2nd	0.000000	-0.782983	0.782983
7	2nd	0.782983	-0.782983	0.000000
8	2nd	0.000000	0.782983	-0.782983
9	2nd	0.782983	0.000000	-0.782983
10	2nd	0.782983	0.782983	0.000000
11	2nd	-0.782983	0.000000	-0.782983
12	2nd	0.000000	-0.782983	-0.782983
13	2nd	-0.782983	-0.782983	0.000000
14	2nd	-0.782983	0.782983	0.000000
15	2nd	-0.782983	0.000000	0.782983
16	2nd	0.000000	0.782983	0.782983

Hexagonal diamond

Variant 1:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	-0.553653	0.319652	-0.226028
2	1st	0.000000	-0.639303	-0.226028
3	1st	0.553653	0.319652	-0.226028
4	1st	0.000000	0.000000	0.678083
5	2nd	-1.107306	0.000000	0.000000
6	2nd	-0.553653	0.319652	-0.904111
7	2nd	-0.553653	0.958955	0.000000
8	2nd	0.553653	-0.958955	0.000000
9	2nd	0.000000	-0.639303	-0.904111
10	2nd	-0.553653	-0.958955	0.000000
11	2nd	0.553653	0.319652	-0.904111
12	2nd	0.553653	0.958955	0.000000
13	2nd	1.107306	0.000000	0.000000
14	2nd	0.000000	-0.639303	0.904111
15	2nd	0.553653	0.319652	0.904111
16	2nd	-0.553653	0.319652	0.904111

Variant 2:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	-0.553653	-0.319652	-0.226028
2	1st	0.553653	-0.319652	-0.226028
3	1st	0.000000	0.639303	-0.226028
4	1st	0.000000	0.000000	0.678083
5	2nd	-0.553653	-0.958955	0.000000
6	2nd	-0.553653	-0.319652	-0.904111
7	2nd	-1.107306	0.000000	0.000000
8	2nd	1.107306	0.000000	0.000000
9	2nd	0.553653	-0.319652	-0.904111
10	2nd	0.553653	-0.958955	0.000000
11	2nd	0.000000	0.639303	-0.904111
12	2nd	-0.553653	0.958955	0.000000
13	2nd	0.553653	0.958955	0.000000
14	2nd	0.553653	-0.319652	0.904111
15	2nd	0.000000	0.639303	0.904111
16	2nd	-0.553653	-0.319652	0.904111

Variant 3:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.000000	-0.639303	0.226028
2	1st	-0.553653	0.319652	0.226028
3	1st	0.553653	0.319652	0.226028
4	1st	0.000000	0.000000	-0.678083
5	2nd	-0.553653	-0.958955	0.000000
6	2nd	0.000000	-0.639303	0.904111
7	2nd	0.553653	-0.958955	0.000000
8	2nd	-0.553653	0.958955	0.000000
9	2nd	-0.553653	0.319652	0.904111
10	2nd	-1.107306	0.000000	0.000000
11	2nd	0.553653	0.319652	0.904111
12	2nd	1.107306	0.000000	0.000000
13	2nd	0.553653	0.958955	0.000000
14	2nd	-0.553653	0.319652	-0.904111
15	2nd	0.553653	0.319652	-0.904111
16	2nd	0.000000	-0.639303	-0.904111

Variant 4:

Index	Neighbor shell	x	y	z
0	central atom	0.000000	0.000000	0.000000
1	1st	0.553653	-0.319652	0.226028
2	1st	-0.553653	-0.319652	0.226028
3	1st	0.000000	0.639303	0.226028
4	1st	0.000000	0.000000	-0.678083
5	2nd	0.553653	-0.958955	0.000000
6	2nd	0.553653	-0.319652	0.904111
7	2nd	1.107306	0.000000	0.000000
8	2nd	-1.107306	0.000000	0.000000
9	2nd	-0.553653	-0.319652	0.904111
10	2nd	-0.553653	-0.958955	0.000000
11	2nd	0.000000	0.639303	0.904111
12	2nd	0.553653	0.958955	0.000000
13	2nd	-0.553653	0.958955	0.000000
14	2nd	-0.553653	-0.319652	-0.904111
15	2nd	0.000000	0.639303	-0.904111
16	2nd	0.553653	-0.319652	-0.904111

Graphene

Variant 1:

Index	Neighbor shell	x	y
0	central atom	0.000000	0.000000
1	1st	0.000000	0.672028
2	1st	0.581993	-0.336014
3	1st	-0.581993	-0.336014
4	2nd	-0.581993	1.008042
5	2nd	0.581993	1.008042
6	2nd	1.163986	0.000000
7	2nd	0.581993	-1.008042
8	2nd	-0.581993	-1.008042
9	2nd	-1.163986	0.000000

Variant 2:

Index	Neighbor shell	x	y
0	central atom	0.000000	0.000000
1	1st	-0.581993	0.336014
2	1st	0.581993	0.336014
3	1st	0.000000	-0.672028
4	2nd	-1.163986	0.000000
5	2nd	-0.581993	1.008042
6	2nd	0.581993	1.008042
7	2nd	1.163986	0.000000
8	2nd	0.581993	-1.008042
9	2nd	-0.581993	-1.008042