XND lattice symmetry.

Rules and identification of lattice Symetry in xnd.

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We have choosen in xnd to read the symmetry operation rather than to generate the space group operation. This allows more flexibility in the symetry definition using by example exotic choice of the origin, introduction of non crystalline symmetry or changing the setting if the identity operation is not given.

3D Bravais lattices

This table recall the 3D Bravais lattice and their extinction rule. They are encoded in the definition on the symmetry read in xnd. This table is only recalled here as a basis for the 4D lattices.

Bravais extinction rule
P
A k + l = 2n
B h + l = 2n
C h + k = 2n
I h + k + l = 2n
F k + l = 2n, h + l = 2n, (h + k = 2n)
R h - k + l = 3n
S k - h + l = 3n
H h - kl = 3n

Bravais	extinction rule
P
A	k + l = 2n
B	h + l = 2n
C	h + k = 2n
I	h + k + l = 2n
F	k + l = 2n, h + l = 2n, (h + k = 2n)
R	h - k + l = 3n
S	k - h + l = 3n
H	h - kl = 3n

4D Bravais lattices

In this case the lattice depend not only of the 3D Bravais lattice but also on the component of the centering vector in the 4th dimension. To generate the 4D symmetry, xnd needs to known the rational components of the modulation vector (q_x, q_y, q_z) and the 3D Bravais lattice : this allow to generate the following extinction rules ...

q_x q_y q_z 3D lattice extinction rule listing code
0 0 0 all using 3D rules
1/2 0 0 P h + m = 2n A4P
A h + m = 2n, k + l = 2n A4A
0 1/2 0 P k + m = 2n B4P
B k + m = 2n, h + l = 2n B4B
0 0 1/2 P l + m = 2n C4P
C l + m = 2n, h + k = 2n C4C
0 1/2 1/2 orthorhombic k + m = 2n, l + m = 2n U4O
tetragonal k + l + m = 2n U4Q
1/2 0 1/2 orthorhombic h + m = 2n, l + m = 2n V4O
tetragonal h + l + m = 2n V4Q
1/2 1/2 0 orthorhombic h + m = 2n, k + m = 2n W4O
tetragonal h + k + m = 2n W4Q
1 0 0 P L4P
A k + l = 2n L4A *
B h + l + m = 2n L4B
C h + k + m = 2n L4C
I h + k + l + m = 2n L4I
F h + k + m = 2n, k + l = 2n L4F
0 1 0 P M4P
A k + l + m = 2n M4A
B h + l = 2n M4B *
C h + k + m = 2n M4C
I h + k + l + m = 2n M4I
F h + k + m = 2n, h + l = 2n M4F
0 0 1 P N4P
A k + l + m = 2n N4A
B h + l + m = 2n N4B
C h + k = 2n N4C *
I h + k + l + m = 2n N4I
F h + k = 2n, k + l + m = 2n N4F
1 1 0 P LM4P
A k + l + m = 2n LM4A
B h + l + m = 2n LM4B
C h + k = 2n LM4C +
I h + k + l = 2n LM4I +
F h + k = 2n, k + l + m = 2n LM4F
0 1 1 P MN4P
A k + l = 2n MN4A
B h + l + m = 2n MN4B
C h + k + m = 2n MN4C
I h + k + l + m = 2n MN4I
F h + k + m = 2n, k + l = 2n MN4F
1 0 1 P LN4P
A k + l = 2n LN4A
B h + l + m = 2n LN4B
C h + k + m = 2n LN4C
I h + k + l + m = 2n LN4I
F h + k + m = 2n, k + l = 2n LN4F
1/3 1/3 0 R h - k + m = 3n R4

q_x	q_y	q_z	3D lattice	extinction rule	listing code
0	0	0	all	using 3D rules
1/2	0	0	P	h + m = 2n	A4P
A	h + m = 2n, k + l = 2n	A4A
0	1/2	0	P	k + m = 2n	B4P
B	k + m = 2n, h + l = 2n	B4B
0	0	1/2	P	l + m = 2n	C4P
C	l + m = 2n, h + k = 2n	C4C
0	1/2	1/2	orthorhombic	k + m = 2n, l + m = 2n	U4O
tetragonal	k + l + m = 2n	U4Q
1/2	0	1/2	orthorhombic	h + m = 2n, l + m = 2n	V4O
tetragonal	h + l + m = 2n	V4Q
1/2	1/2	0	orthorhombic	h + m = 2n, k + m = 2n	W4O
tetragonal	h + k + m = 2n	W4Q
1	0	0	P		L4P
A	k + l = 2n	L4A	*
B	h + l + m = 2n	L4B
C	h + k + m = 2n	L4C
I	h + k + l + m = 2n	L4I
F	h + k + m = 2n, k + l = 2n	L4F
0	1	0	P		M4P
A	k + l + m = 2n	M4A
B	h + l = 2n	M4B	*
C	h + k + m = 2n	M4C
I	h + k + l + m = 2n	M4I
F	h + k + m = 2n, h + l = 2n	M4F
0	0	1	P		N4P
A	k + l + m = 2n	N4A
B	h + l + m = 2n	N4B
C	h + k = 2n	N4C	*
I	h + k + l + m = 2n	N4I
F	h + k = 2n, k + l + m = 2n	N4F
1	1	0	P		LM4P
A	k + l + m = 2n	LM4A
B	h + l + m = 2n	LM4B
C	h + k = 2n	LM4C	+
I	h + k + l = 2n	LM4I	+
F	h + k = 2n, k + l + m = 2n	LM4F
0	1	1	P		MN4P
A	k + l = 2n	MN4A
B	h + l + m = 2n	MN4B
C	h + k + m = 2n	MN4C
I	h + k + l + m = 2n	MN4I
F	h + k + m = 2n, k + l = 2n	MN4F
1	0	1	P		LN4P
A	k + l = 2n	LN4A
B	h + l + m = 2n	LN4B
C	h + k + m = 2n	LN4C
I	h + k + l + m = 2n	LN4I
F	h + k + m = 2n, k + l = 2n	LN4F
1/3	1/3	0	R	h - k + m = 3n	R4

As it can be seen in the upper table, the case of rational components (1/2,1/2,0) generates two kind of 4D extinction which are not related directly to the 3D Bravais lattice but to the symmetry of this lattice (tetragonal or orthorhombic). Some cases (*) have been generated for encoding reasons but are not associated with real crystallographic cases; the cases(+) are associated with non classical settings.
The systematic relation between the rational part of q and the 4D extinction setting was introduced in xnd by D. Grebille (ISMRA, Caen) and coll. as an efficient way to describe the extinction rules for 4D structure. This seems to have been a reasonable choice as no standard name remains for 4D Bravais lattice.

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