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--------------- p1 ----------------Consider the object R+.
No other symmetry equivalent object
exists in the unit cell.
Thus the equivalent positions are:
(x, y, z)
--------------- p-1 ----------------The origin has been chosen on 1 and the
object
has been located.
Apply the operator 1 to R+.
Apply the P operator in a and b
directions to the 1 operator and to the
pair of enantiomorph objects R+ RThe objects 1 and 8 (2 and 7) are related
by an inversion center at (½ 0 0).
The objects 1 and 4 (2 and 3) are related
by an inversion center at (0 ½ 0).
The objects 1 and 6 (2 and 5) are related
by an inversion center at (½ ½ 0).
Apply.
If we also apply the P operation in the c
direction we will obtain a total of 8
inversion centres.
(0 0 0) (½ 0 0) (0 ½ 0) (½ ½ 0)
(0 0 ½) (½ 0 ½) (0 ½ ½) (½ ½ ½)
In the cell we have two equivalent
positions:
(x, y, z) (-x, -y, -z)
The location of the symmetry element
can be seen on the next image.
--------------- p2 ----------------The origin has been chosen on the binary
axis (the b axis) and the object
has
been located.
Apply the operator 2 to .
Apply the operator P in the a and b
directions to the binary axis and to the
pair R+ R-.
The pair (1,3) is referred to the pair
(8,6) by the two-fold axis (½ y 0) .
The application of P also along the c
direction defines a total of 4 two-fold
axis:
(0 y 0) (½ y 0) (0 y ½) (½ y ½).
Apply.
The two equivalent positions in the unit
cell are:
(x, y, z)
(-x, y, -z)
The location of the symmetry elements
can be seen in the next image.
--------------- c2 ----------------The origin has been chosen on the binary
axis (the b axis) and the object
has
been located.
Apply the operator 2 to .
Apply the operator C in the a and b
directions to the binary axis and to the
pair R+ R-.
The object 1 is referred to object 10 by a
21 axis along b at ¼ of a.
Object 8 is referred to object 9 by a 21
axis along b at ¾ of a.
The periodicity along the direction c
defines a total of 4 two-fold axis:
(0 y 0) (½ y 0) (0 y ½) (½ y ½)
and 4 screw axis :
(¼ y 0) (¾ y 0) (¼ y ½) (¾ y ½)
Apply.
The four equivalent positions in the unit
cell are:
(x, y, z)
(-x, y, -z)
(x, y, z) + ( ½, ½, 0) = (x+½, y+½, z)
(-x, y, -z) + ( ½, ½, 0) = (-x+½, y+½, -z)
The location of the symmetry elements
is thus accomplished.
Note: the space group C2 coincides with
C21.
--------------- pca21 ----------------The origin has been chosen at the
intersection of the glide plane c normal
to a and of the glide plane a normal to b.
The object
has been located.
To apply the operator c click on Next.
Apply the operator a to the pair of
located objects.
Apply the operator P to the located
objects.
The symmetry elements are localized as
in the next figure.
For the space group diagram (objects
omitted) click on Next.
The International Tables prefer to
choose the origin on the screw axis.
To obtain the conventional diagram click
on Next.
--------------- p4 ----------------The origin has been chosen on the fourfold axis
(along the c axis) and the
object
has been located.
Apply the operator 4 to .
Apply the operator P in the a and b
directions to the four-fold axis and to the
four objects R+.
The objects 1, 2, 3, 4 are referred to the
objects 11, 12, 9, 10 respectively by the
binary axes (0 ½ z).
Apply also for the other pairs of objects.
The four equivalent positions in the unit
cell are:
(x, y, z) (-x, y, -z)
(-y, x, z) ( y, -x, z)
The location of the symmetry elements
can be seen in the next image.