In this section we

1
In this section well consider space groups Pm,
Cm, Pc and Cc. All four belong to crystal class
m. Glide planes (more details to follow) involve
a reflection ? to the plane followed by a
specified translation. In the monoclinic system
with b as the unique axis, the possible glide
planes are "a", "c" and "n". The "a" and "c"
glide planes have translational components of ½
along the a and c directions, respectively, while
an "n" glide plane has a translation along the ac
diagonal, viz., (½, 0, ½).
2
Now we will build Pm, space group number 6.
Well put in an m ? to b, thus in the plane of
the page, and again use that m and the
translation operators to derive the complete
mathematical group. Well use as the
symbol for an m in the page.
Again, we can examine the effect of mis-placing
the mirror if we place it in the ab or bc planes


And if this array is reflected through the mirror
shown, the result is incompatible with the
translation repetition of the monoclinic lattice.
3

• Since the mirror plane is in the plane of the
  screen (viz., the ac plane), we need a device or
  trick to remind ourselves that the reflected
  molecule
• is directly behind the molecule at (x,y,z)
• is of opposite chirality to the molecule at
  (x,y,z)
• Well do this by taking our original

After applying the mirror operation, the new
molecule appears behind the old one, and we
indicate this using a vertical bar (splitting the
original group), a comma, and a - to show
chirality and position
4
,
-
Z2
Is this space group enantiomorphous or
non-enantiomorphous?
Non-enantiomorphous
5
Z2
Is this space group centrosymmetric or
non-centrosymmetric?
Non-centrosymmetric
6
The two independent mirror planes at y 0 and ½
are the two special positions for Pm with Msp 1
and coordinates
Msp Element Coordinates
Mirror planes gen'd at intervals of b/2
1 m (x, 0, z) (x, ½, z)
Note that (x, 0, z) represents a plane rather
than a point!
7
Hopefully you have discerned that a mirror plane
is generated at y ½. This is hard to see at
this stage we need a lesson to make ourselves
aware of it! Lets do a quick derivation of Pm
again, but this time well use an ab projection
And, now we see the mirror at y ½. Often it is
very useful to redraw a space group in a more
user-friendly orientation!
8
How could we have found the mirror plane at y ½
only from the Pm diagram in the ac plane?
One unit cell above the molecules in the top left
corner of the ac diagram lies another pair at (x,
1y, z) and (x, 1-y, z). The location of the
mirror planes is easy to deduce by averaging the
coordinates of the corresponding pairs of
molecules. Thus, (x, y, z) and (x, -y, z) are
related by a mirror plane at y (-y) / 2
0, and (x, y, z) and (x, 1-y, z) are related by a
mirror plane at y1-y / 2 ½ !!!
9
Next well build space group Pc, number 7. The
c-glide plane is placed in the ac plane, ? b,
just as we did with the mirror plane in the case
of space group Pm. A c-glide plane, like the 21
screw axis, is a two-step operation. In the
first step, the atom or molecule is reflected ?
b, and then is translated in the c-direction by
c/2. (This operation, also like the 21, is
commutativethe order in which we do the two
steps makes no difference to the final
result). Unlike screw axes, which never change
the chirality of the molecule under
consideration, glide planes always produce a
molecule of opposite chirality.
10
Let's look at the glide operation in a
perspective view. The overall result is shown
above. The unfilled atom is first reflected
through a mirror in the ac plane, followed by a
translation of c/2. Click ahead to see the
stepwise process.
11
First, the atom is reflected through the ac
plane.
12
Then, the molecule is translated by c/2.
13
Finally, note that the reflected molecule is of
opposite chirality.
We can now attempt our derivation of space group
Pc.
14
Z2
Is this space group enantiomorphous or
non-enantiomorphous?
Non-enantiomorphous
15
a



c

Z2
Is this space group centrosymmetric or
non-centrosymmetric?
Non-centrosymmetric
16
It should be clear, once again, that space group
symmetry elements which combine translations with
rotation or reflection will not be special
positions in the unit cell. A point
corresponding to a location on a glide plane will
not be mapped onto itself by any further
operation of the group (cf. slide 16 in the first
Chapter!). Thus, in Pc, if we place an atom or
molecule on the glide plane at (x, 0, z), another
is generated at (x, 0, ½z). No symmetry
restrictions are placed on the molecule, and it
does not have a reduced multiplicity. The
characteristics of the point (x, 0, z) are no
different from those of the point (x, y, z).
17
Recall again that we can obtain the crystal
class by removing the translations from the
symbol thus a c glide plane becomes an m. To get
the crystal class for any space group, we remove
the translational symmetry, and ignore the
lattice type. Thus, space groups Pm and Cm
belong to crystal class m, as do space groups Pc
and Cc.
As we move forward, recall that the C-centering
operation involves a (½, ½, 0) translation
applied to all primitive lattice points.
18
To construct a diagram for Cm, well start by
regenerating our previous Pm diagram, and then
add the molecules generated by the C centering
operation. At the end of the process, what extra
symmetry will be generated?look for it just
after the last molecule is added.
Erwin Félix Lewy-Bertaut
Sue N. Smart
19
a


c

Z4
Is this space group enantiomorphous or
non-enantiomorphous?
Non-enantiomorphous
20
This is very nice! There is indeed an a glide
plane generated. Hmmmif I think about this
intuitively, I could say that the C centering
interacts with the mirror plane at (x, 0, z) to
generate another mirror at (½x, ½, z)but an a
glide plane at y ¼ would also do exactly that!
Sue N. Smart
21
a


c

Z4
Is this space group centrosymmetric or
non-centrosymmetric?
Non-centrosymmetric
22
a


c

Z4
Msp Element Coordinate
2 m (x, 0, z)
23
Aha, this will generate an n glide plane at y
¼. Ill have to watch for that!
A. Genius
24
a


c

Z4
Special Positions none
25
End of Section 3, Pointgroup m