# This six 6 is responsible for the crystal coming under the hexagonal crystal system even though there is no usual 6 fold axis Q by HC12032002193

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```									                               Hexagonal Close Packed Crystal
 While SC, FCC, BCC are Lattices; HCP, DC (Diamond Cubic) are Crystals
 HCP Crystal = Hexagonal Lattice + two identical spheres as motif {at (0,0,0) and (2/3, 1/3, 1/2)}
 Note: there is only one type of Hexagonal Lattice  (Simple Hexagonal Lattice)
(If the spheres are of different size then the crystal is not HCP)
 Like CCP Crystal (CCP Crystal = FCC lattice + single sphere motif) HCP is not
just close packed but is closest packed! (Packing fraction of 0.74  74% of
space is filled with spheres)

Points to be noted:
   HCP has a packing fraction of 74% assuming a hard sphere packing.
   This packing also implies an ideal c/a ratio (see below).
   As real crystals do not have an ideal c/a ratio (table below), this also implies that they are not closest packed in the
‘truest sense’.                                                                                    Metal    c/a
Cd     1.886
Zn     1.856
Mg     1.624
c 2h    2                                                                         Zr    1.590
IDEAL c/a                 2    1.632...
a a     3                                                                         Ti    1.588
Be     1.586
HCP crystal
Hexagonal Lattice

=

+

Motif

   As we have seen that HCP crystal is constructed starting with a hexgonal close packed layer, with ABABAB.. packing.
   The 6-fold axis (present in a single layer) is lost on ABABAB.. packing and only a 3-fold remains.
Two atom Motif      Though 3 unit cells are often shown in diagrams- the rhombic prism (blue outline) is the unit cell.
Spheres at: (0,0,0), (⅔, ⅓,½)

Note: Atoms are coloured differently but are the same
Q: If the HCP crystal has only a 3-fold axis why does it come under the ‘Hexagonal class*’ of crystals?

A: The HCP crystal has ABAB.. packing. The C site is vacant! A 63 Screw axis passes through
the C-sites (along [0001] direction). This six (6) is responsible for the crystal coming under
the hexagonal crystal system (even though there is no usual 6-fold axis)

* As hexagonal word would imply 6-fold

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