This document discusses voids in different crystal structures including face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP). It explains that voids exist between atoms due to incomplete packing. In FCC crystals, the voids take the shape of tetrahedra and octahedra. BCC crystals contain distorted tetrahedral and distorted octahedral voids. HCP crystals contain tetrahedral and octahedral voids identical to FCC. The size and positions of voids in each structure are analyzed. Carbon is shown to prefer the smaller octahedral void in BCC iron over the larger tetrahedral void, affecting its solubility.

1. Part of
MATERIALS SCIENCE
& A Learner’s Guide
A Learner’s Guide
ENGINEERING
AN INTRODUCTORY E-BOOK
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh
http://home.iitk.ac.in/~anandh/E-book.htm

2. Voids
 We have already seen that as spheres cannot fill entire space → the packing
fraction (PF) < 1 (for all crystals)
 This implies there are voids between the atoms. Lower the PF, larger the volume
occupied by voids.
 These voids have complicated shapes; but we are mostly interested in the largest
sphere which can fit into these voids→ hence the plane faced polyhedron version
of the voids is only (typically) considered.
 The size and distribution of voids in materials play a role in determining aspects
of material behaviour → e.g. solubility of interstitials and their diffusivity
 The position of the voids of a particular type will be consistent with the
symmetry of the crystal
 In the close packed crystals (FCC, HCP) there are two types of voids →
tetrahedral and octahedral voids (identical in both the structures as the voids are
formed between two layers of atoms)
 The octahedral void has a coordination number 6 (should not be confused with 8 coordination!)
 In the ‘BCC crystal’ the voids do NOT have the shape of the regular tetrahedron
or the regular octahedron (in fact the octahedral void is a ‘linear void’!!)

3. SC
 The simple cubic crystal (monoatomic decoration of the simple cubic lattice) has large void in the
centre of the unit cell with a coordination number of 8.
 The actual space of the void in very complicated (right hand figure below) and the polyhedron version
of the void is the cube (as cube is the coordination polyhedron around a atom sitting in the void)
rx
= ( 3 − 1) = 0.732
r
True Unit Cell of SC crystal
Polyhedral model (Cube)
Video: void in SC crystal
Video: void in SC crystal
Actual shape of the void (space)!
 Later on we will talk about tetrahedral and octahedral
voids in FCC, BCC & HCP crystals:
 note that there are NO such tetrahedral and octahedral
voids in SC crystals and the only polyhedral void is
CUBIC (i.e. coordination number of 8)

4. FCC

Actual shape of the void is as shown below. This shape is very completed and we use the polyhedral version of the
void- i.e. tetrahedral and octahedral voids.
Actual shape of void
Position of some of the atoms w.r.t to the void

5. FCC

The complicated void shown before is broken down in the polyhedral representation into two shapes the octahedron
and the tetrahedron- which together fill space.
Octahedra and tetrahedra in an unit cell
Quarter of a octahedron which belongs to an unit cell
Video of the construction
Video of the construction
shown in this slide
shown in this slide
4-Quarters forming a full octahedron
Central octahedron in view- this is a full octahedron
4-tetrahedra in view

6. VOIDS
FCC = CCP
Tetrahedral
TV
¼ way along body diagonal
{¼, ¼, ¼}, {¾, ¾, ¾}
+ face centering translations
Vtetrahedron
1
= Vcell
24
rvoid / ratom = 0.225
Note: Atoms are coloured differently but are the same
Octahedral OV
At body centre
{½, ½, ½}
+ face centering translations
1
Voctahedron = Vcell
6
rVoid / ratom = 0.414
Video: voids CCP
Video: voids CCP
Video: atoms forming the voids
Video: atoms forming the voids

7. More views
Tetrahedral
TV
Octahedral OV

8. Once we know the position of a void then we can use
the symmetry operations of the crystal to locate the
other voids. This includes lattice translations
FCC- OCTAHEDRAL
Site for octahedral void
{½, ½, ½} + {½,
½, 0} = {1, 1, ½} ≡ {0, 0, ½}
Face centering translation
Note: Atoms are coloured differently but are the same
Equivalent site for an octahedral void

9.  There are 8 tetrahedral voids per cell and 4 octahedral voids per cell. The location of the voids and
number of voids per atom in the unit cell are to be noted from the table below.
FCC voids
Position
Voids / cell
Voids / atom
Tetrahedral
¼ way from each vertex of the cube
along body diagonal <111>
→ ((¼, ¼, ¼))
8
2
4
1
• Body centre: 1 → (½, ½, ½)
Octahedral
• Edge centre: (12/4 = 3) → (½, 0,
0)

10. Now let us calculate the largest size sphere which can fit into these voids.
Size of the largest atom which can fit into the tetrahedral void of FCC
The distance from the vertex of the tetrahedron to the centroid (DT) is the distance spanned by radius of the atom and the
radius of the interstitial sphere.
DT = r + x
Radius of the interstitial atom (sphere)
If ‘e’ is the edge length of the tetrahedron then CV = (√6/4)e → see below in triangle ABC
DT =
6
e=r+x
4
6
r=r+x
2
e = 2r
x  6 
=
 2 − 1÷ ~ 0.225
÷
r 

In tetrahedron ABCD
In triangle ABC
e2
e = + AM 2
4
2
AM =
AO =
3
e
2
2
2 3
e
AM =
e=
3
3 2
3
AD 2 = e2 = AO 2 + DO 2
2
e
e = + DO 2
3
2
DT =
3
DO
4
DT =
3 2
6
e=
e
4 3
4
DO = e
2
3

11. Size of the largest atom which can fit into the Octahedral void of FCC
2r + 2x = a
2a = 4r
x
=
r
(
)
2 − 1 ~ 0.414
Thus, the octahedral void is the bigger one
and interstitial atoms (which are usually
bigger than the voids) would prefer to sit
here

12. VOIDS
HCP
TETRAHEDRAL
OCTAHEDRAL
This void extends across 3
conventional unit cells and
hence is difficult to visualize
Coordinates : (0,0, 3 8 ), (0,0, 5 8 ), ( 2 3 , 1 3 , 18 ), ( 2 3 , 13 , 7 8 )
Coordinates: (⅓ ⅔,¼), (⅓,⅔,¾)
 These voids are identical to the ones found in FCC (for ideal c/a ratio)
 When the c/a ratio is non-ideal then the octahedra and tetrahedra are distorted (non-regular)
Important Note: often in these discussions an ideal c/a ratio will be assumed (without stating the same explicitly)
Note: Atoms are coloured differently but are the same

13. The other orientation of the tetrahedral void
Octahedral voids occur in 1 orientation, tetrahedral voids occur in 2 orientations
Note: Atoms are coloured differently but are the same

14. Further views
This void extends across 3
conventional unit cells and hence
is difficult to visualize
Note: Atoms are coloured differently but are the same

15. Further views
Octahedral voids
Tetrahedral void
Note: Atoms are coloured differently but are the same

16. Voids/atom: FCC ≡ HCP
→ as we can go from FCC to HCP (and viceversa) by a twist of 60° around a central atom of
Central atom
two void layers (with axis ⊥ to figure)
Atoms in HCP crystal: (0,0,0), (⅔, ⅓,½)
Check below
HCP voids
Position
Voids /
cell
Voids / atom
Tetrahedral
(0,0,3/8), (0,0,5/8), (⅔, ⅓,1/8),
(⅔,⅓,7/8)
4
2
Octahedral
• (⅓ ⅔,¼), (⅓,⅔,¾)
2
1

17. Further views
Various sections along the c-axis
of the unit cell
A
B
Octahedral void
Tetrahedral void
A

18. Further views with some models
Visualizing these voids can sometimes be difficult especially in the HCP
crystal. ‘How the tetrahedral and octahedral void fill space?’ is shown in the
accompanying video
Video: Polyhedral voids filling space
Video: Polyhedral voids filling space

19. Voids in BCC crystal
 There are NO voids in a ‘BCC crystal’ which have the shape of a regular
polyhedron (one of the 5 Platonic solids)
 The voids in BCC crystal are: distorted ‘octahedral’ and distorted tetrahedral
→ the correct term should be non-regular instead of distorted.
 However, the ‘distortions’ are ‘pretty regular’ as we shall see
 The distorted octahedral void is in a sense a ‘linear void’
→ an sphere of correct size sitting in the void touches only two of the six atoms surrounding it
 Carbon prefers to sit in this smaller ‘octahedral void’ for reasons which we shall
see soon

20. VOIDS
BCC
Distorted TETRAHEDRAL
Distorted OCTAHEDRAL**
a√3/2
a
a
a√3/2
Coordinates of the void:
{½, 0, ¼} (four on each face)
Coordinates of the void:
{½, ½, 0} (+ BCC translations: {0, 0, ½})
Illustration on one face only
rvoid / ratom = 0.29
Note: Atoms are coloured differently but are the same
rVoid / ratom = 0.155
** Actually an atom of correct size touches only
the top and bottom atoms

21. TV
OV
{0, 0, ½})
Illustration on one face only
BCC voids
Distorted
Tetrahedral
Non-regular
Octahedral
Position
• Four on each face: [(4/2) × 6 = 12] → (0, ½, ¼)
• Face centre: (6/2 = 3) → (½, ½, 0)
• Edge centre: (12/4 = 3) → (½, 0, 0)
Voids /
cell
Voids
/ atom
12
6
6
3

22. alculation of the size of the distorted tetrahedral void
a
BCC: Distorted Tetrahedral Void
a√3/2
a2 a2
5
From the right angled triange OCM: OC =
+
=
a=r+x
16 4
4
For a BCC structure:
3a = 4r ( a =
4r
3
5 4r
x  5 

=r+x ⇒ =
 3 − 1 = 0.29
4 3
r 

)

23. alculation of the size of the distorted octahedral void
Distorted Octahedral Void
a√3/2
a
* Point regarding ‘Linear Void’
 Because of this aspect the OV along the 3 axes can be
differentiated into OVx, OVy & OVz
 Similarly the TV along x,y,z can be differentiated
OB =
a
= 0.5a
2
OA =
2a
= .707 a
2
As the distance OA > OB the atom in the
void touches only the atom at B (body
centre).
⇒ void is actually a ‘linear’ void*
This implies:
a
OB = r + x =
2
4r
r+x=
2 3
BCC : 3a = 4r
x 2 3 

=
 3 − 1 = 0.1547
r 


24. Where does the carbon atom sit in the BCC and FCC forms of iron? How does it
affect the solubility of carbon in these forms of Fe?
Surprising facts!
 C dissolves more in the close packed structure (FCC, γ-Fe) (albeit at higher temperatures
at 1 atm. pressure− where FCC is stable) than in the open structure (BCC-Fe).
 C sits in the smaller octahedral void in BCC in preference to the larger tetrahedral void in
BCC.
 Fe carbon alloys are important materials and hence we consider them next.
 The octahedral void in FCC is the larger one and less distortion occurs when
carbon sits there → this aspect contributes to a higher solubility of C in γ-Fe.
 The distorted octahedral void in BCC is the smaller one → but (surprisingly)
carbon sits there in preference to the distorted tetrahedral void (the bigger one) (we shall see the reason shortly).
 Due to small size of the voids in BCC the distortion caused is more and the
solubility of C in α-Fe is small
 this is rather surprising at a first glance as BCC is the more open structure
 but we have already seen that the number of voids in BCC is more than that in
FCC → i.e. BCC has more number of smaller voids.
See next slide for figures

25. Relative size of voids, interstitials and Fe atom
Spend some time over this slide
FCC
Fe
FCC
r
Size of Fe atom
CCP crystal
x
Size of the OV
Size of Carbon atom
Fe
FCC
Void (Oct) Void (Tet)

= 1.292 A
FeFCC
C

(oct ) = 0.534 A
r = 0.77 A
r = 0.71 A
H

C

N
r = 0.46 A
N
H

Relative sizes of voids w.r.t to atoms
Note the difference in size of the atoms
BCC

Size of Fe atom
BCC crystal
Fe
rBCC = 1.258 A
Size of the TV
FeBCC
Size of the OV
x
x
Fe
BCC
Fe
BCC

(d .tet ) = 0.364 A

(d .oct ) = 0.195 A
Fe
xBCC (d .tet )
= 0.29
Fe
rBCC
Fe
xBCC (d .oct )
= 0.155
Fe
rBCC

26.  We had mentioned that the octahedral void in BCC is a linear one
(interstitial atom actually touches only two out of the 6 atoms surrounding it)
 In the next slide we make a approximate calculation to see till what size will it
continue to touch only two Fe atoms
(these are ‘ideal’ simplified geometrical calculations and in reality other complications will have to be considered)

27. Ignoring the atom sitting at B and assuming the interstitial atom touches the atom at A
2a
OA = r + x A =
2
2 6r
r + xA =
3
Fe
BCC
r

= 1.258 A
BCC : 3a = 4r
xA  2 6 
=
− 1÷ = 0.6329
÷
r  3



OX = x A = 0.796 A

OY = xB = 0.195 A
x
Fe
BCC

(d .tet ) = 0.364 A

28.  This implies for x/r ratios between 0.15 and 0.63 the interstitial atom has to push
only two atoms
 (xcarbon/rFe)BCC ~ 0.6
 This explains why Carbon preferentially sits in the apparently smaller octahedral
void in BCC

29. DC
 In the DC structure out of the family of 8 (¼, ¼, ¼) type positions only 4 are occupied
[(¼, ¼, ¼), (¾, ¾, ¼), (¼, ¾, ¾), (¾, ¼, ¾)].
 The other four are like void positions- which are all tetrahedral in nature.

30. Summary of void sizes
rvoid / ratom
SC
BCC
FCC
DC
Octahedral
(CN = 6)
Not
present
0.155
(non-regular)
0.414
Not present
Tetrahedral
(CN = 4)
Not
present
0.29
(non-regular)
0.225
1
(½,½,½) & (¼, ¼, ¼)
Cubic
(CN = 8)
0.732
Not present
Not present
Not present

31. Funda Check
Some points and checks on
voids!
 Voids should not be confused with vacancies- vacancies are due to missing atoms or ions
in crystals.
 Holes should also not be confused with voids- holes are ‘missing electrons’ from the
valence band of a solid.
 Voids have complicated shapes- we usually use a polyhedral version – the coordination
polyhedron around a sphere of ‘correct size’.
 Sometimes, as in the case of ‘octahedral void’ in the BCC- the second nearest neighbours
are also included in constructing the coordination polyhedron.
 In ionic crystals, unlike metallic crystals the cation ‘does not sit’ in the void formed by the
anions- the cation is bigger than the anion. The void size calculation is to demarcate the
regimes of various coordination structures.
 If an interstitial atom wants to jump from one metastable equilibrium position to anotherit has to cross an energy barrier.