voids in crystals

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MATERIALS SCIENCE
& A Learner’s Guide
ENGINEERING
Part of
AN INTRODUCTORY E-BOOK
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: [email protected], URL: home.iitk.ac.in/~anandh
http://home.iitk.ac.in/~anandh/E-book.htm
Interested readers may refer to this excellent article
“Space Subdivision and Voids Inside Body-Centred Cubic Lattices”
Claudio Giomini and Giancarlo Marrosu
Chem. Educator 16 (2011) 232–237.
Interatomic Voids*
 When an atom of an alloying element/impurity is added to a pure crystal, the atom
added may ‘sit’ in a lattice/sublattice site in place of the host atom (e.g. Ag added
to Au sits in a FCC lattice site→ Ag is a substitutional alloying element) or ‘go
into’ the space between atoms (e.g. C added to Fe→ C is a interstitial alloying
element). In some rare cases the atoms may be present both in the lattice and
interstitial sites (e.g. B in steel).
 If the ‘fit’ of the added atom in the ‘available space’ is not too bad, then the
solubility of the added element in the host crystal is expected to be good.
 In the hard sphere model of atoms, atoms are visualized as spheres. We have
already seen that as spheres cannot fill entire space. This implies that the packing
fraction (PF) < 1 (for all crystals).
 This further implies there is void space between the atoms. Lower the PF, larger
the volume occupied by the void space.
 We are mostly interested in the largest sphere which can fit into these voids.
 The void space forms a continuous network across the whole solid. Part of this
void space lies within the unit cell (which when translated by the lattice translation vectors gives rise to the entire
void space).
* Do not confuse these voids with microscopic/macroscopic voids/cracks in crystals or holes (semiconductors) or vacancies (missing atoms).
Note: in some cases the void within a crystal can be large enough to accommodate small/large molecules (e.g. in Metal Organic Frameworks)
→ we are not going to describe these voids in this set of slides.
 This void space within the unit cell has a complicated shape, but (typically) we
only consider the plane faced polyhedron version of the voids.
 There may be more than one type of such void polyhedra in a single unit cell.
These void polyhedra when put together fill space; however, in some cases only a
part of a given polyhedron may lie within a unit cell.
 The size and distribution of voids* in materials plays 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 (if a void of a particular type is located at (x,y,z), then all similar
voids can be obtained by the symmetry operations 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’!!)
* Common way of referring to the void polyhedron/polyhedra.
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).
 Voids in SC crystal are not often described in detail in text books as the only element crystallizing SC
structure is Polonium.
rx
 ( 3  1)  0.732
r
True Unit Cell of SC crystal
Polyhedral model (Cube)
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).
CCP
 Actual shape of the void is as shown below. This shape is very complicated and we use the
polyhedral version of the void. The polyhedra involved are the regular tetrahedron and the
regular octahedron (referred to as the tetrahedral and octahedral voids).
Actual shape of the void
Ignore the nonidealities in the wax
casting process!
Position of some of the atoms w.r.t to the void


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).
Note that a given tetrahedron (pink colour) is fully present within the unit cell, while only the octahedron (blue
colour) at the body centre is present within the unit cell. The octahedron with its centroid at centre of the edge of the
unit cell is shared by 4 unit cells (the cut faces are shown in green colour).
 Note that regular tetrahedra cannot
fill space. Similarly, regular
octahedra cannot fill space. But, a
combination of tedrahedra and
octahedra can fill space (in the
ratio 2:1).
Video of the construction
shown in this slide
Octahedra and tetrahedra in an unit cell
Quarter of a octahedron which belongs to an unit cell
4-Quarters forming a full octahedron
Central octahedron in view- this is a full octahedron
4-tetrahedra in view
VOIDS
Tetrahedral
TV
Location of the void:
¼ way along body diagonal {¼, ¼, ¼}, {¾, ¾, ¾}
+ face centering translations
Octahedral OV
Location of the void:
At body centre {½, ½, ½}
+ face centering translations
Calculation shown later
rvoid / ratom = 0.225
Vtetrahedron
1
 Vcell
24
Note: Atoms are coloured differently but are the same
rVoid / ratom = 0.414
1
Voctahedron  Vcell
6
Video: voids CCP
Video: atoms forming the voids
More views
Tetrahedral
TV
Octahedral OV
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
 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
Octahedral
• Body centre: 1  (½, ½, ½)
• Edge centre: (12/4 = 3)  (½, 0, 0)
4
1
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
e  2r
6
rrx
2
x  6 
 
 1 ~ 0.225
r  2

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
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
HCP
VOIDS
TETRAHEDRAL
Note: Atoms are coloured differently but are the same
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 , 13 , 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).
If c/a ratio is not the ideal one then the voids will not be ‘regular’ (i.e. regular octahedron and regular tetrahedron).
Further views
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
This void extends across 3
conventional unit cells and hence is
difficult to visualize
Further views
Octahedral voids
Tetrahedral void
Note: Atoms are coloured differently but are the same
Voids/atom: FCC  HCP
 as we can go from FCC to HCP (and viceversa) by a twist of 60 around a central atom of
two void layers (with axis  to figure)
Central atom
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
Further views
Various sections along the c-axis
of the unit cell
A
B
Octahedral void
Tetrahedral void
A
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
Voids in BCC crystal
 Let us start with some surprising facts (which we shall list later as well):
(i) In a CCP crystal the octahedral void can host a bigger sphere than the
tetrahedral void– the reverse is true in BCC crystals.
(ii) In the case of Fe the solubility of C in close packed CCP structure is more than
the non-close packed BCC structure.
(ii) In BCC Fe, C sits in the octahedral void (which can host a smaller sphere) in preference to
the tetrahedral void (which can host a larger sphere).
 The solubility of C in Fe (both FCC & BCC) is much lower than the available
interstitial void space.
 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 (the ‘distortions’ are
‘pretty regular’ as we shall see shortly).
 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.
BCC
VOIDS
Distorted OCTAHEDRAL**
Distorted* TETRAHEDRAL
a3/2
a
a
a3/2
Coordinates of the void:
{½, 0, ¼} (four on each face)
rvoid / ratom = 0.29
Coordinates of the void:
{½, ½, 0} (+ BCC translations: {0, 0, ½})
Illustration on one face only
Note: Atoms are coloured differently but are the same
rVoid / ratom = 0.155
* Non-regular is a better term as compared to distorted.
** Actually an atom of correct size touches only the top and
bottom atoms
TV
OV
{0, 0, ½})
Illustration on one face only
(due to symmetry all faces will be identical)
BCC voids
Position
Voids /
cell
Voids /
atom
Distorted
Tetrahedral
• Four on each face: [(4/2)  6 = 12]  (0, ½, ¼)
12
6
Non-regular
Octahedral
• Face centre: (6/2 = 3)  (½, ½, 0)
• Edge centre: (12/4 = 3)  (½, 0, 0)
6
3
Calculation of the size of the distorted tetrahedral void
a
BCC: Non-regular 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   
 1  0.29
4 3
r  3 
)
Calculation of the size of the distorted octahedral void
Non-regular 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 
 
 1  0.1547
r  3

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, -Fe) (albeit at higher temperatures at
1 atm. pressure  where FCC is stable) than in the open structure (BCC-Fe).
 Solubility CCP Fe 2.06 wt.% at 1147ºC, BCC Fe 0.008 wt.% at RT & 0.025% at
723C.
 C sits in the smaller octahedral void in BCC in preference to the larger tetrahedral void.
 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 -Fe.
 The distorted (non-regular) 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 -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
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
Relative size of voids, interstitials and Fe atom

Void (Oct)
 1.292 A
0.77 A
FeCCP
C
Void (Tet)
0.534 A

(oct )  0.534 A

r  0.77 A
C
N
r N  0.71 A
H
r H  0.46 A
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

Fe
xBCC
(d .tet )  0.364 A
x
Fe
BCC

(d .oct )  0.195 A
Fe
xBCC
(d .tet )
 0.29
Fe
rBCC
Fe
xBCC
(d .oct )
 0.155
Fe
rBCC
 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).
 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.
Calculations in this regard are shown in the next page
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  xA  0.796 A
OY  xB  0.195A
x
Fe
BCC

(d .tet )  0.364 A
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.
 Just because there is a large void space available, this does not imply that an atom can
actually occupy these void spaces– this depends on other factors including the type of
bonding in the crystal.
 Other types of voids can also be envisaged for the DC crystal (not considered here for now).
Largest sphere which ‘can’ sit in the
void in DC
Summary of void sizes
rvoid / ratom
SC
BCC
FCC
DC
Not
present
0.155
(non-regular)
6 voids/cell
0.414
4 voids/cell
Not present
Tetrahedral
(CN = 4)
Not
present
0.29
(non-regular)
12 voids/cell
0.225
1
8 voids/cell (½,½,½) & (¼, ¼, ¼)
Cubic
(CN = 8)
0.732
Not present
Octahedral
(CN = 6)
Not present
Not present
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.
 In other contexts a ‘void’ could also imply a larger void* (of the size of nanometers or microns) and
not the void between atoms in a crystal structure.
 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 anionsthe 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 another- it has to
cross an energy barrier.
 Diffusivity of interstitial atoms (like C in Fe) is expected to be faster at a given temperature, as
compared to substitutional atoms. This is because, typically most of the interstitial sites are vacant
and hence an interstitial atom can jump from one site to neighbouring site.
 The solubility of an interstitial element is often much smaller than the available void space. This is
due to the overall strain energy cost of introducing an interstitial solute.
* Schematic of a ‘large void’ implied under some circumstances 
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