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crystal consist of voids ......

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.

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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
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’!!)
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)
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
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
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
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 4 1 • Body centre: 1 → (½, ½, ½) Octahedral • Edge centre: (12/4 = 3) → (½, 0, 0)
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
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
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
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
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
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 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
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 Video: Polyhedral voids filling space
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
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
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
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   )
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  
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
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
 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)
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
 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
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.
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
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.