Geometrical Theory of Crystalline interfaces A boundary surface ...

Geometrical Theory of Crystalline interfaces
A boundary surface between any two crystals will take up such a position that
the crystals exhibit more or less optimum matching
Formulate the notion of the "matching" of two periodic structures in a correct
mathematical manner.
1. The coincidence-site lattice
Consider two interpenetrating translation lattices (lattice 1, lattice 2)
usually lattice 1 : fixed
lattice 2 : all changes such as translation, rotation etc. are
performed.
At a given relative orientation of two lattices, lattice 2 is translated in such a
way that one of its points coincides with a point in lattice 1 "lattice
coincidence site" "point of best fit" atoms located in this position is in an
unstrained position in both lattices
other point of lattice coincidence within the interpenetrating lattice
- none
- one
- infinite number

example 1
Σ CSL van be constructed by rotating around [00] axis as follows
θ (36.87°)
Σ
( are integers)
All of the CLS formed by rotations arround [001] can be obtained by the
generating equations
Σ ( ) odd
Σ ( ) even ( ) : odd
θ

epresentation of CSL
(1) reciprocal density of common lattice points, Σ
(2) angle of rotation around the axis with the lowes Miller indices
Determination of Σ
rotation θ about [hkl]
θ
and are integers
Σ
and : any positive integers
Σ : the reciprocal of the fraction of atomic sites which are in coincidence
λ : length of unit vector of CSL

λ Σ
CLSs are formed only at certain specific rotations
However, an infinite number of CSL's exist at essentially all misorientations with
unit cell dimensions which change in a highly discontinuous manner with small
change in θ. Therefore, it is always possible to find a CSL corresponding to a
misorientation which is arbitrary close to the actual misorientation of the two
lattices.
The densest CSLs (with Σ < 100 ) formed by [001] rotations are listed below
13, 1
11, 1
9, 1
8, 1
7, 1
6, 1
5, 1
9, 1
13, 3
4, 1
11, 3
7, 2
3, 1
8, 3
13, 5
5, 2
θ Σ
8.80 85
10.39 61
12.68 41
14.25 65
16.26 25
18.92 37
22.62 13
25.06 85
25.99 87
28.07 19
30.51 65
31.89 53
36.87 5
41.11 73
42.08 97
43.60 29

CSLs by rotation around other rotation axes possible
an example of Σ
CSL can be obtained by rotation around [111]
rotation around [111]
θ = 38.21°
Σ
Ranganathan's formula
In cubic crystals for a general rotation around [h,k,l] axis
Σ
θ
are integers
Σ are odd number

choose two lattice vectors and on (hkl) such that
The CSL can be constructed by choosing and
: parallel to ( are integers)
The and are orthogonal vector sets on (hkl) which are shortest vectors in
each direction
Then and have and directions respectively
example 1. rotation around [001]. simple cubic
Σ
θ from above equation
If , , Σ
ˆ ˆ ˆ
For a simple cubic, volume of the primitive cell in the lattice is and
that in the CSL ( )
And the same is true of the FCC according to the Ranganathan formula
When are taken as basis vectors
Then voulme of the corresponding primitive cell in CSL is
Σ
again the fraction of atoms in coincidence is 1/5.

For BCC
and Σ
example 2. rotation axis [110], FCC
Σ
θ
If Σ
ˆ ˆ ˆ
example 3. rotation axis [110], FCC
Thus Σ
( is not shortest vector in FCC )
: number of atoms in

Number of atoms in the primitive cell of CSL is 18 because volume/atom in
FCC is . Since the volume of the primitive cell in , there are 2 atoms
in the p.c of the lattice 1.
Thus, Σ
example 4. rotation axis [111], FCC
Σ
θ
If Σ
ˆ ˆ ˆ
Grain boundary configuration
Starting with the two lattice on the CSL, we may construct any GB by the
following operation
(1) pass a plane through the two interpenetrating lattices and discard all atoms
of lattice 1 on one side of the plane and all atoms of lattice 2 on the other
side. This produces a rigid lattice GB. structure having a periodicity which is
identical to that of the corresponding plane in the CSLs.
(2) Allow the rigid lattice GB. configuration to relax to produce the final
equilibrium structure of minimum energy.
In this process, the periodicity of the arrangement of atoms across the boundary
Σ

is unchanged, and the periodicity of the relaxed pattern of atoms in the
boundary is therefore the same as that of the corresponding of the CSL.
position of nuclei determines positions of the electrons (electron moves much
faster than nuclei)
electron nucleus interactions inter-nuclear potential function
(This is basis of Born-Oppenheimer approximation)
If Born-Oppenheimer approximation is valid it defines many body internuclei
potential. (potential where contribution due to electron distribution is a function
of all the nuclei coordinate)
Hellman-Feyman theorem
force acting on each nucleus may be computed frome the change in electron
charge density which occurs upon the motion of the nuclei + direct inter nuclei
Coulombic forces.
Each nuclei or ion core influences its neighbor through direct ion-ion interaction
indirect and ion-electron-ion interaction
ionic crystal : ion-ion interaction is dominant
metal : ion-electron-ion interaction is more dominant
Net effect of interaction in metal is a volume dependent contribution to the
lattice energy.
Central pair-wise interaciton is over simplification.
( Failure to obey Cauchy relations between elastic constants confirms this in
cubic metals)
Selection of interatomic potential is most important for GB. energy calculation.
1) pair potential method.
for simplicity this was assumed.
1-1) Basesd on geometrical assumptions, construct a translation lattice by
changing the atomic arrangement of the adjacent matrix in the vicinity of the
interface
this will produce elastic deformation of both crystals
associated stored energy has been calculated by elastic continuum and regular

crystalline array
1-2) More recent method - minimization of the total bicrystal energy by
relaxation method
Total energy of the atomic array at a bicrystal interfaces
ψ
ψ : interaction potential between th and th atoms
is summed over all atoms in the array
is summed only over those atoms separated by distances less
than 2nd or 3rd nearest neighbors.
1/2 : correction for double counts ( ψ )
ψ
Morse potential
ψ
Δ Δ
: sublimation energy
: equilibrium separation
:crystal constants
α α
: numerical value close to the nearest neighbor distance (not
necessary)
: dissociation energy of bonds
α : positive constant
"spline-fit" potential (emperical developed by Jones (1864))
ψ
: determined by various experimental observation

Calculated GB energy 2×measured GB energy (at hig temperature)
The calculated value :0 K ( Δ
The difference will be reduced
)
Calculated grain boundary entropy
Total GB entropy
ν λ
ν : vibrational entropy (harmonic)
: configurational entropy
λ : anharmonic contribution
Einstein model : each atom triharmonic oscillator
entropy per atom
λ
ν
ν : Einstein frequency
ν ν ν
θ
π θ
Ω
ε
ε
ν : matrix atom vibrational frequency
ν : boundary atom vibrational frequency

Problems of CSL
: anharmonic factor
: atomic mass
θ : Einstein temperature
ε ε : bond energy of interior and boundary atom respectively
1. Position of coincidence site tells nothing about its surroundings.
except that the pattern of lattice point must be periodic.
(periodic pattern may exist without coincidence site)
2. In more complicated pattern, it is impossible to identify a coincidence site.
No. measure of the spacing between two points mathematically
it does not tell if two points coincide or are 0.0001 apart?
3. There are points where no CSL lattice exist
(for cubic case θ is irrational)
Generalization of concept of CSL is necessary.
Consider more general and mathematical method
Each of the two lattice can be understood as a translation group
constructed by 3 linearly independent basic vectors
: lattice point
: rotation in this case but could be general after transformation
(expansion, rotation, shear, etc)
Whole set of the lattice points may be called "equivalence class" or "class"
Above equation means the two lattices are two related classes of points.

O-lattice
Example : choose the crystal coordinate system as being given by the unit
vectors of lattice 1.
all lattice points ate indicated by integral coordinates. Consider crystal
coordinates of an arbitrary point ( 12.138, 7.248, -4.431 )
designate - external coordinates integer ( 12, 7, -5 ) : unit cell
internal coordinates ( 0.138, 0.243, 0.539 ) within unit cell
In crystal coordinate system all the points of a coset have the same internal
coordinates
Energy coset is represented inside every unit cell
Infinite number of cosets forms a partition by translation group having internal
coordinates (000)
--by
this equation every point in crystal 1 is related one in crystal 2.
Define "O" points
generalized lattice coincidence sites.
as "coincidences of elements of related equivalence classes
regardless the values of the internal coordinates of these classes"
(or coincidences of internal coordinates )

lattice coincidence site special O-points with internal coordinates of [000]
Derivation of equation for O-points
: arbitrary point in crystal 1. ( with arbitrary external and internal
coordinates )
: equivalence class of that point
corresponding point in the related class
alternatively start from , find other elements of the class
.
: translation vector of lattice 1 to
If the point defined by and coincides, designate that coincidence point by
together with equation
or ---
identity = unit transformation
All the possible translation vector
All the O-points are solutions of eq
---
---
---

This equation is the basis of the whole geometrical theory of crystlline interfaces
General procedure for solving the equation
1) choice of the coordinate system
2) formulation of the transformation which relates the two lattices
3) Formulation of the basic equation by matrix operators
4) if the determinant , the solution is

example : refer to the following fig.
consider 2D cubic crystal
rotation counterclockwise θ crystal 2 with respect to crystal 1
θ ˚ ˚
Σ
transformation matrix
θ θ
θ θ
θ θ
θ θ
if , solution will be given by equation
inverse matrix
θ
θ
θ

As the two basic translation vectors are given by the coordinates (1,0) and (0,1)
( due to the crystal coordinate system), unit vectors of O-lattice are column
vectors of the above matrix.
θ and
Geometrical interpretation of this result, in fig. 12.3/3
For the rotation around [001] { θ ˚ Σ }
The same as the fig. in the previous page.
Other special case of coincidence site lattice
θ
The unit vectors of the O-lattice are and
Return
θ
θ

It can be seen that the coordinates of the unit cell of lattice 1 and 2 indeed
coincide exactly on each mode of the O-lattice. On the other hand , if we
regard the transformation as -36.87˚.
From these, we find:
(1) there are more than one O-lattice for each crystal misorientation. this is a
result of the fact that there is more than one transformation [A] which can
be used to describe the misorientatioin
(2) The CSL is a sublattice of the O-lattice
(3) Each O-lattice point can be used as an origin for the transformation A, ie.
the O-lattice can be regarded as a "lattice of origins".

(4) The lattice spacings of the O-lattice vary continuously as the rotation varies.
The particular O-lattice which is of great physical significance is the O-lattice
corresponding to the transformation which relates atoms which are nearest
neighbors in the vicinity of these O-lattice points.
The regions in the vicinity of these O-lattice points are then the regions of best
matching. In the previous example [simple cubic [001] tilt Σ ,
θ ˚ ], the O-lattice based on the 36.87˚ rotation gives better
description of the regions of good fit. In general, the coarser the O-lattice is, the
better it describe the degree of good local fit.
Grain boundary configuration
When we construct a grain boundary, we pass the desired plane of the boundary
through the O-lattice and discard all atoms of lattice 1 on one side and all
atoms of lattice 2 on the other. Then there will be a region around each of the
intersections of the O-lattice with the boundary plane where the two lattices are
almost matched and this is not changed by a translation of lattice 2 with respect
to lattice 1 in a subsequent relaxation.
In describing low angle boundaries, the regions of good fit in the boundary are
regions of perfect lattice between the dislocations which make up the boundary.
Thus, dislocations are arranged so that they run between the O-lattice points and
the O-lattice define the geometry of the dislocation network. For a small angle
twist boundary ([001] twist, θ ˚) region of good fit around each O-lattice
point are easily visible, and dislocations run between them. This is precisely the
result obtained from the Frank equation of the PGBDS.