1. Crystal Structure NaCl Well defined surfaces
reflects some internal order

2. Crystal Structure In 1913, William L. Bragg NaCl
discovered ordered diffraction pattern of x-ray
going through crystals
The symmetry of the diffraction pattern reflects the symmetry of the crystal
NaCl

3. Crystal Structure Transition from single molecule to a crystal
NaCl
Transition from single molecule to a crystal
From single ionic NaCl molecule
Into a lattice

4. Crystal Structure 1D lattice of alternating ions The binding energy
(N – number of molecules)
is the interaction energy of the ion i with all other ions
nearest neighbors
otherwise
1D lattice of alternating ions

5. Crystal Structure The binding energy (N – number of molecules)
is the interaction energy of the ion i with all other ions
nearest neighbors
otherwise
z – number of nearest neighbors
Madelung constant

6. Determining crystal parameters
Madelung constant should be positive
1D lattice of alternating ions

7. Nomenclature
Ideal crystal – infinite repetition of identical structural units in space
Lattice – regular periodic array of points in space (called a net, in 2D)
Translation vectors, a1 a2 a3
r’ = r + n1a1 + n2a2 + n3a3
Lattice looks the same from every r’

8. Basis – set of atoms associated with each lattice point
general crystal = lattice + basis

9. Primitive translation vectors – every point in lattice reached using integer ni
There are many different possible choices
Primitive cell
parallelpiped of primitive vectors
minimum volume cell that fills space
one lattice point
Volume of Cell:

10. construction: intersection of bisectors of lines to
Wigner-Seitz cell –
region of space about a lattice point closer to that point than to any other
construction: intersection
of bisectors of lines to
neighboring lattice points
BCC
Of particular significance when discussing phonon and electron wavevectors
FCC

11. Bravais Latice

12. Symmetry Operations
Symmetry operation – brings crystal back into itself
Besides translation, point symmetry operations (point operation – leave one point fixed):
Rotation (possible: 2p, 2p/2, 2p/3, 2p/4, 2p/6; there is no 2p/5, 2p/7)
Reflection (mirror planes)
Inversion
Compound operations (rotation-reflection, rotation-inversion)

13. Symmetry Operations

14. Point Group Nomenclature
What distinguishes the crystal systems is point group symmetries
To discuss symmetries we need to know how they are classified
Two different nomenclature systems
Schoenflies
International (Hermann-Mauguin)
Considering rotation, reflection, rotation-reflection
Space group adds symmetry operations of screw axis and glide plane.
Screw axis – rotation plus translation along axis
Glide plane – reflection in a plane plus translation parallel to plane

15. Schoenflies / international Notation

16. Any symmetry operation on lattice can be made from translation + point operation
(see Ashcroft, 113)
Point groups – set of point operations
Space groups – set of point operations + translations (i.e. full symmetry group)

17. Bravais Lattices
Based on symmetry, can classify lattices into archetypes known as Bravais lattices
In 2D, 5 types
In 3D, 14 types
14 space groups – Bravais lattices
(M.L. Frankenheim, 1842; corrected by A. Bravais, 1845)

18. Lattice: 7 distinct point groups 14 distinct space groups
Lattice + basis:
32 point groups
230 space groups
For visualization and examples:

19. Bravais Latice

21. Naming vectors and planes
Lattice direction:
The direction defined by the lattice point r = n1a1 + n2a2 + n3a3 is
[n1 n2 n3]
Lattice planes: Miller indices (h k l)
a) Intercepts of plane with axes: x1, x2, x3
b) Miller indices: smallest three integers proportional to 1/ x1, 1/ x2, 1/ x3
h:k:l = 1/ x1:1/ x2:1/ x3
For negative numbers use:

22. Lattice planes: Miller indices (h k l)

24. (hkl) define family of lattice planes
[parallel, equally spaced, contain all the points of the lattice]
{hkl} means (hkl) and all (hkl) equivalent in terms of the crystal symmetry
Example: for cubic crystal
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