Bravais lattice
a lattice is a regular periodic array of points in space
a (3D) Bravais lattice consists of all points with position vectors R of the form
R = n1a1 + n2a2 + n3a3 where a1, a2, a3 are any vectors not all in the same plane
and ni range through all the integer values
ai – primitive vectors
primitive vectors generate the lattice
the Bravais lattice specifies the periodic array in which the repeated units of the crystal are arranged
the units themselves may be single atoms, groups of atoms, molecules, ions, etc.
2D Bravais lattice
P = a1 + 2a2
Q = - a1 + a2
a simple cubic
3D Bravais lattice
the set of primitive vectors is not
unique – there are infinitely many
nonequivalent choices
primitive cell
a volume of space that, when translated through all
the vectors in a Bravais lattice, just fills all the space
without overlapping itself or leaving voids is called
a primitive cell or primitive unit cell of the lattice
several possible choices of primitive cell
there is no unique way of choosing a primitive
cell for a given Bravais lattice
a primitive cell must contain exactly one lattice point
the volume of the primitive cell v = V/N = 1/n,
where n is the density of points in the lattice
the volume of the primitive cell is independent
of the choice of cell
Wigner-Seitz cell
a primitive cell with the full
symmetry of the Bravais lattice
crystal structure consists of identical points of the same physical unit,
called the basis, located at all the points of a Bravais lattice
crystal structure is a lattice with a basis
examples of crystal structures
body-centered-cubic (bcc) Bravais lattice
a set of primitive vectors
a
( y  z  x)
2
a
a 2  (z  x  y )
2
a
a3  (x  y  z )
2
a1 
bcc = scA + scB
a – lattice constant
primitive and unit cell or
conventional unit cell
generally chosen to
have the lattice
symmetry
bcc lattices are
described
by cubic unit cell
the Wigner-Seitz cell of
the bcc Bravais lattice
a “truncated octahedron”
face-centered-cubic (fcc) Bravais lattice
fcc Bravais lattice
a set of primitive vectors
a
a1  (y  z )
2
a
a 2  ( z  x)
2
a
a3  (x  y )
2
Volume of unit cell
Vc=l(a1xb1).c1l
primitive cell and
unit cell
fcc lattices are
described
by cubic unit cell
the Wigner-Seitz cell of
the fcc Bravais lattice
a “rhombic dodecahedron”
the surrounding cube is not
the conventional unit cell
but one in which lattice points
are at the center of the cube
diamond structure
zincblende structure
formed by the carbon
in a diamond crystal
the diamond lattice consists of two
interpenetrating fcc Bravais lattices
displaced by (a/4)(x + y + z)
diamond lattice is fcc lattice with the
two point basis 0 and (a/4)(x + y + z)
ZnS consists of equal number of zinc and sulfur
ions distributed on a diamond lattice so that
each Zn has 4 S as its nearest neighbors
= fcc lattice with a basis consisting
of Zn at 0 and S at (a/4)(x + y + z)
hexagonal close-packed (hcp) structure
ideal c 
8
a
3
c-axis
simple hexagonal Bravais lattice
2D triangular nets stacked
above one another
a set of primitive vectors
a1  ax
a
3a
a2  x 
y
2
2
a3  cz
hcp – …ABABAB…
fcc - …ABCABCABC…
hcp structure consists of two interpenetrating
simple hexagonal lattices displaced by a1/3 + a2/3 +a3/2
truly close-packed structure with the ideal
value of c/a – an ideal hcp structure
infinitely many other closed
packing arrangements, e.g.
…ABACABACABAC…
The Sodium Chloride structure
NaCl consists of equal number of Na and Cl ions
placed in alternative points of sc lattice so that
each Na has 6 Cl as its nearest neighbors
= fcc Bravais lattice with a basis consisting
of Na at 0 and Cl at (a/2)(x + y + z)
The Cesium Chloride structure
CsCl consists of equal number of Cs and Cl ions
placed at the points of bcc lattice so that
each Cs has 8 Cl as its nearest neighbors
= sc Bravais lattice with a basis consisting
of Cs at 0 and Cl at (a/2)(x + y + z)
coordination number – the number of the nearest
neighbors to a given point in the lattice
diamond lattice
sc
bcc
hcp
fcc
coordination
number
4
6
8
12
12
diamond
simple cubic
Not necessary to have long range periodicity to have long range order
Two examples I can think of
1)Quasicrystals (Shechtman et al (Physical Review Letters V53 (1984)
(Stephens and Goldman, Physics Today April 1991)
You can fill space with tiles that have rotational symmetry. In general, the
vertex angles of a regular polygon of n sides are given by Π(n-2)/n
Eg
n=2 →0 degrees
n=3 →60 degrees
n=4→90 degrees
n=5→108 degrees
n=6→120 degrees
Periodic tilings are possible if these angles are integral fractions of 2Π
i.e. 2n/(n-2) is an integer
PENROSE TILING fills space (Roger Penrose) in a non-periodic way
Fibonicci Series
Another way to create long range order without periodicity.
Fibonacci Series. There are many ways to to construct a fibonacci series but
one simple way is to build a 1d lattice of two bond lengths using the following rules
A long bond creates a long and short bond L→LS
A short bond creates a long bond
S→L
Start with a long bond
L
LS
LSL
LSLLS
LSLLSLSL
LSLLSLSLLSLLS
LSLLSLSLLSLLSLSLLSLSL…………..
A classical Fibonnici sequence results in a ratio τ= 2cos 36° = (1+sqrt5)/2=1.618034
This is called “the golden mean” and occurs naturally in many circumstances
point defects
interstitial impurity
self-interstitial
substitutional impurity
vacancy
point defects are present even in the thermal equilibrium crystals
E
e.g. for a vacancy:
 0
k T
in thermal equilibrium n  Ne B
E0 – potential energy required
to remove one atom
for E0 = 1 eV n ~ 10-15 N at room T
line defect: dislocations
plastic deformation - permanent and irreversible deformation
the model of a solid as a perfect crystal cannot account for
the force necessary to deform a crystal plastically:
the force required to deform real crystals is orders of magnitude lower
than that for a perfect crystal → the mystery of weakness of crystals
slip of a crystal occurs via motion of dislocations
simplest kinds:
edge dislocation
screw dislocation
1 – along a dislocation the crystal is
distorted locally and additional distortion
required to move the dislocation sideways
requires relatively little force
2 – step by step moving a dislocation
through many lattice constants results to
displacement of the whole half of a crystal
by a lattice constant → moving a carpet
is easier by the passage of a linear ripple
across the carpet
dislocation densities depend on the sample preparation n ~ 102 – 1012 cm-2 and
annealing reduces dislocation densities
annealing
perfect crystal is hard
purified crystal is soft
poorly prepared crystal is hard
too many dislocations and defects number of dislocations and defects is reduced no dislocations
unimpeded flow of dislocations makes
they impede each other flow
plastic deformation of the crystal easy
work hardening: repeated bending lead to creation of many dislocations
until they are so many that they impede each other flow
→ the crystal is uncapable of further plastic deformation
and breaks under subsequent stress
definition of reciprocal lattice
consider set of points R constituting a Bravais lattice
a plane wave
eiKr
the set of all wavevectors K that yield plane waves with the periodicity of
a given Bravais lattice is its reciprocal lattice
K belongs to the reciprocal lattice of a Bravais lattice of points R,
iK ( r  R )
 eiK r holds for any r, and for all R in the Bravais lattice
provided that the relation e
the reciprocal lattice is the set of wave vectors K satisfying eiK R  1 for all R in the Bravais lattice
K  R  2 n
1D
linear crystal lattice
a
● ● ● ● ●
a
reciprocal lattice
○
2
K  2
a
○
2
K 
a
○
0
direct lattice
b
○
2
K
a
○
K 2
2
a
the reciprocal lattice is a Bravais lattice
a1, a2, a3 is a set of primitive vectors for the direct lattice
then reciprocal lattice can be generated by a set of primitive vectors
b1  2
a 2  a3
a1   a 2  a3 
b 2  2
a3  a1
a1   a 2  a3 
a1  a2
b3  2
a1   a2  a3 
bi a j  2 ij
 ij  0, i  j
 ij  1, i  j
for any
k  k1b1  k2b 2  k3b3
R  n1a1  n2a2  n3a3
k  R  2  k1n1  k2n2  k3n3 
eikR  1 if ki are integers
the reciprocal to the reciprocal lattice is the original direct lattice → K and R come
symmetrically to
eiK R  1
the volume of a Bravais lattice primitive cell is v  a1   a2  a3 
the volume of the reciprocal lattice primitive cell is (2)3/v
examples
sc Bravais lattice with lattice cell a →
reciprocal lattice is a sc lattice with primitive cell of side 2/a
primitive vectors
a1  ax, a2  ay, a3  az
2
2
2
b1 
x, b 2 
y, b3 
z
a
a
a
b1  2
a 2  a3
a1   a 2  a3 
b 2  2
a3  a1
a1   a 2  a3 
b3  2
a1  a2
a1   a2  a3 
bi a j  2 ij
fcc Bravais lattice with conventional unit cell of side a →
reciprocal lattice is a bcc lattice with conventional unit cell of side 4/a
a
4 1
a1 
(y  z )
2
a
a 2  ( z  x)
2
a
a3  (x  y )
2
b1 
a
4
b2 
a
4
b3 
a
( y  z  x)
2
1
(z  x  y)
2
1
(x  y  z)
2
bcc Bravais lattice with conventional unit cell of side a →
reciprocal lattice is a fcc lattice with conventional unit cell of side 4/a
4 1
a
a1 
( y  z  x)
2
a
a 2  (z  x  y )
2
a
a3  (x  y  z )
2
b1 
a
4
b2 
a
4
b3 
a
( y  z)
2
1
(z  x)
2
1
(x  y )
2
examples
b1  2
a 2  a3
a1   a 2  a3 
b 2  2
a3  a1
a1   a 2  a3 
b3  2
a1  a2
a1   a2  a3 
bi a j  2 ij
simple hexagonal Bravais lattice with lattice constants c and a →
reciprocal lattice is a simple hexagonal Bravais lattice with lattice constants 2/c and 4 / 3a
rotated through 300 about c-axis with respect to the direct lattice
primitive vectors
first Brillouin zone
the Wigner-Seitz primitive cell of the reciprocal lattice
is the first Brillouin zone
the surrounding cube is not
the conventional unit cell
but one in which lattice points
are at the center of the cube
the Wigner-Seitz cell of
the bcc Bravais lattice
a “truncated octahedron”
the first Brillouin
zone for the
bcc lattice
the Wigner-Seitz cell of
the fcc Bravais lattice
a “rhombic dodecahedron”
fcc lattice
lattice planes
lattice plane – any plane containing at least 3 noncollinear Bravais lattice points
points in such a plane form 2D Bravais lattice
family of lattice planes – a set of parallel, equally spaced lattice planes,
which together contain all the points of the 3D Bravais lattice
sc Bravais lattice
represented as a family
of lattice planes
for any family of lattice planes separated by a distance d, there are reciprocal
lattice vectors perpendicular to the planes, the shortest of which have a length of 2/d.
conversely,
for any reciprocal vector K, there is a family of lattice planes normal to K and separated by
a distance d, where 2/d is the length of the shortest reciprocal lattice vector parallel to K
reciprocal vector K of length 2/d
iK R
family of lattice planes separated by d
1
definition of K → e
iK R
 1 has the same value at all points lying in a family of planes
plane wave e
that are perpendicular to K and separated by an integer number of its wavelengths d = l = 2/K
Miller indices of lattice planes
the Miller indices of a lattice plane are the coordinates of the
shortest reciprocal lattice vector normal to that plane
(hkl) plane
a plane with Miller indices h, k, l is normal to the reciprocal lattice vector K = hb1 + kb2 + lb3
K r  A
K  ( xi ai )  A
K  a1  2 h
r
K  a2  2 k
K  a3  2 l
x1  A  2 h 
x2  A  2 k 
x3  A  2 l 
geometrical interpretation
of the Miller indices
h:k :l 
1 1 1
: :
x1 x2 x3
(h,k,l) plane
[abc] square brackets are
used to specify directions
in the direct lattice
(abc) parentheses are
used to specify directions
in the reciprocal lattice
= Miller indices
direction in the direct lattice [n1n2n3] = n1a1 + n2a2 + n3a3 is normal to (n1n2n3) plane for cubic lattice
probing crystal structure
interatomic distances in a solid ~ Å
electromagnetic probe of the crystal structure must have l ~ Å = 10-8 cm or shorter
→ X-rays or electrons
X-rays
photon energy   ck  c
2
l

hc
 12 keV
8
10 cm
Techniques for determining structure
• X-rays,neutrons,light atoms,electrons
Interatomic distances are on the scale of
Angstroms
For photons 1Å
E=10KV
For neutrons 1Å E=100mV
For He atoms 1Å E=20mV
For electrons 1Å E=100V
electrons
Louis de Broglie 1925: a particle with momentum p possess a wavelength l = h/p
Clinton J. Davisson and Lester H. Germer, 1927: direct experimental proof
by diffraction experiments
When an electron is accelerated through a potential difference V, it gains a kinetic energy
1 2
mv  eV
2
v
h
h
l 

p mv
h
 V 
 (1.23 nm) 

2meV
 1 Volt 
For V=50 V l 
2eV
m
mv  2emV
1/ 2
Accelerating electrons in a voltage
readily produces a beam of electrons
with a sub-nanometer wavelength
h
1/ 2
 (1.23 nm)  50
 0.17 nm  1.7 Å
2meV
Conditions for constructive interference: Constructive interference will occur for the rays
scattered from atoms if the difference in path length is a whole number of wavelengths
Scattering of waves from a plane of atoms
Scattering of waves from successive planes of atoms
1. Condition for constructive interference
for the rays scattered from neighboring
atoms separated by a distance d’
a e  c b  d'cos   d'cos   ml
2. Condition for constructive interference
for the rays scattered from successive planes
separated by a distance d
a b  b c  d sin   d sin   nl
These conditions can be satisfied simultaneously if  = . In that case m = 0 satisfies the first
condition for any d’, and nl  2d sin satisfies the second condition.

 
nl  2d sin 
X-ray Bragg diffraction:
intense peaks of scattered radiation are observed for certain wavelengths and directions
specular reflection by a plane
implies constructive interference
of rays scattered by individual
ions within the plane
Bragg peaks
incident ray
reflected ray
William Henry Bragg and William Lawrence Bragg, 1912:
the path difference 2dsin
the conditions for a sharp peak in the intensity of scattered wave
1 – the X-ray should be specularly reflected by the ions in one plane
2 – the reflected waves from successive planes should interfere
constructively
Bragg condition nl  2d sin
 – Bragg angle
2 – the angle by which the
n – order of the corresponding reflection
incident beam is deflected
the same lattice,
the same incident ray
but different direction
and l of the reflected ray
any family of planes
produces reflections
-3
-5
each
eachplane
planereflects
reflects1010-3––1010-5
ofofthe
theincident
incidentradiation
radiation
the Ewald construction
k-space
for incident wave vector k draw a sphere in k-space
there will be some wave vector k’ satisfying the Laue condition
K = k – k’ if and only if
some reciprocal lattice point lies on the surface of the sphere
in this case there will be a Bragg reflection from the family of
direct lattice planes perpendicular K
for a fixed incident wave vector k, i.e. for the fixed k = 2/l and the direction of k,
there will be in general no diffraction peak at all
to search experimentally for Bragg peaks either k = 2/l or the direction of k should be varied
the Laue method
the Laue method uses fixed incident direction and
nonmonochromatic X-ray beam containing
wavelengths from l1 to l0
the Bragg peaks will be observed
corresponding to any reciprocal lattice vectors K
lying between the spheres determined by
2
2
k0 
nˆ and k1 
nˆ
l0
l1
i.e. lying within the shaded region
the spread of l should be sufficiently large to find some Bragg reflections
and not too large to avoid too many Bragg reflections
if the incident direction lies along the symmetry axis of the crystal
the pattern of spots produced by the Bragg reflected rays will have the same symmetry
the rotating-crystal method
the rotating-crystal method uses monochromatic X-rays
and allows the angle of incidence to vary
by the crystal rotation
the Ewald sphere is fixed while the reciprocal lattice
rotates about the axis of rotation of the crystal
each reciprocal lattice point traverses a circle and
each intersection of such a circle with the Ewald sphere
gives the wave vector of a Bragg reflection ray
the powder or Debye-Scherrer method
Ewald sphere
K = k’ – k
1
K  2k sin 
2
k = 2/l
measured 
determine K
K = n2/d → determine d
incident ray k
uses monochromatic X-rays
the axis of rotation is varied over all possible orientations
by using polycrystalline sample or powder with randomly oriented grains
each reciprocal vector K generates a sphere of radius K about the origin
such a sphere will intersect the Ewald sphere in a circle
Bragg reflection will occur for any wavevector k’ connecting any point
on the circle of intersection to the tip of the vector k
the scattered rays lie on the cone that opens in the direction opposite to k
each K generates the diffraction ring (if K < 2k)

film
Synchrotron Radiation emits a broad spectrum of x-rays
X-rays
Monochrometer
Sample
Rotating
detector
electrons
Louis de Broglie 1925: a particle with momentum p possess a wavelength l = h/p
Clinton J. Davisson and Lester H. Germer, 1927: direct experimental proof
by diffraction experiments
When an electron is accelerated through a potential difference V, it gains a kinetic energy
1 2
mv  eV
2
v
h
h
l 

p mv
h
 V 
 (1.23 nm) 

2meV
 1 Volt 
For V=50 V l 
2eV
m
mv  2emV
1/ 2
Accelerating electrons in a voltage
readily produces a beam of electrons
with a sub-nanometer wavelength
h
1/ 2
 (1.23 nm)  50
 0.17 nm  1.7 Å
2meV
Atomic beams
• Limited to surface scattering because of
scattering cross sectionT
Typically a thermal He beam