1
Q
crystals have a
A
regular repetition (lattice) of a basic structural unit (basis)
2
Q
lattice
A
mathematical
array of points
3
Q
the basis can be
A
an atom or group of atoms (eg molecule or virus)
4
Q
elemental solids tend to be
A
crystalline
5
Q
inorganic technological materials are often crystalline. eg:
A
metals, steels, alloys (often polycrystalline)
Si, GaAs semiconductors
magnetic materials
superconductors like YBCO
6
Q
what materials are generally not crystalline
A
glasses, plastics and organic solids
BUT concepts from the study of crystals v useful for their study
7
Q
what is central to materials science
A
the link between structure and function
(function meaning magnetic, electric etc)
8
Q
most functionality depends on
A
structure
9
Q
to determine structure and functionality, we need to know:
A
- symmetry and the arrangement of atoms in crystals.
- the properties that emerge from this.
- how they are studied
10
Q
perfect crystals are
A
rare
why diamonds are so £££
11
Q
crystalline
A
perfect translation symmetry
12
Q
polycrystalline
A
crystalline regions
does have regions of perfect periodicity but lots of different regions
function depends more on the gaps than the atoms themselves
like grains of sand squashed together
13
Q
amorphous
A
no long range order
cannot see very much from electron microscopy
14
Q
translational symmetry and regular repeat pattern is described by
A
a lattice - a regular arrangement of points (not atoms!) in 2 or 3 dimensional space
15
Q
all lattice points are separated by an
A
integer multiple of lattice vectors
16
Q
a 2D lattice is defined by
A
two non co-linear basis vectors, a1 and a2
T=u1a1 + u2a2
u1,u2 integers
17
Q
can get from point 1 to 2 in a lattice with
A
linear combinations of the basis vectors
18
Q
lattice vectors define
A
cells and unit mesh, given by a1xa2
19
Q
unit cells translated by lattice vectors
A
completely fill space
20
Q
which options are best for choosing lattice vectors
A
no unique choice
shortest options or simple symmetry are best
21
Q
primitive cell
A
one lattice point per cell
22
Q
reasons for choosing cell with more lattice points
A
may give higher symmetry
eg simpler maths is larger rectangle than smaller parallelogram
23
Q
cell area scales with
A
lattice points
24
Q
symmetry is used to
A
distinguish lattices and will affect material properties
25
rotational symmetry
rotation about a lattice point gives an identical lattice
rotation by 2pi/n for n=1,2,3,4 or 6
26
mirror symmetry
reflection in a plane gives an identical lattice
27
inversion symmetry
inversion through a point gives an identical lattice
(x,y,z) --> (-x,-y,-z)
in 2D this is identical to a 180 degree rotation
28
symmetry - to get an identical lattice
1. rotation about lattice point
2. reflection in a plane
3. inversion through a point
29
how many distinct lattice types in two dimensions
5 - the Bravais lattices
30
what defines each lattice type
symmetry
31
the 5 Bravais lattices
oblique
rectangular primitive
rectangular centred
square
hexagonal (rhombic)
32
can describe everything with an oblique lattice BUT
will miss details of additional symmetries
33
oblique
|a1| does not = |a2|
angle does not =90 or 120
2 fold rotation
no reflection
34
rectangular primitive
|a1| does not = |a2|
angle =90
2 fold rotation
2 refelctions
35
rectangular centred
|a1| does not = |a2|
angle =90
2 fold rotation
2 refelctions
(step down obique but include extra step for rectangular = easier maths)
36
square
|a1|= |a2|
angle =90
4 fold rotation
4 refelctions
37
hexagonal (rhombic)
|a1|= |a2|
angle =120
6 fold rotation
6 refelctions
38
not all lattices are Bravais lattices - eg
honeycomb lattice adopted by graphene
this has a basis of two atoms per lattice point
39
by does graphene have a basis of two atoms per lattice point
**see diagram on slide 12 for visual**
points at each end of the hexagon have bonds going in opposite directions so not the same
so label A and B and have two atom basis
40
usually take unit cells with
corners on lattice sites
but don't need to
41
the Wigner-Seitz unit cell
same unit area as a primitive cell
centred on a lattice point
42
sketching Wigner-Seitz cell steps
1. draw line connecting nearest lattice points
2. draw the perpendicular bisector (exactly half way along step 1 lines and at right angles0
3. connect all bisectors to form cell
43
the Winger-Seitz unit cell will be used in analysis in reciprocal space of
k-states
phonons (lattice vibrations)
motion of charge carriers
44
a 3D lattice is defined by
three (non co-planar) basis vectors, a1,a2 and a3 so that the lattice vector (ie the vector between any two lattice points) is
T=u1a1 +u2a2 +u3a3 (u1,u1,u3 integers)
45
often write T=
[u1 u1 u3]
46
square brackets imply a
lattice vector
47
as in 2D, for 3D there is no unique set of vectors but usually look for
orthogonal, short and/or symmetry selected vectors
48
alternative method for defining 3D lattices
define the lattice using lattice vector lengths a1,a2,a3 and their mutual angles
alpha between a2 and a3
beta between a3 and a1
gamma between a1 and a2
49
vector lengths a1,a2,a3 are known as
lattice constants
50
3 dimensional lattices - two methods so make sure
can convert between
51
how many types of symmetries for 3D crystal lattices
7
52
the 7 3D crystal lattices
cubic (isometric)
hexagonal
trigonal
tetragonal
orthorhombic
monoclinic
triclinic
53
3D cubic
|a1|a2|=|a3|
alpha=beta=gamma=90
54
3D hexagonal
|a1|=|a2| does not=|a3|
alpha=beta=90
gamma=120
55
3D trigonal
|a1|=|a2|=|a3|
alpha=beta=gamma <120 (not=90)
56
3D Tetragonal
|a1|=|a2| does not =|a3|
alpha=beta=gamma=90
57
3D orthorhombic
|a1|does not =|a2|does not=|a3|
alpha=beta=gamma=90
58
3D monoclinic
|a1|does not=|a2|does not=|a3|
alpha=gamma=90
beta does not=90
59
3D triclinic
|a1| does not =|a2| does not=|a3|
alpha does not = beta does not = gamma
60
combining 7 crystal systems with 4 cell types gives
14 Bravais lattices:
cubic: P,I,F
hexagonal: P
trigonal/rhombohedral: R (primitive rhombohedral)
tetragonal: P,I
orthorhombic: P,C,I,F
monoclinic: P,C
triclinic: P
61
most metals and common semiconductors have lattices of
high symmetry
62
just as in 2D, primitive cells contain
just one lattice point
63
primitive cells - each of the 8 corners are shared between
8 neighbouring cells
each corner contributes an 1/8
x8 corners=1
64
using non-primitive cells can
simplify calculations
eg a lattice point centred in the face of a cell is shared between two cells
65
3D cells types - P
corner points only
8x1/8=1 lattice point/cell
66
3D cell types - F
corners plus centre of each face
8x1/8 +6x1/2 = 4 lattice points/cell
67
3D cell types - I
(body-centred, latin?)
corners plus centre of cell
8x1/8 +1=2 lattice points/cell
68
3D cell types - A,B or C (base-centred)
corners plus one set of face-centred points
8x1/8 +2x1/2=2 lattice points/cell
69
the lattice is an array of
delta function like points in space that describes crystal symmetry
70
the atoms associated with each lattice point are
the basis
71
mathematically, the crystal is
a convolution of the lattice and the basis
72
convolving f(x) with a delta function is like
duplicating f(x) at each delta location
73
lattice convolved with basis=
crystal
74
the basis may be
an atom, group of atoms or larger structures such as proteins and DNA
75
elemental metals often have a basis of a single atom
eg: fcc
Al, Ni, Cu, Rh, Au,...
76
elemental metals often have a basis of a single atom
eg: bcc
Li, Na, K, Rb, Cs, Fe,...
77
compounds may have a basis of a
formula unit
eg NaCl
or larger groups of atoms
eg molecules, proteins, viruses,...
78
bcc unit cells
commonly adopted by metals
body centred cubic cell has two lattice points per cell
79
bcc unit cells - for a basis on atom per lattice site
atoms touch along the bcc body diagonal
each atom has 8 nearest neighours
lattice constant a=4/root3 r for atomic radius r
80
a rhombohedral primitive cell can be constructed using
three vectors from an origin to adjacent 'centre' points
eg lattice vectors 1/2[1 -1 -1], 1/2[1 1 1] and 1/2[-1 -1 1] with mutual angles 109.5 degrees
81
bcc - the Wigner-Seitz primitive cell is a
truncated octahedron
82
fcc unit cells
another common metal structure
face centred cubic cell has four lattice points per cell
83
fcc unit cells for a basis of one atom per lattice site
atoms touch along the fcc face diagonal
each atom has 12 nearest neighbours - close packing
lattice constant, a=2 root2 r, derived from Pythagoras
84
a rhombohedral primitive cell can be constructed using
vectors from an origin to adjacent 'centre' points
eg lattice vectors lattice vectors 1/2[1 1 0], 1/2[1 0 1] and 1/2[0 1 1] with mutual angles 60 degrees
85
mutual angles are calculated using
vector dot and cross products
86
fcc Wigner-Seitz primitive cell is a
rhombic dodecahedron
87
optimal space filling
close packing
88
atoms in an fcc lattice in (111) planes
arrange in hexagonal patterns
6 neighbours in each layer plus 3 neighbours in each of the 2 surrounding layers = 12 neighbours
89
fcc close packing - the 3 repeating layers
ABCABCABCABC
next layer, either middle of upward or downward facing triangle
third layer the other that was not chosen
90
HCP cells
ABABABABAB
hcp does not have a cubic cell
described using a hexagonal lattice
91
real life cells
combination of ABAB and ABCABC
eg ABCABABABCABC
92
common industrial semiconductors in industry have a
diamond structure
93
epitaxy
diamond like structure allows crystals to grow on top of one another
94
C (diamond), Si, Ge and GaAs all adopt an
fcc structure with a 2-atom basis
all have tetrahedral network of covalent bonds
95
Miller indices - integers h,k,l - are used to
define crystal planes and vectors
96
lattice vectors are specified as
[h k l]
97
planes are specified (using their surface normal) as
(h,k,l)
98
miller indices - h is the
reciprocal of the intercept of the first plane away form the origin with the a1 axis
99
miller indices - k is the
reciprocal of the intercept of the first plane way from the origin with the a2 axis
100
miller indices - l is the
reciprocal of the intercept of the first plane away from the origin with the a3 axis
101
the (h,k,l) plane connects points
(a1/h 0 0), (0 a2/k 0) and (0 0 a3/l)
102
the (1 1 0) plane never intercepts the
z axis
1/inf = 0
103
lattice vectors are specified as
[h k l]
square brackets
where h,k,l are miller indices
104
h bar means
-h
use bars instead of - signs
105
a set of similar vectors is denoted
106
for a cubic system includes
cyclic permutations of h k l and their negatives
eg <210> includes [2 1 0], [1 0 2], [0 2 1bar] etc
107
planes are specified (using their surface geometry) as
(h,k,l)
108
{h,k,l} specifies
a family of planes
eg all the cyclic permutations and their negatives
109
**NOTE**
[h,k,l] vectors align along cartesian axes only for
cubic, tetragonal and orthorhombic systems
110
crystal planes - analogy to cinema seats/graveyard
looking down different angles, sometimes appears more transparent (different electron scattering events)
some angles will have easy propagation, others lots of scattering
111
in a 3D crystal, we will consider sets of
parallel planes
112
eg (100) planes all have
(100) surface normal vectors and cut the x-axis at integer values
(think of as stack of planes - like mirrors - give rise to scattering/interference effects)
113
plane (h,k,l) has normal vector n_khl=
1/sqrt(h^2+k^2+l^2) (h,k,l)
114
the spacing between planes (h,k,l) is
d_hkl
in a cubic system of lattice constant a,
=a/sqrt(h^2+k^2+l^2)
for orthonormic systems
=1/sqrt((h/a)^2+(k/b)^2+(l/c)^2)
gets much more complicated when alpha,beta, gamma do not =90 - not to worry in this course
115
diffraction from a grating - recap from W&D
plane waves incident
huygens wavelets emitted other side
116
W&D recap
diffraction peak for constructive interference
Δ=nλ=d sin(θ)
117
W&D recep
treat grating as
convolution of a single slit with an array of delta functions
118
W&D recap
diffraction pattern given by
convolution theorem
the multiplication of a single slit function and an array function
119
W&D recap
the diffraction pattern was loosely described as
the fourier transform of the grating with the pattern drawn in reciprocal space
120
W&D recap
spot position depended on
grating
121
W&D recap
spot intensity depended on
slit
122
slit function is a small thing in real space so
broad envelope in reciprocal space
123
lattice gives rise to
position
124
basis gives rise to
intensity
125
why is Bragg's law not enough
only has one angle
so cannot fully describe
126
crystals act as
3D gratings
127
atomic spacing in a crystal is around an angstrom so need to choose
radiation with matching wavelength
x-rays
electrons
neutrons
atoms and ion beams (not used as much)
128
diffraction occurs when the incident wavelength is
comparable to the sample periodicity
atoms are a few angstroms in dimension so optical radiation is not suitable
129
crystal diffractive techniques
x-rays
diffract strongly from crystals
scatter from sample electron density
penetrative so need around 100 micrometer samples
(oscillating E field excites electrons in the sample - gives rise to scattering event)
130
crystal diffractive techniques
thermal energy neutrons
scatter from nuclei and magnetic moment (since have spin associated)
very penetrative so mm samples
131
crystal diffractive techniques
electron beams
scatter via coulomb force
strong scattering
small samples (100nm) or surface reflection
132
crystal diffractive techniques
can diffract from crystal surfaces via Pauli repulsion forces
(less common technique)
133
x-ray sources - standard lab source is
an x-ray tube
(also used for medical radiography)
134
x-ray tube
an electron beam is fired into a metal target (Cu, Al etc)
excited atoms in target relax to emit x-rays
spectrum has characteristic peaks and broad background
135
x-ray tube in medical setting
rotating copper so hot spot changes to avoid the Cu melting
leaded glass vacuum tube to protect the user
136
x-ray sources
electron transitions
fire electrons
on occasion will ionise Cu
now have a core hole (unstable)
e- from higher drops down to fill hole
energy difference emitted as a photon
137
spectrum from x-ray sources
Bremsstrahlung background usually filtered out
very monochromatic
138
synchrotrons
brighter x-ray sources available at national facilities:
high energy electron beam is steered around a large circular vacuum tube
arrays of magnets - 'wigglers' force oscillating trajectory
can tune wavelength with magnets
139
x-rays emitted from either source are not monochromatic
how to fix this
1. using filters and/or diffraction from crystals at particular angles
2. synchrotrons can also produced polarised beams and pulses for more specialised experiments
140
x-rays interact with
'core' electrons - dense electron clouds surrounding nuclei
141
total integrated scattering for an atom is proportional to
atomic number
light elements are difficult to detect
heavy elements detected a lot more
142
why does most of a sample appear transparent?
penetration depth is high
need 100 micrometers-1mm samples for reasonable signal
143
θ-2θ or 'Bragg-Brentano' geometry
typically single crystal experiment
crystal and detector rotate concentrically
to maintain 2θ detection angle, tilt sample by dθ and detector by 2dθ
sharp peaks allow crystal lattice structure and lattice constant to be determined
144
powder diffraction techniques use a
powdered crystalline sample
multitude of crystals at random angles to x-ray
145
powder diffraction
observed pattern is
summation over all crystal orientations at once
individual crystals' diffraction spots merge into rings
146
powder diffraction
each Bragg angle θ produces a
diffraction cone of semi-angle 2θ
147
powder diffraction
collecting image
a camera/photograph can collect 2D image of Debye-Scherrer rings
or a detector can scan across detector plane to give series of peaks
148
what is super important to remember for x-ray diffraction
to convert to 2θ
lots of people lose marks for using θ instead of 2θ
**always sketch**
149
thermal energy neutrons have a de Broglie wavelength of
angstrom scale
150
kinetic energy relates to de Broglie wavelength by
E=p^2/2m (p=h/lambda)
151
neutron diffraction - neutrons scatter and diffract from
sample nuclei
152
neutron diffraction - H vs D
strong variations between isotopes
H has much larger neutron cross section that D
153
neutron diffraction - v weak interactions require
large (cm) samples
154
scattering from magnetic moments is useful for
determining magnetic structures
155
nuclear reactors are used to produce
fast neutrons
(typically large scale international facilities)
156
neutron production - 'spallation sources'
smaller scale facilities
eg ISIS facility at Rutherford Appleton Labs in Oxfordshire
157
neutron production steps
1. hit a heavy neutron-rich target with high energy protons
2. protons are accelerated and fired into a tungsten target
3. each proton collision releases 15 neutrons from a W atom
158
neutrons are typically produced with relatively high energies so must be moderated by
scattering from a material with light atoms (ie low Z)
159
production and moderation of neutrons produces a
broad spectrum of energies
ie not monochromatic
but signal is too low to 'throw away' neutrons by monochromation
160
neutron production - beam is usually
collimated and pulsed using shutters
161
neutron production - what is used to separate neutrons of different energy
'time of flight' techniques
162
most common diffraction techniques
x-ray
then electron diffraction
163
approx monochromatic electrons can be conveniently produced by
thermionic or field emission and can be accelerated to any desired energy
164
electron diffraction
approx eV
low energy electron diffraction
(LEED)
165
electron diffraction
keV to MeV
high voltage electron microscopy
(HVEM)
166
electron diffraction
modern lab instruments operate at
40-300keV
(ie at relativistic stage)
167
electron diffraction
need relativistic corrections for what energies?
anything >10keV
168
electron diffraction
relativistic correction term
(1+eV/2M0C^2)
169
how to adjust electron diffraction for required wavelength
changing voltage
changes energy
changes wavelength
170
electrons scatter strongly from materials
they scatter from
electrostatic potential of atoms (Coulomb interaction)
171
electron scattering depends on
atomic number, Z
(scattering scales roughly with Z: heavy elements scatter more)
172
electron diffraction
for transmission experiments, need
thin/small samples
typically <100nm
173
how does electronic charge make the manufacture of lenses easy
can focus and manipulate the electron beam with electric and magnetic field
174
electron diffraction is commonly implemented within an
electron microscope
multiple lenses to form condenser, objective and projector systems
can obtain real images and diffraction patterns of the same sample
175
comparison of optical and electron microscope
glass directs optical light using reflection/refraction
E and B fields direct the electrons
176
electron diffraction imaging
variety of imaging and diffraction modes
most data sets collected as 2D images
direct visualisation of atoms is possible
2D diffraction patterns collected via cameras - direct visualisation of the reciprocal lattice
177
Bragg scattering assumes
crystal planes to act as mirrors
with i=r
178
Bragg scattering - the diffracted beam is deviated through
2 theta
179
Bragg scattering - constructive interference arises when
path difference is an integer multiple of wavelength
n lambda = 2 d sin (theta)
180
Bragg scattering - multiple sets of planes (with different spacings, d) give
multiple diffraction peaks
181
Bragg scattering - von Laue approach reaches same conclusion without
specular reflection
182
generalised scattering - set up
consider two scattering sites, O and P, within a sample with position-dependent electron density n(r)
site P is at position r with respect to the origin, O
183
generalised scattering - incident and scattered wavevectors
incident wavevector k with |k|=2pi/lambda
scattered wavevector k' with |k'|=2pi/lambda (elastic scattering)
184
generalised scattering - scattered amplitude from
point P (prop to n(r) dV - ie prop to electron density)
185
generalised scattering - scattered phase will depend on
path difference
186
generalised scattering - scattered intensity
from integrating over whole specimen
187
generalised scattering - path difference here is
Δ = AP-OB
(little extra at top - little extra at bottom)
188
generalised scattering - using vectors, AP and OB=
AP=k.r/|k| and OB=k'.r/|k'|
subbing into Δ = AP-OB gives Δ=K.r/|k| where K=k-k' and |k|=|k'|
189
geometric way to check Δ=K.r/|k|
trig
triangle with corners O,A,P
k.r=|k||r|cos(alpha)
k.r/|k|=|r|cos(alpha) = AP
190
K
scattering vector
central to much of the rest of the course
191
phase difference between scattered waves
Φ = 2pi/lambda Δ = 2pi/lambda k.r-k'.r/|k| = K.r
192
total scattered amplitude
integrating over sample
Ψ(K) = ∫sample n(r)e^iK.r dV
193
atomic scattering factor
what we get for the scattered amplitude for a single atom
f(K) = ∫atom n(r)e^iK.r dV
194
both x-rays and electrons scatter from a
sample's electron density
195
atomic scattering plots for x-rays and electrons
functional form similar, with greater high-angle scattering for x-rays
strongly peaked plot with x-ray above electrons
196
the scattering factor is effectively
the Fourier transform of a sample or atom's electron density
(will get similar result for neutrons, ions etc where n(r) is exchanged for appropriate scattering distribution)
197
structure factor
by extending the integration across the unit cell
198
n(r) for the unit cell
close to that for the um of individual atom terms (can write as sum of integrals over atoms)
we know each cell is identical - phase will change but amplitude will remain same
ie bonding effects are slight
199
position vectors need a
common origin: r=ri+pi for p the position within atom wrt nucleus, r distance from origin
200
scattering from unit cell, final equation
F(K) = Σ over j of fj exp(iK.rj)
summing over j atoms in cell
fi amplitude, exp(iK.rj) phase
201
total scattered amplitude, Ψ(K) is the
fourier transform of n(r)
202
since n(r) is periodic with a well-defined period a, we expect Ψ(K) to
have sharp, delta function like features
ie diffraction spots
203
nr and Ψ(K) FT - only takes tiny deviations from constructive interference to get
back to destructive
multiplying billions of times if billions of cells - even the tiniest difference will eventually result in a large difference
204
a diffraction grating of spacing d and slit width of a is described by
f(x)*g(x)
the convolution of a grating function g(x) (array of delta func)
and a slit function f(x)
205
2D diffraction grating - diffracted amplitude is
Ψ(k) = F(k)G(k)
where F(k) and G(k) are FT of f(x) and g(x)
206
|Ψ(k)|^2 has a series of
sharp peaks, G(k)^2, with an F(k)^2 envelope
207
n(r) is identical for each unit cell with scattered waves from each cell only differing in phase
so Ψ(k) =
F(K)Σ exp(iK.Tn)
summing over n cells
(each term in summation has magnitude 1 but varying phase)
208
Tn is
translation vector linking lattice points
T=u1a1+u2a2+u3a3
us are integers, as are lattice basis vectors
209
scattering from whole sample - constructive interference occurs when
contributions to summation have phase difference of 2pi
ie K.Tn = 2mpi
for integer m
(equivalent of integer no of wavelengths for W&D)
210
when K.Tn=2mpi for integer m across n cells, Ψ(k) =
n F(K)
where F(K) is the structure factor
211
when K.Tn=2mpi, the intensity of diffracted beam is
I prop to |Ψ(k) |^2 = n^2F(K)^2
212
reciprocal lattice - looking for vectors G that satisfy
G.Tn = G. (u11a1+u2a2+u3a3) = 2mpi
G will have dimensions of inverse length
G will be an array of discrete points since T is
213
possible G vectors define a
reciprocal lattice in reciprocal- or k-space
214
reciprocal lattice basis vectors
b1.b2.b3
for h,k,l integers, G=hb1+kb2+lb3
215
each point in the reciprocal lattice corresponds to
a vector and unique h,k,l
216
reciprocal lattice - it can be shown that we will need ai.bj=
2piδij
where δij is the Kronecker delta
ie a1.b1=a2.b2=a3.b3=2pi and a1.b1=a2.b2=a3.b3=0
217
ai.bj=2piδij implies that b1 will be
perpendicular to both a2 and a3
218
b1=
2pi a2xa3/a1.(a2xa3)
=2pi/V a2xa3
(same approach for b2 and b3)
V is the volume of the primitive unit cell in terms of a
eg Vfcc=a^3/4
219
we have Ψ(k) =F(K)Σ exp(iK.Tn) approx nF(G)
what does this mean for diffraction?
strong diffraction for K=G
strong diffraction when the scattering vector corresponds to a point on the reciprocal lattice
220
final form of G=
2pi / d_hkl n_hkl
n_hkl is the normal vector to the planes producing diffraction
d_hkl is the separation of those planes
221
plane (khlP intersects crystal axes at
A=a1/h
B=a2/lk
C=a3/l
222
vectors AB and BC
lie in the plane
AB=a2/k-a1/h
BC=a3/l-a2/k
223
the plane normal then lies along
ABxBC
find using cross product
=V/2pi hkl G_hkl
this is surface normal parallel to the G vector
224
(hkl) planes are spaced by
d_hkl
where d_hkl is the length of a line that is perpendicular to the plane (khl) and passes through the origin
ie d_hkl=|OP|
225
geometrically, how to find d_hkl=|OP|
triangle - points O,A,P
|OP|=|OA|cos phi
use vector dot product
226
d_hkl=|OP|=
a1/h . G_hkl/|G_hkl| = 2pi/|G_hkl|
(G_hkl/|G_hkl| is a unit vector)
227
G_hkl has length
2pi/d_hkl
and lies perpendicular to the (hkl) planes
228
we had K=k-k' and |k|=|k'| so for scattering through 2 theta ,|K|=
|k-k'|=2|k| sin (theta)
229
general diffraction condition is
K=G
230
looking at just magnitudes, |K|=|G| gives
2|k| sin theta = 2pi/d_hkl
inserting |k|=2pi/lambda and rearranging yields lambda=2d_hkl sin(theta)
231
what is super important to remember for Braggs law
almost always n=1 as have effectively combined order with d
232
the Bragg condition
G=K=k-k'
233
start with Bragg condition
using consine rule and as |k|=|k'| we get
|G|^2 = 2G.k
or Ghat.k=|G|/2
where Ghat is the unit vector in the direction of G
234
interpret |G|^2 = 2G.k or Ghat.k=|G|/2 as
the equation of a plane in k space, perpendicular to G, distance |G|/2 from origin
235
any k-vector starting at the origin and finishing on this plane will satisfy
the Bragg condition and undergoes strong diffraction
236
the first Brillouin Zone
the smallest volume enclosed by a set of planes in k space, perpendicular to G, distance |G|/2 from origin in reciprocal space
the reciprocal lattice Wigner-Seitz cell
237
the reciprocal lattice must have the form of
one of the 14 Bravais lattices
238
general form of the reciprocal lattice can be determined using
bi=2pi/V ajxak
239
points to note when calculating reciprocal lattices
b1 perpendic to a2 and a3
b2 perpendic to a3 and a1
|a1|>|a2| means |b1|<|b2| (ie long axis in real space=short axis in reciprocal space)
240
calculating reciprocal lattices - for a primitive cubic system, the lattice vectors are
a1=ax, a2=ay, a3=az
so b1=2pi/V a2xa3 = 2pi/a yxz = 2pi/a x
241
reciprocal lattices represent
where diffraction spots might be observed
(but we get systematic absences etc)
242
the bcc reciprocal lattice
the primitive cell vectors take the form
a/2 (xyz)
ie corner of cell to 3 of the nearest body centres
243
bcc reciprocal lattice
cross product calculation from bcc primitive cell vectors givnes
vectors from the origin to centre of face
ie FCC lattice in reciprocal space
244
a real fcc lattice has a reciprocal
bcc lattice
245
what causes the fcc systematic absences
adding in atoms in the middle of face
more scattering
more destructive interference
absences
246
allowed G vectors relating to an fcc real lattice will be given by
2pi/a (h,k,l)
247
F(G) calculation for fcc
all atoms the same so can extract fi from the summation
simplify to get
F(G)= f(G)(1+exp(ipi(h+k)+exp(ipi(h+l)+exp(ipi(k+l)
248
F(G)= f(G)(1+exp(ipi(h+k)+exp(ipi(h+l)+exp(ipi(k+l)
if h,k,l are all even
F(G)=f(G)[1+1+1+1]=4f(G)
(h,k,l integers so k+k, h+l, k+l integers)
249
F(G)= f(G)(1+exp(ipi(h+k)+exp(ipi(h+l)+exp(ipi(k+l)
if h,k,l are all odd
F(G)=f(G)[1+1+1+1]=4f(G)
250
F(G)= f(G)(1+exp(ipi(h+k)+exp(ipi(h+l)+exp(ipi(k+l)
if h,k,l are a mix of even and odd
F(G)=f(G)[1+-1+1+-1]=0
(2 of the combinations negative)
this is the extinction condition, producing systematic absences arising from destructive interference of scattering from atoms within the cell
251
for determining allowed fcc diffraction spots, consider 0 to be
an even number
252
since |G| is prop to sqrt(h^2+k^2+l^2) enumerating spots vs sqrt(h^2+k^2+l^2) can be useful as
increasing radius in k space
ie further from origin
253
fcc reciprocal lattice
what integer for the sum of the squares is missing
7
254
reciprocal lattice for fcc requires
(h,k,l) to be all odd or even
255
systematic absences for ordered systems with more than a 1 atom basis
instead of no spot get a very weak spot
the different atoms will have different form factors so now taking the difference so left with a small number
256
diffraction ring intensity also depends on
the number of planes in the {hkl} family
eg intense spot at 210 due to multiplicity (lots of permutations)
257
diffraction patterns are
slices through the 3D reciprocal lattice as if viewed along the incident vector
258
diffraction patterns are 2D - so need to
rotate sample
(particularly important for x-ray diffraction)
259
Bragg condition |G|^2=2G.K is a scalar equation - it gives
radial distance but not 3D position
260
the Ewald sphere
a 3D visualisation in reciprocal space
261
steps for drawing the Ewald sphere
1. draw vector -k back from lattice origin to define source S
2.draw a sphere of radius |k| centred on the start of the incident k-vector
262
the Ewald sphere is the locus of all
points satisfying |k|=|k'|
263
Ewald sphere - if sphere intersects another lattice point then
K=G is satisfied
264
Ewald sphere - reciprocal lattice points on sphere are observed as
diffraction spots
265
Ewald sphere for x-rays
very few spots observed so rotate to get more
(need more scattering angles)
266
for x-rays, the Ewald radius is comparable to the reciprocal lattice spacing so
have large diffraction angles
but few diffraction peaks
typically very few diffraction spots are formed so crystal is rotated to explore many scattering angles
267
Ewald sphere for fast electrons - wavelength a factor of 100 smaller than d
so ewald r = 100 x reciprocal lattice spacing
this means
small diffraction angles
many diffraction peak excited
central part is almost flat - almost planar when it intercepts
much less rotation needed
268
at higher angles, the Ewald sphere intersects with other reciprocal lattice planes to produce
rings of diffraction spots known as Higher Order Laue Zones
269
electron diffraction samples typically very thin - reciprocal lattice spots are
streaks that are easier to intersect
P4 Solid State Physics
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