3.012 Fund of Mat Sci: Structure – Lecture 19
FROM DIFFRACTION TO STRUCTURE
Images removed for copyright reasons.
3-fold symmetry in silicon along the [111] direction.
Forward (left) and backward (right) scattering.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Mon Nov 28
• Study: 3.3 Allen-Thomas (Symmetry
constraints)
• Read all of Chapter 1 Allen-Thomas
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time:
1.
2.
3.
4.
Laue condition in 3 dimensions
Ewald construction
Bragg law, and equivalence to Laue condition
Powder diffraction, Debye-Scherrer
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Atoms as spherical scatterers
Images removed for copyright reasons.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Huygens construction
(Left-pointing 1st, 2nd, and
3rd order, and all higher order
beams not shown.)
0th Order
1st Order
2nd Order
3rd Order
Groove
Incident Plane Wave (Lambda = 2/11 * Grating Pitch)
Diffraction Grating
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
All three Laue conditions
r r
S − S0 = integer multiple of λ
r r
S − S0 = integer multiple of λ
r r
S − S0 = integer multiple of λ
(
(
(
r
a1 ⋅
r
a2 ⋅
r
a3 ⋅
)
)
)
S0
a.S0
a.S
r
a2
a
S
Figure by MIT OCW.
r
a1
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Ewald construction
Diffracted
Beam
2π S/λ
Incident
Beam
2π/λ
θ
(hkl)
ν
θ
ν
θ
2π S0/λ
(
(
*
S-S0 2π
dhkl
=
λ
Reciprocallattice Origin
Ewald Sphere
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Equivalence to Laue condition
( nλ = d hkl 2sin θ )
Diffracted
Beam
2π S/λ
Incident
Beam
2π/λ
θ
(hkl)
ν
θ
ν
θ
2π S0/λ
(
(
*
S-S0 2π
dhkl
=
λ
Reciprocallattice Origin
Ewald Sphere
Figure by MIT OCW.
r r
⎛ S − S0 ⎞ 2π
2π 2π
∗
2π ⎜
cosν = d hkl =
2sin θ
=
⎟=
λ
d hkl
⎝ λ ⎠ λ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Laue condition needs “white” spectrum
202
201
Reflected
Beam
Incident
Beam
102
201
200
200
Reflected
201
Beam
201
Reflected
Beam
202
Reflecting Sphere for
Smallest Wavelength
002
001
101
100
2π/λmax
000
2π/λmin
101
001
102
002
Reflecting Sphere for
Largest Wavelength
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Powder diffraction
Image removed for copyright reasons.
Please see the diagram at http://capsicum.me.utexas.edu/ChE386K/html/powder_diffraction_3.htm.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Interplanar spacings
Cubic: d
2
hkl
2
a
= 2
2
2
h +k +l
⎛ ∗
2π ⎞
⎜ d hkl =
⎟
d hkl ⎠
⎝
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Debye-Scherrer camera
nλ = d hkl 2sin θ
2
Cubic: d hkl
a2
= 2
h + k2 + l2
Figure by MIT OCW.
⎛ 2sin θ ⎞
h +k +l = a ⎜
⎟
⎝ nλ ⎠
2
2
2
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
2
2
Tables removed for copyright reasons. See http://www.matter.org.uk/diffraction/x-ray/indexing_powder_pattern.htm
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Systematic
absences
Image removed for copyright reasons.
Please see the table at http://capsicum.me.utexas.edu/ChE386K/html/systematic_absences.htm.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Effects of symmetry on diffraction
Images removed for copyright reasons.
Please see the images at http://capsicum.me.utexas.edu/ChE386K/html/diffraction_symmetry1.htm.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Structure Factor
Images removed for copyright reasons.
Please see the graph at http://capsicum.me.utexas.edu/ChE386K/html/scattering_factor_curve.htm.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Friedel’s law
• The diffraction pattern is always
centrosymmetric, even if the crystal is not
centrosymmetric
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Point symmetry + inversion = Laue
Image removed for copyright reasons.
Please see the table at http://capsicum.me.utexas.edu/ChE386K/html/diffraction_symmetry2.htm.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
X-ray powder diffraction images removed for copyright reasons.
• X-ray powder diffraction for silver,
aluminum, gold, and copper
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
X-ray powder diffraction images removed for copyright reasons.
•
Detail, back-scattering direction, showing the line splitting that takes place due
to the presence of the K-alpha-1 and K-alpha-2 lines of the copper spectrum
which the x-ray machine produced. Measurements of the diffraction angles for
these lines can yield very accurate values for the crystal unit cell size.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Rocksalt
X-ray powder diffraction images removed for copyright reasons.
Source: Wikipedia
•
Detail of the small-angle end of the film, showing that the NaCl and KCl patterns do not look the
same, and that the crystal sizes are different. The KCl appears almost exactly as if it were a simple
cubic, due to the fact that the K and Cl ions, very close to each other on the periodic table, are almost
exactly alike. The Na and Cl ions are not so close on the periodic table, thus their ions are not the
same, thus the powder diffraction pattern does not look like the pattern from a simple cubic crystal.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Physical properties and
their relation to symmetry
•
•
•
•
•
Density (mass, from a certain volume)
Pyroelectricity (polarization from temperature)
Conductivity (current, from electric field)
Piezoelectricity (polarization, from stress)
Stiffness (strain, from stress)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Scalar, vector, tensor properties
• Mass (0), polarization (1), strain (2)
Z
Undeformed
Deformed
P u P'
Y
Z
X
P
R
o
u = R' - R
P'
R'
Y
X
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Transformation of a vector
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Orthogonal Matrices
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Transformation of a tensor
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Neumann’s principle
• the symmetry elements of any physical
property of a crystal must include all the
symmetry elements of the point group of the
crystal
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Tensor properties of materials
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Symmetry constraints
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Curie’s Principle
• a crystal under an external influence will
exhibit only those symmetry elements that
are common to the crystal without the
influence and the influence without the
crystal
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)