3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

3.012 Fund of Mat Sci: Structure – Lecture 19

FROM DIFFRACTION TO STRUCTURE

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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. Laue condition in 3 dimensions2. Ewald construction3. Bragg law, and equivalence to Laue condition4. 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

Groove

Incident Plane Wave (Lambda = 2/11 * Grating Pitch)Diffraction Grating

0th Order1st Order

2nd Order3rd Order

(Left-pointing 1st, 2nd, and3rd order, and all higher orderbeams not shown.)

Figure by MIT OCW.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

All three Laue conditions( )( )( )

1 0

2 0

3 0

integer multiple of

integer multiple of

integer multiple of

a S S

a S S

a S S

λ

λ

λ

⋅ − =

⋅ − =

⋅ − =

r rr

r rr

r rr

S0

a.S0

a.Sa

S

2ar

1ar

Figure by MIT OCW. Figure by MIT OCW.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Ewald construction

θθ

θ

ν

ν

(hkl)

2π/λ

2π S/λ

Diffracted Beam

Reciprocal-lattice Origin

Ewald Sphere

Incident Beam 2π S0/λ

λdhkl 2πS-S0=( (*

Figure by MIT OCW.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Equivalence to Laue condition

0 2 2 22 cos 2sinhklhkl

S S dd

π π ππ ν θλ λ λ

∗⎛ ⎞−= = = =⎜ ⎟

⎝ ⎠

r r

( )2sinhkln dλ θ=

θθ

θ

ν

ν

(hkl)

2π/λ

2π S/λ

Diffracted Beam

Reciprocal-lattice Origin

Ewald Sphere

Incident Beam 2π S0/λ

λdhkl 2πS-S0=( (*

Figure by MIT OCW.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Laue condition needs “white” spectrum

202201 Reflected Beam

Incident Beam 200

Reflected Beam

102

201 001101

1002π/λmax

2π/λmin

200 000

001

002

101201

002102202

201Reflected Beam

Reflecting Sphere forSmallest Wavelength

Reflecting Sphere forLargest 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

22

2 2 2

2Cubic: = hkl hkl

hkl

ad dh k l d

π∗⎛ ⎞= ⎜ ⎟+ + ⎝ ⎠

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Debye-Scherrer camera

2sinhkln dλ θ=

22

2 2 2Cubic: hklad

h k l=

+ +

22 2 2 2 2sinh k l a

nθ

λ⎛ ⎞+ + = ⎜ ⎟⎝ ⎠

Figure by MIT OCW.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

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 forthese lines can yield very accurate values for the crystal unit cell size.

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)

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)

Y

Und

efor

med

X

DeformedZ

X

P P'u

R

PP'

R'

u = R' - R

o

Z

Y

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