UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures

$: UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures$
Other diffraction techniques and common crystal structures of real materials

Shyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Other diffraction techniques¡ XRD is the most common (and typically most economical)

means of determining crystal structures.

¡ Electrons and neutrons are also widely used scattering techniques.¡ Employs wave-particle duality of electrons and neutrons¡ Transmission electron microscopes (TEM) are more expensive than

XRD equipment, but still common¡ Neutron diffraction is typically performed at national and

international reactor facilities

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 13

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Comparison of different diffraction techniques

X-rays

λ ~ 0.1nm

Uncharged

No magnetic dipole

Scattered by atomic charge densities

(~0.1nm)

Electrons

λ ~ 0.002nm

Charged

Possesses magnetic dipole

Scattered by electronic and nuclear charge densities (~0.1nm)

Neutrons

λ ~ 0.1nm

Uncharged

Possess magnetic dipole

Scattered by atomic nucleus (~0.0001nm)


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Neutron diffraction¡Neutrons are scattered by crystalline solids

¡Uncharged: Penetrates deeply into most materials (allows for sample with large volumes ~ cm3)

¡ Interacts with atomic nuclei and magnetic dipole moment of nuclei – can provide information on magnetic point and space group symmetry.

¡No interaction with electron cloud


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Applications of Neutron Scattering¡ Elastic nuclear scattering (Bragg scattering)

¡ Ability to locate light atomic species in presence of heavy atoms (no simple dependence on Z)

¡ Elastic magnetic scattering¡ Probe magnetic structure

¡ Inelastic scattering¡ Interaction of low-energy neutrons with vibrating crystal lattice and spin waves¡ Probe magnetic phase transitions

¡ Isotopic substitution¡ Isotopes can have different neutron scattering¡ Tailor scattering factors to be more or less sensitive to a particular element


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Neutron atomic scatter factors¡ No simple dependence

on Z (unlike X-rays!)

¡ Variations in b due to resonance absorption in compound nucleus formation

¡ Negative scattering lengths for some elements (H, Li, Ti, V, Mn)

¡ V has smallest absolute value of b (i.e., mostly transparent to neutrons and is typically used as sample holder.


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Comparison of different diffraction techniques

X-rays

λ ~ 0.1nm

Uncharged

No magnetic dipole

Scattered by atomic charge densities

(~0.1nm)

Electrons

λ ~ 0.002nm

Charged

Possesses magnetic dipole

Scattered by electronic and nuclear charge densities (~0.1nm)

Neutrons

λ ~ 0.1nm

Uncharged

Possess magnetic dipole

Scattered by atomic nucleus (~0.0001nm)


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Electrons as a diffracting beam¡ Electrons can be accelerated by electric fields

¡ Velocities can be significant proportion of speed of light –need relativistic corrections


λ =hmv

=h

2m0eV (1+e

2m0c2 V )

=1226.39

V + 0.97845×10−6V 2

Note that wavelengths are in pm!

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Interaction of electrons with crystal lattices¡ Strong interactions with both electron clouds and positively charged

nuclei

¡ Probability of scattering of electron beam is ~ four times higher than for X-rays

¡ Electron scattering factor can be related to the X-ray scattering factor by the Moth-Bethe formula:

¡ High probability of scattering leads to multiple scattering events (dynamic scattering) è electron scattering must be treated with dynamic scattering theory


f el (s) =e

16π 2ε0 s2 [Z − f X (s)] where s = sinθ

λ

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Electron diffraction geometry¡ Wavelengths for electrons are much smaller¡ Diffraction angles are much smaller (typically on the order of milliradians)¡ Bragg condition can be approximated by

¡ Need to only look close to incident beam to find diffracted beam¡ Ewald spheres are ~ 100-1000 times larger than X-rays (possible to orient

crystal such that whole plane of reciprocal lattice points are tangent to sphere)


2dhklθ = λ

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Sample preparation¡ Strong interaction necessitates the use of very thin or

small samples (~ few hundred nm)

¡ Unlike neutrons and X-rays, electrons are absorbed by matter

¡ Implications:¡ Sample preparation is time-consuming and difficult¡ Infinite crystal assumption is no longer valid and reciprocal lattice

points can no longer be treated as zero-volume mathematical points (reciprocal point shape is the reciprocal of the extent of crystal – for thin films, this means a cylinder extended in direction of thin film)

¡ Reciprocal point volume can intersect Ewald sphere even when actual point is not on Ewald sphere – increases probability of diffraction


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Transmission electron microscope (TEM)


Heating W filament emits electrons

Focuses and directs electron beam using magnetic fields

Main image forming lens

Fairly large- Can be > 3 stories

high!- Vacuum is maintained

as electrons cannot travel very far in air

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Synchrontron X-ray sources¡ X-rays produced by circular motion of charged particles,

e.g. electrons (recall that X-rays are produced by acceleration of charged particles)

¡ Properties¡ High beam intensity: continuous source with five or more orders of

magnitude higher intensities than X-ray tubes¡ Broad radiation spectrum¡ Strong polarization: Nearly 100% linearly polarized, circular or

elliptical polarizations possible


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Example applications of SynchrotonXRD

¡ In-situ study of crystallization with heating¡ High-flux allows for diffraction patterns to be taken in short

amount of time

¡Study of superlattice orderings, e.g., ordered α’-FeCo¡ Superlattice reflections can be several orders of magnitude

less intense than fundamental reflections


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Common crystal structures


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Concept of tessellation¡Tessellation – the filling of space using geometric

shapes with no overlap and no gaps¡ 2D - Tiling using polygons¡ 3D – Space filling using polyhedra

¡Frequently used in the way that materials scientists think about crystal structures

¡Topic is vastly and mathematically beautiful, but in this course, we will focus only a few simple concepts and build some common crystal structures


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Regular polygons and polyhedra¡Regular polygons and polyhedra have identical sides

and angles.¡2D¡ Equilateral triangle¡ Square¡ Regular hexagon

¡3D (also known as platonic solids)¡ Tetrahedron¡ Cube¡ Octahedron¡ Icosahedron¡ Pentagonal dodecahedron


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2D tilings¡ Regular tilings (monohedral tilings)

¡ Labeled using the Schläfli symbols (nm, where n is number of the sides of the polygon and m is the number of polygons meeting at a vertex)

¡ Archimedean tilings¡ Still uses regular polygons, but more

than one type allowed.¡ All vertices remain identical.¡ Only 8 additional possible tilings

¡ K-uniform and other tilings¡ Further relaxation of number of

unique vertices and other conditions.


36 44 63

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Stackings of 36 tilings¡ 3D crystals can be constructed by stackings of 2D tilings.

¡ An important class of structures – the closed-packed structures – comprise of stacking of 36 tilings


• A 36 tiling has plane group p6mm• If we consider a single layer of atoms

occupying “A” sites, two kinds of interstitial sites are created – “B” sites and “C” sites.

• Subsequent layers can occupy either B or C sites to form close-packed structures, and the process can be repeated to generate infinite variations of stacking

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Two common forms of closed packing¡ Hexagonal close-packed¡ ABAB stacking

¡ Face-centered cubic¡ ABCABC stacking


Fcc perspective views

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Interstitials in close-packed structures¡ Tetrahedral interstitial

¡ Four-fold coordinated¡ Formed by having an atom atop three atoms

¡ Octahedral interstitial¡ Formed when there are no atoms sitting directly on top of

the interstitial¡ Six-fold coordinated¡ Larger than tetrahedral site¡ Interstitial can be labeled as α, β, or γ depending on

whether the site is formed by B-C, A-C or A-B planes respectively.

¡ (Recall that NaCl can be thought of as having an fcc lattice of Na with Cl sitting in the oct interstitial)


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Notations for close-packed structures¡ ABC notation¡ Can be modified to describe compounds (e.g., CdI2 with Cd occupying

tetrahedal interstices between close-packed I planes can be denoted as α BC…

¡ E.g., SiC has many polytypes that differ simply in the stacking sequence of Si (C simply occupies the tetrahedral interstitials).

¡ Ramsdell notation¡ Total number of close-packed layers followed by a letter that indicates

whether the lattice type is cubic (C), hexagonal (H), or rhombohedral (R). If two or more structures have the same lattice type and the same repeat period, a subscript a, b, c, or 1, 2, 3 is used to distinguish between structures.

¡ E.g., SiC has two hexagonal polytypes with stacking sequences ABCACB . . .and ABCBAB . . . for the Si atoms. They are distinguished by their subscripts as 6H1 and 6H2, respectively.

¡ Does not specify actual stacking sequence


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Crystal structures of elements


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Parent, derivative and superlatticestructures

¡Derivative structures are those that are derived from simpler structures by substitution of one atom for another

¡ Interstitial structures are obtained by ordered occupation of subsets of interstitial sites in simple structures


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Definition of Superlattice¡ Consider two lattice’s whose basis vectors are related by:

¡ If Mij are integers and det(M) = 1, then the two lattices coincide.¡ If Mij are integers and det(M) > 1, then the lattice defined by A’ is a

superlattice of that defined by A.¡ If Mij are integers and det(M) < 1, then the lattice defined by A’ is a

sublattice of that defined by A.


!A =

a1!

a2!

a3!

"

#

$$$$$

%

&

'''''

=

M11 M12 M13

M21 M22 M23

M31 M32 M33

"

#

$$$$

%

&

''''

a1a2a3

"

#

$$$$

%

&

''''=MA

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Classification of structures¡ Strukturbericht symbols¡ Letter followed by number¡ Sequential numbering by order of

discovery, e.g., A2 refers to bcc structure.¡ Frequently used in materials science

literature

¡Pearson symbol¡ Bravais lattice symbol (a, m, o, t, h,

c) + centering (P, C, I, F) + # of atoms in unit cell.

¡ E.g., NaCl has Pearson symbol cF8.

¡ Not unique.


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Structure of metals¡ Derivative structures are based on fcc, bcc, and hcp

parent structures¡ fcc (A1) and hcp (A3) – Stackings of 36 tiles.¡ bcc (A2) – Stacking of 44 tiles.¡ fcc and bcc are derivatives of the simple cubic structure, which is

rarely observed for elements (though common in compounds) ¡ fcc and hcp are close-packed – packing fraction of 74.05%. bcc is

not close-packed – packing fraction of 68.02%.


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Atomic sizes¡ Some metals have more than one allotrope¡ Fe exists in bcc and fcc structures¡ Bcc lattice constant = 2.8664 Angstroms¡ Fcc lattice constant = 3.6468 Angstroms

¡ If we treat Fe atoms as touching spheres, we can derive associated atomic radii for the two structures¡ Bcc spheres are touching along [111]

¡ Fcc spheres are touching along [110]

¡ ~4% larger radius for fcc


4rbcc = a2 + a2 + a2 → rbcc =3a4

=1.242nm

4rfcc = a2 + a2 → rfcc =2a4

=1.289nm

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Interstitial sites in fcc and bcc¡ How many of each

type of interstitials are there in the bcc and fcc unit cells?¡ Bcc¡ # oct sites = 6 x 0.5 +

12 * 0.25 = 6¡ # tet sites = 6 x 4 x

0.5 = 12¡ Fcc¡ # oct sites = 1 + 3 x 4

x 0.24 = 4¡ # tet sites = 8


bcc

fcc

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Packing fractions¡ Ratio of the volume occupied by atoms to the total volume of

unit cell

¡ Kepler’s conjecture: No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements – recently proved with help of computers by Thomas Hale


pfcc =4× 43πrfcc

3

a3= 74.05%

pbcc =2× 43πrbcc

3

a3= 68.02%

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Alloys¡ Substitutional solid solutions¡ Same crystal structure as components¡ Lattice constant governed by Vegard’s law (with some deviations)

¡ Interstitial alloys¡ One component atom sits in interstitial sites of another large

component atom¡ Lattice constant determined by strain effects


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Hume-Rothery rules1. Atomic size factor: The range of solid solubility will be restricted if

the atomic radii differ by more than about 15%.

2. Electronegativity valency effect: Large electronegativity differences between components of a binary alloy can promote charge transfer and differences in the covalency, ionicity, or metallicity of the bonds. This leads to bond energy differences between A−A, A−B, and B−B bond energies in the alloy. A strong proclivity for A−B bond formation can lead to the formation of stable compounds.

3. Relative valency effect: A metal of lower valency is more likely to dissolve in a metal of higher valency than vice versa. This rule is not universally obeyed.


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Derivative and superlattice structures in alloys¡ At high temperatures, alloys are typically disordered. As temperature

is decreased, some alloys undergo a disorder-order phase transition resulting in an ordered solid solution or superlattice or superstructure.

¡ fcc (Fm-3m) derivatives:


L10: CuAu, FePt, FePd L12: Cu3Au, Au3Cd, AlCo3

Tetragonal P4/mmm Cubic Pm-3m

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XRD patterns


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Fcc- interstitial substitution¡Diamond cubic (A4)¡ Tet interstial in fcc occupied¡ Common for many Group IV

semiconductors (Si, Ge)

¡Rocksalt (B1)¡ Oct interstitial occupied by a different

atom¡ Common for many ionic compounds

(NaCl, LiF, etc.)


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Bcc derived structures¡ CsCl structure

¡ 2x2x2 bcc superlattices (characteristics of both fcc and bcc)


D03: BiF3, Fe3Si, AlF3 L21: A2BC – Cu2MnAl

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Diamond cubic derived structures

¡Zincblende (B3)¡ III-V and II-VI compounds¡ ZnS, AsGa, InSb

¡Fluorite – AB2 (C)¡ CaF2

¡Anti-fluorite¡ Similar to fluorite, but with

cations and anions reversed¡ K2O


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Hcp derived structures¡ Interstitial occupation in

tet or oct sites¡ Wurtzite ZnO (B4) structure

– occupation of one of tetsites in hcp

¡ B81 NiAs structure –occupation of oct sites


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Types of ceramics¡ Restrict our discussion to binary ceramics (two

components)

¡ Ionic ceramics¡ Compounds of an electropositive cation M with an electronegative

anion X, e.g., Al2O3, MgO, ZrO2, NaCl¡ Packing and coordination governed by Pauling’s rules as discussed

earlier in the course

¡ Covalent ceramics¡ Covalent bonding¡ Examples: GaAs, SiC, ZnO, SiO2

¡ Note that the ionic/covalent classification is not a strict one – all structures have some ionic and covalent character


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Halide salts¡ Monovalent cations and anions

¡ Two main types of structure¡ Rocksalt or NaCl structure (fcc derivative)¡ CsCl structure (bcc derivative)

¡ Significantly larger radius of Cs compared to Na responsible for the difference in structures

¡ CsCl in fact exists in both the α-CsCl form, as well as a rocksalt structure at higher temperatures


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Oxides of Fe¡FeO (wustite)¡ Rocksalt (NaCl) structure

¡Fe3O4 (Magnetite)¡ Spinel structure

¡Fe2O3¡ α (hematite)¡ γ (maghemite)¡ β and δ are two other allotropes


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Close-packed sulfides and oxides

¡Zincblende ZnS structure discussed previously

¡Corundum Al2O3¡ Mineral name for ruby / emerald / sapphire¡ Almost hcp O2- with Al3+ occupying 2/3 of oct sites


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Ternary oxide structures¡ Perovskite ABO3 (E21)¡ Prototype CaTiO3

¡ Large family of crystalline ceramics¡ Idealized perovskite is cubic, but more typically is tetragonal or

orthorhombic due to tilting or distortion¡ Many technological applications rely on perovskite structures¡ Solar¡ Ferroelectrics¡ Catalysts¡ Superconductors¡ ….


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Ternary oxide structures¡Spinel AB2O4¡ Prototype MgAl2O4

¡ Fcc sublattice of O2-, with A occupying a fraction of tetsites and B occupying a fraction of oct sites.


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UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures

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Transcript of UCSD NANO106 - 13 - Other Diffraction Techniques and Common Crystal Structures