Introduction Crystallography

7/31/2019 Introduction Crystallography

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X-Ray Diffraction is used to study

crystalline materials X-rays scatter off of the

atoms in a sample

If those atoms aresystematically ordered, the

scattered X-rays tell us:what atoms are present

how they are arranged


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First we need some basicsof crystallography

A crystal consists of a periodic arrangement of the unit cell into alattice. The unit cell can contain a single atom or atoms in a fixedarrangement.

a,b and c (length) and , and angles between a,b and c are latticeconstants or parameters which can be determined by XRD.


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Crystalline materials are characterized by the

orderly periodic arrangements of atoms.

Parallel planes of atoms intersecting the unit cell are used to definedirections and distances in the crystal.

These crystallographic planes are identified by Miller indices.

The (200) planes

of atoms in NaClThe (220) planes

of atoms in NaCl


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Examples of Miller Planes

a

b

h=1, k=-1

h=1, k=0

h=1, k=-2

h=4, k=-1

h=2, k=1

Note that the 2, 2 and-2, -2 planes are identical


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The diffraction process occurswhen the Braggs law (condition)is satisfied. It is expressed as:

Where is the wavelength of x-rays

d is the interplanar spacing

is the x-ray angle

n is an integer

Braggs law


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Lattices

In 1848, Auguste Bravais demonstratedthat in a 3-dimensional system there arefourteen possible lattices

A Bravais lattice is an infinite array ofdiscrete points with identical environment

seven crystal systems + four latticecentering types = 14 Bravais lattices

Lattices are characterized by translationsymmetry

Auguste Bravais(1811-1863)


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Crystal System External Minimum Symmetry Unit Cell Properties

Triclinic None a, b, c, al, be, ga,Monoclinic One 2-fold axis, || to b (b unique) a, b, c, 90, be, 90

Orthorhombic Three perpendicular 2-folds a, b, c, 90, 90, 90

Tetragonal One 4-fold axis, parallel c a, a, c, 90, 90, 90

Trigonal One 3-fold axis a, a, c, 90, 90, 120

Hexagonal One 6-fold axis a, a, c, 90, 90, 120

Cubic Four 3-folds along space diagonal a, a, ,a, 90, 90, 90

triclinictrigonal

hexagonal

cubic tetragonal

monoclinicorthorhombic

Specifically 3-D crystals may bearranged in 7 crystal systems


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Rotation axes (by 60, 90, 120, or 180)Notated: 6, 4, 3, and 2, respectively

Screw Axes (translation and rotation)

Notated: 61, 62, 63, 64, 65; 41, 42, 43; 31, 32; and 21(Translation)

Reflection planes

Glide reflection planes (reflection and translation)Inversion pointsRotary inversion axes (rotation and inversion)

Proper symmetry elements in 3D


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http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture1/Bravais.gif

Categories of Space Groups


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The combination of all available symmetry operations (32 pointgroups), together with translation symmetry, within the allavailable lattices (14 Bravais lattices) lead to 230 Space Groups

that describe the only ways in which identical objects can bearranged in an infinite lattice. The International Tables list thoseby symbol and number, together with symmetry operators, origins,reflection conditions, and space group projection diagrams.

An interactive tutorial on Space Groups can be found on-line in Bernhard RuppsCrystallography 101 Course: http://www-structure.llnl.gov/Xray/tutorial/spcgrps.htm


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Generating a Crystal Structure from its CrystallographicDescription

Using the space group information contained in the InternationalTables we can do many things. One powerful use is to generate anentire crystal structure from a brief description.

Let us consider the description of the crystal structure of NaCl.

Space Group = Fm3ma = 5.44 Atomic Positions

Using the face centeringgenerators (0,0,0), (,,0), (,

0,), (0,,) together with thecoordinates of each Wyckoffsite we can generate thefractional coordinates of allatoms in the unit cell.

Cl

1:(0.0,0.0,0.0), 2:(0.5,0.5,0.0), 3:

(0.5,0.0,0.5), 4:(0.0,0.5,0.5)Na1:(0.5,0.5,0.5), 2:(0.0,0.0,0.5), 3:(0.0,0.5,0.0), 4:(0.5,0.0,0.0)


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With no knowledge of the symmetry diagram we can identify the crystal systemfrom the space group symbol.

Cubic The secondary symmetry symbol will always be either 3 or -3 (i.e. Ia3,Pm3m, Fd3m)

Tetragonal The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43(i.e. P41212, I4/m, P4/mcc)

Hexagonal The primary symmetry symbol will always be a 6, (6), 61, 62, 63, 64 or

65 (i.e. P6mm, P63/mcm) Trigonal The primary symmetry symbol will always be a 3, (3), 31 or 32 (i.e P31m,

R3, R3c, P312)

Orthorhombic All three symbols following the lattice descriptor will be either mirrorplanes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc21, Pnc2)

Monoclinic The lattice descriptor will be followed by either a single mirror plane,glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P21/n)

Triclinic The lattice descriptor will be followed by either a 1 or a (1).

Summary:


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Wyckoff Sites The Wyckoff positions tell us where the atoms in a crystalcan be found.

Wyckoff position denoted by a number and a letter. Numberis called multiplicity of the site and letter is called Wyckoffsite.

Multiplicity tells us how many atoms are generated by symmetry if weplace a single atom at that position.

The letter is simply a label and has no physical meaning. They are assignedalphabetically from the bottom up.

The uppermost Wyckoff position (with highestmultiplicity), corresponding to an atom at an arbitraryposition never resides upon any symmetry elements. This

Wyckoff position is called the general position. All ofthe remaining Wyckoff positions are called specialpositions. They correspond to atoms which lie upon oneof more symmetry elements, because of this they alwayshave a smaller multiplicity than the general position.


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The reciprocal lattice why bother?

The reciprocal lattice is important in all phases ofsolid state physics, so an understanding of thisconcept is useful in and of itself

Vector algebra is very convenient for describingotherwise complicated diffraction problems

The reciprocal lattice offers a simple approach tohandling diffraction in terms of vectors

Use of the reciprocal lattice permits the analysis ofdiffraction problems that cannot be accessed byBraggs Law (example: off-peak scattering)


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Approaches for placing a reciprocallattice point onto the Ewald sphere

move the reciprocal lattice through the

Ewald sphere (the rotating crystal

method)

change the radius of the Ewald sphere

by changing the wavelength (the Laue

method)

Note: S/used in place of k


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The Powder Diffraction Pattern

Powders (i.e., polycrystalline aggregates) arebillions of tiny crystallites in all possibleorientations.

When placed in an x-ray beam, all possibleinteratomic planes will be seen

By systematically changing the experimentalangle, we will produce all possible diffractionpeaks from the powder


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Geometric relationship between the 2D reciprocal lattice[d*(hk) vector] and its 1D projection |d*| = 2sinq/l

Indexing: Determination of hkl indices for eachpeak and lattice parameters evaluation, i.e.,

reconstruction of the 3-D geometry


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We do not need to collect the rings from cones butjust the ring intersection on an equatorial film

If the scattering power is low, the integration of

intensity in the cones may be required.


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In the early times this was done with aDebye-Scherrer camera


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The relationships between a film intensity and

ordinary powder diffraction pattern


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FilmFilm is the oldest, and in some cases, still the best method

(certainly the least expensive) for recording X-rayintensities optical density:

light X-rays

logE

D

E

D

light-- requiries many photons totransform one AgI grain (D logE)

X-rays -- one photon transformsmany AgI grains (DE)

gamma


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Which Wavelength to Use in the home lab?

Generally use Cu 1.5418 or Mo 0.71073The longer the wavelength the farther apart

the diffraction spots (lines) are in space. Forlarge unit cells like macromolecules biologistsshould prefer Cu.

Cu produces x-rays efficiently and thedetectors have a higher efficiency inmeasuring them.

Mo is not as absorbed as Cu. Best for heavyelement problems. However metals have ingeneral small unit cells.


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Phase identifications (crystalline and amorphous)

Crystal structure determination

Crystal structure refinements

Quantitative phase analysis (and crystallinity determination)

Microstructural analyses (crystallite sizes - microstrain)

Texture analysis

Residual stress analysis

Order-disorder transitions and compositional analyses

Thin films


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The issue of indexing apowder pattern

was treated in a monograph

2002By the InternationalUnion of Crystallography


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To which programs should we address (the list is notso numerous) ???

We had a period of productivity in the past 12-16

years


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Ab initio structure solutionIndexingStructure solution

Structure databasesSearch match

Crystal structure databases


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Collection of a very good experimental patternaccurate peak positions

low noise, good intensities, sharp peaks

Indexing (to determine symmetry, cell parameters and

possible space groups)Atoms location

usual:

structure factors extraction

structure solving by specialized programsdirect:

atoms location from fitting

Structure refinement (Rietveld)

Structure validation (Platon?)


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Indexing:smoothing, background subtraction and peak locations

peak list export for classical indexing programs (Dicvol, Treoretc.)or genetic indexing or full pattern indexing (when no one else

program succeed)Dicvol indexing results import and space group sorting (space

group identification)Le Bail fitting for cell refinement and structure intensities extraction(with export for structure solution programs)Structure solutions:

Genetic algorithmElectron density mapsExport to other programs (superflip, sir20xx, shelx....)


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monoclinic

Rw(%)=7.5a=19.87820(5)

b=8.19454(2)c=11.24154(3)

=106.0656(2)


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Direct methods (Shelx, Sir2004...):

Some algorithms are used to solve the phase problemThey work best with single crystal structure factors

Electron density maps, difference electron density maps and Patterson mapsusing Fourier transform or maximum entropy methods

from electron density maps atom should then be located (not easy)

Charge flipping algorithm: special technique to obtain electron density maps,as for the previous lot of reflections are required (ex. superflip)

Direct atom locations:

genetic algorithmssimulated annealingMontecarlo methods

Special methods were developed for particular cases: envelopes, energyprinciples....


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First use the search-match (next)

Check for every similar compound

Check literature and structural databases

Repeat the experiment in case

If nothing similar is found then go for the ab initio structure

solution

After indexing better to do the search again


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Open access on the Web :

PDB (proteins)

NDB (nucleic acids)

AMCSD (minerals)

COD (small/medium crystal structures, based on donations)

Toll databases :

CSD (organic, organometallic)

ICSD (inorganic, minerals)

CRYSTMET (metals, intermetallics)

ICDD (powder patterns)


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For all metallic/intermetallic phases: The Pearson books (inpaper format)

For some simple structures: Wyckoff, Crystal Structures(book)

Literature: all IUCr journals like Acta Cryst., J. Appl.Cryst. etc.


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Maud has a small structure database that can be expandedby the user

Phases, Instruments can be stored in the databases in CIFformat

The CIF format is the Crystallographic Information Formatdeveloped by the IUCr

Maud can import from any file containing phases orinstruments in CIF format

A special function can submit a solved or refined structuredirectly to the COD database online

Documents

Introduction Crystallography