Introduction to Crystallography and Mineral Crystal Systems UNIT-1.
Introduction Crystallography
Transcript of 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