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283 Cards in this Set
- Front
- Back
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crystals are solids
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but solids are not necessarily crystalline
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crystals
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have symmetry (kepler) and long range order
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Lattice points
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point which have identical environments, connected to form unit cells
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unit cell
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small shape, which can be repeated regularly over infinte area
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tetrangonal
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a=b, not equal to c, all angles 90
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orthohormbic
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a not equal to b not equal to c, all angles 90
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hexagonal
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a=b, not equal to c, alfa and beta = 90, gamma = 120
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difference between Rhombohedral and Monoclinic
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Rhombohedrual has a=b (min) and Monoclinic has not cell parameter equalities
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crystal system is defined
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in terms of symetry, not by crystal shape
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triclinic
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nothing equal to each other
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orthogonal
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when all angles = 90
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non-orthogonal
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when one of the angles are not 90
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mirror line
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reflect across line, an object can be bisected by mirror line
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n fold rotation
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360/n
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Combinations that make 10 plane point groups
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1,2,3,4,6,m,mm,3m,4m,6m
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Cubic essential symetry
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4 3 fold axes, along body diagonals
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Tetrangonal essential symetry
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1 4 fold axis, parallel to c, in center of ab
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Orthorhomic essential symetry
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3 mirrors or 3 2 fold axes, perpendicular to each other
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Hexagonal essential symetry
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1 6 fold axis, down c
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Trigonal essential symetry
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1 3 fold axis, down the long diagonal
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Monoclinic essential symetry
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1 n fold axis, along the "unique" axis
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Triclinic essential symetry
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none
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3 types of centering/structuring or lattices
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body centered, face centered, primitive. also side centering, but not very common
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coordination number
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(example) how many Na are in contact with Cl (6)
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distance between adjacent lattice planes
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d spacing
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to tranlate fractional into miller index
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take reciporocals, multiply to get integers
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In reverse, a plane with miller index (hkl) has intercepts at:
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a/h, b/k, c/l
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planes parallel to faces:
intercepts: infinity 1 infinity, infinity, 1/2, infinity |
(010), (020)
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note that (100) not equal to (001)
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when not cubic, in tetragonal, c not equal to a or b
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optical diffraction grating
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1 dimensional analogue of X-ray diffraction, coherent incident light impinges upon an evenly spaced grating: the parallel lines in the grating act as secondary light sources
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optical diffraction grating
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coherent indicent light goes through slits and you get diffracted light
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planes in crystals are
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considered to be reflecting planes
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Braggs Law
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2dsin (theta) = n (lambda)
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in packing two dimentions: two packing arrangements
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hexagonal and square
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in three dimentions, three packing arrangments are common
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heagonal close packing (hcp)
body centered cubic (bcc) face centered cub or cubic close packing (fcc) |
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Hexagonal close packing
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ABABAB
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Cubic Close Packing/fcc
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ABCABC
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Tetrahedral intersital sites
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CN 4
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Octrahedral intersitial sites
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CN 6
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two types of tetrahedral sites
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T+ means hole is above 3 molecules
T- means hole is under 3 molecules |
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packing fraction
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space occupied by atoms/available space
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Diamond and Zinc Blend, both important in semiconductor industry
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Carbon at face centered and tetrahedral positions in diamonds, S in face centered, tetrahedral positions is Zn
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Antifluorite, Na2O
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oygen atoms in all fcc positions, sodium atoms in all tetrahedral sites
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Flourite CaF2, ZrO2
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Ca atoms in all fcc positions, F atoms in all tetrahedral sites
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In both flourite and antiflorite
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there is 4 fcc atoms and 8 filled tetrahedral sites in a unit cell
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With ccp anion array
Rock Salt NaCl Zinc Blend ZnS Antiflourite Na2O |
O occupied
T+ or T- occupied T+ and T- occupied |
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With hcp anion array
Nickle Arsenide NiAs Wurtzite ZnS |
O occupied
T+ or T- occupied |
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With ccp cation array
Flourite ZrO2 |
T+ and T- occupied
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Peroskite ABX3
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can be polorizable
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Laue model takes in
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account the 3-d nature of the crystal lattice
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X rays are
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electromagnetic radiation
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X ray photon energy
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E=hv
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X ray wavelength vary from
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.01-10 Anstroms
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X ray diffraction
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X ray source, Sample, Detector
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First assumption
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X rays elastically scattered by electrons
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Second assumption
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spherical discrete atoms
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interaction is weak
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so need many electrons
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Laue Method, fixed crystal
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Uses: alignment of single cystal, infomration on unit cell, imformation on imperfections, defects in crystal
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4 circle diffractometer
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Uses: unit cell derermination, crystal structure detirmination
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Powder diffraction
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assume a random orientation of large number of small crystalines so that Braggs condition is always satified,
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"Fingerprinting"
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matches whole patterns with information from a database, ICDD Powder diffraction File, there is also a Inorganic Crystal Structure database
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Why is is listed as D spacing and not 2 theta
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D spacing is constant, different sample of same material
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Uses of Fingerprinting
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identify substance or mixture (could following a reaction), distinquish between different structures (even if chemically similar), distinquish between different polymorphs
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Problems of Fingerprinting
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some rubbish information, not sensitive
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If we use X ray powder diffraction, a crystalline material will lead to
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very sharp diffraction peaks, whereas an amorphous material will give several broad bands
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If the crystals are small,
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the diffraction lines broaden
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B is the line broading, measured in radians
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from the peak width at half maximum (BM) reffered to a standard (BS)
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peaks are
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connected to unit cell dimentions
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Rietveld method
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refinement technique, not structural detirmination, used when there is a similarity to an existing structure, basis behind qualntitaive X ray diffraction
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Systematic Absences P
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P: no restrictions, all allowed
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Systematic Absenses I
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h+k+l = 2n
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Systematic Absenses F
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h,k,l, all odd or all even
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even though mocules have similar structure
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behavior is different
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ionic radii
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hard to measure, estimated, composmise between attractive and repulsive
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random dipoles
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paraelectric
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aligned dipole
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ferroelectric
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Solids
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single crystal, glass, polymer, inorganic, compsite material
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Macro
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overall structure
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micro
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grains, defects
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Nano
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crystal structure, atomic level, focus of XRD
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surface v bulk
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at surface, structure is not perfect, bulk is better
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properties like mechanical, magnetic, chemical
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not investigated by XRD
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Different kinds of diffraction
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single crystal, powder, X ray, Neutron, Electron
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For XRD, we must have
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constructive interferance, exactly in phase
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Laue Equations
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basic idea is still constructive interference and interger wavelength
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why not use microscope
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can't focus x rays, costly, electron miscroscope are only 2 d
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Electrons stopped at target
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converted into X rays
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Incident electrons displace inner shell electrongs
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each target has a characteristic wavelength
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Scattering will depend on
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Z and angle
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Laue Method
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white source is weak, very slow, each spot represents a crystal plane
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4 circle method
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primary method to find crystal structure
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(222)
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all intersect at 1/2 1/2 1/2
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when miller indices increase
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plane numbers increase, d spacing decrease
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X ray diffraction by crstal is anologous to
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light difraction through a gradting. Braggs law is essential in describing x ray diffraction
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Detector
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as x ray detector scans through a range of theta values, peak intensities are noted, allowing calculation of d for different spacings
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For CCP and FCC
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they have same coordination numbers
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tetrahedral sites
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8 tetrahedral sites in a fcc structure, 1/4, 1/4, 1/4
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in cubic unit cell
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octrahedral bond length is a/2
tetrahedral is 1/4 a square root 3 |
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In FCC and HCP
For primitaive |
.74
.524 |
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E=hv
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for phontons
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c=lamda v
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another equation
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lamda = hc/E
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another equation
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each element
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has characteristic wavelenth
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K beta
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is transition from 3p-1s or from M-K
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K alpha
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transition from L-k or 2p - 1s
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To find energy between levels
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use E=hv and V=c/lamda
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the whole pattern is what matters
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not just one peak
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Powder diffraction
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main use is fingerprinting
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2 theta
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depends on lamda used
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ICSD
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Inorganic crystal structure database, free to acadmeics, can specify elements, will list artiles, draw pictures, can be exported into ATOMS
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Advantages to Powder Diffraction
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quick and easy, non destruvie for forensics
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Problems with Powder diffraction
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need reliable standards, some entries not good, not very sensivitve, not good for organics
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Prefered orientation
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if crystals are platey or needle shaped, orentiation wil not be random, matchen then relies on a peak possition rather than intersites
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Peak Broadening
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don't get perfect lines in reality, radiation not perfectly monochromatic, heisenburg uncertainy, focussing geometry
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Schere Equation
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t=.9lamda/Bcostheta, B is line boundary by reference to a standard, measure in radians
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B=
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Bm^2-Bs^2
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convert from degrees to radians
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multiply by pi/180
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start with larges d spacing
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7 and 15 cannot be obtained
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due to symmetry, certain reflections cancel each other out. Destructive interfearance
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systematic absenses
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Bravais Lattice
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P-no restrictions
I h+k+l=2n allowed R (face centered) hkl, all odd or all even |
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face centered is sparcer of reflections
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so first reflection not necessarily 110
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sum of scattering
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is sum of all waveds diffracted from crystal
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centrosymetric
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if you have an atom at x y z, you also have one at -x, -y, -z
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2pi (hx+ky+lz)
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phase difference/geometric structure factor
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Intensity proportional to FF*
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F* is complex conjugate
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Scattering factor depends on
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Z, but electron density is the important factor
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structure factor is
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fourier transform of electron density, can switch back and forth using a computer program
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direct method
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make a sensible guess, then test and repeat as needed
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Patterson Method-
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uses peaks corresponding to vectors between atoms
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Heavy atom method
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heavy atoms dominate, easy to locate, Patterson to find others
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neutron and electron diffraction
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matterwave-particles rather than EM
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De Broglie,
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everything has wavelenth
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neutrons scatter by
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interaction with nucleus, interaction with spin of unpaired electrons
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2 types of scattering
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elastic-diffraction
inelastic-spectrometer, we concentrate on elastic scatering |
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isotypes detirmined easly,
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wide range of d spacing, more hkl
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some atoms good neutron absorber
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such as CD, Gd, LI6
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Some atoms
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near zero scattering, can be cases where you don't want scattering
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source
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need reactor and accelerator, expensive to set up, not routine but more common
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ISIS
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being developed even further, HPRD, on the highest resolution instraments in world, HPRD2 will be the next one, currently being built
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ILL in Genenoble France
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Not a synchroton
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IPNS Argonne Chicago
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one of best in US, 20 worldwide
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At many sources, neturons are produced by fission, then selected by wavelength
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losses many neutrons
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Alternative, H- produced at source, plused at 50 hz
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electrons stripted, protons 3*10^13 for pluse, proton beam hits target, each proton-25 neturons
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electrongs move, creating a dipole
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inducing opposite dipole in adjacent atom, dipole dipole puls together
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Ti almost always octrahedral
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SI almost alwasy tetrahedtral
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Ionic radii increase
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with CN
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Ionic radii increse
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going down a group (more electron shellss added)
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Anions bigger than
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cations
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Neutrons possess of magnetic
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dipole moment
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4 types of magneticm
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para, ferro, anti ferro, ferri
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Paramagneticm
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distorted
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Ferromagneticim
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aligned
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Antiferomagnet
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opposed in ordered way, net mag zero
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Ferrimagnetion
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inequally opposed, net mag not zero
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Why solids
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most elements solid at room temperature, atoms in fixed position
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classical x rays
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interact with electrons in atom, caustng them to oscillate with x ray beam
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Compton Scattering
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results in loss of energy, classically, no frequency shift
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acceleration in a circle
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by magnet, large spectural range, focused and intense x rays produced
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booster and synchoront
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ring of electromagnets, high count rates, wavlength variable, sharp peaks, incident beam is usally monochormatic and parallel
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synchroton diffraction uses
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high resolution, x ray power diffraction, resonate x ray power diffraction, sample environment, surface xrd, small cystals
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photoelectron ejected with energy equal to that of the incoming photons-minus binding energy
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what the heck does that mean?
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symetry element
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symmetry axis
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many inorganic structures may be described as
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space filling polyhedra, or tetrahedrah or octahedrah
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CN 3-D
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12, both types
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cations are smaller
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and hense fill the intersitial sites
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set volume in terms of A and R,
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and solve for R
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CsCl
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CN=8
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NaCl, NiAs
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CN=6
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wurtzite
ZnS sphalerite |
CN = 4
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Ion radii for given element increase with CN
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Ion radii for given element decrease with increasing oxidation state/positive charge
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Ion radii for given element increase with increasing CN and with decreasing oxidation state
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yep
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Solids are
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Powder
Single crystal Glass/amorphous Polymer Inorganic/Organic Composite material |
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what scale are we interested in?
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What scale are we interested in?
Bulk/Macro – overall structure Micro (microstructure) – grains, defects Nano – crystal structure |
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Diffraction
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to find atom level structures
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Laue, Bragg and Bragg
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responsible for advances in XRD
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closer the slits
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further diffraction lines
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With X-rays, the interaction with matter is very weak – most pass straight through
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Therefore we need many (100-1000s) of waves
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Laue's Equation
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These work well and describe the interactions
Basic idea is still the constructive interference which occurs at an integer no. of wavelengths However, not routinely used |
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If we draw the Bragg construction in the same way as the optical grating, we can clearly see that the diffracted angle is 2. The plane of “reflection” bisects this angle.
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Thus we measure 2 in the experiment – next section…
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X-rays - electromagnetic waves
So X-ray photon has E = hv |
X-ray wavelengths vary from .01 - 10Å; those used in crystallography have frequencies 2 - 6 x 1018 Hz
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use Kev
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so you get Anstroms out
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2 things happen with X ray emission
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Electrons stopped by target; kinetic energy converted to X-rays
Incident electrons displace inner shell electrons, intershell electron transitions from outer shell to inner shell vacancy |
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x ray spectrum
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mixture of line and not
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2p (L) - 1s (K), known as the Kalpha line
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3p (M) - 1s (K), known as the Kbeta line
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sample can be
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solid or powder
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X ray assumptions
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First assumption: X-rays elastically scattered by electrons.
Second assumption: Spherical, discrete atoms |
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More electrons means more scattering ( Z)
Scattering per electron adds together, so helium scatters twice as strongly as H |
We define an atomic (X-ray) scattering factor, fj, which depends on:
the number of electrons in the atom (Z) the angle of scattering |
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f varies as a function of angle , usually quoted as a function of (sin )/
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The more diffuse the electron cloud, the more rapid the reduction in the scattering function with scattering angle.
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theta = 0, f is equal to the total number of electrons in the atom, so
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ftheta=0 = Z, but ions have the same number of electrons,But as increases, Cl- has smaller f as it has a more diffuse electron cloud
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Changes d-spacing and atoms within the planes
So we need to either (a) rotate the crystal or (b) have lots of crystals at different orientations simultaneously |
you must
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Laue method: Each spot corresponds to a different crystal plane
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USES:
alignment of single crystal info on unit cell info on imperfections, defects in crystal |
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4-circle Method
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Crystal can be oriented so that intensities for any (hkl) value can be measured
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area detector
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area detector which removes one circle.
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uses of XRD
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Unit cell determination
Crystal structure determination (primary method) |
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Powder Diffraction
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By “powder”, we mean polycrystalline, so equally we can use a piece of metal, bone, etc.
We assume that the crystals are randomly oriented so that there are always some crystals oriented to satisfy the Bragg condition for any set of planes |
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Debye-Scherrer Camera
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no longer used
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Single crystal is a primary technique for structure determination
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Powder diffraction relies on a random orientation of (small) crystallites
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XRD
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fingerprinting
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ICDD
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(International Centre for Diffraction Data)
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Remember: we rely on a random orientation of crystallites.
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When crystals are platey or needle-shaped (acicular) they will pack in a non-random fashion, preferentially exposing some planes to the incident radiation.
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Intensity mismatch –
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due to using single crystal
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Even if two structures are the same (and they are chemically similar) differences can be observed:
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Peak positions (unit cell changes) and relative intensities (atoms)
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In an X-ray diffraction pattern, peak width depends on
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the instrument
radiation not pure monochromatic Heisenberg uncertainty principle focussing geometry the sample… - a crystalline substance gives rise to sharp lines, whereas a truly amorphous material gives a broad “hump”. |
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If crystal size < 0.2 m, then peak broadening occurs
At <50nm, becomes significant. |
At slightly higher than the Bragg angle, each plane gives a “lag” in the diffracted beam.
For many planes, these end up cancelling out and thus the net diffraction is zero. In small crystals, there are relatively fewer planes, so there is a “remanent” diffraction |
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Particle size determinaton
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An estimate, rather than an absolute value - also will be dominated by smallest particles.
Good for indication of trends. A useful complement to other measurements such as surface area, electron microscopy etc. |
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Amorphous / micro-crystalline?
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It can be difficult to distinguish between an amorphous material and a crystalline sample with very small particle size.
BUT the idea of such a small size “crystal” being crystalline doesn’t make sense! 5nm = 50Å = e.g. 10 unit cells Is this sufficient for long range order?? |
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if you heat a sample up
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you can change unit cell parameters
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Structure refinement –
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the Rietveld method
Here there was a similarity between the powder pattern of this phase and an existing one – also chemical composition similar. |
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Caveat Indexer
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Other symmetry elements can cause additional systematic absences in, e.g. (h00), (hk0) reflections.
Thus even for cubic symmetry indexing is not a trivial task Have to beware of preferred orientation (see previous) Often a major task requiring trial and error computer packages Much easier with single crystal data – but still needs computer power! |
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This is obviously wavelength dependent
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Hence in principle using a smaller wavelength will access higher hkl values
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Indexing a powder pattern means correctly assigning the Miller index (hkl) to the peak in the pattern.
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If we know the unit cell parameters, then it is easy to do this, even by hand.
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Due to symmetry, certain reflections cancel each other out.
These are non-random – hence “systematic absences” |
For each Bravais lattice, there are thus rules for allowed reflections:
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So a plot of (h2 +k2 + l2) against sin has slope 2a/
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yep
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It’s a fundamental equation in crystallography.
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structure factor equation
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Structure factor equation
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Builds on concepts we have encountered already:
Miller index fj Z Unit cells Positions of atoms (x,y,z) Symmetry (Wave equations) |
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What makes a diffraction pattern?
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Positions of peaks/spots
entirely due to size and shape of unit cell a,b,c, ,, which gives d ( 2) Intensities of peaks following section: why all different? Sample, instrumental factors |
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Intensities depend on…
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scattering power of atoms ( Z)
position of atoms (x,y,z) vibrations of atoms - “temperature” factor B Polarisation factor (function of sin /) (see previous) Lorentz factor (geometry) absorption extinction preferred orientation (powders) multiplicities (i.e. 100=010=001 etc) |
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Scattering
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From before: “the scattering from the plane will reflect which atoms are in the plane”. The scattering is the sum of all waves diffracted from the crystal.
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Atomic scattering factor
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Again, from before:
The atomic scattering factor, fj, depends on: the number of electrons in the atom (Z) the angle of scattering f varies as a function of angle , usually quoted as a function of (sin )/ f=0 = Z |
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Summing the waves
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The overall scattering intensity depends on
Atom types (as above) - “electron density” Their position relative to one another. |
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This is the sum of the (cos) waves, where:
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- fj is the atomic scattering factor for atom j
- hkl are the Miller indices - xj, yj, zj are the atomic (fractional) coordinates |
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The expression 2(hx+ky+lz) =
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phase difference
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Thus we get:
I fj2 as the cos (or exp) terms cancel out. So something quite complex becomes simple, but…. |
measure intensity which is proportional to FF*
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Thus, the odd reflections are systematically absent
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Generally true for all body centred structures
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The phase problem
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We can calculate the diffraction pattern (i.e. all Fhkl) from the structure using the structure factor equation
Each Fhkl depends on (hkl) (x,y,z) and fj fj depends primarily on Z, the number of electrons (or electron density) of atom j The structure factor is thus related to the electron density, so if we can measure the structure factor, we can tell where the atoms are. |
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We measure intensity I = F.F*
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so we know amplitude of F.….but phases lost.
Several methods to help – complex but briefly |
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Direct methods
(Nobel Prize 1985 - Hauptmann and Karle) |
Statistical trial and error method. Fhkl’s are interdependent so by “guessing” a few we can extrapolate
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Patterson Methods
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Uses an adapted electron density map where peaks correspond to vectors between atoms - peak height Z1Z2
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Heavy Atom Methods
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High Z atoms will dominate the electron density - “easy” to locate
Use Patterson vectors to find other atoms. |
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Limitations of X-ray Structure determination
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gives average structure
light atoms are difficult to detect (f Z) e.g. Li, H difficult to distinguish atoms of similar Z (e.g. Al, Si) need to grow single crystals ~ 0.5mm time for data collection and analysis (?) new instruments mean smaller crystals, shorter collection times! So in fact – data can pile up…. |
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De Broglie
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Extended the idea of wave-particle duality
1923 – particles can be wavelike Idea that everything has a wavelength! |
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E = mc2 = (mc)c but momentum, p=mv
and for a photon v=c |
E = pc = p f but E=hf (Planck/Einstein)
hf = p f |
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Neutron scattering
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Neutron can be scattered by atoms by:
interaction with nucleus interaction with spin of unpaired electrons - magnetic interaction, magnetic scattering. This happens because the neutron has a magnetic moment. (later) |
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More Neutron Scattering
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Also the interaction can be:
elastic (diffractometer) structural studies inelastic (spectrometer) loss of energy on scattering gives information on phonon dispersion (effect of vibrations in lattice) and stretching of bonds |
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Scattering from neutrons
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X-rays: fj Z - can be calculated
Neutrons: small dependence of fj on Z but major part Z independent. fj must be determined experimentally |
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Good points to neutron scattering
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Can detect light atoms
Can often distinguish between adjacent atoms Can distinguish between isotopes Can accurately find atoms in presence of very high Z atoms Covers a wide range of d-spacings - more hkl - BUT excellent complimentary to XRD |
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bad points to neutron scattering
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Some atoms/isotopes good neutron absorbers (e.g. Cd, Gd (Gadolinium), 6Li (so use 7Li)
V has very low, ~0 scattering (but..) need neutron source VERY expensive (~£10,000 per DAY!) |
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Clifford Schull
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for the development of the neutron diffraction technique
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Bertram Brockhouse
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"for the development of neutron spectroscopy
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IPNS, Argonne, Chicago IL
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Intense Pulsed Neutron Source
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Other neutron sources are also available…
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Los Alamos Neutron Science Center (New Mexico, US)
Lucas Heights (Sydney Australia) Oak Ridge (Tennessee, USA) KENS (Tsukuba, Japan) Chalk River (Ontario, Canada) Risø (Roskilde, Denmark) |
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The experiment
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At many sources (e.g. ILL at Grenoble) neutrons are produced by fission in a nuclear reactor and then selected by wavelength - but with neutrons there are no “characteristic” wavelengths:
..so by selecting a wavelength we lose neutrons and lose intensity |
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Alternative
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UK neutron source at Rutherford Appleton Laboratory uses “time of flight” neutron diffraction
Electrons stripped protons (~3 x 1013) H- produced at source (pulsed) |
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Time-of-flight neutron diffraction
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We are measuring d, so two variables, and
In lab X-ray powder diffraction, is constant, variable In time-of-flight (t.o.f), is constant, variable This takes advantage of the full “white” spectrum |
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Errors
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The biggest error in the experiment is where the neutrons originate
This gives an error in the flight path, L typical value ~5cm Hence as L increases, error in d is reduced - resolution of the instrument is improved e.g. instrument at 10m compared to instrument at 100m 100m = HRPD, currently highest resolution in the world |
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Magnetic Diffraction
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Neutrons possess a magnetic dipole moment
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Heavy equipment
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Furnaces, cryostats, pressure cells, magnets, humidity chambers, etc.
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Electron Diffraction
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Similar principle – matter waves, but me = 9.109 x 10-31 kg
Also applied “accelerating potential” V such that: |
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G. P. Thomson
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Experiments performed at Marischal College in the late 1920's
(also Lester Werner and Clinton Davisson at Bell labs in New York) Picture of electron diffraction taken by Thomson |
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Bragg’s Law redux
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Since l is very small, q is also very small, so we can rewrite Bragg’s law as:
l = 2d q As previously, we can derive: d ~ L/D |
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Uses (of what)
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Can be used to look at individual crystallites: must be thin (why?)
Useful to help determine unit cell parameters; need many orientations (see animation here) Shape of spots: streaking can give information on crystal size and shape Can identify packing defects (see later) Added extra: EDX for elemental analysis: Electrons knock out inner shell electrons Characteristic X-rays emitted as outer shell electron drops down to fill gap |
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Both neutron and electron diffraction are very useful complementary techniques to X-ray diffraction
Neutron diffraction has a number of advantages over X-ray diffraction – but cost is a major disadvantage! Both fission and spallation sources are used Magnetic diffraction is possible due to the dipole present with neutrons |
Electrons can be focussed, allowing high resolution imaging as well as diffraction
Information on defects (see later) and unit cells |
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Classical vs quantum
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In the classical treatment, X-rays interact with electrons in an atom, causing them to oscillate with the X-ray beam.
The electron then acts as a source of an electric field with the same frequency Electrons scatter X-rays with no frequency shift |
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Compton Scattering
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Some radiation is also scattered, resulting in a loss of energy [and hence, E=h, shorter frequency and, c= , longer wavelength].
The change in frequency/wavelength depends on the angle of scattering. This effect is known as Compton scattering It is a quantum effect - remember classically there should be no frequency shift. |
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Moseley’s Law
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Moseley corrected anomalies:
Also identified a gap at Z=43 (Tc) |
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Absorption
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X-ray photons absorbed when E is slightly greater than that required to cause a transition
- i.e. wavelength slightly shorter than K So, as well as characteristic emission spectra, elements have characteristic absorption wavelengths |
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Uses of absorption
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We want to choose an element which absorbs K [and high energy/low white radiation] but transmits K
e.g. Ni K absorption edge = 1.45 Å As a general rule use an element whose Z is one or two less than that of the emitting atom |
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Monochromator
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Choose a crystal (quartz, germanium etc.) with a strong reflection from one set of lattice planes, then orient the crystal at the Bragg angle for K1
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Synchrotron X-rays
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When charged particles are accelerated in an external magnetic field (according to Lorentz force), they will emit radiation (and lose energy)
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Synchrotron X-rays
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Acceleration in a circle…
Electrons are kept in a narrow path by magnets Emit e.m. radiation ahead Large spectral range Very focussed and intense X-rays produced (GeV) (also applications in particle, medical physics amongst other things) |
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SchematicSynchrotron X-rays
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electron gun (2) linear accelerator
(3) booster synchrotron (4) storage ring (5) beamlines (6) experiment stations. |
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Inside the synchrotron
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Electrons emitted from cathode ~1100° C.
Accelerated by high-voltage alternating electric fields in linac. Accelerates the electrons to 450 MeV - relativistic Electrons injected into booster synchrotron (a ring of electromagnets); accelerated to 7 GeV 7 GeV electrons injected into the 1 km storage ring Circle of > 1,000 electromagnets etc. |
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Synchrotron vs lab data
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Much higher count rates signal to noise better
Wavelengths are variable. Incident beam is usually monochromatic and parallel. Very sharp peaks (smaller instrumental contribution) – FWHM can be 10 times narrower – better resolution |
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Synchrotron Diffraction - Uses
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High resolution X-ray powder diffraction
“Resonant” X-ray powder diffraction (can select wavelength) Analysis of strain (see later) Sample environment (as with neutrons) Surface XRD Diffraction on very small single crystals (0.0001 mm3) |
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Back to absorption
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Photoelectron ejected with energy equal to that of the incoming photon minus the binding energy.
Characteristic of element. The ejected photoelectron then interacts with the surrounding atoms |
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Absorption energies
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Energies of K edges Z2
Elements with Z>18 have either a K or L edge between 3 and 35 keV |
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Interference effects
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The ejected photoelectron then interacts with the surrounding atoms
This gives information on the local environment round a particular element within the crystal structure |
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XAS
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X-ray Absorption spectroscopy complements diffraction
Diffraction gives you information on average 3d structure of crystalline solids XAS gives you localised environment in solids (including glasses), liquids, gases. Info on bonds, coordination, valence |
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XANES/EXAFS
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X-ray Absorption – near edge structure
Extended X-ray Absorption – Fine Structure |
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Summary
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The interaction of X-rays with matter produces a small wavelength shift (Compton scattering)
The wavelength of X-rays varies as a function of atomic number - Moseley’s law Filters can be used to eliminate K radiation; monochromators are used to select K1 radiation. Synchrotrons can produce high intensity beams of X-rays suitable for structural studies Absorption can be exploited to give localised information on elements within a crystal structure. |
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Close Packing
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Each atom excludes others from the space it occupies
Attraction? Electrons are moving so that, at some instant, distribution is uneven Positively (electron-deficient) and negatively (electron-rich) charged regions electrical dipole Dipole induces an opposing dipole in neighbouring atom attraction |
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Attractive force is known as:
van der Waals interaction London interaction induced dipole-induced dipole interaction |
yep
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The total potential energy for two atoms a distance, r, apart can be written, sum of forces U[] equation
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This is called the Lennard-Jones (12,6) potential function
First term is repulsive, second term is attractive. We want to find a minimum - so differentiate w.r.t. r |
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2^1/6=r
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This is the van der Waals radius, the distance between the atoms that minimises their energy
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subsitute van dervals r into equation
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U min = - crazy E
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Ionic radius
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Previously we looked at trends in ionic radius of atoms
The “absolute” value is important, but relative values also give useful information Can give information on likely coordination (see following sections) and also viability of a structure (e.g. “Goldschmidt tolerance factor” for perovskites |
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Radius ratio rules
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Rationalisation for octahedral coordination: R= radius of large ion, r=radius of small ion
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Radius ratio rules
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Rationalisation for 8-fold coordination
Unit cell edge a = 2R Atoms touch along diagonal (if small ion fits perfectly into space) so a3 = 2(R+r) Divide: 3 = (R+r)/R Multiply out 3R = R+r R(3 -1) = r r/R = 3 -1 = 0.732 |
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Other ways of classifying structures
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1) Structure Field Maps
e.g. for AxByOz compounds, plot radius of A against radius of B and note trends of structure as rA and rB change. 2) Mooser-Pearson plots Focuses on the covalent character of bonds. Plot of difference in electronegativity versus average principal quantum number of atoms involved. |
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at higher temperature
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paramagetim
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high penetration and flux of neturons
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so used heavy instruments
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if ration < square root of two+
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then CN is less than 6
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pressure
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can reduce bond length
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