QuickTime and a decompressor are needed to see this picture. Basic Crystallography for X-ray Diffraction QuickTime QuickTime and and aa decompressor decompressor are are needed

needed to to see see this this picture. picture. Earle Ryba What's this weird thing in the database??? Mi l l e s c i

r n g i n b o [220 i I cmub n d i ce t i

i t s ] o s ic set o h r p o Bravais latticeoff

d h t n r 2 k o c Wy And what are these guys??? In X-ray diffraction, use repetition of atom

arrangement to get diffraction pattern Repetition = Symmetry Repetition = Symmetry Types of repetition: Rotation Translation Rotation What is rotational symmetry? I can rotate this object

rotate Please close your eyes while I rotate (maybe) this object rotate Did I rotate it? rotate The object is obviously symmetricit has

symmetry The object is obviously symmetricit has symmetry Can be rotated 90 w/o detection so symmetry is really doing nothing Symmetry is doing nothing - or at least doing something so that it looks like nothing was done!

What kind of symmetry does this object have? Another example: And another: What about translation? Same as rotation What about translation? Same as rotation

Ex: one dimensional array of points What about translation? Same as rotation Ex: one dimensional array of points Translations are restricted to only certain values to get symmetry (periodicity) 2D translations Example This block can be represented by a point

Each block is represented by a point This array of points is a LATTICE Lattice - infinite, perfectly periodic array of points in a space Not a lattice: Not a lattice - becuz not just points

.some kind of STRUCTURE Lattice - infinite, perfectly periodic array of points in a space each point has identical surroundings use this as test for lattice points Cs Cl

Cs Cl CsCl structure lattice points Combining periodicity and rotational symmetry What types of rotational symmetry allowed? Combining periodicity and rotational symmetry

Suppose periodic row of points is rotated through Combining periodicity and rotational symmetry To maintain periodicity, S t t

vector S = an integer x basis translation t S t t

vector S = an integer x basis translation t t cos = S/2 = mt/2 m cos axis 2 1 0 -1 -2

1 1/2 0 -1/2 -1 0 2 1 /3 5/3 6 /2 3/2 4 2/3 4/3 3 -

2 m cos 2 1 0 -1 -2 1

1/2 0 -1/2 -1 axis 0 1 /3 5/3 6

/2 3/2 4 2/3 4/3 3 - - 2 Only rotation axes consistent with lattice periodicity in 2-D or 3-D What about 5-fold axes? Can fill space OK with square (4-fold) by translating But with pentagon (5-fold). We abstracted points from the block shape:

We abstracted points from the block shape: Now we abstract further: (every block is identical) Now we abstract further: This is a UNIT CELL Represented by two lengths and an angle .or, alternatively, by two vectors

Basis vectors and unit cells a b T T=t a+t b a b a and b are the basis vectors for the lattice

In 3-D: [221] direction c b a T T=t a+t b+t c a b c

a, b, and c are the basis vectors for the 3-D lattice Different types of lattices Lattices classified into crystal systems according to shape of unit cell (symmetry) In 3-D c

b a Lengths a, b, c & angles , , are the lattice parameters Crystal systems System Triclinic Interaxial Angles

Axes 90 abc Monoclinic = = 90 Orthorhombic = = = 90 abc abc

Tetragonal = = = 90 a=bc Cubic = = = 90 a=b=c

Hexagonal = = 90, = 120 a = b c Symmetry characteristics of the crystal systems System Triclinic Monoclinic Orthorhombic Tetragonal Cubic Hexagonal Trigonal

Minimum symmetry 1 or 1 2 or 2 three 2s or 2s 4 or 4 four 3s or 3s 6 or 6 3 or 3 Stereographic projections Show or represent 3-D object in 2-D Procedure:

1. Place object at center of sphere 2. From sphere center, draw line representing some feature of object out to intersect sphere 3. Connect point to N or S pole of sphere. Where sphere passes through equatorial plane, mark projected point 4. Show equatorial plane in 2-D this is stereographic projection S

Stereographic projections of symmetry groups Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points

Stereographic projections of symmetry groups Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points

Stereographic projections of symmetry groups Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points Stereographic projections of symmetry groups

Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) All objects, structures with i symmetry are centric symmetry elements equivalent points

Stereographic projections of symmetry groups Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points

Stereographic projections of symmetry groups Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points Stereographic projections of symmetry groups

More than one rotation axis - point group 222 symmetry elements equivalent points Stereographic projections of symmetry groups More than one rotation axis - point group 222 symmetry elements equivalent points

Stereographic projections of symmetry groups More than one rotation axis - point group 222 symmetry elements equivalent points orthorhombic Stereographic projections of symmetry groups More than one rotation axis - point group 222 [100]

Stereographic projections of symmetry groups More than one rotation axis - point group 222 [010] [100] Stereographic projections of symmetry groups More than one rotation axis - point group 222 [001] [010]

[001] [100] [010] [100] Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm [001]

Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm [100] Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm [001] [110] [010]

[100] [110] Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm symmetry elements equivalent points tetragonal

Stereographic projections of symmetry groups Rotation + mirrors - point group 2/m [010] Stereographic projections of symmetry groups Rotation + mirrors - point group 2/m symmetry elements equivalent points monoclinic

Combining point groups with Bravais lattices to form crystal (need consider only one unit cell) A space group is formed (3-D) Pmm2 Combining point groups with Bravais lattices to form crystal (need consider only one unit cell) Note: if coordinates (x,y,z) of one atom known, then, because of

symmetry, all other atom coordinates known Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 3-D example: body-centered cubic (bcc, or I cubic) (two lattice pts./cell) The primitive unit cell is not a cube Within each crystal system, different types of

centering consistent with symmetry System Allowed centering Triclinic Monoclinic Orthorhombic Tetragonal Cubic Hexagonal Trigonal

P (primitive) P, I (innerzentiert) P, I, F (flchenzentiert), A (end centered) P, I P, I, F P P, R (rhombohedral centered) The 14 Bravais lattices 230 space groups (see Int'l Tables for Crystallography, Vol. A) Combine 32 point groups (rotational symmetry) with

a. 14 Bravais lattices (translational symmetry) b. glide planes (rotational + translational symmetry) a, b, c, n, d, e QuickTime QuickTime and and aa decompressor decompressor are are needed needed to to see see this this picture.

picture. c. screw axes (rotational + translational symmetry) 21, 31, 32, 41, 42,43, 61, 62, 63, 64, 65 QuickTime QuickTimeand andaa decompressor decompressor are are needed needed to to see

see this this picture. picture. Screw axis example - 42 Space groups Combine all types of translational and rotational symmetry operations (230 possible combinations) Some examples: P 4mm (tetragonal) P 6/m (hexagonal)

I 23 (cubic) F 4/m 3 2/m (cubic) P 2 21 21 1(orthorhombic) P 6 3/mmc (hexagonal) screw axis glide plane QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. QuickTime QuickTime and and aa decompressor decompressor

are are needed needed to to see see this this picture. picture. CrN: Pmmn a = 2.9698, b = 4.1318, c = 2.8796

Cr in 2a, z = 0.24 QuickTime and a decompressor are needed to see this picture. N in 2b, z = 0.26 QuickTime and a decompressor are needed to see this picture. Axes settings

Unit cells can be chosen various ways - particularly, a problem in monoclinic & orthorhombic c a b

acb c b a QuickTime and a decompressor

are needed to see this picture. cba Example from a database Authors list compd as Ibam Ibma Database interchanged b and c, lists space group as Ibma not possible combination of symmetry operations Interchanging b and c gives Icma For given lattice, infinite number of

unit cells possible: When choosing unit cell, pick: Simplest, smallest Right angles, if possible Cell shape consistent with symmetry Must be a parallelepiped When cell chosen, everything is fixed for lattice. For ex., diffracting planes

Infinite number of sets of reflecting planes Keep track by giving them names - Miller indices (hkl) Miller indices (hkl) Choose cell, cell origin, cell axes: origin b a Miller indices (hkl)

Choose cell, cell origin, cell axes Draw set of planes of interest: origin b a Miller indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin: origin b a

Miller indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes: origin 1,1, b 1 a 1

Miller indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes origin 1,1, Invert these to get (hkl) (110) b 1 a

1 Miller indices (hkl) If cell is chosen differently, Miller indices change origin b 1/3,1, Inverting (310) (110)

a 1/3 1 Reciprocal Reciprocal lattice lattice Real Real space space lattice lattice

Reciprocal Reciprocal lattice lattice Real Real space space lattice lattice -- basis basis vectors vectors a

a Reciprocal Reciprocal lattice lattice Real Real space space lattice lattice -- choose choose set set of of planes planes

(100) planes n100 Reciprocal Reciprocal lattice lattice Real Real space space lattice lattice -- interplanar

interplanar spacing spacing dd (100) planes 1/d100 d100 n100 Reciprocal

Reciprocal lattice lattice Real Real space space lattice lattice > > the the (100) (100) recip recip lattice lattice pt pt

(100) planes d100 n100 (100) Reciprocal Reciprocal lattice lattice The The (010)

(010) recip recip lattice lattice pt pt n010 (010) planes d010 (010) (100)

Reciprocal Reciprocal lattice lattice The The (020) (020) recip recip lattice lattice pt pt n020

(020) planes (010) (020) (100) d020 Reciprocal Reciprocal lattice lattice The

The (110) (110) recip recip lattice lattice pt pt (110) planes d110 (010) (020) (100)

(110) n110 Reciprocal Reciprocal lattice lattice Still Still more more recip recip lattice lattice pts

pts (010) (020) (100) (230) the reciprocal lattice Reciprocal Reciprocal lattice lattice Recip

Recip lattice lattice notation notation Reciprocal Reciprocal lattice lattice Hexagonal Hexagonal real real space space lattice lattice

Reciprocal Reciprocal lattice lattice Hexagonal Hexagonal real real space space lattice lattice Reciprocal Reciprocal lattice lattice Hexagonal

Hexagonal real real space space lattice lattice Reciprocal Reciprocal lattice lattice Hexagonal Hexagonal real real space space lattice lattice

Reciprocal Reciprocal lattice lattice Reciprocal Reciprocal lattice lattice vectors: vectors: Ewald Ewald construction construction Think

Think of of set set of of planes planes reflecting reflecting in in x-ray x-ray beam beam Center Center sphere sphere on on specimen

specimen origin origin x-ray x-ray beam beam is is aa sphere sphere diameter diameter Construct Construct lines lines as as below below

Ewald Ewald construction construction Ewald Ewald construction construction Ewald Ewald construction construction

Ewald Ewald construction construction Ewald Ewald construction construction Ewald Ewald construction construction Ewald

Ewald construction construction Most Most common common in in single single crystal crystal studies studies is is to to move move (usually

(usually rotate) rotate) crystal crystal Consider Consider crystal crystal placed placed at at sphere sphere center center oriented oriented w/ w/ planes

planes of of points points in in reciprocal reciprocal lattice lattice as as below below Ewald Ewald construction construction

Looking Looking down down on on one one plane plane of of points.... points.... the the equatorial equatorial plane: plane:

Ewald Ewald construction construction Looking Looking down down on on one one plane plane of of points.... points.... the the equatorial

equatorial plane plane No No points points on on sphere sphere (here, (here, in in 2-D, 2-D, aa circle); circle); must must rotate

rotate reciprocal reciprocal lattice lattice to to observe observe reflections. reflections. rotate rotatearound aroundaxis axishere, here, perpendicular

to screen perpendicular to screen Ewald Ewald construction construction Looking Looking down down on on one one plane plane of

of points.... points.... the the equatorial equatorial plane plane Must Must rotate rotate reciprocal reciprocal lattice lattice to to observe

observe reflections. reflections. rotate rotatearound aroundaxis axishere, here, perpendicular to screen perpendicular to screen

Ewald Ewald construction construction Looking Looking down down on on one one plane plane of of points.... points.... the the equatorial

equatorial plane plane Must Must rotate rotate reciprocal reciprocal lattice lattice to to observe observe reflections. reflections. rotate

rotatearound aroundaxis axishere, here, perpendicular to screen perpendicular to screen Ewald Ewald construction construction Looking

Looking down down on on one one plane plane of of points.... points.... the the equatorial equatorial plane plane Must Must rotate

rotate reciprocal reciprocal lattice lattice to to observe observe reflections. reflections. rotate rotatearound aroundaxis axishere, here,

perpendicular to screen perpendicular to screen Ewald Ewald construction construction hk0 hk0 reflected reflected rays rays all all lie

lie in in the the equatorial equatorial plane. plane. Ewald Ewald construction construction hk0 hk0 reflected reflected rays rays all

all lie lie in in the the equatorial equatorial plane. plane. hk1 hk1 reflected reflected rays rays lie lie on on aa cone. cone.

Ewald Ewald construction construction hk0 hk0 reflected reflected rays rays all all lie lie in in the the equatorial equatorial plane.

plane. hk1 hk1 reflected reflected rays rays lie lie on on aa cone. cone. Ewald Ewald construction construction Sheet

Sheet of of film film or or image image paper paper wrapped wrapped cylindrically cylindrically around around crystal.... crystal.... looks looks like like this

this after after x-ray x-ray exposure exposure of of oscillating oscillating crystal crystal .....when .....when flattened: flattened: Ewald

Ewald construction construction To To see see reflections: reflections: move move sphere sphere move move crystal crystal change

change sphere sphere size size use polycrystalline sample reciprocal space real space only one set of planes one (hkl) Ewald Ewald construction construction

Ewald sphere reciprocal lattice representation Ewald Ewald construction construction X-ray X-ray powder

powder diffractometer diffractometer rarely used now film or image paper X-ray X-ray powder powder diffractometer diffractometer radiation

counter Crystal structures Ex: YCu2 is Imma, with a = 4.308, b = 6.891, c = 7.303 , Y in 4e, z = 0.5377, B = 0.82 2 and Cu in 8h, y = 0.0510, z = 0.1648, B = 1.13 2 Intensities

Now Ihkl = scale factor p LP A |Fhkl|2 e2M(T) e2M(T) = temperature factor (also called Debye-Waller factor) 2M(T) = 162 ((T))2 (sin )2/2 2 = mean square amplitude of thermal vibration of atoms direction normal to planes (hkl) I(high T) e2M(high T)

1 = = I(low T) e2M(low T) e2M(high T) - 2M(low T) Intensities > crystal structure So, OK, how do we do it? Outline of procedure: Measure reflection positions in x-ray diffraction pattern index, get unit cell type and size, possible space groups Measure density, if possible, to get number formula units/unit cell (N)

density = N x formula wt/(cell volume x Avogadro's no.) Measure reflection intensities, get F-values, calculate electron density distribution from Intensities > crystal structure Electron density distribution tells where the atoms are (XYZ) is plotted and ) is plotted and contoured to show regions of high

electron density (atom positions) anthracene Intensities > crystal structure But WAIT!!! Ihkl = K |Fhkl|2 = K Fhkl* x Fhkl = K (Ahkl - iBhkl) (Ahkl + iBhkl) = K (Ahkl2 + Bhkl2) Ihkl/K = (Ahkl2 + Bhkl2) So, can't use Ihkls directly to calculate Fhkls and (XYZ) is plotted and )!! Many techniques for using Ihkls to determine atom

positions have been developed, most of which, at some stage, involve formulating a model for the crystal structure, and then adjusting it to fit the intensity data