S , , What is crystallography? TrentoSchool,Sept15th-20th2012 Crystallography is the branch of science, which is dedicated to the study of Associazione Italiana di Cristallografia matter on the atomic scale. and 10th School "Paolo Giordano Orsini" Crystallography is primarily interested in the spatial distribution of atoms or groups of atoms.

There is an important relation between physical and chemical properties of I : matter and its structural characteristics. , ,

Therefore, we can claim that the structure-property relations are part of our definition of crystallography Rita Berisio, Istituto di Biostrutture e Bioimmagini, Consiglio Nazionale delle Ricerche, Napoli. [email protected]

http://escher.epfl.ch/eCrystallography/ (by Gervais Chapuis) HOW TO IMMAGINE A CRYSTAL A crystal is an ordered repetion of an object in space: 1.DRAW ONE MOLECULE

2.MOVE IN SPACE STARTING FROM TWO DIMENSIONS

3-dimensional c lattice a' b a

Unit cell 0 b

a

3. NOW MOVE MOLECULES IN THE THIRD DIMENSION After you have drawn it, you will realise that the organisation of molecules in the crystal can be described YOU WILL BE ABLE TO DESCRIBE THE WHOLE SET OF MOLECULES IN SPACE IN TERMS OF A CRYSTAL LATTICE by its UNIT CELL a

b six parameters cell lengths a, b and c CONVENTION: c isoppositetotheaaxis, 0 the angles , and . isoppositetothebaxisetc.

With thisconvention,allthe c angles are unambigously determinedoncetheaxesare b labelled. a

Point group symmetry Rotation 2 /1 2 /2 2 /3 2 /4 2 /6 Molecules in a crystals are related by SYMMETRY OPERATIONS Symbol for the 1 2 3 4 6 What is a symmetry operation? symmetryoperationsleavingatleastonepoint rotation unchangedduringtheoperation. Symbol for the 1 2 3 4 6 symmetry operation: a displacement, which maps the object onto itself such that the roto-inversion mapped object cannot be distinguished from the object in the original state.

ROTO-INVERSION

ROTATION

Immagine thatthe originison the centre of symmetry x, y, z -x, -y, -z Helicoidal (screw) axis Point symmetry groups represent only a subset of all possible symmetry For each rotations axis,we can have a number of operations observed in crystals and possibilitieswhichisequaltotheorderoftherotationsaxis-1 defined by the SPACE GROUP

Table of possible cases All symmetry operations in crystals are defined by the Axis Translations Symbols SPACE GROUP

2 1 2 21 3 1 3, 2 3 3 , 3 Anexampleofspacegroupsymmetry 1 2 4 1 4, 1 2, 3 4 4 , 4 , 4 Helicoidal(screw)axis 1 2 3

6 1 6, 1 3, 1 2, 2 3, 5 6 61, 62, 63, 64, 65

A molecular example: The number of SPACE GROUPS is quite large, 230, each Maltose binding protein numbered from 1 to 230.

Screw axis 41

International Tables of crystallography 1. ROTATION OF 90 ALONG THE AXIS are a UNIQUE SOURCE of information!! 2. FOREACH ROTATION: TRANSLATION OF A QUARTER OF THE CELL AXIS

UNITCELLLENGHT

Independent Systems Crystal classes Bravaislattices lattice parameters No worry, of the 230 space groups, most frequent are a small a=b=c Cubic m3m, m3, 43m, 432, 23 P, I, F subset = = =90 a=b, c Tetragonal 4/mmm, 42m, 4mm, 422, 4/m, 4, 4 P, F = = =90 Inproteins,onlyafewspace groupsfrequentlyoccur: a=b, c Hexagonal 6/mmm, 6mm, 6m2, 622, 6/m, 6, 6 = =90 , =120 P a=b, c Trigonal 3m, 3m, 32, 3, 3 R = =90 , =120

a, b, c Orthorhombic mmm, mm2, 222 P, C (or A or B), I, F A STATISTICS = = =90

a, b, c Monoclinic = =90 , 2/m, m, 2 P, C (+cyclic perm.)

a, b, c Triclinic 1, 1 P , ,

International Tables of crystallography are a UNIQUE SOURCE of information!! Given a sample, find its atomic structure Protein crystallography in one slide diffraction techniques Over-expression

X-ray,neutronorelectrondiffractionarethemostimportanttoolstostudythe structureofmatteranddeterminethearrangementofatomsinmatter.

Purification

DIFFRACTION Data collection Density map interpretation Crystallisation

proteina + reservoir

H2O

Theinterferenceisconstructivewhenthephaseshiftisproportionalto2 ;this reservoir conditioncanbeexpressedby Bragg'slaw: Structure solution and density map calculation

We give the name atomic scattering factor to the ratio between the amplitude scattered by an at Scattering by atomic electrons Scattering by atoms By scattering we will refer here to the changes of direction suffered by the Wegivethenameatomicscatteringfactortotheratiobetweenthe incident radiation amplitudescatteredbyanatomandasingleelectron. Theincidentradiationonly"sees"anaverageelectroniccloud,which Electron scattering factor ischaracterizedbyanelectrondensityofcharge (r).Ifthis distributionisconsideredsphericallysymmetric,itwilljustdependon fe(r*) = e (r) exp(2 r* r)dr thedistancetothenucleus,sothat,with:

Duetothemovementoftheatomicthermalvibrationswithinthematerial,theeffectivevolumeof theatomappearslarger,leadingtoanexponential decreaseofthescatteringpower, characterizedbyacoefficientB

f= fe(r*)exp [ -B sin2 / 2 ]

B= temperature factor

Scattering by a crystal

Resultantoftheinteractionbetweentheelecromagneticradiationandelectronsinthecrystal. WAVESWILLCOMBINETOPRODUCEINTERFERENCES(eitherconstructiveordisruptive)

Electronsbecomesourcesofelectromagneticradiation SCATTERINGIN ALLDIRECTIONS scattered waves produced by all atoms in the unit cell, NOT IN ALL DIRECTIONS!!!

INTERFERENCE: COMBINATION OF WAVES SCATTERED BY ELECTRONS Thus,wegivethename structurefactor, F,totheresultantwave fromallscatteredwavesproducedinonedirectionbyallatomsin theunitcell:

F(h,k,l) = j fj exp 2 i (hxj + kyj + lzj)

CONSTRUCTIVE = |F(hkl)|expi (hkl) =A(hkl) +iB(hkl) INTERFERENCE |F(hkl)| is the amplitude and (hkl) is the phase Vectorial sum of scattering factors of each atom

2 i(hxj+kyj+lzj) fje F(hkl) DIFFRACTION ACCORDING TO Sir BRAGG In a typical diffraction experiment: Single crystals made of planes reflecting x-ray radiation which is d, which , which

Diffracted x- ray Incident x- 2 d ray

Goniometer f or crystal orientation Detector

X-rays reflected by adjacent planes are IN PHASE when Diffraction WHEN the Bragg s law is respected. BUT: How to predict the position of spots on the detector??? 2d sin = n Bragg Law We will use two geometric constructions called THE RECIPROCAL LATTICE and THE EWALD SPHERE

TheEwaldsphere RECIPROCAL LATTICE: its main features Maincharacteristics Reflecting Planes: Millerindices: described by the Miller inverse intercepts Indices h,k,l along the lattice 1.Centeredonthecrystal vectors

2.Ofradiusr=1/ Features of the scattering vector s1 S= ( s1 -s0 ) AMPLITUDE: Thereciprocalspaceisageometricconstruct Reflecting planes hkl Scattering vector S = 1/ dhkl s0

s0 Maincharacteristics S direction: dhkl perpendicular to the family of reflecting 1.ItsoriginisontheEwaldsphereANDontheincomingbeam planes hkl

Diffraction conditions: |S| = 1/dhkl = 2sen / 2.ItisalittlemorecomplicatedtobuildthantheEWALDsphere, but Iwillmanagetoconvinceyouthat itmakesourlifeEASIER! Families of scattering-vectors set of points lattice planes hkl

RECIPROCAL LATTICE: HOW TO BUILD IT Let s use both Ewald sphere AND the reciprocal lattice General, simple rules The orthorombic case: S a* b, c c 1/dhkl = 2 x 1/ x sin 1 a*=1/a; 2 b b*=1/b; c* b* aa* ab* ac* 1 0 0 b* a, c c*=1/c ba* bb* bc* = 0 1 0 X-ray beam ca* cb* cc* 0 0 1 c* a, b a* a d A little more complicated, but not too much. Reciprocal lattice Direct Lattice 2 sen = 1/dhkl 2 sen = 1/dhkl Bragg s Law!!!

WHAT WE LEARNED: When the reciprocal lattice crosses the Ewald sphere, the All the reciprocal lattice points can be described by a linear combination of multiples of two Bragg s Law is respected! basis vectors a* b* and c*. h = ha* + kb* + lc*

THE INFORMATION WE DERIVE FROM OUR DIFFRACTION IMAGES Simulation of a diffraction experiment

crystal de t e ct

Spacingbetweendiffractionspots or Space group symmetry X ray source (after projecting back on the Unit cell parameters Ewaldsphere)definesunitcell

Distribution of atoms in the Spot Intensities (Ihkl) unit cell Ewald sphere

Proportional to the amplitude of the diffracted wave (Fhkl )

3 2 2 Ihkl=K I0 L P A Vcryst. Fhkl / V cell Reciprocal lattice Simulation of a diffraction experiment Indexing: unit cell parameters Therealcellparametersaredeterminedbytherelativepositionsofthereciprocallattice Rotation method points. Thecrystalreciprocallatticecannotbe Diffracted x-ray Reciprocalcellparametersaredeterminedduringaprocessknownas indexing the diffractionpattern.Fromthereciprocalcellparameterstherealcellparametersarethen sampledinonerotationasthiswould Incident x- producemanyspotsononeimage ray 2 calculatedaccordingtotherelationsbelow. (overlaps) a = b* c* = (b*c* sin *) / V* b = c* a* = (c*a* sin *) / V* series of rotation images c = a* b* = (a*b* sin *) / V* Goniometer for crys tal orientation Detector V* = 1/V = a*b*c* (1-cos2 *-cos2 *-cos2 * + 2 cos * cos * cos *) HOW MANY IMAGES? This depends on the crystal symmetry cos = (cos * cos * -cos ) / (sin * sin *) cos = (cos * cos * -cos ) / (sin * sin *) cos = (cos * cos * -cos ) / (sin * sin *)

Indexing: crystal symmetry Example:spacegroupsinthemonocliniccrystalsystem(e.g.P2,P21,C2)must haveasingle2-foldrotationorscrewaxisalongtheb-axisoftheunitcell(by In real space,symmetryisa unique pattern of symmetry elements convention). combinationofrotationsandtranslations calledthe spacegroup. Eachspace groupisamemberofaBravaisLattice, P2, P21 are PrimitiveMonoclinic, andaCrystalClass. C2 is C-facecenteredMonoclinic.

In the reciprocal space, the Theyallhavea2-foldaxisparalleltotheunitcellb-axis.Whentranslationcomponentsofthe translationalcomponentsofsymmetriesare symmetriesareremoved,theyallhavethesamesymmetryindiffractionspace,i.e.that of not relevant to the symmetry of the severaldifferentspacegroupscanhave pointgroup2,oractually2/m ifyoutakeintoaccountFriedel'sLaw diffractionpattern.Onlytherotationalparts thesamediffractionsymmetry oftheoperatorcausesymmetryindiffraction Crystal System Point Group Laue Class Friedel'sLaw Theequalityrelationshipbetweentheintensitesof space. Triclinic 1 -1 Monoclinic 2 2/m (hkl)and(-h-k-l).

Orthorhombic 222 mmm

Tetragonal 4 4/m

422 4/mmm

Trigonal 3 -3 Example: 32 (312 and 321) -3m thesymmetryoperator(-x,y,-z)asfoundinthespacegroup P2 hasthesameeffectindiffractionspaceasthesymmetryoperator(-x,y+1/2,- Hexagonal 6 6/m For P2, P2 and C2 theintensityofreflection(h,k,l) z)asfoundinthespacegroup P2 . P2andP21haveexactlythesamediffractionpatternsymmetries (symmetrybetweenreflections 1 1 622 6/mmm (h,k,l)and(-h,k,-l)) isidenticaltothatofreflection(-h,k,-l). Cubic 23 m-3 IfoneincludesFriedel'sLaw(h,k,l)isalsorelatedto 432 m-3m (-h,-k,-l)and(h,-k,l)

Systematic absences and space group determination Diffraction spot integration: Somesymmetryoperationscanbereadilyidentifiedbyspecificinformationintheintensities For each spot (diffraction in that specific direction) Intensity can be derived ofthediffractionpattern.Inparticular,cellcentering,screwaxes,andglideplaneoperations by integrating the peak (e.g. Gaussian method) canbeidentifiedbythefactthattheycausecertaingroupsofdiffractionpointstobe systematicallyabsent. Then,foreachpeak,youwillderive|F|fromI:

A centered hkl k + l = 2n 3 2 2 B centered h + l = 2n Glide reflecting in a 0kl Ihkl=K I0 L P A Vcryst. Fhkl / V cell C centered h + k = 2n b glide k = 2n F centered k + l = 2n, h + l = 2n, h + k = 2n c glide l = 2n I centered h + k + l = 2n n glide k + l = 2n R (obverse) -h + k + l = 3n d glide k + l = 4n R (reverse) h - k + l = 3n Glide reflecting in b h0l a glide h = 2n c glide l = 2n n glide h + l = 2n ThestructurefactorFisstrictlyconnectedtotheelectrondensityvia Screw || [100] h00 d glide h + l = 4n 21, 42 h = 2n Glide reflecting in c hk0 anoperationcalledFOURIERTRANSFORM 41, 43 h = 4n b glide k = 2n Screw || [010] 0k0 a glide h = 2n 21, 42 k = 2n n glide k + h = 2n 41, 43 k = 4n d glide k + h = 4n Screw || [001] 00l 1 21, 42, 63 l = 2n x v z F h k l exp 2 i kx ky lz 31, 32, 62, 64 l = 3n V h k l 41, 43 l = 4n 61, 65 l = 6n

Representation of the structure Factor F Representation of the structure Factor F (1) (2)

F(h,k,l) = j fj exp 2 i (hxj + kyj + lzj)

Unit cell = |F(hkl)|expi (hkl) = A(hkl) +iB(hkl) |F(hkl)| is the amplitude and (hkl) is the phase 1 1 1 Vectorial sum of scattering F ( h , k , l ) V ( x , y , z ).exp[ 2 i( hx ky lz )] dxdydz factors of each atom x 0 y 0 z 0

2 i(hxj+kyj+lzj) fje F from each reflection Electron density

F(hkl) The structure Factor is a vector, with an amplitude The importance of phases: the Karleman and a phase experiment

Karle Hauptman

|F(hkl)|expi (hkl)

2 i(hxj+kyj+lzj) fje

The complex plane F(hkl)

ELECTRON DENSITY

1 (x, y, z) | F(hkl) | exp i (hkl) exp 2 i(hx ky lz) hkl STRUCTURE SOLUTION V amplitude phase Electron density = Structure Factor

DETERMINATION OF STRUCTURE FACTOR Onceyouhavephases calc(hkl) You have a rough 3D model PHASES REFINEMENT