Chemical Crystallography CHEM 6181
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Structure solution The electron density distribution in a solid and hence it structure can be determined by Fourier synthesis if the the values of a large number of Fhkl are known 7 but only thkll are obtained from measured intensities 7 the phases are unknown Solving a crystal structure involves determining these unknown phases X7y7Z7 F p Fhkl 7 2 t eXp2139cihXikyilzi 7 Ahkl iBhkl pXyzlVEEEFhkleXp21tihxkylz 7 or pXyZlVZZZFhkllCos2nhXkylz 0Lhkl Recovering the phase information If the electron density in a crystal could take any value trying to estimate what these lost phases should be would be hopeless However the electron density distribution is positive everywhere and tends to occur in balls the atoms This knowledge allows us to recover the missing phases Constraint on pXyz ondmnns rmummg rr lollVI39 mm mmusgnhvlly of demoquot den WW Choosing phases For a centrosymmetric structure the phase angle 0L can be 0 or 1800 7 For N Fs there are 2N possible sets of phases gtgt only one of which gives the right answer For a noncentrosymmetric structure the phase angle 0L can take any value between 0 and 3600 7 For N Fs there are 74N distinct sets of phases gtgt considering 90 increments is usually OK An example of phase choice quot by HOW DO I comma THE ELECTRONDENSITY fl WAVES x mm r WNW m Wm H 7 my W W 7 My 3m mm w er uh m Approaches to structure solution Direct methods 7 common for small organics Patterson methods 7 need a heavy atom lsomorphous replacement and MAD 7 for large biopolymers Direct methods The constraint that pXyz is positive and has well defined peaks allows the development of statistical methods for phase determination There are many slightly different approaches to this 7 still an active area theoretically E values Calculations are done using normalized structure factors E rather than Fs 39 Ehk1 Fhk1 SEfJZGIkIHIZ 7 where fjo eXpBjsin267t2 Removes thermal motion and form factor fall off 7 behaves like point atoms Point atoms and scattering 20 point atom atomic scattering factor 0 normal atom 0 l l 0 1 2 sinHA Intensity statistics Examining the distribution of E values can be informative A 10 a Avcragv value iluncenlmsymmelnc cenlrnsymmcinc hypercenlnc K lt E 0886 0798 0 718 lt E1 gt 1000 1000 1000 lt E2 71gt 0 730 0968 1145 lt E2 71 gt 10 20 ltE 7l3gt 20 30 Triplets Direct methods make use of Triplets 7 three re ections hlklll h2k212 h3k3l3 related by h1h2h3 7 k1k2k3 7 111213 7 0 7 three re ections are often written a say h h1 and h where h implies hkl 456 135 and 5211 form atriplet If the three re ections have large Es and two of them have known phases the third phase can usually be estimated reliably es ea lvtgt t 5 the used bcnause they halt br tht mam thhttthatm tn the a map phhm are assigned th a hmttttt Il ltmlllt r 1 selected re ections These are tht starting set and Include those that de ne the Walton athe angm vrxazibde mny phases and thus that tiethe athteh 39nllnlmml ph hf the crystal strmrtltrt 2 any 25 thheeh enantmmorphrde ntng phases Sets uf three ahead reflertzmls are selected with mdtces that satisfy the tnpte product htgn 739Elatwllshlp 22 formula ah z and hth 7 an 81 ahm 11 means the retatwt phasc angle ofthc entrosymmetrzc Bragg Tc eclwn hkt and a means quotis probably equal to quot In other words It or 11 h k I Dr 1139 h h I k I l or 11 h are found This 2 called the 2 sigma2 listing and 1s prepared by eahttdmag each Ehk1 gt 15 and searching far all possible tnteraetmns of h uttth h and h h Some Bragg re ettmhe hm many mtmmtmns uthxlc rtthers hm anly a few Direct methods in outline 2 Probabilities are calculated that each triple product 15 pos39itzve untl that each E value is positive The tangent formula Eh I EWEII a taintuh that T mm N Eh IEh Eh h39 t costnh that 810 is usut fur the refinement of nunccrttrttsymmetrtc structures to give a consistent set of relative phases The mast probably current best phase set 13 used tu culnulttte an E map This map is calculated the same way as an electrondensity map but uses lEhlcl I instead afFhlcl l as the Fourier cae iczent The area around each peak in the E map is settreltett far evitlenue 0 atomic connectivity and hence muletulttr fragrrtenta The coordinates af peaks selected from this map represent atomic positions and are used as an xnztzal trial structure Patterson maps Patterson maps indicate all the interatomic vectors that occur in a structure For large numbers of atoms they can get very complicated N2 peaks Peak heights are proportional to Z2 so heavy atoms stand out Puvw ZEZFhkl2cos2nhu kvlw What is the Patterson function The Patterson is a convolution of the electron density at all points Xyz with the electron density at points xuyVzw Puvw ipxyzpxuy vz wdV Note the Patterson function has its largest peak at 000 and that they are always centrosymmetric A graphical illustration oligui 0 lt7 9 We Or A 11h tOO D 0 on r thy tar r o uuernrnrnu vector Interatomic vectors in a centrosymmetric structure origin y Aminrit l 11 t lz Jars at Alumtit glyr i 72 0 Jim n 01m rugas not w n H ilili39mtnmu wrtors u u litlogyilmiunl Mnliiun n Tin rimmlt nnnupennnt tutu u u I Atlerwu ureu Patterson men crystal structure FIGURE 3 14 a and p Crystal rtrueture of potassium dihydwgen phosphate sud e the Patterson map The potnsrrurn ions and phosphorus atom he over each L pr 39Notembquot1 39 r groups so that there are eight rather than our P70 veotors around the origin Ref 57 Copper sulfate FIGURE 8 15 3 Copper sulfate pcntahydrate Patterson map with b CuCu and Cues vectors shown and c the erystai structure Ref 58 Cu black circles 8 open circles O stippled circles H atoms omitted for Clarity Hexachlorobenzene on Hot ms t it in i he Patterson limp mi h the rrystai structure oflitxaciilmobenr zeiw Ref w c lhlougit 1g utow trees afilm crystal strutturo sunorrmhnsed on i u Prtlivrcon map in outer to aliow whmt the Patterson punks Lame mm Types of Patterson map If E2 or E2l are used as coefficients different types Patterson maps can be calculated 7 EV gives super sharpened maps 7 lEZll gives a map with no origin peak Harker planes and lines Peaks in the Patterson map tend to concentrate on planes or along lines depending on the space group symmetry Sharpening and origin removal lo m lage m a Asymmelric pm at u now Ionian o m Pulgmu mp ol nnmnl munsnnucm ermmmmudmmm Harker lines Consider space group Pm 7 Xyz and Xyz are equivalent gtgt you will get peaks at 02y0 in the Patterson gtgt along the Harker line 0u0 Consider Pc 7 Xyz and Xyzl2 are equivalent 7 peaks in Patterson occur at 02yl2 7 on the Harker line 0vl2 Harker planes Consider P21 7 Xyz and Xyl2z are equivalent 7 peaks in Patterson occur at 2Xl22z 7 on the Harker plane ul2w Consider P2 7 Xyz and Xyz are equivalent 7 peaks in Patterson occur at 2X02z 7 on the Harker plane u0w Using a Patterson Pattersons are usually used for heavy atoms However once a heavy atom is located phases can be estimated for each re ection 7 assumes heavy atom dominates scattering 7 use phases to calculate a density map and look for peaks in the density map gtgt locate lighter atoms this way Does not work well for complicated structures Isomorphous replacement Consider a structure that contains just one heavy atom l3971 FM FR If we have two structures that are identical except that the heavy atom is a different element F2FM2FR iandFlF2FM1FM2 Isomorphous replacement If we have located the heavy metal position using a Patterson we know phase and amplitude FM1 and FM2 We know F1 and F2 from the diffraction data For a centrosymmetric structure we can uniquely determine the phases of F1 and F2 Example calculation FMi F i F2 i J P3 1 Possibilities Deductions 011 M2 I Fi I Sign of F and F2 4 3 9 12 yes 3912 b 73 9 6 no 7396 c 73 9 12 yes 379712 7 7 d 3 9 6 no 37976 Isomorphous replacement and macromolecules Isomorphous replacement is commonly used to solve protein structures Protein crystals contain a lot of water Replace some water with a heavy atom You now have native protein and a heavy atom derivative that are hopefully isomorphous FPHFPFH Locate heavy atoms Locate heavy atom in derivative using a difference Patterson coeffs llFPHl39lFPH2 This Patterson is dominated by vectors between the heavy atoms Phase ambiguity with one deriv radius IFP protein c phase a pm em Baitt phase a pim eahH a39h g c min i F i mi i Fp Interpreting atomic coordinates A crystal structure refinement leads to estimates of model parameters 7 coordinates temperature factors and occupancies What matters chemically is atomic connectivity bonding and packing Need interatomic distances and structural drawings Atomic connectivity lnteratomic distances can be used to determine the presence of bonds 7 chemical bonds often lead to well defined interatomic distances gtgt Csp3 C sp3 153A gtgt Csp2 Csp2 7 1 32A Distances are readily calculated from atomic fractional coordinates The calculation of bond lengths Position of atom in unit cell 7 r X21 yb zc Vector between two atoms 7 ArAxaAybAzc Calculate length from 7 ArAr lArlZ Symmetry equivalent atoms In most space groups there is more than one symmetry equivalent of an atom in the unit cell 7 you need to take this into account when calculating bond lengths Benzene Space group Pbca Each molecule has its center at an inversion center Errors on bond lengths Refinement give estimate of errors on Xyz 7 this information can be used to estimate errors on derived quantities such as bond lengths For two atoms each with a positional standard deviation of 0005A 7 62bond length 0 005212 7 assumes atomic positions are not correlated Bond length calculation programs take into account correlations Structural diagrams Many different types 7 ball and stick PLUTO 7 thermal ellipsoids ORTEP 7 coordination polyhedra STRUPLO Many different styles 7 perspective 7 projection 7 stereo ORTEP ORTEP plots are standard for small molecules The orientation and size of the ellipse tells you about the presence of thermal motion static disorder and errors in the model ORTEP plots FeCsHs Coordination polyhedra Monoclinic ZrMoZO8 Trigonal ZrMoZO8 Bond valence sums In general there is a correlation between bond length and bond strength Pauling recognized that for some types of structure you could associate a valence with each bond between a central atom and its neighbors and that these bond valencies should sum to give a number equal to the atomic valence of the central atom Bond strength bond length o o m a 0M Valence o l v m 10 30 0x nmunc K Fig 1 n a I quot0 bands 0 vainmes and lengms found in normal asymmemc hydrogen bonds me mm sulld lme Indlmlcs he vzlnnw and length found n symmzlncal hydrogen bonds Why measure dif acted beams The measurement of diffracted beam intensities allows the calculation of the structure factor moduli 1hk1 OL 10 x3vXLpAmv2Fhk12 7 Vx volume ofcrystal in beam 7 V volume of unit cell 7 L Lorentz correction 7 A absorption correction 7 p polarization correction 7 m angular velocity Methods of data collection Monochromatic camera methods 7 Rotation Oscillation Weissenberg Polychromatic camera methods 7 Laue photography Automated diffractometer and point detector Diffractometer and area detector How many re ections For each parameter you wish to determine you need to have 10 observed reflections N 47t38V1 1L3 7 11 number of lattice points in the unit cell Mon CuKn CrKa 071 A 1 54 A 2 29 A Radiation sources Xrays tubes 7 low ux line source With a White background synchrotron radiation 7 more ux White radiation use a monochromator Nuclear reactor or spallation source for neutrons 7 expensive but useful Xray tubes mme focusing tup Producing Synchrotron Radiation sEnELHan mm sscvu um Iis runno uuucuun c vchLBENnmG quot u GNEV WINNING SYAHON 52m mvs m cum szcvas I quotJ maul I Q t x mm I V 4 sicvowvn a quot 5mm quot lt45 4 Lu MP l mu tunnoIDLECULAR iwwmn 5mm The Advanced Photon Source mun m Synchrotron radiation High intensity Plane polarized Intrinsically collimated Wide energy range Has well de ned time structure Insertion deVice beam lines Fm r O m handwmmlm mu m antIOJKN at too a min p mg Neutron scattering lengths X CY 9J8quot sin 9 o5 U39 z u Potential xcut terinq contribution Scattering amplitude 10quot2cm N O Powder neutron diffraction data for cubic ZrMoZO8 min2cm neutzur VT HisL Bun 1 2Thutu Mata LrS yclu 188 ohm and Diff Fru ile I l I l I I a N CA a to Cuunts Unuseu Spallation neutron source Neutrons are produced by bombarding a metal target With pulses of protons Gives different wavelength distribution to reactor Peak uX from pulsed source is high but time average uX is not so good 7 Need to use time structure of source to make up for this no man1 s lnnf l na mtm l rumquot Reactor Pulsed source Timeof ight diffraction L Sample 29 L1 source Detector Time from source to detector is determined by neutron wavelengt vLL1I and mv h1 so t Can measure IQ without scanning detector 472mL L1sin a Q ht Use many separate detectors and sum the counts recorded in each to measure IQ with good counting statistics in less time mL L1t SEPD Special Environment Powder Diffractometer 2 theta Solid angle Str i 1450 0 086 SLIP D i 90 0 086 i 600 0 052 300 0 017 150 0 017 Only small fraction of total solid angle covered Detectors Film 7 poor sensitivity high background low dynamic range Scintillation counters 7 good sensitivity low background high dynamic range Imaging plates 7 good sensitivity low background high dynamic range CCDs and Multiwire detectors 7 fast readout good sensitivity low background good dynamic range very efficient data collection Preparing the sample For small molecule crystallography crystals gt0 lmm are usually needed Quality matters 7 Use microscope polarizers to examine sample No twins or split crystals Crystals are either glued to a glass ber or wedged inside a capillary tube 7 capillary excludes air and can contain mother liquor Sample mounts 4 crystal plug k adhesive 4 glass ber mother 4 adhesive liquor mounting pin 77 crystal Goniometer heads arc adjustment 1mm I adjustment NT latex 1 l Cb Preliminary screening of crystals After a crystal has been selected and mounted it is usually photographed 7 rotate crystal in beam and record diffraction pattern Typically this is done on the diffractometer Low temperature data collection In many laboratories Xray diffraction data is routinely collected at low temperature Slows down decompositionwater loss Allows collection of data to higher angles due to suppression of thermal motion Done by passing a stream of cold nitrogen gas over the crystal A four circle diffractometer Two common diffractometer geometries 7 Eularian cradle and Karma geometrV Setting up for a data collection Determine unit cell and orientation matrix using a photo or a blind search Orientation matrix describes the orientation of the crystal axes relative to the diffractometer coordinate system Check quality by scanning some re ections 7 reject crystals with split peaks or broad pro les Check Laue symmetry Orientation matrix mmmr mm LII mu 1 sun Is 21 u mm mm rmms m n mrrk u TmlLlLli 1005mm 1 x sxsn u murmur L WM mmc qurVTnU mum H41 r m H u 9 le mum I ller Peak shapes Data collection FIGURE 7 18 a A good and b a had peak pro le Often automatic With a point detector you can collect 100 re ections an hour May need WOO20000 re ections for a small molecule structure 7 12 hours to 15 weeks Periodically remeasure some standard re ections to check all is well Collection using oscillation photography Used with equipment that makes use of Imaging plates or Film as a detector Rotate crystal by a small amount in the X ray beam and record a photograph 7 may be 050 60O Repeat until the crystal has been rotated through 1800 or more Often requires user intervention Laue Photography Laue picture make use of polychromatic radiation 7 fast data acuuisition eligin Elvlllll sphcm tar lmlgm wavolLllgth Laue Photograph Vital information about your crystal a General inrannatien an the erystal How it was grown the faces that it developed and its dimensions in mm chlwtjnillio6 chemical formula M 44542 formula weight Monocliulc crystal system P2rr space on a 17 57210 A ullibccll dimensions with earls in parentheses I7 7 7274 Al is 73984 A p 101904 pr 3 7 only listed if not 90 by space7group symmetry 1 98299 A3 unit rell volume Number and 20 ranges of Bragg re ections used to measure unibcell dimensions Z 2 number of asymmetric units in the unit ce DI 1503 Mg nn3 density calculated from V formula weight and 2 pm measured density A Mo ke 071069 A radiation used and value used for the wavelength is 0100 mind linear absorption eoe irient T 293 K temperature or study Space group symmetry Periodic solids have lattice symmetrypurely translational point symmetry no translational component and possibly glide and or screw axes partly translational Together all the symmetry operations make up the space group for the material Lattice centering o i 39 E o o O l go a l I IKI O 1 r r O P C I F Hermann Maugmn system followed by the Schoen ies system of notation Bravais lattices cFacnremeud nlounclmlc C h Face0 mede I h b B Bodymucmi All facerempmi onhulhombic 1 gimp mnmiwl 10ml at the Ceuh r of cfacm I V 3 quotquot1 quot i 1 f ordmrhnmhir My Point a comer of cad Earr t t I V Tritium monorbnic m Sinp10 wrumuw vrlhn hmnblr J bigl liat r t t 01 0 P0quot In mm ofrcll No additional poian urn ci mg bz a Emly ccaulm l utmgunnl il 1 BodyrenIIaml uulnc I All fuemmmml mlm 15 Simple primilive Hexagonal Rhmnlmlmdml Simpio gpnmixiws rubir in Point in rmmr of vol Pain in L uuh39r ur rEll Paint m rmwr of path In lclmgunal P1 Nn mmuoml pmnh No addiluma pmm two fold screw translation aZ Higher order screw axes The combination of a rotation axis and a translation parallel to the axis produces a screw axis The direction of such an axis is usually along a unit cell edge and the translation must be a subintegral fraction of the unit translation in that direction Screw axes are designa 39 a subscript m where n between successive points of 13 mn of a unit translation Fig 320 Point 2 is generated from point 1 by rotating 360 3 and advancing 13 o the unit translation Point 3 is generated from point 2 by another rotation of 360 3 and an advance of 13 advancing 23 Point 339 arises from an additional rotation of 360 l3 and an additional translation of 23 The rest of the points in Fig 321 1 239 and 3 are obtained from points 1 2 and 3 by a unit translation Comparison of Figs 320 and 321 will indicate that 3 and 32 are related in the same way as right and lefthanded screws that is they are enantiomorphs It is characteristic of an nfold screw axis that the position of the nth point laid down differs from the initial point by an integral number of unit translations that is the positions of these points within their respective cells are identical Unit translation Figure 320 Screw axis 3 Unit trunslunon um tru slnhon l l Figure 311 Screw axis 32 Higher order screW axes 2 L If we draw a helicoidal trajectory joi a 3 axis related b a 3 and by and 6 and 4 Equot 9315 Screwaxesjnangementolsymmetryr also a 3 and a 2 equivalent objects 4V is also a 2 axis 62 is also a 2 and a 3 n is also a 2 and a 3 and 6J is ning the centres of all the objects we will obtain in the rst case a right handed helix and in the second a leftrhandcd one the two bellies are enuntiomorphous The same applies to the pairs 4 and 41 6 and 5 6 An a glide translation 112 glide plane 7 FIGURE 412 A 4 a 39 2 r plane nln a trail r n of half the unit cell edge translation a2 A lefthanded molecule is converted to a righthanded molecule by this symmetry operation Two such operations give the original molecule translated to the next unit cell translation 2 ml Other glide operations a b c n and d glides occur a glide has translational component of 12a n glide has translational component 12 a 12b or 12a 12c or d glide has translational component of the type 14a 14b Space group symbols P21 in its standard setting 7 has a 21 along the b aXis P212121 7 has 21 screws along each aXis ana 7 has an n glide plane perpendicular to a 7 a mirror plane perpendicular to b 7 and an a glide plane perpendicular to c 7 and other implied elements Combining symmetry elements For three dimensions 32 point groups l4 Bravais lattices but only 230 space groups For two dimensions 5 lattices 10 point groups but only 17 plane groups The 17 plane groups 1 SE ME zlzzzzezzzil 92 my I v l 1l 1 Se H4b 1 p2gy l 39 39 l r r 4 39 I I 39 The 17 plane groups 2 4r 4 E K e quotquotquot A 4 leg Pl 2 a a0 4 394 I FIGURE 222 um cells of the 17 plane groupsl In each pair of drawings um showing the general equivalenl positions uses a scalcnc man motif instead of the usual circle p 2 m g 2 m m Rectangular NO 7 p 2 m 3 Patterson symmetry p 2mm 0 O Origin at 2 lg Asymmetric unit OSXSt OSySl Symmetry operations 1 l 2 2 00 3 m Ly 4 a x0 Generators selected I tl0 I0l 239 3 Positions Coordinates Reflection conditions General 4 d I 1xy 2 3ily only M h2n Special as above plus c Ly U no extra conditions 2 2 b 2 2 a 2 00 90 hk h2n OJ Ll hk h2n Maximal nonisomorphic subgroups l 2p211p2 12 le in l p m 21p 1 15w Ila none In 2p Zgg 2b Maximal isomorphic subgroups oi lowest index k 31p 2mga 32 2p 2m gb39 2b 3 l4 Minimal nonisomorphic supergroups I none I Zlc2mm2p2mm2a n Oblique Patterson symmetry p2 Origin at 2 Asymmetric unit ngSi Oiysl Symmetry operations 1 I 2 2 00 Generators selected I tl0 r0l Positions 2 t 1 May 2 l d 2 Ll l 1 2 90 l b 2 01 l u 2 0 0 Maximal non somorphie subgroups I 2pl 1 Coordinates I 2 Maximal isomorphic subgroups oi lowest index lIc 2p2a3921 or b39 2 or a abb oab Minimal nonisomorphic mpergroups I 21p2mm 2192m31l2lp238 2162mm21p4 31196 I none 0 p2 No2 0 Reflection conditions General no conditions Special no extra conditions P21C can No 14 UNIQUE AXIS b DIFFERENT CELL CHOICES P I 21 C 1 UNIQUE AXIS b CELL CHOICE 1 Origin at l Asymmetric uni OSXSI OSySQ 052 Generatnrs selected 1 1000 10l0 00l 2 3 Positinns Coordinates Sn symmetry 39 4 e l I xyz 2 LyLZ 3 x12 4 XJHJH 2 d 1 HM H1 2 c 1 009 090 Monoclinic Re ection conditions Geneml Special as above plus MI 1 1 2n hkl kl 2n lZ39Z39Zd intersecting pairs of parallel 2 axes 2 2 2 Orthorhombic Palterson symmetry Pmmm 4 2I00 XI0 CONTINUED N0 14 P2C P 1 217 1 UNIQUE AXIS b CELL CHOICE 2 1 Origin al 1 Asymmetric nnit 03x31 OSySi OSZSI ana m 21 m m m Orthorhombic Generators selected 1 1000 010 00 2 3 N0 PZIn Zlm Zla Positions Patterson symme y Pmmm Mul Coordinates Refleclion conditions Wy 39 I Site symmcuy General 222 PE JZ Pliigfx i k quot 4 2 l 1xvyz 2i y zi 3191 4xiyi zi N F K F r Special as above plus N1 Ni 15 quot 1 9 di 100 0M hklhkl2n n t 9 L0 090 hkl hkl 2n 003 i0 hk12hk12n m a NNNN h T 090 LM hklhk2n l 40 Q 0 0 0 UNIQUE AXIS b CELL CHOICE 3 12 Oi39 i O Oi Origin at i O 40 Asymmetric nnit OSXSI OSySi OSle 3 Q 3 I l Generatorsselected l Il00 t010 t00l 2 5 Positions Coordinates Reflection conditions Origin at 1 on 121 snesynuneuy39 General 4 e 1 xyz ZHHJHJ 3111 4xiiivz hot h2n Symmetry operations 2 I II Asymmetric unit 03x39 OSySi OSZSI Special as above plus 1 l 2 200J L01 3 20 0 0y0 4 20110 x i i 1 H yz 5 1 MW 6 a LN 7 m xz s quot0 2 d 1 00i LL hklzhk2n 2 c T LOX 090 hk1hk2n ana E 5 E 3 u quot0 s 9 a g iquot a s quot 39 396 E E 2 N a 250 in 8 N a 2 II n a a 9 n n o 4 4 a H NNNNN m 9 gala 2 39 llll g 2quot g 5 ENlt4 a c ltao v 2 2 agequot g quot s quot 0 8 Li 3 E amp Oggxoo m E t 3 lt2 N u o 3 i A N o a z Z A 5 1 m 4 x V 1 RR g G u A A E m 5A N V00 v N Emu 35 quot m quota 1 a A 39 r V E 3 W i s g as m DA 8 1 H a g H 5 r wag amp my u quot1 lt Ell3quot 3 x3 8 3quot o no 5 E v VV 2 w quot 1 3 us 6 g 7quot aquot Eamp t g a n E a 8D a O m V 5 EN 4 8 1 39 U 5 e 3 FT q U 5 T lt5 o m m s u m o 0 N 39 1a mE m 2 a Q a u v K O 4 a cm 3 SA AA s m hm Mk NW O O Av N z 2 3 A 2 EA 2 G a a a V 5 a E II N c E s e 5 n o 5 3 1 c o 8 E 33 E v3 3 K o c at 2 E 3 g a N l I quot a s s a 3 l 9 AQ c g E a 5 D E 5 a 2 225 z 2 39 w 5 H 7 i 2 2 3 739 g as i z Emit E E Z a u a a x g E E 3quot w an g 8 3 V v rm 2 2 2 Symmetry in the diffraction pattern 0 Laue symmetry diffraction pattern symmetry O The symmetry of a diffraction pattern weighted reciprocal lattice tells you about the symmetry of the crystal 0 Laue symmetry is the point symmetry of the solid plus an additional center of symmetry 91hkl Ihkl Friedel s law Assignment of point group 91f possible the Laue symmetry should be used for point group assignment not the metric symmetry OBecause of Friedel s law the point group symmetry can not always be unambiguously assigned Distinguishing orthorhombic from monoolinio b n t o clini a horhombic cases in addition tn Fr edel symmetry HUI IGITIH a Orthorhombic 1hkl hFI 1ch1 nic 1hk1 10031 HMO with B 90 and c with 7 9 l h I 39 ensities in 1 indicates that the crystal i FIGURE 417 Laue symme ry in In no c nd ort Systematic absences The absence of certain re ections from a diffraction pattern tells you about lattice centering SCI39CW axes glide planes Note systematic absences only tell you about symmetry elements that involve translation Table of absences TABLE 45 Examples of crystal symmetry determinations from systematic absences in Bragg re ections All Bragg re ections of the type listed below must be absent a Bravais lattice absent re ections deduction on symmetry symbol translation no restrictions primitive P none k 1 odd centered on Aface 100 A 22 02 l h odd centered on Bface 010 B 02 12 h 1 odd centered on C face 001 C 112 62 h kl not all odd centered on all faces F a b2 or all even b c2 and a c2 h 1 1 odd bodycentered I a b cQ 1 Symmetry elements absent re ections deduction on symmetry symbol translation hOO 11 odd 2fold screw azis along a 2 112 010 1 odd 2fold screw axis along 19 21 b2 001 l odd 2fold screw axis along 6 2 02 001 I 3n 1 3fold screw arts along 6 31 or 32 cS 203 or 311 2 Okl 10 odd glide plane perpendicular to a b b2 Okl 1 odd glide plane perpendicular to a c Assigning space groups 3 mm so onuuuuuu o h kw Puma b Puma h h c 1222 3V4 DEDUCTION 0F POSSIBLE SPACE GROUPS Point group mmmZm Zm Zml Table 32 cum g 2 3 lt2 3 a ORTHORHOMBIC Laue class mmm12m xx ixa xrxdxxxatd l4 GRAPHIC U SYMBOLS FOR SYMMH39RY ELEMENTS d Symmetry axes normal to the plane of projection three dimensions and symmetry points in the plane of the gure two dimensions Screw vectort quota righthanded Printed symbol I V s V r y t screw rolalto n units ofihe suhetemenis Symmcuy m m symmul pmm bldpmcdl ymbol shortest lattice translation vector in parenthe 5 parallel to the axis ldcnlll None 1 Twoiold mutton axis 1 Twoi old rotation point lino dimensions 5 Twoi old screw Threcl39old miaiion axts 1 Threeiolrl rotation point two dimensions l Threei old screw axis 393 sub 1 Threeroid screw axis 3 sub 239 None 2 Fourfold rotation iith Fourfold rotation point I two dimensions Fourfold screw mus 4 sub 139 Fouriold screw axis 4 sub 2 Fourfold screw axrsv 394 sub 339 Sixlold rotation axis Sixiold rotation point tiwo dimensions Sixfold screw axis 6 sub 1 nil1hr Sixiold screw axis 6 sub 239 Stxiold screw axis 6 sub 3 Sixiold screw axrs 6 sub 4 Sixioltl screw axis 396 sub 539 Centre of symmetry inversion centre 1 1 har 0 Re ection pointr mirror point tune dimension lnyersion axis 3 bar go 4 4lt okooe hovj Imersion axis 4 bar39 Inversion ax39 6 bar39 Twofold rotation axis wtth centre of symmetry Twoiold screw axis with centre of symmetry 6 3 Fouriold rotation aXis Will centre 0 ll None of symmetry 4 sub 2 row axts with centre of symmetry I 2 41m Sixlold rotation axrs vtith centre 0 None at symmetry 6 sub 3 screvt XmS with centre of symmetry 2 0 Z s lt o slim 6132511 tug It tli sttttinctty pt Is 311mm 4 is iii symmetry 1 rttiii d5 wiii ti ticmun tomb i and s tvt 4s mti tom pitrttl it with only one i 10ml orctir tit pairs 1 Itetglits l means it U ind i in ith spitt groups him it is gttct i a li tnds int i tntl is it mt including it t and t lllt immtitniirw Lllul l illlcrnJU i t CL nlch l SYMBOLS AND TERMS USED IN THIS VOLUME b Symmetry planes plrnllel to the plane of projection Symmetry plane Re ection plane mirror plane Axial glide plane Axial39 glide plane in centred cells only 39Diagonal39 glide plane Diamond glide plane Graphical symbol 7 l V 1 F pair of planes in centred l t 2 cells only r Glide vector in units of lattice translation vectors parallel to the projection plane pnme j symb l None m i n the direction of the arrow a b or c in either of the directions of the two a by or c arrows gin the direction of the arrow rt 4 in the direction of the arrow the glide 1 vector is always hall of a centring vector ie one quarter Ola diagonal of the conventional facegcentred cell amp stmimn gut minit t ill it lelt enrii til tht xitriccvgtuiip l grim A t wtpmic intuit slttll int i L4tntl rm citim ll 1quotth ca 0 1 symbol in Lllttlt mm 1 1 1ttitl i st net Hlll ll lCS Nit yinmetr t t 26 e Symmetry planes inclined to the plane of projection in cubic space groups of classes 43m and mjm onlv Symmetry plane Re ection plane mirror plane Axial glide plane Axial glide plane Diagonal glide plane Diamond glide plane pair of planes in centred cells only ti iiiii n Ht Graphical symbol for planes normal to Glide vector in units ollattice transla tion vectors for planes normal to Printed 011and 01139 In iii it Hit by 101 and 101 0 from 25 011and011 101and101 None None m along 100 along 010 a or b along 01139 lalon 101 or or along 011 2 g 1 along 101 Jalonglll or ialongUll or n iong 111139 iong l 1 l39 1along 111 or 1along 111 or iong 111x thong 1112 along ill or along 111 or along 11 lx along 111 ms complete tirlltiigmlihlt printtutu tit llIL39 symmetry iitmtttii cc lritllsltllttm CClUrs Form factors the FT of atomic p 47r1392pr electronsA scattering factor felectrons Fourier transform I 0 2 U LU distance to atomic center sin 9 3 1 Perfect and ideally imperfect crystals Most real crystals are not perfectly ordered 7 they are made up of blocks displaced relative to one another gtgt angular displacement is the mosaic spread For perfect crystals 1 0L F For ideally imperfect crystals 1 0L F2 7 most crystals are near the ideally imperfect limit Why does mosaic spread matter In perfect crystals Xray photons may be scattered more than once 7 leads to primary extinction In perfect crystals a significant number of photons are scattered by strong re ections 7 this reduces the number of photons making it to the back of the crystal gtgt secondary extinction Structure Factors Fhkl Fhklexpiochk1 Ahkl iBhkl Ahkl Z cos2nhXJkyjlzj Bhkl Z sin2nhXJkyjlzj Thermal motion Thermal motion of atoms in crystals smears the electron density of the crystal out This can be modeled using a temperature factor or thermal parameter I I 2 2 fJ fJO eXp BJs1n 67 BJ 8712 meanu2 gtgt meanu2 is the mean square amplitude of Vibration Typical temperature factors Temperature factors vary with 7 sample type 7 and temperature For inorganic solids B N l3A2 For organics B N 26A2 For biopolymers B N 30A2 Polar materials Many solids have a polar axis 7 LiNbO3 ZnS etc L C 3 J J 25 1 HI quotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquot quot p013 l 0 J l sense 4 Zn 2 SM 99 J Chiral materials Chiral solids can arise in a number of different ways 7 optically pure chiral molecules will crystallize to give a chiral solid 7 racemic chiral molecules may crystallize as chiral solids gtgt one enantiomer in any given crystal 7 achiral units may organize to give a chiral solid gtgt quartz Space groups and chirality For a solid to be chiral 7 NO mirror planes inversion centers glide planes Any material that crystallizes in a space group that contains only rotation and screw axes is chiral However some space groups are enantiomorphous 7 For example P61 and P65 Absolute structure When you have a polar solid what sense does the polar axis have When a solid is chiral which enantiomer do you ave Both of the above are questions of absolute structure Absolute structure is difficult to establish 7 chemical and spectroscopic methods usually determined relative structure Absolute structure and crystallography The primary technique for establishing absolute structure is crystallography The determination of absolute structure makes use of anomalous scattering and the resultant breakdown of Friedel s law Anomalous scattering When anomalous scattering is important 7 f fo f E if E Consider the structure factor expression 7 Fhkl A iB F if A1 iBa Inverting the absolute structure xyz to xyz g1ves 7 Fhk1 7 A iB r 19an iBa The magnitudes of these structure factors are not the same Waves Fourier series syntheses and transform s Waves and Argand diagrams The magnitude and phase of a wave can be represented on an Argand diagram 7 F lFlCosot i Sinot Interpreting Argand Diagrams Phase angle 0L Amplitude 7 tan0L 7 BA 7 Fl 7 A2 132 112 00 090quot 045 0225 3mm Bow mm BUzkl e r Atriklgllrx m Avgmlgiljlolkl ffv 44 J g 39 gt1 firm0 7m umplll ude ampmuda Amth udc amplmlda Phase angles The phases of scattered waves Xrays in crystallography are chosen relative to the unit cell origin phase relative to wave at Origin wave scattered from origin Summing waves Scattered Xrays in a diffraction experiment have the same frequency 7 summing can be represented on a vector diagram The electron density The electron density in a crystal is periodic This can be represented as a sum of cosine and sine waves 7 wavelengths are related to the lattice periodicity Breaking down a periodic function into sine and cosine waves is a Fourier analysis Constructing a periodic function by summing cosine and sine waves is a Fourier synthesis Fourier analysis Fourier synthesis Fourier gt analysis Fourier syntheses using lenses Fourier synthesis and electron density 3 1s The electron density in a crystal is related ANAL 0F DIFFRACTED to the diffraction pattern of the crystal through a Fourier synthesis like procedure pogyJ g lpwk l cos27rhx ky lz om or usually 1 E d 3 px y z Fhkl I exp 13905 eXp i27rhx ky 12 a n di m m a 231510 lens beam I 9 periodic brought 3 5 Object separately P 39 t focus Fourier synthes1s of p Bragg peaks and electron dens1ty Duo A A Intensity of a diffraction pattern peak is related to the L smmi um l j A m m H on magnitude of an electron density wave in the crystal pm i w new 5quot y 39W 2m 11 V L fC mm to 900 The direction of the wave is given by the Miller indices xxVzvao 000 to 700 for the re ection I l r s g 4 F39Fquot quot quot 39w quot m j 0quot lo 600 m 7 m 97 Tm A 000 La 50a ifquot 39 Vx vaMl yuu X x 000 to 400 I cw n wan N can to 300 m V Xx g 000 to 200 L iJrrJ7G r7 one to 100 If Electron density waves and lattice planes am Bragg rcllcclmu K I 0 0 2 0 U Bragg re ection Bragg matLuau Y Y 144m 13 3 1 u 0 density wave 2 u u dunsuy MM 3 o 0 density me i i summation or dumle m ulcclmxxrdcmlly map Fourier Transforms f x ZeXp27dxygydy gy Zexp 27rixyfxdx Example Fourier transforms l vl39H n M NH FTs of electron density waves Fourier transforms and crystals R39STL LATTICE I39VIT CELL CON39HN I S mmvolm Ion cm smL STRUCTURE outz VALUES DIFFRACTION PA I J EHN lF21lt39ll L139ES mumplmiun Rl lFm L LATTICE MOLECULAR TRANSFORM Example lD FT W Iv39nm Mm Thuuni wng WWW mum g m usan m a unit uu rcpta nx as A Urinure n I o o o o u I o w n Mer munquot 011M WWW m Inlluumg mum unwliludes I u u a 5 o 0 45 2 u u 57 7 u u 4 1 u a w a a a mm 4 n u an 9 u u n 3 u o m In 0 u l Hany m rum K m vrlallv znh l u u 52 5 v n m 2 u o Jquot n n M 5 n u 94 5 n a mm 4 n o 7w x n o 41 a n a my 10 n n a l i 71 um may in O m shurlmv swim mm mmpm w m w mum sum m a m Dif action A basic understanding of diffraction physics is required if crystal structure solution and refinement is to be understood Wave characteristics pnasu w relative to origin wamlvngth Interference between waves Double slit experiment 200 l i 300 ll 42 I 0gt n IUU 300 Too 72000 l l l w a b 200 c Diffraction at a single slit The envelope function envelope envelope 4 a narrow slit b wide 5m Dif action and sampling Optical transforms DUDE BIB BEEQ El n DIE Scattering of Xrays and neutrons by atoms Xrays are scattered electrons in atoms 7 the electron cloud is about the same size as the wavelength of the Xrays Neutrons are scattered by nuclei 7 nuclei are much smaller than the neutron wavelength 7 for magnetic materials electron spin interacts With neutron spin and gives scattering Phase change on scattering hva brain Illh39anml boom 180 phin change Xray scattering by atoms Xray and neutron form factor The form factor is related to the envelope function for an atom m Scattering amplitude 12 10 2 c mm scattering factor sin WA Neutron scattering lengths TABLE 3 2 Some scattering factors for neutrons and x rays Elmmnt 1mm x rags X 1 ys Neutrona Nautrmxsquot sin A smoA b10quotquot rm normalzzed 117 0 05 100 or H 11 11 10 0 07 038 7100 21111 10 0 07 005 171 Li 61 3 0 10 01s0 025i 071o006i L1 30 10 025 7 0 015 r C 6 0 17 0 66 174 C 0 0 17 0 60 1 48 o 60 8 0 23 0 48 1 03 396 Me 200 115 042 111 6Fe 21 0 11 5 101 2 66 71390 260 115 023 061 co 27 0 122 0 za 0 66 U mU 92 0 53 0 0 21a 2 24 Diffraction from crystals A crystal is a three dimensional diffraction grating The lattice periodicity of the crystal determines the sampling regions of the diffraction pattern The unit cell contents give you the envelope function Laue equations Laue first mathematically described diffraction from crystals 7 consider Xrays scattered from every atom in every unit cell in the crystal and how they interfere With each other 7 to get a diffraction spot you must have con structive interference 7 Laue equations PD1 7 h171PD2 7 1121 PD3 7 11371 Laue and Bragg diffraction The Bragg equation Bragg discovered that you could consider the diffraction to have arisen from re ection from lattice planes Reformulated Laue equations 7 Zdhklsinehkl n7 The orientation of lattice planes It is possible to describe certain directions and planes With respect to the crystal lattice using a set of three integers referred to as Miller Indices 1139 0 s 7H n pyhuc 1s nkA Ienaoo Miller indices hkl Miller Indices are the reciprocal intercepts of the plane on the unit cell axes Identify plane adjacent to origin 7 can not determine for plane passing through origin v1 Find intersection of plane on all three axes Take reciprocal of intercepts prlane runs parallel to axis intercept is at 00 so Miller index is 0 Examples of Miller indices Families of planes Dif action Geometry Miller indices describe the orientation and My A spacing of a family of planes 7 The spacing between adjacent planes in a family is referred to as the dspacing crystal mm H z p13 w Three different 7 u Note all families of planes 100 Planes d mm dspacing between are members Lilystnl mm 300 planes is one of the 300 plane W third ofthe 100 family Spamg 100 2001 300 Unit cells and dhkl Real space and reciprocal space CRYSTAL SPACE DIFFRACTION SPACE Crystal lattice lt v 7 7 r 7 7 7 r 7 7 7 r r r A gt reciprocal lattice unit cell V a r A structure factors li rzuition crystal H pattern of crystal contents The reciprocal lattice It is convenient When talking about diffraction to use the concept of a reciprocal lattice The reciprocal lattice is related to the real space lattice by b i b i 17 27 alla2an alazxa3 a2 xa3 a3 X611 all X a2 b 3 alazxa3 a1 a2 213 are the vectors of the real space lattice alternatively abc and b1b2b3 are the vectors of the reciprocal lattice alternatively abc Note a1azxa3 is the unit cell volume Properties of the reciprocal lattice Note alum51 So a1b1 l but a1b2 0 and a1b3 0 etc 7 This is the origin of the term reciprocal lattice 7 The reciprocal lattice and real space lattice are orthonormal Any point on the reciprocal lattice can be speci ed by a vector Hhkl hlo1 kb2 lb3 hkl are integers 7 This vector is perpendicular to the plane in real space with Miller indices hkl 7 The length of this vector Hhkl ldhkl where dhkl is the interplanar spacing in real space 7 We get to represent a whole family of planes in real space by a single point in reciprocal space The geometrical construction of the reciprocal lattice 9quot y H 1 Geometrical relationship between real and reciprocal space Note reciprocal lattice vector is always perpendicular to the corresponding real space plane Only in orthogonal axis systems are the real and reciprocal lattice vectors parallel A precession photograph Sampling DNA Fourier Maps Electron density distribution is given by pXyzlVEZEFhkleXp21tihxkylz 7 or pXyZlVZZZFhkllCos21IhxkylZcxhkl Density maps are often calculated using different coefficients 7 lFol 39 Fobs map 7 chl 39 Foalc map 7 lFolchl difference map etc Building up a 2D Fourier map in Fobs maps Measured F are used along with phases calculated from some structural model may show peaks that do not correspond to atoms in the current model allows location of missing atoms 1calc maps I Fcalc map 7 lFls and phases calculated from some structural model are used 7 the map only has peaks corresponding to atom positions in the model 7 not very informative Difference maps Both measured and calculated Fs are combined with calculated phases Map emphasizes the incorrect features of your current model 7 should see peaks where there are missing atoms 7 should see valleys if atoms have been incorrectly placed The representation of maps 3D Fourier maps can be presented as 7 tables numbers 7 contoured sections or projections 7 chicken wire drawings 7 peak lists gtgt software searches for peaks in a map and lists them Contoured sections Projections Chicken wire An example difference map Hydrogen location from a difference map HCCCH2 COOH oc 0 O H O Map resolution A Fourier synthesis in principle involves all Fhkl an in nite number of terms However usually a limited amount of data is ava able Neglect of high resolution data has two effects 7 structural details can not be seen 7 there may be unwanted ripples in the map due to termination errors Maps at different resolution 777 W W a quotquot K JV xUn l l 3K Hi i l a