a * a (2,1) 1,1 0,1 1,1 2,1 hkl 1,0 1,0 2,0 O 2,1 0,1 1,1 0,2 1,2 2,2

Transcription

1 The Reciprocl Lttice The inverse of the intersections of plne with the unit cell xes is used to find the Miller indices of the plne. The inverse of the d-spcing etween plnes ppers in expressions for the diffrction ngles, Eqs nd The ppernce of the inverse of unit cell dimensions is repeting pttern in diffrction experiments. The reson is tht X-rys diffrct from plnes of toms in the crystl. X-rys experience different view of the crystl lttice thn is depicted in direct representtions of the lttice structure, Figure The wy X-rys interct with the lttice is est represented using the concept of the reciprocl lttice. In the rel lttice, which is lso clled the direct lttice, lttice points re occupied y toms, ions, or molecules. On the other hnd, points in the reciprocl lttice correspond to plnes within the crystl. The symmetry of the reciprocl lttice is directly displyed in the diffrction pttern. Distnce within the reciprocl lttice is given y the inverse of the corresponding plne spcing. The reciprocl lttice is mp of the plnes in the direct lttice. Ech point in the reciprocl lttice represents fmily of plnes in the direct lttice. By convention, properties of the reciprocl lttice re shown with n sterisk, *. The reciprocl lttice hs the repet spcings,, *, nd c *, s given y the inverse of the direct lttice unit cell dimensions: = 1/ * = 1/ c * = 1/c The coordintes of reciprocl lttice point correspond to the Miller indices of the corresponding plne. The direction of the vector from the origin to the reciprocl lttice point is perpendiculr to the corresponding plnes in the direct lttice. Consider s n exmple the fmily of (2,1)- plnes depicted in the two-dimensionl unit cell in Figure d hkl (). (2,1) 1,2 0,2 1,2 2,2 1,1 0,1 1,1 2,1 d hkl * = 1/d hkl 1,0 1,0 2,0 O 1,1 0,1 1,1 2,1 * 1,1 0,2 1,2 2,2 (). Figure : Distnces in the reciprocl lttice re the inverse of the d-spcing in the direct lttice. The origin of the reciprocl lttice is chosen s the point leled O. Points in the reciprocl lttice re leled with the Miller indices of the corresponding diffrction plne. The point representing the (2,1)-plnes in the direct lttice is leled s (2,1) in the reciprocl lttice, Figure The distnce of the reciprocl lttice point from the reciprocl lttice origin is the inverse of the corresponding d-spcing in the direct lttice: d * hkl = 1/dhkl The Reciprocl Lttice hs the Sme Symmetry s the Rel Lttice: If the direct lttice is orthorhomic, then the reciprocl lttice is orthorhomic, Figure If the direct lttice is hexgonl then the reciprocl lttice is hexgonl. The symmetry of the reciprocl lttice is

2 19 reflected in the diffrction pttern. However, if the orthorhomic lttice is tll nd nrrow, then the reciprocl lttice is wide nd short, ecuse of the inversion of the unit cell dimensions. For monoclinic lttices the reciprocl lttice ngle is 180, with the direct lttice ngle. Orthorhomic * Hexgonl * Monoclinic * Figure : The reciprocl lttice hs the sme symmetry s the direct lttice. In ech exmple the direct lttice is on the left nd the corresponding reciprocl lttice is on the right. The Ewld Sphere Mps the Reciprocl Lttice: The Ewld construction gives simple visul pproch for understnding the formtion of the diffrction pttern from the reciprocl lttice, Figure The Ewld sphere is centered on the rel lttice t point C nd hs rdius of 1/. F Detector P d * hkl = 1/d hkl sin = OP OX X C 2 O D sin = 1 /d hkl 2/ 2d hkl sin = 1/ M Figure : Reflections occur t ngles tht correspond to n intersection of the Ewld sphere with reciprocl lttice point. The rdius of the Ewld sphere is 1/. The X-ry em reflects from the rel lttice t C. The incident em intersects the Ewld sphere t X nd O.

3 20 The X-ry em line intersects the Ewld sphere t points X nd O. The origin of the reciprocl lttice is plced t point O. The ngle of incidence of the X-rys with the rel nd reciprocl lttices is. As the rel lttice is rotted the reciprocl lttice lso rottes y the sme ngle. A reflection occurs if reciprocl lttice point intersects the Ewld sphere. The reflection ngle t 2 corresponds to the line from the crystl origin to the reciprocl lttice point, CP. The reciprocl lttice point tht corresponds to the d-spcing of the plnes tht give the reflection is t d * hkl = 1/dhkl, Eq How does the Ewld construction give the reflection ngle? Consider the right tringle with hypotenuse OX with length equl to the dimeter of the Ewld sphere 2(1/). The side-opposite, OP, hs the length of the reciprocl lttice distnce d * hkl = 1/dhkl. The side-djcent is XP. The interior ngle of this right-tringle is. The sine of the ngle is the rtio of the side-opposite to the hypotenuse: sin = OP OX = 1 /dhkl 2/ Rerrnging this lst eqution gives Brgg s Lw: 2dhkl sin = (34.2.3) In other words, the Ewld construction is clever geometricl representtion tht gives the reflection ngles s the crystl is rotted out its xes. The reflection ngles re determined y the lttice symmetry, while the intensities of the reflections re determined y the composition of the unit cell, which we discuss next. Plnr Arry Cmers in X-ry Diffrction: The cmer used in typicl single crystl X-ry diffrctometer is plnr multi-element solid-stte rry, similr to ut much lrger thn the solid-stte detector in cell phone digitl cmer. Ech individul detector, or pixel, in the rry cn detect individul X-rys. An rry detector llows the intensity of multiple reflections to e cquired simultneously. The typicl size of the cmer is 10x14 cm. A computer reds out the X-ry intensity of ech individul pixel in series of exposures, or frmes. The crystl is rotted to orient the crystl so s to collect the diffrction pttern over the full sphere of reflection in multiple frmes. Exmple : Determining Lttice Prmeters from Reciprocl Lttice Projections Two two-dimensionl projections of the reciprocl lttice re required to find ll three unit cell dimensions, Figure The mgnifiction of the oserved reciprocl lttice is determined y the crystl-detector distnce, M or line CD in Figure The distnce etween the crystl nd the reflection spot corresponding to d * hkl on the detector is CF. Compring corresponding sides of tringles COP nd CDF gives: 2 OP OC = DF CD or d hkl * 1/ = DF M Solving for the direct lttice spcing gives: d = 1 d * = M hkl DF where DF is the mesured distnce on the reciprocl lttice projection

4 21 The mesured distnces etween spots three rows or columns prt on the reciprocl lttice projections for 2-dimethylsufurnylidene-1,3-indnedione re shown on Figure The detector distnce is 70.7 mm. A Mo X-ry source ws used with = Å. Clculte the unit cell dimensions. +h to right, k = 0, +l up h = 0, +k up, +l to right c * 25.5 mm 8.2 mm * c * 8.2 mm 16.7 mm Figure : Reciprocl lttice projections. The intensity t ech point is proportionl to the spot size. The missing reflection tht is oscured y the em stop is shown s, which corresponds to the direction of the incident X-ry em. Answer: The reciprocl lttice spcing in the direction, tken from the (h,0,l) projection, is 25.5 mm/3 = 5.57 mm. This single reciprocl lttice spcing corresponds to d100 = in lttice with ll 90 ngles. Using Eq : = mm ( Å) d hkl * = = 5.91 Å 8.50 mm The reciprocl lttice spcing in the * direction, tken from the (0,k,l) projection, is 16.7 mm/3 = 5.57 mm. Using Eq : = mm ( Å) d hkl * = = 9.02 Å 5.57 mm The reciprocl lttice spcing in the c * direction tken from the (h,0,l) or (0,k,l) projections, which should give the sme result, is 8.2 mm/3 = 2.73 mm. Using Eq : c = mm ( Å) d * = = 18.4 Å hkl 2.73 mm More ccurte vlues result if greter numers of rows or columns thn four re mesured in the reciprocl lttice imge Moleculr Structure is Determined from Scttering Intensities The Inverse Fourier Trnsform Determines the Electron Density: X-rys sctter primrily from the electrons in n ion or molecule. For sphericl tom or ion the scttering power is determined y the tomic structure fctor, fi, which is proportionl to the numer of electrons in the tom or ion. The intensity of reflection is then determined y the structure fctor,

5 22 Eq This reltionship is pplicle for sphericl toms or ions. However, if the lttice is composed of polytomic ions or molecules, the lttice positions re not occupied y sphericl ojects. Insted we must consider the electron density s function of position within the unit cell, (x,y,z), Figure For nottionl simplicity, the electron density t position (x, y, z) is often simply denoted s (r). Eq must then e generlized to give the reflection intensity from this electron density distriution: Fhkl = (r) e 2i (hx + ky + lz) dx dy dz Figure : Electron density mp of the six-memered ring portion of 2-dimethylsufurnylidene-1,3-indnedione. Atom positions re not directly determined. The electron density contours re truncted t high vlues of the electron density. The summtion in Eq must e replced y n integrl over the unit cell volume ecuse the scttering centers cn no longer e considered s loclized prticles. Comprison with Eq shows tht this reltionship is the three-dimensionl Fourier trnsform of the electron density distriution. The structure fctors re determined y collection of the reflection intensities, Ihkl = Fh * kl Fhkl. The electron density distriution is determined y clculting the inverse Fourier trnsform of the structure fctors. Since the Miller indices re integers, the integrls in the inverse trnsform reduce to summtions: (r) = h = k = Fhkl e 2i (hx + ky + lz) l = The process of moleculr structure determintion then is ccomplished y collection of the structure fctors of the reflections with Miller indices (h,k,l) nd the clcultion of the electron density using the inverse Fourier trnsform. In prcticl ppliction n infinite numer of reflections re not ville. As result, the resolution of the resulting electron density mp, Figure , is determined y the numer of reflections tht re used in the inverse Fourier trnsform. Incresing the numer of detected reflections increses the specil resolution of the resulting electron density mp. Notice tht tom positions re not determined in this process.

6 23 Once the electron density distriution is determined, lest squres procedure is used to find tom elementl identities nd positions tht est reproduce the experimentlly determined electron distriution. Hydrogen toms hve smll tomic structure fctor. As result, the positions of hydrogen toms re poorly defined in X-ry sed electron density mps. The positions of the hydrogen toms re usully inferred using moleculr mechnics clcultions. As result, the positions of hydrogens in the finl structure solution re lrgely uncertin. Neutron diffrction is sensitive to hydrogen tom positions. Correspondingly, if precise hydrogen loctions re needed then neutron diffrction must e used. The Phse Prolem: One importnt difficulty in X-ry structure determintion is tht the reflection structure fctors, Fhkl, re not determined directly. Ech structure fctor is complex numer, which is given y mgnitude nd phse ngle. In other words, reflected rys rrive t the detector with different reltive phses, ecuse of differing pth lengths within the unit cell. The knowledge of the different pth lengths is required for the fithful sptil reproduction of the contents of the unit cell. The experimentl intensity of reflection, Ihkl = Fh * kl Fhkl, does not determine the phse of the reflection, rther only the mgnitude of the reflection. The resulting miguity is clled the phse prolem. In smll molecule diffrction, severl useful itertive methods hve een developed for solving the phse prolem, which we must leve to your further study. X-ry Diffrction nd Moleculr Motion: Therml Ellipsoids: Just s in spectroscopy, the width of reflection is mesure of moleculr motion. The lest squres procedure tht fits tom positions to the experimentlly determined electron density lso clcultes the uncertinties of the tom positions. The result of the lest squres fitting is displyed with the uncertinties of the non-hydrogen tom positions given s therml ellipsoids, Figure O S O Figure : Lest squres fit of toms to the electron density, Figure , gives tom positions nd uncertinties. Positionl uncertinties re displyed s therml ellipsoids. Lrge therml ellipsoids correspond to lrge mplitude motion. Therml ellipsoids cnnot e determined for hydrogens. Hydrogen positions re usully determined y moleculr mechnics. The principle xes of the therml ellipsoids give the positionl uncertinties. The minimum sizes of the therml ellipsoids for given moleculr structure re determined y the sptil resolution

7 24 resulting from the numer of reflections ville in the inverse Fourier trnsform. Therml ellipsoids lrger thn the minimum correspond to either lrge mplitude virtionl motion or disorder within the crystl. Lrge mplitude virtions cn hve n importnt effect on the properties of crystl. For exmple, lrge mplitude motions my llow smll molecules to diffuse into the crystl. Lrge mplitude motions lso increse the het cpcity of the solid sustnce. Extremely lrge mplitude virtionl motion results in disorder within the crystl tht in some instnces prevents the position of the corresponding tom from eing determined Summry Looking Ahed X-ry diffrction is the primry sis of moleculr structure determintion in inorgnic chemistry, orgnic chemistry, iochemistry, nd structurl iology. An extensive dt se of X- ry derived smll molecule crystl structures is mintined y the Cmridge Crystllogrphy Dt Center, CCDC, which is primry resource of moleculr structure nd crystl structure informtion. An extensive dt se of X-ry nd NMR derived protein nd nucleic cid structures is ville from the RCSB Protein dt nk. These structures re commonly clled pd files. One disdvntge of X-ry crystllogrphy is the requirement of stle well-formed crystls. An extensive NSF progrm for the determintion of protein crystl structures is underwy. Reserch res in this progrm include the development of high-throughput prllel methods for producing pure proteins using moleculr cloning techniques nd then forming crystls suitle for diffrction studies. Synchrotron light sources re the preferred X-ry source for protein crystllogrphy, ecuse of the higher X-ry em intensity s compred to the metl trget X-ry tues used in smll molecule studies. The determintion of X-ry crystl structures hs clerly een identified s n importnt interntionl reserch priority. The resulting moleculr structures then ecome the sis of structure-function studies designed to conquer mny of the chllenges tht we fce s society.