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Br39anislav K Nikoli Department of Physics and Astronomy University of Deaware U5A PHYS 624 Introduction to Solid State Physics httpwww physicsudel edubnikolicteachingphy5624phy5624 html Principles of Diffraction 0 How do we learn about crystalline structures Answer Diffraction Send a beam of particles of de Broglie wavelength 111p or radiation with a wavelength ha comparable to characteristic length scale of the lattice a twice the atomic or molecular radii of the constituents O EXPERIMENT Identify Bragg peaks which originate from a coherent addition of scattering events in multiple planes within the bulk of the solid PHYS 624 Experimental Determination of Crystal Structures waves or panicks K Azlatlorlazl V Figure 1 Scattering ofwaves or particles with wavelength of roughly the same size as the lattice repeat distance allows us to learn about the lattice structure Coherent addition of two particles or waves requires that 2dsin Z the Bragg condition and yields a scattering maximum on a distant screen PHYS 624 Experimental Determination of Crystal Structures Xemys Neutmns Electrmls Charge 0 0 e Mass 0 LETlU ET kg 9111041 kg Typical Energy 10 ke v 003 allquot 100 KEV Typical wavelength Lil l 3 00331 Typical lt llllall n length 100 p111 5 m l pm Typlml atomic farm facial f 10 3 A lllll 4 El 10 El 4 PHYS 624 Experimen ral Defermina rion of Cr ys ral S rr uc rur es 0 Not all particles with de Broglie wavelength ha will work for this application gt For example most charged particles cannot probe the bulk properties of the crystal since they lose energy to the scatterer very quickly dE N 47an262 In myv3 N q2 2 2 dx mv qewo v 0 For nonrelativistic electron scattering into a solid with 6132A a1 1231WWJEE5V o The distance at which initial energy is lost is 5E E n 21023 cm393 gt 5x 2100A 0 NOTE Low energy electron diffraction can be used to study the surface of extremely clean samples 5 PHYS 624 Experimental Determination of Crystal Structures O O O r 0 a r 0 r v ltgt 0 O o O Q U Q C O O C O O Q Q a 0 Q w 0 o O O 0 O O o i 3 v O o o o o o C Q o Q Q 3 Q r 0 O O O O Q 0 Q o J 00 Q o o O O 0 O C OXYECU O Q 0 O 3 Q 0 C O O O 0 O O O O C 0 Figure 2 An electron about to scatter from a typical material However at the surface of the material oxidation and surface reconstruction distort the lattice If the electron scatters from this region we cannot learn about the structure of the bulk PHYS 624 Experimen fql Defermina rion of Cr ys l al Sfr uc rur es sca r rer ing allows one To s rudy Ia r rice vibra rions 39 w W m 22 Anhfer39r39omagnef E IN I 021 5 x f 7 F g mm m 3114441 g MK v HM am IOU Like NaCl A 0 2 4 N Scdtturiug auglt 29 ldcgtm PHYS 624 Expemmerrm Derermmamn of Crys ra S rruc rures EINeLI rr ons sca r rer almos r comple rely iso rr opic Elastic sca r rer ing gives precise informa rion abou r The s ra ric Ia r rice s rr uc rur e while Inellasfic SpinSpin In remc rions MnO ha 0 X Ray Tube Ins m7 u m7 n um CIIrrrzr39clc rion d2nsi1 yvibr crrzs 1139 The frequency of The incoming radicrhon x 4 Vu uh A of X rays wifh condznszd ma zr charged particle zlchr on PHYS 624 Expzmmzmal Determinantn qurysml Structures 0 Three basic assumptions 1 The operator which describes the coupling of the target to the scattered quotobjectquot in this case the operator is the density commutes with the Hamiltonian gt realm of classical physics 2 Huygens principle Every radiated point of the target will serve as a secondary source spherical waves of the same frequency as the source and the amplitude of the diffracted wave is the sum of the wavelengths considering their amplitudes and relative phases 3 Resulting spherical waves are not scattered again For example in the fully quantum theory for neutron scattering this will correspond to approximating the scattering rate by Fermi golden rule ie the socalled firstorder Born approximation 9 PHYS 624 Experimental Determination of Crystal Structures obsewer or screen target R gtgt r gt AB Aoe quot Rrquot b incident wave ezkRLr R r Ezruvkm AB R olt A0g1kuRkRwntJdrpl W j l PHYS 624 Experimental Determination of Crystal Structures Hygens ABR olt j drAPpm I At very large R ie in the socalled radiation or far zone z39k0RkR a0t I e z39 r ABROCA0TIdrpre k k0 2 In terms of the scattered intensnty B OC AB le OC l2 2 I CamryWW A 2 A 390 idrpltrgtequot ngllpltKgt2 K Fourier transform of the density of scatterers R 2 11 B PHYS 624 Experimental Determination of Crystal Structures Phase Information is Lost 1KocpKl MK Idrpmew 2 pKezQK IltKgt pltrgt a From a complete experiment measuring intensity for all scattering angles one does not have enough information to get density of scatterers by inverting Fourier ns tead guess for one of the 74 Bravais lattices and the basis Fourier 39 I data t form a lns transform this fit parameters to compare to experimenta From the Fourier uncertainty principle AxAk z 77 Resolution of smaller structures requires larger values ofK some combination of large scattering angles and short wavelenght of the incident light PHYS 624 Experlmemal Determination of Crystal Structures 1K oc pK2 oc IpreiKrdr jpr e Kr dr rar39r 1K oc jeiKr drprpr rdr 4 The Patterson function is the autocorrelation function of the scattering density it has maximum whenever 139 corresponds to a vector between atoms in the structure Pr J prpr rdrl 13 PHYS 624 Experimental Determination of Crystal Structures PattersonPr Nfz oy NJ pr r rdr 71quot Pair Correlation f2gr Iprl r r dr Structure Factor 1K cc SK1J grk39K39rdr 6 Pair correlation function gr N 0 J3 o U1 r Si liquid Fe morphous Si ry quotAFELJFm nearest next naarnsi and lhird ne I neighbors in the crystalline phases r2SI rSigt r and v Lienule he Iislances O Ihe aurasl Radius r A PHYS 624 Experimenful Defer39minufion of Crysful Sfr ucfur39es 0 Density of periodic crystal 030 me ma W 2 me ane s 16 x ma ti iGma pxmaZne quot EZJne quote quot px 1gtG 22 n thma e a PHYS 624 Experimental Determination of Crystal Structures O Generalization to threedimensional structures prrnprrma1nza2nsa3wlpns Z Grn 2m7rm Z Gzhg1kg2 Zg3 hg1kg2 Zg339 1312m me Z g13931227z39ag239alzg3393120 K J 16 PHYS 624 Experimen ral Defermina rion of Cr ys ral S rr uc rur es CI The orthonormal set reciprocal lattice glg2g3forms the basis of the B Keach other with point group symmetry 0 PHYS 624 Experimental Determination of Crystal Structures 212 X213 33 X31 a1gtlta2 g1 2 9 2 9 3 a1 39a2 X513 a1 39a2 X513 a1 39a2 Xa3 httpIwwwmatterorgukldiffractionlgeometrylsperpositionofwavesexerciseshtm 23 2703 932 a g1g2gtltg3l I 31 3932 Xa3 QPUC CI Realspace and reciprocal lattice have the same point group symmetry but do not necessarily have the same ravais lattice example FCC and BCC are reciprocal D 17 2 0r Z oGe Gr 3 B K oc 3917le Jdrz pGe iK Gr G G iK Gr V G 2 K J6 dr V lattlce volume 0 G i K IBUQ oc lel 2 2 RI IOGI V26GK This is called Laue cgndition for scattering The fact that this is proportional to V rather than V indicates that the diffraction spots in this approximation are infinitely bright for a sample in thermodynamic limit gt when real broadening is taken into account B K oc V j 18 PHYS 624 Experimental Determination of Crystal Structures 2 1th C phkll pre R gt pa pic 1 h k Z E m 1th For every spot at k k0 G there will be one atk k0 G Thus for example if we scatter from a crystal with a 3fold symmetry axis we will get a 6fold scattering pattern The scattering pattern always has an inversion center ven if none is present in the target PHYS 624 Experimental Determination of Crystal Structures G gt G 19 If and only ifthe three vectors involved form a closed triangle is the Laue cond n met If the Laue condi on is not met incoming wave Just moves through the lattice and emerges on the other side of the crystal nnglccting absorption PHVS 624 Expemmenmx Determmmmn uf Crystal Structures Ewald Sphere Figure 1 The Ewald Construction to determine if the conditions are correct for obtaining a Bragg peak Select a point in k space as the origin Draw the incident wavevector k0 to the origin From the base of k 0 spink remember thatfor elastic scattering lkol in allpossible directions toform a sphere At each point where this sphere intersects a lattice point in k space there will be a Bragg peak with In the example above we find 8 Bragg peaks If however we change k 0 by a small amount then we have none 21 PHYS 624 Experimental Determination of Crystal Structures O O h k I plane J O o o a O o o o 3 O dh k l 49h O O lt39Y 0 al 0 O 0 o O o a n O 1 1 1 hklgthkl hEh v w g1gzg3hkl gt1VlilerindioesltgtG 211 plane Grn 2 mltgtqnl Pg331 012 Pg3az 713 P81 gz33 Gq2 m wzz quot Conventions hid th hkl hid May 22 PHYS 624 Experimenful Deferminufion of Crysful Sfrucfures Bragg vs Laue Recirocal vs Real 5 ace Anal sis Laue Condition Bragg Condition in reciprocal space in real space k 39 k0 Gina Zdnleine 7 iKiK ik k0th1 K 2k0sine4 sine 2 gt 12dmsin6 A dhkl PHYS 624 Experimental Determination of Crystal Structures EIWe wanT To know which parTicLIlar wave vecTors ouT of many an infiniTe seT in facT meeT The diffracTion Bragg amp Laue condiTion for a given crysTal laTTice plane EIIf we consTrLIcT Wigner52in cells in The reciprocal laTTice all wave vecTors ending on The WignerSeiTz cell walls will meeT The Bragg condiTion for The seT of laTTice planes represenTed by The cell wall manilme m o n im G hkl GWk 0 2 k2 k02 GW k2 do k22GW k PHYS 624 ExperimenTal DeTermina Tian cf CrysTal STrueTures 14 Constructing Brillouin zones is a good example for the evolution of complex systems from the repeated application of simple rules to simple starting conditions any 12year old can do it in two dimensions but in 3D PhD thesis in 1965 25 PHYS 624 Experimental Defer mina rion of Crystal Structures Endpoints for wave venom BVK I nkr I nkr Njaj k z LXNla LX Ly L2 Ur Ur R 3Ur ZUr Gr G Reciprocal lattice points DAr biTr ar y wave vecTor k can be wr39iTTeh as a sum of some reciprocal IaTTice vecTor 6 plus a suiTabIe wave vecTor k39 e we can always wr39iTe k 6 k39 and k39 can always be confined To The first Brillouih zone ie The elemenTar y cell of The reciprocal IaTTice PHYS 524 Expemmenm bevermmmmn uf Crysm svrucmres EIPeriodic Function wi rh The same per39iodic ry as The Ia r rice expanded in Fourier series Ur UrR gtUr ZUre Gr G CIArbi rr39ar39y wave function expanded in plane waves Tm qu kquot EIBloch wavefunc rions eigens ra res of crystal Hamil ronian expansion km 2rR K I kl q eiW Wacky 27 PHYS 624 Experimen ral De rer mina rion of Cr ys ral S rr uc rur es Density of valence electrons is rather smoothly varying Minimum at holes in structure Peaked at bonds quotham h quot quot Reasonable to consider as a perturbation starting from uniform system just like the 1d problem that we solved Calculated valence electron density in a 110 plane in a Si crystal Cover of Physics Today 19TH 28 PHYS 624 Experimental Determination of Crystal Structures 451 insl Free eledron gas Free elemon gas and 39ltrsc on at Brmouin znnes All wave vectors that end on a 32 will fulfill the Bragg 7crondition and thus are diffracted h states with is Bragg reflected into state wi and vice versa never get diffracted they move pretty much as if the potential would be constant i t Wave vectorsacompletely in the interior of the 1 32 orawell in between any two BZs will hey ehave very close to the solutions of the free electron gas J 29 PHYS 624 Experimental Determination of Crystal Structures Interpretation of Standing waves at Br loujn Zone boundar Bragg scattering at k na leads to the two possible combinations of the right and Ie going waves 1 exp i 1rXa exp i 1rxla 2 cos1rxla y H exp inxla exp i1rxla 2 Slnnxia with denstty n 4 cosZ7era n 4 sin27rxa quot3 El39owener y 39 high eneby Ygtltlt v 39 39 39 V msattractive J 39negativepotential PHYS 624 Expemmemm Defermmuhnn nf Crysm Svmcmres EB EB E3 0 Body Centered Cubic f Need structure factor 8 and form factorf respectively f PHYS 624 Experimental Determination of Crystal Structures 1th X Ipth 1 j r 1 139 r 2 a 0th Z Jdrp k thl L drpre Gh d cells 1 1G rnrar 0th E L drpre Slnce 1 1 ra r V N1N2N3 1 STr ucTur e FacTor 0th V 26 dl par e cell a ATomic lG r 2 ScaTTer39ing For39m fa Idr paO 6 W Z gtI cx Z FacTor 1 S iG rd W S f 2 One aTom per39 uni l cell 0th 726 W fa C a C 32 PHYS 624 Experimen ral Defermina rion of Cr ys ral S rr uc rur es r Maya1 voza2 ona3 SW ZfoX exp 2m hugK hva wu 1 2 A V l l 112000 F2 55 0 hklodd P i7rhkl 2fhkleven Inte Wf1e PHYS 624 Experimental Determination of Crystal STruc rures fX my x ray Fe on on 1 if Unlt Cell ofBCC ordered FeCo Fe Co all ition of Bragg reflection Shape and dimension of the unit cell my 33 nsities of reflections Con the unit cell Example Diffraction of electron on crystalline potential la10 mgtE01W ikr ik r e 1 Ur Z Ua r ra gt 6 transition from Pk e to Pk W QuantumMechanical Probability Amplitude for this transition Aklk lt Pk1 I Pkgt idr l erwarrka 7 1 Aklk Eat W Je kk1rr Uar radr UaGSG UaG i Idr e Grr Uar ra 1 G V0 SG Ze Fa Structure factor is completely determined by geometrical N 0 properties of the crystal 34 PHYS 624 Experimental Determination of Crystal Structures 0 Any matrix element that describes a transition between two electronic states under the action of crystalline potential will contain a structure factor 0 Crystal potential does not have to be necessarily expressed in terms of sum of the atomic potentials furthermore the transition do not necessarily involve external electrons gt everything is valid also for transition between electronic states of a crystal itself 0 Extracting of structure factor reflects how spatial distribution of ions affects dynamics of processes in crystals Example pr 2m ra gt 0k Skp0 k for electrons Arp11gtSkp ltpkfa mme PHYS 624 Experimental Determination of Crystal Structures A Vrquot llwmuing Bcam q PHVS 624 Experimental Determination ofCrysml Structures 5 5111 Kj39Ek j and the mdms r 0 him 31 thi scattrzlfmg nng due 0 Ifeclplmal lattice vector is 5 r Gian 37 PHYS 624 Experimen ral Defermina rion of Cr ys ral S rr uc rur es