Dynamical Diffraction

Transcription

1 Dynamical versus Kinematical Di raction kinematical theory valid only for very thin crystals Dynamical Diffraction excitation error s 0 primary beam intensity I 0 intensity of other beams consider di racted beam of lattice planes S = h i a i recall expression obtained for one column: F[S] 2 = F 0 [S] 2 N 2 Sin[ ts]2 ( ts) 2 di racted intensity can be written as I S = 2 2 S Sin[ ts] 2 ( s) 2, 2 Dynamical versus Kinematical Di raction Amplitude Phase Diaram for Di racted Amplitude / Intensity where the extinction distance S of the reflection S is defined as S := V F (hi ) F (hi ) F 0 [S]: structure amplitude; V : volume of one unit cell; Brolie wavelenth. : de if Bra condition exactly satisfied, s = 0, I S = 2 t 2 / 2 S, I 0 = I S can only hold if t S /0 if Bra condition not satisfied, s î 0, I S oscillates with increasin t, reaches maximum values of 2 S s 2 3 4

2 Double Di raction Double Di raction s 0 reflections of hih intensity possibility of double di raction: stron reflections behave like new primary beams can also be di racted by crystal scatterin vectors add: S = P i S i possible e ect: additional reflections example: Si 0 F 200 = reflections are kinematically forbidden however, 200 reflections appear if {} reflections are stron and di racted themselves with S = {,, } 5 6 consider incident wave of amplitude 0 simplifyin assumption: only one di racted beam can form reflectin planes: = h i a i amplitude of the di racted beam: conceive direct beam and di racted beam transmitted throuh layer with thickness dz of the crystal chane of amplitude: d 0 + d 7 8

3 Howie Whelan Equations amplitude chanes can be calculated from Fresnel di raction theory, usin the column approximation: dz Exp[2 ikr] d = 0 F V (hi ) da R A R 2 dz = 0 F V (hi ) Exp[2 ikr]dr R 0 F (hi ) = i 0 Exp[2 ikr 0 ]dz V = i 0 Exp[2 ikr 0 ]dz k: wave number (k s = s 0, renamin avoids confusion with the excitation error s); V : volume of unit cell; A: area of first Fresnel zone (main contribution to di raction), da = 2 RdR. 9 two di erential equations for variation of 0 and with z: d 0 dz d dz = i 0 + i Exp[2 isz] 0 = i 0 Exp[ 2 isz] + i 0 Howie Whelan equations second term of first equation results from scatterin of di racted wave back into primary beam sin is opposite or scatterin in opposite sense (first term of second equation) 0 Howie Whelan Equations Plane Waves ultimately, only beam intensity matters transformation: 0 := 0 Exp[ i z/ 0 ]; = Exp[2 isz i z/ 0 ] additional phase factor cancels on calculatin intensity substitution into Howie Whelan equations: d 0 dz = i d dz = i is 2

4 Bloch Waves dynamical di raction as eienvalue problem solve Schrödiner equation with potential V [r ] correspondin to superposition of all atomic Coulomb potentials V [r ] has same periodicity as crystal lattice V [r ] can be expressed as Fourier sum: V [r ] = V Exp[2 i r ] = h2 U Exp[2 i r ], 2m where REL, thus = h i a i, h i Z a value V can be attributed to each 3 4 V [ev] and U [cm 2 ] are related to structure amplitude: 2 E E + 2E 0 V = F 2 V E + E (hi ) = h2 0 2 mv F (h i ); U = F (h i ) V with V and U, extinction distance can be written as V F (hi ) = E E + 2E 0 = h2 = 2V E + E 0 2m V U solvin the relativistic Schrödiner equation 2m 2 + E V [r ] = 0; E := E E + 2E 0 2(E + E 0 ) for periodic potential V [r ] = V Exp[2 i r ] = h2 U Exp[2 i r ] 2m 5 leads to solutions known as Bloch waves : b () [k, r ] = 6 Exp[2 i(k () 0 + ) r ] thus to describe electron wave in crystal and fulfill boundary conditions at vacuum crystal interface requires summation over n REL points =,..., n =,..., n Bloch waves with wave vectors k () 0 + two-beam case: =, 2 amplitude factors recallin a i a = i shows that Bloch waves possess translation symmetry of crystal lattice

5 relation between wave number and enery: k = p h = 2mE h replacin E by sum of kinetic enery E and coe cient V 0 = eu i (inner potential) and applyin relativistic correction yields wave number K inside the crystal: K = h 2m 0 E( + E 2E ) + 2m 0 0 V 0 ( + E E ) 0 insertin K, periodic potential V [r ], and Bloch wave solution into relativistic Schrödiner equation yields the followin system of equations: K 2 holds for all coe (k () 0 + )2 + U h Exp[2 h 0 ih r ] Exp[2 i(k () 0 + ) r ] = 0 cients of all identical exponential terms must vanish collect terms containin factor Exp[2 yields: K 2 8 i(k () 0 + ) r ] 2 k () 0 + C () + U h C() h = 0; =,..., n h 0 Modification of the Ewald Construction k () 0 + : wave vectors of Bloch waves; k() 0 + K recall Ewald construction: attach tips of vectors k () 0 at REL point correspondin startin points M ()? express k () 0 as k () 0 = k () z + k x = (K z + () )u z + k x u x u x, u z : unit vectors in xand z direction, respectively. k x depends on tilt anle : = k x = s K z 9 20

6 Modification of the Ewald Construction Modification of the Ewald Construction 2 22 since K ; K + k () z 2K; () = (k () z K z ), K 2 (k () 0 +)2 = (K+ k () 0 + )(K k() 0 + ) 2K(s ) (s < 0 if is outside the Ewald sphere) system of equations (eienvalue problem!): A A 2...A n A 2 A 22...A 2n 2 = () A n A n2...a nn C n () for =,..., n and with the matrix elements 2 n A = 0, A = s, A h = A h = U h 2K = 2 h () a iven matrix A has n di erent eienvalues () ( =,..., n) correspondin eienvectors: ( =,..., n ) define two further matrices: C: matrix whose columns are the eienvectors : C = : diaonal matrix of eienvalues: i := () i with these definitions, the system of equations is A C = C C A C = 23 24

7 the eienvectors of di erent eienvalues are mutually orthoonal and fulfill the orthoonality relationships: C (i) C () = i C () h = h the n eienvalues () correspond to n Bloch waves with wave vectors k () 0 + startin points M discussed above di erent specimen tilts di erent k x M [k x ] lie on dispersion surface boundary condition: tanential components of the waves must be continuous at crystal surfaces construction for obtainin the startin points of the vectors k () 0 + on the dispersion surface construction details: K 0 incident beam determines point M draw straiht line parallel to crystal normal and throuh M intersection points with n-fold dispersion surface define the excitation points M normal incidence M vertically alined total wave function corresponds to linear combination of Bloch waves with coecients () : = () b () [k, r ] = () amplitude of particular di racted beam: Exp[2 i(k () 0 + ) r ] summation over all =,..., n waves from excitation points M to the correspondin REL point thus = () Exp[2 i(k () 0 + ) r ] or, if omittin a constant phase factor: = () Exp[2 i () z] excitation coe cients () of the Bloch waves follow from boundary conditions at crystal surface: at z = 0, the phase factors Exp[2 i () z] are all unity the plane wave in the vacuum and the Bloch wave field in the crystal must be continuous this requires 0 [0] = [0] = () 0 = () =

8 the same relation in matrix formulation: C() 0 C (2) 0 C () C (2) extension to n dimensions: () (2) = C = [0] compare first orthoonality relation (above) C (i) C () = i this means () = [0] = [0] 0 thus: 0[0] = [0] = eneral case (n components): 0 C() 0 = 0 C() = 0 0 = C [0] orthoonality of eienvectors implies C T : transposed matrix. C = C T 30 consider ust primary beam and one di racted beam (quite unrealistic!) recall kinematical theory: tip of incident wave vector K 0 at REL point 0 Ewald sphere has radius / K 0 around startin point M of if direction of incident beam is varied: various startin points M on sphere of radius / around 0 if di racted beam becomes stron, it can be rearded as new primary beam draw a second Ewald sphere around S = h i a i, h i Z fundamental equations of dynamical theory reduced to two-beam case: U 2K C() 0 + ( () C () 0 + U 2K C() = 0 () + s)c () = 0 system of homoeneous linear equations for C (), C (2) 0 solution only if determinant of coe () U 2K U 2K () + s cient matrix vanishes: = ( () ) 2 s () U 2 4K 2 = 0 quadratic equation for eienvalues (), (2) 3 32

9 Relation to the Howie Whelan Equations recall d 0 dz d dz = i = i is substitute and d in second equation by expressions obtained from the first equation: d dz 2 2 is d 0 dz + 0 = 0 we look for solutions of the form 0 = A Exp[2 thus d 0 dz = 2 i () 0, d 2 0 dz 2 = 4 2 ( () ) 2 0 insertin and recallin that = K/U yields ( () ) 2 + s () U 2 4K 2 = 0 i () z], the Howie Whelan equations lead to the same solution as the Bethe theory a correspondin equation is obtained for Anpassun solvin the quadratic equation for the eienvalues yields () = s + ( ) s 2 + U2 2 K 2 s ( ) s () = 2 usin w := s to characterize the deviation from the Bra condition yields () = 2 w ( ) + w

10 two circles around O and G do not intersect approach each other most closely if Bra condition is fulfilled, thus if w = 0 their separation is then min[ k z ] = () (2) = U K = by use of eienvalues () and C () C (2) 0 2 =, the equations for the C () can be solved: normal incidence four Bloch waves with wave vectors k () 0 + amplitude () = 0 C() 37 solution: 0 C() 0 = 2 + ( w ) + w 2 ( ) 0 C() = 2 + w 2 for w = 0 (Bra condition), all four waves have the amplitude /2 intensity of the primary beam ( transmission ): T := I 0 = 0 0 intensity of the di racted beam ( reflection ): T := I = 38 substitute specimen thickness t for z component of vector r : 0 [t] = [t] = substitutin 0 [t] = [t] = Cos 2 0 C() 0 = 2 = Exp[2 ik() z t] 0 C() Exp[2 ik () z t] Exp[2 ix] (), 0 C() 0, C() 0 C() + w 2 i +w 2 Sin from above yields: iw +w 2 Sin + w 2 + w 2 Exp[2 ix] for the correspondin beam intensities, this implies = 0 0 = + w 2 Sin2 + w 2 t R T recall w = s for w 0 (lare tilt away from the Bra anle), the above solution for the dynamical two-beam case corresponds to the kinematical theory: I = = 2 Sin 2 [ ts] 2 ( s) 2 for w = 0, on the other hand, the classical theory predicts R t 2 ; R > wron! 39 40

11 for near-bra orientation (w 0), only dynamical theory is correct: t R = T = Sin clarifies sinificance of the extinction distance depth periodicity of two-beam intensity for w = 0 ( Bra condition fulfilled) even for w = 0, increasin specimen thickness causes the electron intensity to oscillate between primary beam and di racted beam Pendel-Lösun (pendulum solution analoy to enery transfer between two sprin-coupled mechanical pendulums) compare depth periodicity: kinematical theory: t = s dynamical theory: t = s dynamical di raction: e ective excitation error s s Extinction Distance Examples Example for Pendel-Lösun. Kinematic Case extinction distances of low-indexed reflections of some face-centered cubic materials (in nm): Element () (200) (220) Al Cu Ni A Pt Au

12 Example for Pendel-Lösun 2. Dynamic Case Example for Pendel-Lösun 3. Multibeam Case: A