Elctron nry in crystal potntial r r p c mc mc mc Expand: r r r mc mc mc r r p c mc mc mc r pc m c mc p m m m m r E E m m m r p E m r nr nr whr: E V mc E m c
Wav quation Hamiltonian: Tim-Indpndnt Schrodinr Eqn: Rarran: Prviously dfind: Structur Function: Wav Equation: H ˆ pˆ m r m m me m 4 r h h r nr U r E nr h me nr m h r k Ur r 4 k r r r r menr h
Th structur function is priodic: So: Bloch-wav solutions U rr Ur uvw r U U i r Th cofficints of th structur function ar: U m h Th wav quation looks lik: i k U r r 4 onsidr solutions of th form: ik r r r u whr: u r i r Ths solutions ar Bloch wavs: r i k r
Structur Function Th cofficints of th structur function ar: Lt s rlat ths to th structur factors. U m m X h h m atoms m m id //Fourir transform of unit cll f m dr 3 h r //atomic form factors m m ir r m m f h m //sphrical symmtry m F f m atoms id m //crystal structur factors F m h U F X
Proprtis of Bloch wavs (I) Th Bloch wavs ar quantum stats:... * * *... W can normaliz thm: Any two Bloch stats ar orthoonal: ;
Proprtis of Bloch wavs (II) Th Bloch wavs form an orthonormal st: V V VV I Also: T T V V I T V T V Th -componnts of th Bloch wavs also form an orthonormal st: *,
Total wav function Th total wav function is a linar combination of Bloch wavs: r r Th ar th xcitation amplituds. Thy ar found usin th boundary conditions at th top (ntranc) surfac. In trms of diffractd bams: r i k r r r ik r i s r iks r Whr: k k r i s r
Boundary condition(s) (I) At th foil ntranc surfac, w must hav: i) r continuous and r ii) nˆ continuous W can pick th foil ntranc surfac to b th plan z=, with Assumin that abov th sampl: r ikr nˆ -zˆ w ar only abl to satisfy condition i). (To satisfy both conditions, includ back-scattrd wavs.) onsidr condition i) only (nlctin back-scattrd wavs): i ikr k r z z i r z
Boundary ondition(s) (II) ondition i) is satisfid if: and i z i r r z W must hav ˆ n so that r z i r z ir z * In othr words: r * r
Diffractd bam amplituds r r r i i k ks r s r r i k++s r W can pick: s s nˆ r z i s z r z s r s z
Solvin for th Bloch-wav cofficints Find th particular Bloch wavs ar solutions of: Th first trm is: i K U r r 4 K k U k r 4 k i k r W can find th by solvin: i i k hr k k U h h r
Rarranin th sums Brak up th sum into two trms: i i k r h k U h r h Rindx: h h h=h- h=h = i i U h r U hr h h h h i r k k U ihr h h Now w can roup trms that hav th sam xponntial function: k -h h k U h
Rwritin th sum W want to simplify: k k U-hh h I. II. k k k k k k k k k++s ks s k k s I. - II. k So now w hav: k k s s s U-hh k s h
Hih-nry approximation k k k s s k s knˆ k cos k k s cos s Assum This ivs: -h h k s U h
Extinction distanc W can now writ: Dfin th xtinction distanc: U s h k -h h k v h U F -h -h Th charactristic lnth ovr which % of bam diffracts from on particular channl into anothr (if no othr ffcts ar prsnt). v F s h -h h
Einvalu Problm This has th form of an invalu problm. A... Whr A is a matrix: s. s. A..... s n n n n n This systm of quations can b solvd for th () and th
Two-bam condition Only and ar sinificant: r z z i +s r ikr So w only nd two Bloch wavs: r r r ik r i k r ir i z ikr r ik r i k r ir i z ikr r * * Assum th structur factor is ral: * F F F Thn: U U s s Th problm bcoms: A s
Solvin th two-bam condition W nd to solv a invalu problm: s Hr s how to do it: A dt s s s Two invalus: and two infunctions:, s s,,,,, s
Spcial cas, stron bam (s=) Now find th s:,,, Finally, find th () s:,
Th stron, two-bam solution r r r r r ir iz ikr ir iz ikr ir r cos z isin z ikr z z z isin z cos z z cos z sin z
Th nral two-bam solution Th amplituds ar rlatd by:,,, i) ii) s Dfin: Normaliz:,,,, w,, s, w w s s, w w, Now dfin:, ombin: W can pick: cot w sin cos sin cos sin cos
Findin th Bloch wavs Now find th cofficint of th Bloch wav:,,, Th factor in front can b writtn as:, cos sin sin cos cos sin r r sin r cos r cos sin ir issff z ikr sin cos r ir issff z ikr cos sin
Two-bam rsult Dfin: sff Udisn diffractd bams: s, r z z s s ff i +s r ikr s ff ff ff z sff z i sff z sin cos cos cos sin is sin ff z is ff z isz isz sin sin isz z i sff z is z is z isz isz w cot s cos sin w s cot ff w w s s tan ff s zcossff zi sin sff z s ff z isin sff z s ff isz isz
Now find th diffractd intnsity: I Two-bam intnsity s z sin ff sff sff s
Howi-Whlan Equations Diffractd Bam Amplitud: z i s z Tak th Drivativ: d dz i s i s z ondition on Block-Wavs: s h h -h Substitut and Group: d i h dz h h-h i s z i s h s z Substitut aain: d dz h i -h is h s z h Bam amplituds ar xprssd without any rfrnc to Bloch wavs.