Diffraction from Crystals Shape Factor

Transcription

1 Letue 3 Diffation fom Cystals Shape Fato - Fultz & Howe, Chap. 5 - Williams & Cate, Chap. 7 - Reime, Chap. 7

2 Satteed wave: Shape Fato Cystal size effet k F k ep i k Shape fato: S k ep ik z Unit ell (/w a, b, paametes) Cystal (N N y N z unit ells) N y b N N N N z z y N a y z ep n y ny z nz S k i k a k b k n 0n 0n 0 y z N N N - Loation of eah unit ell: n an byn z y z - Δk omponents: k k k yk z y z y z ep i kan ep i kybny ep i kznz n 0 n 0 n 0 y z

3 Shape Fato N N N y z ep n ep y ny ep z nz S k i k a i k b i k n 0 n 0 n 0 y z Eah sum is a tunated eometi seies of the fom: S... N n 0 N 3 4 N ep ik a ep ik an n ep ik a whee, e.., ep ik a * ep ik an ep ik an SSk ep ik a ep ik a * SS k ka N ka

4 Shape Fato The denominato vaies slowly with espet to the numeato, appoimation leads to: * SS k ka kan kan ka The envelop funtion: E k ka As N beomes lae: - Heiht of main peak N - Width of main peak (an) - - Heiht of satellite peaks - Satellite peaks et lose

5 Reipoal Lattie Rods (Relod) Shape Fato Thin foil effet Relod - Steaks may also ou alon o y dietion if the dimension is small. - Steaks ae the eason why eat oientation of a ystal is not possible with spot diffation patten.

6 Shape Fato k k ka N F k k a k bn kn k yb k z y y z z

7 Shape Fato Eample: Guinie-Peston (GP) zones - Al-4%Cu alloy - Fe-.9%Mo alloy disk k FAl Cu k FAl k i k F whole Al k ik

8 Deviation Veto Epess Δk as the diffeene of an eat eipoal veto,, and a deviation veto, s: k s ks * ss s ys z Effet on Shape Fato: S k ep i k ep i s ( intee) S k ep i intee ep is ep is S S k s Effet on Stutue Fato: F k F k ( s k is small) * k s, moe eneal

9 Kinematial ntensity k F s a N s a sy b Ny sz Nz s yb sz Shape fato, S(s), depends only on s. Stutue fato, F(), depends only on.

10 Deviation fom Eat Ba Condition s k k 0 Eat Ba ondition: Deviation, s: k k 0 k k 0 s Sine is an intee k F k ep i s s F ep is S Definition: - if s < 0: eipoal point outside Ewald sphee - if s > 0: eipoal point inside Ewald sphee

11 Kikuhi Lines sample Diffation one β α d hkl θ=β - α dak biht seen

12 Kikuhi Lines Pai of a biht and dak line ) ( ) ( K ) ( K

13 Kikuhi Lines The dashed line iht between the eess line and the defiient line maks the intesetion of the efletin planes with the plane of the diffation patten The spain D hkl of the two lines oesponds to θ, B thus D hkl = Lθ B = L/d hkl D hkl equals the spain between the spot of the tansmitted beam and the spot of the (hkl) Ba efletion when the speimen is oiented eatly fo Ba efletion at the (hkl) planes, the defiient Kikuhi line intesets the tansmitted beam, while the eess Kikuhi line intesets the Ba efletion

14 Kikuhi Lines Pais of paallel lines onsistin of one biht and one dak lines in diffation mode Eletons ae satteed elastially (diffation spots) o inelastially (diffuse in all dietions, with maimum intensity alon inident beam dietion and deea intensity it with inea i anle) The intensity deeases with inea sattein anle. nelastially satteed eletons (assumption: neliible eney-loss) at as a new pimay beam whih an undeo Ba diffation (in all thee dimensions) au diffation ones Kikuhi lines eist only in thik samples Can be used fo auate detemination of oientation (a. 0.deee instead of typially 4 deees in spot pattens)

15 Diffation Condition position of plane Eat Ba (ston two-beam ondition) s =0 (dynami theoy equied!) Diffation patten Symmeti ase (Laue ondition) s<0 s < 0 with s = -θ (kinemati theoy!) position of plane

16 Kinematial vs. Dynamial Kinematial theoy - d << 0 (weak sattein) kfk epik - Thin samples - Small deviation fom eat Ba ondition (equied): satteed veto S must diffe by an eitation ti eo s fom any eipoal Δk= + s - Amplitude A of the diffated beams ae summed up takin phase shifts into aount Dynamial theoy - d 0 (multiple sattein) A3 A A s F ep is - Thik samples - Eat Ba ondition - Consides diffation fom beam diffated bak into the pimay beam - Uses Shödine equation (Howie-Whelan Eqn)

17 Home Wok Fultz & Howe book (3 d ed.): Poblems 5., 5.4, 5.0, 5.6, 5.8. Due date: Apil 6 th.