Grdute School of Engineering Ngo Institute of Technolog Crstl Structure Anlsis Tkshi Id (Advnced Cermics Reserch Center) Updted Nov. 3 3 Chpter 5 Diffrction condition In Chp. 4 it hs been shown tht the verge structure fctor of crstl is given b the following expressions even if the therml vibrtion of toms cnnot be neglected. F totl ( K) G( K)F( K) : verge structure fctor of crstl (5.) G( K) exp π i K ξ + ηb + ςc (5.) ξ ηζ F( K) M f ( K)T ( K)exp π i K r : crstl structure fctor (5.3) f ( K) T ( K) ρ ( r )exp( π i K r )d v : tomic scttering fctor (5.4) R 3 g ( r )exp( π i K r )d v : tomic displcement fctor (5.5) R 3 where it is ssumed tht M toms re included in unit structure nd ρ ( r ) is the electron densit of the -th tom g ( r ) is the probbilit densit of the loction of the -th tom round the verge position. Note tht simplified expression: R 3 d v is used instd of d xd d z in Eqs. (5.4) & (5.5). The smbol R 3 mens tht the function should be integrted over three dimensionl rel spce nd dv d x d d z mens the volume element in the simplified expression. In this chpter the diffrction condition given b Eq. (5.) is discussed. It will be shown tht it is lmost equivlent to the Brgg s lw though it m look quite different from the expression of the Brgg s eqution: nλ d sinθ.
5- Lue function & Lue condition Nb b Nc C N b c Fig. 5. A crstl with prllelepiped shpe Note tht the expression of Eq. (5.) does not full determine the formul bout the diffrction condition G( K). The concrete formul of G( K) should depend on the size nd shpe of the crstl through the rnges of the subscripts ξ η ζ to locte ech unit cell. Assume tht the shpe of the crstl is prllelepiped with three edges given b the repetition numbers of N N b N c long the unit cell vectors b c respectivel. (It is known tht crstl of sodium chloride NCl certinl tends to hve cubic shpe.) In this prticulr cse we cn full determine the formul of G( K) b the following eqution N N b N c G( K) exp π i K ξ + ηb +ζ c N ξ η ζ N b exp( π iηk b N c ) exp( π iζ K c ) exp π iξ K. (5.6) ξ η It is not difficult to solve the bove eqution. For exmple the sum : N exp π iξ K + exp π i K ξ ζ + exp( 4π i K ) ++ exp N π i K is nothing but the sum of geometric progression with the first term of nd common rtio of exp π i K nd ppling the formul : n x xn x the solution is given b N ξ exp π iξ K exp π i N K exp π i K. (5.7)
As the energ of wve is proportionl to the squred mplitude the intensit scttered b crstl is proportionl to F totl ( K) G( K) F( K). (5.8) So the intensit should be proportionl to the squred bsolute vlue nd the formul for intensit is given b exp( π iξ K N ) exp π i N ( K ) ξ exp( π i K ) exp π i N ( K ) exp π i N ( K ) exp( π i K ) exp ( π i K ) cos πn ( K ) cos πk cos πn K ( ) ( cos πk sin πn K ) sin πk (5.9) for exmple. Finll the following formul cn be derived G( K) sin πn K sin πk sin ( πn bk b ) sin πk b sin πn ck c sin πk c. (5.) This function is clled the Lue function. The Lue function is defined s three-dimensionl function but the min chrcteristics of the function cn be understood through one-dimensionl prt of the function. ( Wht chnge is expected in the vlue of sin πn K ) sin ( πk on chnging the length or direction of ) the scttering vector K? The profile of the function is shown in Fig. 5. where K is tken s the horizontl xis. Min pek(s) of the function re locted t K h ( h : integer) the intensit becomes zero t ±/ N ± / N ±3/ N... nd smll sub-peks re locted between them. The height of the min pek is given b N tht is sin lim x ( πn x) sin πx N lim x πn x sin πn x πx sin( πx) N (5.)
N sin ( πn K ) sin ( π K ) ~ N -.5. ( Fig. 5. Lue function sin πn K ) sin πk.5. K for the cse N.5..5 nd the full width t the hlf mximum is bout / N. On incresing the vlue of N the height of the min pek becomes higher nd the width becomes nrrower. For the cse of N the height of the min pek is N while the height of the st sub-pek is bout sin πn 3/ N 4.85 sin π( 3/ N ) sin π( 3/ N ) nd the height of the nd sub-pek is bout sin πn 5 / N. sin π( 5 / N ) sin π( 5 / N )... nd so on nd we cn expect ll of the intensities of the smll sub-peks become negligible for lrge number of N tpicll bout 3 ~ 5. In the cse of n ordinr crstl which hs lrge vlues of N N b N c the Lue function in Eq. (5.) returns significnt vlues onl when K h K b k K c (h k l : integer) (5.) l nd the mximum vlue should be given b G( K) N N b N c N (5.3) The vlue N N b N c N is the totl number of unit cells in the crstl. Since the width of the pek is proportionl to / N the integrted intensit should be proportionl to N s expected. The condition given b Eq. (5.) is clled the Lue condition. An pproximte formul for the Lue function for the rnge ner one of the mxim is given b
sin πn K + ΔK sin π( K + ΔK ) ( πk ) sin ( πδkd) sin ( πδk) sin N G K + ΔK sin πn ΔK πδkd / N π ( ΔK ) D sin πn h + πn ΔK sin πh + πδk N sin πδkd where D N is the dimension of the crstl long the -direction. Another formul : f Lue D ( ΔK ) sin πδkd π ΔK stisfing the normliztion condition: f Lue ( ΔK )d ΔK (5.4) m sometimes be more convenient. The pek-top vlue of the normlized formul is given b lim ΔK ΔK f Lue D. The formul given b (5.4) is lso clled the Lue function. The reltion between the scttering vector nd scttering ngle : K sinθ λ leds the following reltion ( ΔK Δθ )cosθ λ which will be discussed gin in Chp. 6. 5- Lttice vectors nd reciprocl lttice vectors The three vectors b c to represent the periodicit of the crstl re clled the lttice vectors or the unit trnsltionl vectors. The three vectors b c defined b the following equtions re clled the reciprocl lttice vectors. b c b b b b c c c b c c (5.5) The vector is perpendiculr to b nd c nd the inner product with is for exmple. B using the reciprocl lttice vectors the Lue condition defined b Eq. (5.) is exctl equivlent to tht the scttering vector K cn be expressed b K h + kb + lc (h k l : integer). (5.6) When we ssume tht the x z components of the lttice vectors b c nd the reciprocl lttice vectors b c re given b b c (5.7)
b c (5.8) The reltion given b Eq. (5.5) is equivlent with the following eqution (5.9) which mens the inverse mtrix of the mtrix defined b the reciprocl lttice vector b c b c is equivlent with the trnsposed mtrix of the mtrix defined b the lttice vectors. The outer product p q for rbitrr two three-dimensionl vectors p is defined b p q p p z p z p x p x p From the definition we obtin p p q q p x p p z ( p q ) p p z p x p p z q. (5.) + p p z p x + p z p x + p z p x p + p x p (5.) (5.) nd cn confirm tht the vector p q is perpendiculr to both p nd q. B compring the following three equtions p q p x + p + p z q x + q ( + ) p x + p x + p x + p + p + p z + p z + p z + p z (5.3) ( p q ) ( p x + p + p z ) p x + p + p z + p x p + p p z + p z p x (5.4) p q ( p p z ) + ( p z p x ) + ( p x p ) p + p p z + p z + p z p z p x + p x + p x p x p + p (5.5) we obtin p q p q + p q. (5.6) When the ngle between the vectors p nd q is θ tht is p q p q cosθ the following reltion is derived
p q p q p q cos θ p q sinθ (5.7) tht is p q is the vector perpendiculr to p q sinθ [Eq. (5.7)]. p nd q [Eq. (5.) Eq. (5.)] hving the length of The prllelepiped defined b the lttice vectors b c is trditionll clled the unit cell in the field of crstllogrph. The outer product b is the vector orthogonl to nd b hving the length equl to the re of the prllelogrm formed b nd b b. As b is the vector perpendiculr to nd b hving the length of unit ( b ) c b is the length of proection of c on to the direction perpendiculr to nd b. The the unit cell volume is given b V ( b ) c. Similr reltions hold for the combintion of ( b nd c ) nd ( c nd ) s summrized b V ( b ) c ( b c ) ( c ) b. (5.8) The following reltions between the lttice vectors nd the reciprocl lttice vectors re lso stisfied b c V b c c V b V (5.9) (5.3) (5.3) nd the reciprocl lttice vector cn be clculted from the lttice vectors b the bove equtions. It is not necessr to use the bove formul on evlution of the reciprocl lttice vectors. All we should do is evlution of the 3-b-3 inverse mtrix. But the coding (computer progrmming) bsed on Eqs. (5.9) - (5.3) is recommendble becuse of unmbiguit nd efficienc on relistic computing. 5-3 Lttice constnts The reltions between the lttice constnts b c α β γ nd the lttice vectors b c re following : length of b : length of b c : length of c α : ngle between b nd c
β : ngle between c nd γ : ngle between nd b It is es to evlute the lttice constnts bcαβγ ( ) x + + z b c cosα bc + + b x + b + c x + c ( + ) from the components of the lttice vectors for exmple. In contrst it is little complicted to evlute the lttice vectors from the lttice constnts prtl becuse of rbitrriness bout the choice of direction of the coordinte sstem. One unmbiguous selection of the coordinte sstem is (i) ssume prllel to the X xis (ii) ssume b is on the upper XY plne (Y > ). In this cse it is es to find tht the lttice vector should be nd the lttice vector b should be given b b bcosγ bsinγ. It is ssumed tht the lttice vector c is given b c. (5.3) (5.33) (5.34) The condition : the ngle between c nd is β is expressed b c ccosβ (5.35) nd the following reltion is derived from Eqs. (5.3) nd (5.34) c + +. (5.36) Then from Eq. (5.35) the x-component of the lttice vector c is determined b ccosβ. (5.37) Next the reltion : the ngle between b nd c is α gives b c bccosα (5.38) nd from Eqs. (5.33) nd (5.34) b c b cosγ + b sinγ (5.39) b ccosβ cosγ + sinγ
nd then c cosα cosβ cosγ sinγ. (5.4) Finll the condition : the length of c is c determines the z-component of the lttice vector c b c (5.4) The coordintes derived b this method belongs to the right-hnded sstem. The unit cell volume V cn be clculted from the components of the lttice vectors determined in the bove w simpl b V The components of the reciprocl lttice vectors cn be clculted b nd so on. 5-4 Lttice plne b c V V (5.4) The Lue condition restricts the ppernce of shrp diffrction peks for the scttering (diffrction) vector K to stisf K h + kb + lc ( h k l : integer). On the other hnd the vector defined b h + kb + lc (5.43) mens the vector with the length of reciprocl interplnr spcing long the direction orthogonl to the lttice plne indexed b h k l (-plne). The index is clled Miller index. The orthogonl direction nd interplnr spcing of the -plne re equivlent with the orthogonl direction of flt plne pssing through the three points defined b the three vectors b k c l (for h k l ) nd the distnce of the plne from the origin respectivel. The vector p locting n rbitrr point on this plne is expressed b p h + x b k c h + l h ( x : rbitrr rel number) (5.44) nd the plne tht is prllel to the bove plne nd psses through the origin should be expressed b b p x k h + c l h ( x : rbitrr rel number). (5.45) h Exmine the inner product of the vectors p nd d defined b Eq. (5.45) nd Eq. (5.43). You will find tht the reltion: p d is lws stisfied for n x. It mens tht the vector
is directed long the orthogonl direction of the ( ) plne. The interplnr spcing should be given b the inner product of p nd cn conclude tht the interplnr spcing is equivlent to points. As the reltion: p d. holds for n x nd we Next let us exmine the cse of l. The hk -plne mens tht it psses through the two h nd b k nd is prllel to c. The vector p to express this plne is given b p h + x b k h + c (x : rbitrr rel number) (5.46) nd the plne tht is prllel to the bove nd psses through the origin is given b b p x k h + c (x : rbitrr rel number) (5.47) nd the reltions : p d nd p d re stisfied. The sitution will exctl be sme for the cse of k or h. Finll exmine the cse of k l. The h -plne psses through nd is prllel to h both b nd c. The vector p to express this plne is given b p h + x b + c (x : rbitrr rel number) (5.48) nd the prllel plne pssing through the origin is given b p x b + c (x : rbitrr rel number) (5.49) nd ou will find the reltions p d cse of h l or h k will be similr. nd p d We cn conclude tht the length of the vector interplnr spcing of the -plne. gin. Of course the sitution bout the is lws equl to the reciprocl of the If ll the components ( ) of the reciprocl lttice vectors b c re given the reciprocl interplnr distnce of the -plne cn be clculted b d h x + kb ( x + l ) + h + kb ( + l ) + h z + kb ( z + l ). (5.5) Note tht the interplnr distnce of the lttice plne in the Brgg s lw is generll clculted s the reciprocl of the reciprocl interplnr distnce d. 5-5 Crstl structure fctor nd Miller indices The verge position of the -th tom in the unit cell r is expressed b r x + b + z c (5.5)
where x z re the frctionl coordinte hving vlues from to. As the diffrction pek onl ppers when the scttering vector K is given b K h + kb + lc (h k l : integer) the crstl structure fctor denoted b F( K) so fr cn be expressed b M F f d T d exp π i hx + k + lz (5.5) where d Here f h + k b + l c. d is the tomic scttering fctor nd it depends onl on the interplnr distnce when the electron densit distribution of the tom is ssumed to hve sphericl smmetr. The tomic scttering fctor cn be expressed b f θ stisfing the Brgg s eqution λ sinθ. And T d sinθ λ using the Brgg ngle is the tomic displcement fctor generll representing nisotropic displcement of n tom from the verge position. 5-6 Anisotropic tomic displcement fctor The probbilit densit function bout the displcement of n tom from the verge position cn be modeled b g ( r ) (π) 3/ U / U / U exp X Y Z / 3 U U U 3 (5.53) where it is ssumed tht the displcement of the tom is expressed b the nisotropic (ellipsoidl) Gussin function nd the vector locting the tomic position r x is expressed b z r Xp X + Yp Y + Zp Z (5.54) using the unit vectors long the three principl xes of the ellipsoid p Xx p X p X p Yx p Y p Y p Zx p Z p Z. p Xz p Yz p Zz The probbilit densit function bout the tomic displcement given b Eq. (5.53) corresponds to the modeling of therml vibrtion of toms b independent hrmonic oscilltors. The reltion given b Eq. (5.54) is rewritten b using mtrix s P ( p py pz X ) p Xx p Yx p Zx p X p Y p Z p Xz p Yz p Zz (5.55)
x z P X Y Z. (5.56) The mtrix P is n orthogonl mtrix the trnspose mtrix of which is equivlent to the inverse mtrix tht is X Y Z P x z Pt x z P Xx P X P Xz P Yx P Y P Yz P Zx P Z P Zz x z. (5.57) The tomic displcement fctor bsed on the probbilit distribution of tomic displcement shown in Eq. (5.53) is given b T ( K) g ( r )exp( π i K r )d v R 3 g ( r )exp π i K r d X dy d Z exp X Y Z (π) 3/ U / U / / U 3 U U U 3 exp π i ( K X + K Y + K Z X Y z ) d X dy d Z exp π K X U + K Y U + K Z U 3 exp π K X K Y K Z exp π ( K x K K z )P When we define mtrix U xx U xz U U x U U z U zx U z U zz P U U U 3 U U U 3 U U U 3 P t the tomic displcement fctor of the -th tom is given b T ( K) K x exp π ( K x K K z )U K K z exp π K t U K P t K X K Y K Z K x K K z. (5.58) nd the expnsion results in the following formul
T ( K) exp π K x K K z exp π (( U xx ) K x + U { + Define the mtrix U U U 3 U U U 3 U 3 U 3 U 33 K + U zz ( U xx ) ( ) U ( zx ) ( ) U ( ) U ( z ) ( U zx ) ( U z ) U zz K z K x K + ( U z ) K K z + U zx x / / z / b x / b b / b b z / b c x / c c / c c z / c b c nd the following reltion will be derived U xx U zx U U z U zx U z U zz b c K z K x U xx U zx U U z U zx U z U zz U xx U zx U U z U zx U z U zz U U U 3 U U U 3 U 3 U 3 U 33 U U b U 3 c U b U b U 3 b c U 3 c U 3 b c U 33 c K x K K z }. (5.59) x / b x / b c x / c / b / b c / c / / b / c b c which leds the formul of the tomic displcement fctor for the scttering vector K h + k b + l c T ( K) exp π U ( h + ( U ) k b + ( U 33 ) l c b c (5.6) (5.6) +( U ) hk b + ( U 3 ) hl c + ( U 3 ) klb c ). (5.6)
The six independent elements of the mtrix { U U U 33 U U 3 U 3 } is clled nisotropic tomic displcement prmeters. When we define nother mtrix B B B 3 z U xx U zx B B B 3 π b z U U z B 3 B 3 B 33 c z U zx U z U zz the following reltion is derived U xx U zx U U z π U zx U z U zz B B B 3 B B B 3 B 3 B 3 B 33 (5.63) (5.64) nd the formul of tomic displcement fctor for the scttering vector K h + k b + l c should be T ( K) exp B h + ( B ) k + ( B 33 ) l + ( B ) hk + ( B 3 ) hl + ( B 3 ) kl { }. (5.65) The formul given b Eq. (5.65) is little more convenient for clcultion thn the formul given b Eq. (5.6). The six independent elements of the mtrix { B B B 33 B B 3 B 3 } is lso clled nisotropic tomic displcement prmeters. The nisotropic U prmeters nd nisotropic B prmeters re relted b the following equtions B π U B π b U B 33 π c U 33 B π b U B 3 π c U 3 B 3 π b c U 3. (5.65) Note tht the bove reltions between the nisotropic B nd U prmeters re different from the reltion between the isotropic displcement prmeters B nd U B 8π U s shown in Chp. 4. It is difficult to find wht nisotrop is expected from the vlue of nisotropic B prmeters while the prmeter U hs the mening of men squre tomic displcement long U is the men squre tomic displcement long b nd U 33 is the men squre tomic displcement long c. As the definition of B prmeters m be introduced ust for slight convenience on clcultion. The vlues of nisotropic displcement prmeters re sometimes restricted b the smmetr of the tomic positions. Let us exmine the restriction b smmetr of crstl structure belonging to hexgonl sstem s n exmple. In hexgonl crstl sstem the c xis is trditionll tken to be prllel to the six-fold rottion xis. If n tom is locted on the six-fold rottion xis one of the principl xes of the ellipsoidl tomic displcement is uniquel ssigned to the c-xis (z-xis). There remins rbitrriness in the selection of the direction of nd b xes or the other two principl xes of the ellipsoid but we cn ssume tht the -xis (x-xis) is prllel to one of the principl xes nd the lst principl xis is on the b-plne nd orthogonl to the -xis which mens tht the lst
principl xis cn be tken prllel to the -xis. Then we should ssume U xx U U U zz U 3 U xz U z. When we ppl the three lttice vectors s b / 3 / c c we find the reciprocl lttice vectors / / 3 b / 3 c / c nd the unit vectors long the direction prllel to the reciprocl lttice vectors 3 / / b c b c nd then we hve the formul for the nisotropic displcement for the tom U U U 3 U U U 3 / / U 3 U U 3 U 3 U 33 3 / / U 3 3 / / 3U / U / U U 3 U U / U / U U 3 which implies the restrictions: U U U U / nd U 3 U 3. Exercise Assume the lttice constnts 6. A b 5. A c 4.A α. β. γ.. Evlute ll the components of the lttice vectors b c nd lso the components of the corresponding reciprocl lttice vectors. Next clculte the interplnr distnce of the 3-plne (the lttice plne with h k nd l 3)