X-ray Diffraction from Materials

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Darren Perkins
5 years ago
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1 X-ay ffato fo Matal 8 Spg St Lt; Yag Mo Koo Moday ad Wdday :5~6:

2 . X-ay ff Sattg. Thal ff Sattg d to to Vbato. Odg ad ff Sattg d to Stat plat Howo

3 . X-ay ff Sattg ffato of x-ay: pod aay of ato Ba podty by o ao dffato pa dff Thal vbato of ato Log ag odg th alloy Loal pptato th alloy Thal vbato of ato Th dplat of ato a b wtt a ( t o( wt δ wh ad δ th avag poto vto th pha dff btw wav ( tav wav ( logtdal wav *Poto th avag tt a how dottd

4 . X-ay ff Sattg Phoo dpo v Bllo zo boday Fo a vto - logtdal wav - tav wav Log ag odg th alloy Log ag od paat: S -ado ato aagt: S -pftly odd ato aagt: S

5 . X-ay ff Sattg Loal pptato th alloy Loal odg : - hot ag od paat of ghbog ato α l : l dat th ghbotg g ato - loal tato dta d to ato z dff B l

6 . Thal ff Sattg d to to Vbato Th Ft Sattg Eqato f f xp wh of { ( } vbato fqy of ato th od. S th hght o a ot dtt th ato poto of th x - ay attg ba of hg fqy dff btw th (x -ay : ~ 8 Hz. O a oly dtt th t avag of th x - ay ttg f f xp { ( } xp ( H < > dbd th t avag. { } T avag t of th abov qato a b plfd a xp { ( } xp { ( } { ( }...!!

7 . Thal ff Sattg d to to Vbato dplat all Fo ( { } ( { } xp xp p ( ( ( ( [ ] xp xp xp th qato d th attg tty had d of th ght two t of Th ft ( ( M dotd M. fato Wall by - fato. Wall whh alld th by - y g q g ( ( [ ] ( ( ( ( t a b xpadd a Th thd xpotal xp ( ( [ ] ( ( ( ( ( (!! xp fo. a b wtt a aalytal oo - ato ytal th attg qato of Th

8 . Thal ff Sattg d to to Vbato M ( f xp { ( } f! ( ( ( ( ( ( M ( ( ( (! f ( ( ( M ( : Bagg pa tty d by th thal vbato : Th ft od thal dff attg at attg vto whh atfd th Bagg odto wth ad of th phoo wav vto. : Th od od thal dff attg at attg vto whh atfd th Bagg odto wth ad of two otv th phoo wav vto. : Th thd od thal dff attg at attg vto. Th agtd of th thal dff attg tty vy all oapd wth that of Bagg pa. Typally l tha %. Th agtd of th od od dff attg tty alo a fw pt of that of th ft od dff attg. Thfo th ft od thal dff attg tty ta to aot th aalyof x -ay attg xpt. Callato of th od od thal dff attg tty dd fo hgh tpat xpt ba thod tty ot glgbl athgh tpt.

9 . Thal ff Sattg d to to Vbato Callato of th Thal ff Sattg tty Phoo : Cofd th Cytal: Spl Cb - t Bllo zo N N N N t ll Nb of Phoo : N - Tav Wav (N - Logtdal Wav (N. to dplat E ( t o ( w t δ Kt gy & ω ( w t δ

10 . Thal ff Sattg d to to Vbato vag t gy of th phoo E & N ω Total gy of ato th of th t gy ad th pottal gy. Th avag gy of ato tw of th avag t gy o that E tot wh N ω E xp hω ( / T hω Fo th hgh tpat (oo tpat E T Nω Nω E fo a hao ollato. E T.

11 . Thal ff Sattg d to to Vbato Th Ft Od Thal ff Sattg (TS Th t avag of th ft od th of M ( f xp ( th qato bo { }( ( f M xp M f xp { ( } ( o{ ( } { ( } ( ( o{ ( } wh t vto alog ah phoo opot. x -x o x ( / th ft od thal M ( f ( ( [ xp{ ( / ( } xp{ ( / ( } ] Ug th th latohp f M ( ( ( / ( / { } dff tty bo

12 . Thal ff Sattg d to to Vbato ( ± / th how th fg / th dffato tty of whh th vto ( ± / vto. th Bag odto wth ad of th phoo vto ± / at a attg Lt allat ( ± / followg th th a of (. ( ± / xt wh ± / loatd a thod fo at th vty of th H poal pot fo. o at wh S p ( ± / p ( ± / hl whh th lot Th dffato oly * * / H p a p b p p hl a vy all val. agfl fo all val of ( ( a b ad a a otat. * p p p

13 . Thal ff Sattg d to to Vbato S N vy lag th ov of th qato a b plavd by a tgato ov th ft Bllo zo. Th dty pot tgal th ft od dff tty bo * N/ a. tal pot of th / vto xpd by dv M ( f ( ( N t BZ f { ( / ( / } M M ( f NT ( ( o ω ( N a * a tg th avag apltd t to th qato * d a vol lt * / a dp dp dp dp dp dp o a gt. fo th Expd a a

14 . Thal ff Sattg d to to Vbato f V th vloty of vto w ( o M NT ( f V V / thal dff attg by th followg qato;. Th o a allatd th ft od Th Sod Od Thal ff Sattg (TS ( ± / ± ' / Th fto wll hav hav hap axa at poto poal pa h that ' ± ± H hl Th dffato oly o ± ± wh p p p a vy all val. ' * * * Hhl pa pb p

15 . Thal ff Sattg d to to Vbato TS a b pd b Th od od ( ( { } xp f M TS a b xpd by Th od od ( ( ( ( ( ( ' o o ' ' ' ' ' ( ( ( ( ( ( ( ( ( f o o M B o B o oob Ug th o lato ( ( ( ( ( ( ( 8 ' ' xp xp f ' ' ' ' ' M ( ( ' ' xp xp ( ( ( ( 8 ' ' ' ' f ' ' ' ' ' M

16 . Thal ff Sattg d to to Vbato ± ± xt at th vty of th Hhl ad th tgato fo all poal pa bo N. S dffato o at ± ± H hl th wold b o th o hl whh atfy th Bagg odto a how th fg. M ( N f ( ( ( ' ' ( ' H ' ' hl Th hadd aa of th fg th ato of all fo hl poal pot whh atfy th Bagg odto. Covtg th hadd aa to tgal fo M ( N f ( ( ( ( ' ' ' OBZ * H ' ( hl wh OBZ dot th ovlappd - t Bllo Zo. ( o ( V N T o M 8 f OBZ ω ω wh V th vol of th t ll. H hl ' ' ' a N dv Th qato a b wtt a ' dv

17 . Thal ff Sattg d to to Vbato xapl of th thal dff attg tty t of C-B alloy how blow. Th ft od TS qt bg lo to Bagg pa both oo tpat ad hgh tpat. Howv th od od TS tty oly gfat at hgh tpat. pa

18 . Thal ff Sattg d to to Vbato Mat of th Elat Cotat of th Cytal Fo th dfto of σ op wh ε opq opq q opq th t x a th lat tff. q Fo all vol lt o t th ltat fo qal to th a t th alato. σ x op p opq x p x q ρ t Th ato vbto a b xpd a { ( t } xp ω tg to th dfftal qato ρω opq p q Lt df th dtoal o of th wav vto a f ad df th pha vloty a V ω ρv opq f p f q o. Th f ad f

19 . Thal ff Sattg d to to Vbato Rwttg th qato ( ρv ( ρv ( ρv of. f Th qato oly hav olto wh th dtat of th th offt o od th b yt th Ug th lat tff th dtat of f opq a b wtt a th qato bo ( f f ρ V ( f f ( f f ( f f f ( f f ρv ( f f ( f f ( f f f ( f f ρv

20 . Thal ff Sattg d to to Vbato Fo xapl ( a wav popagat alog []. S f f f th followg olto a b obtad fo th abov qato ad V l V t ρ V t Fo xapl ( ρ ( : logtdal wav ( : tav wav a wav popagat alog []. S followg olto a b obtad fo th abov qato V l V t V t ρ ρ f ( ( : logtdal wav ( ( : ρ : tav wav f ( : tav wav f th

21 . Thal ff Sattg d to to Vbato Fo xapl ( a wav popagat alog []. S followg olto a b obtad fo th abov qato V V l ρ ( ( : logtdal wav ρ f f ( ( : tav wav t V t f th Mt of th ft od thal dff attg tty; ( tg ( NT M f ω o ω V o ( o ( o ( o ( V V V Th - t od TS at P oly otbtd by th logtdaal wav vto. V

22 . Thal ff Sattg d to to Vbato Th abov qato bo /V. f o a th - od TS V a b Th ( allatd. Th V l ρ l l Fo th pot Q φ o φ Vl ( Vt ( Vt ( Wh obtad a b allatdg l th abov qato. Fo th at of th dfftty at Q a alo b allatd. Mathatal dvato bad o th ot odl whh ad that <<d. Ba of th ao th a lag o fo lag. odfd odl popod by Bad ad Powll; V V ( d / ( d /

23 . Thal ff Sattg d to to Vbato tato of t-ato Fo Cotat that th ato vbato a hao ad that th tog fo atg btw ato a b dbd a Hoo law th fo o ato l popotoal to th latv dplat of all th oth ato. Th opot of fo alog th th othogoal ax x y ad z a b wtt F F F wh ( l x { ( l xx ( x lx ( l xy ( y ly ( l xz ( z lz } l ( l y { ( l yx( x lx ( l yy ( y ly ( l yz ( z lz } l ( l z { ( l zx ( x lx ( l zy ( y ly ( l zz ( z lz } l pt th -th opot of th dplat of ato xtdd ov all ato xpt ato l. Th qatt ( l dt th fo o ato dplat th - dto. th todg addtoal otat ll ad wh th th tato fo otat atg btw ato ad ato l l -dto wh ato gv a t latv ( ( l wh odd a otat dtg th fo o ato tlf dplad th -dto. l x y ad z. Th ay b l th -th dto wh ato l

24 . Thal ff Sattg d to to Vbato Eqatg th fo o a ato to th podt of t a ad t alato th F d l dt { } l ( l ( l ( l ( l O tt to odg oly latt wth o ato p to dodg ato dplat th qato bo - ω o x x y y z z ( t o( w t δ ptv ll ad ( w t δ { ( l ( l ( l } o( w t δ l x y z Fo plfato ag that th ytal to - yt - ω x { ( l ( l ( l } o ( x x todg th w vaabl { xx( ω } x xy ( y xz ( yx ( x { yy ( ω } y yz ( ( ( ( ω zx x y y z y z { } { } ( ( ( l o ( { } ( z { } yz y zz z l z z l

25 . Thal ff Sattg d to to Vbato To hav o tval olto th dtat of th offt t vah; xx ( ω xy ( xz ( yx ( yy ( ω yz ( ( ( ( zx yz zz Th olto gv th fq of th th od of vbato fo th lat wav wth th wav vto. ω a xapl FCC ytal h ( a a a a l C t. paalll l to ptvly to th latt vto a a x l a * y a * z a * Ug th qato ( ( o( x y Th th ad a. z Lt othgoal ax x y ad z ot Th ov xtd ov ay ato l ad all t ghbo. fo ( ± ( ± ( ± ( ± ( ± (. th - t at ghbo wold b ±

26 . Thal ff Sattg d to to Vbato Cod th tto that th yty of FCC po o th tato fo otat. Ct of yty ( ( Rflto pla ppdla to z ax; Fo ato a dplat whh l wth flto pla a ot pod a z - opot of { ( ( }. zx zy fo o ato Th dplat of z dto a ot pod o ato a opot fo whh ( ( th lto pla { }. of xz yz xy-o pla

27 . Thal ff Sattg d to to Vbato pplyg flto yty to th oth bo ato o a gt ( xz ( yz ( zx ( ( xy ( yz ( yx ( ( ( ( ( yx zx zy yz ( xz ( yz ( zx ( zy ( xy ( yz ( yx ( yz ( ( ( ( yx zx xy xy xz xz Th z - ax a ax of - fold otatoal yty ad th fo o ato d to dplat of ato ad t how th a otatoal yty. y Fo xapl th fo o ato th y - dto d to a dplat of ato th x - dto t b qal to th fo o ato th x - dto d to a qal dplat of ato th - y dto. To atfy th - fold yty th followg lato t xt

28 . Thal ff Sattg d to to Vbato ( yx ( xy ( xy ( yx ( xx ( yy ( yy ( yy ( zz ( ( xz ( zx ( zx ( xz ( xx ( yy ( yy ( yy ( zz ( zz ( xz ( zx ( zx ( xz ( ( ( ( ( ( xx yy yy yy zz to ad l a flto pla whh ppdla to th [ ]. S th fo opot o ato d to dplat ato t how th yty. Slay fo all {} o pla t follow that ( xx ( yy ( xy ( yx ( zz ( xx ( zx ( xz ( yy ( zz ( yz ( zy. ( ( ( ( ( ( xy ( ( ( ( ( ( xz ( ( zz yz xy xx. xx zz xx zz zz yx yy yz zz zz

29 . Thal ff Sattg d to to Vbato a lt of yty tto FCC th tog fo btw a ato ad t twlv at ghbo a b pfd by oly th offt : ( α ( β ( xx yy zz β γ O a apply th a yty odato to th fo otat fo o dta. by two offt; by fo Th fo otat t fo ( ( yy xx α yy β. offt; ( xx α ( ( γ ( x od at ghbo a b xpd Th fo otat fo thd at ghbo a b xpd yy β yz xy δ. Th ft od ad thd aghbo fo otat a gv th Tabl. S th bdg gy of th ato popotal to -6 ~ all ato byod th thd ghbo wll b ad to b glgbl. tato wth

30 . Thal ff Sattg d to to Vbato

31 . Thal ff Sattg d to to Vbato Th qatt ad ( a b allatd g th tabl. [ ( ] α o o o ( o o β ( α ( ( o o o β [ α ( ] β o o o o o ( γ γ o ( δ o wh xy ad z wth Sbttto of oplatd. (. th qatt to th la qato th fq ω a b obtad. Th a b qato ω ad t olto fo a abttay lat wav a b Howv th olto bo qt pl fo a dto of Fo xapl lt olv th qato fo th wav popagatg th [] dto; ad fo all. y z hgh yty.

32 . Thal ff Sattg d to to Vbato S th dt dtat t of th la qato o a gt th fq g th abov qato; 6 x ω ( ( x ( xx α β α α 8 x ω yy α β α β β β ω ω ( ( ( t tth podt of t dagoal t whω a logtdal wav ad ω ad ω a tav wav. Ug th qato o a obta th dpo v of th phoo alog[] dto. x Fo popagatg th [] dto to - ad z x ad th ft Bllo zo lt x. Fo qato all xz ( yz ( zx ( zy ( xy ( yx( xx ( yy (. Th th la qato bo ( ( ω ω ω [ ][ ( ( ][ ( ] xx xy xx xy zz y x

33 . Thal ff Sattg d to to Vbato Solto of th qato; ω xx ω ( ( xy x ( α β α β δ ( α α β β γ γ xx ω ( α β δ ( ( xy x x ( α ( β α β δ α α β β γ γ ( α β δ x x δ x ( ( α β ( β β α zz x x x Fo popagatg th [] dto to - xx x ad th ft Bllo zo lt x. Fo qato xy ( yz ( zx ( yz ( zy ( xz ( ( ( (. Th th la qato bo yy zz y z x ad

34 . Thal ff Sattg d to to Vbato ( ( [ ] ( ( [ ] ( ( [ ] ( ( th qato; Solto of xy xx xy xx xy xx xy xx ω ω ω ω ( ( ] x y δ γ β α γ γ β α β α β α ( ] ( ( ( x xy xx x γ γ β α β α β α ω ω δ γ β α ( ( ] x δ γ β α γ γ β β β hav th pl fo wav popagatg alog hgh yty dto fo Th dpo lato x ω th tato fo otat. obato of a la wh ω

35 . Thal ff Sattg d to to Vbato Elat wav wth agla fqy ω ad wav vto ( ω/. vloty V Th pha vlot of yty dto of th tt oby th lato x V th lt of V vy log wavlgth 6 x ( th latty. opao of tavl wth a pha th wav popagatg alog th th th ta th fo Th vlot t qal to th opodg vlot dtd by a gltd o obta a α α 6α a α β β two gv a t of 8β lato btw th lat otat ad th tato otat. f tato byod thd ghbo a ( γ 8γ δ. wh a th latt paat. α β

36 . Thal ff Sattg d to to Vbato Callato of th by-wall Fato by Wall fato otag oly o d of M ( o ( Rwttg th qato g tpat t h M o N ω xp( ω / T h Followg th thod of by o a that ll wav a p logtdal ad p tav Th avag vlot of th wav a otat ato xpd by ( Th Bllo zo ad a hp of ad of who vol qal to that of th Bllo zo V ( /. Th dty of th pot th ph N/V tg o ( BZ. / ad BZ θ / λ to th qato o a obta h θ N M N λ ω xp( ω h VBZ d

37 . Thal ff Sattg d to to Vbato g th vlot of wav a otat. Fo V ω / d ω dω /V adv BZ ( / h θ ω M d N ω ω λ ω xp( hω / T Lt hω hω ε x T Ug th fto Φ x εdε Φ( x x ε M bo T Θ ω / T /V ( x todd by by h T θ x M Φ( x λ Θ Lt paat haatt tpat ad offt x logtdal ad th tav wav : Θ l hω / Θ hω l t l / x l Θ l. / T x t Θ t / T. fo th

38 . Thal ff Sattg d to to Vbato todg th lato h T θ x M Φ l Θ λ l Θt By xpadg th xpotal t x x x Φ( x 6 6 Fo x < Φ x x / Φ x x fato [ ( ( ] [ ( ( / ]. l l dfd by ( /Θ ( /Θ ( /Θ M M t l xt t t adly how that l ( x Φ( x h T θ x M Φ( x Θ M λ Th Θ ot a a th Θ of th pf hat. Th fft of thal vbato to obta a val fo M fo th lat otat. t t. Now tod a avag Θ M fo th M Th dff howv ot gat ad to a ft appoxato o a fo Θ M th tablatd by tpt Θ to d th tgatd tt. t alo pobl.

Noise in electronic components.

Noise in electronic components. No lto opot5098, JDS No lto opot Th PN juto Th ut thouh a PN juto ha fou opot t: two ffuo ut (hol fo th paa to th aa a lto th oppot to) a thal at oty ha a (hol fo th aa to th paa a lto th oppot to, laka