Quantitative Texture Analysis From X-Ray Spectra - J Appl Cryst

7/27/2019 Quantitative Texture Analysis From X-Ray Spectra - J Appl Cryst

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4 4 3J . A p p l . C r y s t . ( 1 9 9 7 ) . 3 0 , 4 4 3 - 4 4 8

Q u a n t i ta t i v e T e x t u r e A n a l y s i s f r o m X - r a y D i f f r a c t io n S p e c t r aY. D . W ANG, a L . Z u o f l Z . D . LIANG, a C. LARUELLE, A. VADON b AND J. J. HEIZMANN b

a D e p a r t m e n t o f M a t e r i a l s S c i e n c e a n d E n g i n e e r i n g , N o r t h e a s t e r n U n i v e r s it y , S h e n y a n g 1 1 0 0 0 6 , P e o p l e ' s R e p u b l i co f C hi n a, a n d b L E T A M / I S G M P , C N R S U R A 2 0 9 0 , U n i v er s it d d e M e t z, 5 7 0 4 5 M e t z C E D E X 0 1, F ra n c e .E - m a i l : l a r u e l l e @ l e t a m . u n i v - m e t z . f r(Received 18 June 1996; accepted 2 January 1997)

A bs tra c tA method to ob ta in t he o r i en t a t ion d i s t r i bu t ion func t ion( O D F ) o f a p o l y c ry s t a l l in e m a t e r i a l d i r e c tl y f r o m X - r a yd i f f rac t ion spec t ra i s p re sen ted . I t uses t he maximum-t e x t u r e - e n t ro p y a s s u m p t i o n t o r e d u c e t h e d i f f r a c ti o n d a t an e e d e d f o r th e O D F a n a l y s i s . T h e v a l i d it y o f t h i s n e wm e t h o d i s i l l u s t r a t e d t h r o u g h t w o m o d e l e x a m p l e s .

I . Introduct ionThe or i en t a t ion d i s t r i bu t ion fun c t ion (OD F) of a po ly-c rys t a l l i ne ma te r i a l i s gene ra l ly de t e rmined f rom po le -f ig u r e d a t a b y t h e v a r i o u s p o l e - f ig u r e i n v e r s i o n m e t h o d ss u c h a s t h e h a r m o n i c m e t h o d ( H M ) ( B u n g e , 1 9 6 5 ; R o e ,1 9 6 5 ) , t h e v e c t o r m e t h o d ( V M ) ( R u e r & B a r o , 1 9 7 7 ;V a d o n , 1 9 8 1 ) a n d t h e W I M V m e t h o d ( M a t t h i e s &Vine l , 1982) . Recen t ly , X- ray (o r neu t ron ) d i f f rac t ions p e c t r a h a v e b e e n u s e d to d e t e r m i n e t h e O D F s o f s o m epolyc rys t a l l i ne ma te r i a l s (Wenk , Ma t th i e s & Lut t e ro t t i ,1994) . Such an approach i s expec ted to p rov ide no t on lyan e f f i c i en t means fo r t he quan t i t a t i ve t ex ture ana lys i s o fc o m p l e x m a t e r i a l s ( s u c h a s i n t e r m e t a l l i c s , c e r a m i c s a n dd u p l e x p h a s e a l l o y s ) w i t h s i n g l e o r / a n d o v e r l a p p i n gdi f f rac t ion peaks , bu t a l so a s ign i f i can t t oo l fo r t he fa s tt e x t u r e m e a s u r e m e n t s o f tr a d i ti o n a l m a t e r i a ls .I n t h e l a s t d e c a d e , t h e m a x i m u m - e n t r o p y c o n c e p t h a sb e e n a p p l i e d to t h e O D F d e t e r m i n a t i o n f r o m p o l e - f ig u r eda ta fo r cub ic , hexagona l and t e t ragona l ma te r i a l s(W ang , Xu & Liang , 1987; Schaeben , 1988) . I t has beend e m o n s t r a t e d t h a t f e w e r p o l e - f i g u r e d a t a m a y b erequ i red b y the use o f a re f ined a lgor i thm, i .e . the so-c a l l e d m o d i f i e d m a x i m u m - e n t r o p y m e t h o d ( M M E M )( W a n g , X u & L i a n g , 1 9 9 6 ) . T h e n e w a l g o r i t h m i n t r o -d u c e s d i r e c t l y t h e m a x i m u m - e n t r o p y c o n c e p t i n t o t h el eas t - squa res-equ a t ion o f t he po le - f igure i nve rs ion .Thus , i t i s poss ib l e t o pe r form the pa r t i a l ODF ana lys i sw i t h o n l y a l i m i t e d n u m b e r o f i n p u t p o l e - f i g u r e d a t an o r m a l i z e d a c c o r d i n g t o a s t a n d a r d t e x t u r e l e s s s a m p l e( W a n g , V a d o n , H e i z m a n n & X u , 1 9 9 6 ). I n t h e p r e s e n tp a p e r t h e M M E M i s e x t e n d e d t o t h e d e t e r m i n a t i o n o ft h e c o m p l e t e O D F s d i r e c t l y f r o m X - r a y d i f f r a c t i o n

@ , 1 9 9 7 I n t e rn a t i o n a l U n i o n o f C r y s t a l l o g r a p h yP r i n t e d i n G r e a t B r i t a i n - a l l r i g h t s r e s e r v e d

spec t ra . I t s va l id i ty i s i l l us t ra t ed th rough two s imula t ede x a m p l e s .

2. Mathematica l a lgori thmL e t u s c o n s i d e r a s i n g l e - p h a s e p o l y c r y s t a l l i n e s a m p l ewh ose c rys t a l s t ruc tu re i s known . In t he case o f arandom gra in -or i en t a t ion d i s t r i bu t ion , t he i n t egra t edin t ens i ty , l~ l , o f t he {h} Brag g d i f f rac t ions i s g iven b yt h e k i n e m a t i c a p p r o x i m a t i o n ( T a y l o r , 1 9 6 1 ) .

R = K S ( h ) ( 1 ){h}w i t h

S ( h ) = P ( O ) N { h } IF{h}12 ex p ( - 2 B s in 2 0 /A2) , (2 )where K i s a cons t an t , P ( O ) i s t he Loren tz po la r i za t ionfac to r , wh ich i s a func t ion o f t he B ragg d i f f rac t ion ang le0, F{h} is the the oret ica l s t ructu re factor , N{h} is them u l t i p l i c it y f a c to r , B i s t h e D e b y e - W a l l e r f a c t o r o f t h em e a s u r e d s a m p l e a n d 2 i s t h e w a v e l e n g t h o f th e i n c i d e n tX-rays .For a t ex tured ma te r i a l t he i n t egra t ed in t ens i ty , I~} , o fthe {h} Bragg d i f f rac t ion a t a ce r t a in sample d i rec t ion ym u s t b e e x p r e s s e d a s ( H e d e l , B u n g e & R e c k , 1 9 9 4 )

I{~I(Y) = l~h}P{hl(Y ' (3 )wh ere P Ih /(Y) i s t he norm a l i zed po le de ns i ty o f t hec rys t a l p l anes {h} .T h e d i s c r e te v a l u e s o f th e O D F , f (g j) ( j = l , 2 . . . . . J ) ,a n d t h e p o l e - f ig u r e d e n s i t i e s a r e r e l at e d b y ( W a n g , V a d o n ,H e i z m a n n & Z u o , 1 9 9 6 )

/max l l W { h } , . ,P I h } ( Y ) - ~ ~ ~ lm, e lm, (Y) (4 )/ = 0 m = - I n = - Iwi th

JW~m = Y~. O~m, (g j ), (5 )j = lwh ere the ~ , , , , , a re t he m a t r i ce s re l a t i ng the t ex-h {h}ure coeff ic ients Wl, , , and f ( g j ) and t e elm,,(y are the

J o u r n a l o f A p p l i e d C ~ s t a l l o g r ap h yI S S N 0 0 2 1 - 8 8 9 8 ~(',~ 1 9 9 7


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4 4 4 Q U A N T I T A T I V E T E X T U R E A N A L Y S I S F R O M X - R A Y D I F F R A C T I O N S P E C T R Amatr ices re la t ing Wlmn and the pole-f igure densities.{Illetm,(Y) read

Ih) l m n ( Y ) : 2rt[2/(21 + 1)]l/2p~/(cos Z)x exp (- imr/ ) /~/(c os 69) exp ( indP) , (6 )

wh ere (Z, r /) and (O , q)) are the spherical pola r coord i-na tes of y and h , respec t ive ly , and the P ' ~ ( x ) are theassoc ia ted Legendre func t ions .On the a s sum pt ion o f m a x im um te x tu r e e n t r opy, t heODF may be wr i t ten as (Wang, Xu & Liang, 1987)( )( g y ) = e x p - 1 - 2 0 - Y ] y ] ~ 21 , m , n ,OJl,m,n ,1 '=1 r a ' = - l ' n = - l '

(7 )where 2 o and 2t,m, , a re the unkno wn Lagrangian mul-t ip l ie r s tha t can be de te rmined e i ther f rom the texturecoef f ic ients (Wang, Xu & Liang, 1987) or d i rec t ly f romthe pole - f igure invers ion equa t ion (Wang, Xu & Liang,1996). Here, a ll the unknown multipliers are to beeva lua ted di rec t ly f rom the X-ray di f f rac t ion spec t ra .Introducing (4)-(7) into (2) results in

/max 1' l'/ i r l ( y ) = K S ( h ) y ~ y ~ ~ g'lmn(y){h}1 = 0 m = - l n = - lx ~ dZm,,ex - 1 - 2 oj = l

I ' ma x 1 ' I ' X- - Z Z Z 21,m,n,O~,m,n, ( 8 )l ' = 1 m ' = - l ' n ' = - l '

Thu s, if the real intensities, I~I(Y), o f several {h} Brag gdif f rac t ion peaks a t the di f fe rent sam ple di rec t ions y a remeasured, the Lagrangian mul t ip l ie r s in (7) can beder ived by solut ion o f the fol lowing leas t -squaresequa t ion:lmax 1 lT Y] I I~I (Y) KS(h) y] Z y] Ih l: - - e . l m n( Y )h ,y I = 0 m = - I n = - I

J = ll ' m ax 1 ' 1 ' ) ] 2

- - Z Z Z 2l 'm 'n 'O~'m'n ' = m i n . ( 9 )1 '=1 m ' = - l ' n = l 'To obta in a convergent solut ion o f the normal ized fac torK and the unknown mul t ip l ie r s 2t, m,,, the gradientme thod (Bro usse, 1988) is sugges ted. I t proceed s asfol lows:

(i) the initial valu es o f K, 2 o and 21'm',, are set asK() = Y~ /~I(Y )/Y~ S(h)h , y h,y , (10 )k () = 0

where k is the vec tor express ion of unknow n 2 o andAl, m,n,;

(ii) in the first iterative stage, while K is kept fixed, kis approached by the i terative processesk ( i + l ) = ~ ( i ) __ O~ V T [ K ( ) ' ~ ( i ) ]

i iVT[K< O) ,k ( i ) ] l I (11)the op timal step leng th 0e is determin ed in each iterationso as to minimize the objec t ive func t ion T;( ii i) af ter several i teration steps, a new iterative pro-cess is used

K ( i + ] ) = E l ~ l ( Y ) I ~ h ~ r ( Y ) / ~ - , [I~h))r(Y)]h , y h , y~ ( i + 1 ) = ~ ( i ) _ _ ~ V T [ K ( i + l ), ~-(i)] , (12)

IIV T [K (i+~), k (] IIwhere i[~]r(y) are the recalculated intensities of somediffraction peaks in the i th i teration.In general, the selection of lmax in (8) depends mainlyon the to ta l numb er of the input I~}(y) used for the ODFanalys is and the c rys ta l s t ruc ture of the measuredsample . The lma va lue in (8) should cor respond to thesharpness of texture. In this paper lm a and l~ax arechosen to be 22 and 10, respectively

3 . M o d e l e x a m p l e sThe method descr ibed above is tes ted by the two modelexamples : for the case of cubic -or thorhomb ic sym metryand for tha t of te t ragona l-or thorhombic sym metry . Ineach case a model ODF is const ruc ted wi th the ce r ta inGaussian peak- type texture components (Wagner ,Wenk, Esl ing & Bunge , 1981) . A to ta l number of 15X-ray di f f rac t ion spec t ra a re prede te rmined by the HM(/max = 22). For the 15 mo delle d spectra the corre-sponding sample di rec t ions a re shown in F ig . 1 . As thepola r ang les Z of the input d i f f rac t ion spec t ra a re notla rger than 50 , no d efocusing cor rec t ions to the di f -f rac tion in tensi t ies w ould be required i f the Schulzre f lec t ion metho d is used to obta in X -ray di f f rac tion da ta .

7 . = 5 0

7 . = 25

~ = 5

' ~ ' 5 o

: ..!iii"..... : . . , , ," r 1 = 6 7 .5 "iiiii!ii,,-.,,, .......i . . . . . . , . - , " ' " ; " , - " I

F i g . I . I l l u s t ra t i o n o f t h e p o s i t i o n s o f 1 5 m e a s u r i n g p o i n t s o n t h e p o l es p h e r e s e l e c t e d f o r t h e O D F a n a l y s is .


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Y. D . WANG e t a l . 4453 . 1 . C u b i c - o r t h o r h o m b i c s y m m e t r y

For the cubic -or thorhombic symmetry case , theassu med d iffraction spectra cons ist of f ive single peaks,i .e . {200}, {220}, {222}, {311} and {511 } ( the pow derdif f rac t ion in tensi t ies a re input according to the ASTMcard o f Cu samp le) . The co nstant p sections of them ode l ODF , c om pose d o f {111 }(112) and {110} (110)texture components , a re shown in F ig . 2(a ) . The samesec t ions obta ined by the present method f rom the 15X-ray diffraction spectra are displayed in Fig. 2(b) . I t isseen tha t the ODF reproduced f rom the di f f rac t ionspec tra i s in very good agreement wi th the model one .The com par ison be tw een the m odel led {200} pole f igure(Fig . 3a) and tha t reca lcula ted f rom the der ived ODF(Fig 3b) confirms that i t is possible to obtain quantitativetexture information from the diffraction spectra. Fig.2(c) gives the constant q~ sections o f the O DF that isobta ined f rom the sam e di f f rac tion spec t ra by the c lass ic

HM. I t i s evident tha t , compared wi th the model ODF,the much lower or ienta t ion in tensi t ies a t the peak posi -t ions of main texture compon ents a re obviouslyobserved. This is due to the fact that in the classic HMno posi t iv i ty condi t ion is involved and the use o f fewerdif f rac t ion da ta genera l ly leads to a smal le r number ofdeterminable texture coeff icients.The de ta i led compar isons be tween the numer ica lresul ts o f the OD F ana lys is f rom d if f rac t ion spec t ra , bythe present method and the c lass ic HM, a re summar izedin Tab le 1. The accu racy criter ion o f recalculatedintensi ty i s g iven by the parameter R (Ruer & Baro,1977),

1 /2R ( h '~ y/ l ~ lY ) ~~ - I ~ - ~ Y ~ - / I ~ , ( y ) ] 2 ~ /h , y , ] '

x 100% . (13)

\D 0 9

b b 0b@

( a )

b b )

\

b(b )

>

l O ~i

} } ) ,) o

D l 0 ~ 1 1(( ')

F i g . 2 . T h e c o n s t a n t ~ o ( 0, 5 , 1 0 . . . ) s e c t i o n s o f t h e O D F s ( c u b i c - o r t h o r h o m b i c s y m m e t r y ) ( a ) m o d e l l e d b y t h e { 1 1 1 } ( 1 1 2 ) a n d { 1 1 0 } ( 1 1 0 )G a u s s i a n p e a k - t y p e t e x t u r e c o m p o n e n t s w i t h t h e f u l l w i d t h a t h a l f - m a x i m u m ( F W H M ) o f 1 5 ( l e v e l s : ! , 4 , 7 , 1 3 , 1 6 ) ( b ) c a l c u l a t e d b y t h ep r e se n t m e tho d ( l e ve l s : 1 , 4 , 7 , 13 , 16 ) a nd ( c ) c a l c u l a t e d by t he c l a s s i c H M ( l e ve l s : 2 , 4 , 6 , 8 , 10 ) .


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4 4 6 Q U A N T I T A T I V E T E X T U R E A N A L Y S I S F R O M X - R A Y D I F F R A C T I O N S P E C T R AT h e a v e r a g e e r r o r b e t w e e n t h e mo d e l a n d r e c a l c u l a t e dO D Fs i s d e f i n e d a s ( W r i g h t & A d a ms , 1 9 9 0 )

3 = ~ _ [ f m ( g j) _ fC ( g j )] 2 , (14)

w h e r e t h e s u m i s t a k e n o v e r a l l t h e r a n g e o f t h e E u l e rspace a n d f m ( g j ) a n d f C ( g j ) a r e t he mo d e l a n d c a l c u l a t e dor ien ta t ion dens i t ie s a t ce r ta in o r ien ta t ion g j , r e s p e c -t ive ly . The r e la t ive o r ien ta t ion dens i t ie s a t the peakpos i t ions and the vo lume f rac t ions ,4 V / V ( t n i ) o f t h e t w oma i n t e x t u r e c o mp o n e n t s a r e a l s o l i s t e d i n T a b l e 1 .Here , the vo lume f r ac t ion ,4 V / V ( o g i ) i s de f ined a s ani n t e g ra l i n t e n s i t y o f o r i e n t a t i o n d e n s i t i e s a b o u t t h e p e a kreg ion up to a f ixed or ien ta t ion d is tance co f rom thepeak pos i t ion g i (Zuo , Mul le r & Es l ing , 1993) .As show n in Table 1 , bo th the R va lue and the 6 va lue ,for the p re sen t me thod , a re ve ry sma l l . Also , the o f ten-

'RD

6 6 3 3 7L

(a)~RI)

~ ~ M a x : 6.796

(b)Fig. 3. {200} pole figure (a) mod elled and (b) recalculated from F ig.2(b). Levels 1, 2, 3, 4, 5, 6.

t a t i o n d e n s i t i e s a t t h e p e a k p o s i t i o n a n d t h e v o l u mef r a c t io n s o f th e t w o m a i n t e x t u r e c o mp o n e n t s p r o d u c e db y t h e n e w a l g o r i t h m a r e v e r y c l o s e t o th e m o d e l v a l u e s .H o w e v e r , i n s p i te o f t h e r e b e i n g a l mo s t t h e s a me R v a l u efor the c la ss ic HM, i t s ~ va lue i s obvious ly la rge r . Asc o mp a r e d w i t h t h e mo d e l O D F, t h e c l a s s i c H M p r o d u c e st h e r e l a t i v e l y l o w e r o r i e n t a t i o n d e n s i t i e s a t t h e p e a kp o s i t i o n s a n d t h e s ma l l e r v o l u me f r a c ti o n s o f th e m a i nt e x t u r e c o mp o n e n t s .

3.2. T e t r a g o n a l - o r t h o r h o m b i c s y m m e t r yW e r e f e r to t h e p o w d e r X - r a y d i f f ra c t i o n d a t a o f T i A1a l l o y w i t h L Io s t r u c tu r e (W a n g , Su n , Ch e n & H e , 1 9 9 2) .

T h e mo d e l t e x t u r e i s c o n s t r u c t e d w i t h t h e t w o p e a k -type G auss ian com pone nts , (111)[11"2] and (110) [110] .Accord ing to the geometr ic se t in Fig . 1 , 15 s imula ted' e x p e r i me n t a l ' d i f f r a c t i o n s p e c t r a , e a c h c o n s i s t i n g o fs e v e n s i n g l e o r / a n d o v e r l a p p i n g p e a k s , i .e. (001) , (110) ,( 1 1 1 ) , ( 0 0 2 ) + ( 2 0 0 ) , ( 2 0 1 ) , ( 2 0 2 ) + ( 2 2 0 ) a n d(113) + (311) , a re u sed a s the input da ta fo r the O DFana lys is .

Fig . 4 (a ) g ives the cons tan t tp sec t ions o f the mode lO D F. T h e s a m e s e c t i o n s o f t h e O D F o b t a i n e d f r o m th e1 5 X - r a y d i f f r a c t i o n s p e c t r a b y t h e p r e s e n t me t h o d a r ed i s p l a y e d i n F i g . 4 (b ) . T h e r e i s g o o d a g r e e m e n t b e t w e e nthe two O DFs. I t ind ica te s tha t i t is poss ib le to pe r fo rm aq u a n t i t a t i v e O D F a n a l y s i s w i t h a l i m i t e d n u mb e r o fd i f f r ac t ion spec t r a fo r te t r agona l ma te r ia l s . In con t ra s t ,w h e n t h e O D F i s d e t e r mi n e d d i r e c t l y f r o m t h e i n c o m-ple te po le f igure s , usua l ly more input d i f f r ac t ion da taw i t h t h e p o l a r a n g l e l a r g e r t h a n 5 0 a r e r e q u i r e d ( W a n g ,X u & L i a n g , 1 9 9 6) . T h e o b v i o u s r e d u c t i o n o f th e i n p u td i f f ra c t i o n d a t a i n t h is e x a m p l e c o me s f r o m t h e f a c t t h a tmo r e c o n s t r a i n t s , i .e. t h e q u a n t i t a t i v e r e l a t i o n s h i p samong the d i f f e ren t {h} d i f f r ac t ion p lanes , a re in t ro -d u c e d i n t o t h e t e x tu r e a n a l y s i s .F i g . 5 s h o w s t h e s i mu l a t e d a n d r e c a l c u l a t e d d i f f r a c -t i o n s p e c tr a at X = 5 0 , r / = 0 a n d a t Z = 5 0 , r / - 9 0 .A s c a n b e s e e n , g o o d a g r e e me n t b e t w e e n t h e s i mu l a t e da n d r e c a l c u l a t e d s p e c t r a w i t h t h e r a t h e r s ma l l r e s i d u a le r ror s o f the i r d i f f rac t ion in tens i t ie s fo r the s ing le a ndove r lapp ing d i f f r ac t ion peaks i s ev iden t . Th is sugges tst h a t t h e O D F ma y a l s o b e d e t e r mi n e d d i r e c t l y f r o m

t h o s e d i f f r a c t i o n s p e c t r a w i t h s o me o v e r l a p p i n g p e a k s .Fu r t h e r c o mp a r i s o n s b e t w e e n t h e n u me r i c a l r e s u l t s o ft h e O D F a n a l y s i s b y o u r n e w a l g o r i t h m a n d t h e c l a s s i cH M a r e g i v e n i n T a b l e 2 . S a t i s f a c t o ry r e s u lt s h a v e b e e no b s e r v e d f o r t h e p r e s e n t me t h o d , w h e r e a s r e l a t i v e l yl a r g e r R a n d 6 v a l u e s a n d c o n s i d e r a b l y s ma l l e r v o l u mef r a c ti o n s o f t h e ma i n t e x t u r e c o mp o n e n t s c o m p a r e d w i t ht h e mo d e l t e x t u r e a r e a s s o c i a t e d w i t h t h e c l a s s i c H M .T h e a b o v e t w o e x a mp l e s i n d i c a t e t h a t o u r a l g o r i t h mma y a l l o w t h e q u a n t i t a t i v e t e x t u r e i n f o r ma t i o n t o b eobta ined d i r ec t ly f rom the d i f f r ac t ion spec t r a . I t shouldb e m e n t i o n e d t h a t t h e a c c u r a c y o f t h e O D F a n a l y s i s is


5/6

Y . D . W A N G et al .Tab l e 1 . S umma r y o f n um e r i c a l r e su l t s o f th e OD F a n a l y si s o r t h e c u b i c - or t h o r h om b i c symme t r y

4 4 7

{ 1 11 } ( 1 1 2 ) c o m p o n e n t { 1 1 0 } ( 1 1 0 ) c o m p o n e n tza / v (%) a v / v (%)

R ( % ) a f m ax ( g ) 0 9 = 5 0 9 = 1 0 a~ = 1 5 f m ax ( g ) ~o = 5 ~ = 1 0 to = 1 5 M o d e l t e x t u r e - - 8 . 3 0 1 . 3 2 8 . 8 3 2 2 . 8 6 1 6 . 5 9 1 . 3 4 9 . 1 9 2 4 . 7 5P r e s e n t m e t h o d 7 . 6 6 0 . 3 9 8 . 6 5 1 . 3 6 8 . 7 3 2 0 . 9 3 i 7 . 4 6 1 . 3 7 8 . 9 1 2 2 . 4 0C l a s s i c H M 8 . 4 8 0 . 7 7 7 . 1 8 1 . 1 6 8 . 1 6 2 2 . 1 8 1 0 . 0 3 0 . 8 2 5 . 8 5 1 6 . 4 4

a f f e c t e d b y t h e t e x t u r e c h a r a c t e r o f m e a s u r e d s a m p l e sa n d t h e g e o m e t r y o f s e l e c t e d d i f f r a c t i o n s p e c t r a . H o w -e v e r , a s c o m p a r e d w i t h t h e c o n v e n t i o n a l p o l e - f i g u r ei n v e r s i o n m e t h o d s , t h e p r e s e n t m e t h o d m a y c o n -

} ,L b,o

(a )

> 3 b~ 0

(b )F i g . 4 . T h e c o n s t a n t t p ( 0 , 5 , 1 0 . . . ) s e c t i o n s o f t h e O D F s ( t e t r a g o n a l -

o r t h o r h o m b i c s y m m e t r y ) ( a ) m o d e l l e d b y t h e ( 1 1 1 ) [ 1 1 2 ] a n d( 1 1 0 ) [ 1 1 0 ] G a u s s i a n p e a k - t y p e t e x t u re c o m p o n e n t s w i t h t h e fu l lw i d t h a t h a l f - m a x i m u m ( F W H M ) o f 1 5 ( le v e l s : 3 , 7 , I I , 1 5 , 1 9 , 2 3 )a n d ( b ) c a l c u l a t e d b y t h e p r e s e n t m e t h o d ( l e v e l s : 2 , 4 , 1 0 , 1 4 , 1 8 , 2 2 ,2 6 ) .

s i d erab l y red u ce th e d i f f rac t i on d ata n eed ed for a p o l y-crys ta l l i n e ma ter i a l w i th k n ow n crys ta l s t ru ctu re . Th i s i sd u e to th e fac t th at th e i n t en s i ty re l a t i on sh i p s amon gd i f f eren t {h } d i f f rac t i on p l an es are e f f ec t i ve l y u sed forth e ODF an al ys i s d i rec t l y f rom th e d i f f rac t i on sp ec tra .T h u s , t h e p r e s e n t m e t h o d m a y p r o v i d e a u s e f u l m e a n sf o r t h e q u a n t i t a t i v e t e x t u r e a n a l y s i s o f c o m p l e x m a t e r i -a l s w i th over l ap p i n g d i f f rac t i on p eak s .

4 . C o n c l u d i n g r e m a r k sT h e n e w m e t h o d d e s c r i b e d h e r e p r o v i d e s a c o n v e n i e n ta n d e f fe c t i v e m e a n s o f q u a n ti ta t i v e O D F a n a l y s i s f r o mX - r a y d i f f r a c t i o n s p e c t r a f o r m a t e r i a l s w i t h a k n o w n

tr ,X r , " ,

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3 -

% = 5 0 , q = 0

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15 21

a s s u m e d. . . . . . . r e c a l c u l a t e d. . . . r e s i d u a l e r r o r s

~ +

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2 0 ( o )( a )

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t~ ,

e-

% = 5 0 , r l = 9 0 - - ~ a s s u m e dt . . . . . . . r e c a l c u la t e d,.~[ ~ . . . . . . r e s i d u a l e r r o r sI i ~ = =II + ~ ~,I I ' ~ + - - r

% / "i . . i i ~ . I .

~ 1 5 2'1 2 6 3 2 3 7 4 3 4 8 5 4 5 9 6 5 7 0 7 62 0 ( )

( b )F i g . 5 . T h e s i m u l a t e d a n d r e c a l c u l a t e d X - r a y d i f f r a c t i o n s p e c t r a o f

T i A I a l l o y w i t h th e t e x t u r e s a s s u m e d i n F i g . 4 : ( a ) X = 5 0 , r / = 0 ;( b ) X = 5 0 , r / = 9 0 . T h e u p p e r t w o l i n e s a r e t h e s i m u l a t e d a n dr e c a l c u l a t e d i n t e n s i t y c u r v e s ; t h e l o w e r l i n e i s th e d i f f e r e n c eb e t w e e n t h e m .


6/6

4 4 8 Q U A N T I T A T I V E T E X T U R E A N A L Y S I S F R O M X - R A Y D I F F R A C T IO N S P E C T R AT a b l e 2 . S u m m a r y o f n u m e r i c a l r e s u lt s o f th e O D F a n a l y s i s f o r t h e t e t ra g o n a l - o r t h o r h o m b i c s y m m e t r y

(111)[112] comp onent (110)[170] comp onentz~V/V(%) ~V/V(%)

R(%) 6 fma~ (g) to = 5 to = 10o to = 15 fma~ (g) to = 5 to = 10 to = 15Mo del texture s - - 24.90 1.30 8.56 21.03 24.90 1.31 8.56 21.02Prese nt me thod 6.86 0.51 25.70 1.32 8.03 18.31 26.28 1.35 8.43 19.00Class ic HM 11.10 2.53 17.16 0.89 6.45 18.35 12.12 0.60 4.50 13.54

c r y s ta l s t r u ct u re , e s p e c i a l l y f o r l o w - s y m m e t r y m a t e r i a l sw i t h c e r t a i n o v e r l a p p i n g d i f f r a c ti o n p e a k s. B y i n t ro d u -c i n g th e m a x i m u m - t e x t u r e - e n t r o p y c o n c e p t i n t o t h ek i n e m a t i c e q u a t i o n o f d i f f r a c t i o n i n t e n s i ty , it n e e d s o n l ya r e d u c e d n u m b e r o f d i f f r a c t i o n s p ec t r a . T h e i t e r a t i v ep r o c e d u r e p r o p o s e d i n th e p r e s e n t p a p e r e n a b l e s a c c e s st o a s ta b l e s o l u t i o n t o t h e n o n l i n e a r l e a s t - s q u a r e s p r o -b l e m s o f t h e O D F a n a l y s i s f r o m t h e d i f f r a c t io n s p e c tr a .

T h i s w o r k i s s u p p o r t e d b y t h e S t at e E d u c a t i o n C o m -m i s s i o n o f C h i n a a n d t h e M i n i s t ry o f M e t a l lu r g i c a lI n d u s t ry o f C h i n a .

Re f e r e n c e sBrousse , P. (1988) . Optimization in Mechanics: P roblem s andMethods. A mst e r dam: E l s ev i e r Sc i ence .Bunge, H. J . (1965) . Z Metallkd. 56 , 8 7 2 - 8 7 4 .Hedel , R . , Bunge, H. J . & Reck, G. (1994) . Mater. Sci. Forum,157 / 162 , 2067- 2074 .Mat thies , S. & Vinel , G. W. (1982) . Phys. Status Solidi B, 112,

1 1 1 - 1 2 0 .Roe, R. J. (1965). J. Appl. Phys. 36 , 2024- 2031 .Ruer , D. & Baro, R . (1977) . Adv. X-ray Anal. 20 , 187- 200 .Schaeben, H. (1988) . Phys. Status Solidi B, 148, 63-72.

Taylor , A. (1961) . X-ray Metallography, pp . 233- 288 . N ewY or k : John Wi l ey .V adon , A . ( 1981) . T hes i s , U n i ve r s i t y o f Me t z , F r ance .Wagner , F. , Wenk, H. R. , Es l ing, C . & Bunge, H. J . (1981) .Phys. Status Solidi A, 67 , 269- 285 .Wang, Y. D. , Sun, Z . Q. , Chen, G. L . & He, C . Z . (1992) .Trans. Nonferrous Met. Soc. Chin. 2 , 47 - 50 .Wang, Y. D. , Vadon, A. , Heizmann, J . J . & Xu, J . Z . (1996) .Scr. Mater. 35 , 905- 911 .Wang, Y. D. , Vadon, A. , Heizmann, J . J . & Zuo, L . (1996) .The Eleventh International Conference on Textures ofMateria ls ( ICOTOM 11) , edi ted by Z . D. L iang, L . Zuo &Y. Y. Chu , pp. 197 -204. Intern at ional Aca dem ic Press ,B e i j i ng , C h i na .Wang, F. , Xu, J . Z . & Liang, Z . D. (1987) . The Eighth Inter-national Conference on Textures o f Materia ls ( ICO TOM 8) ,edi ted by J . S. Kel lend & G. Got t s te in , pp. 111-114. TheMet a l l u r g i ca l Soc i e t y , San te Fe , U SA .Wang, Y. D. , Xu, J . Z . & Liang, Z . D. (1996) . TexturesMicrostruct. 26 , 103- 110 .Wenk, H. R. , Mat thies , S. & Lut terot t i , L . (1994) . Mater. Sci.Forum, 157 / 162 , 473- 480 .Wright , S. I . & Adams, B. L . (1990) . Textures Microstruct. 12,

6 5 - 7 6 .Zuo, L . , Mul ler , J . & Esl ing, C . (1993) . J. Appl. Cryst. 26,4 2 2 - 4 2 5 .

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