CALCULATED POWDER X-RAY DIFFRACTION LINE PROFILES VIA ABSORPTION

Transcription

1 16 17 CALCULATED POWDER X-RAY DFFRACTON LNE PROFLES VA ABSORPTON Keji Liu nd Heifen Chen School of Mteril Science nd Engineering, Shnghi nstitute of Technology, Shnghi, Chin 2233 ABSTRACT We hve clculted powder X-ry diffrction (PXRD) line profiles bsed on the bsorption of X-rys in pssing through mterils. The results showed tht the line profile function depends on the bsorption coefficient of the smple. The observed luminum PXRD ptterns re in excellent greement with the clculted line profile functions derived from bsorption. This work lso demonstrted tht both Cuchy nd Guss profile fitting functions re the pproximtion of the clculted line profile obtined in this pper. NTRODUCTON PXRD pttern provides lot of informtion of crystlline smple. t is well known tht, if the crystl structure is known, the interplnr spcing, d, of crystlline smple cn be clculted vi the Brgg s eqution nd the pek intensity cn be clculted by the intensity theory. However, the line profile, or the line shpe, cnnot be predicted by ny theory even for n idel cse. This is probbly due to the XRD theory tht the line profile of n idel smple, collected on n idel instrument, does not hve defined shpe, but the Dirc delt function, i.e., the intensity line would hs n infinitely smll ngle distribution nd very strong intensity t exct Brgg ngle. However, so fr, there is no experimentl evidence showing the delt-function line profile. n other words, ll observed diffrction peks hve the non-zero width; usully, for most common XRD instruments, n observed pek hs the width of.1 degree in full width t hlf mximum (FWHM). Up to our knowledge, the observed delt-function like pek hs not been reported. About one hundred yers go, Scherrer (Scherrer, 1918) discussed line profile medited by the finite size of the powder smple, nd the fmous Scherrer formul provides the reltionship between the observed FWHM nd the crystllite size of the crystlline smple. Shortly fter Scherrer s work, Lue (Lue, 1926) considered the cse of crystllites hving the form of prllelepiped nd introduced the generl form of the integrl bredth, nd lso used Cuchy nd Gussin functions to model line profiles for the first time.

2 This document ws presented t the Denver X-ry Conference (DXC) on Applictions of X-ry Anlysis. Sponsored by the nterntionl Centre for Diffrction Dt (CDD). This document is provided by CDD in coopertion with the uthors nd presenters of the DXC for the express purpose of educting the scientific community. All copyrights for the document re retined by CDD. Usge is restricted for the purposes of eduction nd scientific reserch. DXC Website CDD Website -

3 17 18 n the rel cses, the PXRD line profile will be brodened by the imperfect experiment nd smple conditions. These conditions cn be the imperfection of crystlline smple, such s its smll size (less thn 25 microns, for exmple) nd the disloctions in the crystls; the smple residul stress hs been shown to cuse the brodened s well (Guinier, 1963). t is well known tht the imperfectly designed diffrctometer is nother source for PXRD line brodening. However, in principl, ll imperfections cn be eliminted; therefore, the delt function PXRD line profile would hd been observed. We propose tht, in relity, some hidden effects my prevent ny PXRD line profile forms in the shpe of delt function even in the idel cse, nd these hidden effects led the formtion of the chrcteristic line profile of rel PXRD peks. We define this profile the intrinsic profile in this pper. We ssume, in this work, tht the PXRD pek lwys hs its prticulr profile, cused by the bsorption of X-rys in pssing through the mterils. As ll mterils bsorb the X-rys, the bsorption generted profile is of course the intrinsic profile. To the uthor s knowledge, there is no such study correlting PXRD line profile to X-ry bsorption. Some works on line profile nlysis (Mittemeijer, 24) hs been done. However, they discussed the brodening of the line nd how to fitting the profile, not relted to the intrinsic profile s we proposed in this pper. n this pper, we will clculte the non-trivil PXRD line profile which is generted by the bsorption of X-rys, nd then compre the clculted line profile with tht of observed. The devition of clculted nd observed PXRD line profiles is estimted to demonstrte the greement lso. At the end of this pper, some comprisons between our clculted line profile nd some common profile fitting functions, such s Cuchy fitting nd Gussin function fitting, will be discussed. THE NTENSTY CALCULATON FOR THE ABSORBED XRD BEAM The intensity of diffrction line cn be obtined by superposing of the mplitude of the wve bems. Assume smple hs N+1 crystlline plnes with d s its interplnr spcing, which reflects the incidence X-ry, s shown in Fig.1. By setting A n s the reflecting mplitude of the X-ry by the n th plne, which my be the function of the bsorption coefficient. Letting l n be the pth difference between two close bems, nd exp(i n ) be the reflecting phse fctor, where is the phse ngle, which cn be obtined by l s where is the wve length. = 2 l/ = 2 2d( sin)/, The totl reflecting mplitude of reflecting bem, denoted by A Tot, is the superposition of ll wve mplitudes, thus

4 18 19 A Tot N n A n i n exp, (1) Fig.1. Schemtic digrm of X-ry reflection by crystlline plnes nd the diffrcting intensity is = (A Tot ) * A Tot = A Tot 2. (2) By considering tht the bsorption hppens in the medi where the bem pssing through, we hve = exp(- s) (Guinier, 1963), where s is the minimum unit of the bem length pssing from the medi, i.e., from zeroth plne to the first plne, nd then reflecting from first plne to the zeroth plne (see Fig. 1), we thus obtin A n exp f ( s) n, (3) where is the reflected intensity, nd f(s) is function of s nd to be determined. We cn clculte s (see Fig.1) by d s s =2(d/sin). Note tht s is of the order of d, i.e., ~1-1 m (or 1-8 cm), except for the smll ngle ( <<.1rd) diffrction. t cn be seen from Eq.(1) tht the series is geometric progression with the proportionl fctor exp(-f( s)) exp(i), thus, A Tot f ( s) N N 1 e exp[ in] A exp i A n n n. f ( s) 1 e exp[ i] We therefore obtin the diffrction intensity with the bsorption considered 2 f ( s) N f ( s) N * 1 e 2e cos( N) Tot Tot 2 f ( s) f ( s) A A 1 e 2e cos( ) When the smple is very big, i.e., N, Eq.(4) cn be simplified s. (4)

5 e 2e cos( ) 2 f ( s) f ( s) (for N>>1). (5) The simplest cse of Eq.(5) is tht f( s=, i.e., the intensity is independent of the bsorption. n this cse the intensity is just the distribution function of diffrcting grting, sin N / (for f( s= sin / 2 This formul implies tht the intensity of lrge perfect crystl (N ) will be delt function. Therefore, the distribution function of the diffrcting grting is just the non-bsorption cse of Eq.(4). EXPERMENTAL DEMONSTRATON AND APPLCATON n order to obtin the line profile concretely, we firstly determine the form of the function f(s) s following. Assuming s<<1, nd by expnding the function exp(f(s)) to Tylor series t the Brgg ngle (i.e., in Eq.(5)) i.e., exp( f(s)) = 1 f(s) + ( f(s)) 2 /2! + O(s), one hs f ( s) f ( s) 2 f ( s) / 2! ( f ( s)) 2 2 On the other hnd, the X-ry bsorption theory of PXRD shows tht the diffrction intensity is inversely proportionl to the bsorption coefficient, i.e., by compring Eq.(6) nd (7), the following reltionship is obtined, (6) 1, (7) f(s)=(s) 1/2. (8) n order to estimte the order of s, one cn consider copper (Cu) smple with Cu K rdition. The mssive bsorption coefficient of Cu is m = 52.7 (cm 2 /g), nd the density is 8.9(g/cm 3 ). As s is the order of 1-1 m, we hve s (= m s) is of the order of 1-6, so tht the higher order of s could be omitted. We therefore obtin the pek of the line profile t the exctly Brgg s ngle (= B ): mx 2 s 1 e. s As the quntity of s is of the order of 1-6,we cn estimte tht mx is of the order of 1 6. We finlly obtin,

6 sn s N * 1 e 2e cos( N) ATot ATot. (9) 2 s s 1 e 2e cos( ) For N, we hve the pproximtion Eq.(9) is the min results of this work. mx. (1) s 2 s 2 s 4e sin ( ) 1 4e sin ( ) / s 2 2 The experimentl verifiction for the XRD line profile of Eq.(9) ws crried out with of nneled Aluminum (Al) powder smple. The selected reflection plne is (2) with m = 48.7 (cm 2 /g), nd the density is = (g/cm 3 ). The rdition wvelengths re doublet K lines, 1 = Å nd 2 = Å,with the intensity rtio 1:.45. The diffrctometer used is the Rigku D/mx 22PC. The experimentl curve is not smoothed. One cn see in Fig.2, tht the difference between the clculted intensity nd the experimentl intensity ( cl - exp ) is less thn 1%. Therefore, the clculted result is in good greement with the experimentl results. We therefore conclude tht the line profile is minly shped by the X-ry bsorption, rther thn brodening cused by instrumentl or by the imperfection of smple. Fig.2. X-ry diffrction line-profiles of the experiment (red) nd the theoreticl (blck) one for Aluminum powder (2) plne, with Copper (Cu) rdition of K1 nd K2. n order to compre our results with the other line profile fittings, we expnd Eq(1) in the vicinity ofè B. For <<1,one hs sin( B +) sin B + cos B, such tht Eq.(1)

7 21 22 cn be re-written s mx, (11) where = 4(2 d/) 2 exp(-( s) 1/2 ) cos 2 ( ). By compring Eq.(11) with the Cuchy fitting function, mx. 2 1 C 2 We cn see tht the Cuchy fitting function is just n pproximtion of Eq.(1) with <<1. Moreover, we cn lso theoreticlly obtin tht the fitting coefficient C =. 1 8 () theoriticl result (b) Cuchy fitting (c) Gusin fitting Fig.3. the line profile for Al (2) with K 2 /K 1 =.45, () the clculted curve by Eq.(9); (b) the Cuchy fitting function with C = 1.56 nd (c) the Gussin fitting with = Another fitting function, the Gussin one, cn lso be obtined by the pproximtion of Eq.(9). With the Tylor expnsion exp ( u 2 ) = 1 u 2 + O(u 4 ), Eq.(11) cn be written s the Gussin fitting function = mx exp( 2) 2 ). Therefore, both the Cuchy fitting function nd the Gussin function re included in our clculted line profile. Although the difference between our clculted results nd the Cuchy fitting one is not disprte by Fig.3, the physicl mening is totlly different. Eq.(9) comes from the theoreticl derivtion which depends on the bsorption of the smple, while the Cuchy fitting is only fitting nd the coefficient is chosen to fit the curve which hs no physicl mening t ll. REFERENCES Guinier, A., (1963). X-Ry Diffrction in Crystls, (W. H. Freemn & Co., New York). Mittemeijer, E. J. nd Scrdi, P., (24). Diffrction nlysis of the microstructure of mterils, (Springer-Verlg, Berlin)

8 22 23 Lue, von, M.,(1926). Z. Krystllogr. 64, 115. Scherrer, P., (1918). Nchr. Ges. Wiss. Göttingen. 26, 98.