The work deals with the problem of preferential orientation
of crystallites in quantitafrwe X-ray analysis. A n ew method has
been worked out theoretically. and practically for the correction
of reflection intensities from \u27polycrystalline samples w ith preferentially
oriented crystallites. Equations are included for the
determination of correction parameters. Practical a pplication of
the new method is demonstrated on quantitative X-ray analysis of
calcium hydroxide. The work includes a calibration graph for the
determination of coefficient a, comparing corrected and uncorrecte
d values of analytical intensities of Ca(OH)2

A. Bezjak

I. Jelenić

English

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CR 0 AT IC A CHEMIC A ACT A 37 (1965) 255 CCA-391 545.824 :546.41-36 Original Scientific Paper Correction of Preferential Crystallite Orientation in X-ray Quantitative Analysis. Quantitative X:ray Determination of Ca (OH)2 A. Bezjak and I. Jelenic Research Department, Association of Yugoslav Cement Producers, Zagreb, Croat'ia, Yugoslavia R eceived October 15, 1965 The work deals with the problem of preferential orientation of crystallites in quantitafrwe X-ray analysis. A new method has been worked out theoretically. and practically for the correction of reflection intensities from 'polycrystalline samples with prefe-rentially oriented crystallites. Equations are included for the determination of correction parameters. Practical application of the new method is demonstrated on quantitative X-ray analysis of calcium hydroxide. The work includes a calibration graph for the determination of coefficient a, comparing corrected and uncor-rected values of analytical intensities of Ca(OH)2. Quantitative X-ray analysis has become rather important 'in the determina-tion of constituents in multiphase systems. Methods have lbeen wo·rked out for the application of quantitave X-ray analysis to special systems with the inten-tion of obtaii.ning more precise data. Regardless of the considerable number of works published on the general and special application of quantitative X-ray analysis there are still some unsolved or only partially solved problems, so sometimes this extremely important and unique method is not accurate enough. Such a crucial problem which has yet to be solved for the quarntitative X-ray analysis of the phase composition in multiphase systems is the question of preferential orientation of crystallites in the examined sample. This proiblem is more important in spectrometric analysis than in the photographic technique As today due to its considerable advantages deff.racilometric analysis is gaining ground, an answer should be found on the phenomenon of preferential orien-tation of crystallites irn quantitative X-ray analysis. Preferentlial orientatiion in the specimen leads to an increased irntensity of basal reflections and a dimi-nution of other reflections. This diminutiorn or increase depends on the degree of the preferential crystallite mientation in the specimen and as the degree is influenced by various agents it has not always the same value and so the intensity data are not reproducible. So far the problem of preferen1Jial orientation has mainly been wlved lby a special technique of sample preparation which consists in pulverizing the specimen to a size of the order < 5 ~t and in a special technique of mounting the specimen in the holder. Yet, when crystaillites have a specially expressed plane or lathlike habit and a pronounced cleavage the preferential orientation in the sample cannot be avoided nor a satisfactory reproducibility of intensity 256 A. BEZJAK AND I. JELENIC data for particular diffraction maxima achieved. This is the reason why such specimens are unsuitaible fo,r quantitative X-ray determination. This work presents theoretically and practically a method which offers a correctl!on of the intensity data for particular reflechons in a polycrystalline specimen exhibiting a preferred orientation. The calculations are made from experimental data obtaiined from the intensity of reflections, considering boith those increased and those diminished by the preferential odentation. The worked out theory has been applied in practice to quantitative X-ray analysis of calcium hydroxide where the crystallites are characterized by a hexagonal plane habit and therefore have a tendency to orient preferentially which leads to an eror in qua!Iltitative determination. Copeland and Bragg' who developed a methor for quantitative analysis of calcium hydroxide in hydrated coment, tried to avoid the preferential orientation of crystallite's by a special technique of specimen preparation, i.e. by attempting to eliminate this orientation in the sample. They checked the accuracy of single analytical data by comparing the results obtained from increased reflections to those from diminished reflections, both !brought about by preferentially rnriented crystallites in the specimen. This work describes a more general method which can find wider , applica-tions ~n practice. THEORETICAL Quantitative X-ray analysis is based on the known Alexander and Klug2 equation: Ii Is = aX; (1) where Ii intensity of the i-th reflection of the quantJitatively examined suJb-stance I8 intensity of the standard reflection a -- coefficient of proportionality X; = concentration of i-th component This equation refers to the method of ·interna•l standard. In practice, quantitative analysis by means of this equation becomes impossible if during the prepara-tion of specime the I; changes because of the preferentially oriented crystallites. However, for such preferentially oriented specimens another equation ana-logous to equation (1) can be derived: (Ip)o = a X I I s (2) In equation (2) (Ip) 0 stands for the intensity of the p-th reflection for all speci-mens demonstrating the same degree of preferred orientation. Mostly, however, in specimen prepara•tion the same degree of preferential or.ientation is not always achieved and this results in the change of intensity I' r-We can take that (3) X-RAY QUANTITATIVE ANALYSIS 257 In this relation I' v s·ignifices the intensity of a selected reflection, if the degree of preferen tial orientation in the specimen differs from the degree for which the intensity of the selected reflection · is expressed as (Ip) 0 • In equation (3) f is the factor term for reciprocal dependence of the mentioned intensities, i. e. the factor by which I '" should be murtipl1ied to dbtain (Ip) 0 • This factor f is not constant but dependent ·on the degree of the preferred orientation of crystallites. In order to solve the problem of determiining the f fac1Jo,r following considera-tions have been made. In X-ray analysis according to Bragg-Brentano principle on a flat and mounted powder sample those planes only aJre reflected which lie parallel to the sample plane. In the usual way of preparation crystallrites w ith laminated (plate-like) structures such as caloium hydroxide and clay minerals are prefe-ren1Jially oriented in such a way that their basal planes come parallel to the sample plane. As a result the ·intensity of basal reflections 1becomes .intensified whilst that of all the other reflections weakens. This consequently indicates that there is an interdependence between the intensification of basal refle-ctions and the weakening of a ll the other reflections. Therefore the intensities of all . the reflections in the specimen containing preferentially oriented crystallites ·~ '::tn be devided into two groups. The inten-sities of the first group become stronger if the degree of preferential orientation of crystallites in the specimen increases (they are mostly 1basal reflections). The inten si1Jies of these reflec1Jions are expressed by I 11 if the degree of preferen1Jial orientation of crystals in the specimen equals zero and if the degree is bigger thaJn zero they are expressed by I'w For the second group of reflections the intensities becom e less with the increased degree of preferred .orientation. They are expressed by I np if the degree of preferential orientation equals zero and by I' np 'if this degree is bigger , than zero. In the case when an x-th part of the total number of crystals has a preferred orientation then I' r = (1 - x) · IP + B xI" = (1 - x + B x ) · Iv (4) and I' np = (1 - x) · Inp (5) where x = part by weight of the specimen crystals which have oriente& pre-ferentially IP and I 11 P = intensities of r~flec1Jions olbtaiined cm a specimen with randomly oniented crystals, i.e. intensi1Jies conesponding 1Jo nonnal Debye-intensities. In a specimen with preferred orientation the n o1rmal Debye intensities are diminished for the factor (1 - x) (x being the m easurement of the degree of preferential orientation of crystals), because the part of ra~domly oriented cristals also diminished fo·r the same factor. The intensity I' P in a specimen with preferentially oriented crystals increases for the member B x i v- The coef-ficient B is the factor for converti:ng the 1inten sity of the Debye reflection IP into the inten sity of the coresponding reflection which results from a large number of preferentially oriented small monocrystals. B depends on the re-flection angle and the X-ray characteristics of the substance so it must be separately detennined for every component and ever y reflection. 258 A. BEZJAK AND I. JELENIC By eliminating x in the equaUons (4) and (5) (Appendix-I) we have: _!p__ = B + (1 - B> . _!_n_!l_ Iv Inv This linear dependence can be written in another form (Appendix-II). I' nv where q = --I' v _1- = aq + b I'P I I' Equation (7) shows the linear dependence of _ r_ upon the quotient _llll_. I' I' , p p (6) (7) I' __!1P_ is the ratio of the reflection intensities of the two different groups for the I' v I same specimen and _1'_ the ratio of reflection intensities of the first group I'P in a randomly and a preferentially oriented specimen. If the basic intensity is taken as the one with a definite degree of preferred oriientation of crystallites (Ip) 0 , then the equation (7) lby substituting can be transformed into (IP)Q... = Kb + Kaq I' v or if A = Ka and B = Kb then or where (Ip),, = Aq + B I' v (Ip)o = (Aq + B) I 'v Aq + B = f (8) (9) (10) (11) The conversion of the intensity I' v into (Ip)o is made by mea1I1s of equation (10) . F,or this conversion the constants A and B must be known and they are found by a sepamte previous determination. An analogous correction can be made for (Inv>o too. where (Inp)o = A' + B' q' I'nv (Inp)o = (A' + B' q') . I' np , 1 I' v q = - = -q I'nv (12) (13) X-RAY QUANTITATIVE ANALYS!S 259 The constants A' and B' are also obtained experimentally. E XPERIMENTAL The explained theory was applied in practice for quantitative X-ray analysis of calcium hydroxide. Calcium hydroxide was prepared by the r eaction of calcium oxide with water at the temperature of 100° C. Thus obtained precipitate of calcium hydroxide was separated by quick filtration and dried in a desicator over KOH to constant w eight. The diffraction patterns were obtained with Philips Geiger counter diffracto-meter. The radiation was CuK a. X-ray tube operated at 36 KV and 18 mA, X-ray beam was collimated by means of 1° divergent and scatter slits and a 0.2 mm receiving slit. Scanning speed was 1;,o 2 0 per minute and the chart paper speed 400 mm per hour. To determine the coefficients A and B in equation (9) and (10) a mixture of calcium carbonate and calcium hydroxide was prepared at the ra tio of 1 : 1. Calcium carbonate was used as standara. Five X-ray pictures of the prepared sample were made with a different degree of preferred orientation of crystallites. The difference in the degree of the preferred orientation was achieved by different techniques in specimen preparation. In the analysis the followin g reflections were used : for calcium hydroxide index 001 101 d 4.90 A 2.628 and for calcium carbonate the reflection belonging to d = 3.035 A. Table I shows the ratios of intensity for calcium hydroxide to standard intensity of calcium carbonate. I' p/ I s (I' 001/I caco) x 0.05 0.05 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.25 0.30 0.30 0.866 0.495 1.090 0.576 0.616 I' p/18 0.354 0.300 0.740 0.720 1.070 1.040 1.520 1.260 1.550 2.060 2.240 2.140 TABLE I I ' np/ I s (I' 101/ Icacog) 0.624 0.738 0.578 0.720 0.708 TABLE II I 'np q = I'p 0.720 1.490 0.530 1.250 1.150 I ' np/I s q = l 'nv_ l 'p 0.336 0.950 0.308 1.030 0.536 0.726 0.491 0.682 0.838 0.778 0.750 0.715 0.956 0.630 1.068 0.850 1.281 0.825 1.270 0.620 1.520 0.679 1.500 0.701 260 A. BEZJ AK AND I. JELENIC The calibration graph for the determination of coefficient a with the method of internal standard was made by means of calibration mixtures of calcium hydro-xide, the standard substance and a dilutiens . Sodium chloride was used as internal standard and white cement as dilutiens . Mixtures were prepared with 5, 10, 15, 20, 25 and 'CJO/o of calcium hydroxide and 100/o of the standard. Two samples were made from every mixture and two independent diffraction patterns were obtained. Table II Ioo1 shows the ratios of intensities --, l NaCI 1101 1101 and --I NaCI 1001 RESULTS The constants A and B were determined from . data quoted in Table I. The value of intensity I' P taken for (Ip)o is the intensity of the lowest degree of preferred or:ientation, i . e. q = 1.490. To avioid errors likely to happen because of differences in absorption and in the compactness of sample the inten-(lp)o IP 0'2 0'6 o·a ro 1'2 11. lnp Tp Fig. 1. Calibration graph to d etermine A and B of Ca(OH)z. sities were compared to the strongest intensity of calcium carbonate. For such comparisron the equation (9) takes the form (Ip)o/Is = Aq + B I'v/Is Table III gives the data for the calculation of coefficients A and B. (14) The least square method was used for the determination of the coefficents A and B; A = 0.561 and B = 0.153. Figure 1 shows the line from which coefficients A and B were determined. x 0.05 0.05 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.25 0.30 0.30 (lp)o Is l5 ro X-RAY QUANTITATIVE ANALYSIS (Ip)o/I s I' p/ I s 0.570 1.000 0.455 0.860 0.803 0.950 1.030 0.726 0.682 0.778 0.715 0.630 0.850 0.825 0.620 0.679 0.701 TABLE III TABLE IV f = Aq + B 0.686 0.730 0.561 0.535 0.589 0.555 0.507 0.631 0.615 0.501 0.535 0.547 I )( I'np q = I'o 0.720 1.490 0.530 1.250 1.150 0.354 0.300 0.740 0.720 1.070 1.040 1.520 1.260 1.550 2.060 2.240 2.140 )( )( )( )( (lp)o/I s 0.244 0.219 0.416 0.385 0.632 0.582 0.770 0.790 0.960 1.020 1.210 1.170 0'05 0'10 0'15 0'20 0'25 0'30 X C aCOHl 2 Fig. 2. Calibration graph for quantitative determination of Ca(OH),. o - corrected intensities X - uncorrected intensities 261 262 A. BEZJ AK AND I. JELENIC Equa1tfrorn (10) was used for the correction of experimentally obtaiined in-tensities when preparing the cailibration graph. Table IV cootains data for the concentration of calcium hydroxide (X) q, f, I't,/I, and (Iv)0 /I5 • Fig. 2 shows the caLibratiion graph for the quantitative determination of Ca(OH) 2 • The diagram was obtained from data of corrected intensities. Coef-ficient a is 4.0. This value is also found by the least square method. DISCUSSION As demonstrated on Figure 2 satisfactory results in quantitative X-ray analysis can lbe obtained by correcting the intensities caused through preferred orientation of crystals rin the sample. Without such correction the calibration line can be determined only with a very erase standaTd deviation. The cor-rection is specially important for quantitative analysis of carlcium hydroxide in unknown mixtures, the · error caused 1by preferred orientation of crystallites being hard to define in such a case. The presented method has another advantage, namely the possibility to analyze a sample with a high degree of preferred orientation which gives more precise daita on the intensities. This is specially importarnt when there are low concentratJions of calcium hydrmd de because preferred orientation increases its analytical 001 reflection. At the present stage the only serious disadViantage of the proposed ·method occured in case when I' nv cannot be determ~ned. When diiscussing this method it should be noted that all the oriented crystallites do not contribute to the total intensity as measured lby spectrometric analysis. This is due to the statistical orientation of crystallites oriented round the most faV'Ourable posi biorn. It can be assumed that this statistical distribution 'Of o rienta1tion is not necessary the same for all substances, but it will always be nearly identica1l for the same materfal. The influence of this statistical distribution on the intensity is included in the coefficient B so that this coefficient has a wider significance than it has been pointed out in the theoretical part. New methods are in progress to ehminate the mentJioned difficulties which may occur under certain circumstances. APPENDIX I. Two fundamental equations for correction of preferred orientation are Ip' = (1 - x + B xl Ip and Inp' = (1 - xl Inp from A.2 we get Inp' . x = 1-Inp and by substituting x in (A.ll ~ = I'np + B ( 1- I'np ) = B + (1- Bl I'np lp Inp Inp Inp I II . Multipleing equation (A.4l by _ P_ I ' p Ip I ' np 1 = B - + c1 - Bl -I ' p Inp -Ip Ip Ip = B- +Cl - Bl -I ' p I ' p Inp (A.ll (A.2l (A.3) (A.4l (A.5l X-RAY QUANTITATIVE ANALYSIS and dividing (A.5) by B 1 f1 or when we obtain and finally Il;p + ..!_(.! B _IIp P np B - 1 . .!r._ = a and B l np _!g_ = a I'np + b I' P I ' P ~ = aq + b I' p I' np where --- = q I' p 1 B III. Multipleing equation (9) (Theoretical part) or when (Ip),, = k (I 11 p) 0 we have 1 . by -q 1 1 (Ip) c, - = A + B -I'p q q (Ip)o I ' 1 _ P_ = A -1- B -l 'p I'np q (Ip)o = A + B~ I ' np q (I11p)o A B 1 - - = - + -I'np k k q B 1 = b A k = B' and -- = A' are constants. If - = q' k q we finally get (Inp )o = A' q' + B' I 'nn REFERENCES 1. L . E. Cope 1 and and R. H . Br a g g, Anal. Chem. 30 (1958) 196. 263 (A.6) (A .7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.14) (A.15) 2. H. P . K 1 u g and L. E. A 1 ex and er, The X-ray Diffraction Procedures, John Wiley, New York 1954. p. 412. IZVOD Korekcija s obzirom na preferiranu orijentaciju kristalita u rendgenskoj kvantitativnoj analizi. Rendgenska kvantitativna analiza Ca(OH)2 A. Bezjak i I. Jelenic Rad obraduje problem preferirane orijentacije kristaliteta u kvantitativnoj rendgenskoj analizi. Teoretski i prakticki je razradena nova metoda za korekciju intenziteta refleksa s polikristalinicnih uzoraka koj i sadr:le preferirarno orijentirane 264 A. BEZJAK AND I. JELENIC kristalite . Izvedene su jednadzbe za odredivanje korekcionog faktora. Prakticna pri-mjena nove m etode prikazana je na kvantitativnoj rendgenskoj analizi kalcijevog hidroksida. Dobiven je bazdarni dijagram za odredivanje koeficijenta a uz usporedbu korigiranih i nekorigiranih vrijednosti analitickog intenziteta. ISTRAZIV ACK! ODJEL, UDRUZENJA PROIZVOE>ACA CEMENTA ZAGREB Primljeno 15. studenoga 1965. 

Correction of Preferential Crystallite Orientation in X-ray Quantitative Analysis. Quantitative X:ray Determination of Ca (OH)2

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