BIJDRAGEN TOT; DE THEORIE EN PRAKTIJK
VAN RONTGENOGRAFISCHE QUANTITATIEVE
BEPALINGEN MET DE
POEDERDIFFRACTIE-METHODE
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECH-NISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE WAARNEMEND RECTOR MAGNIFICUS, J. M. TIENSTRA, HOOGLERAAR IN DE AFDELING DER WEG- EN WATERBOUWKUNDE, VOOR EEN COM-MISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG, 14 FEBRUARI 1951, DES NAMIDDAGS
. ' , •. TE 4 UUR • •
• DOOR
PIETER MAARTEN DE WOLFF
NATUURKUNDIG INGENIEUR, - • GEBOREN TE BANDOENG
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
Dit proefschrift is goedgekeurd door de promotor Prof. Dr H. B. DORGELO
I
I
Aan myn vrouw
In latter years, the use of X-ray diffraction technique for quantitative determinations has become more and more frequent. The principle is simply this: the intensity of a diffraction ring from the material under examination in the unknown sample, is compared with the corresponding intensity from a known amount of it in a second, synthetic sample. Their ratio, if corrected for the difference in the two exposures and effective sample volumes and in the absorption by the specimens, is equal to the ratio of known and desired concentration.
The accuracy of such determinations is not comparable with t h a t obtained by chemical analysis. Moreover the field of application is limited by the condition t h a t the constituent under examination should have a well-defined structure, and t h a t it be neither too coarse nor too fine-grained. On the other hand, the X-ray method is unequalled in t h a t it reveals structural differences between chemically identical materials, thus allowing the determination of each separate modification or compound. The deter-mination of quartz in mixtures containing silicates, for instance, puts the chemist before the difficult problem to discern quartz from chemically bound silica; but it is easily solved by X-ray diffraction, even when chemical methods are completely ruled out by the presence of other crystalline or amorphous forms of SiO^. Generally speaking the X-ray method provides a unique tool for the analysis of mixtures containing different modifications, hydrates or ionic compounds of the same sub-stance.
For this reason a systematic investigation of various conditions influ-encing the accuracy of the results or the efficiency of the method seems to be worth while. Such an investigation will be presented in the following two chapters of this thesis. The first deals with methods, while the second is consecrated especially to the systematical error due to absorption in the powder particles. Another error, caused by irreproducible impurities and structure defects in the relevant constituent, probably occurs more often. However, this source of difficulties will be mentioned only inci-dentally. I t lacks a theoretical background allowing quantitative treat-ment and must be considered for each case separately.
In the third chapter a photographic technique is described, based on the conclusions of the preceding ones. The fourth and last chapter contains a discussion of various experimental details, as well as a des-cription of some preliminary experiments on particle absorption.
C H A P T E R I PRINCIPLES § 1. Theory of powder diffraction
In this chapter we will give a survey of principles of quantitative deter-mination and a discussion of their application in previous investigations. As a base for the discussion of these and other items, a schematical analysis of the intensity of a diffraction line will be presented first.
Properly speaking, the term "intensity" does not apply to the result of a photographic measurement. In fact, this result possesses the dimension of an energy. Both a real intensity measurement made with an ionization chamber or a GM-counter, and a photographical. energy measurement are, however, meaningless without a reference intensity, c.q. energy, determined by the same measuring device, owing to the fact t h a t otherwise, it would be impossible to eliminate the primary intensity, c.q. energy.
Consequently, the only significant quantity to be derived from either way of measurement is a ratio of two readings. Hence we impose no limit on the scope of the discussion when considering intensities onty. Finally, though our derivation relates to crystalline powders (or micro-crystalline solids) this does not mean t h a t quantitative determinations must be confined to such. If a single amorphous constituent occurs, it may be determined as well and on the same principles, as shown by the very accurate determinations of amorphous rubber mentioned in § 3. Suppose, then, a powder sample is placed in a monochromatic X-ray beam, the intensity of which was a function IQ{X, y, z) of coordinates
X, y and z before introduction of the sample.
We shall evaluate first the integrated intensity 7™ of a diffraction ring jdelded by the m-th constituent of the sample. The integration is to be performed along a circle of radius R, situated in a plane through the direct beam, and with its centre in the specimen.
The theoretical estimation of Im is essentially a statistical problem, since the actual value of this quantity depends on the fortuitous arrangement of powder particles in the sample.
As always in statistics, an imaginary set of experiments must be found, resembling as closely as possible an actual series of repeated experiments, and still allowing computation of the average value of the result. To this end we imagine an infinite set of /m-measurements which are identical in every respect, except t h a t the sample is thoroughly mixed every time in between, in such a way t h a t the centre of each crystal particle occurs with equal frequency in any two equal volumes within the sample
boun-dary. The distribution of orientation is also supposed to be homogeneous. The ensuing "expected value" will be called 7 ^ .
Then we may write Im as the average sum of the contributions of all m-crystalUtes. Reversal of the two operations is allowed, so t h a t
^ m 2-, 'mi 2^ 'mj> V-*-^
J 3
where rmj is the contribution of the j-th. m-crystallite.
Now it is a well-established fact, t h a t the contribution of any particle is proportional to the intensity incident upon it. This intensity is only a fraction of ZQ (^,. 2/,> ^j) because of absorption of the direct beam in the sample. The scattered rays are also absorbed before leaving the sample. Combining both reductions to a factor Amu we write
^m^^ 2. 9.mi A^j J Q (Xj, y^, Z-) , ( 2 )
i
where qmi is the scattered intensity, per unit incident intensity, of Cm; (the ^-th m-crystallite), disregarding the absorption Ami by other particles. The averaging of (2) may be performed in three stages:
a. Averaging over all positions and orientations of the other particles,
t h a t of Cm,- remaining constant. Then only Ami varies, and the other factors may be left out from the operation. The resulting mean value will in general depend on the dimensions and on the orientation of Cmj. I n fact, Ami is related to the p a t h of incident and scattered rays occupied by other particles, which is clearly in negative correlation with the length of p a t h occupied by C»,- itself.
For the present purpose this correlation will be disregarded alltogether. We shall p u t
M „ , - = e x p ( - ^ Z ) , (3) where /x is the mean absorption coefficient of the sample as a whole, and
I the total length which is occupied by the sample of the path of primary
and scattered rays through the centre of Cmj-.
I n chapter I I a theoretical investigation will be presented, from which it follows t h a t this approximation is justified only if the absorption by particles of any constituent is negligible. (This condition does not neces-sarily exclude the presence of extinction. I n fact, the proportion of par-ticles in which extinction occurs is quite negligible in a powder without preferential orientation).
However, postponing the discussion of absorbing particles, we assume the validity of (3) and pass on to the second stage:
b. Averaging over all positions of C™,-, its orientation being kept
con-stant. Now /Q *1SO varies; but as a function of the coordinates only, just as "^Ami. By virtue of our suppositions
I I
the integration being carried out over the volume V of the sample. c. Lastly, the orientation of Cm» is varied, which affects only g^,-. If the crystallites consisting of component m are of the ideal mosaic type, the following relation holds (see, for example, COMPTON and ALLISON, X-rays in Theory and Experiment, p . 415):
''l^i=1mVmilR, (5)
where Vmi is the volume of C™,-, and g™ a function of Bragg angle, wave length and crystal structure parameters. The case of extinction occurring in the m-particles must be excluded for reasons to be given in I I § 2. Inserting (4) and (5) in (2) we obtain:
J„=qmCmKm ; Km= (lIB) ! h dV BXl^ {-fll), (6)
V
with Cm = 2 ^m)7 V for the volume concentration of the m-th component
)•
in the sample.
Now in most cases the maximum value im of the intensity in the diffraction ring is measured instead of the integrated intensity. The two quantities are coimected by means of the integral breadth b^, defined by
-'m = bm im- ( ' )
Unless an appreciable hne broadening of physical origin (e.g. colloidal size of the crystals) occurs, the integral breadth may be supposed to depend only on the geometry of camera and sample and on the absorption coefficient of the latter. The same obtains for Km in (6).
Combining again all factors of this kind for the case of peak values being measured, we write:
im=Cm<lmK I K= iVK^) Ï h^V Q^^ {—nl). (8)
If both geometrical and physical line broadening are of the same order of magnitude, the use of peak values is not recommendable because a strict separation of geometrical influences (combined in km) and those characteristic for m (g^) is necessary in most of the cases to be discussed. If, on the other hand, bm is chiefly of physical origin, there is no objection against peak values. In fact, successful measurements have been made on amorphous materials using the intensity belonging to a fixed value of Ö, which is equivalent to peak intensity measurement (§ 3).
With regard to quantitative analysis (determination of Cm), it follows from (6) c.q. (8) t h a t at least one additional exposure must be made in order to eliminate the unknown qmkm- Evidently the second sample should have a known m-concentration. The corresponding peak intensity is:
12
SO t h a t
''m ^ ^m *m "'m/^m "'m ' ( l ^ ) accents indicating quantities related t o the second exposure.
§ 2. Method of direct comparison
It has been mentioned above t h a t any measurement of i is
meaning-less without a reference measurement. Previously published quantitative determinations may be classified in three groups according to the kind of reference intensity which has been chosen in each case.
This choice has been governed, of course, by the wish to obtain in the simplest way an accurately known value of k'mjkm. The object of the discussion in the next three §§ consists chiefly in showing how far the different methods have succeeded in doing so.
The first group is characterised by the use of i and i' as a reference for each other; hence the title of this §. Though apparently simple, this procedure requires the greatest care owing to the difficulty of eliminating
k'jk in (10).
NAVIAS [1925], ZwETSCH and STUMPEN [1929], and G R E B E [1930] have used it in various forms. Equality of IQ and /Ó they tried to obtain by taking simultaneous exposures, Navias with a double camera, and the others by using both windows of the same X-ray tube. The former arrangement is essentially better, as it yields a double pattern on one film, thus ehminating processing conditions. Still it suffers severely from unavoidable misadjustments. The absorption in the sample is either disre-garded or eliminated by extrapolation of the results towards vanishing sample thickness.
Recently, the method has been recommended by WILCHINSKY [1947] for use with the GM-counter spectrometer designed by Norelco. This unit contains a stabilized X-ray tube, on which the slit system is fastened, so t h a t a constant IQ is indeed ensured. The absorption factor, however, remains difficult to account for. The method suggested by Wilchinsky (calculation of A based on an absorption coefficient which must be deter-mined by elemental analysis of each sample), though correct and simple enough as regards the calculation for this sample form (case a of § 4), cannot be called an elegant one; at least not for general purposes. § 3 . Use of the direct beam as a reference
In this group, the primary beam is most often measured after it has
traversed the specimen. Before reaching the film, it passes through a fixed foil whereby it is reduced, either by absorption or by diffraction, to t h e same order of magnitude as the simultaneously measured diffraction intensity im or i'm.
13 in this way, because the primary beam is not absorbed in the specimen to the same extent as the diffracted rays. The only exception is the case of a sample in the form of a thin plane parallel slab, the normal of which bisects the angle between direct and diffracted rays. If the direct beam is measured as the peak intensity ir (the suffix r will be used for all quan-tities related to the reference intensity) of a spot or ring with an "integral surface" ST (being the ratio between integrated and peak intensities),
i.,= (IjSf) ilodO exp {—[xl)— (cos ö/te,) ƒ ZQC^F exp (— /utjcoad)
0 V
where 0 is the cross section of the direct beam, and t the sample's thick-ness. Now for this special position and form of the specimen, clearly the latter integral is equal to the one, occurring in the corresponding equation
for im, so t h a t ,
imlir=S,tCmqmlKIi^O^^
If the sjTithetic sample yields diffraction Unes of the same breadth (6m = 6m), a n d if Sr= s',
For all other sample forms and positions, the absorption coefficients enter in the final formula.
NAHMIAS [1932] and AGAFONOVA [1937] apphed a weighted quantity of
the substance under examination on an Al-wire, using the Al-inter-ferences as a reference. The absorption must be neghgible, since the ideal case described above is far from being reahzed, the reference intensity being absorbed twice in the specimen.
Determinations of this variety have also been made by F I E L D [1940] and GoppEL [1946] on rubber. They have measured the degree of crystalU-nity on stretching or on freezing by careful determination of the amor-phous fraction, which, in contrast to the crystalline one, is also obtainable in pure form. This application is particularly interesting since the internal standard method (to be discussed hereafter) could not be used for special reasons, thus leaving the present way of measurement as the only suitable one. Both authors used a flat sample normal to the direct beam. For the small values of Ö which are of interest here, the ideal situation is thus reaUzed to a considerable extent. The degree of approximation obtained by using (11) has been examined in detail by Goppel, who pointed out also some severe mistakes in Field's manner of accounting for absorption. Reduction of the primary intensity was obtained with a lead foil b y Field, thus allowing the determination of white radiation only. I n Goppel's camera a standard sample is placed at a short distance from the fllm, yielding small diffraction rings of a conveniently measurable intensity 'in the centre of the pattern. In this way very satisfactory results have
14
Another way of using the direct beam is to let it be diffracted by a standard sample before it hits the sample. Even as in the former case, there exists an exceptional sample arrangement in which several factors of k'm and km cancel out in their quotient. Only this time the remaining factors are the respective absorption coefficients of unknown and synthetic sample, taking the place of t and t' in (11).
This ideal case consists in back reflection from the flat surface of a quasi-inflnitely thick sample. I t has been realized by SEKITO [1931] who measured
the austenite content of steel by covering it with a thin gold layer and using the gold reflections for reference. I n these investigations, equality of the absorption coefficients certainly obtains. In general, however, the same difficulty would be encountered as in the direct comparison method of Wilchinsky (§ 2).
§ 4. Internal standard
Both unknown and synthetic sample are mixed with a known amount of a suitable standard substance, the interferences of which are used for reference. The desired concentration can be whoUy expressed in known concentrations and intensities in two "ideal" cases, almost identical with those of the two methods described in the previous §; they are illustrated in fig. 1.
Fig. 1. The two cases of internal standard technique in -which the ratio bet-ween the intensities diffracted by the constituent under investigation {m) and the standard . substance (r) is independent of the absorption coefSoient of the mixture.
a. Back reflection from the flat surface of a sample of quasi infinite
thickness. If /x and ju' represent the absorption coefficients of unknown and synthetic sample, the following equation is easily demonstrated:
ƒ /o dV exp ( - /xlm) = EJfi [I + sin a/sin (26m-a)] (12)
V
where EQ is the total incident energy per second, and a the angle of inci-dence. Corresponding equations obtain for the reference diffraction hne (with öj. instead of 0^) and for the synthetic sample (with ju' instead of ^M).
15
I t follows t h a t
provided t h a t
bmK=b,lb[ (14)
If this is not the case, eq. (13) still holds true for the integrated intensities. 6. Transmitted scattering from samples in the form of a thin slab, the normal of which bisects the angle between rays diffracted in the refe-rence- and in the m-diffraction line respectively. Then r- and m-rays diffracted in a given point evidently traverse the sample over the same length. Eq. (13) obtains under the same condition.
The case a. occurs in cameras of the Bragg-Brentano type as well as in Seemann-Bohlin cameras. Recently, R E D M O N D S [1947] used the dis-position a. with a Norelco spectrometer for the study of tungsten alloys.
BKENTANO [1949] also describes some applications.
Other apphcations of the internal standard method have been made with wedge-formed specimens extending halfway in the direct beam a t a grazing angle of incidence, a device which shows no clear advantages for this purpose. In this way, CLARK and R E Y N O L D S [1936] have obtained important results on the quartz content of mine dusts and lungs of silicosis victims. I n another paper, CLARK, K A Y E and PARKS [1946] describe determinations of 2.4 dinitrophenylhydrazones with cylindrical samples. This specimen form makes the ehmination of absorption effects equally doubtful.
GROSS and MARTIN [1946] determined various minerals in mixtures, with NaCl as a standard substance. They used wedgeformed samples of the kind described above. An ingenious measuring principle allowed them, however, t o account for this non-ideal sample form as well as for the condition (14) eventually not being exactly fulfilled.
In fact, they obtained a third "ideal" variety of internal standard deter-minations :
c. Interpolation method of Gross and Martin.
Their reasoning, when reduced to its simplest form, comes down to t h i s : The quantity km defined by (9) is a smooth function of the glancing angle of the diffraction line to which it apphes. The sample enters in km only with its overall dimensions and absorption coefficient. Hence the proper notation is: k{Qm), and correspondingly: k'{dm), k{dr), k'(dr)- The same obtains for the quotient k'jk, a smooth function of 9, of which all values corresponding to r-diffraction rings are known:
(k'lk)e=c,i'Ki,. (16)
l é
(k'lk)Q^ may be read as the ordinate for 6 = dm- Inserting the result in
(10a) we obtain the following symboUcal expression for c»:
- ••• --^ •' ; • cm=c'm^^\S-] ::-• '•'.'••• in)
*m Cr L ^ r J f m
Because k{d) and k'{d) are both arbitrary, the exposures of unknown and synthetic sample may be performed with entirely different cameras, or even with different wave lengths, if anomalous absorption is avoided; Ö should then be replaced by sin Ö/A.
The cases a. and 6. may be interpreted as special arrangements yielding the same value of k'jk for dm and dr, in the former case even a value inde-pendent of 9.
Of the case 6., which may be reahzed in a Guinier camera, the work reported by F A I V R E [1947] contains some examples regarding the CaCOg content of minerals. The author, when performing a number of unpu-blished analyses of quartz in mine dusts, has arrived independently a t exactly the same technique as the one described by Faivre. So far as typical properties of the Guinier camera are concerned, the reader is referred to Ch. I l l ; but there is also a more general feature to be mentioned. In fact, this technique differs from the general procedure in t h a t it consists of two determinations ' • • : • ' ' • •
a. a preliminary one, yielding a first value of Cm with a precision of,
say, 5 %, . . • . ,
.-6. a definitive measurement using a mixture which according to the result of a. should render the intensities of unknown and standard dif-fraction line equal. Actually one obtains easily a ratio between 0.9 and 1.1. This circumstance greatly facilitates a precise comparison, since it eliminates various film and photometer characteristics.
Faivre mentions a precision of better t h a n 1 %, obtained in this way in a favorable case. The author has sometimes reached 1 %. He would like to point out a variant of the method, consisting in the use of different standard substances for the first and the second sample. In this way the absence of stray diffraction hnes coinciding with the definitive standard line can be ascertained without a third exposure.
In the final formulas (13) or (17) of the internal standard method, the diverse concentrations occur in a double ratio allowing them to be replaced by weight concentrations. In this form they are, of course, much more convenient for use.
Finally, it may be mentioned t h a t the characteristic constant g^ can be measured most effectively with an internal standard for which it is known. Only a single exposure of a sample with known Cm and c, is needed.
B R I N D L E Y [1936] has performed measurements of this kind with a view to the determination of the structure factor included in qm- To this end he used the disposition a. -.' ,- •_ r • ; • '> -^ '.; > ;
ï 7 The interpolation method c. is also easily adapted t o this purpose, giving t h e general result
<ir.=i.^m,^ . (18)
§ 5. Conclusions
I n t h e previous §§ we have encountered two sample arrangements pos-sessing "ideal" properties with regard t o one or more of the methods discussed. A slight difference occurs between the definition of the trans-mitted-scattering-arrangement as given in § 3 and t h a t of arrangement b. in § 4. I t is not essential, however, since the direct beam may be regarded also as a reference line.
The following table summarizes t h e properties of each method when used in these "ideal" conditions, t h a t is, in such a way t h a t as many as possible of the irrelevant circumstances (primary intensity, dimensions of camera and sample, absorption coefficient of the latter) cancel in t h e final formula for Cm- I t is supposed t h a t t h e measured intensities are peak values.
Method
Direct comparison Primary beam used for reference before
After hitting the
sample Internal standard Sample to be used in ar-rangement a a b a b arbitrary Conditions bm ^^ b m> ^0^-^ 0 /^'//j. k n o w n bm/Sr = b'm/s'r, /l'//l kno-wn bm/Sr^b'mls'r, t'jt known j hmlbr = Vmlb'r requires measurement of various r-lines and
interpolation Recommendable for solid samples (e.g. alloys) solid samples light materials not allowing admix-ture of a standard
powders
a = reflection from t h e flat surface of a quasi-infinitely thick specimen. b = transmission of diffracted r- and wi-rays through a thin slab making equal
angles -with both.
The indications in the last column are based on the following consider-ations.
Generally the internal standard method, used in either of the three "ideal" ways, involves b y far the least trouble in obtaining accurate results. I n fact, the preparation of mixtures with a standard can be performed much
i8 . .
sooner and more easily t h a n either t- or ^-measurements of the same accuracy. Also, it is the only method yielding weight concentrations directly, as well as t h e only one allowing correct results to be obtained from peak intensity measurements (viz. by the interpolation method). •There are cases, however, in which the use of an internal standard is excluded, because the unknown must for some reason or other be examined as a solid sample. This occurred, for instance, in the investigation of rubber mentioned in § 3; it might occur also when analysing aUoys, t h e tendency of which to develop lattice distortions on filing is well known. I n situations like these, t h e next best procedure consists generally in the use of the primary beam as a reference. If, as in t h e former one of the cases mentioned, the unknown is little absorbing, a transmission type (b) of sample is to be preferred because a thick sample of type (a) would give rise t o broad lines. For heavily absorbing solids, t h e latter type might be preferred on account of t h e difficulty of preparing suffi-ciently thin slabs.
Finally, absorbing solids may be analysed by direct comparison if t h e apparatus allows it (as in the determinations with a Norelco unit, cited a t t h e end of § 2).
R E F E R E N C E S • BBENTANO, J . C. W., J . Appl. Phys. 20 (1949), 1215.
AGAFONOVA, T . N . , C . R . Moskou (N.S.) 16 (1937), 367. BRINDLEY, G . W . , Phil. Mag. (7) 21 (1936), 778.
CLAEK, G . L . , W . I. K A Y E and T. D. PARKS, Ind. Eng. Chem. An. Ed. 18 (1946), 310.
CLARK, G . L . , and D. H . REYNOLDS, Ind. Eng. Chem. An. Ed. 8 (1936), 36.
FAIVRE, R . , Métaux et Corr. 22 (1947), 21.
F I E L D , J . E., J . Appl. Phys. 12 (I94I), 23. v GOPPEL, J . M., Thesis Delft (1946). . ' . ,
GREBE, L . , Z . Techn. Phys. 67 (1930), 583.
GROSS, S . T . , and D. E . MARTIN, Ind. Eng. Chem. An. Ed. 16 (1944), 95.
NAHMIAS, M . E . , Z . Krist. 83 (1932), 329.
NA-VIAS, L . , J . Am. Ceram. Soc. 8 (1925), 296. " REDMONDS, J . C , Ind. Eng. Chem. An. Ed. 19 (1947), 773. SEKITO, S., Sci. Rep. Tohoku Un. 20 (1931), 312, 369. WILCHINSKY, Z . W . , J . Appl. Phys. 18 (1947), 929.
Z-WBTSCH, A., and H . STUMPEN, Ber. Deut. Keram. Ges. 10 (1929), 561.
/
C H A P T E R I I
ERRORS CAUSED BY ABSORBING PARTICLES § 1. Introduction
The simple relations derived in the preceding chapter are strictly valid if the absorption in the individual particles, either in the unknown sample or in the synthetic one, is neghgible.
The case of absorbing particles has received only little attention up to now, though it is frequently met with in practice. Even if we p u t the hmit at 10 % absorption, many cases can be found where the powder cannot be reduced to a grain size corresponding to this value, or in which such a reduction would introduce undesirable distortions. Lowering of absorption by the use of smaller wave lengths, on the other hand, may cause fluorescence, while in any case it reduces the resolving power, thereby increasing the possibility of overlapping diffraction lines. An estimation of the error, caused by application of the preceding formulas to mixtures with absorbing particles, will be useful in choosing a wave length which yields a suitable compromise. I n some cases, however, it will be very difficult to obtain a balance between this systematical error and the other, chiefly unsystematical ones. Then the error must be corrected for, or it must be minimized in another way; it is with a view to this case t h a t we have undertaken the examination of as general as situation as we were able to handle.
An attempt to calculate the influence of particle size on the intensity of scattered radiation has been made by B R I N D L E Y [1945] who gives a complete survey of former Uterature concerning this problem. A rather important error in the derivation of his formule will be dealt with in some detail at the end of § 2. I n particular, we will show t h a t RUSTERHOLZ
[1931] has at once given the right formula for the simple powder model used also b y later authors.
I n a paper by the present author [1947] an approximative formula was derived for a somewhat less simplified model of a powder. I n contrast to former publications, much consideration was given to the effect of steric hindrance. Als the case of absorption by a known thickness of powder was discussed in a general way, and the dependance of the mean absorption coefficient on grain size was expressed in a formula. I t has since proved to be possible to derive also the exact expressions for these effects as they occur in the above-mentioned model, as weU as for another without and one with exaggerated steric hindrance. As a result of this exact treatment, the original approximative formulas
— which, in contrast to the exact ones, are of a simple form — follow in a much more direct way t h a n in the paper. Obversely, a certain anomaly occurring at the specimen's surface can be studied by the approximative method, whereas the exact one fails on this point. , § 2. Restriction to cases in which no extinction occurs
The very general method used in Ch. I is capable of handling the dif-fraction by particles both with and without extinction. In fact, the scattering contribution of each particle, and the absorption Ami by other particles occur as separate factors in the basic equations. By taking due account of the correlation existing between these factors, a very general formula could be arrived at. However, so many unknown parameters and implicit relations would be contained in it as to render it perfectly useless.
We shall, therefore, have to content ourselves with the treatment of a less general situation. Since, anyhow, it is impossible either to calculate extinction in an unknown sample, or to reproduce it in a synthetical one, we hardly loose anything by excluding this phenomenon from our dis-cussion.
Then the kinematical theory applies, according to which the integrated diffraction intensity does not depend on the dimensions of the coherently scattering domains. Since the results of the foregoing chapter refer prima-rily to integrated intensities, this property entitles us to choose the said dimensions at will. We shall take at once the most extreme point of view, and assume all diffracting crystallites to consist of a great many mosaic blocks, equal in size and shape, and of a dimension which can be neglected in comparison with the particle dimensions.
Now a very important change can be made in the interpretation of equation (2) of I, § 1. In fact, it retains fully its meaning if mosaic blocks are allowed to assume the role of particles. This is so, because the contributions from mosaic blocks can be treated in exactly the same additive way as those from particles. Also the statistical model used in deriving eq. I, (4) and (5) is implicite in the one, obtained by transferring its definition to mosaic blocks.
Consequently, the averaging of equation I, (2), can be performed sub-stantially on the same hnes. We may again discern three stages
variable surroundings position orientation constant a. position; orientation b. orientation c. —
I n the former treatment, the words "position" etc. referred to a particular particle. In the present one, however, there is no reason why we should
2 Ï
not average right at the beginning over all m-mosaic blocks, since they have lost every trace of individuality. Instead of the sum occurring in equation I, (2), we simply get
T — N n 4 F
^m -^'m f m ^^jn-^0 '
Nm = total number of m-mosaic blocks.
On account of the latter's alleged properties, the individual suffix j has been dropped. As before, the letter m is used to denote the constituent of which a particular diffraction hne is considered.
Then Am means: the factor by which an X-ray pencil, pointing to an m-mosaic block (that is, a point P situated in an m-particle) and leaving it in the secondary direction, is reduced in intensity.
W h a t we are mostly interested in now is the average value J"„, resulting from aU possible arrangements of other mosaic blocks, for a given situa-tion of P with respect to the boundaries of the sample. This way of avera-ging corresponds to stage a. of I, § 1; the conditions have been hterally translated in the new version. Putting them in an equivalent but more practical form, we may say t h a t all particles are to form all the arrangements which are possible, provided a given point P within the sample lies in a m-particle.
We may as well add at once, t h a t this m-particle should be orientated so as to reflect in the given secundary direction, since the contribution of particles with a different orientation is nil. B y means of this extra condition, any correlation which in stage c. of the averaging (variation of only the orientation of the mosaic block in P) might otherwise occur between g^ and A^, is fully accounted for. Stage c. can then be treated in exactly the same way as in I, § 1, so it needs not be alluded to any further. Stage 6., on the contrary, will offer several new aspects and will be dealt with a t large in § 9.
Now Am may be devided in two factors as foUows
A^=E-exY){—fimdm). (1)
The last factor accounts for the absorption by the grain conditionnally given at P . Referring to fig. 2, dm=S'P -\- P T ; and /Um is the absorption
Fig. 2. Schematic drawing of a particle in which a mosaic block at P diffracts radiation in the direction PU.
22
coefficient of constituent m. The first factor E is the combined "trans-parency" of all the other particles intercepted by the ray R P U . If we assign a mean absorption coefficient [i to the powder it is clear t h a t , t o a rough approximation, we m a y assume it to be operative along t h e p a t h l~dm,(l = EP+PU). We then obtain:
"^m = [exp —/i{l—dm)] • exp (—/<„ dm) = exp [— (^„—/^) c?J • (exp—^i).
This is the formula proposed by RUSTERHOLZ [1931]. Comparing it with
-• equation I, (3), we observe t h a t an extra factor Pm = exp [— (Hm—lA dm}
has been added, for which we shall adopt the name coined by Brindley: particle absorption factor, and the definition
' ' - 4 „ = ^ m e x p ( - / ^ Z ) .
Brindley has tried to derive a formula for P on exactly the same assumpt-ions as Rusterholz, but in a slightly more comphcated manner. He went wrong, however, in supposing the total length of path occupied by particles to be cl, while it is (still to the same rough approximation)
C (I — dm) + dm,
(c is the mean volume fraction of solid material in the powder) on account of the m-particle conditionally given at P . Substituting the right value in Brindley's equation, one finds the result of Rusterholz again. We propose to show in the next six paragraphs t h a t a better approximation t h a n Rusterholz' leads to
P „ = C [ e x p — ( / ^ „ — l . l ^ ) ( i j , • ; in which formula the unknown parameter C has the same value for all constituents.
§ 3. Steric hindrance
I t has been pointed out already in I, § 1 t h a t there exists a correlation between the factors of eq. (1). The larger the value of dm is chosen, the larger will the corresponding average value of E turn out to be, for the obvious reason t h a t a smaller length is left for the path RS -\- T U to which E refers.
The essential cause of this correlation is the simple fact, t h a t no particle can occupy any element of space already occupied by another particle. In fact, suppose the obverse would be true, particles being allowed to interpenetrate each other freely. In t h a t case the presence of an m-particle a t P would not have the least infiuence on the arrangement of other particles, and the respective absorption factors would be completely independent.
2}
The impossibility of interpenetrating particles will be called: steric
hindrance. From the above example it appears t h a t this property is very
closely related to the absorption effects we are considering.
Now the model set of experiments, used in I, § 1 in order to define an ideally well-mixed powder, can be imagined quite as well with as with-out steric hindrance. I n order to account for t h a t property we shall, therefore, have to add another characteristic to our model of powder structure.
When looking for such a feature, the actual steric hindrance is soon observed to withstand every attempt to describe it in an exact way. However, a rather simple approximative treatment presents itself by-considering the intervals, which the particles intercept of the X-ray pencil. This method is the more appropriate since it is precisely the sequence of intervals which determines Am.
I n the ideally well-mixed powder, clearly the distribution of interval
lengths only depends on t h a t of grain sizes and forms. Also this
distri-bution will be considered as predetermined and known. I t is not in the least interfered with by any condition as to the sequence of intervals. Hence we may freely t r y to adapt the latter to the situation in an actual powder.
§ 4. Three models
To begin with the most simple case: absence of steric hindrance, as discussed above, clearly corresponds to freely overlapping intervals. The case may seem too unreal; still it is worth considering since it yields the end member of a series of models, between which the actual powder is to be fitted in. I t will be aUuded to as model M^.
I n constructing a more realistic model, we should obviously begin by not allowing intervals to overlap. The model M^ is further defined b y the following conditions:
The length of an interval, and thé material in which it is situated, are not correlated with either of the corresponding characteristics of the neighbouring intervals, nor with the lengths of the adjoining interstices. The distribution of interstice lengths needs not to be specified for the present purpose. In spite of the resulting flexibihty, it is easily realised t h a t model Mj can never afford an adequate description of a real powder. For instance, consider a powder of equal spherical particles, radius R. If the X-ray pencil just touches one of the spheres, the next interval cannot have a length 2R if the interstice does not exceed R{V3 — 1) (see fig. 3), which is precisely the kind of correlation we have excluded in the last part of the above definition. Quite generally we observe t h a t other intervals are kept a t a considerable distance from 1.
On the other hand, when a second compound is present in the form of very small grains, these will undergo steric hindrance by the large grain
24
only if they almost touch the interval 1. Hence the possibihty for the next interval to be intercepted by a small grain will be greater than normal, and the first part of our definition does not hold either. Both failures occur chiefly in these rather extreme cases of grazing
inci-Fig. 3. Example of steric hindrance.
dence. They may be interpreted by saying t h a t intervals, which are small compared with the particles to which they belong behave as if they were larger t h a n they really are, or as if they were bordered by empty spaces, which should not be overlapped by at least a part of the other intervals. The whole situation is too complicated to be treated rigourously but we may get an impression of the effect on our results by taking the following exaggerated view.
We imagine each particle to be surrounded by a "screening cylinder": a Active cylinder of which both mantle- and endfaces are tangent to the particle, while the generatrices are parallel to the X-ray pencil. Since
Fig. 4. Illustration of three models MQ (free overlapping of intervals), Mj (inter-vals not allo-wed to overlap) and M (lengthened inter(inter-vals not allowed to overlap).
25
only particles intersected by the latter are relevant, the d«finition is unequivocal except for the particle containing P . For this particular particle a special screening body is created as illustrated in fig. 4. Now suppose the condition defining model M^ is imposed, not on the actual intervals, b u t on those intercepted by the screening cylinders. The resulting model Me has just the required property, viz. empty spaces, bordering the intervals, and increasing in length with the excentricity of intersection. Only it is rather "overdone", as appears from an inspec-tion of fig. 4. I t has the advantage, however, t h a t absorpinspec-tion effects can be readily computed at least for particles of sperical shape, thus yielding the necessary figures for an a t t e m p t to interpolate the corresponding effects in an actual powder.
§ 5. The mean absorption coefficient for model Mj
Since the averaging of eq. (1), in model MQ, is a very trivial matter, we shall at once turn to the corresponding task for model M^.
As a preparation for further separation of Am into independent factors, it will be necessary to deal at first with a more general problem. What is the average reduction A* in intensity of an X-ray beam, traversing a given length of p a t h t in the powder?
In a recent paper the author [1947] has made it plausible t h a t the function
A*{t), after starting in a decidedly non-exponential manner, tends very
soon to an exponential asymptote. The anomalous behaviour practically vanishes within a few particle diameters from the origin {t= 0). Leaving aside this anomaly for the moment, we shall boldly assume t h a t a value of t exists, beyond which A* is exponential in t, and ask which value // of the corresponding absorption coefficient is consistent with the given powder structure. The calculation leading to the answer will be preceded by the stipulation of two important points:
First, the material in which the particles are embedded is considered as transparent. This is not a serious restriction, since a medium of absorption coefficient /<g can easily be accounted for b y subtracting fi^ from all absorption coefficients and adding it again to the resulting fi.
Second, only one of the end points of the p a t h t is subjected to the con-dition of being independent of the powder structure. This point will be referred to as Q. The other end point may, for instance, be bound to ly in a particle, or on the surface of a particle.
Leaving the position of the last-mentioned end point untouched, we shall start by analyzing quite formally the increase dA'^ of A*, caused by an infinitesimal displacement dt of Q in the direction of the ray. Introducing conditional mean values A*m, each of these being obtained in case Q is situated in an m-particle, we write down the following disjunctions:
(2) (3) A* — 2 ^m ^m
i6
Here Cm is the relative frequency with which the condition is realised in our model set of experiments. From the definition of this set it follows t h a t Cm is the mean volume concentration of the material m. The sum-mations are to include the embedding substance, for which we shall always use the index m. = 0; so t h a t fx^ = 0.
According to eq. (3), the differential absorption coefficient /x becomes
• •'" " [X = dA*lA*dt = -J^CmlXmAljA* (4) If we now suppose t h a t jx is independent of t, eq. (2) and (4) prove t h a t ,
for a powder of one material 1 embedded in substance 0, A* and A* are proportional to exp (— ^t). This being ascertained, specification of the interstice length distribution will lead to a definite equation for fi (by the method which will be demonstrated below for the general case). If, however, more t h a n one material is present, it becomes impossible to solve the said equations for AmjA*. In t h a t case we have to assume the exponential behaviour of A*, while t h a t of A* (otherwise stated, the Constance of fx) will be shown to follow therefrom.
I t is characteristic for the stubborn nature of the problem t h a t , in order to estabhsh even the existence of a /^-value consistent with the given model, such an artificial assumption has to be made. In the discussion (§ 6), however, we shall demonstrate t h a t the assumption relating to A* is equivalent to the one concerning A*.
Consider fig. 5, representing a one of the cases leading to the average value At, b one to which A% refers, and c a third case, defined by the
Fig. 5. The three possible situations of the end point Q of a given path with f .. " : respect to the powder particles.
condition t h a t Q lies on the surface of a particle to the right of Q. The corresponding mean absorption will be denoted by A%. The case a can be readily reduced to c as follows. Let us keep constant a t first the length
27 d^, which is common to the path t and the m-grain at Q. Then upon
averaging under this condition. ., ;
A*m{t)=A*{t~dmi)eXT^{—/l^dmi). (5)
The case 6, however, cannot be reduced so easily. Here the interstice length distribution has to be resorted to.
We shall assume, then, t h a t interstice lengths are distributed as the distances between succeeding points in a random array of points on a straight line. I t is a well known fact that, under these circumstances, the distance from Q to the end point R of the preceding interval has the same distribution in case b as in case c ("Weglangenparadox). Other-wise stated, it does not interfere with this particular interstice length distribution if in case 6 we consider Q as an infinitesimally small interval. Case 6 is then a special form of case c. On account of the definition of model Mj, on the other hand, the length of the interval which in case c begins in Q, is not correlated with the arrangement of intervals to the left of Q. Hence both cases will lead to exactly the same average value
AUt) = At(t). (7)
Expressing the assumed exponential behaviour of At as
At'(t+ x) = At (t) exp (— Mx),
we obtain for a given dmi from (5),
Atit) = A* it) exp [ - if^m-M) dmr], (7a)
and in general
Al = A* exp [ - ifim-M) dmil (8)
From the equation (8), together with (2) and (4), all A*'s with and without suffix are easily ehminated.
The result is
^ ^ - ^ = i" = 2 Cm /"m e x p [— (jUm — fi) <^mi]/2 ^m « x p [— (^„ —/<) dmi] (9) in which expression dg^ must be defined as doi = 0 in order to allow sum-mation over all m values.
We may write also
2 Cm (A«m —/*) e x p [— (fXm — fl) dmi] = 0 (10) This equation invites us to perform a t least a part of the averaging, viz.
over all possible positions of Q within a given interval dm (cf. fig. 5a). The relative frequency of (Zmi-values between xdm and (x -\- dx)dm being
dx, this conditional average amounts to
'. 1 . . .
e x p [— (fim—/^) dmi] = ! dx e x p [— (/Xm—f^) xdm] =
0
M)dm-28
This procedure may be apphed to all terms for which m 9^ 0 (it is meaning less for m = 0). We obtain with the help of 2'Cm=l,
(l—c)fX=2Cm[^ — exp--{fXm — fi)dm]ldm ', C = 1 — Co, ( H )
m=i=0
or
(1—c)/^=(l—c)a—2 c„[exp—(//„—/^)CZJ/É^„ ; a=2cJcZm(l—c)- (12)
m*0
When the average values of functions of dm (as indicated by the bar) are t o be formed actually, care should be taken to use the proper weight factors. For instance, if only two interval lengths a and b occur with equal frequency, the average dm is {a^ -f 6^)/(a + 6) and not \{a -\- b); but (lldm)=2l(a+b).
The quantity a is the reciprocal value of the average interstice length. I n fact, 1/a is defined in (12) as the concentration (1 —c) of interstitial space devided b y t h e total number Z Cmjdm of intervals (or interstices) per cm.
§ 6. Discussion
Let us begin b y summarizing the results of the previous §. The existence of a value of /x=^ din A*jdt consistent with model Mj was established
a. for particles of one material only: generally.
6. for mixtures: If d In Aojdt does not depend on t it can be shown to be equal to ^, and its value can be expressed in powder structure para-meters. The reverse, however, could not be proved.
Now one might conceive the idea t h a t the above, highly artificial assump-tion is not valid in reality. I n other words, t h a t with a constant /x-value corresponds a non-exponential behaviour of At, in which case eq. (9) for fx would be founded on nothing at aU.
T h a t it is not so can be shown as follows. The mean values A* and A* both apply to a path t, of which one end point may still be conditioned a t wiU. The other end point, Q, should lie in an interstice if A* is to be obtained, and it should be independent of the particle arrangement in order to yield A*. Now imagine a p a t h t of which one end point hes in the interstitial space, whue the other is not correlated with the particle arrangement. The mean absorption corresponding to this path then m a y be called A* SiS well as A*, depending on which end point is chosen as Q. So any possible value of the decrement of A* must be equally possible for t h a t of At; and if the latter is unique, the former cannot have more t h a n one and the same value either.
I n both cases a. and 6. a " r a n d o m " distribution of interstice lengths had t o be supposed. I t is probable t h a t other distributions would also allow t h e computation of /x, though not in so simple a manner and not always.
29 I n fact, it will appear from the subsequent discussion t h a t a constant ^-value is possible only for a certain class of distributions.
The "random" interstice length distribution will, henceforth, be incorporated in the definition of model M^, and of model Me, as far as it is derived from M^.
The net result then amounts to the following.
Only one value of /x is compatible with the now completely defined powder model Mj^. This value is expressed in given functions, without any approxi-mation, by equation (9), (10), (11) or (12).
These equations, being implicit in /x, are not readily solved in an arbitrary case. In some very simple cases, however, the solution is easy. Then they can be used to test the accuracy of approximations.
Of these, two are important for weakly absorbing particles. They are obtained by expanding the exponentials in (11) in power series. The hnear term yields
/X=fl* = J,Cm^m,. (13)
which is only the usual formula for homogeneous mixtures. The second term leads to
/X= IX* — ^^Cmdm(/Xm — /^*)^, (14)
Eq. (14) had been derived previously by the author [1947], though on quite different hnes; he proved t h a t it is exact if the distribution of In ^ * is a Gaussian one. His prediction, t h a t the correction term would not be in error more t h a n 10 % (of the correction) if no /Xmdm exceeds unity, wiU be tested below in two simple cases.
Another approximation is found from eq. (12) for heavily absorbing particles. In fact, we find t h a t , if all particles are intransparent,
fi= a.
This result is only obvious, since the chance of a ray to penetrate through a p a t h t is, in this case, given by the interstice length distribution
^••(i) = const. exp(—a<). (15) At the same time, it follows from this argument t h a t a constant ^-value
can be expected only for interstice length distributions obeying (15) above a certain finite length; we have aUuded to this circumstance already before.
For strongly absorbing particles, an exphcit approximation is obtained from (12) by replacing [x in the exponent by a:
{l — c)[i= (1 —c) a— 2 c» [exp—(ƒ*„—a) c i j / d „ , (16)
m*0
the accuracy of which can again be judged from the exact calculations. These have been performed, firstly, for particles of a single material
30
with only one interval length d. (One might object against a constant dm-value, since it could result only from oriented prismatic particles. However, the absence of preferred orientations, prescribed for the original model set of experiments, is not essential in this section. I t may, there-fore, be disregarded temporarily).
Then eq. (11) apphes. I t m a y be written for this case as
px -\- exp {x — g) = 1,
with
x = /xd g = /Xjd p= (l — Ci)/ci.
The following table compiles values of x corresponding to various values of g (up to ca. 5), and of the volume concentration c^. The a;-values are round since it was easier to compute g from x than to go the observe and more logical way. Various approximated values have also been computed, using eq. (13), (14) and (16).
A comparison shows t h a t the common formula (13), which is not corrected for particle absorption, is already 10 % in error for fXyd=^ 0.5. The next approximation is fairly good u p t o /x^d = 1.0. The one in the last column, calculated from eq. (16) for strongly absorbing particles, appears to be vahd down to /x-jd = 3 and becomes extremely poor for lower values.
TABLE I C i = l / 2 Cl = 1/3 c = 1/6 fid .2054 .657 1.516 3.203 5.595 .155 .507 1.216 2.753 5.100 .125 .417 1.036 2.483 4.803 e q . (13) 1st a p p r . .1027 "^ .329 .758 1.60 2.80 .05018 .1690 .405 .918 1.700 .0209 .0695 .173 .414 .800 x= ij,d a c eq. (14) 2 n d a p p r . .1001 .301 .614 .95 .89 .05003 .1500 .304 .540 -.01998 .0595 .111 .055 -j o r d i n g t o eq. (11) e x a c t .1000 .300 .600 .900 .990 .05000 .1500 .300 .450 .495 .02000 .0600 .1200 .1800 .1980 eq. (16) a p p r . — — .40 .89 .990 — — • .255 .447 .495 — — .084 .1803 •1980
As a conclusion it may be said t h a t the /^^d-regions below 1 and above 3 are fairly well covered by eq. (14) and (16), respectively. In the inter-mediate range, however, there is no weU-fitting approximation. The upper limit of the range in which eq. (14) satisfies, shifts to lower values with decreasing concentration. If we put t h a t limit at 1 % error, it is
31 reached at fXid= 1.1 for C i = 1/2 or 1/3 but already for iXjd= 0.6 when Cj= 1/6. This means that the Gaussian distribution of the total interval length, (which had to be assumed in the paper cited above in order to arrive at eq. (14)) is much better reahsed for high concentrations than for small ones.
Secondly, we have computed /^-values for an M^-model of spherical particles of equal diameter D and absorption coefficient fx^. The average values occurring in eq. (11) are then easily found by integration, and the ensuing exact value of [x can be expressed as
\px=\—[\ — {\ + q — x) exp — (g — x)]l(q — x)^,
with
x = /xD q = fx^D 2J = (1 — Ci)/Ci
The results can be compared with the 1st and 2nd order approximations following from eq. (13) and (14), respectively, in table I I .
TABLE II c = 1/2 c = 1/3 c = 1/6 fi,D .3858 .9174 1.7076 3.055 5.330 .2929 .7087 1.354 2.527 4.665 .3537 .8092 1.682 3.246 4.266 X = eq. (13) 1st appr. .1929 .4587 .8538 1.5027 2.665 .0976 .2362 .4513 .8423 1.555 .0589 .1347 .2803 .541 .711 = nD according eq. (14) 2nd appr. .1859 .4192 .7176 1.089 1.331 .0923 .2083 .3493 .488 .34 .0535 .1069 .157 .07 -to eq. (11) exact .1858 .4174 .7076 1.055 1.330 .0923 .2087 .3538 .527 .665 .0537 .1092 .182 .246 .266
The agreement between the last two colums up to /x^D = 1 is about equally good as in table I. A comparison with eq. (16) — for strongly absorbing particles — has not been made because t h a t equation is difficult to handle in this case.
§ 7. Application to the models M^ and MQ
Suppose every interval in a M^-powder (exaggerated steric hindrance, fig. 4) is elongated till it equals the height of the screening cyhnder. Also a new absorption coefficient is ascribed to it, equal to the actual one
32
divided by the elongation factor e. Obviously the new arrangement wül have the same differential absorption coefficient ^ as the original one. On the other hand, it is a Mj-arrangement on account of the definition of Me. So the value of fx, compatible with the original Me-powder can be obtained by inserting the modified parameters d'm = edm and /Xm = jUmje instead of fXm and dm in eq. (11). Keeping in mind t h a t e depends on dm, we obtain finally
{l — ëc)/X={l—c)a — 2Cm[^^V — {l^m—efi)dm]ldm- (17)
The mean value ë occurring in (17) is the ratio of the total volume of aU screening cylinders to t h a t of all particles.
As regards model Mg, it is not difficult to derive an expression for fx in a, direct way. By far the easiest way to find it, however, is the reduction to Mj which ensues if here also an elongation factor is introduced. In fact, by choosing e infinitesimally small ("contraction factor" would be a more appropriate name here), a random arrangement of point-like inter-vals is produced which satisfies in an excellent way the conditions defining model Mj.
Equation (17), then, appears to cover all three of our models. If we define, according to the screening cylinders and particle forms,
e = e(dm), the /x of model M^ is found;
if p — 1 M •
and if e = 0 , „ „ „ ,, Mg ,,
Also it foUows from the above argumentation, t h a t (17) adequately yields the /x of any model, derived from M^ with the aid of a screening body, provided e{dm) is defined according to the form thereof. I n the case of MQ, for instance, this screening body is only a plane at right angles to the beam and containing the particle centre.
An impression of the effect of screening cylinders on ju can be obtained b y explicite approximations. On developing the exponential terms in (17) in power series, the first order term yields eq. (13) again. The next approximation becomes
/X= fX* — ^2Cmdm{l^m—e/X*)^. ' (18)
For spherical particles the average values of the coefficients t u r n out to be as in
M=fl*-hlCmdm(/^l-2-i/Xmf^* + 2/X*^) (19)
(while dm amounts to | times the diameter).
The maximum value to which /x tends if all /Xm's are increased beyond any hmit, is now
33
Using this value in the exponents in (17), another approximation for strongly absorbing particles may be obtained. However, it is difficult to handle and will not be mentioned in detail.
As a numerical example, the case of spherical particles of equal diameter
D and equal composition will be considered. The average values in (17)
are readily found by integration. Using dimensionless parameters we write the result as
px=\ — 2 exp a;. [1 — (1 + g) exp (— q)]lq^ (22)
with
X = fxD q = ix^D p^ (\ — êc)lëc; ê = |
The ensuing exact values of /xD are compared with those following from the approximative equation (19) in table I I I . Again we observe a good agreement up to jUiD-v&lues of about 1.
TABLE III C i = l / 2 C i = l / 3 C i = l / 6 fiD .308 .628 1.31 2.97 5.3 .313 .656 1.03 1.96 4.56 .189 .705 1.82 2.81 5.1 X = eq. (13) 1st appr. .154 .314 .655 1.49 2.65 .104 .219 .343 .653 1.52 .0314 .1175 .303 .467 .850 = /iD according eq. (19) 2nd appr. .151 .302 .601 1.21 1.77 .100 .200 .299 .493 .65 .0300 .0985 .177 .163 — to eq. (17) exact .150 .300 .600 1.200 1.800 .100 .200 .300 .500 .800 .0300 .100 .200 .250 .300
§ 8. Final discussion of the mean absorption coefficient
The existence of a fairly accurate and simple formula for /x being
esta-bhshed, we have yet to decide in which way the actual powder shall be fitted in between the models Mj^ and Me. I t appears both from the approxi-mative formulas (14) and (19), respectively, and from the graphical representation of the exact values (fig. 6), th*t the difference is conside-rable. I t is even much larger t h a n t h e error of the second-order approxima-tion in its useful range (fx^D < 1).
Fig. Ga. Exact values offiD/Ci = qx /i/fi* as a function of /i^D = q; i.e. the average absorption coefficient ft of a one-component powder, as a function of the /ijD-value of its particles. The dimensionless ordinate nD/Ci has been chosen instead of /i so as to obtain curves starting with a slope 45° at the origin (where /j,= /j,*). Curves have been drawn for volume concentration c^ = ^ and ^ , and the following cases:
I I oriented prismatic particles each intercepting a length D of the path of X-rays. O spherical particles, diameter D, arranged in model Mj (no overlapping of inter-vals on the infinitesimal beam under consideration).
El same particles arranged according to model M^ (no overlapping of lengthened intervals).
Fig. 66. The ratio of the average absorption coefficients of a one component powder, and of the same considered as a homogeneous substance, as a function of the i^iD- (or q-) value of its particles. Curves are drawn for three values of the volume concentration Cj. They apply only to the case indicated by O in fig. 6a. The second approximation obtained from eq. (14) has been represented by broken lines, which are straight since the right hand side of eq. (14) becomes linear in/*i after division by/<*.
35 t h a n on the exact formula (17), since we have practical applications in view for which t h a t formula would be a prohibitively intricate one. Also the useful range of the approximations is already large, compared with the range in which the formula (13) for quasi-homogeneous material apphes with equal accuracy.
I t has already been intimated in § 4, t h a t the actual effect of steric hind-rance is much exaggerated b y the screening body of model Me. With a powder of spherical particles, however, it is very difficult to obtain an idea as to the extent of exaggeration. We shall, therefore, consider a case which allows at least a very rough quantitative treatment, viz. a powder consisting of identical prismatic particles, orientated with their square end faces parallel to the beam, and mantle faces a t 45° with it. I n fig. 7 a cross section is drawn. . '
Fig. 7. Powder of particles with square cross sections, illustrating the internaediate position of a real powder between the models Mj and M^ (cf. fig. 4).
If we look at two neighbouring particles cut by the X-ray pencil, it strikes us t h a t the actual arrangement wiU offer many opportunities for a close approach, which are forbidden by the screening bodies (cf. fig. 4). Now in this simple case there exists a 1,1-correspondence between the length of an interval, and the distance from the centre of the corresponding particle to the beam.
A given set of intervals thus determines the ^-coordinates of particle centres (the beam being defined hjy= 0), apart from their sign. Accor-dingly we are now able to give the following reahstic touch t o the model: we impose the condition t h a t interstices may not decrease below the value corresponding to direct contact between the adjoining particles. This condition may be accounted for by the use of elongation factors as follows. Consider, for instance, particle a in fig. 7. On moving the corres-ponding interval t o the left, it can be made to touch its neighbour without interpenetration of particles. If it is moved to the right, however, a and b come into contact before the intervals do; hence the screening triangle drawn to the right of a. Generally, a particle must be provided with screening triangles pointing to those neighbours (if any), whose centres are on the same side of the beam and further from it.
Moreover we shall suppose the resulting arrangement of elongated inter-vals to be an M^ —one. Then the fraction of all interinter-vals of length aD ( Z ) = length of diagonal), who have for instance their left neighbour at
}6
the same side of, as well as further off the beam, amounts to ^a. Conse-quently the elongation factor is
1 for a fraction (1 — | a ) ^ 1/3 ,, „ a(l — | a ) (2 a)/a ,, ,, ,, ^a
of these intervals. Substituting the above factors in (18), and taking the average with the proper weight factors, we obtain
fi=/x*-ilCmdm(A-2.^,,Xmf^* + ^fx*^) ; J , „ = | i ) , (24)
for the best approach to an actual powder, as compared with
/* = /** — i 2 c», d^ (/<m— 2 • i/Xmfi* + 3/<*2) (25) for an Me-arrangement as in fig. 4.
Now these expressions, even as equation (19), much resemble the form
fl=/X* — i2Cmdm(/^m — ef^*)^ (26)
so long a,s ju* <^ jUm. I n t h a t case e is given b y half the value of the second coefficient:
equal spheres, model Me: e = •§• equal squares, model Me: e = f equal squares, improved model: e =
-J-In adopting a value of e for use with actual powders, it should be kept in mind t h a t even the improved model is scarcely more than one-dimen-sional. For instance, in dense powder specimens there wUl undoubtedly exist a tendency for particle centres to lie alternately on both sides of the beam, which greatly lowers the value of e. Generally, we think t h a t a value of 1.1 would be the most appropriate to be chosen.
If fi* > jUm, the diversity in the last coefficient of (19), (24) and (25)
demonstrates the impossibility of finding a useful approximation for this case.
§ 9. The effect of particle absorption on scattered intensity
Resuming the treatment of the main effect where we left it in § 2, we observe t h a t the situation of fig. 2 contains twice t h a t of fig. 5a namely one time on the primary beam SP, and again on the secondary beam P T . Hence
^™(RPU) = ^ * ( R P ) . ^ : ( P U ) . (27) Averaging first for constant lengths of SP and P T and making use of the
37
absence of correlation, which otherwise exists between the factors of (27), we obtain for model Mj from eq. (7a)
J „ ( R P U ) = At (RP). At (PU). exp[-{fXm-fi)dm].
Since this intermediate result appears to depend only on SP + P T = dm, we may proceed immediately to take the average on varying
dmAm (RPU) = At (RP). At (PU). exp{fxmf^)dm
-The case of vanishing dm teaches us t h a t
. A,(RFV) = At(RF)-At(FV) • so t h a t we may write
Am=AoeXTp~{/Xm—fx)dm. (28)
Here AQ refers to the mean absorption undergone by the same ray R P U , if P is situated between particles. I t should be stressed t h a t both Am and
AQ have a wider meaning here t h a n At and A* in § 5, since the conditions
defining them now relate to an arbitrary point P , instead of the end point Q, of the total path. Also dm is not quite identical with the former
dm, since it consists of two segments in different directions. This, however,
is not a very important change for the small deviation angles commonly used in quantitative determinations. Another difference is, t h a t the average in (28) should be formed only for dm's, relating to m-particles in reflecting position (cf. § 2).
I n the model Me, a factor e must be put before the fx in (28). An argument, similar t o the one followed in the previous §, makes plausible a working formula •
Am=Aoexp[—{fi^—e/x*)dm] for (/^,„ —e/^*) rfm< 1, (29)
where £ = 1 . 1 will generally be the best choice. •
In fact, since both this £ and the one occurring in the previous paragraph result from the averaging of a term linear in edm, they must be identical and subject to the same considerations. Only the regions in which they can be used successfully are quite different (cf. the condition in (29)) owing to the essentially different problems to which they are related. I t is very important to note t h a t (29) obtains only if the mean absorption may be supposed to be an exponential function of both R P and P U . Then the fx occurring in (29) is indeed uniquely determined, as the foregoing ample discussions have shown. Since A^ is then also exponential in the total path R P U = I, say
