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The separation of coherent and incoherent Compton X-ray scattering

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BRIT. J. APPL. PHYS., 1964, VOL. 15

The separation of coherent and incoherentCsmpton X-ray scatteringW. RULANDUnion Carbide European Research Associates, sa., 95 rue Gatti de Gamond,Bruxelles 18, BelgiumM S . received 29th May 1964Abstract. A theoretical and experimental study of the possibility of separatingcoherent and incoherent Compton x-ray scattering by monochromatization of thescattered radiation is carried out. It is shown that this possibility depends on theprofile of the Compton line as well as on the resolution of the monochromator.An approximate treatment of the problem is developed and an application to carbonand organic materials is given.

1. IntroductionX-ray studies of disorder and thermal motion involve an accurate determination of thediffuse coherent scattering. Using counter techniques, monochromatization of the scat-tered radiation can be carried out with relative ease which should, in principle, enable aseparation of th e coherent scattering fro m C omp ton and fluorescence scattering. Th elatter can generally be eliminated without difficulty; the elimination of the form er, however,poses some problem s which ar e due to the effect of the finite resolution of a monochromatorand the width of the Com pton line. The presen t report deals with the theoretical andpractical aspec ts of these problems with special attention to carbon and organic materials.

2. TheoreticalThe monochromatization of scattered radiation can be defined by

I(e)=J M ( A ) (e, A) d/\where I ( 0 ) is the intensity of the monochromatized scattering, h(0, A) the distribution ofwavelengths scattered under the Bragg angle 0 and M(A) a distribution representing thepass-band\of the monochrom ator. If h(0, A) is given by the wavelength distribution ofCompton scattering the elimination of this Scattering can be assessed by the a ttenuationO(8) defined bv

For Compton energies Ec large compared with the binding forces of the electrons in theirradiated material, the wavelength distribution is completely determined by the momen tumdistribution J (q ) (DuM ond 1933). These distributions have been calculated fo r a numberof elements (Kirkpatrick, Ross and Ritland 1936, Hicks 1937, Duncanson and Coulson1945, March 1954). The observed Com pton profle s (Burkhardt 1936, Kappeler 1936,Kirkpatrick, Ross and Ritland 1936) show that there a re considerable differences betweenthe proses of free and bound atoms.For the purpose of the present work, the DuMond approach is not strictly valid, sincethe basic cond ition for its applicab ility, high values of Ec, is in general no t fidlilled in therange of 0 and A of interest in x-ray diffraction work . If EC is comparable with the bindingforces of the electrons, the C om pton p ro fl e is composed of a line spectrum (the Com pton-Raman effect) an d a continuous spectrum. This effect has been discussed by Schnaidt

1301

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1302 W. Ruland(1934), Som merfe ld (1937), Franz (1937) an d recently by Das Gupta (1959). The linespectrum is d ue t o discrete energy transitions below the ionization energy and is in principleequivalent to the &e structure of the x-ray abso rption edges. In solids there is a numberof low energy transition probab ilities due to the band structure, which can produce Comptonscattering of nearly the same wavelength as the coh erent Scattering at small angles. Thismeans that i t is difficult to separate coherent and incoherent scattering at relatively smallangles even with monochromators of high resolution.Since the calculation of the Com pton profile for solids as well as its accurate determina.tion is rather difficult the following approximation is made for the evaluation of equa-tion (1).Fo r large values of E c, the integral width of the Compton profde d q is a constant whenthe profile is taken as a function of c Aq = - -2 A sin 0where c is thevelocity of light (generally given in atomic units), A =A - c, Ac the averagevalue of the Compton wavelength distribution and A, the incident wavelength. Theintegral width A h of the wavelength distribution is thus given by

2A h =A sin 0 - Aq.For smaller values of Ec, A q can be considered as a function of Ec, which means that itis a function of s =2 sin O/hosince

C

s?.h sin2 0Ec =-- =-c h,2 2m cOne can thus write generally

We introduce furthermore the assumption t hat f or th e determination of Q (equation (1))the Compton profile can be approximated by a unique type of distribution h which iscentred on d h c , the Compton shift, and whose width is proportional to AA . This gives

if h is considered to be normalized. For a given wavelength of t he incident beam and agiven material, Q can thus be calculated as a function of s with the knowledge of dq(s),M ( h ) and Ahc.3. Determination of the Compton prome

A Philips fluorescence spectrometer has been used f or the determination of the Comptonprofde of graphite , diamond an d some polymers. Since the scattering angle (defined bythe positions of the focus, the sample and the analyser crystal) cannot be varied over awide range, a series of different wavelengths fo r the incident beam have been used: chromiumK,Band K a tungsten La, gold La, molybdenum K,B and Ka. A typical recording is shownin figure 1. The advantage of the fluorescence spectrometer is a high intensity yield, thedisadvantage is a low resolntion f or the Com pton profile due t o the rather large divergenceof the incident beam. It is thus no t possible to observe the C ompton-Raman effect, buttaking the divergence into account by appropriate corrections for t he integral width of theCompton profYe A&) values can be obtained with relatively high accuracy. An exampleis given in figure 2 which shows clearly tha t A q is not constant in the range of s generallyoccurring in x-ray diffraction work. A number of functions have been tested in an attempt

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The separation of coherent and incoherent Compton x-ray scattering 1303

x A,Figure 1. Distribution of wavelength scatteredunder 0 =47" with I. , = 1.4763 A (WLa,)for polyethylene.

Figure 3 . Observed and calculated intensitydifference between a Y and an Sr filter,l o=0.7107 A, polyethylene.

Figure 2. Integral width dq of the Compton proae asfunction of s for polyethylene.

to approximate the results by an analytical expression. Since A&) should tend towardsa constant for high values of s, a function of the type

Seems to be a reasonable approach.mental data are given in the table together with some A h values.Values of dqmax nd a obtained by a least-squares fitting of such a function to the experi-

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1304 W.RuIandA A e =450)(4 (4

Sample 4 a-.0 =1.542 >bo =0.711

Diamond 3.39 0.63 0.037 0.023Graphite 3.05 0.53 0.036 0.021Polyethylene 2.86 0.52 0.034 0.020hTylon6 3.00 0.54 0.035 0.020Teflon 3.55 0.55 0.041 0.024A comparison of the Aqmax values with the theoretical Aq values given by Duncansonand Coulson (1945) (carbon 1.57, fluorine 2.84) for free atoms shows the expected mag-nitude of the effect of the binding forces on the Compton proae.The results of the measurements with the spectrometer were checked with Woflans(1934) a t e r technique. Figure 3 shows a comparison of the observed and Calculated

intensity difference between a Y and an Sr filter for MO a as incident radiation usingAq(s) as given by equation (4)and a Cauchy distribution to represent the Compton profilefo r the calculated curve. Th e filters were produced from t he oxides embedded in poly-ethylene as already described (Ruland 1959). The thickness of the filters was chosen suchthat the pass-band defined by the difference of the absorption functions has a constantvalue between the Y an d the Sr absorp tion edge a nd is effectively zero outside the edges.Except for a small shift in the position the two curves are very similar, which shows thatthe interpolation of the d q values as well as the approximation of the profile are consistentwith the experiment.Th e difference n position is exp lained by the effect of the binding forces on the Comptonshift as discussed by Ross and Kirkpatrick (1934) and Bloch (1934). If we consider thiseffect to be given bywe find a value of about 1 . 5 x ascalculated by Bloch (1934) an d 1e48 x as m easured by Ross and Kirkpatrick (1934)for carbon.Equation (5) is of course only valid fo r high Comp ton energies aild therefore not generallyapplicable in the h and 0 range used in x-ray diffraction. Fo r copper radiation, for example,Dho2would have the value 0.0036 A and Ahc would become negative for 0

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The separation of coherent and incoherent Compton x-ray scattering 1305fines of the impurities in the target material. Three types of monoch rom atizatio n havebeen studied: flat Siilgle crystals in reflection position, Ross filters and pulse-height dis-,imination. F or the first type, a LiF an d a pentaerythritol crystal have been chosen, fo rthe second type Ni, CO and Fe filters for Cu radiation and Zr, Y and Sr iiIters for MOradiation have been produced fro m the oxides embedded i n polyethylene. Pulse-heightdiscrimination was studied with a xenon-filled proportional counter in the Cu region only.

A-A, (A !

I n

L A , A iFigure 4. Pass-bands of a flat LiF mono- Figure 5 . Pass-bandsof a flat pentaerythritolchromator crystal, (200) reflection. monochromator crystal, (002) reflection.

Figure 6. Pass-band of a Ni-Co filter set. Figure 7. Pass-bands related to channel( N O and Co,O, in polyethylene.) widths (in v) for the discrimination of apulse-height distribution from a xenon-Hledproportional counter. 1.54 A equals 50 v onthe pulse-height scale.Typical results are shown in figures 4, 5, 6 and I. A number of pass-band functions areobtained for a given crystal monochromator by varying the slit system. The relativelyhigh yield of the f lat-crystal monochromators ( e 1 5 %) is due to the small divergence ofthe beam diffracted by the analyser crystal. Un der norm al diffractone ter conditions theyield of the monochromators is much lower.The ripples o n the pass-band functions for Ross filte rs a re due to the h e t ructure o fthe absorp tion edges. Owing to the density fluctuations in th e oxide filters the effectiveabsorption is somewhat different from that for filters of uniform thickness. This effectlowers the optimum yield, but allows a better zero fit a t both absorption edges.

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1306 W.Ruland5. Calculation of the attenuation function Q

Inspection of equation (3) shows that Q akes the form of a convolution of & and h.It has been shown that h ca n be app roximated by a distribution of the Cauchy type. porkf s given by c rystal monoc hrom ators o r nar row chann els in pulse-height discriminationan approximation by a Cauchy distribution is also reasonable, whereas for Ross filters Mcan be approximated by a ste p function.If th e integral width of M as expressed by a Cauchy distribution is b and its value forthe incident wavelength is taken as unity, Q is given by

Since A h and Ahc are both functions of A,, and s (equations (2 )and (5)), Q can be CoEputerJas function of s fo r an y given A ,. Figures 8 and 9 show Q func tions calculated for poly-ethylene, Cu o r M O radiation and a variety of b values.

2

Figure 8. Attenuation Q calculated as func- Figure 9. Attenuation Q calculated as func-t ion o fs for 3.0 = 1.5418 A, Cauchy type Pass- tion ofs for j ., =0.7107 A, Cauchy type pass-bands (integral widths b) and polyethylene: bands (integral widths b ) and polyethylene.---- bserved Q values for LiF mono-chromator; - - - bserved Q values forpentaerythritol monochrom ator; . . . .. . . . .observed Q values for pulse-height discrimina-tion, 4 channel.If M is a step function (Ross filters) a nd un ity a t A,

10-0 :

0.O S L \ r-Y

1 \0 O 5 s&, I O 0 0 5 1 0 ~ r 8 - , 1 5 2 0 2 5

Figure 10. Attenuation Q calculated as func- Figure 11. Attenuation Q calculated as func-tion of s or A. =1.5418 A, Ni-Co and Ni-Fe tion of s fo r i., =0.7107 A, Zr-Y and Zr-Srfilter sets, and polyethylene. filter sets, and polyethylene.

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The separation of coherent and incoherent Compton x-ray scattering 1307where x c =ho+A h nd Ax is the position of the absorption edge of the second filter.Figures 10 and 11 show calculations for polyethylene, Ni-Co and Ni-Fe iilter sets for CUradiation an d Zr-Y and Zr-Sr filter sets for MO radiation.6 , Experimental check O f attenuation functionsFor a given monochromatization the ob( :ved scatte ring intensity is

Iobs = n f Q Iincoh)where k is a normalization factor an d I c o h and I iocoh the normalized coherent and incoherentintensities. If the observed intensity is Il for one kind of monochromatizationand I , for another, and k l and kz are the corresponding normalization factors, one finds

If el is not expected to vary considerably in the region where Q, is non-zero, one canmeasure Ql y a comparison of normalized scattering intensities.This measurement has been carried o ut on scattering, monochromatized by flat LiF andpentaerythritol crystals, and pulse-height discrimination. The results are shown in figure8(broken lines). The results agree fairly well with the calculations; fo r the higher resolution(LiF, b U 0.007 A) the observed Q values are somewhat higher than the calculated ones.7. Discussion

The observed and calculated attenua tion functions show the importance of the Comptonprofle for the separation of coherent and incoherent scattering. Altho ugh a completeseparation is not possible crystal monoc hromators of high resolution can effectively attenu-ate the medium an d wide angle Com pton scattering. Since the correction for the remainingCompton scattering at smaller angles involves information on the Compton proae it willoften be preferable to use a combination of ROSSilters and pulse-height discriminaticn,which are chosen such tha t the Com pton scattering is not affected by the monochrom atiza-tion, and to eliminate the Compton scattering by calculation.Acknowledgments

The author is indebted to Mr. J. M. Gilles for stimulating discussions during the courseof this work and to Mr. J. P. Pauwels for technical assistance. Thanks are also due toDr. J. L. de Vries an d M r. L. Caeymaex,N.V. Philips, Gloeilampenfabrieken, Eindhoven,for providing part of the w ork facilities.ReferencesBLOCH,F., 1934, Phys. Rev., 46,674.BURKHARDT,G., 1936, Ann. Phys., Lpz., 26, 567.DASGUPTA,K., 1959, Phys. Rev. Letters, 3, 38.DUMOND, . W. M., 1933, Rev. Mod. Phys., 5, 1.DUNCANSON,W. E., and COVLSON,C. A ., 1945, Proc. Phys. Soc., 57, 190.FRANZ, ., 1937, Ann. Phys., Lpz., 29, 721.HrCKS, B., 19:7, Phys. Rev., 52, 436.~ P E L E R , . , 1936, Ann. Phys., Lpz., 27,129.K ~ ~ ~ ~ A T N c K ,., Ross, P. A ., and -AND, H. ., 1936, Phys. Rev., 50, 928.MARCH,N.H., 1954, Proc. Phys. Soc., A67, 9.Ross, P. A,, and KIRKPATRICK,P., 1934, Phys. Rev., 46, 668.RULAND,W., 1959, Acta Cryst., 12, 679.SCHNPIDT,F., 1934, Ann. Phys., Lpz., 21,89.SOMMERFELD, A ., 1937, Ann. Phys.,Lpz., 29,715.WOLLAN, E. O., 1934, Phys. Z . , 35, 353.