Particle Size Distributions from SANS Data Using the Maximum Entropy Method. By J. A. POTTON, G. J. DANIELL AND B. D. RAINFORD

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1 3 J. Appl. Cryst. (1988). 21,3-8 Particle Size Distributins frm SANS Data Using the Maximum Entrpy Methd By J. A. PTTN, G. J. DANIELL AND B. D. RAINFRD Physics Department, The University, Suthamptn S9 5NH, England (Received 2 ctber 1987; accepted 2 April 1988) Abstract The extractin f particle size distributins frm small-angle neutrn scattering data is an example f a practical linear inverse prblem. Additinal assumptins are necessary t btain a unique slutin. The applicatin f the maximum entrpy methd t select a realistic size distributin is discussed. Principal features f the methd include a prper treatment f experimental errrs, n interplatin r smthing f data, n fitting t empirical mdels, and guaranteed psitivity f the slutin everywhere in spite f statistical nise in the data. The resulting slutin is the mst unifrm cnsistent with the data. Mdel data results are presented t shw that the maximum entrpy criterin prves very useful in prblems f this type. 1. Intrductin Small-angle scattering f either X-rays r neutrns is ne f the mst direct ways f extracting particle size infrmatin. The scattering intensities are given by It= ~ St(r)p(r ) dr, j = 1,..., m (1) where St(r) is the scattering functin fr a single particle f linear dimensin r, calculated fr the jth scattering vectr qt, and p(r) is the unknwn prbability distributin sught. There are M bservatins. We assume that there are n interparticle interference effects; hwever, any knwn effects may be included in St(r). The determinatin f p(r) frm the measured scattering pattern is an example f a linear inverse prblem. In practice it is impssible t extract a unique slutin since nt nly are the bservatins subject t experimental errr but, mre imprtantly, nly a finite number M f bservatins ver a limited range f scattering vectrs is available. Bth f these limitatins restrict the infrmatin cntent in the data. An imprvement t existing analyses t extract a feasible p(r) slutin will include n interplatin r smthing f data, explicit use f experimental errrs and n need fr a mdel functin, which wuld prejudice the slutin. Facilities t decnvlve instrumental reslutin and t crrect fr pssible systematic errrs intrduced via backgrund subtractin wuld be a benefit. The maximum entrpy methd (MEM) determines a unique slutin f the inverse prblem, which pssesses the useful prperty f being everywhere psitive. The resulting histgram f p(r) has the maximum cnfiguratinal entrpy cmpatible with the available data, and is als the mst unifrm. This apprach has been widely used in gephysical prblems and image recnstructin (Parker, 1977) and mre recently in the analysis f particle sizes by the technique f magnetic granulmetry (Pttn, Daniell & Melville, 1984). 2. The maximum entrpy methd The prblem is t determine a set f numbers xi, i= 1,..., N, which are values f an intrinsically psitive physical quantity, given the bservatins d t, j = 1,..., M. The bservatins are measurements f anther quantity Yt, which is linearly related t xi by N Yt = ~ tixi, J= 1,..., M. (2) i=1 Since the number f unknwns is greater than the number f data pints, and the matrix ji may have n inverse, a unique slutin is impssible. Even where a slutin frmally exists it is frequently unstable and large amplificatin f the errrs in the data may ccur in cmputing xi, leading t physically impssible negative values. Additinal assumptins are necessary t btain sensible values fr xi, and the maximum entrpy criterin has prved a very useful ne. The entrpy f a set f numbers x1 is defined as N S = - ~ xi lg xi/b i (3) i=1 where b1 are a prescribed set f values which give a scale t the magnitude f xi. The MEM selects the set x1 fr which S is a maximum subject t the values yj being cnsistent with the available data dj. It is dangerus t cnsider slutins that fit the data exactly. Such slutins will have features that are /88/ Internatinal Unin f Crystallgraphy

2 4 CNFERENCE PRCEEDINGS necessary nly t accmmdate bservatinal errrs. A measure f the misfit between experiment (d j) and Yi predicted by (2) is given by the X z statistic M )~2= ~ (dj- yj)zfiry (4) j=l where ej is the standard deviatin f the measurement dj. Each pint cntributes unity n average, and thus the mean f the distributin is M, the number f data pints. The MEM selects the set f xi fr which the cnfiguratinal entrpy S is a maximum, subject t the cnstraint that X2= M. Details f the search prcedure are given by Skilling & Bryan (1984). The matrix j~ specifies the particular prblem under cnsideratin. Fr the small-angle prblem it takes the frm f a pretabulated matrix f scattering functins Sj(r) calculated fr cmbinatins f linear dimensin r~ and scattering vectr qj. bviusly we must define the scattering functin fr a particular shape. If the particles can be assumed t be spherical then the spherical particle scattering functin is applicable, S(q, r) = 12rc(p - p3 2 (sin qr- qr cs qr)z/q (5) where p and Ps are the scattering-length densities f the particles and the substrate respectively. If the errrs f all the bservatins were infinite, the data wuld in n way cnstrain the slutin. The uncnstrained maximum entrpy slutin is then simply x~ = bile = Bi where e is the number As the errr bars are reduced frm infinity t their true values, a pint in the maximum entrpy map x~ wuld deviate frm its default value B~ prvided there is infrmatin in the data abut this particular x~. Departures frm the default are necessary in rder t fit the experimental data and cannt be artefacts intrduced by experimental errrs r bad data prcessing. We may chse the default B~ t reflect knwn physical reasns why the slutin must take particular values in particular ranges f i, even thugh the data may cntain little infrmatin t supprt this. The physical limits f p(r) are p(r)~ as r~ p(r)~ asr~. () Since we are t determine a prbability distributin p(r), each particle size is equally likely in the absence f data. We culd therefre chse Bi = 1/N as the starting pint fr the iterative prcedure. Hwever, we knw that p() = and p(rmax) ~ S in practice the default level is chsen t be as small as pssible (greater than zer) withut hindering cnvergence. Cnvergence can be slw fr very small defaults since the initial value f )~z is enrmus. Since the MEM appraches the frward prblem Pi--* I~ by generating trial distributins which are then cmpared with bservatin, it is straightfrward t take int accunt the effects f instrumental gemetry and reslutin where these are imprtant. These refinements require nly mdificatin f the matrix ~i. In sme cases we must deal with data which include systematic errrs. Fr small-angle neutrn scattering measurements the magnitude f the backgrund is uncertain. We may minimize )~2 and iterate t btain a gd estimate f the backgrund crrectin. In the next sectin we present mdel data results t shw the applicability f the MEM t small-angle scattering data. 3. Investigatin f mdel data Small-angle scattering was simulated fr nninteracting spherical particles with varius distributins f particle sizes. A mdel experiment cnsisted f 5 intensity measurements, in equal steps in full scattering angle frm 1 t a maximum f 1. This is equivalent t a minimum scattering vectr f.183 and a maximum f "1825 A-1 if we assume a neutrn wavelength typically f A [the standard definitin fr q is 4n (sin /2)/2 where is the full scattering angle]. The minimum value was chsen t reflect beamstp practicalities. Statistical errrs were included. Randm errrs with a Gaussian distributin f zer mean and chsen standard deviatin were added t the scattering data. The maximum entrpy methd was used t analyse the resulting data. A default f.1 was used thrughut. We present a selectin f the results t illustrate the behaviur f resultant slutins. All the fllwing figures depict particle size distributins. The input distributin is cntinuus and the maximum entrpy slutin is shwn as discrete. The size f the errrs f the crrespnding scattering data is indicated at the bttm right f each figure. Initially we investigated unimdal size distributins f similar Gaussian shape but varying in peak psitin up t a maximum f 1.~. The errrs in the scattering data were fixed at 1%. All peaks in the distributins were reprduced well, bth in respect f psitin in radius and height. Figs. l and shw the results fr small particles (1 A peak) and large particles (7 A peak) respectively. Next we examined the results as a functin f the size f the errrs in the simulated data. A Gaussian particle size distributin with peak at 2 A radius was chsen. Statistical errrs in the data varied frm as little as.2% up t a maximum f 1%. Figs. 2 and shw the results fr 1% and 5% errrs respectively. Fr errrs better than 2%, the distributin was reprduced almst perfectly. As the errrs are increased we bserve a gradual reductin in the peak height, althugh its psitin in radius is crrect.

3 .. CNFERENCE PRCEEDINGS 5 Fr 5% errrs the discrepancy between input and slutin is 13%. In reality we wuld hpe t measure intensities t better than 5%, s this effect is acceptable. Indeed it is reassuring that we can reprduce the distributin s well fr data with errrs as large as 5 r even 1%. The width f the Gaussian distributin functin was fund nt t affect the resultant slutin. A wide distributin was reprduced as well as a delta-like functin. Fig. 3 shws the result fr a distributin with a half-peak width f -,~ 5 A, twice that f Fig. 2. A serius questin arises when errr size is being cnsidered. The MEM relies n the experimenter knwing the standard deviatin f his data. Althugh this is nt an unreasnable request, ften errr estimatin is inaccurate. The maximum entrpy algrithm selects that slutin with maximum cnfiguratinal entrpy which cnfrms t the cnstraint that ~2- M, the number f data pints. This ptimum value f Z 2 relies n the errrs being knwn de EL precisely. We addressed the prblem f ver- and underestimated errrs. Again a Gaussian particle size distributin with a peak at 2 A radius was chsen. Simulated scattering data were generated with 1% errrs included. The data were subsequently analysed fr varius assumed errr bars using the MEM. bservatinal errrs were verestimated by as much as a factr f 1 and underestimated by as much as a factr f 1. Fig. 2 shws that the distributin is reprduced well fr crrectly predicted errrs (1%). Fr verestimated errrs, we bserve an increasing discrepancy in the peak height as the discrepancy in the errrs increases. c. ~ --- "J n I 1 I I i 1 2 RADIUS (/~) 17= I.L I I I 1 I I I I RADIUS (~,) [3_. ry EL_ r, ~ (D 4 A A A A A A A 7= I I --I -- --I ~ I J RAD I US (h) Fig. 1. This and all the fllwing figures depict particle size distributins. The input distributin is cntinuus and the maximum entrpy slutin is shwn as discrete. The size f the errrs in the crrespnding scattering data is indicated at the bttm right f each figure. Here is shwn the agreement between input and maximum entrpy slutin f unimdal distributins with peaks at 1 A and 7 A. The errrs in the scattering data were 1%. ~. ~.5% A AAAAAAAA A AAAII, IA.A t i ~ I J T I I RADIUS Fig. 2. Agreement between input and maximum entrpy slutin f a unimdal distributin with a peak at 2 A fr errrs in the scattering data f 1% and 5%. cy v f7 AAAA A ~..r~a ~ ~ k A A A A A A A A AA A I I -- I I I I RADIUS (~,) Fig. 3. Agreement between input and maximum entrpy slutin fr a brad distributin (t be cmpared with Fig. 2 which has a half-peak width a factr f tw smaller).

4 CNFERENCE PRCEEDINGS The psitin f the peak in radius remains crrect. verestimated errrs result in a maximum entrpy slutin with an increased value f final Z 2. We have necessarily lst infrmatin and wuld expect a brader distributin than reality t result. ur bservatins agree with cmmn sense. Fig. 4 shws the slutin btained fr errr bars f 5%, a factr f five larger than reality. We bserve a small reductin in the peak height. The questin f underestimated errrs can be mre serius. Fr significant underestimatin f errrs, cnvergence f 2 t M is nt attained. Instead )(2 cnverges t a value greater than M. The resultant slutin will have maximum cnfiguratinal entrpy subject t )~2 being as small as pssible (but greater than M). It is pssible that the algrithm can reduce )~2 by interpreting the randm nise in the data. We may therefre bserve small instabilities in the slutin. Fig. 4 shws the slutin attained fr errr bars f.2%, a factr f five smaller than reality. We still bserve very gd agreement between input and slutin. Hwever, we bserve an indicatin f a discrepancy at ~. The value f )~2 has been reduced by including this feature. t 13. ~-'~ 13_,=; 1 2 RADI US (#) d p,-)-- AAakkAkAA~kAA 5% 1% I I % The MEM des nt fail fr inaccurate errrs. verestimated errrs lead t underfitting f the data and a cnsequent lss f particle size infrmatin (brader distributin than reality). Underestimated errrs can lead t verfitting f the data and sme randm nise interpretatin. A mre stern test f the technique wuld be t attempt t reprduce bimdal distributins f particle sizes. Such distributins were investigated bth as a functin f peak psitin and errrs in the scattering data. Fig. 5 shws the result fr peaks at 1 and 2/k, and errrs f 1% in the intensities. We bserve a slight reductin in the height f the 1 A peak, thugh bth maxima are in the crrect psitins in radius. Fig. 5 shws hw the slutin may be imprved by reducing the errrs in the data t.2%. Fig. 5(c) illustrates the results fr peaks at significantly larger radii, at and 7 A, and errrs f 1% in the simulated data. We bserve gd agreement between input and slutin fr bth peaks. The agreement is marginally better than Fig. 5. It appears t be easier t reprduce bimdal distributins at higher radii. Indeed, Fig. 5(d) shws that data with as much as 5% errrs can result in reasnable peak reprductin. Finally, scattering data representing a trimdal particle size distributin were investigated. The initial distributin cnsisted f peaks at 1, 2 and 3/k, with errrs f 1% included in the crrespnding mdel data. Fig. illustrates the results. We bserve best agreement fr the "3 A peak, reasnable agreement fr the 2 A peak and sme agreement fr the 1 A peak. The maximum f each peak appears at the crrect radius value. Again, verall agreement can be imprved by reducing the size f the errrs in the bservatins. Fig. shws the result fr errrs f.2%. Reprductin is very gd. Fig. (c) shws the result at crrespndingly larger radii. Peaks are psitined at 5, and 7 A. Fr 1% data all three peaks are reprduced well, bth in respect f psitin in radius and height. Data with errrs as large as 5% can prduce three reasnable peaks, as illustrated in Fig. (d). t t"t". "--s 13_ AAAAAAA& AAA AA <~ I i i I 1 2 RADIUS ' k.... 1% ~AAAAAAAA I I I 3 4 Fig. 4. Agreement between input and maximum entrpy slutin fr ver- and underestimated errrs f 5% and.2%. The true errrs in the scattering data are 1%. Fig. 2 shws the result fr crrect errr estimatin. Cncluding remarks We have shwn hw the apprach f maximizing cnfiguratinal entrpy t chse a unique slutin f a practical linear inverse prblem may be applied t small-angle neutrn scattering. It must be stressed that the MEM is n substitute fr gd data; it simply makes the best f available data. The algrithm successively generates trial slutins fr the prbability functin p(r) and cmpares these with bservatin using X 2 statistics. Cnvergence is attained when Z 2 is equal t the number f data pints and the cnfiguratinal entrpy is a maximum. The

5 CNFERENCE PRCEEDINGS 7 ~d iv-)-. t CE "--' _ k_ (3._ t g t LI ~' 1% I I I I 1 I I RADIUS 2Z AAAA A A A A A A A A A A A A A ~ I I 1 I I 1 I I RADI US A ry CL CL It:: "-~(9 CL t {D J IV')" c I I I I I I RADI US..2Z AA~AA A A AAAA I I i I T I I I RADI US 4 k AAAAAAAAAAA,~ A A ~ ~ ~AAA A I I I I ~ I I RADIUS (c) 4 I 1 I I 3" I I RADIUS (/~) (c) cy 13_ g.sz AA AAAAAAAA &A I k" I ~AAA I I --I I RADIUS (/~) (c/) Fig. 5. Agreement between input and maximum entrpy slutin fr large- and small-radius bimdal distributins fr varying errrs in the scattering data. Peaks at 1 and 2/~ with errrs in the data f 1% then "2%. Peaks at and 7/~, with errrs f(c) 1% in the data and then (d) 5%. ry CL AA*AA A * i I 1 m I T 4 Z 5 7 RADI US I 8 Fig... Agreement between input and maximum entrpy slutin fr large- and small-radius trimdal distributins fr varying errrs in the intensity data. Peaks at 1, 2 and 3,~ with errrs in the data f 1% and "2%. Peaks at 5, and 7,~ with errrs in the data f(c) 1% and (d) 5%.

6 8 CNFERENCE PRCEEDINGS resultant p(r) will be the mst unifrm slutin cnsistent with the data, and has the attractive prperty! being everywhere psitive. The methd may be extended t include crrectins fr any reslutin r slit smearing since these effects enter as a cnvlutin f a reslutin functin with the calculated scattering results, which are then cmpared with bservatin. This is mre straightfrward than attempting t decnvlve the reslutin functin frm the data befre appraching the inverse prblem t extract a particle size distributin. The mdel results are very encuraging. Applicatin f the MEM t experimental small-angle scattering data has been carried ut successfully (Pttn, Daniell & Rainfrd, 198). Mre results tgether with a full descriptin f a cmplementary technique which enables errr bars t be assigned t each pint n a maximum entrpy slutin are published in this vlume (Pttn, Daniell & Rainfrd, 1988). References PARKER, R. L. (1977). Rev. Earth Planet. Sci. 5, PTTN, J. A., DANIELL, G. J. & MELVILLE, D. (1984). J. Phys. D, 17, PTTN, J. A., DANIELL, G. J. & RAINFRD, B. D. (198). Inst. Phys. Cnf. Set. 8, PTTN, J. A., DANIELL, G. J. & RAINFRD, B. D. (1988). J. Appl. Cryst. 21, SKILLING, J. BRYAN, R. K. (1984). Mn. Nt. R. Astrn. Sc. 211,

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