Hauptma ad Karle Joit ad Coditioal Probability Distributios Robert H Blessig HWI/UB Structural Biology Departmet Jauary 00 ormalized crystal structure factors are defied by E h = F h F h = f a hexp ihi h f a h It follows that E h h = sice the squared ormalizatio factor F h = E h exp i* h = h f a h is the expectatio value Wilso 949 for the itesity of the Bragg reflectio labled h fo structure of atoms per uit cell with the atoms distributed idepedetly ad uiformly at positio vectors The factors h are symmetry degeeracy factors with kow values Iwasaki ad Ito 977 that accout for multiple ehacemet of certai classes of reflectio itesity averages due to space group symmetry: I space groups P ad P h = foll reflectios; i space groups of higher symmetry if the lattice is primitive h = for geeral reflectios ad h > for some sets of zoal oxial special reflectios due to atomic overlap i projectio; if the lattice is cetered h > foll the allowed reflectios due to extictio of the forbidde reflectios Hauptma ad Karle 958 ad Karle ad Hauptma 958 applied probability theory to derive joit ad coditioal probability distributios for structure factor triplets E h E h E h i order to estimate the three-phase structure ivariats h + h + h with h + h + h = 0 I outlie the basic priciples of joit ad coditioal probability distributios are: b P a < x < b a < x < b b = p J x x dx dx p J - C u u / x x = a a b b exp i u j x j * C u u du du a a j= ++ ++ = exp +i u j x j + + = exp i u j x j * j= j= x x = p J x x m a m+ a * p J x x dx dx p C x x m a m+ a p M a m+ a - ++ ++ p M x m+ x = p J x x m x m+ x m dx dx + + m /
where the fuctio C u = C u u is called the characteristic fuctio of the joit x = p J x x ; the fuctio p C x x m a m+ a is probability desity fuctio p J the coditioal probability desity fuctio fo subset of m of the radom variables give particular values a m+ a for the other m variables; ad the fuctio p M x m+ x is the margial probability desity fuctio for the subset of m of the radom variables irrespective of the values of the other m variables The joit probability desity fuctio ad the characteristic fuctio are related by ad uiquely defie oe aother by Fourier trasformatio Thus Fˆ C u = ˆ C u p J x F ˆ p J x = F ˆ C u For the joit distributio of three structure factors that correspod to a uiform radom distributio of atomic positios fo = atoms per uit cell Hauptma ad Karle showed that the joit probability desity fuctio p J x is p J R R R = + F p J x + d 6 d d d d d R R R 0 0 0 * exp i + j= + R j j cos j j - / C where R R R ad deote respectively the radom variable amplitudes E h E h E h ad phases h h h of the three structure factors ad the = ˆ = ˆ characteristic fuctio C u C is F p J x F p J R R R = dr dr dr d d d R R R 0 0 0 exp +i - j= ++ *+ ++ *+ ++ *+ R j j cos j * j / p J R R R = R R R exp +i- R j j cos j * j / j= I space group P the characteristic fuctio becomes = exp-i C / j= f a h j R R R f a h j j cos *h j i + j 0
ad if we itroduce the simplifyig otatios we have C = C ad The usig the elemetary relatioships exp x + y we obtai C = exp i c a j j= = exp i c a j j= ad usig the series expasio f a h j f a h j C = exp i c a j j= = exp x exp y l xy = l x + l y = exp l x = x l exp x x + y = x + y x + iy = r exp i l x + iy r = = l r exp i = exp i c a j j= = c a j cos *h i r + j j a j y x + y = ta x ad = lr + i = exp l + exp i c a j j= * + - = exp i c a j r a r j= = exp i c a j r j= a a exp ix = + ix + i x + i x + the complex expoetial factors i the cotiued product C = exp i c a j r a ca be expressed as sums of averages of cosie products exp i c a j r a = + i c + i c ak ak c al j= k = j= k = l = Usig the elemerary relatioship cos xcos y = cos x + y + i c ak c al c am + ra k = l = m= + cos x y
the averages of cosie products ca be show to be expressible i terms of averages of cosie ad squared-cosie fuctios The by the hypothesis of a uiform radom distributio of atomic positios the cosie argumets spa the whole iterval [0 π ad the averages cos x = 0 ad cos x = < 0 x mod allow simplificatio ad termwise sixfold itegratio over the dummy variables to obtai the fial result for the joit distributio fuctio p J R R R = * exp R + R + + R - + R R R * + f a h f a h f a h / - 0 * + - f a h f a h f a h R R R cos + + f a h f a h f a h / 0 Specifyig give values R = E h R = E h R = E h for the amplitudes ad expadig the cosie expoetial fuctio i modified Bessel fuctios of the secod kid + = I x exp xcos exp i = I x = + exp xcos cos d leads from the joit distributio fuctio to the coditioal distributio fuctio p C + + R R R = I 0 A where f a h f a h exp A cos + + f a h A = E h E h E h + f a h f a h f a h * - The coditioal distributio has the form of a vo Mises or circular ormal distributio I crystallography it is kow as the Cochra distributio because Cochra 955 discovered that it was the distributio fuctio for the origi-idepedet triplet or three-phase structure ivariats h + h + h with h + h + h = 0 / 0 4
Coditioal Distributio for the Triplet Ivariats I customary otatio the coditioal distributio fuctio for the three-phase structure ivariat hk h + k + hk is writte as where I 0 p C hk = I 0 exp A cos hk hk x deotes the zeroth order modified Bessel fuctio ad the amplitudes factor f a k f a h k f a h = E h E k E hk f a h f a k f a h k * + - is a iverse measure of the variace or spread of the distributio I the limit of large the coditioal distributio approaches the ormal distributio I other words the vo Mises circular ormal distributio has the property exp cos µ p µ = lim p µ I 0 * µ + + = Sice atomic scatterig factor curves all have approximately the same shape f a h Z a f h Z the amplitudes factor ca be approximated as = Z a f h Z f k Z f h k Z f h Z f k Z f h k Z Z a Z a E h E k E hk = E h E k E hk Z a which uder the approximatio of equal poit-atoms at rest or with equal mea-square atomic displacemets simplifies to E h E k E hk The latter expressio makes it clear that the spread or flatess of the coditioal distributio icreases ad hece its predictive or estimatio power decreases with icreasig strucure size ad decreasig structure factors magitudes 5
Coditioal Expectatio Values from the Triplet Distributio The coditioal distributio of the three-phase structure ivariat gives expectatio values y = + p x dx y x = y hk hk = 0 si hk = 0 cos hk = I I 0 + d hk p C hk var y = y y = y y var hk = hk var si hk = si hk = I I 0 = cos hk cos hk = si hk cos hk var cos hk = I I 0 I A hk I 0 * + 6
The Taget Formula The coditioal distributio of the three-phase or triplet structure ivariats also leads to the taget formula Give a set of coditioal probability distributios for pairs of kow phases with fixed h p C hki A hki for { ki hki i = } the combied distributio is give by the product of the distributios P = K = K = K p i p C hki A hki I 0 i + exp - i cos h + ki + *h*ki - where K is a ormalizatio costat At the most probable value of h ad sice we obtai or P = 0 A hki si h + ki + hki h si h i cos ki + hki = 0 si x + y = si xcos y + cos xsi y + cos A h hki ta h = si h cos h = ta h = si ki + hki i si ki + hki i cos ki + hki E k E hk si k + hk k E k E hk cos k + hk k 0 / 0 = 0 7
Refereces Cochra W 955 Relatios betwee the Phases of Structure Factors Acta Cryst A8 47-477 Hauptma HA ad Karle J 958 Phase Determiatio from ew Joit Probability Distributios: Space Group P Acta Cryst 49-57 Iwasaki H ad Ito T 977 Values of ε for obtaiig ormalized structure factors Acta Cryst A 7-9 Karle J ad Hauptma HA 956 A theory of phase determiatio for the four types of o-cetrosymmetric space groups P P P P Acta Cryst 9 65-65 8
Karle J ad Hauptma HA 958 Phase Determiatio from ew Joit Probability Distributios: Space Group P Acta Cryst 64-7 9
Modified Bessel fuctios i polyomial approximatio t = x 75 I x = + exp xcos cos d I 0 + 56t + 090t 4 + 07t 6 + 0660t 8 + 00607t 0 + 000458t 75 x +75 x = e x x 0989 + 009t + 0005t 000576t + 00096t 4 00058t 5 + 0066t 6 00648t 7 +00094t 8 + 75 x < I x = x + 08789t + 0550t 4 + 0508t 6 +00659t 8 + 00005t 0 + 00004t 75 x +75 e x x 0989 00988t 00060t + 00068t 0000t 4 + 008t 5 00895t 6 + 00787t 7 +00040t 8 + 75 x < Ratios of modified Bessel fuctios x / 0 < x < ; I x I 0 x 05658 x 004 x + 0006 x 0 < x < 6 ; x > 6 I x I x I 0 x = x I 0 x 0